text stringlengths 14 5.77M | meta dict | __index_level_0__ int64 0 9.97k ⌀ |
|---|---|---|
pcpo
====
Post Correspondence Problem (PCP) in miniKanren
Written with Tony Tuttle.
Links describing the PCP:
http://en.wikipedia.org/wiki/Post_correspondence_problem
http://webdocs.cs.ualberta.ca/~games/PCP/
http://www.cis.upenn.edu/~jean/gbooks/PCPh04.pdf | {
"redpajama_set_name": "RedPajamaGithub"
} | 4,833 |
using PICSimulator.Helper;
namespace PICSimulator.Model.Commands
{
/// <summary>
/// The W register is subtracted (2's comple-
/// ment method) from the eight bit literal 'k'.
/// The result is placed in the W register.
/// </summary>
class PICCommand_SUBLW : PICCommand
{
public const string COMMANDCODE = "11 110x kkkk kkkk";
public readonly uint Literal;
public PICCommand_SUBLW(string sct, uint scl, uint pos, uint cmd)
: base(sct, scl, pos, cmd)
{
Literal = Parameter.GetParam('k').Value;
}
public override void Execute(PICController controller)
{
uint a = Literal;
uint b = controller.GetWRegister();
bool carry;
bool dc = BinaryHelper.getSubtractionDigitCarry(a, b);
if (carry = a < b)
{
a += 0x100;
}
uint Result = a - b;
controller.SetUnbankedRegisterBit(PICMemory.ADDR_STATUS, PICMemory.STATUS_BIT_Z, (Result % 0x100) == 0);
controller.SetUnbankedRegisterBit(PICMemory.ADDR_STATUS, PICMemory.STATUS_BIT_DC, dc);
controller.SetUnbankedRegisterBit(PICMemory.ADDR_STATUS, PICMemory.STATUS_BIT_C, !carry);
Result %= 0x100;
controller.SetWRegister(Result);
}
public override string GetCommandCodeFormat()
{
return COMMANDCODE;
}
public override uint GetCycleCount(PICController controller)
{
return 1;
}
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 5,856 |
{"url":"http:\/\/theoretical-physics-digest.wikia.com\/wiki\/Ekman_Layer","text":"FANDOM\n\n154 Pages\n\nLet us consider a wind blowing parallel to the surface of a sea. Due to viscosity, deeper layers of the sea will move, with the magnitude of the velocity decreasing with depth. In a rotating frame, like on a spinning planet, the velocity is also affected by the Coriolis force. The governing equations for the two components of the velocity parallel to the surface $u$, $v$ are\n\n$-f v = D \\frac{d^2 u}{d z^2}$\n\n$f u = D \\frac{d^2 v}{d z^2}$\n\nwhere $D$ is the viscosity or diffusion coefficient, $f = 2 \\Omega \\sin \\phi$ is the Coriolis parameter, $\\Omega$ is the spin frequency and $\\phi$ is the latitude. We can express the two components of the velocity as a single complex variable $w = u + i v$\n\n$D \\frac{d^2 w}{d z^2} = i f w \\Rightarrow w = w_0 \\exp \\left(- z \\sqrt{\\frac{f}{2 D}} \\right) \\left(1+i \\right)$\n\nHence the velocity rotates and declines as the depth $z$ increases. The characteristic distance over which this happens is $\\sqrt{D\/f}$.","date":"2018-10-22 18:16:50","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\": 0.8077546954154968, \"perplexity\": 188.15582789460805}, \"config\": {\"markdown_headings\": false, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2018-43\/segments\/1539583515375.86\/warc\/CC-MAIN-20181022180558-20181022202058-00272.warc.gz\"}"} | null | null |
package com.sharingapples.sync.resource;
import com.sharingapples.sync.resource.ResourceException;
/**
* Created by ranjan on 12/12/15.
*/
public class ParseException extends ResourceException {
public ParseException(String message) {
super(message);
}
public ParseException(Throwable cause) {
super(cause);
}
public ParseException(String message, Throwable cause) {
super(message, cause);
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 8,552 |
Q: Button JQuery event only working once, why? <script>
$("button").on("click", function() {
$.getJSON("http://quotesondesign.com/wp-json/posts?filter[orderby]=rand&filter[posts_per_page]=1", function(json) {
$(".author").html(json[0].title);
$(".quote").html('"'+json[0].content+'"');
});
});
</script>
Situation: I click, the data is loaded. I click again, nothing changes.
Codepen: http://codepen.io/anon/pen/vxGgaZ
Reference: https://quotesondesign.com/api-v4-0/
A: The problem is because the first response is being cached. You can solve that by adding a $.ajaxSetup() call which stops the caching:
$.ajaxSetup({
cache: false
})
Updated Codepen
Alternatively, use $.ajax() and set cache directly in the settings:
$("button").on("click", function() {
$.ajax({
url: 'http://quotesondesign.com/wp-json/posts',
type: 'get',
cache: false,
data: 'filter[orderby]=rand&filter[posts_per_page]=1',
success: function(json) {
$(".author").html(json[0].title);
$(".quote").html('"' + json[0].content + '"');
}
});
});
A: Try Like This:
Maybe it's help you
<script>
$(document.body).on("click", 'button', function() {
$.getJSON("http://quotesondesign.com/wp-json/posts?filter[orderby]=rand&filter[posts_per_page]=1", function(json) {
$(".author").html(json[0].title);
$(".quote").html('"'+json[0].content+'"');
});
});
</script>
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 5,727 |
\section{Introduction}
\label{sec:intro}
Let $f_U : \mathsf{U} \to [0,\infty)$ be an intractable target
density, and suppose that
$f : \mathsf{U} \times \mathsf{Z} \to [0,\infty)$ is a joint density
whose $u$-marginal is the target, i.e.,
$\int_{\mathsf{Z}} f(u,z) \, dz = f_U(u).$ Think of $U$ as the
parameters in a Bayesian model, and $Z$ as latent data. If
straightforward sampling from the associated conditional densities is
possible, then we can use the data augmentation (DA) algorithm to
explore $f_U.$ Of course, running the algorithm entails alternating
between draws from $f_{Z | U}$ and $f_{U | Z},$ which simulates the
Markov chain whose Markov transition density (Mtd) is
\[
k_{\text{DA}} (u' | u) = \int_{\mathsf{Z}} f_{U | Z} (u'
| z) f_{Z|U}(z | u) \, dz \;.
\]
It's easy to see that $k_{\text{DA}} (u' | u) f_U(u)$ is
symmetric in $(u, u'),$ so the DA Markov chain is reversible with
respect to $f_U$.
To fix ideas, we introduce a simple example. Let $W_1, \ldots, W_m$
be a random sample from the location-scale Student's $t$ distribution
with known degrees of freedom, $\nu > 0$, and consider a Bayesian
model with prior density given by
$\pi(\mu,\sigma^2) \propto \sigma^{-2} I_{\mathbb{R}_+}(\sigma^2)$,
where $\reals_+ := (0,\infty)$. This is a standard diffuse prior for
location-scale problems. We assume throughout that $m \geq 2$, which
is necessary and sufficient for posterior propriety. The resulting
posterior density is an intractable bivariate density characterized by
\begin{equation}
\label{eq:toyposterior}
f_U(\mu, \sigma^2) \propto (\sigma^2)^{- \frac{m+2}{2}}
\prod_{i=1}^m \left( 1 + \frac{(w_i - \mu)^2}{\nu \sigma^2}
\right)^{-(\nu + 1)/2} I_{\mathbb{R}^+}(\sigma^2) \;.
\end{equation}
So here the role of $U$ is being played by $(\mu,\sigma^2)$, and, in
order to keep the notation consistent, we are suppressing the
dependence on the data, $w = (w_1,\ldots,w_m)$. There is a simple DA
algorithm for this problem that is based on the standard
representation of a Student's $t$ variate in terms of normal and
$\chi^2$ variates. Conditional on $(\mu,\sigma^2)$, let
$(W_1,Z_1),\ldots,(W_m,Z_m)$ be independent and identically
distributed (iid) pairs such that, for $i=1,\ldots,m$,
\begin{align*}
& W_i | Z_i , \mu , \sigma^2 \sim \text{N}(\mu, \sigma^2 / z_i) \\ &
Z_i |\mu, \sigma^2 \sim \text{Gamma}(\nu / 2, \nu / 2) \;.
\end{align*}
Letting $z = (z_1,\ldots,z_m) \in \reals^m_+$, the joint density of
$\{ (W_i,Z_i) \}_{i=1}^m$ is given by
\[
p(w,z | \mu, \sigma^2) = \prod_{i=1}^m \frac{\sqrt{z_i}}{\sqrt{2 \pi
\sigma^2}} \exp \left\lbrace -\frac{z_i}{2 \sigma^2} (w_i - \mu)^2
\right\rbrace \frac{\left( \frac{\nu}{2}
\right)^{\frac{\nu}{2}}}{\Gamma \left( \frac{\nu}{2} \right)}
z_i^{\frac{\nu}{2} - 1} \exp \left\lbrace - \frac{\nu z_i}{2}
\right\rbrace I_{\mathbb{R}^+}(z_i)\;.
\]
Now define $f(\mu,\sigma^2,z) \propto p(w,z | \mu, \sigma^2)
\pi(\mu,\sigma^2)$. Then it's easy to see that
\[
\int_{\reals^m_+} f(\mu,\sigma^2,z) \, dz = f_U(\mu, \sigma^2) \;,
\]
so $z$ is indeed latent data. It is straightforward to sample from
$f_{Z|U}(z|\mu,\sigma^2)$ since the $Z_i$ are conditionally
independent, each with a gamma distribution. It's also easy to sample
from $f_{U|Z}(\mu,\sigma^2|z)$ (sequentially) because both
$f_{\sigma^2|Z}(\sigma^2|z)$ and $f_{\mu|\sigma^2,Z}(\mu|\sigma^2,z)$
have simple forms \citep[see, e.g.,][Section
10.1]{hobert2011handbook}. We will return to this simple example
several times in order to illustrate certain ideas without having to
wade through the heavy notation associated with a more sophisticated
model. Now back to the general case.
There are many situations in which useful latent data exist, but the
DA algorithm is not directly applicable. Specifically, it is often
the case that it \textit{is} possible to draw from $f_{Z|U},$ but it
is not possible to draw from $f_{U|Z}.$ On the other hand, in such
cases, one can sometimes break $u$ into two pieces, $u = (x,y)$, where
$x \in \mathsf{X}, y \in \mathsf{Y}, \mathsf{U} = \mathsf{X} \times
\mathsf{Y},$ in such a way that one is able to draw from $f_{X|Y,Z}$
and from $f_{Y|X,Z}$. In such cases, we can run a three-block Gibbs
sampler based on $f_{X|Y,Z}$, $f_{Y|X,Z}$ and $f_{Z|X,Y}$. Of course,
the random scan (RS) version of this Gibbs sampler is reversible,
while the systematic scan (SS) version is not.
Consider again our simple Student's $t$ example, and suppose we change
the prior to
\[
\pi^*(\mu,\sigma^2) \propto \sigma^{-2} \exp \Big \{ -\frac{1}{2} (\mu -
\gamma)^2 \Big \} I_{\mathbb{R}^+}(\sigma^2) \;,
\]
where $\gamma \in \mathbb{R}$ is fixed. In this case,
$(\mu,\sigma^2)$ is playing the role of $(x,y)$. Under the new prior,
drawing from $(\mu,\sigma^2)|z$ is no longer straightforward, because
the distribution of $\sigma^2|z$ is nonstandard. Hence, while the DA
algorithm is still technically implementable (using, say, a rejection
sampler for $\sigma^2|z$), it is much less attractive under the new
prior. On the other hand, the conditional densities of
$\mu|(\sigma^2,z)$, $\sigma^2|(\mu,z)$, and $z|(\mu,\sigma^2)$ all
have standard forms, so the three-block Gibbs sampler would be easy to
run.
In this paper, we consider an alternative to the SS and RS three-block
Gibbs samplers. We call it the \textit{hybrid scan Gibbs sampler}.
Fix $r \in (0,1)$ to play the role of a selection
probability. Consider a Markov chain $\{(X_n,Y_n) \}_{n=0}^{\infty}$
with state space $\mathsf{X} \times \mathsf{Y}$ that evolves as
follows. If the current state is $(X_n,Y_n) = (x,y),$ then we simulate
the new state, $(X_{n+1},Y_{n+1}),$ using the following two-step
procedure.
\vskip .2truecm\hfill \hrule height.5pt \vskip .2truecm \vspace*{2mm}
\noindent {\rm Iteration $n+1$ of the hybrid scan Gibbs sampler:}
\begin{enumerate}
\item Draw $Z \sim f_{Z|X,Y}(\cdot|x,y),$ call the result $z,$ and,
independently, draw $W \sim \text{Uniform}(0,1).$
\item
\begin{enumerate}
\item If $W \le r$, draw $X^* \sim f_{X|Y,Z}(\cdot|y,z),$ and set
$(X_{n+1},Y_{n+1}) = (X^*,y)$.
\item Otherwise, draw $Y^* \sim f_{Y|X,Z}(\cdot|x,z),$ and set
$(X_{n+1},Y_{n+1}) = (x,Y^*)$.
\end{enumerate}
\end{enumerate}
\vspace*{-3mm}
\vskip -.1truecm\hfill \hrule height.5pt \vskip .4truecm
\bigskip
A standard SS Gibbs sampler based on $f_{X|Y,Z}$, $f_{Y|X,Z}$ and
$f_{Z|X,Y}$ updates all three components (in the same prespecified
order) at each iteration. To run the RS version, we first fix three
selection probabilities $r_1, r_2, r_3 \in (0,1)$ such that
$r_1+r_2+r_3=1$. Then, at each iteration, we draw from exactly one of
the three full conditionals according to the probabilities $r_1$,
$r_2$ and $r_3$, and leave the remaining two components fixed. So
hybrid scan (HS) Gibbs can be viewed as a compromise between these
standard Gibbs samplers in the sense that, at each iteration of HS
Gibbs, exactly two of the three full conditionals are sampled. The
idea of including both systematic and random scan ingredients in a
single Markov chain Monte Carlo (MCMC) algorithm is not new
\citep[see, e.g.,][]{levine2005note}, but we believe that this is the
first concentrated study of this particular algorithm.
The reader will note that we have yet to demonstrate that the HS Gibbs
sampler is actually valid. In fact, it follows directly from one of
the general results in Appendix~\ref{app:theory} (Proposition 1) that
the Markov chain associated with HS Gibbs is reversible with respect
to $f_{X,Y}(x,y)$ for any $r \in (0,1)$, which implies that the
algorithm is valid. Proposition 1 is rather technical, and its proof
is based on operator theory. Fortunately, there is a much simpler way
to establish the desired reversibility. Indeed, we now show that HS
Gibbs is equivalent to a RS, variable-at-a-time Metropolis-Hastings
(MH) algorithm (in which every proposal is accepted). It then follows
immediately from basic MCMC theory that the Markov chain associated
with HS Gibbs is reversible with respect to $f_{X,Y}(x,y)$ \citep[see,
e.g.,][]{geyer2011handbook}. First, it's clear by inspection that a
single iteration of HS Gibbs can be recast as follows: Suppose the
current state is $(X_n,Y_n) = (x,y)$. Flip an ``$r$-coin.'' If the
coin comes up heads, then set $(X_{n+1},Y_{n+1}) = (X',y)$, where $X'$
is drawn from the density
\[
c_1(x'|x;y) = \int_{{\mathsf{Z}}} f_{X|Y,Z}(x'|y,z) f_{Z|X,Y}(z|x,y) dz \;.
\]
If, on the other hand, the coin comes up tails, then set
$(X_{n+1},Y_{n+1}) = (x,Y')$, where $Y'$ is drawn from the density
\[
c_2(y'|y;x) = \int_{{\mathsf{Z}}} f_{Y|X,Z}(y'|x,z) f_{Z|X,Y}(z|x,y) dz \;.
\]
Now consider a MH algorithm in which, at each iteration, with
probability $r$ we (keep $y$ fixed and) perform a MH update that
leaves $f_{X|Y}(\cdot|y)$ invariant using candidate density
$c_1(x'|x;y)$, and with probability $1-r$ we (keep $x$ fixed and)
perform a MH update that leaves $f_{Y|X}(\cdot|x)$ invariant using
candidate density $c_2(y'|y;x)$. The Hastings ratio for the update
that leaves $f_{X|Y}(\cdot|y)$ invariant is given by
\[
r(x,x') = \frac{f_{X|Y}(x'|y) c_1(x|x';y)}{f_{X|Y}(x|y) c_1(x'|x;y)}
\;.
\]
The numerator of $r(x,x')$ can be written as
\begin{equation*}
\frac{1}{f_Y(y)} f_{X,Y}(x',y) \int_{{\mathsf{Z}}} f_{X|Y,Z}(x|y,z)
f_{Z|X,Y}(z|x',y) dz = \frac{1}{f_Y(y)} \int_{{\mathsf{Z}}} \frac{f(x,y,z)
f(x',y,z)}{f_{Y,Z}(y,z)} dz \;,
\end{equation*}
which is clearly a symmetric function of $(x,x')$. Hence, $r(x,x')
\equiv 1$, so the candidate is never rejected. A similar argument
shows that, for fixed $x$, the MH update for $f_{Y|X}(\cdot|x)$ using
candidate density $c_2(y'|y;x)$ also never rejects. Therefore, the HS
Gibbs sampler is, in fact, a RS, variable-at-a-time
Metropolis-Hastings (MH) algorithm, and reversibility follows.
As mentioned above, the HS algorithm can be viewed as a compromise
between SS and RS Gibbs. Now, if it were known that one of the
standard scans always produces a superior Gibbs sampler, then it might
not make sense to consider such a compromise. However, as we now
explain, this is far from being the case. There are two standard
criteria for comparing MCMC algorithms. The first is based on the
convergence rates of the underlying Markov chains, and the second is
based on the asymptotic variances of ergodic averages.
(Appendix~\ref{app:theory} contains some general theory on this topic
for reversible chains.) It is known that neither of the standard scan
Gibbs samplers dominates the other in terms of convergence rate.
Indeed, there are examples in the literature of SS Gibbs samplers that
converge faster than their RS counterparts, and others where the
opposite is true \citep[see,
e.g.,][]{roberts2015surprising,robe:sahu:1997,he:desa:mitl:re:2016}.
On the other hand, there is some general theory suggesting that the SS
Gibbs sampler is better when the criterion is asymptotic variance, but
these results are rather limited in scope. For example,
\citet{greenwood1998information} show that the asymptotic variance
under the RS algorithm is no more than twice that under the SS
algorithm, and \citet{andrieu2016random} proves that, when the Gibbs
sampler has exactly two blocks, the SS algorithm is always better.
(See also \citet{maire2014comparison}, \citet{liu1995covariance} and
\citet{latuszynski2013adaptive}.) So, based on what is currently
known, there is no clear cut winner between the SS and RS versions of
the Gibbs sampler when there are more than two blocks.
The HS Gibbs sampler has important advantages over the two standard
scan Gibbs samplers. Firstly, it can be much easier to establish the
geometric ergodicity of a HS Gibbs Markov chain than to do the same
for the corresponding systematic and random scan Gibbs chains. We
provide examples of this in Sections~\ref{sec:glmm} and~\ref{sec:smn}.
(Of course, the important practical benefits of basing one's MCMC
algorithm on a geometrically ergodic Markov chain have been
well-documented by, e.g., \citet{roberts1998markov},
\citet{jones2001honest}, \citet{flegal2008markov} and
\citet{latuszynski2013nonasymptotic}.) Secondly, as we explain in
Section~\ref{sec:adding}, the sandwich methodology of
\citet{hobert2008theoretical} can be applied to the HS Gibbs algorithm
(but not to the standard scan Gibbs samplers). This allows for the
addition of up to two extra steps at each iteration that can
potentially speed up the convergence rate without adding much to the
computational complexity. Moreover, because HS Gibbs is reversible,
we are able to prove that, under weak regularity conditions, adding
sandwich steps always results in an improved algorithm in terms of
both convergence rate and asymptotic variance. Another advantage that
HS Gibbs has over SS Gibbs (but not over RS Gibbs) is that, if
specific information about the target distribution is known, the
practitioner may vary the selection probability $r \in (0,1)$ to cause
one set of parameters to be updated more frequently than the other.
Lastly, note that the $Z$ component, which is typically used only to
facilitate sampling and is not itself of inferential interest, is
\textit{not} part of the HS Markov chain. The same is true of the
basic DA algorithm. While it is possible to marginalize over the $Z$
component in the SS Gibbs chain and still have a bona fide Markov
chain, such marginalization is not possible with the RS Gibbs
algorithm.
It is straightforward to extend the HS Gibbs sampler to situations in
which there are more than three blocks. Indeed, suppose that breaking
$U$ into two components is not enough. That is, suppose that we are
unable to identify a partition $u = (x,y)$ such that sampling from
$f_{X|Y,Z}$ and $f_{Y|X,Z}$ is straightforward, but we are able to
find an $s$-component partition, $u=(x_1,x_2,\ldots,x_s)$, such that
it is possible to sample from each $f_{X_j | X_{-j}, Z}$, for
$j=1,\dots,s$, where, as usual,
$X_{-j} = (X_1,\dots,X_{j-1},X_{j+1},\dots,X_s)$. It is
straightforward to extend the HS algorithm (and all the results that
we discuss in this paper) to this more general case. For example, at
each iteration of the (generalized) HS algorithm, we update $Z$ and
one randomly chosen element from the random vector $(X_1,\dots,X_s)$.
The only MCMC methods that have been considered so far in this paper
are the DA algorithm and the Gibbs sampler, which could be considered
``classical'' MCMC techniques. In particular, we have not mentioned
any ``state of the art'' MCMC techniques, such as particle MCMC
\citep{andr:douc:hole:2010} or Hamiltonian Monte Carlo
\citep{neal:2011}. There are two reasons for this. Firstly, these
methods are \textit{much} more complex than the classical ones, and
even describing them accurately requires the introduction of a great
deal of notation. Secondly, and perhaps more importantly, these more
sophisticated methods are often not required to solve a given problem.
Indeed, there are plenty of Bayesian models with posterior
distributions that, while intractable, are perfectly amenable to
classical MCMC methods such as the Gibbs sampler and the
Metropolis-Hastings algorithm. (Several such examples are detailed in
this paper.) In such situations, there is no need to consider more
sophisticated MCMC methods, which can be \textit{much} more difficult
to design, code, and analyze than the classical methods. As an analogy, consider a situation where we have a posterior
distribution that is complex, but from which we can make iid draws (in a
reasonably efficient manner). In such a case, there would be no need to
resort to MCMC since we could effectively explore the posterior using
classical (iid) Monte Carlo.
The remainder of this paper is organized as follows.
Section~\ref{sec:glmm} contains our first serious example of a HS
Gibbs sampler. The target is the posterior distribution associated
with a Bayesian shrinkage model with random effects. This algorithm
was first introduced by \citet{abrahamsen2019}, and we restate their
main result, which provides easily checked sufficient conditions for
geometric ergodicity of the underlying Markov chain. The section ends
with a description of a small empirical study comparing SS, RS and HS
Gibbs. The topic of Section~\ref{sec:adding} is the \textit{hybrid
scan sandwich} (HSS) algorithm, which is the result of adding
sandwich steps to a HS Gibbs sampler. We illustrate the construction
of HSS algorithms by adding sandwich steps to the HS algorithm for our
Student's $t$ example, and to the algorithm described in
Section~\ref{sec:glmm}. Section~\ref{sec:smn} deals with the
development and analysis of a HS Gibbs sampler for a Bayesian linear
regression model with scale mixtures of normal errors. A general
result providing sufficient conditions for geometric ergodicity is
stated and applied to two specific mixing densities. We close with a
discussion in Section~\ref{sec:discussion}. The Appendix contains
important theoretical results for the general HSS algorithm, as well
as a proof of the convergence rate result stated in
Section~\ref{sec:smn}.
\section{The General Linear Mixed Model with a Continuous Shrinkage Prior}
\label{sec:glmm}
The general linear mixed model takes the form
\begin{equation}
\label{eq:glmm}
Y = X\beta + Zu + e \;,
\end{equation}
where $Y$ is an $N \times 1$ data vector, $X$ and $Z$ are known
matrices with dimensions $N \times p$ and $N \times q$, respectively,
$\beta$ is an unknown $p \times 1$ vector of regression coefficients,
$u$ is a random vector whose elements represent the various levels of
the random factors in the model, $e \sim
\mbox{N}_N(0,\lambda_0^{-1}I),$ and the random vectors $e$ and $u$ are
independent. Suppose that the model contains $m$ random factors, so
that $u$ and $Z$ may be partitioned as $u = (u_1^T \; u_2^T \cdots
u_m^T)^T$ and $Z = (Z_1 \; Z_2 \cdots Z_m),$ where $u_i$ is $q_i
\times 1$, $Z_i$ is $N \times q_i$, and $q_1+ \cdots + q_m = q.$ Then
$Zu = \sum_{i=1}^m Z_i u_i.$ It is assumed that $u \sim
\mbox{N}_q(0,D),$ where $D = \bigoplus_{i=1}^m \lambda_i^{-1}
I_{q_i}.$ Finally, let $\lambda$ denote the vector of precision
components, i.e., $\lambda = (\lambda_0 \; \lambda_1 \cdots
\lambda_m)^T.$
A Bayesian version of the general linear mixed model requires
specification of a prior distribution for the unknown parameters
$\beta$ and $\lambda$. A popular choice is the proper (conditionally)
conjugate prior that takes $\beta$ to be multivariate normal, and
takes each of the precision components to be gamma. However, in the
increasingly important situation where $p$ is larger than $N,$ we may
wish to use a so-called Bayesian shrinkage prior on $\beta$ (see,
e.g., \citealt{griffin2010inference}). Indeed, \citet{abrahamsen2019}
considered the following Bayesian shrinkage version of the general
linear mixed model which incorporates the normal-gamma prior due to
\citet{griffin2010inference}:
\begin{align*}
& Y|\beta,u,\tau,\lambda \sim \mbox{N}_N(X\beta+Zu,\lambda_0^{-1}I_N)\;\\
& u|\beta,\tau,\lambda \sim \mbox{N}_q(0,D) \; \\
& \beta|\tau,\lambda \sim \mbox{N}_p(0,\lambda_0^{-1}D_{\tau}) \;
\end{align*}
where $D_{\tau}$ is a diagonal matrix with $\tau = (\tau_1 \; \tau_2
\cdots \tau_p)^T$ on the diagonal. Finally, all components of $\tau$
and $\lambda$ are assumed \textit{a priori} independent with
$\lambda_i \sim \text{Gamma}(a_i,b_i),$ for $i=0,1,\ldots,m,$ and
$\tau_j \sim \text{Gamma}(c,d),$ for $j=1,\ldots,p.$ There is evidence
(both empirical and theoretical) suggesting that values of $c$ in
$(0,1/2]$ lead to a posterior that concentrates on sparse $\beta$
vectors \citep{bhattacharya2012bayesian,bhattacharya2015dirichlet}.
Define $\theta = (\beta^T \; u^T)^T$ and $W = [X \; Z]$, so that
$W\theta = X\beta + Zu$. The vector $\tau$ is treated as latent data,
and the distribution of interest is the posterior distribution of
$(\theta,\lambda)$ given the data, $Y=y$. In terms of the notation
used in the Introduction, the role of $Z$ is being played here by
$\tau$, and the role of $(x,y)$ is being played by $(\theta,\lambda)$.
(Ideally, we would keep the notation consistent with that used in the
Introduction, but given how entrenched the roles of $X$, $y$ and $Z$
are in mixed linear regression models, adherence to the notation from
the Introduction would make this section rather difficult to read.)
Here is the full posterior density:
\begin{align}
\begin{split}
\label{eq:fullposterior}
\pi(\theta,\tau,\lambda|y) &\propto \lambda_0^{N/2} \exp \left\{
-\frac{\lambda_0}{2} (y-W\theta)^T (y-W\theta) \right\} \\ &\times
\lambda_0^{p/2} \left[ \prod_{j=1}^p \tau_j^{-1/2} \right] \exp
\left\{ -\frac{\lambda_0}{2} \beta^T D_{\tau}^{-1} \beta \right\}
\left[ \prod_{i=1}^m \lambda_i^{q_i/2} \right] \exp \left\{
-\frac{1}{2} u^T D^{-1} u \right\}\\ &\times \left[ \prod_{j=1}^p
\tau_j^{c-1} e^{-d\tau_j} I_{\reals_+}(\tau_j) \right] \left[
\prod_{i=0}^m \lambda_i^{a_i-1} e^{-b_i \lambda_i}
I_{\reals_+}(\lambda_i) \right] \;.\\
\end{split}
\end{align}
In order to use the basic DA algorithm, we would need to be able to
sample from $\pi(\tau | \theta, \lambda, y)$ and from $\pi(\theta,
\lambda | \tau, y)$. The former is not a problem, as we now explain.
We write $V \sim \text{GIG}(\zeta,\xi,\psi)$ to mean that $V$ has a
generalized inverse Gaussian distribution with density
\begin{equation}
\label{eq:gig}
\frac{\xi^{\zeta/2}}{2 \psi^{\zeta/2} \; \text{K}_{\zeta}(\sqrt{\xi \psi})} v^{\zeta - 1} e^{-\frac{1}{2}(\xi v + \frac{\psi}{v})}I_{\reals_+}(v),
\end{equation}
where $\xi > 0, \psi > 0,$ and $\text{K}_{\zeta}(\cdot)$ denotes the
modified Bessel function of the second kind. Conditional on $(\theta,
\lambda, y),$ the components of $\tau$ are independent with $$\tau_j
\sim \text{GIG}(c-1/2, 2d, \lambda_0 \beta_j^2).$$ Unfortunately, it
is not straightforward to make draws from $\pi(\theta, \lambda | \tau,
y)$. Thus, the DA algorithm is not applicable. On the other hand, the
conditional density of $\theta$ given $(\lambda, \tau, y)$ is
multivariate normal, and, given $(\theta, \tau, y),$ the components of
$\lambda$ are independent gammas. Hence, the HS Gibbs algorithm is
applicable.
We now state the conditional densities, beginning with $\lambda.$
First, $$\lambda_0 | \theta, \tau, y \sim \text{Gamma} \left(
\frac{N+p+2a_0}{2}, \frac{\norm{y-W\theta}^2}{2} + \frac{\beta^T
D_{\tau}^{-1} \beta}{2} + b_0 \right).$$ Now, for $i =
1,2,\ldots,m,$ we have $$\lambda_i | \theta, \tau, y \sim \text{Gamma}
\left( \frac{q_i + 2a_i}{2}, \frac{\norm{u_i}^2}{2} + b_i \right).$$
Now, define $T_{\lambda,\tau} = \lambda_0 (X^T X + D_{\tau}^{-1}),
M_{\lambda,\tau} = I - \lambda_0 X^T T_{\lambda,\tau}^{-1} X^T,$ and
$Q_{\lambda,\tau} = \lambda_0 Z^T M_{\lambda,\tau} Z + D^{-1}.$
Conditional on $(\lambda,\tau,y), \theta$ is $(p+q)$-variate normal
with mean
\begin{equation*}
\Theta := \text{E}[\theta | \tau, \lambda, y]
=
\begin{bmatrix}
\lambda_0 T_{\lambda, \tau}^{-1} X^T y - \lambda_0^2 T_{\lambda,
\tau}^{-1} X^T Z Q_{\lambda, \tau}^{-1} Z^T M_{\lambda,\tau} y
\\ \lambda_0 Q_{\lambda, \tau}^{-1} Z^T M_{\lambda,\tau} y
\end{bmatrix},
\end{equation*}
and covariance matrix
\begin{equation*}
\Sigma := \text{Var}[\theta | \tau, \lambda, y]
=
\begin{bmatrix}
T_{\lambda, \tau}^{-1} + \lambda_0^2 T_{\lambda, \tau}^{-1} X^T Z
Q_{\lambda, \tau}^{-1} Z^T X T_{\lambda, \tau}^{-1} & -\lambda_0
T_{\lambda, \tau}^{-1} X^T Z Q_{\lambda, \tau}^{-1} \\ -\lambda_0
Q_{\lambda, \tau}^{-1} Z^T X T_{\lambda, \tau}^{-1} & Q_{\lambda,
\tau}^{-1}
\end{bmatrix}.
\end{equation*}
The HS Gibbs sampler is based on the Markov chain $\Phi = \{(\theta_n,
\lambda_n)\}_{n=0}^{\infty}$ with state space $\mathsf{X} =
\reals^{p+q} \times \reals_+^{m+1}$ and fixed selection probability $r
\in (0,1)$. If the current state is $(\theta_n,
\lambda_n)=(\theta,\lambda),$ then we simulate the new state,
$(\theta_{n+1}, \lambda_{n+1}),$ using the following two-step
procedure.
\vskip .2truecm\hfill \hrule height.5pt \vskip .2truecm \vspace*{2mm}
\noindent {\rm Iteration $n+1$ of the HS Gibbs sampler:}
\begin{enumerate}
\item Draw $\{\tau_j\}_{j=1}^p$ independently with $\tau_j \sim
\text{GIG}(c-1/2, 2d, \lambda_0 \beta_j^2)$, let $\tau = (\tau_1 \;
\tau_2 \cdots \tau_p)^T,$ and, independently, draw $W \sim
\text{Uniform}(0,1).$
\item
\begin{enumerate}
\item If $W \le r,$ draw $(\lambda^*_0,\ldots,\lambda^*_m)$
independently with $$\lambda^*_0 \sim \text{Gamma} \left(
\frac{N+p+2a_0}{2}, \frac{\norm{y-W\theta}^2}{2} + \frac{\beta^T
D_{\tau}^{-1} \beta}{2} + b_0 \right),$$ and for
$i=1,\dots,m$, $$\lambda^*_i \sim \text{Gamma} \left( \frac{q_i +
2a_i}{2}, \frac{\norm{u_i}^2}{2} + b_i \right),$$ and let
$\lambda^* = (\lambda^*_0 \; \lambda^*_1 \cdots \lambda^*_m)^T.$ Set
$(\theta_{n+1}, \lambda_{n+1}) = (\theta, \lambda^*)$.
\item Otherwise if $r < W \le 1,$ draw $$\theta^* \sim \mbox{N}_{p+q}
\left( \Theta, \Sigma \right),$$ and set $(\theta_{n+1},
\lambda_{n+1}) = (\theta^*, \lambda).$
\end{enumerate}
\end{enumerate}
\vspace*{-3mm}
\vskip -.1truecm\hfill \hrule height.5pt \vskip .4truecm
\bigskip
We know from the discussion in the Introduction that the Markov chain
driving this algorithm is reversible with respect to
$\pi(\theta,\lambda|y)$. Furthermore, it is straightforward to show
that this chain is Harris ergodic (i.e., irreducible, aperiodic and
Harris recurrent). \citet{abrahamsen2019} analyzed this HS Gibbs
sampler, and proved that it is geometrically ergodic under mild
regularity conditions. Here is their main result.
\begin{theorem}
\label{thm:glmm}
The HS Gibbs Markov chain, $\{(\theta_n,\lambda_n)\}_{n=0}^{\infty}$
is geometrically ergodic for all $r \in (0,1)$ if
\begin{enumerate}
\item $Z := (Z_1 \; Z_2 \; \cdots \; Z_m)$ has full column rank.
\item $a_0 > \frac{1}{2} \big( \hbox{rank}(X) - N + (2c+1)p+2 \big)$,
and
\item $a_i > 1$ for each $i \in \{1,2,\ldots,m\}$.
\end{enumerate}
\end{theorem}
The conditions of Theorem~\ref{thm:glmm} are quite easy to check, and
the result is applicable when $p > N$. Moreover, there are no known
convergence rate results for the corresponding SS and RS Gibbs
samplers. Indeed, \citet{abrahamsen2019} contend that HS Gibbs is
much easier to analyze than the other two, despite being no more
difficult to implement.
We now compare the HS, SS, and RS Gibbs samplers in the context of a
numerical example. (No numerical results were presented in
\citet{abrahamsen2019}.) We also include in the comparison the hybrid
scan sandwich algorithm, which is developed in
Subsection~\ref{ssec:glmm}. We consider three simulation settings
corresponding to the situations where $N > p$, $N = p$, and $N < p$,
respectively, in order to account for the effects of the shrinkage
prior. The elements of the design matrix $X$ were chosen by
generating iid $N(0,1)$ random variables. There is one random effect
with 5 levels, i.e., $m=1$ and $q_1=q=5$, and we use the standard cell
means model structure for the matrix $Z$. Recall from
Theorem~\ref{thm:glmm} that there are several restrictions on the
hyperparameters that must be adhered to in order for the HS Gibbs
Markov chain to be geometrically ergodic. This sometimes requires
$a_0$ to be large. We mitigate this by setting $b_0 = a_0$ in each
simulation setting to give the corresponding prior distribution a mean
of 1. We set $a_1 = 1.5$ and $b_1 = 1$ for all three simulations.
Also, recall that there is empirical and theoretical evidence
suggesting that values of $c$ in $(0,1/2]$ lead to a posterior that
concentrates on sparse $\beta$ vectors. Accordingly, we set $c=1/4$
and $d=1$ throughout. Here is a summary of the simulation settings
considered.
\begin{center}
Table 1:
\textit{Hyperparameter settings}
\end{center}
\begin{center}
\begin{tabular}{ccccccccccc}
\hline
\textbf{Setting} & $\boldsymbol{N}$ & $\boldsymbol{p}$ & $\boldsymbol{m}$ & $\boldsymbol{q}$ & $\boldsymbol{a_0}$ & $\boldsymbol{b_0}$ & $\boldsymbol{a_1}$ & $\boldsymbol{b_1}$ & $\boldsymbol{c}$ & $\boldsymbol{d}$ \\ \hline
1 & 100 & 10 & 1 & 5 & 1 & 1 & 1.5 & 1 & 0.25 & 1 \\
2 & 100 & 100 & 1 & 5 & 77 & 77 & 1.5 & 1 & 0.25 & 1 \\
3 & 100 & 200 & 1 & 5 & 152 & 152 & 1.5 & 1 & 0.25 & 1 \\ \hline
\end{tabular}
\end{center}
\medskip
\noindent In each case, the data were simulated according to the model
using a ``bottom up'' strategy, i.e., the hyperparameters were
randomly drawn from their priors, and so on, up the hierarchy.
We fix the selection probability at $r = 1/2$ for the HS and HSS
algorithms. For RS Gibbs, we fix the selection probabilities at
$r_1=r_2=r_3=1/3$. We wish to compare the algorithms using
autocorrelation plots, but the four algorithms make different numbers
of updates per iteration. Indeed, the SS, HS, HSS and RS algorithms
make 3, 2, 2 and 1 updates/iteration, respectively. So an adjustment
must be made in order to perform an ``apples to apples'' comparison.
If $k$ is a positive integer, then it seems fair to compare the lag
$2k$ autocorrelation for SS algorithm with the lag $3k$
autocorrelation for the HS and HSS algorithms, and the lag $6k$
autocorrelation for the RS algorithm.
In each of the three separate simulations, we ran the SS, HS, HSS, and
RS algorithms for 40,000 iterations, 60,000 iterations, 60,000
iterations, and 120,000 iterations, respectively. We then discarded
the first half of each run as burn-in, and computed the
autocorrelations based on the remaining data as described above. We
used the function $(y - W \theta)^T(y - W \theta) + \lambda_0 +
\lambda_1$ because it involves both parameters of interest ($\theta$
and $\lambda$). The results are summarized in
Figure~\ref{fig:simplot}. (Just to be clear, as an example, what is
plotted above the abscissa at the value 6 for the SS, HS, HSS, and RS
algorithms is the estimated autocorrelation for lag 6, 9, 9, and 18,
respectively.) We can clearly see that for all three simulations, the
magnitude of the autocorrelations for SS Gibbs is the lowest, while
the other three are all a bit higher and quite close to each other.
The performances of the HS Gibbs sampler and the HSS algorithm are
similar.
\begin{figure}[H]
\includegraphics[width=\linewidth]{Updated_Plot_v2.png}
\caption{Autocorrelations for SS, HS, HSS, and RS algorithms}
\label{fig:simplot}
\end{figure}
While it is true that the SS Gibbs algorithm seems to be marginally
better than the others in this particular case, recall that it remains
unknown whether the corresponding Markov chain is geometrically
ergodic. On the other hand, the HS chain (and the HSS chain - see
Subsection~\ref{ssec:HSS}) are both known to be geometrically ergodic.
Thus, in order to ensure reliability, we recommend the two ``known
quantities.''
\section{The Hybrid Scan Sandwich Algorithm}
\label{sec:adding}
In this section, we explain how to add sandwich steps to the HS Gibbs
sampler to form the \textit{hybrid scan sandwich} (HSS) algorithm.
There are four subsections. The basic sandwich algorithm of
\citet{hobert2008theoretical} is described in
Subsection~\ref{ssec:basics}. A generic description of the HSS
algorithm is provided in Subsection~\ref{ssec:HSS}. In
Subsection~\ref{ssec:toy}, we illustrate the techniques using the
Student's $t$ example from the Introduction. Finally, in
Subsection~\ref{ssec:glmm}, we develop a HSS algorithm for the
intractable posterior associated with the mixed model discussed in
Section~\ref{sec:glmm}.
\subsection{The sandwich algorithm}
\label{ssec:basics}
In keeping with the notation of the Introduction, recall that the
transition associated with a single iteration of the DA algorithm may
be represented as
\[
\begin{tikzcd}
U \arrow[r] & Z \arrow[r] & U' \;.
\end{tikzcd}
\]
Building on ideas in \citet{liu1999parameter}, \citet{meng1999seeking}
and \citet{van2001art}, \citet{hobert2008theoretical} introduced an
alternative to the DA algorithm that employs an extra move on the
$\mathsf{Z}$ space that is ``sandwiched'' between the two conditional
draws. If the extra move is chosen carefully, it can break the
correlation between consecutive iterates of the DA algorithm, thereby
speeding up the algorithm. Again, using notation from the
Introduction, let $f_Z$ denote the $z$-marginal of $f(u,z)$, and
suppose that $R(z,dz')$ is \textit{any} Markov transition function
(Mtf) that is reversible with respect to $f_Z$, i.e., $R(z,dz') f_Z(z)
dz = R(z',dz) f_Z(z') dz'$. The \textit{sandwich algorithm} simulates
the Markov chain whose Mtd is
\[
k_{\text{S}}(u' | u) = \int_{\mathsf{Z}} \int_{\mathsf{Z}}
f_{U|Z}(u'|z') R(z,dz') f_{Z|U}(z|u) \, dz \;.
\]
It's easy to see that $k_{\text{S}}(u' | u) f_U(u)$ is symmetric in
$(u,u')$, so the sandwich Markov chain is reversible with respect to
$f_U$. Also, the sandwich algorithm reduces to DA if we take $R$ to
be the trivial Mtf whose chain never moves from the starting point.
To run the sandwich algorithm, we simply run the DA algorithm as
usual, except that after each $z$ is drawn, we perform the extra step
$z' \sim R(z,\cdot)$ before drawing the new $u$. Hence, the sequence
of steps in a single iteration of the sandwich algorithm looks like
this:
\[
\begin{tikzcd}
U \arrow[r] & Z \arrow[r] & Z' \arrow[r] & U' \;.
\end{tikzcd}
\]
We now explain how a sandwich step can effectively break the
correlation between $U$ and $U'$ in the context of a toy example.
Suppose the target density is
\[
f_U(u) = \int_{\mathbb{R}} \frac{1}{\sqrt{8\pi}} e^{-\frac{1}{2}
(u-z)^2 -|z|} \; dz \;.
\]
In order to construct a DA algorithm, we require a joint density whose
$u$-marginal is the target. Here's an obvious candidate:
\[
f_{U,Z}(u,z) = \frac{1}{\sqrt{8\pi}} e^{-\frac{1}{2} (u-z)^2 -|z|} \;.
\]
Note that $f_Z(z) = \frac{1}{2} e^{-|z|}$, so the marginal
distribution of $Z$ is standard Laplace (or double exponential). In
order to run the DA algorithm, we need the full conditionals.
Clearly, $U|Z \sim \mbox{N}(Z,1)$, but the distribution of $Z$ given
$U$ is non-standard:
\[
f_{Z|U}(z|u) \propto e^{-\frac{1}{2} (u-z)^2 -|z|} \;.
\]
It's a simple matter to simulate from this density using a rejection
sampler with a Laplace candidate. We now construct a sandwich
algorithm. Define a Mtf on $\mathbb{R}$ as follows:
\[
R(z,dz') = r(z'|z) dz' = e^{-|z'|} \big[ I_{\mathbb{R}_+}(z')
I_{\mathbb{R}_+} (z) + I_{\mathbb{R}_-}(z') I_{\mathbb{R}_-} (z)
\big] dz' \;,
\]
where $\mathbb{R}_- := (-\infty,0]$. It's clear that $R(z,dz')$ is
reversible with respect to $f_Z(z)$. Note that the Markov chain
defined by $R$ is \textit{not} irreducible. In fact, the chain
remains forever on whichever side of zero it is started. We now
provide some intuition about how the extra step breaks the
correlation between $U$ and $U'$. Imagine for a moment that
$r(z'|z)$ were just $f_Z(z')$. Then $U'$ would be a perfect draw
from $f_U$ (independent of $U$), and the Markov chain would simply
be an iid sequence from the target distribution. Of course,
$r(z'|z)$ is not $f_Z(z')$, but it actually isn't that far from it.
First, $r(z'|z)$ depends on $z$ \textit{only through its sign}.
Now, when $z>0$, $r(z'|z)$ is nothing but $f_Z(z')$ truncated to the
positive half-line, and when $z \le 0$, $r(z'|z)$ is just $f_Z(z')$
truncated to the negative half-line. So we can interpret the extra
step as follows: Once $Z$ is drawn from $f_{Z|U}(\cdot|u)$, the
extra step then draws $Z'$ from a truncated version of $f_Z$.
Intuitively, it seems clear that the correlation between $U$ and
$U'$ should be quite a bit weaker under the sandwich dynamics, than
under the DA dynamics. In order to test this empirically, we ran
stationary versions of each chain for one million iterations, and
constructed the autocorrelation plot in Figure~\ref{fig:toyacf}
using the function $V(u)=u^2$. Clearly, the autocorrelation of the
sandwich Markov chain decays to zero much more rapidly than that of
the corresponding DA chain.
\begin{figure}[H]
\includegraphics[width=\linewidth]{toy_acf_plot_v3.png}
\caption{Autocorrelations for the DA and sandwich algorithms}
\label{fig:toyacf}
\end{figure}
Of course, a sandwich algorithm is a useful alternative to the
underlying DA algorithm only if the computational burden of drawing
from $R$ is small relative to the improvement it provides. Consider,
for example, the Mtf $R(z,dz') = r(z'|z)dz'$ where $r(z'|z) =
\int_{\mathsf{U}} f_{Z|U}(z'|u) f_{U|Z}(u|z)du$. This $R$ leads to a
sandwich algorithm that is nothing but two consecutive iterations of
the DA algorithm. Thus, whatever is gained by adding the extra step
is offset exactly in increased computational effort. Fortunately, it
is often possible to find an $R$ that leads to a significant
improvement, while adding very little to the overall computational
cost. This is typically accomplished by choosing $R(z,dz')$ such that,
for fixed $z$, the (reducible) chain driven by $R(z,\cdot)$ lives in a
low dimensional subspace of $\mathsf{Z}.$ (Note that such an $R$ would
typically not have a Mtd with respect to Lebesgue measure on
$\mathsf{Z}$, and this is the reason why it is defined via its Mtf,
instead of a Mtd.)
There are a couple of simple techniques for constructing sandwich
moves \citep[see, e.g.,][]{hobert2008theoretical,liu1999parameter},
and the resulting Mtfs can often be simulated with relatively little
computational effort. In such cases, there is nothing to lose by
adding the step. In other cases, where simulation of the extra step
requires substantial computational effort, one must decide if the
trade-off is worthwhile. There are many examples of sandwich
algorithms that drastically outperform their DA counterparts in
empirical studies, see, e.g., \citet{liu1999parameter} and
\citet{meng1999seeking}. Moreover, the superiority of the sandwich
algorithm has also been established theoretically. Indeed, results in
\citet{hobert2008theoretical} and \citet{khare2011spectral} show that,
under mild regularity conditions, the sandwich algorithm converges at
least as fast as the DA algorithm, and is at least as good in the
sense of asymptotic variance.
\subsection{The HSS algorithm}
\label{ssec:HSS}
We now explain how to add up to two different sandwich steps to the HS
Gibbs sampler. Recall that the transition mechanism for each iteration of HS Gibbs with selection probability $r$ is given by
\[
\begin{tikzcd}
& & (X^*,Y) \\
(X,Y) \arrow[r] & Z \arrow[ur,"r"] \arrow[dr,"1-r"] & \;\\
& & (X,Y^*)
\end{tikzcd}
\]
For fixed $y \in \mathsf{Y},$ let $R_1(z,dz';y)$
denote a Mtf on $\mathsf{Z}$ that is reversible with respect to
$f_{Z|Y}(z|y),$ so that
\begin{equation}
\label{eq:reverse}
R_1(z,dz';y) f_{Z|Y}(z|y) dz = R_1(z',dz;y) f_{Z|Y}(z'|y) dz' \;.
\end{equation}
Define
\[
k_1(x'|x;y) = \int_{\mathsf{Z}} \int_{\mathsf{Z}} f_{X|Y,Z}(x'|y,z')
R_1(z,dz';y) f_{Z|X,Y}(z|x,y) \, dz \;.
\]
A routine calculation shows that $k_1(x'|x;y)f_{X|Y}(x|y)$ is
symmetric in $(x,x')$, so $k_1(x'|x;y)$ is reversible with respect to
$f_{X|Y}(x|y)$. Analogously, for fixed $x \in \mathsf{X}$,
define
\[
k_2(y'|y;x) = \int_{\mathsf{Z}} \int_{\mathsf{Z}} f_{Y|X,Z}(y'|x,z')
R_2(z,dz';x) f_{Z|X,Y}(z|x,y) \, dz \;,
\]
where $R_2(z,dz';x)$ is reversible with respect to $f_{Z|X}(z|x)$.
The HSS algorithm is simply a RS algorithm which, at each iteration,
employs either $k_1(x'|x;y)$ or $k_2(y'|y;x)$. In particular, fix $r
\in (0,1)$, and consider a Markov chain $\{(\tilde{X}_n,\tilde{Y}_n)
\}_{n=0}^{\infty}$ with state space $\mathsf{X} \times \mathsf{Y}$
that evolves as follows. If the current state is
$(\tilde{X}_n,\tilde{Y}_n) = (x,y)$, then we simulate the new state,
$(\tilde{X}_{n+1},\tilde{Y}_{n+1})$, using the following two-step
procedure.
\vskip .2truecm\hfill \hrule height.5pt \vskip .2truecm \vspace*{2mm}
\noindent {\rm Iteration $n+1$ of the HSS algorithm:}
\begin{enumerate}
\item Draw $Z \sim f_{Z|X,Y}(\cdot|x,y)$, call the result $z$, and,
independently, draw $W \sim \text{Uniform}(0,1)$.
\item
\begin{enumerate}
\item If $W \le r$, draw $Z' \sim R_1(z,\cdot;y)$, call the result
$z'$, draw $X^* \sim f_{X|Y,Z}(\cdot|y,z')$, and set
$(\tilde{X}_{n+1},\tilde{Y}_{n+1}) = (X^*,y)$.
\item Otherwise if $r < W \le 1$, draw $Z' \sim R_2(z,\cdot;x)$, call
the result $z'$, draw $Y^* \sim f_{Y|X,Z}(\cdot|x,z')$, and set
$(\tilde{X}_{n+1},\tilde{Y}_{n+1}) = (x,Y^*)$.
\end{enumerate}
\end{enumerate}
\vspace*{-3mm}
\vskip -.1truecm\hfill \hrule height.5pt \vskip .4truecm
\bigskip
\newpage
Thus, the HSS algorithm makes the following transition at each
iteration.
\[
\begin{tikzcd}
& & Z' \arrow[r] & (X^*,Y) \\
(X,Y) \arrow[r] & Z \arrow[ur,"r"] \arrow[dr,"1-r"] & \\
& & Z' \arrow[r] & (X,Y^*)
\end{tikzcd}
\]
If we take both $R_1$ and $R_2$ to be trivial, then the HSS algorithm
collapses back into the HS Gibbs sampler. In
Appendix~\ref{app:theory}, we develop theoretical results for the HSS
algorithm. We begin by showing that the HSS algorithm is reversible,
which allows us to prove analogues for the HSS algorithm of the strong
theoretical results that have been established for the basic sandwich
algorithm. In particular, we prove that the HSS algorithm is
\textit{always} at least as good as HS Gibbs in terms of asymptotic
variance, and that the HSS Markov chain converges at least as fast as
the HS Gibbs chain as long as the Markov operators associated with
$R_1$ and $R_2$ are both \textit{positive}. (All of the $R$s employed
in this paper, including the trivial $R$, yield positive Markov
operators - see \citet{hobert2008theoretical}.) One important consequence of the convergence rate result is that, when
$R_1$ and $R_2$ are both positive operators, geometric ergodicity of the
HS Gibbs Markov chain implies that of the HSS chain.
This is extremely useful in practice because the HS Gibbs algorithm is
much simpler than the HSS algorithm, and hence much easier to analyze.
We should point out that \citet{pal2015improving} also developed an
alternative to SS and RS Gibbs for Bayesian latent data models that is
based on the sandwich methodology of \citet{hobert2008theoretical}.
Unfortunately, it is difficult to obtain theoretical results for their
algorithm because the corresponding Markov chain is not reversible.
Recall that near the end of the Introduction we considered a
generalization in which $U$ is partitioned into three or more pieces,
and we wrote the corresponding conditional densities as $f_{X_j |
X_{-j}, Z}$ for $j=1,\dots,s$. It is a simple matter to extend the
methodology described above to this more general case. Indeed, for
fixed $X_{-j} = x_{-j}$, let $R_j(z,dz';x_{-j})$ denote a Mtf on
$\mathsf{Z}$ that is reversible with respect to $f_{Z |
X_{-j}}(z|x_{-j})$. Define
\[
k_j(x'_j | x_j; x_{-j}) = \int_{\mathsf{Z}} f_{X_j | X_{-j},
Z}(x'|x_{-j},z') R_j(z,dz';x_{-j}) f_{Z|X_j, X_{-j}}(z|x_j,x_{-j})
\, dz \;.
\]
At each step of the generalized version of HSS, we choose among
$k_1,\ldots,k_s$ according to positive probabilities $a_1,\dots,a_s$
in the usual way, and apply the chosen $k_j$. All of the theoretical
results that we establish for the HSS algorithm in
Appendix~\ref{app:theory} can be easily extended to this
generalization.
\subsection{Student's $t$ example}
\label{ssec:toy}
Consider again the first Student's $t$ model from the Introduction
(with prior $\pi(\mu,\sigma^2) \propto \sigma^{-2}
I_{\mathbb{R}^+}(\sigma^2)$). We now develop a HSS algorithm for this
model. Of course, we already know that this model can be handled by
the usual DA algorithm, so our HSS algorithm would never be used in
practice. However, we believe that it is instructive to demonstrate
the construction of a HSS algorithm in a simple context where the
details of the model are not themselves overwhelming.
The first step is to identify the distributions of $Z | \mu$ and $Z |
\sigma^2$. Let $z_{\cdot} = \sum_{i=1}^m z_i$. It's easy to show
that
\begin{align*}
f_{Z|\mu}(z|\mu) & \propto \bigg( \sum_{i=1}^m z_i (w_i - \mu)^2
\bigg)^{-\frac{m}{2}} \Bigg[ \prod_{i=1}^m z_i \Bigg]^{\frac{\nu -
1}{2}} \exp \bigg \{ \! -\frac{z_{\cdot} \nu}{2} \bigg \}
\prod_{i=1}^m I_{\mathbb{R}^+}(z_i) \;.
\end{align*}
Let $g \in \reals_+$. It follows from the group theoretic arguments
in \citet{hobert2008theoretical} that the move $z \mapsto g z$ for
$z=(z_1,\ldots,z_m)$ is reversible with respect to $f_{Z|\mu}(z|\mu)$
if $g$ is drawn from the density proportional to $f_{Z|\mu}(gz | \mu)
g^{m-1}$. (This is a low-dimensional move since, for fixed $z \in
\reals_+^m$, the points $gz$ all lie on a ray emanating from the
origin and passing through the point $z$.) As a function of $g$, we
have
\begin{align*}
f_{Z|\mu}(gz | \mu) g^{m-1} & \propto g^{\frac{m \nu}{2} - 1} \exp
\bigg \{ \! -\frac{g \nu z_{\cdot}}{2} \bigg \} \Bigg[ \prod_{i=1}^m
I_{\mathbb{R}^+}(z_i) \Bigg] I_{\mathbb{R}^+}(g) \;,
\end{align*}
which is a $\text{Gamma}\big( \frac{m \nu}{2}, \frac{\nu z_{\cdot}}{2}
\big)$ density. Now, it's easy to show that
\begin{align*}
f_{Z|\sigma^2}(z|\sigma^2) & \propto \frac{1}{\sqrt{z_{\cdot}}} \exp
\bigg \lbrace \! -\frac{z_{\cdot} v(z,w)}{2\sigma^2} \bigg \rbrace
\Bigg[ \prod_{i=1}^m z_i \Bigg]^{\frac{\nu - 1}{2}} \exp \bigg \{ \!
-\frac{z_{\cdot} \nu}{2} \bigg \} \prod_{i=1}^m I_{\mathbb{R}^+}(z_i)
\;,
\end{align*}
where
\[
v(z,w) = \frac{1}{z_{\cdot}} \sum_{i=1}^m z_i \big( w_i - \theta(z,w)
\big)^2 \;,
\]
and $\theta(z,w) = \frac{1}{z_{\cdot}} \sum_{i=1}^m z_i w_i$. Using
the same transformation, $z \mapsto g z$, we need to sample $g$ from
the density proportional to $f_{Z|\sigma^2}(gz | \sigma^2) g^{m-1}$.
A straightforward calculation shows that, as a function of $g$, we
have
\begin{align*}
f_{Z|\sigma^2}(gz | \sigma^2) g^{m-1} & \propto g^{\frac{m(\nu
+1)-3}{2}} \exp \left\lbrace -g z_. \bigg(
\frac{v(z,w)}{2\sigma^2} + \frac{\nu}{2} \bigg) \right \rbrace \;,
\end{align*}
which is a $\text{Gamma} \Big( \frac{m(\nu + 1) - 1}{2} , z_. \big(
\frac{v(z,w)}{2\sigma^2} + \frac{\nu}{2} \big) \Big)$ density.
Fix a selection probability $r \in (0,1)$ and consider the Markov
chain $\{(\tilde{\mu}_n,\tilde{\sigma}^2_n)\}_{n=0}^{\infty}$ with
state space $\reals \times \reals_+$. The HSS algorithm proceeds as
follows. If the current state is $(\tilde{\mu}_n,\tilde{\sigma}^2_n)
= (\mu,\sigma^2)$, then we simulate the next state,
$(\tilde{\mu}_{n+1},\tilde{\sigma}^2_{n+1}),$ by performing the
following two steps:
\vskip .2truecm\hfill \hrule height.5pt \vskip .2truecm \vspace*{2mm}
\noindent {\rm Iteration $n+1$ of the HSS algorithm for the Student's
$t$ example:}
\begin{enumerate}
\item Draw $Z_1, \ldots, Z_m$ independently, with $$Z_i \sim
\text{Gamma} \left( \frac{\nu + 1}{2}, \frac{1}{2} \left( \frac{(w_i
- \mu)^2}{\sigma^2} + \nu \right) \right) \;,$$ call the observed
values $z=(z_1,\ldots,z_m),$ and, independently, draw $W \sim
\text{Uniform}(0,1).$
\item
\begin{enumerate}
\item If $W \le r,$ draw $$g \sim \text{Gamma} \bigg( \frac{m \nu}{2},
\frac{\nu z_{\cdot}}{2} \bigg) \,,$$ then draw $$\sigma^{*2} \sim
\text{IG} \bigg( \frac{m}{2}, \frac{1}{2} \sum_{i=1}^m g z_i (y_i -
\mu)^2 \bigg),$$ and set $(\tilde{\mu}_{n+1},\tilde{\sigma}^2_{n+1})
= (\mu,\sigma^{*2}).$
\item Otherwise if $r < W \le 1,$ draw $$g \sim \text{Gamma} \bigg(
\frac{m(\nu + 1) - 1}{2} , z_. \Big( \frac{v(z,w)}{2\sigma^2} +
\frac{\nu}{2} \Big) \bigg) \;,$$ and then draw $$\mu^* \sim \text{N}
\left( \theta(z,w), \frac{\sigma^2}{g z_.} \right),$$ and set
$(\tilde{\mu}_{n+1},\tilde{\sigma}^2_{n+1}) = (\mu^*,\sigma^2)$.
\end{enumerate}
\end{enumerate}
\vspace*{-3mm}
\vskip -.1truecm\hfill \hrule height.5pt \vskip .4truecm
\bigskip
In terms of computation, the difference between running one iteration
of this HSS algorithm versus one iteration of the HS Gibbs sampler
upon which it is based is a single draw from the gamma distribution.
Thus, if $m$ is even moderately large, this extra draw would add
relatively little to the overall computational effort of the HS Gibbs
algorithm.
\subsection{General linear mixed model example}
\label{ssec:glmm}
\citet{abrahamsen2019} introduced and analyzed the HS Gibbs sampler
described in Section~\ref{sec:glmm}, but they did not consider adding
sandwich steps to their algorithm. In this subsection, we develop a
HSS algorithm with a single sandwich step based on the conditional
density $\pi(\tau|\theta,y)$. (It is much more difficult to construct
a sandwich step based on $\pi(\tau|\lambda,y)$.) A routine
calculation shows that
\[
\pi(\tau|\theta,y) \propto \left( \frac{\norm{y-W\theta}^2}{2} +
\frac{\beta^T D_{\tau}^{-1} \beta}{2} + b_0 \right)^ {-
\left(\frac{N}{2}+\frac{p}{2}+a_0 \right)} \prod_{j=1}^p
\tau_j^{c-\frac{3}{2}} e^{-d \tau_j} I_{\reals_+}(\tau_j) \,.
\]
As in the previous subsection, the move $\tau \mapsto g \tau$ is
reversible with respect to $\pi(\tau|\theta,y)$ if $g$ is drawn from
the density proportional to $\pi(g \tau|\theta,y) g^{p-1}
I_{\reals_+}(g)$. Now, as a function of $g$,
\[
\pi(g \tau | \theta, y) \propto \left( \frac{\norm{y-W\theta}^2}{2} +
\frac{g^{-1} \beta^T D_{\tau}^{-1} \beta}{2} + b_0 \right)^ {-
\left(\frac{N}{2}+\frac{p}{2}+a_0 \right)} g^{p(c-\frac{3}{2})}
e^{-g \left( d \sum_{j=1}^p \tau_j \right)} \prod_{j=1}^p
I_{\reals_+}(\tau_j) \;,
\]
so the density from which $g$ must be drawn is given by
\[
h(g ; \tau, \theta, y) \propto
\frac{g^{\frac{N}{2}+cp+a_0-1-s}}{\left( \frac{\beta^T D_{\tau}^{-1}
\beta}{2} + g \left( \frac{\norm{y-W\theta}^2}{2} + b_0 \right)
\right)^{\frac{N}{2}+\frac{p}{2}+a_0}} \; \left[ g^s e^{-g \left( d
\sum_{j=1}^p \tau_j \right)} \right] I_{\reals_+}(g) \;,
\]
where $s > 0$ is a free parameter. So,
\begin{equation}
\label{eq:hg}
h(g ; \tau, \theta, y) \propto \frac{g^{\frac{N}{2}+cp+a_0-1-s}}{(1 +
C g)^{\frac{N}{2}+\frac{p}{2}+a_0}} \; \left[ g^s e^{-g \left( d
\sum_{j=1}^p \tau_j \right)} \right] I_{\reals_+}(g) \;,
\end{equation}
where
\[
C = \frac{\norm{y-W\theta}^2 + 2 b_0}{\beta^T D_{\tau}^{-1}\beta} \;.
\]
If we choose $s \in \Big( \max \Big\{ 0, \; p \big(c - \frac{1}{2}
\big) \Big \} , \frac{N}{2} + cp + a_0 \Big)$, then two things happen:
(1) the first term on the right-hand side of ~\eqref{eq:hg} is
proportional to a scaled $F$ density, and (2) the second term is
bounded. In fact, the second term achieves its maximum at $\hat{g} = s
\big( d \sum_{j=1}^p \tau_j \big)^{-1}$. Thus, we can use a simple
accept/reject algorithm with an $F$ candidate to draw from $h.$ In
particular, let $\nu_1 = N + 2cp + 2a_0 -2s$ and $\nu_2 = p(1-2c) +
2s.$ Here's the algorithm.
\vskip .2truecm\hfill \hrule height.5pt \vskip .2truecm \vspace*{2mm}
\noindent {\rm Accept/Reject algorithm for $h$:}
\begin{enumerate}
\item Draw $V^* \sim F(\nu_1,\nu_2),$ set $V = (V^* \nu_1) / (C
\nu_2),$ and independently draw $U \sim \text{Uniform}(0,1).$
\item If $$U \le \left( \frac{d V \sum_{j=1}^p \tau_j}{s} \right)^s \;
e^{s - d V \sum_{j=1}^p \tau_j},$$ then accept $V$ as a draw from
~\eqref{eq:hg}, otherwise return to 1.
\end{enumerate}
\vspace*{-3mm}
\vskip -.1truecm\hfill \hrule height.5pt \vskip .4truecm
\bigskip
If $r \in (0,1)$ is the selection probability, then our HSS algorithm
proceeds as follows. Let the current state of the chain be
$(\theta_n, \lambda_n) = (\theta, \lambda)$. First, draw $\tau \sim
\pi(\tau | \theta, \lambda, y)$, and then flip an $r$-coin. If the
coin comes up heads, we move to $(\theta_{n+1}, \lambda_{n+1}) =
(\theta, \lambda^*)$ by first drawing $g \sim h(\cdot ; \tau, \theta,
y)$ and then drawing $\lambda^* \sim \pi(\lambda| \theta, g \tau, y)$.
If the coin comes up tails, we move to $(\theta_{n+1}, \lambda_{n+1})
= (\theta^*, \lambda)$ by drawing $\theta^* \sim \pi(\theta | \lambda,
\tau, y)$. Another, perhaps simpler, way to describe the HSS
algorithm is via a simple modification of the HS Gibbs algorithm described
in Section~\ref{sec:glmm}. Step 1 remains exactly the same. In step
2, if $r < W \le 1$, then, again, nothing changes. However, if $W \le
r$, then, instead of using $\tau$ from step 1, we draw $g \sim h(\cdot
; \tau, \theta, y)$, and use $g \tau$ in place of $\tau$.
It follows from Proposition~\ref{prop:main_comp} in
Appendix~\ref{app:theory} that, whenever the HS Gibbs sampler of
Section~\ref{sec:glmm} is geometrically ergodic, so is our HSS
algorithm. Recall that some empirical results for this HSS algorithm
are depicted alongside the results for the HS, SS, and RS Gibbs
samplers in Figure~\ref{fig:simplot} of Section~\ref{sec:glmm}. In
that example, the rejection sampler is quite efficient, with an
acceptance probability of more than 70\% in each of the three
simulations settings considered. The per iteration computational cost
of HS Gibbs obviously grows with $p$ while the extra cost associated
with rejection sampling is basically constant in $p$. As a result, in
the second and third simulation settings, the HSS algorithm was only
about 2\% slower than HS Gibbs, while in the first setting, the HSS
algorithm is substantially slower than HS Gibbs. Note that the
performance of the rejection sampler is a function of $C$ and
$\sum_{j=1}^p \tau_j$. For these simulations, we developed a table in
a preliminary offline investigation to decide the appropriate value of
the free parameter $s$ for a given $\big( C,\sum_{j=1}^p \tau_j \big)$
pair.
\section{Bayesian Linear Regression with Scale Mixtures of Normal Errors}
\label{sec:smn}
In this section, we provide another example of a Bayesian model that
leads to a highly intractable posterior distribution that lends itself
to the HS Gibbs sampler. Let $Y_1,\ldots,Y_m$ be independent random
variables from the linear regression model
\begin{equation}
\label{eq:smnmodel}
Y_i = x_i^{T}\beta + \sigma\epsilon_i \;,
\end{equation}
where $x_i$ is a $p \times 1$ vector of known covariates associated
with $Y_i$, $\beta$ is a $p \times 1$ vector of unknown regression
coefficients, $\sigma \in (0,\infty)$ is an unknown scale parameter,
and $\epsilon_1,\ldots,\epsilon_m$ are iid errors. The standard
assumption that the errors are Gaussian is often inappropriate, e.g.,
when the data contain outliers. Various heavy-tailed alternatives can
be constructed as scale mixtures of the Gaussian density. Consider an
error density of the form
\begin{equation}
\label{eq:smndensity}
f_H(\epsilon) = \int_0^\infty
\frac{\sqrt{z}}{\sqrt{2\pi}}\exp\left\lbrace -
\frac{z}{2}\epsilon^2\right\rbrace \, dH(z) \;,
\end{equation}
where $H$ is the distribution function of some non-negative random
variable. By varying the mixing distribution $H$, many symmetric and
unimodal distributions can be constructed. Thus, datasets with
various types of tail behavior (particularly with heavier tails than
the normal) are often modeled by choosing a distribution from this
class. In this section, we consider a Bayesian analysis of the linear
regression model ~\eqref{eq:smnmodel} when the errors
$\epsilon_1,\ldots,\epsilon_m$ are iid random variables with the
general scale mixture of normals density $f_H$ given in
~\eqref{eq:smndensity}. There are several different prior
distributions available that lead to conditional distributions with
standard forms. \citet{hobert2018convergence} consider a standard
improper prior and show that a DA algorithm is available. A DA
algorithm is also available in the case where we specify a proper
conditionally conjugate prior on $(\beta,\sigma^2)$ by setting $\beta
| \sigma^2 \sim \mbox{N}_p(\mu,\sigma^2 \Sigma)$ and $\sigma^2 \sim
\mbox{IG}(\alpha,\gamma)$. Throughout this section, we will instead
consider the proper prior which takes $\beta$ and $\sigma^2$ to be
\textit{a priori} independent with $\beta \sim \mbox{N}_p
(\mu,\Sigma)$ and $\sigma^2 \sim \mbox{IG}(\alpha,\gamma)$. This
slight change to the prior makes the DA algorithm difficult to
implement, but the HS Gibbs sampler is a viable alternative. We now
provide the details.
Let $y = (y_1,\ldots,y_m)$ denote the observed data. Let $X$ denote
the $m \times p$ matrix whose $i$th row is $x_i^{T}$. We assume
throughout that $m \ge \max\{2,p\}$. We also assume that $H$ has a
density, $h$, with respect to Lebesgue measure on $\mathbb{R}_+$.
Letting $p_H(y|\beta,\sigma^2)$ denote the joint density of the data
from the linear regression model, the posterior density is given by
\begin{align*}
\pi(\beta,\sigma^2|y) & \propto p_H(y|\beta,\sigma^2)
\pi(\beta,\sigma^2) \\ & \propto \Bigg[ \prod_{i=1}^m
\frac{1}{\sigma}f_H\left(\frac{y_i - x_i^{T} \beta}{\sigma}\right)
\Bigg] \pi(\beta,\sigma^2) \\ & \propto \Bigg[ \prod_{i=1}^m
\int_{\reals_+} \frac{\sqrt{z_i}}{\sqrt{2\pi\sigma^2}}
\exp\left\lbrace
-\frac{z_i}{2}\frac{(y_i-x_i^{T}\beta)^2}{\sigma^2}\right\rbrace
h(z_i) \, dz_i \Bigg] \\ & \hspace*{8mm} \times
(\sigma^2)^{-\alpha-1}\exp \left\lbrace -\frac{\gamma}{\sigma^2}
\right\rbrace \exp \left\lbrace
-\frac{(\beta-\mu)^T\Sigma^{-1}(\beta-\mu)}{2} \right\rbrace
I_{\mathbb{R}^+}(\sigma^2) \;.
\end{align*}
Define the complete data posterior density as
\begin{align*}
\pi(\beta,\sigma^2,z|y) &= \prod_{i=1}^m
\frac{\sqrt{z_i}}{\sqrt{2\pi\sigma^2}} \exp\left\lbrace
-\frac{z_i}{2}\frac{(y_i-x_i^{T}\beta)^2}{\sigma^2}\right\rbrace
h(z_i) \\ & \times (\sigma^2)^{-\alpha-1}\exp \left\lbrace
-\frac{\gamma}{\sigma^2} \right\rbrace \exp \left\lbrace
-\frac{(\beta-\mu)^T\Sigma^{-1}(\beta-\mu)}{2} \right\rbrace
I_{\mathbb{R}^+}(\sigma^2) \;,
\end{align*}
and note that $\int_{\reals_+^m} \pi(\beta,\sigma^2,z|y) \, dz =
\pi(\beta,\sigma^2|y)$, so that $z = (z_1,\ldots,z_m)$ constitutes
latent data. We now state the conditional densities needed for the HS
Gibbs sampler. First, conditional on $(\beta, \sigma^2, y)$, $z_1,
\ldots, z_m$ are independent, and the conditional density of $z_i$
given $(\beta, \sigma^2, y_i)$ is given by
\begin{equation}
\label{eq:zcond}
\pi(z_i|\beta,\sigma^2,y_i) \propto z_i^{\frac{1}{2}} \exp \left\{ -
\frac{z_i}{2} \frac{(y_i - x_i^T \beta)^2}{\sigma^2} \right\} h(z_i)
\;.
\end{equation}
In some cases, this density turns out to be standard. For example,
when $h$ is a gamma density, then so is $\pi(z_i|\beta,\sigma^2,y_i)$,
and when $h$ is inverted gamma, then $\pi(z_i|\beta,\sigma^2,y_i)$ is
generalized inverse Gaussian. Even when it's not a standard density,
as long as one can make draws from $h$, then $h$ can be used as the
candidate in a simple rejection sampler.
Next, let $Q$ be an $m \times m$ diagonal matrix whose $i$th diagonal
element is $z_i^{-1}$. We have
\[
\sigma^2 \; | \; \beta, z, y \sim \text{IG}\left( \frac{m}{2} +
\alpha, \frac{(y-X\beta)^T Q^{-1} (y-X\beta) + 2\gamma}{2} \right) \;.
\]
Finally, $\beta \; | \; \sigma^2, z, y \sim \mbox{N}_p \left( \mu',
\sigma^2 \Sigma' \right)$, where
\[
\mu' = \left( X^T Q^{-1} X + \sigma^2 \Sigma^{-1} \right)^{-1} \left(
X^T Q^{-1} y + \sigma^2 \Sigma^{-1} \mu \right) \hspace{5mm}
\mbox{and} \hspace{5mm} \Sigma' = \left( X^T Q^{-1} X + \sigma^2
\Sigma^{-1} \right)^{-1} \;.
\]
The HS Gibbs sampler is based on the Markov chain $\Phi = \{
(\beta_n,\sigma_n^2)\}_{n=0}^{\infty}$ with state space $\mathsf{X} =
\reals^p \times \reals_+$ and selection probability $r \in (0,1).$ The
dynamics of $\Phi$ are defined by the following two step procedure for
moving from $(\beta_n,\sigma_n^2) = (\beta,\sigma^2)$ to
$(\beta_{n+1},\sigma_{n+1}^2).$
\newpage
\vskip .2truecm\hfill \hrule height.5pt \vskip .2truecm \vspace*{2mm}
\noindent {\rm Iteration $n+1$ of the hybrid scan Gibbs sampler:}
\begin{enumerate}
\item Draw $Z_1, \ldots, Z_m$ independently with $$Z_i \sim \text{the
density proportional to} \; z_i^{\frac{1}{2}} \exp \left\{ -
\frac{z_i}{2} \frac{(y_i - x_i^T \beta)^2}{\sigma^2} \right\} h(z_i)
\;,$$ call the observed values $z = (z_1,\ldots,z_m),$ and,
independently, draw $W \sim \text{Uniform}(0,1).$
\item
\begin{enumerate}
\item If $W \le r,$ draw $$\sigma^{*2} \sim \text{IG}\left(
\frac{m}{2} + \alpha, \frac{(y-X\beta)^T Q^{-1} (y-X\beta) +
2\gamma}{2} \right),$$ and set $(\beta_{n+1},\sigma_{n+1}^2) =
(\beta,\sigma^{*2}).$
\item Otherwise if $r < W \le 1,$ draw $$\beta^* \sim \mbox{N}_p
\left( \mu', \sigma^2 \Sigma' \right),$$ and set
$(\beta_{n+1},\sigma_{n+1}^2) = (\beta^*,\sigma^2).$
\end{enumerate}
\end{enumerate}
\vspace*{-3mm}
\vskip -.1truecm\hfill \hrule height.5pt \vskip .4truecm
\bigskip
We now provide convergence rate results for this HS algorithm and the
corresponding SS Gibbs sampler. Let $\hat{\Phi} = \{
(\hat{\beta}_n,\hat{\sigma}_n^2)\}_{n=0}^{\infty}$ denote the Markov
chain defined by the following Mtd:
\[
k_{\text{G}} (\beta, \sigma^2 | \hat{\beta},\hat{\sigma}^2) =
\int_{\mathbb{R}_+^n} \pi(\beta|\sigma^2,z,y)
\pi(\sigma^2|\hat{\beta},z,y) \pi(z|\hat{\beta},\hat{\sigma}^2,y) \,
dz \;.
\]
Of course, this is just the Markov chain that one is left with when
one runs the three-block SS Gibbs sampler and ignores the latent data.
It is well known that this chain has exactly the same convergence rate
as the SS Gibbs chain. The following result, which is proven in
Appendix~\ref{app:proof}, provides sufficient conditions for each of
these algorithms to be geometrically ergodic.
\begin{theorem}
\label{thm:smn}
The following results hold for any mixing density $h.$
\begin{enumerate}[(i)]
\item Suppose there exist constants $0 \leq \psi_1 < 1$ and $L_1 \in
\reals$ which do not depend on $\beta$ or $\sigma^2$ such that
\begin{equation}
\label{ine:driftgibbs}
\frac{\sum_{i=1}^m E[z_i|\beta,\sigma^2,y](y_i - x_i^T \beta)^2}{m + 2
\alpha - 2} \leq \psi_1 \left[ \sum_{i=1}^m (y_i - x_i^T \beta)^2 +
\beta^T \Sigma^{-1} \beta + \sigma^2 + \frac{1}{\sigma^2} \right] +
L_1 \;
\end{equation}
for every $\beta \in \reals^p, \sigma^2 \in \reals_+$. Then
$\hat{\Phi}$ is geometrically ergodic.
\item Suppose there exist constants $\psi_2 \in \reals_+$, $0 \leq
\psi_3 < 1$, and $L_2 \in \reals$ which do not depend on $\beta$ or
$\sigma^2$ such that
\begin{equation}
\label{ine:drifthybrid}
\frac{\sum_{i=1}^m E[z_i|\beta,\sigma^2,y](y_i - x_i^T \beta)^2}{m + 2
\alpha - 2} \leq \psi_2 \Bigg[ \sum_{i=1}^m (y_i - x_i^T \beta)^2 +
\, \beta^T\Sigma^{-1}\beta \Bigg] + \psi_3 \left( \sigma^2 +
\frac{1}{\sigma^2} \right) + L_2 \;
\end{equation}
for every $\beta \in \reals^p, \sigma^2 \in \reals_+$. Then $\Phi$ is
geometrically ergodic for all $r \in (0,1)$.
\end{enumerate}
\end{theorem}
\begin{remark}
Note that if \eqref{ine:driftgibbs} holds, then
\eqref{ine:drifthybrid} holds with $\psi_2 = \psi_3 = \psi_1$, and
$L_2 = L_1$. So the sufficient condition for geometric ergodicity of
the HS Gibbs algorithm is weaker than the corresponding sufficient
condition for the SS Gibbs sampler. Of course, we are dealing with
sufficient conditions here, so by no means does Theorem~\ref{thm:smn}
imply that HS Gibbs algorithm is geometrically ergodic more often than
the SS Gibbs sampler. On the other hand, in a given situation, if it
is known that the HS algorithm is geometrically ergodic, and it is
unknown whether or not the same is true of the SS Gibbs sampler, then
one should probably use the HS algorithm.
\end{remark}
In order to actually apply Theorem~\ref{thm:smn}, we must specify $h$
so that we can calculate (or at least bound)
$E[z_i|\beta,\sigma^2,y]$. For example, suppose that $h$ is a
$\text{Gamma}(\frac{\nu}{2},\frac{\nu}{2})$ density, which leads to a
Student's $t$ distribution with $\nu$ degrees of freedom for the
regression errors. In this case, $z_i|(\beta,\sigma^2,y)$ is
\[
\mbox{Gamma} \bigg( \frac{\nu+1}{2}, \frac{(y_i - x_i^T \beta)^2 + \nu
\sigma^2}{2\sigma^2} \bigg) \;,
\]
and
\[
E[z_i|\beta,\sigma^2,y] = \frac{\sigma^2 (\nu+1)}{(y_i - x_i^T
\beta)^2 + \nu \sigma^2} \;.
\]
It follows that \eqref{ine:drifthybrid} is satisfied since
\begin{align}
\label{eq:ia}
\frac{\sum_{i=1}^m E[z_i|\beta,\sigma^2,y](y_i - x_i^T \beta)^2}{m + 2
\alpha - 2} & = \frac{1}{m + 2 \alpha - 2} \sum_{i=1}^m
\frac{\sigma^2 (\nu+1)(y_i - x_i^T \beta)^2}{(y_i - x_i^T \beta)^2 +
\nu \sigma^2} \nonumber \\ & \le \frac{\nu+1}{\nu(m + 2 \alpha - 2)}
\sum_{i=1}^m (y_i - x_i^T \beta)^2 \;.
\end{align}
Thus, Theorem~\ref{thm:smn} implies that the HS Markov chain is
geometrically ergodic (without any additional assumptions).
Unfortunately, this argument doesn't work for the SS Gibbs sampler.
Indeed, \eqref{eq:ia} doesn't establish that \eqref{ine:driftgibbs} is
satisfied unless we make the additional assumption that that $\nu >
1/(m+2\alpha-3)$. However, another upper bound on the left-hand side
of \eqref{eq:ia} is as follows:
\begin{equation}
\label{eq:ib}
\frac{\sum_{i=1}^m E[z_i|\beta,\sigma^2,y](y_i - x_i^T \beta)^2}{m +
2 \alpha - 2} \le \frac{m(\nu+1)}{m + 2 \alpha - 2} \sigma^2 \;.
\end{equation}
Now, \eqref{eq:ib} will establish \eqref{ine:driftgibbs} if $\nu <
(2\alpha-2)/m$. So Theorem~\ref{thm:smn} implies that the SS Gibbs
chain is geometrically ergodic if either $\nu > 1/(m+2\alpha-3)$ or
$\nu < (2\alpha-2)/m$. Of course, if $1/(m+2\alpha-3) <
(2\alpha-2)/m$, then at least one of these two inequalities must hold.
However, when $\alpha$ is small, this is not the case.
Consider a second example where $h$ is taken to be an
$\text{IG}(\alpha,1)$ density. Under this mixing density, the
regression errors have a generalized hyperbolic distribution, which
has tails that are heavier than Gaussian, but lighter than Student's
$t$ \citep[see, e.g.,][]{jung:hobe:2014}. In this case,
$z_i|(\beta,\sigma^2,y)$ is
\[
\mbox{GIG} \bigg( \frac{1}{2} - \alpha, \frac{(y_i - x_i^T
\beta)^2}{\sigma^2}, 2 \bigg) \;,
\]
and
\[
E[z_i|\beta,\sigma^2,y] = \frac{\sqrt{2 \sigma^2}}{\sqrt{(y_i - x_i^T
\beta)^2}} \frac{\mbox{K}_{- \alpha + 3/2} \Big( \sqrt{\frac{2(y_i -
x_i^T \beta)^2}{\sigma^2}} \Big)}{\mbox{K}_{- \alpha + 1/2}
\Big( \sqrt{\frac{2(y_i - x_i^T \beta)^2}{\sigma^2}} \Big)} \;.
\]
\citet[][p. 62]{jungdiss} shows that
\[
\frac{\mbox{K}_{- \alpha + 3/2} \Big( \sqrt{\frac{2(y_i - x_i^T
\beta)^2}{\sigma^2}} \Big)}{\mbox{K}_{- \alpha + 1/2} \Big(
\sqrt{\frac{2(y_i - x_i^T \beta)^2}{\sigma^2}} \Big)} \le 1 +
\frac{\sqrt{\sigma^2}}{\sqrt{2(y_i - x_i^T \beta)^2}} \;.
\]
Hence, letting $C>0$ be an arbitrary positive constant, we have
\begin{align}
\label{eq:iga}
\frac{\sum_{i=1}^m E[z_i|\beta,\sigma^2,y](y_i - x_i^T \beta)^2}{m + 2
\alpha - 2} & \le \frac{1}{m + 2\alpha - 2} \sum_{i=1}^m \bigg[
\frac{\sqrt{2 \sigma^2}}{C} \sqrt{C(y_i - x_i^T \beta)^2} + \sigma^2
\bigg] \nonumber \\ & \le \frac{1}{m + 2\alpha - 2} \sum_{i=1}^m
\bigg[\frac{C (y_i - x_i^T \beta)^2}{2} + \frac{\sigma^2}{C} +
\sigma^2 \bigg] \nonumber \\ & = \frac{C}{2(m + 2\alpha - 2)}
\sum_{i=1}^m (y_i - x_i^T \beta)^2 + \frac{m(C+1)}{C(m + 2\alpha - 2)}
\sigma^2 \;.
\end{align}
If $\alpha>1$, then we can find $C>0$ such that $\frac{m(C+1)}{C(m +
2\alpha - 2)} < 1$. Therefore, Theorem~\ref{thm:smn} implies that
the HS Gibbs chain is geometrically ergodic whenever $\alpha>1$.
Now, if we can find a single value of $C>0$ such that $m(C+1) < C(m +
2\alpha - 2)$ \textit{and} $C<2(m + 2\alpha - 2)$, then \eqref{eq:iga}
will imply that \eqref{ine:driftgibbs} holds. The existence of such a
$C$ is equivalent to $\alpha$ and $m$ satisfying the following
inequality
\[
8 \alpha^2 + \alpha(4m-16) + 8 - 5m > 0 \;.
\]
Thus, Theorem~\ref{thm:smn} implies that the SS Gibbs chain is
geometrically ergodic if $\alpha > \big( 4-m+\sqrt{m(m+2)} \big)/4$.
This inequality holds for all $\alpha \ge \frac{5}{4}$, regardless of
the value of $m$, and it does hold for smaller values of $\alpha$ when
$m$ is fixed. For example, if $m=2$, then we only need $\alpha > (1 +
\sqrt{2})/2 \approx 1.21$.
Once $h$ is specified, HSS algorithms can be created by adding
sandwich steps to the HS Gibbs sampler. \citet{grantdiss} develops a
HSS algorithm with two sandwich steps for the case where $h$ is a
$\text{Gamma}(\frac{\nu}{2},\frac{\nu}{2})$ density.
\section{Discussion}
\label{sec:discussion}
We have introduced generic forms of the hybrid scan Gibbs sampler and
the hybrid scan sandwich algorithm, and we have shown that, under weak
regularity conditions, the latter is theoretically better than the
former. Moreover, we have developed and studied specific versions of
these algorithms in the context of two different realistic Bayesian
hierarchical models. It is clear that the hybrid scan algorithms are
quite flexible, and can be used in conjunction with a variety of
practical Bayesian models. As another example, consider a
generalization of the model in~\eqref{eq:smnmodel} in which the error
density has both heavy tails and \textit{skewness}.
\citet{da2011skew} define a skew scale mixture of normal densities,
$f_{H,\lambda}(\epsilon)$, by
\begin{equation}
f_{H,\lambda}(\epsilon) = 2 f_H(\epsilon) \Phi(\lambda \epsilon)
\;,
\end{equation}
where $f_H(\epsilon)$ is the scale mixture of normal densities defined
at~\eqref{eq:smndensity}, $\Phi(\cdot)$ is the standard normal
cumulative distribution function, and $\lambda \in \mathbb{R}$ is a
fixed parameter that controls the skewness. Combining the associated
likelihood with the same conjugate normal/inverse-gamma prior employed
in Section~\ref{sec:smn} gives rise to a posterior distribution that
is even more unwieldy than the one studied in Section~\ref{sec:smn}.
However, \citet{jungdiss} shows that there exist two sets of latent
variables, $z=(z_1,\ldots,z_m)$ and $t=(t_1,\ldots,t_m)$,
conditionally independent of one another given $(\beta,\sigma^2,y)$,
that give rise to a complete data posterior with the following
conditionals. Conditional on $(\beta,\sigma^2,y)$, $z_1,\ldots,z_m$
are independent, and the density of $z_i$ given $(\beta,\sigma^2,y)$
is the same as~\eqref{eq:zcond}. Also, conditional on
$(\beta,\sigma^2,y)$, $t_1,\ldots,t_m$ are independent, and the
density of $t_i$ given $(\beta,\sigma^2,y)$ is truncated normal.
Finally, $\sigma^2$ given $(\beta, z, t, y)$ is inverse-gamma, and
$\beta$ given $(\sigma^2,z,t,y)$ is multivariate
normal. Unfortunately, the distribution of $\sigma^2$ given $(z,t,y)$
is not available in closed form, so that the DA algorithm is not
straightforward to apply. Each iteration of the HS Gibbs algorithm
proceeds, as usual, by updating (all of) the latent data, and updating
either $\beta$ or $\sigma^2$, depending on the outcome of the flip of
an $r$-coin.
Lastly, we reiterate that, as far as theoretical convergence rates go,
it is now generally accepted that one should, if possible, use a Monte
Carlo Markov chain that converges at a geometric rate, or at least a
rate fast enough to ensure that the corresponding MCMC estimators are
asymptotically normal \citep[see, e.g.,][]{roberts1998markov}. Hence,
even if an alternative MCMC algorithm (such as SS or RS Gibbs) appears
marginally better than a geometrically ergodic HS Gibbs sampler
according to empirical measures, that algorithm should not be favored
over the HS Gibbs algorithm \textit{unless} it is known that the
alternative has an acceptably fast convergence rate. At present, it
appears that the convergence rates of alternative MCMC algorithms for
the family of posteriors considered in Section~\ref{sec:glmm} (and, to
some extent, those considered in Section~\ref{sec:smn}) are not known.
In situations such as these, we recommend HS Gibbs for practical use
and remind the reader that it is no more difficult to implement than
its SS or RS counterparts.
\vspace*{6mm}
\noindent {\bf \large Acknowledgment}. The second and fourth authors
were supported by NSF Grant DMS-15-11945.
\vspace*{8mm}
\newpage
\noindent {\LARGE \bf Appendices}
\begin{appendix}
\vspace*{-3mm}
\section{Theory for the HSS Algorithm}
\label{app:theory}
We begin with some requisite background material on Markov
operators. In keeping with the notation in the Introduction, the
target density, $f_{X,Y}(x,y),$ can be used to define an inner
product
\[
\langle g_1, g_2 \rangle_{L^2_0} = \int_{\mathsf{X}} \int_{\mathsf{Y}}
g_1(x,y) g_2(x,y) f_{X,Y}(x,y) \, dy \, dx \;,
\]
and norm $\norm{g} = \sqrt{\langle g,g \rangle}$ on the Hilbert
space
\[
L^2_0 = \left\{ g: \mathsf{X} \times \mathsf{Y} \to \reals :
\int_{\mathsf{X}} \int_{\mathsf{Y}} g^2(x,y) f_{X,Y}(x,y) \, dy \, dx
< \infty \; \text{and} \; \int_{\mathsf{X}} \int_{\mathsf{Y}} g(x,y)
f_{X,Y}(x,y) \, dy \, dx = 0 \right\} \;.
\]
To keep things simple, we assume throughout that $f_{X,Y}(x,y)$ is a
density with respect to Lebesgue measure, but we note that the results
actually hold much more generally - see, e.g., the set-up in
\citet{khare2011spectral}. The Mtd $k_1$ corresponds to a Markov
operator $K_1: L^2_0 \to L^2_0$ that takes $g \in L^2_0$ into
\[
(K_1 g)(x,y) = \int_{\mathsf{X}} g(x',y) k_1(x'|x;y) \, dx' \;.
\]
Now, if we define $K_2$ using $k_2$ in an analogous way, then it is
clear that the Markov operator associated with the HSS algorithm,
$K_{\text{HSS}} : L^2_0 \to L^2_0,$ is given by $K_{\text{HSS}} = r
K_1 + (1-r) K_2$, where $r \in (0,1)$ is the selection
probability. Here is our first result.
\begin{proposition}
The Markov chain underlying the HSS algorithm is reversible.
\end{proposition}
\begin{proof}
It suffices to show that $K_{\text{HSS}}$ is a self-adjoint
operator. We start by showing that $K_1$ is self-adjoint. First, it's
easy to see that $f_{X,Y}(x,y) k_1(x'|x;y) = f_{X,Y}(x',y)
k_1(x|x';y).$ It follows that
\begin{align*}
f_{X,Y}(x,y)(K_1 g)(x,y) &= f_{X,Y}(x,y) \int_{\mathsf{X}} g(x',y)
k_1(x'|x;y) \, dx' \\ &= \int_{\mathsf{X}} g(x',y) f_{X,Y}(x,y)
k_1(x'|x;y) \, dx' \\ &= \int_{\mathsf{X}} g(x',y) f_{X,Y}(x',y)
k_1(x|x';y) \, dx'
\end{align*}
Thus,
\begin{align*}
\langle K_1 g, h \rangle_{L^2_0} &= \int_{\mathsf{X}}
\int_{\mathsf{Y}} (K_1 g)(x,y) h(x,y) f_{X,Y}(x,y) \, dy \, dx \\ &=
\int_{\mathsf{X}} \int_{\mathsf{Y}} h(x,y) \left[ \int_{\mathsf{X}}
g(x',y) f_{X,Y}(x',y) k_1(x|x';y) \, dx' \right] \, dy \, dx \\ &=
\int_{\mathsf{X}} \int_{\mathsf{Y}} \left[ \int_{\mathsf{X}} h(x,y)
k_1(x|x';y) \, dx \right] g(x',y) f_{X,Y}(x',y) \, dy \, dx' \\ &=
\langle g, K_1 h \rangle_{L^2_0} \;,
\end{align*}
where the third equality follows from Fubini's theorem. Now an
analogous argument shows that $K_2$ is self-adjoint, and it follows
immediately that $r K_1 + (1-r) K_2$ is also self-adjoint.
\end{proof}
We now look more closely at the two criteria for comparing MCMC
algorithms that were mentioned in the Introduction: rate of
convergence and asymptotic variance. Let $\Phi =
\{(X_n,Y_n)\}_{n=0}^{\infty}$ denote a generic Markov chain on
$\mathsf{X} \times \mathsf{Y}$ that is reversible with respect to
$f_{X,Y}$. Assume further that $\Phi$ is Harris ergodic; that is,
aperiodic, irreducible and Harris recurrent. Let $K$ denote the
corresponding Markov operator on $L^2_0$. Let $L^2_{0,1} \subset
L^2_0$ denote the functions for which
\[
\int_{\mathsf{X}} \int_{\mathsf{Y}} g^2(x,y) f_{X,Y}(x,y) \, dy \, dx
= 1 \;.
\]
The norm of the operator $K$ is defined as
\[
\norm{K} = \sup_{g \in L^2_{0,1}} \norm{K g} \;.
\]
(Since $K$ is self-adjoint, we also have $\norm{K} = \sup_{g \in
L^2_{0,1}} |\langle K g , g \rangle_{L^2_0}|$.) The quantity
$\norm{K},$ which takes values in $[0,1]$, represents the convergence
rate of $\Phi,$ with smaller values associated with faster
convergence. In fact, $\Phi$ is geometrically ergodic if and only if
$\norm{K} < 1$ \citep{roberts1997geometric}. One way to choose between
two MCMC algorithms for the same problem is to favor the one whose
Markov operator has smaller norm.
Now let $g : \mathsf{X} \times \mathsf{Y} \to \reals$ be (non-constant
and) such that
\[
\int_{\mathsf{X}} \int_{\mathsf{Y}} g^2(x,y) f_{X,Y}(x,y) \, dy \, dx
< \infty \;.
\]
Let $\theta = \int_{\mathsf{X}} \int_{\mathsf{Y}} g(x,y) f_{X,Y}(x,y)
\, dy \, dx$, and let $\overline{g}_n = \frac{1}{n} \sum_{i=0}^{n-1}
g(X_n,Y_n)$. If $\Phi$ is geometrically ergodic, then the Markov
chain CLT implies that there exists $\sigma^2_{g,K} \in (0,\infty)$
such that, as $n \to \infty,$ $\sqrt{n} (\overline{g}_n - \theta)
\cvgindist \text{N}(0,\sigma^2_{g,K}).$ If $g$ is square integrable
with respect to $f_{X,Y},$ but the CLT does not hold, then set
$\sigma^2_{g,K} = \infty.$ Suppose $\Phi^*$ is a second Markov chain
(with corresponding operator $K^*$) that satisfies all the properties
we have assumed $\Phi$ satisfies. If $\sigma^2_{g,K^*} <
\sigma^2_{g,K}$ for all square integrable $g,$ then we say that $K^*$
is more efficient than $K,$ and we write $K^* \succeq_E K$.
Before we can state the main result, we must define a few more
operators. First, let $\hat{L}^2_0$ denote the space of functions that
are square integrable and have mean zero with respect to
$f_{Y,Z}(y,z).$ We denote the inner product on this space by $\langle
\cdot, \cdot \rangle_{\hat{L}^2_0}.$ The Mtf $R_1(z,dz';y)$ defines an
operator $R_1 : \hat{L}^2_0 \to \hat{L}^2_0$ that takes $h \in
\hat{L}^2_0$ to $$(R_1 h)(y,z) = \int_{\mathsf{Z}} h(y,z')
R_1(z,dz';y) \;.$$ It follows immediately from ~\eqref{eq:reverse}
that $R_1$ is self-adjoint (with respect to $f_{Y,Z}$). Of course,
$R_1$ is a positive operator if $\langle R_1 h, h
\rangle_{\hat{L}^2_0} \geq 0$ for all $h \in \hat{L}^2_0.$ Let $R_2$
denote the analogous operator corresponding to the Mtf $R_2$, and let
$K_{\text{HS}}$ denote the Markov operator (on $L^2_0$) corresponding
to the HS Gibbs sampler.
\begin{proposition}
\label{prop:main_comp}
Suppose the Markov chains associated with $K_{\text{HSS}}$ and
$K_{\text{HS}}$ are both Harris ergodic. Then $K_{\text{HSS}}
\succeq K_{\text{HS}}$. If, in addition, $R_1$ and $R_2$ are both
positive operators, then $\norm{K_{\text{HSS}}} \leq
\norm{K_{\text{HS}}}.$
\end{proposition}
\begin{proof}
Fix $g \in L^2_0$ and define $$g^*(y,z) = \int_{\mathsf{X}} g(x,y)
f_{X|Y,Z}(x|y,z) \, dx \;.$$ It's easy to see that $g^* \in
\hat{L}^2_0.$ Now
\begin{align}
&\langle K_1 g , g \rangle_{L^2_0} \nonumber \\ &= \int_{\mathsf{X}}
\int_{\mathsf{Y}} (K_1 g)(x,y) g(x,y) f_{X,Y}(x,y) \, dy \, dx
\nonumber \\ &= \int_{\mathsf{X}} \int_{\mathsf{Y}} \left[
\int_{\mathsf{X}} g(x',y) \int_{\mathsf{Z}} \int_{\mathsf{Z}}
f_{X|Y,Z}(x'|y,z') R_1(z,dz';y) f_{Z|X,Y}(z|x,y) \, dz \, dx'
\right] g(x,y) f_{X,Y}(x,y) \, dy \, dx \nonumber \\ &=
\int_{\mathsf{X}} \int_{\mathsf{Y}} \int_{\mathsf{X}}
\int_{\mathsf{Z}} \int_{\mathsf{Z}} g(x',y) f_{X|Y,Z}(x'|y,z')
R_1(z,dz';y) f_{Z|X,Y}(z|x,y) g(x,y) f_{X,Y}(x,y) \, dz \, dx' \, dy
\, dx \nonumber \\ &= \int_{\mathsf{Y}} \int_{\mathsf{Z}} \left[
\int_{\mathsf{Z}} g^*(y,z') R_1(z,dz';y) \right] g^*(y,z)
f_{Y,Z}(y,z) \, dy \, dz \nonumber \\ &= \langle R_1 g^*, g^*
\rangle_{\hat{L}^2_0} \;.
\label{eq:cov}
\end{align}
Note that $\langle R_1 g^*, g^* \rangle_{\hat{L}^2_0}$ is the
covariance of $g^*(Y_0,Z_0)$ and $g^*(Y_1,Z_1)$ where
${(Y_n,Z_n)}_{n=0}^{\infty}$ is the \textit{stationary} version of the
Markov chain driven by $R_1$ (so $(Y_0,Z_0) \sim f_{Y,Z}$). Let
$\tilde{K}_1$ denote $K_1$ when $R_1$ is trivial. Then $\langle
\tilde{K}_1 g , g \rangle_{L^2_0} = \langle g^*, g^*
\rangle_{\hat{L}^2_0},$ which is the variance of $g^*(Y_0,Z_0)$ when
$(Y_0,Z_0) \sim f_{Y,Z}.$ Hence by Cauchy-Schwarz, $$\langle K_1 g , g
\rangle_{L^2_0} = \langle R_1 g^*, g^* \rangle_{\hat{L}^2_0} \leq
\langle g^*, g^* \rangle_{\hat{L}^2_0} = \langle \tilde{K}_1 g , g
\rangle_{L^2_0} \;.$$ An analogous argument shows that $\langle K_2 g
, g \rangle_{L^2_0} \leq \langle \tilde{K}_2 g , g \rangle_{L^2_0},$
where $\tilde{K}_2$ denotes $K_2$ with a trivial $R_2$. Of course,
$K_{\text{HS}} = r \tilde{K}_1 + (1-r) \tilde{K}_2$. Therefore, for
any $g \in L^2_0,$ we have
\begin{equation}
\label{dsinequality}
\langle K_{\text{HSS}} g , g \rangle_{L^2_0} = \langle (r K_1 + (1-r)
K_2) g , g \rangle_{L^2_0} \leq \langle (r \tilde{K}_1 + (1-r)
\tilde{K}_2) g , g \rangle_{L^2_0} = \langle K_{\text{HS}} g , g
\rangle_{L^2_0} \;,
\end{equation}
and it now follows from results in \citet{mira1999ordering} that
$K_{\text{HSS}} \succeq_E K_{\text{HS}}$.
Now, if $R_1$ is positive, then it follows immediately from
~\eqref{eq:cov} that $K_1$ is also positive. Of course, in an
analogous manner, positivity of $R_2$ implies that of $K_2.$ Then
since $K_{\text{HSS}}$ and $K_{\text{HS}}$ are both self-adjoint, it
follows from \eqref{dsinequality} that $\norm{K_{\text{HSS}}} \leq
\norm{K_{\text{HS}}}$.
\end{proof}
\begin{remark}
As explained in \citet{mira1999ordering}, generally fast convergence
and small asymptotic variance are conflicting goals. Indeed, a Markov
chain has a small norm when the spectrum of its operator is
concentrated near zero, whereas small asymptotic variance is
associated with a spectrum that is concentrated near -1. When $R_1$
and $R_2$ are both positive operators, then $K_{\text{HSS}}$ and
$K_{\text{HS}}$ are also positive, which implies that their spectra
are both subsets of $[0,1]$. In this context, fast convergence and
small asymptotic variance are both associated with a spectrum
concentrated near zero, and are no longer conflicting goals.
\end{remark}
\section{Proof of Theorem~\ref{thm:smn}}
\label{app:proof}
We begin with several lemmas. The following lemma is proved in
\citet{khare2011spectral}.
\begin{lemma}
\label{lemma:kh}
Fix $m \in \{2,3,\ldots\}$ and $p \in \mathbb{N},$ and let
$t_1,\ldots,t_m$ be vectors in $\reals^p.$ Then
\[
C_{p,m}(t_1 ; t_2, \ldots, t_m) := \sup_{c \in \reals^m_+} t_1^T
\left( t_1 t_1^T + \sum_{i=2}^m c_i t_i t_i^T + c_1 I \right)^{-2}
t_1
\]
is finite.
\end{lemma}
For a symmetric matrix $M$, let $\lambda^*\{M\}$ denote the largest
eigenvalue of $M$, and define the matrix norm as follows
\[
\norm{M} = \sup_{\norm{x}=1} \norm{Mx} = \sup_{\norm{x}=1} \sqrt{x^T
M^2 x} \;.
\]
The following result is easily established.
\begin{lemma}
\label{lemma:matrix}
If $A$ is a symmetric, non-negative definite matrix, then
\[
\norm{(I + A)^{-1}} = \lambda^* \big\{ (I + A)^{-1} \big\} \le 1 \;,
\]
and $I - (I + A)^{-1}$ is non-negative definite.
\end{lemma}
Let $\{y_i\}_{i=1}^m$ and $\{x_i\}_{i=1}^m$ be the data and the
covariates, respectively, from the model in Section~\ref{sec:smn}.
\begin{lemma}
\label{lemma:drift}
Define $V: \mathbb{R}^p \times \mathbb{R}_+ \rightarrow (0,\infty)$ as
follows
\[
V(\beta, \sigma^2) = \sum_{i=1}^m (y_i - x_i^T \beta)^2 + \beta^T
\Sigma^{-1} \beta + \sigma^2 + \frac{1}{\sigma^2} \;.
\]
The function $V$ is unbounded off compact sets, i.e., the sublevel
sets of $V$ are compact.
\end{lemma}
\begin{proof}
We must show that for every $d \ge 0$, the set
\[
S_d = \bigg \{ (\beta,\sigma^2) \in \reals^p \times \reals_+ :
V(\beta, \sigma^2) = \sum_{i=1}^m (y_i - x_i^T \beta)^2 + \beta^T
\Sigma^{-1} \beta + \sigma^2 + \frac{1}{\sigma^2} \le d \bigg\}
\]
is compact. Since $V$ is continuous, it suffices to show that
$|\beta_i|$ is bounded for all $i \in \{1,2,\ldots,p\}$ and that
$\sigma^2$ is bounded away from $0$ and $\infty.$ Since $\Sigma$ is
positive definite, $\beta^T \Sigma^{-1} \beta \le d$ implies that
$|\beta_i|$ is bounded for all $i \in \{1,2,\ldots,p\}.$ Also,
$\sigma^2 + \frac{1}{\sigma^2} \le d$ implies that $\sigma^2$ is
bounded away from $0$ and $\infty$.
\end{proof}
\begin{lemma}
\label{lemma:hybridge}
If the hybrid scan Gibbs sampler is geometrically ergodic for some
selection probability $r^* \in (0,1),$ then it is geometrically
ergodic for every selection probability $r \in (0,1)$.
\end{lemma}
\begin{proof}
The Mtf of the HS chain (with selection probability $r$) is given by
\begin{align*}
K_{\text{HS, $r$}}((x,y),A) &= r \int_{\mathsf{X}} I_A(x',y)
\int_{\mathsf{Z}} f_{X|Y,Z}(x'|y,z) f_{Z|X,Y}(z|x,y) \, dz \, dx'
\\ &+ (1-r) \int_{\mathsf{Y}} I_A(x,y') \int_{\mathsf{Z}}
f_{Y|X,Z}(y'|x,z) f_{Z|X,Y}(z|x,y) \, dz \, dy'.
\end{align*}
It is easy to show that
\[
K_{\text{HS, $r$}}((x,y),A) \geq \text{min} \left( \frac{r}{r^*},
\frac{1-r}{1-r^*} \right) K_{\text{HS, $r^*$}}((x,y),A) \;,
\]
and thus
\[
K_{\text{HS, $r$}}((x,y),A) \geq \delta' K_{\text{HS, $r^*$}}((x,y),A)
\]
for all measurable sets $A$ and all $(x,y) \in \mathsf{X} \times
\mathsf{Y}$, where $\delta' = \text{min} \left( \frac{r}{r^*},
\frac{1-r}{1-r^*} \right) > 0$. Since the HS chain is reversible,
Theorem 1 in \citet{jones2014convergence} implies the result.
\end{proof}
\begin{proof}[Proof of Theorem~\ref{thm:smn}]
In view of Lemma~\ref{lemma:drift} above and Lemma 15.2.8 of
\citet{meyn2012markov}, in each case it suffices to verify the
geometric drift condition for the function
\[
V(\beta, \sigma^2) = \sum_{i=1}^m (y_i - x_i^T \beta)^2 + \beta^T
\Sigma^{-1} \beta + \sigma^2 + \frac{1}{\sigma^2} \;,
\]
i.e., we must show that
\[
E(V(\beta, \sigma^2) | \hat{\beta}, \hat{\sigma}^2) \le \lambda
V(\hat{\beta}, \hat{\sigma}^2) + L
\]
for some constants $\lambda \in [0,1)$ and $L \in \reals$, where for
part (i) of the theorem the expectation is taken with respect to the
Mtf of the SS Gibbs chain, and for part (ii) of the theorem the
expectation is taken with respect to the Mtf of the HS chain. We
begin with the SS Gibbs algorithm.
\begin{align*}
E(V(\beta, \sigma^2) | \hat{\beta}, \hat{\sigma}^2) & =
\int_{\reals_+} \int_{\reals^p} V(\beta, \sigma^2) \bigg[
\int_{\reals^m_+} \pi(\beta | \sigma^2, z, y) \pi(\sigma^2 |
\hat{\beta}, z, y) \pi(z| \hat{\beta}, \hat{\sigma}^2, y) \, dz
\bigg] \, d\beta \, d\sigma^2 \\ &= \int_{\reals^m_+} \left[
\int_{\reals_+} \left\{ \int_{\reals^p} V(\beta, \sigma^2) \pi(\beta
| \sigma^2, z, y) \, d\beta \right\} \pi(\sigma^2 | \hat{\beta}, z,
y) \, d\sigma^2 \right] \pi(z| \hat{\beta}, \hat{\sigma}^2, y) \,
dz.
\end{align*}
We have
\begin{align*}
\sum_{i=1}^m (y_i - x_i^T \beta)^2 + \beta^T \Sigma^{-1} \beta &=
\norm{y-X\beta}^2 + \beta^T \Sigma^{-1} \beta \\ &\leq 2\norm{y}^2 +
2\norm{X\beta}^2 + \beta^T \Sigma^{-1} \beta \\ &= 2\norm{y}^2 +
2\norm{X \Sigma^{\frac{1}{2}} \Sigma^{-\frac{1}{2}} \beta}^2 + \beta^T
\Sigma^{-1} \beta \\ &\leq 2\norm{y}^2 + (2 \norm{X
\Sigma^{\frac{1}{2}}}^2 + 1) \norm{\Sigma^{-\frac{1}{2}} \beta}^2.
\end{align*}
Let $\tilde{X} = X\Sigma^{\frac{1}{2}},$ let $\tilde{x}_i$ be the
$i$th column of $\tilde{X}^T,$ and let $\tilde{Q}$ be an $m \times m$
diagonal matrix whose $i$th diagonal element is $\sigma^2 z_i^{-1}$.
Then, given $(\sigma^2,z,y),$ $\Sigma^{-\frac{1}{2}} \beta$ is a
multivariate normal random vector with mean $(\tilde{X}^T
\tilde{Q}^{-1} \tilde{X} + I)^{-1} (\tilde{X}^T \tilde{Q}^{-1} y +
\Sigma^{-\frac{1}{2}} \mu)$, and covariance matrix $(\tilde{X}^T
\tilde{Q}^{-1} \tilde{X} + I)^{-1}.$ It follows from
Lemma~\ref{lemma:kh} that for each $i \in \{1,2,\ldots,m\}$ and for
all $z \in \reals^m_+$,
\[
\tilde{x}_i^T \left( \tilde{x}_i \tilde{x}_i^T + \sum_{j \neq i}
\frac{z_j}{z_i} \tilde{x}_j \tilde{x}_j^T + \frac{\sigma^2}{z_i} I
\right)^{-2} \tilde{x}_i \leq C_i(\tilde{X}) \;,
\]
where $C_i(\tilde{X})$ is a finite constant. Recall that if $A$ and
$B$ are symmetric matrices of the same dimension such that $A - B$ is
non-negative definite, then $\text{tr}(A) \geq \text{tr}(B).$ Then, we
have
\begin{align*}
&E\left[ \norm{\Sigma^{-\frac{1}{2}} \beta}^2 \Bigm| \sigma^2, z, y
\right] \\ &= \norm{(\tilde{X}^T \tilde{Q}^{-1} \tilde{X} +
I)^{-1} (\tilde{X}^T \tilde{Q}^{-1} y + \Sigma^{-\frac{1}{2}}
\mu)}^2 + \text{tr}((\tilde{X}^T \tilde{Q}^{-1} \tilde{X} +
I)^{-1}) \\ &\leq 2 \norm{(\tilde{X}^T \tilde{Q}^{-1} \tilde{X} +
I)^{-1} \tilde{X}^T \tilde{Q}^{-1} y}^2 + 2 \norm{(\tilde{X}^T
\tilde{Q}^{-1} \tilde{X} + I)^{-1} \Sigma^{-\frac{1}{2}} \mu}^2 +
\text{tr}((\tilde{X}^T \tilde{Q}^{-1} \tilde{X} + I)^{-1}) \\ &\leq
2 \norm{(\tilde{X}^T \tilde{Q}^{-1} \tilde{X} + I)^{-1} \tilde{X}^T
\tilde{Q}^{-1} y}^2 + 2\norm{(\tilde{X}^T \tilde{Q}^{-1} \tilde{X}
+ I)^{-1}}^2 \norm{\Sigma^{-\frac{1}{2}} \mu}^2 +
\text{tr}((\tilde{X}^T \tilde{Q}^{-1} \tilde{X} + I)^{-1}) \\ &\leq
2 \norm{(\tilde{X}^T \tilde{Q}^{-1} \tilde{X} + I)^{-1} \tilde{X}^T
\tilde{Q}^{-1} y}^2 + 2\norm{\Sigma^{-\frac{1}{2}} \mu}^2 +
\text{tr}(I) \\ &= 2 \left\Vert\sum_{i=1}^m \left( \sum_{j=1}^m
\frac{z_j \tilde{x}_j \tilde{x}_j^T}{\sigma^2} + I \right)^{-1}
\frac{z_i \tilde{x}_i y_i}{\sigma^2}\right\Vert^2 +
2\norm{\Sigma^{-\frac{1}{2}} \mu}^2 + p \\ & \leq 2 \left(
\sum_{i=1}^m \left\Vert\left( \frac{z_i \tilde{x}_i
\tilde{x}_i^T}{\sigma^2} + \sum_{j \neq i} \frac{z_j \tilde{x}_j
\tilde{x}_j^T}{\sigma^2} + I \right)^{-1} \frac{z_i \tilde{x}_i
y_i}{\sigma^2}\right\Vert \right)^2 + 2\norm{\Sigma^{-\frac{1}{2}}
\mu}^2 + p \\ &= 2 \left( \sum_{i=1}^m |y_i| \left\Vert \left(
\tilde{x}_i \tilde{x}_i^T + \sum_{j \neq i} \frac{z_j}{z_i}
\tilde{x}_j \tilde{x}_j^T + \frac{\sigma^2}{z_i} I \right)^{-1}
\tilde{x}_i \right\Vert \right)^2 + 2\norm{\Sigma^{-\frac{1}{2}}
\mu}^2 + p \\ &= 2 \left( \sum_{i=1}^m |y_i| \sqrt{\tilde{x}_i^T
\left( \tilde{x}_i \tilde{x}_i^T + \sum_{j \neq i} \frac{z_j}{z_i}
\tilde{x}_j \tilde{x}_j^T + \frac{\sigma^2}{z_i} I \right)^{-2}
\tilde{x}_i} \right)^2 + 2\norm{\Sigma^{-\frac{1}{2}} \mu}^2 + p
\\ &\leq 2 \left( \sum_{i=1}^m |y_i| \sqrt{C_i(\tilde{X})} \right)^2
+ 2\norm{\Sigma^{-\frac{1}{2}} \mu}^2 + p,
\end{align*}
where the third inequality follows from Lemma~\ref{lemma:matrix}.
Therefore, $E\left[ \norm{\Sigma^{-\frac{1}{2}} \beta}^2 \Bigm|
\sigma^2, z, y \right]$ is bounded above by a finite constant that
we will call $D$. Therefore, we have
\begin{equation}
\label{eq:t1p1}
\int_{\reals^p} \bigg[ \sum_{i=1}^m (y_i - x_i^T \beta)^2 + \beta^T
\Sigma^{-1} \beta \bigg] \pi(\beta | \sigma^2, z, y) \, d\beta \le
2\norm{y}^2 + D \big( 2 \norm{X \Sigma^{\frac{1}{2}}}^2 + 1 \big)
\;.
\end{equation}
Now, recall that $\sigma^2 | \beta, z, y \sim \mbox{IG} \Big(
\frac{m}{2} + \alpha, \frac{(y-X\beta)^T Q^{-1} (y-X\beta) +
2\gamma}{2} \Big)$. It follows that
\begin{equation}
\label{eq:t1p2}
E\big[ (\sigma^2)^{-1} | \hat{\beta},z,y \big] =
\frac{m+2\alpha}{(y-X\hat{\beta})^T Q^{-1} (y-X\hat{\beta}) +
2\gamma} = \frac{m+2\alpha}{\sum_{i=1}^n z_i (y_i - x_i^T
\hat{\beta})^2 + 2\gamma} \leq \frac{m+2\alpha}{2\gamma} \;.
\end{equation}
And since $\frac{m}{2} + \alpha > 1$, we have
\begin{equation*}
E(\sigma^2 | \hat{\beta},z,y) = \frac{(y-X\hat{\beta})^T Q^{-1}
(y-X\hat{\beta}) + 2\gamma}{m+2\alpha-2} = \frac{\sum_{i=1}^m z_i
(y_i - x_i^T \hat{\beta})^2 + 2\gamma}{m+2\alpha-2} \;.
\end{equation*}
Our assumption then implies that
\begin{equation}
\label{eq:t1p3}
\int_{\reals^m_+} \left[ \int_{\reals_+} \sigma^2 \pi(\sigma^2 |
\hat{\beta}, z, y) \, d\sigma^2 \right] \pi(z| \hat{\beta},
\hat{\sigma}^2, y) \, dz \le \frac{2\gamma}{m+2\alpha-2} + \psi_1
V(\hat{\beta},\hat{\sigma}^2) + L_1 \;,
\end{equation}
where $\psi_1 \in [0,1)$ and $L_1 \in \mathbb{R}$.
Combining~\eqref{eq:t1p1}, ~\eqref{eq:t1p2}, and ~\eqref{eq:t1p3} we
have
\begin{equation*}
E(V(\beta, \sigma^2) | \hat{\beta}, \hat{\sigma}^2) \le \psi_1
V(\hat{\beta},\hat{\sigma}^2) + C \;,
\end{equation*}
where
\[
C = 2\norm{y}^2 + D \big( 2 \norm{X \Sigma^{\frac{1}{2}}}^2 + 1 \big)
+ \frac{m+2\alpha}{2\gamma} + \frac{2\gamma}{m+2\alpha-2} + L_1 \;,
\]
and hence the SS Gibbs Markov chain is geometrically ergodic.
Now for the HS algorithm, we have
\begin{align*}
E(V(\beta, \sigma^2) | \hat{\beta}, \hat{\sigma}^2) = r
\int_{\reals^m_+} \int_{\reals_+} & V(\hat{\beta}, \sigma^2)
\pi(\sigma^2 | \hat{\beta}, z, y) \pi(z| \hat{\beta}, \hat{\sigma}^2,
y) \, d\sigma^2 \, dz \\ & + (1-r) \int_{\reals^m_+} \int_{\reals^p}
V(\beta, \hat{\sigma}^2) \pi(\beta | \hat{\sigma}^2, z, y) \pi(z|
\hat{\beta}, \hat{\sigma}^2, y) \, d\beta \, dz \;.
\end{align*}
Equation~\eqref{eq:t1p1} implies that
\begin{align*}
\int_{\reals^m_+} \int_{\reals^p} V(\beta, \hat{\sigma}^2) \pi(\beta |
\hat{\sigma}^2, z, y) & \pi(z| \hat{\beta}, \hat{\sigma}^2, y) \,
d\beta \, dz \\ & \le 2\norm{y}^2 + D \big( 2 \norm{X
\Sigma^{\frac{1}{2}}}^2 + 1 \big) + \hat{\sigma}^2 +
\frac{1}{\hat{\sigma}^2} \;.
\end{align*}
Equations~\eqref{eq:t1p2} and ~\eqref{eq:t1p3} imply that
\begin{align*}
\int_{\reals^m_+} \int_{\reals_+} & V(\hat{\beta}, \sigma^2)
\pi(\sigma^2 | \hat{\beta}, z, y) \pi(z| \hat{\beta}, \hat{\sigma}^2,
y) \, d\sigma^2 \, dz \le \sum_{i=1}^m (y_i - x_i^T \hat{\beta})^2 +
\hat{\beta}^T \Sigma^{-1} \hat{\beta} \; + \\ &
\frac{m+2\alpha}{2\gamma} + \frac{2\gamma}{m+2\alpha-2} +
\frac{1}{m+2\alpha-2} \sum_{i=1}^m (y_i - x_i^T \hat{\beta})^2
\int_{\reals_+} z_i \, \pi(z_i| \hat{\beta}, \hat{\sigma}^2, y) \,
dz_i \;.
\end{align*}
By assumption, we have $\psi_2 \in \mathbb{R}_+$, $\psi_3 \in [0,1)$
and $L_2 \in \mathbb{R}$ such that
\begin{align*}
\int_{\reals^m_+} \int_{\reals_+} & V(\hat{\beta}, \sigma^2)
\pi(\sigma^2 | \hat{\beta}, z, y) \pi(z| \hat{\beta}, \hat{\sigma}^2,
y) \, d\sigma^2 \, dz \le \sum_{i=1}^m (y_i - x_i^T \hat{\beta})^2 +
\hat{\beta}^T \Sigma^{-1} \hat{\beta} \; + \\ &
\frac{m+2\alpha}{2\gamma} + \frac{2\gamma}{m+2\alpha-2} + \psi_2
\sum_{i=1}^m (y_i - x_i^T \hat{\beta})^2 + \psi_2 \hat{\beta}^T
\Sigma^{-1} \hat{\beta} + \psi_3 \Big( \hat{\sigma}^2 +
\frac{1}{\hat{\sigma}^2} \Big) + L_2 \;.
\end{align*}
Putting all of this together, we have
\begin{align*}
E(V(\beta, \sigma^2) | \hat{\beta}, \hat{\sigma}^2) \le r (\psi_2 + 1)
\bigg( \sum_{i=1}^m (y_i - x_i^T \hat{\beta})^2 \bigg) + & r (\psi_2
+1) \hat{\beta}^T \Sigma^{-1} \hat{\beta} \; + \\ & \Big[ r \psi_3 +
(1-r) \Big] \Big( \hat{\sigma}^2 + \frac{1}{\hat{\sigma}^2} \Big) +
C' \;,
\end{align*}
where
\begin{equation*}
C' = (1-r) \Big[ 2\norm{y}^2 + D \big( 2 \norm{X
\Sigma^{\frac{1}{2}}}^2 + 1 \big) \Big] +
\frac{r(m+2\alpha)}{2\gamma} + \frac{2r\gamma}{m+2\alpha-2} + r L_2
\;.
\end{equation*}
Hence,
\begin{equation*}
E(V(\beta, \sigma^2) | \hat{\beta}, \hat{\sigma}^2) \le \max \Big \{
r(1+\psi_2), r \psi_3 + (1-r) \Big \} V(\hat{\beta}, \hat{\sigma}^2) +
C' \;,
\end{equation*}
and, since $r \psi_3 + (1-r) < 1$ for all $r \in (0,1)$, we have a
valid geometric drift condition as long as $r < (1+\psi_2)^{-1}$.
Finally, an appeal to Lemma~\ref{lemma:hybridge} completes the proof.
\end{proof}
\end{appendix}
\bibliographystyle{ims}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 6,410 |
\section{Overview}
Robots and social artificial intelligence systems can support children's development and well-being by providing automatic, real-time and personalised feedback in the context of natural interactions ~\cite{irfan2019personalization}. These technological advances have the potential to positively transform various areas related to children's development, such as the formal education, of neurotypical and neurodiverse children. However, for a meaningful and successful personalised intervention, these systems need to develop an understanding of the user, a mental model ~\cite{tabrez2020survey} ~\cite{thill2012robot} that integrates the child's individual characteristics, their needs and preferences, their current reactions as well as their development over the course of a long term interaction. As such, modelling children's behaviour is a challenging endeavour particularly if we consider children's intrinsic motivation for exploration, play, creativity and curiosity, necessary faculties for their holistic development, that often make them distinct from adults.
Decades of research in developmental science have produced an increasingly detailed characterisation of learning in children and provide the fundamentals for our understanding of their behaviour and development. More recently, the use of computational approaches in developmental science provides new insights in children's behaviours and cognitive processes which sometime is in contrast with previously established beliefs about children's development. Recent research, for example, criticises the picture of human life history as involving a linear transition from more curious in early childhood to less curious with age. Instead, exploration appears to become more elaborate throughout human childhood ~\cite{pelz2020elaboration}. Furthermore, research with neurodiverse children has shed light to our understanding of atypical cognitive processes and social and emotional engagement, language learning and play ~\cite{pellicano2022annual} ~\cite{gargot2022automatic}.
At the same time, recent developments in artificial intelligence have resulted in breakthrough improvements in areas like machine learning, computer vision and speech processing. These developments have made it increasingly easy to develop applications in robotics, too; for example, libraries such as OpenCV and OpenPose allow for simplified image processing and generation of data sets of child-robot interactions ~\cite{lemaignan2018pinsoro}.
At the same time, these new solutions raise new challenges, for example with respect to biases in data sets and algorithms \cite{mehrabi2021surveyhr} that mean any solution developed using those might not work for aspects that are not considered as much, such as specific needs of neurodiverse children. Attempts to quantify and assess personality traits such as trust are also very problematic ~\cite{spanton2022measuring} in this context, especially since they might lead to a misguided notion that automated assessment and classification of children might be possible. More generally, policy initiatives and guidelines indicate that in parallel with the emerging opportunities robotics and social AI bring for children, there are risks that need to be addressed during the whole life-cycle of an AI product ~\cite{unicef2021} ~\cite{JRC127564}.
Overall, while progress in AI and ML opens vast new opportunities for deploying truly novel solutions (robotic or not) that can have a positive impact on the lives of children, the road there is far from straightforward and there remains an urgent need to raise awareness of the challenges that lie along it as well as discussing how to overcome them.
There also remain fundamental technical challenges; for example, it is still not clear how to design algorithmic (and possibly robotic) approaches towards Theory of Mind despite decades of attempts ~\cite{scassellati2002theory}. Here, research and developments in the field of social robotics and child-robot interaction also brings new insights but also new questions regarding personalisation in children's development. Robots have proven effective in cognitive and socio-emotional support for second language learning ~\cite{vogt2019second}, problem-solving ~\cite{charisi2020child} and story-telling \cite{park2017telling}. In addition, researchers are trying to understand the impact of robots on child-child social interaction ~\cite{charisi2021effects}, perspective taking ~\cite{yadollahi2022motivating} and inter-generational interaction ~\cite{joshi2019robots} as well as a tool to study flexibility in human social cognition ~\cite{wykowska2020social}. However, often current research focuses on very controlled tasks, and modelling children's processes in complex real-life environments remains still a challenge.
As such, the main questions we aim to tackle with this workshop are the following: How can we leverage the insights from developmental science to create social artificial agents able to model children's complex behaviour for meaningful and effective interventions? How might these models help us understand even better children's behaviour especially in the context of their interaction with artificial agents? How can we ensure that resulting technological solutions are in line with children's needs in a fair and non-discriminatory way and mitigate any emerging risk related to AI? How can we take advantage by the involvement of relevant stakeholders such as educators and industry?
The overarching aim of this workshop is to gain interdisciplinary insights into these questions and to promote a common ground and a shared understanding among the relevant scientific communities through a program consisting of paper presentations, interactive sessions, invited talks and a panel discussion.
\subsection{Activities, Schedule and Format}
Submission of 4-6 pages papers describing preliminary results or work in progress relevant to the workshop is encouraged. Submitted papers will undergo a review process. We propose a full day workshop that will include two sessions of short paper presentations from accepted submissions (child's development, AI algorithms and robot solutions), oral presentations from the invited speakers, two interactive sessions and to stimulate interaction among participants, we will culminate the workshop with a plenary discussion and potentially social robot demonstrations.
We propose a full day workshop on the 16th of December with the following tentative schedule:
\textbf{Tentative Schedule\\}
9:00 Welcome by the organizers\\
9:15 Invited talk 1\\
10:00 Short papers presentations 1\\
10:20 Enlightening talk by educator 1 \\
10:30 Coffee break \\
10:45 Short papers presentations 2\\
11:15 Invited talk 2 \\
12:00 Interactive session 1 \\
12:30 Lunch break \\
13:30 Invited talk 3 \\
14:15 Enlighting talk by educator 2 \\
14:25 Enlighting talk by educator 3\\
14:35 Interactive session 2\\
15:15 Invited talk 4 \\
16:00 Plenary discussion with the participation of the audience\\
16:45 Wrap-up \\
17:00 End of the workshop \\
\subsection{List of Possible Topics}
The planned discussion topics within this workshop will include but will not be limited to the following:
\begin{itemize}
\item User modelling
\item Artificial Intelligence in HRI
\item Adaptive robots for children
\item Evaluation methods founded on cognitive, developmental or comparative psychology
\item Child-robot interaction
\item Child speech recognition
\item Eye-tracking techniques
\item Children's perspectives about robots
\item Ethical considerations
\item Privacy and data minimization
\end{itemize}
\subsection{Target Audience and Pre-requisites}
The workshop addresses a broad range of researchers within the fields of Social Robotics, HRI, HCI, Design, Developmental Robotics, that work with neurotypical and neurodiverse children neighboring disciplines.
\subsection{Recruitment and expected number of participants}
Participants will be recruited by means of a call for papers distributed via relevant mailing lists, social media, the organizers professional network, the IEEE society for cognitive and developmental systems and via the International Consortium of Socially Intelligent Robotics (https://mypersonalrobots.org/).
\subsection{Plan for Documenting the work}
The accepted papers will be published as proceedings at the http://ceur-ws.org. A special issue in a suitable journal will be organised depending on the themes of the submissions received and the final outcomes of the workshop discussions.
\section{Invited Speakers}
Confirmed invited speakers include:
\begin{itemize}
\item Thomas Weisswange (Principal Scientist, HONDA Research Institute EU)
\item Mohamed Chetouani (Professor, Sorbonne Université, ISIR-UPMC, CNRS)
\item Agnieszka Wykowska (Principal Investigator, Italian Institute of Technology)
\end{itemize}
In addition to the above mentioned invited speakers, we aim to stimulate a discussion by introducing three short talks by practitioners and educators, with the following confirmed speakers:
Enlighting talks by educators:
\begin{itemize}
\item Tiija Rinta, University College London, UK
\item Tomoko Imai, Jiyugaoka Gakuen High School, Tokyo, Japan
\item Chris Zotos, Arsakeio Lyceum, Patras, Greece
\end{itemize}
\section{Organizers}
\begin{itemize}
\item Serge Thill is an associate professor of artificial intelligence at the Donders Institute, Radboud University. He is a cognitive roboticist and cognitive scientist with a background in computational modelling and computational neuroscience. He is interested in human cognition in the context of interaction with different artificial cognitive systems such as robots. His spans a range of disciplines, from theoretical cognitive science (in particular theories of embodiment and how these relate to machine intelligence) over language and concept grounding to (neuro)-computational models of cognitive mechanisms and practical applications in, for example, autonomous vehicles and robots for therapy for children with autism spectrum disorder. Most recently, he is a PI on the recently funded Horizon Europe Project EMPOWER, which develops technological support for neurodiverse children in educational settings.
\item Vicky Charisi (vasiliki.charisi@ec.europa.eu) \\ is Research Scientist at the Joint Research Centre of the European Commission with a focus on the impact of AI on human behaviour, with a particular interest on child-robot interaction. Her research includes topics such as child's cognitive and socio-emotional development in the context of their interaction with social robots. In parallel, her work informs policy-oriented discussions in relation to AI and children's rights and she is interested in bringing different stakeholders, including researchers, policymakers, children, developers etc together to create common understaning in temrs of AI and children's rights. She is an Associate Editor at the International Journal of Child-Computer Interaction and she serves as a Chair of the IEEE Computational Intelligence Society for Cognitive and Developmental Systems TF for Human-Robot Interaction.
\item Tony Belpaeme is Professor at Ghent University and Visiting Professor of Cognitive Systems and Robotics at Plymouth University. He is a member of IDLab – imec at Ghent and is associated with the Centre for Robotics and Neural Systems at Plymouth. His research interests include social systems, cognitive robotics, and artificial intelligence in general.
\item Ana Paiva focuses on the problems and techniques for creating social agents that can simulate human-like behaviours, be transparent, natural and eventually, give the illusion of life. Over the years she has dealt with this problem by engineering agents that exhibit specific social capabilities, including aspects such as emotions, personality, culture, non-verbal behaviour, empathy, collaboration, and others.
\end{itemize}
\section{Funding}
This workshop is supported in part by the Horizon Europe project EMPOWER (www.project-empower.eu), grant agreement No 101060918, funded by the European Commission.
\bibliographystyle{splncs04}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 1,616 |
Краевски — фамилия:
(род. 1949) — бывший немецкий политик (СДПГ); с 1990 по 1999 год работала министром в Сааре, с 2001 по 2002 годы — сенатором в Берлине.
(род. 1988) — конный тренер (секция обучения верховой езде) и с ноября 2016 года является национальным тренером соревнований по юниорам в Федерации конного спорта Германии, а также активным наездником на соревнованиях. | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 4,147 |
Free Essays for Students
History: People
People in the Mid East
Submitted by: SPB123
Category: History: People
Date Submitted: 03/07/2010 06:41 AM
Report this Essay
triinik,
firstly decide if you like/dislike Kavanagh's poetry. Then focus on 4/5 features of Kavanagh's poetry and say why you like/dislike them, to back up your overall answer of whether you like/dislike the poetry, and use 3+ examples to give evidence for your liking/disliking of the poetry (show how why you said you like/dislike it is true). Below is an example of what you could do, with three sample paragraphs:
Overall answer: I like the poetry of Patrick Kavanagh.
The poetry gives an insight into rural Irish life: Kavanagh's poetry shows what life was like in rural Ireland, during his time. This provides with a glimpse of Ireland, but a very different one from which we lived in, which is not only interesting but also shows how much our country has developed over time.
Examples: Iniskeen Road/ Shancoduff/ Epic/ Advent
His poetry celebrates the ordinary and familiar world: Kavanagh's poetry focuses on and celebrates the ordinary world. This means his poetry focuses on things/places that I am used to, meaning that I don't get lost in his poetry, which happens with other poets when they deal with specific, out-dated subjects, that I have no knowledge of.
Examples: Advent/ Lines Written On a seat on the Grand Canal, Dublin/ A Christmas Childhood/ Epic/ Iniskeen Road/ Shancoduff
Use of I: Kavanagh's poetry often focuses on his life, his desires, wishes, wants, feelings etc. This makes his poetry of worth as it shows that it is meaningful and important as it is relevant to Kavanagh, as well as giving us a centre to explore the poetry and see what it is about, as we consider why Kavanagh wrote his poems.
Examples: Lines Written On a seat on the Grand Canal, Dublin/ Epic/ Iniskeen Road/ Shancoduff/ Advent
http://www.ryjolc.weebly.com
http://www.allhonours.ie/notes/
http://www.allhonours.ie/lc-jc-english-grinds-online-weekly-newsletter-free-online-help-answers-to-every-question-ever/
http://www.gumtree.ie/dublin/97/47948397.html...
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Події
11 травня — Впровадження журналу GameWeek.
З 11 по 16 травня у Лос-Анджелесі відбувається перша виставка Electronic Entertainment Expo.
5 листопада — GameFAQs дебютує в Інтернеті, як архів відеоігор: запитання та відповіді.
Релізи
Пристрої
Nintendo випускає Virtual Boy — першу домашню гральну консоль, здатну відображати «справжню» тривимірну графіку.
SEGA представила приставку Sega Saturn в Європі та Північній Америці (у Японії ця приставка вийшла в листопаді 1994)
у вересні Sony випускає PlayStation в Європі та Північній Америці
Компанії
Випускники Альбертського університету Рей Музика, Грег Зещук та Августин Їп створили компанію BioWare.
Засновано компанію TalonSoft — майбутніх видавців Jagged Alliance 2.
Див. також
Інші події цього року
Примітки
Роки у відеоіграх
1995
1990-ті | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 9,127 |
Q: save a same value using unique constraint with different foreign key id in grails I have a domaing named Thana where I save thanaName with the parent id of district. I have used unique constraint for thanaName. When I save thana with a name suppose 'Thana A' with the district foreign key value 1 it's saved. When I want to save the 'Thana A' again with district foreign key value 2 it would not save it because of unique constraint. But I need to do it cause here district are different although the thanaName are same. Can anyone please help me on this please ? Thanks a lot.
class Thana {
String thanaName
District district
static constraints = {
thanaName unique: true // each instance must have a unique name.
}
static mapping = {
table('thana')
version(false)
district column: 'district_id'
}
}
A: You can use multi-column unique constraints, here is the link of the documentation:
Unique
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 1,543 |
Файзабад () — округ в индийском штате Уттар-Прадеш. Административный центр — город Файзабад. Площадь округа — 2765 км². По данным всеиндийской переписи 2001 года население округа составляло 2 088 928 человек. Уровень грамотности взрослого населения составлял 56,28 %, что ниже среднеиндийского уровня (59,5 %).
Округа Уттар-Прадеш | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 335 |
Mac experts all agree -- if you want to keep an eye on your Mac's performance, there's probably no single app that can do the job better than Drive Genius 5. We want you to get the benefit of this powerful utility app without paying full price, so our deal on this lovely spring Friday will get you a standard license for Drive Genius 5 for just $39 - 60% off of the regular $99 price tag.
Drive Genius 5 consists of 19 powerful utilities that work behind the scenes to keep your Mac running at top performance. What can it do for you?
How good is Drive Genius? It's been used at Apple Genius Bars for the past 7 years. Now you can get your hands on the same powerful software used by Apple Geniuses for only $39. Buy it today! | {
"redpajama_set_name": "RedPajamaC4"
} | 6,623 |
using Stormancer.Common.Helpers;
using Stormancer.Plugins.Authentication;
using System;
using System.Collections.Generic;
using System.Linq;
using System.Text;
using System.Threading.Tasks;
namespace Stormancer
{
public class AuthenticationService
{
private readonly Client _client;
private bool _authenticated = false;
private string _loginRoute = "login";
private string _authenticationSceneName = "authenticator";
private Task<Scene> _authenticationScene;
public string UserId { get; private set; }
public string UserName { get; private set; }
public GameConnectionState ConnectionState { get; set; }
public AuthenticationService(Client client)
{
this._client = client;
}
public Task<Scene> SteamLogin(string steamTicket)
{
var authContext = new Dictionary<string, string> { { "provider", "steam" }, { "ticket", steamTicket } };
return Login(authContext);
}
public Task<Scene> DeviceIdentifierLogin()
{
var identifier = UnityEngine.SystemInfo.deviceUniqueIdentifier;
#if UNITY_EDITOR
identifier = identifier + "editor";
#endif
UnityEngine.Debug.Log(identifier);
var authContext = new Dictionary<string, string> { { "provider", "deviceidentifier" }, { "deviceidentifier", identifier } };
_client.DependencyResolver.Resolve<ILogger>().Log(Diagnostics.LogLevel.Debug, "authenticationservice", "Logging in with identifier " + identifier);
return Login(authContext);
}
public Task<Scene> Login(Dictionary<string, string> authContext)
{
if (_authenticated)
{
return TaskHelper.FromException<Scene>(new InvalidOperationException("Already authenticated."));
}
return GetAuthenticationScene().Then(authScene =>
{
ConnectionState = GameConnectionState.Authenticating;
return authScene.RpcTask<Dictionary<string, string>, LoginResult>(_loginRoute, authContext);
})
.Then(loginResult =>
{
if (loginResult.Success)
{
ConnectionState = GameConnectionState.Authenticated;
UserId = loginResult.UserId;
UserName = loginResult.UserName;
return _client.GetScene(loginResult.Token);
}
else
{
throw new Exception(loginResult.ErrorMsg);
}
});
}
public Task<Scene> GetPrivateScene(string sceneId)
{
return GetAuthenticationScene()
.Then(authScene => authScene.RpcTask<string, string>("sceneauthorization.gettoken", sceneId))
.Then(token => _client.GetScene(token));
}
private Task<Scene> GetAuthenticationScene()
{
if (_authenticationScene == null)
{
lock (this)
{
if (_authenticationScene == null)
{
_authenticationScene = _client.GetPublicScene(_authenticationSceneName, "")
.Then(authScene =>
{
return authScene.Connect().Then(() => authScene);
});
}
}
}
return _authenticationScene;
}
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 7,262 |
Tarzan e os Super 7 (Tarzan and The Super Seven, no original em inglês) é um show de animação, criado pelos estúdios Filmation. Estreou nos EUA em 9 de setembro de 1978.
Faziam parte deste show os seguintes desenhos:
Tarzan (Tarzan, Lord of the Jungle)
The New Adventures of Batman
Força da Liberdade
Manta e Moray (Manta and Moray)
Homem Elástico e Mini Mini
Mulher Aranha (Web Woman)
Jasão do Comando Estelar (Jason of Star Command): o único que não era desenho, e sim seriado, com atores
Elenco
Atores
Tarzan: Olan Soule
Batman / Bruce Wayne: Adam West
Robin / Dick Grayson: Burt Ward
Batgirl / Barbara Gordon: Sherry Alberoni
Alfred Pennyworth: Casey Kasem
Comissário James Gordon: Frank Welker
Mulher Gato: Shannon Farnon
Coringa: Danny Dark
Pinguim: Michael Rye
Charada: Ted Knight
Sr. Frio: Norman Alden
Vozes
Estúdio: Herbert Richers
Tarzan: Márcio Seixas
Batman / Bruce Wayne: Celso Vasconcelos
Robin / Dick Grayson: Rodney Gomes
Batgirl / Barbara Gordon: Ângela Bonatti
Alfred Pennyworth: Waldir Fiori
Comissário James Gordon: Newton da Matta
Mulher Gato: Sumara Louise
Coringa: Nilton Valério
Pinguim: Pádua Moreira
Charada: Júlio Cézar
Sr. Frio: Jorgeh Ramos
Ver também
Filmation
Ligações externas
1978 na televisão
Desenhos animados da década de 1970
Filmation
Séries de televisão de Tarzan | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 3,333 |
{"url":"http:\/\/clay6.com\/qa\/25605\/www.clay6.com\/qa\/25605\/beh-2-and-mgh-2-are-prepared-using-the-reagent","text":"Comment\nShare\nQ)\n\n# $BeH_2$ and $MgH_2$ are prepared using the reagent\n\n$(a)\\;LiH\\qquad(b)\\;AlH_3\\qquad(c)\\;LiAlH_4\\qquad(d)\\;All$\n\nComment\nA)\n$BeH_2$ and $MgH_2$ are prepared using the reagent $LiAlH_4$","date":"2019-02-16 11:26:03","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\": 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\": 0.34817036986351013, \"perplexity\": 2257.1737536842547}, \"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-2019-09\/segments\/1550247480272.15\/warc\/CC-MAIN-20190216105514-20190216131514-00547.warc.gz\"}"} | null | null |
Italian Carrara Circle Mosaic is made up of small circular pieces of Carrara marble, a classic Italian white stone with grey cloudy veins in a delicate mosaic pattern.
Supplied ready mounted on mesh backing sheets*, Carrara Circle Mosaic has a polished finish and is suitable for use on floors and walls. Priced at £589.00m² Ex VAT with a lead time of 4 to 6 weeks.
* Italian Carrara Circle Mosaic is made up of 110 pieces of circular marble per one sheet. All sheets are un-grouted. | {
"redpajama_set_name": "RedPajamaC4"
} | 7,750 |
Homeless Man Saves Girl from Robbery, She Sees His Photo on Her Father's Desk That Evening – Story of the Day
By Roshanak Hannani
https://amodays.com/288612-homeless-man-saves-girl-robbery-she-sees.html
Linda's purse was stolen when she left a coffee shop, but luckily, a homeless man collided with the robber and returned her things. He rejected her offer for money in exchange and left. That night, she told her father what happened and noticed the homeless man in a picture on his desk. Then she realized the shocking truth.
"Bye, everyone!" 16-year-old Linda told her friends while exiting a café. She rummaged through her purse for her phone to call her driver, although the whole idea of not being able to drive on her own still made her angry. Her father, Frank, was overprotective and had hired someone instead of giving her a car.
But before Linda could make that call, a young man on a bicycle snatched her purse and drove off. "OH! STOP THAT THIEF!" she yelled at no one in particular but pointed her finger.
A homeless man did something remarkable. | Source: Shutterstock
People turned to see the robber, and suddenly, a man in tattered clothing launched himself at the bicycle. They collided on the floor, and several officers who had been at the same café caught the robber.
"Who is this?" Linda said, turning the frame around so her dad could see it.
The man approached Linda with her purse and gave it to her with an awkward smile. She looked him up and down, realizing he had to be homeless, and thanked him for his bravery. "Please, take this," she offered, taking a $50 bill out of her purse. "Would you like something to eat? I can buy you something."
The man stared at her for a long time. He was looking at her face as if he knew her. His eyes squinted, and Linda smiled, encouraging him to take her offer. But eventually, he snapped out of his thoughts, shook his head, and rejected her request.
"No, young lady. I have no place among wealthy people," he muttered and walked away.
Linda tried to call out to him, but he continued walking, and his words made her frown. "'No place among wealthy people.' What does that mean?" she asked herself, raising her phone to her ear and calling her driver at last.
Linda decided to tell her dad what happened. | Source: Pexels
During the car ride home, she mulled over the homeless man's words and concluded that he must have been treated poorly by wealthier people. He probably noticed how well-dressed Linda was and thought she was like that. The idea made her sad, but there was nothing she could do about it.
She resolved to find him again someday and give him something to eat as a thank you for returning her purse. The car arrived at her house, and she got out, forgetting all about that issue for a while.
That night, she decided to enter her father's office and tell him what happened. Maybe he was right to be overprotective. Perhaps, instead of just a driver, she needed a bodyguard. Her dad was a well-known businessman in Houston, and her family had always been affluent.
Frank raised his eyes from his computer and saw Linda entering his office. "Darling!" he said, standing up from his desk and extending his arms while approaching her for a big hug.
Linda returned the gesture and spoke up. "Dad, I need to tell you something…."
Frank was worried about his daughter's safety. | Source: Pexels
By the time she finished recounting what happened, her dad was pacing the office, agitated. "I knew this! I knew it was dangerous for you to go out on your own. See? Never complain about that driver again, and I'm going to ask him if he wants more responsibilities. He might be a good bodyguard for you."
Linda sat down at his desk and watched him as he continued pacing. He was still talking about her safety, but it was more to himself than for her benefit. She stared at all the paperwork on his desk, and something caught her attention. There were several picture frames in his work area, but Linda grabbed one in particular.
"Dad! Who is this?"
"What?" he asked, stopping his pacing and frowning at her.
"Who is this?" Linda said, turning the frame around so he could see it.
"Oh… that's Simon, my brother. I told you about him… briefly," Frank answered, pain lacing his voice.
Linda frowned, turned the frame back, and stared intensely at the picture. She knew it was him. "Dad, this is the man that got my purse back! The homeless man!"
"Darling, that's impossible. Simon disappeared long ago. It couldn't be him," Frank replied, shaking his head.
Frank finally agreed to go see this man. | Source: Pexels
"I swear it's him, Dad. Hear me out. Tomorrow, you can come with me to the café, and hopefully, he'll be there," Linda suggested, standing up from his desk. "Please! I swear. My uncle saved me today. We have to do something."
Frank looked at his daughter and finally nodded, making her happy.
Frank and Linda were sipping coffee at the local shop, waiting to see if the man who had saved her from a robbery yesterday would come by at some point. She turned towards the entrance and saw his figure sauntering around. She rose quickly from her chair and urged her father to follow.
"It's him, Dad! Come on!" she almost yelled at him, rushing out of the shop. Frank followed as quickly as he could.
"Sir! Sir! Stop, please!" she called out to the man who stopped and turned. He looked at her for a second, recognition in his eyes. Then his eyes veered towards Frank, and they widened in shock.
"Simon?" Frank uttered breathlessly.
"Frank?" the man asked, and his eyes revealed a terrible pain. Linda wondered what happened all those years ago. Suddenly, her dad did something unexpected.
They brought him home and settled him in a room. | Source: Pexels
Frank rushed toward Simon and gave him a tight hug. "Bro!" he said as if he was a still kid, and Simon embraced his big brother reluctantly.
After that touching reunion, Frank insisted on bringing Simon to their home, although he tried to refuse several times. However, Linda's dad was insistent, and no one could say no to him.
They settled Simon in a room, gave him fresh clothing, and he had a shower before dinner.
All of them, including Frank's wife, Lara, sat down to eat. It was quiet at first, but Linda finally broke the ice. "I need to know what happened. The curiosity is killing me. I've heard a few things about it, but the story was that you disappeared."
"Honey," her mother scolded gently.
"No, she's right. We all have a right to know the truth. What happened, Simon? Where were you all these years?" Frank asked, curiosity and fear in his voice. Perhaps, he didn't want to know the answer at all.
Simon put down his cutlery and stared at all of them. "I left for Virginia when you two started dating," he answered.
"What?" Frank blurted, almost spitting his food.
Simon got angry and disappeared because of jealousy. | Source: Pexels
"I was so jealous that you started dating a wonderful woman like Lara, and I couldn't deal with it. Maybe I was also jealous that she stole you away. But those feelings mixed with resentment because our family name had haunted me our entire lives. I wanted to be my own man. Make my way in the world. Do something for myself. So I left. I wasn't successful. One thing led to another, and I ended up on the streets," Simon explained.
"That doesn't make any sense! You didn't have to leave home for that, and why didn't you come back when things got rough?" Frank asked, shocked at the insane story Simon was telling them.
"I don't know. I think it was pride. I was convinced you guys would turn me away after I disappeared without a word. I was stupid. I returned to Houston because I had a friend here who offered me his couch for a while. But he kicked me out eventually," Simon continued.
Linda couldn't believe that story either. Their family was wealthy, and he had been gone for decades at that point. He could've returned at any moment, but his pride prevented it.
Homeless Man Finds Photo of His Ex & a Child Who Looks Like Him in Abandoned Hospital – Story of the Day
Woman Who Drives a Taxi to Raise Money for Dad's Surgery Finds a Black Bag in the Back Seat — Story of the Day
Frank stared at him thoughtfully, as if weighing his next questions. "Are you ready to come home now? Or are you going to disappear again?"
"I'm ready," his little brother answered, nodding his head. "I'm so sorry. It was the stupidest thing I've ever done. I can't believe I didn't come home as soon as I returned to Houston. I don't know what was going in my head."
They continued eating dinner because everything was alright now. | Source: Pexels
Frank patted his shoulder, pursing his lips as if he was holding his tears. "It's fine, Bro. Everything's alright now," he comforted.
They changed the subject after that and continued eating. Linda was sure that her father probably yelled at Simon later when they were alone. Their entire family thought he was dead or had been kidnapped. That's why Frank was so paranoid about Linda's safety.
But he had disappeared as an adult, and the police never did much despite how influential their family could be. They now had the answers even if his actions didn't make sense. Sometimes, people hold too much pride and don't know how to rectify their mistakes.
You might regret impulsive actions all your life. Simon did something stupid and regretted it for decades, but he didn't dare to make things right until Frank found him.
Your family's safety is a priority. Frank believed that his brother was kidnapped or worse, so he was so concerned about his daughter's safety. But any parent thinks about his family's safety above everything else.
Share this story with your friends. It might brighten their day and inspire them.
If you enjoyed this story, you might like this one about a homeless man who raised a little girl until she was taken away.
Millionaire's Wife Sees Husband Visiting Woman with Triplets, Learns Kids Will Inherit Their Mansion — Story of the Day
A woman loses trust in her marriage after catching her husband red-handed meeting a woman with three triplet girls and later discovering he's named their mansion after the toddlers. But, things take a very different course in the end.
Every Day, Janitor Walks 2.5 Hours on Foot to Get To Work until Colleagues Find Out about It – Story of the Day
Jason walked 2.5 hours every day to get to work because he couldn't afford a new car. After many years, a colleague offered him a ride and a few days later, a strange woman did the same. Then, she and his colleague gave him the shock of a lifetime.
It's a Pity Your Late Dad's Not Here,' Says Bride's Mom, a Familiar Man Approaches Soon After — Story of the Day
A widow giving away her daughter in marriage talks about her late husband's wishes during her speech, and immediately after, a familiar figure approaches her.
Frank was out fishing with his dad when he wandered on the trail and eventually got lost. He decided to seek shelter from the pouring rain in an abandoned old house in the middle of the forest, only to realize that a mother and child had been living there with no adequate supplies.
Every Day, Crying Old Lady Waits at Bus Stop and Walks Away When Bus Arrives – Story of the Day
Eric moved to a small town and loved being a bus driver. He noticed a crying old lady at a specific bus stop one day. She was there every day almost at the same time but never got in. He met her one day at the end of his shift and asked her why she cried. Her answer was staggering.
Little Girl Always Meets Brother When He Returns from School, One Day He Doesn't Get off the Bus – Story of the Day
Gloria waited for her big brother, Anthony, to get home from school everyday single day. But one day, Anthony, didn't get off the bus. Their mother, Lynn, got worried but Anthony eventually appeared. However, a new bus driver showed up the following day, and Lynn started getting suspicious.
Restaurant Owner Pretends to Be Homeless after Seeing Waitress Taking Out Leftovers – Story of the Day
After two failed marriages, Michael had lost all hope of finding a woman who wouldn't want him for his money. But one day, an employee at his restaurant did something extraordinary. He decided to test her, and something remarkable happened.
After His Wife Passes Away, Man Finds an Old Chest in Her Closet with a Letter Inside – Story of the Day
When Mary discovered she was going to die soon, she told her husband, Frank, and started enjoying the rest of her life. After her passing, Frank discovered an old box in her closet that revealed a shocking secret about their son, Anthony.
Bride Gets into Car Crash on Wedding Day, Meets Man Who Caused the Accident a Year Later – Story of the Day
Bridget ran away from her forced wedding and got into a taxi. She told the driver to go faster and they accidentally got into a crash with another car. A year later, she had moved to a small town and found a lost dog. When she called the owner, someone shockingly showed up at her door.
40 Numb Quotes to Help Remove Your Emotional Armor
50 'Milk and Honey' Quotes: Food for Thought to Empower You
47 Eyelashes Quotes for Fresh, Fun and Fabulous Perspectives | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 4,264 |
\section{Boundary points of quasi-attractors}\label{s.boundary}
We discuss the properties of chain-hyperbolic homoclinic classes as in the previous section
that are furthermore quasi-attractors. In particular, we conclude the proof of proposition~\ref{p.position}. The following slightly more general setting will be considered.
\begin{itemize}
\item[--] Let $V\subset M$ be an invariant open set which is a trapping region $f(\overline V)\subset V$.
\item[--] Assume that the maximal invariant set in $V$ is endowed with a partially
hyperbolic splitting $E^s\oplus E^c\oplus E^u$ such that $\dim(E^c)=1$.
\item[--] Let $H(p)\subset V$ be a chain-hyperbolic homoclinic class with the splitting
$E^{cs}\oplus E^{cu}=(E^s\oplus E^c)\oplus E^u$ and containing the
unstable manifold of $p$.
\end{itemize}
In particular, $H(p)$ is saturated by the unstable leaves, tangent to $E^{u}$, and
$U$ is foliated by a forward invariant foliation which extends the strong stable lamination tangent to $E^s$.
\subsection{Comparison of unstable leaves through the strong stable holonomy}\label{su-intersections}
Let us assume that $H(p)$ satisfies the following property.
\begin{description}
\item[Strong intersection property:] \emph{there exist $x,y\in H(p)$ with $y\in W^{ss}(x)\setminus \{x\}$.}
\end{description}
As explained in section~\ref{ss.reduction}, this property prevents the class
to be contained in a lower dimensional submanifold tangent to $E^c\oplus E^u$.
For any point $x\in H(p)$, we fix arbitrarily some plaque $\mathcal{D}} \def\cJ{\mathcal{J}} \def\cP{\mathcal{P}} \def\cV{\mathcal{V}$
transverse to $W^{ss}_{loc}(x)$ and define for any $z$
close to $W^{ss}_{loc}(x)$ the projection $\Pi^{ss}(z)\in \mathcal{D}} \def\cJ{\mathcal{J}} \def\cP{\mathcal{P}} \def\cV{\mathcal{V}$
through the strong stable holonomy.
When $z$ belongs to $H(p)$, the map $\Pi^{ss}$ is a homeomorphism from a neighborhood of $z$
in $\cW^{cu}_z$ to a neighborhood of $\Pi^{ss}(z)$ in $\mathcal{D}} \def\cJ{\mathcal{J}} \def\cP{\mathcal{P}} \def\cV{\mathcal{V}$.
Hence, the projection $\Pi^{ss}(W^u_{loc}(z))$
is a one-codimensional topological submanifold of $\mathcal{D}} \def\cJ{\mathcal{J}} \def\cP{\mathcal{P}} \def\cV{\mathcal{V}$.
In particular, in a neighborhood of $z$, the set
$\mathcal{D}} \def\cJ{\mathcal{J}} \def\cP{\mathcal{P}} \def\cV{\mathcal{V}\setminus \Pi^{ss}(W^u_{loc}(z))$ has locally two connected components.
\begin{definition}\label{definition-cases}
Let us fix $\varepsilon_0>0$ small. The following situations can occur.
\begin{description}
\item[-- The transversal case.] There exists $x,y\in H(p)$ with $y\in W^{ss}_{loc}(x)\setminus \{x\}$ such that
$\Pi^{ss}(W^u_{loc}(y))$ intersects both components of
$\Pi^{ss}(B(x,\varepsilon_0))\setminus \Pi^{ss}(W^u_{loc}(x))$.
\item[-- The jointly integrable case.] There exists $x,y\in H(p)$ with $y\in W^{ss}_{loc}(x)\setminus \{x\}$
such that \\ $\Pi^{ss}(W^u_{loc}(x))$ and $\Pi^{ss}(W^u_{loc}(y))$ coincide in $\Pi^{ss}(B(x,\varepsilon_0))$.
\item[-- The strictly non-transversal case.]
For any $x,y\in H(p)$ with $y\in W^{ss}_{loc}(x)\setminus \{x\}$,
the projection $\Pi^{ss}(W^u_{loc}(y))$
intersects one of the components of
$\Pi^{ss}(B(x,\varepsilon_0))\setminus \Pi^{ss}(W^u_{loc}(x))$ and is disjoint from the other.
\end{description}
\end{definition}
\noindent Note that these definitions do not depend on the choice of the plaque $\mathcal{D}} \def\cJ{\mathcal{J}} \def\cP{\mathcal{P}} \def\cV{\mathcal{V}$.
Clearly one of these three cases happen.
The transversal and the jointly integrable cases may occur at the same time.
The strictly non-transversal case is quite particular.
\begin{lemma}\label{l.boundary}
Let us assume that $H(p)$ does not satisfy the transversal case and consider
two points $x,y\in H(p)$ with $y\in W^{ss}_{loc}(x)\setminus \{x\}$.
For $\varepsilon$ small, if $\Pi^{ss}(W^u_{loc}(y))$ intersects
$\Pi^{ss}(B(x,\varepsilon))\setminus \Pi^{ss}(W^u_{loc}(x))$,
then $x$ and $y$ are not accumulated by $H(p)$
in the same component of $\cW^{cs}_x\setminus W^{ss}_{loc}(x)$.
\end{lemma}
\begin{proof}
Note that if $\varepsilon$ is small enough
and if $\Pi^{ss}(W^u_{loc}(y))$ intersects
$\Pi^{ss}(B(x,\varepsilon))\setminus \Pi^{ss}(W^u_{loc}(x))$,
then $\Pi^{ss}(W^u_{loc}(x))$ intersects
$\Pi^{ss}(B(y,\varepsilon_0))\setminus \Pi^{ss}(W^u_{loc}(y))$.
We denote by $U^+_x,U^-_x$ the local
connected components of $\Pi^{ss}(B(x,\varepsilon))\setminus \Pi^{ss}(W^u_{loc}(x))$
such that $\Pi^{ss}(W^u_{loc}(y))$ meets $U^-_x$ and is disjoint from $U^+_x$.
We also denote by $U^+_y,U^-_y$ the local
connected components of $\Pi^{ss}(B(y,\varepsilon_0))\setminus \Pi^{ss}(W^u_{loc}(y))$
such that $\Pi^{ss}(W^u_{loc}(x))$ meets $U^+_y$ and is disjoint from $U^-_y$.
In particular, $U^+_x\subset U^+_y$.
Let us assume by contradiction that $y$ is accumulated by $H(p)$
from the side of $\cW^{cs}_x\setminus W^{ss}_{loc}(x)$ which projects in $U^+_x$.
Let us consider a point $z\in H(p)$ close to
$y$ and which projects inside $U^+_x$.
Its local unstable manifold is close to the unstable manifold of $y$, hence
$\Pi^{ss}(W^u_{loc}(z))$ meets $U^-_x$ also.
This implies that we are in the transversal case which is a contradiction.
Similarly if $x$ is accumulated by $H(p)$ from the side of
$\cW^{cs}_x\setminus W^{ss}_{loc}(x)$ which projects in $U^-_y$, we find a contradiction.
One deduces that $x$ and $y$ can not be accumulated by $H(p)$
on the same side of $\cW^{cs}_x\setminus W^{ss}_{loc}(x)$.
\end{proof}
\subsection{Structure of the stable boundary points}\label{ss.structure}
For quasi-attractors not in the transversal case, we prove
that the stable boundary points (see section~\ref{ss.one-codim})
belong to the unstable manifold of a periodic orbit.
\begin{proposition}\label{p.boundary}
Let $H(p)$ be a homoclinic class such that
\begin{itemize}
\item[--] $H(p)$ is a quasi-attractor endowed with a partially hyperbolic structure $E^s\oplus E^c\oplus E^u$ such that
$E^c$ is one-dimensional and $E^{cs}=E^s\oplus E^c$ is thin trapped,
\item[--] for any periodic points $q,q'\in H(p)$ homoclinically related to the orbit of $p$, the manifolds
$W^{ss}(q)\setminus \{q\}$ and $W^u(q')$ are disjoint,
\item[--] the transversal case does not hold.
\end{itemize}
Then any stable boundary point of $H(p)$ belongs to the unstable manifold of a periodic point.
\end{proposition}
\begin{proof}
Let $x$ be a stable boundary point of $H(p)$.
Let us assume by contradiction that the point $x$ does not belong to the unstable manifold of a periodic point.
In particular, the unstable manifolds $W^u(f^n(x))$ for $n\in \ZZ$
are all distinct.
Let us consider a point $\zeta$ in the $\alpha$-limit set of $x$.
By considering a plaque transverse to $W^{ss}_{loc}(\zeta)$,
the holonomy $\Pi^{ss}$ is well defined in a neighborhood of $\zeta$.
Since $E^{cs}$ is thin trapped, the plaques of the family $\cW^{cs}$
can be chosen small and one may thus assume that one of the components of
$\cW^{cs}_{x}\setminus W^{ss}_{loc}(x)$ is disjoint from $H(p)$.
Let us introduce two backward iterates $x_1=f^{-n}(x)$ and $x_2=f^{-m}(x)$,
of $x$ close to $\zeta$. By the trapping property, one of the components of
$\cW^{cs}_{x_i}\setminus W^{ss}_{loc}(x_i)$ is also disjoint from $H(p)$ for $i=1$ and $i=2$.
Since $x_1$ and $x_2$ are close, it makes sense to compare the orientations of $E^c_1$ and $E^c_2$.
Choosing different iterates $x_1$ and $x_2$ if necessary, one may assume
that the tangent map $Df^{n-m}\colon E^c_{x_1}\to E^c_{x_2}$ preserves the orientation.
\begin{claim}
Exchanging $x_1$ and $x_2$ if necessary, $W^{ss}_{loc}(x_2)$
meets $W^u_{loc}(x_1)$.
\end{claim}
\begin{proof}
Observe that the plaque $\cW^{cs}_{x_2}$ meets $W^u_{loc}(x_1)$ at a point $x'_1\in H(p)$.
One chooses a small path $t\mapsto x_1(t)$ inside $W^u_{loc}(x_1)$
between $x_1=x_1(0)$ and $x'_1=x_1(1)$. Since $H(p)$ is a quasi-attractor
this path is contained in $H(p)$.
Each plaque $\cW^{cs}_{x_1(t)}$ meets $W^u_{loc}(x_2)$ at a point $x_2(t)$,
defining a path $t\mapsto x_2(t)$ inside $W^u_{loc}(x_2)\cap H(p)$.
For any $t\in [0,1]$, the plaques $\cW^{cs}_{x_1(t)}$ and $\cW^{cs}_{x_2(t)}$
projects by $\Pi^{ss}$ on a $C^1$ curve $\gamma(t)$ which is
topologically transverse to $\Pi^{ss}(W^u_{loc}(x_1))$ and $\Pi^{ss}(W^u_{loc}(x_2))$.
The set $\mathcal{D}} \def\cJ{\mathcal{J}} \def\cP{\mathcal{P}} \def\cV{\mathcal{V}\setminus \Pi^{ss}(W^u_{loc}(x_1))$ has locally two connected components $U^+,U^-$. Hence, $\gamma(t)\setminus \Pi^{ss}(x_1)$ has two connected components $\gamma^+(t)\subset U^+$ and $\gamma^-(t)\subset U^-$ for each $t$.
Let us consider the components $\gamma_1^\pm:=\gamma^\pm(0)$.
By lemma~\ref{l.NGSHI} and since $x_1$ is a stable boundary point,
$\Pi^{ss}(H(p)\cap \cW^{cs}_{x_1})$ meets one of them, $\gamma_1^-$,
and is disjoint from the other one, $\gamma_1^+$.
Similarly, we define $\gamma^-_2,\gamma^+_2$ the connected components of
$\gamma(1)\setminus \Pi^{ss}(x_2)$, such that
$\Pi^{ss}(H(p)\cap \cW^{cs}_{x_2})$ meets the first
and is disjoint from the second.
One deduces that $\gamma^+_2$ is contained in $U^+$ or in $U^-$.
Recall that $\gamma^+_1\subset U^+$.
Since $Df^{n-m}$ preserves the local orientation of $E^c$,
the orientations on $\gamma^+_1$ and $\gamma^+_2$ match and $\gamma_2^+$
is contained in $U^+$.
As a consequence $\Pi^{ss}(W^u_{loc}(x_2))$ is disjoint from
$\gamma^+_1:=\gamma^+(0)$ and from $\gamma^-_2:=\gamma^-(1)$. Since we are not in the transversal case,
one deduces that $\Pi^{ss}(W^u_{loc}(x_2))$ contains $\Pi^{ss}(x_1)$ or
$\Pi^{ss}(x_1')$. Exchanging $x_1$ and $x_2$ if necessary, one has
$W^{ss}(x_1')=W^{ss}(x_2)$.
\end{proof}
\smallskip
Let us denote $x'_1$ the intersection point between
$W^{ss}_{loc}(x_2)$ and $W^u_{loc}(x_1)$.
Since $x_2$ is a boundary point, one connected component of
$\cW^{cs}_{x_2}\setminus W^{ss}_{loc}(x_2)$ is disjoint from $H(p)$.
By lemma~\ref{l.NGSHI} the other component contains sequences of points
of $H(p)$ that accumulate on $x_2$ and $x_1'$.
One deduces from the lemma~\ref{l.boundary} that
the projections of $W^u_{loc}(x_1)$ and $W^u_{loc}(x_2)$ through
the strong stable holonomy match.
Consequently, there exists a periodic point $q\in H(p)$ such that
$W^u_{loc}(x_1)$ and $W^u_{loc}(x_2)$
project on $W^u(q)$ by the strong stable holonomy.
Note that when $x_1,x_2$ are arbitrarily close to $\zeta$,
the point $q$ is also close.
If $q$ and $\zeta$ are distinct, one may consider
backward iterates $x'_1,x'_2$ closer to $\zeta$.
One builds another periodic point $q'\in H(p)$.
All the local unstable manifolds of $x_1,x_2,x'_1,x'_2,q,q'$
have the same projection through the strong stable holonomy.
By lemma~\ref{l.linked}, $q$ and $q'$ are homoclinically related to the orbit of $p$.
This proves that $W^{ss}_{loc}(q)$ and $ W^{u}_{loc}(q')$ intersect,
contradicting our assumption.
If $q$ and $\zeta$ coincide, one can consider higher backward iterates
$f^{-n}(x)$ in a neighborhood of $\zeta$. They all have distinct local unstable plaques
whose projection by the strong stable holonomy coincide.
One deduces that one can find a sequence of such backward iterates which accumulates on
a point $\zeta'\in W^{ss}_{loc}(\zeta)$ different from $\zeta$.
Repeating the construction near $\zeta'$,
one builds a periodic point $q'\in H(p)$
distinct from $q$ and as before $W^{ss}_{loc}(q)$ and $ W^{u}_{loc}(q')$ intersect,
giving again a contradiction.
This ends the proof of the proposition.
\end{proof}
\subsection{The transversal case}
When $H(p)$ is a quasi-attractor, the lemma~\ref{l.cont-quasi-attractor} ensures that
for diffeomorphisms $g$ close to $f$ the unstable manifold
$W^u(p_g)$ is still contained in $H(p_g)$.
\begin{lemma}\label{l.transversal}
Let us assume that $H(p)$ is a quasi attractor and consider $f'$, $C^1$-close to $f$,
such that the transversal case holds for a pair of points $x\neq y$ in $H(p_{f'})$.
Then, for any two different hyperbolic periodic points $p_x,p_y$ homoclinically related
to the orbit of $p_{f'}$ and close to $x$ and $y$ respectively,
and for any diffeomorphism $g$ that is $C^1$-close to $f'$
there exist $x'\in W^u(p_{x,g})$ and $y'\in W^u(p_{y,g})$ in $H(p_g)$ satisfying $W^{ss}(x')=W^{ss}(y')$.
\end{lemma}
\begin{proof}
Let $x, y\in H(p_{f'})$ with $y\in W^{ss}_{loc}(x)\setminus \{x\}$
such that the intersection between $\Pi^{ss}(W^{u}_{loc}(x))$
and $\Pi^{ss}(W^{u}_{loc}(y))$ is topologically transversal.
Consider two periodic points $p_x,p_y$ homoclinically related to $p_{f'}$
and close to $x$ and $y$ respectively, so that
the local unstable manifolds of $p_x$ and $p_y$ are close to the local
unstable manifold of $x$ and $y$.
This implies that $\Pi^{ss}(W^u_{loc}(p_x))$ and $\Pi^{ss}(W^u_{loc}(p_y))$
intersect topologically transversally. By continuity of the local unstable manifolds and the local strong stable holonomy this property still holds for any
$g$ close to $f'$:
there are points $x'\in W^u_{loc}(p_{x,g}), y'\in W^u_{loc}(p_{y,g})$ such that
$W^{ss}(x')=W^{ss}(y')$. By lemma~\ref{l.cont-quasi-attractor}, the local unstable manifolds of $p_{x,g},p_{y,g}$ remain in $H(p_g)$ and therefore the points $x', y'$ are in $H(p_g).$
\end{proof}
\subsection{The jointly integrable case}
The next lemma states that in the jointly integrable case either a heterodimensional cycle is created by a $C^r-$perturbation or for any point in the class there is a well defined continuation.
\begin{lemma}\label{joint.int.continuation}
Let us assume that $H(p)$ is a quasi-attractor whose periodic orbits
are hyperbolic, that $E^{cs}$ is thin trapped
and that the jointly integrable case holds.
Then for any $r\geq 1$ such that $f\in \operatorname{Diff}^r(M)$, one of the following cases occurs.
\begin{itemize}
\item[--] There exists $g$ that is $C^r$-close to $f$
such that $H(p_g)$ exhibits a strong homoclinic intersection.
\item[--] There exists a hyperbolic periodic point $q$ homoclinically related to the orbit of $p$, two maps $g\mapsto x_g,y_g$ defined on a neighborhood $\cV$ of $f$ in $\operatorname{Diff}^r(M)$ and continuous at $f$ such that for any diffeomorphism $g\in \cV$ the points $x_g,y_g$ belong to $H(p_g)\cap W^{s}(q_g)$ and are continuations of $x_f,y_f$. Moreover
$y_g$ belongs to $W^{ss}_{loc}(x_g)$.
\end{itemize}
\end{lemma}
\begin{proof}
Note that by our assumptions the results of sections~\ref{s.weak-hyperbolicity} and~\ref{s.continuation} apply. In particular
for $g$ $C^1$-close to $f$ the class $H(p_g)$ is still chain-hyperbolic and
contains $W^u(p)$. Let us assume that the first item of the proposition does not hold:
on a $C^r$-neighborhood $\cV$ of $f$, there is no diffeomorphism whose homoclinic
class $H(p_g)$ has a strong homoclinic intersection.
Recall that all the periodic orbits are hyperbolic. Since $E^s\oplus E^c$ is thin trapped,
they have the same index and by lemma~\ref{l.linked}, they are all homoclinically related.
There is no periodic points $q,q'\in H(p)$ such that
$W^{ss}(q)\setminus \{q\}$ and $W^u(q')$ intersect: otherwise, one gets a strong homoclinic intersection by using lemma~\ref{joint-int-easy}.
In particular, the proposition~\ref{p.boundary} can be applied.
As in definition~\ref{definition-cases}, let $x,y\in H(p)$ be two close points with disjoint local unstable manifolds
such that for any $z\in W^u_{loc}(x)\cap B(x,\varepsilon_0)$
we have $W^{ss}_{loc}(z)\cap W^u_{loc}(y)\neq \emptyset.$
Observe that there exists a periodic point $q\in H(p)$ close to $x$
whose local stable manifold intersects both the local unstable manifold of $x$ and $y$. Without lose of generality, we can assume that $x,y$ belong to $W^s_{loc}(q)$.
The point $x, y$ do not belong both to the unstable manifold of some periodic points $p_x, p_y$:
otherwise, we would get a strong connection by applying lemma~\ref{joint-int-easy}.
We can thus now assume that $x$ does not belong to the unstable manifold of a periodic point.
In particular, by proposition~\ref{p.boundary} it is not a stable boundary point
and it is accumulated by points in $H(p)$ from both connected components of $\cW^{cs}_x\setminus W^{ss}_{loc}(x).$
The corollary~\ref{c.continuation} (in the orientation preserving case) implies that there exist two maps
$g\mapsto x_g,y_g$ on $\cV$ satisfying $(x_f,y_f)=(x,y)$ and
for any $g$ close to $f$, the points $x_g,y_g$ belong to $H(p_g)$ and have the same strong stable manifold. The points $x_g,y_g$ are accumulated by $H(p_g)$ in the same component
of $\cW^{cs}_{x_g}\setminus W^{ss}_{loc}(x_g)$.
Let us prove the continuity.
Since the point $x$ is accumulated from both sides, it has two continuations
$g\mapsto x_g,x'_g$. By lemma~\ref{l.cont-central},
for any $g$ one has $x'_g\in \cW^{cs}_{x_g}$.
One can choose an orientation of $E^c_x$ and by lemma~\ref{l.ordering}
assume that for any $g$, the point $x'_g$ does not meet
$\cW^{cs,+}_{x_g}$.
By lemma~\ref{l.continuite}, the map $g\mapsto x'_g$ is semi-continuous at $f$:
when $(g_n)$ is a sequence that converges to $f$, then
any limit $\bar x'$ of $(x'_{g_n})$ does not meet $\cW^{cs,-}_{x'_f}=\cW^{cs,-}_{x}$.
One deduces that any limit $\bar x$ of $(x_{g_n})$ does no meet
$\cW^{cs,-}_{x}$ either.
Since the map $g\mapsto x_g$ is also semi-continuous,
the limit $\bar x$ does not meet $\cW^{cs,+}_{x}$.
One deduces that $\bar x$ belongs to $W^{ss}_{loc}(x)$.
The orbit of $\bar x$ is shadowed by the orbit of $x$, hence one
deduces that $\bar x=x$.
Let us now consider any limit point $\bar y$
of $(y_{g_n})$. By construction it has to belong to $W^{ss}_{loc}(x)$
and $W^{ss}_{loc}(y)$,
and so $\bar y=y$. We have thus proved that the maps
$g\mapsto x_g,y_g$ are continuous at $f$.
\end{proof}
\subsection{The strictly non-transversal case}
In the strictly non-transversal case, roughly speaking is proved that either by perturbation is created a strong homoclinic connection, or for a diffeomorphisms nearby the strong stable leaves contains at most one point in the class or there are two periodic points such that for any diffeomorphisms nearby their unstable manifolds intersects some strong stable leaves (see lemma \ref{l.strictly-transversal}).
\begin{lemma}\label{l.boundary1}
Let us assume that $H(p)$ satisfies the strictly non-transversal case.
Then, any close points $x\neq y$ in $H(p)$ satisfying $y\in W^{ss}_{loc}(x)$
are stable boundary points. Moreover they are not accumulated by
$H(p)$ in the same component of $\cW^{cs}_{x}\setminus W^{ss}_{loc}(x)$.
\end{lemma}
\begin{proof}
Since $H(p)$ satisfies the strictly non-transversal case and $x,y$ are close,
there exists $y'\in W^u_{loc}(y)$ and $x'\in W^u_{loc}(x)$ such that
$y'\in W^{ss}_{loc}(x')$ and for any $\varepsilon>0$, the manifolds
$\Pi^{ss}(W^{u}_{loc}(y'))$ intersects
$\Pi^{ss}(B(x',\varepsilon))\setminus \Pi^{ss}(W^u_{loc}(x'))$.
By lemma~\ref{l.boundary}, they are not accumulated by $H(p)$
in the same component and in particular both are stable boundary points.
\end{proof}
\smallskip
For the points $(x,y)$ as in the previous lemma
the following property obviously holds
(the open region considered below is then empty):
\begin{itemize}
\item[(**)]
\it $\cW^{cs}_x$ contains $y$. The open region in $\cW^{cs}_x$ bounded by $W^{ss}_{loc}(x)\cup W^{ss}_{loc}(y)$
does not meet $H(p)$.
\end{itemize}
Note that this property already appeared in corollary~\ref{c.continuation}.
The next lemma states that the set of such pairs $(x,y)$ is quite small.
\begin{lemma}\label{l.boundary2}
Let $H(p)$ be a quasi-attractor such that $E^{cs}$ is thin trapped,
the strictly non-transversal case holds and for any periodic points
$q,q'\in H(p)$ the manifolds $W^{ss}(q)\setminus \{q\}$ and $W^{u}(q)$ are disjoint.
Let us fix $\delta>0$.
Then, there exist $N\geq 1$ and finitely many periodic points $p_1,\dots,p_s$ such that
any points $x\neq y$ in $H(p)$ satisfying (**) and $d(x,y)\geq \delta$
belong to the union of the $f^N(W^u_{loc}(p_i))$, $i\in\{1,\dots,s\}$.
\end{lemma}
\begin{proof}
We fix $\delta>0$ small.
We first note that by lemma~\ref{l.boundary1} and proposition~\ref{p.boundary},
any $x,y$ as in the statement of the lemma are stable boundary points and there exists
some periodic points $p_x,p_y\in H(p)$ such that
$x$ belongs to $W^u(p_x)$ and $y$ to $W^u(p_y)$.
Let $P$ be the (closed) set of pairs $(x,y)\in H(p)^2$ satisfying~(**) and $d(x,y)\geq \delta$.
We have to prove that if two pairs $(x,y)$ and $(x',y')$ in $P$ are close, then
$x'\in W^u_{loc}(x)$ and $y'\in W^u_{loc}(y)$. This is done by contradiction:
we consider a sequence $(x_n,y_n)_{n\geq 0}$ in $P$ that converges toward $(x,y)$
and assume that all the leaves $W^{u}_{loc}(x_n)$ are distinct.
One may assume that $x$ is accumulated by $H(p)$ inside $\cW^{cs,+}_{x}$.
First we claim that $W^{u}_{loc}(x_n)$ does not cut $W^{ss}_{loc}(x)$.
Otherwise, we denote by $z_n$ the intersection point. The plaque $\cW^{cs}_{z_n}$ coincides
with $\cW^{cs}_{x}$ in a neighborhood of $z_n$ by lemma~\ref{l.uniqueness-coherence}, hence
$z_n$ is not accumulated by $H(p)\cap \cW^{cs,-}_{z_n}$ for $n$ large.
One deduces that $z_n$ and $x$ belongs to the same local strong stable leaf and are accumulated
by points of $H(p)\cap \cW^{cs,+}_{z_n}$ and $H(p)\cap \cW^{cs,+}_{x}$ respectively,
contradicting the definition of the strictly non-transversal case.
Let $\Pi^{ss}$ be the projection along the strong stable holonomy on a disk $\mathcal{D}} \def\cJ{\mathcal{J}} \def\cP{\mathcal{P}} \def\cV{\mathcal{V}$ transverse
to $W^{ss}_{loc}(x)$. The projections $\Pi^{ss}(W^u_{loc}(x_n)), \Pi^{ss}(W^u_{loc}(x)), \Pi^{ss}(W^u_{loc}(y)), \Pi^{ss}(W^u_{loc}(y_n))$ are one codimensional manifolds of $\mathcal{D}} \def\cJ{\mathcal{J}} \def\cP{\mathcal{P}} \def\cV{\mathcal{V}$:
by our assumptions, the one-dimensional curve $\gamma=\Pi^{ss}(\cW^{cs}_x)$ meets them in this order.
Since we are in the strictly non-transversal case, the order is the same on any other curve
$\gamma'=\Pi^{ss}(\cW^{cs}_{x'})$ where $x'\in W^u_{loc}(x)$ is close to $x$.
In particular, when $x'$ is the intersection point between $W^u_{loc}(x)$ and
$\cW^{cs}_{x_n}$, one finds a contradiction since $W^{u}_{loc}(x)$ and $W^{u}_{loc}(y)$
cannot intersect the open region of $\cW^{cs}_x$ bounded by $W^{ss}_{loc}(x)\cup W^{ss}_{loc}(y)$
and by the same argument as above, $W^{u}_{loc}(x)\cap W^{ss}_{loc}(x_n)$ and
$W^{u}_{loc}(y)\cap W^{ss}_{loc}(y_n)$ are empty.
This concludes the proof of the lemma.
\end{proof}
\begin{lemma}\label{l.strictly-transversal}
Let us assume that $H(p)$ is a quasi-attractor whose periodic orbits
are hyperbolic, that $E^{cs}$ is thin trapped
and that the strictly non-transversal integrable case holds.
Then for any $r\geq 1$ such that $f\in \operatorname{Diff}^r(M)$, one of the following cases occurs.
\begin{itemize}
\item[--] There exists $g$, $C^r$-close to $f$
such that $H(p_g)$ exhibits a strong homoclinic intersection.
\item[--] There exists $g$, $C^r$-close to $f$
such that for any $x\neq y$ in $H(p_g)$ one has $W^{ss}(x)\neq W^{ss}(y)$.
\item[--]
There exist two hyperbolic periodic points $p_x,p_y$ homoclinically related
to the orbit of $p$ and an open set $\cV\subset \operatorname{Diff}^r(M)$
whose closure contains $f$, such that for any $g\in \cV$ the class $H(p_g)$ contains
two different points $x\in W^u(p_{x,g})$ and $y\in W^u(p_{y,g})$
satisfying $W^{ss}(x)=W^{ss}(y)$.
\end{itemize}
\end{lemma}
\begin{proof}
As in the proof of lemma~\ref{joint.int.continuation},
for $g$ that is $C^1$-close to $f$ the class $H(p_g)$ is still chain-hyperbolic and contains $W^u(p)$. Moreover, one can assume that for any periodic points $q,q'\in H(p)$
the manifolds $W^{ss}(q)\setminus \{q\}$ and $W^u(q)$ do not intersect.
Let us fix $\delta<0$ small.
One can consider the periodic points $p_1,\dots,p_s$ and the integer $N\geq 1$
provided by the lemma~\ref{l.boundary2}.
These points are hyperbolic, homoclinically related to $p$ by lemma~\ref{l.linked}
and have a continuation for any $g$ that is $C^1$-close to $f$.
One may also assume there is no $g$ in a $C^r$-neighborhood of $f$
such that $H(p_g)$ has a strong homoclinic intersection.
One can then consider the continuation given by proposition~\ref{p.continuation}.
We also introduce the period $\tau_i$ of each periodic point $p_i$.
In a small neighborhood of $f$ in $\operatorname{Diff}^r(M)$,
consider for each pair $(p_i,p_j)$ the (closed) subset $D_{i,j}$ of diffeomorphisms $g$ such that
the class $H(p_g)$ contains some distinct points $x\in f^{N+\tau_i}(\overline{W^u_{loc}(p_{i,g})})$
and $y\in f^{N+\tau_j}(\overline{W^u_{loc}(p_{j,g})})$ with $y\in \overline{W^{ss}_{loc}(x)}$.
The diffeomorphisms in the interior of $D_{i,j}$ are in the third case of the lemma.
If the sets $D_{i,j}$ have empty interior,
there exists an open set $\cV$ in $\operatorname{Diff}^r(M)$
whose closure contains $f$ such that for any $g\in \cU$,
any $p_i,p_j$ and any distinct points
$x\in f^{N+\tau_i}(W^u_{loc}(p_{i,g}))$
and $y\in f^{N+\tau_j}(W^u_{loc}(p_{j,g}))$ one has $y\not\in W^{ss}_{loc}(x)$.
To conclude, we have to prove that for $g\in \cU$ close to $f$
and any distinct points $x, y\in H(p_g)$ one has $W^{ss}(x)\neq W^{ss}(y)$,
giving the second case of the lemma.
This is done by contradiction: one considers a pair $(x,y)$ such that
$y\in W^{ss}_{loc}(x)$ and
up to consider a backward iterate, one can require that
the points $x,y$ satisfy $d(x,y)>2\delta$.
Having chosen $g$ close enough to $f$, one deduces (lemma~\ref{l.cont-central})
that any continuations $x_f,y_f$ for $f$ still satisfy
$d(x_f,y_f)>\delta$.
If $x,y$ are accumulated in the same component
of $\cW^{cs}_{x}\setminus W^{ss}_{loc}(x)$, then by corollary~\ref{c.continuation}
(in the orientation preserving case) the same holds for the continuations $x_f,y_f$ for $f$.
This contradicts lemma~\ref{l.boundary1}.
If $x,y$ are accumulated in different components
of $\cW^{cs}_{x}\setminus W^{ss}_{loc}(x)$,
then by corollary~\ref{c.continuation} (in the orientation reversing case)
the continuations $x_f,y_f$ for $f$
satisfy (**). Since their distance is bounded from below by $\delta$,
lemma~\ref{l.boundary2} implies that $x_f,y_f$
belong to $f^N(W^u_{loc}(p_i))$ and $f^N(W^u_{loc}(p_j))$ respectively.
By lemma~\ref{l.cont-unstable}, one deduces that for the diffeomorphism $g$ close, the points $x,y$
belong to $g^{N+\tau_i}(W^u_{loc}(p_{i,g}))$ and $g^{N+\tau_j}(W^u_{loc}(p_{j,g}))$ respectively.
This contradicts our assumption on $g$.
\end{proof}
\subsection{Proof of proposition~\ref{p.position}}
Let us consider a diffeomorphism $f\in \operatorname{Diff}^{1+\alpha}(M)$, $\alpha\geq 0$, and a homoclinic class $H(p)$ as in the statement of theorem~\ref{t.position}
and assume that the two first cases of the proposition do not occur.
If the jointly integrable case holds,
the lemma~\ref{joint.int.continuation} gives the third case of the proposition.
If the transversal or the strictly non-transversal case holds,
the lemmas~\ref{l.transversal} and~\ref{l.strictly-transversal}
give the fourth case of the proposition.
\section{Chain-recurrence classes far from homoclinic bifurcations}
\label{s.classes}
We introduce in sections~\ref{ss.trapped} and~\ref{ss.homoclinic}
the notion of trapped plaque families and chain-hyperbolic homoclinic classes.
Their basic properties will be studied systematically later in section~\ref{s.weak-hyperbolicity},
but we will derive before (sections~\ref{ss.homoclinic}, \ref{ss.aperiodic}
and~\ref{ss.finiteness}) important consequences for the generic dynamics far
from homoclinic bifurcations.
We also present (sections~\ref{ss.extremal} and~\ref{ss.quasi-attractor})
the main results of the paper that are proved in the next sections
and explain how they imply the main theorem.
In the last part (section~\ref{ss.consequences}) we give other consequences of our techniques.
We start this section by recalling some classical definitions.
\medskip
In all the paper $M$ denotes a compact boundaryless manifold.
\begin{defi}
We say that $f\in \operatorname{Diff}^1(M)$ exhibits a \emph{homoclinic tangency} if there is a hyperbolic periodic orbit $O$
and a point $x\in W^s(O)\cap W^u(O)$ with $T_xW^s(O)+ T_xW^u(O)\neq T_xM$.
\end{defi}
\begin{defi}
We say that $f\in \operatorname{Diff}^1(M)$ exhibits a \emph{heterodimensional cycle} if there are
two hyperbolic periodic orbits $O$ and $O'$ of different stable dimension,
such that $W^u(O)\cap W^s(O')\neq \emptyset$ and $W^u(O')\cap W^s(O)\neq \emptyset$.
\end{defi}
\begin{defi} From now on, with $\overline{\operatorname{Tang}\cup\operatorname{Cycl}}$ it is denoted the set of diffeomorphisms
that can be $C^1-$approximated by one exhibiting either a homoclinic tangency or a heterodimensional cycle.
We say that a diffeomorphisms $f$ is \emph{$C^1-$far from cycles and tangencies}
if $f\in \operatorname{Diff}^1(M)\setminus\overline{\operatorname{Tang}\cup\operatorname{Cycl}}$
\end{defi}
\medskip
The global dynamics of a diffeomorphism may be decomposed in the following way.
The \emph{chain-recurrent set} is the set of points that belong to a periodic $\varepsilon$-pseudo orbit
for any $\varepsilon>0$. This compact invariant set
breaks down into invariant compact disjoint pieces, called the \emph{chain-recurrence classes}:
two points belong to a same piece if they belong to a same periodic $\varepsilon$-pseudo orbit
for any $\varepsilon>0$.
An invariant set is \emph{chain-transitive} if it contains a $\varepsilon$-dense
$\varepsilon$-pseudo-orbit for any $\varepsilon>0$.
\begin{defi} A \emph{quasi-attractor} is a chain-recurrence class which is Lyapunov stable,
i.e. which admits a basis of neighborhoods $U$ satisfying $f(U)\subset U$.
\end{defi}
\medskip
For any diffeomorphism, we define another notion of ``piece of the dynamics''.
Associated to a hyperbolic periodic point $p$, one introduces
its \emph{homoclinic class} $H(p)$ which is the closure of the transverse
intersection points between the unstable and the stable manifolds $W^u(O),W^s(O)$
of the orbit $O$ of $p$.
It also coincides with the closure of the set of hyperbolic points $q$ that are
\emph{homoclinically related} to the orbit of $p$, i.e. such that
$W^u(q)$ and $W^s(q)$ have respectively a transverse intersection point
with the stable and the unstable manifolds of the orbit of $p$.
Note that for diffeomorphisms $g$ that are $C^1$-close to $f$, the periodic point $p$
has a hyperbolic continuation $p_g$. This allows to consider the homoclinic class $H(p_g)$.
For a $C^1$-generic diffeomorphism, the periodic points are hyperbolic and
\cite{BoCr} proved that a chain-recurrence
class that contain a periodic point $p$ coincides with the homoclinic class $H(p)$.
The other chain-recurrence classes are called the \emph{aperiodic classes}.
Those classes are treated in subsections \ref{ss.homoclinic} and \ref{ss.aperiodic}.
We state two other consequences of Hayashi's connecting lemma and~\cite{BoCr}.
\begin{lemma}\label{l.prelim} For any $C^1$-generic diffeomorphism $f$ and any homoclinic class $H(p)$,
\begin{itemize}
\item[--] if $H(p)$ contains periodic points with different stable dimensions, then
$f$ may be $C^1$-approxi\-mated by diffeomorphisms having a heterodimensional cycle;
\item[--] $H(p)$ is a quasi-attractor if and only if it contains the unstable manifold of $p$.
\end{itemize}
\end{lemma}
\medskip
Quasi-attractor always exist but for a $C^1$-generic diffeomorphism they attract most orbit.
\begin{theorem}[\cite{MP,BoCr}]
Let $f$ be a diffeomorphism in a dense G$_\delta$ subset of $\operatorname{Diff}^1(M)$.
Then the $\omega$-limit set of any point $x$ in a dense G$_\delta$ subset of $M$
is a quasi-attractor.
\end{theorem}
According to this result, the main theorem is a consequence of two independant properties
of $C^1$-generic diffeomorphisms that are $C^1$-far from cycles and tangencies:
\begin{itemize}
\item[--] the union of the quasi-attractors is closed (see proposition~\ref{p.finiteness});
\item[--] each quasi-attractor is a hyperbolic set (see theorem~\ref{t.position}).
\end{itemize}
Indeed by the shadowing lemma, any quasi-attractor which is hyperbolic is transitive
and attracts any orbit in a neighborhood. In particular,
the quasi-attractors are isolated in the chain-recurrence set. Since their union is closed,
they are finite.
\subsection{Trapped tangent dynamics}\label{ss.trapped}
Let $f$ be a diffeomorphism and $K$ be an invariant compact set.
\medskip
A \emph{dominated splitting} on $K$ is a decomposition $T_KM=E\oplus F$ of its
tangent bundle into two invariant linear sub-bundles such that, for some integer
$N\geq 1$, any unitary vectors $u\in E_x,v\in F_x$ at points $x\in K$ satisfy
$$2\|Df^N.u_x\|\leq \|Df^N.v_x\|.$$
This definition does not depend on the choice of a Riemannian metric on $M$.
In the same way, one can define dominated splittings
$T_KM=E_1\oplus \dots \oplus E_s$ involving more than two bundles.
When the bundle $E$ is uniformly contracted (i.e. when there exists $N\geq 1$
such that for any unitary vector $u\in E$ one has $\|Df^N.u\|\leq 2^{-1}$),
the stable set of each point $x$ contains an injectively embedded sub-manifold
$W^{ss}(x)$ tangent to $E_x$ called the \emph{strong stable manifold of $x$},
which is mapped by $f$ on the manifold $W^{ss}(f(x))$.
A \emph{partially hyperbolic splitting} on $K$
is a dominated splitting $T_KM=E^s\oplus E^c\oplus E^u$ such that
$E^s$ and $E^u$ are uniformly contracted by $f$ and $f^{-1}$ respectively.
\medskip
\begin{defi}
A \emph{plaque family tangent to $E$} is a continuous map $\cW$ from the linear bundle $E$
over $K$ into $M$ satisfying:
\begin{itemize}
\item[--] for each $x\in K$, the induced map $\cW_x\colon E_x\to M$ is a $C^1$-embedding
which satisfies $\cW_x(0)=x$ and whose image is tangent to $E_x$ at $x$;
\item[--] $(\cW_x)_{x\in K}$ is a continuous family of $C^1$-embeddings.
\end{itemize}
The plaque family $\cW$ is \emph{locally invariant} if there exists $\rho>0$
such that for each $x\in K$ the image of the ball $B(0,\rho)\subset E_{x}$ by $f\circ \cW_{x}$
is contained in the plaque $\cW_{f(x)}$.
\end{defi}
We often identify $\cW_x$ with its image.
The plaque family theorem~\cite[theorem 5.5]{HPS} asserts that a locally invariant plaque family
tangent to $E$ always exists (but is not unique in general).
\begin{defi}\label{d.tt}
The plaque family is \emph{trapped} if for each $x\in K$, one has
$$f(\overline{\cW_x})\subset \cW_{f(x)}.$$
It is \emph{thin trapped} if for any neighborhood $S$ of the section $0$ in $E$ there exist:
\begin{itemize}
\item[--] a continuous family $(\varphi_x)_{x\in K}$ of $C^1$-diffeomorphisms of the spaces $(E_x)_{x\in K}$ supported in $S$;
\item[--] a constant $\rho>0$ such that for any $x\in K$ one has
$$f(\overline{\cW_x\circ \varphi_x(B(0,\rho))})\subset \cW_{f(x)}\circ \varphi_{f(x)}(B(0,\rho)).$$
\end{itemize}
\end{defi}
If a plaque family $\cW$ is thin trapped, then it is also the case
for any other locally invariant plaque family $\cW'$ tangent to $E$
(moreover there exists $\rho>0$ such that for each $x\in K$,
the ball $B(0,\rho)\subset E_{x}$ is sent by $\cW'_{x}$ into $\cW_{x}$, see lemma~\ref{l.uniqueness-coherence}).
One thus say that \emph{$E$ is thin trapped}.
\begin{remark}\label{r.nested}
Note also hat when $E$ is thin trapped, there exist nested families of trapped plaques
whose diameter are arbitrarily small.
\end{remark}
The two following properties are classical (see for instance~\cite[Lemma 2.4]{C2}).
On a small neighborhood of $K$, we introduce a cone field $\mathcal{C}} \def\cI{\mathcal{I}} \def\cO{\mathcal{O}} \def\cU{\mathcal{U}^E$ which is a thin neighborhood of the bundle $E$.
\begin{lemma}\label{l.uniqueness-coherence}
Let $K$ be a compact invariant set endowed with a dominated decomposition $T_KM=E\oplus F$.
There exists $r>0$ such that if there exists a trapped plaque family $\cW^{cs}$ tangent to $\mathcal{C}} \def\cI{\mathcal{I}} \def\cO{\mathcal{O}} \def\cU{\mathcal{U}^E$
whose plaques have a diameter smaller than $r$, then the following properties hold.
\begin{itemize}
\item[--] If $\widehat{\cW}^{cs}$ is another locally invariant plaque family tangent to $E^{cs}$,
then there exists $\rho>0$ such that
for each $x\in H(p)$ the image of the ball $B(0,\rho)\subset E_{x}$ by $\cW^{cs}_{x}$
is contained in $\widehat \cW^{cs}_{x}$.
\item[--] There exists $\varepsilon>0$ such that for any points $x,x'\in H(p)$ that are
$\varepsilon$-close with $\cW^{cs}_x\cap \cW^{cs}_{x'}\neq \emptyset$, then $f(\overline{\cW^{cs}_{x'}})\subset \cW^{cs}_{f(x)}$.
\end{itemize}
\end{lemma}
\subsection{Homoclinic classes}\label{ss.homoclinic}
Far from homoclinic bifurcations, the homoclinic classes of a generic diffeomorphism
satisfy some weak form of hyperbolicity.
\begin{defi}\label{d.chain-hyperbolic}
A homoclinic class $H(p)$ is said to be \emph{chain-hyperbolic} if:
\begin{itemize}
\item[-] $H(p)$ has a dominated splitting $T_{H(p)} M= E^{cs}\oplus E^{cu}$ into
center stable and center unstable bundles;
\item[-] there exists a plaque family $(\cW^{cs}_x)_{x\in H(p)}$ tangent to $E^{cs}$
which is trapped by $f$ and a plaque family $(\cW^{cu}_x)_{x\in H(p)}$ tangent to $E^{cu}$
which is trapped by $f^{-1}$;
\item[-] there exists a hyperbolic periodic point $q_s$ (resp. $q_u$) homoclinically related to the orbit of $p$
whose stable manifold contains $\cW^{cs}_{q_s}$
(resp. whose unstable manifold contains $\cW^{cu}_{q_u}$).
\end{itemize}
Such a class is \emph{topologically hyperbolic} if its
center stable and center unstable plaques are thin trapped by $f$ and $f^{-1}$ respectively.
\end{defi}
One will see (lemma~\ref{l.chain-stable} below) that for any point $x\in H(p)$, the plaque $\cW^{cs}_{x}$
is contained in the chain-stable set of $H(p)$.
This justifies the name ``chain-hyperbolicity":
this definition generalizes the hyperbolic basic sets endowed with
families of stable and unstable plaques
(in this case the plaques $\cW^{cs}$ are the images of
local stable manifolds by a backward iterate $f^{-n}$).
With additional assumptions, the chain-hyperbolicity is a robust property:
if $H(p)$ is chain-hyperbolic for $f$, coincides with its chain-recurrence class and
if $E^{cs}, E^{cu}$ are thin trapped by $f$ and $f^{-1}$ respectively,
then for any $g$ that is $C^1$-close to $f$ the homoclinic class $H(p_g)$
associated to the continuation $p_g$ of $p$ is also chain-hyperbolic (see lemma~\ref{l.robustness}).
\begin{theo}[\cite{C2}]\label{t.homoclinic}
Let $f$ be a diffeomorphism in a dense G$_\delta$ subset of $\operatorname{Diff}^1(M)\setminus \overline{\operatorname{Tang}\cup\operatorname{Cycl}}$.
Then, any homoclinic class of $f$ is chain-hyperbolic.
Moreover, the central stable bundle $E^{cs}$ is thin trapped.
If it is not uniformly contracted,
it decomposes as a dominated splitting $E^{cs}=E^s\oplus E^c$
where $dim(E^c)=1$ and $E^s$ is uniform; and there exist periodic orbits
homoclinically related to $p$ and whose Lyapunov exponents along $E^c$ are arbitrarily close to $0$.
The same holds for the central unstable bundle $E^{cu}$ and $f^{-1}$.
\end{theo}
\begin{proof}
The statement in~\cite{C2} is slightly different and we have to justify why the center stable bundle $E^{cs}$
is thin trapped. When $E^{cs}$ is uniformly contracted, this is very standard. When $E^{cs}$
is not uniformly contracted, \cite[section 6]{C2} asserts that there exists a dominated splitting $E^{cs}=E^s\oplus E^c$
such that $\dim(E^c)=1$, $E^s$ is uniformly contracted and that the bundle $E^c$ has ``type (H)-attracting":
there exists a locally invariant plaque family $\mathcal{D}} \def\cJ{\mathcal{J}} \def\cP{\mathcal{P}} \def\cV{\mathcal{V}$ tangent to $E^c$ and arbitrarily small open neighborhoods
$I$ of the section $0$ in $E^c$ satisfying $f(\overline{\mathcal{D}} \def\cJ{\mathcal{J}} \def\cP{\mathcal{P}} \def\cV{\mathcal{V}_x(I_x)})\subset \mathcal{D}} \def\cJ{\mathcal{J}} \def\cP{\mathcal{P}} \def\cV{\mathcal{V}_{f(x)}(I_{f(x)})$ for each $x\in H(p)$.
The neighborhood $I$ may be chosen as a continuous family of open intervals $(I_x)_{x\in H(p)}$.
Let us now consider a locally invariant plaque family $\cW$ tangent to $E^{cs}$.
Since $I$ is small, one has
$\mathcal{D}} \def\cJ{\mathcal{J}} \def\cP{\mathcal{P}} \def\cV{\mathcal{V}_x(I_x)\subset \cW_x$ for any $x\in H(p)$ (see~\cite[lemma 2.5]{C2}).
One then builds for each $x$ a small open neighborhood $V_x$ of
$\mathcal{D}} \def\cJ{\mathcal{J}} \def\cP{\mathcal{P}} \def\cV{\mathcal{V}_x(I_x)$ in $\cW_x$ which depends continuously on $x$:
this can be obtained by modifying a tubular neighborhood of $\mathcal{D}} \def\cJ{\mathcal{J}} \def\cP{\mathcal{P}} \def\cV{\mathcal{V}_x(I_x)$ in $\cW_x$.
Since $E^s$ is uniformly contracted one can still require the
trapping property $f(\overline{V_x})\subset V_x$.
Let $U_x\subset E^{cs}_x$ be the backward image of $V_x$ by $\cW^{cs}$.
Since $U_x$ can be obtained by modifying the tubular neighborhood of a $C^1$-curve,
it can be chosen diffeomorphic in $E^{cs}$ to an open ball
through a diffeomorphism as stated in definition~\ref{d.tt}.
\end{proof}
One deduces that the tangent bundle over
a non-hyperbolic homoclinic class as in theorem~\ref{t.homoclinic}
has a dominated splitting $TM=E^s\oplus E^c\oplus E^u$ or $E^s\oplus E^c_1\oplus E^c_2\oplus E^u$
where each bundle $E^c$ or $E^c_1,E^c_2$ is one-dimensional, $E^s$ is uniformly contracted and $E^u$
is uniformly expanded (however, one of them can be trivial).
Note that under perturbations the homoclinic class $H(p_g)$
is still chain-hyperbolic but its center stable bundle $E^{cs}$ is a priori not thin trapped.
\medskip
We will focus on the invariant compact sets $K$ that are \emph{Lyapunov stable},
i.e. that have a basis of neighborhoods $U$ that are invariant by $f$
(i.e. $f(U)\subset U$).
\begin{coro}\label{c.homoclinic}
Let $f$ be $C^1$-generic in $ \overline{\operatorname{Tang}\cup\operatorname{Cycl}}^c$.
Then, for any Lyapunov stable homoclinic class of $f$ the center unstable bundle is uniformly expanded.
\end{coro}
\begin{proof}
For any open set $U\subset M$ and any integer $d\geq 0$,
one considers the following property:
\begin{description}
\item[$P(U,d)$:] There exists a hyperbolic periodic orbit $O\subset U$
whose stable dimension equals $d$.
\end{description}
This property is open: if $P(U,d)$ is satisfied by $f$, then
so it is by any diffeomorphism $g$ that is $C^1$-close to $f$.
Let us fix a countable basis of open sets $\mathcal{B}} \def\cH{\mathcal{H}} \def\cN{\mathcal{N}} \def\cT{\mathcal{T}$, i.e. for any compact set
and any open set $V$ satisfying $K\subset V\subset M$, there exists $U\in \mathcal{B}} \def\cH{\mathcal{H}} \def\cN{\mathcal{N}} \def\cT{\mathcal{T}$
such that $K\subset U\subset V$.
Then, for any diffeomorphism $f$ in dense G$_\delta$ subset $\cR_0\subset \operatorname{Diff}^1(M)$,
for any open set $U\in \mathcal{B}} \def\cH{\mathcal{H}} \def\cN{\mathcal{N}} \def\cT{\mathcal{T}$ and any $d\geq 0$,
if there exists a perturbation $g$ of $f$ such that $P(U,d)$ holds for $g$, then the same holds for $f$.
We denote by $\cR\subset \cU$ a dense G$_\delta$ subset of
$\operatorname{Diff}^1(M)\setminus \overline{\operatorname{Tang}\cup\operatorname{Cycl}}$ whose elements satisfy theorem~\ref{t.homoclinic}
and have hyperbolic periodic orbits are.
Let us consider $f\in \cR$ and a homoclinic class $H(p)$ of $f$ whose
center unstable bundle $E^{cu}=E^c_2\oplus E^u$ is not uniformly expanded. Hence $\dim(E^{c}_2)$
is one-dimensional, $p$ is not a sink (and apriori $E^u$ could be degenerated).
By the theorem~\ref{t.homoclinic}, there exists a hyperbolic periodic orbit $O$
homoclinically related to $p$ having some Lyapunov exponent along $E^{cu}$ arbitrarily close to $0$.
By Franks lemma, one can find a perturbation $g$ of $f$ such that $O$ becomes a hyperbolic periodic orbit
whose stable space contains $E^{c}_2$.
Since $f\in \cR_0$, one deduces that
any neighborhood of $H(p)$ contains a periodic orbit whose stable dimension is $d^s+1$,
where $d^s$ denotes the stable dimension of $p$.
Let us consider a locally invariant plaque families $\cW$ tangent to $E^{cs}$
over the maximal invariant set in a neighborhood of $H(p)$.
Let us consider a periodic orbit $O$
contained in a small neighborhood of $K$,
with stable dimension equal to $d^s+1$.
As a consequence, using the domination $E^{cs}\oplus E^{cu}$,
the Lyapunov exponents along $E^{cs}$ of $O$ is smaller than some uniform constant $-C<0$.
If the plaques of the family $\cW$ are small enough,
the lemma~\ref{l.largestable} and the remark~\ref{r.large} below then ensure that at some $q\in O$
one has $\cW_{q}\subset W^s(q)$.
By lemma~\ref{l.contper} below, $q$ is close to a hyperbolic periodic point
$z$ homoclinically related to $p$
whose plaque $\cW^{cu}_z$ is contained in the unstable set of $z$.
The plaque
$\cW_{q}$ intersects transversally the plaque $\cW^{cu}_z$.
This proves that the stable manifold of $q$ also intersects transversally
the unstable manifold of the orbit of $p$.
Since $H(p)$ is Lyapunov stable, it contains $W^u(z), q$ and $W^u(q)$.
As for $H(p)$, the point $q$ is not a sink. This proves that $E^u$ is non trivial.
Let $y\in W^u(q)\setminus \{q\}$.
Since $y$ belongs to $H(p)$, the stable manifold of the orbit of $p$ accumulates on $y$,
hence by a $C^1$-small perturbation produced by Hayashi's connecting lemma, one can create an intersection between the unstable manifold of $q$ and
the stable manifold of the orbit of $p$. The intersection between $W^u(p)$ and $W^s(q)$ persists hence
we have built a heterodimensional cycle, contradicting our assumptions.
We have proved that if $H(p)$ is Lyapunov stable, the bundle $E^{cu}$ is uniformly expanded.
\end{proof}
\subsection{Aperiodic classes}\label{ss.aperiodic}
Far from homoclinic bifurcations, the aperiodic classes have also a partially hyperbolic structure.
\begin{theo}[\cite{C2}]\label{t.aperiodic}
Let $f$ be a diffeomorphism in a dense G$_\delta$ subset of $\operatorname{Diff}^1(M)\setminus \overline{\operatorname{Tang}\cup\operatorname{Cycl}}$.
Then, any aperiodic class of $f$ is a minimal set
and holds a partially hyperbolic structure $E^s\oplus E^c \oplus E^u$.
Moreover, there exists a continuous familly of center stable plaques
$\cW^{cs}$ tangent to $E^{cs}=E^s\oplus E^c$
which are trapped by $f$.
Similarly, there exists a continuous family of center unstable plaques
$\cW^{cu}$ tangent to $E^{cu}=E^c\oplus E^u$ which are trapped by $f^{-1}$.
\end{theo}
\medskip
\begin{coro}\label{c.aperiodic}
Let $f$ be generic in $\operatorname{Diff}^1(M)\setminus\overline{\operatorname{Tang}\cup\operatorname{Cycl}}$. Then,
for any aperiodic class, the bundles $E^u$ and $E^s$ are non-degenerated.
The strong unstable manifolds of points of the class are not contained in the class.
In particular, the class is not Lyapunov stable.
\end{coro}
\begin{proof}
Let us consider an aperiodic class $K$ and
a locally invariant plaque family $\cW$ tangent to
$ E^{cs}$ over the maximal invariant set in a small neighborhood of $K$.
There exists a sequence of periodic orbits that accumulate on $K$.
A trapped plaque family $\cW^{cs}$ over $K$ whose plaques have small diameters
are contained in the plaques $\cW$ by lemma~\ref{l.uniqueness-coherence} below.
One deduces that one can extend the plaque family $\cW^{cs}$
over the maximal invariant set in a small neighborhood of $K$
as a trapped plaque family.
Since $K$ is a minimal set and $f$ is $C^1$-generic, Pugh's closing lemma
(the general density theorem) implies that $K$ is the Hausdorff limit of a sequence of periodic orbits.
For any $\tau$-periodic point $p$ whose orbit is close to $K$,
the plaque $\cW^{cs}_p$ is mapped into itself by $f^\tau$.
Since the plaque $\cW^{cs}$ is tangent to the bundle $E^{cs}= E^s\oplus E^c$
where $E^c$ has dimension $1$ and $E^s$ is uniformly contracted,
the orbit of any point in $\cW^{cs}_p$ accumulates in the future on a periodic orbit.
If $E^u$ is degenerate, the union of the plaques $\cW^{cs}_p$
cover a neighborhood of $K$, hence the orbit of any point in $K$
converges towards a periodic orbit, which is a contradiction.
If $E^u$ is not degenerate, the strong unstable manifold $W^{uu}(x)$ tangent to $E^u$
of any point $x\in K$ intersects the plaque $\cW^{cs}_p$ of a periodic point $p$.
One deduces that theres exists an orbit that accumulates on $K$ in the past and on
a periodic orbit $O$ in the future. If $W^{uu}(x)$ is contained in $K$,
the periodic orbit $O$ is contained in $K$, contradicting the fact that $K$ is an aperiodic class.
\end{proof}
\begin{remark}
Actually, a stronger result can be proved.
\noindent
\emph{For any $C^1$-generic diffeomorphism and any aperiodic class $K$
endowed with a partially hyperbolic structure
$T_KM=E^s\oplus E^c\oplus E^u$ with $\dim(E^u)=1$,
the class is not contained in a locally invariant submanifold tangent to $E^s\oplus E^c$.}
\noindent
Indeed, otherwise, one could work in this submanifold and get a contradiction as in the previous proof.
See also section \ref{ss.reduction}.
\end{remark}
\subsection{Reduction of the ambient dimension}\label{ss.reduction}
Let us consider an invariant compact set $K$
with a dominated splitting $T_KM=E^s\oplus F$ such that $E^s$ is uniformly
contracted. The dynamics on $K$ may
behave like the dynamics inside a manifold of smaller dimension.
This motivates the following definition.
\begin{definition}
A $C^1$-submanifold $\Sigma$ containing $K$ and tangent to $F$ is \emph{locally invariant}
if there exists a neighborhood $U$ of $K$ in $\Sigma$ such that
$f(U)$ is contained in $\Sigma$.
\end{definition}
More generally,
when $K$ admits a partially hyperbolic splitting $T_KM=E^s\oplus E^c\oplus E^u$
one may define the notion of locally invariant submanifold tangent to $E^c$. The next proposition state that the property
defined above is robust by $C^1-$perturbations.
\begin{proposition}[\cite{BC2}]\label{p.whitney}
Let $K$ be an invariant compact set endowed with a dominated splitting
$T_KM=E^s\oplus F$ such that $E^s$ is uniformly contracted.
If $K$ is contained in a locally invariant submanifold tangent to $F$,
then the same holds for any diffeomorphism
$C^1$-close to $f$ and any compact set $K'$ contained in a small neighborhood
of $K$.
\end{proposition}
There exists a simple criterion for the existence of a locally invariant submanifold.
\begin{theo}[\cite{BC2}]\label{t.whitney}
Let $K$ be an invariant compact set with a dominated splitting $E^s\oplus F$
such that $E^s$ is uniformly contracted.
Then $K$ is contained in a locally invariant submanifold tangent to $F$
if and only if the strong stable leaves for the bundle $E^s$
intersect the set $K$ in only one point.
\end{theo}
One can deduce a generic version of previous theorem.
\begin{corollary}\label{c.whitney}
Let $f$ be $C^1$-generic and $H(p)$ be a homoclinic class having a dominated splitting $E^s\oplus F$ such that $E^s$ is uniformly contracted.
Then, either $H(p)$ is contained in a locally invariant submanifold
tangent to $F$ or for any diffeomorphism $g$ that is $C^1$-close to $f$,
there exist two different points $x\neq y$ in $H(p_g)$ such that $W^{ss}(x)=W^{ss}(y)$.
\end{corollary}
\begin{proof}
By~\cite{BoCr},
there exists a dense G$_\delta$ subset $\cR\subset \operatorname{Diff}^1(M)$
of diffeomorphisms whose homoclinic classes are chain-recurrence classes.
In particular, for any $f\in \cR$ and any homoclinic class $H(p)$ for $f$,
the class $H(p_g)$ for $g$ $C^1$-close to $f$ is contained in a small neighborhood of $H(p)$. By proposition~\ref{p.whitney}, one deduces that
if $H(p)$ has a dominated splitting $E^s\oplus F$ and
is contained in a locally invariant submanifold tangent to $F$, then the same holds for the classes $H(p_g)$.
As a consequence, for any $f$ in a dense G$_\delta$ subset of $\operatorname{Diff}^1(M)$,
and any homoclinic class $H(p)$ of $f$, either for any diffeomorphism $g$
close to $f$ the class $H(p_g)$ is contained in a locally invariant submanifold tangent to $F$
or for any diffeomorphism $g$ close to $f$ the class $H(p_g)$ is not contained in such a manifold.
The theorem~\ref{t.whitney} ends the proof.
\end{proof}
\medskip
The previous result raises an important question for us:
\begin{question*}
When $H(p)$ is not contained in a locally invariant submanifold tangent to $F$,
is it possible to find a periodic point $q$ homoclinically related to the orbit of $p$
whose strong stable manifold $W^{ss}(q)\setminus \{q\}$ intersects $H(p)$?
\end{question*}
Such an intersection is called a generalized strong homoclinic intersection in the next section. We will provide answers for this problem in some particular cases, see
theorems~\ref{t.tot-discontinuity} and~\ref{t.position} below.
\subsection{Strong homoclinic intersections}
Inside a homoclinic class, some periodic points exhibit
a transverse intersection between their stable and unstable manifolds.
If this intersection holds along strong stable and unstable manifolds of
the periodic orbit we say that there is a strong
homoclinic connection. More precisely, we introduce the following definition:
\begin{defi}\label{strong-int}
Given a hyperbolic periodic orbit $O$ with a dominated splitting $T_OM=E\oplus F$
such that the stable dimension of $O$ is strictly larger (resp. strictly smaller) than $\dim(E)$
it is said that $O$ exhibits a \emph{strong stable homoclinic intersection}
(resp. a \emph{strong unstable homoclinic intersection}) if the invariant manifold of $O$
tangent to $E$ and the unstable manifold of $O$ (resp. the invariant manifold of $O$
tangent to $F$ and the stable manifold of $O$) have an intersection point outside the orbit $O$.
\end{defi}
This definition can be generalized for homoclinic classes.
\begin{defi}
A homoclinic class $H(p)$ has a \emph{strong homoclinic intersection}
if there exists a hyperbolic periodic orbit orbit $O$ homoclinically related to $p$
which has a strong homoclinic intersection.
\end{defi}
The strong homoclinic intersections allow sometimes to create heterodimensional cycles.
The following statement generalizes~\cite[proposition 2.4]{Pu1}. The proof is similar and we only sketch it.
\begin{prop}\label{p.strong-connection}
Let $H(p)$ be a homoclinic class for a diffeomorphism $f$ such that:
\begin{itemize}
\item[--] $H(p)$ has a dominated splitting $E\oplus F$ and the stable dimension of $p$
is $\dim(E)+1$;
\item[--] there exist some hyperbolic periodic orbits homoclinically related to $p$
having some negative Lyapunov exponents arbitrarily close to $0$.
\end{itemize}
If there exist some diffeomorphisms $g$ $C^1$-close to $f$ such that $H(p_g)$
has a strong homoclinic intersection, then there exist some $C^1$-close perturbations
of $f$ that have an heterodimensional cycle
between a hyperbolic periodic orbit homoclinically related to $p$ and
a hyperbolic periodic orbit of stable dimension $\dim(E)$.
\end{prop}
Before proving this proposition, we explain how it is possible by a $C^r$-perturbation to
transport the strong homoclinic intersection to another periodic orbit.
\begin{lemma}\label{l.change-connection}
Let $H(p)$ be a homoclinic class for a $C^r$-diffeomorphism $f$ with $r\geq 1$ such that:
\begin{itemize}
\item[--] $H(p)$ has a dominated splitting $E\oplus F$ and the stable dimension of $p$
is $\dim(E)+1$;
\item[--] $H(p_g)$ has a strong homoclinic intersection.
\end{itemize}
Then for any periodic point $q$ homoclinically related to $p$
there exist some $C^r$-close perturbations of $f$ that have a periodic point $q'$
homoclinically related to the orbit of $p$ which exhibit a strong homoclinic intersection
and whose minimal Lyapunov exponents along $F$ are close to the one of $q$.
\end{lemma}
\begin{proof}
Let us consider a transverse intersection point
$z_s$ between $W^{s}(O)$ and $W^u(q)$ and a transverse intersection point $z_u$ between $W^u(O)$
and $W^s(q)$ where $O$ is the orbit of $p$. There exists a transitive hyperbolic set $K$ which contains $z_s,z_u,O$
and which is included in a small neighborhood $U$ of $\overline{\{f^n(z_s)\}_{n\in \ZZ}
\cup\{f^n(z_u)\}_{n\in \ZZ}}$.
One deduces that there exists a sequence of periodic points $(q_n)$ converging to $p$
and whose orbit is contained in $U$ and homoclinically related to $p$.
One may choose these orbits in such a way that they spend most of their iterates
close to the orbit of $q$. Note that $K$ has a dominated splitting of the form
$E\oplus E^c\oplus F'$ where $E^c$ is one-dimensional
and $E\oplus E^c, F'$ respectively coincide with the stable and the unstable bundle.
As a consequence the minimal Lyapunov exponents of $q_n$ along $E$ are arbitrarily close to
the corresponding exponent of $q$ when $n$ is large.
For a small $C^r$ perturbation $g$ supported in a small neighborhood of $\zeta$
(hence disjoint from $K$), one can first ensure that $T_\zeta W^u(O)\oplus E_\zeta$ is one-codimensional
and then consider a small arc of diffeomorphisms $(g_t)$ which coincides with $g$ when $t=0$ and
which unfolds the strong intersection:
in a neighborhood of $\zeta$ the strong homoclinic intersection has disapeared for $t\neq 0$.
The local unstable manifold and the local manifold tangent to $E$
for $q_n$ accumulate on the local unstable manifold and the local manifold tangent to $E$ for $O$
respectively. One thus deduces that for a diffeomorphism $C^r$ close to $g$
and $n$ large enough, the strong stable and the unstable manifolds of the orbit of $q_n$ intersect.
This gives the conclusion for $q'=q_n$.
\end{proof}
\begin{proof}[Sketch of the proof of proposition~\ref{p.strong-connection}]
Let us fix $\varepsilon>0$ and a periodic point $q$ homoclinically related to the orbit of $p$
and whose minimal Lyapunov exponent along $F$ belongs to $(-\varepsilon,0)$.
Let $g$ be a diffeomorphism $C^1$-close to $f$ and
$O$ be a periodic orbit homoclinically related to the continuation
$p_{g}$ of $p$ for $g$ which exhibits a strong homoclinic intersection $y$ between its unstable manifold
and its invariant manifold tangent to $E$.
By lemma~\ref{l.change-connection}, one can find a small $C^1$-perturbation $g_1$
having a periodic point $q_1$ homoclinically related to $p_{g_1}$, whose minimal
Lyapunov exponent along $F$ belongs to $(-\varepsilon,0)$ and which exhibits a strong homoclinic intersection.
Let us consider a local stable manifold $\mathcal{D}} \def\cJ{\mathcal{J}} \def\cP{\mathcal{P}} \def\cV{\mathcal{V}$ of $q_1$.
Since $q_1$ has a stable exponent close to $0$, one can by $C^1$-perturbation $g'$
(as small as one wants if one chooses $\varepsilon$ and $q$ accordingly) create inside $\mathcal{D}} \def\cJ{\mathcal{J}} \def\cP{\mathcal{P}} \def\cV{\mathcal{V}$ a hyperbolic periodic point
$q'$ of stable dimension $\dim(E)$. Since $\mathcal{D}} \def\cJ{\mathcal{J}} \def\cP{\mathcal{P}} \def\cV{\mathcal{V}$ has dimension $\dim(E)+1$, one can also require
that $\mathcal{D}} \def\cJ{\mathcal{J}} \def\cP{\mathcal{P}} \def\cV{\mathcal{V}$ contains finitely many periodic points of stable dimension $\dim(E)+1$,
close to $q_1$, whose stable sets cover a dense subset
of $\mathcal{D}} \def\cJ{\mathcal{J}} \def\cP{\mathcal{P}} \def\cV{\mathcal{V}$. If the perturbation is realized in a small neighborhood of $q_1$,
the manifold $W^u(p_{g'})$ intersects transversally $\mathcal{D}} \def\cJ{\mathcal{J}} \def\cP{\mathcal{P}} \def\cV{\mathcal{V}$, hence one can ensure that
the unstable manifold of $p_{g'}$ intersects transversally the stable manifold of a periodic
point $q''$, so that $q''$ and $p_{g'}$ are homoclinically related.
The stable manifold of $q''$ intersects the unstable manifold of $q'$ along an orbit
contained in $\mathcal{D}} \def\cJ{\mathcal{J}} \def\cP{\mathcal{P}} \def\cV{\mathcal{V}$. Since the local invariant manifolds of $q',q''$ are close to those
of $q_1$, one can by a small perturbation close to the strong homoclinic intersection of $q_1$
create an intersection between $W^u(q')$ and $W^{s}(q'')$.
This gives a heterodimensional cycle associated to the periodic orbit $q''$
that is homoclinically related to $p_{g'}$.
\end{proof}
\medskip
If a homoclinic class $H(p)$ contains two hyperbolic periodic points $q,q'$ homoclinically related
to $p$ such that the strong stable manifold $W^{ss}(q)\setminus \{q\}$ and the unstable manifold $W^u(q')$ intersect, one can create
a strong homoclinic intersection by a $C^r$-perturbation, for any $r\geq 1$.
We have a more general result.
\begin{lemma} \label{joint-int-easy} Let $f$ be a $C^r$-diffeomorphism, $r\geq 1$ and
let $q, p_x, p_y$ be three periodic points whose
orbits are homoclinically related such that
\begin{itemize}
\item[--] the homoclinic class $H(q)$ has a dominated splitting $T_{H(q)}M=E^s\oplus F$ and
$\dim(E^s)$ is strictly smaller than the stable dimension of $O$;
\item[--] there are two distinct transversal intersection points $x\in W^u(p_x)\mbox{~$|$\hspace{ -.46em}$\cap$}~ W^s(q)$, $y\in W^u(p_y)\mbox{~$|$\hspace{ -.46em}$\cap$}~ W^s(q)$ sharing the same strong stable leaf.
\end{itemize}
Then for any $r\geq 1$, there is $g$ $C^r$-close to $f$ such that $H(q_g)$ has a strong homoclinic intersection.
\end{lemma}
\begin{proof}
One can assume that $y$ is distinct from $q$.
There is a transitive hyperbolic set $\Lambda$
that contains $p_x$, $p_y$, $x$ and $q$ but not $y$.
So, it follows that there is a periodic point $\hat q$ homoclinically related to $p$ arbitrarily close to $x$
and whose orbit is close to $\Lambda$ in the Hausdorff topology.
One deduces that the local strong stable manifold of $\hat q$ and the local
unstable unstable manifold of the orbit of $\hat q$ are close to $y$.
By a $C^r$-perturbation, one can thus create an intersection at $y$, hence a strong connection between these manifolds,
keeping the transverse homoclinic orbits with $p$.
This shows that $H(q_g)$ has a strong homoclinic intersection for this new
diffeomorphism $g$.
\end{proof}
\medskip
We generalize again the definition of strong homoclinic intersection.
\begin{defi}
A homoclinic class $H(p)$ has a \emph{generalized strong homoclinic intersection}
if there exists a hyperbolic periodic orbit orbit $O$ homoclinically related to $p$,
having a dominated splitting $T_OM=E\oplus F$
such that the stable dimension of $O$ is strictly larger (resp. strictly smaller) than $\dim(E)$,
and whose invariant manifold tangent to $E$ (resp. to $F$) contains a point $z\in H(p)\setminus O$.
\end{defi}
Using the $C^1-$connecting lemma due to Hayashi, the following result holds immediately.
\begin{prop}\label{p.generalized-strong-connection}
Let $H(p)$ be a homoclinic class for a diffeomorphism $f$
which has a generalized strong homoclinic intersection.
Then, there exist some $C^1$-close diffeomorphisms $g$ such that
$H(p_g)$ has a strong homoclinic intersection.
\end{prop}
One may wonder if this last result still holds in $C^r$-topologies for $r>1$.
We have a result in this direction under stronger assumptions.
The proof is much less elementary than the previous ones and will be obtained
as a corollary of theorem~\ref{t.stable} at the end of section~\ref{proofjointint}.
\begin{prop}\label{p.generalized-strong-connectionCr}
For any diffeomorphism $f_0$ and any homoclinic class
$H(p)$ which is a chain-recurrence class
endowed with a partially hyperbolic structure $E^s\oplus E^c\oplus E^u$, $\dim(E^c)=1$,
such that $E^s\oplus E^c$ is thin trapped,
there exists $\alpha_0> 0$ and a $C^1$-neighborhood $\cU$ of $f_0$
with the following property.
For any $\alpha\in [0,\alpha_0]$ and any $C^{1+\alpha}$-diffeomorphism $f\in \cU$ such that $H(p_f)$
has a generalized strong homoclinic intersection,
there exists a diffeomorphism $g$ arbitrarily $C^{1+\alpha}$-close to $f$ such that
$H(p_g)$ has a strong homoclinic intersection.
\end{prop}
\subsection{Total disconnectedness along the center-stable plaques}
Let us consider a chain-hyperbolic homoclinic class $H(p)$. In certain part of the proof of the main theorem, we need a better understanding on the geometrical properties of the class in order, for instance,
to build analogs of Markov partitions. To do that, we
need to ensure that the intersection of $H(p)$ with its center-stable plaques is totally
disconnected. By lemma~\ref{l.uniqueness-coherence} this property does not depend on the choice of a center-stable plaque family.
It is provided by the following result proved in section~\ref{s.2D-central}.
\begin{theo}\label{t.tot-discontinuity}
Let $f$ be a diffeomorphism and $H(p)$ be a chain-hyperbolic homoclinic class
with a dominated splitting $E^{cs}\oplus E^{cu}=(E^{ss}\oplus E^c_1)\oplus E^c_2$ such that $E^c_1,E^c_2$ are one-dimensional and $E^{cs}$ and $E^{cu}$ are thin trapped.
Then, one of the following cases holds.
\begin{itemize}
\item The strong stable manifolds (tangent to $E^s$) intersect the class in at most one point.
\item There exists a periodic point $q$ in $H(p)$ whose strong stable manifold
$W^{ss}(q)\setminus\{q\}$ intersects $H(p)$.
\item The class is totally disconnected along the center-stable plaques.
\end{itemize}
\end{theo}
Under this general setting the point $q$ is not necessarily homoclinically related to $p$.
Note that this theorem also applies and may be interesting for locally maximal hyperbolic sets $K$ having
a dominated splitting $T_KM=E^s\oplus E^u=(E^{s}\oplus E^c)\oplus E^u$ such that
$E^c,E^u$ are one-dimensional.
\subsection{Extremal bundles}\label{ss.extremal}
Theorems~\ref{t.homoclinic} and~\ref{t.aperiodic}
show that the chain-recurrence classes $K$ of a $C^1$-generic
diffeomorphism far from homoclinic bifurcations have a partially hyperbolic
splitting $T_KM=E^s\oplus E^c\oplus E^u$ with $\dim(E^c)\leq 2$.
We now prove that the extremal bundles are non-degenerated.
This will ensure that the diffeomorphisms considered in the main theorem have
only finitely many sinks.
For aperiodic classes this has already been obtained with corollary~\ref{c.aperiodic}.
For homoclinic classes one can apply the following result.
\begin{teo}\label{t.extremal}
Let $f$ be a diffeomorphism in a dense G$_\delta$ subset of $\operatorname{Diff}^1(M)$
and let $H(p)$ be a homoclinic class endowed with a partially hyperbolic splitting
$T_{H(p)}M=E^s\oplus E^c_1\oplus E^c_2\oplus E^u$, with $\dim(E^c_1)\leq 1$ and $\dim(E^c_2)\leq 1$.
Assume moreover that the bundles $E^s\oplus E^c_1$ and $E^c_2\oplus E^u$
are thin trapped by $f$ and $f^{-1}$ respectively and that the class is contained in a locally invariant
submanifold tangent to $E^s\oplus E^c_1\oplus E^c_2$.
Then one of the two following cases occurs:
\begin{itemize}
\item[--] either $H(p)$ is a hyperbolic set,
\item[--] or there exists diffeomorphisms $g$
arbitrarily $C^1$-close to $f$ with a periodic point $q$ homoclinically related
to the orbit of $p_g$ and exhibiting a heterodimensional cycle.
\end{itemize}
\end{teo}
\begin{remark}
We will see in section~\ref{ss.consequences} that the result can be improved: the second
case of the theorem never appears.
\end{remark}
The proof relies on techniques developed in \cite{PS1, PS2, PS4} for $C^2$-diffeomorphisms
that extend a result in \cite{mane-contribution-stabilite} for one-dimensional endomorphisms.
We list different settings that have been already studied.
\paragraph{ a) The surface case.}
For $C^2$-maps, the non-hyperbolic transitive sets which have a dominated splitting
contain either a non-hyperbolic periodic point
or a curve supporting the dynamics of an irrational rotation.
\begin{theo}[\cite{PS1}]\label{t.KS1}
Let $f$ be a $C^2$ diffeomorphism of surface and $K$ be a compact invariant set
having a dominated splitting $T_KM=E\oplus F$, $\dim(F)=1$ whose periodic orbits are all hyperbolic. Then, one of the following cases occur.
\begin{itemize}
\item[--] $K$ contains a sink or a compact invariant one-dimensional submanifold
tangent to $F$.
\item[--] $F$ is uniformly contracted by $f^{-1}$.
\end{itemize}
\end{theo}
One deduces the following generic result.
\begin{corollary}\label{c.PS1}
Let $f$ be a $C^1$-generic diffeomorphism and $K$ be a
partially hyperbolic set endowed with a dominated splitting $T_KM=E^s\oplus E^c_1\oplus E^c_2\oplus E^u$, with $\dim(E^c_1)=\dim(E^c_2)=1$.
If $K$ is contained in a locally
invariant surface tangent to $E^c_1\oplus E^c_2$
and does not contain a periodic orbit of stable dimension
$\dim(E^s)$ or $\dim(E^s)+2$, then $K$ is hyperbolic.
\end{corollary}
\noindent
Note that a periodic orbit of stable dimension $\dim(E^s)$ or $\dim(E^s)+2$
is a source or a sink in the surface.
If $K$ is transitive and non trivial, it does not contain such a periodic orbit.
\begin{proof}
By proposition~\ref{p.whitney} and theorem~\ref{t.whitney},
the property for a partially hyperbolic set to be contained in a locally invariant
surface tangent to $E^c_1\oplus E^c_2$ is robust.
It is thus enough to consider open sets $\cU\subset \operatorname{Diff}^1(M)$,
$U\subset M$ and a (non necessarily invariant) compact set $\Lambda\subset U$
such that for each $f\in \cU$ any invariant compact set $K$ contained in $U$ has a dominated splitting $T_KM=E^s\oplus E^c_1\oplus E^c_2\oplus E^u$ and is contained in a locally invariant surface tangent to $E^c_1\oplus E^c_2$: we have to obtain the conclusion of the theorem for an open and dense subset of diffeomorphisms in $\cU$ and invariant compact sets contained in $\Lambda$.
A standard Baire argument then concludes that the theorem holds for $C^1$ generic diffeomorphisms.
Let us fix a diffeomorphism $f_0\in \cU$ and consider the maximal invariant set $K_0$ in a small closed
neighborhood of $\Lambda$.
By assumption it is contained in a locally invariant surface $\Sigma_0$
tangent to $E^c_1\oplus E^c_2$. One can conjugate $f_0$ by a diffeomorphism which sends $\Sigma_0$
on a smooth surface $\Sigma$ and approximate the obtained diffeomorphism $f_1$ by a smooth diffeomorphism.
By this new diffeomorphism, the smooth surface $\Sigma$ is mapped on a smooth surface $f_1(\Sigma)$
which is $C^1$-close to $\Sigma$. As a consequence, there exists a smooth diffeomorphism $f_2$
that is $C^1$-close to $f_1$ which preserves $\Sigma$.
One deduces that the maximal invariant set $K_2$ for $f_2$ in a small neighborhood of $\Lambda$ is contained in
$\Sigma$.
One can perturb the restriction of $f_2$ to a neighborhood of $K_2$
in $\Sigma$ and obtain a smooth Kupka-Smale diffeomorphism
without any invariant one-dimensional submanifold supporting the dynamics of an irrational rotation.
This perturbation can be extended to a smooth diffeomorphism of $M$: indeed the compactly supported diffeomorphism close to the identity
in $\Sigma$ are isotopic to the identity and can be extended in a trivializing neighborhood of $\Sigma$
as a compactly supported diffeomorphism close to the identity.
At this point we have built a smooth diffeomorphism $f_3$ that is $C^1$-close to $f$
and an invariant smooth surface $\Sigma$ which contains the maximal invariant set $K_3$ of $f_3$
in a small neighborhood of $\Lambda$. Moreover all the periodic orbits in $K_3$ are hyperbolic
and the dynamics inside any invariant one-dimensional submanifold of $K_3$ is Morse-Smale.
Theorem~\ref{t.KS1} then shows that any orbit in $K_3$ accumulates on a hyperbolic set.
Now, for any diffeomorphism $C^1$-close to $f_3$, the dynamics contained in a small neighborhood of
$\Lambda$ is hyperbolic: it contains a hyperbolic set $L$ of stable dimension $\dim(E^s)+1$,
a finite collection of hyperbolic periodic orbits $O_1,\dots,O_s$ of stable dimension $\dim(E^s)$ or $\dim(E^s)+2$ and
any other orbit accumulates in the future and in the past on $L\cup O_1\cup\dots O_s$.
\end{proof}
\paragraph{ b) The one-codimensional case.}
This has been considered for homoclinic classes.
\begin{theo}[\cite{PS4}]\label{codimension-one}
Let $f$ be a $C^2$ diffeomorphism and $H(p)$ be a homoclinic class
endowed with a partially hyperbolic splitting $E^s\oplus E^c$ with $\dim(E^c)=1$
whose periodic orbits are hyperbolic. Then $H(p)$ is hyperbolic.
\end{theo}
As before, this gives the following generic result
(which is a particular case of theorem~\ref{t.extremal}).
\begin{corollary}\label{c.codim-one}
For any $C^1$-generic diffeomorphism,
any homoclinic class $H(p)$ that is
\begin{itemize}
\item[--] endowed with a partially hyperbolic splitting $E^s\oplus E^c\oplus E^u$, $\dim(E^c)=1$,
\item[--] contained in a locally invariant submanifold tangent to $E^s\oplus E^c$,
\end{itemize}
is hyperbolic.
\end{corollary}
\begin{proof}
Consider a $C^1$-generic diffeomorphism $f$
and a homoclinic class $H(p)$ as stated in the corollary
and $\Sigma$ the locally invariant submanifold tangent to $E^s\oplus E^c$
containing $H(p)$.
By genericity, one can suppose that
the class $H(p)$ is a chain-recurrence class and that for any diffeomorphism
$g$ close to $f$, the class $H(p_g)$ is contained in a small neighborhood of $H(p)$.
Moreover, if for some arbitrarily close diffeomorphisms $g$ the chain-recurrence class
containing $p_g$ is hyperbolic, then the class $H(p)$ for $f$ is also hyperbolic.
Let us consider a $C^2$-diffeomorphism
$g$ arbitrarily close to $f$ in $\operatorname{Diff}^1(M)$ and whose periodic orbits
are hyperbolic. By proposition~\ref{p.whitney}, the chain recurrence class $\Lambda$ containing $p_g$ is still contained in a locally invariant submanifold $\Sigma_g$.
As in the proof of corollary~\ref{c.PS1}, one may have chosen $g$ so that
$\Sigma_g$ is a smooth submanifold.
Let us assume by contradiction that $\Lambda$ is not hyperbolic:
there exists an invariant compact set $K\subset \Lambda$
that is not hyperbolic and that is minimal for the inclusion.
Since $K$ coincides with the support of an ergodic measure
whose Lyapunov exponent along $E^c$ is non-positive,
the set $K$ is transitive.
The set $K$ cannot be a sink, nor contain
an invariant one-dimensional submanifold tangent to $ E^c$,
since by transitivity the set $K$ would be reduced to a sink or
a union of normally attracting curves in $\Sigma_g$, contradicting the fact that $\Lambda$ is chain-transitive and contains $p_g$.
One can thus apply~\cite[lemma 5.12]{PS4} and conclude that $K$ is contained in a homoclinic
class $H(q)$. Since $H(q)$ is contained in a small neighborhood of $H(p)$,
it is contained in $\Sigma_g$. By theorem~\ref{codimension-one} applied for $g$
inside $\Sigma_g$, one deduces that $H(q)$ is a hyperbolic set. This
contradicts the fact that $K$ is non hyperbolic.
As a consequence, the chain-recurrence class containing $p_g$ is hyperbolic,
hence coincides with $H(p)$.
This proves that the homoclinic class $H(p)$ is hyperbolic.
\smallskip
\end{proof}
\paragraph{ c) The $2$-codimensional case.} For homoclinic classes
with two-codimensional strong stable bundle, one can replace the uniformity of the center stable
bundle by the thin trapping property and the total disconnectedness along the center stable plaques.
This theorem is proved in section~\ref{s.2D-central2}.
\begin{theo}\label{t.2D-central}
Let $f_0$ be a diffeomorphism and $H(p_{f_0})$ be a chain-recurrence class which is
a chain-hyperbolic homoclinic class endowed with a dominated splitting $E^{cs}\oplus E^{cu}$ such that $E^{cu}$ is one-dimensional and $E^{cs},E^{cu}$ are thin trapped (for $f_0$ and $f_0^{-1}$ respectively).
Assume moreover that the intersection of $H(p_{f_0})$ with its center-stable plaques is totally disconnected.
Then, for any $C^2$ diffeomorphism $f$ that is close to $f_0$ in $\operatorname{Diff}^1(M)$
and for any $f-$invariant compact set $K$ contained in a small neighborhood of $H(p_{f_0})$ and whose
periodic orbit are hyperbolic, one of the following cases occurs.
\begin{itemize}
\item[--] $K$ contains a sink or a compact invariant one-dimensional submanifold
tangent to $E^{cu}$.
\item[--] $E^{cu}$ is uniformly contracted by $f^{-1}$.
\end{itemize}
\end{theo}
\medskip
We can now prove that for $C^1$-generic diffeomorphisms far from
homoclinic bifurcations, the extremal sub-bundles of the homoclinic classes
are non-degenerated.
\begin{proof}[Proof of theorem~\ref{t.extremal}]
As before, one can assume that, for $g$ close to $f$,
the class $H(p_g)$ is contained in a small neighborhood of $H(p)$.
Moreover, if for some arbitrarily close diffeomorphisms $g$ the chain-recurrence class
containing $p_g$ is hyperbolic, then the class $H(p)$ for $f$ is hyperbolic.
The following several cases have to be considered.
\smallskip
Note first that when the bundle $E^{c}_1$ or $E^c_2$ is degenerated,
corollary~\ref{c.codim-one} implies that $H(p)$ is a hyperbolic set.
\smallskip
When the strong stable leaves intersect the class in at most one point,
theorem~\ref{t.whitney} implies that the class is contained in a
locally invariant submanifold tangent to $E^c_1\oplus E^{c}_2$.
By corollary~\ref{c.PS1} the class is then hyperbolic.
\smallskip
When the intersection of the class with the center stable plaques
is totally disconnected, one can apply theorem~\ref{t.2D-central}.
For any $C^2$ diffeomorphisms $g$ $C^1$-close to $f$ in $\operatorname{Diff}^1(M)$
with hyperbolic periodic orbits, the chain-recurrence class containing
$p_g$ is hyperbolic. As a consequence $H(p)$ is hyperbolic.
\smallskip
It remains the case that both bundles $E^c_1,E^{c}_2$ are one-dimensional,
$E^c_1$ is not uniformly contracted,
the class contains two different point in a same strong stable leaf
and the intersection of the class with the center stable plaques is not
totally disconnected.
One can then apply theorem~\ref{t.tot-discontinuity} when the dynamics is restricted to a
locally invariant submanifold tangent to $E^s\oplus E^c_1\oplus E^{c}_2$
and one deduces that the class has a generalized strong homoclinic intersection.
By lemma~\ref{l.weak} and remark~\ref{r.weak} the class contains hyperbolic periodic orbits homoclinically related to $p$
and whose Lyapunov exponent along $E^c_1$ is arbitrarily close to zero.
One concludes applying the propositions~\ref{p.generalized-strong-connection}
and~\ref{p.strong-connection} and creating a heterodimensional cycle
associated to a periodic orbit homoclinically related to $p$.
\end{proof}
\subsection{Finiteness of quasi-attractors}\label{ss.finiteness}
We now consider the \emph{quasi-attractors}
and prove one part of the main theorem.
\begin{prop}\label{p.finiteness}
For any $C^1$-generic diffeomorphism that is far from homoclinic tangencies and heterodimensional cycles, the union of all the quasi-attractors is closed.
\end{prop}
\begin{proof}
Consider a sequence of quasi-attractors $(A_n)$ which converges towards a (chain-transitive) set $L$.
By theorem~\ref{t.homoclinic}, they are homoclinic classes $A_n=H(p_n)$
and one can assume that all the periodic orbits $p_n$ have the same dimension.
\medskip
\noindent{\it Claim 1.
$L$ is a contained in a homoclinic class $H(p)$.}
\begin{proof}
If $L$ is contained in an aperiodic
class, by theorem~\ref{t.aperiodic}
it has splitting $T_LM=E^s\oplus E^c\oplus E^u$ with $\dim(E^c)=1$.
So, this is the same for the classes $A_n$.
Since the classes $A_n$ are quasi-attractor,
they are saturated by strong unstable leaves,
and therefore the same holds for $L$.
This contradicts corollary~\ref{c.aperiodic}.
\end{proof}
\medskip
By theorem~\ref{t.homoclinic}, the class
$H(p)$ has a dominated splitting $E^s\oplus E^c_1\oplus E^c_2\oplus E^u$
where $E^c_1$ and $E^c_2$ have dimension $0$ or $1$.
We assume by contradiction that $H(p)$ is not a quasi-attractor.
\medskip
\noindent{\it Claim 2.
The stable dimension of the periodic points $p_n$ is strictly
larger than the stable dimension of $p$.}
\begin{proof}
Let us consider some plaque families $\cW^{cs},\cW^{cu}$
over the maximal invariant set in a neighborhood of $L$
and tangent to $E^s\oplus E^c_1$ and $E^{c}_2\oplus E^u$ respectively,
as in the definition of chain-hyperbolic class.
Let us assume by contradiction that the stable dimension of $p_n$ is smaller or equal to
the stable dimension of $p$.
We claim that for any periodic point $q_n$ homoclinically related to the orbit of $p_n$,
one has $\cW^{cu}_{q_n}\subset W^u(q_n)$.
Indeed if it is not the case, using that $\cW^u$ is trapped by $f^{-1}$,
there would exists a periodic point $q'_n\in \cW^{cu}_{q_n}$,
in the closure of $W^u(q_n)$ and whose stable dimension
is $\dim(\cW^{cu}_{q_n})-1$. Since $H(p_n)$ is a quasi-attractor it contains $q'_n$
and by lemma~\ref{l.prelim}, there exist $C^1$-perturbations of $f$ which exhibit a
heterodimensional cycle. This is a contradiction.
In particular, one has $\cW^{cu}_{q_n}\subset H(p_n)$ and, passing to the limit,
the set $L$ contains all the plaques $\cW^{cu}_x$, $x\in L$.
Let us consider any periodic point $q$ homoclinically related to the orbit of $p$
and close to $L$ such that $\cW^{cs}_q\subset W^s(q)$.
This exists by lemma~\ref{l.contper}.
The plaques $\cW^{cs}_q$ and $\cW^{cu}_x$ for some $x\in L$ intersect transversally,
hence the forward iterates of $\cW^{cu}_x$ accumulate $W^u_q$.
One thus deduces that $H(p)$ contains $W^u(q)$.
By lemma~\ref{l.prelim} $H(p)$ is thus a quasi-attractor, contradicting our assumption.
\end{proof}
\medskip
We are thus reduced to consider the case that on the union of the $A_n$ and $H(p)$
there exists a dominated splitting $TM=E^{cs}\oplus E^c\oplus E^u$
such that $E^c$ is one-dimensional, $E^{cs}\oplus E^c$ is thin trapped by $f$ over each quasi-attractor
$A_n$ and $E^{c}\oplus E^{cu}$ is thin trapped by $f^{-1}$ over $H(p)$.
We also fix a point $z\in L$ and a small neighborhood $U$ of $z$.
\medskip
\noindent{\it Claim 3.
In each set $A_n$ there exists a periodic orbit $O_n$
which avoids $U$.}
\begin{proof}
By theorem~\ref{t.extremal}, the bundle $E^u$ is non-degenerated
and the set $A_n$ is saturated by strong unstable leaves.
By a standard argument (see for instance~\cite[lemma 5.2]{mane-contribution-stabilite}),
each class $A_n$
contains an invariant compact set $K_n$ which avoids $U$.
Then one can reduce $K_n$ and assume that it is minimal.
Let us consider two plaque families $\tilde \cW^{cs}, \tilde \cW^u$
tangent to $E^{cs}\oplus E^c$ and $E^u$
with arbitrarily small diameter, above the maximal invariant set
in a small neighborhood of $A_n$
and whose restriction to $A_n$ satisfy the definition of chain-hyperbolic classes.
By the closing lemma, there exists a periodic orbit $\tilde O_n$
arbitrarily close to $K_n$ in the Hausdorff topology.
By lemma~\ref{l.contper}, there exists a point $q$ homoclinically related to the orbit of $p$
such that $\tilde \cW^{u}_q$ is contained in $W^u(q)$ and
intersects $\tilde \cW^{cs}_y$ for some $y\in \tilde O_n$ at a point $\zeta$.
One deduces that $\zeta$ converges toward a periodic orbit $O_n$ contained in the plaques of the family $\tilde \cW^{cs}$ above $\tilde O_n$.
Since $A_n$ is a quasi-attractor, it contains $\zeta$ and $O_n$.
By construction $O_n$ is included in an arbitrarily small neighborhood of $K_n$,
as required.
\end{proof}
\medskip
\noindent{\it Claim 4.
There exists $N\geq 1$ such that
$f^N(W^u_{loc}(p))$ intersects transversally $W^s_{loc}(O_n)$
for each $n$ large.
Moreover, this property is stable under $C^1$-perturbations
with supports avoiding a neighborhood of $O_n$.}
\begin{proof}
Since the stable space of $O_n$ is $E^{cs}\oplus E^c$
and since $E^c$ is non-degenerate,
all the exponents of $O_n$ along $E^{cs}$ are bounded away from zero.
By lemma~\ref{l.largestable} and remark~\ref{r.large}, the orbit
$O_n$ contains a point $q_n$ such that
$\cW^{cs}_{q_n}\subset W^s_{loc}(q_n)$.
For $N$ large, $f^N(W^u_{loc}(p))$
is close to any point of $L$.
For $n$ large, $O_n$ is contained in a small neighborhood of $L$.
One thus deduces that $f^N(W^u_{loc}(p))$ and $W^s_{loc}(q_n)$ intersect transversally
and this property is robust under perturbations
with supports avoiding a neighborhood of $O_n$.
\end{proof}
\medskip
\noindent{\it Conclusion.}
Since $W^u(O_n)$ is dense in $A_n$,
and $A_n$ converges towards $L$, the unstable manifold of $O_n$
has a point close to $z$ for $n$ large.
Since $z$ is in $H(p)$, the stable manifold of $p$ has a point close to $z$. Observing that the orbits of $O_n$ are far from the neighborhood $U$ of $z$,
there exists small perturbations given by the connecting lemma
in a small neighborhood of a finite number of iterates of $z$, such that
$W^s(p)$ and $W^u(O_n)$ intersect. The orbit of $O_n$ has been preserved
and the intersection $W^s(O_n)\cap W^u(p)$ is still non empty.
This gives a heterodimensional cycle and therefore a contradiction.
As a consequence $H(p)$ is a quasi-attractor.
\end{proof}
\begin{remark}
In the case the quasi-attractors $A_n$ are non-degenerated, $E^u$ coincides with the
unstable dimension of the periodic points in the sets $A_n$; hence we already know that
$E^u$ is non-degenerated. Theorem~\ref{t.extremal} is thus needed only to guarantee that
the sinks of $f$ accumulate on quasi-attractors.
\end{remark}
\subsection{Hyperbolicity of quasi-attractors: proof of the main theorem}
\label{ss.quasi-attractor}
It remains now to prove that
for any $C^1$-generic diffeomorphism that is far from homoclinic tangencies and heterodimensional cycles,
the quasi-attractors are hyperbolic.
The proof is independent from proposition~\ref{p.finiteness}.
Let us consider a quasi-attractor and let us assume by contradiction that it is not hyperbolic.
From sections~\ref{ss.homoclinic} and~\ref{ss.aperiodic}, the quasi-attractor is a homoclinic class $H(p)$
with a splitting $E^s\oplus E^c\oplus E^u$ where $E^c$ is one-dimensional, $E^s\oplus E^c$
is thin-trapped and it contains arbitrarily weak periodic orbits homoclinically related to $p$.
From theorem~\ref{t.extremal} and corollary~\ref{c.whitney}, for any diffeomorphism $g$ that is $C^1$-close to $f$
the homoclinic class $H(p_g)$ associated to the continuation of $p$ for $g$
contains two different points $x,y$ such that $W^{ss}(x)=W^{ss}(y)$.
The end of the proof is based on the next theorem. The first case of the dichotomy
is not satisfied in our setting
and in the second case, one can create a heterodimensional cycle in $H(p)$, by proposition~\ref{p.strong-connection}.
This contradicts our assumptions on the diffeomorphism $f$ and
concludes the proof of the main theorem.
\begin{teo}\label{t.position}
Let $H(p)$ be a homoclinic class of a diffeomorphism $f$ which is a quasi-attractor
endowed with a partially hyperbolic structure $E^s\oplus E^c\oplus E^u$
such that $\dim(E^c)=1$ and $E^{cs}=E^s\oplus E^c$ is thin trapped.
Assume also that all the periodic orbits in $H(p)$ are hyperbolic.
Then, there exists $\alpha\geq 0$ (which is positive if $f$ is $C^r$ for some $r>0$)
and $C^{1+\alpha}-$small perturbations $g$ of $f$ such that the homoclinic class associated to the continuation
$p_g$ of $p$ satisfies one of the following cases.
\begin{itemize}
\item[--] Either one has $W^{ss}(x)\neq W^{ss}(y)$ for any $x\neq y$ in $H(p_g)$ and therefore
the class $H(p_g)$ is contained in a $C^1$-submanifold $N\subset M$ tangent to $E^c\oplus E^u$
which is locally invariant.
\item[--] Or one has $W^{ss}(x)= W^{ss}(y)$ for some hyperbolic periodic point $x$ homoclinically
related to the orbit of $p_g$ and some $y\neq x$ in $H(p_g)\cap W^u(x)$ and therefore
the class $H(p_g)$ has a strong homoclinic intersection.
\end{itemize}
\end{teo}
\begin{obs}\label{r.position} We want to emphasize some features of theorem~\ref{t.position}.
\begin{enumerate}
\item The result does not require any generic assumption.
\item It holds in the $C^{1+\alpha}-$category for $\alpha>0$ small.
\item The theorem can also be applied to the context of hyperbolic attractors
whose stable bundle has a dominated splitting $E^s=E^{ss}\oplus E^c$ such that $\dim(E^c)=1$.
This can have important consequences in terms of the Hausdorff dimension of the attractor: if the the attractor is contained in a submanifold, the Hausdorff dimension is smaller than $1+u$ (where $u=dim(E^u)$); if there is a strong connection, the dimension could jump close to $1+u+s$ (where $s=dim(E^{ss})$)
(see~\cite{BDV-hausdorff}).
Note that the proof in the hyperbolic case is simpler
since we can use the hyperbolic continuation of any point in the attractor.
\item\label{i.position} In the case the bundle $E^c$ is not uniformly contracted, one can assume that
the periodic point $x$ has an arbitrarily small Lyapunov exponent.
Indeed by lemma~\ref{l.weak} and remark~\ref{r.weak}, for any $\varepsilon>0$, there exists a
periodic point $q$ homoclinically related to the orbit of $p$ and whose central Lyapunov exponent is contained
in $(-\varepsilon,0)$. Let us consider a perturbation $g$ having a periodic point $x$ homoclinically related to $p_g$ and exhibiting a strong homoclinic intersection. By another $C^r$-small perturbation
(see lemma~\ref{l.change-connection}), one can obtain a periodic point $x'$ homoclinically related to the orbit of $p$, with a strong connection and a central Lyapunov exponent close to the exponent of $q$.
\end{enumerate}
\end{obs}
The proof of theorem~\ref{t.position} is based on the following proposition
whose proof is postponed to section~\ref{s.boundary}
and uses the notion of \emph{stable boundary point} (see section~\ref{s.weak-hyperbolicity})
and of \emph{continuation of a homoclinic class} (see section~\ref{s.continuation}).
As before $W^{ss}_{loc}(x)$ and $W^u_{loc}(x)$ denote local stable and unstable
manifolds tangent to $E^s_x$ and $E^u_x$ respectively for the points $x\in H(p_g)$.
Note that this result holds in any $C^r$-topology, $r\geq 1$.
\begin{prop}\label{p.position}
Given a $C^r$ diffeomorphisms under the assumptions of theorem~\ref{t.position}, for any $\alpha\in[0,r-1]$
one of the following cases occurs.
\begin{enumerate}
\item There exists $g$, $C^{1+\alpha}$-close to $f$, such that
for any $x\neq y$ in $H(p_g)$, one has $W^{ss}(x)\neq W^{ss}(y)$.
\item There exists $g$, $C^{1+\alpha}$-close to $f$, such that $H(p_g)$
exhibits a strong homoclinic intersection.
\item There exist a neighborhood $\cV\subset \operatorname{Diff}^{1+\alpha}(M)$ of $f$ and
some hyperbolic periodic points $q$ and $p^x_n, p^y_n$ for $n\in \NN$
such that:
\begin{itemize}
\item[--] the continuations $q_g,p^x_{n,g},p^y_{n,g}$ exist on $\cV$ and
are homoclinically related to the orbit of $p_g$;
\item[--] $(p^x_{n,g})$, $(p^y_{n,g})$ converge towards two distinct points $x_g,y_g$ in $H(p_g)\cap W^s_{loc}(q_g)$ for any $g\in \cV$;
\item[--] the map $g\mapsto x_g,y_g$ is continuous at $f$;
\item[--] $y_g\in W^{ss}_{loc}(x_g)$ for any $g\in \cV$.
\end{itemize}
\item There exist two hyperbolic periodic points $p_x,p_y$ homoclinically related to the orbit of $p$ and an open set $\cV\subset \operatorname{Diff}^{1+\alpha}(M)$
whose closure contains $f$, such that for any $g\in \cV$ the class $H(p_g)$ contains
two different points $x\in W^u(p_{x,g})$ and $y\in W^u(p_{y,g})$
satisfying $W^{ss}(x)=W^{ss}(y)$.
\end{enumerate}
\end{prop}
\medskip
One concludes the proof of theorem~\ref{t.position} by discussing the two last cases of the proposition~\ref{p.position}.
The two following theorems, proved in sections~\ref{proofjointint} and~\ref{p-nontransversal}
give a strong homoclinic intersection.
In the first case, the points $x,y$ belong to the stable manifold of a periodic point $q$.
\begin{theo}\label{t.stable}
For any diffeomorphism $f_0$ and any homoclinic class $H(p)$ which is a chain-recurrence class
endowed with a partially hyperbolic structure $E^s\oplus E^c\oplus E^u$, $\dim(E^c)=1$,
such that $E^s\oplus E^c$ is thin trapped, there exists $\alpha_0> 0$ and
a $C^1$-neighborhood $\cU$ of $f_0$ with the following property.
For any $\alpha\in [0,\alpha_0]$, any diffeomorphism $f\in \cU$
and any $C^{1+\alpha}$-neighborhood $\cV$ of $f$,
if there exist:
\begin{itemize}
\item[--] some hyperbolic periodic points $q_f$ and $p_{n,f}^x,p_{n,f}^y$ with $n\in \NN$ for $f$
whose hyperbolic continuations $q_g,p_{n,g}^x,p_{n,g}^y$
exist for $g\in\cV$ and are homoclinically related to the orbit of $p_g$,
\item[--] two maps $g\mapsto x_g,y_g$ defined on $\cV$, continuous at $f$,
such that for any $g\in \cV$ the points $x_g,y_g$ belong to $W^{s}(q_g)$,
are the limit of $(p_{n,g}^x)$ and $(p_{n,g}^y)$ respectively
and satisfy $y_g\in W^{ss}_{loc}(x_g)$,
\end{itemize}
then, there exist $C^{1+\alpha}$-small perturbations $g$ of $f$
such that the homoclinic class $H(p_g)$ exhibits a strong homoclinic intersection.
\end{theo}
In the second case, the points $x,y$ belong to the unstable manifold of periodic points
$p_x,p_y$.
\begin{theo}\label{t.unstable}
For any diffeomorphism $f_0$, for any homoclinic class $H(p)$ which is a chain-recurrence class
endowed with a partially hyperbolic structure $E^s\oplus E^c\oplus E^u$, $\dim(E^c)=1$,
such that $E^s\oplus E^c$ is thin trapped
there exists $\alpha_0> 0$ and for any hyperbolic periodic points $p_x$, $p_y$
homoclinically related to the orbit of $p$,
there exists a $C^1$-neighborhood $\cU$ of $f$ with the following property.
Given any $\alpha\in [0,\alpha_0]$ and any $C^{1+\alpha}$-diffeomorphism $f\in \cU$, if
there exist two different points $x\in W^u(p_{x,f})$ and $y\in W^u(p_{y,f})$ in $H(p_{f})$
satisfying $W^{ss}(x)=W^{ss}(y)$,
then, there exist $C^{1+\alpha}$-small perturbations $g$ of $f$
such that the homoclinic class $H(p_g)$ exhibits a strong homoclinic intersection.
\end{theo}
\medskip
Some weaker results similar to theorems \ref{t.unstable} and \ref{t.stable} were obtained in \cite{Pu2} for attracting homoclinic classes in dimension three and assuming strong dissipative properties.
\subsection{Other consequence on quasi-attractor. Main theorem revisited} \label{ss.consequences}
As it was mentioned in the introduction, for $C^1$-generic diffeomorphisms
one obtains a stronger version of theorem~\ref{t.position}.
We point out that what follows in this section is not used in the proof of our main theorem.
\begin{theorem}\label{t.consequences}
Let $f$ be a diffeomorphism in a dense G$_\delta$ subset of $\operatorname{Diff}^1(M)$
and let $\Lambda$ be a quasi-attractor endowed with a partially hyperbolic splitting
$T_\Lambda M=E^s\oplus E^c\oplus E^u$ with $\dim(E^c)=1$.
If $E^c$ is not uniformly contracted and not uniformly expanded, then $\Lambda$ is a homoclinic class which contains hyperbolic periodic points of both stable dimensions $\dim(E^s)$ and $\dim(E^s)+1$.
\end{theorem}
The proof uses the following result from \cite{BDKS}.
\begin{theorem}[\cite{BDKS}]\label{t.BDK}
Let $f$ be a diffeomorphism that exhibits a heterodimensional cycle
between two hyperbolic periodic points $p,q$
whose stable dimensions differ by $1$.
Then, there exist a $C^1$-perturbation $g$ of $f$ and two transitive hyperbolic sets $K,L$ -
the first contains the hyperbolic continuation $p_g$, the second has same stable dimension as $q$ -
that form a robust cycle: for any diffeomorphism $h$ that is $C^1$-close to $f$,
there exists heteroclinic orbits that join the continuations $K_g$ to $L_g$
and $L_g$ to $K_g$.
\end{theorem}
A consequence of this result is that for any $C^1$-generic diffeomorphism
and any hyperbolic point $p$ of stable dimension $i\geq 2$,
if there exists some small perturbations $g$ of $f$ which exhibits a heterodimensional cycle between a periodic point homoclinically related to $p_g$
and a periodic orbit of stable dimension $i-1$, then the homoclinic class $H(p)$ for $f$
contains periodic points of indices $i-1$.
\begin{proof}[Proof of theorem~\ref{t.consequences}]
The existence of the dominated splitting implies that there is no diffeomorphism $C^1$-close to $f$
which exhibits a homoclinic tangency in a small neighborhood of $\Lambda$.
\medskip
\noindent
\emph{Step 1.} We first prove that $\Lambda$ is a homoclinic class $H(p)$ which contains periodic orbits whose central exponents are arbitrarily close to $0$.
This uses the following.
\smallskip
\noindent
{\it Claim. If $\Lambda$ contains an invariant compact set $K$
such that any invariant measure supported on $K$ has a Lyapunov exponent along $E^c$ equal to $0$,
then $\Lambda$ contains periodic orbits whose central Lyapunov exponent is arbitrarily close to $0$.}
\begin{proof}
The proof is similar to the proof of theorem 9.25 in \cite{C3} and uses proposition 9.23 also in \cite{C3}. See also~\cite{Y}.
\end{proof}
\medskip
Since $E^c\oplus E^u$ is not uniformly expanded, the trichotomy given by~\cite[theorem 1]{C2}
and the previous claim imply that
the class $\Lambda$ contains periodic orbits whose central exponent is negative or arbitrarily close to $0$.
Similarly, since $E^s\oplus E^c$ is not uniformly contracted,
the class $\Lambda$ contains periodic orbits whose central exponent is positive or arbitrarily close to $0$.
In any case $\Lambda$ is a homoclinic class $H(p)$
which contains for any $\delta>0$
some periodic orbits $O^-_{\delta},O^+_{\delta}$ whose central exponent
is respectively smaller than $\delta$ and larger than $-\delta$. From the results in
\cite{ABCDW} follows that that $H(p)$ contains periodic orbits
whose central exponents are arbitrarily close to $0$.
\medskip
\noindent
\emph{Step 2.} We then show that one can find a diffeomorphism
$C^1$-close to $f$ and a periodic orbit homoclinically related to $p_g$
which exhibits a heterodimensional cycle.
Using the center models introduced in~\cite{C1}, the dynamics along the central bundle $E^c$ can be classified into {\em chain-recurrent, chain-neutral, chain-hyperblic and chain-parabolic} (see~\cite[section 2.2]{C2} for details).
Since $H(p)$ contains hyperbolic periodic orbits, some types can not occur
(the neutral and the parabolic ones).
Note that since $H(p)$ contains periodic orbits whose central exponent is close to $0$ and since $f$ is $C^1$-generic,
the class $H(p)$ is the limit of periodic orbits of both indices $\dim(E^s)$ and $\dim(E^s)+1$
for the Hausdorff topology.
When the central dynamics has the chain-recurrent type, \cite[proposition 4.1]{C2},
this implies that these periodic orbits are contained in $H(p)$, hence both indices appear in the class.
It reminds to consider a central dynamics which has the chain-hyperbolic type:
equivalently two cases are possible: either $E^s\oplus E^c$ is thin trapped by $f$ or $E^c\oplus E^u$
is thin-trapped by $f^{-1}$. In any case it follows that there exists a diffeomorphism $g$ that is $C^1$-close to $f$
and a periodic point homoclinically related to the continuation $p_g$ which exhibits a heterodimensional cycle:
in the first case, this is a direct consequence of theorem~\ref{t.position}, corollary~\ref{c.whitney},
theorem~\ref{t.extremal} and proposition~\ref{p.strong-connection};
in the second case, one argues as on the proof of corollary~\ref{c.homoclinic}.
\medskip
\noindent
\emph{Step 3.} We then concludes with theorem~\ref{t.BDK} that the class
$H(p)$ contains hyperbolic periodic points of different stable dimension.
\end{proof}
\bigskip
Theorem~\ref{t.extremal} can be combined with theorem~\ref{t.BDK} to get the following improvement.
\begin{theorem-extremal}
Let $f$ be a diffeomorphism in a dense G$_\delta$ subset of $\operatorname{Diff}^1(M)$
and let $H(p)$ be a homoclinic class endowed with a partially hyperbolic splitting
$T_{H(p)}M=E^s\oplus E^c_1\oplus E^c_2\oplus E^u$, with $\dim(E^c_1)\leq 1$ and $\dim(E^c_2)\leq 1$.
Assume moreover that the bundles $E^s\oplus E^c_1$ and $E^c_2\oplus E^u$
are thin trapped by $f$ and $f^{-1}$ respectively and that the class is contained in a locally invariant
submanifold tangent to $E^s\oplus E^c_1\oplus E^c_2$.
Then $H(p)$ is a hyperbolic set.
\end{theorem-extremal}
\begin{proof} Arguing by contradiction, from theorem 6 it would be possible to create a heterodimensional cycle involving points of different indexes and from
theorem \ref{t.BDK} it is get a robust heterodimensional cycle, then for generic diffeomorphisms the center dynamics it is not trapped neither for $f$ nor for $f^{-1};$ a contradiction.
\end{proof}
\section{Uniform hyperbolicity of the extremal bundles: proof of theorem~\ref{t.2D-central}}\label{s.2D-central2}
In this section we end the proof of theorem~\ref{t.2D-central}.
We consider:
\begin{enumerate}
\item\label{a1} a diffeomorphism $f_0$ and a chain-hyperbolic homoclinic class $H(p_{f_0})$ which is a
chain-recurrence class endowed with a dominated splitting
$E^{cs}\oplus E^{cu}$ such that:
\begin{itemize}
\item[1a.] $E^{cu}$ is one-dimensional and $E^{cs},E^{cu}$ are thin trapped by $f$ and $f^{-1}$ respectively.
\item[1b.] The intersection of $H(p)$ with the center-stable plaques is totally disconnected.
\end{itemize}
\item\label{a2} a $C^2$-diffeomorphism $f$ that is $C^1$-close to $f_0$,
\item\label{a3} a chain-recurrence class $K$ for $f$ contained in a small neighborhood of $H(p_{f_0})$ such that:
\begin{itemize}
\item[3a.] All the periodic points of $K$ are hyperbolic.
\item[3b.] $K$ does not contain a sink, nor a closed curve $\gamma$ tangent to $E^{cu}$,
invariant by some iterate $f^n$, $n\geq 1$,
such that $f^n_{|\gamma}$ is conjugated to an irrational rotation.
\end{itemize}
\item\label{a4} a transitive invariant compact set $\Lambda\subset K$ for $f$ such that
the bundle $E^{cu}$ is uniformly expanded on any proper invariant compact subset of $\Lambda$.
\end{enumerate}
We prove here the following proposition.
\begin{propo}\label{propoE^{cu}ishyp}
Let us consider some diffeomorphisms $f_0$, $f$, some chain-recurrence classes
$H(p_{f_0})$, $K$ and a subset $\Lambda\subset K$ satisfying the assumptions~\ref{a1})-\ref{a4}) above. Then the bundle $E^{cu}$ is uniformly expanded on any proper invariant compact subset of $\Lambda$.
\end{propo}
Let us explain how to conclude the proof of the theorem~\ref{t.2D-central}.
\paragraph{Proof of theorem~\ref{t.2D-central}}
Under the hypothesis of the theorem, the assumptions~\ref{a1}) and~\ref{a2}) above are clearly satisfied.
Note that since $K$ is contained in a small neighborhood of $H(p_{f_0})$, the same holds for any
chain recurrence class $K'$ which meets $K$.
If for any such chain-recurrence class $K'$, the bundle $E^{cu}$ is uniformly expanded,
the same holds for $K$, hence the conclusion of the theorem holds.
Note that if $K'$ contains a curve $\gamma$ tangent to $E^{cu}$ such that
$f^n$ preserves $\gamma$ and is conjugated to an irrational rotation for some $n\geq 1$,
then from the domination $E^{cs}$ is uniformly contracted on the union $X$ of the iterates of $\gamma$
and consequently $X$ is an attractor. Since $K'$ is chain-transitive,
$K'$ coincides with $X$ and is contained in $K$; this gives theorem~\ref{t.2D-central} in this case.
The same holds if $K'$ contains a sink.
We will now assume by contradiction that the conclusion of the theorem does not hold and
hence that $K'$ satisfies~\ref{a3}) and that the bundle $E^{cu}$ is not uniformly expanded by $f$ on $K'$.
One can then consider an invariant compact set $\Lambda\subset K'$ whose bundle $E^{cu}$ is not uniformly expanded and that is minimal for this property. Such a set exists by Zorn's lemma
since if $\{\Lambda_\alpha\}_{\alpha\in \mathcal{A}} \def\cG{\mathcal{G}} \def\cM{\mathcal{M}} \def\cS{\mathcal{S}}$ is a family of invariant compact sets totally ordered by the inclusion and if $E^{cu}$ is uniformly expanded on the intersection $\cap_{\alpha\in \mathcal{A}} \def\cG{\mathcal{G}} \def\cM{\mathcal{M}} \def\cS{\mathcal{S}}\Lambda_\alpha$, then the same holds on the $\Lambda_{\alpha}$ for $\alpha$ large enough.
By minimality, for any proper invariant compact set of $\Lambda$, the bundle $E^{cu}$ is uniformly expanded.
Since $E^{cu}$ is one-dimensional and not uniformly expanded on $\Lambda$, there exists
an invariant measure $\mu$ supported on $\Lambda$ and whose Lyapunov exponent along $E^{cu}$
is non-positive. One can assume that $\mu$ is ergodic and by minimality of $\Lambda$
its support coincides with $\Lambda$. This implies that $\Lambda$ is transitive and
satisfies~\ref{a4}).
By applying proposition~\ref{propoE^{cu}ishyp} to $f,\Lambda,K'$, the bundle $E^{cu}$
is uniformly expanded on $\Lambda$ which is a contradiction.
Consequently the conclusion of theorem~\ref{t.2D-central} holds.
\hskip 445pt \rule{2mm}{2mm}
\bigskip
In the following we are in the setting of proposition~\ref{propoE^{cu}ishyp}
and prove that $E^{cu}$ is uniformly expanded on $\Lambda$. The proof follows the strategy
of~\cite{PS1} (see also~\cite{PS3,PS4,Pu1} for more general contexts). The new difficulty is to work with a
non-uniformly contracted bundle $E^{cs}$ having dimension larger than $1$;
the summability arguments and the construction of Markovian rectangles become more delicate.
\paragraph{Strategy.}
Our goal is to find a non-empty open set $B$ of $\Lambda$
which satisfy:
\begin{itemize}
\item[(E)] \emph{For any $x\in B$ and $n\geq 1$ such that $f^{-n}(x)\in B$
we have $\|Df^{-n}_{|E^{cu}}(x)\|<\frac 1 2$.}
\end{itemize}
This concludes the proof of the proposition~\ref{propoE^{cu}ishyp}.
Indeed if one considers any point $x\in \Lambda$, then:
\begin{itemize}
\item[--] either its backward orbit intersects $B$ and property (E) applies,
\item[--] or the $\alpha$-limit set of $x$ is a proper invariant compact subset of $\Lambda$
whose bundle $E^{cu}$ is uniformly contracted by $f^{-1}$.
\end{itemize}
In both cases, the point $x$ has a backward iterate $f^{-n}(x)$ such that
$\|Df^{-n}_{|E^{cu}}(x)\|<1$.
By compactness one deduces that there is some $k\geq 1$ such that
for any $x\in \Lambda$ the derivative $\|Df^{-k}_{|E^{cu}}(x)\|$ is smaller than $1/2$,
concluding the proof that $E^{cu}$ is uniformly expanded on $\Lambda$.
\medskip
\subsection{Topological hyperbolicity on $\Lambda$}\label{ss.topo}
We begin with preliminary constructions and recall some results from~\cite{PS1}
which only use the one-codimensional domination $E^{cs}\oplus E^{cu}$
and the fact that $f$ is $C^2$.
We introduce (in this order) the following objects satisfying several properties stated
in this section:
\begin{itemize}
\item[--] some constants $\kappa,\lambda,\mu,\chi$ related to the domination,
\item[--] two transverse cone fields $\mathcal{C}} \def\cI{\mathcal{I}} \def\cO{\mathcal{O}} \def\cU{\mathcal{U}^{cs},\mathcal{C}} \def\cI{\mathcal{I}} \def\cO{\mathcal{O}} \def\cU{\mathcal{U}^{cu}$ on a neighborhood of $H(p_{f_0})$: they are thin neighborhoods
of the bundles $E^{cs}$ and $E^{cu}$ over $H(p_{f_0})$ and they are invariant by $f^{-1}_0$ and $f_0$ respectively.
\item[--] two continuous trapped $C^1$-plaques families
$(\cW^{cs}_f)$, $(\cW^{cu}_f)$ provided by the lemma~\ref{l.robustness},
defined for diffeomorphisms $f$
that are $C^1$-close to $f_0$ and tangent to
the bundles $(E^{cs}_f)$ and $(E^{cu}_f)$ over the maximal invariant sets
in a small neighborhood of $H(p_{f_0})$: the plaques are small and
tangent to $\mathcal{C}} \def\cI{\mathcal{I}} \def\cO{\mathcal{O}} \def\cU{\mathcal{U}^{cs},\mathcal{C}} \def\cI{\mathcal{I}} \def\cO{\mathcal{O}} \def\cU{\mathcal{U}^{cu}$.
\item[--] some constants $\varepsilon,\widetilde \varepsilon$
which control the geometry of the center-stable plaques under backward iterations,
their coherence and their intersections,
\item[--] some small neighbourhood $U$ of $H(p_{f_0})$:
for any diffeomorphism $f$ we then denotes $K_f$ the maximal invariant set of $f$
in $U$.
\item[--] a continuous family of trapped $C^1$-plaques $(\widehat \cW^{cs}_f)$
tangent to $E^{cs}$ over the maximal invariant set in a small neighborhood of $H(p_{f_0})$:
they have a small diameter so that $\widehat \cW^{cs}_x$ is contained in $U$
for each $x\in K$; moreover for each $x\in K$, the plaque
$\widehat \cW^{cs}_x$ is contained in $\cW^{cs}_x$.
This family is obtained by remark~\ref{r.nested}.
It will be used in order to define holes at section~\ref{ss.rectangle}.
\item[--] a scale $\rho$ smaller than the diameter of the plaques $\widehat \cW^{cs}$
and which control the size of Markovian rectangles,
\item[--] a $C^2$-diffeomorphism $f$, a chain-recurrence class $K$ and a chain-transitive set $\Lambda$
satisfying the conditions of the proposition~\ref{propoE^{cu}ishyp}:
the $C^1$-distance between $f$ and $f_0$
and the size of the neighborhood of $H(p_{f_0})$ containing $K$
are chosen small enough in order to satisfy further conditions
that will appear in section~\ref{ss.construction}.
\item[--] a scale $r>0$ which depends on the $C^2$-diffeomorphism $f$ and on the set $\Lambda$,
where the plaques $\cW^{cu}$ are nicely controled.
\end{itemize}
Now we list a series of properties that are used (and refered to) in the proof of proposition \ref{propoE^{cu}ishyp}.
\paragraph{a) Dominated splitting.}
We first state some consequences of the domination $E^{cs}\oplus E^{cu}$.
To simplify the presentation, one can change the Riemannian metric (see~\cite{gourmelon}) and
find $\kappa\in (0,1)$ such that for each point $x\in H(p_{f_0})$,
and each unitary vectors $u\in E^{cs}_x$ and $v\in E^{cu}_x$, one has
$\|Df_0.u\|\leq \kappa \|Df_0.v\|$.
One then chooses some $\lambda,\mu\in (0,1)$ such that
$\lambda\mu>\kappa$. This implies that:
\begin{itemize}
\item[] \emph{For any $x\in K_f$ one has
$$\|D{f}_{|E^{cs}}(x)\|\geq \lambda \;\; \Rightarrow \;\; \|D{f}_{|E^{cu}}(x)\|> \mu^{-1}.$$}
\end{itemize}
\medskip
Since $E^{cu}$ is not uniformly expanded on $\Lambda$, there exists $\zeta\in \Lambda$
such that $\|Df^n_{|E^{cu}}(\zeta)\|\leq 1$ for all $n\geq 1$.
Note that since $E^{cu}$ is uniformly expanded on any invariant compact subset, the forward orbit of $\zeta$ is dense in $\Lambda$.
With the domination $E^{cs}\oplus E^{cu}$ one deduces:
\begin{itemize}
\item[(i)] \emph{There exists a point $\zeta$ with dense forward orbit in $\Lambda$
such that for each $n\geq 1$ one has
$$\prod_{i=0}^{n-1}\|Df_{|E^{cs}}(f^i(\zeta))\|\leq \kappa^n.$$}
\end{itemize}
\medskip
We fix some small constant $\chi>0$ such that $(1+\chi)\kappa<\lambda$.
Choosing $\mathcal{C}} \def\cI{\mathcal{I}} \def\cO{\mathcal{O}} \def\cU{\mathcal{U}^{cs}$ thin enough one gets:
\begin{itemize}
\item[(ii)] \emph{For any points $x,y$ that are close and contained in a small neighborhood of $H(p_{f_0})$ and for any unitary vector $u\in \mathcal{C}} \def\cI{\mathcal{I}} \def\cO{\mathcal{O}} \def\cU{\mathcal{U}^{cs}_x$, one has
$$\|Df_x.u\|\leq (1+\chi)\; \sup\left\{\|Df_y.v\|,\; v\in \mathcal{C}} \def\cI{\mathcal{I}} \def\cO{\mathcal{O}} \def\cU{\mathcal{U}^{cs}_y, \|v\|=1\right\}.$$}
\end{itemize}
\paragraph{b) Center stable and unstable plaques.}
Assuming that the plaques are small and the cones thin, one deduces from
our choice of $\lambda,\mu$:
\begin{itemize}
\item[(iii)] \emph{If for some point $x\in K_f$ and any $n\geq 0$ one has
$$\prod_{i=0}^{n-1}\|Df_{|E^{cs}}(f^{i}(x)\|\leq \lambda^n,$$
then $\cW^{cs}_x$ is contained in the stable set of $x$, i.e.
the diameter of $f^n(\cW^{cs}_x)$ goes to $0$ as $n\to +\infty$.}
\item[(iv)] \emph{If for some point $x\in K_f$ and some $n\geq 0$ one has
$$\prod_{i=0}^{n-1}\|Df_{|E^{cs}}(f^{i}(x)\|\geq \lambda^n,$$
then the norm of the derivative of $f^{-n}$ along
the plaque $\cW^{cu}_{f^n(x)}$ is smaller than $\mu^n$.}
\end{itemize}
\medskip
The center-stable discs do not degenerate under
backward iterations: let us fix $\varepsilon>0$ small; then
there is $\widetilde \varepsilon>0$ small such that
choosing $f$ close to $f_0$ and $U$ small the following holds.
\begin{itemize}
\item[(v)]\emph{Consider any segment of orbit $(z,\dots,f^{n}(z))$ in $U$ and any disc $D$ tangent to $\mathcal{C}} \def\cI{\mathcal{I}} \def\cO{\mathcal{O}} \def\cU{\mathcal{U}^{cs}$,
containing a ball centered at $f^n(z)$ of radius $\widetilde \varepsilon$.
Then the preimage $f^{-n}(D)$ contains a ball $B$ centered at $z$ and of radius $\varepsilon$,
whose iterates $f^i(B)$, $i\in \{0,\dots,n\}$, have radius bounded by $\widetilde \varepsilon$.}
\end{itemize}
Indeed each point $f_0^{i}(z)$ is close to a point $x_i\in H(p_{f_0})$.
Each disc $D$ in the plaque $\cW^{cs}_{f_0^n(z)}$ at $f_0^n(z)$ can be viewed as the graph of a Lipschitz map above a domain $\Delta_n$
of $\cW^{cs}_{x_n}$. The invariance of the cones $\mathcal{C}} \def\cI{\mathcal{I}} \def\cO{\mathcal{O}} \def\cU{\mathcal{U}^{cs},\mathcal{C}} \def\cI{\mathcal{I}} \def\cO{\mathcal{O}} \def\cU{\mathcal{U}^{cu}$
and the fact that the bundle $E^{cs}$ is thin trapped shows that
$f_0^{-k}(D)$, for $k\in \{0,\dots,n\}$ contains the graph of a Lipschitz map
above a domain $\Delta_{n-k}$ of $\cW^{cs}_{x_{n-k}}$ whose radius is uniformly bounded
from below. The property extends to any diffeomorphism $f$ that is $C^1$-close.
\medskip
The coherence of the plaques (lemma~\ref{l.uniqueness-coherence}) gives:
\begin{itemize}
\item[(vi)] \emph{For any points $x,y\in K_f$ that are $\varepsilon$-close,
if $\cW^{cs}_x\cap \cW^{cs}_y\neq \emptyset$ then $f(\cW^{cs}_{y})\subset \cW^{cs}_{f(x)}$.
If $\cW^{cu}_x\cap \cW^{cu}_y\neq \emptyset$ then $f^{-1}(\cW^{cu}_{y})\subset \cW^{cu}_{f^{-1}(x)}$.}
\end{itemize}
\medskip
The holonomy along the center-stable plaques can be chosen to ``preserve the order":
\begin{itemize}
\item[(vii)] \emph{For any points $x,y\in K_f$ that are $\varepsilon$-close, the plaques $\cW^{cs}_{x}$ and
$\cW^{cu}_{y}$ intersect in a unique point.}
\item[(viii)] \emph{For any points $x^-,x^+,y,z\in K_f$ that are $\varepsilon$-close,
assume that $y$ belongs to a subinterval of $\cW^{cu}_{y}$ bounded by $x^-,x^+$
and denote $\tilde x^-,\tilde x^+,\tilde y$ the intersections of
the plaques $\cW^{cs}_{x^-}, \cW^{cs}_{x^+}, \cW^{cs}_{y}$ with $\cW^{cu}_z$.
Then $\tilde y$ belongs to the subinterval of $\cW^{cu}_z$ bounded by $\tilde x^-,\tilde x^+$.}
(This is a consequence of the coherence of the $\cW^{cs}$-plaques given by the property (vi).)
\end{itemize}
\medskip
\paragraph{c) Smoothness and stability of the center-unstable plaques.} We now use the following result which is based on a Denjoy argument.
(The proof in~\cite{PS1} is written for surface diffeomorphisms but as it is noticed in~\cite{PS3}
this does not make any difference.)
\begin{lemma}[\cite{PS1}, lemma 3.3.2, item1)]\label{l.smoothness}
Let $f$ be a $C^2$-diffeomorphism and $K$ be an invariant compact set endowed with a dominated
splitting $E^{cs}\oplus E^{cu}$ such that $E^{cu}$ is one-dimensional, $K$ does not contain sinks and all its periodic points hyperbolic. Then, there exists
a locally invariant plaque family $\gamma$ tangent to $E^{cu}$ such that
\begin{itemize}
\item[--] the maps $\gamma_x\colon E^{cu}_x\to M$, $x\in K$,
define a continuous family of $C^2$-embeddings;
\item[--] for any $r_0>0$, there exists $r_1>0$ such that for any $x\in K$ and $n\geq 0$
the image of the curve $\gamma_{x,r_1}:=\gamma_x(B(0,r_1))$ by $f^{-n}$
is contained in $\gamma_{f^{-n},r_0}$.
\end{itemize}
\end{lemma}
For the $C^2$-diffeomorphism $f$ and the chain-recurrence class $K$ one deduces that
the plaques $\cW^{cu}$ are $C^2$ in a neighborhood of the section $0\in E^{cu}$
which remains small by backward iterations. Indeed, the
coherence (lemma~\ref{l.uniqueness-coherence}), gives $r>0$ such that
$\cW^{cu}_x(B(0,r))$ is contained in $\gamma_{x}$ for any $x\in K$.
\paragraph{d) Topological expansion along the center-unstable plaques.}
The following result, whose proof is identical to the surface case~\cite{PS1}, asserts that the center-unstable curves
$\gamma$ in the center-unstable direction are unstable manifolds.
\begin{lemma}[\cite{PS1}, lemma 3.5.2]\label{l.top-expansion}
Under the setting of lemma~\ref{l.smoothness}, for any transitive invariant compact set
$\Lambda\subset K$ such that on any proper invariant compact sets the bundle $E^{cu}$ is uniformly expanded, there exists $r>0$ such that
\begin{itemize}
\item[] for any $x\in \Lambda$, the length of
$f^{-n}(\gamma_{x,r})$ decreases uniformly to $0$ as $n\to+\infty$.
\end{itemize}
\end{lemma}
\medskip
In the following we fix $r>0$ small and depending on $\Lambda$,
as given by the previous lemma, and we denote by
$W^{cu}_{loc}(x)$ the $C^2$-curve $\gamma_{x,r}$ for $x\in K$.
By lemma~\ref{l.smoothness}, the family of unstable curves $(W^{cu}_{loc}(x))_{x\in K}$
is continous for the $C^2$ topology.
For points $x\in \Lambda$ we sometimes write $W^{u}_{loc}(x)=W^{cu}_{loc}(x)$.
\subsection{Adapted rectangles}\label{ss.rectangle}
\paragraph{a) Rectangles.}
The set $B$ in condition (E) will be obtained from a geometry adapted to the splitting $E^{cs}\oplus E^{cu}$.
A rectangle\footnote{The name refers to the rectangles of Markov partitions.
For general hyperbolic sets $K$ the rectangles are subsets of $K$ but on surfaces one can also consider
geometrical Markov partitions~\cite[Appendix 2]{PT} whose rectangles are subsets of the surface diffeomorphic to $[0,1]^2$.
In higher dimensions, when the unstable bundle is one-dimensional,
one can build rectangles that are laminated by curves as in definition~\ref{d.rectangle}.}
of $\Lambda$ will be a union of local unstable leaves of points of $K$.
\begin{defi}\label{d.rectangle} A \emph{rectangle} $R$ is a union $\bigcup_{x\in X}\gamma_x$ with $X\subset K$
such that for each $x\in X$ the set $\gamma_x$ is an open interval of $W^{cu}_{loc}(x)$ bounded by two distinct points
$x^-_R,x^+_R$ in $K_f$ and such that the following properties hold:
\begin{enumerate}
\item\label{rectangle1} $R$ has diameter smaller than $\rho$,
\item $R\cap \Lambda$ is open in $\Lambda$,
\item\label{rectangle3} for any $x,y\in X$, the point $y^-_R$ belongs to $\cW^{cs}_{x^-_R}$
and the point $y^+_R$ belongs to $\cW^{cs}_{x^+_R}$.
\end{enumerate}
The sets $\{x^-_R,\; x\in X\}$ and $\{x^+_R,\; x\in X\}$ are called the \emph{boundaries} of $R$.
\medskip
By item~\ref{rectangle3}) and the property (vi),
any two curves $\gamma_x,\gamma_{x'}$ with $x,x'\in X$ are either disjoint or coincide.
For any $z\in X$ or $z\in R\cap\Lambda$, one can thus denote by
$W^{cu}_R(z)$ the curve $\gamma_x$ containing $z$.
\end{defi}
\medskip
\begin{defi}
A rectangle $S$ is a \emph{subrectangle} of $R=\bigcup_{x\in X}\gamma_x$ if it is a union
$\bigcup_{x\in X}\gamma'_x$ over the same set $X$ as $R$ and if one has $\gamma_x'\subset \gamma_x$ for each $x\in X$.
\end{defi}
\begin{remark}\label{r.uniqueness}
Note that if $S,T$ are two subrectangles of $R$ and if
$x^-_S=x^-_T$ for some $x\in X$, then it holds for all $x$.
Indeed for any $y\in X$, the point $y^-_{T}$ is the intersection of
$\cW^{cs}_{x^{-}_T}=\cW^{cs}_{x^{-}_S}$ with $W^{cu}_{loc}(y)$.
In particular if $W^{cu}_{S}(x)=W^{cu}_{T}(x)$ for some $x\in X$, then $S=T$.
\end{remark}
\paragraph{b) Adapted rectangles.}
We introduce for rectangles a kind of Markov property.
\begin{defi}\label{d.adapted} A rectangle $R$ is \emph{adapted} if for any $x,y\in X$ and $n\geq 0$,
\begin{enumerate}
\item[--] the curve $W^{cu}_R(y)$ is either disjoint from or contained in $f^n(W^{cu}_R(x))$,
\item[--] in the case $W^{cu}_R(y)\subset f^n(W^{cu}_R(x))$ there exists a subrectangle $S$ of $R$
such that for each $z\in X$ the image $f^n(W^{cu}_S(z))$ is an unstable curve of $R$
and such that $f^n(S)$ contains $W^{cu}_R(y)$.
\end{enumerate}
This subrectangle $S$ is called a \emph{return} and $n$ is called a \emph{return time} of $R$.
In the case $f^k(S)$ is disjoint from $R$ for any $0<k<n$, one says that $S$ and $n$
are a \emph{first return} and a \emph{first return time} of $R$.
\end{defi}
The next lemma shows that returns of adapted rectangles are adapted (take $S=R$).
\begin{lemma}\label{l.adapted}
Let $R$ be an adapted rectangle and $S$ be a subrectangle of $R$.
Let also $R'$ be a return of $R$ with return time $n$.
Then $S'=R'\cap f^{-n}(S)$ is a subrectangle of $R'$. If $S$ is adapted, $S'$ is adapted.
\end{lemma}
\begin{proof}
Note that $S'$ has diameter smaller than $\rho$ and $S'\cap \Lambda$ is open in $\Lambda$.
For $x'\in X$, we consider the point $x\in X$ such that $f^n(W^{cu}_{R'}(x'))=W^{cu}_{R}(x)$ and we define $\gamma_{x'}=f^{-n}(W^{cu}_{S}(x))$. By construction and since $R$ is adapted, $S'$ is the union
$\bigcup_{x'\in X}\gamma_{x'}$.
In order to prove that $S'$ is a rectangle it remains to check the item~\ref{rectangle3} of the definition.
For $x',y'\in X$ , we consider $x,y\in X$ such that $f^n(W^{cu}_{R'}(x'))=W^{cu}_{R}(x)$
and $f^n(W^{cu}_{R'}(y'))=W^{cu}_{R}(y)$.
We then denote $x_{S'}^-=f^{-n}(x_S^-)$ and $y_{S'}^-=f^{-n}(y_S^-)$.
We have to prove that $y^-_{S'}$ belongs to $\cW^{cs}_{x^-_{S'}}$.
Let $z$ by the intersection
between $\cW^{cs}_{x_{S'}^-}$ and $W^{cu}_{R'}(y_{S'}^-)$.
The image $f^{n}(z)$ is the intersection between $\cW^{cs}_{x_{S}^-}$ and $W^{cu}_{R}(y_{S}^-)$.
Since $S$ is a subrectangle of $R$, $f^{n}(z)$ and $y_{S}^-$ coincide, hence $z$ and $y_{S'}^-$ coincide, as required.
\medskip
We now assume that $S$ is adapted and prove that $S'$ is adapted too.
Let us suppose that $f^m(W^{cu}_{S'}(x'))$ intersects $W^{cu}_{S'}(y')$ for some $m\geq 0$.
Taking the image by $f^m$,
one deduces that $f^m(W^{cu}_{S}(x))$ intersects $W^{cu}_{S}(y)$.
Since $S$ is adapted, one has $W^{cu}_{S}(y)\subset f^m(W^{cu}_{S}(x))$.
This implies that $W^{cu}_{S'}(y')$ is contained in $f^m(W^{cu}_{S'}(x'))$,
proving the first item of definition~\ref{d.adapted}.
Since $R$ is adapted, there exists a subrectangle $R''$ of $R$
such that, for each $z'\in X$, the image
$f^m(W^{cu}_{R''}(z'))$ is an unstable curve of $R$ and such that
$f^m(W^{cu}_{R''}(x'))=W^{cu}_{R}(y')$. By the first part of the lemma, the intersection
$T'=R''\cap f^{-m}(S')$ is a subrectangle of $R''$.
Note that $W^{cu}_{T'}(x')$ is contained in $W^{cu}_{S'}(x')$. By property (viii)
this implies that for any $z\in X$ one has $W^{cu}_{T'}(x')\subset W^{cu}_{S'}(x')$
proving that $T'$ is a subrectangle of $S'$ such that
$W^{cu}_{T'}(x')$ is mapped on $W^{cu}_{S'}(y')$. Hence $S'$ is adapted.
\end{proof}
\bigskip
\paragraph{c) Holes.} In general, $\Lambda\cap R$ is smaller than $R$ and one can introduce the notion of hole.
\begin{defi}
A \emph{hole} of a rectangle $R$ is a subrectangle that is disjoint from
$\Lambda$ and that is maximal for the inclusion and these properties.
A hole has \emph{aperiodic boundary} if
its boundary $\bigcup_{x\in X}\{x^-_S,x^+_S\}$ is disjoint from its forward iterates.
\end{defi}
\begin{lemma}\label{l.hole}
\emph{1.} If $S$ is a hole of $R$ then
either for any unstable curve $W^{cu}_R(x)$ of $R$ one has $x^-_S=x^-_R$
or there exists a sequence $(x_n)$ in $R\cap \Lambda$ such that
$d(x_n,x_{n,S}^-)$ goes to zero as $n\to +\infty$.
\smallskip
\noindent
\emph{2.} Holes of adapted rectangles are adapted.
\smallskip
\noindent
\emph{3.} For any adapted rectangle $R$, any hole $S$ with aperiodic boundary and any $\tau\geq 1$, there exists $N\geq 1$
such that for any $x\in \Lambda\cap R$ and any $n\geq N$ satisfying
$f^{-n}(W^{cu}_S(x))\subset S$, the iterates $f^{-n-k}(W^{cu}_S(x))$
for $k\in\{1,\dots,\tau\}$ are disjoint from $S$.
\end{lemma}
\begin{proof}
Let $S$ be a hole of $R$ and $W^{cu}_R(x)$ be an unstable curve.
We suppose that $x^-_S\neq x^+_R$.
The points $y\in K_f\cap \overline{R}$ can be ordered by considering the projections
$\cW^{cs}_y\cap \overline{W^{cu}_R(x)}$ on $\overline{W^{cu}_R(x)}$ in such a way that $x^-_S<x^+_S$.
The union of the curves $\gamma'_y\subset W^{cu}_R(y)$ for $y\in X$,
bounded by $y^-_R$ and $y^+_S$, is a rectangle. Thus, since $S$ is a hole
and $x^-_R<x^-_S$, there exists points $y\in \Lambda\cap R$
such that $x^-_R< y \leq x^-_S$.
If there exists an increasing sequence $(x_n)\in \Lambda\cap R$
whose projections on $W^{cu}_R(x)$ converge towards $x^-_S$,
then the distance $d(x_n,x^-_{n,S})$ goes to zero and we are done.
So we assume by contradiction that this is not the case.
There exists a point $\bar x\in \Lambda\cap \overline R$
which is the limit of points $y\in \Lambda\cap R$ and such that
there is no point $y\in \Lambda\cap R$ satisfying $\bar x<y\leq x^-_S$.
Since $R$ has diameter smaller than $\rho$, which has been chosen smaller than
the size of the plaque $\widehat \cW^{cs}$,
the plaque $\widehat \cW^{cs}_{\bar x}$ intersects each curve
$W^{cu}_R(y)$ at a point $y^-_T$.
The union of the curves $\gamma'_y\subset W^{cu}_R(y)$ for $y\in X$,
bounded by $y^-_T$ and $y^+_R$ is a rectangle
whose intersection with $\Lambda$ is empty. This contradicts
the maximality of $S$. We have thus proved the first item of the lemma.
\medskip
Let us assume that $R$ is adapted and that $W^{cu}_S(y)$ intersects $f^n(W^{cu}_S(x))$
for some $n>0$ and some $x,y\in X$.
We have to show that $f^n(x^-_S)$ and $f^n(x^+_S)$ do not belong to the open curve
$W^{cu}_S(y)$.
Since $R$ is adapted, there exists a return $T$ of $R$
such that $f^n(W^{cu}_T(x))=W^{cu}_R(y)$.
By property (viii), the rectangle $T$ contains $S$.
In the case $z^-_S$ and $z^-_R$ coincide for $z\in X$, the point
$f^n(z^-_S)=f^n(z^-_T)$ does not belong to the interior of the curves of $R$, as required.
Otherwise, there exists by the first item a sequence $(x_k)$ in $\Lambda\cap R$
such that $d(x_k,x_{k,S^-})$ goes to $0$ as $k$ goes to $+\infty$.
Hence $f^n(x_k)$ is close to $f^n(x_{k,S}^-)$ and belongs to $R$.
We have thus proved that $\cW^{cs}_{f^n(x^-_S)}$ is accumulted by points of $\Lambda\cap R$.
As a consequence, $f^n(x^-_S)$ can not belong to the interior of an unstable curve of $S$.
This gives the second item of the lemma.
\medskip
Note that $S$ has only finitely many returns with return time smaller or equal to $\tau$.
If $S$ has aperiodic boundary, its boundary is disjoint from the boundary of each of its returns:
there exists $\delta>0$ such that for any return $T$ with return time smaller or equal to $\tau$,
one has $d(x^-_S,x^-_T)>\delta$ and $d(x^+_S, x^+_T)>\delta$.
For $n$ larger than some constant $N$, the unstable curves $f^{-n}(W^{u}_{loc}(x))$ of points $x\in \Lambda$
have a size smaller than $\delta$. If $x\in R$ and $f^{-n}(W^{cu}_S(x))\subset S$, then
the iterate $f^{-n}(x)\in \Lambda$ belongs to $R\setminus S$. One deduces that $f^{-n}(W^{cu}_S(x))$
belongs to a return of $S$ with return time larger than $N$.
This gives the third item of the lemma.
\end{proof}
\subsection{Construction of adapted rectangles}\label{ss.construction}
The assumptions~\ref{a1}) and~\ref{a2}) are now used for the construction of adapted rectangles.
The proof is strongly based on proposition~\ref{p.box}.
\begin{proposition}\label{p.construction}
There exists an adapted rectangle $R$ such that $R\cap \Lambda$ is non-empty.
Moreover one can choose $R$ in such a way that one of the following cases occur.
\begin{enumerate}
\item For any $\tau\geq 0$, there is a first return $S$ of $R$ with return time larger than $\tau$
such that $\Lambda\cap S\neq \emptyset$.
\item There exists a hole $S$ of $R$ with aperiodic boundary.
\end{enumerate}
\end{proposition}
\noindent
The section continues with the proof of this proposition.
\paragraph{a) The construction.}
We have to require further assumptions on $f$ and $\Lambda$
needed to perform the following construction.
We first choose $\eta>0$ small. In particular one has $\eta<\rho<\varepsilon$
and the $10\; \eta$-neighborhood of $H(p_{f_0})$ is contained in $U$.
Let us apply the proposition~\ref{p.box}: one gets a smaller open neighborhood
$\widetilde U$ of $H(p_{f_0})$ such that for any diffeomorphism $f$ that is close enough
to $f_0$ in $\operatorname{Diff}^1(M)$, there exists a continuous family
of $C^1$-plaques $\widetilde \cW^{cs}$ tangent to $E^{cs}$
over the maximal invariant set $\widetilde K_f$ of $f$ in $\widetilde U$
which satisfies the following properties:
\begin{itemize}
\item[--] If two plaques $\cW^{cs}_x$ and $\widetilde \cW^{cs}_y$ have an intersection in the $\rho$-ball centered at $x$
then $\widetilde \cW^{cs}_y\subset \cW^{cs}_x$.
\item[--] Any $\widetilde\cW^{cs}$-connected set of $K\cap \cW^{cs}_x$ containing $x$ has radius smaller than $\eta$.
\end{itemize}
Since $f$ is close to $f_0$, if the chain-transitive set $\Lambda$ for $f$ is contained in a small neighborhood
of $H(p_{f_0})$, then the chain-recurrence class $K$ that contains $\Lambda$
is also contained in $\widetilde U$.
We thus have the inclusions $\Lambda\subset K\subset \widetilde K_f\subset K_f$.
\paragraph{\it Approximation by periodic orbits.}
We build a sequence of periodic points $(p_k)$ in $K$ such that
\begin{itemize}
\item the orbit of $p_k$ converges toward $\Lambda$ for the Hausdorff topology,
\item for each iterate $f^n(p_k)$, the plaque $\cW^{cs}_{f^n(p_k)}$ is contained in
the stable manifold of $f^n(p_k)$.
\end{itemize}
Let us fix $\zeta\in \Lambda$ satisfying the property (i).
With the property (iii), the plaque $\cW^{cs}_\zeta$ is contained in the stable set of $\zeta$.
Note that $\zeta$ is not periodic since otherwise $\Lambda$ would be a periodic orbit: by our assumptions, either it would be a sink or the bundle $E^{cu}$ would be uniformly expanded,
contradicting our assumptions.
This ensures that all the plaques $\cW^{cs}_{f^n(p_k)}$ and $\cW^{cs}_\zeta$ are disjoint.
\begin{claim}\label{l.shadowing}
For any $\alpha>0$, there exists $\delta>0$ with the following property.
For any forward return $y=f^n(\zeta)$ that is $\delta$-close to $\zeta$,
there exists $x\in W^{cu}_{loc}(\zeta)\cap K$ such that
$d(f^k(x),f^k(\zeta))$ is smaller than $\alpha$ for each $0\leq k\leq n$ and
the image $f^n(\cW^{cs}_{x})$ is contained in $\cW^{cs}_{x}$.
In particular for any $k\geq 0$ one has
$$\prod_{i=0}^{k-1}\|Df_{|E^{cs}}(f^i(x))\|\leq \lambda^k.$$
\end{claim}
\begin{proof}
From lemma~\ref{l.top-expansion}, there exists $r_0$ such that for any
point $z\in \Lambda$, the backward iterates of the ball centered at $z$ and of radius $r_0$ in
$W^{u}_{loc}(z)$ have a length smaller than $\alpha$.
For $\delta_0$ small enough and any point $y,x\in K$ that are $\delta_0$-close to $\zeta$,
the plaque $\cW^{cs}_{x}$ intersects $W^{cu}_{loc}(y)$ at a point $y'$ which belongs
to the ball centered at $y$ and of radius less than $r_0$ in $W^{cu}_{loc}(y)$.
For $n$ large enough, the length of any curve $f^{-n}(W^{cu}_{loc}(y))$ with $y\in \Lambda$
is smaller than $\delta_0$.
We choose $\delta\in (0,\delta_0)$ so that the returns $f^n(\zeta)$
that are $\delta$-close to $\zeta$ occur for $n$ large.
We define inductively a sequence of points $x_i\in K\cap W^{cu}_{loc}(\zeta)$
that are $\delta_0$-close to $\zeta$ and satisfy:
\begin{itemize}
\item[--] $d(f^{k}(x_i),f^k(\zeta))<\alpha$ for any $0\leq k\leq n$,
\item[--] $f^n(\cW^{cs}_{x_{i+1}})$ is contained in $\cW^{cs}_{x_{i}}$ and $x_0=\zeta$.
\end{itemize}
With properties (i) and (ii), this implies that
\begin{itemize}
\item[--] For any $k\geq 0$ one has $\prod_{j=0}^{k-1}\|Df_{|E^{cs}}(f^j(x_i))\|\leq \lambda^k$.
\end{itemize}
The construction is done in the following way.
Let us assume that $x_i$ has been defined.
Then the plaque $\cW^{cs}_{x_i}$ intersects $W^{u}_{loc}(y)$ in a point
$y_i$. By property (iii) the point $y_i$ belongs to the stable and the unstable set of $\Lambda$,
hence belongs to $K$. Moreover the distances $d(f^{-k}(y_i),y)$
are smaller than $\alpha$ for any $k\geq 0$.
We then define $x_{i+1}=f^{-n}(y_i)$ and by construction
this point is $\delta_0$-close to $\zeta$ and belongs to $W^{u}_{loc}(\zeta)$.
The map $x_i\mapsto x_{i+1}$ is continuous and monotonous, hence converges to
a fixed point $x\in W^{u}_{loc}(\zeta)\in K$. The construction and properties (i), (ii)
give the announced conclusions on $x$.
\end{proof}
\medskip
Since $\zeta$ is recurrent, the lemma~\ref{l.shadowing}
gives a sequence of points $(x_k)$ in $W^{u}_{loc}(\zeta)\cap K$
which converges toward $\zeta$ and such that each plaque $\cW^{cs}_{x_k}$
is mapped into itself by an iterate $f^{n_k}$.
The contraction along the bundle $E^{cs}$ at $x_k$ shows that
each plaque $\cW^{cs}_{x_k}$ is contained in the stable manifold of a periodic point
$p_k\in \cW^{cs}_{x_k}\cap K$.
By construction, the orbit $(x_k,f(x_k),\dots,f^{n_k-1}(x_k))$ is contained in an arbitrarily
small neighborhood of $\Lambda$. With the contraction along the bundle $E^{cs}$ at $x_k$
and the fact that $f^{n_k}(\cW^{cs}_{x_k})\subset \cW^{cs}_{x_k}$, one deduces that the whole
forward orbit of $x_k$ and the orbit of $p_k$ are close to $\Lambda$ for the Hausdorff
topology. Since the plaques $\cW^{cs}$ are trapped, each plaque $\cW^{cs}_{f^n(p_k)}$
is contained in the stable set of the orbits of $p_k$.
\paragraph{\it The boundary $\cW^{cs}_{p^-},\cW^{cs}_{p^+}$.}
We fix some periodic point $p_k$ for $k$ large and consider the set $P$ of all iterates
$p'$ of $p_k$ such that $d(p',x)<5\; \eta$.
\medskip
We choose $x_0\in \Lambda$ close to $\zeta$ and $p^-,p^+\in P$
so that the open interval $I\subset W^{u}_{loc}(\zeta)$ bounded by
$\cW^{cs}_{p^-}$ and $\cW^{cs}_{p^+}$ has the following properties:
\begin{itemize}
\item[--] for any point $p'\in P$ the intersection of
$\cW^{cs}_{p'}$ with $W^{u}_{loc}(\zeta)$ does not belong to $I$,
\item[--] $\cW^{cs}_{x_0}$ intersects $I$.
\end{itemize}
The plaques $\cW^{cs}_{f^n(p^\pm)}$ close to $\zeta$ are controled:
\begin{claim}\label{c.iterate}
For any $n\geq 0$, either the iterate $f^n(\cW^{cs}_{p^+})$ does not meet the ball centered at $\zeta$ of radius $2\eta$,
or $\cW^{cs}_{f^n(p^+)}$ does not intersect $I$. The same holds with the iterates of $\cW^{cs}_{p^-}$.
\end{claim}
\begin{proof} Let us fix a large integer $N$.
Since $\zeta$ is non-periodic and $\cW^{cs}_\zeta$ is contained in its
stable set, the iterates $f^{n}(\cW^{cs}_\zeta)$ are pairewise disjoint.
From the construction, having chosen $\cW^{cs}_{p^+}$ close to $\cW^{cs}_\zeta$
and $I$ close to $\zeta$, the plaques $f^n(\cW^{cs}_{p^+})$ do not meet $I$ for $n\leq N$.
When $n=N$, the radius of the plaque $f^n(\cW^{cs}_{p^+})$ is small, and the plaque is contained
in $\widetilde \cW^{cs}_{f^{n}(p^+)}$. By the trapping property, any iterate
$f^n(\cW^{cs}_{p^+})$ with $n\geq N$ is thus contained in $\widetilde \cW^{cs}_{f^{n}(p^+)}$
and has a radius smaller than $\eta$.
One deduces that if $f^n(\cW^{cs}_{p^+})$ meets the ball centered at $\zeta$ of radius $2\eta$,
then the distance between $f^n(p^+)$ and $\zeta$ is smaller than $3\eta$.
Consequently, $f^n(p^+)$ belongs to $P$ and $\cW^{cs}_{f^n(p^+)}$ does not meet $I$.
\end{proof}
\medskip
\paragraph{\it The rectangle $R$.}
Let us consider in the $2\eta$-neighborhood of $\zeta$ the set $X_0$ of points $z\in \cW^{cs}_\zeta$ that belong
to a forward iterate $f^j(W^{u}_{loc}(y))$ associated to a point $y\in \Lambda$. Then we define $X$ as
the largest $\widetilde \cW^{cs}$-connected subset of $X_0$ containing $\zeta$. By the choice of $\widetilde \cW^{cs}$,
the set $X$ has diameter bounded by $\eta$.
We define $R$ as the union of curves
$\gamma_z\subset W^{cu}_{loc}(z)$, $z\in X$, bounded by
the intersections $z^-,z^+$ between $W^{cu}_{loc}(z)$
and $\cW^{cs}_{p^-},\cW^{cs}_{p^+}$.
By the choice of $\eta$ and the construction, the points $z^-,z^+$ belong to $K_f$.
With property (vi), one deduces that the items~\ref{rectangle1})
and~\ref{rectangle3}) of the definition~\ref{d.rectangle} are satisfied.
Consider any close points $x,y\in \Lambda$ with $x\in R$.
The intersections of $W^{u}_{loc}(x)$ and $W^{u}_{loc}(y)$ with $\cW^{cs}_\zeta$
are close, hence belong to the same $\widetilde \cW^{cs}$-component of $X_0$.
As a consequence, $W^{u}_{loc}(y)\cap \cW^{cs}_\zeta$ belongs to $X$.
This shows that $y$ belongs to $R$. We have proved that $\Lambda\cap R$ is open in $\Lambda$ and that $R$ is a rectangle. By construction it contains the point $x_0$ and the intersection $R\cap \Lambda$
is non-empty.
\medskip
\paragraph{b) $R$ is adapted.}
Let us assume that for some $x,y\in X$,
a forward iterate $f^n(\gamma_{x})$ intersects $\gamma_y$.
Considering a large backward iterate, the two curves $f^{n-m}(\gamma_x)$
and $f^{-m}(\gamma_y)$ are small and contained in local unstable curves
$W^{u}_{loc}(x')$ and $W^{u}_{loc}(y')$ for some points $x',y'\in \Lambda$.
By property (vi), one deduces that $f^{n-m}(\gamma_x)$
and $f^{-m}(\gamma_y)$ are contained in a same unstable curve $W^{u}_{loc}(x')$.
In particular, if $f^n(\gamma_x)$ intersects $\gamma_y$ but does not contain $\gamma_x$,
then the image of an endpoint $f^n(x^-)$ (or $f^n(x^+)$) of $\gamma_x$
is contained inside $\gamma_y$. One deduces that $\cW^{cs}_{f^n(p^-)}$
intersects $I$. Since $f^n(x^-)$ is $2\eta$-close to $x$, this contradicts the claim~\ref{c.iterate} above.
We have thus proved the first item of definition~\ref{d.adapted}.
\medskip
Assume now that $f^n(\gamma_x)$ contains $\gamma_y$.
One can define the subrectangle $S$ of $R$ whose unstable curves are bounded by
$\cW^{cs}_{x^-_S}$ and $\cW^{cs}_{x^+_S}$, with $x^\pm_S=f^{-n}(y^\pm_R)$.
It remains to prove that $f^n(S)$ is contained in $R$.
Let us consider the set $X^+_S$ of points $z^+_S$ for $z\in X$, i.e. the intersection
of $\cW^{cs}_{x^+_S}$ with the unstable curves $W^{cu}_{loc}(z)$.
Since $z^+_S$ and $z$ are close, the set $X^+_S$ is connected for the larger plaque family
$f^{-1}(\widetilde \cW^{cs})$ containing the plaques $f^{-1}(\widetilde\cW^{cs}_{f(x)})$ for $x\in \widetilde K_f$.
Note that $n$ is larger than $2$.
As a consequence, the set $f^n(X^+_S)$ is $f(\widetilde \cW^{cs})$-connected.
One thus deduces that the set $X'$ of intersections of the curves $W^{cu}_{loc}(z)$, $z\in X^+_S$,
with $\cW^{cs}_\zeta$ is $\widetilde \cW^{cs}$-connected. Since it contains $y\in X$, this set is contained
in the $\widetilde \cW^{cs}$-component $X$. This proves the second item of definition~\ref{d.adapted}
and $R$ is adapted.
\paragraph{c) Periodic center-stable plaques.}
Let us assume that there exist $x\in \Lambda$ and $n\geq 1$ such that
the plaque $\cW^{cs}_x$ is mapped into itself by $f^n$.
The set $\Lambda$ is not contained in the orbit of the plaque $\cW^{cs}_x$:
otherwise the property (i) would imply that $\zeta$ is a sink of $\cW^{cs}_x$,
contradicting the fact that $\Lambda$ is non-periodic.
Since $\cW^{cs}_\zeta$ is contained in the stable manifold of $\zeta$, the closure of $\cW^{cs}_\zeta$
and of $\cW^{cs}_x$ are disjoint.
Note that the rectangle $R$ can have been constructed arbitrarily thin in the center unstable direction,
hence it is contained in an arbitrarily small
neighborhood of $\cW^{cs}_\zeta$. In particular, the closure of $R$
and the closure of the orbit of $x$ are disjoint.
Since $\Lambda$ is transitive the first return time on $\Lambda\cap R$ is unbounded,
giving the first case of proposition~\ref{p.construction}.
\paragraph{d) Non-periodic center-stable plaques.}
Let us assume the opposite case: there does not exist $x\in \Lambda$ and $n\geq 1$ such that
the plaque $\cW^{cs}_x$ is mapped into itself by $f^n$.
Let $R_0$ be a rectangle as obtained in paragraphs a) and b).
One can assume also that the first case of the proposition~\ref{p.construction} does not hold.
Since $R_0\cap \Lambda$ is non empty and since $\Lambda$ is the Hausdorff limit of periodic points,
there exists a periodic point $p\in K_f$ whose plaque $\cW^{cs}_p$ intersects
all the unstable leaves $W^{cu}_{R_0}(z)$ of $R_0$ at a point $z_p$ which is not in $\Lambda$.
As in the proof of lemma~\ref{l.hole}, the points $K_f\cap R_0$ are ordered by
their projection on an unstable curve of $R_0$.
There exist two points $x^-,x^+\in K_f\cap \overline R_0$
such that $x^-<z_p<x^+$, any point $y\in \Lambda\cap R_0$ satisfies $y\leq x^-$
or $y\geq x^+$ and such that there is no $\bar x^-<x^-$ or
$\bar x^+>x^+$ with the same properties.
One checks easily that the collection of curves
$\gamma'_z\subset W^{u}_{R_0}(z)$ bounded by points
in $\cW^{cs}_{x^-}$ and $\cW^{cs}_{x^+}$ is a rectangle and a hole $S_0$ of $R_0$.
\medskip
We then explain how to modify $R_0$ in order to obtain a hole with aperiodic boundary.
Since $R_0\cap \Lambda$ is non-empty, one can assume by lemma~\ref{l.hole} that
there exists a sequence $x_n\in \Lambda\cap R_0$ such that $d(x_n,x^-_{n,S_0})$ goes to zero
as $n$ goes to $+\infty$. Let us denote by $X^-=\{z^-_{S_0},\;z\in X\}$
one of the boundaries of $S_0$. By construction there exists $x^-\in\overline{X^-}\cap \Lambda$
such that the plaque $\cW^{cs}_{x^-}$ contains $X^-$.
One deduces that $X^-$ is disjoint from its forward iterates.
One can choose the points $x_n$ to have a dense forward orbit.
In particular they return to $R_0$. Since the return time is bounded,
one can also assume that they all have the same return time, hence belong
to a same return $T$ of $R_0$. The set $X^-$ belong to $T$: otherwise it would be
contained in the boundary of $T$ and mapped by a forward iterate into the boundary of $R_0$;
since the closure of $X^-$ meets $\Lambda$ and since the boundary of $R_0$ is contained
in the stable set of a periodic orbit, this would imply that the $\cW^{cs}$-plaque of a point of $\Lambda$
is mapped into itself, contradicting our assumption.
Note that the set $X^-$ is still one of the boundaries of a hole of $T$
and that the boundary of $T$ is still contained in the stable set of periodic orbits.
One can thus replace $R_0$ by $T$ and repeat the same argument.
Doing that several times, one gets a deeper return $R$ of $R_0$
which contains the set $X^-$.
The rectangle $R$ is arbitrarily thin in the unstable direction,
hence it contains a hole $S$ whose boundaries are $X^-$ and a boundary of $R$.
By construction the boundaries of $R$ are disjoint from their iterates.
This implies that $S$ has aperiodic boundaries.
\bigskip
The proof of the proposition~\ref{p.construction} is now complete.
\hskip 445pt \rule{2mm}{2mm}
\subsection{Summability}
For any point $x\in K$ we denote by
$\ell(J)$ the length of any interval $J$ contained in its local unstable manifold $W^{cu}_{loc}(x)$.
This section is devoted to the proof of the next proposition.
\begin{proposition}\label{p.summability}
For any adapted rectangles $S,R$, where $S$ is a subrectangle of $R$,
there exists $K(S)>0$ satisfying the following: for any $x\in \Lambda\cap R$,
and any $n\geq 0$ such that the curves $f^{-k}(W^{cu}_{S}(x))$, $0<k<n$,
are disjoint from $S$, we have
$$\sum_{k=0}^{n-1}\ell(f^{-k}(W^{cu}_{S}(x)))<K(S).$$
Moreover, there is $K_0>0$ which only depends on $R$ such that
$K(S)<K_0$ when $S$ is a return of $R$.
\end{proposition}
\bigskip
\paragraph{a) Summability for first returns.}
The first case corresponds to~\cite[lemma 3.7.1]{PS1}.
\begin{lemma}\label{ll.sum}
For any adapted rectangle $R$ with $R\cap \Lambda\neq \emptyset$,
there are $C_1>0$, $\sigma_1\in (0,1)$ as follows.
For any unstable curve $W^{cu}_{R}(x)$ of $R$ with $W^{cu}_{loc}(x)\cap \Lambda\neq \emptyset$,
any interval $J\subset W^{cu}_R(x)$
and any $n\geq 0$ such that the iterates $f^{-j}(W^{cu}_{R}(x))$ with $0<j<n$
are disjoint from $R$ we have
$$\ell(f^{-n}(J))\leq C_1\; \sigma_1^n\; \ell(J).$$
\end{lemma}
\begin{proof}
Let us consider a point $z\in \Lambda\cap R$.
Since $\Lambda\cap R$ is open, one can choose a small open neighborhood $V$ of $z$.
The maximal invariant set
$$\Lambda_1= \bigcap_{n\in \ZZ}{f^n(\Lambda - V)}$$
in $\Lambda\cap (M\setminus V)$ is compact and proper in $\Lambda$.
By assumption $E^{cu}$ is expansive on $\Lambda_1$.
It is thus possible to get a neighborhood of $\Lambda_1$ such that while the iterates remain
in this neighborhood the subbundle $E^{cu}$ is uniformly expanded by $Df$.
Moreover, the number of iterates that an orbit of $\Lambda$
remains in the complement of the mentioned neighborhood of $\Lambda_1$
and $V$ is uniformly bounded.
Since $W^{cu}_{loc}(x)\cap \Lambda\neq \emptyset$,
one can always assume that $x$ belongs to $\Lambda$.
By lemma~\ref{l.top-expansion}, choosing $n_0$ large enough (and independant from $x,J,j$), the curves $f^{-j}(W^{u}_{loc}(x))$
for $j\geq n_0$ are small. If $j<n$, the segment $f^{-j}(J)$ is disjoint from $R$,
hence $f^{-j}(x)$ is disjoint from $V$. Moreover $x$ belongs to $\Lambda$.
One deduces that there exist uniform constants $\sigma\in (0,1)$ and $C>0$
such that $\|Df^{-j}_{|E^{cu}}(x)\|< C\sigma^{j}$ for all $0<j<n$.
Since for $n_0$ large enough the curves $f^{-j}(W^{u}_{loc}(x))$
are small, the derivatives $\|Df_{E^{cu}}(f^{-j}(x))\|$
and $\|Df_{E^{cu}}(f^{-j}(y))\|$ for $y\in W^{u}_{loc}(x)$ are close.
One deduces that for any $0<j<n$ and $y\in W^{u}_{loc}(x)$ one also has
$\|Df^{-n}_{|E^{cs}}(y)\|< C_1\sigma_1^{n}$ for other constants $C_1>0$, $\sigma_1\in (0,1)$.
The conclusion of the lemma follows.
\end{proof}
\medskip
\paragraph{b) Distortion along center-stable holonomies and contracting returns.}
We will need to compare the unstable curves.
\begin{defi}
A rectangle $R$ has \emph{distortion bounded by $\Delta>0$} if for any unstable
curves $W^{cu}_{R}(x)$, $W^{cu}_{R}(y)$ one has:
$$\frac 1 {\Delta}\leq \frac{\ell(W^{cu}_R(x))}{\ell(W^{cu}_R(y))}\leq {\Delta}.$$
\end{defi}
We will also need to consider returns that contract along the center-stable bundle.
\begin{defi}\label{d.contracting}
Let us fix $N\geq 0$. A point $z_0\in K_f$
is \emph{$N$-contracting} in time $n$ if there exists
$m\leq N$ in $\{0,\dots,n\}$ such that for each
$i\in \{m,\dots,n\}$ one has
$$\prod_{k=m}^{i}\|Df_{|E^{cs}}(f^{k}(z_0))\|\leq \lambda^{i-m}.$$
A return $S$ of a rectangle $R$ with returning time $n$
is \emph{$N$-contracting} if there $z_0\in K_f\cap \overline S$
which is $N$-contracting in time $n$.
\end{defi}
The following lemma is similar to~\cite[lemma 3.4.1]{PS1}.
\begin{lemma}\label{l.distortion-holo}
For any adapted rectangle $R$ and any $N\geq 0$,
there is $\Delta_1>0$ such that
any $N$-contracting return of $R$ has distortion bounded by $\Delta_1$.
\end{lemma}
\begin{proof}
One chooses a $C^1$-foliation $\mathcal{F}} \def\cL{\mathcal{L}} \def\cR{\mathcal{R}} \def\cX{\mathcal{X}$ tangent to the cone field $\mathcal{C}} \def\cI{\mathcal{I}} \def\cO{\mathcal{O}} \def\cU{\mathcal{U}^{cs}$
and containing the plaques $\cW^{cs}_{x^-}$, $\cW^{cs}_{x^+}$ of $R$.
For any unstable curves of $R$ with basepoints $x,y\in X$,
one gets a diffeomorphism $\Pi_{x,y}\colon W^{cu}_{R}(x)\to W^{cu}_{R}(y)$,
whose derivative is bounded from above and below uniformly in $x,y\in X$.
Let $S$ be a $N$-contracting return of $R$.
For any unstable curves of $S$ with basepoints $x',y'$, their images by $f^n$
coincide with some curves $W^{cu}_{R}(x)$, $W^{cu}_{R}(y)$ of $R$.
This allows us to define a diffeomorphism $\Pi^n_{x',y'}\colon W^{cu}_{S}(x')\to W^{cu}_{S}(y')$ by
$$\Pi^n_{x',y'}=f^{-n}\circ\Pi_{x,y}\circ f^n.$$
The distortion of $S$ is thus controled by the following quantity, for any $z\in W^{cu}_{S}(x')$:
$$\|Df^n_{|TW^{cu}_{S}(z)}\|/\|Df^n_{|TW^{cu}_{S}(\Pi^n_{x',y'}(z))}\|.$$
Using that the forward iterates of any vector tangent to $\mathcal{C}} \def\cI{\mathcal{I}} \def\cO{\mathcal{O}} \def\cU{\mathcal{U}^{cu}$ in $U$
converge towards $E^{cu}$ (uniformly) exponentially fast and that the bundle $E^{cu}$
is H\"older (see~lemma~\ref{l.bundle}), one can argue as in~\cite[lemma 3.4.1]{PS1}
and show that there exist some uniform constants $C>0$ and $\alpha\in (0,1)$ such that
$$\frac{\|Df^n_{|TW^{cu}_{S}(z)}\|}{\|Df^n_{|TW^{cu}_{S}(\Pi^n_{x',y'}(z))}\|}
\leq \exp\left(C+C\sum_{i=0}^{n-1}d(f^i(z),f^i(\Pi^n_{x',y'}(z)))^\alpha\right).$$
It remains to estimate $d(f^i(z),f^i(\Pi^n_{x',y'}(z)))$
and it is clearly enough to consider the case $i\geq N$.
Using the property (v) stated in section~\ref{ss.topo}, there exists a disc $B$
centered at $z$ tangent to $\mathcal{C}} \def\cI{\mathcal{I}} \def\cO{\mathcal{O}} \def\cU{\mathcal{U}^{cs}$ of radius larger than $\varepsilon$,
whose iterates $f^{i}(B)$, $i\in \{0,\dots,n\}$ have a radius smaller than $\widetilde \varepsilon$
and such that $f^{n}(B)$ is contained in a leaf of the foliation $\mathcal{F}} \def\cL{\mathcal{L}} \def\cR{\mathcal{R}} \def\cX{\mathcal{X}$.
One deduces that $B$ contains the point $\Pi^n_{x',y'}(z)$.
From property (ii), the distance $d(f^i(z),f^i(\Pi^n_{x',y'}(z))$ is thus bounded
by
\begin{equation*}
\begin{split}
d(f^i(z),f^i(\Pi^n_{x',y'}(z)))&\leq
d(f^m(z),f^m(\Pi^n_{x',y'}(z)))\;
(1+\chi)^{i-m}\;\prod_{k=m}^{i}\|Df_{|E^{cs}}(f^k(z_0))\|\\
&\leq \widetilde\varepsilon\;(1+\chi)^{i-m}\;\lambda^{i-m},
\end{split}
\end{equation*}
where $z_0\in K_f\cap \overline S$ is a point which satisfies the definition~\ref{d.contracting}
for some interger $m\geq N$.
We have assumed that $(1+\chi)\lambda<1$ (recall section~\ref{ss.topo}),
hence the sum $\sum_{i=0}^{n-1}d(f^i(z),f^i(\Pi^n_{x',y'}(z)))^\alpha$
is bounded uniformly. This concludes the proof of the lemma.
\end{proof}
With the same proof,
the lemma~\ref{l.distortion-holo} generalizes to the following setting.
\begin{lemma}\label{l.distortion-holo2}
For any adapted rectangles $S,R$ such that $S$ is a subrectangle of $R$ and for any $N\geq 0$,
there exists $\Delta_1(S)$ such that for any $N$-contracting return $R'$ of $R$ with return time $n$,
the subrectangle $S'=R'\cap f^{-n}(S)$ has distortion bounded by $\Delta_1(S)$.
\end{lemma}
\paragraph{c) Summability between contracting returns}
One now obtains the summability for returns which do not satisfy lemma~\ref{l.distortion-holo}.
\begin{lemma}\label{l.deep}
For any adapted rectangle $R$ and any $N\geq 1$ large enough, there is $K_1>0$ as follows.
Consider $x\in \Lambda\cap R$ and $0\leq k<l$ such that:
\begin{itemize}
\item[--] $f^{-k}(W^{cu}_R(x))\subset R$ and $f^{-k}(x)$ is $N$-contracting in time $k$,
\item[--] for any $k<j<l$, either $f^{-j}(W^{cu}_R(x))\cap R=\emptyset$
or $f^{-j}(x)$ is not $N$-contracting in time $j$.
\end{itemize}
Then for any curve $J\subset W^{cu}_R(x)$ one has
$$\sum_{j=k}^{l}\ell(f^{-j}(J))\leq K_1\; \ell(f^{-k}(J)).$$
\end{lemma}
\begin{proof}
We let $\cR\subset \{k,\dots,l\}$ be the set of integers $n$ such that $f^{-n}(W^{u}_R(x))\subset R$. Since $R$ is adapted the other integers satisfy $f^{-n}(W^{u}_R(x))\cap R=\emptyset$.
The lemma~\ref{ll.sum} can be restated as:
\begin{claim}\label{l.sum1} There exists $K_2>0$ satisfying the following.
For any integers $r<p$ in $\{k,\dots,l\}$ such that
$r\in \cR$ and $\{r+1,r+2,\dots p-1\}\cap \cR=\emptyset$, one has
$$\sum_{j=r}^{p}\ell(f^{-j}(J))<K_2\; \ell(f^{-r}(J)).$$
\end{claim}
\medskip
We also introduce the set $\cP\subset \{0,\dots,l\}$
of integers $n$ such that for each
$0\leq i < n$ one has
$$\prod_{j=i+1}^n\|Df_{|E^{cs}}(f^{-j}(x))\|\leq \lambda^{n-i}.$$
The summability between iterates in $\cP$ is ensured by the next classical argument.
\begin{claim}\label{l.sum2}
For any integers $p<r$ in $\{k,\dots,l\}$
such that $p\in \cP$ and $\{p+1,p+2,\dots,r-1\}\cap \cP=\emptyset$, one has
$$\ell(f^{-r+1}(J))<\mu^{r-p-1}\ell(f^{-p}(J)).$$
\end{claim}
\begin{proof}
Using that the integers $n\in \{p+1,\dots,r-1\}$ are not in $\cP$,
one proves inductively that
\begin{equation}\label{e.pliss}
\prod_{j=p+1}^n\|Df_{|E^{cs}}(f^{-j}(x))\|> \lambda^{n-p}.
\end{equation}
Indeed, if one has $\|Df_{|E^{cs}}(f^{-p-1}(x))\|\leq \lambda$,
then using that $p$ belongs to $\cP$, one deduces that $p+1$ also, which is a
contradiction. Moreover if the inequatlity~\eqref{e.pliss}
holds for all the integers $p+1,\dots,n-1$
and is not satisfied for $n$, then for all $i\in\{p,\dots,n-1\}$ one gets
$$\prod_{j=i+1}^{n}\|Df_{|E^{cs}}(f^{-j}(x))\|\leq \lambda^{n-i}.$$
Since $p$ belongs to $\cP$ this
implies that $n$ also which is a contradiction.
This proves that~\eqref{e.pliss} holds.
The property~\eqref{e.pliss} for $n=r-1$ together with (iv) in section~\ref{ss.topo}
imply that the norm of $Df^{p-r+1}_{|W^{u}_S(f^{-p}(x))}$
along the plaque $W^{cu}_S(f^{-p}(x))$ is smaller than $\mu^{r-p-1}$,
giving the required conclusion.
\end{proof}
\medskip
We can now prove the lemma.
Let $C_f>1$ be an upper bound of $\|Df\|$.
We choose $N$ large enough so that one has $\mu^N K_2 C_f<\frac 1 2$.
Let us consider $p_s<p_{s-1}<\dots<p_0$ in $\cP$
and $k\leq r_s<r_{s-1}<\dots<r_0\leq l$ in $\cR$
which satisfy:
\begin{enumerate}
\item[--] For each $i\in \{0,\dots,s\}$ one has $p_i\leq r_i$ and
for $i\in \{1,\dots,s\}$ one has $r_{i}\leq p_{i-1}$.
\item[--] There is no $r\in \cR$ such that $r_{i}<r<p_i$.
There is no $p\in \cP$ such that $p_i<p<r_{i-1}$.
\item[--] $p_s\leq k$ and when $s\geq 1$ one has $k<p_{s-1}$. There is no $r\in \cR$ such that $r_0<r\leq l$.
\end{enumerate}
These sequences are defined inductively:
$r_0$ is the largest integer in $\cR$ smaller or equal to $l$ and
$p_0$ is the largest integer in $\cP$ smaller or equal to $r_0$.
Assume that $p_i\leq r_i$ have been constructed.
If $p_i\leq k$ we set $s=i$ and the construction stops.
Otherwise we let $r_{i+1}$ be the largest integer in $\cR$ that is smaller or equal
to $p_i$ and smaller than $r_{i}$.
By assumption $p_{i}$ is larger than $n$, hence $r_{i+1}$ is larger or equal to $n$.
Then $p_{i+1}$ be the largest integer in $\cP$ smaller or equal to $r_{i+1}$ and smaller than $p_i$.
\medskip
Since $f^{-k}(x)$ is $N$-contracting in time $k$, we have $p_s\geq k-N$.
One deduces
$$\ell(f^{-p_s}(J))\leq C_f^{N}\; \ell(f^{-k}(J)).$$
Using claims~\ref{l.sum1} and~\ref{l.sum2}, for each $i\in\{1,\dots,s\}$ one has
$$\sum^{p_{i-1}}_{k=p_i} \ell(f^{-k}(J))\leq ((1-\mu)^{-1}+K_2C_f)\; \ell(f^{-p_i}(J)),$$
$$\sum_{k=p_0}^{l} \ell(f^{-k}(J))\leq ((1-\mu)^{-1}+K_2C_f)\; \ell(f^{-p_0}(J)),$$
$$\ell(f^{-p_{i-1}}(J))\leq \mu^{r_{i}-p_i}K_2C_f\; \ell(f^{-p_{i}}(J)).$$
By our assumptions, when $i$ satisfies $0<i<s$ the point $f^{-r_i}(x)\in R$
is not $N$-contracting. As a consequence $r_i-p_i\geq N$. This implies by our choice of $N$,
$$\ell(f^{-p_{i-1}}(J))\leq \mu^N K_2 C_f\; \ell(f^{-p_i}(J)) \leq \frac 1 2 \ell(f^{-p_i}(J)).$$
Putting all these estimates together one gets the conclusion:
$$\sum_{j=k}^{l}\ell(f^{-j}(J))<((1-\mu)^{-1}+K_2C_f)(1+2K_2C_f)C_f^N\; \ell(f^{-k}(J)).$$
\end{proof}
\bigskip
\begin{proof}[\bf d) Proof of the proposition~\ref{p.summability}]
Let us choose $N\geq 1$ large and consider the constant $K_1$ given by lemma~\ref{l.deep}.
The lemma~\ref{l.distortion-holo2} applied to the rectangle $S$ gives a bound $\Delta_1(S)$.
We fix an unstable curve $W^{cu}_R(x_0)$ of $R$.
We set $K(S)=2\Delta_1(S) K_1\ell(W^{cu}_R(x_0))$.
We also set $n_S=0$ (in the case $S$ is a return we will obtain a better result taking
$n_S$ equal to the return time).
Let $x\in \Lambda\cap R$ and $J=W^{u}_S(x)$.
We introduce the set $\cR_{\cP}\subset \{-n_S,\dots,n\}$
of integers $i$ such that
$f^{-i}(J)\subset R$ and $f^{-i}(x)$ is $N$-contracting in time $i+n_S$.
Since $R$ is adapted, the lemma~\ref{l.adapted} shows that for each $i\in \cR_{\cP}$, there exists a subrectangle $S_i$ of $R$ such that
\begin{itemize}
\item[--] $f^{-i}(J)$ is an unstable curve of $S_i$,
\item[--] for each unstable curve $W^{cu}_{S_i}(z)$ of $S_i$ the image
$f^{i}(W^{cu}_{S_i}(z))$ is an unstable curve of $S$.
\end{itemize}
\begin{lemma}\label{l.sum4}
For any $i'< i$ in $\cR_\cP\cap \{1,\dots,n\}$, the rectangles $S_i,S_{i'}$ are disjoint.
\end{lemma}
\begin{proof}
Assume by contradiction that some unstable curves $f^{-i}(W^{cu}_{S}(y))$ and $f^{-i'}(W^{cu}_{S}(y'))$ of $S_i$ and $S_{i'}$ intersect.
Then $W^{cu}_{S}(y')$ intersects $f^{i'-i}(W^{cu}_{S}(y))$ and since $S$ is adapted,
there exists a return $T$ of $S$ with returning time $i-i'$ such that
$f^{i'-i}(W^{cu}_{S}(y))$ is an unstable curve of $T$.
One deduces from remark~\ref{r.uniqueness} and lemma~\ref{l.adapted}
that $f^{i'}(S_{i})$ is contained in $T$, hence in $S$.
This contradicts the assumption that $f^{i'-i}(W^{cu}_S(x))$ is disjoint from $S$.
\end{proof}
Let $i_0$ be the largest integer in $\cR_\cP$ which is smaller or equal to $n_S$.
(When $n_S=0$, one has $i_0=0$).
We now end the proof of the proposition~\ref{p.summability}.
The lemma~\ref{l.deep} implies that
\begin{equation*}\begin{split}
\sum_{k=0}^{n-1}\ell(f^{-k}(W^{cu}_{S}(x)))
&\;\leq\; \sum_{k=i_0-n_S}^{n-1}\ell(f^{-k}(W^{u}_{S}(x)))\\
&\;\leq\;K_1 \left(\ell(f^{-i_0}(J))+\sum_{i\in \cR_\cP,\; i> n_S}\ell(f^{-i}(J))\right).
\end{split}
\end{equation*}
Since $f^{-i}(x)$ is $N$-contracting in time $i+n_S$,
the lemma~\ref{l.distortion-holo} implies that for each $i\in \cR_\cP$
$$\ell(f^{-i}(J))\leq \Delta_1(S)\; \ell(W^{cu}_{S_i}(x_0)).$$
The lemma~\ref{l.sum4} implies that the intervals $W^{cu}_{S_i}(x_0)$ for $i\in \cR_\cP$
with $i> n_S$ are disjoint.
As a consequence
$$\sum_{i\in \cR_\cP,\; i> n_S}\ell(W^{cu}_{S_i}(x))\leq \ell(W^{cu}_R(x_0)).$$
Putting together these last three inequalities, one concludes the proof of the proposition~\ref{p.summability}
in the general case $S$ is an adapted subrectangle:
$$\sum_{k=0}^{n-1}\ell(f^{-k}(W^{cu}_{S}(x)))\leq 2\Delta_1(S) K_1\; \ell(W^{cu}_R(x_0))=K(S).$$
When $S$ is a return, we take $n_S$ equal to the return time so that $f^{n_S}(J)$
is an unstable curve of $R$.
The constant $\Delta_1$ is given by lemma~\ref{l.distortion-holo2} and as before we set
$K_0=2\Delta_1 K_1\ell(W^{cu}_R(x_0))$.
We repeat the same proof, noting that the subrectangles $S_i$ are returns of $R$, so that
for each $i\in \cR_\cP$ we have the better estimate
$$\ell(f^{-i}(J))\leq \Delta_1\; \ell(W^{cu}_{S_i}(x_0)).$$
The conclusion of the proposition~\ref{p.summability} thus holds with the uniform constant $K_0$.
\end{proof}
\subsection{Proof of the proposition~\ref{propoE^{cu}ishyp}}
In order to conclude the proof of proposition~\ref{propoE^{cu}ishyp} we
consider a rectangle $R$ as given by the section~\ref{ss.construction} and we
distinguish between two cases described by the proposition~\ref{p.construction}.
\paragraph{a) Distortion along unstable curves.}
Since by lemma~\ref{l.smoothness}, the unstable curves the set $K$ are contained in a continuous $C^2$-plaque family,
the classical distortion estimates hold (see for instance~\cite[lemma 3.5.1]{PS1}).
\begin{itemize}
\item[(D)] \emph{There is $\Delta_2>0$ such that for any $z\in K$, any $x,y$ in an interval $J\subset W^{cu}_{loc}(z)$,
and any $n\geq 0$,
$$\frac{\|Df^{-n}_{|E^{cu}}(x)\|}{\|Df^{-n}_{|E^{cu}}(y)\|}\leq \exp\left(\Delta_2\sum_{k=0}^{n-1}\ell(f^{-k}(J))\right).$$
In particular,
\begin{equation}\label{e.disto}
\|Df^{-n}_{|E^{cu}}(x)\|\leq \frac{\ell(f^{-n}(J))}{\ell(J)}\; \exp\left(\Delta_2\sum_{k=0}^{n-1}\ell(f^{-k}(J))\right).
\end{equation}}
\end{itemize}
As a consequence we also get the following.
\begin{itemize}
\item[(D')] \emph{For any $C>0$ there is $\eta>0$ such that for any $z\in K$, for any
intervals $J\subset \widehat J\subset W^{cu}_{loc}(z)$ and for any $n\geq 0$ satisfying
$\ell(\widehat J)\leq (1+\eta)\; \ell(J)$ and $\sum_{k=0}^{n-1}\ell(f^{-k}(J))\leq K$, then one has
$$\sum_{k=0}^{n-1}\ell(f^{-k}(J))\leq 2\; C.$$
In particular for any $x\in \widehat J$ one has
\begin{equation*}
\|Df^{-n}_{|E^{cu}}(x)\|\leq \frac{\ell(f^{-n}(J))}{\ell(J)}\; \exp\left(2\; \Delta_2\sum_{k=0}^{n-1}\ell(f^{-k}(J))\right).
\end{equation*}}
\end{itemize}
\bigskip
\paragraph{b) Adapted rectangles with unbounded first returns.}
We conclude proposition~\ref{propoE^{cu}ishyp} in the first case
of the proposition~\ref{p.construction}.
(The end of the proof corresponds to~\cite[lemma 3.7.4]{PS1}.)
\begin{lemma}\label{l.unbounded2}
For any adapted rectangle $R$, there exists $\tau\geq 0$ as follows.
If there exists a first return $S_0$ of $R$ with return time larger than $\tau$
and such that $S_0\cap \Lambda\neq \emptyset$, then, there also exists a return $S$
of $R$ such that $S\cap \Lambda\neq \emptyset$ which has the following property:
for any $x\in S\cap \Lambda$ and $n\geq 1$ such that
$f^{-n}(x)\in S$ we have $\|Df^{-n}_{|E^{cu}}(x)\|<\frac 1 2$.
\end{lemma}
In particular the property (E) holds with $B=S\cap \Lambda$.
\begin{proof}
Let $K_0,K_1,N,\Delta_2$ be some constants associated to $R$ so that proposition~\ref{p.summability} and lemmas~\ref{l.distortion-holo} and~\ref{l.deep} hold.
Let $L$ be a lower bound for the length of unstable curves $W^{cu}_R(z)$ of $R$
and $l$ be an upper bound for all the backward iterates $f^{-j}(W^{cu}_R(z))$ with $j\geq 0$.
Recall that $\Delta_2>0$ is a constant such that~\eqref{e.disto} holds.
We also set
$$\delta=\frac L {\Delta_1}\exp(-\Delta_2\;(K_0+K_1\;l))/3$$
and choose $\tau\geq 1$ so that for any $z\in \Lambda$ the backward iterates $f^{-k}(W^{cu}_{loc}(z))$
with $k\geq \tau$ have a length smaller than $\delta$
(see lemma~\ref{l.top-expansion}).
We then consider a return $S_0$ of $R$ with return time $n_0$
larger than $\tau$ such that $S_0\cap \Lambda\neq \emptyset$. Two cases occur.
\paragraph{\it Case 1: no contracting backward iterate.}
We assume first that for any $x\in S_0\cap \Lambda$, there is no backward iterate $f^{-j}(x)$ with
$j\geq 0$ which belongs to a $N$-contracting return of $R$ with return time $j$.
In this case, we set $S=S_0$. For any point $x\in S\cap \Lambda$
and any $j\geq 1$ such that $f^{-j}(x)\in S$ one can apply
the lemma~\ref{l.deep} to $x$ and the integers $k=0$ and $l=j$.
One deduces that one has
$$\sum_{i=0}^j \ell(f^{-j}(W^{cu}_R(x)))\leq K_1\ell(W^{cu}_R(x))\leq K_1\;l.$$
Note that $j\geq n_0\geq \tau$. Since $z$ belongs to $\Lambda$, one deduces that
$f^{-j}(W^{cu}_R(z))$ is smaller than $\delta$. With property (D), one gets
\begin{equation*}
\begin{split}
\|Df^{-j}_{|E^{cs}}(x)\|&\leq \frac{\ell(f^{-j}(W^{cu}_R(z)))}{\ell(W^{cu}_R(z))}\exp(\Delta_2\;K_1\;l)\\
&\leq \frac{\delta}{L}\exp(\Delta_2\;K_1\;l)<1/2.
\end{split}
\end{equation*}
The lemma is thus proved in this case.
\medskip
\paragraph{\it Case 2: contracting backward iterates exist.}
We first build the return $S$.
\begin{claim}\label{c.select} There exists a $N$-contracting return $S$ of $R$ with return time $n_1\geq \tau$
such that $\Lambda\cap S\neq \emptyset$
and such that for each $z\in \Lambda\cap S$ one has
$$\sum_{j=0}^{n_1}\ell(f^j(W^{cu}_{S}(z)))<K_1\;\ell(W^{cu}_{R}(f^{n_1}(z))).$$
\end{claim}
\begin{proof}
There exists a point $x\in f^{n_0}(S_0)\cap \Lambda$ and a backward iterate $f^{-n_1}(x)$ with $n_1> n_0$ which belongs to a $N$-contracting $S$ return of $R$ with return time $n_1$. One can assume that
$n_1$ is minimal: consequently for any $i\in \{1,\dots,n_1-n_2\}$ the iterate $f^{i}(S)$ does not intersect a
$N$-contracting return of $R$ with return time $n_1-i$.
Since $S_0$ is a first return, the iterates $f^i(S)$ for $i\in \{n_1-n_0+1,\dots,n_1-1\}$
do not intersect $R$. The lemma~\ref{l.deep} can thus be applied to the points $z\in\Lambda\cap f^{n_1}(S)$ and the integers
$k=0$ and $l=n_1$. In particular, for any $z\in\Lambda\cap S$ one gets the announced inequality.
\end{proof}
We now prove that the return $S$ given by the claim~\ref{c.select}
satisfies the conclusions of the lemma~\ref{l.unbounded2}.
It is enough to consider a point $x\in S\cap \Lambda$ and $n\geq 1$ such that
$f^{-n}(x)\in S$ and $f^{-k}(x)\notin S$ for $0<k<n$. By lemma~\ref{l.adapted}, the rectangle $S$ is adapted,
hence $f^{-k}(W^{cu}_S(x))$ is disjoint from $S$ for any $0<k<n$.
One deduces by proposition~\ref{p.summability} that
$$\sum_{k=0}^{n-1} \ell(f^{-k}(W^{cu}_{S}(x)))<K_0.$$
By our choice of $S$ one has
$$\sum_{j=0}^{n_1} \ell(f^{j}(W^{cu}_{S}(f^{-n}(x))))<K_1 \ell(f^{n_1}(W^{cu}_{S}(f^{-n}(x)))=
K_1\ell(W^{cu}_{R}(f^{n_1-n}(x))).$$
In particular, the property (D) gives
$$\|Df^{n_1-n}_{|E^{cu}}(x)\|\leq \frac{\ell(f^{n_1-n}(W^{cu}_S(x)))}{\ell(W^{cu}_S(x))}\; \exp\left(\Delta_2\;K_0\right).$$
$$\|Df^{-n_1}_{|E^{cu}}(f^{n_1-n}(x))\|\leq \frac{\ell(W^{cu}_S(f^{-n}(x)))}{\ell(W^{cu}_R(f^{n_1-n}(x)))}\; \exp\left(\Delta_2\;K_1\;l\right).$$
Since $S$ is an $N$-contracting return of $R$, the lemma~\ref{l.distortion-holo} gives
$$\frac{\ell(W^{cu}_S(f^{-n}(x)))}{\ell(W^{cu}_S(x))}\leq \Delta_1.$$
We also have
$$\ell(f^{n_1-n}(W^{cu}_S(x)))=\ell(f^{-n}(W^{cu}_R(f^{n_1}(x))))<\delta.$$
Combining these inequalities, one gets the required estimate:
$$\|Df^{-n}_{|E^{cu}}(f(x))\|\leq \frac{\delta}{L\; \Delta_1}\exp\left(\Delta_2\;(K_0+K_1\;l)\right)<1/2.$$
\end{proof}
\paragraph{c) Adapted rectangles with holes.} We obtain a stronger summability result for holes.
This is similar to~\cite[lemma 3.7.7]{PS1}.
\begin{lemma}\label{l.sum-hole}
Let $R$ be an adapted rectangle and $S$ be a hole of $R$ with aperiodic boundary.
Then, there exists $K_3>0$ such that for any $x\in R\cap \Lambda$, we have
$$\sum_{k\geq 0} \ell(f^{-k}(W^{cu}_{S}(x)))<K_3.$$
\end{lemma}
\begin{proof}
By lemma~\ref{l.hole}, $S$ is an adapted rectangle.
Let $(n_i)$ be the sequence of returns of $W^{cu}_S(x)$ into $S$, that is the integers such that
$f^{-n_i}(W^{cu}_S(x))\subset S$.
Let us consider two consecutive returns $n_i,n_{i+1}$. By the proposition~\ref{p.summability}, we have
$$\sum_{k=n_i}^{n_{i+1}}\ell((f^{-k}(W^{cu}_{S}(x)))<K(S).$$
It is enough to bound uniformly the sum $\sum_{i\geq 0} \ell(f^{-n_i}(W^{cu}_{S}(x)))$.
From (D) we have
$$\frac{\ell(f^{-n_{i+1}}(W^{cu}_{S}(x)))}{\ell(f^{-n_{i}}(W^{cu}_{S}(x)))}
\leq \frac{\ell(f^{n_i-n_{i+1}}(W^{cu}_{S}(f^{-n_i}(x))))}{\ell(W^{cu}_{S}(f^{-n_i}(x)))}
\exp(\Delta_2K(S)).$$
By lemma~\ref{l.hole},
there exists $N\geq 1$ such that for $n_i\geq N$ the difference $n_{i+1}-n_i$
is large and by lemma~\ref{l.top-expansion}, the length $\ell(f^{n_i-n_{i+1}}(W^{cu}_S(f^{-n_i}(x))))$ is smaller than $\ell(W^{cu}_S(f^{-n_i}(x)))\exp(-\Delta_2K(S))/2$.
In particular $\ell(f^{-n_{i+1}}(W^{cu}_{S}(x)))$
is smaller than $\ell(f^{-n_{i}}(W^{cu}_{S}(x)))/2$ for any $n_i\geq N$.
The corollary follows.
\end{proof}
\medskip
It remains to conclude proposition~\ref{propoE^{cu}ishyp} in the second case
of the proposition~\ref{p.construction}.
\begin{lemma}\label{l.hole-conclusion}
For any adapted rectangle $R$ having a hole $S$ with aperiodic boundary and such that
$R\cap \Lambda\neq \emptyset$, there exists a non-empty open subset
$B\subset R$ of $\Lambda$ such that property (E) holds.
\end{lemma}
\begin{proof}
Let $K_3,\Delta_2$ be the constants given by lemma~\ref{l.sum-hole}
and the property (D)
and let $\eta$ be the constant given by the property (D') and associated to $C=K_3$.
Since $R\cap \Lambda$ is non-empty $S$ is proper in $R$.
Up to exchange the boundaries $x^-,x^+$ of $R$ and $S$, one deduces
by lemma~\ref{l.hole} that
there exists a sequence $(x_n)$ in $R\cap \Lambda$ such that
$d(x_n,x^-_{n,S})$ goes to zero as $n\to +\infty$.
Since $\Lambda$ is transitive and is not a single periodic orbit,
one can assume that the points $x_n$ are not periodic.
We fix such a point $x$ so that
$d(x,x^-_{S})<\eta\; \ell(W^{cu}_S(x))$.
Let $L$ be a lower bound for the length of the curves $W^{cu}_S(z)$ of $S$
and let
$\delta=L\exp(-2\;\Delta_2\;K_3)/3$. We choose $\tau$ large enough such that for any
$z\in \Lambda$ the curves $f^{-n}(W^{cu}_{loc}(z))$ for $n\geq \tau$
have a length smaller than $\delta$.
Since $x$ is not periodic, one can find a small neighborhood $B$
of $x$ in $\Lambda$ such that $B$ is disjoint from its first $\tau$ iterates
and for any $y\in B$ one has $d(y,y^-_{S})<\eta\; \ell(W^{cu}_S(y))$.
For any return $f^{-n}(y)$ in $B$ one has $n\geq \tau$.
Lemma~\ref{l.sum-hole} and property (D') thus give:
\begin{equation*}
\begin{split}
\|Df^{-n}_{|E^{cu}}(y)\|&\leq \frac{\ell(f^{-n}(W^{cu}_S(y)))}{\ell(W^{cu}_S(y))}\; \exp\left(2\; \Delta_2\sum_{k=0}^{n-1}\ell(f^{-k}(W^{cu}_S(y)))\right)\\
&\leq \frac{\delta}{L}\exp(2\;\Delta_2\; K_3)<1/2.
\end{split}
\end{equation*}
\end{proof}
\bigskip
The proof of the proposition~\ref{propoE^{cu}ishyp} is now complete.
\section{Continuation of chain-hyperbolic homoclinic classes}\label{s.continuation}
Let $H(p)$ be a homoclinic class of a diffeomorphism $f$ and
assume that it is a chain-recurrence class endowed with
a partially hyperbolic structure $E^s\oplus E^c\oplus E^u$ such that $\dim(E^c)=1$ and
the bundle $E^{cs}=E^s\oplus E^c$ is thin trapped.
By lemma~\ref{l.robustness}, the homoclinic class $H(p_g)$ is still chain-hyperbolic for the
diffeomorphisms $g$ close to $f$. We explain here, how in certain sense, the points in $H(p)$
can be continued in $H(p_g)$. If $f$ is far from strong homoclinic intersections, proposition~\ref{p.continuation}
shows that the points of $H(p_g)$
are in correspondence with the continuation of the points of $H(p)$ up to some identifications
and blow-ups in the central direction (that can be compared with the blow-up of an Anosov diffeomorphism during
the construction of a ``derived from Anosov'' map).
\subsection{Preliminary constructions}
\paragraph{Local central orientation.}
The bundle $E^c$ on $H(p)$ is one-dimensional and locally trivial.
Moreover it depends continuously on the dynamics $f$.
One deduces that for any $g,g'$ close to $f$, the orientations of $E^c_{g,x}$ and $E^c_{g',x'}$
for two points $x\in H(p_g)$ and $x'\in H(p_{g'})$ can be compared provided $x$ and $x'$ are close
(say at distance less than $\varepsilon$).
To make this precise, one can cover a neighborhood of $H(p)$ by
a finite number of open sets $U_i$ endowed with non-singular one-forms $\alpha_i$
such that $\alpha_i$ never vanishes on the bundle $E^c$.
Two close points $x,x'$ belong to a same open set $U_i$.
Two orientations on $E^c_{g,x}$ and $E^c_{g',x'}$ match if they both coincide with the class
of $\alpha_i$ or the class of $-\alpha_i$. If $x,x'$ are close enough, this does not depend on the open set $U_i$
containing $\{x,x'\}$. If one considers another collection of pairs $(U'_i,\alpha'_i)$,
the orientations on $E^c_{g,x}$ and $E^c_{g',x'}$ still match if the distance between $x$ and $x'$
is small and $g$ is close enough to $f$.
\paragraph{Plaque families.}
In the following one fixes $\delta>0$ small which is a lower bound for the modulus of the Lyapunov exponents
of $p_g$ for $g$ close to $f$.
One chooses some continuous collections of plaque families $(\cW^{cs}_{g})$
for the diffeomorphisms $g$ close to $f$ as given by lemma~\ref{l.robustness}.
Since $E^{cs}$ is thin trapped, the plaques may be chosen with a small diameter
so that the properties stated in lemma~\ref{l.uniqueness-coherence} hold.
Also, by lemma~\ref{l.largestable}, for $g$ that is $C^1$-close to $f$
and for any periodic point $q\in H(p_g)$
whose Lyapunov exponents along $E^{cs}$ are smaller than $-\delta/2$,
the plaque $\cW^{cs}_{g,q}$ is contained in the stable set of $q$.
One will consider local manifolds $W^ {ss}_{g,loc}(x)$ and $W^ {u}_{g,loc}(x)$
for $x\in H(p_g)$ with a small diameter so that $W^ {u}_{g,loc}(x)$ intersects a plaque
$\cW^ {cs}_{g,y}$ in at most one point and the intersection is always transversal.
\paragraph{Shadowing.}
Then, one chooses $\varepsilon>3\varepsilon'>0$ so that the following lemma holds and
so that for any $g,g'$ close to $f$ and any $x\in H(p_g)$ and $y\in H(p_{g'})$
satisfying $d(x,y)<\varepsilon$
the local manifold $W^{u}_{g,loc}(x)$ intersects $\cW^{cs}_{g',y}$.
\begin{lemma}\label{l.expansivity}
There exists $\varepsilon>3\varepsilon'>0$ small such that any diffeomorphisms $g,g'$ close to $f$ satisfy:
\begin{itemize}
\item[--] if $x,y\in H(p_g)$ are two points such that the forward orbit of $x$ is $\varepsilon$-shadowed
by the forward orbit of $y$, then $y\in \cW^{cs}_{g,x}$;
\item[--] if $x,y\in H(p_g)$ are two points $\varepsilon'$-close
such that $y$ belongs to $\cW^{cs}_{g,x}$, then
the forward orbit of $x$ is $\frac \varepsilon 3$-shadowed by the forward orbit of $y$;
\item[--] for any periodic orbit $O\subset H(p_g)$ of $g$
whose central Lyapunov exponent is smaller that $-\delta$,
any periodic orbit of $g'$ that $\varepsilon$-shadows $O$
also $\varepsilon'$-shadows $O$, has a central Lyapunov exponent smaller than $\delta/2$
and is homoclinically related to $p_{g'}$; moreover any point $x\in H(p_g)$ whose backward orbit $\varepsilon$-shadows $O$ belongs to the unstable manifold of $O$.
\end{itemize}
\end{lemma}
\begin{proof}
We prove the first item. Let us consider the intersection point $z$ between $\cW^{cs}_{g,x}$ and $W^u_{g,loc}(y)$.
By uniform local invariance of $\cW^{cs}_{g}$, one checks inductively that
the point $g^n(z)$ is the intersection point between
$\cW^{cs}_{g,g^n(x)}$ and $W^u_{g,loc}(g^n(y))$ for $n\geq 0$.
If $z\neq y$, since $z$ and $y$ belong to the same unstable leaf, the distance
$d(g^n(z),g^n(y))$ increases exponentially and becomes much larger than $\varepsilon$, contradicting
that the distance between $g^n(x)$ and $g^n(y)$ is bounded by $\varepsilon$.
One deduces that $y=z$, hence $y$ belongs to $\cW^{cs}_{g,x}$.
Now we choose $\varepsilon'\ll \varepsilon$ and prove the second item.
Since $E^{cs}$ and $E^{u}$ are thin trapped by $f$ and $f^{-1}$,
lemma~\ref{l.robustness} associates some continuous trapped plaque families $\widehat \cW^{cs}_g$ and $\widehat \cW^{cu}_g$
over $H(p_g)$ for $g$ close to $f$ with diameter smaller than $\varepsilon/3$.
From lemma~\ref{l.uniqueness-coherence} if $\varepsilon'$ is small enough, then
for any $x,y\in H(p_g)$ such that $y\in \cW^{cs}_{g,x}$ and $d(x,y)<\varepsilon'$, the point $y$ belongs to $\widehat \cW^{cs}_{g,x}$. By the trapping property, $g^n(y)$ belongs to $\widehat \cW^{cs}_{g,g^n(x)}$
for any $n\geq 0$, hence $d(g^n(x),g^n(y))<\varepsilon/3$ as required.
We then prove the properties of the third item.
We first note that if $g,g'$ are close to $f$ and $\varepsilon$ is small enough, then
any periodic orbit $O'$ of $g'$ that $\varepsilon$-shadows a periodic orbit $O$ of $H(p_g)$
still has a partial hyperbolic structure and has Lyapunov exponents close to those of $O$.
This proves that the central Lyapunov exponent of $O'$ is smaller than $-\delta/2$.
One deduces from lemma~\ref{l.largestable} that for some point $q'\in O'$
the stable manifold of $q$ has uniform size inside $\cW^{cs}_{g',q'}$.
From lemma~\ref{l.contper}, there exists a dense set of periodic points $x\in H(p_{g'})$
whose stable manifold has a uniform size.
If $\varepsilon$ is small enough
and $g,g'$ close enough to $f$, one thus deduces that
$q'$ is close to a point of $H(p_{g'})$.
From the uniformity of the invariant manifolds, we deduce that
the stable and unstable manifolds of $q'$ intersect the stable and unstable manifolds of a hyperbolic
periodic orbit homoclinically related to $p_{g'}$. In particular, $O'$ is homoclinically related to $p_{g'}$.
Let us consider again, as given by lemma~\ref{l.robustness},
some continuous plaque families $\widehat \cW^{cs}_h$ and $\widehat \cW^{cu}_h$
over $H(p_h)$ for $h$ close to $f$ with diameter much smaller than $\varepsilon'$.
From lemma~\ref{l.uniqueness-coherence} there exists $\rho>0$ such that
for any $g$ close to $f$ and any $x\in H(p_g)$, the ball $B(x,\rho)$
in $\cW^{cs}_{g,x}$ is contained in $\widehat \cW^{cs}_{g,x}$.
From the trapping property,
the following holds for $g,g'$ close to $f$:
if $x\in H(p_g)$ and $y\in H(p_{g'})$ such that $d(x,y)<\varepsilon$
satisfy that $W^u_{g',loc}(y)$ intersects $\widehat \cW^{cs}_{g,x}$, then the same holds for $g(x)$ and $g'(y)$.
Using the estimate~(\ref{e.unif}) in the proof of lemma~\ref{l.largestable},
there exists a uniform integer $N\geq 1$
and an iterate $q\in O$ such that $g^N(\cW^{cs}_{g,q})$ has radius smaller than $\rho$, hence is contained
in $\widehat \cW^{cs}_{g,g^N(q)}$. Let us choose $q'\in O'$ such that $d(g'{}^n(q'),g^n(q))<\varepsilon$
for each $n\in \ZZ$.
Provided that $g,g'$ have been chosen close enough to $f$, the intersection $z_n$
between $W^u_{g',loc}(g'{}^n(q'))$ and $\cW^{cs}_{g,g^n(q)}$ for $0\leq n\leq N$ are close
to the $N$ first iterates of $z_0$ under $g$, hence $z_N$ is contained in $\widehat \cW^{cs}_{g,g^N(q)}$.
By our construction, one deduces that $W^u_{g',loc}(g'{}^n(q'))$
intersects $\widehat \cW^{cs}_{g,g^n(q)}$ for any $n\geq N$, hence any $n\in \ZZ$.
With the same argument, $\cW^{cs}_{g',g'{}^n(q')}$
intersects $\widehat \cW^{cu}_{g,g^n(q)}$, for any $n\in \ZZ$.
Since the diameter of the plaques $\widehat \cW^{cu}$ and
$\widehat \cW^{cs}$ is much smaller than $\varepsilon'$, one deduces that $g^{n}(q)$ and $g'{}^n(q')$
are at distance smaller than $\varepsilon'$. We have proved that $O$ is $\varepsilon'$-shadowed by $O'$.
Let us now consider a point $x\in H(p_g)$ whose backward orbits $\varepsilon$-shadows
the backward orbit of a point $q\in O$.
Let us introduce for each $n\geq 0$ the intersection point $z_n$
between $W^u_{g,loc}(g^{-n}(x))$ and $\cW^{cs}_{g,g^{-n}(q)}$.
By construction one has $g(z_{n+1})=z_n$ and in particular $z_0$
is contained in the intersection of the $g^n(\cW^{cs}_{g,g^{-n}(q)})$.
By assumption $\cW^{cs}_{g,g^{-n}(q)}$ is contained in the stable manifold of
$g^{-n}(q)$. This proves that $z_0$ coincides with $q$. As a consequence
$z_0$ belongs to $W^u_{g,loc}(q)$.
\end{proof}
\subsection{Continuation of uniform periodic points}
The periodic points with uniform Lyapunov exponents have a uniform hyperbolic continuation.
\begin{lemma}\label{l.cont-periodic}
There exists a simply connected open neighborhood $\cU\subset\operatorname{Diff}^1(M)$ of $f$ such that:
\begin{itemize}
\item[--] The hyperbolic continuation of $p$ exists for any $g\in \cU$
and the class $H(p_g)$ is chain-hyperbolic.
\item[--] For any $g\in \cU$ and any periodic orbit $O\subset H(p_g)$ of $g$
whose central Lyapunov exponent is smaller than $-\delta$,
the hyperbolic continuation $O_{g'}$ of $O$ exists for any $g'\in \cU$
and is homoclinically related to $p_{g'}$.
Moreover its central Lyapunov exponent is still smaller than $-\delta/2$,
and $O_{g'}$ is $\frac \varepsilon 3$-shadowed by $O$.
\end{itemize}
\end{lemma}
\begin{proof}
Lemma~\ref{l.robustness} gives the existence of an open set $\cU$ satisfying the first item.
Let us consider a path $(\gamma_t)_{t\in [0,1]}$ in $\cU$ between $g$ and $g'$
and the maximal interval $I$ containing $0$ where the hyperbolic continuation $O_t$ of $O$
is defined and $\varepsilon/2$-shadows $O$.
If $I=[0,t_0)$ with $t_0<1$, one can consider a periodic orbit $O_{t_0}$ for $g_{t_0}$
that is the limit of a sequence of orbit $O_t$ for $t<t_0$. By construction $O_{t_0}$
$\varepsilon$-shadows $O$, hence $O_{t_0}$ has a central Lyapunov exponent smaller than $-\delta/2$
and also $\varepsilon'$-shadows $O$ by lemma~\ref{l.expansivity}.
Since $\varepsilon'<\varepsilon/3$, we have contradicted the definition of $t_0$.
Hence, the orbit $O$ has a hyperbolic continuation $O_{g'}$ for $g'$. Since $\cU$ is simply connected,
this continuation is unique. We have shown that $O$ is $\varepsilon'$-shadowed by $O_{g'}$,
hence by lemma~\ref{l.expansivity}, $O_{g'}$ is homoclinically related to $p_{g'}$, has
a central Lyapunov exponent smaller than $-\delta/2$. Since $\varepsilon'<\varepsilon/3$,
all the properties stated in the second item are satisfied.
\end{proof}
\medskip
This justifies the following definition.
\begin{definition} Let us
denotes with $\cP$, the set of hyperbolic periodic points $q\in H(p)$ homoclinically related to the orbit of $p$
whose continuation $q_g$ exists for any diffeomorphism
$g\in\cU$ and such that for some $g\in \cU$ the central Lyapunov exponent of $q_g$
is smaller than $-\delta$.
\end{definition}
\noindent
Since for any $g\in \cU$ the central Lyapunov exponents of $p_g$ is smaller than $-\delta$, there exists a dense set of periodic points in $H(p_g)$
whose central Lyapunov exponent is smaller than $-\delta$.
By lemma~\ref{l.cont-periodic}, one deduces that the continuations $q_g$ of points in $q\in\cP$ are dense in $H(p_g)$.
Note also that by lemma~\ref{l.cont-periodic} the central Lyapunov exponent of $q_g$ for $q\in \cP$
is smaller than $-\delta/2$; hence the plaque $\cW^{cs}_{g,q_g}$ is contained in $W^s_{g}(q_g)$.
\subsection{Pointwise continuation of $H(p)$}
\begin{definition}\label{d.continuation}
For any $g,g'\in \cU$, one says that two points $x\in H(p_g)$
and $x' \in H(p_{g'})$ \emph{have the same continuation} if there exists a sequence of
hyperbolic periodic points $(p_n)$ in $\cP$ such that
$(p_{n,g})$ and $(p_{n,g'})$ converge toward $x$ and $x'$ respectively.
\end{definition}
\noindent
This implies that
$g^k(x)$ and $g'^k(x')$ have the same continuation for each $k\in \ZZ$.
\medskip
By compactness and density of the points $q_g$ with $q\in \cP$,
one sees that, for any $g,g'\in \cU$, any point $x\in H(p_g)$
has the same continuation as some $x'\in H(p_{g'})$.
In general $x'$ is not unique.
The following implies that if $x'_1,x'_2\in H(p_{g'})$ have the same continuation as $x$,
then $x'_2$ belongs to $\cW^{cs}_{g',x'_1}$.
\begin{lemma}\label{l.cont-central}
For any $g,g'\in \cU$, let us consider $x\in H(p_g)$
and $x'\in H(p_{g'})$ such that $x$ and $x'$ have the same continuation.
Then, the orbits of $x$ by $g$ is $\frac \varepsilon 3$-shadowed by the orbit of $x'$ by $g'$.
As a consequence, if $x_1,x_2\in H(p_g)$ are $\varepsilon'$-close and satisfy
$x_2\in \cW^{cs}_{g,x_1}$, then for any $x'_1,x'_2\in H(p_{g'})$ such that $x_i,x'_i$ have the same continuation
for $i=1,2$, one still has $x'_2\in \cW^{cs}_{g',x'_1}$.
\end{lemma}
\begin{proof}
Let us consider a sequence $(p_n) \in \cP$ whose continuations $(p_{n,g})$, $(p_{n,g'})$
for $g$ and $g'$ converges toward $x$ and $x'$ respectively.
From lemma~\ref{l.cont-periodic}, the orbit of $(p_{n,g})$ by $g$ is
$\frac \varepsilon 3$-shadowed by the orbit of $(p_{n,g'})$ by $g'$. Taking the limit, one deduces that
the orbit of $x$ by $g$ is $\frac \varepsilon 3$-shadowed by the orbit of $x'$ by $g'$.
If $x_1,x_2\in H(p_g)$ are $\varepsilon'$-close and satisfy
$x_2\in \cW^{cs}_{g,x_1}$, by lemma~\ref{l.expansivity} the forward orbit of $x_2$ is
$\frac \varepsilon 3$-shadowed by the forward orbit of $x_1$.
By the first part of the lemma, one deduces that
for any $x'_1,x'_2\in H(p_{g'})$ such that $x_i,x'_i$ for $i=1,2$ have the same continuation, then
the forward orbit of $x'_1$ by $g'$
is $\varepsilon$-shadowed by the forward orbit of $x'_2$ by $g'$.
By lemma~\ref{l.expansivity}, this implies that $x'_2\in \cW^{cs}_{g',x'_1}$.
\end{proof}
One then shows that
if $x$ is a hyperbolic periodic point in $\cP$,
then $x'$ coincides with its hyperbolic continuation (hence is unique).
This is also true for the unstable manifold of points in $\cP$.
\begin{lemma}\label{l.cont-unstable}
For any $g\in \cU$, let $q_g$ be the hyperbolic continuation of some point $q\in \cP$
and let us consider some point $x\in W^u(q_g)\cap H(p_g)$.
Then, for any $g'\in \cU$, there exists a unique point $x'\in H(p_{g'})$
which has the same continuation as $x$; moreover $x'$ belongs to $W^u(q_{g'})$ and varies continuously with $g'$.
In particular the hyperbolic continuation $q_{g'}$ of $q_g$ is the unique point in $H(p_{g'})$
such that $q_g$ and $q_{g'}$ have the same continuation (in the sense of the definition~\ref{d.continuation}).
\end{lemma}
\begin{proof}
Let us consider any $x'\in H(p_{g'})$ which has the same continuation as $x$.
From lemma~\ref{l.cont-periodic}, the orbit of $q_{g'}$ by $g'$ is $\frac \varepsilon 3$-shadowed by
the orbit of $q_g$ by $g$ and from lemma~\ref{l.cont-central}, the orbit of $x$ by $g$ is $\frac \varepsilon 3$-shadowed by
the orbit of $x'$ by $g'$.
There exists $N\geq 1$ such that the backward orbit of $g^{-N}(x)$ is
$\frac \varepsilon 3$-shadowed by the backward orbit of $g^{-N}(q_g)$.
Hence the backward orbit of ${g'}^{-N}(x')$ is $\varepsilon$-shadowed by
the backward orbit of ${g'}^{-N}(q_{g'})$. By lemma~\ref{l.expansivity},
$x$ belongs to the unstable manifold of $g^{-N}(q_g)$.
It remains to prove that $x'$ is the only point in $H(p_{g'})$ which has the same
continuation as $x\in H(p_g)$.
Let $x'_1,x'_2\in H(p_{g'})$ be two points that have the same continuation as $x\in H(p_g)$.
By lemma~\ref{l.cont-central} their orbits under $g'$ both $\frac \varepsilon 3$-shadow the orbit of $x$ under $g$.
By lemma~\ref{l.expansivity},
${g'}^n(x'_2)$ belongs to $\cW^{cs}_{g',{g'}^n(x'_1)}$ for each $n\in \ZZ$.
When $n$ goes to $-\infty$, the points ${g'}^n(x'_2)$ and ${g'}^n(x'_1)$
are contained in a small local unstable manifold of the orbit of $q_{g'}$.
Since the plaques $W^u_{loc}$ and $\cW^{cs}$ intersect in at most one point,
this implies that $x'_1=x'_2$.
Let us denote by $x_{g'}$ the point which has the same continuation as $x$.
In order to prove the continuity of the map $g'\mapsto x_{g'}$,
one considers any limit point $x'$ of points $x_{g'}$ when $g'$ goes to $g$.
As before, the orbit of $x$ by $g$ is $\varepsilon$-shadowed by the orbit of $x'$,
so that ${g}^n(x')$ belongs to the unstable manifold of the orbit of $q$ and to
$\cW^{cs}_{g,{g}^n(x)}$ for each $n\in \ZZ$. This implies $x=x'$.
\end{proof}
\smallskip
\begin{remark}\label{r.cont-unstable}
Lemma~\ref{l.cont-unstable} also implies that definition~\ref{d.continuation} does not depend on the choice
of $\delta$ and $\cU$. Indeed, if one considers $\widetilde \delta\in (0,\delta)$ and $\widetilde \cU\subset \cU$
another neighborhood of $f$, then one gets two sets of periodic points
$\cP\subset \widetilde \cP$.
Let us consider $g,g'\in \widetilde{\cU}$ and two points $x\in H(p_g)$,
$x' \in H(p_{g'})$ which have the same continuation on $\widetilde \cU$ with respect to $\widetilde \cP$;
we claim that they also have the same continuation with respect to $\cP$.
Indeed one considers a sequence $(\widetilde {p}_n)$ in $\widetilde \cP$
such that $(\widetilde {p}_{n,g})$ converges toward $x$.
Then, for each $n$ there exists $p_n\in \cP$ such that
$p_{n,g}$ is close to $\widetilde {p}_{n,g}$. By lemma~\ref{l.cont-unstable},
$p_{n,g'}$ is close to $\widetilde {p}_{n,g'}$, hence one can obtain a sequence
$(p_n)$ in $\cP$ such that $(p_{n,g})$ converges toward $x$ and $(p_{n,g'})$ converges toward $x'$,
as wanted.
\end{remark}
\subsection{Continuations far from strong homoclinic intersections}
For $g\in \cU$ we define $\widetilde{H(p_g)}$
as the set of pairs $\tilde x=(x,\sigma)$ where $x\in H(p_g)$ and $\sigma$
is an orientation of $E^c_{g,x}$, such that
$x$ is accumulated in $H(p_g)\cap \cW^{cs,+}_{g,x}$ where $\cW^{cs,+}_{g,x}$
is the component of $\cW^{cs}_{g,x}\setminus W^{ss}_{loc}(x)$ determined by the orientation $\sigma$
as introduced in section~\ref{ss.one-codim}.
One can view $\widetilde{H(p_g)}$ as a subset of the unitary bundle associated to $E^c_g$ over $H(p_g)$.
The dynamics of $g$ can thus be lifted to $\widetilde{H(p_g)}$ and defines a map $\tilde g$.
One also defines the projection $\pi_g\colon \widetilde{H(p_g)} \to H(p_g)$ such that $\pi_g(x,\sigma)=x$.
\begin{proposition}\label{p.continuation}
Let $H(p)$ be a homoclinic class of a diffeomorphism $f\in \operatorname{Diff}^r(M)$ such that
\begin{itemize}
\item[--] it is not a periodic orbit,
\item[--] is a chain-recurrence class endowed with
a partially hyperbolic structure $E^s\oplus E^c\oplus E^u$ such that $\dim(E^c)=1$ and
$E^{cs}=E^s\oplus E^c$ is thin trapped.
\end{itemize}
In a $C^1$-small neighborhood $\cU$ of $f$ in $\operatorname{Diff}^1(M)$ we consider
a $C^r$-open connected set $\cV\subset \cU$ such that there is no diffeomorphism $g\in \cV$
whose homoclinic class $H(p_g)$ has a strong homoclinic intersection.
Then, for each $g,g'\in \cV$, the following holds:
\begin{description}
\item[a)] \emph{(Lifting).} The map $\pi_g\colon \widetilde{H(p_g)}\to H(p_g)$ is surjective
and semi-conjugates $\tilde g$ to $g$.
\item[b)] \emph{(Continuation of the lifting).}
For any $\tilde x_g=(x_g,\sigma)\in \widetilde{H(p_g)}$,
there is a unique $\tilde x_{g'}=(x_{g'},\sigma')\in \widetilde{H(p_{g'})}$
such that $x_g=\pi_g(\tilde x_g)$ and $x_{g'}=\pi_{g'}(\tilde x_{g'})$ have the same continuation
and such that the orientations $\sigma$ on $E^c_{g,x_g}$ and $\sigma'$ on $E^c_{g',x_{g'}}$ match;
this defines a bijection $\Phi_{g,g'}\colon \widetilde{H(p_g)}\to \widetilde{H(p_{g'})}$.
We denote $\Phi_g:=\Phi_{f,g}$.
\item[c)] \emph{(Continuation of the projection).}
For any $x_g\in H(p_g)$ and $x_{g'}\in H(p_{g'})$ having the same continuation,
there exists $\tilde x\in \widetilde{H(p)}$ such that $\pi_g(\Phi_g(\tilde x))=x_g$
and $\pi_{g'}(\Phi_{g'}(\tilde x))=x_{g'}$.
\end{description}
\end{proposition}
\begin{remarks}\label{r.continuity}
One may consider on $\widetilde{H(p_g)}$ the topology induced by $E^c_g$.
This set is in general not compact since a sequence of points $x_n\in H(p_g)$ that are accumulated
in $H(p_g)\cap \cW^{cs,+}_{g,x_n}$ may converge toward a point $x\in H(p_g)$ which is not accumulated
in $\cW^{cs,+}_{g,x}$.
One can show however that the map $(g,\tilde x)\mapsto \Phi_g(\tilde x)$ is semi-continuous.
\end{remarks}
The next lemma is used in the proof of the proposition~\ref{p.continuation} and of lemma~\ref{l.continuite}.
\begin{lemma}\label{l.ordering}
Let us consider $q_1,q_2\in \cP$ and $g,g'\in \cV$ such that
$d(q_{1,g},q_{2,g})<\varepsilon/3$.
If $W^u_{g,loc}(q_{1,g})$ intersects $\cW^{cs,+}_{g,q_{2,g}}$, then
$W^u_{g',loc}(q_{1,g'})$ does not intersect $\cW^{cs,-}_{g',q_{2,g'}}$.
\end{lemma}
\begin{proof}
By lemma~\ref{l.cont-periodic} and our choice of $\varepsilon$,
one has $d(q_{1,h},q_{2,h})<\varepsilon$ for any $h\in \cU$, hence
$W^u_{h,loc}(q_{1,h})$ intersects $\cW^{cs}_{h,q_{2,h}}$.
Now, $W^{ss}_{h,loc}(q_{2,h})$ is one-codimensional in $\cW^{cs}_{h,q_{2,h}}$
and varies continuously with $h$.
Let us assume that $W^u_{g,loc}(q_{1,g})$ intersects $\cW^{cs,+}_{g,q_{2,g}}$ and that
$W^u_{g',loc}(q_{1,g'})$ intersects $\cW^{cs,-}_{g',q_{2,g'}}$.
By connectedness of $\cV$,
one deduces that for some $h_0\in \cV$ the local manifolds $W^u_{h_0,loc}(q_{1,h_0})$
and $W^{ss}_{h_0,loc}(q_{2,h_0})\setminus \{q_{1,h_0}\}$ intersect.
By using lemma~\ref{joint-int-easy} one gets a diffeomorphism $h\in \cV$ having a strong
homoclinic intersection in $H(p_h)$, giving a contradiction.
\end{proof}
\begin{lemma}\label{l.continuite}
Under the setting of proposition~\ref{p.continuation},
if $(g_n)$ converges in $\cV$ toward $g$ and
$(\tilde x_n)$ toward $\tilde x$ in $\widetilde {H(p)}$, then
any limit $\bar x$ of $(\Phi_{g_n}(\tilde x_n))$ satisfies
$\pi_g(\bar x)\in \cW^{cs}_{g,x_g}\setminus \cW^{cs,+}_{g,x_g}$ where $x_g=\pi_g(\tilde x)$.
\end{lemma}
\begin{proof}
Let us assume by contradiction that $\bar x$ belongs to $\cW^{cs,+}_{g,x_g}$.
There exists a sequence $(p_n)$ in $\cP$ that converges toward $x_f=\pi_f(\tilde x)$
such that $W^u_{loc}(p_n)$ intersects $\cW^{cs,+}_{x_f}$
and $(p_{n,g_n})$ converges towards $\bar x$.
By proposition~\ref{p.continuation}, the sequence $(p_{n,g})$ converges toward $x_g$.
Hence, one can consider $n$ large such that $p_{n,g}$ is close to $x_g$.
By continuity of the map $h\mapsto p_{n,h}$, the point $p_{n,h}$ is still close to $x_h$ for a diffeomorphisms $h$ nearby. For $m$ large enough,
$p_{n,g_m}$ is close to $x_g$ and $p_{m,g_m}$ is close to $\bar x$.
One deduces that $W^u_{loc,g_m}(p_{n,g_m})$ meets $\cW^{cs,-}_{g_m,p_{m,g_m}}$.
On the other hand, since $W^u_{loc}(p_n)$ meets $\cW^{cs,+}_{x_f}$,
the local manifold $W^u_{loc}(p_{n,g})$ meets $\cW^{cs,+}_{g,p_{m}}$.
The lemma~\ref{l.ordering} below contradicts our assumption that $f$ is far from homoclinic intersections.
\end{proof}
\begin{proof}[Proof of proposition~\ref{p.continuation}]
We introduce the open set $\cU$ and the collection of periodic points $\cP$ as
in the previous sections.
The item a) of the proposition is a direct consequence of lemmas~\ref{l.NGSHI}
and~\ref{joint-int-easy}.
The item b) is first proved in the case $x_g$ is the hyperbolic continuation $q_g$
of a periodic point $q\in \cP$. In this case there is only one possible continuation $x_{g'}$.
We are thus reduced to prove.
\medskip
\noindent{\it Claim 1.
Consider any periodic point $q\in \cP$ and an orientation $\sigma$ on $E^c_q$. If
$q_g$ is accumulated by $H(p_g)\cap \cW^{cs,+}_{g,q_g}$ for some $g\in \cV$, then the same
holds for any $g$.}
\begin{proof}
Let us consider $g\in \cV$ such that $q_g$ is accumulated by $H(p_g)\cap \cW^{cs,+}_{g,q_g}$.
In particular, there exists a sequence $(p_n)$ in $\cP$ such that $(p_{n,g})$ converges toward $q_g$
and $W^u_{g,loc}(p_{n,g})$ intersects $\cW^{cs,+}_{g,q_g}$.
By lemma~\ref{l.cont-unstable}, the sequence $(p_{n,g'})$ converges toward $q_{g'}$.
Moreover $W^u_{g',loc}(p_{n,g'})$ does not intersect $W^{ss}_{g',loc}(q_{g'})$
since this would contradict our assumptions
by lemma~\ref{joint-int-easy}.
Also by lemma~\ref{l.ordering}, $W^u_{g',loc}(p_{n,g'})$ does not intersects $\cW^{cs,-}_{g',q_{g'}}$.
One thus deduces that $W^u_{g',loc}(p_{n,g'})$ intersects $\cW^{cs,+}_{g',q_{g'}}$.
The intersection point belongs to $H(p_{g'})$ by lemma~\ref{l.bracket0}, hence $q_{g'}$ is accumulated
by $H(p_{g'})\cap \cW^{cs,+}_{g',q_{g'}}$.
\end{proof}
\medskip
We now prove the item b) in the general case.
\medskip
\noindent{\it Claim 2.
Let us consider $x_g\in H(p_g)$ and $x_{g'}\in H(p_{g'})$ and a sequence $(p_n)$ in $\cP$
such that $(p_{n,g})$ converges toward $x_g$ and $(p_{n,g'})$ converges toward $x_{g'}$.
If the local unstable manifolds $W^u_{g,loc}(p_{n,g})$ intersect $\cW^{cs,+}_{g,x_g}$,
then there exists another sequence $(\bar p_n)$ in $\cP$ having the same properties as $(p_n)$
and which satisfies furthermore that
the local unstable manifolds $W^u_{g',loc}(\bar p_{n,g'})$ intersect $\cW^{cs,+}_{g',x_{g'}}$.}
\begin{proof}
We first remark that each point $(p_{n,g})$, with $n$ large enough,
is accumulated by $H(p_{g})\cap \cW^{cs,+}_{g,p_{n,g}}$.
Indeed, $W^u_{g,loc}(p_{1,g})$ intersects $\cW^{cs,+}_{g,p_{n,g}}$ for $n$ large at some point
$y_n$ which belongs to $H(p_g)$ by lemma~\ref{l.bracket0}.
By lemma~\ref{l.largestable}, $\cW^{cs,+}_{g,p_{n,g}}$ is contained in the stable manifold of $p_{n,g}$,
hence the forward orbit of $y_n$ accumulates the orbit of $p_{n,g}$, proving the announced property.
From claim 1, the points $p_{n,g'}$ are also accumulated by
$H(p_{g'})\cap \cW^{cs,+}_{g',p_{n,g'}}$.
Now we note that $W^u_{g',loc}(p_{n,g'})$ does not intersect $\cW^{cs,-}_{g',x_{g'}}$.
Indeed, if this occurs, one would deduce that for $m\gg n$
the manifold $W^u_{g',loc}(p_{n,g'})$ intersects $\cW^{cs,-}_{g',p_{m,g'}}$
and that $W^u_{g,loc}(p_{n,g})$ intersects $\cW^{cs,+}_{g,p_{m,g}}$.
By lemma~\ref{l.ordering} this would contradict our assumptions.
If $W^u_{g',loc}(p_{n,g'})$ intersects $\cW^{cs,+}_{g',x_{g'}}$ for a subsequence $(\bar p_n)$ of $(p_n)$,
the claim holds. We thus reduced to consider the case
$W^u_{g',loc}(p_{n,g'})$ intersect $W^{ss}_{g',loc}(x_{g'})$. We denote by $z_n$ the intersection.
Since $p_{n,g'}$ is accumulated by $H(p_{g'})\cap \cW^{cs,+}_{g',p_{n,g'}}$,
lemma~\ref{l.integrability} implies that there exists $\bar p_n\in \cP$ such that
\begin{itemize}
\item[--] $\bar p_{n,g'}$ is close to $p_{n,g'}$ (hence $(\bar p_n)$ has the same properties as $(p_n)$),
\item[--] $W^u_{g',loc}(\bar p_{n,g'})$ intersects $\cW^{cs,+}_{g',x_{g'}}$ as announced.
\end{itemize}
\end{proof}
The last claim implies the existence statement of the item b): if $\tilde x_g$ belongs to $\widetilde{H(p_g)}$,
one may approximate the points of $H(p_g)\cap \cW^{cs,+}_{g,x_g}$ by periodic points that are the continuations
for $g$ of points in $\cP$. Hence, there exists a sequence $(p_n)$ in $\cP$ such that
$W^u_{g,loc}(p_{n,g})$ intersects $\cW^{cs,+}_{g,x_g}$ for each $n$. Taking a subsequence, one may also assume that
the points $p_{n,g'}$ converge toward a point $x_{g'}\in H(p_{g'})$. One defines $\tilde x_{g'}=(x_{g'},\sigma)$
such that $\sigma$ is the orientation with matches with the orientation of $\tilde x_g$.
By the previous claim, one can replace the sequence $(p_n)$ by another one $(\bar p_n)$ such that
$(\bar p_{n,g'})$ still converges toward $x_{g'}$ and furthermore
$W^u_{g',loc}(\bar p_{n,g'})$ intersects $\cW^{cs,+}_{g',x_{g'}}$ for each $n$.
The intersection point belongs to $H(p_{g'})$ by lemma~\ref{l.bracket0}, hence $\tilde x_{g'}$ belongs to
$\widetilde {H(p_{g'})}$, as required.
\medskip
\noindent{\it Claim 3.
For any $g_1,g_2\in \cV$,
let us consider $x_1\in H(p_{g_1})$ and $x_2\in H(p_{g_2})$ having the same continuation.
Then, there exist two matching orientations on $E^c_{g_1,x_1}$, $E^c_{g_2,x_2}$
and a sequence $(p_n)$ in $\cP$ such that $(p_{n,g_i})$ converges toward $x_i$ and
the local unstable manifolds $W^u_{g_i,loc}(p_{n,g_i})$ intersects $\cW^{cs,+}_{g_i,x_i}$
for $i=1,2$.}
\begin{proof}
By assumption, there exists a sequence $(p_n^0)$ in $\cP$ such that
$(p^0_{n,g_i})$ converges toward $x_i$ for $i=1,2$.
We first replace $(p_n^0)$ by a sequence $(p_n^1)$ so that $W^u_{g_1,loc}(p_{n,g_1}^1)$ does not intersect $W^{ss}_{g_1,loc}(x_1)$:
if there exists a subsequence of $(p_n^0)$ which has this property, we get the subsequences $(p_n^1)$;
otherwise, one can assume that $W^u_{g_1,loc}(p_{n,g_1}^0)$ intersects $W^{ss}_{g_1,loc}(x_1)$ for each $n\geq 0$.
From lemma~\ref{l.integrability},
there exists $y\in \cW^{cs}_{g,p_{n,g_1}^0}$ arbitrarily close to $p_{n,g_1}^0$
such that its unstable manifold intersects $\cW^{cs}_{x_1}\setminus W^{ss}_{g_1,loc}(x_1)$.
One can approximate $y$ by a point $p_{n,g_1}^1$ with $p_n^1\in \cP$.
Doing this for each $n$,
one gets a required sequence $(p_n^1)$ such that $(p^1_{n,g_1})$ still converges toward $x_1$.
By lemma~\ref{l.cont-unstable}, one can ensure that the sequence $(p^1_{n,g_2})$ converges toward $x_2$.
By choosing the orientations on $E^c_{g_i,x_i}$, one can now assume that
$W^u_{g_1,loc}(p^1_{n,g_1})$ intersects $\cW^{cs,+}_{g_1,x_1}$.
By the claim~2, one can modify again the sequence $(p_n^1)$ and replace it by a sequence $(p_n)$
having the required properties.
\end{proof}
One can now conclude the uniqueness part of the item b).
Let us assume by contradiction that $\tilde x_g\in \widetilde{H(p_g)}$ has two
distinct continuations $\tilde x_{g'}^1,\tilde x_{g'}^2$ in $\widetilde{H(p_{g'})}$ as stated in item b).
By lemma~\ref{l.cont-central}, one may assume that $x_{g'}^1$ belongs to $\cW^{cs,-}_{g',x^2_{g'}}$.
Claim~3 provides us with two sequence $(p_n^i)$, $i=1,2$.
On the one hand $W^u_{g,loc}(p^i_{n,g})$ intersects $\cW^{cs,+}_{g,x_g}$,
hence for $n\geq 1$ and $m\gg n$, $W^u_{g,loc}(p_{n,g}^1)$ intersects
$\cW^{cs,+}_{g,p^2_{m,g'}}$.
On the other hand $x_{g'}^1\in \cW^{cs,-}_{g',x^2_{g'}}$,
hence $W^u_{g',loc}(p^1_{n,g'})$ intersects $\cW^{cs,-}_{g',p^2_{m,g'}}$.
By lemma~\ref{l.ordering}, this contradicts our assumptions.
\medskip
The item c) is a direct consequence from the claim~3.
\end{proof}
\bigskip
\begin{corollary}\label{c.continuation}
Under the assumptions of proposition~\ref{p.continuation},
let us consider $g\in \cV$ and $\tilde x,\tilde y\in \widetilde{H(p_{g})}$ such that
the projections $x=\pi_{g}(\tilde x)$ and $y=\pi_{g}(\tilde y)$ are $\varepsilon'$-close and satisfy $y\in W^{ss}_{g,loc}(x)$.
Then, for any $g'\in \cV$ the projections $x'=\pi_{g'}(\tilde x')$ and $y'=\pi_{g'}(\tilde y')$,
associated to the continuations $\tilde x',\tilde y'\in \widetilde{H(p_{g'})}$ of $\tilde x,\tilde y$, still satisfy $y'\in \cW^{cs}_{g',x'}$ and the open region in $\cW^{cs}_{g',x'}$ bounded by
$W^{ss}_{loc}(x)\cup W^{ss}_{loc}(y')$ does not meet $H(p)$.
When the orientations of $\tilde x$ and $\tilde y$ match, one also has $y'\in W^{ss}_{g',loc}(x')$.
\end{corollary}
\begin{proof}
Let us consider two points $\tilde x,\tilde y$ whose projections are $\varepsilon'$-close
and satisfy $y\in \cW^{cs}_{g,x}$.
Then, the same holds for $g'$ and the continuations $x',y'$ by lemma~\ref{l.cont-central}.
The point $x$ is the limit of a sequence $(p_{n,g})$ with $p_n\in\cP$ such that $W^u_{loc}(p_{n,g})$
intersects $\cW^{cs,+}_x$. We claim that $y'$
does not belong to $\cW^{cs,+}_{x'}$.
Let us assume by contradiction that this is not the case.
On the one hand $y$ does not meet $\cW^{cs,+}_{g,x}$
whereas $W^u_{loc}(p_{n,g})$ intersects $\cW^{cs,+}_x$: this implies that
$W^u_{loc}(p_{n,g})$ intersects the component of $\cW^{cs}_{g,y}\setminus W^{ss}_{g,loc}(y)$
corresponding to the orientation of $\tilde x$.
On the other hand for $m$ large $p_{m,g'}$ is close to $x'$, hence
$W^u_{loc}(p_{m,g'})$ intersects the component of $\cW^{cs}_{g',y'}\setminus W^{ss}_{g',loc}(y')$
corresponding to the reversed orientation of $\tilde x$.
There exists $q\in \cP$ such that $q_g$ and $q_{g'}$ are arbitrarily close to $y$ and $y'$ respectively.
Hence, $W^{u}_{g,loc}(p_{n,g})$ intersects one component of $W^{cs}_{g,q}\setminus W^{ss}_{g,loc}(q_g)$
and $W^{u}_{g',loc}(p_{n,g'})$ intersects the component of $W^{cs}_{g',q}\setminus W^{ss}_{g',loc}(q_{g'})$
which corresponds to the other orientation.
From lemma~\ref{l.ordering}, this implies that there exists $h\in \cU$ such that $H(p_h)$
has a strong homoclinic intersection, contradicting our assumptions.
Similarly, $x'$ does not belong to $\cW^{cs,+}_{g',y'}$ for the orientation
on $E^c_{y'}$ induced by $\tilde y$.
When the orientations of $\tilde x$ and $\tilde y$ match, this
implies that $y'$ belongs to $W^{ss}_{g',loc}(x')$.
When the orientations differ, $W^{ss}_{g',loc}(x')$ and $W^{ss}_{g',loc}(y')$
bound the open region $\cW^{cs,+}_{g',x'}\cap \cW^{cs,+}_{g',y'}$.
If there exists a point $\tilde z\in \widetilde{H(p)}$ whose projection by $\pi_{g'}$
belongs to this region, the discussion above proves that
the projection of its continuation for $g$ also belongs to
$\cW^{cs,+}_{g,x}\cap \cW^{cs,+}_{g,y}$. But for $g$ this open region
is empty since $y\in W^{ss}_{g,loc}(x)$.
This is a contradiction. Hence the open region bounded by $W^{ss}_{g',loc}(x')$ and $W^{ss}_{g',loc}(y')$
in $\cW^{cs}_{g',x'}$ does not meet $H(p_{g'})$.
\end{proof}
\begin{corollary}\label{c.continuation2}
Under the assumptions of proposition~\ref{p.continuation},
let us consider a diffeomorphism $g\in \cV$ and a hyperbolic periodic point $q_g$
whose hyperbolic continuation $q_{g'}$ is defined and homoclinically related to the orbit of
$p_{g'}$ for each $g'\in \cV$.
Then, for $g'\in \cV$, $q_{g'}$ is the unique point in $H(p_{g'})$ which has the same
continuation as $q_g$.
\end{corollary}
\begin{proof}
It is enough to prove that in a small neighborhood of $g$, the point $q_{g'}$ is the unique point in $H(p_{g'})$ which has the same continuation as $q_g$.
Let us consider a sequence $(p_n)$ in $\cP$ such that $p_{n,g}$
accumulates on $q_{g}$ and $W^{u}_{g,loc}(p_{n,g})$ intersects $\cW^{cs,+}_{g,q_{g}}$.
One may also choose the $p_n$ such that
$W^{u}_{g,loc}(q_{g})$ intersects $\cW^{cs,-}_{g,p_{n,g}}$.
By lemma~\ref{r.continuity}, for any $g'\in \cV$,
the limit $\bar q_{g'}$ of $(p_{n,g'})$ is a periodic point in $\cW^{cs}_{g,q_{g'}}\setminus
\cW^{cs,-}_{g,q_{g'}}$.
Also $W^{u}_{g',loc}(\bar q_{g'})$ intersects $\cW^{cs,-}_{g',p_{n,g'}}$.
For $n$ large and $g'$ close to $g$, the points $p_{n,g'}$ and $q_{g'}$ are close:
this implies that $\bar q_{g'}$ is contained in a small neighborhood of $q_{g'}$.
Since $q_{g'}$ is uniformly hyperbolic for any $g'$ close to $g$, this implies that
$\bar q_{g'}$ and $q_{g'}$ coincide, as claimed.
\end{proof}
\section{Structure in the center-stable leaves}\label{s.2D-central}
In this section we prove theorem~\ref{t.tot-discontinuity}
on the geometry of chain-hyperbolic classes.
It is used in the proof of theorems~\ref{t.extremal} and~\ref{t.2D-central}.
As a consequence (see proposition~\ref{p.box}),
for some chain-hyperbolic classes, one can
replace the plaques $\cW^{cs}_x$ by submanifolds $V_x$ whose
boundaries are disjoint from $H(p)$.
In the whole section, $H(p)$ is a chain-recurrence class with a dominated splitting $E^{cs}\oplus E^{cu}=(E^{s}\oplus E^{c}_1)\oplus E^c_2$ such that
$E^c_1,E^c_2$ are one-dimensional and $E^{cs},E^{cu}$ are thin-trapped.
We assume moreover that for each periodic point $q\in H(p)$, the set
$W^{ss}(q)\setminus \{q\}$ is disjoint from $H(p)$.
\subsection{Geometry of connected compact sets}
One can obtain connected compact sets as limit of $\varepsilon$-chains, i.e.
finite sets $\{x_0,\dots,x_m\}$ such that $d(x_i,x_{i+1})<\varepsilon$ for each $0\leq i<m$.
This idea is used to prove the following lemma.
\begin{lema}\label{l.connex}
For any $n\geq 1$, any distance on $\RR^n$ which induces the standard topology,
any closed connected set $K\subset \RR^n$, any point $x\in K$
and any $0\leq D\leq\mbox{Diam}(K)$, there exists a compact connected set $K(D)\subset K$ containing $x$ and whose diameter is equal to $D$.
\end{lema}
\begin{proof}
For $\varepsilon>0$, one can choose a finite set
$X_\varepsilon=\{x_0,x_1,\dots,x_m\}\subset K$
such that
\begin{itemize}
\item[--] $x$ belongs to $X_\varepsilon$;
\item[--] for each $0\leq i< m$, the open balls $B(x_i,\varepsilon)$
and $B(x_{i+1},\varepsilon)$ intersect;
\item[--] the diameter of $X_\varepsilon$ belongs to $[D,D+2\varepsilon]$.
\end{itemize}
Let $K_\varepsilon$ be the closed $\varepsilon$-neighborhood of $X_\varepsilon$.
It is a connected compact set contained in the $\varepsilon$-neighborhood of $K$.
Up to considering an extracted sequence, $(K_\varepsilon)$
converges for the Hausdorff topology towards a compact set $K(D)$
which contains $x$, is connected and has diameter $D$ as required.
\end{proof}
\medskip
Recall that for $x\in H(p)$, the submanifold
$W^{ss}(x)$ is diffeomorphic to $\RR^{d-2}$, where $d=\dim(M)$.
\begin{lema}\label{l.converge}
Consider a sequence $(z_n)$ in $H(p)$ which converges to a point
$z$ and for each $n$ a compact connected set $C_n\subset W^{ss}(z_n) \cap H(p)$
which converges for the Hausdorff topology in $M$ towards a (compact connected) set
$C\subset W^{ss}(z) \cap H(p)$. Then the restriction of the
intrinsic distance of $W^{ss}(z_n)$ to the set
$C_n$ converges towards the intrinsic distance of $W^{ss}_z$ to $C$.
\end{lema}
\begin{proof}
Let $U$ be a bounded neighborhood of $K$ inside $W^{ss}(z)$
which is diffeomorphic to $\RR^{d-2}$.
For $z_n$ close to $z$, there exists an open set $U_n\subset W^{ss}_{z_n}$,
containing $z_n$, diffeomorphic to $\RR^{d-2}$ and which is close to $U$ for the $C^ 1$-topology on immersions
of $\RR^{d-2}$. In particular, $U$ and $U_n$ are diffeomorphic by
a map whose Lipschitz constant is arbitrarily close to $1$.
Since $K_n$ is connected and contains $z_n$, it is included in $U_n$.
This gives the conclusion.
\end{proof}
\subsection{Structure in the strong stable leaves}
We are aimed first to prove total discontinuity in the strong stable leaves.
\begin{prop}\label{ss-tot-discontinuity}
Let $f$ be a diffeomorphism and $H(p)$ be a chain-hyperbolic class
satisfying the assumptions of theorem~\ref{t.tot-discontinuity}.
If for any periodic point $q\in H(p)$ the set $W^{ss}(q)\setminus \{q\}$ is disjoint from $H(p)$, then, for each $x\in H(p)$, the set $W^{ss}_{loc}(x)\cap H(p)$ is totally disconnected.
\end{prop}
At any points, one considers the plaques $\cW^{cu}_x\subset f(\cW^{cu}_{f^{-1}(x)})$.
We choose the plaques $\cW^{cs},\cW^{cu}$ with a diameter small enough so that
for each $x,y\in H(p)$ the intersection
$\cW^{cs}_x\cap f(\cW^{cu}_y)$ is transversal and contains at most one point
(which belongs to $H(p)$ by lemma~\ref{l.bracket0}).
For this proof we will endow $H(p)$ with the \emph{center-stable topology}:
two points $x,y\in H(p)$ are close if the distance $d(x,y)$ is small
and $x\in \cW^{cs}_y$ (or equivalently $y\in \cW^{cs}_x$ by lemma~\ref{l.uniqueness-coherence}).
The \emph{center-stable distance} on $H(p)$
is the distance between $x$ and $y$ inside $\cW^{cs}_x$.
Since $\cW^{cs}$ is trapped, $W^{ss}(x)\cap \cW^{cs}_x$ is contained inside $W_{loc}^{ss}(x)$ and the center-stable topology induces on $W^{ss}(x)\cap H(p)$ the intrinsic topology of $W^{ss}(x)$.
\paragraph{Local holonomy.} We fix $\rho>0$ and define the ball $B^{cs}(x)$
centered at $x\in H(p)$ of radius $\rho$ contained in $\cW^{cs}_x$.
If $\rho$ is small, for any points $x_0\in H(p)$ and $y_0,z_{0}\in \overline{\cW^{cu}_{x_0}}\cap H(p)$
the \emph{local holonomy} $\Pi^{cu}$ along the center-unstable plaques
$f(\cW^{cu}_{f^{-1}(x)})$, ${x\in \cW^{cs}(x_0)\cap H(p)}$,
is defined from $B^{cs}(z_0)\cap H(p)\subset \cW^{cs}_{z_0}$ to $\cW^{cs}_{y_0}$.
\paragraph{Global holonomy.}
We now try to extend globally the holonomy. A strong stable leaf may intersect a plaque of $\cW^{cu}$
in several points, hence the global holonomy may be multivalued.
A \emph{global holonomy along the plaques $\cW^{cu}$} is a closed connected set
$\Delta\subset H(p)\times H(p)$ (endowed with the product center-stable topologies)
such that for any $(x,y)\in \Delta$ one has
$y\in \overline{\cW^{cu}_{x}}$ and $x\in \overline{\cW^{cu}_{y}}$.
The sets $\pi_1(\Delta)$ and $\pi_2(\Delta)$ denote the projections on the first and the second factors.
\medskip
One can obtain global holonomies from connected sets contained in a strong stable leaf.
\begin{lema}\label{l.extension}
Let $\Delta_0$ be a global holonomy along the center-unstable plaques,
and $C\subset H(p)$ be a set which is closed and connected
for the center-stable topology and which contains $\pi_1(\Delta_0)$.
Then, there exists a global holonomy $\Delta$ along the center-unstable plaques
containing $\Delta_0$, such that $\pi_1(\Delta)\subset C$ and satisfying one of the following cases.
\begin{enumerate}
\item\label{i.extension1} $\pi_1(\Delta)=C$;
\item\label{i.extension2} $\Delta$ is non-compact;
\item\label{i.extension3} there exists $(x,y)\in \Delta$ such that $y\in \overline{\cW^{cu}_{x}}\setminus \cW^{cu}_x$ or
$x\in \overline{\cW^{cu}_{y}}\setminus \cW^{cu}_y$.
\end{enumerate}
\end{lema}
\begin{proof}
If $\{\Delta_n\}$ is a family of global holonomies along the center-unstable plaques
that is totally ordered by the inclusion, then the closure of the union $\overline{\cup_n \Delta_n}$
is also a global holonomy. By Zorn's lemma one deduces that there exists a global holonomy
$\Delta$ containing $\Delta_0$, satisfying $\pi_1(\Delta)\subset C$ and maximal with these properties for the inclusion. We prove by contradiction that $\Delta$ satisfies one of the properties above.
We fix a pair $(x_0,y_0)\in \Delta_0$.
If $\pi_1(\Delta)\neq C$, then there exists $r_1>0$ and for each $\varepsilon_1>0$
there exists a sequence $(x_0,\dots,x_s)$ in $C$ such that
\begin{itemize}
\item[--] for each $0 < i \leq s$, the points $x_{i-1},x_{i}$ are at distance less than $\varepsilon_1$
and $x_{i}\in B^{cs}(x_{i-1})$;
\item[--] the point $x_s$ and the set $\pi_1(\Delta)$ are at distance exactly $r_1$ inside $\cW^{cs}_{x_s}$.
\end{itemize}
If $\Delta$ does not satisfies the items~\ref{i.extension2}) or \ref{i.extension3}), then
for any $(x,y)\in H(p)\times H(p)$ close to $\Delta$
and any $x'\in H(p)$ close $x$ (for the center-stable topology),
$B^{cs}(y)$ meets $\cW^{cu}_{x'}$ at a point $y'\in H(p)$ which also satisfies
$x'\in \cW^{cu}_{y'}$.
This allows to build inductively a sequence $(y_0,\dots,y_{\ell})$ for some $0\leq \ell\leq s$
and associated to $(x_0,\dots,x_\ell)$ such that, for each $i$, the pair $(x_i,y_i)$ is at
a small distance from $(x_{i-1},y_{i-1})$ for the center-stable distance.
More precisely, there exists $r>0$ and for each $\varepsilon>0$ there exists
a sequence $(x_0,y_0),\dots,(x_\ell,y_\ell)$ such that for the product center-stable distance
on $H(p)\times H(p)$ the following holds:
\begin{itemize}
\item[--] for each $0 < i \leq \ell$, one has $x_i\in \cW^{cu}_{y_i}$ and
$y_i\in \cW^{cu}_{x_i}$;
\item[--] for each $0 < i \leq \ell$, the pairs $(x_{i-1},y_{i-1})$ and $(x_{i},y_i)$ are at distance less than $\varepsilon$;
\item[--] the pair $(x_\ell,y_\ell)$ and the set $\Delta$ are at distance exactly $r$.
\end{itemize}
When $\varepsilon$ goes to $0$ and up to consider a subsequence,
the set $\Delta\cup \{(x_0,y_0),\dots,(x_\ell,y_\ell)\}$ converges for the Hausdorff distance
towards a compact connected set $\Delta'$ which is a global holonomy, strictly contains $\Delta$
and satisfies $\pi_1(\Delta')\subset C$. This contradicts the maximality of $\Delta$ and proves the lemma.
\end{proof}
\medskip
The strong stable leaves are preserved under global holonomies along center-unstable plaques.
\begin{addendum}\label{a.extension}
In the case each set $C$ and $\pi_2(\Delta_0)$ is contained in a strong stable leaf,
one can ensure furthermore that $\pi_2(\Delta)$ is also contained in a strong stable leaf.
\end{addendum}
\begin{proof}
We repeat the proof of lemma~\ref{l.extension} requiring furthermore
that the projection $\pi_2(\Delta)$ of the global holonomies are contained in the strong stable leaf $W^{ss}(y_0)$.
Indeed if $\{\Delta_n\}$ is totally ordered family of such global holonomies, then
the closure of the union $\overline{\cup_n\Delta_n}$ projects in $W^{ss}(y_0)$ by $\pi_2$: this is due to the choice of
the center-stable topology.
Let us consider a maximal global holonomy $\Delta$ satisfying
$\pi_2(\Delta)\subset W^{ss}(y_0)$ and given by Zorn's lemma.
Assume by contradiction that $\Delta$ does not satisfies the three items of
lemma~\ref{l.extension}. In particular, it is compact and
one may fix $(x,y)\in \Delta$ and an $n\geq 1$ such that
$f^n(\pi_2(\Delta))$ is contained in $\cW^{cs}_{f^n(y_0)}$.
One repeats the same construction as above and builds a global holonomy
$\Delta'$ that contains strictly $\Delta$. If $\pi_2(\Delta')$ is contained in $W^{ss}(y_0)$, one has contradicted the maximality of $\Delta$. One will thus assume that the set $f^n(\pi_2(\Delta'))\subset \cW^{cs}_{f^n(y_0)}$ is not contained in a strong stable leaf. Since it is connected, it contains a point $z$
such that both local components of $\cW^{cs}_z\setminus W^{ss}_{loc}(z)$ at $z$
meet $\Pi^{cu}(C)$. If one considers a hyperbolic periodic orbit $O$ homoclinically related to $p$ having a point $q_0$ close to $z$, the local holonomy $\Pi^{cu}$ along the plaques of $\cW^{cu}$ allows to project $f^n(\pi_2(\Delta'))$ on a connected compact subset of
$\cW^{cs}_{q_0}$ which meets $W^{ss}(q_0)$.
Since $W^{ss}(q_0)\setminus\{q_0\}$ is disjoint from $H(p)$,
one deduces that the projection contains $q_0$.
Consequently the unstable manifold of some point $q\in O$ meets $C$ at some point $x$.
By lemma~\ref{l.connex} and since $E^{ss}$ is uniformly contracting,
there exists $\varepsilon>0$ such that any backward iterate
$x_{-n}=f^{-n}(x)$ is contained in a connected compact set $C_{-n}\subset W^{ss}_{loc}(x_{-n})\cap H(p)$
which has a radius equal to $\varepsilon$. Since $x$ belongs to the unstable set of some point $f^k(q)$ in the orbit of $q$,
the backward iterates of $x$ and $q$ become arbitrarily close.
Let $\tau$ be the period of $q$.
One gets that the projection $\Pi^{cu}(C_{-n\tau})$ by holonomy on $\cW^{cs}_q$
converges to a compact connected set contained in $W^{ss}_{loc}(q)$
with diameter equal to $\varepsilon$.
This contradicts our assumption that $W^{ss}(q)\setminus\{q\}$ is disjoint from $H(p)$.
In all the cases we have found a contradiction and the lemma is proved.
\end{proof}
\paragraph{Triple holonomy.} The previous results on holonomies extend to
connected set of triples.
\begin{lema}\label{l.triple}
Let $\Delta$ be a global holonomy along the center-unstable plaques,
$(x_0,y_0)$ be a pair in $\Delta$ and $z_0\in H(p)$ be a point
which belong to the connected component of $\overline{\cW^{cu}_{x_0}}\cap \overline{\cW^{cu}_{y_0}}$ bounded by $x_0$ and $y_0$.
Then there exists a set $X\subset H(p)\times H(p)\times H(p)$ containing $(x_0,y_0,z_0)$
such that
\begin{itemize}
\item[--] $X$ is closed and connected for the center-stable topology,
\item[--] for each triple $(x,y,z)\in X$ one has $(x,y)\in \Delta$ and
$z\in \overline{\cW^{cu}_{x_0}}\cap \overline{\cW^{cu}_{y_0}}$,
\item[--] one of the two following cases holds:
\begin{enumerate}
\item the set of pairs $(x,y)$ for $(x,y,z)\in X$ coincides with $\Delta$,
\item $X$ is non-compact.
\end{enumerate}
\end{itemize}
Moreover if $\pi_1(\Delta)$ and $\pi_2(\Delta)$ are contained in strong stable leaves, then the same holds for $\pi_3(X)$.
\end{lema}
\begin{proof}
The proof is the same as for lemma~\ref{l.extension} and addendum~\ref{a.extension} but the third case of lemma~\ref{l.extension} has not to be considered since
for all the triples $(x,y,z)\in X$, the point $z$ belongs to the connected component of $\overline{\cW^{cu}_{x}}\cap \overline{\cW^{cu}_{y}}$ bounded by $x$ and $y$
and its distance to $x$ and $z$ is thus controlled.
\end{proof}
\begin{remark}\label{r.triple}
If one projects the set $X$ obtained in lemma~\ref{l.triple} on any
pair of coordinates, for instance as $\pi_{1,3}(X)=\{(x,z),(x,y,z)\in X\}$,
one gets a set which is connected. Hence the closure of $\pi_{1,3}(X)$
for the center-stable topology is a global holonomy.
\end{remark}
\paragraph{Non compact holonomy.}
We now build non bounded holonomies.
\begin{lema}\label{l.unbounded}
If for some $x\in H(p)$ the set $W^{ss}(x)\cap H(p)$ is not totally disconnected,
then there exists a global holonomy $\Delta$ along the center-unstable plaques which is non-compact, non trivial (i.e. there exists $(x_0,y_0)\in \Delta$ such that $x_0\neq y_0$)
and such that both $\pi_1(\Delta)$ and $\pi_2(\Delta)$ are contained in strong stables leaves.
\end{lema}
\begin{proof}
One considers a non trivial compact connected set $\Gamma\subset H(p)$ contained in some strong stable leaf
and the accumulation set $\Lambda$ of the backward iterates $f^{-n}(\Gamma)$ (which is invariant by $f$).
The uniform expansion along $E^{ss}$ and the lemmas~\ref{l.connex} and~\ref{l.converge}
above imply that for any $x_0\in \Lambda$ the strong stable leaf $W^{ss}(x_0)$,
contains a closed connected set $C_0\subset \Lambda$ which is not compact and contains $x_0$.
There exist some points $y_0\in H(p)$ distinct from $x_0$ such that $x_0\in \cW^{cs}_{y_0}$
and $y_0\in \cW^{cs}_{x_0}$ hold.
Indeed, $x_0$ is accumulated by periodic points $q\in H(p)$ whose period is arbitrarily large.
Consequently the sets $\cW^{cs}_q$ are pairwise disjoint. Hence, there exists $q$ close to $x_0$
whose plaque $\cW^{cs}_q$ intersects $\cW^{cu}_{x_0}$ at a point $y_0$ which belongs to
$H(p)\setminus \{x_0\}$ from lemma~\ref{l.bracket0}.
Assuming that the conclusion of the lemma does not hold one builds a sequence of compact holonomies
$(\Delta_n)$ such that $\pi_1(\Delta_n)$ is contained in $\Lambda$, both $\pi_1(\Delta_n), \pi_2(\Delta_n)$ are contained in strong stable leaves, and the diameter of
$\pi_1(\Delta)$ in the strong stable leaf goes to infinity with $n$.
The holonomy $\Delta_0$ is just the initial pair $(x_0,y_0)$.
One constructs $\Delta_{n+1}$ from $\Delta_n$ in the following way.
In the strong stable leaf that contains $\pi_1(\Delta_n)$,
one considers a closed non-compact connected set $C_n\subset \Lambda$.
One then applies lemma~\ref{l.extension} and its addendum~\ref{a.extension}
and finds a global holonomy $\Delta_n'\supset \Delta_n$ such that again $\pi_1(\Delta_n')$ is contained in $C_n$ and both $\pi_1(\Delta_n'), \pi_2(\Delta_n')$ are contained in strong stable leaves.
By assumption $\Delta_n'$ is compact and in particular $\pi_1(\Delta'_n)$ is strictly contained
inside $C_n$. As a consequence there exists $(x'_n,y'_n)\in \Delta_n'$ such that
$x'_n\in \overline{\cW^{cu}_{y'_n}}\setminus \cW^{cu}_{y'_n}$
or $y_n\in \overline{\cW^{cu}_{x'_n}}\setminus \cW^{cu}_{x'_n}$.
Using the fact that for each $x\in H(p)$ we have
$$f^{-1}(\overline{\cW^{cu}_x})\subset \cW_{f^{-1}(x)}^{cu},$$
the set of images $(f^{-1}(x),f^{-1}(y))$ for $(x,y)\in \Delta'_n$ is still a compact global holonomy:
this is $\Delta_{n+1}$. We also define $(x_{n+1},y_{n+1})=(f^{-1}(x_n),f^{-1}(y_n))$.
By construction $\pi_1(\Delta_1)$ is a non-trivial compact connected set.
Since $E^{ss}$ is uniformly contracted, the projection
$\pi_1(\Delta_n)$, which contains $f^{-n}(\pi_1(\Delta_1))$, has a diameter
(for the distance inside $W^{ss}(x_n)$) which increases exponentially.
This ends the construction of the sequence $(\Delta_n)$.
Up to considering a subsequence, one can assume that
the sequence $(x_n,y_n)$ converges towards a pair $(x,y)\in H(p)\times H(p)$
for the classical topology on $M$. By construction $x_n,y_n$ are at a bounded distance,
hence $x$ and $y$ are distinct.
For each $n$, one endows $W^{ss}(x_n)\times W^{ss}(y_n)$ with the supremum distance
between the intrinsic distances inside $W^{ss}(x_n)$ and $W^{ss}(y_n)$.
Let us fix $D>0$.
By lemma~\ref{l.connex}, for each $n$ large one can find a compact connected set
$\Delta_n^D$ contained in $\Delta_n$ of diameter $D$ and containing $(x_n,y_n)$.
One can assume that the sequence $(\Delta_n^D)$ converges for the Hausdorff topology
towards a compact connected set $\Delta^D\subset W^{ss}(x)\times W^{ss}(y)$.
By lemma~\ref{l.converge}, this set has diameter $D$.
Now the closure of the union of the $\Delta^D$ over $D$ is a global holonomy which is non-compact and
whose projections by $\pi_1,\pi_2$ are both contained in strong stable leaves.
\end{proof}
\paragraph{Unbounded projections of holonomies}
Non-compact holonomies allow to obtain non-compact connected sets inside strong stable leaves.
\begin{lema}\label{l.projection}
Let $\Delta$ be a non-compact holonomy such that $\pi_1(\Delta),\pi_2(\Delta)$ are contained in strong stable leaves. Then the closure of $\pi_1(\Delta)$ for the center-stable topology
is non-compact.
\end{lema}
\begin{proof}
First notice that one can replace $\Delta$ by $f^{-1}(\Delta)$.
By the trapping of the center-unstable plaques this allows to have
$x\in \cW^{cu}_y$ and $y\in \cW^{cu}_x$ for each $(x,y)\in \Delta$
and to work with the plaques of the family $\cW^{cu}$.
The set of pairs $(x,y)\in \Delta$ such that $x=y$ is closed.
By the choice of the central-stable topology it is also open. Hence two cases occurs:
either $x=y$ for each $(x,y)\in \Delta$ and $\pi_1(\Delta)=\pi_2(\Delta)$ is non-compact;
or for each $(x,y)\in \Delta$ one has $x\neq y$ and this is the case one considers now.
For any pair $(x,y)\in \Delta$, we denote by $[x,y]$ the closed segment of $\cW^{cu}_{x}$ bounded by $x,y$.
\medskip
Let us assume by contradiction that the closure of $\pi_1(\Delta)$ is compact.
One can find a finite
collection of points $\{x_j\}\subset \pi_1(\Delta)$ which satisfies that for any $x\in \pi_1(\Delta)$ there exists $x_j$
such that
\begin{itemize}
\item[--] $x$ belongs to $B^{cs}(x_j)$;
\item[--] for any $y,z\in H(p)\cap \cW^{cu}_x$ such that $(x,y)\in\Delta$ and $z\in [x,y]$,
the plaque $\cW^{cu}_{x_j}$ intersects $B^{cs}(z)$.
\end{itemize}
In the following we will consider holonomies $D$ with $\pi_1(D)\subset \pi_1(\Delta)$
and we introduce the set of points $x_{j}$ that are ``avoided'' by $D$:
$$\cP(D)=\{x_j, \; \forall (x,y)\in D,\; \forall z\in [x,y]\cap H(p), \; x\notin\cW^{cs}_{x_j}
\text{ or } B^{cs}(z)\cap \cW^{cu}_{x_j}=\emptyset\}.$$
Since the closure of $\pi_1(\Delta)$ is compact and $\Delta$ is not,
one can find $x_i$ with the following property.
\begin{description}
\item[(****)] \it There exists $(x',y'),(x'',y'')\in\Delta$ with $x',x''\in\cW^{cs}_{x_i}$
such that
\begin{itemize}
\item[--] for each $z\in ([x',y']\cup [x'',y''])\cap H(p)$, the plaque $\cW^{cs}_z$ intersects
$\cW^{cu}_{x_i},\cW^{cu}_{x'},\cW^{cu}_{x''}$;
\item[--] $\cW^{cs}_{y'}$ and $\cW^{cs}_{y''}$ intersect $\cW^{cu}_{x_i}$ in two distinct points.
\end{itemize}
\end{description}
Note that in particular the plaques $\cW^{cs}_{y'}$ and $\cW^{cs}_{y''}$ are disjoint.
This allows us to build a compact holonomy $D\subset \Delta$ which ``almost fails'' to be a graph above its first projection.
\begin{claim}\label{c.holonomy}
There exists a compact holonomy $D$ having the following properties:
\begin{enumerate}
\item\label{i.zero} $\pi_1(D)\subset \pi_1(\Delta)$;
$\pi_2(D)$ is contained in a strong stable leaf;
\item\label{i.one} $D$ is a continuous graph over its first factor;
\item\label{i.two} there is $x_i\in \cP(D)$ satisfying (****).
\end{enumerate}
\end{claim}
\begin{proof}
Let us first notice that since $\Delta$ is non-compact it contains compact holonomies
$\Delta'$ with arbitrarily large diameter by lemma~\ref{l.connex}.
One can thus assume that for such a compact holonomy $\Delta'$, there
exists $x_{i}$ and two pairs $(x',y'),(x'',y'')\in \Delta'$ satisfying (****).
Working with $\varepsilon$-chains as in the proof of lemma~\ref{l.connex}, one can build a compact connected set $D_0\subset \Delta'$ such that~\ref{i.two}) is satisfied for $x_i$. More precizely for any $\varepsilon>0$ one builds a finite set
$X_{\varepsilon}=\{(x(0),y(0)),\dots,(x(s),y(s))\}$ contained in $\Delta'$ such that
\begin{itemize}
\item[--] $(x(k),y(k))$ and $(x(k+1),y(k+1))$ are $\varepsilon$-close for each $0\leq k<s$;
\item[--] the pairs $(x',y')=(x(0),y(0))$ and $(x'',y'')=(x(s),y(s))$ and the point $x_i$
satisfy (****);
\item[--] for any pair $(x(k),y(k))$ with $x(k)\in \cW^{cs}_{x_i}$,
and for any point $z\in [x(k),y(k)]\cap H(p)$ the intersection $B^{cs}(z)\cap \cW^{cu}_{x_i}$ is empty.
\end{itemize}
The compact holonomy $D_0$ is obtained as limit of the sets $X_\varepsilon$.
Repeating the construction with the other points $x_j$, one gets a new compact global holonomy
$D\subset D_0$ such that~\ref{i.one}) is satisfied. Note that~\ref{i.two}) is still satisfied but for a new point $x_i$. Since $D\subset \Delta$, the condition~\ref{i.zero}) holds also.
\end{proof}
\medskip
We now fix a compact holonomy $D$ satisfying the properties~\ref{i.zero}), ~\ref{i.one}) and \ref{i.two}) above.
We do not assume that it is contained in $\Delta$. However we choose it so that the cardinal of $\cP(D)$ is maximal.
\medskip
Let us consider the points $x_i,x',x''$ in property~\ref{i.two}) and (****) and consider the plaques $\cW^{cs}_{x_i},\cW^{cs}_{x'},$ $\cW^{cs}_{x''}$
and the ordering of their intersection on $\cW^{cu}_{x_i}$.
Then $\cW^{cs}_{x_i}$ is not ``in the middle'' of $\cW^{cs}_{x'}$ and $\cW^{cs}_{x''}$.
\begin{claim}\label{c.ordering}
The point $x_i$ does not belong to the connected component
of $\cW^{cu}_{x_i}\setminus (\cW^{cs}_{x'}\cup \cW^{cs}_{x''})$
bounded by $\cW^{cs}_{x'}$ and $\cW^{cs}_{x''}$.
\end{claim}
\begin{proof}
Let us define the compact connected set $C:=\pi_1(D)$.
For each $x\in C$, there exists a unique pair $(x,y)\in D$; moreover
$x\neq y$. One can thus consider
the orientation on $E^{cu}_x$ determined by the component of $\cW^{cu}_x\setminus \{x\}$
which contains $y$. This defines a continuous orientation of the bundle
$E^{cu}_{|C}$.
One can compare the orientations of $E^{cu}_{x'}$ and $E^{cu}_{x''}$
as transverse spaces to the one-codimensio\-nal plaque $\cW^{cs}_{x_i}$. By the trapping property, for any $k\geq 0$ the forward iterates $f^{k}(x')$
and $f^{k}(x'')$ still belong to the same plaque $\cW^{cs}_{f^k(x_i)}$,
hence the orientations comparison will be the same for $k=0$ or $k$ large.
Since $C$ is a compact subset of a strong stable leaf,
for $k\geq 1$ large $f^k(C)$ is contained in $\cW^{cs}_{f^k(x_i)}$; so for any continuous orientation of $E^{cu}_{|f^k(C)}$, the orientations on $E^{cu}_{f^k(x')}$ and $E^{cu}_{f^k(x'')}$ match.
One deduces that for the orientation on $E^{cu}_{|C}$ considered above,
the orientations on $E^{cu}_{x'}$ and $E^{cu}_{x''}$ match.
By definition of the orientation on $E^{cu}_{|C}$, this implies the claim.
\end{proof}
\medskip
Let $\gamma'=[x',y']$ and $\gamma''=[x'',y'']$.
One now defines a homeomorphism $\varphi\colon\gamma'\cap H(p)\to\gamma''\cap H(p)$.
For $z'\in \gamma'\cap H(p)$, one can use lemma~\ref{l.triple}
and find a closed connected set $X_{z'}\subset H(p)\times H(p)\times H(p)$
containing $(x',y',z')$ and such that for all $(x,y,z)\in X_{z'}$ one has
$z\in \cW^{cu}_x\cap \cW^{cu}_y$ and $(x,y)\in D$.
\begin{claim}
There exists a unique map $\chi\colon D\to H(p)$ which is continuous for the center-stable topology, sends $(x',y')$ on $z'$
and satisfies $\chi(x,y)\in [x,y]$ for each $(x,y)\in D$. Its graph coincides with $X_{z'}$,
which is thus compact.
\end{claim}
\begin{proof}
By remark~\ref{r.triple}, the closure $\widetilde \Delta$
of $\pi_{1,3}(X_{z'})$ is a gobal holonomy satisfying property 1).
Let us assume by contradiction that the projection map
$\pi_{1,2}\colon X_{z'}\to D$ is not injective: in particular
$\widetilde \Delta$ contains two different pairs $(x,z)$ and $(x,\zeta)$,
having the same projection by $\pi_1$. Let us choose $x_j$ such that $x\in B^{cs}(x_j)$
and $\cW^{cu}_{x_j}$ intersects both $B^{cs}(z)$ and $B^{cs}(\zeta)$.
Repeating the argument of the proof of claim~\ref{c.holonomy},
there exists a compact holonomy $\widetilde D\subset \widetilde \Delta$
satisfying the properties~\ref{i.zero}), \ref{i.one}), \ref{i.two}) above
such that $x_j$ belongs to $\cP(\widetilde D)$.
By construction for each $(x,z)\in \widetilde \Delta$, there exists $(x,y)\in D$ such that
$z$ belongs to $[x,y]$. The definition of the set $\{x_{j}\}$ and the fact that for each $(x,z)\in \widetilde D$ there exists
$(x,y)\in D$ such that $z\in [x,y]$ imply that $\cP(D)\subset \cP(\widetilde D)$.
Since $x_{j}$ belongs to $\cP(\widetilde D)\setminus\cP(D)$, we have contradicted the maximality of $D$.
Hence the map $\pi_{1,2}\colon X_{z'}\to D$ is injective.
Since $D$ is compact, one deduces that $X_{z'}$ is also compact and the first case of lemma~\ref{l.triple} holds.
Consequently, the projection $\pi_{1,2}$ is also surjective $X_{z'}$.
This proves that $X_{z'}$ is the graph of a map $\chi\colon D\to H(p)$. Since $X_{z'}$ is compact, this map is continuous.
The connectedness of $D$ implies that the map $\chi$ is unique.
\end{proof}
\medskip
One deduces that $X_{z'}$ contains a unique triple of the form $(x'',y'',z'')$ and
one sets $\varphi(z')=z''$. The claim implies that $\varphi$ is monotonous
for the ordering on $\gamma',\gamma''$.
One can build similarly a map from $\gamma''$ to $\gamma'$, which is an inverse of $\varphi$.
Consequently $\varphi$ is a homeomorphism which is monotonous for the ordering on $\gamma',\gamma''$.
\medskip
Let $y'_i$ be the intersection between $\cW^{cs}_{y'}$ and $\cW^{cu}_{x_i}$
and $y''_i$ be the intersection between $\cW^{cs}_{y''}$ and $\cW^{cu}_{x_i}$.
Let $\gamma'_i,\gamma_{i}''$ be the segments contained in $\cW^{cu}_{x_i}$ and bounded by $\{x_i,y'_i\}$ and $\{x_i,y''_i\}$ respectively.
One defines two monotonous homeomorphisms
$\psi'\colon \gamma'\cap H(p)\to \gamma_{i}\cap H(p)$ and $\psi''\colon \gamma''\cap H(p)\to \gamma_{i}\cap H(p)$ which send $x'$ and $x''$ on $x_{i}$.
There are obtained by considering local projection throught the center-stable holonomy:
one has $\psi'(z')=z$ when $z\in \cW^{cs}_{z'}$ (and equivalently
when $z'\in \cW^{cs}_{z}$).
One thus obtains a monotonous homeomorphism $\varphi_{i}=\psi'\circ\varphi\circ{\psi''}^{-1}$
from $\gamma_i'\cap H(p)$ to $\gamma_i''\cap H(p)$.
From the claim~\ref{c.ordering} and exchanging $(x',y')$ and $(x'',y'')$ if necessary,
one can assume that $y''_i$ is between $x_i$ and $y'_i$ inside $\cW^{cu}_{x_i}$.
Consequently $\varphi_i$ maps monotonously
$H(p)\cap \gamma_i'$ into itself.
The sequence $z_n=\varphi^n_i(y'_i)$ thus converges to a point $z$
which is fixed by $\varphi_i$ but all the $z_n$ are distinct since by assumption
$z_0=y'_i$ and $z_1=y''_i$ are distinct.
By construction, for each $n$ one associates a compact connected set
$X_n=X_{{\psi'}^{-1}(z_n)}\subset H(p)\times H(p)\times H(p)$ which contains the
triples $(x',y',{\psi'}^{-1}(z_n))$ and $(x'',y'',{\psi''}^{-1}(z_n))$.
Its projection on its third factor is a compact connected set $C_n\subset H(p)$
containing ${\psi'}^{-1}(z_n)$ and ${\psi''}^{-1}(z_n)$ and contained
in a strong stable leaf.
Similarly, let $X=X_{{\psi'}^{-1}(z)}$ and $C$ be its projection on the third factor.
Then, $C_n$ converges towards $C$ for the Hausdorff topology on compact sets of $M$,
whereas the points ${\psi'}^{-1}(z_n), {\psi''}^{-1}(z_{n+1})\in C_n$
converge towards ${\psi'}^{-1}(z), {\psi''}^{-1}(z)\in C$.
Since $z$ is fixed by $\varphi_i$, the center-stable plaques
of the points ${\psi'}^{-1}(z), {\psi''}^{-1}(z)$ intersect,
whereas since $z_n,z_{n+1}$ are distinct, the center-stable plaques
of the points ${\psi'}^{-1}(z_n), {\psi''}^{-1}(z_{n+1})$ are disjoint.
Thus the intrinsic distances between ${\psi'}^{-1}(z), {\psi''}^{-1}(z)$
and ${\psi'}^{-1}(z_n), {\psi''}^{-1}(z_{n+1})$ are bounded away,
contradicting lemma~\ref{l.converge}.
The proof of lemma~\ref{l.projection} is now complete.
\end{proof}
\bigskip
We now finish the proof of the proposition.
\begin{proof}[\bf Proof of proposition~\ref{ss-tot-discontinuity}]
Let us assume by contradiction that for some point $x\in H(p)$
the set $H(p)\cap W^{ss}(x)$ is not totally disconnected.
We will build a periodic point $q\in H(p)$, a point $z_0\in W^s(q)\cap H(p)$
and a set $C\subset W^{ss}(z_0)$ which is closed connected and non-compact
for the intrinsic topology on $W^{ss}(z_0)$.
In the stable manifold of the orbit of $q$,
the iterates $f^n(C)$ accumulate a non-trivial subset of $W^{ss}(q)$,
contradicting the assumption that $W^{ss}(q)\cap H(p)= \{q\}$.
In order to build $q$ and $C$, we apply lemma~\ref{l.unbounded}
and consider a non-compact holonomy $\Delta$
and a pair $(x_0,y_0)\in \Delta$ such that $x_0\neq y_0$.
The sets $\pi_1(\Delta),\pi_2(\Delta)$ are contained in strong stable leaves
and by lemma~\ref{l.projection} their closures in the leaves are not compact.
Let us remind that $\cW^{cu}_{x_0}$ is a one-dimensional curve
and consider the open connected subset $U\subset\cW^{cu}_{x_0}$
bounded by $\{x_0,y_0\}$.
Two cases have to be studied.
If $H(p)$ does not meet the set $U$,
then $x_0$ is an unstable boundary point of the
chain-hyperbolic class $H(p)$ (see definition~\ref{d.boundary}).
By lemma~\ref{p.boundary0},
there exists a periodic point $q$ in $H(p)$
whose stable set contains $\pi_1(\Delta)$.
We define $z_0=x_0$ and the set $C$ as the closure of $\pi_1(\Delta)$ in $W^s(q)$,
finishing the proof in this case.
Let us assume now that there exists a point $\zeta\in U\cap H(p)$.
We introduce a hyperbolic periodic point $q$ homoclinically related to $p$
and close to $\zeta$ such that $\cW^{cs}_q\subset W^{s}(q)$ as given by lemma~\ref{l.contper}.
The plaques $\cW^{cs}_q$ and $\cW^{cu}_{x_0}$ intersect at a point $z_0\in U\cap H(p)$.
By lemma~\ref{l.triple}, there is a closed connected set $X\subset H(p)\times H(p)\times H(p)$
which contains $(x_0,y_0,z_0)$, such that for each $(x,y,z)\in X$ one has
$z\in\cW^{cu}_x\cap \cW^{cu}_y$ and $(x,y)\in \Delta$.
Moreover the projection $\pi_3(X)$ is contained in a strong stable leaf of $W^s(q)$
and $X$ is non-compact. We want to show that the closure of $\pi_3(X)$ in
$W^s(q)$ is non-compact.
We know that the closure of
one of the three projections $\pi_1(X),\pi_2(X),\pi_3(X)$ is non-compact.
If for instance this happens for $\pi_1(X)$, the closure of
$\pi_{1,3}(X)$ is a non-compact holonomy by remark~\ref{r.triple}.
Hence by lemma~\ref{l.projection}, the closure of $\pi_3(X)$ is non-compact also.
One concludes that in any case the closure $C$ of
$\pi_3(X)$ is non-compact: we have found a non-compact connected
set contained in $H(p)\cap W^{ss}(z_0)$ as claimed, concluding the proof of the proposition in the second case.
\end{proof}
\subsection{Structure in the center-stable leaves: proof of theorem~\ref{t.tot-discontinuity}}
By the trapping property, the iterates of each plaque $\cW^{cs}_x$, $x\in H(p)$,
remain in a small neighborhood of $H(p)$, hence is covered by a strong stable foliation.
We call \emph{strong stable plaques} the connected components of the strong stables leaves of
$\cW^{cs}_x$.
\begin{lema}\label{l.graph}
For any $x\in H(p)$,
let us consider a connected compact set $\Gamma\subset H(p)\cap \cW^{cs}_x$.
Then $\Gamma$ intersects each strong stable plaque of $\cW^{cs}_x$ in at most one point.
In particular this is a curve.
\end{lema}
\begin{proof}
Let us assume by contradiction that $\Gamma$ intersects some strong stable leaf $L$ of
$\cW^{cs}_x$ in at least two distinct points $z,z'$.
Let us consider two small closed neighborhoods $U$ and $U'$ of $z,z'$ in $\cW^{cs}_x$,
such that that $U\setminus L$ and $U'\setminus L$ have two connected components.
We introduce the connected components $\Gamma_z,\Gamma_{z'}$ of $\Gamma\cap U$ and $\Gamma\cap U'$
containing $z$ and $z'$ respectively.
These two sets are not reduced to $z$ and $z'$ and, by proposition~\ref{ss-tot-discontinuity}
$\Gamma_{z}\cap L$ and $\Gamma_{z'}\cap L$ are totally disconnected.
In one of the connected components $V$ of $U\setminus L$, all the strong stable plaques
close to $z$ are met by $\Gamma_z$. The same holds for $\Gamma_{z'}$ and a component
$V'$ of $U'\setminus L$.
We claim that one can reduce to the case both components $V,V'$ are on the same side
of $L$. Indeed if this is not the case, the connected set $\Gamma$ intersects $L$
at another point $z''$. One can thus define three sets $V,V',V''$; among them, two
are on the same side of $L$.
Let $\tilde L$ be a strong stable plaque close to $z$ and $z'$ which intersects
$V$ and $V'$: all the plaques close to $\tilde L$ meet both $\Gamma_z$ and $\Gamma_{z'}$.
Let $q$ be a periodic point homoclinically related to $p$
and close to a point in $\Gamma_{z'}\cap \tilde L$.
The local strong stable manifold $W^{ss}_{loc}(q)$ is close to $\tilde L$
and the projection of $\Gamma_{z}$ by the center-unstable holonomy
on $\cW^{cs}_q$ is a connected compact set that intersects both sides
of $W^{ss}_{loc}(q)$. One deduces that this projection meets $\Gamma_{z}$
at a point $y\in H(p)\cap W^{ss}(q)$ which is distinct from $q$.
This contradicts our assumption.
\end{proof}
\medskip
Let us call~\emph{graph} of a plaque $\cW^{cs}_x$ a connected compact set of $\cW^{cs}$
which intersects each strong stable leaf of $\cW^{cs}_x$ in at most one point.
\begin{lema}\label{l.graph-uniform}
If for some point $x_0\in H(p)$,
the set $\cW^{cs}_{x_0}\cap H(p)$ is not totally disconnected,
then for each $x\in H(p)$, there exists a graph $\Gamma_x\subset \cW^{cs}_x\cap H(p)$
containing $x$ which meets all the strong stable plaques of $\cW^{cs}_x$ that intersect a small neighborhood of $x$.
\end{lema}
\begin{proof}
Let us consider a non trivial connected compact set $\Gamma\subset \cW^{cs}_{x_0}$.
By lemma~\ref{l.graph} this is a graph.
Let us consider a point $z\in \Gamma$ which is not an endpoint.
One also chooses a trapped plaque family $\mathcal{D}} \def\cJ{\mathcal{J}} \def\cP{\mathcal{P}} \def\cV{\mathcal{V}$ above $H(p)$ tangent to $E^{cs}$
whose plaques have a small diameter and are contained in the plaques of $\cW^{cs}$.
Consequently the connected component $\Gamma_z$
of $\Gamma\cap \mathcal{D}} \def\cJ{\mathcal{J}} \def\cP{\mathcal{P}} \def\cV{\mathcal{V}_z$ contains $z$ and has its endpoints inside $\overline{\mathcal{D}} \def\cJ{\mathcal{J}} \def\cP{\mathcal{P}} \def\cV{\mathcal{V}_z}\setminus\mathcal{D}} \def\cJ{\mathcal{J}} \def\cP{\mathcal{P}} \def\cV{\mathcal{V}_z$. We are aimed to build at each point $x\in H(p)$ a similar graph $\Gamma_x\subset \mathcal{D}} \def\cJ{\mathcal{J}} \def\cP{\mathcal{P}} \def\cV{\mathcal{V}_x$.
This will imply the conclusion of the lemma.
Let us first choose a periodic point $q$ homoclinically related to $p$ and close to $z$.
By projecting $\Gamma$ inside $\cW^{cs}_{q}$ along the center-unstable holonomy,
one deduces that $\mathcal{D}} \def\cJ{\mathcal{J}} \def\cP{\mathcal{P}} \def\cV{\mathcal{V}_q$ contains a graph $\Gamma_q\subset H(p)$
whose endpoints are inside $\overline{\mathcal{D}} \def\cJ{\mathcal{J}} \def\cP{\mathcal{P}} \def\cV{\mathcal{V}_q}\setminus \mathcal{D}} \def\cJ{\mathcal{J}} \def\cP{\mathcal{P}} \def\cV{\mathcal{V}_q$.
It contains a point close to $q$. Since $W^{ss}_{loc}(q)\setminus \{q\}$
is disjoint from $q$, this proves that $\Gamma_q$ contains $q$.
By the trapping property, for each $n\geq 0$, the connected component $\Gamma_{f^{-n}(q)}$
of $f^{-n}(\Gamma_q)\cap \mathcal{D}} \def\cJ{\mathcal{J}} \def\cP{\mathcal{P}} \def\cV{\mathcal{V}_{f^{-n}(q)}$ has also
its endpoints inside the boundary of $\mathcal{D}} \def\cJ{\mathcal{J}} \def\cP{\mathcal{P}} \def\cV{\mathcal{V}_{f^{-n}(q)}$.
As a consequence $H(p)$ contains a dense set of periodic points $y$ and inside each plaque
$\mathcal{D}} \def\cJ{\mathcal{J}} \def\cP{\mathcal{P}} \def\cV{\mathcal{V}_y$ there exists a graph $\Gamma_y$ containing $y$ whose endpoints belong to
$\overline{\mathcal{D}} \def\cJ{\mathcal{J}} \def\cP{\mathcal{P}} \def\cV{\mathcal{V}_y}\setminus \mathcal{D}} \def\cJ{\mathcal{J}} \def\cP{\mathcal{P}} \def\cV{\mathcal{V}_y$.
For any point $x\in H(p)$ there exists a sequence of periodic points $(y_n)$
converging towards $x$ such that the sequence of graphs $(\overline{\Gamma_{y_n}})$ converges
towards a connected compact set $\overline{\Gamma_x}$: by lemma~\ref{l.graph} this is a graph
and by construction its endpoints belong to $\overline{\mathcal{D}} \def\cJ{\mathcal{J}} \def\cP{\mathcal{P}} \def\cV{\mathcal{V}_y}\setminus \mathcal{D}} \def\cJ{\mathcal{J}} \def\cP{\mathcal{P}} \def\cV{\mathcal{V}_y$
as required.
\end{proof}
\bigskip
We are now able to finish the proof of the theorem.
\begin{proof}[\bf Proof of theorem~\ref{t.tot-discontinuity}]
We assume that the conclusion of the theorem does not holds: in particular, the lemma~\ref{l.graph-uniform} applies.
By theorem~\ref{t.whitney}, there exists two distinct points $x,y\in H(p)$ with $y\in W^{ss}(x)$.
By iterations one may assume that $y$ belongs to the strong stable plaque of $x$
in $\cW^{cs}_x$. By lemma~\ref{l.graph-uniform}, there exists a graph
$\Gamma_x\subset \cW^{cs}_x$ which contains $x$ and meets all the strong stable
plaques of points close to $x$ in $\cW^{cs}_x$.
One now argues as at the end of the proof of lemma~\ref{l.graph}:
if $q$ is a periodic point close to $y$, the projection of $\Gamma_x$
to $\cW^{cs}_q$ has to intersect $W^{ss}_{loc}(q)$ at a point close to $x$, hence different from $q$.
This contradicts the assumptions.
\end{proof}
\subsection{Construction of adapted plaques}
We now give a consequence of theorem~\ref{t.tot-discontinuity} giving plaques
adapted to the geometry of the classes along the center-stable plaques.
Let us consider an invariant compact set $K$ with a dominated splitting $E\oplus F$
and a trapped family tangent to $E$ such that the coherence holds for some constant $10\; \varepsilon>0$
(see lemma~\ref{l.uniqueness-coherence}).
Let $\widetilde \cW$ be another trapped family tangent to $E$ whose plaques have a small diameter
and such that for each $x\in K$ one has $\widetilde \cW_x\subset \cW_x$.
The coherence ensures that any plaque $\widetilde \cW_y$ that intersects the $5\varepsilon$-ball centered at $x$
inside $\cW_x$ is contained in $\cW_x$.
\begin{definition}
In this setting, a set $X\subset K$ that is contained in the $\varepsilon$-ball centered at a point $x\in K$ inside the plaque $\cW_x$
is said to be \emph{$\widetilde \cW$-connected} if the union of the plaques $\cW^{cs}_y$ for $y\in X$
is connected.
\end{definition}
\medskip
When the diameters of the plaques $\widetilde \cW^{cs}$ are small, the $\widetilde \cW^{cs}$-connected sets
have a small diameter.
\begin{prop}\label{p.box}
Let $f_0$ be a diffeomorphism, $H(p_{f_0})$ be a chain-recurrence class which is chain-hyperbolic
such that the bundles $E^{cs},E^{cu}$ are thin trapped and consider some neighborhoods $U$ of $H(p_{f_0})$, $\cU$ of $f_0$ in $\operatorname{Diff}^1(M)$ and a plaque family $(\cW_{f,x}^{cs})_{f\in \cU,x\in K_f}$ as provided by lemma~\ref{l.robustness}.
If for each $x\in H(p_{f_0})$, the set $H(p_{f_0})\cap \cW^{cs}_{x}$ is totally disconnected, then
for any $\eta>0$ small,
there exist smaller neighborhoods $\widetilde U\subset U$ of $H(p_{f_0})$ and $\widetilde \cU$ of $f_0$
and there are other plaque families $(\widetilde \cW_{f,x}^{cs})_{f\in \widetilde \cU,x\in \widetilde K_f}$
defined on the maximal invariant sets $\widetilde K_f$ in the closure of $\widetilde U$ for $f$,
satisfying the following properties for each $f\in \widetilde \cU$ and $x\in \widetilde K_f$:
\begin{itemize}
\item[--] The plaque $\widetilde \cW^{cs}_{f,x}$ is contained in $\cW^{cs}_{f,x}$.
\item[--] Any $\widetilde \cW^{cs}_f$-connected set of $K_f\cap\cW^{cs}_{f,x}$ which contains $x$
has diameter smaller than $\eta$.
\end{itemize}
\end{prop}
\begin{proof}
One considers a constant $\varepsilon>0$, two neighborhoods $U_\varepsilon$ of $H(p_{f_0})$
and $\cU_\varepsilon$ of $f_0$ which decrease to $H(p_{f_0})$ and $f_{0}$ as $\varepsilon$ goes to zero,
and a continuous collection of plaque family
$(\cW^{cs}_{\varepsilon,f})_{f\in \cU_\varepsilon}$
defined on the maximal invariant set $K_{\varepsilon,f}$ in the closure of $U_\varepsilon$.
We assume that these families are trapped, that each plaque
$\cW^{cs}_{\varepsilon,f,x}$ has diameter smaller than $\varepsilon$
and that for each $x\in K_{\varepsilon,f}$, the plaque $\cW^{cs}_{\varepsilon,f,x}$ is contained
$\cW^{cs}_{\varepsilon,f}$. Such plaque families are given by remark~\ref{r.nested}.
For $f\in \cU_{\varepsilon}$,
one makes the union $\Delta_f$ of the sets $\overline{\cW^{cs}_{\varepsilon,f,x}}$.
We claim that when $\varepsilon$ goes to zero, the supremum of the diameter of the connected components
of $\Delta_f$ (with respect to the center-stable topology) goes to zero.
Indeed, if this is not the case, one finds as limit set a non-trivial connected component of $H(p_{f_0})$
for $f_0$ and the center-stable topology, which contradicts our assumption.
The plaque family $(\widetilde \cW^{cs}_{f})$ is thus chosen to be $(\cW^{cs}_{\varepsilon,f})$
for some $\varepsilon$ small enough.
\end{proof}
\section{Introduction}
\subsection{Mechanisms classifying the dynamics}
In the direction to describe the long range behavior of trajectories
for ``most" systems (i.e. in a subset of the space of dynamics
which is residual, dense, etc.), a crucial goal is to identify any generic dynamical behavior.
It was briefly thought in the sixties that this could be realized
by the property of \emph{uniform hyperbolicity}. Under this
assumption, the limit set decomposes into a
finite number of disjoint (hyperbolic) transitive sets and the asymptotic
behavior of any orbit is described by the dynamics in those finitely many transitive sets (see \cite{S}). Moreover, under
the assumption of hyperbolicity one obtains a satisfactory (complete)
description of the dynamics of the system from a topological and statistical point of view.
Hyperbolicity was soon realized to be a less universal
property than what one initially thought: the space of dynamics
contains open sets of non-hyperbolic systems. We are now aimed to understand
how the space of systems can be organized according to the different kinds of
dynamical behavior they exhibit.
\paragraph{a- Characterization of non-hyperbolic systems.}
Dynamicists were lead to look for obstructions to hyperbolicity.
For instance any non-hyperbolic diffeomorphism can be approximated in the $C^1$-topology
by a system having a non-hyperbolic periodic orbit (see~\cite{M}, \cite{Aoki}, \cite{H1}).
Since Poincar\'e we know that some very simple configurations (such that the existence of a homoclinic orbit) could be the source of a widely complex behavior.
One has identified two simple obstructions for hyperbolicity which generate rich dynamical phenomena and they have played a crucial role in the study of generic non-hyperbolic behavior:
\begin{enumerate}
\item {\it heterodimensional cycle}: the presence of two
periodic orbits of different stable dimension linked through the
intersection of their stable and unstable manifolds (see \cite{AS}, \cite{Sh}, \cite{D1});
\item {\it homoclinic tangency}: the existence of a non-transversal
intersection between the stable and unstable manifolds of a periodic orbit (see \cite{N1}, \cite{N2}, \cite{PT}, \cite{PV}, \cite{BD}).
\end{enumerate}
These obstructions are relevant due to several dynamical consequences that they involve:
the first one is related to the existence of {non-hyperbolic robustly transitive systems} (see \cite{D1}, \cite{BDPR}, \cite{BDP});
the second one generates cascade of bifurcations, is related to
the existence of residual subsets of diffeomorphisms displaying infinitely many periodic attractors (see \cite {N3}) and to the local variations of entropy for surface diffeomorphisms (see~\cite{PS2}).
Another important property is that these obstructions are not isolated in the $C^1-$topology, and
sometimes, there are not isolated in a strong way: $i)$ among $C^2$-surface diffeomorphisms, any system with a homoclinic tangency is limit of an open set of diffeomorphisms having homoclinic
tangencies associated to hyperbolic sets (see \cite{N3}); $ii)$ among $C^1$-diffeomorphisms, any system with a heterodimensional cycle is limit of an open set of diffeomorphisms having heterodimensional cycles associated to hyperbolic sets of different indexes (see \cite{BDKS} and section \ref{ss.consequences}).
\medskip
In the 80's Palis conjectured (see \cite{P}, \cite{PT}) that these two bifurcations are the main
obstructions to hyperbolicity:
\begin{palis-conjecture}
Every $C^r$ diffeomorphism of a compact manifold can be $C^r$
approximated by one which is hyperbolic or by one exhibiting a
heterodimensional cycle or a homoclinic tangency.
\end{palis-conjecture}
This conjecture may be considered as a starting point to obtain a generic description
of $C^r$-diffeomorphisms. If it turns out to be true, we may focus on the two bifurcations
mentioned above in order to understand the dynamics.
\paragraph{b- Mechanisms \emph{versus} phenomena.}
To elaborate the significance of this conjecture, we would like to recast it in terms of mechanisms and dynamical phenomena.
\smallskip
By a \emph{mechanism}, we mean a simple dynamical configuration for one diffeomorphism
(involving for instance few periodic points and their invariant manifolds)
that has the following properties:
\begin{itemize}
\item[--] it {\em ``generates itself''}: the system exhibiting this configuration is not isolated.
In general the mechanism is a co-dimensional bifurcation, but it produces a cascade of diffeomorphisms sharing the same configuration;
\item[--] it {\em ``creates or destroys''} rich and different dynamics for nearby systems
(for instance horseshoes, cascade of bifurcations, entropy's variations).
\end{itemize}
Following this definition, homoclinic tangencies and heterodimensional cycles are mechanisms in any $C^r-$topology for $r\geq 1.$
\medskip
In our context a {\em dynamical phenomenon} is any dynamical property
which provides a good global description of the system
(like hyperbolicity, transitivity, minimality, zero entropy, spectral decomposition) and which occurs on a ``rather large" subset of systems.
We relate these notions and say that
{\em a mechanism is a complete obstruction to a dynamical phenomenon} when:
\begin{itemize}
\item[--] it is an {\em obstruction}: the presence of the mechanism prevents the phenomenon to happen;
\item[--] it is {\em complete}: each system that does not exhibit the dynamical phenomenon is approximated by another displaying the mechanism.
\end{itemize}
In other words, a mechanism (or a dynamical configuration) is a complete obstruction to a dynamical phenomena, if it not only prevents the phenomenon to happen but it also generates itself creating rich dynamics and it is common in the complement of the prescribed dynamical phenomenon. Following this approach, Palis's conjeture can be recasted:
\begin{palis-conjecture-recast}
Heterodimensional cycles and homoclinic tangencies are a complete obstruction to hyperbolicity.
\end{palis-conjecture-recast}
Let us give some examples where a dichotomy mechanism / phenomenon has been
proved or conjectured.
\begin{itemize}
\item[--] \emph{Homoclinic bifurcations / hyperbolicity}. This corresponds to the previous
conjecture and is known in dimensions 1 and 2 for the $C^1$-topology, see~\cite{PS1}.
\item[--] \emph{Transverse homoclinic intersection / robust zero topological entropy}.
It has been proved in any dimension for the $C^1$-topology, see~\cite{BGW}, \cite{C1}.
\item[--] \emph{Trapping region / residual transitivity}. Any $C^1$-generic diffeomorphism
$f$ is either transitive or sends a compact set into its interior, see~\cite{BoCr}.
\item[--] \emph{Homoclinic tangency / global dominated splitting}.
After a $C^1$-perturbation any diffeomorphism exhibits a homoclinic tangency or its limit dynamics holds a (robust) dominated splitting with one-dimensional central bundles, see~\cite{CSY}.
\end{itemize}
\paragraph{c- Main result.}
In the present paper, we prove the mentioned conjecture in the $C^1-$topology for
a weaker notion of hyperbolicity.
\begin{definition*}
A diffeomorphism is {\em essentially hyperbolic} if it has
a finite number of transitive hyperbolic attractors and if the union of their basins of attraction is open and dense in the manifold.
\end{definition*}
The essential hyperbolicity recovers the notion of Axiom A: most of the trajectories (in the Baire category) converge to a finite number of transitive attractors that are well described from a both topological and statistical point of view. Moreover, the dynamics in those hyperbolic attractors, govern the dynamics of the trajectories that converge to them.
In fact, in an open and dense subset the forward dynamics does not distinguish the system
to an Axiom A diffeomorphism.
\medskip
Now, we state our main theorem:
\begin{main-theorem}
Any diffeomorphism of a compact manifold can be $C^1-$approximated by another diffeomorphism which:
\begin{enumerate}
\item either has a homoclinic tangency,
\item or has a heterodimensional cycle,
\item or is essentially hyperbolic.
\end{enumerate}
\end{main-theorem}
Roughly speaking we proved that {\em homoclinic tangencies and heterodimensional cycles are the $C^1-$complete obstructions for the essential hyperbolicity.}
\begin{remark}
The proof gives a more precise result: inside the open set of diffeomorphisms that are not
limit in $\operatorname{Diff}^1(M)$ of diffeomorphisms exhibiting a homoclinic tangency or a heterodimensional cycle,
the essentially hyperbolic diffeomorphisms contain a G$_\delta$ dense subset.
As a consequence, one may also require that these diffeomorphisms are also essentially hyperbolic for $f^{-1}$.
\end{remark}
\paragraph{d- Mechanisms \emph{associated to} phenomena.}
In contrast to the previous dichotomies, a mechanism could also be the key for a rich (semi-global) dynamics.
We say that {\em a mechanism is associated to a dynamical phenomenon} if the following holds:
\begin{itemize}
\item[--] the systems exhibiting the dynamical phenomenon can be approximated
by ones displaying the mechanism;
\item[--] the ones exhibiting the mechanism generate (at least locally) the dynamical phenomenon.
\end{itemize}
As in the notion of complete obstruction, a mechanism is associated to a dynamical phenomenon not only if it generates it but if any time that the phenomenon appears by small perturbations the mechanism is created. Thus a goal would be to establish a dictionary between mechanisms
and (semi-global) dynamical phenomena.
\medskip
Let us mention some known examples.
\begin{itemize}
\item[--] \emph{Transverse homoclinic intersections / non-trivial hyperbolicity.}
On one hand, systems exhibiting a transversal homoclinic point of a hyperbolic periodic point has horseshoes associated to them; on the other hand horseshoes displays transversal homoclinic points (see for instance \cite{B} and \cite{S}).
\item[--] \emph{Heterodimensional cycles / non-hyperbolic $C^1$-robust transitivity.}
On the one hand, systems displaying heterodimensional cycles are
$C^1-$dense in the interior of the set of non-hyperbolic transitive diffeomorphisms
(see for instance~\cite{GW});
on the other hand, the $C^r-$unfolding of a (co-index one) heterodimensional cycles creates maximal invariant robustly transitive non-hyperbolic sets (see \cite{D1}).
\item[--] \emph{Homoclinic tangencies / residual co-existence of infinitely many independent pieces.} On the one hand, the existence of a homoclinic tangency
for $C^2$ surface diffeomorphisms, sectionally dissipative tangencies in higher dimension or the existence of a homoclinic tangencies
combined with heterodimensional cycles for $C^1$ diffeomorphisms may imply
locally residually the co-existence of infinitely many attractors (Newhouse phenomenon),
see~\cite{N3},~\cite{PV} and~\cite{BD}. On the other hand, it is conjectured that
any diffeomorphism exhibiting infinitely many attractors can be approximated by a diffeomorphism
which exhibits a homoclinic tangency (see for instance \cite{Bo}).
\end{itemize}
Related to the above conjecture in \cite{Bo}, it was proved in \cite{PS4} that for smooth diffeomorphisms, the co-existence of infinitely many attractors in a ``sectionally dissipative region of the manifold'' implies the creation of sectionally dissipative tangencies by $C^1$ perturbations (see corollary 1.1 in \cite{PS4} for details). In a more general framework as a byproduct of the proof of the main theorem, we prove the following.
\smallskip
\noindent{\bf Theorem. }{\em The co-existence of infinitely many attractors implies that either heterodimensional cycles or homoclinic tangencies can be created by $C^1$ perturbations.}
\smallskip
\noindent See item c- in section~\ref{itinerary} for details and proof.
\paragraph{e- Robust mechanisms}
The mechanisms we presented are simple configurations of the dynamics but
as bifurcations are also one-codimensional.
From the deep studies of the role of cycles and tangencies, Bonatti and Diaz have proposed to enrich Palis's conjecture and introduced the notion of
{\em robust heterodimensional cycles} and \emph{robust homoclinic tangencies},
meaning that now the mechanisms involve non-trivial transitive hyperbolic sets
instead of periodic orbits so that the cycles and tangencies may occur on an open set of diffeomorphisms.
From~\cite{BD2} the main theorem can be restated in the following way:
\medskip
\noindent{\bf Main theorem revisited. }{\em Any diffeomorphism of a compact manifold
can be $C^1-$approximated by another diffeomorphism which either is essentially hyperbolic,
or has a homoclinic tangency, or has a robust heterodimensional cycle.}
\medskip
We also refer to~\cite{Bo} for a complementary program about the dynamics of $C^1$-diffeomorphisms.
\subsection{Itinerary of the proof}\label{itinerary}
The proof focuses on diffeomorphisms far from homoclinic bifurcations and
consists in three parts.
\begin{itemize}
\item We first conclude that the quasi attractors (the Lyapunov stable chain-recurrence classes) are ``topologically hyperbolic'': they are partially hyperbolic homoclinic classes with a one-dimensional ``stable'' center bundle and
the union of their basin of attraction is dense in the manifold.
\item We then develop a series of perturbation techniques which ensure that topologically hyperbolic quasi-attractors are uniformly hyperbolic attractors.
\item At the end we prove that the union of the quasi-attractors is closed. With the second point this gives the finiteness of the hyperbolic attractors.
\end{itemize}
A diffeomorphism which satisfies the first and the third property could be called ``essentially topologically hyperbolic''.
\paragraph{a- Topological hyperbolicity.} From the start, we concentrate the study on quasi-attractors. Following \cite{C1,C2} (see theorems \ref{t.homoclinic} and \ref{t.aperiodic} below), it is concluded that $C^1-$far from homoclinic bifurcations, the aperiodic chain-recurrent classes are partially hyperbolic with a one-dimensional central bundl, and
the homoclinic classes are partially hyperbolic with their central bundles being at most two-dimensional (however the hyperbolic extremal subbundles may be degenerated).
Moreover, a special type of dynamics has to hold along the central manifolds:
the center stable is chain-stable and the center unstable is chain-unstable.
We define a weak notion of topological hyperbolicity that we call \emph{chain-hyperbolicity}:
this is suitable for our purpose since in some cases the chain-hyperbolicity is robust under perturbations.
(See definition \ref{d.chain-hyperbolic} for details and justification of the names topological hyperbolicity and chain-hyperbolicity).
From corollary \ref{c.aperiodic} it is concluded that aperiodic classes can not be attractors and therefore they are out of our picture. For homoclinic classes, whenever the partially hyperbolic splitting has two extremal hyperbolic subbundles, corollary \ref{c.homoclinic} concludes that the central bundle is one-dimensional subbundle and chain-stable otherwise a heterodimensional cycle is created.
\paragraph{b- Uniform hyperbolicity.}
At this step, a first dichotomy is presented (see corollary \ref{c.whitney}): either the quasi-attractor is contained in a normally hyperbolic submanifold (and from there one concludes the hyperbolicity, see corollary \ref{c.codim-one}) or the strong stable foliation is non-trivially involved in the dynamic, meaning that at least two different points $x,y$ in the class share the same local strong stable leaf. In this second case (see theorem \ref{t.position}), we will perturb the diffeomorphism in order to
obtain a {\em strong connection} associated to a periodic point, i.e. a periodic point whose strong stable and unstable manifolds intersect, see definition \ref{strong-int}; in particular, assuming that the quasi-attractor is not hyperbolic, a heterodimensional cycle can be created (see proposition \ref{p.strong-connection}).
To perform the perturbations, one has to discuss the relative position between two unstable leaves after projection by the strong stable holonomy: the position types are introduced in definition~\ref{definition-cases}. In particular, by analyzing the geometry of quasi-attractors one can reduce to the case the points $x,y$ belong to stable or to unstable manifolds of some periodic orbits.
Improving~\cite{Pu1} and~\cite{Pu2},
three different kinds of perturbations may be performed. They correspond to the following cases:
\begin{itemize}
\item[--] $x,y$ belong to unstable manifolds and their forward orbits have fast returns close to
$x$ or $y$.
\item[--] $x,y$ belong to unstable manifolds and their forward orbits have slow returns close to $x$ or $y$.
\item[--] $x,y$ belong to a stable manifold.
\end{itemize}
The two first cases are covered by theorem \ref{t.unstable} and the last one by theorem \ref{t.stable}.
To perform these perturbations one needs to control how the geometry of the class changes for any perturbed map; we prove (see proposition~\ref{p.continuation}) that whenever the perturbation of the homoclinic class does not display strong connection associated to periodic points then it is possible to get a well defined continuation for the whole class.
\paragraph{c- Finiteness of the attractors.}
The delicate point is to exclude the existence of an infinite number of sinks.
This is done by proving that for any non-trivial chain-recurrence classes, the extremal subbundles
are hyperbolic. We thus consider the splittings $E^s\oplus E^{cu}$ or $E^s\oplus E^{cs}\oplus E^{cu}$,
where $E^{cs},E^{cu}$ are one-dimensional, and in both cases we prove that $E^{cu}$ is hyperbolic.
The first case follows from results in \cite{PS4}. In the second case, the hyperbolicity of the center unstable subbundle follows for a more detailed understanding of the topological and geometrical structure of the homoclinic class (see theorem \ref{t.2D-central}). In fact, from being far from heterodimensional cycles, it is concluded that the the class is totally disconnected along the center stable direction (see theorems \ref{t.tot-discontinuity}) and from there a type of geometrical Markov partition is constructed (see proposition \ref{p.box}); this allows to use $C^2-$distortion arguments to conclude hyperbolicity of $E^{cu}$ as in~\cite{PS1} and \cite{PS4}.
After it is concluded that the chain-recurrence classes are partially hyperbolic with non-trivial extremal hyperbolic subbundles, the finiteness follows quite easily (see section \ref{ss.finiteness}).
\paragraph{Structure of the paper.}
In section \ref{s.classes} it is proved that the chain-recurrence classes for systems far from homoclinic bifurcations are ``topologically hyperbolic''.
Moreover, we stated there all the theorems (proved in the other sections)
needed to conclude the main theorem, which is done in subsection \ref{ss.quasi-attractor}.
In section \ref{s.weak-hyperbolicity} we give a general study of the chain-hyperbolic classes and their topological and geometrical structures.
This allows to obtain the continuation of some partially hyperbolic classes (done in section \ref{s.continuation}), and to introduce the notion of boundary points for quasi-attractors (done in section \ref{s.boundary}). In sections \ref{proofjointint} and \ref{p-nontransversal} are stated and proved the new perturbations techniques that hold in the $C^{1+\alpha}-$topology. In sections \ref{s.2D-central} and \ref{s.2D-central2} are studied partially hyperbolic homoclinic classes with a two-codimensional strong stable bundle, first analyzing their topological and geometrical structure and latter their hyperbolic properties.
\subsection{Some remarks about new techniques and $C^r-$versions of the main theorem}
We would like to highlight many of the new techniques developed in the present paper and that can be used in other context.
\paragraph{\rm\em 1- Chain-hyperbolicity.}
We introduce the notion of chain-hyperbolic homoclinic class which generalizes the locally maximal hyperbolic sets. It allows to include some homoclinic classes having hyperbolic periodic points
with different stable dimensions, provided that at some scale, a stable dimension is well-defined.
We recover some classical properties of hyperbolic sets: the local product structure, the stability under perturbation, the existence of (chain) stable and unstable manifolds. See section~\ref{s.weak-hyperbolicity}.
\paragraph{\rm\em 2- Continuation of (non necessarily hyperbolic) homoclinic classes.}
It is well known that isolated hyperbolic sets are stable under perturbation and have a well
defined and unique continuation. We extend this approach to certain partially hyperbolic sets
which are far from strong connections.
This is done by extending the continuation of their hyperbolic periodic points to their closure,
a technique that resembles to the notion of holomorphic motion. See section~\ref{s.continuation}.
\paragraph{\rm\em 3- Geometrical and topological properties of partially hyperbolic attractors.}
We study the geometrical structure of partially hyperbolic attractors
with a one-dimensional central direction in terms of the dynamics of the strong stable foliation.
For instance:
\begin{itemize}
\item[--] It is presented a dichotomy proving that a homoclinic class is
either embedded in a submanifold of lower dimension of the ambient space or
one can create a strong connection (maybe after a perturbation). See theorems~\ref{t.tot-discontinuity} and~\ref{t.position}.
\item[--] In certain cases it is introduced the notion of stable boundary points of a partially hyperbolic homoclinic class (extending a classical notion for hyperbolic surfaces maps) which permits us to control the bifurcations that holds after perturbations. See proposition \ref{p.boundary} and lemma \ref{l.boundary1}.
\item[--] If they are no (generalized) strong connection,
it is proved that the homoclinic class is totally disconnected along its stable leaves. See theorem~\ref{t.tot-discontinuity}.
\item[--] The total disconnectedness mentioned above, allows us to introduce kind of
Markov partitions for non-hyperbolic partially hyperbolic classes. See proposition~\ref{p.box}.
\end{itemize}
\paragraph{\rm\em 4- Hyperbolicity of the extremal subbundles.}
For invariant compact sets having a dominated splitting $E\oplus F$ with $\dim(F)=1$,
\cite{PS1} and \cite{PS4} have developed a technique which allows to prove that $F$ is hyperbolic
provided $E$ is either uniformly contracted or one-dimensional.
We extend this result for partially hyperbolic systems with a $2$-dimensional central bundle,
that is when $E$ is only ``topologically contracted". See section~\ref{s.2D-central2}.
\paragraph{\rm\em 5- New perturbation techniques.} It is developed new perturbation techniques suitable for partially hyperbolic sets with one-dimensional central directions. See theorems~\ref{t.unstable} and~\ref{t.stable}. We want to point out, that these perturbations hold in the $C^{1+\alpha}-$topology. Those perturbation resemble the $C^1-$connecting lemma but since in the present context a better understanding of the dynamic is available, then the perturbation can be perform in the $C^{1+\alpha}-$topology.
\paragraph{\rm\em 6- Consequences for hyperbolic dynamics.} Previous highlighted techniques can be formulated for hyperbolic attractors and have consequences
in terms of topological and geometrical structure. See theorems~\ref{t.tot-discontinuity} and~\ref{t.position}.
\paragraph{\rm\em 7- Generic structure of partially hyperbolic quasi-attractors.}
A byproduct of the proof shows (see theorem~\ref{t.consequences}) that for $C^1$-generic diffeomorphisms,
any quasi-attractor which has a partially hyperbolic structure with a one-dimensional
central bundle contains periodic points of different stable dimension.
\bigskip
We want to emphasize that many of the results contained in the present paper work in the $C^r-$category
for any $r\geq 1$ or for $r=1+\alpha$ with $\alpha\geq 0$ small.
For instance, theorems \ref{t.position}, \ref{t.unstable} and \ref{t.stable} hold in the $C^{1+\alpha}-$topology.
This allows to prove (see the remark~\ref{r.position}, item \ref{i.position}) a partial version of Palis conjecture in the $C^{1+\alpha}-$category
when one restricts to partially hyperbolic attractors with one-dimensional center direction).
\begin{theorem*}
For any $C^2$ diffeomorphism $f$ of a compact manifold and any ``topologically hyperbolic attractor" $H(p)$ (i.e. which satisfies the assumptions stated in theorem~\ref{t.position}), there exists $\alpha>0$ with the following property.
For any $\delta>0$, there exists $C^{1+\alpha}$-perturbations $g$ of $f$ such that
\begin{itemize}
\item[--] either the homoclinic class $H(p_g)$ associated to the continuation $p_g$
of $p$ is hyperbolic,
\item[--] or there exists a periodic orbit $O$ of $g$ which has a strong homoclinic intersection
and one of its Lyapunov exponents has a modulus smaller than $\delta$.
\end{itemize}
\end{theorem*}
We don't know however if under the conclusions of this theorem it is possible to create a heterodimensional cycle by a $C^{1+\alpha}$-perturbation of
the diffeomorphism.
\section{Periodic stable leaves: proof of theorem \ref{t.stable}}
\label{proofjointint}
In this section we prove theorem~\ref{t.stable} and proposition~\ref{p.generalized-strong-connectionCr}.
Let us consider:
\begin{itemize}
\item[1)] A diffeomorphism $f_0$
and a homoclinic class $H(p_{f_0})$ which is a chain-recurrence class
endowed with a partially hyperbolic splitting $E^s\oplus E^c\oplus E^u$ where
$E^c$ is one-dimensional and $E^s\oplus E^c$ is thin-trapped.
\item[2)] Some $\alpha\in [0,1)$, a $C^{1+\alpha}$-diffeomorphism $f$ that is $C^1$-close to $f_0$,
an open neighborhood $\cV\subset \operatorname{Diff}^{1+\alpha}(M)$ of $f$ and some collections of hyperbolic periodic points $q_{f}$, $\{p_{n,f}^x\}_{n\in \NN}$ and $\{p^{y}_{n,f}\}_{n\in \NN}$ for $f$
such that the following properties hold.
\begin{itemize}
\item[--] For $g\in \cV$, the continuations $q_{g}$, $p_{n,g}^x$, $p_{n,g}^y$
exist and are homoclinically related to $p_{g}$.
\item[--] For each $g\in \cV$, the sequences $(p_{n,g}^x)$ and $(q_{n,g}^y)$ converge
towards two distinct points $x_g,y_g$ in $H(p_g)\cap W^s_{loc}(q_g)$
such that $y_g$ belongs to $W^{ss}_{loc}(x_g)$.
\item[--] The maps $g\mapsto x_g,y_g$ are continuous at $f$.
\end{itemize}
\end{itemize}
We will show that if $\alpha\geq 0$ is small, then
there exists a diffeomorphism $g\in \cV$ whose homoclinic class $H(p_g)$ has a strong homoclinic intersection.
\begin{propo}\label{p.jointintegrable}
For any diffeomorphism $f_0$ and any homoclinic class $H(p_{f_0})$ satisfying the assumption 1) above, there exists $\alpha_0\in (0,1)$ and a $C^1$-neighborhood $\cU$ of $f$
with the following property.
\noindent
For any $\alpha\in [0,\alpha_0]$, any diffeomorphism $f$,
any neighborhood $\cV\subset \operatorname{Diff}^{1+\alpha}(M)$ and any maps $g\mapsto x_g,y_g$ satisfying the assumption 2), there exists a transverse intersection $z\in W^s(q_{f})\cap W^{u}_{loc}(q_{f})\setminus \{q_{f}\}$
and an arc of diffeomorphisms $(g_t)_{t\in [-1,1]}$ in $\cV$ such that
\begin{itemize}
\item[--] for each $t\in [-1,1]$, considering the
(unique) continuation $z_t$ of $z$ for $g_t$,
the center stable plaque $\mathcal{D}} \def\cJ{\mathcal{J}} \def\cP{\mathcal{P}} \def\cV{\mathcal{V}^{cs}_{z_t}$ intersects $W^u_{loc}(x_{g_t})$
and $W^u_{loc}(y_{g_t})$ at some points $\hat x_t$ and $\hat y_t$;
\item[--] considering an orientation of the central bundle in a neighborhood of $q$,
one has $$\hat y_{-1}\in \mathcal{D}} \def\cJ{\mathcal{J}} \def\cP{\mathcal{P}} \def\cV{\mathcal{V}^{cs,-}_{\hat x_{-1}} \text{ and } \hat y_{1}\in \mathcal{D}} \def\cJ{\mathcal{J}} \def\cP{\mathcal{P}} \def\cV{\mathcal{V}^{cs,+}_{\hat x_{1}}.$$
\end{itemize}
\end{propo}
\medskip
Let us conclude the proof of theorem~\ref{t.stable}.
By construction and lemma~\ref{l.bracket0}, for each $t\in [-1,1]$ the points
$z_t,x_t,y_t$ belong to the homoclinic class $H(p_{g_t})$.
Moreover one can find for each $n\in \NN$ two hyperbolic periodic points
$\hat p_{n,g}^x$ and $\hat p_{n,g}^y$ whose continuations exists for every $g\in \cV$,
are homoclinically related to $p_{g}$ and are arbitrarily close to the intersections
$\hat x_g,\hat y_g$
between $W^{s}_{loc}(z_{g})$ and $W^u_{loc}(x_g)$ or $W^u_{loc}(y_g)$ respectively.
By corollary~\ref{c.continuation2}, one can assume that the hyperbolic points
$\hat p_{n,g}^x$ and $\hat p_{n,g}^y$ are the hyperbolic continuations of points of $\cP$.
For $n$ large, $W^u_{loc}(\hat p^y_{n})$ intersects $\cW^{cs,-}_{\hat p^x_{n}}$ for $g_{-1}$
and $\cW^{cs,+}_{\hat p^x_{n}}$ for $g_{1}$.
One can thus apply lemma~\ref{l.ordering} and obtain a diffeomorphism $g\in \cV$ which has a strong
homoclinic intersection.
Note that the neighborhood $\cV$ of $f$ can be taken arbitrarily small. As a consequence
the perturbation $g$ is arbitrarily $C^{1+\alpha}$-close to $f$.
Hence the proposition implies theorem \ref{t.stable}.
\subsection{An elementary $C^{1+\alpha}$-perturbation lemma}
\label{elementary}
The perturbations in sections~\ref{proofjointint} and~\ref{p-nontransversal} will be realized through the following lemma.
\begin{lemma}\label{l.perturbation}
Let us consider a $C^{1+\alpha}$ map $v_0\colon \RR^d\to \RR^\ell$,
and two numbers $\widehat D>2 D>0$.
Then, there exists a $C^{1+\alpha}$-map $v\colon \RR^d\to \RR^\ell$
which coincides with $v_0$ on the ball $B(0,D)$ and with $0$
outside the ball $B(0,\widehat D)$
and whose $C^{1+\alpha}$-size is arbitrarily small if
the $C^{1+\alpha}$-size of $v_0$ and the quantity
$\widehat D^{-(1+\alpha)} \sup_{B(0,\widehat D)}\|v_0\|$ are small.
\end{lemma}
\begin{proof}
One chooses a smooth bump map $\rho\colon \RR^d\to [0,1]$
which coincides with $0$ outside $B(0,\frac 2 3 \widehat D)$ and with $1$ inside
$B(0,D)$. The map $v$ is then defined by $v=\rho.v_0$.
When $\alpha>0$, we define $\operatorname{Lip}_\alpha(h)$ the $\alpha$-H\"older size of a map $h$, that is $$\operatorname{Lip}_\alpha(h)=\sup_{x\neq y}\frac{\|h(x)-h(y)\|}{\|x-y\|^\alpha}.$$
We then denote by $A, A'$ the $C^0$ norm of $v_0,Dv_0$ and by
$A_\alpha,A'_{\alpha}$ the $\alpha$-H\"older sizes of $v_0,Dv_0$ on $B(0,\widehat D)$. There exists a universal constant $C>0$ such that for any $\alpha\in (0,1]$ one has
$$\operatorname{Lip}_\alpha(\rho)\leq C.\widehat D^{-\alpha},$$
$$\operatorname{Lip}_\alpha(D\rho)\leq C.\widehat D^{-(1+\alpha)}.$$
From inequalities above, it is easily to check that when the $C^{1+\alpha}$ size of $v_0$ is small,
the $C^{1+\alpha}$-size of $v$ is controlled by $A \widehat D^{-(1+\alpha)}$:
\begin{itemize}
\item[--] The $C^0$ norm of $v$ is smaller than $A$.
\item[--] The $C^0$-norm of $Dv$ is bounded by $A\operatorname{Lip}_1(\rho)+A'\leq CA\widehat D^{-1}+A'$.
\item[--] When $\alpha>0$, the $\alpha$-H\"older constant of $Dv$ is bounded by
\begin{equation}\label{e.control}
A_\alpha\operatorname{Lip}_1(\rho)+A\operatorname{Lip}_\alpha(D\rho)+A'_\alpha+\sup_{B(0,\frac 2 3 \widehat D)}\|Dv_0\|\;\operatorname{Lip}_\alpha(\rho).
\end{equation}
\end{itemize}
Observe that the three first terms in~(\ref{e.control}) are small when $A'_\alpha$
and $A\widehat D^{-(1+\alpha)}$ are small.
Indeed the usual convexity estimate gives
$$A_\alpha\leq C A^{1/(1+\alpha)}{A'_\alpha}^{\alpha/(1+\alpha)}.$$
For any $x\in B(0,\frac 2 3 \widehat D)$ one has
\begin{equation*}
\begin{split}
\|Dv_0(x)\|&\leq C.\left[ \sup_{\|u\|=\widehat D/3}
\frac {\|v_0(x+u)-v_0(x)\|}{\|u\|}+\sup_{y\in B(x,\widehat D/3)}
\|Dv_0(y)-Dv_0(x)\|\right]\\
&\leq 3C.(A\widehat D^{-1}+A'_\alpha\widehat D^\alpha).
\end{split}
\end{equation*}
The last term in~(\ref{e.control}) is thus smaller than $A\widehat D^{-(1+\alpha)}+A'_\alpha$.
When the $C^{1+\alpha}$-size of $v_0$ is small, $A'_\alpha$ is small and the lemma follows.
\end{proof}
\begin{remark}\label{r.perturbation}
When $v_0(0)=0$, for proving that the quantity
$\widehat D^{-(1+\alpha)} \sup_{B(0,\widehat D)}\|v_0\|$ is small
it is enough to show that
$\widehat D^{-\alpha} \sup_{B(0,\widehat D)}\|Dv_0\|$ is small.
\end{remark}
\subsection{Preliminary constructions}\label{ss.prel}
To simplify the presentation, one will assume that $q_0$ coincides with $p_0$
and is fixed by $f_0$.
\paragraph{The smoothness bound $\alpha_0$.}
We denote also by $\lambda_c\in (0,1)$ an upper bound for the contraction along $E^c$ and by $\lambda_u>1$ a lower bound for the expansion along the bundle $E^u$.
We choose $\alpha_0>0$ small so that
$$\lambda_u^{\alpha_0}\max(\lambda,\lambda_c)<1,$$
$$\|Df_0^{-1}\|^{\alpha_0}\lambda<1.$$
In particular, one can consider $\rho\in (0,1)$ such that
$$\lambda^{1/\alpha_0}<\rho<\|Df_0^{-1}\|^{-1}.$$
\paragraph{The neighborhoods $V_1,V_2$ of $x$.}
Once the smoothness $\alpha\in[0,\alpha_0]$ and the neighborhood $\cV$
have been fixed, one introduces a continuity point $f'\in \cV$
for both maps $g\mapsto x_g,y_g$.
Let $z_{f'}\in W^s(p_{f'})\cap W^u_{loc}(p_{f'})$ be a transverse homoclinic point of the orbit of $p_{f'}$ that does not belong to the orbit of $x_{f'}$ or $y_{f'}$.
We choose two small open neighborhoods $V_1,V_2$
of $x_{f'}$, such that $\bar V_2\subset V_1$.
Choosing them small enough,
the orbit of the intersection $\overline{V_1}\cap W^s_{loc}(p_{f'})$
is disjoint from the orbit of $y_{f'}$ and $z_{f'}$.
Since $f'$ is a continuity point of $g\mapsto x_g,y_g$,
for any diffeomorphism $g\in \cV$ close to $f'$,
the point $x_g$ still belongs to $V_2$ and
the orbit of the intersection $\overline{V_1}\cap W^s_{loc}(p_{g})$
is still disjoint from the orbit of the continuations $y_g,z_g$.
\paragraph{The diffeomorphism $f$.}
We choose a diffeomorphism $f\in \cV$ arbitrarily close to $f'$.
One can require that $f$ is of class $C^\infty$ and that there is no resonance
between the eigenvalues of the linear part associated to the orbit of $p_f$.
As a consequence of Sternberg linearization theorem, the dynamics in a neighborhood
of the orbit of $p_f$ can be linearized by a smooth conjugacy map.
In order to simplify the notations we will denote $p=p_f$, $q=q_f$, $x=x_f$, $y=y_f$.
\paragraph{Local coordinates.}
One can find a small neighborhood $B$ of $p$ and a $C^r$-chart $B\to \RR^d$
which linearizes the dynamics and maps $p$ on $0$ and
the local manifolds $W^{ss}_{loc}(p), W^s_{loc}(p), W^u_{loc}(p)$ inside the coordinate planes $\RR^s\times \{0\}^{u+1}$, $\{0\}^s\times \RR\times \{0\}^u$
and $\{0\}^{s+1}\times \RR^u$, where $s,u,d$ denotes the dimension of $E^{ss}, E^u$
and $M$ respectively. The coordinates in the chart are written $(\bar x,\bar y, \bar z)\in \RR^s\times \RR\times \RR^u$.
The map $f$ viewed in the chart
is thus a linear map $A=A_s\times A_c\times A_u$ of $\RR^d$ which preserves these coordinate planes.
Replacing $x,y$ by iterates, one can assume that their forward orbits
are contained in $B$.
\paragraph{The local stable disk $D$.}
Let $z_0$ be the transverse homoclinic point of the orbit of $p$ for $f$
that is the continuations of $z_{f'}$. For $n\geq 0$ we also define $z_{-n}=f^{-n}(z_0)$.
We can thus choose a small neighborhood $D$ of $z_0$ in $W^s(p)$ whose orbit
is disjoint from the orbits of $x$ and $y$.
Replacing $z_0$ by an iterate, one can assume that its backward orbit belongs to $B$.
The disk $D$ (or one of its backward iterates) endowed with its strong stable foliation
can then be linearized.
\begin{lemma}
By a $C^{1+\alpha}$-small perturbation of $f$ one may assume
furthermore that in the chart at $p$,
\begin{itemize}
\item[--] $D$ is contained in an affine plane parallel to
the local stable manifold $W^{s}_{loc}(p)$,
\item[--] the strong stable manifolds inside $D$ coincide with the affine planes
parallel to $W^{ss}_{loc}(p)$.
\end{itemize}
\end{lemma}
\begin{proof}
We choose a large integer $n\geq 1$.
The ball $W$ centered a $z_{-n}$ of radius $r=\lambda_u^{-n}$
does not intersect the local stable manifold of the orbit of $p$,
neither the iterates $z_{-k}$ for $k\neq n$.
We first rectify the disc $D$:
in the chart, the disc $f^{-n}(D)$ can be seen as the graph
of a map whose derivative has norm smaller than $\lambda^n$.
By the $\lambda$-lemma,
this graph is arbitrarily $C^{1+\alpha}$-close to
the linear plane $W^{s}_{loc}(p)$.
One can thus apply lemma~\ref{l.perturbation}: by a diffeomorphism supported inside
$W$ which fixes $z_{-n}$,
one can send a neighborhood of $z_{-n}$ inside $D$ in an affine plane
parallel to $W^s_{loc}(p)$.
By remark~\ref{r.perturbation}, this diffeomorphism is $C^{1+\alpha}$-close to the identity provided
that $\lambda^nr^{-\alpha}=(\lambda\lambda_u^{\alpha})^n$ is small, which is the case if $\alpha<\alpha_0$ and our choice of $\alpha_0$.
Assuming now that $D$ is contained in an affine plane parallel to $W^{s}_{loc}(p)$,
we denote by $W^{c}_{loc}(z_0)$ the affine space containing $z_0$ parallel to $\{0\}^s\times \RR\times\{0\}^u$.
We rectify the strong stable foliation inside $D$:
this is the image of the affine foliation parallel to $W^{ss}_{loc}(p)$
by a diffeomorphism $\Phi$ of the form
$$\Phi\colon(\bar x,\bar y,\bar z)\mapsto (\bar x, \varphi(\bar x,\bar y),\bar z),$$
which fixes $z_0$ and $W^{c}_{loc}(z_0)$.
Let us again consider $n\geq 1$ large.
Inside $f^{-n}(D)$, the strong stable foliation is the image of the affine foliation by
the map $\Phi_n=A^{-n}\circ \Phi\circ A$ where $A=(A_s,A_c,A_u)$ is the linear map of $\RR^d$
which coincides with $D_pf$.
The components $\Phi_{n,\bar x},\Phi_{n,\bar z}$ of $\Phi$ along the coordinates
$\bar x,\bar z$ coincide with the identity of the planes $\RR^s\times \{0\}$
and $\{0\}\times \RR^u$.
The derivative of the component $\Phi_{n,\bar y}$ at a point $\zeta$ is
$$D\Phi_{n,\bar y}(\zeta)= A^{-n}_c\;\partial_{\bar x} \varphi(A^n.\zeta) \; A^n_s+ \partial_{\bar y}\varphi(A^n.\zeta).$$
When $n$ goes to infinity, the first term $A^{-n}_c\;\partial_{\bar x} \varphi \; A^n_s$
goes to zero as $\lambda^n$ since the contraction $A_s$ is stronger than $A_c$.
Since $f$ is assumed to be smooth, $\partial_{\bar x}\varphi(\zeta), \partial_{\bar y}\varphi(\zeta)$
are Lipschitz in $\zeta$.
The map $A^n$ sends a uniform neighborhood of $z_{-n}$ in $f^{-n}(D)$ inside
a ball of radius $\lambda_c^n$ of $D$; hence
if one restricts $D\Phi_{n,\bar y}$ to a small neighborhood of $p$,
the second term $\partial_{\bar y}\varphi(A^n.\zeta)$ is $\lambda_c^n$-close to $\partial_{\bar y}\varphi(z_0)$.
One deduces that $D\Phi_{n,\bar y}$ converges uniformly to the identity and that
$$\|D\Phi_{n,\bar y}-\operatorname{Id}\|\leq \lambda^n+\lambda_c^n.$$
The same argument shows that the Lipschitz constant
of $D\Phi_{n,\bar y}$ goes to zero as $n$ goes to infinity.
One can thus apply lemma~\ref{l.perturbation}, in order to rectify the strong stable foliation
on a small neighborhood of $z_{-n}$ in $f^{-n}(D)$, by a map supported on the ball
$B(z_{-n},\lambda_u^{-n})$. The perturbation is small in topology $C^{1+\alpha}$,
provided that
$$\|D\Phi_{n,\bar y}-\operatorname{Id}\|\lambda_u^{n\alpha}\leq (\lambda^n+\lambda_c^n)\lambda_u^{n\alpha}$$
is small, which is the case when $n$ is large since $\alpha<\alpha_0$ by the choice of $\alpha_0$.
\end{proof}
\paragraph{The perturbation support}
Let us denote by $D^m$ the connected component of $f^{-m}(D)\cap B$ which contains $z_{-m}$.
We choose two small open neighborhoods $U_1,U_2$
of $x$ in $W^s_{loc}(p)$, such that $\bar U_2\subset U_1$:
they are obtained as the intersection of $V_1,V_2$ with $W^s_{loc}(p)$.
By construction, their orbit is disjoint from the orbit of $z_0$ and $y$.
For each $n\geq 0$ and $s>0$, we introduce $R_1^n(s)$ the product (in the coordinates of the chart at $p$)
$$R_1^n(s)=f^{n}(U_1)\times \{|\bar z|<s\},$$
and similarly we define $R_2^n(s)$. See figures~\ref{rectji} and~\ref{staji}.
\begin{figure}[subsection]
\begin{center}
\input{pert1-new.pstex_t}
\end{center}
\caption{The perturbation support. \label{rectji}}
\end{figure}
\begin{figure}[subsection]
\begin{center}
\input{pert2-new.pstex_t}
\end{center}
\caption{The local stable disks $D^m$. \label{staji}}
\end{figure}
\subsection{The perturbation}\label{ss.twist}
Let us choose a linear form $L$ on $\RR^u$ and recall that $\rho\in (0,1)$ has been chosen smaller than $\|Df^{-1}\|^{-1}$.
The perturbation $g$ of $f$ will be obtained as the composition $T\circ f$ where $T$
in the chart around $p$ coincides with a map $T_n$, for $n$ large, given by the following lemma.
See figure~\ref{pertji}.
\begin{figure}[subsection]
\begin{center}
\input{pert3-new.pstex_t}
\end{center}
\caption{The perturbation. \label{pertji}}
\end{figure}
\begin{lemma}\label{l.twist} There exists a sequence of smooth diffeomorphisms $T_n$ of $\RR^d$ such that
\begin{itemize}
\item[--] $T_n$ coincides with the identity outside $R_1^n(\rho^n)$ and on $W^{s}_{loc}(p)$,
\item[--] $DT_n$ coincides on $R_2^n(\rho^{n+1})$ with the linear map
$$B:(\bar x, \bar y, \bar z)\mapsto (\bar x,\bar y + \rho^{\alpha_0 n}.L(\bar z), \bar z),$$
\item[--] $(T_n)$ converges to the identity in topology $C^{1+\alpha}$.
\end{itemize}
\end{lemma}
\begin{proof}
Let us choose a smooth map $\varphi\colon \RR^{s+1}\to [0,1]$
supported on $U_1$ which takes the value $1$ on $\overline{U_2}$
and a smooth map $\psi\colon \RR^u\to [0,1]$ supported on the unit ball
and which coincides with $1$ on the ball $B(0,\rho)$.
We then define
$$T_n\colon (\bar x,\bar y,\bar z)\mapsto (\bar x,\bar y+t_n(\bar x,\bar y,\bar z), \bar z),$$
$$t_n(\bar x,\bar y,\bar z)=\rho^{\alpha_0 n} \; \varphi\circ f^{-n}(\bar x,\bar y,0)\; \psi(\rho^{-n}.\bar z)\; L(\bar z).$$
The two first properties are clearly satisfied.
On $R_1^{n}(\rho^n)$, the factor $L(\bar z)$ is bounded
(up to a constant) by $\rho^n$.
Since $f^{-n}$ is linear
and (by our choice of $\rho$) has a norm smaller than $\rho^{-n}$, as before the $C^{1+\alpha}$
size of the perturbation $T$ can be easily computed: it is smaller
than $(\rho^{\alpha_0-\alpha})^n$ and goes to zero as $n$ gets larger.
\end{proof}
\begin{remark}
After the perturbation, the orbits of $z_0$ and $p$ are unchanged.
The local manifold $W^s_{loc}(p)$ and its strong stable foliation
are also the same. For $m$ large and $s>0$ small, the forward orbit of $D^m\cap R_1^{n+1}(s)$
does not intersect the support of the perturbation,
hence the strong stable foliation on $D^m\cap R_1^{n+1}(s)$ still
coincides with the linear one.
\end{remark}
\subsection{Proof of proposition~\ref{p.jointintegrable}}
Recall that $z_{-m}$ is the image of $z_0$ by the linear map $A^{-m}$.
Let us choose a linear form $L$ on $\RR^u$ and a constant $c>0$ such that
for infinitely many values of $m\geq 0$ one has
\begin{equation}\label{e.choixL} L(z_{-m})>c\|z_{-m}\|.\end{equation}
We define $L_t=-tL$ for any $t\in [-1,1]$.
The construction of section~\ref{ss.twist}
associates to $n\geq 1$ large, a perturbation $g_t=T_{n,t}\circ f$.
We also consider a large integer $m\geq 1$ so that the distance of $z_{-m}$
to $p$ is smaller than $\rho^{n+1}$ and~\eqref{e.choixL} is satisfied.
The point $z$ announced in the statement of the proposition wil be $z_{-m}$.
We introduce the continuations $x_t=x_{g_t},y_t=y_{g_t}$ of $x,y$ for $g_t$
and the intersection $\hat x_t,\hat y_t$
of the local unstable manifold at $g_t^{n+1}(x_t),g_t^{n+1}(y_t)$ with the disc $D^m$.
By lemma~\ref{l.largemanifold}, the points $\hat x_t,\hat y_t$ belong to $H(p_t)$.
Since for each considered perturbation, the disc $D^{m}$ is still contained in $W^s_{loc}(p)$
and is endowed with the same linear strong stable foliation,
it is enough to introduce the projection $\pi_c$ on the central coordinate $\bar y$
and to show that
\begin{equation}\label{e.trans}
\pi_c(\hat x_{1})<\pi_c(\hat y_{1}) \text{ and } \pi_c(\hat x_{-1})>\pi_c(\hat y_{-1}).
\end{equation}
\medskip
First we notice that
since $g_t$ is close to $f$ and $f'$ in $\cV$, the continuations $x_t$ and $y_t$
of $x,y$ are still contained in $U_2$ and in $W^s_{loc}(p)\setminus \overline{U_1}$.
Since $g_t$ coincides with $A$ outside $R_1^n(\rho^n)$,
the local unstable manifolds of $g_t^n(x_t)$ and $g_t^{n+1}(y_t)$ are tangent to the cone
$$\mathcal{C}} \def\cI{\mathcal{I}} \def\cO{\mathcal{O}} \def\cU{\mathcal{U}^u_n=\{(v^{cs},v^u)\in \RR^{s+1}\times \RR^u, \; \|v^{cs}\|\leq \lambda^n\|v^u\|\}.$$
By construction the local unstable manifold of $g_t^{n+1}(x_t)$ in $f(R_2^n(\rho^{n+1}))$ is
tangent to the cone $B_t(\mathcal{C}} \def\cI{\mathcal{I}} \def\cO{\mathcal{O}} \def\cU{\mathcal{U}^u_{n+1})$, where $B_t$ is the linear map
associated to $L_t$ as in lemma~\ref{l.twist}.
The points $\hat x_t,\hat y_t$ are contained in the intersection of
these cones with the affine plane parallel to $\RR^{s+1}\times \{0\}$ containing $A^{-m}(z_0)$.
One deduces that
$$\pi_c(\hat x_t)\in B(\pi_c(x_t)-t\rho^{\alpha_0 n}L(z_{-m}),\lambda^n\|z_{-m}\|),$$
$$\pi_c(\hat y_t)\in B(\pi_c(y_t),\lambda^n\|z_{-m}\|).$$
By assumption we have $\pi_c(x_t)=\pi_c(y_t)$ and by our choice of $\rho$ one has $\lambda<\rho^{\alpha_0}$.
In particular, by~\eqref{e.choixL}, for $n$ large enough and $t=-1$ or $t=1$, these two balls are disjoint.
One also controls the sign of $\pi_c(\hat y_t)-\pi_c(\hat x_t)$ and gets~\eqref{e.trans} as wanted.
\subsection{Proof of proposition~\ref{p.generalized-strong-connectionCr}}
The number $\alpha_0>0$ is given by theorem~\ref{t.stable}.
The open set $\cU$ is chosen to satisfy theorem~\ref{t.stable} and
proposition~\ref{p.continuation}.
We then consider $\alpha\in [0,\alpha_0]$
and a diffeomorphism $f$ as in the statement of the proposition.
Let us assume by contradiction that in a $C^{1+\alpha}$-neighborhood $\cV$ of $f$,
there is no diffeomorphism $g$ such that $H(p_g)$ has a strong homoclinic intersection.
The proposition~\ref{p.continuation} applies.
By assumption there exists a hyperbolic periodic point $q_f$ homoclinically related to the
orbit of $p$ and a point $x_{f}\in H(q_{f})\cap W^{ss}(q_{f})\setminus \{q_{f}\}$.
By lemma~\ref{l.NGSHI}, $x_{f}$ is accumulated by points of the class $H(p_{f})$ in
$\cW^{cs}_{x_{f}}\setminus W^{ss}_{loc}(x_{f})$.
Considering the forward orbit of these points,
one deduces that $x_{f}$ and $q_{f}$ are accumulated by points of $H(q_{f})$
inside the same component of $\cW^{cs}_{x_{f}}\setminus W^{ss}_{loc}(x_{f})$.
By corollary~\ref{c.continuation}, there exists a continuation $g\mapsto x_g$ such that
$x_g$ belongs to $W^{ss}_{loc}(q_g)$ for each $g\in \cV$. Since $q_g$ and $W^{ss}_{loc}(q_g)$
vary continuously, one can argue as in the proof of lemma~\ref{l.cont-unstable} and conclude that
$g\mapsto x_g$ is continuous.
Now the theorem~\ref{t.stable} applies to the diffeomorphisms $f_0,f$ and to the points $q=y=p$ and $x$. One gets a strong homoclinic intersection for some $g\in \cV$ and the class $H(p_g)$.
This is a contradiction, concluding the proof of the proposition.
\section{Periodic unstable leaves: proof of theorem \ref{t.unstable}}
\label{p-nontransversal}
Now we continue with the proof of theorem \ref{t.unstable}.
In this section we consider:
\begin{itemize}
\item[1)] A diffeomorphism $f_0$ and a homoclinic class $H(p_{f_0})$
which is a chain-recurrence class endowed with a partially hyperbolic splitting $E^s\oplus E^c\oplus E^u$ where $E^c$ is one-dimensional and $E^s\oplus E^c $ is thin-trapped.
\item[2)] Two hyperbolic periodic points $p_{x,f_{0}}$ and $p_{y,f_{0}}$
homoclinically related to the orbit of $p_{f_0}$ and a diffeomorphism
$f$ that is $C^1$-close to $f_0$ such that
there exists two points $x\in W^{u}(p_{x,f})$, $y\in W^{u}(p_{y,f})$ in $H(p_f)$ whose strong stable manifold coincide.
\end{itemize}
Note that by lemma~\ref{l.robustness}, the homoclinic class
associated to the hyperbolic continuation $p_f$ of $p_{f_0}$ is still chain-hyperbolic. Moreover the continuations of $x,y$ are well defined and unique (lemma~\ref{l.cont-unstable}).
We will show that $f$ is the limit of diffeomorphisms $g$ such that $H(p_g)$ has a strong homoclinic intersection.
The results of this section are sum up in the next proposition.
\medskip
\begin{propo}\label{p.unstable-nt}
For any diffeomorphism $f_0$ and any homoclinic class $H(p_{f_0})$
satisfying the assumption 1) above, there exists $\alpha_0\in (0,1)$
with the following property.
For $\alpha\in [0,\alpha_0]$ and any hyperbolic periodic points $p_{x,f_0},p_{y,f_0}$
homoclinically related to the orbit of $p_{f_0}$,
any $C^{1+\alpha}$-diffeomorphism $f$ that is $C^1$-close to $f_0$
and satisfies 2) can be $C^{1+\alpha}$-approximated
by a diffeomorphism $g$ such that:
\begin{itemize}
\item[--] Either there exists a periodic point $q$ homoclinically related to the orbit of $p_g$
such that $$W^{ss}_{loc}(q)\cap W^u(p_{y,g})\neq \emptyset.$$
\item[--] Or the continuations of $x,y$ satisfy
$x_g \notin W^{ss}_{loc}(y_g)$ and also $x_g$ belongs to an arbitrary previously selected component of $\cW^{cs}_x\setminus W^{ss}_{loc}(x)$.
\end{itemize}
\end{propo}
\medskip
Note that by using corollary \ref{c.continuation} in both cases of the conclusion of the proposition,
$f$ is $C^{1+\alpha}$-approximated by a diffeomorphism whose homoclinic class
$H(p)$ exhibits a strong homoclinic intersection.
The proposition thus clearly implies theorem \ref{t.unstable}.
In what follows, in subsection \ref{ss.holonomy} are introduced the fake holonomies and it is explained the H\"older regularity. In subsection \ref{ss.localization} it is shown that the recurrences to the point $x$ in proposition \ref{p.unstable-nt} hold along the center direction and in subsection \ref{recurrence-dichotomy} it is a presented a dichotomy related to the recurrence time. Related to this dichotomy, two different perturbations are introduced in lemma \ref{nontransversal1} and \ref{nontransversal2-bis} proved in sections \ref{nont-proof} and \ref{nont2-bis-proof} respectively.
\subsection{Strong stable holonomy}\label{ss.holonomy}
\paragraph{Plaques.}
Using an adapted metric if needed, we can assume that there exist
constants $\lambda>1$ and $0<\lambda_s<1<\lambda_u$ such that for any $x\in H(p_{f_0})$
and any unitary vectors $u\in E^s_x$, $v\in E^c_x+E^u_x$ and $w\in E^u$, one has
$$\lambda.\|D_xf_0.u\|<\|D_xf_0.v\|, \quad \|D_xf_0.u\|\leq \lambda_s\quad
\text{and} \quad \|D_xf_0.w\|\geq \lambda_u.$$
Let us introduce a strong stable cone field $\mathcal{C}} \def\cI{\mathcal{I}} \def\cO{\mathcal{O}} \def\cU{\mathcal{U}^s$ above $H(p_{f_0})$:
one can choose $a>0$ small and define at each point $x$ the set
$$\mathcal{C}} \def\cI{\mathcal{I}} \def\cO{\mathcal{O}} \def\cU{\mathcal{U}^s_x=\{(u^s,u^c,u^u)\in E^s_x+E^c_x+E^u_x,\quad
\|u^s\|\geq a.\|u^c+u^u\|\}.$$
The cone field extends continuously to a neighborhood $U$ of $H(p_{f_0})$
such that at any $x\in U\cap g(U)$,
$$D_xf_0^{-1}.\mathcal{C}} \def\cI{\mathcal{I}} \def\cO{\mathcal{O}} \def\cU{\mathcal{U}^s_{g(x)}\subset \mathcal{C}} \def\cI{\mathcal{I}} \def\cO{\mathcal{O}} \def\cU{\mathcal{U}^s_{x}.$$
For some $r_0>0$, at any point $x\in U$ there exists a plaque of radius $r_0$
tangent to $\mathcal{C}} \def\cI{\mathcal{I}} \def\cO{\mathcal{O}} \def\cU{\mathcal{U}^s$.
Similarly, one can define a center-unstable cone field $\mathcal{C}} \def\cI{\mathcal{I}} \def\cO{\mathcal{O}} \def\cU{\mathcal{U}^{cu}$ and
an unstable cone-field $\mathcal{C}} \def\cI{\mathcal{I}} \def\cO{\mathcal{O}} \def\cU{\mathcal{U}^u$ on $U$ close to the bundles $E^{c}\oplus E^u$ and $E^u$ respectively. All these properties remain valid for any diffeomorphism that is $C^1$-close to $f_0$.
\paragraph{Strong stable holonomy.}
It is a classical fact that the strong stable holonomies are H\"older.
The proof extends to more general objects, that we call
\emph{fake holonomies}. For more references see~\cite{burns-wilkinson}.
Let us consider a small constant $\delta>0$ that is used to measure how orbits separate.
For any diffeomorphism $f$ that is $C^1$-close to $f_0$, let us consider two different points $z\in H(p_f)$ and
$z'\in W^{u}_{loc}(z)$ close to each other. Note that there exists a smallest integer
$N=N(z,z')\geq 1$ such that $f^N(z)$ and $f^N(z')$ are at distance larger than $\delta$.
\begin{defi}\label{fake-holonomy}
Two points $\widehat{\Pi^{ss}}(z), \widehat{\Pi^{ss}}(z')$ are called
\emph{fake strong stable holonomies} of $z,z'$ if they satisfy the following properties.
\begin{itemize}
\item[--] There exists a center-unstable plaque of radius $r_0$ containing $\widehat{\Pi^{ss}}(z)$ and $\widehat{\Pi^{ss}}(z')$.
\item[--] There exists two plaques of radius $r_0$ at $f^N(z)$ and $f^N(z')$
that are tangent to $\mathcal{C}} \def\cI{\mathcal{I}} \def\cO{\mathcal{O}} \def\cU{\mathcal{U}^s$ and contain $f^N(\widehat{\Pi^{ss}}(z))$ and
$f^N(\widehat{\Pi^{ss}}(z'))$ respectively.
\item[--] For $0\leq k \leq N$, the distances $d(f^k(z), f^k(\widehat{\Pi^{ss}}(z)))$, $d(f^k(z),f^k(\widehat{\Pi^{ss}}(z)))$ are smaller than $r_0$.
\end{itemize}
\end{defi}
Note that by invariance of the cone field $\mathcal{C}} \def\cI{\mathcal{I}} \def\cO{\mathcal{O}} \def\cU{\mathcal{U}^s$ under backward iterations
the point $f^k(\widehat{\Pi^{ss}}(z))$ belongs to a plaque
at $f^k(z)$ tangent to $\mathcal{C}} \def\cI{\mathcal{I}} \def\cO{\mathcal{O}} \def\cU{\mathcal{U}^s$ and whose radius is smaller than $\lambda_s^k.r_0$.
\medskip
The choice for the plaques tangent to $\mathcal{C}} \def\cI{\mathcal{I}} \def\cO{\mathcal{O}} \def\cU{\mathcal{U}^s$ is of course not unique: one can consider for instance
the local strong stable manifold (in this case, the fake holonomies coincide with the usual
strong-stable holonomies) but one can also choose the local strong stable manifold of a diffeomorphism
$C^1$-close to $f$. In fact the fake holonomies allow us to compare the holonomies when the diffeomorphism is changed.
\paragraph{H\"older regularity.}
We now sketch how the classical result about H\"older regularity adapts for the fake holonomies.
\begin{lemma}\label{l.holonomy}
If $\delta>0$ has been chosen small enough,
then there exists $\alpha_s>0$ such that for any diffeomorphism $f$
that is $C^1$-close to $f_0$,
for any $z\in H(p_f)$ and $z'\in W^u_{loc}(z)$ close,
and for any fake holonomies $\widehat{\Pi^{ss}}(z), \widehat{\Pi^{ss}}(z')$, one has
$$d(\widehat{\Pi^{ss}}(z), \widehat{\Pi^{ss}}(z'))\leq d(z,z')^{\alpha_s}.$$
\end{lemma}
\begin{proof}[Sketch of the proof]
Observe that if $N$ is sufficiently large (provided that $z'$ is close enough to $z$), the distances $d(f^N(\Pi^{ss}(z)),f^N(z))$ and $d(f^N(\Pi^{ss}(z')),f^N(z'))$
are exponentially small. Hence
$d(f^N(\widehat{\Pi^{ss}}(z)), f^N(\widehat{\Pi^{ss}}(z')))$
is of the same order than $d(f^N(z), f^N(z'))$ and close to $\delta$.
The distance $d(\widehat{\Pi^{ss}}(z), \widehat{\Pi^{ss}}(z'))$
is bounded by $\|Df^{-1}\|^Nd(f^N(\Pi^{ss}(z)), f^N(\widehat{\Pi^{ss}}(z')))$
and the distance $d(z, z')$ is bounded from below by
$\|Df\|^{-N}d(f^N(\Pi^{ss}(z)), f^N(\widehat{\Pi^{ss}}(z')))$.
This proves that there exists $\sigma>0$ (which only depends on $f_0$)
such that
$$d(\widehat{\Pi^{ss}}(z), \widehat{\Pi^{ss}}(z'))\leq \sigma^N. d(z,z').$$
On the other hand, since the distance along the unstable manifolds
growth uniformly, there exists another constant $C>0$ such that
$$N\leq C.\log d(z,z').$$
The result follows from these two last inequalities.
Observe that the exponent $\alpha_s$ only depends on $C$ and $\sigma$ which are uniform on
a $C^1$-neighborhood of $f_0$.
\end{proof}
\paragraph{Regularity of the strong stable bundle.}
The regularity of the strong stable bundle needs more smoothness on the diffeomorphism.
Note that the strong stable bundle is defined at any point whose forward orbit is contained in a
small neighborhood $U$ of $H(p_{f_0})$.
\begin{lemma}\label{l.bundle}
There exists $\alpha'_s$ such that for any diffeomorphism $f$ that is $C^1$-close to $f_0$
and of class $C^{1+\alpha}$ for some $\alpha\in (0,\alpha'_s)$, there exists a constant $C>0$
with the following property.
At any points $z,z'$ close having their forward orbit contained in $U$, one has
$$d(E^{ss}_z,E^{ss}_{z'})\leq C.d(z,z')^{\alpha}.$$
\end{lemma}
\begin{proof}[Sketch of the proof]
Let us choose $K>\|Df\|_\infty$ and as before denote by $\lambda\in (0,1)$ a bound for the domination between $E^{ss}$ and $E^c\oplus E^u$. We choose $\alpha'_s>0$ such that
$K^{\alpha'_s}\lambda<1$.
By working in charts, one has for some constant $C>0$,
\begin{equation*}
\begin{split}
d(E^{ss}_z,E^{ss}_{z'})&\leq d( Df^{-1}_{f(z)}(E^{ss}_{f(z)}), Df^{-1}_{f(z)}(E^{ss}_{f(z')}))+
d(Df^{-1}_{f(z)}(E^{ss}_{f(z')}),Df^{-1}_{f(z')}(E^{ss}_{f(z')}))\\
&\leq \lambda d(E^{ss}_{f(z)},E^{ss}_{f(z')}) + C.d(f(z),f(z'))^\alpha.
\end{split}
\end{equation*}
By induction one gets for any $k\geq 1$,
$$d(E^{ss}_z,E^{ss}_{z'})\leq C.\sum_{j=0}^{k-1} \lambda^j d(f^{j+1}(z),f^{j+1}(z'))^\alpha
+ \lambda^{k}d(E^{ss}_{f^k(z)},E^{ss}_{f^k(z')}).$$
One can bound $d(f^{j}(z),f^{j}(z'))$ by $K^jd(z,z')$.
Since $\lambda K^{\alpha'_s}<1$,
this gives
$$d(E^{ss}_z,E^{ss}_{z'})\leq C (d(z,z')^\alpha + \lambda^k).$$
By choosing $k$ large enough, one gets the estimate.
\end{proof}
\subsection{Localization of returns to $p_x$}\label{ss.localization}
We now fix a diffeomorphism $f$ that is $C^1$-close to $f_0$.
We assume that is $C^{1+\alpha}$ for some $\alpha\in [0,1)$.
In order to simplify the notations we will now set $p=p_f$, $p_x=p_{x,f}$ and $p_y=p_{y,f}$. Let $\tau_x$ be the period of $p_{x}$
and let us consider a local central manifold $W^c_{loc}(p_x)$.
\smallskip
We will use the following assumption:
\begin{itemize}
\item[(***)] \emph{The intersection between $W^{ss}(p_x)\setminus \{p_x\}$ and
$H(p)$ is empty.}
\end{itemize}
The orbit of any point $z\in W^s(p_{x})\setminus \{p_{x}\}$ meets the fundamental
domain $f^{\tau_x}(W^s_{loc}(p_{x}))\setminus W^s_{loc}(p_{x})$.
The next lemma states, that if $H(p)$ and $p_x$ satisfy (***),
and if $z$ belongs to $H(p)\cap W^s_{loc}(p_x)$ then its orbit meets a kind of ``fundamental center domain"
of $p_x$.
\begin{lema}\label{l.localization1}
\label{local1} If (***) is satisfied,
there are points $z_0, z_1$ contained in $ W^c_{loc}(p_x)\setminus \{p_x\}$
such that if $z\in W^{s}_{loc}(p_x)\cap H(p)$ then there is $k\in \ZZ$ verifying that
$$f^{k}(z)\in W^{ss}_{loc}([f^{2\tau_x}(z_0),z_0)])\cup W^{ss}_{loc}([f^{2\tau_x}(z_1),z_1)])$$
where $[f^{2\tau_x}(z_i),z_i]$, for $i\in \{0,1\}$
is the connected arc of $W^c_{loc}(p_x)$ whose extremal points are $z_i,$ $f^{2\tau_x}(z_i)$
and (see figure~\ref{f.funda})
$$W^{ss}_{loc}([f^{2\tau_x}(z_i),z_i])= \bigcup_{\{z'\in [f^{2\tau_x}(z_i),z_i]\}} W^{ss}_{loc}(z').$$
\end{lema}
\begin{figure}[subsection]
\begin{center}
\input{per-new.pstex_t}
\end{center}
\caption{Fundamental center domains. \label{f.funda}}
\label{center domain}
\end{figure}
\begin{proof}
Let us consider two points $z_0^0$ and $z_1^0$ in two different connected
components of $W^c_{loc}(p_x)$ and set $z^n_i=f^{i\tau_x}(z_i^0)$.
Note that the image of $W^{ss}_{loc}([f^{2\tau_x}(z_i^n),z_i^n])$ by $f^\tau$
is contained in $W^{ss}_{loc}([f^{2\tau_x}(z_i^{n+1}),z_i^{n+1}])$.
The union of the $W^{ss}_{loc}([f^{2\tau_x}(z_i^n),z_i^n])$ over $n\geq 0$
and of $W^{ss}_{loc}(p_x)$ contains a neighborhood of $p_x$.
If the thesis of the lemma does not hold, it follows that
for arbitrarily large $n\geq 0$, there exists a point $\zeta_n\in H(p_x)\cap W^s_{loc}(p_x)$
which belongs to $W^{ss}_{loc}([f^{2\tau_x}(z_i^{n+1}),z_i^{n+1}])$ and whose
preimage by $f^{\tau}$ does not belong to $W^{ss}_{loc}([f^{2\tau_x}(z_i^n),z_i^n])$.
An accumulation point $\zeta$ of $\{\zeta_n\}$ belongs to
$W^{ss}_{loc}(p_x)\setminus\{p_x\}]\cap H(p_x)$,
contradicting the assumption of the lemma.
\end{proof}
\bigskip
We now describe the returns of the forward orbit of $x$ in the neighborhood $W$ of the orbit of $p_x$. We need to take into account the orbits
that follow the orbit of $x$ during some time.
For that we let $\lambda_s\in (0,1)$ be an upper bound for the
contraction along $E^s$, we let $\lambda>1$ be a lower bound for the
domination between $E^s$ and $E^c\oplus E^u$ as in section~\ref{ss.holonomy}
and we let $\mu_c>\mu_s$ in $(0,1)$ be the modulus of the center eigenvalue at $p_x$
and the maximal modulus of the strong stable eigenvalues at $p_x$.
We also choose $\rho>1$ such that
$$\rho<\min (\lambda,\lambda_s^{-1/2},\mu_c/\mu_s).$$
We then introduce some ``forward dynamical balls" centered at $x$:
we fix $k_0\geq 1$ and for $n\geq 0$ we define the set
$$B_n(x)=\left\{z\in M,\; \forall\, 0\leq k\leq n, \, d(f^k(z),f^k(x))<\rho^{k-k_0}.\prod_{\ell=0}^{k-1}\|Df_{|E^s}(f^\ell(x))\|\right\}.$$
Note that:
\begin{itemize}
\item[(i)] By our choice of $\rho$,
the intersection of all the balls $B_n(x)$ coincides with a local strong stable manifold of $x$
and the image $f^n(B_n(x))$ has diameter smaller than $\sqrt \lambda_s^{n-k_0}$.
\item[(ii)] By taking $k_0$ large enough, the point $y$ belongs to the balls $B_n(x)$ and its forward iterates
satisfy the stronger estimate
\begin{equation}\label{i.2}
d(f^k(y),f^k(x))<\frac 1 3\rho^{k-k_0}.\prod_{\ell=0}^{k-1}\|Df_{|E^s}(f^\ell(x))\|.
\end{equation}
\end{itemize}
\noindent Let us now assume that (***) holds.
\begin{itemize}
\item[(iii)] For $n$ large enough, $f^n(B_n(x))$ does not intersect $W^u_{loc}(p_x)$. Otherwise
$B_n(x)$ would intersect a large backward iterate of $W^u_{loc}(p_x)$: this would imply that the strong stable manifold of the orbit of $p_x$ contains $x$
and contradicts our assumptions that $W^{ss}(p_x)\cap H(p)=\{p_x\}$. In fact, by first item if for $n$ large enough, $f^n(B_n(x))$ intersects $W^u_{loc}(p_x)$ it follows that $p_x\in W^{ss}_{loc}(x)$.
\item[(iv)] One can choose the neighborhood $W$ of the orbit of $p_x$ so that
the backward orbit of $x$ is contained in $W$ and $x\not \in \overline W$.
The lemma~\ref{l.localization1} above implies that the forward iterates of $x$ close to $p_x$
are close to the central manifold of $p_x$. Consequently, their distance to the local unstable manifold of the orbit of $p_x$ decreases by iteration by a factor close to the central eigenvalue of $p_x$. One thus gets the following.
\end{itemize}
\begin{lemma}\label{l.localization2}
Let us fix $\eta>0$ small.
If (***) holds,
any large iterate $f^n(B_n(x))$ which intersects $B(x,\delta)$
has the following property.
Let $m$ be the largest integer such that $m<n$ and
$f^{m}(B_n(x))$ is not contained in $W$. Then the distance between the points of $f^n(B_n(x))$
and $W^u_{loc}(p_x)$ belongs to $[\mu_c^{(1+\eta).(n-m)},\mu_c^{(1-\eta).(n-m)}]$, where
$\mu_c$ denotes the modulus of the center eigenvalue associated to $p_{x}$.
\end{lemma}
\begin{itemize}
\item[(v)] For any forward iterate $f^\ell(x)$ close to $p_x$, the quantity
$\rho \|Df_{|E^s}(f^\ell(x))\|$ is smaller than $\mu_c$ by our choice of $\rho$.
We thus obtain another version of the estimate of item (i).
\end{itemize}
\begin{lemma}\label{l.localization3}
If $f^n(B_n(x))$ intersects $B(x,\mu_c^N)$ for some $N\geq 1$ large,
then the diameter of $f^n(B_n(x))$ is smaller than
$\sqrt\lambda_s^{n-N}\mu_c^N$.
\end{lemma}
\subsection{Recurrence time dichotomy}\label{recurrence-dichotomy}
As before we denote by $\lambda_u,\lambda>1$ the lower bounds for the expansion along $E^u$ and the domination between $E^s$ and $E^c\oplus E^u$.
By lemma~\ref{l.holonomy}, there exists $\alpha_s\in (0,1)$
such that the strong stable fake holonomies are $\alpha_s$-H\"older.
The lemma~\ref{l.bundle} gives $\alpha'_s\in (0,1)$ which control the smoothness of the strong stable bundle.
Recall that by $\mu_c\in (0,1)$ we denote the modulus of the center eigenvalue associated to $p_{x}$ for $f$.
We also denote by $\bar \alpha_0$ the bound on the smoothness associated to
$H(p_{f_0})$ in proposition~\ref{p.generalized-strong-connectionCr}.
Let $\chi,K_1,K_2$ be some positive constants defined by
$$\chi=\frac{\log \lambda_u}{\log\lambda_u+\|Df^{-1}_0\|},$$
$$K_1= \frac{|\log\mu_c|}{\chi\;\log\lambda},\,\,\,\,\,K_2=\frac{(1-\alpha_s)|\log\mu_c|}{\alpha_s\log\lambda_u}.$$
\medskip
Let us consider again the $C^{1+\alpha}$-diffeomorphism $f$ that is $C^1$-close to $f$.
If $\alpha$ belongs to $[0,\bar \alpha_0]$ and condition (***) does not hold, then
the proposition~\ref{p.generalized-strong-connectionCr} implies that there exist
$C^{1+\alpha}$-perturbation $g$ of $f$ such that $H(p_g)$ has a strong homoclinic intersection,
concluding the proof of the proposition.
\medskip
In the following we assume that condition (***) holds for $f$ and as in the statement of
proposition~\ref{p.unstable-nt}, that there exist two different points $x\in W^u(p_x)$
and $y\in W^u(p_y)$ whose strong stable manifolds coincide.
For any $N$ large, we take $V$ a neighborhood of size $\mu_c^N$ around $f^{-1}(x).$
We define $n= n(N)$, the smallest element of $\NN\cup \{\infty\}$
such that $f^n(B_n(x))$ intersects $V$.
By the property (iii) of section~\ref{ss.localization}, the sequence $\{n(N)\}$ increases and goes to $+\infty$ as $N$ increases.
\medskip
We fix a constant $K>\max(1,K_0,K_1,K_2)$ and we are going to consider two cases:
\begin{enumerate}
\item {\it Fast returns.} There exists arbitrarily large $N$ such that
\begin{eqnarray}\label{no-opttr}
n(N)\leq K. N. \end{eqnarray}
\item {\it Slow returns.} There exists arbitrarily large $N$ such that
\begin{eqnarray}\label{opttr} n(N)> K.N . \end{eqnarray}
\end{enumerate}
One of these two conditions (maybe both) occur.
If the first option holds, we prove the following.
\begin{lemma}\label{nontransversal1}
Assume that (***) and (\ref{no-opttr})
hold for some $K>0$, and that $\alpha<\inf(\frac 1 {K-1},\alpha'_s)$.
Then there exists a diffeomorphism $\varphi\in \operatorname{Diff}^{1+\alpha}(M)$
that is $C^{1+\alpha}$-close to the identity such that $g=\varphi\circ f$ has a hyperbolic periodic point $q$
homoclinically related to the orbit of $p_{x,g}$ and whose strong stable manifold
$W^{ss}(q)\setminus \{q\}$ intersects $W^{u}(p_{y,g})$.
\end{lemma}
\noindent If the second option holds, we prove the following.
\begin{lemma}\label{nontransversal2-bis}
Assume that~(\ref{opttr})
holds for some $K>\max(K_1,K_2)$,
and that $1+\alpha<\frac{K}{\max(K_1,K_2)}$.
Then, there exists a diffeomorphism $\varphi\in \operatorname{Diff}^{1+\alpha}(M)$ that is $C^{1+\alpha}$-close to the identity such that $g=\varphi\circ f$ satisfies the second option of
proposition \ref{p.unstable-nt}: if one fixes an orientation on $E^c_{y}$,
there exist two such diffeomorphism $g^+,g^-$ such that
$x_{g^+}$ (resp. $x_{g^-}$) belongs to $\cW^{cs,+}_{g^+,y_{g^+}}$
(resp. $\cW^{cs,-}_{g^-,y_{g^-}}$).
\end{lemma}
\noindent Both lemmas and the proposition~\ref{p.generalized-strong-connectionCr} conclude the proof of proposition~\ref{p.unstable-nt}.
\medskip
Note that for proving proposition~\ref{p.unstable-nt}
one can choose $K$ independently from $\mu_c$, for instance any
$$K=\|Df_0\|\max\left(\frac 3{\log\lambda_s},\frac {2\;\chi} {\log \lambda}, \frac{2\;(1-\alpha_s)}{\alpha_s\log \lambda_u}\right).$$
In this way we obtain a bound
$$\alpha_0=\inf\left(\bar \alpha_0, \frac 1 {K-1},\alpha'_s, \frac{K}{\max(K_1,K_2)}\right)$$
for the smoothness exponent $\alpha$ in proposition~\ref{p.unstable-nt},
which only depends on $f_0$ as announced.
\subsection{Fast returns: proof of lemma \ref{nontransversal1}}\label{nont-proof}
Let us assume that condition~(\ref{no-opttr}) holds for some large values of $N$
and some $K>0$ such that $\alpha<\inf\left(\frac{1}{K-1},\alpha'\right)$.
We also assume that (***) holds so that the lemma~\ref{l.localization2} applies.
\begin{lemma}\label{l.ab}
There are $a>b$ in $(K^{-1},1)$ such that some arbitrarily large $N$ and $n=n(N)$ satisfy:
\begin{enumerate}
\item $f^n(B_n(x))\cap B(x,\mu_c^{a\,n})\neq \emptyset$ and
\item $f^m(B_m(x))\cap B(x,\mu_c^{b\,n})= \emptyset$ for any $k_0<m<n.$
\end{enumerate}
Moreover $\frac a b$ can be chosen arbitrarily close to $\frac K {K-1}$.
\end{lemma}
\begin{proof}
We introduce the integers $N_i$ and $n_i=n(N_i)$ satisfying
for any $i$,
$$ N_i<N_{i+1},\;\; n_i<n_{i+1}, \text{ and }\,
\forall\, N_{i-1} < N \leq N_{i},\;\; n(N)=n_i.$$
We will prove that there are positive constants $b'<a'$ in $(K^{-1},1)$
and there is $n_i$ sufficiently large such that $N_i> a'.n_i$ and $N_j< b'.n_i$ for $0\leq j<i$. We then choose any $b<a$ in $(b',a')$.
One can check easily that for these large $n=n_i$ the result holds:
\begin{itemize}
\item[--] We have $f^n(B_n(x))\cap B(x,\mu_c^{N_i})\neq\emptyset$ with $N_i>a'.n$, hence
$f^n(B_n(x))\cap B(x,\mu_c^{a'.n})$ is non-empty.
By lemma~\ref{l.localization3}, the diameter of $f^n(B_n(x))$ is bounded
by $\sqrt \lambda_s^{n-N_i}\mu_c^{N_i}$ which is much smaller than $\mu_c^{a'.n}$.
As a consequence $f^n(B_n(x))$ is contained in $B(x,\mu_c^{a.n})$.
\item[--] By definition of the sequence $(N_j)$, for any $m<n=n_i$, one has $f^m(B_m(x))\cap B(x,\mu_c^{N_{i-1}+1})=\emptyset$ and
$b.n>b'.n+1>N_{i-1}+1$, implying the second condition of the lemma.
\end{itemize}
\medskip
Let us now prove the existence of the constants $a'<b'$.
We denote by $m_i$ the smallest integer such that
$$\forall m_i\leq m < n_i,\;\; f^m(B_{n_i}(x))\subset W.$$
By lemma~\ref{l.localization2} if one chooses $\varepsilon>0$ small
and if $N_i$ is large enough, one has
$$(1+\varepsilon).(n_i-m_i)\geq N_i.$$
Let us define
$$R=\limsup_{j\to +\infty} \frac{N_j}{n_j}.$$
By~(\ref{no-opttr}), $R$ belongs to $[K^{-1},1]$.
For any $j$ larger than a constant $j_0$
we have $\frac{N_j}{n_j}< (1+\varepsilon)R.$
For some $i$ sufficiently large we also have $\frac{N_i}{n_i}> (1-\varepsilon)R$.
If $j<i$ we have $n_j\leq m_i\leq n_i-(1+\varepsilon)^{-1}N_i$ and so for $j_0<j<i$ we have
$$N_j \leq (1+\varepsilon)R n_j\leq (1+\varepsilon)R (n_i-(1+\varepsilon)^{-1}N_i)\leq R[1-(1-\varepsilon)R+\varepsilon] n_i.$$
Since $R$ belongs to $[K^{-1},1]$, then $[1-(1-\varepsilon)R+\varepsilon] <(1-\varepsilon)$ for $\varepsilon$ small and therefore taking $a'= (1-\varepsilon)R$ and $b'= R[1-(1-\varepsilon)R+\varepsilon] $ the result holds. To check that it also holds for $j<j_0$ it is enough to take $i$ sufficiently large.
Observe that the quantity $\frac a b$ is close to $\frac {1-\varepsilon}{1-(1-\varepsilon)R+\varepsilon}$. Since $R\geq K^{-1}$,
when $\varepsilon$ goes to $0$
the limit is larger or equal to $\frac{K}{K-1}$.
\end{proof}
\bigskip
We can now conclude the proof of lemma~\ref{nontransversal1}.
\begin{proof}[\it Proof of lemma~\ref{nontransversal1}]
We fix $a,b$ and a large integer $n$ as in lemma~\ref{l.ab}.
By assumption $\alpha<(K-1)^{-1}$ and $\frac a b$ can be chosen close
to $\frac K {K-1}$. One can thus ensure that $1+\alpha$ is smaller than $a/b$.
Let $D\subset W^{ss}_{loc}(x)$ be the smallest disc containing $y$.
By construction it is contained in the ball $B_n(x)$, hence its image
by $f^n$ is contained in $B(x,\mu_c^{an})$.
We consider a $C^{1+\alpha}$-diffeomorphism $\varphi$ supported in
$B(x,\mu_c^{bn})$ which sends $f^n(D)$ into $D$
and define $g=\varphi\circ f$.
By construction the support of the perturbation $g$
is disjoint from $D$ and its $n-1$ first iterates.
\begin{claim}
If $1+\alpha<\inf(\frac a b,\alpha'_s)$, by choosing $n$ large the
diffeomorphism $\varphi$ can be taken arbitrarily close to the identity
in $\operatorname{Diff}^{1+\alpha}(M)$.
\end{claim}
\begin{proof}
Let us consider a $C^{1+\alpha}$-chart $U\to \RR^d$ of a neighborhood $U$ of $x$ such that
$x$ coincides with $0$ and $W^{ss}_{loc}(x)$ coincides with the plane
$\RR^k\times \{0\}$, where $k=\dim(E^{ss})$.
For $n$ large, the plaque $W^{ss}_{loc}(f^n(x))$ is close to $W^{ss}_{loc}(x)$
and coincides in the chart with the graph of a map $\chi_0\colon \RR^k\to \RR^{d-k}$.
We introduce the map
$$(z_1,z_2)\mapsto (0,-\chi_0(z_1))$$
which is close to $0$ in the $C^{1+\alpha}$ topology and satisfies $\|v_0(0)\|\leq e^{-an}$
by construction.
One can thus apply the lemma~\ref{l.perturbation} in order to build
a map $v\colon \RR^d\to \RR^{d-k}$ which
coincides with $v_0$ on the ball $B(0,e^{-an})$ and with $0$ outside $B(0,e^{-bn})$.
The map $\varphi\colon (z_1,z_2)\mapsto (z_1,z_2-v(z_1,z_2))$ is the announced diffeomorphism.
In order to prove that $\varphi$ is close to the identity in $\operatorname{Diff}^{1+\alpha}(M)$,
one has to check that $e^{(1+\alpha)bn}\sup_{B(0,e^{-bn})}\|v_0\|$ is small.
Since $\alpha<\alpha'_s$, by lemma~\ref{l.bundle} we have
$$\|Dv_0(0)\|\leq C\|v_0(0)\|^\alpha\leq Ce^{-\alpha an}.$$
Since $v_0$ is close to the identity in $\operatorname{Diff}^{1+\alpha}$,
there exists an arbitrarily small constant $\varepsilon>0$ such that
$$\sup_{B(0,e^{-bn})}\|Dv_0\|\leq \|Dv_0(0)\|+ \varepsilon e^{-\alpha bn}\leq 2\varepsilon e^{-\alpha bn}.$$
This gives
$$\sup_{B(0,e^{-bn})}\|v_0\|\leq \|v_0(0)\|+e^{-bn}\sup_{B(0,e^{-bn})}\|Dv_0\|\leq e^{-an}+2\varepsilon e^{-(1+\alpha)bn}.$$
Since $a>(1+\alpha)b$, this shows that $e^{(1+\alpha)bn}\sup_{B(0,e^{-bn})}\|v_0\|$ is small
when $n$ is large.
\end{proof}
\medskip
To continue with the proof of lemma~\ref{nontransversal1}, we note that
the map $\varphi\circ f^n$ is a contraction on $D$, hence
the diffeomorphism $g$ has a $n$-periodic point $q$
whose strong stable manifold contains $D$.
Since the backward orbit of $W^u_{loc}(p_y)$
is disjoint from the support of the perturbation,
the manifolds $W^{ss}(q)$ and $W^u_{loc}(p_{y,g})$
intersect.
In particular $W^{s}(q)$ and $W^u_{loc}(p_{y,g})$
have a transversal intersection.
On the other hand the orbit of $q$ has a point close to $p_x$,
hence $W^{s}(p_x)$ and $W^u(q)$ have a transversal intersection.
One deduces that $q$ is homoclinically related to the orbits
of $p_{x,g}$ and $p_{y,g}$.
This concludes the proof of lemma~\ref{nontransversal1}.
\end{proof}
\subsection{Slow returns: proof of lemma \ref{nontransversal2-bis}}\label{nont2-bis-proof}
Let us fix a center-unstable plaque $\mathcal{D}} \def\cJ{\mathcal{J}} \def\cP{\mathcal{P}} \def\cV{\mathcal{V}$ at $x$ and for diffeomorphisms $g$ close to $f$
we consider the strong stable holonomy $\Pi^{ss}_g$ to $\mathcal{D}} \def\cJ{\mathcal{J}} \def\cP{\mathcal{P}} \def\cV{\mathcal{V}$.
Since the map $f^\tau_x$ is linear in a neighborhood of $p_x$, one can choose $\mathcal{D}} \def\cJ{\mathcal{J}} \def\cP{\mathcal{P}} \def\cV{\mathcal{V}$
in the linear plane corresponding to the sum of the central and unstable eigenspaces.
Observe that it contains the manifold $W^{u}_{loc}(p_x)$ for $f.$
Under condition~(\ref{opttr}), we are going to get a perturbation $g$ of $f$ such that
$\Pi^{ss}_g(x_g)\neq \Pi^{ss}_g(y_g)$, proving that $W^{ss}_{loc}(x_g)$ and $W^{ss}_{loc}(y_g)$ are disjoint.
Since $x_g,y_g$ belong to a same center-stable plaque $\cW^{cs}_{g,y_g}$, the projections
$\Pi^{ss}_g(x_g),\Pi^{ss}_g(y_g)$ are contained in a $C^1$-curve of $\mathcal{D}} \def\cJ{\mathcal{J}} \def\cP{\mathcal{P}} \def\cV{\mathcal{V}$
that is tangent to a central cone field.
Moreover, one will be able to choose the perturbation to satisfy either $x_g\in\cW^{cs,+}_{g,y_g}$ or $x_g\in\cW^{cs,-}_{g,y_g}$.
\paragraph{Description of the perturbation.}
We recall that we have fixed a large integer $N\geq 1$ and that $V$ denotes the ball $B(x,\mu^N_c)$.
Let us fix two small constants $\widehat r=\frac 1 2 \mu_c^N$ and $r<\widehat r$ in $(0,1)$.
We perform the perturbation $g$ of $f$ in the ball $B(f^{-1}(x),\widehat r)$, in such a way that
$W^{u}_{loc}(p_x)$ is still contained in the coordinate subspace and the distance
between $x=f(f^{-1}(x))$ and $g(f^{-1}(x))$ is $r$ along the central coordinate.
This can be realized by a small perturbation of $f$ in $\operatorname{Diff}^1(M)$ provided
$\widehat r$ and $r/\widehat r$ are large enough. Moreover one can require that the $C^0$ size of the perturbation
is equal to $r$.
Later, in item $7$, we explain how the perturbation can be adapted to be $C^{1+\alpha}$-small.
Note that the point $x$ can be pushed to $g(f^{-1}(x))$ along $E^c_x$ in any of the two
central directions at $x$.
To get the conclusion, we choose a small constant $\varepsilon>0$
(independent from $N$) and show that
the distances $d(\Pi^{ss}_g(y_g), x)$ and $d(x_g, g(f^{-1}(x)))$ are smaller than $\varepsilon.r$,
which is much smaller than $d(x,g(f^{-1}(x)))$.
\bigskip
\noindent {\bf 1- Estimating $d(y_g, y)$.}
Observe that $y_g$ does not necessarily coincide with $y$ since the forward orbit of $y$ may intersect the region of
perturbation. However by lemma~\ref{l.cont-unstable} the point $y_{g}$ belongs to the local unstable manifold of $p_{y,g}=p_{y,f}$
which coincides for $f$ and $g$.
We will consider the distance $dist$ along the unstable plaques (which is locally comparable in a uniform way to the distance in the ambient space).
We also introduce a constant $C\gg\frac{1}{\lambda_u-1}$
independent from $N$.
\begin{lemma}\label{l.upper}
If for some positive integer $m$ the two points $f^m(y), g^m(y_g)$ belong to a same
unstable plaque, then their distance satisfies $dist(f^m(y), g^m(y_g))< C.r$.
\end{lemma}
\begin{proof}
Let us assume by contradiction that the estimate does not hold.
Observe that the distance by the action of $f$ growth by a factor $\lambda_u$ and the $C^0$ distance between $f$ and $g$ is at most $r$,
which is much smaller than $C.r$. One deduces that the points $f^{m+1}(y), g^{m+1}(y_g)$ still belong to a same unstable plaque.
Denoting $\gamma=\frac{\lambda_uC-1}C>1$, their distance now satisfy
\begin{equation*}
\begin{split}
dist(f^{m+1}(y),g^{m+1}(y_g))&> \lambda_u \; dist(f^{m}(y),g^{m}(y_g))-r\\
&>(\lambda_u-C^{-1}) \;dist(f^{m}(y),g^{m}(y_g)) =\gamma.dist(f^{m}(y),g^{m}(y_g)).
\end{split}
\end{equation*}
Therefore after $k$ iterates the distance become larger than
$ \gamma^k.C.r$ and so increasing to infinity. This is a contradiction with the fact $y_g$ is a continuation of $y$.
\end{proof}
\begin{lemma}\label{l.compare1}
The $n(N)$ first iterates of $y$ and $y_{g}$ coincide for $f$ and for $g$.
\end{lemma}
\begin{proof}
Since $y$ belongs to the dynamical balls $B_n(x)$,
the segment of orbit $(y,\dots,f^{n(N)}(y))$ is also a segment of orbit for $g$.
Let us consider the first integer $m\geq 1$ such that $g^m(y_g)=f^m(y_g)$ enters in the region of perturbation and let us assume by contradiction
that $m< n(N)$.
As for $y,y_g$, one knows that $f^m(y_g)$ and $f^m(y)$ belong to a same unstable plaque: by lemma~\ref{l.upper} they are at distance smaller than $C.r$.
If $r$ has been chosen small enough one has $C.r<\frac 1 2 \mu_c^N=\widehat r$.
By definition of $m$ one also has
$d(f^m(y_g),x)<\hat r=\frac 1 2 \mu_c^N$.
As a consequence $f^m(y)$ belongs to $V$, hence $m\geq n(N)$.
This contradicts our assumption.
This shows that the orbit $(y_g,\dots,g^{n(N)}(y_g))$
coincides for $f$ and for $g$.
\end{proof}
Since $y,y_g$ belong to an unstable plaque,
and since by lemma~\ref{l.compare1}
their $n(N)$ first iterates are the same by $f$ and by $g$,
the points $f^{n(N)}(y)$ and $g^{n(N)}(y_g)$ belong to
a same unstable plaque and by lemma~\ref{l.upper}, their distance
is smaller than $C.r$. For any $0\leq m\leq n(N)$ we obtain
\begin{equation}\label{l.dy}
d(g^m(y_g),f^m(y))<\lambda_u^{m-n(N)}C.r.
\end{equation}
\bigskip
\noindent {\bf 2- Estimating $d(x_g, g(f^{-1}(x)))$.}
Arguing as in lemma~\ref{l.upper}, one shows that
for any positive integer $m$, if the two points $f^m(x), g^m(x_g)$ belong to a same
unstable plaque, then their distance satisfy $dist(f^m(x), g^m(x_g))<C r$.
Let us denote by $\lambda'>1$ a lower bound for the domination between the bundles $E^c$ and $E^u$
and consider two large constants $k\ll\ell$ (independent from $N$)
such that $\lambda_{u}^{\ell}.(\lambda')^{-k}>C$.
If $N$ has been chosen large, the $\ell$ first iterates of $x,x_g,g(f^{-1}(x))$ are the same by $f$ and by $g$.
Let us assume by contradiction that the distance $dist(g(f^{-1}(x)),x_{g})$ inside $W^u_{loc}(p_{x,g})$ is larger than
$(\lambda')^{-k}.r$. Since the distance between $x$ and $g(f^{-1}(x))$ in the central direction is
equal to $r$, one deduces that the distance from $f^{\ell}(g(f^{-1}(x)))$ to $f^{\ell}(x_{g})$ is much larger than its distance
to $f^{\ell}(x)$. In particular $f^{\ell}(x)$ and $f^{\ell}(x_{g})$ are contained in a same unstable plaque and by our choice of $k,\ell$, their distance
is larger than $C.r$, which is a contradiction. Consequently
$$dist(g(f^{-1}(x)),x_{g})<(\lambda')^{-k}.r.$$
Taking $k$ large enough, one has $d(g(f^{-1}(x),x_g)<\varepsilon. r$ as wanted.
\bigskip
\noindent {\bf 3- Estimating $d(\widehat{\Pi^{ss}_f}(y_g), \Pi^{ss}_f(y))$.}
Since $y_{g}$ belongs to the unstable manifold $W^u_{loc}(y)$ for $f$, one can
introduce some fake holonomies $\widehat{\Pi^{ss}_f}(y_g), \widehat{\Pi^{ss}_f}(y)=\Pi^{ss}_f(y)$ for $f$.
By~(\ref{l.dy}) and lemma~\ref{l.holonomy}, one gets
$$d(\widehat{\Pi^{ss}_f}(y_g), \Pi^{ss}_f(y))< d(y, y_{g})^{\alpha_s}<[\lambda_u^{-n(N)}C.r]^{\alpha_s}.$$
\bigskip
\noindent {\bf 5- Estimating $d(\Pi^{ss}_g(y_g), \widehat{\Pi^{ss}_f}(y_g))$.}
As before we first compare the iterates of $f$ and $g$.
\begin{lemma}\label{l.compare2}
The $\chi.n(N)$ first iterates of $y_{g}$, $\Pi^{ss}_g(y_g)$ and $\widehat{\Pi^{ss}_f}(y_g)$ coincide for $f$ and for $g$, where $\chi=\frac{\log\lambda_u}{\log \lambda_u+\log\|Df_0^{-1}\|}.$
\end{lemma}
\begin{proof}
By lemma~\ref{l.compare1} we already know that
the $n(N)$ first iterates of $y_g$ under $f$ and $g$ coincide.
Since $\chi\in (0,1)$ and from the estimate~(\ref{l.dy}),
the points $y$ and $y_g$ do not separate by $f$ during the time $\chi.n(N)$
and by definition of the fake holonomies, the $\chi.n(N)$ first iterates of the points $\widehat{\Pi^{ss}_f}(y_g)$ and $y_g$ remain in a same strong stable plaque.
From~(\ref{l.dy}) and the definition of $\chi$,
we also have that for $0\leq m\leq \chi.n(N)$,
\begin{equation}\label{e.sep}
d(f^{m}(y_g),f^{m}(y))<\lambda_u^{-n(N)+m}.C.r<\|Df_0^{-1}\|^{-m}<\frac 1 3 \rho^{m-k_0}.
\prod^{m-1}_{\ell=0}\|Df_{|E^s}(f^\ell(x))\|.
\end{equation}
With~(\ref{i.2}), this shows that $y_g$ belongs to the dynamical ball $B_{\chi.n(N)}(x)$.
We will prove by induction on $m\leq \chi.n(N)$ that
$\Pi^{ss}_g(y_g)$ and $\widehat{\Pi^{ss}_f}(y_g)$ by $f$ also belong to the dynamical ball
$B_m(x)$. This will imply that their $m^\text{th}$ iterates by $f$ and $g$ coincide
and conclude the proof of the lemma.
Let us choose $\eta>0$ small and $m_0\geq 0$ large.
If $N$ has been chosen large enough, the point $y_g$ is close to $y$
and the points $\Pi^{ss}_g(y_g)$ and $\widehat{\Pi^{ss}_f}(y_g)$ are close to $x$;
as a consequence, the three points belong to the dynamical balls $B_m(x)$
with $0\leq m\leq m_0$. When $m$ is larger than $m_0$, the diameter of
$f^{m-1}(B_{m-1}(x))$ is small, hence
$$d(f^{m}(\Pi^{ss}_g(y_g)),f^{m}(y_g))
\leq e^\eta.\|Df_{|E^s}(f^{m-1}(x))\|.d(f^{m-1}(\Pi^{ss}_g(y_g)),f^{m-1}(y_g)),$$
$$d(f^{m}(\widehat{\Pi^{ss}_f}(y_g)),f^{m}(y_g))
\leq e^\eta.\|Df_{|E^s}(f^{m-1}(x))\|.
d(f^{m-1}(\widehat{\Pi^{ss}_f}(y_g)),f^{m-1}(y_g)).$$
With~(\ref{e.sep}), (\ref{i.2}), this gives the required estimate and gives the conclusion.
\end{proof}
Since the points $\Pi^{ss}_g(y_g), \widehat{\Pi^{ss}_f}(y_g)$
belong to a same center-unstable plaque and since their $\chi.n(N)$ first iterates by $f$ remain close,
one deduces that for any $0\leq m\leq \chi.n(N)$, the points
$f^m(\Pi^{ss}_g(y_g))$ and $f^m(\widehat{\Pi^{ss}_f}(y_g))$ are still contained in a center-unstable plaque,
whereas the pairs of point $f^m(\Pi^{ss}_g(y_g))$, $f^m(y_g)$ and $f^m(\widehat{\Pi^{ss}_f}(y_g))$, $f^m(y_g)$
are contained in strong-stable plaques.
This shows that
$$d(\Pi^{ss}_g(y_g), \widehat{\Pi^{ss}_f}(y_g))<\lambda^{-\chi.n(N)},$$
where $\lambda>1$ is the lower bound for the domination between the bundles $E^{s}$
and $E^c\oplus E^u$.
\bigskip
\noindent {\bf 6- Estimating $d(\Pi^{ss}_g(y_g), x)$.}
From the estimates we obtained, we get
$$dist(\Pi^{ss}_g(y_g), x)< d(\Pi^{ss}_g(y_g), \widehat{\Pi^{ss}_f}(y_g)) + d(\widehat{\Pi^{ss}_f}(y_g), \Pi^{ss}_f(y))
< \lambda^{-\chi.n(N)}+[\lambda_u^{-n(N)}C.r]^{\alpha_s}.$$
In order to conclude, the perturbation should thus satisfies:
$$ \lambda^{-\chi.n(N)}+[\lambda_u^{-n(N)}C.r]^{\alpha_s}<\varepsilon.r.$$
Since $\chi,\alpha_s, C, \varepsilon$ are constants independent from $N$,
this inequality holds if $N$ large enough and the following are satisfied:
$$\alpha_s(n(N)\log \lambda_u+|\log r|)>|\log r|+c,$$
$$n(N)\log\lambda>|\log r|+c,$$
where $c>0$ is independent from $N$.
From the definition of $\widehat r$ and since $n(N)> K.N$, one gets the following condition
\begin{equation}\label{e.est1}
|\log r|< B.|\log \widehat r|-c,
\end{equation}
where
$$B= \inf\left(\chi\log\lambda,\frac{\alpha_s}{1-\alpha_s}\log\lambda_u\right)\frac{K}{|\log\mu_c|}.$$
Note that by our choice of $K$, the factor $B$
is larger than $1$.
\bigskip
\noindent {\bf 7- Realization of the $C^{1+\alpha}$ perturbation.}
By lemma~\ref{l.perturbation},
in order to be able to realize a $C^{1+\alpha}$ perturbation
supported on a ball of radius $\widehat r$ such that $d(g(f^{-1}(x)),x)=r$,
one has to check that for some $A>\alpha$
one can choose $r,\widehat r$ arbitrarily small satisfying
\begin{equation}\label{e.perturb}
|\log r|>(1+A)|\log \widehat r|.
\end{equation}
Note that this also implies the estimate
$C.r<\widehat r=\frac 1 2 \mu_c^N$ that we used in paragraph 1.
By our choice of $K$, both conditions~(\ref{e.est1}) and~(\ref{e.perturb}) can be realized
simultaneously provided $1+\alpha$ is smaller than $B$.
\section{Properties of chain-hyperbolic homoclinic classes}\label{s.weak-hyperbolicity}
Let $H(p)$ be a homoclinic class which is chain-hyperbolic for a diffeomorphism $f$.
We consider as in the definition~\ref{d.chain-hyperbolic}
the two periodic points $q_s,q_u\in H(p)$
and the plaque families $\cW^{cs},\cW^{cu}$ respectively tangent to the bundles $E^{cs},E^{cu}$.
\subsection{Periodic points with large stable manifold}
We first give an immediate consequence of the trapping property.
\begin{lemma}\label{l.largemanifold}
Let $O$ be a periodic orbit in $H(p)$.
If there exists a point $q_0\in O$ such that $\cW^{cs}_{q_0}$ is contained in the stable manifold of $q_0$,
then this property holds for any point $q\in O$ and more generally for any point
$z\in W^s(q_0)\cap H(p)$.
\end{lemma}
\begin{proof}
Any point $q\in O$ can be written as $q=f^{-n}(q_0)$ with $n\geq 0$.
By the trapping property, $\cW^{cs}_q$ is contained in $f^{-n}(\cW^{cs}_{q_0})$, hence
in $f^{-n}(W^{s}(q_0))=W^s(q)$.
Any point $z\in W^s(q_0)$ has large forward iterates $f^n(z)$, $n\geq n_0$ which remain close to $O$.
By continuity and the coherence (lemma~\ref{l.uniqueness-coherence})
one deduces that $\cW^{cs}_{f^n(z)}$ is also contained in the stable manifold of $O$.
By the trapping property this also holds for $z$.
\end{proof}
The homoclinic class $H(p)$ contains a dense set of ``good'' periodic points, in the sense which is defined in the next lemma:
\begin{lemma}\label{l.contper}
For any $\delta>0$ small, there exists a dense set $\cP_0\subset H(p)$ of periodic points homoclinically
related to the orbit of $p$ with the following property.
\begin{itemize}
\item[--] The modulus of the Lyapunov exponents of any point $q\in \cP_0$ are larger than $\delta$.
\item[--] The plaques $\cW^{cs}_q$ and $\cW^{cu}_q$ for any point $q\in \cP_0$
are respectively contained in the stable and in the unstable manifolds of $q$.
\end{itemize}
\end{lemma}
\begin{proof}
Let us choose $\delta>0$ such that the modulus of the Lyapunov exponents of $q_s$ and $q_u$
are larger than $2\delta$.
Let $U_s$ and $U_u$ be some small disjoint neighborhoods
of the orbits of $q_s$ and $q_u$ respectively:
there exist some constant $j\geq 1$ such that for any segment of orbit $\{x,\dots, f^{jn}(x)\}$ contained in
$H(p)\cap U_s$ or in $H(p)\cap U_u$, one has for any $u\in E_x$ and $v\in F_x$,
$$\prod_{i=0}^{n-1}\|Df^j_{f^{ij}(x)}.u\|\leq e^{-2\delta nj}\|u\| \text{ and }
\prod_{i=0}^{n-1}\|Df^j_{ij}.v\|\geq e^{2\delta nj}\|v\|.$$
We fix $\varepsilon>0$ small and
consider the periodic orbits $O$ that are homoclinically related to the orbit of $p$ with the following
combinatorics:
there are at least $\frac 1 2(1- \varepsilon).\tau$ consecutive iterates in $U_s$
and at least $\frac 1 2(1- \varepsilon).\tau$ consecutive iterates in $U_u$, where $\tau$ is the period of $O$.
In particular, the maximal Lyapunov exponent of $O$
along $E^{cs}$ is smaller than $-\delta$ and the minimal Lyapunov exponent of $O$
along $E^{cu}$ is larger than $\delta$.
Let us write the orbit $O=\{z,\dots,f^{\tau-1}(z)\}$ as the concatenation
of a segment of orbit $\{z,\dots, f^{m-1}(z)\}$ in $U_s$,
a segment of orbit $\{f^{m+\ell_1}(z),\dots, f^{2m+\ell_1-1}(z)\}$ in $U_u$,
and two other segments of orbit
$\{f^{m}(z),\dots,f^{m+\ell_1-1}(z)\}$ and
$\{f^{2m+\ell_1}(z),\dots,f^{2m+\ell_1+\ell_2-1}(z)\}$,
such that $m\geq \frac 1 2(1- \varepsilon).\tau$,
and $\ell_1,\ell_2\leq \frac \varepsilon 2 \tau$.
Provided $\varepsilon$ is small, at any iterate $z_k=f^k(z)$ with
$0\leq k <m/2$, one has for any $u\in E_{f^k(z)}$ and any $n\geq 0$,
$$\prod_{i=0}^{n-1}\|Df^j_{f^{ij}(z_k)}.u\|\leq e^{-\delta nj}.\|u\|.$$
One deduces that there exists $\rho>0$ such that the ball centered at $z_k$ with radius $\rho$
in $\cW^{cs}_{z_k}$ is contracted by forward iterations so that it is contained in the stable set of $z_k$.
Since the stable set of $q_s$ contains $\cW^{cs}_{q_s}$, there exists $N\geq 2$
such that $f^N(\cW^{cs}_{q_s})$ has a radius smaller than $\rho/2$.
If $\tau$ is large enough, since $\{z,\dots,f^{m-1}(z)\}$ is contained in the neighborhood
$U_s$ of the hyperbolic orbit of $q_s$, there exists an iterate $z_k=f^{k}(z)$,
$0\leq k <\frac m 2 -N$ arbitrarily close to $q_s$.
By continuity of the plaque family $\cW^{cs}$, one deduces that
$f^{N}(\cW^{cs}_{z_k})$ has radius smaller than $\rho$, hence is contained in the stable set of
$f^{N}(z_k)$. Consequently the plaque $\cW^{cs}_{z_k}$ is contained in the stable manifold of $z_k$.
By lemma~\ref{l.largemanifold} for any point $q$ in the orbit $O$, the plaque $\cW^{cs}_q$
is contained in the stable manifold of $q$. Similarly the unstable manifold of $q$ contains the plaque
$\cW^{cu}_q$.
In order to prove the lemma, it remains to show that the union of the orbits $O$ we considered is dense
in $H(p)$:
Indeed any point $x$ in $H(p)$ can be approximated by a hyperbolic periodic point
$q$ whose orbit is homoclinically related to the orbit of $q_s$ and $q_u$. Then there exists a transitive
hyperbolic set which contains the points $q,q_s,q_u,p$. One deduces by shadowing
that there exists a hyperbolic periodic orbit $O$ having a point close to $x$
which is homoclinically related to the orbit of $p$ and has the required combinatorics.
\end{proof}
When the central bundles are one-dimensional, one can control the size of the
invariant manifolds of the periodic orbits whose Lyapunov exponents are far from $0$.
\begin{lemma}\label{l.largestable}
Let us assume that there is a dominated splitting $E^{cs}=E\oplus E^c$ such that
$E^c$ has dimension $1$.
For any $\delta>0$, there exists $\rho>0$ with the following property:
let $O\subset H(p)$ be a periodic orbit whose Lyapunov exponents along $E^{cs}$ are smaller than $-\delta$.
Then, there exists $q\in O$ whose stable set contains the ball centered at $q$ with radius $\rho$.
\end{lemma}
\begin{proof}
Let $O\subset H(p)$ be a hyperbolic periodic orbit whose Lyapunov exponents along $E^{cs}$ are smaller than $-\delta$: since $E^c$ is one-dimensional this implies that there exists $q_0\in O$ such that for each $n\geq 0$ one has
$\|Df^n_{|E^c}(q_0)\|=\prod_{i=0}^{n-1}\|Df_{|E^{c}}(f^{i}(q_0))\|
\leq e^{-n.\delta}$.
The domination $E\oplus E^{c}$ then implies that for each $n\geq 0$, one has
\begin{equation}\label{e.unif}
\prod_{i=0}^{n-1}\|Df^N_{|E^{cs}}(f^{i.N}(q_0))\|\leq C.e^{-n},
\end{equation}
where $C,N>0$ are some uniform constants given by the domination.
One deduces from~(\ref{e.unif}) that a uniform neighborhood of $q_0$ in $\cW^{cs}_{q_0}$
is contained in $W^s(q_0)$.
\end{proof}
\begin{remark}\label{r.large}
The previous lemma still holds if one replaces $g$ by a diffeomorphism $C^1$-close to $f$
and if one considers a periodic orbit $O$ of $g$ contained in a small neighborhood of $H(p)$ and
a locally invariant plaque family of $g$ over $O$ whose plaques are $C^1$-close to the plaques of $\cW^{cs}$.
\end{remark}
\medskip
\begin{lemma}\label{l.linked}
Let us assume that $E^{cs}$ and $E^{cu}$ are thin trapped by $f$ and $f^{-1}$ respectively.
Then, all the hyperbolic periodic orbits contained in $H(p)$ are homoclinically related together.
\end{lemma}
\begin{proof}
First, observe that all the hyperbolic periodic points in $H(p)$
have the same stable index.
Let us take a periodic point $q$ in the class.
By lemma~\ref{l.contper}, there exists a periodic orbit $O$
homoclinically related to $p$ and having a point $q'$
arbitrarily close to $q$ such that $\cW^{cs}_{q'}\subset W^s(q')$ and $\cW^{cu}_{q'}\subset W^u(q')$.
One deduces that the plaques $\cW^{cs}_{q'}$ intersects $W^u(q)$ and
$\cW^{cu}_{q'}$ intersects $W^s(q)$. As a consequence $O$ and $q$ are homoclinically related.
\end{proof}
\subsection{Local product stability}
For any invariant compact set $K$, we define the \emph{chain-stable set} of $K$
as the set of points $x\in M$ such that for any $\varepsilon>0$, there exists a $\varepsilon$-pseudo-orbit
that joints $x$ to $K$. The chain-unstable set of $K$ is the chain stable set of $K$ for the map $f^{-1}$.
\begin{lemma}\label{l.chain-stable}
For any point $x\in H(p)$, the plaque $\cW^{cs}_x$ (resp. $\cW^{cu}_y$)
belongs to the chain-stable set (resp. the chain-unstable set) of $H(p)$.
\end{lemma}
\begin{proof}
By lemma~\ref{l.contper}, the point $x$ is the limit of periodic points $p_n\in \cP_0$
such that $\cW^{cs}_{p_n}$ is contained in the stable set of $p_n$ for each $n\geq 0$.
By definition of a plaque family, any point of $\cW^{cs}_{x}$ is limit
of a sequence of points $x_n\in \cW^{cs}_{p_n}$, proving that $x$ is contained in the
chain-stable set of $H(p)$.
\end{proof}
\begin{lemma}\label{l.bracket0}
For any points $x,y\in H(p)$, any transverse intersection point
between $\cW^{cs}_{x}$ and $\cW^{cu}_{y}$ is contained in $H(p)$.
\end{lemma}
\begin{proof}
By lemma~\ref{l.contper}, there exist two periodic points $p_x$ and $p_y$
close to $x$ and $y$ respectively whose orbits are homoclinically related to $p$
such that $\cW^{cs}_{p_x}\subset W^s(p_x)$
and $\cW^{cu}_{p_y}\subset W^u(p_y)$.
By continuity of the plaque families $\cW^{cs}$ and $\cW^{cu}$, one deduces that
$\cW^{cs}_{p_x}$ and $\cW^{cu}_{p_y}$ intersect transversally at a point $z'\in H(p)$ close to $z$.
Hence $z$ belongs to $H(p)$.
\end{proof}
\subsection{Robustness}
Let us consider a compact set $K$ having a dominated splitting $T_KM=E\oplus F$ for $f$.
If $U\subset M$ and $\cU\subset \operatorname{Diff}^1(M)$ are some small neighborhoods of $K$ and $f$,
then for each $g\in \cU$
the maximal invariant set $K_g=\bigcap_{n\in \ZZ} g^n(\overline U)$ has a dominated splitting
$E_g\oplus F_g$ such that $\dim(E_g)=\dim(E)$. Moreover the maps $(g,x)\mapsto E_{g,x}, F_{g,x}$ are continuous.
Hence one may look for a plaque family tangent to the continuation $E_g$ of $E$ for $g$.
A collection of plaque families $(\cW_g)_{g\in \cU}$ tangent to the bundles $(E_g)_{g\in \cU}$
over the sets $(K_g)_{g\in \cU}$ is \emph{continuous} if
$(\cW_{g,x})_{g\in\cU,x \in K_g}$ is a continuous family of $C^1$-embeddings.
It is \emph{uniformly locally invariant} if there exists $\rho>0$ such that for each $g\in \cU$ and $x\in K_g$,
the image of the ball $B(0,\rho)\subset E_{g,x}$ by $g\circ \cW_{g,x}$
is contained in the plaque $\cW_{g,g(x)}$.
\begin{lemma}\label{l.extension0}
Let $K$ be an invariant compact set for a diffeomorphism $f$
having a dominated splitting $E\oplus F$. Then,
there exist some neighborhoods $U$ of $K$ and $\cU\subset \operatorname{Diff}^1(M)$ of $f$
and a continuous collection of plaque families $(\cW_g)_{g\in \cU}$
tangent to the bundles $(E_g)_{g\in \cU}$ over the maximal invariant sets $(K_g)_{g\in \cU}$ in $\overline{U}$,
which is uniformly locally invariant.
\end{lemma}
\begin{proof}
Let $\exp$ be the exponential map from a neighborhood of the section $0$ in $TM$ to $M$.
Each diffeomorphism $g$ close to $f$ induces a diffeomorphism $\hat g$ on $TM$,
which coincides for each $x\in K$ with the map $\exp^{-1}_{g(x)}\circ g\circ \exp_x$
on a small neighborhood of $0\in T_xM$ and with the linear map $T_xg$ outside another small
neighborhood of $0$; moreover,
$\hat g$ is arbitrarily close to the linear bundle automorphism $Tg$ over the map $g$.
The proof of the plaque family theorem~\cite[theorem 5.5]{HPS} associates to each $x\in K_g$
the graph $\Gamma_{g,x}$ in $T_xM$ of a $C^1$ map $\psi_{g,x}\colon E_{g,x}\to F_{g,x}$ tangent to $E_{g,x}$ at $0\in T_xM$ and satisfying
\begin{equation}\label{e.invariance}
\hat g(\Gamma_{g,x})=\Gamma_{g,g(x)}.
\end{equation}
The graphs $\Gamma_{g,x}$ are uniformly Lipschitz and are characterized for some constant $C>0$ by
$$\Gamma_{g,x}=\bigcap_{n\geq 0}
\hat g^{-n}
(\{(y_1,y_2)\in E_{g,g^{-n}(x)}\times F_{g,g^{-n}(x)}, C\|y_1\|\geq \|y_2\|\}).$$
One thus deduces that they depend continuously on $(g,x)$ for the $C^0$-metric.
On the other hand, each map $\hat g$ has a dominated splitting $\hat E^{cs}\oplus\hat E^{cu}$
inside the spaces $T_x M$ and each graph $\Gamma_{g,x}$ is tangent to the bundle $\hat E^{cs}$.
The bundle $\hat E^{cs}$ depends continuously on $(g,x)$, hence the graphs $\Gamma_{g,x}$
depend continuously on $(g,x)$ also for the $C^1$-metric.
The plaque $\cW_{g,x}$ is defined as the image by the exponential
$\exp_x$ of a uniform neighborhood of $0\in \Gamma_{g,x}$.
For instance, one may choose $\varepsilon>0$ small and define for any $z\in E_{g,x}$,
$$\cW_{g,x}(z)=\exp_x(y,\psi_{g,x}(y)),\text{ where } y=\varepsilon.\frac{\arctan\|z\|}{\|z\|}.z.$$
By construction and the invariance~(\ref{e.invariance}),
the plaque families $(\cW_{g})$ are uniformly locally invariant.
\end{proof}
\begin{lemma}\label{l.robustness}
Let us assume that $E^{cs}$ and $E^{cu}$ are thin trapped by $f$ and $f^{-1}$ respectively
and that $H(p)$ coincides with its chain-recurrence class.
Then, there exist some neighborhoods $U$ of $K$ and $\cU\subset \operatorname{Diff}^1(M)$ of $f$
and two continuous collections of plaque families $(\cW^{cs}_g)_{g\in\cU},(\cW^{cu}_g)_{g\in \cU}$
tangent to the bundles $(E^{cs}_g),(E^{cu}_g)$ over the maximal invariant sets $(K_g)_{g\in \cU}$ in $U$,
which are trapped by $g$ and by $g^{-1}$ respectively.
The plaques may be chosen arbitrarily small. As a consequence, for any diffeomorphism $g$ that is $C^1$-close to $f$, the homoclinic class $H(p_g)$ of $g$ associated to the continuation $p_g$ of $p$ is still chain-hyperbolic.
\end{lemma}
\begin{proof}
Let us consider a continuous collection of plaque families $(\cW_{g})$ tangent to the bundles $(E^{cs}_g)$
over the sets $(K_g)$ as given by lemma~\ref{l.extension0}.
Since $E^{cs}$ is thin trapped for $f$ over $H(p)$,
there exists a constant $\rho>0$ and a continuous family of embedding $(\varphi^0_x)$ of $(E^{cs}_x)$ supported in a small neighborhood $S$
of the section $0$ in $E^{cs}$ and
satisfying for each $x\in H(p)$,
\begin{equation}\label{l.trapped}
f(\overline{\cW_{f,x}\circ\varphi^0_x(B(0,\rho)}))\subset \cW_{f,f(x)}\circ\varphi^0_{f(x)}(B(0,\rho)).
\end{equation}
One may find a continuous family of embeddings $(\varphi_x)$ that is close to $(\varphi^0_x)$
for the $C^1$-topology and that extends to any point $x$ in a neighborhood of $H(p)$:
one fixes a finite collection of points $x_i$ in $H(p)$
and using a partition of the unity one defines $\varphi_x$ as a barycenter between
$\varphi^0_{x_i}$ associated to points $x_i$ that are close to $x$.
One deduces that there exist a neighborhood $U$ of $H(p)$ in $M$,
and a continuous family of embeddings
$(\varphi_x)$ of $(E^{cs}_x)$ over $U$,
such that (\ref{l.trapped}) still holds for $g$, $x\in K_g$
and $(\varphi_x)$.
One can thus define $\cW^{cs}_{g,x}$ as the embedding
$$z\mapsto \cW_{g,x}\circ \varphi_{g,x} \left(\frac{\rho.\arctan\|z\|}{\|z\|}.z\right).$$
By construction the plaque family $\cW^{cs}_g$ is trapped by $g$
and the collection $(\cW^{cs}_g)_{g}$ is continuous.
The plaques $\cW^{cs}_{g,x}$ may have been chosen arbitrarily small
and in particular much smaller than the stable manifold $W^s(q_{s,g})$
of the continuation $q_{s,g}$ of $q_s$.
The trapping property thus implies that $\cW^{cs}_{q_{s,g}}$ is contained in the stable manifold of $q_{s,g}$.
One builds similarly the plaques $\cW^{cu}_{g,x}$ and proves that $\cW^{cu}_{q_{u,g}}$
is contained in the unstable manifold of $q_{u,g}$ for any $g$ close to $f$.
We have thus shown that $H(p_g)$ is chain-hyperbolic.
\end{proof}
\begin{remark}\label{r.coherence}
Under the setting of lemma~\ref{l.robustness}.
One can check that the numbers $r,\rho,\varepsilon$ that appear in lemma~\ref{l.uniqueness-coherence}
for the coherence and the uniqueness of the plaque families can be chosen uniform in $g$.
\end{remark}
\subsection{Quasi-attractors}
\begin{lemma}\label{l.cont-quasi-attractor}
If the chain hyperbolic class $H(p)$ is a quasi-attractor and if the bundle $E^{cu}$
is uniformly expanded, then
for any diffeomorphism $g$ $C^1-$close to $f$ and
any hyperbolic periodic point $q$ homoclinically related to the orbit of $p_g$,
the unstable manifold $W^u(q)$ is contained in the homoclinic class $H(p_g)$.
\end{lemma}
\begin{proof}
Since $H(p)$ is a homoclinic class, there exists a dense set of points $x\in H(p)$
that belong to the stable manifold of $p$. Moreover by the trapping property,
$\cW^{cs}_x$ contains $f^{-n}(\cW^{cs}_{f^n(x)})$ for any $n\geq 0$, hence is contained in the
stable manifold of the orbit of $p$.
If $H(p)$ is a quasi-attractor and $E^{cu}$ is uniformly expanded,
it is the union of the unstable manifolds $W^u(x)$
of the points $x\in H(p)$. If one fixes $\rho>0$ then any disk
$D$ of radius $\rho$ contained in an unstable manifold $W^u(x)$ intersects transversally
the stable manifold of $p$. Hence, by compactness there exists $N\geq 1$ uniform such that
$f^N(D)$ intersects transversally the local stable manifold $W^s_{loc}(O)$ of the orbit $O$ of $p$.
This property is open: since $H(p)$ is a chain-recurrence class, for any $g$ close to $f$,
the class $H(p_g)$ is contained in a small neighborhood of $H(p)$,
hence for any disk $D$ of radius $\rho$
contained in $W^u(x)$ for some $x\in H(p)$,
the iterate $f^N(D)$ intersects transversally $W^s_{loc}(O_g)$.
Moreover since $H(p)$ is a quasi-attractor, there exists an arbitrarily small open neighborhood $U$
of $H(p)$ such that $f(\overline U)\subset U$. Hence for $g$ close to $f$ one still
has $g(\overline U)\subset U$ and the unstable manifold $W^u(O_g)$ is contained in $U$.
Since $U$ is a small neighborhood of the set $H(p)$, the partially hyperbolic structure extends
to the closure of $W^u(O_g)$; in particular the dynamics of $g$ uniformly expands along the manifold
$W^u(O_g)$.
One deduces that for any $g$ close to $f$,
for any point $x\in W^u(O_g)$, for any neighborhood $V$ of $x$ inside $W^u(O_g)$,
there exists an iterate $g^n(V)$ with $n\geq 1$ which contains a disk of radius $\rho$,
so that $g^{n+N}(V)\subset W^u(O_g)$ intersects transversally $W^s_{loc}(O_g)$.
One deduces that $H(p_g)$ meets $g^{n+N}(V)$, hence $V$. Since $V$ can be chosen arbitrarily small
and $H(p_g)$ is closed, the point $x$ belongs to $H(p_g)$. We have proved that $\overline{W^u(O_g)}\subset H(p_g)$.
Let $q$ be any hyperbolic periodic point homoclinically related to $p_g$.
The unstable manifolds of the orbit of $p$ and $q$ have the same closure.
In particular $W^u(q)\subset H(p_g)$.
\end{proof}
\subsection{Stable boundary points}\label{ss.one-codim}
We now discuss the case the center stable bundle has a dominated decomposition
$E^{cs}=E^s\oplus E^c$ with $\dim(E^c)=1$ and $E^s$ is uniformly contracted.
\paragraph{Half center-stable plaques.}
Any point $x\in H(p)$ has a uniform strong stable manifold which is one-codimensional inside $\cW^{cs}_x$.
A neighborhood of $x$ intersects $\cW^{cs}_x\setminus W^{ss}_{loc}(x)$ into two connected components.
The choice of an orientation on $E^c_x$ allows to denote them by $\cW^{cs,+}_x$
and $\cW^{cs,-}_x$.
One can then consider if $x$ is accumulated inside $\cW^{cs}_x\setminus W^{ss}_{loc}(x)$
by points of $H(p)$ in one or in both components.
Note that this does not depend on the choice
of the plaque family $\cW^{cs}$. Note also that the same case will occur all along the orbit of $x$.
\medskip
If one considers a point $y\in H(p)\cap W^{cu}_x$ close to $x$,
one gets an orientation of $E^c_y$ that matches with the orientation of $E^c_x$.
The points of $H(p)\cap \cW^{cs}_x$ close to $x$ projects on $\cW^{cs}_y$ through the holonomy
along the center unstable plaques, but there is no reason that the projection of the points
in $H(p)\cap \cW^{cs,+}_x$ are contained inside $\cW^{cs,+}_y$. However the following can be proved.
\begin{lemma}\label{l.integrability}
Consider any periodic point $q$ homoclinically related to $p$ and any point $x\in H(p)$ close to $q$
such that $W^u_{loc}(q)$ intersects $W^{ss}_{loc}(x)$ at a point $z$.
If $q$ is accumulated by $H(p)\cap \cW^{cs,+}_q$ then $z$ is accumulated by $H(p)\cap \cW^{cs,+}_x$.
More precisely, there exists $y\in H(p)\cap \cW^{cs,+}_q$ arbitrarily close to $q$
such that $W^u_{loc}(y)$ intersects $H(p)\cap\cW^{cs,+}_{x}$ close to $z$.
\end{lemma}
\begin{proof}
Let us consider a point $y_0\in \cW^{cs,+}_q$ close to $q$; it belongs to $W^s(q)$.
Let $D$ be a neighborhood of $z$ in $\cW^{cs}_{x}$ and $D^+$ a neighborhood of $z$ in $\cW^{cs,+}_{x}$.
By the $\lambda$-lemma, the sequence $f^{-n}(D)$, $n\geq 0$ converges toward $W^s(q)$.
Observe that the strong stable manifolds of $x$ and $z$ coincide.
By continuity of the strong stable lamination, the sequence $f^{-n}(W^{ss}_{loc}(x))$ converges toward
$W^{ss}(q)$. Hence $\cW^{cu}_{y_0}$ intersects $f^{-n}(D^+)$ close to $q$ for $n$ large enough.
The intersection is transversal, hence belongs to $H(p)$ by lemma~\ref{l.bracket0}. One thus deduces that $D^+$ intersects $H(p)$.
By taking $D^+$ arbitrarily small, one has proved that $z$ is accumulated by $H(p)\cap \cW^{cs,+}_x$.
Also the local unstable manifold $W^u_{loc}(y)$ of the point $y=f^n(y_0)$ intersects $D^+$, giving the conclusion.
\end{proof}
\begin{lemma}\label{l.NGSHI}
Let us assume that $H(p)$ does not contains periodic points $q,q'$ homoclinically related
to the orbit of $p$ such that $W^{ss}(q)\setminus\{q\}$ and $W^u(q')$ intersect.
Then any point $x\in H(p)$ is accumulated by $H(p)$ in $\cW^{cs}_x\setminus W^{ss}_{loc}(x)$.
\end{lemma}
\begin{proof}
Let us assume by contradiction that there exists a point
$x\in H(p)$ which is not accumulated by points in $(\cW^{cs}_{x}\cap H(p))\setminus W^{ss}_{loc}(x)$.
Let $q\in H(p)$ be a periodic point close to $x$ and homoclinically related to the orbit of $p$.
Its unstable manifold intersects transversally
$\cW^{cs}_x$ at a point $z\in H(p)$. Since $z$ can be chosen arbitrarily close to $x$,
it belongs to $W^{ss}_{loc}(x)$ and
it is not accumulated by points in $(\cW^{cs}_{x}\cap H(p))\setminus W^{ss}_{loc}(x)$.
By lemma~\ref{l.integrability}, $W^s_{loc}(q)\setminus W^{ss}(q)$ is disjoint from $H(p)$.
In particular the point $q$ is not accumulated by points in $(\cW^{cs}_{q}\cap H(p))\setminus W^{ss}_{loc}(q)$.
One can thus repeat for $q$ the argument we have made for $x$ and find a periodic point $q'\neq q$
homoclinically related to the orbit of $p$ such that $W^u(q')$ intersects $W^{ss}(q)$.
This contradicts our assumption.
\end{proof}
We now introduce the definition of the stable boundary points, generalizing the notion
of stable boundary points for uniformly hyperbolic set whose stable bundle is one-dimensional
(see~\cite[appendix 2]{PT}). This notion plays an important role
and it is extensively studied in section \ref{s.boundary}.
\begin{defi}\label{d.boundary}
A point $x\in H(p)$ is a \emph{ stable boundary point} if it is not accumulated inside
both components of $\cW^{cs}_x\setminus W^{ss}_{loc}(x)$ by points of $H(p)$.
\end{defi}
Observe that if $x$ is a stable boundary point, then any iterate of $x$ is also.
Note that if $E^{cs}$ is one-dimensional, a stable boundary point $x\in H(p)$
is a point which is not accumulated by points of $H(p)$ in both components of $\cW^{cs}_x\setminus \{x\}$.
Naturally in the same way, if the center unstable subbundle split $E^{cu}=E^c_2\oplus E^u$,
where $E^{c}_2$ is one-dimensional and $E^u$ is uniformly expanded, it can be defined the notion of
{\em unstable boundary point}.
\medskip
The next lemma about stable boundary points is a version of a classical one for hyperbolic systems.
A more general proposition about stable boundary points is provided in section \ref{s.boundary}.
\begin{lemma}\label{p.boundary0} Let $f$ be a diffeomorphism and $H(p)$ be a chain-recurrence class which is
a chain-hyperbolic homoclinic class endowed with a dominated splitting $E^{cs}\oplus E^{cu}=E^{cs}\oplus (E^c_2\oplus E^{u})$ such that $E^{cs},E^{c}_2$ are one-dimensional, $E^{cs},E^{cu}$ are thin trapped (for $f$ and $f^{-1}$ respectively) and $E^u$ is uniformly expanded.
Then any stable boundary point of $H(p)$ belong to the
unstable set of a periodic point.
\end{lemma}
\begin{proof}
Let $x$ be a stable boundary point of $H(p)$.
Let us introduce three backward iterates $x_1=f^{-k}(x)$, $x_2=f^{-l}(x)$ and $x_3=f^{-m}(x)$ arbitrarily close
with $k<l<m$.
If the center-unstable plaques of two of those three points (for instance $x_1,x_2$) intersect,
from the coherence (lemma~\ref{l.uniqueness-coherence}) it follows that the center-unstable plaque $\cW^{cu}_{x_1}$
is mapped into itself by $f^{k-l}$. Since $E^{cu}$ splits as $E^{c}_2\oplus E^u$,
one deduces that the backward orbit of $x_1$ belongs to the unstable set of a periodic point
of $\cW^{cu}_{x_1}$ (this point is not necessarily hyperbolic).
If the center unstable plaques of the three points do not intersect, we can assume that the center stable plaque of namely $x_2$ intersects the center unstable plaques of the other two points in different connected components of $\cW_{x_2}^{cs}\setminus \{x_2\}.$ By lemma \ref{l.bracket0} those points of intersection belong to $H(p)$ and using that $E^{cs}$ is thin trapped, the forward orbits of those points remain arbitrarily close to $x$ (provided that the points $x_1, x_2, x_3$ were sufficiently close) and contained in different components of $\cW_{x_2}^{cs}\setminus \{x_2\};$ a contradiction.
\end{proof}
The following proposition is not needed in the context of the present paper, however we provide it since it helps to understand the notion of boundary point.
\begin{prop}
Using~\cite[proposition 3.2]{C2}, one can prove that if the homoclinic class $H(p)$ is endowed with a partially hyperbolic
structure $E^s\oplus E^c\oplus E^u$ with $\dim(E^c)=1$ such that $E^{cs}=E^s\oplus E^c$
is thin trapped, then,
\begin{itemize}
\item[--] either any stable boundary point $x\in H(p)$ belongs to the unstable manifold of
a periodic point,
\item[-] or there exists a diffeomorphism $g$ that is $C^1$-close to $f$ and a periodic orbit contained in
a small neighborhood of $H(p)$ which has a strong homoclinic intersection.
\end{itemize}
\end{prop}
\noindent One will use instead a similar result for quasi-attractors, see section~\ref{ss.structure} below.
\begin{proof}[Sketch of the proof]
Let $x$ be a strong boundary point.
Let us take the sequence $\{x_n=f^{-n}(x)\}_{n>0}$.
Since $E^{cs}$ is thin trapped, one may take a small plaque family $\cW^{cs}$ which is trapped
and such that for each $n\geq 0$, one connected component $U_n$ of $\cW^{cs}_{x_n}\setminus W^{ss}_{loc}(x_n)$ is disjoint from $H(p)$.
In particular:
\begin{description}
\item[(*)]\emph{For any two close iterates $x_n,x_m$, the unstable manifold $W^u_{loc}(x_n)$
does not meet $U_m$.}
\end{description}
We consider two cases:
either the orientation of the center manifolds of all close backward iterates is preserved or not.
Equivalently, the tangent map $Df$ preserves or not a continuous orientation of the bundle $E^c$
over $\alpha(x)$, the $\alpha$-limit set of $x$. One can assume that $\alpha(x)$ is not reduced to
a periodic orbit since otherwise, $x$ belongs to the unstable manifold of a periodic orbit
and the statement follows.
\noindent
\emph{- The orientation preserved case.}
From property (*), any two close iterates $x_n,x_m$ are in twisted position
(see~\cite[section 3]{C2}), implying that $\alpha(x)$ is twisted.
If $\alpha(x)$ contains a periodic orbit $O$, it contains points in $W^{ss}(O)\setminus O$
and in $W^u(O)\setminus O$;
as a consequence, one can apply the Hayashi connecting lemma and get a strong
homoclinic intersection for $O$ by an arbitrarily small $C^1$-perturbation.
Otherwise $\alpha(x)$ contains a non-periodic minimal set and from~\cite[proposition 3.2]{C2},
there exists a diffeomorphism $g$ that is $C^1$-close to $f$ and a periodic orbit contained in
a small neighborhood of $H(p)$ which has a strong homoclinic intersection.
\noindent
\emph{- The orientation reversed case.}
Let us consider a sequence of arbitrarily close points $x_{n_k},x_{m_k}$
such that $Df^{m_k-n_k}$ reverse the local orientation on $E^c$ at $x_{n_k}$.
One may assume that they converge toward a point $y\in \alpha(x)$.
Property (*) now implies that $H(p)\cap\cW^{cs}_y$ is contained in $W^{ss}(y)$.
This contradicts lemma~\ref{l.NGSHI} above.
\end{proof}
\subsection{Non-uniformly hyperbolic bundles}
When the bundle $E^{cs}$ is not uniformly contracted, the class may contain
weak periodic orbits.
\begin{lemma}\label{l.weak}
Let us assume that $H(p)$ is a chain-recurrence class
and that there exists a dominated splitting $E^{cs}=E^s\oplus E^c$ where $E^c$
is one-dimensional, $E^{cs}$ is thin-trapped and $E^s$ is uniformly contracted.
Then, there exists some hyperbolic periodic orbits in $H(p)$
whose Lyapunov exponent along $E^c$ is arbitrarily close to zero.
\end{lemma}
\begin{remark}\label{r.weak}
If one assumes that all the periodic orbits in $H(p)$ are hyperbolic, then one
can ensure that the obtained periodic orbits are homoclinically related to $p$.
Indeed, since $E^{cs}$ is thin-trapped, all the periodic orbits in $H(p)$
have the same stable dimension and by lemma~\ref{l.linked} are homoclinically related to $p$.
\end{remark}
\begin{proof}
One can consider an invariant compact set $K\subset H(p)$ such that
the restriction of $E^c$ to $K$ is not uniformly contracted and $K$ is minimal
for the inclusion and these properties.
Since the bundle $E^c_{|K}$ is one-dimensional, thin trapped and not uniformly contracted, $K$ coincides
with the support of an ergodic measure $\mu$ whose Lyapunov exponent along $E^c$ is zero.
The exponent of any other measure supported on $K$ is non-positive.
In the case there exists ergodic measures $\mu$ supported on $K$ whose Lyapunov exponent
along $E^c$ is negative and arbitrarily close to zero, the domination $E^{cs}\oplus E^{cu}$
implies that these measures are hyperbolic and the $C^1$-version of Anosov closing lemma
(see~\cite[proposition 1.4]{C2}) ensures that the chain-recurrence class $H(p)$ contains hyperbolic periodic orbits
whose Lyapunov exponent along $E^c$ is arbitrarily close to zero.
In the case there exists ergodic measures supported on $K$ with negative Lyapunov exponent along $E^c$
but never contained in a small interval $(-\delta,0)$, one can argue as in the proof of
\cite[theorem 1]{C2} and apply Liao's selecting lemma. Once again, the chain-recurrence class $H(p)$ contains hyperbolic periodic orbits
whose Lyapunov exponent along $E^c$ is arbitrarily close to zero.
In the remaining case, all the measures supported on $K$ have a Lyapunov exponent along $E^c$ that is equal to zero.
In particular, $E^{cu}$ is uniformly expanded on $K$.
We have also assumed that $E^{cs}$ is thin trapped.
As a consequence, one can choose over the maximal invariant set in a neighborhood of $K$
some plaques $\mathcal{D}} \def\cJ{\mathcal{J}} \def\cP{\mathcal{P}} \def\cV{\mathcal{V}^{cs}$ and $\mathcal{D}} \def\cJ{\mathcal{J}} \def\cP{\mathcal{P}} \def\cV{\mathcal{V}^{cu}$ with arbitrarily small diameter and that are trapped by $f$
and $f^{-1}$ respectively.
For any $\varepsilon>0$ there exists a periodic $\varepsilon$-pseudo-orbit
$x_0,x_1,\dots,x_n=x_0$ contained in $K$ such that the quantity
$$\frac 1 n \sum_{k=0}^{n-1}\log \|Df_{|E^c}(x_k)\|$$
is arbitrarily close to zero. By the weak shadowing lemma~\cite[lemma 2.9]{C2},
there exists a periodic orbit $O_0$ contained in an arbitrarily small neighborhood of $K$
and whose Lyapunov exponent along $E^c$ is close to zero.
The unstable manifold of a point $x\in K$ close to $O_0$ intersects a center-stable plaque
of $O_0$. Since these plaques are trapped and $E^c$ is one-dimensional,
this implies that the center-stable plaques of $O_0$ contains a periodic orbit $O'$
whose stable manifold intersects $W^u(x)$.
On the other hand $W^u(O')$ intersects a center-unstable plaque
of a point of $H(p)$. As a conclusion $O'$ is contained in the chain-recurrence class of $p$.
Since the plaques $\mathcal{D}} \def\cJ{\mathcal{J}} \def\cP{\mathcal{P}} \def\cV{\mathcal{V}^{cs}$ have a small diameter, the Lyapunov exponent of $O'$ along $E^c$
is close to the Lyapunov exponent of $O$, hence is close to zero.
The conclusion of the lemma has been obtained in al the cases.
\end{proof} | {
"redpajama_set_name": "RedPajamaArXiv"
} | 1,183 |
Кордебале́т:
Кордебалет
Кордебалет (фильм)
Кордебалет (мюзикл)
Кордебалет (песня) — песня из репертуара Филиппа Киркорова. | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 2,149 |
This checkpoint is part of a series that will be conducted through the end of the year as part of the Irvine Police Department's continued commitment to reducing injury and deaths caused by impaired drivers.
According to the National Highway Traffic Safety Administration, in 2011, alcohol-impaired driving was involved in 34 percent of traffic fatalities in California. Research has shown that high visibility enforcement (HVE) can reduce drunken driving fatalities by as much as 20 percent. The Irvine Police Department publicizes these highly visible sobriety checkpoints to deter impaired drivers, encourage the use of sober designated drivers and to bring increased awareness to the consequences of impaired driving.
The checkpoint will be clearly marked and vehicles will be selected to be checked on a pre-set basis to ensure objectivity. Motorists will be greeted and given information about impaired driving. Driver's licenses will be checked and trained officers will direct impaired drivers to a secondary area for further evaluation. Most motorists will experience little delay, if any at all.
Funding for this checkpoint is provided by the California Office of Traffic Safety (OTS) through the National Highway Traffic Safety Administration.
« Who's up for John Campbell's seat? | {
"redpajama_set_name": "RedPajamaC4"
} | 9,035 |
Six more tourists injured in New Zealand crash
Posted by te2ataria on January 27, 2016
"Vanload of tourists crashes 10 metres down West Coast bank"
Two people have been flown to Nelson and Greymouth hospitals after a van carrying six tourists crashed 10 metres down a bank on the West Coast.
Two others were taken to Buller Hospital. The fates of the remaining two tourists were not reported. http://www.stuff.co.nz/the-press/news/west-coast/76313889/vanload-of-tourists-crashes-10-metres-down-west-coast-bank.html
Logging truck driver killed in crash in Far North town of Waipapa
A truck driver was killed after his logging truck went out of control on Koropewa Road and crashed into a tree in the Far North about 3.40am Wednesday morning.
http://www.stuff.co.nz/auckland/local-news/northland/76286660/Logging-truck-driver-killed-in-crash-in-Far-North-town-of-Waipapa
Motorcyclist killed in south Wairarapa crash named
Police have named a motorcyclist killed when he crashed into an oncoming car on a coastal road in Wairarapa south of Martinborough on Sunday. He was Marc Burgiss, 55.
http://www.stuff.co.nz/dominion-post/news/76260847/Motorcyclist-killed-in-south-Wairarapa-crash-named
Man killed after collision with train in Christchurch
Police believed they knew the identity of the victim, a man aged in his early 20s, but fingerprint identification was needed to confirm that, Detective Sergeant Geoff Rudduck said.
http://www.stuff.co.nz/the-press/news/76206577/man-killed-after-collision-with-train-in-christchurch
Woman killed while diving off Wellington's south coast at Owhiro Bay
The death is at least the third reported in the area this summer in Wellington. http://www.stuff.co.nz/national/76262309/Wellington-diver-dead-despite-CPR-effort
Woman choppered from multi-car smash near Napier
A woman [with near-fatal injuries] was airlifted from the scene of a multi-car crash on State Highway 5 near Napier.
http://www.newstalkzb.co.nz/news/emergency/woman-choppered-from-multi-car-smash-near-napier/
Tourist bus crashes in Northland
A tourist bus with at least 20 people on board has crashed north of Auckland. "Initial reports suggested there were 17 patients on board with multiple injuries, however it has now been clarified [after police intervention] that there were 20 passengers with one person injured."
http://www.nzherald.co.nz/northern-advocate/news/article.cfm?c_id=1503450&objectid=11579347
Woman hit by car in Te Kuiti
A woman who was hit by a car in Te Kuiti on Tuesday, remains in a serious condition.
http://www.odt.co.nz/news/national/370956/woman-hit-car-te-kuiti
Driver stuck after truck 'tipped over'
Two injured in Paeroa pileup
One was flown to Waikato Hospital in a critical condition and another was taken by ambulance to Thames Hospital in serious condition.
Man suffers critical injuries in [yet another] quad bike incident
The victim was pinned under quad bike for several hours.
Driver's 'lucky' escape after car plunges off bridge in South Taranaki
http://www.stuff.co.nz/taranaki-daily-news/news/76306039/drivers-lucky-escape-after-car-plunges-off-bridge-in-south-taranaki
Missing climber was probably killed in fall – Coroner
Missing engineer Simon Bell (33) was probably killed in a mountaineering fall, Coroner David Crerar has ruled.
Warning after Australian toddler drowns in a Wellington hotel bath
An Australian toddler drowned in a Wellington hotel in March 2014 after her grandfather left her unattended in a bath, an inquest has found. 23-month-old Leila Sofia Riquelme accidentally drowned when she was left alone in a filled bath at the Rydges Hotel.
Thousands of children treated for drug and alcohol abuse
NZ's Fucked Generation: Every year thousands of teenagers and adolescents, even children as young and five, are treated for drug or alcohol addiction.
http://www.stuff.co.nz/national/health/76035795/Thousands-of-children-treated-for-drug-and-alcohol-abuse
Ministry of Transport (heavily doctored) Road Toll: Today
Road deaths year to date
As at 27 January 2016 28
Same time last year 19
Fatal crashes year to date
Ministry of Transport (heavily doctored) Road Toll: Yesterday
http://www.transport.govt.nz/research/roadtoll/
No of fatalities NOT reported so far this year: probably as many as 31 [Blog estimate: At least 59 road fatalities have occurred in NZ since 1 January 2016.]
Posted in Tourist Deathtrap | Tagged: Australian toddler drowns, driver killed, killed in NZ, Motorcyclist killed, Mt Earnslaw, road fatalities, road toll, Simon Bell, young addicts | Leave a Comment »
Another toddler critically injured
Toddler cyclist critical after struck on footpath
A 6-year-old Raglan boy is fighting for his life after a reversing driver struck him as he cycled past on the footpath. http://www.stuff.co.nz/national/76304562/Raglan-child-cyclist-critical-after-struck-on-footpath
British backpacker dies in hostel
Kristina Renner, 21, died in hospital after being found unconscious at the Sunset Backpackers hostel in Mildura, northestern Victoria. An autopsy was unable to find a cause of death.
http://www.stuff.co.nz/world/australia/76306254/Backpacker-stuck-between-two-sofas-dies
Woman attacked on walk to work
Hamilton police say they are seeking a man wearing hi-vis orange overalls after he attacked a woman [raped/molested/physically assaulted her?] as she walked to work on Hood St shortly after 6am yesterday.
Posted in Tourist Deathtrap | Tagged: critically injured, driver killed, driveway incident, Hamilton, Kristina Renner, Paeroa, quad bike, toddler injured, Waipapa | Leave a Comment » | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 8,343 |
#include "lib.h"
#include "array.h"
#include "str.h"
#include "strescape.h"
#include "hash.h"
#include "mail-user.h"
#include "mailbox-list.h"
#include "acl-global-file.h"
#include "acl-cache.h"
#include "acl-api-private.h"
struct acl_letter_map {
char letter;
const char *name;
};
static const struct acl_letter_map acl_letter_map[] = {
{ 'l', MAIL_ACL_LOOKUP },
{ 'r', MAIL_ACL_READ },
{ 'w', MAIL_ACL_WRITE },
{ 's', MAIL_ACL_WRITE_SEEN },
{ 't', MAIL_ACL_WRITE_DELETED },
{ 'i', MAIL_ACL_INSERT },
{ 'p', MAIL_ACL_POST },
{ 'e', MAIL_ACL_EXPUNGE },
{ 'k', MAIL_ACL_CREATE },
{ 'x', MAIL_ACL_DELETE },
{ 'a', MAIL_ACL_ADMIN },
{ '\0', NULL }
};
struct acl_object *acl_object_init_from_name(struct acl_backend *backend,
const char *name)
{
return backend->v.object_init(backend, name);
}
struct acl_object *acl_object_init_from_parent(struct acl_backend *backend,
const char *child_name)
{
return backend->v.object_init_parent(backend, child_name);
}
void acl_object_deinit(struct acl_object **_aclobj)
{
struct acl_object *aclobj = *_aclobj;
*_aclobj = NULL;
aclobj->backend->v.object_deinit(aclobj);
}
int acl_object_have_right(struct acl_object *aclobj, unsigned int right_idx)
{
struct acl_backend *backend = aclobj->backend;
const struct acl_mask *have_mask;
unsigned int read_idx;
if (backend->v.object_refresh_cache(aclobj) < 0)
return -1;
have_mask = acl_cache_get_my_rights(backend->cache, aclobj->name);
if (have_mask == NULL) {
if (acl_backend_get_default_rights(backend, &have_mask) < 0)
return -1;
}
if (acl_cache_mask_isset(have_mask, right_idx))
return 1;
if (mailbox_list_get_user(aclobj->backend->list)->dsyncing) {
/* when dsync is running on a shared mailbox, it must be able
to do everything inside it. however, dsync shouldn't touch
mailboxes where user doesn't have any read access, because
that could make them readable on the replica. */
read_idx = acl_backend_lookup_right(aclobj->backend,
MAIL_ACL_READ);
if (acl_cache_mask_isset(have_mask, read_idx))
return 1;
}
return 0;
}
const char *const *
acl_backend_mask_get_names(struct acl_backend *backend,
const struct acl_mask *mask, pool_t pool)
{
const char *const *names;
const char **buf, **rights;
unsigned int names_count, count, i, j, name_idx;
names = acl_cache_get_names(backend->cache, &names_count);
buf = t_new(const char *, (mask->size * CHAR_BIT) + 1);
count = 0;
for (i = 0, name_idx = 0; i < mask->size; i++) {
if (mask->mask[i] == 0)
name_idx += CHAR_BIT;
else {
for (j = 1; j < (1 << CHAR_BIT); j <<= 1, name_idx++) {
if ((mask->mask[i] & j) == 0)
continue;
/* @UNSAFE */
i_assert(name_idx < names_count);
buf[count++] = p_strdup(pool, names[name_idx]);
}
}
}
/* @UNSAFE */
rights = p_new(pool, const char *, count + 1);
memcpy(rights, buf, count * sizeof(const char *));
return rights;
}
static int acl_object_get_my_rights_real(struct acl_object *aclobj, pool_t pool,
const char *const **rights_r)
{
struct acl_backend *backend = aclobj->backend;
const struct acl_mask *mask;
if (backend->v.object_refresh_cache(aclobj) < 0)
return -1;
mask = acl_cache_get_my_rights(backend->cache, aclobj->name);
if (mask == NULL) {
if (acl_backend_get_default_rights(backend, &mask) < 0)
return -1;
}
*rights_r = acl_backend_mask_get_names(backend, mask, pool);
return 0;
}
int acl_object_get_my_rights(struct acl_object *aclobj, pool_t pool,
const char *const **rights_r)
{
int ret;
if (pool->datastack_pool)
return acl_object_get_my_rights_real(aclobj, pool, rights_r);
T_BEGIN {
ret = acl_object_get_my_rights_real(aclobj, pool, rights_r);
} T_END;
return ret;
}
const char *const *acl_object_get_default_rights(struct acl_object *aclobj)
{
return acl_backend_mask_get_names(aclobj->backend,
aclobj->backend->default_aclmask,
pool_datastack_create());
}
int acl_object_last_changed(struct acl_object *aclobj, time_t *last_changed_r)
{
return aclobj->backend->v.last_changed(aclobj, last_changed_r);
}
int acl_object_update(struct acl_object *aclobj,
const struct acl_rights_update *update)
{
return aclobj->backend->v.object_update(aclobj, update);
}
struct acl_object_list_iter *acl_object_list_init(struct acl_object *aclobj)
{
return aclobj->backend->v.object_list_init(aclobj);
}
int acl_object_list_next(struct acl_object_list_iter *iter,
struct acl_rights *rights_r)
{
if (iter->failed)
return -1;
return iter->aclobj->backend->v.object_list_next(iter, rights_r);
}
void acl_object_list_deinit(struct acl_object_list_iter **_iter)
{
struct acl_object_list_iter *iter = *_iter;
*_iter = NULL;
iter->aclobj->backend->v.object_list_deinit(iter);
}
struct acl_object_list_iter *
acl_default_object_list_init(struct acl_object *aclobj)
{
struct acl_object_list_iter *iter;
const struct acl_rights *aclobj_rights;
unsigned int i;
pool_t pool;
pool = pool_alloconly_create("acl object list", 512);
iter = p_new(pool, struct acl_object_list_iter, 1);
iter->pool = pool;
iter->aclobj = aclobj;
if (!array_is_created(&aclobj->rights)) {
/* we may have the object cached, but we don't have all the
rights read into memory */
acl_cache_flush(aclobj->backend->cache, aclobj->name);
}
if (aclobj->backend->v.object_refresh_cache(aclobj) < 0)
iter->failed = TRUE;
aclobj_rights = array_get(&aclobj->rights, &iter->count);
if (iter->count > 0) {
iter->rights = p_new(pool, struct acl_rights, iter->count);
for (i = 0; i < iter->count; i++)
acl_rights_dup(&aclobj_rights[i], pool, &iter->rights[i]);
}
return iter;
}
int acl_default_object_list_next(struct acl_object_list_iter *iter,
struct acl_rights *rights_r)
{
if (iter->failed)
return -1;
if (iter->idx == iter->count)
return 0;
*rights_r = iter->rights[iter->idx++];
return 1;
}
void acl_default_object_list_deinit(struct acl_object_list_iter *iter)
{
pool_unref(&iter->pool);
}
struct acl_mailbox_list_context *
acl_backend_nonowner_lookups_iter_init(struct acl_backend *backend)
{
return backend->v.nonowner_lookups_iter_init(backend);
}
int acl_backend_nonowner_lookups_iter_next(struct acl_mailbox_list_context *ctx,
const char **name_r)
{
return ctx->backend->v.nonowner_lookups_iter_next(ctx, name_r);
}
void
acl_backend_nonowner_lookups_iter_deinit(struct acl_mailbox_list_context **_ctx)
{
struct acl_mailbox_list_context *ctx = *_ctx;
*_ctx = NULL;
ctx->backend->v.nonowner_lookups_iter_deinit(ctx);
}
int acl_backend_nonowner_lookups_rebuild(struct acl_backend *backend)
{
return backend->v.nonowner_lookups_rebuild(backend);
}
void acl_rights_write_id(string_t *dest, const struct acl_rights *right)
{
switch (right->id_type) {
case ACL_ID_ANYONE:
str_append(dest, ACL_ID_NAME_ANYONE);
break;
case ACL_ID_AUTHENTICATED:
str_append(dest, ACL_ID_NAME_AUTHENTICATED);
break;
case ACL_ID_OWNER:
str_append(dest, ACL_ID_NAME_OWNER);
break;
case ACL_ID_USER:
str_append(dest, ACL_ID_NAME_USER_PREFIX);
str_append(dest, right->identifier);
break;
case ACL_ID_GROUP:
str_append(dest, ACL_ID_NAME_GROUP_PREFIX);
str_append(dest, right->identifier);
break;
case ACL_ID_GROUP_OVERRIDE:
str_append(dest, ACL_ID_NAME_GROUP_OVERRIDE_PREFIX);
str_append(dest, right->identifier);
break;
case ACL_ID_TYPE_COUNT:
i_unreached();
}
}
const char *acl_rights_get_id(const struct acl_rights *right)
{
string_t *str = t_str_new(32);
acl_rights_write_id(str, right);
return str_c(str);
}
static bool is_standard_right(const char *name)
{
unsigned int i;
for (i = 0; all_mailbox_rights[i] != NULL; i++) {
if (strcmp(all_mailbox_rights[i], name) == 0)
return TRUE;
}
return FALSE;
}
int acl_rights_update_import(struct acl_rights_update *update,
const char *id, const char *const *rights,
const char **error_r)
{
ARRAY_TYPE(const_string) dest_rights, dest_neg_rights, *dest;
unsigned int i, j;
if (acl_identifier_parse(id, &update->rights) < 0) {
*error_r = t_strdup_printf("Invalid ID: %s", id);
return -1;
}
if (rights == NULL) {
update->modify_mode = ACL_MODIFY_MODE_CLEAR;
update->neg_modify_mode = ACL_MODIFY_MODE_CLEAR;
return 0;
}
t_array_init(&dest_rights, 8);
t_array_init(&dest_neg_rights, 8);
for (i = 0; rights[i] != NULL; i++) {
const char *right = rights[i];
if (right[0] != '-')
dest = &dest_rights;
else {
right++;
dest = &dest_neg_rights;
}
if (strcmp(right, "all") != 0) {
if (*right == ':') {
/* non-standard right */
right++;
array_append(dest, &right, 1);
} else if (is_standard_right(right)) {
array_append(dest, &right, 1);
} else {
*error_r = t_strdup_printf("Invalid right '%s'",
right);
return -1;
}
} else {
for (j = 0; all_mailbox_rights[j] != NULL; j++)
array_append(dest, &all_mailbox_rights[j], 1);
}
}
if (array_count(&dest_rights) > 0) {
array_append_zero(&dest_rights);
update->rights.rights = array_idx(&dest_rights, 0);
} else if (update->modify_mode == ACL_MODIFY_MODE_REPLACE) {
update->modify_mode = ACL_MODIFY_MODE_CLEAR;
}
if (array_count(&dest_neg_rights) > 0) {
array_append_zero(&dest_neg_rights);
update->rights.neg_rights = array_idx(&dest_neg_rights, 0);
} else if (update->neg_modify_mode == ACL_MODIFY_MODE_REPLACE) {
update->neg_modify_mode = ACL_MODIFY_MODE_CLEAR;
}
return 0;
}
const char *acl_rights_export(const struct acl_rights *rights)
{
string_t *str = t_str_new(128);
if (rights->rights != NULL)
str_append(str, t_strarray_join(rights->rights, " "));
if (rights->neg_rights != NULL && rights->neg_rights[0] != NULL) {
if (str_len(str) > 0)
str_append_c(str, ' ');
str_append_c(str, '-');
str_append(str, t_strarray_join(rights->neg_rights, " -"));
}
return str_c(str);
}
int acl_rights_parse_line(const char *line, pool_t pool,
struct acl_rights *rights_r, const char **error_r)
{
const char *id_str, *const *right_names, *error = NULL;
/* <id> [<imap acls>] [:<named acls>] */
if (*line == '"') {
line++;
if (str_unescape_next(&line, &id_str) < 0 ||
(line[0] != ' ' && line[0] != '\0')) {
*error_r = "Invalid quoted ID";
return -1;
}
if (line[0] == ' ')
line++;
} else {
id_str = line;
line = strchr(id_str, ' ');
if (line == NULL)
line = "";
else
id_str = t_strdup_until(id_str, line++);
}
memset(rights_r, 0, sizeof(*rights_r));
right_names = acl_right_names_parse(pool, line, &error);
if (*id_str != '-')
rights_r->rights = right_names;
else {
id_str++;
rights_r->neg_rights = right_names;
}
if (acl_identifier_parse(id_str, rights_r) < 0)
error = t_strdup_printf("Unknown ID '%s'", id_str);
if (error != NULL) {
*error_r = error;
return -1;
}
rights_r->identifier = p_strdup(pool, rights_r->identifier);
return 0;
}
void acl_rights_dup(const struct acl_rights *src,
pool_t pool, struct acl_rights *dest_r)
{
memset(dest_r, 0, sizeof(*dest_r));
dest_r->id_type = src->id_type;
dest_r->identifier = p_strdup(pool, src->identifier);
dest_r->rights = src->rights == NULL ? NULL :
p_strarray_dup(pool, src->rights);
dest_r->neg_rights = src->neg_rights == NULL ? NULL :
p_strarray_dup(pool, src->neg_rights);
dest_r->global = src->global;
}
int acl_rights_cmp(const struct acl_rights *r1, const struct acl_rights *r2)
{
int ret;
if (r1->global != r2->global) {
/* globals have higher priority than locals */
return r1->global ? 1 : -1;
}
ret = r1->id_type - r2->id_type;
if (ret != 0)
return ret;
return null_strcmp(r1->identifier, r2->identifier);
}
void acl_rights_sort(struct acl_object *aclobj)
{
struct acl_rights *rights;
unsigned int i, dest, count;
if (!array_is_created(&aclobj->rights))
return;
array_sort(&aclobj->rights, acl_rights_cmp);
/* merge identical identifiers */
rights = array_get_modifiable(&aclobj->rights, &count);
for (dest = 0, i = 1; i < count; i++) {
if (acl_rights_cmp(&rights[i], &rights[dest]) == 0) {
/* add i's rights to dest and delete i */
acl_right_names_merge(aclobj->rights_pool,
&rights[dest].rights,
rights[i].rights, FALSE);
acl_right_names_merge(aclobj->rights_pool,
&rights[dest].neg_rights,
rights[i].neg_rights, FALSE);
} else {
if (++dest != i)
rights[dest] = rights[i];
}
}
if (++dest != count)
array_delete(&aclobj->rights, dest, count - dest);
}
bool acl_rights_has_nonowner_lookup_changes(const struct acl_rights *rights)
{
const char *const *p;
if (rights->id_type == ACL_ID_OWNER) {
/* ignore owner rights */
return FALSE;
}
if (rights->rights == NULL)
return FALSE;
for (p = rights->rights; *p != NULL; p++) {
if (strcmp(*p, MAIL_ACL_LOOKUP) == 0)
return TRUE;
}
return FALSE;
}
int acl_identifier_parse(const char *line, struct acl_rights *rights)
{
if (strncmp(line, ACL_ID_NAME_USER_PREFIX,
strlen(ACL_ID_NAME_USER_PREFIX)) == 0) {
rights->id_type = ACL_ID_USER;
rights->identifier = line + 5;
} else if (strcmp(line, ACL_ID_NAME_OWNER) == 0) {
rights->id_type = ACL_ID_OWNER;
} else if (strncmp(line, ACL_ID_NAME_GROUP_PREFIX,
strlen(ACL_ID_NAME_GROUP_PREFIX)) == 0) {
rights->id_type = ACL_ID_GROUP;
rights->identifier = line + 6;
} else if (strncmp(line, ACL_ID_NAME_GROUP_OVERRIDE_PREFIX,
strlen(ACL_ID_NAME_GROUP_OVERRIDE_PREFIX)) == 0) {
rights->id_type = ACL_ID_GROUP_OVERRIDE;
rights->identifier = line + 15;
} else if (strcmp(line, ACL_ID_NAME_AUTHENTICATED) == 0) {
rights->id_type = ACL_ID_AUTHENTICATED;
} else if (strcmp(line, ACL_ID_NAME_ANYONE) == 0 ||
strcmp(line, "anonymous") == 0) {
rights->id_type = ACL_ID_ANYONE;
} else {
return -1;
}
return 0;
}
static const char *const *
acl_right_names_alloc(pool_t pool, ARRAY_TYPE(const_string) *rights_arr,
bool dup_strings)
{
const char **ret, *const *rights;
unsigned int i, dest, count;
/* sort the rights first so we can easily drop duplicates */
array_sort(rights_arr, i_strcmp_p);
/* @UNSAFE */
rights = array_get(rights_arr, &count);
ret = p_new(pool, const char *, count + 1);
if (count > 0) {
ret[0] = rights[0];
for (i = dest = 1; i < count; i++) {
if (strcmp(rights[i-1], rights[i]) != 0)
ret[dest++] = rights[i];
}
ret[dest] = NULL;
if (dup_strings) {
for (i = 0; i < dest; i++)
ret[i] = p_strdup(pool, ret[i]);
}
}
return ret;
}
const char *const *
acl_right_names_parse(pool_t pool, const char *acl, const char **error_r)
{
ARRAY_TYPE(const_string) rights;
const char *const *names;
unsigned int i;
/* parse IMAP ACL list */
while (*acl == ' ' || *acl == '\t')
acl++;
t_array_init(&rights, 64);
while (*acl != '\0' && *acl != ' ' && *acl != '\t' && *acl != ':') {
for (i = 0; acl_letter_map[i].letter != '\0'; i++) {
if (acl_letter_map[i].letter == *acl)
break;
}
if (acl_letter_map[i].letter == '\0') {
*error_r = t_strdup_printf("Unknown ACL '%c'", *acl);
return NULL;
}
array_append(&rights, &acl_letter_map[i].name, 1);
acl++;
}
while (*acl == ' ' || *acl == '\t') acl++;
if (*acl != '\0') {
/* parse our own extended ACLs */
if (*acl != ':') {
*error_r = "Missing ':' prefix in ACL extensions";
return NULL;
}
names = t_strsplit_spaces(acl + 1, ", \t");
for (; *names != NULL; names++) {
const char *name = p_strdup(pool, *names);
array_append(&rights, &name, 1);
}
}
return acl_right_names_alloc(pool, &rights, FALSE);
}
void acl_right_names_write(string_t *dest, const char *const *rights)
{
char c2[2];
unsigned int i, j, pos;
c2[1] = '\0';
pos = str_len(dest);
for (i = 0; rights[i] != NULL; i++) {
/* use letters if possible */
for (j = 0; acl_letter_map[j].name != NULL; j++) {
if (strcmp(rights[i], acl_letter_map[j].name) == 0) {
c2[0] = acl_letter_map[j].letter;
str_insert(dest, pos, c2);
pos++;
break;
}
}
if (acl_letter_map[j].name == NULL) {
/* fallback to full name */
str_append_c(dest, ' ');
str_append(dest, rights[i]);
}
}
if (pos + 1 < str_len(dest)) {
c2[0] = ':';
str_insert(dest, pos + 1, c2);
}
}
void acl_right_names_merge(pool_t pool, const char *const **destp,
const char *const *src, bool dup_strings)
{
const char *const *dest = *destp;
ARRAY_TYPE(const_string) rights;
unsigned int i;
t_array_init(&rights, 64);
if (dest != NULL) {
for (i = 0; dest[i] != NULL; i++)
array_append(&rights, &dest[i], 1);
}
if (src != NULL) {
for (i = 0; src[i] != NULL; i++)
array_append(&rights, &src[i], 1);
}
*destp = acl_right_names_alloc(pool, &rights, dup_strings);
}
bool acl_right_names_modify(pool_t pool,
const char *const **rightsp,
const char *const *modify_rights,
enum acl_modify_mode modify_mode)
{
const char *const *old_rights = *rightsp;
const char *const *new_rights = NULL;
const char *null = NULL;
ARRAY_TYPE(const_string) rights;
unsigned int i, j;
if (modify_rights == NULL && modify_mode != ACL_MODIFY_MODE_CLEAR) {
/* nothing to do here */
return FALSE;
}
switch (modify_mode) {
case ACL_MODIFY_MODE_REMOVE:
if (old_rights == NULL || *old_rights == NULL) {
/* nothing to do */
return FALSE;
}
t_array_init(&rights, 64);
for (i = 0; old_rights[i] != NULL; i++) {
for (j = 0; modify_rights[j] != NULL; j++) {
if (strcmp(old_rights[i], modify_rights[j]) == 0)
break;
}
if (modify_rights[j] == NULL)
array_append(&rights, &old_rights[i], 1);
}
new_rights = &null;
modify_rights = array_count(&rights) == 0 ? NULL :
array_idx(&rights, 0);
acl_right_names_merge(pool, &new_rights, modify_rights, TRUE);
break;
case ACL_MODIFY_MODE_ADD:
new_rights = old_rights;
acl_right_names_merge(pool, &new_rights, modify_rights, TRUE);
break;
case ACL_MODIFY_MODE_REPLACE:
new_rights = &null;
acl_right_names_merge(pool, &new_rights, modify_rights, TRUE);
break;
case ACL_MODIFY_MODE_CLEAR:
if (*rightsp == NULL) {
/* ACL didn't exist before either */
return FALSE;
}
*rightsp = NULL;
return TRUE;
}
i_assert(new_rights != NULL);
*rightsp = new_rights;
if (old_rights == NULL)
return new_rights[0] != NULL;
/* see if anything changed */
for (i = 0; old_rights[i] != NULL && new_rights[i] != NULL; i++) {
if (strcmp(old_rights[i], new_rights[i]) != 0)
return TRUE;
}
return old_rights[i] != NULL || new_rights[i] != NULL;
}
static void apply_owner_default_rights(struct acl_object *aclobj)
{
struct acl_rights_update ru;
const char *null = NULL;
memset(&ru, 0, sizeof(ru));
ru.modify_mode = ACL_MODIFY_MODE_REPLACE;
ru.neg_modify_mode = ACL_MODIFY_MODE_REPLACE;
ru.rights.id_type = ACL_ID_OWNER;
ru.rights.rights = aclobj->backend->default_rights;
ru.rights.neg_rights = &null;
acl_cache_update(aclobj->backend->cache, aclobj->name, &ru);
}
void acl_object_rebuild_cache(struct acl_object *aclobj)
{
struct acl_rights_update ru;
enum acl_modify_mode add_mode;
const struct acl_rights *rights, *prev_match = NULL;
unsigned int i, count;
bool first_global = TRUE;
acl_cache_flush(aclobj->backend->cache, aclobj->name);
if (!array_is_created(&aclobj->rights))
return;
/* Rights are sorted by their 1) locals first, globals next,
2) acl_id_type. We'll apply only the rights matching ourself.
Every time acl_id_type or local/global changes, the new ACLs will
replace all of the existing ACLs. Basically this means that if
user belongs to multiple matching groups or group-overrides, their
ACLs are merged. In all other situations the ACLs are replaced
(because there aren't duplicate rights entries and a user can't
match multiple usernames). */
memset(&ru, 0, sizeof(ru));
rights = array_get(&aclobj->rights, &count);
if (!acl_backend_user_is_owner(aclobj->backend))
i = 0;
else {
/* we're the owner. skip over all rights entries until we
reach ACL_ID_OWNER or higher, or alternatively when we
reach a global ACL (even ACL_ID_ANYONE overrides owner's
rights if it's global) */
for (i = 0; i < count; i++) {
if (rights[i].id_type >= ACL_ID_OWNER ||
rights[i].global)
break;
}
apply_owner_default_rights(aclobj);
/* now continue applying the rest of the rights,
if there are any */
}
for (; i < count; i++) {
if (!acl_backend_rights_match_me(aclobj->backend, &rights[i]))
continue;
if (prev_match == NULL ||
prev_match->id_type != rights[i].id_type ||
prev_match->global != rights[i].global) {
/* replace old ACLs */
add_mode = ACL_MODIFY_MODE_REPLACE;
} else {
/* merging to existing ACLs */
i_assert(rights[i].id_type == ACL_ID_GROUP ||
rights[i].id_type == ACL_ID_GROUP_OVERRIDE);
add_mode = ACL_MODIFY_MODE_ADD;
}
prev_match = &rights[i];
/* If [neg_]rights is NULL it needs to be ignored.
The easiest way to do that is to just mark it with
REMOVE mode */
ru.modify_mode = rights[i].rights == NULL ?
ACL_MODIFY_MODE_REMOVE : add_mode;
ru.neg_modify_mode = rights[i].neg_rights == NULL ?
ACL_MODIFY_MODE_REMOVE : add_mode;
ru.rights = rights[i];
if (rights[i].global && first_global) {
/* first global: reset negative ACLs so local ACLs
can't mess things up via them */
first_global = FALSE;
ru.neg_modify_mode = ACL_MODIFY_MODE_REPLACE;
}
acl_cache_update(aclobj->backend->cache, aclobj->name, &ru);
}
}
void acl_object_remove_all_access(struct acl_object *aclobj)
{
static const char *null = NULL;
struct acl_rights rights;
memset(&rights, 0, sizeof(rights));
rights.id_type = ACL_ID_ANYONE;
rights.rights = &null;
array_append(&aclobj->rights, &rights, 1);
rights.id_type = ACL_ID_OWNER;
rights.rights = &null;
array_append(&aclobj->rights, &rights, 1);
}
void acl_object_add_global_acls(struct acl_object *aclobj)
{
struct acl_backend *backend = aclobj->backend;
const char *vname, *error;
if (mailbox_list_is_valid_name(backend->list, aclobj->name, &error))
vname = mailbox_list_get_vname(backend->list, aclobj->name);
else
vname = "";
acl_global_file_get(backend->global_file, vname,
aclobj->rights_pool, &aclobj->rights);
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 3,912 |
{"url":"http:\/\/mathoverflow.net\/questions\/107891\/can-eigenvector-be-found-without-computing-the-eigenvalue","text":"# can eigenvector be found without computing the eigenvalue [closed]\n\nIs there any ways to compute the eigen vector without computing explicitly the associated eigenvalue?\n\nActually, I'd like to compute the largest eigenvalue of a positive matrix from its eigen vector, so I have to know its eigenvector first.\n\n-\n\n## closed as off topic by Fernando Muro, Suvrit, Lee Mosher, Andres Caicedo, Igor RivinSep 23 '12 at 18:47\n\nQuestions on MathOverflow are expected to relate to research level mathematics within the scope defined by the community. Consider editing the question or leaving comments for improvement if you believe the question can be reworded to fit within the scope. Read more about reopening questions here. If this question can be reworded to fit the rules in the help center, please edit the question.\n\n\u2013\u00a0 Goldstern Sep 23 '12 at 8:43\nObviously no. If you manage to compute an eigenvector $\\mathbf{v}$ of a matrix $A$ then to automatically get the associated eigenvalue $\\lambda$ by the formula $A\\mathbf{v}=\\lambda\\mathbf{v}$. \u2013\u00a0 Fernando Muro Sep 23 '12 at 9:53\n@Fernando: the comment is a bit tautological, since this is exactly what the OP is proposing to do. \u2013\u00a0 Igor Rivin Sep 23 '12 at 18:45\n@Igor: you're right, it seems I didn't pay attention to the question's second paragraph. Anyway, everything looks tautological in this post. \u2013\u00a0 Fernando Muro Sep 28 '12 at 8:22\n\nIf you pick a random vector $v$ and look at $v_n=A^n v\/\\| A^n v\\|,$ that will converge to the dominant eigenvector.\n\n-\nSmall correction: pick a positive vector. If you just pick at random you might get very unlucky and pick one that is orthogonal to the eigenvector. \u2013\u00a0 Felix Goldberg Sep 24 '12 at 10:20\nGood point (in the OP's stochastic context...) \u2013\u00a0 Igor Rivin Sep 25 '12 at 4:09\n\nWhat do you know about the matrix?\n\nIf we know that the rows all have the same sum (but not what that sum is) then we would essentially find it by multiplying by the corresponding eigenvector, $\\mathbb{j}$, the all $1$'s vector. This will be the largest eigenvalue provided that the entries are non-negative.\n\nOne way this could happen (but not the only one) is if the rows are identical or merely each is a permutation of the first.\n\nIf certain rows are equal then we know that $0$ is a eigenvalue although we never \"computed\" it. Then we do know an eigenvector.\n\nIn a circulant matrix we know all the eigenvectors (not just $\\mathbb{j}$) and we essentially use them to compute the corresponding eigenvalues.\n\n-\nAaron, Actually I'd like to compare the spectral radius of 2 matrices, say A1 and A2. The information about the matrices are: - They are doubly sub-stochastic matrix, with one row sum = a<1 ; their determinant are the same; their traces are the same. we know that they both have spectral radius between a and 1, but I'd like to compare them without computing them explicitly. E.g: A1 = [1\/4 1\/4 1\/4 0; 1\/4 3\/4 0 0; 1\/4 0 1\/2 1\/4; 0 0 1\/4 3\/4] A2 = [2\/4 1\/4 0 0; 1\/4 1\/4 1\/4 1\/4; 0 1\/4 3\/4 0; 0 1\/4 0 3\/4] We get, rho(A1) < rho(A2), but can we infer w\/o computing it? \u2013 hayu 0 secs ago \u2013\u00a0 hayu Sep 23 '12 at 13:57","date":"2015-07-04 10:38:02","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.9244633316993713, \"perplexity\": 524.7962717643828}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2015-27\/segments\/1435375096686.2\/warc\/CC-MAIN-20150627031816-00283-ip-10-179-60-89.ec2.internal.warc.gz\"}"} | null | null |
Money is the leading cause of stress among Americans. Sixty-four percent of us said money was a "somewhat" or "very significant" source of stress in a 2015 survey by the American Psychological Association. The top causes are paying for essentials, unexpected expenses and increasing debt.
Not long ago, I was among that 64 percent, but a money management system changed my life. Today, I want to show you how to minimize your stress by building a budget.
I love budgeting because it helps me get more of what I want. I'm sure a budget can do the same for you.
Level 1: Essentials: These are the basic necessities: rent/mortgage, utilities, groceries and gas for your car. If you are spending more than 25 to 35 percent of your take-home pay on your rent or mortgage, you are either overspending on your home or you have an income problem.Level 2: Obligations: These are expenses such as loans, credit card debt, student loans or other financial agreements that carry consequences if they go unpaid. Over time, eliminate as many of these obligations as possible so you have more money for Level 3 and 4 expenses.Level 3: Less-than-monthly expenses: When I was stressed out about money, I used to call these "unexpected" expenses. The truth was, they were expenses I knew were coming but hoped wouldn't. Many people fall into debt – and become stressed – by not accumulating money for these predictable but irregular expenses, such as car repairs, medical bills, home maintenance and holiday gifts. To prepare, realistically consider those future expenses and set aside money for them each month. For example, if you need to spend $400 on snow tires in November, that gives you eight months to save $50 each month. The more you can accumulate for less-than-monthly expenses, the less stress you'll have when they come due.Level 4: Nice-to-haves: Now, it's time to budget money for having fun, such as restaurants, entertainment and miscellaneous spending. Be sure to include spending money too, even if it's just a little bit. Having some money for which you don't have to be accountable is the lubrication that makes this system work.Once you've established your budget, look for places to cut back so you can tackle that stressful Level 2 debt. Start by paying off the smallest debt first and then move on to the next.
A systematic approach to managing your money can lift the weight of financial stress, help you relax and allow you to get more enjoyment out of life.
Durango resident and personal finance coach Matt Kelly owns Momentum: Personal Finance. Visit his website, www.personalfinancecoaching.com. | {
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{"url":"http:\/\/math.stackexchange.com\/questions\/546380\/evaluating-lim-n-rightarrow-inftyx-n1-x-n","text":"# Evaluating $\\lim_{n\\rightarrow\\infty}x_{n+1}-x_n$\n\nLet $f(x)$ be continuously differentiable on $[0,1]$ and\n\n$$x_n = f\\left(\\frac{1}{n}\\right)+f\\left(\\frac{2}{n}\\right)+f\\left(\\frac{3}{n}\\right)+\\ldots+f\\left(\\frac{n-1}{n}\\right)$$\n\nFind $$\\lim_{n\\rightarrow\\infty}\\left(x_{n+1}-x_n\\right)$$\n\nConfusion: I just found a subtle problem in my solution.By the definition of definite integration, like $\\lim_{n\\rightarrow\\infty}\\sum_{i=1}^{n}f(x_i^*)\\Delta x$, the sample point $x_i^*$ should be included in its $ith$ subinterval $[x_{i-1},x_i]$. In this case, I am not sure whether the subinterval $[\\frac{i-1}{n(n+1)},\\frac{i}{n(n+1)}]$ includes the sample point $i$ and $\\xi_i$ or not.I can not deduce it from the given condition,\n\n$$\\xi_i \\in [\\frac{i}{n+1},\\frac{i}{n}] \\implies \\frac{i-1}{n(n+1)}\\lt \\frac{i}{n+1}\\lt\\xi_i\\lt?\\lt\\frac{i}{n}$$\n\nIf those sample points is not always included in the corresponding subinterval, I might not apply the definite integration here. Hope somebody can take a look at this solution.\n\nUpdate: I rewrite part of my solution and fixed the problem I have before. Thanks very much for all the help!\n\n-\n\nBecause $f(x)$ is continuously differentiable on $[0,1]$, tben by the mean value theorem:\n\n$$f\\left(\\frac{i}{n+1}\\right)-f\\left(\\frac{i}{n}\\right) = f'(\\xi_i)(\\frac{i}{n+1} - \\frac{i}{n}) \\text{ where } \\xi_i \\in \\left[\\frac{i}{n+1},\\frac{i}{n}\\right] \\tag{1}$$\n\nThen, by the given formula of $x_n$, we have \\begin{align*} \\ x_{n+1} - x_n &= \\left[f\\left(\\frac{1}{n+1}\\right)-f\\left(\\frac{1}{n}\\right)\\right]+\\dots +\\left[f\\left(\\frac{n-1}{n+1}\\right)-f\\left(\\frac{n-1}{n}\\right)\\right]+\\left[f\\left(\\frac{n}{n+1}\\right)-f\\left(1\\right)\\right] + f\\left(1\\right) \\\\&=f(1) - \\sum_{i=1}^{n}i\\cdot f'(\\xi_i)\\left(\\frac{1}{n(n+1)}\\right) \\end{align*}\n\nHence, by the defintion of definite integration, we have:\n\\begin{align*} \\\\\\lim_{n\\rightarrow\\infty}\\left(x_{n+1}-x_n\\right) &= f(1) - \\lim_{n \\rightarrow \\infty}\\sum_{i=1}^{n}i\\cdot f'(\\xi_i)\\left(\\frac{1}{n(n+1)}\\right) \\\\&=f(1) - \\lim_{n\\rightarrow\\infty}\\sum_{i=1}^{n}\\frac{i}{n+1}\\cdot f'(\\xi_i)\\cdot\\frac{1}{n} \\\\&=f(1) - \\int_{0}^{1}xf'(x)dx \\space\\space\\space\\dots\\text{See Note} \\\\&=f(1) - \\left[xf(x)|_{0}^{1}-\\int_{0}^{1}f(x)dx\\right] \\\\ &=\\int_{0}^{1}f(x)dx \\end{align*}\n\nNote that:: \\begin{align*} \\\\ \\sum_{i=1}^{n}i\\cdot f'(\\xi_i)\\left(\\frac{1}{n(n+1)}\\right) = \\sum_{i=1}^{n}\\frac{i}{n+1}\\cdot f'(\\xi_i)\\cdot\\frac{1}{n} \\end{align*} by (1), it is obvious that: $$\\frac{i-1}{n} \\lt \\frac{i}{n+1} \\lt \\xi_i \\lt \\frac{i}{n}\\space\\space\\space\\text{(where } i \\le n) \\tag{2}$$ which means $\\xi_i, \\frac{i}{n+1} \\in \\left[\\frac{i-1}{n},\\frac{i}{n}\\right]$. Also by $(2)$, we know: $$\\sum_{i=1}^{n}\\frac{i}{n+1}\\cdot f'(\\xi_i)\\cdot\\frac{1}{n} \\le \\sum_{i=1}^{n}\\xi_i\\cdot f'(\\xi_i)\\cdot\\frac{1}{n}$$ Because $\\frac{i}{n+1}$ and $\\xi_i$ share the same subinterval, it implies that if we let $x_{i}^* = \\xi_i$(sample points) and $\\Delta x=\\frac{1}{n}$, then: $$\\lim_{n\\rightarrow\\infty}\\sum_{i=1}^{n}\\frac{i}{n+1}\\cdot f'(\\xi_i)\\cdot\\frac{1}{n}=\\lim_{n\\rightarrow\\infty}\\sum_{i=1}^{n}\\xi_i\\cdot f'(\\xi_i)\\cdot\\frac{1}{n}=\\int_{0}^{1}xf'(x)dx$$\n\nHence, it is ok to apply the definition of definite integration here.\n\n-\nQuite clear. UpVote $0$k. \u2013\u00a0Felix Marin Oct 31 '13 at 7:45\n@FelixMarin I just updated some confusion to my answer, hope you could check that :) \u2013\u00a0SundayCat Oct 31 '13 at 8:59\n$0$k. I'll check tomorrow because it's too late ( 4:30 in the morning ). Thanks. \u2013\u00a0Felix Marin Oct 31 '13 at 9:02\n@FelixMarin Thanks for the help, I rewrite my solution, making it more convincing. Maybe you could take a look at it. \u2013\u00a0SundayCat Nov 1 '13 at 0:41\nTO be fair, you should denote $\\xi_{i,n}$, since this element depends both of $i$ and $n$. \u2013\u00a0Pedro Tamaroff Nov 1 '13 at 1:16\n\nHeuristically, $x_n \\sim n \\int_0^1 f$, so $x_{n+1}-x_n \\sim \\int_0^1 f$ and thus $$\\lim_{n \\rightarrow \\infty} x_{n+1}-x_n= \\int_0^1 f$$\n\n-\nIt would be better if you explained $x_n\\sim\\int_{0}^1f$ in detail. That is almost everything in this problem, all the others are just obvious \u2013\u00a0SundayCat Oct 31 '13 at 3:28\n\nNote that: $$\\lim_{n\\rightarrow\\infty}x_n=\\lim_{n\\rightarrow\\infty}\\sum_{i=1}^{n-1}f \\left(\\frac in\\right)=\\lim_{n\\rightarrow\\infty}n\\cdot\\sum_{i=1}^{n-1}f\\left(\\frac in\\right)\\frac1n=\\lim_{n\\rightarrow\\infty}n\\cdot\\sum_{i=1}^{n-1}f\\left(x_i\\right)\\Delta x$$ where $x_i=\\frac in\\in\\left(0,1\\right)$ and $\\Delta x=\\frac1n$.\n\n-\nBut how do you deal with $n$ here? \u2013\u00a0SundayCat Oct 31 '13 at 19:01\nI just show $x_n$ part. Adding $x_{n+1}$ part, $n$ would be canceled. \u2013\u00a0Shuchang Nov 1 '13 at 0:39\nbut you can not transform the limit to integration here because you can not get rid of $n$. \u2013\u00a0SundayCat Nov 1 '13 at 0:48\n@sundaycat $\\lim_{n\\rightarrow\\infty}(x_{n+1}-x_n)=\\lim_{n\\rightarrow\\infty}(n+1-n)\\sum_{i=\u200c\u200b1}^{n-1}f(x_i)\\Delta x$. Actually $n$ doesn't appear. \u2013\u00a0Shuchang Nov 1 '13 at 1:14\nIt seems like a nice way to solve this problem. It would be perfect if you could write down the whole solution here.:) \u2013\u00a0SundayCat Nov 1 '13 at 3:34","date":"2016-07-02 07:42:04","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\": 0.9965552687644958, \"perplexity\": 864.6700222941577}, \"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-2016-26\/segments\/1466783408840.13\/warc\/CC-MAIN-20160624155008-00048-ip-10-164-35-72.ec2.internal.warc.gz\"}"} | null | null |
{"url":"https:\/\/www.sarthaks.com\/203354\/write-the-sum-of-the-order-and-degree-of-the-differential-equation-d-2y-dx-2-2-dy-dx-3-x-4-0","text":"# Write the sum of the order and degree of the differential equation (d^2y\/dx^2 )^2 + (dy\/dx)^3 + x^4 = 0\n\n54.7k views\n\nWrite the sum of the order and degree of the differential equation (d2y\/dx2\u00a0)2 + (dy\/dx)3 + x4 = 0\n\n+1 vote\nby (30.5k points)\nselected\n\nGiven differential equation is\u00a0(d2y\/dx2\u00a0)2\u00a0+ (dy\/dx)+ x4\u00a0= 0\n\nHere, we see that the highest order derivative is\u00a0d2y\/dx2\u00a0, whose degree is 2.\n\nHere,order = 2 and degree = 2\n\nSum of the order and degree =\u00a02 + 2 = 4","date":"2022-09-28 06:30:02","metadata":"{\"extraction_info\": {\"found_math\": false, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.8474862575531006, \"perplexity\": 863.4323467040202}, \"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-2022-40\/segments\/1664030335124.77\/warc\/CC-MAIN-20220928051515-20220928081515-00392.warc.gz\"}"} | null | null |
\section{Introduction}\label{sec:I}
We have recently shown in Ref.~\cite{GravityLandau} that gravity has a nontrivial effect on the quantum Landau energy levels (see, e.g., Ref.~\cite{LandauLifshitz}) of a particle moving around a spherical static mass and surrounded by a constant and uniform magnetic field. We found that the degeneracy of the Landau levels is removed by splitting the energy of each of the Landau orbitals. We have also pointed out that the gravitational splitting of the levels could be used to test departures from the inverse square-law of gravity using quantum particles.
It is worth recalling, and emphasizing, here that the investigation done in Ref.~\cite{GravityLandau} belongs actually to two different classes of research involving gravity and quantum particles. The first class of investigations aims at bringing into light the gravitational effects on a quantum particle. Noticeable among such investigations are those studying the behavior of cold neutrons inside a gravitational field \cite{UCNinEarth1,UCNinEarth2,UCNinEarth3,BookNeutronsInterfro,Kulin,Abele,Landry1,Landry2}. The central idea behind such an approach is to substitute the gravitational potential for the usual electric potentials frequently used in non-relativistic quantum mechanics of point particles. Just like with the familiar electric potentials, the gravitational one is indeed expected to induce a quantization of the energy of the particle immersed inside it, or even lead to interference patterns as the particle moves inside such a potential. From this point of view, the work we presented in Ref.~\cite{GravityLandau} indeed subscribes to this class of research by showing that the effect of the additional potential due to gravity modifies the familiar Landau energy levels of a charged particle inside a magnetic field.
Besides showing how gravity disturbs the Landau levels of a charged particle, however, the work in Ref.~\cite{GravityLandau} allowed to explicitly expose the effect a departure from the inverse square-law for gravity would have on a quantum particle. As such, our previous work can also be assigned to the second class of investigations the aim of which is to {\it test} the gravitational field itself in the same spirit as the works such as the one in Ref.~\cite{Biedermann}. Investigating the effect of gravity on Landau levels becomes thus a fundamental approach in the sense that it consists in using the {\it quantum} theory to probe the {\it classical} gravitational field.
In the present paper, our goal is to explore even more this second category of investigations while still providing evidence for the effect of more complicated spacetimes on a quantum particle. More specifically, we are going to study the fate of Landau's energy levels of a charged particle inside a uniform and constant magnetic field, first (i) when the particle is moving within the Levi-Civita spacetime then (ii) when the particle is moving within the Kerr spacetime. The goal of the first investigation is, above all, to contribute to the existing efforts in the literature towards devising tests for that elusive and much debated spacetime of general relativity. In fact, although the so-called Levi-Civita spacetime was discovered exactly now a century ago \cite{Weyl,Levi-CivitaPaper}, such a metric still holds many mysteries and is less often used in the literature compared to the more famous ones of the Schwarzschild and Kerr spacetimes.
The Kerr metric, in contrast, is indeed very well known, for it is used mainly to describe the spacetime of a rotating black hole. As such, our present investigation based on the Kerr metric does actually more than just test a special solution to the gravitational equations. It provides a novel way --- {\it quantum mechanical} in nature --- for testing the frame-dragging effect of general relativity which, hitherto, has only been tested through the famous Lense-Thirring effect (see e.g., Ref.~\cite{Iorio} and the references therein). The Lense-Thirring effect consists of the precession of a gyroscope, or any spinning body, in the vicinity of a rotating mass, like the Earth. In this paper, we show how frame-dragging creates a specific signature on the splitting of the quantum Landau levels. For approaches relying instead on the effect of rotating frames on the quantum spin of particles and on their internal clocks, as well as on quantum interferences, see Refs.~\cite{Dowker,Mashhoon1,Lammerzahl,Herrera,Tartaglia,Mashhoon2,Ruggiero,Okawara1,Okawara2,Okawara3,Interference}.
The remaining sections of this paper are organized as follows. In Section~\ref{sec:II}, we build the curved-spacetime Klein-Gordon equation for a charged particle minimally coupled to the electromagnetic field and moving in the full Levi-Civita spacetime created by an infinitely long massive cylinder. We then solve the equation in the Newtonian approximation, with the goal of making contact with laboratory experimental tests, and then we evaluate the splitting of the Landau levels. In Section~\ref{sec:III}, we repeat the same analysis as the one done for the Levi-Civita spacetime in Section~\ref{sec:II}, but using the Kerr metric instead. We compare the Landau levels splitting caused by the latter spacetime to the one obtained in Ref.~\cite{GravityLandau} within the static spherical mass. The frame-dragging effect reveals itself clearly. We conclude this paper with a brief discussion and conclusion section. A short appendix is included in which many of the complicated integrals needed in the text are gathered for reference.
\section{A particle inside a magnetic field in the Levi-Civita spacetime}\label{sec:II}
In both this section and the next, we are going to use a charged spinless particle of mass $m$ as our test particle. For practical purposes, we are going to assume the particle has the elementary charge $e$. This is motivated by the possibility of experimentally implementing the setup by using heavy ions the spin of which can be neglected. Thus, the Klein-Gordon equation for a scalar field in curved spacetime, $\left(\Box+m^2c^2\right)\varphi=0$, will be amply sufficient for our present purposes.
For a minimally coupled particle to the electromagnetic field $A_\mu$ and to the metric $g_{\mu\nu}$ of the spacetime, the Klein-Gordon equation reads (see, e.g., Ref.~\cite{Kiefer}),
\begin{equation}\label{KG}
\left[\frac{1}{\sqrt{-g}}D_\mu\left(\sqrt{-g}g^{\mu\nu}D_\nu\right)+m^2c^2\right]\varphi(x)=0,
\end{equation}
where, the minimal-coupling prescription, $D_\mu=-i\hbar\partial_\mu-eA_\mu$, is assumed. Next, the Levi-Civita metric around an infinitely long cylinder of mass $M$ can be written in the cylindrical coordinates $(t,\rho,\phi,z)$ as follows \cite{Fulling}:
\begin{equation}
\label{LCMetric}
{\rm d}s^2=-c^2\left(\frac{\rho}{\rho_*}\right)^{-2a}{\rm d}t^2+\left(\frac{\rho}{\rho_*}\right)^{-2(a+b)}{\rm d}\rho^2+K^2\rho^2{\rm d}\phi^2+\left(\frac{\rho}{\rho_*}\right)^{-2b}{\rm d}z^2.
\end{equation}
Here, the constants $a,b$ and $K$ are all arbitrary --- with a constraint between $a$ and $b$ --- while $c$ is the speed of light. Note that the form (\ref{LCMetric}) of the Levi-Civita metric we use here is not the one that one often encounters in the literature \cite{Marder,Bonnor,Herrera2,Santos} (see also, the very nice recent review \cite{Bronnikov}.)
In fact, the first difference is that we have introduced here the fixed constant radius $\rho_*$ to allow us to keep inside the metric the radius $\rho$, describing the position of the particle away from the center of the cylinder, with the dimensions of a length. In addition, in view of the approximations we are going to perform in order to be able to solve our differential equations, having a dimensionless ratio is very well suited for expanding the metric in powers of such a ratio as well as for keeping the argument of the logarithms appearing there dimensionless.
Furthermore, as is well-known in the case of logarithmic potentials, in particular the one created by an infinitely long cylinder (see, e.g., Ref.~\cite{BookPotentialCylinder}), one does not have a vanishing potential anywhere. As a consequence, the reference point for potentials cannot be taken to be at infinity anymore (as is the case with a spherical mass). To remedy such an issue, one introduces a fixed radial distance from the center of the source and takes such a point to be the reference for measuring potentials. As we shall see shortly, our fixed radius $\rho_*$ allows us specifically to take it to be the reference point of zero gravitational potential.
The second difference with respect to the usual forms of the metric given in the literature is the presence of the multiplicative constant $K$ in the angular component of the metric. The role of this constant is actually just to avoid rescaling the azimuth angle $\phi$, keeping it instead within the familiar range $[0,2\pi[$ \cite{Fulling}. As such, any possible excess or deficit angle, which would give rise to a conical singularity whenever the spacetime around the cylinder is continued all the way to the center of the latter (the latter becoming then a string), is encoded in the constant $K$. If $K<1$, one has a deficit angle (a wedge is removed from spacetime), whereas for $K>1$ one has an excess of spacetime (a wedge is added)\footnote{The role of this parameter in the Levi-Civita metric in encoding the global topology of the spacetime was first pointed out in Ref.~\cite{Bonnor}.}. As we shall see below, our results make even more transparent the effect of this deficit/excess angle on the energy-spectrum of the particle when we display explicitly the constant $K$ in the metric instead of absorbing it by redefining the angle $\phi$.
The third difference in our choice (\ref{LCMetric}) for the form of the Levi-Civita metric, is the fact that our $z$- and $\rho$-component of the metric acquire different coefficients. This specific choice is merely made here for the sake of simplicity. In fact, had we chosen to use instead the more familiar form of the metric, in which both coordinates acquire the same metric component \cite{Santos,Bronnikov}, the angular component of the metric would also acquire\footnote{See Ref.~\cite{Fulling} for the various coordinate re-definitions that allow one to switch from one form of the metric to the other.} a power function of $\rho$ instead of having the above familiar factor $\rho^2$. This would indeed only render our equations and analysis uselessly complicated.
Let us now focus on the meaning of the remaining two parameters $a$ and $b$ of the metric. First, as mentioned above, the two parameters are not completely arbitrary as they obey a specific constraint in the form of an algebraic relation between them. Such a relation reads, $ab+a+b=0$ \cite{Fulling}. This implies that the total number of independent parameters in the Levi-Civita spacetime is actually just two. The meaning of one of these two parameters, say, $a$, can now be found as follows. For very small $a$, we can expand the $00$-component of the metric to the first order as, $(\rho/\rho_*)^{-2a}\approx 1-2a\ln(\rho/\rho_*)$. Comparing this with the weak-field approximation of general relativity, $g_{00}\approx-1+2U$, reveals what potential $U$ in the post-Newtonian approximation one has; it reads $U=a\ln(\rho/\rho_*)$. This, when compared, in turn, with the well-known Newtonian potential around an infinitely long cylinder \cite{BookPotentialCylinder}, implies that $a$ should be identified with the product $G\lambda/c^2$, where $\lambda$ is the linear mass density of the cylinder. For a finite-radius cylinder, we should then identify $a$ with $\pi G\gamma\rho_0^2/c^2$ when the radius of the cylinder is $\rho_0$ and its volume mass density is $\gamma$. It should be kept in mind, though, that for the infinitely-long cylinder approximation to be accurate in the case of a finite cylinder, the particle should be kept very close to the lateral surface of the long cylinder.
It is worth noting here, however, that, as alluded to in the Introduction, the Levi-Civita metric is not free from ambiguities when it comes to its full interpretation. In fact, it was shown in Refs.~\cite{Herrera2,Santos} that only for the range $0<a<1/4$ of the parameter $a$ does one extract a physically sensible spacetime around the cylinder, for only then do circular time-like geodesics exist. For $a=1/4$ or $a=1$, the circular geodesics are null, whereas for $1/4<a<1$ those geodesics are spacelike. The circular geodesics become timelike only for $0<a<1/4$ or $a>1$. Fortunately, since we are interested here in the case $a\ll1$, such issues do not need to worry us.
Nevertheless, these serious obstacles in the interpretation of the Levi-Civita metric make actually the investigation of the effects of the metric on quantum particles, not only a way for testing cylindrical gravitational fields, but constitutes thus an additional input towards understanding the metric itself.
Let us now substitute the metric (\ref{LCMetric}) into the Klein-Gordon equation (\ref{KG}). As for the vector potential $A_\mu$, we are going to use the usual symmetric gauge adapted to the cylindrical coordinates $(t,\rho,\phi,z)$ in which the only non-vanishing component reads\footnote{We display here the covariant form of the potential vector, as the tetrad form we used in the first version of this manuscript leads to much confusion. In fact, while the usual tetrad form of the vector potential $A_{\hat{\phi}}=\frac{1}{2}B\rho$ has the advantage of displaying the right dimensions for a potential vector, it requires one to be careful when substituting it inside Eq.~(\ref{KG}). By taking such care, the result one finds is, of course, the same with both expressions.}, $A_\phi=\frac{1}{2}KB\rho^2$. Note that with the presence of the magnetic field $\bf B$, one might expect a spacetime metric that is not the one in Eq.~(\ref{LCMetric}), but a metric that would be a solution to the Einstein-Maxwell equations with a massive infinitely long cylinder. However, as explained in detail in Ref.~\cite{GravityLandau}, we assume the magnetic field to be as weak as to allow us to neglect its {\it geometric} effect on the spacetime and, hence, neglect its effect on the particle via geometry. In fact, the correction that arises from taking into account the effect of the magnetic field on geometry is of the order of $G\epsilon_0c^{-2}B^2\rho^2$ and becomes significant only for magnetic fields of the order of $10^{19}{\rm G}$ \cite{DiracInMelvin}. We therefore focus in this paper only on the effect of the magnetic field on the particle due the usual Lorentz force.
Now, because of the time-independence of both the metric and the magnetic field, and thanks to the symmetry of the planar motion of the particle around the $z$-axis, we expect the wavefunction for the particle of energy $E$ to be of the form, $\varphi(t,\rho,\phi,z)=e^{-i\frac{Et}{\hbar}}e^{i\ell\phi}R(\rho)$, with $\ell$ a non-negative integer. For simplicity, we assume that the particle has no momentum along the $z$-direction and that it is moving counterclockwise around the cylinder. Therefore, the Klein-Gordon equation (\ref{KG}) in the curved spacetime (\ref{LCMetric}) takes the form,
\begin{equation}\label{KGinLC1}
\Bigg[\frac{E^2}{\hbar^2c^2}\left(\frac{\rho}{\rho_*}\right)^{2a}-\frac{m^2c^2}{\hbar^2}+\left(\frac{\rho}{\rho_*}\right)^{2a+2b-1}\partial_\rho\left(\frac{\rho}{\rho_*}\,\partial_\rho\right)-\frac{\ell^2}{K^2\rho^2}+\frac{eB\ell}{K\hbar}-\frac{e^2B^2\rho^2}{4\hbar^2}\Bigg]R(\rho)=0.
\end{equation}
In the case of small parameters, $a,b\ll1$, the powers of the ratio $(\rho/\rho_*)$ can be expanded and the above equation then reads, at the first-order approximation in $a$ and $b$, as follows:
\begin{multline}\label{KGinLC2}
\frac{{\rm d}^2R}{{\rm d}\rho^2}+\frac{{\rm d}R}{\rho{\rm d}\rho}+\Bigg[\left(\frac{E^2}{\hbar^2c^2}-\frac{m^2c^2}{\hbar^2}-\frac{\ell^2}{K^2\rho^2}+\frac{eB\ell}{K\hbar}-\frac{e^2B^2\rho^2}{4\hbar^2}\right)\\
+2\left(\frac{m^2c^2}{\hbar^2}+\frac{\ell^2}{K^2\rho^2}-\frac{eB\ell}{K\hbar}+\frac{e^2B^2\rho^2}{4\hbar^2}\right)(a+b)\ln\left(\frac{\rho}{\rho_*}\right)-\frac{2bE^2}{\hbar^2c^2}\ln\left(\frac{\rho}{\rho_*}\right)\Bigg]\,R(\rho)=0.
\end{multline}
Next, performing the change of variable $R(\rho)=\psi(\rho)/\sqrt{\rho}$, and then decomposing the total energy of the test particle as, $E=\mathcal{E}+mc^2$, and using the non-relativistic approximation $E^2\approx2mc^2\mathcal{E}+m^2c^4$, the above equation, in turn, simplifies to,
\begin{equation}\label{KGinLC3}
-\frac{\hbar^2}{2m}\psi''+\Bigg[\frac{e^2B^2\rho^2}{8m}+\frac{\hbar^2}{2m\rho^2}\left(\frac{\ell^2}{K^2}-\frac{1}{4}\right)-\frac{\hbar eB\ell}{2mK}-amc^2\ln\left(\frac{\rho}{\rho_*}\right)\Bigg]\psi=\mathcal{E}\psi.
\end{equation}
We have denoted by a prime a derivative of $\psi(\rho)$ with respect to the radial variable $\rho$. In addition, we have kept here only the leading term $mc^2$ from the second and third lines of Eq.~(\ref{KGinLC2}). Again, this approximation is amply sufficient for our purposes here, for we have indeed $\hbar eB/m\ll mc^2$ for the orders of magnitude of the magnetic fields used in the laboratory. This Schr\"odinger equation will give us the full quantized energy spectrum of the particle.
Now, we argued at length in Ref.~\cite{GravityLandau} (see also the references therein) that there are essentially two practical working methods for extracting the quantized energy levels from such a Schr\"odinger equation containing extra non-trivial central potentials. Our non-trivial extra term here is the logarithmic term inside the square brackets of Eq.~(\ref{KGinLC3}). The first approach relies on the time-independent perturbation theory. The second approach consists simply in approximating the effective potential, contained inside the square brackets, by that of a simple harmonic oscillator. When using the latter approach, one would directly read off the energy levels as given by the familiar energy spectrum of a simple harmonic oscillator. We are going to apply in the rest of this section both methods, starting with the one relying on the time-independent perturbation theory. A short note will be given at the very end of this section about two other less reliable and less practical methods for extracting the quantized levels.
\subsection{Using Perturbation theory}
In Ref.~\cite{GravityLandau}, we have already found the solutions to Eq.~(\ref{KGinLC3}) without the very last term inside the square brackets. Those solutions constitute the unperturbed eigenvalues of the Landau Hamiltonian. Note, however, that now we have the extra parameter $K$ that enters even in the unperturbed equation. Nevertheless, the solutions with such an extra parameter can easily be adapted from the results of Ref.~\cite{GravityLandau}. Indeed, this can be accomplished simply by introducing the reduced orbital quantum number, $\bar{\ell}=\ell/K$. For simplicity, however, and without any loss of generality, we are going to set hereafter $K=1$. The effect of the parameter $K$, when the latter is different from unity, can then be inferred from the results with $K=1$ just by replacing $\ell$ by $\bar\ell$.
The unperturbed eigenfunctions $\psi^{(0)}_{n\ell}(\rho)$ corresponding to Eq.~(\ref{KGinLC3}) without the last term inside the square brackets are then \cite{GravityLandau},
\begin{equation}\label{UnperturbedWF}
\psi^{(0)}_{n\ell}(\rho)=A_{n\ell}\;\rho^{\ell+\frac{1}{2}}e^{-\frac{\beta}{4}\rho^2}\,_1F_1\left(-n;\ell+1;\frac{\beta}{2}\rho^2\right).
\end{equation}
The special functions $\,_1F_1(a,b,z)$ are called Kummer's, or confluent hypergeoemtric, functions \cite{BookKummer}. As usual, $n$ is here a non-negative integer. The normalization constants $A_{n\ell}$ would, in principle, be determined by imposing as usual the completeness condition on the eigenfunctions, $\int_0^\infty \psi_{n\ell}^{(0)*}(\rho)\psi_{m\ell}^{(0)}(\rho) \,{\rm d}\rho=\delta_{nm}.$ However, in contrast to what is assumed in the case of cosmic strings, a cylinder of mass $M$ has a finite nonzero radius $\rho_0$. As a consequence, the test particle's position is necessarily limited to the interval of radii $\rho\in[\rho_0,\infty)$. In addition, our gravitational field is valid only for $\rho>\rho_0$, {\it i.e.}, outside the cylinder.
Because of this particular configuration, we should distinguish two different regions when solving the Schr\"odinger equation. The region outside the cylinder, for which $\rho>\rho_0$, and the region inside the cylinder, for which $\rho<\rho_0$. We shall assume, however, that the cylinder is completely reflective to the test particle. In other words, we take the particle's wavefunction to vanish inside the cylinder, meaning that the particle has zero probability of penetrating inside the latter. In fact, with this assumption we are simply dealing with a semi-infinite potential well, for then our system just consists effectively of a test particle moving around an infinitely long cylinder, inside of which the potential is infinite and outside of which the potential is gravitational and is given by Eq.~(\ref{KGinLC3}). The wavefunction outside the cylinder having the expression (\ref{UnperturbedWF}), all we need to further impose on the latter is its continuity across the surface $\rho=\rho_0$. This condition translates then into the requirement, $\psi^{(0)}_{n\ell}(\rho_0)=0$. Based on expression (\ref{UnperturbedWF}), this requirement is equivalent to the following identity to be imposed on the confluent hypergeometric function:
\begin{equation}\label{FContinuity}
\,_1F_1\left(-n;\ell+1;\frac{\beta}{2}\rho_0^2\right)=0.
\end{equation}
This condition already arose for the case of a spherical mass examined in Ref.~\cite{GravityLandau}. Its physical interpretation is therefore similar to the one proposed in that reference. Indeed, this condition is due to the geometry of the system itself. The condition (\ref{FContinuity}) involves the two unknown integers $n$ and $\ell$ and, hence, implies that the latter are related to the parameter $\beta$, {\it i.e.}, the magnetic field, and to the radius $\rho_0$ of the cylinder. In the absence of the cylinder, all possible Landau levels $n$ and all possible orbital numbers $\ell$ would be accessible to the particle without any restriction. The presence of the cylinder at the center of motion disturbs the motion of the test particle by creating the forbidden region $0\le\rho\leq\rho_0$, implying that, depending on the value of the product $\tfrac{1}{2}\beta\rho_0^2$, a specific correlation emerges between the values of $n$ and $\ell$. This means that only specific combinations of the magnetic field and the radius of the cylinder, sitting at the center of motion of a test particle, would give rise to the quantum numbers $n$ and $\ell$ that the particle could take while moving around the cylinder and avoiding the interior of the latter.
In the case of a string-like mass distribution, {\it i.e.}, for $\rho_0=0$, the requirement $\psi^{(0)}_{n\ell}(\rho_0)=0$ is, of course, automatically satisfied. In that case, the condition (\ref{FContinuity}) does not need to be imposed anymore and, hence, no correlation between the quantum numbers $n$ and $\ell$ and the parameter $\tfrac{1}{2}\beta\rho_0^2$ is implied either. Since we are interested here only in the fate of the Landau energy levels inside the gravitational field, we are going to ignore in the remainder of this paper such a restriction and assume that a specific combination of the radius of the cylinder and of the magnetic field, guaranteeing the appearance of Landau quantum levels and orbitals for the particle, has already been set up.
Because of this forbidden region to the test particle, the normalization condition that we should imposed here is then $\int_{\rho_0}^\infty \psi_{n\ell}^{(0)*}(\rho)\psi_{n\ell}^{(0)}(\rho) \,{\rm d}\rho=1.$
In Ref.~\cite{GravityLandau}, the normalization constants of the wavefunctions $\psi_{n\ell}^{(0)}(\rho)$ implied by such a condition were explicitly found to be $A_{n\ell}=\mathcal{M}_{n\ell}^{-1/2}$. The quantities $\mathcal{M}_{n\ell}$ are infinite series obtained by setting $n=m$ in the infinite series $\mathcal{M}_{mn\ell}$, given explicitly for reference in Eq.~(\ref{AppendixIntegralM}) of appendix \ref{A}.
In addition, the energy eigenvalues corresponding to the unperturbed wavefunctions (\ref{UnperturbedWF}) are given by \cite{GravityLandau},
\begin{equation}\label{Landau}
\mathcal{E}_n^{(0)}=\frac{\hbar eB}{m}\left(n+\frac{1}{2}\right).
\end{equation}
These are the familiar Landau quantized energy levels. The high degeneracy of the levels shows up in the freedom the particle has with the orbital quantum number $\ell$ for each quantum number $n$. Note that, had we kept the parameter $K$ of the Levi-Civita metric (\ref{LCMetric}), these energy levels would not have been modified as the only difference would be the substitution $\ell\rightarrow\ell/K$.
The perturbed Landau energy levels due to the cylindrical gravitational field are now easy to compute at the first order using the time-independent perturbation theory. Although the Landau energy levels are infinitely degenerate, the fact that the gravitational interaction potential $V(\rho)=-amc^2\ln(\rho/\rho_*)$ around the cylinder is rotational symmetric means that the gravitational perturbation does not couple between two different Landau orbitals of quantum numbers $\ell$ and $\ell'$. This implies, as was the case with a spherical mass \cite{GravityLandau}, that the perturbation matrix elements $\braket{n,\ell|V(\rho)|n,\ell'}$ are diagonal. Consequently, the degenerate time-independent perturbation theory yields the first-order correction, $\mathcal{E}_{n\ell}=\mathcal{E}_{n}^{(0)}+\braket{n,\ell|V(\rho)|n,\ell}$, where the term $\mathcal{E}_n^{(0)}$ represents the unperturbed $n^{\rm th}$ Landau level (\ref{Landau}). We have thus the following more explicit first-order correction to the energy of the $n^{\rm th}$ Landau level in the quantum orbital $\ell$:
\begin{align}\label{LCellCorrection}
\mathcal{E}_{n\ell}&=\mathcal{E}_{n}^{(0)}-amc^2\int_{\rho_0}^{\infty} \psi^{(0)*}_{n\ell}(\rho)\psi^{(0)}_{n\ell}(\rho)\ln\left(\frac{\rho}{\rho_*}\right)\,{\rm d}\rho.
\end{align}
In order to evaluate the improper integral in this equation, we have to substitute expression (\ref{UnperturbedWF}) for the unperturbed wavefunctions $\psi_{n\ell}^{(0)}(\rho)$ and replace the normalization constants $A_{n\ell}$ by their expressions $\mathcal{M}_{n\ell}^{-1/2}$ as given by Eq.~(\ref{AppendixIntegralM}). Afterwards, by using the result (\ref{AppendixIntegralLDef}) of appendix \ref{A}, we find,
\begin{equation}\label{ExplicitellCorrection}
\mathcal{E}_{n\ell}=\mathcal{E}_{n}^{(0)}-amc^2\bar{\mathcal{L}}_{n\ell}\bar{\mathcal{M}}_{n\ell}^{-1}.
\end{equation}
As was done in Ref.~\cite{GravityLandau}, we have denoted here by $\bar{\mathcal{L}}_{n\ell}$ and $\bar{\mathcal{M}}_{n\ell}$ the reduced forms of the series (\ref{AppendixIntegralM}) and (\ref{AppendixIntegralL}), obtained by suppressing the constant factor $(2/\beta)^{\ell+1}$ common to both series, and by setting $n=m$ in both. This result shows how the degenerate Landau levels split at the first-order in $amc^2$ due to gravity. Although not explicitly displayed, the dependence of this splitting on the magnetic field $B$ is still present inside the individual series (\ref{AppendixIntegralM}) and (\ref{AppendixIntegralL})
For the sake of concreteness, let us evaluate the explicit correction to the first Landau level by setting $n=1$ in Eq.~(\ref{ExplicitellCorrection}). First, it is obvious from the defining integrals (\ref{AppendixIntegralMDef}) and (\ref{AppendixIntegralLDef}) of $\mathcal{M}_{n\ell}$ and $\mathcal{L}_{n\ell}$, respectively, that for small values of $\ell$, the gravitational correction to the first Landau level is of the order $-amc^2\ln(\rho_0/\rho_*)$. For larger values of $\ell$, however, one cannot easily get a simple physical picture of the effect of the gravitational field on the first Landau level based on the full expression (\ref{M1ell}) of $\mathcal{M}_{1\ell}$ and the full expression (\ref{L1ell}) from which $\mathcal{L}_{1\ell}$ can be found. For this reason, we are going instead to give here an estimate of the perturbation correction for the large-$\ell$ orbitals. In fact, in this case the expressions simplify greatly by using the asymptotic results (\ref{M1ellInfinite}) and (\ref{L1ellInfinite}) for $\mathcal{M}_{1\ell}$ and $\mathcal{L}_{1\ell}$, respectively. We find,
\begin{equation}\label{1ellCorrection}
\mathcal{E}_{1,\ell\gg1}=\mathcal{E}_{1}^{(0)}-amc^2\mathcal{L}_{1,\ell\gg1}\mathcal{M}_{1,\ell\gg1}^{-1}\approx\frac{3\hbar eB}{2m}+\frac{amc^2}{2}\ln\left(\frac{eB\rho_*^2}{2\hbar\ell}\right).
\end{equation}
This result shows that, just like what happens in the case of a spherical static mass \cite{GravityLandau}, the splitting brought to the Landau levels by the gravitational field of the cylinder has, in fact, a simple form for large orbitals $\ell$. This splitting is independent of the radius of the cylinder $\rho_0$ and depends instead on the fixed radius $\rho_*$ we took as a reference for the gravitational potential. In contrast to the case of the spherical mass \cite{GravityLandau}, however, the splitting depends here logarithmically on the magnetic field. For the case of $K\neq1$, the Landau term remains unaffected but the correction term does get affected as the denominator inside the logarithm acquires the multiplicative factor $K^{-1}$. On the other hand, as is the case with the spherical mass, from the general formula (\ref{ExplicitellCorrection}) we see that for large $n$, the first-order correction does not get suppressed.
It is worth noting here also that, like with the case of the spherical mass \cite{GravityLandau}, in the absence of the magnetic field, {\it i.e.} when setting $B=0$ in Eq.~(\ref{ExplicitellCorrection}), the first-order perturbation vanishes together with the zeroth-order levels $\mathcal{E}_{n}^{(0)}$, for both series $\mathcal{M}_{mn\ell}$ and $\mathcal{P}_{mn\ell}$ do not exist in this case as the integrals that gave rise to them vanish for $\beta=0$. A proper treatment of the motion of the particle around the cylinder without the magnetic field consists in solving the Schr\"odinger equation with only the logarithmic potential as the unique potential (see, e.g., Ref.~\cite{LogarithmicPotential}).
At the second order, the corrections to the energy levels would be even more complicated than what was found for the spherical mass case in Ref.~\cite{GravityLandau}. In fact, the correction $\mathcal{E}_{n\ell}^{(2)}=\sum\limits_{k\neq n}|\braket{k,\ell|V|n,\ell}|^2/(\mathcal{E}_{k}^{(0)}-\mathcal{E}_{n}^{(0)})$, which is quadratic in the product $am$, would involve, besides terms logarithmic in the magnetic field, a ratio with the magnetic field in the denominator as well. Suffice it then to note here that, like in the spherical mass case \cite{GravityLandau}, the second-order correction to the energy levels of the particle is quadratic in $G\lambda$, where $\lambda$ is the linear mass density of the long cylinder. Furthermore, because of the presence of the magnetic field in the denominator in such a correction, the latter is not valid anymore without the magnetic field, {\it i.e.}, when $B=0$. In this case, one should instead solve Eq.~(\ref{KGinLC3}) by setting $B=0$ there. In fact, in that case such an equation solves differently from the way the Schr\"odinger equation of the hydrogen atom is solved (see, e.g., Ref.~\cite{BookQM}). Different specific approximation methods can indeed then be applied for that case \cite{LogarithmicPotential}. We are not going here to deal with such a purely gravitational problem, for our main purpose in the present paper is the effect of gravity on the Landau levels.
\subsection{Using the harmonic oscillator approximation}
It is actually possible to also achieve quantization of the energy levels of the particle by starting from the Schr\"odinger equation (\ref{KGinLC3}) and approximating the latter with the equation of a harmonic oscillator. All one needs to do is find the equilibrium radius $\rho_{\rm e}$ around which the particle's effective potential $V_{\rm eff}(\rho)$, as given by the square brackets in Eq.~(\ref{KGinLC3}), reaches a minimum. Such a radius $\rho_{\rm e}$ is thus the solution to the equation ${\rm d}V_{\rm eff}(\rho)/{\rm d}\rho=0$. The latter equation is a quartic equation but its special form (quadratic in $\rho_e^2$),
\begin{equation}\label{VRho*}
\frac{e^2B^2}{4m}\rho^4_{\rm e}-amc^2\rho^2_{\rm e}-\frac{\hbar^2}{m}\left(\ell^2-\frac{1}{4}\right)=0,
\end{equation}
in contrast to the case of the effective potential around the spherical mass \cite{GravityLandau}, is easily solvable. It has indeed two roots; the positive one being,
\begin{equation}\label{RhoEq}
\rho_{\rm e}^2=\frac{2am^2c^2}{e^2B^2}+\sqrt{\frac{4a^2m^4c^4}{e^4B^4}+\frac{4\hbar^2}{e^2B^2}\left(\ell^2-\frac{1}{4}\right)}\approx\frac{2\hbar}{eB}\sqrt{\ell^2-\frac{1}{4}}\left(1+\frac{am^2c^2}{\hbar eB\sqrt{\ell^2-\frac{1}{4}}}\right).
\end{equation}
In the second step we have expanded in powers of $a$ up to the first order as this will allow us shortly to (i) easily see how one recovers the Minkowski case $a=0$, as well as to (ii) extract the first-order correction in $a$ to the Landau levels. The effective potential $V_{\rm eff}(\rho)$ of the particle around this equilibrium position $\rho_{\rm e}$ can now be Taylor-expanded at the second order in $\rho$ and approximated by a quadratic potential as follows:
\begin{equation}\label{VTaylorexpanded}
V_{\rm eff}(\rho)\simeq V_{0}+\frac{1}{2}m\omega^2(\rho-\rho_{\rm e})^2.
\end{equation}
Here, $V_0=V_{\rm eff}(\rho_{\rm e})$ and $m\omega^2={\rm d}^2V_{\rm eff}/{\rm d\rho^2}|_{\rho=\rho_{\rm e}}$. With such a potential, the Schr\"odinger equation (\ref{KGinLC3}) becomes that of a simple harmonic oscillator for which the energy eigenvalues are well-known, and given by,
\begin{equation}\label{EnergySHO}
\mathcal{E}_n=V_0+\hbar\omega\left(n+\frac{1}{2}\right),
\end{equation}
where $n$ is again a non-negative integer. Substituting the value of $\rho_{\rm e}$ from Eq.~(\ref{RhoEq}) into $V_{\rm eff}(\rho_{\rm eq})$ and ${\rm d}^2V_{\rm eff}/{\rm d\rho^2}|_{\rho=\rho_{\rm e}}$, allows us to find the quantized energy levels:
\begin{equation}\label{ApproxHOEnergyLevelsLC}
\mathcal{E}_{n\ell}\approx\,\frac{\hbar eB}{m}\left(n+\frac{1}{2}+\frac{1}{2}\sqrt{\ell^2-\tfrac{1}{4}}-\frac{\ell}{2}\right)-\frac{amc^2}{2\sqrt{\ell^2-\tfrac{1}{4}}}\left[n+\frac{1}{2}+\sqrt{\ell^2-\tfrac{1}{4}}\ln\left(\frac{2\hbar}{eB\rho_*^2}\sqrt{\ell^2-\tfrac{1}{4}}\right)\right].
\end{equation}
This is the expression of the energy levels of the particle --- up to the first order in $a$ --- for each quantum number $n$ and for each quantum orbital $\ell$. It should be recalled, though, that, as mentioned above, for the case of $K\neq1$ one has to replace in this result $\ell$ by $\ell/K$. We clearly see now form this expression that we recover the usual Landau levels plus the first-order correction we obtained using perturbation theory. These, of course, do agree exactly in the large-$\ell$ limit. Indeed, for large $\ell$, Eq.~(\ref{ApproxHOEnergyLevelsLC}) becomes identical to Eq.~(\ref{1ellCorrection}) obtained for the particular case $n=1$. On the other hand, for $a=0$ ({\it i.e.}, by removing the cylinder), formula (\ref{ApproxHOEnergyLevelsLC}) reproduces, in the large-$\ell$ limit, the familiar Landau energy levels of a particle inside a constant and uniform magnetic field within the Minkowski spacetime.
From the full expression (\ref{ApproxHOEnergyLevelsLC}), we also see that the first-order correction depends logarithmically on the magnetic field as well. However, in contrast to the case of the spherical mass \cite{GravityLandau}, and the Kerr spacetime case we are going to see shortly, we do not obtain quantized energy levels when putting $B=0$. In fact, when putting $B=0$ in Eq.~(\ref{ApproxHOEnergyLevelsLC}) the first line vanishes whereas the second line blows up. This is due to the fact that the equilibrium distance $\rho_e$ does not actually exist in the absence of the magnetic field as the logarithmic potential alone does not allow for any equilibrium position of the particle with $\ell\neq0$. In the absence of the magnetic field, we are left with an infinitely long cylinder the gravitational field of which is unable to counterbalance the centrifugal force on the particle due to the circular motion of the latter. The simple harmonic approximation does not therefore work for a pure gravitational field created by an infinitely long cylinder.
It is now enlightening to examine the orders of magnitude involved in such energy levels splittings. For a magnetic field of the order of $10\,$T --- now easily achievable in a laboratory \cite{StrongB} --- and using a $1\,$cm-radius cylinder of pure platinum and $2$ meters in length for the infinitely-long cylinder approximation to hold, leads to a first-order correction to the first Landau levels of the order of $10^{-19}\,$eV. This small energy difference is, unfortunately, still too small for the presently achievable resolution which is of the order of $10^{-15}\,$eV \cite{TodayLimit}. To remedy this, one would just have to increase the size of the cylinder. In fact, using a $1$\,m-radius cylinder would effectively increase such a gravitational correction by four orders of magnitude to easily reach the present sensitivity limit of $\Delta E\sim 10^{-15}\,$eV. The only downside is that one would then have to increase the length of the cylinder accordingly.
Before we move on to the case of a particle inside the Kerr metric, we would like to note here three important facts. The first two are the ones already pointed out in Ref.~\cite{GravityLandau} for the case of the spherical mass and which still apply here. The first is that it is actually possible to rely solely on the solutions to Eq.~(\ref{KGinLC2}) and extract the energy quantization condition without making use of the Schr\"odinger equation (\ref{KGinLC3}). In fact, while it is obvious that the logarithm in Eq.~(\ref{KGinLC2}) makes the latter hardly solvable analytically, by expanding the logarithmic function one might turn the equation into a Heun-like differential equation \cite{BookHeun}. Such a differential equation has well-known solutions, called Heun functions. The procedure then consists in imposing either one of two specific conditions on such a function to guarantee the square-integrability of the latter, and hence to provide it with a physical meaning \cite{GravityLandau}. The problem with such a procedure, as explained in detail in Ref.~\cite{GravityLandau}, is that one of the conditions to impose does not provide a consistent quantization of energy for arbitrary values of the mass-source of the gravitational field, while the other condition does not allow to practically extract a simple answer as it involves finding the zeros of an infinite series. For this reason, we are not going to dwell more on these other two approaches here.
The last point we would like to comment on here is that it would be natural now to attempt to apply the same techniques used above to the case of a rotating cylinder. Unfortunately, however, to deal with such a case one has to use the so-called Lewis spacetime \cite{Cunha}, which is even more complicated than the metric (\ref{LCMetric}). Given that the Lewis spacetime reduces in the limit of zero radius of the cylinder to that of a rotating cosmic string (see, e.g., Ref.~\cite{Bronnikov} and the references therein for more details about such a metric), which, in turn, has extensively been studied in Ref.~\cite{Lewis}, we are going to turn instead into the rotating spacetime represented by the Kerr metric. The latter is indeed much more prone to experimental verification, both at the tabletop experiments level and at the astrophysical level.
\section{A particle inside a magnetic field in the Kerr spacetime}\label{sec:III}
In this section, our test particle is still a charged spinless particle moving in the plane perpendicular to the constant and uniform magnetic field $\bf B$. Now, however, we assume the particle is going around a massive sphere of radius $r_0$, of mass $M$, and of angular momentum $J$ the direction of which is parallel to that of the magnetic field. In the weak-field approximation and slow rotation of the mass source, the Kerr metric around a rotating sphere of mass $M$ and angular momentum $J$ takes the following form in the spherical coordinates $(t,r,\theta,\phi)$ (see, e.g., Ref.~\cite{Kerr}),
\begin{equation}
\label{Kerr}
{\rm d}s^2=-\left(1-\frac{2GM}{c^2r}\right)c^2{\rm d}t^2+\left(1+\frac{2GM}{c^2r}\right){\rm d}r^2+r^2{\rm d}\Omega^2-\frac{4GM\alpha}{cr}\sin^2\theta\,{\rm d}t\,{\rm d}\phi.
\end{equation}
Here, ${\rm d}\Omega^2={\rm d}\theta^2+\sin^2\theta{\rm d}\phi^2$ and $\alpha=J/Mc$ is the specific angular momentum, that is, the angular momentum per unit mass, of the rotating sphere. To the first order in $GM/c^2r$ and in $GM\alpha/c$, at which we expanded this metric, the square root of the determinant of the metric is $\sqrt{-g}\approx cr^2\sin\theta$. On the other hand, because of the time-independence of both the gravitational field and of the magnetic field, and because of the symmetry around the $z$-axis, we expect the wavefunction of the test particle of energy $E$ to be of the form, $\varphi(t,r,\theta,\phi)=e^{-i\frac{Et}{\hbar}}e^{i\ell\phi}R(r,\theta)$. We assume again that the particle has no momentum along the $z$-direction and that it is moving counterclockwise around the sphere. Therefore, in the symmetric gauge, expressed in a covariant form in spherical coordinates, the non-vanishing components of the potential vector in a rotating spacetime read \cite{Wald}, $A_t=\frac{1}{c}B\alpha g_{tt}+\frac{1}{2}Bg_{t\phi}=-cB\alpha[1-\frac{GM}{c^2r}(2-\sin^2\theta)]$ and $A_\phi=\frac{1}{2}Bg_{\phi\phi}+\frac{1}{c}B\alpha g_{t\phi}\approx\tfrac{1}{2}Br^2\sin^2\theta.$\footnote{Note that we displayed here again the covariant form of the potential vector as the tetrad form $A_{\hat{\phi}}=\frac{1}{2}Br\sin\theta$ we used in our previous version of the manuscript leads to much confusion. The advantage of the tetrad expression is that it has the right dimensions for a potential vector and it allows one to straightforwardly extract the magnetic field from the spatial components, $\vec{\bf B}=\nabla\times\vec{\bf A}$. However, one has then to take special care when plugging such an expression inside Eq.~(\ref{KG}). The result one obtains is, of course, the same with both expressions.} Here, we have kept only the first-order in $\alpha$. The effect of the constant term in the time-component $A_t$ consists simply in redefining the energy reference of the charged particle in the spacetime by shifting the energy of the latter by the constant $-ceB\alpha$. Therefore, by redefining the energy reference for $E$ by performing the shift $E\rightarrow E+eB\alpha c$, the constant term in $A_t$ is absorbed. The Klein-Gordon equation for the particle in this curved spacetime then takes, up to the first-order in $\alpha$, the following form,
\begin{align}\label{KGinKerr1}
&\Bigg[\!\left(\frac{E^2}{\hbar^2c^2}+\frac{2GMEeB\alpha(2-\sin^2\theta)}{\hbar^2c^3r}\right)\!\left(1+\frac{2GM}{c^2r}\right)\!-\!\frac{m^2c^2}{\hbar^2}\!+\!\left(1-\frac{2GM}{c^2r}\right)\!\partial_r^2\!
+\!\left(\frac{2}{r}-\frac{2GM}{c^2r^2}\right)\!\partial_r\nonumber\\
&+\frac{\partial^2_\theta}{r^2}
+\frac{\cos\theta\partial_\theta}{r^2\sin\theta}-\frac{\ell^2}{r^2\sin^2\theta}+\frac{eB\ell}{\hbar}
-\frac{e^2B^2r^2\sin^2\theta}{4\hbar^2}-\frac{4GM\alpha E\ell}{\hbar c^3r^3\sin^2\theta}+\frac{2GM\alpha EeB}{\hbar^2c^3r}\Bigg]R(r,\theta)=0.
\end{align}
Further, by having the particle move along the equatorial plane, along which $\theta=\tfrac{\pi}{2}$, the cylindrical symmetry of the system allows us to also expect the radial function $R(r,\theta)$ to depend only on the distance $\rho=r\sin\theta$ of the particle from the $z$-axis which is perpendicular to the plane of motion. This would then make the radial function $R(r,\theta)$ a function of the form $R(r,\sin\theta)=R(r\sin\theta)=R(\rho)$. Therefore, we can use the greatly simplifying identities, $\partial_\theta R=r\cos\theta\partial_\rho R$ and $\partial_\theta^2 R=-r\sin\theta\partial_\rho R+r^2\cos^2\theta \partial_\rho^2R$. In fact, substituting these into the previous equation, the latter takes the following simplified explicit form for $\theta=\tfrac{\pi}{2}$:
\begin{multline}\label{KGinKerr3}
\frac{{\rm d^2} R}{{\rm d}\rho^2}+\frac{1}{\rho}\frac{{\rm d}R}{{\rm d}\rho}+\Biggl[\frac{E^2}{\hbar^2c^2}\left(1+\frac{4GM}{c^2\rho}\right)\\
-\left(1+\frac{2GM}{c^2\rho}\right)\bigg(\frac{m^2c^2}{\hbar^2}+\frac{\ell^2}{\rho^2}-\frac{eB\ell}{\hbar}+\frac{e^2B^2\rho^2}{4\hbar^2}+\frac{4GME\ell\alpha}{\hbar c^3\rho^3}-\frac{4GM EeB\alpha}{\hbar^2c^3\rho}\bigg)\Biggr]R=0.
\end{multline}
By performing the change of variable $R(\rho)=\psi(\rho)/\sqrt{\rho}$, and then decomposing the energy of the test particle as, $E=\mathcal{E}+mc^2$, and using again the usual non-relativistic approximation $E^2\approx2mc^2\mathcal{E}+m^2c^4$, the above equation becomes, after keeping only the leading terms in $\alpha$ and $GM/c^2\rho$, the final Schr\"odinger equation reads,
\begin{equation}\label{SchrodinKerr}
-\frac{\hbar^2}{2m}\psi''+\Bigg[\frac{e^2B^2\rho^2}{8m}+\frac{\hbar^2(\ell^2-\tfrac{1}{4})}{2m\rho^2}-\frac{\hbar eB\ell}{2m}-\frac{GMm}{\rho}\left(1+\frac{2eB\alpha}{mc}\right)+\frac{2\hbar GM\ell\alpha}{c\rho^3}\Bigg]\psi=\mathcal{E}\psi.
\end{equation}
This equation looks very similar to Eq.~(18) of Ref.~\cite{GravityLandau} found for the weak-field limit of the Schwarzschild spacetime. The only difference is, indeed, the presence of the correction term $(1+2eB\alpha/mc)$ multiplying the Newtonian term $GMm/\rho$, as well as the extra term which decreases with the inverse cube of the distance $\rho$ of the particle from the center of the sphere. The reason we kept this latter term, despite the $\hbar$ multiplying it, is, as we shall below, the orbital number $\ell$ could become very large. In fact, it could become as large as $eB\rho_0^2/\hbar$, in which case the last term inside the square brackets of Eq.~(\ref{SchrodinKerr}) becomes of the same order as the first term to its left. Nonetheless, having already obtained the necessary tools for dealing with such extra perturbative terms inside the Schr\"odinger equation in Ref.~\cite{GravityLandau}, we can greatly benefit here from the results in that reference concerning the Newtonian term and its Kerr correction. For the last term, however, a new integral is required and is given in appendix \ref{A}. We are going therefore to apply here also the two different approaches used in Section~\ref{sec:II} to extract the quantized energy levels based on the key results given in Ref.~\cite{GravityLandau}. Note that, similarly to what we discussed in Section \ref{sec:II} for the cylinder, the wavefunctions $\psi_{n\ell}^{(0)}$ around a rotating sphere of finite radius $\rho_0$ have to satisfy the continuity condition (\ref{FContinuity}). However, we are going to simply assume here again that the radius of the sphere and the magnitude of the magnetic field are such that quantum numbers $n$ and $\ell$ are guaranteed for the particle.
\subsection{Using perturbation theory}
Treating Eq.~(\ref{SchrodinKerr}) with the time-independent perturbation theory gives in fact the perturbed Landau energy levels at the first order as \cite{GravityLandau}, $\mathcal{E}_{n\ell}=\mathcal{E}_n^{(0)}+\braket{\psi_{n\ell}^{(0)}|V(\rho)|\psi_{n\ell}^{(0)}}$, where $V(\rho)$ is the perturbing potential given by the last three terms inside the square brackets of Eq.~(\ref{SchrodinKerr}). Thus, by adopting the specific expression (18) found in Ref.~\cite{GravityLandau} --- after inserting the correcting factor $(1+2eB\alpha/mc)$ there --- and adding the contribution of the cubic term in Eq.~(\ref{SchrodinKerr}), we immediately find the first-order correction to the Landau levels as follows:
\begin{align}\label{Reduced1stOrderSplitting}
\mathcal{E}_{n\ell}&=\mathcal{E}_{n}^{(0)}-GMm\left[\left(1+\frac{2eB\alpha}{mc}\right)\mathcal{P}_{n\ell}\mathcal{M}_{n\ell}^{-1}-\left(\frac{2\hbar\alpha}{mc}\ell\right)\mathcal{Q}_{n\ell}\mathcal{M}_{n\ell}^{-1}\right]\nonumber\\
&=\mathcal{E}_{n}^{(0)}-GMm\sqrt{\frac{eB}{2\hbar}}\left[\left(1+\frac{2eB\alpha}{mc}\right)\bar{\mathcal{P}}_{n\ell}\bar{\mathcal{M}}_{n\ell}^{-1}-\left(\frac{eB\alpha}{mc}\ell\right)\bar{\mathcal{Q}}_{n\ell}\bar{\mathcal{M}}_{n\ell}^{-1}\right].
\end{align}
The factor $\mathcal{P}_{n\ell}$ represents an infinite series and is given by expression (\ref{AppendixIntegralP}) of appendix \ref{A} after setting the integers $m=n$ there. It arises from the $1/\rho$ Newtonian term in the potential. The factor $\mathcal{Q}_{n\ell}$ is also an infinite series and is given by expression (\ref{AppendixIntegralQ}). It arises from the $1/\rho^3$ term in the potential. In the second line, we have introduced again the reduced series $\bar{\mathcal{M}}_{n\ell}$ which consists of expression (\ref{AppendixIntegralM}) but without the constant factor $(2/\beta)^{\ell+1}$. Similarly, we introduced the reduced series $\bar{\mathcal{P}}_{n\ell}$ and $\bar{\mathcal{Q}}_{n\ell}$ which consist of expressions (\ref{AppendixIntegralP}) and (\ref{AppendixIntegralQ}), respectively, in which we suppress the constant factors $(2/\beta)^{\ell+\frac{1}{2}}$ and $(2/\beta)^{\ell-\frac{1}{2}}$, respectively. This, in fact, allows us to get explicitly the factor $\sqrt{eB/2\hbar}$ out in Eq.~(\ref{Reduced1stOrderSplitting}). This result shows how the degenerate Landau levels split at the first-order in $GMm$ due to gravity. The ratio $eB\alpha/mc$ contains the frame-dragging effect created by the rotating curved spacetime.
In order to appreciate this result, it is instructive to examine the fate of the first Landau level $n=1$. On the one hand, according to the definitions (\ref{AppendixIntegralMDef}), (\ref{AppendixIntegralPDef}) and (\ref{AppendixIntegralQDef}), for small values of $\ell$, the product $\mathcal{P}_{1\ell}\mathcal{M}_{1\ell}^{-1}$ is simply of the order of $1/\rho_0$ whereas the product $\mathcal{Q}_{1\ell}\mathcal{M}_{1\ell}^{-1}$ is of the order of $1/\rho^3_0$. The last term in the first line in Eq.~(\ref{Reduced1stOrderSplitting}) then becomes suppressed simply because of the presence of the $\hbar$ factor in the numerator. The splitting of the Landau levels in this case reduces to the following first-order correction,
\begin{equation}\label{n=1SplittingEllSmall}
\mathcal{E}_{1\ell}\approx\frac{3\hbar eB}{2m}-\frac{GMm}{\rho_0}\left(1+\frac{2eB\alpha}{mc}\right).
\end{equation}
On the other hand, for large values of $\ell$, substituting the large-$\ell$ limits (\ref{M1ellInfinite}), (\ref{P1ellInfinite}) and (\ref{Q1ellInfinite}) of $\mathcal{M}_{1\ell}$, $\mathcal{P}_{1\ell}$ and $\mathcal{Q}_{1\ell}$, respectively, inside Eq.~(\ref{Reduced1stOrderSplitting}) we find, after using the asymptotic property of the gamma function $\Gamma(z)\sim z^{-\frac{1}{2}}e^{z(\log z-1)}$ \cite{BookKummer}, the following approximation for the energy splitting of the first Landau level:
\begin{equation}\label{n=1SplittingEllInfinite}
\mathcal{E}_{1,\ell\gg1}\approx\frac{3\hbar eB}{2m}-GMm\sqrt{\frac{eB}{2\hbar\ell}}\left(1+\frac{eB\alpha}{mc}\right).
\end{equation}
We see that the Newtonian correction term in Eq.~(\ref{n=1SplittingEllInfinite}) decreases like $1/\sqrt{\ell}$, and therefore becomes gradually suppressed for large $\ell$ as it was the case for a static spherical mass \cite{GravityLandau}. In addition, the frame-dragging correcting factor itself does not depend on the orbital quantum number $\ell$. In contrast, for large $n$ we see from Eqs.~(\ref{AppendixIntegralM}), (\ref{AppendixIntegralP}) and (\ref{AppendixIntegralQ}), giving $\mathcal{M}_{n\ell}$, $\mathcal{P}_{n\ell}$ and $\mathcal{Q}_{n\ell}$, respectively, that the first-order correction (\ref{Reduced1stOrderSplitting}) does not decrease with an increasing $n$. On the other hand, for large $\ell$, we see from Eq.~(\ref{n=1SplittingEllInfinite}) that the correction becomes for $n=1$ insensitive to the radius $\rho_0$ of the rotating sphere. The same remark is valid for $n>1$, though.
It should be kept in mind here, as emphasized in Ref.~\cite{GravityLandau}, that the large-$\ell$ approximation obtained in Eq.~(\ref{n=1SplittingEllInfinite}) is valid for very large values of $\ell$. This is because the large-$\ell$ limit in the appendix was found by taking into account the very large term $\frac{\beta}{2}\rho_0^2$ appearing inside the incomplete gamma functions in Eqs.~(\ref{M1ell}), (\ref{P1ell}) and (\ref{Q1ell}). As a consequence, and contrary to what it might seem at first sight, the correction term obtained on the right-hand side in Eq.~(\ref{n=1SplittingEllInfinite}) is really small as is required for a perturbation.
Similarly, using the results of Ref.~\cite{GravityLandau} for the second order in the perturbation theory, we can also easily deduce the second-order correction to the energy levels. However, given the already complicated first-order expression (\ref{Reduced1stOrderSplitting}), we are not going to display the second order-correction here, suffice it to note that it is going to be quadratic in $GMm$ as was the case for the static spherical mass. The frame-dragging effect will then simply appear as corrections terms proportional to various powers of the ratio $eB\alpha/mc$.
\subsection{Using the harmonic oscillator approximation}
Let us now apply here the method based on approximating the effective potential of the particle by that of a simple harmonic oscillator. Unfortunately, the presence of the last term inside the effective potential in the Schr\"odinger equation (\ref{SchrodinKerr}) renders this method analytically intractable for arbitrary values of the orbital number $\ell$. In fact, the condition ${\rm d}V_{\rm eff}/{\rm d}\rho=0$, that would give the radius $\rho_e$ at which the potential reaches its minimum, becomes in this case a quintic equation. For this reason, the harmonic oscillator approximation becomes really useful only for small values of $\ell$, for then the last term in the effective potential in Eq.~(\ref{SchrodinKerr}) can be neglected.
Therefore, given that for the case of small $\ell$ the only difference between the Schr\"odinger equation of our system as given by Eq.~(\ref{SchrodinKerr}) and that of Ref.~\cite{GravityLandau} resides only in the correcting factor $(1+2eB\alpha/mc)$ that multiplies the Newtonian potential, we are not going to display here the details of the calculations. We are going to content ourselves by displaying the final results after inserting such a correcting term. In addition, since within the perturbation theory we used above we restricted ourselves to the first-order approximation, we are not going to display the second-order correction here either.
Based on the general formula for the perturbed energy levels in the spherical static mass \cite{GravityLandau}, the energy levels for the rotating mass thus split at the first order in the specific angular momentum $\alpha$ as follows:
\begin{align}\label{ApproxHOEnergyLevels}
\mathcal{E}_{n\ell}&\approx\,\frac{\hbar eB}{m}\left(n+\frac{1}{2}+\frac{1}{2}\sqrt{\ell^2-\frac{1}{4}}-\frac{\ell}{2}\right)\nonumber\\&\quad+\frac{GMm}{(\ell^2-\tfrac{1}{4})^{3/4}}\sqrt{\frac{eB}{32\hbar}}\left(1+\frac{2eB\alpha}{mc}\right)\left(n+\frac{1}{2}-4\sqrt{\ell^2-\frac{1}{4}}\right).
\end{align}
We clearly see form this result that we recover again the usual Landau levels plus a similar formal structure for the first-order correction we obtained using perturbation theory. The dependence of the correction on the square root of the magnetic field and on the ratio $eB\alpha/mc$ is remarkable. Of course, despite these similarities between the results of the two methods at this first-order level, the result (\ref{ApproxHOEnergyLevels}) cannot be used for large values of $\ell$, in contrast to the result (\ref{n=1SplittingEllInfinite}) which is specifically found for large $\ell$. This particular case shows the superiority in this investigation of the approach based on perturbation theory over the simple harmonic oscillator approximation.
As was the case with the results obtained in Ref.~\cite{GravityLandau} concerning the static spherical mass, our results here for the rotating spherical mass might {\it a priori} both be applied at the tabletop experiments level and at the astrophysical observations level. Unfortunately, as we shall see, for the latter case our above approximations become too restrictive to be applicable for the wide range of astrophysical situations. In fact, our approximation does show that for the Landau quantization to be significant, the frame-dragging contribution to the effective potential of the particles should not dominate the interaction of the latter with the magnetic field.
Indeed, with protons as the test particles, a $1$\,m-radius spherical mass of platinum, and a laboratory magnetic field of the order of $10\,$T, the first-order correction to the first Landau levels of the protons is, according either to Eq.~(\ref{n=1SplittingEllSmall}) or Eq.~(\ref{n=1SplittingEllInfinite}), of the order of $10^{-8}\,$eV. If the sphere is then rotated at about $100$ revolutions per minute, the frame-dragging effect induces the dimensionless correction to the Newtonian potential, $eB\alpha/mc$, which is of the order of $10^{-7}$. For electrons, this dimensionless factor would be of the order of $10^{-4}$. Of course, due to the presence of the magnetic field, the rotating platinum spherical mass should be grounded in order to avoid any induced electric current.
On the other hand, at the astrophysical level, it is already known in the literature that the strong magnetic fields around rotating neutron stars, magnetars and magnetic white dwarfs could be taken into account to study how the equations of states of the surface (or even the bulk) nuclei matter would be affected by the Landau quantization caused by such strong magnetic fields \cite{Broderick,Chamel,StarsBook}. However, these astrophysical objects could acquire, in addition to the strong magnetic fields, very high rotational speeds that could reach up to $10^4$ revolutions per minute. The contribution to the splitting of the energy levels in Eqs.~(\ref{n=1SplittingEllSmall}) and (\ref{n=1SplittingEllInfinite}) due to the frame-dragging effect becomes then dominant over the contribution due to the Newtonian potential and even over the Landau energy levels themselves. For a $10\,$kilometer-radius neutron star, rotating at such a rate and producing a magnetic field of the order of $10^{10}\,$T, which is also typical of magnetars \cite{McGillCatalogue}, the frame-dragging term $eB\alpha/mc$ is already of the order $10^{15}$ for electrons and of the order of $10^{11}$ for protons. Our weak-field approximation due to a slow rotation of the mass source then breaks down in this case. Actually, such strong magnetic fields combined with a radius of the star that is above one kilometer keeps the frame-dragging effect dominant unless the rotation rate of the star is much smaller than one revolution per year.
\section{Discussion \& Conclusion}\label{sec:V}
We have studied the effect of two different gravitational fields on a charged particle moving inside a uniform and constant magnetic field. The first consists of the field created by an infinitely long cylinder, expressed in the form of the Levi-Civita metric, and the second one was the field created by a rotating spherical mass, expressed in the form of the Kerr spacetime. We found that the infinite Landau degeneracy is removed in both cases as the Landau orbitals of the same Landau level split in energy. As was done in Ref.~\cite{GravityLandau} for the Schwarzschild spacetime case, we used here two independent methods to reach the quantized energy levels implied by the corresponding curved-spacetime Klein-Gordon equations.
The results of the two methods are quantitatively different due to the different degrees of approximation each method relies on. Both methods, however, point towards the same qualitative splitting of the energy levels. In the case of the Levi-Civita metric the splitting is characterized by a logarithmic dependence on the radius of the cylinder and of the radius of the position taken as a reference for the gravitational potential. Our results for this metric would be valid in a realistic setup provided one uses a very long and very thin massive cylinder, with the test particle moving very closely to the surface of the cylinder. This first investigation is more of a gravitational-testing tool. It provides an additional important approach towards testing the century-old and apparently illusive Levi-Civita metric.
The second investigation provided us with a very nice way of testing the famous frame-dragging of general relativity at the level of quantum particles. The larger the specific angular momentum of the rotating massive sphere is, the bigger is the splitting in the energy of the Landau levels. This second investigation is testable at the level of tabletop experiments using strong magnetic fields and rapidly rotating massive grounded spheres. Both investigations have been carried out using, for simplicity, spinless particles. Such a setup can indeed easily be achieved experimentally by using heavy ions the total spin of which is negligible.
At the level of astrophysical observations of rapidly rotating neutron stars, magnetars and magnetic white dwarfs, our investigation showed that for a wide range of realistic astrophysical objects (with fast rotations and strong magnetic fields) the frame-dragging effect cannot constitute a mere perturbation compared to the Newtonian potential neither compared to the Landau levels themselves. We saw that the frame-dragging effect couples to the magnetic field in such a way that the effect of the latter alone on the particles is what actually constitutes a perturbation. Therefore, because of the frame-dragging effect the Landau levels would emerge and dominate on such highly magnetized stars only when the latter are slowly rotating around their axes.
We have based our whole approach in this paper on the combination of the Klein-Gordon equation in curved spacetime and the full spacetime metrics of both the Kerr and Levi-Civita spacetimes. The full equations (\ref{KGinLC2}) and (\ref{KGinKerr3}) have then been approximated into much easier to solve equations by relying on the low-curvature and non-relativistic regime approximations. Such restrictions have been dictated by, respectively, the orders of magnitude of the massive sources and of the magnetic fields in which we are interested in this paper. Our main goal in this paper has indeed been to simply bring into light the effect of more complicated gravitational fields than that due to a static spherical mass on the Landau quantum levels. A fully relativistic treatment of the motion of charged particles in a strong magnetic field and in a curved spacetime, as done in, e.g., Refs.~{\cite{Magnetized1,Magnetized2,Magnetized3,Magnetized4,Magnetized5,Magnetized6,Magnetized7,Magnetized8,Magnetized9,Magnetized10}}, will be the next step. We defer the investigation taking into account the relativistic corrections to the motion of the electrons or neutrons moving on the surface of neutron stars/magnetars/magnetic white dwarfs to forthcoming works. We shall then conduct rigorously a detailed study of the fate of the equation of state on these astrophysical objects caused by the splitting of the Landau levels due to their rotation. In fact, on the one hand, going beyond the non-relativistic regime leads to extra terms of the form $\rho^2\ln\rho$ inside Eq.~(\ref{KGinLC2}) for the Levi-Civita spacetime and might allow one to get to higher order approximations in the parameters $a$ and $b$ of the Levi-Civita metric. On the other hand, allowing for a relativistic regime of the test particle would lead to non-perturbative terms of the form $1/\rho$ and $1/\rho^3$ inside Eq.~(\ref{KGinKerr3}) for the Kerr spacetime. The presence of all these extra terms necessitates different mathematical methods for solving the corresponding differential equations than those adopted here.
\section*{Acknowledgments}
We are grateful to
Bobur Turimov for the helpful comment about our notation for the potential vector and for having pointed out to us Ref.~\cite{Wald} that contains the complete form of the potential vector in rotating spacetimes. We are also grateful to the anonymous referee for the pertinent comment that led us to rectify a previous erroneous version of the condition (\ref{FContinuity}) imposed on the wavefunction. This work is supported by the Natural Sciences and Engineering Research
Council of Canada (NSERC) Discovery Grant (RGPIN-2017-05388).
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import * as types from '../types';
export const dismissAlert = () => (dispatch) => {
return dispatch({type: types.REMOVE_ALERT});
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export default { dismissAlert };
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John Hely-Hutchinson, comte de Donoughmore KP, PC (I) (1787 - ) est un homme politique et pair irlandais.
Biographie
Il est le fils de l'hon. Francis Hely-Hutchinson (d. 1827) (le fils de Christiana Hely-Hutchinson (1re baronne Donoughmore)).
Il représente Tipperary à la Chambre des communes du Royaume-Uni en tant que Whig. À partir de 1832, il siège à la Chambre des lords, ayant hérité des pairies de son oncle, en particulier la vicomté de Hutchinson.
En tant que capitaine du 1st Foot Guards, il contribue à l'évasion de prison du ministre des Postes de Napoléon, le comte de Lavalette. Il est jugé à Paris, avec Robert Thomas Wilson et Michael Bruce, pour avoir aidé le comte à s'évader de prison. Le procès se déroule à la cour d'assises du 22 au 24 avril 1816. Les trois hommes sont reconnus coupables et condamnés à trois mois d'emprisonnement .
Famille
Il épouse l'hon. Margaret Gardiner (fille de Luke Gardiner (1er vicomte Mountjoy)) le 15 juin 1822. Ils ont deux enfants :
Richard Hely-Hutchinson (4e comte de Donoughmore) (né le 4 avril 1823 ; décédé le 22 février 1866)
Margaret Hely-Hutchinson (décédée jeune en 1828)
Il épouse, en secondes noces, Barbara Reynell, fille du lieutenant-colonel William Reynell. Ils ont quatre enfants :
Capt. Hon. John William Hely-Hutchinson (né le septembre 1829 ; décédé le 16 juillet 1855 pendant la guerre de Crimée)
Lady Kathleen Alicia Hely-Hutchinson (décédée le 22 avril 1892), mariée le 3 décembre 1863 à DW Ramsay Carrick Buchanan, de Drumpellier et Corsewall (décédé le 4 mai 1925)
Lady Frances Margaret Hely-Hutchinson (décédée le 11 avril 1866), mariée le 22 septembre 1858 à Arthur Tremayne (d. 14 novembre 1905)
Lady Jane Louisa Hely-Hutchinson (décédée le 29 août 1868)
Références
Liens externes
Député du 8e Parlement du Royaume-Uni
Député du Parti whig (Royaume-Uni)
Comte de la pairie d'Irlande
Membre du Conseil privé d'Irlande
Lord-lieutenant de Tipperary
Chevalier de l'ordre de Saint-Patrick
Décès en septembre 1851
Naissance en 1787
Comte de Donoughmore | {
"redpajama_set_name": "RedPajamaWikipedia"
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package com.amazonaws.services.cognitoidp.model.transform;
import java.util.Map;
import javax.annotation.Generated;
import com.amazonaws.SdkClientException;
import com.amazonaws.services.cognitoidp.model.*;
import com.amazonaws.protocol.*;
import com.amazonaws.annotation.SdkInternalApi;
/**
* AdminRespondToAuthChallengeRequestMarshaller
*/
@Generated("com.amazonaws:aws-java-sdk-code-generator")
@SdkInternalApi
public class AdminRespondToAuthChallengeRequestMarshaller {
private static final MarshallingInfo<String> USERPOOLID_BINDING = MarshallingInfo.builder(MarshallingType.STRING)
.marshallLocation(MarshallLocation.PAYLOAD).marshallLocationName("UserPoolId").build();
private static final MarshallingInfo<String> CLIENTID_BINDING = MarshallingInfo.builder(MarshallingType.STRING).marshallLocation(MarshallLocation.PAYLOAD)
.marshallLocationName("ClientId").build();
private static final MarshallingInfo<String> CHALLENGENAME_BINDING = MarshallingInfo.builder(MarshallingType.STRING)
.marshallLocation(MarshallLocation.PAYLOAD).marshallLocationName("ChallengeName").build();
private static final MarshallingInfo<Map> CHALLENGERESPONSES_BINDING = MarshallingInfo.builder(MarshallingType.MAP)
.marshallLocation(MarshallLocation.PAYLOAD).marshallLocationName("ChallengeResponses").build();
private static final MarshallingInfo<String> SESSION_BINDING = MarshallingInfo.builder(MarshallingType.STRING).marshallLocation(MarshallLocation.PAYLOAD)
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private static final AdminRespondToAuthChallengeRequestMarshaller instance = new AdminRespondToAuthChallengeRequestMarshaller();
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protocolMarshaller.marshall(adminRespondToAuthChallengeRequest.getChallengeName(), CHALLENGENAME_BINDING);
protocolMarshaller.marshall(adminRespondToAuthChallengeRequest.getChallengeResponses(), CHALLENGERESPONSES_BINDING);
protocolMarshaller.marshall(adminRespondToAuthChallengeRequest.getSession(), SESSION_BINDING);
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Categories: Events, Featured, Press Releases / by tipsgeeks January 8, 2022
Post Author: tipsgeeks
Globe, the leader in Mobile, has registered more than 11 million active prepaid SIMs around a month after the law mandating all...
Events, Featured, Press Releases ... , Read More
From left to right: Oamar B. Tabiliran, PLDT Enteprirse Business Head CRB 1 Visayas; Arnold Dosdos, PLDT Enterprise Relationship Manager;...
(Manila, Philippines) January 2023 – Filipino consumers can now use buy now, pay later (BNPL) and card-free installment option to...
To reach more customers across the country, Globe, the leader in Mobile, is opening assisted registration in 30 branches of leading... | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 8,013 |
\section*{Acknowledgements}
\addcontentsline{toc}{section}{Acknowledgements}
We thank Yuliia Tymoshenko (KIT) for providing simulations on~ZnCr$_2$Se$_4$ and~FeP as well as Anup Bera and Bella Lake (both HZB) for the same on~SrCo$_2$V$_2$O$_8$.
Furthermore, we gratefully thank Frank Weber (KIT) who provided the SnTe sample and related information for the neutron experiment at EIGER.
Eventually, we acknowledge the discussions and collaborations with Marcus Noack (LBNL) and Martin Boehm (ILL).
\section*{Funding}
\addcontentsline{toc}{section}{Funding}
This work was supported through the project \textit{Artificial Intelligence for Neutron and X-ray scattering} (AINX) funded by the Helmholtz AI cooperation unit of the German Helmholtz Association.
\section*{Author contributions}
\addcontentsline{toc}{section}{Author contributions}
M.G. and A.S. initiated and supervised the research.
M.T.P. developed the methodology.
M.T.P. and G.B. designed and developed the software components.
M.T.P, G.B., C.F., U.S., and A.S. performed the experiments.
M.T.P. wrote the first draft of the manuscript.
All authors revised drafts of the manuscript.
\section*{Results}
\addcontentsline{toc}{section}{Results}
\subsection*{Problem formulation}
\addcontentsline{toc}{subsection}{Problem formulation}
From a methodological perspective, we aim to discover an intensity function~$i:\mathcal{X}\to[0,\infty)$ on a rectangular set~$\mathcal{X}\subseteq\mathbf{R}^n$, $n\in\mathbf{N}$, with coordinates on a certain hyperplane in four-dimensional $\vec{Q}$-$E$~space~\cite{sivia2011elementary}.
As an example, Fig.~\ref{fig:problem}a displays an intensity function defined on $\mathcal{X}=[2.3,3.3]\times[2.5,5.5]\subseteq\mathbf{R}^2$, where the two-dimensional hyperplane is spanned by the vector~$(0,0,1)$ in $\vec{Q}$~space with offset~$(1,1,0)$ and energy transfer~($E$).
\begin{figure}
\centering
\includegraphics[width=\linewidth]{problem.pdf}
\caption{a) Example of an intensity function on~$\mathcal{X}=[2.3,3.3]\times[2.5,5.5]$ along the $\vec{Q}$ direction~$(0,0,1)$ with offset~$(1,1,0)$ and energy transfer.
b) Inefficient experiment with a majority of measurement locations in the background (dark blue area).
c) More efficient experiment with most measurement locations in the region of signal.}
\label{fig:problem}
\end{figure}
For directions in $\vec{Q}$~space, we use \textit{relative lattice units} (r.l.u.), while energy transfer~$E$ is measured in \textit{micro-electron volts}~(meV).
Note that, due to restrictions of an instrument, the intensity function might only be defined on a subset~$\mathcal{X}^*\subseteq\mathcal{X}$ consisting of all measurement locations reachable.
For more details on the TAS setting, we refer to the Supplementary Information (Sec.~S1{}).
The intensity function~$i$ is accessed by counting scattered neutrons on a detector device for a finite number of measurement locations~$\vec{x}\in\mathcal{X}^*$ yielding noisy observations~$I^+(\vec{x})\sim\text{Pois}(\lambda=i(\vec{x}))$.
Note that detector counts are usually normalized by neutron counts on a monitor device, i.\,e.,~ the corresponding unit of measurement is $\textnormal{detector counts}/(M\textnormal{ monitor counts})$, $M\in\mathbf{N}$.
Since Poisson distributions~$\text{Pois}(\lambda)$ with a sufficiently large parameter~$\lambda>0$ can be approximated by a normal distribution~$\Normdist{\lambda}{\lambda}$, we assume that
\begin{equation}
\label{eq:intensity_noise_physics}
I^+(\vec{x}) = i(\vec{x}) + \sqrt{i(\vec{x})}\eta^+,
\end{equation}
where~$\eta^+\sim\Normdist{0}{1}$.
In the following, we repeatedly refer to \apos{experiments} which are defined as a sequential collection of intensity observations.
\begin{definition}[Experiment]
An \textit{experiment}~$\mathcal{A}$ is an $N$-tuple of location-intensity pairs
\begin{equation}
\label{eq:def_experim}
\mathcal{A} = ((\vec{x}_1,\hat{i}_1),\ldots,(\vec{x}_N,\hat{i}_N))
\end{equation}
where~$\abs{\mathcal{A}} \coloneqq N\in\mathbf{N}$ denotes the number of measurement points, $\vec{x}_j\in\mathcal{X}^*$ are measurement locations, and~$\hat{i}_j$ are corresponding noisy observations of an intensity function~$i$ at~$\vec{x}_j$.
\end{definition}
Since demand for beam time at TAS is high but availability is limited, the goal of an approach is to perform the most informative experiments at the lowest possible cost.
Beam time use can, for example, be optimized when excessive counting in uninformative regions like the background (Fig.~\ref{fig:problem}b) is avoided but focused on regions of signal (Fig.~\ref{fig:problem}c).
We quantify the benefit of an experiment~$\mathcal{A}$ by a benefit measure~$\mu=\mu(\mathcal{A})\in\mathbf{R}$ and its cost by a cost measure~$c=c(\mathcal{A})\in\mathbf{R}$.
For now, it suffices to mention that cost is measured by experimental time (the time used for an experiment) and benefit is defined by reducing a relative weighted error between the target intensity function and a corresponding approximation constructed by the collected intensity observations.
Note that, quantifying benefits this way, their computation is only possible in a synthetic setting with known intensity functions (as is our benchmark setting).
Real neutron experiments do not meet this requirement and thus must be evaluated in a more qualitative way.
In our setting, an approach attempts to conduct an experiment~$\mathcal{A}$ with highest possible benefit using a given cost budget~$C\geq0$, i.\,e.,~ it aims to maximize~$\mu(\mathcal{A})$ while ensuring that $c(\mathcal{A})\leq C$.
The steps of a corresponding general experiment are given in Alg.~$\ref{alg:general}$.
\begin{algorithm}
\caption{General experiment}
\label{alg:general}
\begin{algorithmic}[1]
\Require cost measure~$c$, cost budget~$C\geq0$
\State $\mathcal{A} \gets ()$
\State $J = 0$
\While{$c(\mathcal{A}) < C$}
\State Determine next measurement location~$\vec{x}_{J+1}\in\mathcal{X}^*$ \label{step:next_loc}
\State Observe noisy intensity~$\hat{i}_{J+1}$ at~$\vec{x}_{J+1}$
\State $\mathcal{A} \gets ((\vec{x}_1,\hat{i}_1), \ldots, (\vec{x}_J,\hat{i}_J), (\vec{x}_{J+1},\hat{i}_{J+1}))$
\State $J \gets J+1$
\EndWhile
\end{algorithmic}
\end{algorithm}
Line~\ref{step:next_loc} is most important and crucial for both cost and benefit of the experiment since it decides where to observe intensities, i.\,e.,~ count neutrons, next.
From an algorithmic perspective, if we denote the current step of an experiment by~$J\in\mathbf{N}$, our approach implements the decision for the next measurement location by maximizing an objective function $\phi_J:\mathcal{X}^*\to\mathbf{R}$.
It balances an acquisition function $\textnormal{acq}_J:\mathcal{X}^*\to\mathbf{R}$ and a cost function $c_J:\mathcal{X}^*\to\mathbf{R}$ that both depend on the current step~$J$ and hence can change from step to step.
The acquisition function~$\textnormal{acq}_J$ indicates the value of any~$\vec{x}\in\mathcal{X}^*$ for improving the benefit of the experiment whereas the cost function~$c_J$ quantifies the costs of moving the instrument axes from the current location~$\vec{x}_J\in\mathcal{X}^*$ to~$\vec{x}$.
The metric~$d:\mathcal{X}^*\times\mathcal{X}^*\to[0,\infty)$ used for our particular cost function
\begin{equation}
\label{eq:def_cost_fct}
c_J(\vec{x}) \coloneqq d(\vec{x}_J,\vec{x})
\end{equation}
is formally specified in the Supplementary Information (Sec.~S1{}, Eq.~(10){}).
Our methodology concentrates on developing a useful acquisition function as the crucial component of our approach making it straightforward to find regions of signal.
\subsection*{Log-Gaussian processes for TAS}
\addcontentsline{toc}{subsection}{Log-Gaussian processes for TAS}
We briefly describe here why log-Gaussian processes, our central methodological component, are suitable to identify regions of signal in a TAS experiment.
Methodological details can be found in the Methods section.
Although the intensity function is not directly observable due to measurement noise (Eq.~\eqref{eq:intensity_noise_physics}), we aim to approximate it by the mean function of a log-Gaussian process
\begin{equation}
I(\vec{x}) \coloneqq \exp(F(\vec{x})),
\end{equation}
where~$F$ is a Gaussian process.
That is, after~$J$ steps, we fit logarithmic intensity observations to~$F$ yielding its posterior mean and variance function denoted by~$m_J$ and~$\sigma_J^2$, respectively.
The acquisition function is then defined as the uncertainty of~$I$ given by its posterior standard deviation, i.\,e.,~
\begin{equation}
\label{eq:def_acq_fct_simple}
\textnormal{acq}_J(\vec{x}) = \sqrt{(\exp(\sigma_J^2(\vec{x}))-1) \cdot (\exp(2m_J(\vec{x})+\sigma_J^2(\vec{x})))}.
\end{equation}
Observe the crucial detail that~$m_J$ appears exponentially in this function which is the main reason why our approach is based on \textit{log}-Gaussian processes.
A posterior log-Gaussian process is thus able to find regions of signal just through maximizing its uncertainty.
As illustration, regard Fig.~\ref{fig:gp_loggp} displaying the posterior of the Gaussian process~$F$ together with logarithmic intensity observations (Fig.~\ref{fig:gp_loggp}a) and the corresponding posterior of the log-Gaussian process~$I$ (Fig.~\ref{fig:gp_loggp}b).
\begin{figure}
\centering
\includegraphics[width=0.75\linewidth]{gp_loggp.pdf}
\caption{Transformation of a Gaussian process (a) to a corresponding log-Gaussian process (b) with observations (blue dots) of a one-dimensional intensity function~$i$.
As expected from Eq.~\eqref{eq:def_acq_fct_simple}, maximizing the uncertainty (light blue area) of the log-Gaussian process enables to find regions of signal.}
\label{fig:gp_loggp}
\end{figure}
\subsection*{Intensity threshold and background level}
\addcontentsline{toc}{subsection}{Intensity threshold and background level}
Maximizing the acquisition function from Eq.~\eqref{eq:def_acq_fct_simple} prioritizes regions with high intensities over regions with low intensities.
This poses a problem when there are multiple signal regions with intensities of different magnitudes (Extended Data Fig.~\ref{fig:intens_thresh}a).
Indeed, measurement points are mainly placed in regions with higher intensities whereas regions with less signal are neglected (Extended Data Fig.~\ref{fig:intens_thresh}b).
In TAS, we are interested in each region of signal no matter of which intensity magnitude.
We compensate for this potential problem by introducing an \textit{intensity} \textit{threshold}~$\tau>0$ for observed intensities.
That is, we truncate the observed intensities to a maximum of~$\tau$ before fitting (Extended Data Fig.~\ref{fig:intens_thresh}c).
Consequently, measurement points get more evenly distributed among all signal regions (Extended Data Fig.~\ref{fig:intens_thresh}d) since their placement is not biased due to large differences in their intensity values.
As another problem, intensity observations, in neutron experiments, contain background which is not part of the actual signal, i.\,e.,~ even if there is no actual signal at a certain location, we might nonetheless observe a positive intensity there.
If our approach does not compensate for regions of background, it might not recognize them as parasitic and hence consider them as regions of weak signal which potentially yields uninformative measurement points being placed there.
Therefore, we subtract a \textit{background} \textit{level}~$\gamma\in[0,\tau)$ from already threshold-adjusted intensity observations while ensuring a non-negative value.
Ultimately, the log-Gaussian process actually aims to approximate the modified intensity function
\begin{equation}
\label{eq:intens_fct_backgr_thresh}
i_{\gamma, \tau}(\vec{x}) \coloneqq \max \lbrace \min \lbrace i(\vec{x}), \tau \rbrace - \gamma, 0 \rbrace.
\end{equation}
Observe that, by regarding adjusted intensity observations, the assumption of their noise being normally distributed is violated in general as their noise distribution is asymmetric.
In particular, if~$i(\vec{x}_j)$ substantially exceeds the intensity threshold~$\tau$, the distribution gets rather concentrated at~$\tau$ and thus small in variance.
We, however, do assume noise on adjusted intensities as if they were observed so without adjusting.
This does not change the expected behaviour of getting a useful acquisition function but even ensures numerical stability since noise regularizes the computational problem of solving linear systems in GPR.
It remains to explain how we seek to compute suitable values for the background level~$\gamma$ and the intensity threshold~$\tau$ without knowing the intensity function.
We estimate~$\gamma=\gamma(\mathcal{A}_0)$ and~$\tau=\tau(\mathcal{A}_0)$ by statistics of $J_0\in\mathbf{N}$ initial measurement points
\begin{equation}
\label{eq:init_exp}
\mathcal{A}_0 \coloneqq ((\vec{x}_1,\hat{i}_1), \ldots, (\vec{x}_{J_0},\hat{i}_{J_0}))
\end{equation}
collected to initialize our approach.
The arrangement of initial measurement locations (Extended Data Fig.~\ref{fig:init_locs}) is specified later in the Methods section.
While also the estimation of the background level~$\gamma$ is described later, we define the estimation of the intensity threshold already at this point.
This is necessary to understand the results of the benchmark, for which we set~$\gamma=0$ manually due to its synthetic setting anyway.
We define
\begin{equation}
\label{eq:intens_thresh_param}
\tau(\mathcal{A}_0) \coloneqq \gamma(\mathcal{A}_0) + \beta \cdot (m_{10}-\gamma(\mathcal{A}_0)),
\end{equation}
where~$m_{10}\geq\gamma(\mathcal{A}_0)$ is the median of all intensity observations larger than their $9$-th decile and~$\beta\in(0,1]$ is a parameter controlling the distinction between regions of strong and weak signals that needs to be set before starting an experiment.
Note that this definition is, by the meaning of~$m_{10}$, robust to outliers in~$\mathcal{A}_0$.
\subsection*{Neutron experiment}
\addcontentsline{toc}{subsection}{Neutron experiment}
At SINQ (PSI), we investigated a sample of SnTe (tin telluride) in a real neutron experiment performed at the thermal TAS \textit{EIGER}~\cite{stuhr2017thermal}.
Our general aim is to reproduce known results from~\cite[Fig.~1b]{li2014phonon} using our approach.
Furthermore, we assess the robustness of the experimental result w.r.t.~ changes in the parameters for the background level and the intensity threshold (scenario~1) as well as challenge our approach using a coarser initialization grid on a modified domain~$\mathcal{X}$ with no initial measurement locations directly lying in a region of signal (scenario~2).
The software implementation of our approach communicates with the instrument control system NICOS (\url{nicos-controls.org}) which was configured on site.
For the experimental setting at the instrument, we refer to the Supplementary Information (Sec.~S2{}).
As mentioned, the benefit measure used for benchmarking, involving a known target intensity function, is not computable in this setting of a neutron experiment due to experimental artefacts like background and noise.
We therefore evaluate the results of our approach in a more qualitative way for this experiment.
Also, although the costs for moving the instrument axes contribute to the total experimental time, we do not consider them here for optimizing the objective function since they are approximately constant across the domain~$\mathcal{X}$.
For scenario~1, we adopt the setting from the original results~\cite[Fig.~1b]{li2014phonon} which, in our context, means to investigate intensities on~$\mathcal{X}=[0,2]\times[2,12.4]$ along the vector~$(1,1,0)$ in $\vec{Q}$~space with offset~$(0,0,3)$ and energy transfer.
Initially, we performed measurements in a conventional mode for reference, i.\,e.,~ we mapped~$\mathcal{X}$ with a grid of 11~columns containing 27~measurement points each (bottom row in Fig.~\ref{fig:eiger_scen1_compar}).
\begin{figure}
\centering
\includegraphics[width=\linewidth]{eiger_scen1_compar.pdf}
\caption{Results for scenario~1.
Each row displays the results of a certain approach.
The columns indicate the four stages (I-IV) of the grid approach (top row).
Rows~2-4 correspond to results of our approach in three different settings.
Triangles represent the initialization grid and dots show locations of intensity observations autonomously placed by our approach.
Row~2 corresponds to the default setting ($\gamma_0=58$ and~$\tau_0=94.5$ were computed automatically).
Manually changing the intensity threshold to~$\tau_1=130$ (with~$\gamma_1=\gamma_0$) leads to results depicted in row~3.
Row~4 shows results after changing the background level to $\gamma_2=30$ (with~$\tau_2=\tau_0$).
The bottom row shows the mapping in its conventional order for completeness.}
\label{fig:eiger_scen1_compar}
\end{figure}
These measurements were arranged in columns from bottom to top, i.\,e.,~ from low to high energies, and from left to right, i.\,e.,~ from small to large values for the coordinate along the vector in $\vec{Q}$~space.
A direct comparison with our approach would put this mapping mode in disadvantage since its total experimental time could have been spent more efficiently.
For this reason, the approach that we eventually take for comparison, the grid approach (top row in Fig.~\ref{fig:eiger_scen1_compar}), takes the measurement points from the mapping mode but changes their order into four stages I-IV (Extended Data Fig.~\ref{fig:base_grid_experiment_path}a-d).
The first stage is from bottom to top and from left to right but only consists of every other grid point on both axes.
The second stage is again from bottom to top but from right to left and also consists only of every other grid point on both axes.
The third and fourth stage are analogous but consist of the remaining grid points that were skipped in the first two stages.
With this order, the grid approach can observe intensities on each region of~$\mathcal{X}^*$ already after the first stage and hence can acquire more information in a faster way as with the conventional order.
We ran our approach in the default setting (Tab.~\ref{tab:parameter_values_default}), i.\,e.,~ with automatically computed background level and intensity threshold, as well as with two alternative pairs of corresponding parameter values manually set in order to study the robustness of the results.
In the default setting (row~2 in Fig.~\ref{fig:eiger_scen1_compar}), the background level and intensity threshold were computed to~$\gamma_0=58$ and~$\tau_0=94.5$, respectively.
To study the robustness of these results w.r.t.~ changes in the intensity threshold, the parameters for the first alternative (row~3 in Fig.~\ref{fig:eiger_scen1_compar}) were set to~$\gamma_1=\gamma_0$ and~$\tau_1=130$.
The second alternative (row~4 in Fig.~\ref{fig:eiger_scen1_compar}), for studying the robustness w.r.t.~ changes in the background level, was configured with~$\gamma_2=30$ and~$\tau_2=\tau_0$.
Eventually, the cost budget~$C$ for each instance of our approach was determined by the total experimental time of the grid approach which was $\sim11.5$~hours.
Each respective initialization with the same grid of 61~measurement points took $\sim2.9$~hours of experimental time.
The results displayed in Fig.~\ref{fig:eiger_scen1_compar} show that, after initialization, our approach places measurement points mainly in regions of signal in all three settings.
The grid approach, in contrast, has observed intensities at a considerable amount of uninformative measurement locations.
In scenario~2, we tried to challenge our approach with a more difficult initial setting.
The initialization grid in scenario~1 (triangles in Fig.~\ref{fig:eiger_scen1_compar}) indeed beneficially lies on an axis of symmetry of the intensity function.
Also, partly as a consequence, some initial measurements points are already located in regions of signal.
From a methodological perspective, it is interesting to investigate how our approach behaves after unfavorable initialization, i.\,e.,~ if almost all initial intensity observations are lying in the background and hence almost no useful initial information is available.
Hence, we reduced the number of initial measurement points from~61 to~25 and set~$\mathcal{X}=[0.2,2]\times[2,16]$ to break the axis of symmetry and measure larger energy transfers~(Fig.~\ref{fig:eiger_scen2}a).
\begin{figure}
\centering
\includegraphics[width=0.75\linewidth]{eiger_scen2.pdf}
\caption{Results for scenario~2.
a) Reduced number of initial measurement points (triangles) in a shifted and enlarged domain providing almost no initial information about the intensity function.
b) Intensity observations following the uninformative initialization (dots).}
\label{fig:eiger_scen2}
\end{figure}
Estimating the background level and intensity threshold is not reasonable in this setting since the amount of initial information is too little.
Therefore, we manually set~$\gamma=50$ and~$\tau=80$, respectively.
The cost budget~$C$ for our approach was not set explicitly here.
In fact, we stopped the experiment manually after $\sim7.3$~hours of total experimental time, with $\sim1.1$~hours spent on initialization with the modified grid.
The result is depicted in Fig.~\ref{fig:eiger_scen2}b.
It shows that, even in this challenging situation, our approach keeps making reasonable decisions on measurement locations.
\subsection*{Benchmark}
\addcontentsline{toc}{subsection}{Benchmark}
This section shows results of a benchmark on several two-dimensional (i.\,e.,~ $n=2$) test case intensity functions (Extended Data Fig.~\ref{fig:base_test_cases}) as a quantitative proof of concept.
The benchmarking procedure quantifies the performance of an approach by how much \textit{benefit} it is able to acquire for certain \textit{cost} budgets in the context of predefined \textit{test cases}.
We briefly describe its setting in the following paragraph.
For more details, we refer to the original work~\cite{teixeiraparente2022benchmarking}.
A test case mainly includes an intensity function defined on a certain set~$\mathcal{X}$ and a synthetic TAS used for moving on~$\mathcal{X}^*$ with certain velocities and observing intensities.
As mentioned, the cost measure is chosen to be the experimental time used, i.\,e.,~ the sum of cumulative counting time and cumulative time for moving instrument axes.
The benefit measure is defined by a relative weighted $L^2$~approximation error between the target intensity function and a linear interpolation~$\hat{i}=\hat{i}(\mathcal{A})$ using observed intensities from experiments~$\mathcal{A}$.
It encodes the fact that a TAS experimenter is more interested in regions of signal than in the background which suggests to use~$i$ itself as a weighting function.
However, an important constraint, which is controlled by a benchmark intensity threshold~${\tau^*}>0$ truncating weights to
\begin{equation}
i_{{\tau^*}}(\vec{x}) \coloneqq \min \lbrace i(\vec{x}),{\tau^*} \rbrace,
\end{equation}
as similarly described for the intensity threshold of our approach, is that separate regions of signal with different magnitudes of intensity might be equally interesting.
Formally, we define
\begin{equation}
\label{eq:def_benefit_measure}
\mu(\mathcal{A}) \coloneqq \frac{\norm{i-\hat{i}(\mathcal{A})}}{\norm{i}},
\end{equation}
where $\norm{\cdot} = \norm{\cdot}_{L^2(\mathcal{X}^*,\rho_{i,{\tau^*}})}$ for the weighting function
\begin{equation}
\rho_{i,{\tau^*}}(\vec{x}) \coloneqq \frac{i_{{\tau^*}}(\vec{x})}{\int_{\mathcal{X}^*} i_{{\tau^*}}(\vec{x}') \, \d{\vec{x}'}}.
\end{equation}
For each test case, benefits are measured for several ascending cost budgets, called \apos{milestone values}, to demonstrate the evolution of performance over time.
We note that this synthetic setting includes neither background nor measurement noise as both are artefacts of real neutron experiments.
We run the benchmarking procedure with three approaches for comparison:
\begin{enumerate}
\item random approach,
\item grid approach,
\item our approach with different values for the intensity threshold parameter~$\beta$.
\end{enumerate}
The random approach places intensity observations uniformly at random in~$\mathcal{X}^*$.
The grid approach is adopted from the section on the neutron experiment but using a square grid of dimension~$P\times P$, $P\in\mathbf{N}$.
For our approach, we set a zero background level, i.\,e.,~~$\gamma=0$, manually since background is not included in this synthetic setting.
Our approach is run with four variations of the intensity threshold parameter~$\beta$ (Eq.~\eqref{eq:intens_thresh_param}) in order to study corresponding sensitivities of the benchmark results.
The specific benchmarking procedure involves four milestone values according to the four stages (I-IV) of the grid approach, i.\,e.,~ the $m$-th milestone value represents the experimental time needed to complete stage~$m\in\lbrace\textnormal{I},\textnormal{II},\textnormal{III},\textnormal{IV}\rbrace$.
Observe that they depend on the particular test case.
The specific milestone values used are indicated in the Supplementary Information (Sec.~S4{}).
The number of columns/rows~$P$ for the grid approach is chosen to be the maximum number such that the corresponding experimental time for performing an experiment in the described order does not exceed $9$~hours.
Since both the random and our approach contain stochastic elements, we perform a number of 100~repeated runs with different random seeds for each test case in order to see the variability of their results.
The results for each test case along with its corresponding intensity function are shown in Fig.~\ref{fig:base_results}.
\begin{figure}
\centering
\includegraphics[width=\linewidth]{base_results.pdf}
\caption{Benchmark results.
For each approach, every subfigure plots the decay of a relative approximation error (Eq.~\eqref{eq:def_benefit_measure}) between the target intensity function of the corresponding test case (top right corners) and a linear interpolation of collected intensity observations for four milestone values which are determined by the four stages (I-IV) of the grid approach.
The solid lines show medians of the resulting benefit values, whereas the light color areas indicate the range between their minimum and maximum to visualize their variability caused by stochastic elements.}
\label{fig:base_results}
\end{figure}
Our approach has, on median average, performed significantly better than the random and grid approach in most test cases.
Furthermore, its mostly thin and congruent shapes of variability (light color areas) demonstrate its reproducibility and its robustness w.r.t.~ changes in the intensity threshold parameter~$\beta$.
Examples of particular experiments performed by our approach are additionally provided in Extended Data Fig.~\ref{fig:base_experiments_ariane} for each test case.
\section*{Discussion}
\addcontentsline{toc}{section}{Discussion}
The results of the neutron experiment demonstrate the benefits of our approach.
Indeed, in scenario~1 (Fig.~\ref{fig:eiger_scen1_compar}), our approach identifies regions of strong as well as of weak signal in each setting and even finds isolated relevant signals of small shape at the edge repeatedly.
Therefore, for this experimental setting, the results are shown to be robust w.r.t.~ changes in the estimated background level and intensity threshold, which we view as an important outcome of this experiment.
The choice of these parameters is however directly reflected in the placement of measurement points which indicates a certain aspect of interpretability and explainability~\cite{roscher2020explainable,belle2021principles} of our approach.
An intensity threshold higher in value namely leads to measurement points that are placed on a thinner branch of signal (row~3 in Fig.~\ref{fig:eiger_scen1_compar}), whereas as a lower background level yields more exploratory behaviour, with a risk to measure in regions of background from time to time (row~4 in Fig.~\ref{fig:eiger_scen1_compar}).
Additionally, note that a smaller intensity threshold results in measurement points also being placed in regions of high intensity gradient.
Furthermore, in each setting, our approach (rows~2-4 in Fig.~\ref{fig:eiger_scen1_compar}) has significantly fewer measurement points in the background compared to the grid approach (top row in Fig.~\ref{fig:eiger_scen1_compar}) as expected and thus uses experimental time more efficiently.
The grid approach additionally does not cover small signal regions at the edge.
A simulated intensity between the two regions of signal on the right~\cite[Fig.~1c]{li2014phonon} is not seen by our approach which is, however, in agreement with the original experiment~\cite[Fig.~1b]{li2014phonon}.
In situations like this, a human experimenter can focus on such details and place additional measurement points manually, if necessary.
When applying GPR, we use a common Gaussian kernel as detailed in the Methods section.
The (logarithm of the) intensity function from Fig.~\ref{fig:eiger_scen1_compar} however violates the stationarity of this kernel.
Indeed, using a stationary kernel assumes homogeneous statistical behaviour of the intensity function across the entire domain~$\mathcal{X}$ which is not the case for our particular scenario.
The length scale along the $E$ axis, for instance, is differing for different values on the $\vec{Q}$~axis.
In the middle of the $\vec{Q}$~axis, the length scale is certainly larger than near its edges.
Note that the length scale along the $\vec{Q}$~axis is also non-stationary.
These discrepancies are one of the main reasons for our choice of the material SnTe and the setting mentioned above since it provides an opportunity to demonstrate that a stationary kernel, which is computationally substantially cheaper than non-stationary kernels, is sufficient for identifying regions of signal and hence for performing efficient experiments.
In scenario~2 (Fig.~\ref{fig:eiger_scen2}), we challenged our approach with a difficult setting.
Although the initialization grid and the domain were organized such that no initial measurement point is located in a region of signal, and hence the approach is initialized with little useful information, its behaviour stays reasonable.
It namely keeps placing measurements in regions of signal and can still identify the small region with strong signal on the bottom right.
However, the signal region in the middle of the domain is not fully identified, which can be explained by the relatively short experimental time ($7.3$~hours) as well as by the stationarity of the kernel since the sparse data suggest a short length scale along the $E$~axis leading to the assumption of lower function values in an actual region of signal.
Note that the reduced amount of initial measurement points in scenario~2 is only applied for the purpose of challenging our approach and a demonstration of its robustness.
In productive operation, however, we stay with the larger initialization grid from scenario~1 since placing some measurements in regions of background for a proper determination of the same is a valuable step during an experiment providing relevant information for the experimenter.
Fortunately, this mostly leads to a sufficiently good initialization of our approach, despite using a common stationary kernel.
For further results of neutron experiments in the setting from scenario~1 as well as from an additional scenario (scenario~3) that investigated another phonon of SnTe in the default setting of our approach, we refer to the Supplementary Information (Sec.~S3{}).
The benchmark results (Fig.~\ref{fig:base_results}), as a proof of concept, confirm the outcomes of the neutron experiment quantitatively in a synthetic setting.
Our approach shows a better performance, measured by a relative weighted approximation error (Eq.~\eqref{eq:def_benefit_measure}) between a target intensity function and a linear interpolation of collected intensity observations, compared to the random and grid approach for all test cases.
The improvements are especially substantial for target intensity functions with smaller regions of signal (Fig.~\ref{fig:base_results}a-q).
For intensity functions that cover a substantial part of the domain~$\mathcal{X}$ (Fig.~\ref{fig:base_results}r-t), the competing approaches are also able to place intensity observations in regions of signal early during the experiment and hence it is difficult for any approach to demonstrate its benefit in these scenarios.
The variability in the results of our approach containing stochastic elements, quantified by the median of benefit values and the range between their minimum and maximum, is shown to be small for most test cases and acceptable for the remainder.
Our approach can thus be considered reliable with reproducible results despite starting with a different sequence of pseudo-random numbers for each run.
Moreover, the benchmark results indicate that our approach is robust w.r.t.~ changes in the intensity threshold.
The four variations of the corresponding controlling parameter~$\beta$ (Eq.~\eqref{eq:intens_thresh_param}) yield similar results for most test cases.
It should be noted, however, that the use of a reasonable value for~$\beta$ in real neutron experiments is nevertheless important.
Indeed, it not only eliminates the effect of outliers in initial intensities but also, recall, allows to control the width of signal branches on which the measurement points are placed, and thus the extent to which regions of high gradient are preferred over those of peak intensity.
Finally, in addition to the results of the neutron experiment, the particular experiments performed by our approach in the benchmark setting (Extended Data Fig.~\ref{fig:base_experiments_ariane}) confirm that it, after initialization, autonomously places the majority of measurements in regions of signal for a variety of different intensity distributions.
\section*{Conclusions}
\addcontentsline{toc}{section}{Conclusions}
In the previous sections, we demonstrated that our approach indeed improves the efficiency of TAS experiments and allows to make better use of the experimenter's time.
It maximizes an acquisition function given by approximation uncertainties of a log-Gaussian process in order to find informative measurement points and not waste experimental time in regions such as the background.
Our robust and reproducible results suggest that it is in fact capable of autonomously obtaining a rapid overview of intensity distributions in various settings.
In a real neutron experiment at the thermal TAS EIGER, our approach repeatedly demonstrated its ability to identify regions of signal successfully leading to a more efficient use of beam time at a neutron source of a large-scale research facility.
It was additionally shown that it keeps making reasonable decisions even when being initialized with little information.
Furthermore, substantial performance improvements, in comparison with two competing approaches, and their robustness were quantified in a synthetic benchmark setting for numerous test cases.
Nevertheless, we feel that the automated estimation of algorithmic parameters (background level and intensity threshold) from statistics of initial measurements, despite its good performance in each of the settings discussed, needs to be confirmed in future experiments.
Our approach, so far, solves the problem of \textit{where} to measure.
However, an interesting topic of future research is the major question of \textit{how} to measure at a certain location.
Counting times were assumed to be constant in this work and thus promise to be another possibility to save experimental time if determined autonomously in an advantageous way.
Indeed, large counting times in regions of high intensities and background are actually not needed since the necessary information can also be collected in less time.
Experimenters, in contrast, are often more interested in weaker signals and their comprehensive measurement to reduce corresponding error bars.
Finally, although using a common stationary kernel for GPR has proven to be sufficient for identifying regions of signal, we regard the application of non-stationary kernels with input-dependent hyperparameters~\cite{paciorek2003nonstationary,plagemann2008nonstationary,heinonen2016non,tolvanen2014expectation}, e.\,g.,~ length scales, also modelled by log-Gaussian processes, as another interesting option for future research.
\section*{Methods}
\addcontentsline{toc}{section}{Methods}
Our approach is methodologically based on \textit{Gaussian Process Regression} (GPR)~\cite{rasmussen2006gaussian}, a Bayesian technique for estimating and approximating functions from pointwise data that allows to quantify uncertainties on the approximation itself in the form of normal distributions.
We fit a Gaussian process to logarithmic intensity observations and exponentiate the resulting posterior process yielding a \textit{log-Gaussian process}.
As mentioned, this transformation causes approximation uncertainties to be higher in regions of signal which in turn can be exploited for the definition of a useful acquisition function.
\subsection*{Gaussian Process Regression}
\addcontentsline{toc}{subsection}{Gaussian Process Regression}
We generally intend to approximate a function of interest~$f:\mathcal{X}\to\mathbf{R}$, which becomes the logarithm of the intensity function later.
Using GPR for this, we have to assume that~$f$ is a realization of a Gaussian process~$F$.
\begin{definition}[Gaussian process]
A \textit{Gaussian process}
\begin{equation}
F \sim \mathcal{GP}(m(\vec{x}),\kappa_\theta(\vec{x},\vec{x}'))
\end{equation}
with (prior) mean function $m:\mathcal{X}\to\mathbf{R}$ and parameterized kernel function $\kappa_\theta:\mathcal{X}\times\mathcal{X}\to[0,\infty)$ is a collection of random variables~$(F(\vec{x}))_{\vec{x}\in\mathcal{X}}$ such that for any finite amount of evaluation points~$\vec{x}^{(\ell)}\in\mathcal{X}$, $\ell=1,\ldots,L$, $L\in\mathbf{N}$, the random variables~$F^{(\ell)} \coloneqq F(\vec{x}^{(\ell)})$ follow a joint normal distribution, i.\,e.,~
\begin{equation}
(F^{(1)}, \ldots, F^{(L)})^\top \sim \Normdist{\vec{m}}{\mathcal{K}},
\end{equation}
where
\begin{equation}
\vec{m} = (m(\vec{x}^{(1)}),\ldots,m(\vec{x}^{(L)}))^\top \in \mathbf{R}^L \quad\text{and}\quad \mathcal{K}_{\ell_1,\ell_2} = \kappa_\theta(\vec{x}^{(\ell_1)},\vec{x}^{(\ell_2)}) \geq 0.
\end{equation}
\end{definition}
Note that, for each $\vec{x}\in\mathcal{X}$, it particularly holds that
\begin{equation}
F(\vec{x}) \sim \Normdist{m(\vec{x})}{\sigma^2(\vec{x})},
\end{equation}
where $\sigma^2(\vec{x}) \coloneqq \Var{F(\vec{x})} = \kappa_\theta(\vec{x},\vec{x})\geq0$ is the (prior) variance function.
The kernel function~$\kappa_\theta$ is an important component in GPR since it describes the correlation structure between random variables~$F(\vec{x})$.
Hence, it enables to include assumptions on realizations of~$F$ and determines function properties like regularity, periodicity, symmetry, etc.
In practice, it is crucial to choose a kernel function that matches with properties of the function of interest~$f$.
We acquire knowledge on~$f$ through noisy observations~$\hat{f}_j$ at locations~$\vec{x}_j\in\mathcal{X}^*$ (Eq.~\eqref{eq:def_experim}) in our context.
Therefore, for $j=1,\ldots,J$, we set
\begin{equation}
\label{eq:def_GP_obs}
\hat{F}(\vec{x}_j) = F(\vec{x}_j) + e(\vec{x}_j)\eta_j,
\end{equation}
where $\eta_j\sim\Normdist{0}{1}$ are \textnormal{i.\,i.\,d.}~ random variables independent of~$F(\vec{x}_j)$ and $e(\vec{x})>0$ denotes the noise standard deviation at~$\vec{x}\in\mathcal{X}$.
Note that~$\hat{F}(\vec{x}_j)$ is a normally distributed random variable.
For clear notation, we define
\begin{equation}
X_J \coloneqq
\begin{pmatrix}
| & & | \\
\vec{x}_1 & \cdots & \vec{x}_J \\
| & & |
\end{pmatrix} \in \mathbf{R}^{n\times J}
\quad\textnormal{and}\quad
\hat{\vec{f}}_J \coloneqq (\hat{f}_1,\ldots,\hat{f}_J)^\top \in \mathbf{R}^J,
\end{equation}
and let
\begin{equation}
h(X_J) \coloneqq (h(\vec{x}_1),\ldots,h(\vec{x}_J))^\top \in \mathbf{R}^J
\end{equation}
for any function~$h:\mathcal{X}\to\mathbf{R}$.
After observations have been made, we are interested in the posterior, i.\,e.,~ conditional, Gaussian process.
It holds that
\begin{equation}
F(\vec{x}) \| (\hat{F}(X_J)=\hat{\vec{f}}_J) \sim \Normdist{m_J(\vec{x})}{\sigma_J^2(\vec{x})}
\end{equation}
with the posterior mean function
\begin{equation}
\label{eq:post_mean_fct}
m_J(\vec{x}) = m(\vec{x}) + \kappa_J(\vec{x},X_J)^\top \left[ \kappa_J(X_J,X_J) + \diag{e(X_J)}^2 \right]^{-1} (\hat{\vec{f}}_J-m(X_J))
\end{equation}
and the posterior variance function
\begin{equation}
\label{eq:post_var_fct}
\sigma_J^2(\vec{x}) = \sigma^2(\vec{x}) - \kappa_J(\vec{x},X_J)^\top \left[ \kappa_J(X_J,X_J) + \diag{e(X_J)}^2 \right]^{-1} \kappa_J(\vec{x},X_J).
\end{equation}
If necessary, the hyperparameters~$\theta_J$ of the kernel function~$\kappa_J\coloneqq \kappa_{\theta_J}$ can be optimized using data~$\hat{\vec{f}}_J$.
For this, we compute~$\theta_J$ such that the logarithm of the so-called marginal likelihood, i.\,e.,~
\begin{equation}
\label{eq:marginal_likelihood}
\begin{split}
\log \rho_{\hat{F}(X_J)}(\hat{\vec{f}}_J) &= \log \left( \int_{\mathbf{R}^J} \rho_{\hat{F}(X_J)|F(X_J)}(\hat{\vec{f}}_J|\vec{f}) \, \rho_{F(X_J)}(\vec{f}) \; \d{\vec{f}} \right) \\
&= -\frac{1}{2} \, (\hat{\vec{f}}_J-m(X_J))^\top\left[ \kappa_J(X_J,X_J) + \diag{e(X_J)}^2 \right]^{-1} (\hat{\vec{f}}_J-m(X_J)) \\
&\qquad - \frac{1}{2} \log \left\lvert \kappa_J(X_J,X_J) + \diag{e(X_J)}^2 \right\rvert - \frac{n}{2} \log 2\pi,
\end{split}
\end{equation}
is maximized.
A suitable optimizer is specified below.
Note that the analytical expression for the integral in Eq.~\eqref{eq:marginal_likelihood} is only feasible due to the Gaussian distributions involved.
However, the computational cost of GPR is often hidden in this kernel optimization step since it requires solving linear systems and computing determinants \cite[Sec.~2.3]{rasmussen2006gaussian}.
An appropriate criterion for stopping kernel optimizations that detects stagnant hyperparameters during an experiment is therefore provided below.
Furthermore, observe that, for a fixed non-optimized kernel, $\sigma_J^2$ does not depend on observations~$\hat{\vec{f}}_J$ but only on locations~$X_J$ they have been made at.
The posterior mean function~$m_J:\mathcal{X}\to\mathbf{R}$, incorporating knowledge on $J$~noisy observations of the function of interest~$f$, can now be used as an approximation to~$f$, whereas the posterior variance function~$\sigma_J^2:\mathcal{X}\to[0,\infty)$ quantifies the corresponding uncertainties (Extended Data Fig.~\ref{fig:gpr_demo}).
Note that $\sigma_J^2(\vec{x})<\sigma^2(\vec{x})$ for each~$\vec{x}\in\mathcal{X}$ since $\left[ \kappa_J(X_J,X_J) + \diag{e(X_J)}^2 \right]^{-1}$ is positive-definite.
Since we have $m\equiv0$ later, we can further simplify Eq.~\eqref{eq:post_mean_fct} to
\begin{equation}
m_J(\vec{x}) = \kappa_J(\vec{x},X_J)^\top \left[ \kappa_J(X_J,X_J) + \diag{e(X_J)}^2 \right]^{-1} \hat{\vec{f}}_J.
\end{equation}
\subsection*{Log-Gaussian processes}
\addcontentsline{toc}{subsection}{Log-Gaussian processes}
The Gaussian process, which is fitted to logarithmic intensity observations in our approach, is exponentiated to the original linear scale in this section to become \textit{log-Gaussian}.
It is this transformation that directly yields a useful acquisition function.
Before we describe the details, we need to define a \textit{log-normal} random variable and prove a technical detail that becomes important below.
\begin{definition}[Log-normal distribution~\cite{mathworld2021lognormal}]
Let $\eta\sim\Normdist{0}{1}$.
Then, for parameters~$\mu\in\mathbf{R}$ and~$\sigma>0$, the random variable
\begin{equation}
\label{eq:def_Z}
Z = \exp(\mu+\sigma\eta)
\end{equation}
is said to follow a \textit{log-normal distribution}, denoted by $Z\sim\log$-$\Normdist{\mu}{\sigma^2}$.
\end{definition}
The mean and variance of a log-normal random variable~$Z$ are given by
\begin{equation}
\E{Z} = \exp\left( \mu+\frac{\sigma^2}{2} \right) \quad\text{and}\quad \Var{Z} = (\exp(\sigma^2)-1) \cdot \exp(2\mu+\sigma^2).
\end{equation}
Below, we look at the noise distribution of log-Gaussian processes in order to satisfy our assumption on intensity observations to contain normally distributed noise (Eq.~\eqref{eq:intensity_noise_physics}).
The following result is fundamental for this and proved in the Supplementary Information (Sec.~S5{}).
It investigates the behaviour of log-normal random variables in the small variance limit.
\begin{proposition}[Small variance limit of normalized log-normal random variables]
Let~$Z\sim\log$-$\Normdist{\mu}{\sigma^2}$ as in Eq.~\eqref{eq:def_Z} and define the corresponding normalized random variable
\begin{equation}
\overline{Z} \coloneqq \frac{Z-\E{Z}}{\sqrt{\Var{Z}}}.
\end{equation}
Then, the random variable~$\overline{Z}$ converges pointwise to~$\eta$ as $\sigma\to0^+$, i.\,e.,~
\begin{equation}
\label{eq:lognorm_conv}
\overline{Z}(\omega) \rightarrow \eta(\omega) \;\;\; \textnormal{as $\sigma\to0^+$}
\end{equation}
for each $\omega\in\Omega$, where~$\Omega$ denotes the sample space.
\end{proposition}
Our result for single log-normal random variables gives pointwise convergence.
However, convergence in distribution can also be proven for sums of log-normal random variables~\cite{dufresne2004log}.
As mentioned, the exponentiation of a Gaussian process is log-Gaussian, by definition.
\begin{definition}[Log-Gaussian process]
Let~$F$ be a Gaussian process.
Then, the random process~$(I(\vec{x}))_{\vec{x}\in\mathcal{X}}$ with
\begin{equation}
\label{eq:I_exp_F}
I(\vec{x}) = \exp(F(\vec{x}))
\end{equation}
is called a \textit{log-Gaussian process}.
\end{definition}
Using Eqs.~\eqref{eq:post_mean_fct} and~\eqref{eq:post_var_fct}, it immediately follows for the posterior log-Gaussian process that
\begin{equation}
I(\vec{x}) \| (\hat{F}(X_J)=\hat{\vec{f}}_J) \sim \text{$\log$-}\Normdist{m_J(\vec{x})}{\sigma_J^2(\vec{x})}.
\end{equation}
In particular, its posterior mean function is
\begin{equation}
\E{I(\vec{x}) \| \hat{F}(X_J)=\hat{\vec{f}}_J} = \exp\left( m_J(\vec{x}) + \frac{\sigma_J^2(\vec{x})}{2} \right)
\end{equation}
and its posterior variance function becomes
\begin{equation}
\label{eq:post_var_fct_loggp}
\Var{I(\vec{x}) \| \hat{F}(X_J)=\hat{\vec{f}}_J} = (\exp(\sigma_J^2(\vec{x}))-1) \cdot \exp(2m_J(\vec{x})+\sigma_J^2(\vec{x})).
\end{equation}
\subsection*{Application to TAS}
\addcontentsline{toc}{subsection}{Application to TAS}
In the context of our methodology from the previous sections, we choose the function of interest to be the logarithm of the intensity function from the TAS setting (Supplementary Information, Sec.~S1{}), i.\,e.,~
\begin{equation}
f = \log i.
\end{equation}
If we define
\begin{equation}
\label{eq:def_I_obs}
\hat{F} \eqqcolon \log\hat{I},
\end{equation}
relating to Eq.~\eqref{eq:I_exp_F}, i.\,e.,~ $F = \log I$, Eq.~\eqref{eq:def_GP_obs} gives
\begin{equation}
\label{eq:I_obs_noise}
\begin{alignedat}{3}
&& \log \hat{I}(\vec{x}_j) &= \log I(\vec{x}_j) + e(\vec{x}_j)\eta_j \\
\Leftrightarrow \; && \hat{I}(\vec{x}_j) &= I(\vec{x}_j) \cdot \exp(e(\vec{x}_j)\eta_j)
\end{alignedat}
\end{equation}
for measurement locations~$\vec{x}_j\in\mathcal{X}^*$.
Note that, in contrast to the process~$\hat{F}$ containing additive normally distributed noise (Eq.~\eqref{eq:def_GP_obs}), the noise of~$\hat{I}$ is multiplicative and log-normal, i.\,e.,~
\begin{equation}
\hat{I}(\vec{x}_j) \| (I(\vec{x}_j)=i(\vec{x}_j)) = i(\vec{x}_j) \cdot \exp(e(\vec{x}_j)\eta_j).
\end{equation}
However, referring to Eq.~\eqref{eq:intensity_noise_physics}, the physical assumption on the noise of intensity observations~$I^+$ is to be additive and normally distributed, i.\,e.,~
\begin{equation}
I^+(\vec{x}_j) \| (I(\vec{x}_j)=i(\vec{x}_j)) = i(\vec{x}_j) + e^+(\vec{x}_j)\eta^+_j,
\end{equation}
where $\eta^+_j\sim\Normdist{0}{1}$ and $e^+(\vec{x}_j) = \sqrt{i(\vec{x}_j)}$.
Fortunately, the actual deviation of the two different noise distributions is negligible for sufficiently large intensities~$i(\vec{x}_j)$ as the following explanations demonstrate.
As a first step, let us determine $e(\vec{x}_j)$ from Eq.~\eqref{eq:I_obs_noise} such that the variances of both noise distributions are equal, i.\,e.,~
\begin{equation}
\begin{alignedat}{3}
&& \Var{\hat{I}(\vec{x}_j) \| I(\vec{x}_j)=i(\vec{x}_j)} &= \Var{I^+(\vec{x}_j) \| I(\vec{x}_j)=i(\vec{x}_j)} \\
\Leftrightarrow && \; i(\vec{x}_j)^2 \cdot (\exp(e(\vec{x}_j)^2)-1) \cdot \exp(e(\vec{x}_j)^2) &= i(\vec{x}_j),
\end{alignedat}
\end{equation}
which yields
\begin{equation}
\label{eq:def_e_x_log}
e(\vec{x}_j)^2 = \log\left( \frac{1}{2} \left( \sqrt{4/i(\vec{x}_j) + 1} + 1 \right) \right).
\end{equation}
Note that the intensity term~$i(\vec{x}_j)$ in Eq.~\eqref{eq:def_e_x_log} is not known in practice but can be replaced by the corresponding observation~$\hat{i}_j\approx i(\vec{x}_j)$.
Since we aim to apply the small variance limit for log-normal random variables from above (Eq.~\eqref{eq:lognorm_conv}), we set
\begin{equation}
Z = \hat{I}(\vec{x}_j) \| (I(\vec{x}_j)=i(\vec{x}_j)) \sim \text{$\log$-$\Normdist{\mu}{\sigma^2}$}
\end{equation}
with
\begin{equation}
\mu = \log i(\vec{x}_j) \quad\text{and}\quad \sigma^2 = \log\left( \frac{1}{2} \left( \sqrt{4/i(\vec{x}_j) + 1} + 1 \right) \right).
\end{equation}
Observe that
\begin{equation}
\label{eq:sigma_to_0+}
\sigma \to 0^+ \;\;\; \textnormal{as $i(\vec{x}_j) \to \infty$}.
\end{equation}
Furthermore, note that the term~$\mu$ also depends on the limit~$i(\vec{x}_j)\to\infty$ and hence on~$\sigma\to0^+$, in contrast to the setting above, but disappears in the calculations due to cancellations (see proof in Supplementary Information, Sec.~S5{}).
Using the small variance limit, we immediately get the following result (Extended Data Fig.~\ref{fig:lgp_noise}).
\begin{proposition}
The normalization of the noise random variable~$\hat{I}(\vec{x}_j) \| (I(\vec{x}_j)=i(\vec{x}_j))$ converges pointwise to~$\eta$ as $i(\vec{x}_j)\to\infty$.
In particular, it converges in distribution to a standard normal distribution.
\end{proposition}
If we set
\begin{equation}
\hat{\i}_J \coloneqq (\hat{i}_1,\ldots,\hat{i}_J)^\top \in [0,\infty)^J,
\end{equation}
the acquisition function~$\textnormal{acq}_J$ of our approach can now be defined as
\begin{equation}
\label{eq:def_acq_fct}
\textnormal{acq}_J(\vec{x}) \coloneqq \sqrt{\Var{I(\vec{x}) \| \hat{I}(X_J)=\hat{\i}_J}},
\end{equation}
which eventually gives Eq.~\eqref{eq:def_acq_fct_simple} through Eq.~\eqref{eq:post_var_fct_loggp}.
\subsection*{Consumed areas}
\addcontentsline{toc}{subsection}{Consumed areas}
As placing measurement points too close to each other has limited benefit for an efficient experiment, we mark areas around each measurement location~$\vec{x}_j\in\mathcal{X}^*$ as \apos{consumed} and ignore them as potential locations for new measurement points.
Since the resolution function of a TAS yields ellipsoids in $\vec{Q}$-$E$~space, it is natural to consider consumed areas in~$\mathcal{X}^*$ as ellipses.
An ellipse with center point~$\vec{x}_j\in\mathcal{X}^*$ is defined as
\begin{equation}
\label{eq:ellipse}
\mathcal{E}(\vec{x}_j) \coloneqq \lbrace \vec{x}\in\mathbf{R}^n \| (\vec{x}-\vec{x}_j)^\top E(\vec{x}_j) (\vec{x}-\vec{x}_j) \leq 1 \rbrace
\end{equation}
for a matrix-valued function
\begin{equation}
\label{eq:ellipse_matrix_function}
E(\vec{x}_j) = U(\vec{x}_j)R(\vec{x}_j)^{-\top}R(\vec{x}_j)^{-1} U(\vec{x}_j)^\top = U(\vec{x}_j)R(\vec{x}_j)^{-2}U(\vec{x}_j)^\top \in \mathbf{R}^{n\times n},
\end{equation}
where~$U(\vec{x}_j)\in\mathbf{R}^{n\times n}$ is a rotation matrix and $R(\vec{x}_j)=\diag{r_1(\vec{x}_j),\ldots,r_n(\vec{x}_j)}\in\mathbf{R}^{n\times n}$ with $r_k>0$.
Then, the union of all ellipses at step~$J$ is denoted by
\begin{equation}
\mathcal{E}_J \coloneqq \bigcup_{j=1}^J \mathcal{E}(\vec{x}_j).
\end{equation}
It is included in the objective function~$\phi_J$ which is part of the final algorithm.
\subsection*{Final algorithm}
\addcontentsline{toc}{subsection}{Final algorithm}
Incorporating all of the discussed methodological components, the steps for the final algorithm are listed in Alg.~\ref{alg:final}.
\begin{algorithm}
\caption{Final algorithm}
\label{alg:final}
\begin{algorithmic}[1]
\Require initial measurement locations $\vec{x}_1,\ldots,\vec{x}_{J_0}\in\mathcal{X}^*$,
function~$\gamma=\gamma(\mathcal{A}_0)$ for computing background level,
function~$\tau=\tau(\mathcal{A}_0)$ for computing intensity threshold,
kernel~$\kappa_\theta$ and optimizer for hyperparameters~$\theta$ (Eq.~\eqref{eq:marginal_likelihood}),
cost measure~$c=c(\mathcal{A})$,
cost budget~$C\geq0$,
matrix-valued function~$E(\vec{x})$ for elliptic consumed areas (Eq.~\eqref{eq:ellipse_matrix_function}),
objective function~$\phi_J$,
stopping criterion for kernel optimizations~$P_{\textnormal{KO}}=P_{\textnormal{KO}}(J)$
\State Observe noisy intensities~$\hat{i}_1,\ldots,\hat{i}_{J_0}$ at initial locations~$\vec{x}_1,\ldots,\vec{x}_{J_0}$
\State $\mathcal{A}_0=((\vec{x}_1,\hat{i}_1), \ldots, (\vec{x}_{J_0},\hat{i}_{J_0}))$
\State Optimize hyperparameters~$\theta_{J_0}$ of kernel~$\kappa_{J_0}$
\State Compute background level~$\gamma=\gamma(\mathcal{A}_0)$ and intensity threshold~$\tau=\tau(\mathcal{A}_0)>\gamma$
\State $\mathcal{A}=\mathcal{A}_0$
\State $J=J_0$
\While{$c(\mathcal{A}) < C$}
\State Determine next measurement location~$\vec{x}_{J+1} \coloneqq \argmax_{\vec{x}\in\mathcal{X}^*} \phi_J(\vec{x})$
\State Observe noisy intensity~$\hat{i}_{J+1}$ at~$\vec{x}_{J+1}$
\If{\textit{kernel optimizations stopped}}
\State $\theta_{J+1} = \theta_J$
\Else
\State Optimize hyperparameters~$\theta_{J+1}$ of kernel~$\kappa_{J+1}$
\If{$P_{\textnormal{KO}}(J+1)$}
\State Stop kernel optimizations
\EndIf
\EndIf
\State $\mathcal{A} \gets ((\vec{x}_1,\hat{i}_1), \ldots, (\vec{x}_J,\hat{i}_J), (\vec{x}_{J+1},\hat{i}_{J+1}))$
\State $J \gets J+1$
\EndWhile
\end{algorithmic}
\end{algorithm}
Its requirements are specified in the next paragraphs.
The values used for required parameters are specified below.
The initial measurement locations~$\vec{x}_1,\ldots,\vec{x}_{J_0}\in\mathcal{X}^*$ are deterministically arranged as a certain grid.
It is chosen to be a variant of a regular grid in which every other row (or column) of points is shifted to the center of surrounding points (Extended Data Fig.~\ref{fig:init_locs}).
The intensity observations start in the bottom left corner of~$\mathcal{X}^*$ and then continue row by row.
Initial locations not reachable by the instrument, i.\,e.,~ outside of~$\mathcal{X}^*$, are skipped.
If all initial locations are reachable, we use a total number of
\begin{equation}
\label{eq:def_J0}
J_0 \coloneqq \frac{N_{\textnormal{row}}^2+1}{2},
\end{equation}
where $N_{\textnormal{row}}\in\mathbf{N}$ is the odd number of rows in the grid.
For computing the background level~$\gamma$, we divide the initial intensity observations sorted in ascending order into 10~buckets
\begin{equation}
B_l \coloneqq \lbrace \hat{i}_j \| D_{l-1} < \hat{i}_j \leq D_l, j = 1,\ldots,J_0 \rbrace, \quad l=1,\ldots,10,
\end{equation}
where $D_l$, $l=1,\ldots,9$, denotes the $l$-th decile of initial intensity observations, $D_0\coloneqq-\infty$, and~$D_{10}\coloneqq+\infty$.
The relative and absolute differences of the bucket medians, i.\,e.,~
\begin{equation}
\Delta^\textnormal{rel}_l \coloneqq \frac{m_{l+1}-m_l}{m_l} \quad\textnormal{and}\quad \Delta^\textnormal{abs}_l \coloneqq m_{l+1}-m_l
\end{equation}
with $m_l \coloneqq \median(B_l)$, $l=1,\ldots,9$, are taken to select the first bucket median which has a sufficiently large (relative and absolute) difference to its successor provided the corresponding decile does not exceed a maximum decile.
That is, we define
\begin{equation}
l^* \coloneqq \min\left( \left\lbrace l\in\lbrace 1,\ldots,9 \rbrace \| \Delta^\textnormal{rel}_l > \Delta^{\textnormal{rel}}_{\max} \;\wedge\; \Delta^\textnormal{abs}_l \geq \Delta^{\textnormal{abs}}_{\min} \right\rbrace \cup \lbrace l_{\max} \rbrace \right)
\end{equation}
for parameters $\Delta^{\textnormal{rel}}_{\max}>0$, $\Delta^{\textnormal{abs}}_{\min}>0$, and $l_{\max}\in\lbrace 1,\ldots,9 \rbrace$ and set the function for computing the background level to
\begin{equation}
\label{eq:backgr_level_parameter}
\gamma(\mathcal{A}_0) \coloneqq m_{l^*}
\end{equation}
The function for computing the intensity threshold is already given above (Eq.~\eqref{eq:intens_thresh_param}).
It is selected as a value between the background level~$\gamma$ and the maximum bucket median~$m_{10}$ on a linear scale by the parameter~$\beta$.
The cost measure~$c$ is chosen to represent experimental time, i.\,e.,~ the total time needed to carry out an experiment~$\mathcal{A}$.
Experimental time consists of the cumulative counting time and the cumulative time for moving the instrument axes.
The cumulative counting time is measured by
\begin{equation}
\label{eq:cost_count}
c_{\textnormal{count}}(\mathcal{A}) \coloneqq \sum_{j=1}^{\abs{\mathcal{A}}} {T_{\textnormal{count}}}_{,j}
\end{equation}
where~${T_{\textnormal{count}}}_{,j}\geq0$ denotes the single counting time, i.\,e.,~ the time spent for a single intensity observation, at~$\vec{x}_j$.
The cost measure for the cumulative time spent to move the instrument axes is defined as
\begin{equation}
\label{eq:cost_axes_mov}
c_{\textnormal{axes}}(\mathcal{A}) \coloneqq \sum_{j=1}^{\abs{\mathcal{A}}-1} d(\vec{x}_j,\vec{x}_{j+1}),
\end{equation}
where~$d$ is a metric representing the cost for moving from~$\vec{x}_j$ to~$\vec{x}_{j+1}$.
For details, we refer to Supplementary Information (Sec.~S1{}).
Eventually, we set
\begin{equation}
\label{eq:cost_measure}
c(\mathcal{A}) \coloneqq c_{\textnormal{count}}(\mathcal{A}) + c_{\textnormal{axes}}(\mathcal{A}).
\end{equation}
For simplicity, the single counting times are assumed to be constant on the entire domain~$\mathcal{X}^*$, i.\,e.,~ ${T_{\textnormal{count}}}_{,j} = T_{\textnormal{count}} \geq 0$ yielding
\begin{equation}
c_{\textnormal{count}}(\mathcal{A}) = \abs{\mathcal{A}} \cdot T_{\textnormal{count}}.
\end{equation}
Although NICOS is able to consider an instrument's resolution function for the computation of ellipse matrices~$E(\vec{x}_j)$ from Eq.~\eqref{eq:ellipse_matrix_function}, we decided to use ellipses fixed over~$\mathcal{X}^*$ for both, the neutron experiment and the benchmark.
The details for the matrix-valued function~$E$ are given below.
The parameterized kernel~$\kappa_\theta$ is chosen to be the Gaussian (or radial basis function, RBF) kernel, i.\,e.,~
\begin{equation}
\label{eq:gauss_kernel}
\kappa_\theta(\vec{x},\vec{x}') \coloneqq \sigma^2 \exp\left( -\frac{1}{2} (\vec{x}-\vec{x}')^\top \Lambda^{-1} (\vec{x}-\vec{x}') \right),
\end{equation}
where~$\sigma^2>0$ and $\Lambda=\diag{\lambda_1,\ldots,\lambda_n}\in\mathbf{R}^{n\times n}$ for length scales~$\lambda_k\geq0$.
Hence, the vector of hyperparameters is
\begin{equation}
\label{eq:vec_kernel_hyperp}
\theta=(\sigma^2,\lambda_1,\ldots,\lambda_n)^\top\in\mathbf{R}^{n+1}.
\end{equation}
Recall that the kernel is needed for GPR to fit a Gaussian process to the logarithm of intensity observations.
As mentioned above, we compute optimal hyperparameters by maximizing the logarithm of the marginal likelihood (Eq.~\eqref{eq:marginal_likelihood}).
Since this optimization problem is non-convex in general, it might have several local maxima.
Instead of a global optimizer, we run local optimizations starting with $N_{\cubeHyperparam}\in\mathbf{N}$~different initial hyperparameter values distributed uniformly at random in a hypercube~$\mathcal{H}\subseteq\mathbf{R}^{n+1}$ and choose the one with the largest local maxima.
Note that this introduces stochasticity into our approach.
Recall that the objective function~$\phi_J$, which is supposed to indicate the next measurement location, is composed of the acquisition function~$\textnormal{acq}_J$ from Eq.~\eqref{eq:def_acq_fct} and the cost function~$c_J$ from Eq.~\eqref{eq:def_cost_fct}.
In order to avoid distorting the objective function with physical units of time, we use the normalized cost function
\begin{equation}
\label{eq:cost_function_normalized}
\overline{c}_J(\vec{x}) \coloneqq \frac{c_J(\vec{x})}{c_0},
\end{equation}
where~$c_0>0$ is a normalizing cost value and set to the maximum distance between two initial measurement locations w.r.t.~ the metric~$d$, i.\,e.,~
\begin{equation}
c_0 \coloneqq \max_{1\leq j,j'\leq J_0} d(\vec{x}_j,\vec{x}_{j'}).
\end{equation}
The objective function is then defined as
\begin{equation}
\phi_J(\vec{x}) \coloneqq
\begin{cases}
\textnormal{acq}_J(\vec{x}) / (\overline{c}_J(\vec{x}) + 1) & \textnormal{if $\vec{x}\not\in\mathcal{E}_J$}, \\
-1 & \textnormal{otherwise}.
\end{cases}
\end{equation}
Observe that~$\phi_J$ excludes consumed areas in~$\mathcal{E}_J$ as potential locations for new observations.
Outside~$\mathcal{E}_J$, it reflects the fact that the objective function should increase if the cost function decreases and vice versa.
Also, if there were no costs present, i.\,e.,~ $c_J\equiv0$, then~$\phi_J=\textnormal{acq}_J$ outside~$\mathcal{E}_J$.
Finally, as kernel optimizations are computationally the most expensive part of our methodology, it is reasonable to stop them once a certain criterion is met.
The stopping criterion~$P_{\textnormal{KO}}=P_{\textnormal{KO}}(J)$ is formalized as a predicate, i.\,e.,~ a boolean-valued function, depending on step~$J$.
If $N_{\textnormal{KO}}(J)\in\mathbf{N}$ denotes the number of kernel optimizations performed up until step~$J$, it is defined as
\begin{align}
\label{eq:stop_crit_KO}
\begin{split}
&P_{\textnormal{KO}}(J) \coloneqq N_{\textnormal{KO}}(J) > N_{\textnormal{KO}}^{\max} \\
&\quad\quad \vee \left( N_{\textnormal{KO}}(J) \geq N_{\textnormal{KO}}^{\min} \;\;\wedge\;\; \frac{1}{k_{\textnormal{KO}}-1}\sum_{j=J-k_{\textnormal{KO}}+1}^{J-1} \frac{\norm{\log(\theta_j)-\log(\theta_{j+1})}_2}{\norm{\log(\theta_{j+1})}_2} \leq \eps_{\textnormal{KO}} \right)
\end{split}
\end{align}
for parameters~$N_{\textnormal{KO}}^{\min},N_{\textnormal{KO}}^{\max},k_{\textnormal{KO}}\in\mathbf{N}$ such that
\begin{equation}
2 \leq k_{\textnormal{KO}} \leq N_{\textnormal{KO}}^{\min} \leq N_{\textnormal{KO}}^{\max}
\end{equation}
and~$\eps_{\textnormal{KO}}>0$.
Informally, this predicate indicates that kernel optimizations should be stopped as soon as~$N_{\textnormal{KO}}(J)$ exceeds a given maximum number~$N_{\textnormal{KO}}^{\max}$ or if, provided that~$N_{\textnormal{KO}}(J)$ exceeds a given minimum number~$N_{\textnormal{KO}}^{\min}$, the average relative difference of the last~$k_{\textnormal{KO}}$ hyperparameters falls below a given threshold value~$\eps_{\textnormal{KO}}$, i.\,e.,~ the hyperparameters stagnate and do no longer change substantially.
Note that, in Eq.~\eqref{eq:stop_crit_KO}, the expressions~$\log(\theta)$ for the vector of kernel hyperparameters~$\theta=(\sigma^2,\lambda_1,\ldots,\lambda_n)^\top$ from Eq.~\eqref{eq:vec_kernel_hyperp} are meant componentwise, i.\,e.,~
\begin{equation}
\log(\theta) = (\log(\sigma^2),\log(\lambda_1),\ldots,\log(\lambda_n))^\top \in \mathbf{R}^{n+1}.
\end{equation}
\subsection*{Degenerate cases}
\addcontentsline{toc}{subsection}{Degenerate cases}
An intensity function that is nearly constant along a certain coordinate~$x_k$ in~$\mathcal{X}$, i.\,e.,~ an intrinsically lower-dimensional function, might cause problems for the Gaussian kernel from Eq.~\eqref{eq:gauss_kernel} as the corresponding optimal length scale hyperparameter would be~$\lambda_k=\infty$.
Also, the initial observations~$\mathcal{A}_0$ from Eq.~\eqref{eq:init_exp} might not resolve the main characteristics of the intensity function sufficiently well and hence pretend it to be lower-dimensional.
Most degenerate cases can be identified by kernel optimizations resulting in one or more length scales that are quite low or high relative to the dimensions of~$\mathcal{X}$.
In general, we assess a length scale parameter~$\lambda_k$ as \textit{degenerate} if it violates
\begin{equation}
\label{eq:degen_inequ}
\delta^{-} \leq \frac{\lambda_k}{x_k^+-x_k^-} \leq \delta^{+}
\end{equation}
for two parameters~$0<\delta^{-}<\delta^{+}<\infty$, where~$x_k^-$ and~$x_k^+$ denote the limits of the rectangle~$\mathcal{X}$ in dimension~$k$.
If, after kernel optimization at a certain step, a length scale parameter is recognized to be degenerate, we regard the intensity function on a coordinate system rotated by~$45^\circ$ in order to avoid the mentioned problems.
Of course, the rotation is performed only internally and does not affect the original setting.
In~$n=2$ dimensions, a crystal field excitation, for example, might induce a lower-dimensional intensity function (Extended Data Fig.~\ref{fig:rotat}a).
After rotating the coordinate system, the intensity function becomes full-dimensional (Extended Data Fig.~\ref{fig:rotat}b) allowing non-degenerate kernel optimizations.
\subsection*{Algorithmic setting}
\addcontentsline{toc}{subsection}{Algorithmic setting}
All experiments described in this article are performed in $n=2$~dimensions, i.\,e.,~ $\mathcal{X}\subseteq\mathbf{R}^2$.
If not specified otherwise, we use $N_{\textnormal{row}}=11$ rows corresponding to 61~measurements in the initialization grid (Eq.~\eqref{eq:def_J0}).
For scenario~2 of the neutron experiment, we use~$N_{\textnormal{row}}=7$ rows corresponding to 25~initial measurements.
The default parameter values used for Alg.~\ref{alg:final} in both experimental settings, i.\,e.,~ the neutron experiment and the benchmark, are specified in Tab.~\ref{tab:parameter_values_default}.
\begin{table}
\footnotesize
\centering
\begin{tabular}{l||c|c|c|c|c|c|c|c|c|c|c|c|c}
\textbf{Parameter} & $N_{\textnormal{row}}$ & $\Delta^{\textnormal{rel}}_{\max}$ & $\Delta^{\textnormal{abs}}_{\min}$ & $l_{\max}$ & $\beta$ & $N_{\cubeHyperparam}$ & $\mathcal{H}$ & $N_{\textnormal{KO}}^{\min}$ & $N_{\textnormal{KO}}^{\max}$ & $k_{\textnormal{KO}}$ & $\eps_{\textnormal{KO}}$ & $\delta^{-}$ & $\delta^{+}$ \\ \hline\hline
\textbf{Value} & 11 & 0.5 & 15 & 6 & 0.5 & $100$ & $[10^{-3},10^2]^3$ & $25$ & $75$ & $9$ & $0.025$ & $10^{-3}$ & $1$
\end{tabular}
\caption{Default parameter values used for Alg.~\ref{alg:final}.}
\label{tab:parameter_values_default}
\end{table}
The matrix-valued function~$E$ (Eq.~\eqref{eq:ellipse_matrix_function}), defining ellipses as consumed areas, is chosen to give circles with fixed radius~$r>0$ on a normalized domain, i.\,e.,~ $U(\vec{x}_j)=I$ and
\begin{equation}
\label{eq:ellipse_radius_normalized}
r_k(\vec{x}_j) = \frac{r}{x_k^+-x_k^-}.
\end{equation}
We set~$r=0.02$ for the neutron experiment and~$r=0.025$ for the benchmark.
\section*{Code availability}
\addcontentsline{toc}{section}{Code availability}
The software implementation of our approach is based on the \textsf{GaussianProcessRegressor} class from the Python package \textsf{scikit-learn}~\cite{pedregosa2011scikit}.
All results can be reproduced using our code from the repository \url{jugit.fz-juelich.de/ainx/ariane} (commit SHA: \href{https://jugit.fz-juelich.de/ainx/ariane/-/commit/6ed7de50d8c4f4a2d3714bca3f2aed9f7ca08ecf}{\texttt{6ed7de50}}).
The benchmark results can be reproduced by using code from the repository \url{jugit.fz-juelich.de/ainx/base-ariane-fork} (commit SHA: \href{https://jugit.fz-juelich.de/ainx/base-ariane-fork/-/commit/a7eb3dc950cda034e0a7010e2c575114503fefed}{\texttt{a7eb3dc9}}).
It is a fork, adjusted to our approach, of the benchmark API from \url{jugit.fz-juelich.de/ainx/base} which is part of the mentioned original work on the benchmarking procedure~\cite{teixeiraparente2022benchmarking}.
\section{TAS setting}
\label{si-sec:setting_tas}
In TAS experiments, intensities are observed at a certain point in the $\vec{Q}$-$E$~space of the investigated material by counting scattered neutrons on a detector device.
The momentum space~$\vec{Q}$ is the Fourier transform of an initial periodic spatial lattice depending on the material and provides wavevector coordinates $\vec{q}=(h,k,l)^\top\in\vec{Q}$ (Miller indices) in relative lattice units (r.l.u.) to move within it.
The one-dimensional energy space~$E$ provides a coordinate~$\omega\in E$, mostly measured in units of micro-electron volts (meV), to describe energy transfer.
Neglecting units, $\vec{Q}$-$E$~space can be thought of as a four-dimensional real space, i.\,e.,~ $(\vec{q},\omega)^\top\in\mathbf{R}^r$, $r=4$.
TAS experiments are, however, often not carried out in full-dimensional $\vec{Q}$-$E$~space but on a lower-dimensional (often two-dimensional) hyperplane.
The coordinates on the corresponding hyperplane in $\vec{Q}$-$E$~space are denoted by $\vec{x}=(x_1,\ldots,x_n)^\top\in\mathbf{R}^n$, $n\leq r$, and the transformation $T:\mathbf{R}^n\to\mathbf{R}^r$ is defined as
\begin{equation}
\label{eq:qe_x}
\begin{pmatrix}\vec{q} \\ \omega\end{pmatrix} = W\vec{x} + \vec{b} \eqqcolon T(\vec{x})
\end{equation}
for a full-rank transformation matrix~$W\in\mathbf{R}^{r\times n}$ and an offset~$\vec{b}\in\mathbf{R}^r$.
Most often, the transformation matrix has the shape
\begin{equation}
W = \begin{pmatrix}* & 0 \\ * & 0 \\ * & 0 \\ 0 & 1\end{pmatrix}
\end{equation}
meaning that a two-dimensional hyperplane is spanned by a certain direction in $\vec{Q}$~space and energy transfer.
The re-transformation $T^{-1}:\mathbf{R}^r\to\mathbf{R}^n$, an orthogonal projection onto the hyperplane, is then given by
\begin{equation}
\label{eq:x_qe}
\vec{x} = (W^\top W)^{-1} W^\top\left[ \begin{pmatrix}\vec{q} \\ \omega\end{pmatrix}-\vec{b} \right] \eqqcolon T^{-1}(\vec{q},\omega).
\end{equation}
Note that~$T^{-1}(T(\vec{x}))=\vec{x}$ for each~$\vec{x}\in\mathbf{R}^n$, but $T(T^{-1}(\vec{q},\omega))=(\vec{q},\omega)^\top$ only for~$(\vec{q},\omega)^\top\in T(\mathbf{R}^n)$.
The rectangular set~$\mathcal{X}\subseteq\mathbf{R}^n$, becoming the domain of the intensity function, is defined as
\begin{equation}
\mathcal{X} \coloneqq [x_1^-,x_1^+] \times \cdots \times [x_n^-,x_n^+]
\end{equation}
for given limits of investigation~$x_k^\pm\in\mathbf{R}$, $k\in\lbrace 1,\ldots,n \rbrace$.
For example, in Fig.~1{}, we have $(x_1^-,x_1^+,x_2^-,x_2^+)=(2.3,3.3,2.5,5.5)$.
In order to observe intensities at a certain location in $\vec{Q}$-$E$~space, a TAS needs to move its axes to six related angles.
However, since there exist dependencies among the axes, it is enough to regard a subset of four angles and their corresponding angular velocities~$\vangvelo=(v_1,v_2,v_3,v_4)^\top\in[0,\infty)^4$.
For the connection between points in $\vec{Q}$-$E$~space and their respective angles of the instrument axes, we formally use an angle map
\begin{equation}
\label{eq:angle_map}
\vec{\Psi} : \dom{\vec{\Psi}} \to [0,\pi)^4, \; (\vec{q},e) \mapsto (\Psi_1(\vec{q},e),\ldots,\Psi_4(\vec{q},e))^\top
\end{equation}
with domain~$\dom{\vec{\Psi}}\subseteq\mathbf{R}^r$ containing points in $\vec{Q}$-$E$~space for which~$\vec{\Psi}$ is well-defined, i.\,e.,~ points reachable by the instrument.
Translating this domain to the lower-dimensional coordinates, we define
\begin{equation}
\mathcal{X}^* \coloneqq T^{-1}(\dom{\vec{\Psi}} \, \cap \, T(\mathcal{X}))
\end{equation}
as the set of points~$\vec{x}\in\mathcal{X}$ such that, by abuse of notation, the map
\begin{equation}
\vec{\Psi}(\vec{x}) \coloneqq \vec{\Psi}(T(\vec{x}))
\end{equation}
is well-defined.
We assume that the limits of investigation are set such that $\vec{\Psi}:\mathcal{X}^*\to[0,\pi)^4$ becomes an injective function.
The sample and its orientation induce a scattering function~$s : T(\mathcal{X}) \to [0,\infty)$ describing intensities theoretically present.
For~$\vec{x}\in\mathcal{X}$, we define again
\begin{equation}
s(\vec{x}) \coloneqq s(T(\vec{x})).
\end{equation}
The scattering function is not directly accessible due to limits in the instrument resolution.
If we denote the resolution function by~$\varphi : T(\mathcal{X}^*) \to ( T(\mathcal{X}^*) \to [0,\infty) )$ and again define
\begin{equation}
\varphi(\vec{x}) \coloneqq \varphi(T(\vec{x}))
\end{equation}
for~$\vec{x}\in\mathcal{X}^*$, we eventually get the intensity function~$i : \mathcal{X}^* \to [0,\infty)$ as the convolution of~$s$ with~$\varphi$, i.\,e.,~
\begin{equation}
i(\vec{x}) \coloneqq (s * \varphi(\vec{x}))(\vec{x}).
\end{equation}
Finally, the cost for moving the instrument axes, i.\,e.,~ changing their angles, from a certain location~$\vec{x}\in\mathcal{X}^*$ to another~$\vec{x}'\in\mathcal{X}^*$ is formalized by the metric~$d : \mathcal{X}^* \times \mathcal{X}^* \to [0,\infty)$ defined as
\begin{equation}
\label{eq:def_metric}
d(\vec{x},\vec{x}') \coloneqq \max_{k\in\lbrace1,2,3,4\rbrace} \left\lvert\frac{\Psi_k(\vec{x})-\Psi_k(\vec{x}')}{v_k}\right\rvert,
\end{equation}
where~$\Psi_k$ are components of the angle map from Eq.~\eqref{eq:angle_map} and~$v_k$ denote the corresponding angular velocities of the instrument.
It indicates the maximum time needed changing all angles in parallel.
Note that~$d$ is indeed a metric in the mathematical sense since the angle map is chosen to be injective.
\section{Setup for neutron experiment}
\label{si-sec:setup_neutron_experim}
To prepare the workflow for a neutron experiment at a TAS, we tested real space movements of instrument axes as well as the communication between the software implementation of our approach and the instrument control system NICOS during dry runs, i.\,e.,~ without neutrons, at the cold TAS \textit{PANDA}~(MLZ)~\cite{schneidewind2015panda} for several excitations.
Having a lack of neutron beam time in Europe currently, we are grateful for the granted beam time at the thermal TAS EIGER (PSI) \cite{stuhr2017thermal} making a real experiment possible to finally demonstrate the usefulness and benefits of our approach.
The experimental setup at EIGER was as follows.
We oriented a SnTe sample (space group~225) in the (hhl)~scattering plane and mounted it in a closed cycle cryostat for background decrease, even though we have been measuring at room temperature.
EIGER was operated in constant-$k_f$ mode ($k_f=2.66$~\AA$^{-1}$) with a PG filter on~$k_f$, a double-focusing PG002 monochromator, and a horizontally focusing PG002 analyzer.
In scenarios~1 and~2, we counted neutrons on the detector device at each measurement location in $\vec{Q}$-$E$~space until $100,000$~neutrons were counted on the monitor device, whereas in scenario~3, we counted for $40,000$ (initialization) and $50,000$ (after initialization) monitor counts, respectively.
For scenarios~1 and~2, due to the coupling of~LA and~TO, with an additional TA~mode, the intensity distribution provides an ideal setting including real background, strong and weak signals, and symmetry breaking in intensity along the $\vec{Q}$~direction.
\section{Further results from neutron experiment}
\label{si-sec:results_neutron_experim}
We have additionally tested our approach in the setting of scenario~1 with different values for the background level and intensity threshold.
The results (Supplementary Fig.~\ref{fig:eiger_scen1_compar_2022-05}) further support the claim that our approach identifies regions of signal successfully and that a change in the intensity threshold (from~$\tau_3=90$ to~$\tau_4=130$, both with~$\gamma_3=\gamma_4=45$) does not have significant influence on the final outcome.
\begin{figure}
\centering
\includegraphics[width=\linewidth]{suppl_info/eiger_scen1_compar_2022-05.pdf}
\caption{Further results for scenario~1 in two additional settings of our approach.
Rows and columns have the same meaning as for Fig.~3{}{} (except the last column of the top row since the corresponding total experimental time did not reach the final time of stage~IV).
The top row corresponds to~$\gamma_3=45$ and~$\tau_3=90$ and the bottom row to~$\gamma_4=\gamma_3$ and~$\tau_4=130$.}
\label{fig:eiger_scen1_compar_2022-05}
\end{figure}
On the contrary, we can see again that the particular value of the intensity threshold influences the width of the branches the measurements are placed on, which additionally substantiates our claim of interpretability and explainability (see Discussion).
In scenario~3, we have investigated the material SnTe on~$\mathcal{X}=[0,1]\times[1,12]$ along the $\vec{Q}$ direction~$(1,1,0)$ again but with offset~$(0,0,2)$ (instead of~$(0,0,3)$) and energy transfer.
As the corresponding intensity distribution in this setting has been unknown to us, we have had a scenario that required searching for signals of interest and thus a typical situation for the productive application of our approach in the future.
Note that we used our approach in the default setting (Tab.~1{}) for this scenario, i.\,e.,~ with an automated estimation of the background level ($\gamma=12$) and the intensity threshold ($\tau=54$).
The results (Supplementary Fig.~\ref{fig:eiger_scen3}) demonstrate again that, after initialization, the majority of measurement points is placed in the only region of signal.
\begin{figure}
\centering
\includegraphics[width=0.75\linewidth]{suppl_info/eiger_scen3.pdf}
\caption{Results for scenario~3.
a) Initial measurement points (triangles).
b) Autonomously placed intensity observations (dots).}
\label{fig:eiger_scen3}
\end{figure}
In particular, although the signals vary greatly in magnitude, the measurement points are evenly distributed in this region, which is due to a well-estimated intensity threshold.
\section{Milestone values for benchmark}
\label{si-sec:milestone_values_base}
The milestone values used for the benchmark are given in Supplementary Table~\ref{tab:milestone_values_base}.
Recall that, for each test case, they are determined by the four stages (I-IV) of the grid approach (Extended Data Fig.~8{}).
The intensity functions for all test cases are displayed in Extended Data Fig.~9{} a)-t).
\begin{table}[H]
\centering
\begin{tabular}{c||cccc}
\multirow{2}{*}{Test case} & \multicolumn{4}{c}{Milestone values in hours} \\ \cline{2-5}
& I & II & III & IV \\ \hline\hline
a) & 2.28 & 4.28 & 6.27 & 8.02 \\ \hline
b) & 2.27 & 4.27 & 6.25 & 8.00 \\ \hline
c) & 2.23 & 4.45 & 6.66 & 8.90 \\ \hline
d) & 2.37 & 4.46 & 6.53 & 8.34 \\ \hline
e) & 2.17 & 4.27 & 6.42 & 8.57 \\ \hline
f) & 2.21 & 4.41 & 6.61 & 8.82 \\ \hline
g) & 2.18 & 4.37 & 6.55 & 8.73 \\ \hline
h) & 2.19 & 4.38 & 6.57 & 8.76 \\ \hline
i) & 2.21 & 4.42 & 6.63 & 8.84 \\ \hline
j) & 2.17 & 4.34 & 6.50 & 8.67 \\ \hline
k) & 2.18 & 4.36 & 6.53 & 8.71 \\ \hline
l) & 2.21 & 4.43 & 6.64 & 8.85 \\ \hline
m) & 2.16 & 4.32 & 6.48 & 8.64 \\ \hline
n) & 2.23 & 4.47 & 6.70 & 8.93 \\ \hline
o) & 2.17 & 4.34 & 6.51 & 8.69 \\ \hline
p) & 2.15 & 4.30 & 6.45 & 8.60 \\ \hline
q) & 2.36 & 4.44 & 6.48 & 8.28 \\ \hline
r) & 2.33 & 4.36 & 6.38 & 8.15 \\ \hline
s) & 1.80 & 3.58 & 5.36 & 7.13 \\ \hline
t) & 2.26 & 4.24 & 6.20 & 7.93
\end{tabular}
\caption{Milestone values for each benchmark test case.}
\label{tab:milestone_values_base}
\end{table}
\section{Proof of Proposition on small variance limit for log-normal random variables}
\label{si-sec:proof_prop}
\begin{lemma}
For each~$y\in\mathbf{R}$, it holds that
\begin{equation}
\lim_{\sigma\to0^+} \frac{\exp(\sigma y) - \exp(\sigma^2/2)}{(\exp(\sigma^2)-1)^{1/2} \cdot \exp(\sigma^2/2)} = y.
\end{equation}
\end{lemma}
\begin{proof}
Let~$y\in\mathbf{R}$.
Note that
\begin{equation}
\lim_{x\to0} \frac{\exp(x) - 1}{x} = 1
\end{equation}
using the Taylor expansion of~$\exp(x)$.
With this in mind, we compute
\begin{align}
\frac{\exp(\sigma y) - \exp(\sigma^2/2)}{(\exp(\sigma^2)-1)^{1/2} \cdot \exp(\sigma^2/2)} &= \frac{ \frac{\exp(\sigma y) - 1}{\sigma y} \cdot \sigma y - \frac{\exp(\sigma^{2}/2)-1}{\sigma^{2}/2} \cdot \sigma^2/2} {\left(\frac{\exp(\sigma^2) - 1}{\sigma^2}\right)^{1/2} \cdot \sigma \cdot \exp(\sigma^2/2)}.
\end{align}
Cancelling $\sigma$ from both, the numerator and the denominator, gives
\begin{equation}
\begin{split}
\lim_{\sigma\to0^+} \frac{\exp(\sigma y) - \exp(\sigma^2/2)}{(\exp(\sigma^2)-1)^{1/2} \cdot \exp(\sigma^2/2)} &= \lim_{\sigma\to0^+} \frac{ \frac{\exp(\sigma y) - 1}{\sigma y} \cdot y - \frac{\exp(\sigma^{2}/2)-1}{\sigma^{2}/2} \cdot \sigma/2} {\left(\frac{\exp(\sigma^2) - 1}{\sigma^2}\right)^{1/2} \cdot \exp(\sigma^2/2)} \\
&= \frac{1\cdot y - 1\cdot0}{1\cdot1} \\
&= y.
\end{split}
\end{equation}
\end{proof}
\begin{proof}[Proof of Proposition]
First, we compute
\begin{align}
\frac{Z}{\sqrt{\Var{Z}}} &= \exp\left( \log\left( \frac{1}{\sqrt{\Var{Z}}} \right) +\mu + \sigma \eta \right) \\
&= \exp\left( -\frac{1}{2}\log(\Var{Z}) + \mu + \sigma \eta \right) \\
&= \exp\left( -\frac{1}{2}[\log(\exp(\sigma^2)-1) + 2\mu + \sigma^2] + \mu + \sigma \eta \right) \\
&= \exp\left( -\frac{1}{2}\log(\exp(\sigma^2)-1) - \frac{\sigma^2}{2} + \sigma \eta \right) \\
&= \frac{1}{(\exp(\sigma^2)-1)^{1/2}} \cdot \exp\left( -\frac{\sigma^2}{2} +\sigma \eta \right) \\
&= \frac{\exp(\sigma \eta)}{(\exp(\sigma^2)-1)^{1/2} \cdot \exp(\sigma^2/2)}
\end{align}
and
\begin{align}
\frac{\E{Z}}{\sqrt{\Var{Z}}} &= \frac{\exp\left( \mu+\frac{\sigma^2}{2} \right)}{(\exp(\sigma^2)-1)^{1/2} \cdot \exp(\mu+\sigma^2/2)} \\
&= \frac{\exp(\sigma^2/2)}{(\exp(\sigma^2)-1)^{1/2} \cdot \exp(\sigma^2/2)}
\end{align}
yielding
\begin{equation}
\label{eq:Znorm_explic}
\overline{Z} = \frac{\exp(\sigma \eta) - \exp(\sigma^2/2)}{(\exp(\sigma^2)-1)^{1/2} \cdot \exp(\sigma^2/2)}.
\end{equation}
Applying the lemma above to Eq.~\eqref{eq:Znorm_explic} for~$y=\eta(\omega)$ yields the result.
\end{proof}
Hence, as a corollary, the distribution of~$\overline{Z}$ converges to a standard normal distribution (as $\sigma\to0^+$).
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 5,144 |
Betrayal is book one in the Ann's War Mystery Series written by Hannah Howe, author of the Amazon #1 Sam Smith Mystery Series.
When nine-year-old Eric Danson goes missing in the middle of the night, psychic P.I. Piper Ashwell is reminded of her first case ever, when she was only twelve years old and her ability to read objects first surfaced. And things only get worse when the boy who's been bullying Eric goes missing as well. Can she find the boys before it's too late?
This is a prequel novelette to the Blackwood Bay Witches Mystery Series. On A Witch And A Spell is a short, humorous cozy mystery read filled with magic; a mystery or two; and snarky, no nonsense main character and her friends.
Meet Lori Stockley, the mail order Private Investigator, and Panzer, the black cat. Watch as Lori discovers who's been stealing money from the residents at her Granny's condo facility. This is the first case in the Panzer and the P.I. series.The Lori Stockley mysteries are good, clean puzzles with no blood or murders. (At least not yet!) This is a short-short and can be read on your coffee break.
A village Christmas party is tainted with tragedy after a man fell to his death when straightening the tall tree's angel topper. But why did he care it was crooked? What made him try it without a ladder? And why did no one cancel the party? Luckily Alma Easter is on hand to explain what happened, right after she finishes some chocolate yule log.
This recipe book has a few snippets taken directly from the book Fore! In the Hole along with recipes from the story. At the end is chapter one from Fore! In the Hole as a preview to the book. I hope you enjoy the food as much as I do. The excerpts are in italic and recipes follow. Bon Appetite!
There's a killer lurking backstage at the last night performance of the Goose Meadow Bay Player's Halloween murder mystery. Can Kath keep the show running, the cast from drinking their way through the onstage drinks cabinet and the killer from taking the script literally? Murder and mayhem of the theatrical kind in this cozy mystery short read.
An established landscape artist from South Wales regains consciousness in a London hospital after an attack in the street. On arrival in ER she was murmuring incoherently in Welsh. The young police officer assigned to sit with her also speaks the language and they have a common interest; art and a missing paint box.
This recipe book has a few snippets taken directly from the book Three for Pumpkin Pie? along with recipes from the story. At the end is chapter one from Three for Pumpkin Pie? as a preview to the book. I hope you enjoy the food as much as I do. The excerpts are in italic and recipes follow. Bon Appetite!
This recipe book has a few snippets taken directly from the book A Kayak for One along with recipes from the story. At the end is chapter one from A Kayak for One as a preview to the book. I hope you enjoy the food as much as I do. The excerpts are in italic and recipes follow. Bon Appetite!
This recipe book has a few snippets taken directly from the book Two Buckets of Berries along with recipes from the story. At the end is chapter one from Two Buckets of Berries as a preview to the book. I hope you enjoy the food as much as I do. The excerpts are in italic and recipes follow. Bon Appetite!
A short story featuring characters from The Belinda & Bennett Mysteries. In this short, Jonas gets some help with his wardrobe. Includes a sample of book four in the series, Overkill.
A short story featuring characters from The Belinda & Bennett Mysteries. In this short, between events in Drive-Bye and Overkill, Belinda's grandmother pays her a visit, dropping a surprising piece of information about her grandfather.
Tragedy befalls a jolly December day when a Christmas elf falls from the top floor of a department store to the icy pavement below. But why was he up there and not on his shift? And could it be related to a string of thefts from the store? In this short story, Alma Easter, the cake-loving old-lady detective, thinks she knows the answer.
A merry bit of Christmas shopping takes a more sinister turn when a man is found dead – stabbed to death with a candy cane. Who was the masked assailant who attacked him? And why did they choose a candy cane as a weapon? Luckily Alma Easter, the cake-loving old lady detective, is on hand to investigate.
Beck Nash arrives in the Las Vegas Metropolitan Police Department's Homicide division with a shaky reputation and a lot to prove. Trying to adjust to the mold, Beck's only goal is to make it to week's end. But as shocking as the first day of her first week in Homicide turns out, it will be nothing compared to her last.
Amikor a panelház földszintjén lévő trafik tulajdonosa váratlanul meghal, a kivezényelt nyomozó biztos benne, hogy Keirának van ötlete… Újabb minikrimi az iTunes krimi #1 bestsellere, az "Alattam buli" szerzőjétől!
Murder was never so refined… When her holiday on the coast of Cornwall takes a deadly turn, it is up to Drucilla Winterbourne to uncover the dangerous secrets the inhabitants of Blackridge House will do anything to conceal. But can a proper young lady from London society comprehend the dark motives of a killer? | {
"redpajama_set_name": "RedPajamaC4"
} | 4,031 |
What is Chimerism?
BY Kate Horowitz
Screenshot, TwinzyOKC
Conception is normally a pretty straightforward process: One lucky sperm wiggles its way into one lucky egg. In the case of fraternal twins, each sperm-and-egg pair is just one half of a double date. The resulting fertilized eggs trundle along and begin dividing separately, eventually becoming two separate fetuses. But sometimes, one egg engulfs the other, effectively "eating" its twin.
This isn't as messy as it sounds. At that stage in development, we're all just smooshy single cells. The newly fattened twin-eating cell goes about its business, dividing and dividing, growing and growing, until, many months later, it becomes a person (or a lion, or a rabbit, or a weasel, depending on its parents).
The scientific term is tetragametic chimerism, although the origin of that word is hardly scientific. The ancient Greek Chimera was a mythical monster with the body and fire-breathing head of a lion, a goat head rising from its back, and a snake tail. Over time, the word chimera came to mean any kind of combo-pack animal—including, eventually, people with two sets of DNA.
Because, young as those twin cells are when they become one, each cell has its own set of DNA. After the Big Meal, both sets come to reside in a single cell, which eventually becomes a single animal.
People (and animals) with chimerism may be visibly different. They may have one brown eye and one blue, or one hitchhiker's thumb and one straight one. Their skin may, almost imperceptibly, be a swirl of two different colors. Chimeric animals can sport a two-tone pattern split right down the middle. That appears to be the case for Twinzy, pictured above checking out his reflection in a mirror, a budgie who lives at Rudy's Pet Supply & Feed in Oklahoma City.
Most of the time, though, chimeras look just like everybody else. That was the case for Lydia Fairchild. As a pregnant, single mother of two applying for welfare benefits, in 2002 Fairchild and her children had to undergo paternity testing. The results confirmed that the father of Fairchild's children was, in fact, their father. But, according to the DNA tests, Fairchild wasn't their mother.
The state threatened to take her children away. Fairchild was accused of welfare fraud and illegal surrogacy. The DNA evidence against her seemed airtight, and even her own parents began to doubt her story.
Then her lawyer saw a story in the New England Journal of Medicine that seemed startlingly familiar. A woman named Karen Keegan had been in need of a kidney transplant. To identify a possible donor, Keegan and her family members had their DNA tested. The results came back: Keegan could not be the mother of two of her three sons.
Eventually, researchers at the National Institutes of Health began testing DNA from different parts of Keegan's body. The pieces fell into place: Keegan was a chimera. After the researchers published their findings, Lydia Fairchild's lawyer took their findings to the judge, who granted Fairchild official custody of her own children. After the case was dismissed, scientists determined that Fairchild did, indeed, have two sets of DNA.
So it turns out chimerism isn't all fun and games (and twin-eating). The genetic, physical, and social consequences may be nonexistent. They could be super-cool and make you look like a badass. Or they could be life changing.
biology science | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 5,748 |
A demo Mac OS X app from the BNR Cocoa book.
<https://www.bignerdranch.com/we-write/cocoa-programming/>
| {
"redpajama_set_name": "RedPajamaGithub"
} | 6,055 |
\section{Introduction}
\label{sec:intro}
One of the earliest puzzles in spin physics research was the observation in the 1970s of large asymmetries in single-inclusive reactions where one hadron is transversely polarized~\cite{Bunce:1976yb,Klem:1976ui} -- so-called single transverse-spin asymmetries (SSAs) $A_N$. This eventually was recognized as a signature of multi-parton correlations in hadrons~\cite{Efremov:1981sh,Efremov:1984ip,Qiu:1991pp,Qiu:1991wg,Qiu:1998ia} and has been a source of intense theoretical~\cite{Efremov:1981sh,Efremov:1984ip,Qiu:1991pp,Qiu:1991wg,Qiu:1998ia,Kanazawa:2000hz,Eguchi:2006qz,Kouvaris:2006zy,Eguchi:2006mc,Zhou:2008fb,Koike:2009ge,Metz:2012ct,Kanazawa:2013uia,Beppu:2013uda,Kanazawa:2015ajw,Koike:2017fxr,Koike:2019zxc,Koike:2021awj,Koike:2022ddx,Ikarashi:2022yzg,Ikarashi:2022zeo}, phenomenological~\cite{Qiu:1998ia,Kanazawa:2000kp,Kouvaris:2006zy,Kanazawa:2010au,Kang:2011hk,Metz:2012ui,Beppu:2013uda,Gamberg:2013kla,Kanazawa:2014dca,Gamberg:2014eia,Gamberg:2017gle,Cammarota:2020qcw,Gamberg:2022kdb}, and experimental~\cite{Adams:1991rw,Krueger:1998hz,Allgower:2002qi,Adams:2003fx,Adler:2005in,Lee:2007zzh,Abelev:2008af,Arsene:2008aa,Adamczyk:2012qj,Adamczyk:2012xd,Bland:2013pkt,Adare:2013ekj,Adare:2014qzo,Airapetian:2013bim,Allada:2013nsw,STAR:2020nnl} activity for decades. The collinear twist-3 formalism that underpins this work allows one to explore a rich set of non-perturbative functions, of which SSAs are sensitive to a certain subset. Namely, the na\"{i}ve time-reversal odd (T-odd) nature of SSAs gives access to pole contributions from initial state multi-parton distribution functions (PDFs) (where typically one of the partons' momentum fractions vanishes~\cite{Qiu:1991pp,Qiu:1991wg,Qiu:1998ia,Kanazawa:2000hz,Kouvaris:2006zy,Koike:2009ge,Beppu:2013uda}\footnote{The poles are due to propagators in the hard scattering going on shell. While usually this causes a momentum fraction in the multi-parton PDF to vanish (``soft poles'') there are certain processes that also lead to ``hard poles''~\cite{Eguchi:2006qz,Eguchi:2006mc,Albaltan:2019cyc}, where all parton momentum fractions remain nonzero.}); or to the imaginary part of (non-pole) final-state multi-parton fragmentation functions (FFs)~\cite{Metz:2012ct,Kanazawa:2013uia}.\footnote{We will still refer to initial-state twist-3 functions as parton distribution functions (PDFs) and final-state twist-3 functions as fragmentation functions (FFs), even though they do not have a strict probability interpretation.} For example, $A_N$ in $p^\uparrow p\to \pi\,X$ at forward rapidity is mainly sensitive to the Qiu-Sterman PDF $F_{FT}(x,x)$ (where the two quarks carry the same momentum fraction $x$), as well as $H_1^{\perp(1)}(z)$ (which is the first-moment of the Collins function) and $\tilde{H}(z)$, with $z$ the momentum fraction carried by the produced hadron. The latter two functions are certain integrals over $z_1$ (from $z$ to $\infty$) of the FF $\hat{H}^\Im_{FU}(z,z_1)$~\cite{Kanazawa:2015ajw}, where $\Im$ indicates the imaginary part. There are a plethora of SSA measurements, not only in $p^\uparrow p\to h\,X$ but also semi-inclusive deep-inelastic scattering (SIDIS) $e\,N^\uparrow\to e\,h\,X$~\cite{Airapetian:2009ae, Alekseev:2008aa,Airapetian:2010ds,Qian:2011py,Adolph:2014zba,Zhao:2014qvx,Adolph:2016dvl,HERMES:2020ifk}, electron-positron annihilation $e^+e^-\to h_1\,h_2\,X$~\cite{Seidl:2008xc,TheBABAR:2013yha,Aubert:2015hha,Ablikim:2015pta,Li:2019iyt}, and Drell-Yan $p^\uparrow p\to \{W^\pm,Z,\,{\rm or}\;\ell^+\ell^-\}\,X$~\cite{Adamczyk:2015gyk,Aghasyan:2017jop}. Due to this data, as well as the connection between collinear twist-3 and transverse momentum dependent (TMD) functions~\cite{Ji:2006ub,Ji:2006br,Koike:2007dg,Yuan:2009dw,Zhou:2009jm}, $F_{FT}(x,x)$, $H_1^{\perp(1)}(z)$, and $\tilde{H}(z)$, along with the twist-2 transversity PDF $h_1(x)$, have all been extracted in various phenomenological analyses~(see, e.g., \cite{Kanazawa:2014dca,Echevarria:2014xaa,Kang:2015msa,Echevarria:2020hpy,Bury:2021sue,Cammarota:2020qcw,Gamberg:2022kdb}).
A complimentary observable to study multi-parton correlations in hadrons is the longitudinal-transverse double-spin asymmetry $A_{LT}$ in collisions like $\vec{e}\,N^\uparrow \to \pi\,X$ and $p^\uparrow \vec{p}\to \pi\,X$. These are {\it T-even} reactions that are sensitive to the {\it non-pole} pieces of certain multi-parton PDFs (e.g., $F_{FT}(x,x_1)$ with $x\neq x_1$) and the {\it real part} $\Re$ of certain multi-parton FFs (e.g., $\hat{H}^\Re_{FU}(z,z_1)$). From the theoretical side, $A_{LT}$ has been well studied in electron-nucleon~\cite{Kang:2011jw,Kanazawa:2014tda,Kanazawa:2015ajw} and proton-proton~\cite{Liang:2012rb,Metz:2012fq,Hatta:2013wsa,Koike:2015yza,Koike:2016ura} collisions for various single-inclusive final states (e.g., hadron, jet, or photon), with some limited numerical work performed for the electron-nucleon case~\cite{Kang:2011jw,Kanazawa:2014tda}, but none for proton-proton. The main hindrance to more rigorous predictions has been the lack of input for important non-perturbative functions in $A_{LT}$, which forces one to resort to approximations or the outright neglect of certain terms~\cite{Kang:2011jw,Kanazawa:2014tda}. For example, one of the main PDFs that enters $A_{LT}$ is $g_{1T}^{(1)}(x)$, which is the first-moment of the worm-gear TMD $g_{1T}$, and it has only been extracted recently~\cite{Bhattacharya:2021twu,Horstmann:2022xkk}.\footnote{We mention that the authors of Ref.~\cite{Horstmann:2022xkk} did not directly extract the twist-3 function $g_{1T}^{(1)}(x)$ needed in our analysis.} Previous numerical computations utilizing $g_{1T}^{(1)}(x)$ relied on a Wandzura-Wilczek approximation~\cite{Avakian:2007mv,Accardi:2009au,Kanazawa:2015ajw,Scimemi:2018mmi} that neglects quark-gluon-quark correlators to approximate $g_{1T}^{(1)}(x)$ in terms of an integral of the helicity PDF $g_{1}(x)$:~$g_{1T}^{(1)}(x) = x\int_x^1dy\,g_1(y)/y$. In addition, the twist-3 fragmentation piece to $A_{LT}$ is sensitive to a coupling of the chiral-odd twist-3 FF $E(z)$ with $h_1(x)$~\cite{Koike:2015yza}. No extractions exist of $E(z)$, but recent knowledge obtained about the closely related FF $\tilde{H}(z)$~\cite{Gamberg:2022kdb} allows us for the first time to develop a realistic input for $E(z)$ (in past numerical work, this function had been simply set to zero~\cite{Kanazawa:2014tda}). The potential for future measurements of $A_{LT}$, particularly in electron-nucleon collisions, to provide more direct information about $E(z)$ are intriguing due to the connection of this FF to dynamical quark mass generation in QCD~\cite{Accardi:2017pmi,Accardi:2019luo,Accardi:2020iqn}.
From the experimental side, measurements of $A_{LT}$ in single-inclusive processes like those introduced above are unfortunately lacking. The only data available are from Jefferson Lab 6 GeV (JLab6) on $A_{LT}$ in $\vec{e}\,n^\uparrow \to \pi\,X$~\cite{JeffersonLabHallA:2015vlz}. Therefore, in this paper we give rigorous numerical predictions for $A_{LT}$ in a variety of reactions and kinematic configurations in order to motivate future measurements. Namely, we will present results for $\vec{e}\,N^\uparrow \to \pi\,X$ for JLab 12 GeV (JLab12) with $N=n$, COMPASS with $N=p$, and the future Electron-Ion Collider (EIC) with $N=p$ (along with $\vec{e}\,p^\uparrow \to jet\,X$), as well as for the Relativistic Heavy Ion Collider (RHIC) for $p^\uparrow \vec{p}\to \{\pi, jet, \,{\rm or}\; \gamma\}\,X$. Even with the new information about $g_{1T}^{(1)}(x)$ and $\tilde{H}(z)$ previously mentioned, we still must employ approximations for or neglect certain twist-3 PDFs or FFs due to lack of input for them. Thus, one stands to gain further insight into multi-parton correlations through measurements of $A_{LT}$. Especially with only a few years of running left at RHIC, the world's only polarized proton-proton collider, one may forever lose the chance to measure $A_{LT}$ in $p^\uparrow \vec{p}\to \{\pi, jet,\;{\rm or}\; \gamma\}\,X$.
The paper is organized as follows:~in Sec.~\ref{s:theory} we review the analytical formulas for $A_{LT}$ that have been derived in the literature for the processes of interest along with the twist-3 PDFs and FFs that enter them. We also discuss the inputs and approximations used for these various non-perturbative functions as well as our strategy for computing the average values and uncertainties of our predictions. We examine the main selected results for $A_{LT}$ in electron-nucleon and proton-proton collisions, and their implications for future measurements, in Sec.~\ref{s:results}. The plots themselves can be can be found Appendix~\ref{s:app_a} (for electron-nucleon) and Appendix~\ref{s:app_b} (for proton-proton). In Sec.~\ref{s:concl} we close with our conclusions and outlook.
\section{Theoretical and Computational Background}
\label{s:theory}
In this section we review the analytical formulas for $A_{LT}$ needed for our computational work along with the relevant non-perturbative functions and certain relations between them. The asymmetry itself is generically defined as
\begin{equation}
A_{LT} \equiv \frac{\dfrac{1}{4}\Big\{\!\left[d\sigma_{LT}(+,\uparrow) - d\sigma_{LT}(-,\uparrow)\right]-\left[d\sigma_{LT}(+,\downarrow) - d\sigma_{LT}(-,\downarrow)\right]\!\Big\}} {d\sigma_{unp}}\,, \label{e:ALT}
\end{equation}
where $d\sigma_{LT}$ ($d\sigma_{unp}$) is the longitudinal-transverse spin-dependent (unpolarized) cross section, with $+$ ($-$) indicating a particle with positive (negative) helicity, and $\uparrow$ ($\downarrow$) denoting a particle with transverse spin along the designated positive (negative) transverse axis (e.g., $\pm y$).
We break this section down into the electron-nucleon and proton-proton cases.
\subsection{$\boldsymbol{A_{LT}}$ in Electron-Nucleon Collisions}
We consider the reaction $\vec{e}\,N^\uparrow\to \{\pi\;{\rm or}\;jet\}\,X$, where the produced final-state particle has a transverse momentum $P_T$, which sets the hard scale for the process. We define the +z-axis to be the direction of $N^\uparrow$'s momentum in the electron-nucleon center-of-mass (cm) frame. In addition to $P_T$, the asymmetry also depends on the cm energy $\sqrt{S}$ and rapidity $\eta$ (which can also be written in terms of $x_F=2P_T\sinh(\eta)/\sqrt{S}$). The coordinate system is such that at fixed-target experiments like JLab and COMPASS, the final-state particle is produced in the backward region (i.e., negative rapidity). The two other Mandelstam variables at the hadronic level are
$T= \left(-\sqrt{S}\,\sqrt{P_{T}^2 + x_F^2 S/4}+x_F S/2\right)$ and $U=\left(-\sqrt{S}\,\sqrt{P_{T}^2 + x_F^2S/4}-x_F S/2\right)$.
We can then write $A_{LT}$ for the case of pion production as~\cite{Kanazawa:2014tda,Kanazawa:2015ajw},
\begin{align}
A^{\vec{e}N^\uparrow\!\to\pi X}_{LT} = \frac{\displaystyle\int_{z_{min}}^1\dfrac{dz} {z^3}\!\left(\dfrac{-4P_{T}} {S+T/z}\right)\!\dfrac{1} {x}\displaystyle\sum_a e_a^2\left[\frac{M} {\hat{u}}\,D_1^{\pi/a}(z)\,\mathcal{G}^{a/N}\!(x,\hat{s},\hat{t},\hat{u}) + \dfrac{M_\pi} {z\hat{t}}\,h_1^{a/N}\!(x)\,E^{\pi/a}(z)\left(-\frac{\hat{s}}{\hat{t}}\right)\right]} {\displaystyle\int_{z_{min}}^1 \dfrac{dz} {z^2}\,\dfrac{1} {S+T/z}\,\dfrac{1} {x}\displaystyle\sum_a e_a^2\, f_1^{a/N}\!(x)\,D_1^{\pi/a}(z)\left(\dfrac{\hat{s}^2+\hat{u}^2} {\hat{t}^2}\right)}\,, \label{e:ALTeN}
\end{align}
where
\begin{align}
\mathcal{G}(x,\hat{s},\hat{t},\hat{u})&= \left(g_{1T}^{(1)}\!(x)-x\frac{dg_{1T}^{(1)}\!(x)} {dx}\right)\!\left(\frac{\hat{s}(\hat{s}-\hat{u})} {2\hat{t}^{\hspace{0.025cm}2}}\right)+x\,g_T\!(x)\left(-\frac{\hat{s}\hat{u}}{\hat{t}^2}\right)+x\,g_1\!(x)\left(\frac{\hat{u}(\hat{s}-\hat{u})}{2\hat{t}^{\hspace{0.025cm}2}}\right), \label{e:scriptG}
\end{align}
with $x= -(U/z)/(S+T/z)$,
$z_{min} = -(T+U)/S$, and the partonic Mandelstam variables
$\hat{s} = xS, \hat{t} = xT/z, \hat{u} = U/z$. The sum $\sum_a$ is over all light quark and antiquark flavors ($a=q\;{\rm or}\; \bar{q}$), $e_a$ is the quark or antiquark charge (in units of the positron charge $e$), and $M$ ($M_\pi$) is the nucleon (pion) mass.
The non-perturbative functions in Eqs.~(\ref{e:ALTeN}), (\ref{e:scriptG}) include the (twist-2) unpolarized PDF $f_1(x)$ and FF $D_1(z)$, helicity PDF $g_1(x)$, and transversity PDF $h_1(x)$, along with the kinematical twist-3 PDF $g_{1T}^{(1)}(x)$ (first-moment of the worm-gear TMD), intrinsic twist-3 PDF $g_T(x)$, and (chiral-odd) intrinsic twist-3 FF $E(z)$. We see that Eq.~(\ref{e:ALTeN}) can be separated into two terms:~one involving twist-3 PDFs (what we will call the ``distribution term'') and one involving a twist-3 FF (what we will call the ``fragmentation term''). We note that the case of jet production~\cite{Kang:2011jw} can be readily obtained from Eq.~(\ref{e:ALTeN}) by replacing $D_1(z)$ with $\delta(1-z)$ and setting the fragmentation term to zero.
Some readers may be familiar with the more widely studied/measured $A_{LT}$ asymmetry in inclusive DIS $\vec{e}\,N^\uparrow\to e\,X$~\cite{Anthony:1996mw,Abe:1997qk,Abe:1998wq,Anthony:2002hy,Zheng:2004ce,Kramer:2005qe,Flay:2016wie,Armstrong:2018xgk}, where the scattered electron is detected in the final state instead of a pion. In that process, the entire result depends only on $g_T(x)$, which is connected to the color Lorentz force on a struck quark in DIS~\cite{Burkardt:2008ps}. Already Eq.~(\ref{e:ALTeN}) makes apparent the rich structure of multi-parton correlators one is sensitive to in $A_{LT}$ for $\vec{e}\,N^\uparrow\to \pi\,X$ that cannot be accessed in inclusive DIS. This presents both a challenge, in that one has several unknown twist-3 functions, but also an opportunity to probe different aspects of multi-parton correlations in hadrons.
As alluded to above, there are different categories of twist-3 correlators:~kinematical, intrinsic, and also dynamical~\cite{Kanazawa:2015ajw}. The kinematical twist-3 functions are generically first-moments of twist-2 TMDs ($f^{(1)}(x)\equiv \int d^2\vec{k}_T \,\vec{k}_T^2/(2M^2)\,f(x,\vec{k}_T^2)$); intrinsic use a twist-3 Dirac projection in a quark-quark correlator; and dynamical are quark-gluon-quark or tri-gluon correlators. These twist-3 PDFs or FFs are not independent of each other and can be related through QCD equation-of-motion relations (EOMRs) and Lorentz invariance relations (LIRs). We refer the reader to Ref.~\cite{Kanazawa:2015ajw} (and references therein) for an extensive overview of collinear twist-3 functions, including their correlator definitions, derivations of EOMRs and LIRs, and how to express kinematical and intrinsic twist-3 functions in terms of the dynamical ones. For the PDFs relevant to our study (see Eq.~(\ref{e:scriptG})), we note the following relations~\cite{Tangerman:1994bb,Kotzinian:1995cz,Metz:2008ib,Accardi:2009au,Kanazawa:2015ajw}:
\begin{align}
g_T^{q/N}\!(x) &= g_1^{q/N}\!(x)+\frac{dg_{1T}^{(1)q/N}\!(x)} {dx} - 2\mathcal{P}\!\int_{-1}^1 dy \,\frac{G_{FT}^{q/N}\!(x,y)}{(x-y)^2}\,,\label{e:LIR}\\[0.3cm]
g_{1T}^{(1)q/N}(x) &= xg_T^{q/N}\!(x) -\frac{m_q}{M}\,h_1^{q/N}\!(x)+\mathcal{P}\int_{-1}^1dx_1 \frac{F_{FT}^{q/N}(x,x_1)-G_{FT}^{q/N}(x,x_1)}{x-x_1}\,,\label{e:EOMR}\\[0.3cm]
g_T^{q/N}\!(x)&=\int_x^{\epsilon(x)} dy\,\frac{g_1^{q/N}\!(y)}{y} + \frac{m_q}{M}\left(\frac{h_1^{q/N}\!(x)}{x} + \int_{\epsilon(x)}^x dy\,\frac{h_1^{q/N}\!(y)}{y^2}\right)\nonumber\\
&\hspace{0.2cm}+\,\int_x^{\epsilon(x)}\,\frac{dx_1}{x_1^2}\,\mathcal{P}\!\int_{-1}^1 dx_2\left[\frac{1-x_1\delta(x_1-x)}{x_1-x_2}F_{FT}^{q/N}\!(x_1,x_2) - \frac{3x_1-x_2-x_1(x_1-x_2)\delta(x_1-x)}{(x_1-x_2)^2}G_{FT}^{q/N}\!(x_1,x_2)\right]\,,\label{e:gTdyn}\\[0.3cm]
g_{1T}^{(1)q/N}\!(x)&=x\int_x^{\epsilon(x)} dy\,\frac{g_1^{q/N}\!(y)}{y} + \frac{m_q}{M} x\int_{\epsilon(x)}^x dy\,\frac{h_1^{q/N}\!(y)}{y^2} \nonumber\\
&\hspace{0.3cm} +\,x\int_x^{\epsilon(x)}\,\frac{dx_1}{x_1^2}\,\mathcal{P}\!\int_{-1}^1 dx_2\left[\frac{F_{FT}^{q/N}\!(x_1,x_2)}{x_1-x_2} - \frac{(3x_1-x_2)G_{FT}^{q/N}\!(x_1,x_2)}{(x_1-x_2)^2}\right]\,,\label{e:g1Tdyn}
\end{align}
where $\mathcal{P}$ denotes the principal value prescription, $\epsilon(x)\equiv 2\theta(x)-1$, $m_q$ is the quark mass, and $F_{FT}(x,x_1)$, $G_{FT}(x,x_1)$ are dynamical twist-3 PDFs (with $F_{FT}(x,x_1)$ giving the Qiu-Sterman function when $x=x_1$). The twist-2, kinematical twist-3, and intrinsic twist-3 PDFs all have support $-1\le x \le 1$, where $g_1^{q/N}\!(-x)=g_1^{\bar{q}/N}\!(x)$, $g_T^{q/N}\!(-x)=g_T^{\bar{q}/N}\!(x)$, $g_{1T}^{(1)q/N}\!(-x)=-g_{1T}^{(1)\bar{q}/N}\!(x)$, and $h_1^{q/N}\!(-x)=-h_1^{\bar{q}/N}\!(x)$. The dynamical twist-3 PDFs have support $|x|\le 1$, $|x_1|\le 1$, and $|x-x_1|\le 1$, with $F_{FT}^{q/N}\!(-x_1,-x)=F_{FT}^{\bar{q}/N}\!(x,x_1)$ and $G_{FT}^{q/N}\!(-x_1,-x)=-G_{FT}^{\bar{q}/N}\!(x,x_1)$~\cite{Kanazawa:2015ajw}. The first expression~(\ref{e:LIR}) is a LIR and (\ref{e:EOMR}) is an EOMR, while (\ref{e:gTdyn}), (\ref{e:g1Tdyn}) are the result of solving Eqs.~(\ref{e:LIR}), (\ref{e:EOMR}) for the respective functions~\cite{Kanazawa:2015ajw} so that they only involve dynamical twist-3 correlators (with possibly a twist-2 term, as above with $\int_x^{\epsilon(x)} \!dy \,g_1(y)/y$). Neglecting the quark mass terms and dynamical twist-3 PDFs in Eqs.~(\ref{e:gTdyn}), (\ref{e:g1Tdyn}) leads to the well-known Wandzura-Wilczek (WW) approximations~\cite{Wandzura:1977qf,Tangerman:1994bb,Kotzinian:1995cz,Kotzinian:1997wt,Kotzinian:2006dw,Avakian:2007mv,Metz:2008ib,Accardi:2009au}
\begin{equation}
g_T^{a/N}\!(x)\overset{{\rm WW}}{\approx}\int_x^1 dy\,\frac{g_1^{a/N}\!(y)}{y}\,,\quad\quad\quad
g_{1T}^{(1)a/N}\!(x)\overset{{\rm WW}}{\approx}x\int_x^1 dy\,\frac{g_1^{a/N}\!(y)}{y} \,,\label{e:gTg1TWW}
\end{equation}
where $a=q\;{\rm or}\; \bar{q}$. Until recently, the WW approximation was the only input available for $g_{1T}^{(1)}(x)$. Now with the extraction of $g_{1T}^{(1)}(x)$ in Ref.~\cite{Bhattacharya:2021twu}, we do not necessarily have to resort to the WW approximation. The expression in Eq.~(\ref{e:g1Tdyn}) makes clear there is more structure embedded in $g_{1T}^{(1)}(x)$ than what is accounted for in the WW approximation. Likewise, using the extracted $g_{1T}^{(1)}(x)$ from Ref.~\cite{Bhattacharya:2021twu} in Eq.~(\ref{e:LIR}) in principle inserts information about multi-parton correlators into the expression for $g_T(x)$, which the WW approximation does not encode. Even so, we do not have complete information on $g_T(x)$ because $G_{FT}(x,x_1)$ is not known. In Ref.~\cite{Bhattacharya:2020cen}, $g_{T}^{u-d}(x)$ was extracted for the first time in lattice QCD using the so-called quasi-distribution approach~\cite{Ji:2013dva}. An interesting prospect is one in principle could obtain information on $G_{FT}(x,x_1)$ through a flavor-separated computation of $g_T(x)$ on the lattice (taking $g_1(x)$ and $g_{1T}^{(1)}(x)$ as known functions).
On the fragmentation side we have~\cite{Kanazawa:2015ajw}
\begin{align}
E^{h/q}(z)=-2z\left(\int_z^\infty \frac{dz_1}{z_1^2}\,\frac{\hat{H}_{FU}^{\Re, h/q}(z,z_1)}{\frac{1}{z}-\frac{1}{z_1}}-\frac{m_q}{2M_h}D_1^{h/q}(z)\right), \label{e:E_EOMR}
\end{align}
where $\hat{H}_{FU}(z,z_1)$ is a quark-gluon-quark (dynamical twist-3) FF, and $M_h$ is the hadron mass. The support properties are $0\le z \le 1$ and $z<z_1<\infty$~\cite{Kanazawa:2015ajw}. We mention again that dynamical twist-3 FFs are complex valued because of the lack of a time-reversal constraint in the fragmentation sector and have both real $\Re$ and imaginary $\Im$ parts. Recently, the FF $\tilde{H}(z)$ has been extracted~\cite{Gamberg:2022kdb}, and it is connected to the imaginary part of the same underlying correlator $\hat{H}_{FU}(z,z_1)$ as $E(z)$ depends on~\cite{Kanazawa:2015ajw}:
\begin{equation}
\tilde{H}^{h/q}(z)=2z\int_z^\infty \frac{dz_1}{z_1^2}\,\frac{\hat{H}_{FU}^{\Im,h/q}(z,z_1)}{\frac{1}{z}-\frac{1}{z_1}}\,. \label{e:Htilde}
\end{equation}
We will use $\tilde{H}(z)$ to build up plausible scenarios for $E(z)$ in our numerical work.
\subsection{$\boldsymbol{A_{LT}}$ in Proton-Proton Collisions}
We now consider the reaction $p^\uparrow \vec{p}\to \{\pi, jet,\,{\rm or}\;\gamma\}\,X$. We define the +z-axis to be the direction of $p^\uparrow$'s momentum in the proton-proton center-of-mass (cm) frame. There are three pieces to this observable for the case of pion production, depending on whether the twist-3 effects occur in $p^\uparrow$, $\vec{p}$, or $\pi$ (for $jet$ and $\gamma$, one only has the first two terms). We write $A_{LT}$ for this case as
\begin{equation}
A_{LT}^{p^\uparrow\vec{p}\to \pi X} = \frac{d\sigma_{LT}^{\rm Tdist}+d\sigma_{LT}^{\rm Ldist}+d\sigma_{LT}^{\rm frag}}{d\sigma_{unp}}\,,\label{e:ALTpp}
\end{equation}
where in the numerator we have indicated whether the term contains twist-3 effects from $p^\uparrow$ (transversely polarized distribution -- ``Tdist'')~\cite{Metz:2012fq}, from $\vec{p}$ (longitudinally polarized distribution -- ``Ldist'')~\cite{Koike:2016ura}, or from $\pi$ (fragmentation -- ``frag'')~\cite{Koike:2015yza}. The expression for the unpolarized cross section reads:
\begin{align}
d\sigma_{unp} &= \frac{\alpha_S^2}{S} \int_{z_{min}}^1 dz \int_{x_{min}}^1\frac{dx}{x}\frac{1}{x'z^2(xS+U/z)}\sum_i\sum_{a,b,c}\,f_1^{a/p}(x)\,f_1^{b/p}(x')\,D_1^{\pi/c}(z)\,H_U^i(\hat{s},\hat{t},\hat{u})\,,\label{e:sig_unp}
\end{align}
where $z_{min}=-(T+U)/S$, $x_{min}=-(U/z)/(S+T/z)$, $x' = -(xT/z)/(xS+U/z)$, and the summations are over all channels $i$ and parton flavors $a,b,c$. The hard factors $H_U^i(\hat{s},\hat{t},\hat{u})$ depend on the partonic Mandelstam variables $\hat{s}=xx'S,\hat{t}=xT/z,\hat{u}=x'U/z$, and they can be found in Ref.~\cite{Kouvaris:2006zy}.
We next turn to the longitudinal-transverse polarized cross sections. For $d\sigma_{LT}^{\rm Tdist}$ we have~\cite{Metz:2012fq}
\begin{align}
d\sigma_{LT}^{\rm Tdist} = -\frac{2\alpha_s^2MP_T}{S}\int_{z_{min}}^1 dz\int_{x_{min}}^1 \frac{dx}{x}\frac{1}{x'z^3(xS+U/z)}\sum_i\sum_{a,b,c}\frac{1}{\hat{m}_i}\,\mathcal{G}_i^{a/p^\uparrow}\!(x,\hat{s},\hat{t},\hat{u})\,g_1^{b/\vec{p}}(x')\,D_1^{\pi/c}(z)\,,\label{e:Tdist}
\end{align}
where
\begin{align}
\mathcal{G}_i(x,\hat{s},\hat{t},\hat{u}) &= \left(g_{1T}^{(1)}(x)-x\frac{dg_{1T}^{(1)}(x)}{dx}\right)H^i_{\tilde{g}}(\hat{s},\hat{t},\hat{u})
+ xg_T(x)\,H_{1,G_{DT}}^i(\hat{s},\hat{t},\hat{u}) +\frac{x}{2}\left(g_1(x)-g_T(x)\right)H_{3,G_{DT}}^i(\hat{s},\hat{t},\hat{u})\nonumber\\[0.1cm]
&\hspace{0.3cm}+\,\left[g_{1T}^{(1)}(x)+\mathcal{P}\int_{-1}^1 \frac{dx_1}{x_1}\,\frac{x\left(F_{FT}(x,x_1)+G_{FT}(x,x_1)\right)}{x-x_1}\right]H_{2,G_{DT}}^i(\hat{s},\hat{t},\hat{u})\,.\label{e:scriptGpp}
\end{align}
Some comments are in order about the expressions~(\ref{e:Tdist}), (\ref{e:scriptGpp}). First, the variable $\hat{m}_i$ in Eq.~(\ref{e:Tdist}) is either $\hat{s}$, $\hat{t}$, or $\hat{u}$ depending on the channel $i$, with the specific values found in Table 1 of Ref.~\cite{Metz:2012fq}.\footnote{We note a typo in the last row for the $\hat{t}$ column of Table 1 in Ref.~\cite{Metz:2012fq}, where the channel should read $q\bar{q}\to \bar{q}'q'$.} Second, the original expression in Ref.~\cite{Metz:2012fq} (see Eq.~(17) of that paper) is written in terms of the functions $\tilde{g}(x)$ and $F_{DT}(x,x_1), G_{DT}(x,x_1)$. The former is just a different notation for $g_{1T}^{(1)}(x)$. The latter are ``D-type'' dynamical twist-3 PDFs that use the covariant derivative, whereas we have chosen to write the result in terms of ``F-type'' functions $F_{FT}(x,x_1), G_{FT}(x,x_1)$ that use the field strength tensor. They are related via~\cite{Eguchi:2006qz}
\begin{align}
F_{DT}(x,x_1) &= \mathcal{P}\frac{1}{x-x_1}\,F_{FT}(x,x_1)\,,\\
G_{DT}(x,x_1) &= \mathcal{P}\frac{1}{x-x_1}\,G_{FT}(x,x_1) + \delta(x-x_1)\,g_{1T}^{(1)}(x)\,.
\end{align}
Lastly, we continued to ``optimize'' Eq.~(\ref{e:scriptGpp}) from the original version in Ref.~\cite{Metz:2012fq} so that it is written in terms of a maximal set of functions for which there is input for from the literature. An observation made in Ref.~\cite{Metz:2012fq} was that the hard factors $H^i_{F_{DT}}$, $H^i_{G_{DT}}$ found in Appendix~A\footnote{The hard factors $H^i_{\tilde{g}}$ can also be found in Appendix A of Ref.~\cite{Metz:2012fq}.} of that paper can be broken down into three types of terms, namely, $H^i = H_1^i +H_2^i/(1-\xi)+H_3^i/\xi$, where $\xi=(x-x_1)/x$, with $H_{1,F_{DT}}^i=H^i_{1,G_{DT}}$, $H^i_{2,F_{DT}}=-H^i_{2,G_{DT}}$, and $H^i_{3,F_{DT}}=0$. This insight allows one to use the LIR~(\ref{e:LIR}) and EOMR~(\ref{e:EOMR}) to obtain the final form in Eq.~(\ref{e:scriptGpp}), where now the only non-perturbative functions we lack input for are $F_{FT}(x,x_1),G_{FT}(x,x_1)$, and we will then ignore those terms in our numerical work.
We now give the formulas for the remaining two terms in the numerator of Eq.~(\ref{e:ALTpp}). For $d\sigma_{LT}^{\rm Ldist}$ we have~\cite{Koike:2016ura}
\begin{align}
d\sigma_{LT}^{\rm Ldist} = -\frac{2\alpha_s^2MP_T}{S} \int_{z_{min}}^1 dz\int_{x_{min}}^1 \frac{dx}{x}\frac{1}{z^3(xS+U/z)}\sum_i\sum_{a,b,c}\,h_1^{a/p^\uparrow}\!\!(x)\,\mathcal{H}^{b/\vec{p}}(x',\hat{s},\hat{t},\hat{u})\,D_1^{\pi/c}(z)\,,\label{e:Ldist}
\end{align}
where
\begin{equation}
\mathcal{H}(x',\hat{s},\hat{t},\hat{u}) = h_1(x')\,H_{1L}^i(\hat{s},\hat{t},\hat{u})+h_L(x')\,H_{2L}^i(\hat{s},\hat{t},\hat{u})+\frac{dh_{1L}^{\perp (1)}(x')}{dx'}\,H_{3L}^i(\hat{s},\hat{t},\hat{u})\,. \label{e:scriptH}
\end{equation}
The hard factors $H^i_{\{1,2,3\}L}$ correspond to $\hat{\sigma}_{\{1,2,3\}}$ in Eqs.~(16)--(21) of Ref.~\cite{Koike:2016ura}. The function $h_L(x)$ is an intrinsic twist-3 function while $h_{1L}^{\perp(1)}(x)$ is kinematical twist-3 (first-moment of the other worm-gear TMD function $h_{1L}^{\perp}$). Unlike $g_{1T}^{(1)}(x)$, there are no phenomenological extractions of $h_{1L}^{\perp(1)}(x)$. Therefore, in our numerical work we must use WW approximations that connect $h_L(x)$ and $h_{1L}^{\perp(1)}(x)$ to the twist-2 transversity PDF $h_1(x)$~\cite{Tangerman:1994bb,Metz:2008ib,Kanazawa:2015ajw}:
\begin{equation}
h_L^{a/N}\!(x)\overset{{\rm WW}}{\approx} 2x\int_x^1 dy\,\frac{h_1^{a/N}\!(y)}{y^2}\,,\quad\quad\quad
h_{1L}^{\perp(1)a/N}\!(x)\overset{{\rm WW}}{\approx}x^2\int_x^1 dy\,\frac{h_1^{a/N}\!(y)}{y^2} \,,\label{e:hLh1LperpWW}
\end{equation}
where $a=q\;{\rm or}\; \bar{q}$. Finally, for $d\sigma_{LT}^{\rm frag}$ we have~\cite{Koike:2015yza}
\begin{align}
d\sigma_{LT}^{\rm frag} = \frac{2\alpha_s^2MP_T}{S} \int_{z_{min}}^1 dz\int_{x_{min}}^1 \frac{dx}{x}\frac{1}{x'z^4(xS+U/z)}\sum_i\sum_{a,b,c}\,h_1^{a/p^\uparrow}\!\!(x)\,g_1^{b/\vec{p}}(x')\,E^{\pi/c}(z)\,H_{f}^i(\hat{s},\hat{t},\hat{u})\,,\label{e:frag}
\end{align}
where the hard factors $H^i_{f}$ correspond to $\hat{\sigma}_i$ in Eq.~(15) of Ref.~\cite{Koike:2015yza}, and $E(z)$ is the same dynamical twist-3 FF introduced in the electron-nucleon case (\ref{e:ALTeN}) (see also Eq.~(\ref{e:E_EOMR})).
We mention that the result for $A_{LT}$ in $p^\uparrow \vec{p}\to jet \,X$ can be obtained by replacing $D_1(z)$ by $\delta(1-z)$ in Eqs.~(\ref{e:sig_unp}), (\ref{e:Tdist}), (\ref{e:Ldist}) and setting $d\sigma_{LT}^{\rm frag}$ to zero. We refer the reader to Appendix B of Ref.~\cite{Metz:2012fq} (see also \cite{Liang:2012rb}) for the $d\sigma_{LT}^{\rm Tdist}$ formula for $p^\uparrow \vec{p}\to \gamma \,X$.\footnote{Note that $\hat{m}_i=\hat{u}$ in this case for both channels ($qg\to \gamma q$ and $q\bar{q}\to \gamma g$), which was not explicitly stated in Ref.~\cite{Metz:2012fq}.} To the best of our knowledge, the $d\sigma_{LT}^{\rm Ldist}$ formula for $p^\uparrow \vec{p}\to \gamma \,X$ has not been derived yet in the literature. Since we consider only direct photons, there is no $d\sigma_{LT}^{\rm frag}$ term. The unpolarized cross section $d\sigma_{unp}$ for $pp\to \gamma \,X$ can be found in Ref.~\cite{Kouvaris:2006zy}.
\subsection{Numerical Methodology}\label{s:numerics}
We end this section with a discussion of our strategy for obtaining realistic numerical predictions for $A_{LT}$ given the information set forth in the previous two subsections.
\subsubsection{Non-Perturbative Inputs}
With regard to input for the non-perturbative functions, we use CT18 NLO~\cite{Hou:2019qau} for $f_1(x)$, DSS14 NLO~\cite{deFlorian:2014xna} for $D_1(z)$, NNPDFpol1.1~\cite{Nocera:2014gqa} for $g_1(x)$, and JAM3D-22~\cite{Gamberg:2022kdb} for $h_1(x)$, all via LHAPDF 6.2.3~\cite{Buckley:2014ana}. For $g_{1T}^{(1)}(x)$ and $g_T(x)$ we consider two scenarios:
\begin{itemize}
\item[(1)] {\bf quark-gluon-quark (qgq) scenario:}~We use $g_{1T}^{(1)}(x)$ extracted in Ref.~\cite{Bhattacharya:2021twu}, which in principle implicitly encodes dynamical twist-3 functions (see Eq.~(\ref{e:g1Tdyn})), and Eq.~(\ref{e:LIR}) for $g_T(x)$ with $G_{FT}(x,x_1)$ set to zero (since we have no direct input for it). This is the maximal amount of information about quark-gluon-quark correlations we can include in $g_T(x)$ and $g_{1T}^{(1)}(x)$.
\item[(2)] {\bf WW scenario:}~We use Eq.~(\ref{e:gTg1TWW}) for $g_T(x)$ and $g_{1T}^{(1)}(x)$, which completely neglects quark-gluon-quark correlations.
\end{itemize}
A plot comparing the two different scenarios for $g_{1T}^{(1)}(x)$ is shown in Fig.~\ref{f:g1T_vs_x}, and for $g_T(x)$ is shown in Fig.~\ref{f:gT_vs_x} along with a lattice QCD (LQCD) calculation (for the isovector $u-d$ combination) of the latter~\cite{Bhattacharya:2020cen}.\footnote{We note that the $g_T(x)$ computation in the qgq scenario depends on $g_1(x)$, where we use NNPDF replicas~\cite{Nocera:2014gqa}, and $g_{1T}^{(1)}(x)$, where we use the replicas from Bhattacharya, {\it et al.}~\cite{Bhattacharya:2021twu}. To calculate the central curve and uncertainty band in this case, we use the same bootstrapping method described around Eq.~(\ref{e:Zstat}) below.} We remark that $g_{1T}^{(1)u}(x)$ is larger in the qgq scenario and falls off slower at larger $x$. Both the qgq and WW scenarios are compatible within error bands for $g_{1T}^{(1)d}(x)$. The behavior of $g_T(x)$ in the two scenarios is quite different, mostly due to the $dg_{1T}^{(1)}(x)/dx$ term that enters Eq.~(\ref{e:LIR}) for the qgq case, which causes a change in sign in $g_T(x)$ at moderate $x$ values. For the $d$ quark, the two scenarios are still compatible within error bands, but for the $u$ quark the qgq scenario is generally larger than the WW (in addition to having the aforementioned sign change). The lattice computation for $g_T^{u-d}(x)$ shows agreement with the qgq and WW scenarios up to $x\approx 0.4$. At larger $x$, the WW scenario goes to zero the fastest, while the qgq scenario exhibits a change in sign and slower decrease as $x \to 1$. The lattice calculation at large $x$ must deal with systematic effects in reconstructing the $x$ dependence that make the behavior of $g_T(x)$ in that region unreliable~\cite{Bhattacharya:2020cen}. Once there is a rigorous lattice result of $g_T(x)$ across a wider range of $x$ and for individual $u$ and $d$ flavors, one in principle could use the difference between LQCD and the qgq scenario (taking $g_1(x)$ and $g_{1T}^{(1)}(x)$ as known functions) to extract information on the dynamical twist-3 PDF $G_{FT}(x,x_1)$ (see Eq.~(\ref{e:LIR})).
\begin{figure}[b!]
\includegraphics[width=0.75\textwidth]{g1T_vs_x_plot.pdf}\vspace{-0.2cm}
\caption{Plot of the up ($u$) and down ($d$) quark in a proton kinematic twist-3 PDF $xg_{1T}^{(1)}(x)$ vs.~$x$ at $Q^2 = 4\,{\rm GeV^2}$ for the qgq scenario (blue dashed) and WW scenario (magenta solid) (both with 68\% C.L.~error bands). \vspace{-0.3cm}}
\label{f:g1T_vs_x}
\end{figure}
\begin{figure}[t!]
\includegraphics[width=1\textwidth]{gT_vs_x_plot.pdf}\vspace{-0.2cm}
\caption{Plot of the $u$, $d$, and $u-d$ in a proton intrinsic twist-3 PDF $xg_T(x)$ vs.~$x$ at $Q^2 = 4\,{\rm GeV^2}$ for the qgq scenario (blue dashed), WW scenario (magenta solid), and (for $u-d$) the lattice QCD (LQCD) calculation (green dotted) from Ref.~\cite{Bhattacharya:2020cen} (all with 68\% C.L.~error bands) . \vspace{-0.3cm}}
\label{f:gT_vs_x}
\end{figure}
The last function we need input for is $E(z)$. This intrinsic twist-3 FF was previously given attention in the literature because of its connection to dynamical quark mass generation in QCD~\cite{Accardi:2017pmi,Accardi:2019luo,Accardi:2020iqn}, which can also allow one to probe the transversity PDF $h_1(x)$ in {\it inclusive} DIS~\cite{Accardi:2017pmi}. As explicitly set forth in Eqs.~(\ref{e:E_EOMR}), (\ref{e:Htilde}), $E(z)$ is driven by the same quark-gluon-quark FF ($\hat{H}_{FU}(z,z_1)$) as $\tilde{H}(z)$, which we have input for from the JAM3D-22 analysis~\cite{Gamberg:2022kdb}. Even so, there are some caveats with establishing this connection. $E(z)$ depends on the real part of $\hat{H}_{FU}(z,z_1)$, while $\tilde{H}(z)$ depends on the imaginary part, and the two need not necessarily be related. The functions also obey different sum rules~\cite{Accardi:2020iqn}:
\begin{equation}
\sum_{h}\sum_{S_h} M_h\int_0^1 dz\,E^{h/q}(z)=M_j\,,\quad\quad\quad \sum_{h}\sum_{S_h} M_h\int_0^1 dz\,\tilde{H}^{h/q}(z)=0\,, \label{e:sumrules}
\end{equation}
where the summation is over all hadrons $h$ and their spins $S_h$. The mass $M_j$ is the (gauge-invariant, non-perturbative) ``jet mass'' of a color-screened dressed quark propagating in the vacuum~\cite{Accardi:2019luo,Accardi:2020iqn}, which can be substantially larger than the current quark mass $m_q$.\footnote{The first term in Eq.~(\ref{e:E_EOMR}) can be identified as $\tilde{E}(z)$, which then allows for the decomposition $M_j=m_q+m_q^{corr}$ discussed in Refs.~\cite{Accardi:2019luo,Accardi:2020iqn}, where $M_j$ is broken down into the current quark mass $m_q$ and a term $m_q^{corr}$ that encodes dynamical mass generation due to quark-gluon-quark correlations.} In the next section, we will revisit the possibility of $A_{LT}$ measurements, especially in electron-nucleon collisions, providing direct information about $E(z)$, and, therefore, potentially giving insight into $M_j$. These disclaimers notwithstanding, we think three realistic scenarios to study for $E(z)$ are $E(z)=-\tilde{H}(z)$, $E(z)=0$, and $E(z)=\tilde{H}(z)$. This accounts for $E(z)$ either being the same order of magnitude as $\tilde{H}(z)$ (although we cannot fix its sign) or $E(z)$ being significantly smaller than $\tilde{H}(z)$. A plot for the $E(z)=-\tilde{H}(z)$ scenario is displayed in Fig.~\ref{f:E_vs_z}.
\begin{figure}[b!]
\includegraphics[width=0.75\textwidth]{E_vs_z_plot.pdf}\vspace{-0.2cm}
\caption{Plot of the (favored and unfavored) twist-3 FF $zE(z)$ vs.~$z$ at $Q^2 = 4\,{\rm GeV^2}$ for the $E(z)=-\tilde{H}(z)$ scenario, where $\tilde{H}(z)$ is taken from Ref.~\cite{Gamberg:2022kdb}. \vspace{-0.3cm}}
\label{f:E_vs_z}
\end{figure}
\subsubsection{Computation of Central Curves and Error Bands}
Clearly a numerical calculation of $A_{LT}$ in $\vec{e}\,N^\uparrow\to \{\pi\;{\rm or}\;jet\}\,X$ or $p^\uparrow \vec{p}\to \{\pi, jet,\,{\rm or}\;\gamma\}\,X$ depends on several non-perturbative inputs that have been extracted from various groups. We now discuss our procedure for obtaining the central curves and error bands for the results presented in the next section. To aid in this explanation, we write the asymmetries as
\begin{align}
A^{\vec{e}N^\uparrow\!\to\pi X}_{LT} &=\frac{d\sigma_{LT}^{\rm dist}(g_1,g_{1T}^{(1)},g_T,D_1)+d\sigma_{LT}^{\rm frag}(h_1,E)}{d\sigma_{unp}(f_1,D_1)}\nonumber\\
&\equiv A^{\vec{e}N^\uparrow\!\to\pi X}_{LT,{\rm dist}}(g_1,g_{1T}^{(1)},g_T,f_1,D_1)+A^{\vec{e}N^\uparrow\!\to\pi X}_{LT,{\rm frag}}(h_1,E,f_1,D_1)\,,\\[0.3cm]
A_{LT}^{p^\uparrow\vec{p}\to \pi X} &= \frac{d\sigma_{LT}^{\rm Tdist}(g_1,g_{1T}^{(1)},g_T,D_1)+d\sigma_{LT}^{\rm Ldist}(h_1,D_1)+d\sigma_{LT}^{\rm frag}(h_1,g_1,E)}{d\sigma_{unp}(f_1,D_1)}\nonumber\\
&\equiv A_{LT,{\rm Tdist}}^{p^\uparrow\vec{p}\to \pi X}(g_1,g_{1T}^{(1)},g_T,f_1,D_1) + A_{LT,{\rm Ldist}}^{p^\uparrow\vec{p}\to \pi X}(h_1,f_1,D_1)+A_{LT,{\rm frag}}^{p^\uparrow\vec{p}\to \pi X}(h_1,g_1,E,f_1,D_1)\,,\label{e:ALTppNP}
\end{align}
where we have explicitly indicated for each term which non-perturbative functions it depends on.\footnote{Note for $A_{LT,{\rm Ldist}}^{p^\uparrow\vec{p}\to \pi X}$, the non-perturbative functions that enter are $h_1(x)$, $h_L(x)$, and $h_{1L}^{\perp(1)}(x)$ (see Eqs.~(\ref{e:Ldist}), (\ref{e:scriptH})). However, since we use WW approximations for the latter two, which depend on $h_1(x)$ (see Eq.~(\ref{e:hLh1LperpWW})), we have only denoted a dependence on $h_1(x)$.} For $f_1(x)$ and $D_1(z)$, since they have relatively small uncertainties compared to the other PDFs and FFs, we simply use their central values and do not propagate their error into the computation.
We first focus on the electron-nucleon case. The fragmentation term $A^{\vec{e}N^\uparrow\!\to\pi X}_{LT,{\rm frag}}$ depends on $h_1(x)$ and $E(z)$ (recall we are using $\tilde{H}(z)$ to build our input for $E(z)$). Both $h_1(x)$ and $\tilde{H}(z)$ were extracted simultaneously in JAM3D-22~\cite{Gamberg:2022kdb}, and we use all 450 replicas from that analysis to compute the mean and standard deviation for $A^{\vec{e}N^\uparrow\!\to\pi X}_{LT,{\rm frag}}$. For the distribution term, we are considering the two previously mentioned scenarios (WW and qgq). In the WW scenario, $g_{1T}^{(1)}(x)$ and $g_T(x)$ both depend only on $g_1(x)$. We therefore can use all 100 replicas from NNPDFpol1.1~\cite{Nocera:2014gqa} to determine the mean and standard deviation for $A^{\vec{e}N^\uparrow\!\to\pi X}_{LT,{\rm dist}}$. The qgq scenario is more complicated because it depends on PDFs extracted by completely independent analyses, namely, $g_1(x)$ from NNPDFpol1.1~\cite{Nocera:2014gqa} and $g_{1T}^{(1)}(x)$ from Bhattacharya, {\it et al.}~\cite{Bhattacharya:2021twu} (recall our input for $g_T(x)$ depends on both these functions). For $g_{1T}^{(1)}(x)$ there are 200 replicas, so a complete calculation of $A^{\vec{e}N^\uparrow\!\to\pi X}_{LT,{\rm dist}}$ in the qgq scenario would require computing $100\times 200=20,\!000$ replicas. Instead, we bootstrap the result by randomly sampling replicas for $g_1(x)$ and for $g_{1T}^{(1)}(x)$ (with replacement). We continue to increase the number of replicas sampled and then calculate the (unequal variance or Welch's) $t$-statistic using the current and previous iterations, where~\cite{NumericalRecipes:2007}
\begin{equation}
t = \frac{\mu_1-\mu_2}{\sqrt{\sigma_1^2/N_1+\sigma_2^2/N_2}}\,, \label{e:Zstat}
\end{equation}
with $\mu$ the mean, $\sigma$ the standard deviation, and $N$ the number of ``data points'' (replicas sampled) of the respective distribution of $A_{LT}$ values for a given $P_T$. Once $|t|$ is such that the corresponding $p$-values $\gtrsim 0.1$, then we consider the two distributions statistically equivalent~\cite{NumericalRecipes:2007} and do not proceed with any further iterations.\footnote{For many $P_T$ points, the $p$-values were much greater than 0.1, approaching 1.0 in some cases.} (We also visually inspect the results to confirm the mean and standard deviation of $A_{LT}$ have converged.)
The $t$-statistic, and consequently the number of replicas required for convergence, is kinematic ($\sqrt{S},\,\eta,\,P_T$) and process (initial and final state) dependent. For example, 1500 replicas were needed for JLab12 while 3000 were necessary for the EIC at $\sqrt{S}=29\,{\rm GeV}$. Recall our calculation of $A^{\vec{e}N^\uparrow\!\to\pi X}_{LT,{\rm dist}}$ and $A^{\vec{e}N^\uparrow\!\to\pi X}_{LT,{\rm frag}}$ are totally uncorrelated from each other in that the respective non-perturbative functions that enter each term are from independent analyses by different groups. Thus, once we have the final sample, we determine the central curve and uncertainty ($68\%$ C.L.~error band) as
\begin{equation}
\langle A^{\vec{e}N^\uparrow\!\to\pi X}_{LT}\rangle = \langle A^{\vec{e}N^\uparrow\!\to\pi X}_{LT,{\rm dist}}\rangle + \langle A^{\vec{e}N^\uparrow\!\to\pi X}_{LT,{\rm frag}} \rangle\,, \quad\quad \delta \! A^{\vec{e}N^\uparrow\!\to\pi X}_{LT} = \sqrt{\left(\delta\! A^{\vec{e}N^\uparrow\!\to\pi X}_{LT,{\rm dist}}\right)^{\! 2} + \left(\delta\! A^{\vec{e}N^\uparrow\!\to\pi X}_{LT,{\rm frag}}\right)^{\! 2}}.\label{e:eN_avg_std}
\end{equation}
For the proton-proton case we follow a similar strategy, but there are some new aspects one must consider. The fragmentation term $A_{LT,{\rm frag}}^{p^\uparrow\vec{p}\to \pi X}$ now also depends on $g_1(x)$ (since there is a longitudinally polarized proton involved, not an electron). In addition, $A_{LT,{\rm Ldist}}^{p^\uparrow\vec{p}\to \pi X}$ depends on $h_1(x)$ and, consequently, must be computed simultaneously with $A_{LT,{\rm frag}}^{p^\uparrow\vec{p}\to \pi X}$ using the same replica sampled for $h_1(x)$ in that term. Therefore, we must bootstrap the entire $A_{LT}^{p^\uparrow\vec{p}\to \pi X}$ asymmetry using the replicas from NNPDFpol1.1, Bhattacharya, {\it et al.}, and JAM3D-22, following a similar procedure as outlined for the electron-nucleon case, for both the WW and qgq scenarios.\footnote{Note that even for the WW scenario we need to employ bootstrapping since $g_1(x)$ shows up in $A_{LT,{\rm frag}}^{p^\uparrow\vec{p}\to \pi X}$.} We again calculate the $t$-statistic of our $A_{LT}$ distributions (and visually inspect them) for different iterations to determine the number of samples required for convergence. As before, there is a kinematic and process dependence; for example, RHIC $\sqrt{S}=200\, {\rm GeV}$ at midrapidity ($\eta=0$) needed 2500 samples while 3500 were necessary at forward rapidity ($\eta=3.3$). Since all terms in $A_{LT}^{p^\uparrow\vec{p}\to \pi X}$ are correlated with each other, we determine the central curve and uncertainty using
\begin{equation}
\langle A^{p^\uparrow\vec{p}\to \pi X}_{LT}\rangle = \langle A^{p^\uparrow\vec{p}\to \pi X}_{LT,{\rm Tdist}}\rangle + \langle A_{LT,{\rm Ldist}}^{p^\uparrow\vec{p}\to \pi X} \rangle + \langle A_{LT,{\rm frag}}^{p^\uparrow\vec{p}\to \pi X} \rangle\,, \quad\quad \delta \! A^{p^\uparrow\vec{p}\to \pi X}_{LT} = \delta\!\! \left(A^{p^\uparrow\vec{p}\to \pi X}_{LT,{\rm Tdist}}+A^{p^\uparrow\vec{p}\to \pi X}_{LT,{\rm Ldist}}+A^{p^\uparrow\vec{p}\to \pi X}_{LT,{\rm frag}}\right).
\end{equation}
We mention that for the jet and photon final states in proton-proton collisions, since the fragmentation term does not enter, the transverse and longitudinal distribution terms are uncorrelated. The latter can be calculated using all replicas from JAM3D-22. The former requires bootstrapping for the qgq scenario, but for the WW scenario it can be computed using all replicas from NNPDFpol1.1. The central curve and uncertainty are then found exactly as in Eq.~(\ref{e:eN_avg_std}), with the replacements $(\vec{e}N^\uparrow\!\to\pi X) \longrightarrow (p^\uparrow\vec{p}\to \{jet\;{\rm or}\; \gamma\} X)$, ${\rm dist}\longrightarrow {\rm Tdist}$, ${\rm frag}\longrightarrow {\rm Ldist}$.
\section{Results and Discussion}
\label{s:results}
In this section we report our main results for $A_{LT}$ in electron-nucleon and proton-proton collisions. We mention that, especially at the EIC and RHIC, we extensively studied the ($\sqrt{S},\,\eta,\,P_T$) coverage and are able to provide predictions for any reaction at any kinematics upon request. Here we discuss a selective collection of plots, which can be found in Appendix~\ref{s:app_a} (for electron-nucleon) and Appendix~\ref{s:app_b} (for proton-proton), that highlight the main features of $A_{LT}$ in the single-inclusive processes under investigation. Each plot shows six cases based on the possible combinations of input for $g_{1T}^{(1)}(x)$, $g_T(x)$, and $E(z)$, i.e., qgq or WW scenario for $g_{1T}^{(1)}(x)$, $g_T(x)$, and $E(z)=-\tilde{H}(z)$, $E(z)=0$, or $E(z)=\tilde{H}(z)$. We remark again that the only measurement available of either $\vec{e}\,N^\uparrow \to \{\pi\;{\rm or}\; jet\}\,X$ or $p^\uparrow \vec{p}\to \{\pi, jet, \,{\rm or}\; \gamma\}\,X$ is from JLab6 for $\vec{e}\,n^\uparrow \to \pi\,X$~\cite{JeffersonLabHallA:2015vlz}. There have been a few numerical calculations of $\vec{e}\,N^\uparrow \to \{\pi\;{\rm or}\; jet\}\,X$~\cite{Kang:2011jw,Kanazawa:2014tda}, but only with central curves (no error bands) using the WW approximation for $g_{1T}^{(1)}(x)$, $g_T(x)$ and (for pion production) ignoring the fragmentation term involving $E(z)$. No numerical studies exist for the proton-proton case.
\subsection{Comparison with JLab6 Data} \label{s:JLab6}
The comparison between our predictions and the JLab6 measurement is shown in Fig.~\ref{f:JLab6}. We caution that the data are at $P_T<1\,{\rm GeV}$, so one has to be careful about using a perturbative calculation in this region, and what conclusions to infer from it. (In the computation, for any $P_T$-dependent kinematic quantities we used the actual experimental $P_T$ value, but in the non-perturbative functions we fixed $P_T=1\,{\rm GeV}$.) We see that generally all cases are able to describe the data relatively well, with the distribution term playing a dominant role over the fragmentation term. Nevertheless, there are hints, looking at the $E(z)=\tilde{H}(z)$ row of Fig.~\ref{f:JLab6}, that having a nonzero $E(z)$ with the same sign as $\tilde{H}(z)$ aids in obtaining better agreement with the data. We note that the qgq scenario has larger error bands than the WW scenario because the direct extraction of $g_{1T}^{(1)}(x)$ is much less constrained than $g_1(x)$ (which is used in the WW approximation). This is especially noticeable for $\pi^+$ because $g_{1T}^{d/p}(x)$ has a larger error band than $g_{1T}^{u/p}(x)$~\cite{Bhattacharya:2021twu} (recall JLab6 is for a neutron target).
\subsection{Predictions for JLab12, COMPASS, and the EIC} \label{s:JL_COM_EIC}
We next give predictions for JLab12, COMPASS, and a few sets of EIC kinemtics. We mention that next-to-leading order (NLO) corrections for the electron-nucleon single-inclusive unpolarized cross section ($eN\to \{\pi\,{\rm or}\, jet\}\,X$)~\cite{Hinderer:2015hra} have be shown to be sizeable, and for the double-longitudinal spin asymmetry $A_{LL}$ ($\vec{e}\,\vec{N}\to \{\pi\,{\rm or}\, jet\}\,X$)~\cite{Hinderer:2017ntk} they are also non-negligible. In addition, lower-energy experiments are typically dominated by quasi-real photo-production~\cite{HERMES:2013quo}. These issues should have less impact as one goes to higher $P_T$ ($\gtrsim 2\;{\rm or}\;3\,{\rm GeV}$), but high-precision measurements at the EIC may require NLO calculations.
In Fig.~\ref{f:JLab12} we present results for JLab12 with a neutron target. In all cases, sizeable asymmetries $\sim \!15$-$30\%$ are predicted which grow more substantial with increasing $P_T$. The distribution term gives basically the entirety of $A_{LT}$. The qgq scenario also tends to be larger than the WW scenario, especially at higher $P_T$. Therefore, one may be able to use JLab12 data to test the WW approximation and potentially extract information about dynamical quark-gluon-quark correlations in the nucleon.
The COMPASS results are displayed in Fig.~\ref{f:COMPASS} for a proton target, which are roughly an order of magnitude smaller than JLab12 but still measurable at $\sim \!2$-$4\%$. From the first ($E(z)=-\tilde{H}(z)$) and last ($E(z)=\tilde{H}(z)$) rows of the plot, we see that, unlike JLab12, the $A_{LT}$ fragmentation term can be comparable to the distribution term, at least for $\pi^-$ production. Since the $E(z)=0$ case (middle row) has $A_{LT}$ for $\pi^-$ clearly positive, a measured negative asymmetry would be a likely indication of quark-gluon-quark fragmentation effects. The qgq and WW scenarios may be difficult to distinguish at COMPASS since they give similarly-sized effects.
The low-energy EIC predictions at midrapidity ($\sqrt{S}=29\,{\rm GeV},\,\eta=0$) are shown in Fig.~\ref{f:EIC29_eta0}, where again we notice a further decrease in the size of the asymmetry compared to JLab12 and COMPASS, with $A_{LT}$ now $\sim \!0.5$-$1.5\%$. Similar to COMPASS, a clearly negative signal for $\pi^-$ production would be caused by quark-gluon-quark fragmentation. Since the EIC will also measure jets, we give results for that reaction at higher-energy EIC kinematics and slightly forward rapidity ($\sqrt{S}=63\,{\rm GeV},\,\eta=1$) in Fig.~\ref{f:EIC63jet_eta1}. The asymmetry again decreases, now to $\sim \!0.1$-$0.3\%$, due to the increase in cm energy and the fact that jets are being detected instead of pions.
The general features of $A_{LT}$ in electron-nucleon collisions are that it increases with $P_T$ but decreases significantly with $\sqrt{S}$. However, as $\eta$ increases, and one pushes $P_T$ to the theoretical kinematic limit, the fragmentation term can cause an enhanced growth in $A_{LT}$. A typical example is shown in Fig.~\ref{f:EIC29_eta1}. One sees the asymmetry is basically zero for most of the $P_T$ range and then receives an sizeable enhancement at the largest $P_T$ values. In this region, $z_{min}$ in Eq.~(\ref{e:ALTeN}) is around $0.8$ to $0.9$; one is then integrating at the threshold of producing the pion, where $E(z)$ is not constrained and resummation techniques may be needed~\cite{Anderle:2012rq,Anderle:2013lka,Hinderer:2014qta,Hinderer:2018nkb,Kaufmann:2019ksh}. Whether or not this is a physical effect that would be observed in experiments remains to be seen.
The measurement of $A_{LT}$ in $\vec{e}\,N^\uparrow \to \{\pi\;{\rm or}\; jet\}\,X$ at future experiments has the potential to provide insight into quark-gluon-quark correlations, especially given the precision expected at the EIC. A reduction in the uncertainty of $g_{1T}^{(1)}(x)$ will be key if one is to disentangle dynamical twist-3 effects from the twist-2 WW approximation. More precise measurements of the $A_{LT}^{\cos(\phi_h-\phi_S)}$ modulation in SIDIS at COMPASS, SoLID at JLab, and the EIC will be crucial to achieve this. For example, there are hints in Fig.~\ref{f:EIC29_eta0} that the qgq scenario may differ from the WW scenario by $\sim0.5\%$, but currently the error band in the qgq scenario (that relies on the full extraction of $g_{1T}^{(1)}(x)$) is too large to distnguish the two. A similar statement can be made for jet production in Fig.~\ref{f:EIC63jet_eta1}. Also recall that even in the qgq scenario, we neglected the dynamical twist-3 PDF $G_{FT}(x,x_1)$ in Eq.~(\ref{e:LIR}). Thus, significant differences between the qgq scenario predictions and future data could provide information on this function. Moreover, any significant deviations from the $E(z)=0$ scenario, especially if $g_{1T}^{(1)}(x)$ becomes more constrained, would allow for an extraction of this twist-3 FF. Given its connection to dynamical quark mass generation in QCD (see the discussion around Eq.~(\ref{e:sumrules})), the potential for $A_{LT}$ to give us information on $E(z)$ is another intriguing reason to measure it.
\subsection{Predictions for RHIC} \label{s:RHIC}
We now report on the results for $A_{LT}$ in $p^\uparrow \vec{p}\to \{\pi, jet, \,{\rm or}\; \gamma\}\,X$ at RHIC, the only machine capable of measuring this asymmetry. We focus on $\sqrt{S}=200\,{\rm GeV}$ cm energy at middle and forward rapidities. We remind the reader that there are three pieces to the asymmetry given in Eqs.~(\ref{e:Tdist}), (\ref{e:Ldist}), (\ref{e:frag}) (although the fragmentation term doesn't enter for photon or jet production). Our predictions for charged pion production at midrapidity ($\eta=0$) in Fig.~\ref{f:RHIC_mid_pipm} reach to $\sim \!0.02$-$0.05\%$ for $\pi^\pm$ at the highest $P_T$. The transverse distribution term gives the largest contribution to $A_{LT}$, although the fragmentation term plays a non-negligible role. At forward rapidity ($\eta=3.3$) in Fig.~\ref{f:RHIC_forward_pipm}, the asymmetry has larger error bands for the qgq scenario that are consistent with zero but range from $\sim \!-0.3\%$ to $+0.2\%$. In the WW approximation the uncertainties are much smaller at larger $P_T$ and again consistent with zero. In either case, the transverse distribution term gives the entirety of $A_{LT}$ at forward rapidity. The $\pi^0$ asymmetries (Figs.~\ref{f:RHIC_mid_pi0}, \ref{f:RHIC_forward_pi0}) are similar in size to $\pi^\pm$. For jet or photon production at midrapdity (Fig.~\ref{f:RHIC_mid_jet_gam}), our predictions for $A_{LT}$ are $\lesssim 0.03\%$. We note that at $\sqrt{S}=500\,{\rm GeV}$, the asymmetry (for any final state) is generally an order of magnitude smaller than at $\sqrt{S}=200\,{\rm GeV}$.
The reader may question why $A_{LT}$ in proton-proton collisions is much smaller than $A_N$. Recall that $A_N$ (where one proton is unpolarized and the other is transversely polarized) is another (much more widely studied/measured) twist-3 asymmetry that {\it does} show significant effects, at least in the forward region~\cite{Adams:1991rw,Krueger:1998hz,Allgower:2002qi,Adams:2003fx,Adler:2005in,Lee:2007zzh,Abelev:2008af,Arsene:2008aa,Adamczyk:2012qj,Adamczyk:2012xd,Bland:2013pkt,Adare:2013ekj,Adare:2014qzo,STAR:2020nnl}. We found that there are two driving factors. First, in the $qg\to qg$ channel (which is the dominant channel in the numerator of $A_N$ and $A_{LT}$), the fragmentation term for $A_N$ (which is the main source of the asymmetry~\cite{Kanazawa:2014dca,Gamberg:2017gle,Cammarota:2020qcw,Gamberg:2022kdb}) has hard factors $\sim 1/\hat{t}^3$, whereas in the transverse distribution term (\ref{e:Tdist}) for $A_{LT}$ (which is the main source of that asymmetry) the hard factors $\sim 1/(\hat{t}^2\hat{u})$. Since $\hat{t}\to 0$ in the forward region, this provides an enhancement to $A_N$ not seen in $A_{LT}$. The second difference is $A_N$ has an unpolarized proton, so in the $qg\to qg$ channel, $f_1^g(x)$ multiplies the (twist-3) fragmentation term. On the other hand, $A_{LT}$ has a longitudinally polarized proton, so $g_1^g(x)$ multiplies the (twist-3) transverse distribution term. In the forward region (of the transversely polarized proton), these gluon functions are probed at small $x$; hence, $A_N$ becomes signficantly larger than $A_{LT}$. In fact, we checked that if in the numerator of $A_N$ one replaces $f_1^g(x)$ (in the $qg\to qg$ channel) with $g_1^g(x)$, the asymmetry is nearly as suppressed as $A_{LT}$.
We emphasize that, in addition to the assumptions that underlie our scenarios for $g_{1T}^{(1)}(x),g_T(x)$ and $E(z)$, the proton-proton case has several terms that we are forced to neglect due to lack of input for dynamical twist-3 correlators. Namely, we do not consider the terms in Eq.~(\ref{e:scriptGpp}) involving $F_{FT}(x,x_1), G_{FT}(x,x_1)$. The WW approximation we use for $h_L(x)$ and $h_{1L}^{\perp(1)}(x)$ in Eq.~(\ref{e:scriptH}) sets to zero another dynamical twist-3 PDF called $H_{FL}(x,x_1)$~\cite{Tangerman:1994bb,Metz:2008ib,Kanazawa:2015ajw}.\footnote{We note that there are some model calculations of functions connected to $F_{FT}(x,x_1), G_{FT}(x,x_1)$~\cite{Braun:2011aw}. The worm-gear TMD $h_{1L}^{\perp}$ in the future can be extracted from data on the $A_{LT}^{\sin 2\phi_h}$ modulation in SIDIS~\cite{HERMES:1999ryv,HERMES:2001hbj,HERMES:2002buj,CLAS:2010fns,COMPASS:2016klq,HERMES:2020ifk}.} Therefore, measurements that significantly deviate from our predictions could provide information on these unknown quark-gluon-quark correlators.
\section{Conclusions and Outlook}
\label{s:concl}
We have numerically analyzed the twist-3 asymmetry $A_{LT}$ in single-inclusive electron-nucleon and proton-proton collisions for various final states. This is the first time contributions from all terms entering these asymmetries have been computed. Nevertheless, some approximations/assumptions had to be employed, including ignoring certain dynamical twist-3 PDFs due to a lack of information about them. Using recent extractions of $g^{(1)}_{1T}(x)$~\cite{Bhattacharya:2021twu} and $\tilde{H}(z)$~\cite{Gamberg:2022kdb}, we were able to develop realistic scenarios to investigate for three critical functions in $A_{LT}$:~$g_{1T}^{(1)}(x)$, $g_T(x)$, and $E(z)$. We used bootstrapping to provide a rigorous error quantification of our calculation that accounts for the fact that $A_{LT}$ depends on multiple non-perturbative functions extracted by different groups. We found good agreement with JLab6 data, which is the only $A_{LT}$ measurement available (for single-inclusive observables). We then made predictions for $A_{LT}$ in electron-nucleon collisions at JLab12, COMPASS, and the EIC, as well as proton-proton collisions at RHIC, in order to motivate future measurements. Beyond the results presented in this paper, we are able to provide predictions for any initial/final states and kinematic region $(\sqrt{S},\eta,P_T)$ upon request.
In electron-nucleon collisions, the asymmetry decreases with increasing center-of-mass energy, going from (for $\pi^\pm$ production) $\sim \!15$-$30\%$ at JLab12 to $\sim \!2$-$4\%$ at COMPASS to $\sim \!0.5$-$1.5\%$ for the low-energy EIC configuration (at midrapidity). An intriguing prospect is if significant deviations from the $E(z)=0$ scenario are measured, it could provide direct information on $E(z)$, which is connected to dynamical quark mass generation in QCD~\cite{Accardi:2017pmi,Accardi:2019luo,Accardi:2020iqn}. One may also be able to test the validity of the Wandzura-Wilczek approximation for $g^{(1)}_{1T}(x),g_T(x)$ and probe dynamical twist-3 PDFs, especially with precision measurements at the EIC. The calculation of the proton-proton case at RHIC kinematics showed (for $\pi^\pm$ production) $A_{LT}\sim \!0.02$-$0.05\%$ at midrapidity and can be in the range of $\sim \!-0.3\%$ to $+0.2\%$ at forward rapidity. The asymmetry does not grow rapidly at forward rapidity, in contrast to $A_N$, due to a suppression caused by the other proton being longitudinally polarized instead of unpolarized (where $g_1^g(x)$ then enters the $qg\to qg$ channel in the numerator of the asymmetry instead of $f_1^g(x)$). Since RHIC is the only machine capable of measuring $A_{LT}$ in proton-proton collisions, confirmation or refutation of our predictions would aid in better understanding the role of quark-gluon-quark correlations in hadrons.
\section*{Acknowledgments}
This work has been supported by the National Science Foundation under Grant No.~PHY-2011763. The authors thank S.~Bhattacharya for providing the lattice data of Ref.~\cite{Bhattacharya:2020cen} and for valuable feedback from a careful reading of the manuscript. The authors also thank E.~Aschenauer, A.~Metz, N.~Sato, and R.~Seidl for fruitful discussions about various aspects of this work. The authors are also grateful to C.~Cocuzza for creating LHAPDF tables of the JAM3D-22 functions, to R.~Abdul Khalek for providing the LHAPDF tables of DSS14 created by V.~Bertone, and to Jefferson Lab for access to their computational resources.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 6,834 |
Q: unable to click button with unique button data-order-group-id 's information python+selenium javascript code (repeat order 1):
<div class="col-ea-1 repeat-order repeatable-order"> #common row in other snippets
<button data-order-group-id="a9755447-04ff-4d00-59bf-06ff87f8ead6" #different row
data-restaurant-seo-url="/abc-pizza-bayburt-kirazli-mah"
class="ys-btn ys-btn-primary ys-btn-aa middle repeat-order-button"> REPEAT ORDER </button> ==$0
</div>
In the above snippet javascript code, there is more than one for each order (button data-order-group-id is different for each order) I also want to reach and click the REPEAT ORDER button for each order How can I do this in this piece of code?
A: You can try with the below xpath :
//button[contains(text(), 'REPEAT ORDER') and contains(@class, 'repeat-order-button')]
using find_element for single row, and find_elements for multiple row.
sample code :
for ele in driver.find_elements(By.XPATH, "//button[contains(text(), 'REPEAT ORDER') and contains(@class, 'repeat-order-button')]"):
ele.click()
# do some other stuff
# re-initiate elements here otherwise you will get stale element reference
#break if requires
A: @cruisepandey your code is working well but due to
the website's specific conditions which are help center fixed box and ORDER REPEAT buttons can overlap somehow That's why I used "window.scrollTo" before in addition to your code
i=int(input("choose:"))
if i==0:
driver.execute_script("window.scrollTo(0, 300)") #move the area
driver.find_elements(By.XPATH, "//button[contains(text(), 'REPEAT ORDER') and contains(@class, 'repeat-order-button')]")[0].click()
sleep(7)
elif i==1:
driver.execute_script("window.scrollTo(0, 300)") #move the area
driver.find_elements(By.XPATH, "//button[contains(text(), 'REPEAT ORDER') and contains(@class, 'repeat-order-button')]")[1].click()
sleep(7)
elif i==2:
driver.execute_script("window.scrollTo(0, 600)") #move the area
driver.find_elements(By.XPATH, "//button[contains(text(), 'REPEAT ORDER') and contains(@class, 'repeat-order-button')]")[2].click()
sleep(7)
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 3,666 |
{"url":"https:\/\/www.physicsforums.com\/threads\/definition-of-a-symmetric-connection.416209\/","text":"# Definition of a symmetric connection\n\n1. Jul 15, 2010\n\n### Angelos K\n\nHi, all,\n\nAccording to my script, a connection $$\\nabla_v$$ is symmetric if the following holds (I assume for every pair of vectors):\n\n$$\\nabla_v w - \\nabla_w v =[v,w]$$\n\nWhat is the idea behind that? Why are we interested in that kind of symmetry (not for instance 0 instead of the commutator)?\n\nAngelos\n\n2. Jul 15, 2010\n\n### Ben Niehoff\n\nFirst of all, you can't set the RHS to zero, because it is impossible for that to hold for all vector fields v, w.\n\nThis equation simply states that the torsion vanishes.\n\nIf we travel around a small parallelogram whose sides are geodesics, our intuition says that we ought to end up where we started. If a parallelogram does fail to close, we attribute that to the failure of the vector fields defining its sides to commute; and the remaining gap is given by the commutator.\n\nIn the presence of torsion, this does not hold. Small parallelograms may fail to close even if the vector fields commute! The torsion gives this additional gap.\n\nYour equation expresses the idea that the failure of any parallelograms to close is due precisely to the failure of the vector fields to commute, and that there is no additional gap we need to account for.\n\n3. Jul 15, 2010\n\n### Altabeh\n\nSuppose that $$L$$ is a smooth scalar field then from basic calculus you remember that clearly $$\\partial_a\\partial_b L=\\partial_b\\partial_a L$$. But it is necessary to note that this doesn't follow when the ordinary derivatives are replaced by the covariant derivatives. To wit, $$\\nabla_a\\nabla_b L$$ and $$\\nabla_b\\nabla_a L$$ are not equivalent generally. The reason is that you can simply show there is a tensor $$T_{ab}^c$$ known as the torsion tensor such that for any scalar field of class $$C^\\infty$$ we have\n\n$$(\\nabla_a\\nabla_b -\\nabla_b\\nabla_a) L=T^c_{ab}\\nabla_c L.$$\n\nIf $$T^c_{ab}=0$$, then the connection is said to be torsion-free (torsionless) and obviously it follows that the connection is symmetric because\n\n$$\\nabla_a\\nabla_b L=\\nabla_b\\nabla_a L.$$\n\nBut how does this imply a symmetry of connection in two lower indices? Let us calculate the torsion $$T^c_{ab}$$ in terms of the connection $$\\Gamma^c_{ab}$$. Recalling that $$\\nabla_a U_b=\\partial_a U_b -\\Gamma^c_{ab} U_c$$ for any covariant vector field $$U_b$$. Hence if one sets $$U_b=\\nabla_b L=\\partial_b L$$, we get\n\n$$\\nabla_a\\nabla_b L=\\partial_a \\partial_b L -\\Gamma^c_{ba} \\partial_c L ,$$\n\nand\n\n$$\\nabla_b\\nabla_a L=\\partial_b \\partial_a L -\\Gamma^c_{ab} \\partial_c L .$$\n\nBy subtracting the first from the second we obtain\n\n$$(\\nabla_b\\nabla_a -\\nabla_b \\nabla_a )L =T^c_{ab}\\nabla_c L ,$$\n\nwhere\n\n$$T^c_{ab}=-2\\Gamma^c_{[ab]} .$$\n\nYou must know that the difference of two connections is always a tensor, so is the torsion. Therefore a torsion-free spacetime has this property that its connection is symmetric.\n\nAB\n\n4. Jul 16, 2010\n\n### Angelos K\n\nthanks!\n\nThank you so much both, that really helped, I'm starting to get the idea.\n\nI'll sit down right now and calculate a couple of things about it.\n\n@AB Thanks for mentioning the Christoffel-property. It seems that they use it a lot!","date":"2018-12-12 04:53:48","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\": 0.853102445602417, \"perplexity\": 351.26542555169823}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2018-51\/segments\/1544376823738.9\/warc\/CC-MAIN-20181212044022-20181212065522-00619.warc.gz\"}"} | null | null |
Q: Can hyperledger-fabric get the peer node running status without entering the docker container? Can hyperledger-fabric get the peer node running status without entering the docker container? If so, how should I get it?
A: In docker-compose file, for peer service add following env variable. (You may add a different port for different services)
- CORE_OPERATIONS_LISTENADDRESS=0.0.0.0:9440
expose the port(You may expose port number as per availability). Export different port for different peer
ports:
- 9440:9440
Once all services up hit the following path for specific service(As per port defined)
curl -X GET localhost:9440/healthz
You will get a following response if the service is running.
{
"status": "OK",
"time": "2009-11-10T23:00:00Z"
}
If service is not available, you will get the following response.
{
"status": "Service Unavailable",
"time": "2009-11-10T23:00:00Z",
"failed_checks": [
{
"component": "docker",
"reason": "failed to connect to Docker daemon: invalid endpoint"
}
]
}
A: The Operations Service might be what you are looking for, the simple check is for "Health" and the more complex check is to look the "metrics".
It is covered in the Fabric docs.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 4,382 |
{"url":"https:\/\/ashpublications.org\/blood\/article\/132\/Supplement%201\/2250\/261493\/Impact-of-Myeloproliferative-Neoplasms-on-Patients","text":"Background: Patients with myeloproliferative neoplasms (MPNs), including myelofibrosis (MF), polycythemia vera (PV), and essential thrombocythemia (ET), experience a high symptom burden that may compromise daily functioning and quality of life. The objective of this analysis was to evaluate income loss associated with disease-related employment changes among patients with MPNs in the United States.\n\nMethods: The Living with MPN survey was completed online by adult patients (aged 18-70 years) with MF, PV, or ET in the United States between April and November of 2016. Respondents employed at the time of MPN diagnosis were asked questions about disease-related changes in employment status and salaries occurring between diagnosis and the year of survey (2016). In addition, details related to patient demographics, MPN diagnosis, and MPN-related symptoms were collected. Cumulative income losses as a result of disease-related employment changes up to the time of the survey were calculated based on the timing of employment changes and salaries, which were reported in nominal dollars.\n\nResults: Of the 904 survey respondents, 592 (65%) were employed at the time of MPN diagnosis. Among those employed, mean age was 54.0 years, 70.6% were female, and 72.3% were married or had domestic partners at the time of survey. The average duration of disease was 6.1 years (MF, 4.6; PV, 6.9; ET, 6.3). Approximately half (50.5%) of the employed respondents experienced at least one change in employment status because of their diagnosis. Employment status changes and associated impact on income in patients with MPNs was greatest for those who took early retirement, medical disability leave, or left a job due to their disease ($419,610,$169,048, $168,245, respectively). Respondents who changed from full- to part-time employment, reduced hours, or were reassigned to a lower-paying job because of their disease also reported income losses ($79,492, $47,104,$51,872, respectively; Table).\n\nAmong respondents who were 45-64 years old at the time of the survey (n=383), 18.8% reported retiring early as a result of their disease. In comparison, according to nationally representative data from the Medical Expenditures Panel Survey (MEPS), only 7.8% of individuals aged 45-64 years in excellent or very good health and 9.2% of individuals in poor health reported being retired (longitudinal data set 2014-2015). Moreover, 30.5% (117\/383) of respondents aged 45-64 years in the Living with MPN survey reported leaving a job as a result of their disease. In comparison, 5.5% of MEPS individuals aged 45-64 years in excellent or very good health and 16.4% of individuals in poor health were working at the start of 2014 but not by the end of 2015.\n\nConclusions: About half of employed patients living with MPNs experienced a variety of employment changes as a result of their disease, which in turn had a considerable impact on income. The most frequently reported disease-related employment change was leaving a job, followed by medical disability leave, reduced hours, early retirement, switching from full-time to part-time, and being reassigned to a lower-paying job. Patients 45-64 years old with MPNs were more than twice as likely to have left a job or retired early compared with an age-matched US general population cohort. On average, the foregone income due to disease-related employment changes was greatest for early retirees ($419,610), followed by those who went on medical disability leave ($169,048), and left a job (\\$168,245). Early, effective management of MPNs and associated symptoms may help patients avoid these disease-related changes to their employment status and the subsequent economic and financial impact.\n\nDisclosures\n\nCondliffe:Incyte Corporation: Consultancy. Yu:Incyte Corporation: Employment, Equity Ownership. Paranagama:Incyte: Employment, Equity Ownership. Parasuraman:Incyte: Employment, Equity Ownership.\n\nAuthor notes\n\n*\n\nAsterisk with author names denotes non-ASH members.","date":"2023-02-06 16:36:48","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\": 0.22062264382839203, \"perplexity\": 8172.775212463618}, \"config\": {\"markdown_headings\": false, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2023-06\/segments\/1674764500356.92\/warc\/CC-MAIN-20230206145603-20230206175603-00342.warc.gz\"}"} | null | null |
\section{Introduction}
\label{sec:intro}
Throughout this paper $p$ denotes a prime
number, and $q=p^a$ a power of $p$ with an exponent $a\in \bbN$, the
set of strictly positive integers. The goal of this paper is to calculate explicitly the number of
superspecial abelian surfaces over a finite field ${\bbF}_q$.
This can be regarded as a natural extension of
works of the authors
\cite{xue-yang-yu:num_inv, xue-yang-yu:ECNF} and
the last named author \cite{yu:sp_prime} contributed to
the study of supersingular abelian varieties over finite fields.
Recall that an abelian variety over a field $k$ of
characteristic $p$ is said to be {\it supersingular} if it is
isogenous to a product of supersingular elliptic curves over an
algebraic closure
$\bar k$ of $k$; it is said to be {\it superspecial} if it is
isomorphic to a product of supersingular elliptic curves over $\bar
k$. As any supersingular abelian variety is isogenous to a
superspecial abelian variety, it is very common to study
supersingular abelian varieties through investigating
the classification of superspecial abelian varieties.
For any integer $d\ge 1$, let $\Sp_d(\mathbb F_q)$
denote the set of isomorphism classes of $d$-dimensional
superspecial abelian varieties over the finite field $\mathbb F_q$ of $q$
elements. The case where $d=1$ concerns the classification of
supersingular elliptic curves over finite fields.
The theory of elliptic curves over finite fields
has been studied by Deuring since 1940's and becomes well known.
There are explicit descriptions for each isogeny class;
see Waterhouse \cite[Section 4]{waterhouse:thesis}.
However, the authors could not find an explicit formula for
$|\Sp_1({\bbF}_q)|$ in the literature.
For the sake of completeness we include a formula for
$|\Sp_1({\bbF}_q)|$,
based on the exposition of Deuring's results by Waterhouse
\cite{waterhouse:thesis}.
The goal of the present paper is then to find an explicit
formula for the number $|\Sp_d({\bbF}_q)|$ in the case where $d=2$.
Before stating our main results, we describe a basic method of
counting $\Sp_d({\bbF}_q)$. For simplicity assume that $\mathbb F_q={\bbF}_p$ is
the prime finite field for the moment.
One can divide the finite set $\Sp_d(\mathbb F_p)$ into finitely many subsets
according to the isogeny classes of members.
Therefore, it suffices to classify
all $d$-dimensional supersingular isogeny classes and to count
the number of superspecial members in each supersingular
isogeny class. The Honda-Tate theorem allows us to describe
isogeny classes over ${\bbF}_q$
in terms of multiple Weil $q$-numbers (which are simply finite
nonnegative integral formal sums of Weil $q$-numbers up to conjugate;
see Section~\ref{sec:curve.1}).
If $\pi$ is a supersingular multiple Weil $q$-number,
we denote by $[X_\pi]$ the corresponding supersingular isogeny class
(here $X_\pi$ is an abelian variety in this class),
$H(\pi)$ the number of isomorphism classes of abelian varieties in
$[X_\pi]$ and $H_{sp}(\pi)$ the number of isomorphism classes of
{\it superspecial} abelian varieties in $[X_\pi]$. Then we have
\begin{equation}
\label{eq:intro.1}
|\Sp_d({\bbF}_p)|=\sum_{\pi} H_{sp}(\pi)
\end{equation}
where $\pi$ runs through all supersingular multiple Weil $p$-numbers with
$\dim X_\pi=d$. We classify all possible isogeny
classes $\pi$'s occurring in the sum (see Sections 2--3).
The problem then is to compute each term
$H_{sp}(\pi)$. One should
distinguish the cases according to whether the endomorphism algebra
$\End^0(X_\pi)=\End(X_\pi)\otimes \mathbb Q$ of $X_\pi$
satisfies the Eichler condition \cite[Section~III.4, p.81]{vigneras} or not.
We now focus on the case where $d=2$.
Consider the case where $\pi$ is the Weil $p$-number
$\sqrt{p}$. Correspondingly, $X_\pi$ is a supersingular abelian
surface. It is known (see Tate \cite{tate:eav}) that
the endomorphism algebra $\End^0(X_\pi)$ of $X_\pi$ is isomorphic to
the totally definite quaternion algebra algebra
$D=D_{\infty_1,\infty_2}$ over the quadratic real field
$F=\mathbb Q(\sqrt{p})$ ramified exactly at
the two real places $\{\infty_1,\infty_2\}$ of $F$.
In this case all abelian surfaces in the isogeny
class $[X_{\sqrt{p}}]$ are superspecial,
i.e.~$H(\sqrt{p})=H_{sp}(\sqrt{p})$. When $p=2$ or $p\equiv 3 \pmod
4$, Waterhouse proved that the
number $H(\sqrt{p})$ is equal to the class number
$h(D_{})$ of $D_{}$.
When $p \equiv 1 \pmod 4$, the number
$H(\sqrt{p})$ is equal to the sum of $h(D_{})$
and the class numbers of two other proper $\mathbb Z[\sqrt{p}]$-orders in
$D_{}$ of index $8$ and $16$, respectively
(the descriptions of these orders are made concrete by results of \cite{yu:smf}).
These class numbers are computed systematically in our previous work \cite{xue-yang-yu:ECNF}.
Therefore, we obtain an explicit formula for the term
$H_{sp}(\sqrt{p})$ given below.
In what follows we write $K_{m,j}$ for the number field
$\mathbb Q(\sqrt{m},\sqrt{-j})$ for any square-free integers $m>1$ and $j\ge
1$. If $m\equiv 1 \pmod 4$, then we define
\begin{equation}
\label{eq:varpi_d}
\varpi_m:=3 [O_{\mathbb Q(\sqrt{m})}^\times: \mathbb Z[\sqrt{m}]^\times]^{-1},
\end{equation}
where $O_{\mathbb Q(\sqrt{m})}$ denotes the ring of integers of $\mathbb Q(\sqrt{m})$.
By similar arguments as those in \cite[Lemma~4.1 and
Section~4.2]{xue-yang-yu:num_inv}, we have $\varpi_m\in\{1,3\}$, and
$\varpi_m=3$ if $m\equiv 1\pmod 8$. The class number of a number field
$K$ is denoted by $h(K)$. When $K=\mathbb Q(\sqrt{m})$, we write
$h(\sqrt{m})$ for $h(\mathbb Q(\sqrt{m}))$ instead.
\begin{thm}\label{1.2}
Let $H(\sqrt{p})$ be the number of $\mathbb{F}_p$-isomorphism
classes of abelian
varieties in the simple isogeny class corresponding to the Weil
$p$-number $\pi=\sqrt{p}$, and let $F=\mathbb Q(\sqrt{p})$. Then\\
(1) $H(\sqrt{p})=1,2,3$ for $p=2,3, 5$, respectively. \\
(2) For $p>5$ and $p\equiv 3 \pmod 4$, we have
\begin{equation}
\label{eq:intro.2}
H(\sqrt{p})=\frac{1}{2}h(F)\zeta_F(-1) +
\left(\frac{3}{8}+\frac{5}{8}\left(2-\left(\frac{2}{p}\right)
\right)\right)h(K_{p,1})+\frac{1}{4}h(K_{p,2})+
\frac{1}{3}h(K_{p,3}),
\end{equation}
where $\zeta_F(s)$ is the Dedekind zeta function
of $F$.\\
(3) For $p>5$ and $p\equiv 1 \pmod 4$, we have
\begin{equation}
\label{eq:1.2}
H(\sqrt{p})=
\begin{cases}
8 \zeta_F(-1)h(F)+ h(K_{p,1})+\frac{4}{3}
h(K_{p,3}) & \text{for $p\equiv 1 \pmod 8$;} \\
\frac{1}{2}(15\varpi_p+1) \zeta_F(-1)h(F)+
\frac{1}{4}(3\varpi_p+1) h(K_{p,1})
+\frac{4}{3} h(K_{p,3}) & \text{for $p\equiv 5 \pmod 8$;} \\
\end{cases}
\end{equation}
\end{thm}
The computation in Theorem~\ref{1.2} is based on
the generalized Eichler class
formula \cite[Theorem 1.4]{xue-yang-yu:ECNF}
that the authors developed. This formula allows us to compute the class
number of an arbitrary $\mathbb Z$-order in a totally definite quaternion
over a totally real field $F$. This $\mathbb Z$-order does not necessarily contains the maximal
order $O_F$ of $F$. For a quadratic real field $F$, the special zeta value $\zeta_F(-1)$
can be calculated by
Siegel's formula \cite[Table 2, p. 70]{Zagier-1976-zeta}
\begin{equation}
\label{eq:intro.35}
\zeta_F(-1)=\frac{1}{60}\sum_{\substack{b^2+4ac=\d_F\\ a,c>0}} a,
\end{equation}
where $\d_F$ is the discriminant of $F/\mathbb Q$, $b\in \mathbb{Z}$
and $a,c\in \bbN$.
The first main result of this paper gives the following explicit formula
for $\abs{\Sp_2(\mathbb F_p)}$, the number of isomorphism classes of superspecial abelian surfaces over ${\bbF}_p$.
To obtain this formula, we calculate all terms $H_{sp}(\pi)$ with
$\pi\neq \pm \sqrt{p}$ in
(\ref{eq:intro.1}), and then sum them up together with $H(\sqrt{p})$. The computation of
$H_{sp}(\pi)$ uses a lattice description for superspecial abelian
varieties; see Section 5 for details.
Similar to Theorem~\ref{1.2}, special
attentions have to be paid to the cases with small primes $p$.
\begin{thm}\label{1.3}
We have $|\Sp_2({\bbF}_p)|=H(\sqrt{p})+\Delta(p)$, where the formula for
$H(\sqrt{p})$ is stated in Theorem~\ref{1.2} and $\Delta(p)$ is the
number described as follows.
\begin{enumerate}
\item $\Delta(p)=15,20,9$ for $p=2,3,5$,
respectively.
\item For $p>5$ and $p\equiv 1 \pmod 4$, we have
\begin{equation}
\label{eq:intro.4}
\Delta(p)=(\varpi_p+1) h(K_{p,3})+h(K_{2p,1})
+h(K_{3p,3})+h(\sqrt{-p}),
\end{equation}
\item For $p>5$ and $p\equiv 3 \pmod 4$, we have
\begin{equation}
\label{eq:intro.5}
\Delta(p)=h(K_{p,3})+h(K_{2p,1})+(\varpi_{3p}+1)
h(K_{3p,3})+\left(4-\left (\frac{2}{p}\right ) \right )
h(\sqrt{-p}).
\end{equation}
\end{enumerate}
\end{thm}
A key ingredient of our computation for $\Sp_2({\bbF}_p)$ is
Theorem~\ref{prop:sp.1}, which works only for the prime finite
fields. Centeleghe and Stix \cite{MR3317765} provide a categorical
description of Theorem~\ref{prop:sp.1} (also compare \cite[Theorem
3,1]{yu:sp_prime}). However, their results are also limited to the
prime finite fields. When the base field ${\bbF}_q$ is no longer the prime
finite field, direct calculations via the counting method described
earlier for $\Sp_d({\bbF}_q)$ (even when $d=2$) become more complicated.
Our second main result extends the computations of $\Sp_2({\bbF}_p)$ to
$\Sp_2({\bbF}_q)$ for more general finite fields ${\bbF}_q$ via Galois cohomology.
Observe that if $d>1$, then there is only one isomorphism class of
$d$-dimensional superspecial abelian varieties over $\overline{\bbF}_p$ (see
\cite[Section 1.6, p.~13]{li-oort} or Theorem~\ref{gal.sp}).
Suppose $X_0$ is any $d$-dimensional
superspecial abelian variety over ${\bbF}_p$.
Then there is a bijection of finite sets
\begin{equation}
\label{eq:intro.6}
\Sp_d({\bbF}_p)\simeq H^1(\Gamma_{{\bbF}_p}, G), \quad d>1,
\end{equation}
where $\Gamma_{{\bbF}_p}=\Gal(\overline{\bbF}_p/{\bbF}_p)$ is the absolute Galois group
of ${\bbF}_p$, and $G=\Aut(X_0\otimes \overline{\bbF}_p)$.
Thus, computing the Galois
cohomology would lead to a second proof of Theorem~\ref{1.3}.
However, the complexity of the final formula as in Theorem~\ref{1.3}
suggests that the computation of this Galois cohomology is likely on
the same level of difficulty as the counting method via (\ref{eq:intro.1}).
However, the true advantages of connecting to Galois cohomology are two
folds.
\begin{enumerate}
\item[(a)] It naturally relates $\Sp_d({\bbF}_q)$ and $\Sp_d(\mathbb F_{q'})$ in
the sense of Theorem~\ref{1.4} when the
exponents in $q=p^a$ and $q'=p^{a'}$
have the same parity.
\item[(b)] It gives rise to a lattice description for
$\Sp_d({\bbF}_q)$ when $q=p^a$ is an even power of $p$; see
Theorem~\ref{gal_coh.3}.
\end{enumerate}
\begin{thm}\label{1.4}
Let $q$ and $q'$ be powers of $p$ with same
exponent parity and $d\ge 1$ an integer. Then there is a natural
bijection $\Sp_d({\bbF}_q)\simeq \Sp_d(\mathbb F_{q'})$ preserving isogeny
classes. In particular, the same formulas in Theorem~\ref{1.3} hold
for $|\Sp_2({\bbF}_q)|$ since $|\Sp_d({\bbF}_q)|=|\Sp_d({\bbF}_p)|$ when $q$ is an
odd power of $p$.
\end{thm}
The bijection for the case $d=1$ is handled separately in
Section~\ref{sec:curve} (see Remark~\ref{curve.4}). For $d\geq 2$, the
bijection is established in Theorem~\ref{gal.1}.
Along the way, we prove in Section~\ref{subsec:galois-descent} the
following
general result connecting isogeny classes of abelian varieties
over $\mathbb F_q$ with cohomology classes.
\begin{thm}\label{thm:intro-descent-isog}
Let $[X_0]$ be the $\mathbb F_q$-isogeny class of an arbitrary abelian
variety $X_0$ over $\mathbb F_q$, and $G_\mathbb Q=\End^0(\overline{X}_0)^\times$ where $\overline X_0=X_0\otimes_{\mathbb F_q}\overline{\bbF}_q$. We write $E^0(\overline{\bbF}_q/\mathbb F_q,
[X_0])$ for the set of $\mathbb F_q$-isogeny classes of abelian varieties $[X]$
such that $\overline X$ is isogenous to $\overline X_0$ over $\overline{\bbF}_q$. Then there is a canonical bijection
of pointed sets
\[ E^0(\overline{\bbF}_q/\mathbb F_q,
[X_0]) \stackrel{\sim}{\longrightarrow} H^1(\Gamma_{\mathbb F_q}, G_\mathbb Q) \]
sending $[X_0]$ to the trivial cohomology class.
\end{thm}
Theorem~\ref{1.4} together with Theorem~\ref{prop:sp.1} give a
new lattice description in Theorem~\ref{gal.odd} for $\Sp_d({\bbF}_q)$ when $q$ is an odd power of
$p$. When $q$ is an even power of $p$, a
lattice description of $\Sp_d(\mathbb F_q)$ completely different from the odd
case is given in Theorem~\ref{gal_coh.3}, which paves
the way to explicit formulas of $\abs{\Sp_2({\bbF}_q)}$.
The detailed formulas and computations will be presented in a separated paper \cite{xue-yang-yu:conj_finite}.
The paper is organized as follows. In Section~\ref{sec:par}, we
parameterize simple isogeny classes of supersingular abelian varieties
over $\mathbb F_q$ using Weil $q$-numbers. Their dimensions are
calculated in Section~\ref{sec:dim}. In Section~\ref{sec:curve} we
treat the dimension 1 case and calculate the the number of isomorphism
classes of supersingular elliptic curves over finite fields. The
dimension 2 case is then treated in Section~\ref{sec:sp}, except we work
exclusively over the prime field $\mathbb F_p$, and some arithmetic calculations are
postponed to Section~\ref{sec:arithmetic-results}.
Section~\ref{sec:gal_coh} studies the parity property via Galois
cohomology, thus providing means to extend results of
Section~\ref{sec:sp} to all $\mathbb F_{p^a}$ with $a$ odd. The
aforementioned lattices descriptions are obtained
in this process.
\section{Parameterization of supersingular isogeny classes}
\label{sec:par}
\subsection{}
\label{sec:par.1}
Let $q=p^a$ be a power of a prime number $p$. In this section
we parameterize simple isogeny classes of supersingular abelian varieties
over $\mathbb F_q$.
Let $\overline{\bbQ}\subset \mathbb C$ be the algebraic closure of $\mathbb Q$ in $\mathbb C$.
If two algebraic numbers $\alpha, \beta \in \overline{\bbQ}$ are conjugate over $\mathbb Q$, then
we write $\alpha\sim \beta$. Recall that an algebraic integer $\pi\in \overline{\bbQ}$
is said to be a {\it Weil $q$-number} if $|\iota(\pi)|=q^{1/2}$ for any
embedding $\iota: \mathbb Q(\pi)\hookrightarrow \mathbb C$.
By the Honda-Tate theory, the simple isogeny classes of
abelian varieties over $\mathbb F_q$ are in bijection with the conjugacy
classes of Weil $q$-numbers. A Weil $q$-number is said to be {\it
supersingular} if the corresponding
isogeny class consists of supersingular abelian varieties. Let $W_q^{\rm ss}$ denote the set of conjugacy
classes of supersingular Weil $q$-numbers. We will find a unique
representative for each conjugacy class in $W_q^{\rm ss}$.
Let $\pi$ be a supersingular Weil $q$-number.
It is known (the Manin-Oort Theorem, cf.~\cite[Theorem 2.9]{yu:QMav})
that $\pi=\sqrt{q} \zeta$ for a root of unity $\zeta$.
Let $K:=\mathbb Q(\pi)$ and $L:=\mathbb Q(\sqrt{q}, \zeta)$. Note that
both $L$ and $K$ are
abelian extensions over $\mathbb Q$.
For any $n\in \bbN$ (the set of positive integers), write
$\zeta_n:=e^{2\pi i/n}\in \overline{\bbQ}$.
\begin{lemma}\label{lemma:par.1}
Any supersingular Weil $q$-number $\pi$
is conjugate to $\sqrt{q}\zeta_n$ or
$-\sqrt{q}\zeta_n$ with $n\not \equiv 2 \pmod 4$.
\end{lemma}
\begin{proof}
Let $\pi=\sqrt{q} \zeta_m^\nu$ for some positive integers $\nu$ and
$m$ with $(\nu,m)=1$. Choose an element $\sigma\in \Gal(L/\mathbb Q)$ such
that
$\sigma(\zeta_m^\nu)=\zeta_m$, Then $\sigma(\pi)=\pm \sqrt{q}
\zeta_m$.
If $m \not \equiv 2 \pmod 4$, then we are done. Suppose that $m=2k$
for an odd integer $k=1-2u$. Clearly $(k,u)=1$. Since
$\zeta_{2k}=\zeta_{2k}^{k+2u}=-\zeta_{2k}^{2u}=-\zeta_k^u$, we
have
\[ \pm \sqrt{q} \zeta_{2k}=\mp \sqrt{q} \zeta_{k}^u \sim \epsilon
\sqrt{q} \zeta_k, \quad \text{for some\ } \epsilon\in \{\pm 1\} \]
by the previous argument.
\end{proof}
By Lemma~\ref{lemma:par.1},
there is a unique subset $W$ of $\{\pm \sqrt{q}
\zeta_n ; n\not \equiv 2 \pmod 4\}$ that contains $\{\, \sqrt{q}
\zeta_n ; n\not \equiv 2 \pmod 4\}$ and represents $W_q^{\rm ss}$.
We often identify $W$ with $W_q^{\rm ss}$.
To determine the set $W_q^{\rm ss}$, we need to characterizes
when $\sqrt{q} \zeta_n$ and $-\sqrt{q} \zeta_n$
are conjugate.
As usual, the Galois group $G_n:=\Gal(\mathbb Q(\zeta_n)/\mathbb Q)$ is naturally
identified with $(\mathbb Z/n\mathbb Z)^\times$ by mapping any $r\in (\mathbb Z/n\mathbb Z)^\times$
to the element $\sigma_r\in G_n$ with $\sigma_r(\zeta_n)=\zeta_n^r$.
\subsection{}
\label{sec:par.2}
Let us first assume that $a$ is even, i.e., $\sqrt{q}\in \mathbb Q$. Then
$\sqrt{q} \zeta_n\sim -\sqrt{q} \zeta_n$ if and only if there is an
element $\sigma_r\in G_n$ such that $\sigma_r(\zeta_n)=-\zeta_n$. It
is easy to see that
\begin{equation}
\label{eq:even.1}
\zeta_n^r=-\zeta_n \iff 2|n \text{ and } r=\frac{n}{2}+1,
\end{equation}
and if $4|n$, then $(r,n)=1$. As $n\not\equiv 2 \pmod 4$, this gives
\begin{equation}
\label{eq:even.2}
\sqrt{q} \zeta_n\sim -\sqrt{q} \zeta_n \iff 4|n.
\end{equation}
Thus,
\begin{equation}
\label{eq:W_even}
W_q^{\,\rm ss}\simeq \{\pm \sqrt{q} \zeta_n\ ;\ 2\nmid n\,\} \cup
\{\sqrt{q} \zeta_n\ ;\ 4|n\,\}.
\end{equation}
Alternatively, since $\sqrt{q}\in\mathbb Q$, we have $\sqrt{q}\zeta_n^\nu\sim
\sqrt{q}\zeta_n$ for any $\nu\in \bbN$ with $(\nu, n)=1$.
It follows that
\begin{equation}
\label{eq:W_even.1}
W_q^{\,\rm ss}\simeq \{\sqrt{q} \zeta_n\ ;\ n\in \bbN \}.
\end{equation}
The two descriptions (\ref{eq:W_even}) and (\ref{eq:W_even.1}) match, because when $n$ is odd, $-\zeta_n$ is a primitive $2n$-th root of
unity and hence $-\sqrt{q}\zeta_n$ is conjugate to
$\sqrt{q}\zeta_{2n}$.
\subsection{}
\label{sec:par.3}
We now assume that $a$ is odd. Let $\d_p$ be the
discriminant of $\mathbb Q(\sqrt{p})$. In other words, $\d_p=p$ if $p\equiv 1\pmod{4}$, otherwise $\d_p=4p$. By \cite[Chapter V,
Theorem~48]{ANT-Frohlich-Taylor},
$\sqrt{p}\in \mathbb Q(\zeta_n)$ if and only if $\d_p\mid n$. Suppose this is
the case. Let
\begin{equation}
\label{eq:chi}
\chi: G_n=(\mathbb Z/n\mathbb Z)^\times \to
\Gal(\mathbb Q(\sqrt{p})/\mathbb Q)=\{\pm 1\}, \quad
\sigma_r(\sqrt{p})=\chi(r)\sqrt{p}
\end{equation}
be the associated quadratic character. Clearly, $\chi$ factor through
$G_{\d_p}=\Gal(\mathbb Q(\zeta_{\d_p})/\mathbb Q)$.
\begin{lemma}\label{lemma:par.2}
Let $n$ be a
positive integer with $n\not \equiv 2 \pmod 4$ and $q=p^a$ is an odd
power of $p$.
{\rm (i)} If $\sqrt{p}\not \in \mathbb Q(\zeta_n)$, then
$\sqrt{q}\zeta_n\sim-\sqrt{q}\zeta_n$.
{\rm (ii)} Suppose that $\sqrt{p} \in \mathbb Q(\zeta_n)$, i.e., $n$ is
divisible by $\d_p$.
Then
\begin{equation}
\label{eq:par.4}
\sqrt{q}\zeta_n\sim-\sqrt{q}\zeta_n \iff 4|n \text{
and \ } \chi(n/2+1)=1.
\end{equation}
\end{lemma}
\begin{proof}
(i) As $\sqrt{p}\not \in \mathbb Q(\zeta_n)$, there is an element
$\sigma\in \Gal(L/\mathbb Q)$ such that $\sigma(\zeta_n)=\zeta_n$ and
$\sigma(\sqrt{p})=-\sqrt{p}$. Then $\sigma(\sqrt{q}\zeta_n)=-
\sqrt{q}\zeta_n$.
(ii) First, $\sqrt{q}\zeta_n\sim-\sqrt{q}\zeta_n$ if and only if there
is an
element $\sigma_r\in G_n$ such that
$\sigma_r(\sqrt{q}\zeta_n)=\chi(r) \sqrt{q}
\zeta_n^r=-\sqrt{q}\zeta_n$. If $\chi(r)=-1$, then
$\zeta_n^r=\zeta_n$ and $\sigma_r=1$, which is impossible.
If $\chi(r)=1$, then $\zeta_n^r=-\zeta_n$ and hence $4|n$ and $r=n/2+1$ by
(\ref{eq:even.1}). This concludes our assertion (\ref{eq:par.4}).
\end{proof}
\begin{prop}\label{prop:par.3}
Let $n$ and $q$ be as in Lemma~\ref{lemma:par.2}.
{\rm (a)} Suppose that $p=2$. Then
\begin{equation}
\label{eq:par.6}
\sqrt{q}\zeta_n\sim-\sqrt{q}\zeta_n \iff 8\nmid n \text{ or } 16 |n.
\end{equation}
{\rm (b)} Suppose that $p\equiv 1\pmod 4$. Then
\begin{equation}
\label{eq:par.7}
\sqrt{q}\zeta_n\sim-\sqrt{q}\zeta_n \iff p\nmid n \text{ or }
4p | n.
\end{equation}
{\rm (c)} Suppose that $p\equiv 3\pmod 4$. Then
\begin{equation}
\label{eq:par.8}
\sqrt{q}\zeta_n\sim-\sqrt{q}\zeta_n \iff 4p\nmid n
\text{ or } 8p | n.
\end{equation}
\end{prop}
\begin{proof}
(a) By Lemma~\ref{lemma:par.2}, we have
$\sqrt{q}\zeta_n\sim-\sqrt{q}\zeta_n$ if and only if either $8\nmid n$ or
$8| n$ and $\chi(n/2+1)=1$.
Suppose $8|n$. Note that $\mathbb Q(\zeta_8)=\mathbb Q(\sqrt{2},\sqrt{-1})$
and $\sqrt{2}=\zeta_8+\zeta_8^{-1}$. It follows that
\begin{equation}
\label{eq:par.9}
\chi(r)=
\begin{cases}
1 & \text{if $r\equiv 1, 7\pmod 8$}; \\
-1 & \text{if $r\equiv 3, 5 \pmod 8$}.
\end{cases}
\end{equation}
If $8||n$, then $r=n/2+1\equiv 5 \pmod 8$ and $\chi(r)=-1$.
If $16|n$, then $r=n/2+1\equiv 1 \pmod 8$ and $\chi(r)=1$. Thus,
$\sqrt{q}\zeta_n\sim-\sqrt{q}\zeta_n \iff 8\nmid n \text{ or } 16 |
n.$
(b) By Lemma~\ref{lemma:par.2}, we have
$\sqrt{q}\zeta_n\sim-\sqrt{q}\zeta_n$ if and only if either $p\nmid n$ or
$4p | n$ and $\chi(n/2+1)=1$. If $4p\,|n$, then $\chi(n/2+1)=1$ since
$n/2+1\equiv 1 \pmod p$. Thus,
$\sqrt{q}\zeta_n\sim-\sqrt{q}\zeta_n \iff p\nmid n \text{ or } 4p | n$.
(c) By Lemma~\ref{lemma:par.2}, we have
$\sqrt{q}\zeta_n\sim-\sqrt{q}\zeta_n$ if and only if either $4p \nmid n$ or
$4p\, | n$ and $\chi(n/2+1)=1$.
Suppose that $4p|n$ and write $G_{4p}=G_4 \times G_{p}$. Since
$r=n/2+1\equiv 1 \pmod p$, the image of $\sigma_r$ in $G_p$ is
trivial. In particular, it fixes $\sqrt{-p}\in\mathbb Q(\zeta_p)$.
As $\sqrt{-p}\cdot
\sqrt{-1}=-\sqrt{p}$, one has $\chi(r)=1$ if and only if $r\equiv 1
\pmod 4$. Write $n=4pk$ for some integer $k$. Then $r=2pk+1\equiv 1
\pmod 4$ if and only if $k\equiv 0 \pmod 2$. Therefore, we get
$\sqrt{q}\zeta_n\sim-\sqrt{q}\zeta_n \iff 4p\nmid n
\text{ or } 8p | n$.
\end{proof}
As typical examples, we have
(a) $\sqrt{2}\zeta_8\not\sim -\sqrt{2}\zeta_8$ and
$\sqrt{2}\zeta_{16}\sim -\sqrt{2}\zeta_{16}$,
(b) $\sqrt{5}\zeta_{5} \not\sim -\sqrt{5}\zeta_{5}$ and
$\sqrt{5}\zeta_{20} \sim -\sqrt{5}\zeta_{20}$, and
(c) $\sqrt{3}\zeta_{12} \not\sim -\sqrt{3}\zeta_{12}$ and
$\sqrt{3}\zeta_{24} \sim -\sqrt{3}\zeta_{24}$.
\begin{cor}\label{cor:par.4}
Suppose that $q$ is an odd power of $p$ and $n\not\equiv 2\pmod{4}$.
{\rm (1)} If $p\equiv 1 \pmod 4$, then
\[ W_q^{\rm ss}=\{\sqrt{q}\zeta_n\, ;\, n\not \equiv 2\!\! \pmod 4\,
\}\cup
\{-\sqrt{q}\zeta_n\, ;\, 2\nmid n \, \text{and\ } p|n\, \}.
\]
{\rm (2)} If $p\equiv 3 \pmod 4$ or $p=2$, then
\[ W_q^{\rm ss}=\{\sqrt{q}\zeta_n\, ;\, n\not \equiv 2 \! \! \pmod
4\, \}\cup
\{-\sqrt{q}\zeta_n\, ;\, 4p\mid n \, \text{and\ }
8p \nmid n \, \}.
\]
\end{cor}
\begin{proof}
(1) By Proposition~\ref{prop:par.3}, $\sqrt{q}\zeta_n\not\sim
-\sqrt{q}\zeta_n$ if and only if $p|n$ and $4p\nmid n$,
i.e. $p|n$ and $2\nmid n$. (2) We have $\sqrt{q}\zeta_n\not\sim
-\sqrt{q}\zeta_n$ if and only if $4p|n$ and $8p\nmid n$.
\end{proof}
\begin{defn}
\label{sec:par.5}
Let $\d_q$ be the smallest positive integer such that
$\mathbb Q(\sqrt{q})\subset \mathbb Q(\zeta_{\d_q})$. More specifically, $\d_q=\d_p$
if $q$ is an odd power of $p$, otherwise $\d_q=1$. We say a positive integer $n$
is {\it critical at $q$} if $\d_q|n$ and $2\d_q\nmid n$.
\end{defn}
It is clear from the definition that for a fixed $n\in \bbN$, the condition that $n$ is critical
at $q=p^a$ depends only on $p$ and the parity of $a$.
\begin{prop}\label{prop:par.5}
Let $n\not \equiv 2 \pmod 4$ be a positive integer and $q=p^a$ a
power of a prime number $p$. Then
$\sqrt{q}\zeta_n\sim-\sqrt{q}\zeta_n$ if and only if $n$ is not
critical at $q$.
\end{prop}
\begin{proof}
The proposition reduces to either (\ref{eq:even.2}) or
Proposition~\ref{prop:par.3} according to whether $a$ is even or odd
respectively.
\end{proof}
\begin{cor}\label{cor:par.6}
We have
\[ W_q^{\rm ss}=\{\sqrt{q}\zeta_n\, ;\,
n\not \equiv 2\!\!\! \pmod 4\, \}\cup
\{-\sqrt{q}\zeta_n\, ;\, n\not \equiv 2\!\!\! \pmod 4 \, \text{and
$n$ is critical at $q$} \, \}.
\]
\end{cor}
\section{Dimension of supersingular abelian varieties}
\label{sec:dim}
\subsection{}
\label{sec:dim.1}
Let $q=p^a$ be a power of a prime number $p$, and $\pi$ a
supersingular Weil $q$-number as in the previous section.
Replacing $\pi$ by a suitable conjugate, we may assume that $\pi=\pm \sqrt{q}
\zeta_n$ for a positive integer $n$ with $n\not\equiv 2 \pmod 4$.
Let $X_\pi$ be a simple abelian variety over ${\bbF}_q$
in the isogeny class corresponding to $\pi$.
Its endomorphism algebra
$\mathcal{E}=\mathcal{E}_\pi:=\End^0(X_\pi)$ is a central division algebra over
$K:=\mathbb Q(\pi)$, unique up to isomorphism depending only on $\pi$ and not
on the choice of $X_\pi$. The field $K$ is either a totally real field or a CM
field \cite[Section~1]{tate:ht}.
The goal of this section is to determine the dimension $d(\pi)$ of
$X_\pi$. For each $d\in \bbN$, define
\begin{equation}
\label{eq:Wd}
W^{\rm ss}_q(d):=\{\pi\in W^{\rm ss}_q \mid d(\pi)=d\}.
\end{equation}
According to the
Honda-Tate theory (ibid.), one has
\[ d(\pi):=\frac{1}{2} [K:\mathbb Q] \sqrt{[\mathcal{E}:K]}=\frac{1}{2}\deg_\mathbb Q(\mathcal{E}). \]
(For a semisimple algebra over a field $F$, its $F$-degree is the degree of any of its
maximal commutative semi-simple $F$-subalgebras.)
Moreover, the invariants of $\mathcal{E}$ at a place $v$ of $K$ is given by
\[ \inv_v(\mathcal{E})=
\begin{cases}
1/2 & \text{if $v$ is real;} \\
v(\pi)/v(q) [K_v:{\bbQ}_p] & \text{if $v|p$;} \\
0 & \text{otherwise.}
\end{cases}
\]
Here $K_v$ is the completion of $K$ at the place $v$. Observe that $d(\pi)=d(-\pi)$.
As $v(\pi)/v(q)=1/2$ for all $v|p$, every invariant $\inv_v(\mathcal{E})$ is a
2-torsion.
It follows from the Albert-Brauer-Hasse-Noether theorem that $\mathcal{E}$ is either a
quaternion $K$-algebra or the field $K$ itself (henceforth labeled as case (Q) or (F) respectively).
\subsection{Totally real case}
\label{sec:12}
The case where $K$ is a totally real field is well known.
(a) If $a$ is even, then $K=\mathbb Q$ and $\mathcal{E}$ is the quaternion algebra over
$\mathbb Q$ ramified exactly at $\{p,\infty\}$. One has $\pi=\pm
p^{a/2}$ (two isogeny classes) and $d(\pi)=1$.
(b) If $a$ is odd, then $K=\mathbb Q(\sqrt{p})$
and $\mathcal{E}$ is the quaternion algebra over
$K$ ramified exactly at the two real places $\{\infty_1,
\infty_2\}$ of $K$. One has $\pi=q^{1/2}$ (one isogeny class)
and $d(\pi)=2$.
\subsection{CM case}
\label{sec:dim.3} Consider the case where $K$ is a CM field,
i.e., $n>2$.
Put $L:=\mathbb Q(\sqrt{q}, \zeta_n)\supseteq K$. As $K$ and $L$ are abelian
extensions of $\mathbb Q$,
the degree $[K_v:{\bbQ}_p]$ is even for one $v|p$ if and only if it is so for
all $v|p$. Thus, we have the following two possibilities:
\begin{itemize}
\item [(F)] $[K_v:\mathbb Q_p]$ is even for all $v|p$.
\item [(Q)] $[K_v:\mathbb Q_p]$ is odd for all $v|p$.
\end{itemize}
As $K$ is CM, Condition (F) holds if and only if all invariants
of $\mathcal{E}$ vanish. In this case $\mathcal{E}=K$
and $d(\pi)= [K:\mathbb Q]/2$.
\subsection{The case where $a$ is even.} \label{sec:dim.4}
Suppose that $n>2$. One has $K=\mathbb Q(\zeta_n)$ and
$[K:\mathbb Q]=\varphi(n)$. Thus,
\begin{equation}
\label{eq:dim.1}
d(\pi)=
\begin{cases}
\varphi(n)/2 & \text{if (F) holds}; \\
\varphi(n) & \text{if (Q) holds.}
\end{cases}
\end{equation}
The ramification index of any ramified prime $p$ in $\mathbb Q(\zeta_n)$ is
even, so if $p\mid n$, then (F) holds. When $p\nmid
n$, Condition (F) holds if and only if the order of $p\in \umod{n}$ is
even. In particular, if $[K:\mathbb Q]$ is a power of 2, then Condition (Q)
holds if
and only if $K_v={\bbQ}_p$, or equivalently $p\equiv 1 \pmod n$.
We have the following list, which enables to us to list concretely all
$\pi$ with small values of $d(\pi)$.
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|} \hline
$n\not\equiv 2\pmod{4}$ & $3$ & $4$ & $5$ & $7$ & $8$ & $9$ & $11$ & $12$ &
$15$ & $16$ & $20$ & $21$ & $24$ & rest \\ \hline
$d(\pi)$, (Q) holds & $2$ & $2$ & $4$ & $6$ & $4$ & $6$ & $10$ & $4$ &
$8$ & $8$ & $8$ & $12$ & $8$ & $>8$ \\ \hline
$d(\pi)$, (F) holds & $1$ & $1$ & $2$ & $3$ & $2$ & $3$ & $5$ & $2$ &
$4$ & $4$ & $4$ & $6$ & $4$ & $>4$ \\ \hline
\end{tabular} \\
\end{center} \
\begin{prop}\label{prop:dim.1}
Let $\pi=\pm \sqrt{q} \zeta_n$ be a supersingular Weil $q$-number
with $n\ge 1$ and $n\not \equiv 2\pmod{4}$. Suppose that $q=p^a$ is an even power of $p$.
\begin{enumerate}
\item We have $d(\pi)=1$ if and only if $n=1$, or $n=3, 4$
and $p\not\equiv 1 \pmod n$.
\item We have $d(\pi)=2$ if and only if
\begin{enumerate}
\item $n=3,4$ and $p \equiv 1 \pmod n$, or
\item $n=5, 8, 12$ and $p \not \equiv 1 \pmod n$.
\end{enumerate}
\item We have $d(\pi)=3$ if and only if $n=7$ and $p\not\equiv 1,2,4
\pmod 7$, or $n=9$ and $p\not \equiv 1,4,7 \pmod 9$.
\item We have $d(\pi)=4$ if and only if
\begin{enumerate}
\item $n=5,8, 12$ and $p \equiv 1 \pmod n$, or
\item $n=15,16,20,24$ and $p \not \equiv 1 \pmod n$.
\end{enumerate}
\end{enumerate}
\end{prop}
\subsection{The case where $a$ is odd. }
\label{sec:dim.5}
Suppose that $n>1$ and $n\not \equiv 2\pmod{4}$. Put
\begin{equation}
\label{eq:def-of-m}
m:=
\begin{cases}
n/2 & \text{if $n$ is even,} \\
n & \text{if $n$ is odd,}
\end{cases} \quad \text{and}\quad K:=\mathbb Q(\sqrt{p}\zeta_n).
\end{equation}
We have the following inclusions of number fields.
\def{\rm loc}{{\rm loc}}
\def{\rm mod}{{\rm mod}}
\begin{equation}
\label{eq:dim.5}
\xymatrix{
& L=\mathbb Q(\sqrt{p}\,,\zeta_n) \ar@{-}[ld] \ar@{-}[d] \ar@{-}[rd] & \\
\mathbb Q(\sqrt{p}\,,\zeta_m) \ar@{-}[dr] & K=\mathbb Q(\sqrt{p}\,\zeta_n) \ar@{-}[d] &
\mathbb Q(\zeta_n) \ar@{-}[ld] \\
& E=\mathbb Q(\zeta_m) & }
\end{equation}
Note that the prime $p$ is ramified in $K$ with even
ramification index, and hence Condition (F) always holds. Therefore,
\begin{equation}
\label{eq:dim.12}
\mathcal{E}=K \quad \text{ and } \quad d(\pi)=\frac{1}{2}\, [K:\mathbb Q].
\end{equation}
\begin{lemma}\label{lm:dim.2}
Let $K$ and $E$ be as in (\ref{eq:dim.5}). We have $K=E$ if and only if $n$ is critical at $q$.
\end{lemma}
\begin{proof}
Clearly $[K:E]=1$ or $2$. If $\pi\sim -\pi$, then $\pi\mapsto -\pi$
induces a nontrivial automorphism of $K$ with fixed field $E$.
Thus, $\pi \sim -\pi$ if and only if $[K:E]=2$. By
Proposition~\ref{prop:par.5}, $[K:E]=1$
if and only if $n$ is critical at
$q$. Note that the lemma also holds when $a$ is even with
$K=\mathbb Q(\sqrt{q}\zeta_n)=\mathbb Q(\zeta_n)$.
\end{proof}
\begin{lemma}\label{lemma:dim.3}
Suppose that $a$ is odd and $n>1$ with $4\nmid n$. Then
\begin{equation}
\label{eq:dim.4}
d(\pi)=\frac{1}{2}[K:\mathbb Q]=
\begin{cases}
\varphi(n)/2 & \text{if $p\,|n$ and $p\equiv 1\!\!\! \pmod
4$;} \\
\varphi(n) & \text{otherwise.}
\end{cases}
\end{equation}
\end{lemma}
\begin{proof}
Since $n$ is odd one has $E=\mathbb Q(\zeta_n)$ and $[E:\mathbb Q]=\varphi(n)$.
We have $\d_q=p$ or $4p$ according as $p\equiv 1 \pmod4$ or not. It is easy to see that $n$ is critical at
$q$ if and only if $p\equiv 1 \pmod 4$ and $p|n$. The assertion then
follows from Lemma~\ref{lm:dim.2} and (\ref{eq:dim.12}).
\end{proof}
\begin{lemma}\label{lemma:dim.4}
Suppose that $a$ is odd and $n=4k$ with $k\in \bbN$. Then
\[d(\pi)=\frac{1}{2} [K:\mathbb Q]=
\begin{cases}
\varphi(n)/4 & \text{if $p\not\equiv 1\pmod{4}$, $4p\mid n$ and
$8p\nmid n$}; \\
\varphi(n)/2 & \text{otherwise.}
\end{cases}\]
\end{lemma}
\begin{proof}
Since $4|n$ we have $[E:\mathbb Q]=\varphi(n)/2$.
By Lemma~\ref{lm:dim.2}
we have
$[K:\mathbb Q]=\delta_n \varphi(n)/2$, where $\delta_n=1$ or $2$
depending on
$n$ is critical at $q$ or not. The lemma follows once we note that $n=4k$ is
never critical when $p\equiv 1\pmod{4}$.
\end{proof}
The following is a list of $d(\pi)$ for $\pi=\sqrt{q}\zeta_n$ with
$4\nmid n$ and $4|n$, respectively.
The symbol ($*$) denotes the primes satisfying the
conditions $p\, |n$ and $p\equiv 1\, (4)$,
and ($**$) denotes the primes satisfying the three conditions $p\not\equiv
1\pmod{4}$, $4p\mid n$ and $8p\nmid n$. For the sake of completeness,
the case $n=1$ is included and also marked to make a distinction.
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|} \hline
$n$ odd & $1^\natural$ & $3$ & $5$ & $7$ & $9$ & $11$ & $13$ & $15$
& rest \\ \hline
$\varphi (n)$ & $1$ & $2$ & $4$ & $6$ & $6$ & $10$ & $12$ & $8$
& $>8$ \\ \hline
($*$) & $\emptyset$ & $\emptyset$ & $p=5$
& $\emptyset$ & $\emptyset$ & $\emptyset$ & $p=13$ & $p=5$ &
\\ \hline
$d(\pi) $ & $2$ & $2$ & $2\ (p=5)$ & $6$ & $6$ & $10$ & $6\ (p=13)$ &
$4 \ (p=5)$ & $>4$ \\
& & & $4\ (p\neq 5)$ & & & & $12 \ (p\neq 13)$
& $8 \ (p\neq5)$ & \\ \hline
\end{tabular} \\
\end{center}
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|c|c|} \hline
$n=4k$ & $4$ & $8$ & $12$ & $16$ & $20$ & $24$ & $28$ \\ \hline
$\varphi (n)$ & $2$ & $4$ & $4$ & $8$ & $8$ & $8$ & $12$ \\ \hline
($**$) & $\emptyset$ & $2$ & $3$ & $\emptyset$ & $\emptyset$
& $2$ & $7$ \\ \hline
$d(\pi)$ & $1$ & $1\ (p=2)$ & $1\ (p=3)$ & $4$ & $4$
& $2\ (p=2)$ & $3\ (p=7)$ \\
& & $2\ (p\neq2)$ & $2\ (p\neq3)$ & & & $ 4\ (p\neq 2)$
& $ 6\ (p\neq 7)$ \\ \hline \hline
$n=4k$ & $32$ & $36$ & $40$ & $44$ & $48$ & $56$ & $60$ \\ \hline
$\varphi(n)$ & $16$ & $12$ & $16$ & 20 & $16$ & $24$ & $16$ \\ \hline
($**$) & $\emptyset$ & $p=3$ & $p=2$ & $p=11$ & $\emptyset$ & $p=2$ & $p=3$
\\ \hline
$d(\pi)$ & $8$ & $3\ (p=3)$ & $4\ (p=2)$ & $5\ (p=11)$ & $8$ &
$6\ (p=2)$ & $4\ (p=3)$ \\
& & $6\ (p\neq 3)$ & $8\ (p\neq2)$ & $10\ (p\neq 11)$ & &
$12\ (p\neq 2)$ & $ 8\ (p\neq 3)$ \\ \hline
\end{tabular} \\
\end{center} \
It is easy to see that when $4|n$ and either $n=52$ or $n>60$, the value
$\varphi(n)>16$ and hence $d(\sqrt{q}\zeta_n)>4$.
\begin{prop}\label{prop:dim.5}
Suppose that $q=p^a$ is an odd power of $p$.
\begin{enumerate}
\item $W^{\rm ss}_q(1)$ consists of
\[ \sqrt{q} \zeta_4,\ \pm \sqrt{q} \zeta_8\ (p=2), \
\pm \sqrt{q}\zeta_{12} \ (p=3). \]
\item $W^{\rm ss}_q(2)$ consists of
\[ \sqrt{q},\ \sqrt{q} \zeta_3,\ \pm \sqrt{q} \zeta_5\ (p=5),
\ \sqrt{q}\zeta_8\ (p\neq 2),\ \sqrt{q}\zeta_{12} \ (p\neq3),
\ \pm\sqrt{q}\zeta_{24}\ (p=2). \]
\item $W^{\rm ss}_q(3)$ consists of $\pm \sqrt{q}\zeta_{28}$ if $p=7$, or $\pm\sqrt{q}\zeta_{36}$ if $p=3$.
\item $W^{\rm ss}_q(4)$ consists of
\[ \sqrt{q} \zeta_{5}\ (p\neq 5),
\ \pm \sqrt{q} \zeta_{15}\ (p=5), \ \sqrt{q} \zeta_{16}, \]
\[ \sqrt{q} \zeta_{20},
\ \sqrt{q} \zeta_{24}\ (p\neq 2),
\ \pm \sqrt{q} \zeta_{40}\ (p=2),\ \pm \sqrt{q} \zeta_{60}\ (p=3). \]
\end{enumerate}
\end{prop}
\section{Supersingular elliptic curves over finite fields}
\label{sec:curve}
\def\mathrm{Isog}{\mathrm{Isog}}
\subsection{Isogeny classes over finite fields}
\label{sec:curve.1}
Let ${\mathcal Isog}_q$ denote the set of isogeny classes of abelian
varieties over ${\bbF}_q$, where $q=p^a$ is a power of the prime number
$p$. Let $\mathbb Z W_q$ be the free abelian group (written
multiplicatively) generated by the set $W_q$ of conjugacy classes of
Weil $q$-numbers. A nontrivial
element $\pi\in \mathbb Z W_q$ can be put in the form $\pi_1^{m_1}\times
\dots \times \pi_r^{m_r}$ for some $r\in \bbN$,
where each $\pi_i\in W_q$, $\pi_i\not\sim \pi_j$ if $i\neq j$, and
$m_i\neq 0$ for all $1\leq i\leq r$. Such an element
is called a {\it multiple Weil $q$-number} if $m_i> 0$ for all $i$,
and the set of all these elements is denoted by $MW_q$.
Put $X_\pi:=\prod_i X_{\pi_i}^{m_i}$, where $X_{\pi_i}$ is the simple abelian
variety (up to isogeny) over ${\bbF}_q$ corresponding to $\pi_i$.
The Honda-Tate theorem naturally extends to a bijection $MW_q \simeq
{\mathcal Isog}_q$ which sends each $\pi\in MW_q$ to the isogeny class
$[X_\pi]\in {\mathcal Isog}_q$ of $X_\pi$.
For each $\pi\in MW_q$, we define its {\it dimension} as
\[ d(\pi):=\dim X_\pi= \sum_{i=1}^r m_i d(\pi_i). \]
Let $\mathrm{Isog}(\pi)=\mathrm{Isog}(X_\pi)$
denote the set of ${\bbF}_q$-isomorphism classes of
abelian varieties
isogenous to $X_\pi$ over ${\bbF}_q$, and denote $H(\pi):=|\mathrm{Isog}(\pi)|$.
Let $MW^{\rm ss}_q\subset MW_q$ be the subset of supersingular multiple
Weil $q$-numbers, i.e.~those $\pi\in MW_q$ whose corresponding
abelian varieties $X_\pi$ are supersingular.
For any integer $d\ge 1$, let $MW_q(d)$
(resp. $MW^{\rm ss}_q(d)$) denote
the subset consisting of all elements $\pi$ in $MW_q$
(resp. in $MW^{\rm ss}_q$) of dimension $d$. Let $S_d(\mathbb F_q)$
(resp. ${\rm Sp}_d(\mathbb F_q)$) be the set
of isomorphism classes of
$d$-dimensional supersingular (resp.~superspecial) abelian varieties
over ${\bbF}_q$. When $\pi\in MW^{\rm ss}_q$, we let ${\rm Sp}(\pi)\subset
\mathrm{Isog}(\pi)$ be the subset consisting of superspecial isomorphism classes and denote
$H_{sp}(\pi):=|{\rm Sp}(\pi)|$. Thus,
\begin{equation}
\label{eq:sp.1}
|S_d(\mathbb F_q)|=\sum_{\pi \in MW^{\rm ss}_q(d)} H(\pi), \quad
|{\rm Sp}_d(\mathbb F_q)|=\sum_{\pi \in MW^{\rm ss}_q(d)} H_{\rm sp}(\pi).
\end{equation}
\subsection{Supersingular elliptic curves}
\label{sec:curve.2}
We compute the number $|S_1({\bbF}_q)|$ of isomorphism classes of
supersingular elliptic curves over ${\bbF}_q$, where $q=p^a$ as before.
The method is based almost entirely on the results of Waterhouse
\cite{waterhouse:thesis}, except certain details need to be cleared
up (compare with \cite[Theorems 4.5]{waterhouse:thesis}).
\begin{thm}\label{curve.1}
Let $\pi$ be the Frobenius
endomorphism of an elliptic curve $E_0$ over ${\bbF}_q$, and $K:=\mathbb Q(\pi)$.
Assume that $\pi\not\in\mathbb Q$ so that
$K$ is an imaginary quadratic field. Equivalently, the central $K$-algebra
$\End^0(E_0)$ of the elliptic curve $E_0$ is assumed to be
commutative and thus necessarily an imaginary quadratic field.
\begin{enumerate}
\item Any endomorphism ring $R=\End(E)$ of an elliptic curve $E$ in
the isogeny class $[E_0]$ of $E_0$ contains $\pi$ and is maximal at
$p$, that is, $R\otimes \mathbb Z_p$ is the maximal order in $K\otimes
{\bbQ}_p$. Conversely, any
order $R$ of $K$ satisfying these two properties occurs as an
endomorphism ring of an elliptic curve in this isogeny class.
\item Suppose that $R\subset K$ is a quadratic order as in (1).
Then the Picard
group $\Pic(R)$ of $R$ acts freely on the set $[E_0]_R\subset [E_0]$
of isomorphism
classes of elliptic curves in $[E_0]$ with endomorphism ring
$R$. Moreover, the
number $N$ of orbits is $2$ if $p$ is inert in $K$ and $a$
is even, and $N=1$ otherwise.
\end{enumerate}
\end{thm}
\begin{proof}
Statement (1) is \cite[Theorem 4.2]{waterhouse:thesis}.
We give a proof of the second part of Statement (2)
since it differs from
\cite[Theorem 4.5]{waterhouse:thesis} in some cases.
We assert that the statement of \cite[Theorem
5.1]{waterhouse:thesis} for principal abelian varieties is
directly applicable to this situation. Namely, the number of orbits here
is also given by $N=\prod_{v|p} N_v$,
where $v$ runs through the set of all places of $K$ over $p$,
and each $N_v$ is the number described as follows.
Let $e_v$ and $f_v$ be
the ramification index
and residue degree of $v$, respectively, and set $g_v=\gcd(f_v, a)$
and $m_v:=g_v \mathrm{ord}_v(\pi)/a$. Note that $m_v$ is an integer since $\End^0(E_0)$
is commutative and thus $f_v \mathrm{ord}_v(\pi)/a\in \bbN$. Then $N_v$ is the number of all
$g_v$-tuples $(n_1,\dots, n_{g_v})$ of
integers satisfying $0\le n_j\le e_v$ and $\sum_{j=1}^{g_v} n_j=m_v$.
In the present situation $\End^0(E_0)=K$ is commutative and $R$ is maximal at $p$.
As in the proof of \cite[Theorem
5.1]{waterhouse:thesis}, to find the
the number of orbits for the action of $\Pic(R)$ on $[E_0]_R$, one
needs to classify the Tate-modules $T_\ell E$ at all primes $\ell
\neq p$ and the Dieudonn\'{e} modules at the prime $p$ of $E\in[E_0]_R$. The number
of orbits is then the product of the number of isomorphism classes
of the above modules at each prime.
The Tate-module $T_\ell E$ of each $E\in [E_0]_R$ at a prime $\ell
\neq p$ is naturally an $R_\ell$-module with
$R_\ell=R\otimes_\mathbb Z\Z_\ell$.
Since $R[1/p]$ is a
quadratic order, any fractional $R[1/p]$-ideal $I$ whose order
ring equals $R[1/p]$ must be locally free over $R[1/p]$. Particularly, there is only one isomorphism class
of the prime-to-$p$ Tate modules of $E$ for all
$E\in [E_0]_R$. Thus, $N$ is equal to the
number of isomorphism classes of Dieudonn\'{e} modules occurring in the
isogeny class
$[E_0]$, which is equal to $\prod_v N_v$ as given in the proof of
\cite[Theorem 5.1]{waterhouse:thesis}.
Now it is easy to compute the number $N$ of orbits.
Notice $N_v\neq 1$ only when $g_v>1$. For our
case with $[K:\mathbb Q]=2$ this occurs only when $p$ is inert in $K$ and
$a$ is even.
In this case there is only one place $v$ over $p$, $g_v=2$ and
$e_v=1$. Then $N=N_v$ is the number of pairs $(n_1,n_2)$ with
$0\le n_1, n_2\le 1$ and $n_1+n_2=1$, which is $2$.
\end{proof}
\begin{remark}
In \cite[Theorem 5.1]{waterhouse:thesis} the assumption that
the endomorphism ring $R=\End(A)$ is the maximal order can be replaced
by the weaker assumption that $R$ is
both Gorenstein and maximal at $p$. Indeed, any proper
$R$-lattice of rank one over a Gorenstein order $R$ is locally free
\cite[Theorem 37.16 p.~789]{curtis-reiner:1}, so the same proof of
\cite[Theorem 5.1]{waterhouse:thesis} applies.
\end{remark}
\begin{rem}\label{rem:2-orbits-case}
Suppose that $a$ is even and $p$ is inert in the imaginary quadratic
field $K=\mathbb Q(\pi)$ so that
$N=2$. By the classification of Waterhouse
(\cite[Lemma,~p.537]{waterhouse:thesis}, see also
Proposition~\ref{prop:dim.1}), this occurs only for supersingular
Weil $q$-numbers $\pi$ where
\begin{equation}\label{eq:1-dim-weil-number-2-orbits}
\pi\sim \pm p^{a/2}\zeta_3,\, p\equiv 2\pmod{3} \quad \text{ or } \quad
\pi \sim p^{a/2}\zeta_4, \, p\equiv 3\pmod{4}.
\end{equation}
Then by part (1) of Theorem~\ref{sec:curve.1}, $\End(E)=O_K$ for any
elliptic curve $E$ in the isogeny class corresponding to $\pi$. Since $h(O_K)=1$, part (2) of
Theorem~\ref{sec:curve.1} implies that a complete set of
representatives of $\Sp(\pi)$ consists a pair of
elliptic curves of the form $\{E, E^{(p)}\}$, where
$E^{(p)}:=E\otimes_{\mathbb F_q, \sigma_p} \mathbb F_q$, and $\sigma_p\in \Gal(\mathbb F_q/\mathbb F_p)$ is the Frobenius automorphism of
$\mathbb F_q/\mathbb F_p$. These two elliptic curves are distinguished by the
actions of $O_K$ on the respective 1-dimensional Lie-algebras
$\Lie(E)$ and $\Lie(E^{(p)})$ over
$\mathbb F_q$, which are given by distinct embeddings
$O_K/(p)\simeq \mathbb F_{p^2} \hookrightarrow \mathbb F_q$. This establishes a
natural bijection
$\Sp(\pi)\simeq \Hom(O_K/(p), \mathbb F_q)$ for every $\pi$ in (\ref{eq:1-dim-weil-number-2-orbits}).
\end{rem}
We return to the calculation of $|\Sp_1({\bbF}_q)|$ by the counting
method. The isogeny classes of supersingular
elliptic curves over $\mathbb F_q$ are completely listed by the following Weil numbers
\begin{equation}
\label{eq:curve.2}
\begin{split}
W_q^{\rm ss}(1)&=\{\sqrt{q} \zeta_4,\ \pm \sqrt{q} \zeta_8\ (p=2),
\ \pm \sqrt{q} \zeta_{12}\ (p=3)\, \}, \quad \text{for $a$ odd}; \\
W_q^{\rm ss}(1)&=\{\pm \sqrt{q},\
\pm \sqrt{q} \zeta_3\ (p\not\equiv 1\, (3)),\
\sqrt{q} \zeta_{4}\ (p\not\equiv 1 \,(4))\, \},
\quad \text{for $a$ even.} \\
\end{split}
\end{equation}
For each Weil $q$-number $\pi\in W_q^{\rm ss}(1)$, let $R_0$ be the
smallest quadratic order in $K=\mathbb Q(\pi)$ which contains
$\pi$ and is maximal at $p$.
It is easy to see that $R_0$ is the maximal order except when
$\pi=\sqrt{q} \zeta_4$, $p\equiv 3\pmod{4}$ and $a$ is odd. In the
latter case
$R_0=\mathbb Z[\sqrt{-p}]$ and we have by Theorem~\ref{curve.1} that
\begin{equation}
\label{eq:4odd}
H(\sqrt{q} \zeta_4)=
\begin{cases}
h(O_K) & \text{for $p=2$ or $p\equiv 1 \pmod 4$;} \\
h(R_0)+h(O_K) & \text{for $p\equiv 3 \pmod 4$.}
\end{cases}
\end{equation}
For the other cases the order $R_0$ is maximal and we have
\begin{equation}
\label{eq:curve.4}
H(\pi)=N\cdot h(O_K)
\end{equation}
where $N=2$ if $p$ is inert in $K$ and $a$ is even, and $N=1$ otherwise.
Recall that for a square free $m\in \mathbb Z$, the class
number of $\mathbb Q(\sqrt{m})$ is denoted by $h(\sqrt{m})$.
Suppose first that $a$ is odd. For $p=2$, we have
\begin{equation}
\label{eq:oddp=2}
|\Sp_1(\mathbb F_q)|=H(\sqrt{q}\zeta_4)+2
H(\sqrt{q}\zeta_8)=h(\sqrt{-2})+2h(\sqrt{-1})=3.
\end{equation}
For $p=3$, we have
\begin{equation}
\label{eq:oddp=3}
\begin{split}
|\Sp_1(\mathbb F_q)|&=H(\sqrt{q}\zeta_4)+2
H(\sqrt{q}\zeta_{12}) \\
&=h(\mathbb Z[\sqrt{-3}])+h(\sqrt{-3})+2h(\sqrt{-3})=4.
\end{split}
\end{equation}
For $p>3$, we have (see \cite[Theorem 1.1]{yu:sp_prime})
\begin{equation}
\label{eq:oddp>3}
\begin{split}
|\Sp_1(\mathbb F_q)|&=H(\sqrt{q}\zeta_4) \\
&=
\begin{cases}
h(\sqrt{-p}) & \text{for $p\equiv 1 \pmod 4$;}\\
2h(\sqrt{-p}) & \text{for $p\equiv 7 \pmod 8$ ($2$ splits in
$\mathbb Q(\sqrt{-p})$) ;}\\
4h(\sqrt{-p}) & \text{for $p\equiv 3 \pmod 8$ ($2$ is inert in
$\mathbb Q(\sqrt{-p})$).}
\end{cases}
\end{split}
\end{equation}
Since $\left (\frac{2}{p}\right )=1$ for $p\equiv 1,7 \pmod 8$ and
$\left (\frac{2}{p}\right )=-1$ for $p\equiv 3, 5 \pmod 8$, we can
rewrite (\ref{eq:oddp>3}) as
\begin{equation}
\label{eq:odd_p>3}
|\Sp_1(\mathbb F_q)|=
\begin{cases}
h(\sqrt{-p}) & \text{for $p\equiv 1 \pmod 4$;}\\
\left(3-\left (\frac{2}{p}\right )\right )h(\sqrt{-p}) &
\text{for $p\equiv 3 \pmod 4$}.
\end{cases}
\end{equation}
Suppose now that $a$ is even.
By (\ref{eq:curve.2}), we have
\begin{equation}
\label{eq:curve.9}
|\Sp_1(\mathbb F_q)|=2H(\sqrt{q})+2 \delta_3(p) H(\sqrt{q}\zeta_3)
+\delta_4(p) H(\sqrt{q}\zeta_4),
\end{equation}
where $\delta_m(p)=1,0$ according as $p\not\equiv 1 \pmod m$ or not
for $m=3,4$.
It is well known that $H(\sqrt{q})$ is equal to the class number
$h(B_{p,\infty})$ of the quaternion $\mathbb Q$-algebra
$B_{p, \infty}$
ramified only at $p$ and $\infty$. Thus,
\begin{equation}
\label{eq:curve.10}
H(\sqrt{q})=\frac{p-1}{12}+\frac{1}{3}\left
(1-\left(\frac{-3}{p}\right )\right )+\frac{1}{4}\left
(1-\left(\frac{-4}{p}\right )\right ).
\end{equation}
By Theorem~\ref{curve.1}, we have
\begin{equation}
\label{eq:even3}
\delta_3(p)H(\sqrt{q}\zeta_3)=
\begin{cases}
1 & \text{for $p=3$;}\\
2 & \text{for $p\equiv 2 \pmod 3$;} \\
0 & \text{for $p\equiv 1 \pmod 3$;}
\end{cases}
\end{equation}
and get $\delta_3(p)H(\sqrt{q}\zeta_3)=\left
(1-\left(\frac{-3}{p}\right )\right )$.
Similarly we have $\delta_4(p)H(\sqrt{q}\zeta_4)=\left
(1-\left(\frac{-4}{p}\right )\right )$. Using (\ref{eq:curve.9}) and
(\ref{eq:curve.10}) we get
\begin{equation}
\label{eq:curve.12}
\begin{split}
|\Sp_1(\mathbb F_q)|&=\frac{p-1}{6}+\frac{2}{3}\left
(1-\left(\frac{-3}{p}\right )\right )+\frac{1}{2}\left
(1-\left(\frac{-4}{p}\right )\right )\\
&\quad \quad +2\left
(1-\left(\frac{-3}{p}\right )\right )+\left
(1-\left(\frac{-4}{p}\right )\right ) \\
&= \frac{p-1}{6}+\frac{8}{3}\left
(1-\left(\frac{-3}{p}\right )\right )+\frac{3}{2}\left
(1-\left(\frac{-4}{p}\right )\right ).
\end{split}
\end{equation}
From (\ref{eq:oddp=2}), (\ref{eq:oddp=3}), (\ref{eq:odd_p>3}) and
(\ref{eq:curve.12}), we obtain an explicit formula for the
number $|\Sp_1(\mathbb F_q)|$ of supersingular elliptic curves over ${\bbF}_q$.
\begin{thm}\label{curve.3}
Suppose $q=p^a$ is a power of the prime number $p$.
\begin{enumerate}
\item If $a$ is odd, then
\begin{equation}
\label{eq:curve.13}
|\Sp_1(\mathbb F_q)|=
\begin{cases}
3, 4 & \text{for $p=2,3$, respectively;} \\
h(\sqrt{-p}) & \text{for $p\equiv 1 \pmod 4$;}\\
\left(3-\left (\frac{2}{p}\right )\right )h(\sqrt{-p}) &
\text{for $p\equiv 3 \pmod 4$ and $p>3$}.
\end{cases}
\end{equation}
\item If $a$ is even, then
\begin{equation}
\label{eq:curve.14}
\begin{split}
|\Sp_1(\mathbb F_q)| = \frac{p-1}{6}+\frac{8}{3}\left
(1-\left(\frac{-3}{p}\right )\right )+\frac{3}{2}\left
(1-\left(\frac{-4}{p}\right )\right ).
\end{split}
\end{equation}
\end{enumerate}
\end{thm}
\begin{remark}\label{curve.4}
From the formulas above we observe a phenomenon that the number
$|\Sp_1({\bbF}_q)|$ depends only on the parity of the exponent $a$ of
$q=p^a$. We have already seen in Section 2 that the classification
of supersingular isogeny classes depends only on the parity of
$a$. More explicitly, if the exponents $a$ and $a'$ of $q$ and $q'$ respectively have
the same parity, then a bijective correspondence between
supersingular isogeny classes over
$\mathbb F_q$ and those over $\mathbb F_{q'}$ can be given by matching
$\pi\in W_q^{\rm ss}(1)$ with $\pi'=(-p)^{(a'-a)/2}\pi$ (see
Remark~\ref{rem:parity-isogeny-class}). The parity phenomenon of
$|\Sp_1({\bbF}_q)|$ arises because there is a bijection
$\Sp(\pi)\simeq \Sp(\pi')$ for all pairs $(\pi, \pi')$ as above. Indeed, if $\pi$
and $\pi'$ are of the form in
(\ref{eq:1-dim-weil-number-2-orbits}), then a canonical bijection
$\Sp(\pi)\simeq \Sp(\pi')$ is given by identifying both with
$\Hom(O_K/(p), \mathbb F_q)$ as in Remark~\ref{rem:2-orbits-case}. For the
remaining cases, first suppose that $K=\mathbb Q(\pi)=\mathbb Q(\pi')$ is
imaginary quadratic. Then the endomorphism rings occurring for both
isogeny classes are the same by Theorem~\ref{curve.1}. We partition
$\Sp(\pi)$ into $\coprod_{R} \Sp(\pi, R)$, where $R$ runs over all
possible endomorphism rings, and $\Sp(\pi, R)\subseteq
\Sp(\pi)$ consists of those members with endomorphism ring $R$.
Every $\Sp(\pi, R)$ is
a principal homogeneous
space of $\Pic(R)$. Thus a $\Pic(R)$-equivariant bijection between
$\Sp(\pi, R)$ and $\Sp(\pi', R)$ is established whenever a base point is
chosen respectively in each of them. Lastly, suppose that
$\mathbb Q(\pi)=\mathbb Q(\pi')=\mathbb Q$.
Then $\pi^{a'}=(\pi')^a=p^{aa'/2}$. So we have canonical bijections
$\Sp(\pi)\simeq \Sp(\pi^{a'})\simeq \Sp(\pi')$ by
extending both base fields to $\mathbb F_{p^{aa'}}$ (\cite[
Remark,~p.~542]{waterhouse:thesis}). Equivalently, the bijection
$\Sp(\pi)\simeq
\Sp(\pi')$ can be obtained by matching the $j$-invariants.
\end{remark}
\section{Superspecial abelian surfaces over ${\bbF}_p$}
\label{sec:sp}
In this section we assume that the ground field is the prime
field ${\bbF}_p$; abelian varieties and their morphisms are all
defined over ${\bbF}_p$ unless otherwise stated.
\subsection{Supersingular abelian varieties over ${\bbF}_p$}
\label{sec:sp.1}
We describe a result which allows us to count supersingular
and superspecial abelian
varieties over ${\bbF}_p$, based on a result of
Waterhouse~\cite[Theorem 6.1 (3)]{waterhouse:thesis} (also see an
extension~\cite[Theorem 3.1]{yu:sp_prime} to non-simple isogenies).
Let $X_0$ be a fixed supersingular abelian variety over ${\bbF}_p$ and
let $\pi=\pi_1^{m_1}\times \dots \times \pi_r^{m_r}$ be a multiple Weil
$p$-number corresponding to the isogeny class $[X_0]$. One has
$X_0\sim \prod_{i=1}^r X_i^{m_i}$, where each
$X_i$ is a simple abelian variety with Frobenius endomorphism
$\pi_i$.
The endomorphism algebra $\mathcal{E}=\End^0(X_0)$ of $X_0$ is
equal to
$\prod_{i=1}^r \Mat_{m_i}(\End^0(X_i))$. Let $\pi_0\in \End(X_0)$ be
the Frobenius endomorphism. The $\mathbb Q$-subalgebra $K=\mathbb Q(\pi_0)\subset
\mathcal{E}$ generated by $\pi_0$ is semi-simple and
coincides with the center of $\mathcal{E}$. One has
$K=\prod_i K_i$ and $\pi_0=(\pi_1,\dots, \pi_r)$,
where $K_i=\mathbb Q(\pi_i)$. Let $\calR:=\mathbb Z[\pi_0,p \pi_0^{-1}]\subset
K$ and $\calR_{sp}:=\calR[\pi_0^2/p]\subset K$.
Clearly $\pi_0^2/p$ is an integral element of finite multiplicative order, and
$p/\pi_0=\pi_0\cdot (\pi_0^2/p)^{-1}$, so $\calR_{sp}=\mathbb{Z}[\pi_0,
\pi_0^2/p]\subseteq O_K$, where $O_K=\prod_i
O_{K_i}$ is the maximal order $K$. Observe that the Tate module $T_\ell(X_0)$ (for any prime
$\ell\neq p$), as a $\mathbb Z_\ell[\Gal(\overline{\bbF}_p/{\bbF}_p)]$-module,
is nothing but an $\calR_\ell$-module, and the (covariant) Dieudonn\'{e}
module $M(X_0)$ is simply an $\calR_p$-module, where
$\calR_\ell=\calR\otimes \mathbb Z_\ell$ and $\calR_p=\calR\otimes \mathbb Z_p$.
\begin{thm}\label{prop:sp.1} Let $\pi=\pi_1^{m_1}\times \dots
\pi_r^{m_r}$, and $K$, $\calR$ and $\calR_{sp}$ be as above. Assume that
$K$ has no real place, that is, none of $\pi_i$ is conjugate to
$\sqrt{p}$, and set $V:=\prod_{i=1}^r K_i^{m_i}$.
\begin{enumerate}
\item There is a natural
bijection between the set ${\rm Isog}(\pi)$ and the
set of isomorphism classes of $\calR$-lattices in $V$.
\item Under the above map the subset ${\rm Sp}(\pi)$ is in bijection
with set of isomorphism classes of $\calR_{sp}$-lattices in $V$.
\end{enumerate}
\end{thm}
\begin{proof}
Set $\Lambda:=\prod_{i=1}^r O_{K_i}^{m_i}\subset V$. We view $V$ and $\Lambda$ as
a $K$-module and an $\calR$-lattice, respectively. One chooses an
identification $V\otimes_\mathbb Q \mathbb Q_\ell = T_\ell(X_0)\otimes \mathbb Q_\ell$ for
primes $\ell\neq p$ and $V\otimes_\mathbb Q \mathbb Q_p=M(X_0)\otimes \mathbb Q_p$ such
that $\Lambda_\ell=T_\ell(X_0)$ for almost all primes $\ell$.
Under this identification, any $\calR$-lattice $\Lambda'$
in $V$ gives rise to a unique
quasi-isogeny $\varphi: X\to X_0$ such that
$\varphi_*(T_\ell(X))=\Lambda'\otimes \mathbb Z_\ell$ for $\ell\neq p$ and
$\varphi_*(M(X))=\Lambda'\otimes \mathbb Z_p$.
Two lattices $\Lambda_1$ and
$\Lambda_2$ are isomorphic as $\calR$-modules
if and only if there is an element $g\in
\GL_K(V)$ such that $\Lambda_2=g \Lambda_1$. Two quasi-isogenies are
isomorphic if and only if they differ by an element in
$\mathcal{E}^\times$. Our assumption ensures
that $\GL_K(V)\simeq \mathcal{E}^\times$.
Then the above correspondence induces the desired bijection
(also see \cite[Theorem 3.1]{yu:sp_prime} for a detailed proof).
Note that
the abelian variety $X$ in $[X_0]$ as above
is superspecial if and only if $\pi_0^2 M(X)=pM(X)$, or equivalently,
$M(X)$ is a $(\calR_{sp})_p$-lattice in $M(X_0)\otimes \mathbb Q_p$. That is,
$X$ is superspecial if and only if the corresponding $\calR$-module is
$\calR_{sp}$-stable. The statement (2) then follows from (1).
\end{proof}
\begin{rem}\label{rem:critical-max-order}
Let $\pi=\pi_1^{e_1}$ be a multiple supersingular Weil-$p$ number with $\pi_1=\pm
\sqrt{p}\zeta_n$ and $n$ critical
at $p$. Then by Lemma~\ref{lm:dim.2}, $K=\mathbb Q(\pi_1)=\mathbb Q(\zeta_m)$ and
$O_K=\mathbb Z[\zeta_m]$, where $m$ is define in
(\ref{eq:def-of-m}). Since $\calR_{sp}=\calR[\pi_1^2/p]\ni
\zeta_m$, it follows that $\calR_{sp}$ coincides with the maximal order
$O_K$ in this case.
\end{rem}
\subsection{Proof of the main theorem}
\label{sec:sp.2}
By Section~\ref{sec:dim}, we list the sets $W^{\rm ss}_p(1)$ and
$W^{\rm ss}_p(2)$ of supersingular Weil $p$-numbers of dimension $1$ or
$2$ as follows.
\begin{align} \label{simple_dimension_one}
&W^{\rm ss}_2(1) = \{ \sqrt{2}\zeta_4, \pm \sqrt{2}\zeta_8\}, \notag \\
&W^{\rm ss}_3(1) = \{ \sqrt{3}\zeta_4, \pm \sqrt{3}\zeta_{12} \}, \\
&W^{\rm ss}_p(1) = \{ \sqrt{p}\zeta_4 \}, \quad p \geq 5; \notag
\end{align}
and
\begin{align}\label{simple_dimension_two}
&W^{\rm ss}_2(2) = \{\sqrt{2}, \sqrt{2}\zeta_3, \sqrt{2}\zeta_{12},
\pm \sqrt{2}\zeta_{24} \}, \notag\\
&W^{\rm ss}_3(2) = \{\sqrt{3}, \sqrt{3}\zeta_3, \sqrt{3}\zeta_8 \},\\
&W^{\rm ss}_5(2) = \{\sqrt{5}, \sqrt{5}\zeta_3,
\sqrt{5}\zeta_8, \sqrt{5}\zeta_{12}, \pm \sqrt{5}\zeta_5\}, \notag \\
&W^{\rm ss}_p(2) = \{\sqrt{p}, \sqrt{p}\zeta_3, \sqrt{p}\zeta_8,
\sqrt{p}\zeta_{12}\}, \quad p \geq 7. \notag
\end{align}
Consider the case $\pi\in W^{\rm ss}_p(2)$ or $\pi=\pi_1\times \pi_2$ with
$\pi_1,\pi_2\in W^{\rm ss}_p(1)$. By
(\ref{eq:sp.1}) we have
\begin{equation}
\label{eq:sp.4}
|{\rm Sp}_2({\bbF}_p)|=\sum_{\pi\in W^{\rm ss}_p(2)} H_{sp}(\pi)+
\sum_{\pi_1, \pi_2\in W^{\rm ss}_p(1)} H_{sp}(\pi_1\times \pi_2).
\end{equation}
The number $H_{sp}(\sqrt{p})=H(\sqrt{p})$ has
been calculated in \cite{xue-yang-yu:ECNF}, so
this case will be excluded from our discussion. We refer to
\cite[Section 37]{curtis-reiner:1} for the definition of a Bass
order. Note that when $\pi=\pi_1\times \pi_1$, $\calR_{sp}$ is an order in the quadratic field
$\mathbb Q(\pi_1)$, and such orders are well known to be Bass.
It will be shown in Section~\ref{subsec:order-Rsp-bass} that $\calR_{sp}$
is a Bass order for all $\pi$ considered
(i.e. $\pi\in MW^{\rm ss}_p(2)$).
Thus, when the $K$-module
$V$ is free of rank one
(i.e. in the case where $\pi\neq \pi_1\times \pi_1$),
Theorem~\ref{prop:sp.1} gives
\begin{equation}
\label{eq:sp.5}
H_{sp}(\pi)=\sum_{\calR_{sp}\subset B\subset O_K} h(B).
\end{equation}
In the case when $V$ is free of higher rank (in fact, rank $2$
when $\pi=\pi_1\times \pi_1$) , one can use the results
of Borevich and Faddeev on lattices over orders of cyclic index to
compute $H_{sp}(\pi)$ (cf. \cite[Section 37, p.~789]{curtis-reiner:1}).
In the following, the notation $B_{\pi,j}$ (or $B_j$ for short) with $j\in \bbN$,
will stand for an order $B$ of $K$ with
$\calR_{sp}\subset B\subset O_K$ and $[O_K:B]=j$.
The dependence of $K$, $\calR_{sp}$ and $B_j$ on the choice of the Weil
$p$-number $\pi$ should be understood though
it is omitted from the notation. For any two square-free integers $d>1$ and $j\ge 1$,
we write $K_{d,j}$ for the
CM field $\mathbb Q(\sqrt{d},\sqrt{-j})$. For a finite collection of
algebraic numbers $\alpha_1,\dots, \alpha_n$, the notation
$h(\alpha_1,\dots, \alpha_n)$ denotes the class number of the number field
$\mathbb Q(\alpha_1,\dots, \alpha_n)$. Particularly,
$h(\sqrt{d},\sqrt{-j})$ and $h(K_{d,j})$ have the same meaning. \\
\noindent {\bf Case $\pi=\pi_1\times \pi_1$.}
For $\pi_1=\pm \sqrt{2}
\zeta_8$, one has $K=\mathbb Q(\sqrt{-1})$, $\calR_{sp}=\calR=O_K$, and
$H_{sp}(\pi)=H(\pi)=1$. For $\pi_1=\pm \sqrt{3}
\zeta_{12}$, one has $K=\mathbb Q(\sqrt{-3})$, $\calR_{sp}=\calR=O_K$, and
$H_{sp}(\pi)=H(\pi)=1$.
For $\pi_1=\sqrt{-p}$, we have $K=\mathbb Q(\sqrt{-p})$, $\calR_{sp}=\calR$ and
$[O_K:\calR_{sp}]=2$ or $1$ depending on $p\equiv 3 \pmod 4$ or
not. In this case we have $H_{sp}(\pi)=1,3$ for $p=2,3$, respectively
and
\begin{equation}
\label{eq:self_product}
H_{sp}(\pi)=
\begin{cases}
h(\sqrt{-p}) & \text{for $p\equiv 1 \pmod 4$;} \\
\left(4-\left (\frac{2}{p}\right ) \right ) h(\sqrt{-p})
& \text{for $p\equiv 3 \pmod 4$ and $p>3$;}
\end{cases}
\end{equation}
see \cite[Theorem 1.1]{yu:sp_prime}.
The contribution of the self-product cases is
\begin{equation}
\label{eq:total_self_product}
\sum_{\pi_1\in W^{\rm ss}_p(1)} H_{sp}(\pi_1\times \pi_1)=
\begin{cases}
3, 5 & \text{for $p=2, 3$, respectively;} \\
h(\sqrt{-p}) & \text{for $p\equiv 1 \pmod 4$;} \\
\left(4-\left (\frac{2}{p}\right ) \right ) h(\sqrt{-p})
& \text{for $p\equiv 3 \pmod 4$ and $p>3$.}
\end{cases}
\end{equation}
\noindent {\bf Case $\pi=\pi_1\times \pi_2$, $\pi_1\neq \pi_2$.} This
occurs only when $p=2$ or $3$.
The following are class numbers of $B$ with $\calR_{sp}\subset
B\subset O_K$ obtained in Section~\ref{subsec:supord-Rsp-prod-case}.
\ \\
\begin{tabular}{|c|c|c|c|c|}
\hline
$\pi=\pi_1\times \pi_2$ & $K$ & $[O_K:\calR_{sp}]$ & $\calR_{sp}\subset
B\subset O_K$ & $h(B)$ \\ \hline
$ \sqrt{2}\zeta_4\times \pm \sqrt{2}\zeta_8$ &
$\mathbb{Q}(\sqrt{-2}) \times \mathbb{Q}(\sqrt{-1})$
& 2 & $\calR_{sp}$, $O_K$ & $1,1$ \\ \hline
$\sqrt{2}\zeta_8\times - \sqrt{2}\zeta_8$ &
$\mathbb{Q}(\sqrt{-1}) \times \mathbb{Q}(\sqrt{-1})$ & 8 &
$\calR_{sp},B_4,B_2, O_K$ & $1,1,1,1$ \\ \hline
$ \sqrt{3}\zeta_4\times \pm \sqrt{3}\zeta_{12} $ &
$\mathbb{Q}(\sqrt{-3}) \times \mathbb{Q}(\sqrt{-3})$ & 6 &
$\calR_{sp}, B_3, B_2, O_K$ &
$1,1,1,1$\\ \hline
$ \sqrt{3}\zeta_{12}\times -\sqrt{3}\zeta_{12}$ &
$\mathbb{Q}(\sqrt{-3}) \times \mathbb{Q}(\sqrt{-3})$ & 12 &
$\calR_{sp}, B_4, B_3, O_K$ &
$1,1,1,1$ \\ \hline
\end{tabular} \ \\
The orders $B_j$ are listed here for the convenience of the reader.
\begin{align*}
B_2&=\mathbb Z[(1+\zeta_4, 0), (\zeta_4, \zeta_4)] & \quad \text{ for }
\pi&=\sqrt{2}\zeta_8\times
- \sqrt{2}\zeta_8;\\
B_2&=\mathbb Z[\sqrt{-3}]\times \mathbb Z[\zeta_6] & \quad \text{ for }
\pi&=\sqrt{3}\zeta_4\times \pm \sqrt{3}\zeta_{12};\\
B_3&=\mathbb Z[(\sqrt{-3}, 0), (\zeta_6, \zeta_6)] &\quad \text{ for }
\pi&=\sqrt{3}\zeta_4\times \pm\sqrt{3}\zeta_{12} \text{ or } \sqrt{3}\zeta_{12}\times -\sqrt{3}\zeta_{12};\\
B_4&=\mathbb Z[(2,0), (\zeta_{2p}, \zeta_{2p})] &\quad \text{ for }
\pi&=\sqrt{p}\zeta_{4p}\times -\sqrt{p}\zeta_{4p} \text{ and } p=2,
3.
\end{align*}
The contribution of other non-simple cases is
\begin{equation}
\label{eq:other_product}
\sum_{\pi_1\neq \pi_2} H_{sp}(\pi_1\times \pi_2)=
\begin{cases}
2\times 2+4=8 & \text{for $p=2$;} \\
2\times 4+4=12 & \text{for $p=3$.} \\
\end{cases}
\end{equation}
\noindent {\bf Case $\pi\in W^{\rm ss}_p(2)$.} We have $\pi\in
\{ \pm\sqrt{2}\zeta_{24}, \pm\sqrt{5}\zeta_{5}, \sqrt{p}\zeta_8\
(p\neq 2), \sqrt{p} \zeta_3, \sqrt{p}\zeta_{12}\ (p\neq 3)\}.$
For $\pi=\pm \sqrt{p}\zeta_n$ with $(p,n)=(5,5)$ or
$(2, 24)$, we have $\calR_{sp}=O_K$ by
Remark~\ref{rem:critical-max-order} since $n$ is critical at $p$. For
$\pi=\sqrt{p}\zeta_8$ with $p\neq 2$, we have
$K=\mathbb Q(\sqrt{p}\zeta_8)=\mathbb Q(\sqrt{-1}, \sqrt{2p})$ and
$\calR_{sp}=\mathbb Z[(\sqrt{2p}+\sqrt{-2p})/2, \sqrt{-1}]$, which is the maximal
order in $K$ by Exercise
42(b) of \cite[Chapter 2]{MR0457396}. Therefore,
\begin{equation}
\label{eq:maximal_order}
H_{sp}(\pm\sqrt{2}\zeta_{24})=H_{sp}(\pm\sqrt{5}\zeta_{5})=1,
\quad h(\sqrt{p}\zeta_8)=h(\sqrt{2p},\sqrt{-1}), \quad p\neq 2.
\end{equation}
For $\pi=\sqrt{p}\zeta_3$, we have $K=\mathbb Q(\sqrt{p},\sqrt{-3})$ and
$\calR_{sp}=\mathbb Z[\sqrt{p}, \zeta_3]$.
The suborders $B\subseteq O_K$ containing $\mathbb Z[\sqrt{p}]$ with the property
$[B^\times:\mathbb Z[\sqrt{p}]^\times]>1$
are classified in \cite{xue-yang-yu:num_inv}. We list the
suporders of $\calR_{sp}$ in $O_K$ and their class numbers in the
following table.
\smallskip
\begin{tabular}{|c|c|c|c|}
\hline
$\pi=\sqrt{p}\zeta_3$ & $[O_K:\calR_{sp}]$ & $\calR_{sp}\subset
B\subset O_K$ & $h(B)$ \\ \hline
$p=2$ & 1 & $O_K$ & $1$ \\ \hline
$p=3$ & 3 & $\calR_{sp},\ O_K$ & $1,1$ \\ \hline
$p\equiv 3 \pmod 4,\ p\neq 3$ & 1 & $O_K$ &
$h(K)$\\ \hline
$p\equiv 1 \pmod 4$ & 4 &
$\calR_{sp},\ O_K$ &
$\varpi_p\, h(K), h(K)$ \\ \hline
\end{tabular} \ \\
Thus,
\begin{equation}
\label{eq:zeta3}
H_{sp}(\sqrt{p}\zeta_3)=
\begin{cases}
1, 2 & \text{for $p=2, 3$, respectively;} \\
(\varpi_p+1) h(\sqrt{p},\sqrt{-3}) & \text{for $p\equiv 1 \pmod 4$;} \\
h(\sqrt{p},\sqrt{-3})
& \text{for $p\equiv 3 \pmod 4$ and $p>3$.}
\end{cases}
\end{equation}
For $\pi=\sqrt{p}\zeta_{12}$ ($p\neq 3$), we have
$K=\mathbb Q(\sqrt{-p},\sqrt{-3})$ and $\calR_{sp}=
\mathbb Z[\sqrt{p}\zeta_{12}, \zeta_6]=\mathbb Z[\sqrt{-p}, \zeta_6]$. We have the
following results from
Section~\ref{subsec:supord-Rsp-simple-case}. \\
\begin{tabular}{|c|c|c|c|}
\hline
$\pi=\sqrt{p}\zeta_{12}\ (p\neq 3)$ & $[O_K:\calR_{sp}]$ &
$\calR_{sp}\subset
B\subset O_K$ & $h(B)$ \\ \hline
$p=2$ & 1 & $O_K$ & $1$ \\ \hline
$p\equiv 1 \pmod 4$ & 1 & $O_K$ & $h(K)$\\ \hline
$p\equiv 3 \pmod 4$ & 4 & $\calR_{sp}, O_K$ &
$\varpi_{3p}\, h(K), h(K)$ \\ \hline
\end{tabular} \ \\
Thus,
\begin{equation}
\label{eq:zeta12}
H_{sp}(\sqrt{p}\zeta_{12})=
\begin{cases}
1 & \text{for $p=2$;} \\
h(\sqrt{-p},\sqrt{-3}) & \text{for $p\equiv 1 \pmod 4$;} \\
(\varpi_{3p}+1) h(\sqrt{-p},\sqrt{-3})
& \text{for $p\equiv 3 \pmod 4\ (p\neq 3)$.}
\end{cases}
\end{equation}
The following are the class numbers of the fields
$K=\mathbb Q(\sqrt{p} \zeta_n)$ for $n\in\{3,8,12\}$ and
$p\in\{2,3,5\}$. They are checked using Magma. \\
\begin{tabular}{|c|c|c|c|}
\hline
$h(K)$ & $p=2$ & $p=3$ & $p=5$ \\ \hline
$\mathbb Q(\sqrt{p}\zeta_3)=\mathbb Q(\sqrt{p},\sqrt{-3})$ & $1$ & $1$ & $1$ \\
\hline
$\mathbb Q(\sqrt{p}\zeta_8)=\mathbb Q(\sqrt{2p},\sqrt{-3})$ & $1$ & $2$ & $2$\\ \hline
$\mathbb Q(\sqrt{p}\zeta_{12})=\mathbb Q(\sqrt{-p},\sqrt{-3})$ & $1$ & $1$ &
$2$ \\ \hline
\end{tabular} \ \\
We collect the contribution of simple cases. For $p=2$, we have
\begin{equation}
\label{eq:simple2}
H_{sp}(\sqrt{2} \zeta_3)+H_{sp}(\sqrt{2}\zeta_{12})+2
H_{sp}(\sqrt{2}\zeta_{24})=1+1+2=4.
\end{equation}
For $p=3$, we have
\begin{equation}
\label{eq:simple3}
H_{sp}(\sqrt{3} \zeta_3)+H_{sp}(\sqrt{3}\zeta_{8})=1+2=3.
\end{equation}
For $p=5$, we have
\begin{equation}
\label{eq:simple5}
H_{sp}(\sqrt{5} \zeta_3)+H_{sp}(\sqrt{3}\zeta_{8})+
H_{sp}(\sqrt{5}\zeta_{12})+2
H_{sp}(\sqrt{5}\zeta_{5})=1+2+2+2=7.
\end{equation}
For $p\ge 7$, we have
\begin{equation}
\label{eq:simple}
\begin{split}
& \sum_{\pi\neq \sqrt{p}\in W_p^{\rm ss}(2)} H_{sp}(\pi)=
H_{sp}(\sqrt{p} \zeta_3)+H_{sp}(\sqrt{p}\zeta_{8})+
H_{sp}(\sqrt{p}\zeta_{12}) \\
&=
\begin{cases}
(\varpi_p+1) h(K_{p,3})+h(K_{2p,1})
+h(K_{3p,3}), & \text{for $p\equiv 1 \pmod 4$;} \\
h(K_{p,3})+h(K_{2p,1})+(\varpi_{3p}+1)
h(K_{3p,3}), & \text{for $p\equiv 3 \pmod 4$.}
\end{cases}
\end{split}
\end{equation}
Let $\Delta(p)$ be the number of isomorphism classes of superspecial
abelian surfaces whose Frobenius endomorphism not equal to
$\pm\sqrt{p}$. Then we have
\begin{equation}
\label{eq:Delta}
\Delta(p)=\sum_{\pi\in W^{\rm ss}_p(2), \pi\neq \sqrt{p}}
H_{sp}(\pi)+\sum_{\pi_1\times \pi_2, \pi_1\neq \pi_2}
H_{sp}(\pi_1\times \pi_2)+\sum_{\pi_1\in W^{\rm ss}_p(1)}
H_{sp}(\pi_1\times \pi_1).
\end{equation}
Collecting the results (\ref{eq:total_self_product}),
(\ref{eq:other_product}), (\ref{eq:simple2})
(\ref{eq:simple3}), (\ref{eq:simple5}) and (\ref{eq:simple}),
we obtain the following result.
\begin{thm}\label{sp.2} \
\begin{enumerate}
\item The number $\Delta(p)$ is $15,20,9$ for $p=2,3,5$,
respectively.
\item For $p>5$ and $p\equiv 1 \pmod 4$, we have
\begin{equation}
\label{eq:Delta_1mod4}
\Delta(p)=(\varpi_p+1) h(K_{p,3})+h(K_{2p,1})
+h(K_{3p,3})+h(\sqrt{-p}),
\end{equation}
where $\varpi_{p}$ is defined in (\ref{eq:varpi_d}).
\item For $p>5$ and $p\equiv 3 \pmod 4$, we have
\begin{equation}
\label{eq:Delta_3mod4}
\Delta(p)=h(K_{p,3})+h(K_{2p,1})+(\varpi_{3p}+1)
h(K_{3p,3})+\left(4-\left (\frac{2}{p}\right ) \right ) h(\sqrt{-p}),
\end{equation}
where $\varpi_{3p}$ is defined in (\ref{eq:varpi_d}).
\end{enumerate}
\end{thm}
Theorem~\ref{1.3} then follows from Theorems~\ref{1.2} and \ref{sp.2}.
\begin{remark}\label{sp.3}
Based on our computation we observe that
the endomorphism ring of a
superspecial abelian surface over ${\bbF}_p$ may be a non-maximal order,
or even non-maximal at $p$. For example, when $p=3$ and $\pi=\sqrt{3} \zeta_3$,
the order $\calR_{sp}$, which occurs as the
endomorphism ring of a superspecial abelian surface \cite[Theorem
6.1]{waterhouse:thesis}, has index $3$ in the maximal order.
\end{remark}
\subsection{Asymptotic behavior of $|\Sp_2({\bbF}_p)|$}
\label{sec:sp.3}
\def\mathrm {Mass}{\mathrm {Mass}}
We now determine the asymptotic behavior of the size of $\Sp_2({\bbF}_p)$
as the prime $p$ goes to infinity. For simplicity, let $F=\mathbb Q(\sqrt{p})$.
By Theorem~\ref{1.3}, $|\Sp_2({\bbF}_p)|$ is expressed as a
linear combination of
$\zeta_F(-1) h(F)$, $h(\sqrt{-p})$, and class numbers of certain biquadratic CM
fields. The term $c \zeta_F(-1) h(\sqrt{p})$ (for a suitable
constant $c$) comes from the contribution of
the isogeny class corresponding to the Weil $p$-number
$\pi=\sqrt{p}$. More precisely, it arises from the mass part in the
Eichler class number formula for the calculation of $H(\sqrt{p})$.
We recall from Theorem~\ref{1.2} that the mass part
for $p>5$ is
\begin{equation}
\label{eq:mass}
\mathrm {Mass}(p)=
\begin{cases}
\frac{1}{2} \zeta_F(-1) h(F) & \text{for $p\equiv 3 \pmod 4$;} \\
8 \zeta_F(-1) h(F) & \text{for $p\equiv 1 \pmod 8$;} \\
\frac{1}{2}(15\varpi_p+1)\zeta_F(-1) h(F) & \text{for $p\equiv 5 \pmod
8$}. \\
\end{cases}
\end{equation}
In \cite[Theorem 6.3.1]{xue-yang-yu:ECNF} we showed that the mass part
$\mathrm {Mass}(p)$ is the main term of $H(\sqrt{p})$.
It is expected that $\mathrm {Mass}(p)$ is
also the main term of $|\Sp_2({\bbF}_p)|$. This is true and we have the
following asymptotic formula for the size $\Sp_2({\bbF}_p)$.
\begin{prop}\label{sp.4}
We have
\[ \lim_{p\to \infty} \frac{|\Sp_2({\bbF}_p)|
}{\mathrm {Mass}(p)}=1. \]
\end{prop}
\begin{proof
It is enough to show that $\lim_{p\to
\infty}h(\sqrt{-p})/h(F)\zeta_F(-1)=0$, and for all the biquadratic CM-fields
$K_{d,j}$ appearing in the formula of $|\Sp_2({\bbF}_p)|$, \[\lim_{p\to \infty}
h(K_{d,j})/h(F)\zeta_F(-1)=0.\]
The above limit has been verified for the pairs $(d, j)$ with $d=p$ and
$j=1,2,3$ in \cite[Theorem~6.3.1]{xue-yang-yu:ECNF}, and it remains to
consider the pairs $(2p,1)$ and $(3p, 3)$.
Recall that the
discriminant of $F$ is denoted by $\grd_F$, which is either $p$ or
$4p$. Using the function equation and the trivial inequality
$\zeta_F(2)>1$, we have
$\zeta_F(-1)>c_1 (\grd_F)^{3/2}$ for a constant $c_1>0$.
For any CM-field $K$, let $h^{-}(K)$ be the relative class
number of $K$ defined as $h(K)/h(K^+)$, where $K^+$ is the maximal
totally real subfield of $K$.
By \cite[Lemma~4]{Horie-Horie-1990}, when $K$ range over a sequence of
CM-fields with bounded degree and $\grd_K\to \infty$, we have
\begin{equation}
\label{eq:Horie-Horie}
\lim_{\grd_K\to \infty} (\log
h^-(K))/(\log \sqrt{\grd_K/\grd_{K^+}})=1.
\end{equation}
In particular, applying this to the quadratic imaginary fields
$\mathbb Q(\sqrt{-p})$, we obtain that $h(\sqrt{-p})/\zeta_F(-1)\to 0$ as $p\to \infty$.
Assume $(d,j)=(2p, 1)$ or $(3p,3)$. One calculates that
$\grd_{K_{d,j}}/\grd_{K_{d,j}^+}\leq 32p$. Let $\epsilon_d$ be the
fundamental unit of the quadratic real field $\mathbb Q(\sqrt{d})$. By Siegel's theorem \cite[Theorem
15.4, Chapter 12]{MR665428}, the growth of
$h(K_{d,j}^+)=h(\sqrt{d})$ satisfies the following formula \[\lim_{d\to \infty} (\log
h(\sqrt{d})\log\epsilon_d)/(\log \sqrt{d})=1.\]
Note that $\epsilon_d$ is bounded below by $(1+\sqrt{5})/2$ for all
$d$. Recall that $h(K_{d,j})=h^-(K_{d,j})h(\sqrt{d})$. Combining these bounds yields that $h(K_{d,j})/\zeta_F(-1)\to 0$ as $p$ goes to infinity.
\end{proof}
\section{Galois cohomology of an arithmetic group}
\label{sec:gal_coh}
\subsection{Galois cohomology and conjugacy classes}
\label{sec:gal_coh.1}
We refer to \cite[Section I.5]{Serre-Galois-coh} for the definition of
nonabelian Galois cohomology. Let $\Gamma_{\mathbb F_q}=\Gal(\overline{\bbF}_q/\mathbb F_q)$ be
the absolute Galois group of $\mathbb F_q$, and $G$ a group with discrete
topology on which $\Gamma_{\mathbb F_q}$ acts continuously. Let $\sigma_q$ be the
arithmetic Frobenius automorphism of $\overline{\bbF}_q$, which raises each element
of $\overline{\mathbb F}_q$ to its $q$-th power. The group
$\Gamma_{\mathbb F_q}$ is isomorphic to the profinite group
$\widehat{\mathbb Z}=\varprojlim_{n\in \bbN}\zmod{n}$ with canonical
generator $\sigma_q$. Each $1$-cocycle $(a_\sigma)_{\sigma\in \Gamma_{\mathbb F_q}}$ is uniquely
determined by its value $x=a_{\sigma_q}\in G$ at $\sigma_q$. An element
of $G$ is called a \emph{$1$-cocycle element} if it arises from
a $1$-cocycle in this way. We will identify the set of $1$-cocycles
$Z^1(\Gamma_{\mathbb F_q}, G)$ with the subset of $1$-cocycle elements of $G$. Two $1$-cocycle elements $x, y\in Z^1(\Gamma_{\mathbb F_q},
G)$ define the same cohomology class if and only if they are
$\sigma_q$-conjugate (denoted by $x\sim_{\sigma_q} y$), i.e., there exists
$z\in G$ such that $x=z^{-1}y\sigma_q(z)$. Write $[x]_{\sigma_q}$
for the $\sigma_q$-conjugacy
class of $x\in G$, and $B(G)$ for the set of all $\sigma_q$-conjugacy
classes of $G$. Then
\[ H^1(\Gamma_{\mathbb F_q}, G)= \{[x]_{\sigma_q}\in B(G)\mid
x\in Z^1(\Gamma_{\mathbb F_q},G) \}\subseteq B(G).\]
If the action of $\Gamma_{\mathbb F_q}$ on $G$ is
trivial, then $B(G)$ is reduced to the set $\Cl(G)$ of (the usual)
conjugacy classes of $G$. Define $\Cl_0(G):=\{[x]\in \Cl(G)\mid x \text{ is
of finite order}\}\subseteq \Cl(G)$.
\begin{lemma}\label{lm:gal.1}
Assume that the action of $\Gamma_{\mathbb F_q}$ on $G$ factors through a finite
quotient $\Gal(\mathbb F_{q^N}/\mathbb F_q)$. We have
\begin{gather*}
Z^1(\Gamma_{\mathbb F_q},G)=\{x\in G\mid x\sigma_q(x)\cdots\sigma_q^{N-1}(x) \text{ is of finite order}\,\}.
\end{gather*}
In particular, if the action of $\Gamma_{{\bbF}_q}$ on $G$ is trivial, then $H^1(\Gamma_{\mathbb F_q}, G)=\Cl_0(G)$.
\end{lemma}
\begin{proof}
This follows directly from Exercise~2 of
\cite[Section~I.5.1]{Serre-Galois-coh}.
\end{proof}
\subsection{Abelian varieties over finite fields and twisted forms}\label{subsec:galois-descent}
Let $X_0$ be an abelian variety over $\mathbb F_q$ with Frobenius
endomorphism $\pi_{X_0}\in \End_{\mathbb F_q}(X_0)$. Set $\overline{X}_0=X_0\otimes
\overline{\bbF}_q$, and $G=\Aut(\overline{X}_0)$. The Galois group
$\Gamma_{\mathbb F_q}$ acts on $\End(\overline{X}_0)$ as follows (see \cite[Proposition
4.3]{yu:superspecial})
\begin{equation}
\label{eq:gal.4}
\sigma_q(x)=\pi_{X_0} x \pi_{X_0}^{-1}, \quad \forall\, x\in \End(\overline{X}_0),
\end{equation}
where the conjugation by $\pi_{X_0}$ is taken inside
$\End^0(\overline{X}_0)$. As $\End(\overline{X}_0)$ is a free $\mathbb Z$-module of
finite rank, the action of $\Gamma_{\mathbb F_q}$ factors
through a finite quotient $\Gal(\mathbb F_{q^N}/\mathbb F_q)$, and hence $(\pi_{X_0})^N$
is central in $\End(\overline{X}_0)$.
Recall that an \emph{$(\overline{\bbF}_q/\mathbb F_q)$-form} of
$X_0$ is an abelian varieties $X$ over $\mathbb F_q$ such that $\overline{X}:=X\otimes \overline{\bbF}_q$ is $\overline{\bbF}_q$-isomorphic to $\overline{X}_0$.
Let $E(\overline{\bbF}_q/\mathbb F_q, X_0)$ be the set of $\mathbb F_q$-isomorphism classes of $(\overline{\bbF}_q/\mathbb F_q)$-forms of $X_0$. By
\cite[Section~III.1.3]{Serre-Galois-coh}, there is a canonical
bijection of pointed sets
\begin{equation}
\label{eq:5}
\theta: E(\overline{\bbF}_q/\mathbb F_q, X_0)\stackrel{\sim}{\longrightarrow} H^1(\Gamma_{{\bbF}_q},G),
\end{equation}
sending the $\mathbb F_q$-isomorphism class of $X_0$ to
the trivial class.
The map $\theta$ is induced from mapping each $\overline{\bbF}_q$-isomorphism $f: \overline{X}_0\to \overline{X}$
to the 1-cocycle element $x=f^{-1}\sigma_q(f)\in G$. The injectivity of
$\theta$ follows purely from cohomological formalism, and the
surjectivity is a consequence of Weil's Galois descent.
An isomorphism $f$ of abelian varieties as above induces an
isomorphism
\begin{equation}
\label{eq:isom-end-alg}
\alpha_f:\End(\overline X)\simeq \End(\overline X_0),
\quad y\mapsto f^{-1} y f.
\end{equation}
The Frobenius
endomorphisms $\pi_{X_0}$ and $\pi_X$
are related by the following
commutative diagram (see \cite[(4.2)]{yu:superspecial}):
\begin{equation}
\label{eq:gal_coh.10}
\begin{CD}
\overline X_0 @>{f}>> \overline X \\
@V{\pi_{X_0}}VV @VV{\pi_X}V \\
\overline X_0 @>{\sigma_q(f)}>> \overline X.
\end{CD}
\end{equation}
We compute
\begin{equation}
\label{eq:6}
\alpha_f(\pi_X)=f^{-1} \pi_X f=f^{-1} \sigma_q(f) \pi_{X_0} =x \pi_{X_0}.
\end{equation}
Note that for $x, y, z\in G$,
\[x=z^{-1}y\sigma_q(z)\ \Leftrightarrow\
x\pi_{X_0}=z^{-1}(y\pi_{X_0})z. \]
Hence there is a well-defined injective map
\begin{equation}
\label{eq:7}
\Pi: B(G)\hookrightarrow \End(\overline{X}_0)/G, \quad
[x]_{\sigma_q}\mapsto [x\pi_{X_0}],
\end{equation}
where $\End(\overline{X}_0)/G$ denotes orbits of
$\End(\overline{X}_0)$ under the right action of $G$ by conjugation. In a sense, the image of
$H^1(\Gamma_{\mathbb F_q}, G)$ under $\Pi$ consists of the conjugacy classes
of Frobenius endomorphisms of members of $E(\overline{\bbF}_q/\mathbb F_q, X_0)$.
We can also work in the category of abelian varieties up to
isogeny and study the $(\overline{\bbF}_q/\mathbb F_q)$-forms of
the isogeny class $[X_0]$. Thus we pass from isomorphisms of abelian
varieties to quasi-isogenies, and endomorphism rings to endomorphism
algebras, etc. Let $E^0(\overline{\bbF}_q/\mathbb F_q,
[X_0])$ be the set of $\mathbb F_q$-isogeny classes of abelian varieties $[X]$
such that $\overline{X}$ is isogenous to $\overline{X}_0$ over $\overline{\bbF}_q$, and
$G_\mathbb Q=\End^0(\overline{X}_0)^\times$. Many previous constructions can
be carried over. In particular, both (\ref{eq:gal_coh.10}) and
(\ref{eq:6}) hold true for any quasi-isogeny $f:
\overline{X}_0\to \overline{X}$, and one obtains a 1-cocycle element
$x=f^{-1}\sigma_q(f)\in G_\mathbb Q$ as before. This gives a canonical injective map
\begin{equation}
\label{eq:8}
\theta: E^0(\overline{\bbF}_q/\mathbb F_q, [X_0])\hookrightarrow H^1(\Gamma_{{\bbF}_q},G_\mathbb Q),
\end{equation}
which fits into a commutative diagram
\begin{equation}
\label{eq:9}
\begin{CD}
E(\overline{\bbF}_q/\mathbb F_q, X_0) @>{\cong}>{\theta}> H^1(\Gamma_{{\bbF}_q},G) \\
@VVV @VVV \\
E^0(\overline{\bbF}_q/\mathbb F_q, [X_0]) @>{\theta}>> H^1(\Gamma_{{\bbF}_q},G_\mathbb Q).
\end{CD}
\end{equation}
The left vertical map sends the $\mathbb F_q$-isomorphism class of $X$ to its
$\mathbb F_q$-isogeny class $[X]$, and the right vertical map is induced from the
inclusion of $\Gamma_{\mathbb F_q}$-groups $G\subset G_\mathbb Q$. Thus
(\ref{eq:9}) endows a geometric
meaning for this cohomological map.
We complete the picture by showing that the map $\theta$ in
(\ref{eq:8}) is surjective and thus a bijection of pointed sets as
stated in Theorem~\ref{thm:intro-descent-isog}. Recall that the action of $\Gamma_{\mathbb F_q}$ on
$\End^0(\overline X)$ factors through $\Gal(\mathbb F_{q^N}/\mathbb F_q)$ for a fixed
$N\in \bbN$. Without lose of generality, assume that $X_0$ is
$\mathbb F_{q^N}$-isotypical, i.e., $X_0\otimes\mathbb F_{q^N}$ is isogenous to $(Y_N)^d$, where $Y_N$
is an absolutely simple abelian variety over $\mathbb F_{q^N}$ with
$\End(Y_N)=\End(\overline Y_N)$. Equivalently, we assume
that the multiple Weil $q$-number
$\pi_{0,1}^{t_1}\times \dots \times \pi_{0,u}^{t_u}\in MW_q$
corresponding to the $\mathbb F_q$-isogeny class $[X_0]$ satisfies that
$\pi_{0,1}^N=\pi_{0,2}^N=\cdots= \pi_{0,u}^N$ after suitable
conjugation, and $\mathbb Q((\pi_{X_0})^N)\subset \End^0(X_0)$ is a
field which coincides with $\mathbb Q((\pi_{X_0})^{sN})$
for all $s\in \bbN$.
Then $\End^0(\overline{X}_0)=\Mat_d(\End^0(\overline Y_N))$, and
$\End^0(\overline Y_N)$ is a central division algebra over $\mathbb Q((\pi_{X_0})^N)$. For simplicity, let $D=\End^0(\overline Y_N)$ and
$K_0=\mathbb Q((\pi_{X_0})^N)$. Then $G_\mathbb Q=\End^0(\overline{X}_0)^\times=\GL_d(D)$.
\begin{lem}\label{lem:coh-to-conj-class}
There is a bijection between $H^1(\Gamma_{\mathbb F_q}, G_\mathbb Q)$ and the following
subset of conjugacy classes of $\Cl(G_\mathbb Q)$:
\begin{equation}
\label{eq:1-cocycle-cond}
\mathscr{C}(\pi_{X_0})=\{[\underline\pi]\in \Cl(G_\mathbb Q)\mid \exists M\in \bbN: \
\underline\pi^{NM}=\pi_{X_0}^{NM}\}\subset \Cl(G_\mathbb Q).
\end{equation}
\end{lem}
\begin{proof}
Since $\pi_{X_0}\in G_\mathbb Q$, the map $\Pi$ in (\ref{eq:7})
defines a bijection
\[\Pi: B(G_\mathbb Q)\stackrel{\sim}{\longrightarrow} \Cl(G_\mathbb Q),\quad [x]_{\sigma_q}\mapsto [x\pi_{X_0}]. \]
Let $\pi_x=x\pi_{X_0}$ for each $x\in G_\mathbb Q$. Then
\[x\sigma_q(x)\cdots
\sigma_q^{N-1}(x)=(x\pi_{X_0})^N(\pi_{X_0})^{-N}=(\pi_x)^N(\pi_{X_0})^{-N}. \]
By Lemma~\ref{lm:gal.1}, $x\in Z^1(\Gamma_{\mathbb F_q},G_\mathbb Q)$ if and only
if $(\pi_x)^N=(\pi_{X_0})^N\xi$ for some $\xi\in G_\mathbb Q$ of
finite order, or equivalently, $\pi_x^{NM}=(\pi_{X_0})^{NM}$ for some $M\in \bbN$.
\end{proof}
Any $\underline{\pi}\in G_\mathbb Q$ with $[\underline{\pi}]\in \mathscr{C}(\pi_{X_0})$
is semisimple, as $\underline{\pi}^{NM}=(\pi_{X_0})^{NM}$ lies in the center of the simple
$\mathbb Q$-algebra $\End^0(\overline{X}_0)$. The minimal polynomial of
$\underline{\pi}$ factorizes as a product of distinct irreducible
polynomials over $\mathbb Q$:
\begin{equation}
\label{eq:factor-min-poly}
P(t)=\prod_{i=1}^r P_i(t)\in \mathbb Q[t].
\end{equation}
For all $\underline \pi'$ in the conjugacy class $[\underline \pi]$, the $\mathbb Q$-subalgebra
$K_{\underline\pi'}:=\mathbb Q(\underline\pi')\subset \End^0(\overline{X}_0)$ is canonically
isomorphic to $K:=\mathbb Q[t]/(P(t))$ via the map $\underline \pi'\mapsto
t$. Since
$\pi_{X_0}^{NM}=\underline\pi^{NM}$, the field
$K_0=\mathbb Q(\pi_{X_0}^{NM})$ can be identified with the $\mathbb Q$-subalgebra of $K$
generated by $t^{NM}$, thus providing a
$K_0$-algebra structure on $K$. By (\ref{eq:factor-min-poly}), $K$
factorizes as a products of number fields
\begin{equation}
\label{eq:factor-field}
K=K_1\times \cdots \times K_r, \quad
\text{with}\quad K_i=\mathbb Q[t]/(P_i(t))\supseteq K_0.
\end{equation}
By an abuse of notation, we regard $\pi_{X_0}^N$ as a Weil
$q^N$-number via a embedding $K_0\hookrightarrow \overline{\bbQ}$.
Then for each $1\leq i\leq r$, the roots of $P_i(t)$ in $\overline{\bbQ}$ is a
conjugacy class of Weil $q$-numbers such that one of its
representative $\pi_i$ satisfies $\pi_i^{NM}=\pi_{X_0}^{NM}$. Therefore, given
$\underline\pi\in \mathscr{C}(\pi_{X_0})$, we find $r$
Weil $q$-numbers representing distinct conjugacy classes
\begin{equation}
\label{eq:Weil-num-from-cohomology}
\{\pi_1, \cdots, \pi_r\}\quad \text{with} \quad
\pi_i^{NM}=\pi_{X_0}^{NM}\quad \text{for some } M\in \bbN \text{ and
all } 1\leq i \leq r.
\end{equation}
Next, we fix $P(t)\in \mathbb Q[t]$ as above, and produce a discrete invariant for every conjugacy class
$[\underline\pi]\in \mathscr{C}(\pi_{X_0})$ with minimal polynomial
$P(t)$. Let $V=D^d$ be the right
vector space over $D$ of column vectors. Then $\End_D(V)=\Mat_d(D)$
acts on $V$ from the left by the usual matrix multiplication. We have
a canonical $K_0$-algebra embedding $K\hookrightarrow \End_D(V)$ sending $K$ to
$K_\pi$. Thus $\underline\pi$ endows a
faithful $(K,D)$-bimodule structure on $V$, denoted by
$V_{\underline\pi}$.
By (\ref{eq:factor-field}), there is a decomposition of $V$ into right $D$-subspaces:
\begin{equation}
\label{eq:bimodule-decomposition}
V=\bigoplus_{i=1}^r V_i, \quad d_i=\dim_D V_i.
\end{equation}
The action of $K_i$ on $V_i$ gives rise to a $K_0$-embedding
$K_i\hookrightarrow \End_D(V_i)=\Mat_{d_i}(D)$.
We study each of the embeddings individually first.
\begin{lem}\label{lem:minimal-embedding}
Let $\pi\in W_q$ be a Weil $q$-number such
that $\pi^{NM}=\pi_{X_0}^{NM}$ for some integer $M\in \bbN$, and
$X_\pi$ a simple abelian variety over $\mathbb F_q$ in the isogeny class corresponding to
$\pi$. Let $e=e(\pi)$ be the smallest integer such that there is an
$K_0$-embedding $\mathbb Q(\pi)\hookrightarrow \Mat_e(D)$. Then $\overline
X_\pi=X_\pi\otimes \overline{\bbF}_q$ is
isogenous to $(\overline Y_N)^e$, and $\End^0(X_\pi)$ is isomorphic to the
centralizer $C_\pi$ of $\mathbb Q(\pi)$ in $\Mat_e(D)$.
\end{lem}
\begin{proof}
Since $\pi^{NM}=\pi_{X_0}^{NM}$, there exists an isogeny $\overline X_\pi\to
(\overline Y_N)^e$ for some $e\in \bbN$, which gives an identification of $\End^0(\overline X_\pi)$ with $\Mat_e(D)=\End^0((\overline Y_N)^e)$
in the same way as (\ref{eq:isom-end-alg}). Thus we obtain a $K_0$-embedding $\mathbb Q(\pi)\hookrightarrow \Mat_e(D)$, and $\End^0(X_\pi)$ is
recovered as the $\Gamma_{\mathbb F_q}$-invariants of $\End^0(\overline
X_\pi)$, or equivalently, the centralizer $C_\pi$ of $\mathbb Q(\pi)$ in
$\Mat_e(D)$ by (\ref{eq:gal.4}). On the other hand, $C_\pi$ is also the endomorphism
algebra of the $(\mathbb Q(\pi), D)$-bimodule $D^e$. Now the minimality
of $e$ follows from the fact that $C_\pi=\End^0(X_\pi)$ is a
division algebra.
\end{proof}
Given $e'\in \bbN$, a $K_0$-embedding of $\mathbb Q(\pi)$ into the simple
algebra $\Mat_{e'}(D)$ exist if and only if $e(\pi)$ divides $e'$.
Therefore, every $d_i$ in
(\ref{eq:bimodule-decomposition}) is of the form $m_i e(\pi_i)$
for some positive integer $m_i\in \bbN$ subjecting to the condition
\begin{equation}
\label{eq:admissible-cond}
m_1 e(\pi_1)+\dots+m_r e(\pi_r)=d.
\end{equation}
We shall call the $r$-tuple
$\underline m=(m_1,\dots, m_r)$ the \emph{type} of the $(K, D)$-bimodule
$V_{\underline \pi}$ or simply the \emph{type} of $\underline \pi$.
\begin{lem}\label{lem:bijections-str}
There are natural bijections between the following sets:
\begin{enumerate}
\item the set of conjugacy classes
$[\underline\pi]\in \mathscr{C}(\pi_{X_0})$ with minimal polynomial
$P(t)$;
\item the set of $G_\mathbb Q$-conjugacy classes of $K_0$-embedding
$K\hookrightarrow \End_D(V)$;
\item the set of isomorphism classes of faithful $(K,D)$-bimodule structures on $V$;
\item the set of $r$-tuples
$\underline m=(m_1,\dots, m_r)\in \bbN^r$ satisfying (\ref{eq:admissible-cond}).
\end{enumerate}
\end{lem}
\begin{proof}
The bijection between between (1) and (2) is established by the map sending each $K_0$-embedding
$\phi: K=\mathbb Q[t]/(P(t))\hookrightarrow \End_D(V)$ to $\pi=\phi(t)$. Every faithful $(K,D)$-bimodule structure on $V$ is given by a
$K_0$-embedding $\phi: K\hookrightarrow \End_D(V)$. Two
such embeddings define isomorphic structures if and only if they are
conjugate by an element of $G_\mathbb Q$. Hence (2) is bijective
to (3).
The proof that (2) is bijective to (4) is similar to that of \cite[Proposition~3.2]{shih-yang-yu}
and is omitted.
\end{proof}
\begin{thm}\label{thm:galois-descent-isog-classes}
Each cohomology class $[x]_{\sigma_q}\in H^1(\Gamma_{\mathbb F_q},G_\mathbb Q)$
determines a unique conjugacy class of multiple
Weil $q$-number $\pi_1^{m_1}\times \cdots \times \pi_r^{m_r}\in
MW_q$ such
that
\begin{itemize}
\item $\pi_i^{NM}=\pi_{X_0}^{NM}$ for some
$M\in \bbN$ and all $1\leq i \leq r$;
\item $\underline m=(m_1,\dots, m_r)\in \bbN^r$ satisfies
(\ref{eq:admissible-cond}).
\end{itemize}
In particular, the map $\theta$ in (\ref{eq:8}) is a bijection of pointed sets.
\end{thm}
\begin{proof}
Given $[x]_{\sigma_q}\in H^1(\Gamma_{\mathbb F_q},G_\mathbb Q)$, we produce the
desired multiple Weil $q$-number by combing the type $\underline m=(m_1,
\cdots, m_r)$ of
$[\pi_x]\in \mathscr{C}(\pi_{X_0})$ and the set $\{\pi_1, \cdots, \pi_r\}$ determined by $[\pi_x]$
as in (\ref{eq:Weil-num-from-cohomology}). Let $X=\prod_{i=1}^r (X_{\pi_i})^{m_i}$ be an abelian variety over
$\mathbb F_q$ corresponding to $\pi_1^{m_1}\times \cdots \times \pi_r^{m_r}$. Then $\overline X$ is
isogenous to $\overline{X}_0$ by Lemma~\ref{lem:minimal-embedding} and
(\ref{eq:admissible-cond}). Identify $\End^0(\overline X)$ with $\End^0(\overline X_0)$
via an isogeny $f: \overline{X}_0\to \overline{X}$ as in
(\ref{eq:isom-end-alg}). The conjugacy class of $\alpha_f(\pi_X)\in
G_\mathbb Q$ is independent of the choice of $f$. By the construction,
$\alpha_f(\pi_X)$ is a semisimple element with the same minimal
polynomial and type as $\pi_x=x\pi_{X_0}$. It follows from
Lemma~\ref{lem:bijections-str} that they must lie in the same conjugacy class of $G_\mathbb Q$. We conclude that
$\theta$ is surjective by Lemma~\ref{lem:coh-to-conj-class}.
\end{proof}
\subsection{Superspecial abelian varieties and the parity property}
\label{sec:parity}
We apply the previous construction to the study of superspecial
abelian varieties over finite fields. Let $E_0$ be a supersingular elliptic curve
over the prime finite field ${\bbF}_p$
whose Frobenius endomorphism $\pi_0$ satisfying
$\pi_0^2+p=0$ (Recall that $\sqrt{-p}\in W_p^{\rm ss}(1)$ for all $p$ by
Proposition~\ref{prop:dim.5}). Let $\mathcal{O}:=\End(E_0\otimes \overline{\bbF}_p)$ be
the endomorphism
ring of $E_0\otimes \overline{\bbF}_p$; this is a maximal order in the unique
quaternion $\mathbb Q$-algebra $D=B_{p,\infty}$ ramified exactly
at $\{p,\infty\}$.
Take $X_0=E_0^d$ and $\overline X_0:=X_0 \otimes \overline{\bbF}_p$ for $d\ge 1$.
Then $\End(\overline X_0)=\Mat_d(\mathcal{O})$.
In what follows we denote by
\[ G:=\Aut(\overline X_0)=\GL_d(\mathcal{O}) \]
the automorphism group of $\overline X_0$. Consider $\mathcal{O}$ as a subring of $\Mat_d(\mathcal{O})$ by the diagonal
embedding and view $\pi_0$ as an element in $\Mat_d(\mathcal{O})$. Then
the action of $\Gamma_{{\bbF}_p}$ on $G=\GL_d(\mathcal{O})$ is given by
\begin{equation}
\label{eq:gal-act-ss}
\sigma_p(x)=\pi_0 x \pi_0^{-1}, \quad x\in G.
\end{equation}
We will also write $G_\mathbb Q$ for the group $\GL_d(D)$.
Recall that $\Sp_d({\bbF}_q)$ denotes the set of isomorphism classes of
$d$-dimensional superspecial abelian varieties over ${\bbF}_q$. For the
classification of superspecial abelian varieties over the algebraic
closure $\overline{\mathbb F}_q$ of ${\bbF}_q$,
we have the following result, due to Deligne, Shioda and Ogus
(cf.~\cite[Section 1.6, p.~13]{li-oort}).
\begin{thm}\label{gal.sp}
For any integer $d\ge 2$,
there is only one isomorphism class of $d$-dimensional superspecial
abelian varieties over any algebraically closed field of characteristic $p>0$.
\end{thm}
According to this theorem, any $d$-dimensional
superspecial abelian variety over ${\bbF}_q$
is an $(\overline{\mathbb F}_q/{\bbF}_q)$-form of $X_0\otimes {\bbF}_q$.
Thus we obtain a natural bijection by (\ref{eq:5})
\begin{equation}
\label{eq:gal.3}
H^1(\Gamma_{{\bbF}_q},G)\simeq \Sp_d({\bbF}_q), \quad d>1,
\end{equation}
which sends the trivial class to the isomorphism
class of $X_0\otimes {\bbF}_q$. The set $\Sp_d({\bbF}_q)$ is partitioned
into isogeny classes:
\begin{equation}
\label{eq:partition-ss-class}
\Sp_d({\bbF}_q)=\coprod_{\pi\in MW^{\rm ss}_q(d)} \Sp(\pi).
\end{equation}
\begin{thm}\label{gal.1}
Let $q=p^a$ and $q'=p^{a'}$ be powers of the prime number $p$ such
that $a\equiv a' \pmod {2}$. For any integer $d\ge 1$, there are
natural bijections
\begin{gather}
H^1(\Gamma_{{\bbF}_q},G) \simeq H^1(\Gamma_{\mathbb F_{q'}},G),
\label{eq:gal.parity}\\
\Sp_d({\bbF}_q) \simeq \Sp_d(\mathbb F_{q'}). \label{eq:sp.parity}
\end{gather}
\end{thm}
\begin{proof}
If $d=1$, then (\ref{eq:sp.parity}) has been
proven in Section~\ref{sec:curve.2}; see Theorem~\ref{curve.3} and
Remark~\ref{curve.4}. If $d>1$, then (\ref{eq:sp.parity}) follows
from (\ref{eq:gal.3}) and (\ref{eq:gal.parity}). Therefore, it
remains to
prove (\ref{eq:gal.parity}).
Since $\pi_0^2$ is a central element, the element $\sigma_p^2$ acts
trivially on $G$ by (\ref{eq:gal-act-ss}). Thus $\sigma_q(x)= \sigma_{q'}(x)$ for all $x\in G$.
This together
with the
canonical isomorphism $\Gamma_{{\bbF}_q}\simeq \Gamma_{\mathbb F_{q'}}$ (sending
$\sigma_{q}\mapsto
\sigma_{q'}$) yields
a natural bijection
$H^1(\Gamma_{{\bbF}_q}, G)\simeq H^1(\Gamma_{\mathbb F_{q'}},
G)$.
The theorem is proved.
\end{proof}
\begin{rem}\label{rem:parity-isogeny-class}
By the same token, we have a natural bijection
\begin{equation}
H^1(\Gamma_{{\bbF}_q},G_\mathbb Q) \simeq H^1(\Gamma_{\mathbb F_{q'}},G_\mathbb Q).
\end{equation}
Thus by Theorem~\ref{thm:galois-descent-isog-classes}, there is also a \emph{natural} bijection between the isogeny
classes of supersingular abelian varieties over $\mathbb F_q$ and those over $\mathbb F_{q'}$.
This can be made explicit in terms of multiple Weil numbers.
The Frobenius endomorphism of $X_0\otimes {\bbF}_q$ is $\pi_0^a$.
Hence the Frobenius endomorphisms of the isogeny class
corresponding to a cohomology class $[x]_{\sigma_q}\in H^1(\Gamma_{{\bbF}_q},G_\mathbb Q)$ is given
by the conjugacy class $[x\pi_0^a]$ by
(\ref{eq:6}). Without lose of generality, assume that $a-a'=2s\geq 0$. If $\pi=\pi_1^{m_1}\times
\cdots \times
\pi_r^{m_r}$ is a multiple Weil $q$-number determined by
$[x]_{\sigma_q}$, then the corresponding multiple Weil $q'$-number
is $\widetilde\pi=\widetilde{\pi}_1^{m_1}\times
\cdots \times
\widetilde{\pi}_r^{m_r}$, with $\widetilde{\pi}_i=(-p)^{-s}\pi_i$ for all $1\leq i
\leq r$.
By the commutative diagram (\ref{eq:9}), the bijection
(\ref{eq:sp.parity}) preserves isogeny classes in the sense that there
is a natural bijection
\begin{equation}
\label{eq:bijection-parity}
\Sp(\pi)\simeq \Sp(\widetilde\pi)\qquad \forall\, \pi\in MW^{\rm ss}_q(d).
\end{equation}
\end{rem}
\begin{thm}\label{gal.odd}
Let $q=p^{2s+1}$ be an odd power of $p$. Let $Y_0$ be a fixed
supersingular abelian variety over ${\bbF}_q$ and $\pi=\pi_1^{m_1}\times
\cdots \times
\pi_r^{m_r}$ the corresponding multiple Weil $q$-number. Let $V$ and
$K$ be as in Theorem~\ref{prop:sp.1},
and set $\calR_{sp}:=\mathbb Z[\widetilde \pi_0, p \widetilde
\pi_0^{-1}]\subset K$, where $\widetilde
\pi_0=(-p)^{-s}(\pi_1,\dots,\pi_r)$. Assume that $K$ has no real
places. Then there is a natural bijection between the set $\Sp(\pi)$
of isomorphism classes of superspecial abelian varieties in the
isogeny class $[Y_0]$ and the set of
isomorphism classes of $\calR_{sp}$-lattices in $V$.
\end{thm}
\begin{proof}
This follows from Theorems \ref{prop:sp.1} and \ref{gal.1}.
\end{proof}
The above theorem provides an approach for computing the size of
$\Sp_d({\bbF}_q)$ explicitly in the odd exponent case, subject to
the condition that $K$ has no totally real factors.
For the rest of this section we shall describe
$H^1(\Gamma_{{\bbF}_q},
\GL_d(\mathcal{O}))$ (and hence $\Sp_d({\bbF}_q)$)
when $q$ is an even power of $p$.
\subsection{A description of $H^1(\Gamma_{{\bbF}_q}, \GL_d(\mathcal{O}))$ with
even exponent.}
\label{sec:gal_coh.2}
\def{\rm Emb}{{\rm Emb}}
In what follows we assume that $q=p^a$ is an even power of $p$
and $X_0=E_0^d\otimes \mathbb F_q$ with $d\ge 2$.
The Frobenius endomorphism $\pi_{X_0}=(-p)^{a/2}$ lies
in the center of $\End(\overline X_0)=\Mat_d(\mathcal{O})$.
Hence $\Gamma_{{\bbF}_q}$ acts trivially on
the group $G:=\GL_d(\mathcal{O})$ by (\ref{eq:gal-act-ss}).
Then it follows from Lemma~\ref{lm:gal.1} that $H^1(\Gamma_{{\bbF}_q}, G)$ can be identified with
the set
$\Cl_0(G)$ of conjugacy classes of elements in
$G$ of finite order.
We shall give a lattice
description for $\Cl_0(G)$ and hence for $\Sp_d({\bbF}_q)$ by the previous
correspondence. See Theorem~\ref{gal_coh.3} for details.
Suppose $x\in G$ is an element of finite order, which is necessarily
semi-simple.
The minimal polynomial of $x$ over $\mathbb Q$ has the form
\begin{equation}
\label{eq:gal_coh.5}
P_{\underline n}(t)=\Phi_{n_1}(t) \Phi_{n_2}(t) \cdots \Phi_{n_r}(t), \quad
1\le n_1<n_2<\dots <n_r
\end{equation}
for some $r$-tuple $\underline n=(n_1,\dots, n_r)\in \bbN^r$, where $\Phi_m(t)\in \mathbb Z[t]$ denotes the $m$-th
cyclotomic polynomial. We define
\[ K_{\underline n}:=\frac{\mathbb Q[t]}{\prod_{i=1}^r \Phi_{n_i}(t)} \quad
\text{and}\quad
A_{\underline n}:=\frac{\mathbb Z[t]}{\prod_{i=1}^r \Phi_{n_i}(t)}. \]
The $\mathbb Q$-subalgebras of $\End^0(\overline X_0)=\Mat_d(D)$ generated by $x$ and $\pi_x=x\pi_{X_0}$
coincide and are isomorphic to $K_{\underline n}$. Moreover, the subring
$\mathbb Z[x]\subset \Mat_d(\mathcal{O})$ is canonically isomorphic to $A_{\underline
n}$.
We denote by $C(\underline n)\subset \Cl_0(G)$ the set of conjugacy classes of
$G$ with minimal polynomial $P_{\underline n}(t)$. By Theorem~\ref{thm:galois-descent-isog-classes}, each conjugacy class
$[x]\in C(\underline n)$ determines a (conjugacy class of) supersingular multiple Weil $q$-number
$\pi_1^{m_1}\times \cdots \times \pi_r^{m_r}$, where $\pi_i=(-p)^{a/2}\zeta_{n_i}$, and $\underline m=(m_1, \cdots, m_r)$ is the
type of the faithful $(K_{\underline n},D)$-bimodule structure on $V=D^d$
equipped by $\pi_x\in \Mat_d(D)$. Since
$\mathbb Q(\pi_x)=\mathbb Q(x)\cong K_{\underline n}$, the $(K_{\underline n},D)$-bimodule
structure on $V$ is also equipped \emph{directly} by $x\in\Mat_d(D)$. Thus
$\underline m$ is also called the type of $[x]$, as it depends only on
the conjugacy class. Recall that a $(K_{\underline n},D)$-bimodule $V$ is said to be type
$\underline m$ if the decomposition into $D$-subspaces $V=\oplus_{i=1}^r
V_i$ induced from the decomposition $K_{\underline n}=\prod_{i=1}^r
\mathbb Q(\zeta_{n_i})$ satisfies that $\dim_D V_i=m_ie(\pi_i)$ for all $1\leq i \leq r$,
where $e(\pi_i)$ is defined in Lemma~\ref{lem:minimal-embedding}.
Since $\dim E_0=1$, we have
$e(\pi_i)=d(\pi_i)$, the dimension of the Weil number
$\pi_i$. Note that $d(\pi_i)$ depends
only on the integer $n_i$ as $q=p^a$ is fixed, so we write $d(n_i)$ for it
instead. Equation (\ref{eq:admissible-cond}) becomes
\begin{equation}
\label{eq:gal_coh.8}
m_1 d(n_1)+\dots+m_r d(n_r)=d.
\end{equation}
A pair of $r$-tuples $(\underline n,\underline m)\in \bbN^r\times \bbN^r$
with $1\le n_1<\dots<n_r$ is said to be {\it $d$-admissible} if the
condition~(\ref{eq:gal_coh.8})
is satisfied. Let $C(\underline n, \underline m)\subset
C(\underline n)$ denote the subset of conjugacy classes of type $\underline
m$. An element $x\in G$ or its conjugacy $[x]\in \Cl(G)$ is said to be
type $(\underline n, \underline m)$ if $[x]\in
C(\underline n, \underline m)$.
\begin{lemma}\label{gal.emb}
Fix a faithful $(K_{\underline n},D)$-bimodule $V=D^d$ of type $\underline m$.
There is a natural bijection between the set
$C(\underline n, \underline m)$ and the set of
isomorphism classes of $(A_{\underline n},\mathcal{O})$-lattices in $V$.
\end{lemma}
\begin{proof}
Let $M_0:=\mathcal{O}^d\subset V $ be the standard lattice in $V$.
Every element $x\in G$ of type $(\underline n, \underline m)$ gives rise to an
$(A_{\underline n}, \mathcal{O})$-bimodule structure on $M_0$. Two elements $x,
x'$ determine isomorphism bimodule structures if and only if
they are conjugate in $G$. Therefore, the set $C(\underline n, \underline m)$ is
in bijection
with the set of isomorphism classes of $(A_{\underline n},
\mathcal{O})$-lattices in $V$ that are $\mathcal{O}$-isomorphic to
$M_0$. Since $d\ge
2$, every $\mathcal{O}$-lattice in $V$ is isomorphic to $M_0$. This
follows from a theorem of Eichler \cite{Eichler1938} that the
class number of
$\Mat_d(\mathcal{O})$ is $1$ for $d\ge 2$ (see also
\cite[Theorem~2.1]{IbuKatOort1986}).
\end{proof}
\begin{thm}\label{gal_coh.3}
Let $\Cl_0(G)$ be the set of conjugacy classes of
$G=\GL_d(\mathcal{O})$ of finite order with $d\ge 2$. Then
\begin{equation}
\label{eq:gal_coh}
\Cl_0(G)=\coprod_{(\underline n, \underline m)} C(\underline n, \underline m),
\end{equation}
where $(\underline n, \underline m)$ runs through all $d$-admissible types.
For each fixed $(\underline n, \underline m)$, there are natural bijections between
the following sets:
\begin{enumerate}
\item $C(\underline n,\underline m)$, the set of conjugacy classes of type $(\underline n,
\underline m)$;
\item $\Sp(\pi)$, where $\pi=\pi_1^{m_1}\times \cdots \times
\pi_r^{m_r}$ and $\pi_i=(-p)^{a/2}\zeta_{n_i}$;
\item the set of isomorphism classes of $(A_{\underline n},
\mathcal{O})$-lattices in the $(K_{\underline n}, D)$-bimodule $V$ of
type $\underline m$.
\end{enumerate}
\end{thm}
\begin{proof}
The bijection between (1) and (2) is
established by combining (\ref{eq:gal.3}) and
Theorem~\ref{thm:galois-descent-isog-classes}. The bijection between (1) and (3) follows from
Lemma~\ref{gal.emb}.
\end{proof}
\section{Arithmetic results}
\label{sec:arithmetic-results}
In this section, we prove the arithmetic results used in Section~\ref{sec:sp}
concerning the order $\calR_{sp}$. In the light of
(\ref{eq:sp.5}), our goals are two fold: (1) show that $\calR_{sp}$ is Bass for every supersingular multiple Weil
$p$-number $\pi\in MW_p^{\rm
ss}(2)$ of dimension $2$ distinct from $\pm\sqrt{p}$; (2) classify all suporders of
$\calR_{sp}$ (i.e., orders in $K$ containing $\calR_{sp}$) and calculate their class numbers when $\pi$ is not of
the form $\pi_1\times \pi_1$ with $\pi_1\in
W_p^{\rm ss}(1)$ (The case $\pi=\pi_1\times \pi_1$ has already been
treated in Section~\ref{sec:sp.2}).
\subsection{Orders in products of number fields}
Let $K=\prod_{i=1}^r K_i$ be a product of number fields, and
$\calS$ be an order contained in the maximal order
$O_K=\prod_{i=1}^rO_{K_i}$. We write $\eta_i: K\to K_i$ for the
projection map onto the $i$-th factor.
By a theorem of Borevich and Faddeev \cite{MR0205980}
(see \cite[Section 37, p.~789]{curtis-reiner:1} or
\cite[Theorem 2.1]{MR794792}), $\calS$ is
Bass if and only if $O_K/\calS$ is cyclic as an
$\calS$-module. This leads to the following simple criterion when
$r=2$.
\begin{lem}\label{lem:criterion-Bass-prod}
A suborder $\calS\subseteq O_{K_1}\times O_{K_2}$ that projects
surjectively onto both factors $O_{K_1}$ and $O_{K_2}$ is
Bass.
\end{lem}
\begin{proof}
Each $O_{K_i}$ is equipped
with an $\calS$-module structure via the projection map $\eta_i:
\calS\to O_{K_i}$.
Since $\eta_2(\calS)=O_{K_2}$, the
natural inclusion $O_{K_1}\hookrightarrow O_{K_1}\times
O_{K_2}$ defined by $x\mapsto (x,0)$ induces an isomorphism
of $S$-modules
\begin{equation}
\label{eq:isom-module}
O_{K_1}/(O_{K_1}\cap \calS)\xrightarrow{\simeq} (O_{K_1}\times
O_{K_2})/\calS.
\end{equation}
The left hand side is a cyclic $\calS$-module because
$\eta_1(\calS)=O_{K_1}$.
\end{proof}
We return to the general case with $r\geq 1$.
Let $\a$ be an
$O_K$-lattice (i.e., a fractional $O_K$-ideal that contains a $\mathbb{Q}$-basis of
$K$) contained in $\calS$. There is a one-to-one correspondence
between the orders $B$ intermediate to $\calS \subseteq O_K$ and the
subrings of $O_K/\a$ containing $\calS/\a$. By \cite[Theorem
I.12.12]{MR1697859}, the class number $h(B)$ can be calculated by
\begin{equation}
\label{eq:classNo-order}
h(B)=\frac{h(O_K)[(O_K/\a)^\times:(B/\a)^\times]}{[O_K^\times:B^\times]},
\end{equation}
where $h(O_K)=\prod_{i=1}^r h(O_{K_i})$. A priori, \cite[Theorem
I.12.12]{MR1697859} is only stated for the number field case with $\a$
being the conductor of $B$, but the
same proof applies in the current setting as well.
\begin{lem}\label{lem:classno-suborder=maxorder}
Let $\a\subset O_K$ be an $O_K$-lattice. If the natural map
$O_K^\times\to (O_K/\a)^\times$ is surjective,
then $h(B)=h(O_K)$ for every suborder $B \subseteq O_K$ containing $\a$.
\end{lem}
\begin{proof}
Let $\grK$ be the kernel of $O_K^\times\to (O_K/\a)^\times$. Then
$\grK\subseteq B^\times$ and $[O_K^\times: B^\times]=[O_K^\times/\grK:
B^\times/\grK]$. We identify $O_K^\times/\grK$ with the image of
$O_K^\times\to (O_K/\a)^\times$, and similarly for $B^\times/\grK$.
By \cite[Lemma~2.7]{xue-yang-yu:ECNF},
$B^\times=O_K^\times\cap B$. Hence \[B^\times/\grK=(O_K^\times/\grK)\cap
(B/\a).\]
When $O_K^\times$ maps surjectively onto $(O_K/\a)^\times$, we have
$B^\times/\grK=(O_K/\a)^\times\cap
(B/\a)=(B/\a)^\times$. Therefore, $h(B)=h(O_K)$ by (\ref{eq:classNo-order}).
\end{proof}
\begin{lem}\label{lem:ideal-in-Rsp}
Let $\calS$ be a suborder of $O_K=\prod_{i=1}^r O_{K_i}$, and $\c_1$ be a
nonzero ideal of $O_{K_1}$ contained in $\eta_1(\calS)$. If $x_1\in
O_{K_1}$ is an element such that $(x_1,0, \cdots, 0)\in \calS$,
then $(x_1\c_1,
0, \cdots, 0)$ is an ideal of $O_K$ contained in $\calS$. Similar
results hold for all $1\leq i \leq r$.
\end{lem}
\begin{proof}
Clearly $(x_1\c_1,
0, \cdots, 0)$ is an ideal of $O_K$. For any element $y_1\in
\c_1$, we may find $\mathbf{y}\in \calS$ such
that $\eta_1(\mathbf{y})=y_1$. Then
$(x_1y_1,0,\cdots,0)=(x_1,0,\cdots,0)\cdot \mathbf{y}\in \calS$.
\end{proof}
\subsection{The order $\calR_{sp}$ is Bass when
$d(\pi)=2$.}\label{subsec:order-Rsp-bass}
We recall the definition of $\calR_{sp}$. Suppose that
$\pi=\pi_1^{m_1}\times \cdots \times \pi_r^{m_r}$ is a supersingular
multiple Weil $p$-number with $m_i\in \bbN$ and $\pi_i\not \sim \pi_j$.
Let $K=\prod_i K_i$ with $K_i=\mathbb{Q}(\pi_i)$, and
$\pi_0=(\pi_1, \ldots, \pi_r)\in K$. Then $\calR_{sp}$ is defined to
be the order $ \mathbb{Z}[\pi_0, \pi_0^2/p]\subseteq O_K$. Assume that $\pi$
has dimension 2 and none of $\pi_i$ is conjugate to $\sqrt{p}$. The
case $\pi=\pi_1^2$ with $\pi_1\in W^{\rm ss}_p(1)$ has already been
studied in Section~\ref{sec:sp.2}. It remains to treat the following two
cases:
\begin{enumerate}
\item[(1)] $\pi=\pi_1\times \pi_2$ with both $\pi_1, \pi_2\in
W_p^{\rm ss}(1)$ and $\pi_1\not\sim \pi_2$ (the nonisotypic product case);
\item[(2)] $\pi=\pi_1\in W_p^{\rm
ss}(2)$ and $\pi_1\not\sim \sqrt{p}$ (the nonreal simple case).
\end{enumerate}
The first case occurs only when
\begin{equation}
\label{eq:1}
p=2, 3,\quad\text{ and } \quad \pi=\sqrt{p}\zeta_4
\times (\pm \sqrt{p}\zeta_{4p}),\quad \text{or}\quad \sqrt{p}\zeta_{4p}\times
(-\sqrt{p}\zeta_{4p}).
\end{equation}
In the second case, the supersingular Weil $p$-numbers of
dimension 2 distinct from $\pm \sqrt{p}$ are
\begin{equation}
\label{eq:2}
\sqrt{p} \zeta_3,\ \pm \sqrt{p} \zeta_5\ (p=5),
\ \sqrt{p}\zeta_8\ (p\neq 2),\ \sqrt{p}\zeta_{12} \ (p\neq3),
\ \pm\sqrt{p}\zeta_{24}\ (p=2).
\end{equation}
\begin{lem}\label{lem:Rsp-product-case}
Assume $p=2$ or $3$.
If $\pi=\sqrt{p}\zeta_4
\times (\pm \sqrt{p}\zeta_{4p})$, then
\[\calR_{sp}=\mathbb{Z}[(\sqrt{-p}, 0),
(0, 1+\zeta_{2p})]\subset \mathbb{Q}(\sqrt{-p})\times\mathbb{Q}(\zeta_{2p})=K.\]
If $\pi=\sqrt{p}\zeta_{4p}\times (-\sqrt{p}\zeta_{4p})$, then
\[\calR_{sp}=\mathbb{Z}[(2(1+\zeta_{2p}, 0),
(\zeta_{2p}, \zeta_{2p})]\subset \mathbb{Q}(\zeta_{2p})\times\mathbb{Q}(\zeta_{2p})=K.\]
\end{lem}
\begin{proof}
Note that $\sqrt{p}\zeta_{4p}=1+\zeta_{2p}$ when $p=2$ or $3$. If $\pi=
\sqrt{p}\zeta_4
\times (\pm \sqrt{p}\zeta_{4p})$, then
\[
\begin{split}
\calR_{sp}&=\mathbb{Z}[(\sqrt{-p},\pm \sqrt{p}\zeta_{4p}), (-1,
\zeta_{2p})]=\mathbb{Z}[(\sqrt{-p},\pm \sqrt{p}\zeta_{4p}), (0,
1+\zeta_{2p})]\\
&=\mathbb{Z}[(\sqrt{-p}, 0),
(0, 1+\zeta_{2p})].
\end{split}
\]
If $\pi=\sqrt{p}\zeta_{4p}\times (-\sqrt{p}\zeta_{4p})$, then
\[
\begin{split}
\calR_{sp}&=\mathbb{Z}[(\sqrt{p}\zeta_{4p}, -\sqrt{p}\zeta_{4p}),
(\zeta_{2p}, \zeta_{2p})]=\mathbb{Z}[(1+\zeta_{2p}, -(1+\zeta_{2p})),
(\zeta_{2p}, \zeta_{2p})]\\
&=\mathbb{Z}[(2(1+\zeta_{2p}), 0), (\zeta_{2p}, \zeta_{2p})]. \qedhere
\end{split}
\]
\end{proof}
\begin{prop}
The order $\calR_{sp}$ is a Bass order for every supersingular multiple Weil
$p$-number $\pi \in MW_p^{\rm ss}(2)$ distinct from $\pm\sqrt{p}$.
\end{prop}
\begin{proof} We only need to consider the cases where $\pi$ is not
of the form $\pi_1^2$ with $\pi_1\in W_p^{\rm ss}(1)$.
Suppose that
$\pi=\pm \sqrt{p}\zeta_n\in W_p^{\rm ss}(2)$ is one of the Weil
$p$-numbers listed in (\ref{eq:2}), and $m$ is defined as in
(\ref{eq:def-of-m}).
If $n$ is critical at $p$, then $\calR_{sp}$ equals to the maximal order $\mathbb Z[\zeta_m]$
in $K=\mathbb Q(\zeta_m)$ by Remark~\ref{rem:critical-max-order}.
Otherwise, $[K:\mathbb{Q}(\zeta_m)]=2$ and $\calR_{sp}$ is a
quadratic $\mathbb{Z}[\zeta_m]$-order, and such type of orders are Bass
\cite[Example 2.3]{MR794792}.
If $p=2,3$ and $\pi=\sqrt{p}\zeta_{4p}\times (-\sqrt{p}\zeta_{4p})$,
or $p=2$ and $\pi=\sqrt{2}\zeta_4\times (\pm \sqrt{2}\zeta_8)$, then
$\calR_{sp}$
projects surjectively onto both $O_{K_1}$ and $O_{K_2}$, and hence $\calR_{sp}$ is Bass by Lemma~\ref{lem:criterion-Bass-prod}.
Lastly, suppose that $p=3$ and
$\pi=\sqrt{3}\zeta_4\times (\pm \sqrt{3}\zeta_{12})$. Then
$\eta_1(\calR_{sp})=\mathbb{Z}[\sqrt{-3}]$, a suborder of index 2 in
$O_{K_1}=\mathbb{Z}[\zeta_6]$, while $\eta_2(\calR_{sp})=\mathbb{Z}[\zeta_6]=O_{K_2}$. So by
(\ref{eq:isom-module}), to show that $\calR_{sp}$ is Bass, it
is enough to prove that $O_{K_1}/(O_{K_1}\cap \calR_{sp})$ is a cyclic
$\calR_{sp}$-module. Note that $O_{K_1}\subset O_{K_1}\times O_{K_2}$ is generated
by $(1,0)$ and $(\zeta_6,0)$ over $\mathbb{Z}$, and
\[\calR_{sp}(\zeta_6, 0)\ni (-1+\sqrt{-3},
-1)\cdot(\zeta_6,0)=(1,0)+(\sqrt{-3},0)^2\equiv (1,0)
\pmod{O_{K_1}\cap\calR_{sp}}. \]
Hence $O_{K_1}/(O_{K_1}\cap \calR_{sp})$ is a cyclic
$\calR_{sp}$-module generated by $(\zeta_6, 0)$.
\end{proof}
\subsection{Suporders of $\calR_{sp}$ and class numbers: the
nonisotypic product case}\label{subsec:supord-Rsp-prod-case} Assume that
$p=2$ or $3$ and $\pi=\pi_1\times \pi_2$ is a supersingular multiple
Weil $p$-number of
dimension 2 listed in (\ref{eq:1}). Using
Lemma~\ref{lem:ideal-in-Rsp} and Lemma~\ref{lem:Rsp-product-case}, one
may easily find an $O_K$-lattice $\a$ contained in $\calR_{sp}$ and
compute the quotient rings $O_K/\a$ and $\calR_{sp}/\a$. We obtain the
following
table (For simplicity, we set $i=\zeta_4=\sqrt{-1}$).
\smallskip
\renewcommand{\arraystretch}{1.1}
\begin{tabular}{|c|c|c|c|}
\hline
$\pi=\pi_1\times \pi_2$ & $\a\subset \calR_{sp}$ & $O_K/\a$ &
$\calR_{sp}/\a$ \\ \hline
$ \sqrt{2}\zeta_4\times \pm \sqrt{2}\zeta_8$
& $\sqrt{-2}O_{K_1}\times (1+i)O_{K_2}$ & $(\mathbb F_2)^2$ & $\mathscr{D}_2$ \\ \hline
$\sqrt{2}\zeta_8\times - \sqrt{2}\zeta_8$ &
$(2(1+i)O_{K_1})^2$ &
$(\mathbb Z[i]/(1+i)^3)^2$ & $\mathscr{D}_8$ \\ \hline
$ \sqrt{3}\zeta_4\times \pm \sqrt{3}\zeta_{12} $ &
$(2\sqrt{-3})O_{K_1}\times \sqrt{-3}O_{K_2}$ &
$\mathbb F_4\times (\mathbb F_3)^2$ &
$\mathbb F_2\times \mathscr{D}_3$\\ \hline
$ \sqrt{3}\zeta_{12}\times -\sqrt{3}\zeta_{12}$ &
$(2\sqrt{-3})O_{K_1}\times (2\sqrt{-3})O_{K_2}$ &
$(\mathbb F_4\times \mathbb F_3)^2$ &
$\mathscr{D}_{12}$ \\ \hline
\end{tabular} \\[3pt]
\renewcommand{\arraystretch}{1}
Here $\mathscr{D}_2$, $\mathscr{D}_8$, $\mathscr{D}_3$, and $\mathscr{D}_{12}$ denote
the
diagonal in $(\mathbb F_2)^2$, $(\mathbb Z[i]/(1+i)^3)^2$, $(\mathbb F_3)^2$, and
$(\mathbb F_4\times\mathbb F_3)^2$ respectively.
It is an exercise to show that
$O_K^\times$ maps surjectively onto $(O_K/\a)^\times$ in all the above
cases. By Lemma~\ref{lem:classno-suborder=maxorder}, $h(B)=h(O_K)$
for every order $B$ with $\calR_{sp}\subseteq B \subseteq O_K$. Note
that $h(O_K)=h(O_{K_1})h(O_{K_2})=1$ since both $\mathbb Z[i]$ and
$\mathbb Z[\zeta_6]$ have class number 1. We obtain the following
proposition.
\begin{prop}
Assume that
$p=2$ or $3$ and $\pi=\pi_1\times \pi_2$ is given in
(\ref{eq:1}). Then any suporder $B$ of $\calR_{sp}$ has class number $1$.
\end{prop}
It remains to list all suporders $B$ of $\calR_{sp}$ for each
$\pi$. We recall the convention in Section~\ref{sec:sp.2} that a suporder of
$\calR_{sp}$ with index $j>1$ in $O_K$ is denoted by $B_j$. Our
calculation will show that for those $\pi$ considered in this
subsection, if such an order exists, then it is
unique. So there is no ambiguity in this notation if $\pi$ is clear
from the context. We separate into cases.
\smallskip
\noindent\textbf{Case} $\pi=\sqrt{2}\zeta_4\times \pm
\sqrt{2}\zeta_8$. Since $[O_K:\calR_{sp}]=[O_K/\a:\calR_{sp}/\a]=2$,
there are no other suporders of $\calR_{sp}$ besides $\calR_{sp}$
and $O_K$.
\noindent\textbf{Case} $\pi=\sqrt{3}\zeta_4\times \pm
\sqrt{3}\zeta_{12}$. We have $[O_K:\calR_{sp}]=[\mathbb F_4\times (\mathbb F_3)^2:
\mathbb F_2\times \mathscr{D}_3]=6$. There are two rings properly intermediate to the
inclusion $\mathbb F_2\times \mathscr{D}_3 \subset \mathbb F_4\times (\mathbb F_3)^2$, namely
$\mathbb F_4\times \mathscr{D}_3$ and $\mathbb F_2\times (\mathbb F_3)^2$. Under the inclusion-preserving
correspondence between suborders of $O_K$ containing $\a$ and subrings
of $O_K/\a$, we have
\begin{align*}
B_3:=\mathbb Z[(\sqrt{-3},0), (\zeta_6, \zeta_6)]=\mathbb Z[(1+\zeta_6, 0), (0, 1+\zeta_6)]&\longleftrightarrow \mathbb F_4\times
\mathscr{D}_3,\\
B_2:=\mathbb Z[\sqrt{-3}]\times \mathbb Z[\zeta_6]&\longleftrightarrow \mathbb F_2\times (\mathbb F_3)^2.
\end{align*}
The remaining two cases are best seen in the light of the following
lemma.
\begin{lem}\label{lem:subrings-containing-diagonal}
Let $R$ be a commutative ring, and $\mathscr{D}$ be the diagonal of $R^2$.
Every subring $S$ of $R^2$ containing $\mathscr{D}$ decomposes uniquely as
$\mathscr{D}\oplus (I_S, 0)$, where $I_S$ is an ideal of $R$. In
particular, there is an inclusion-preserving bijective
correspondence between subrings of $R^2$ containing $\mathscr{D}$ and
ideals of $R$.
\end{lem}
\begin{proof}
Every subring of $R^2$ containing $\mathscr{D}$ is naturally an
$R$-submodule of $R^2$. So the intersection $(I_S,0):=S\cap (R, 0)$
is again an $R$-submodule of $R^2$. Equivalently, $I_S$ is an ideal
of $R$. Clearly, we have $S=\mathscr{D}\oplus (I_S,0)$. Conversely, for any ideal $I\subseteq R$, the direct sum
$\mathscr{D}\oplus (I,0)$ is a subring of $R^2$. The correspondence is
established.
\end{proof}
By Lemma~\ref{lem:Rsp-product-case}, if $\pi=\sqrt{p}\zeta_{4p}\times
-\sqrt{p}\zeta_{4p}$ with $p=2$ or $3$, then
$O_K=\mathbb Z[\zeta_{2p}]^2$, and
\[\calR_{sp}=\mathbb Z[(2(1+\zeta_{2p}), 0),
(\zeta_{2p}, \zeta_{2p})]=\mathscr{D}\oplus
(2(1+\zeta_{2p})\mathbb Z[\zeta_{2p}],0).\]
\noindent\textbf{Case} $\pi=\sqrt{2}\zeta_8\times -
\sqrt{2}\zeta_8$. We have $2(1+i)\mathbb Z[i]=(1+i)^3\mathbb Z[i]$. So by
Lemma~\ref{lem:subrings-containing-diagonal}, the suborders of $O_K$
properly containing $\calR_{sp}$ and distinct from $O_K$ are
\begin{align*}
B_4:=\mathbb Z[(i,i), (2,0)] &\longleftrightarrow (1+i)^2\mathbb Z[i]=2\mathbb Z[i],\\
B_2:=\mathbb Z[(i,i), (1+i,0)] &\longleftrightarrow (1+i)\mathbb Z[i].
\end{align*}
\noindent\textbf{Case} $\pi=\sqrt{3}\zeta_{12}\times -
\sqrt{3}\zeta_{12}$. In this case, the ideal
$2(1+\zeta_6)\mathbb Z[\zeta_6]$ factors as the product of the prime ideals
$2\mathbb Z[\zeta_6]$ and $\sqrt{-3}\mathbb Z[\zeta_6]$. The suborders of $O_K$
properly containing $\calR_{sp}$ and distinct from $O_K$ are
\begin{align*}
B_4:=\mathbb Z[(\zeta_6,\zeta_6), (2,0)] &\longleftrightarrow 2\mathbb Z[\zeta_6],\\
B_3:=\mathbb Z[(\zeta_6,\zeta_6), (\sqrt{-3},0)] &\longleftrightarrow \sqrt{-3}\mathbb Z[\zeta_6].
\end{align*}
\subsection{Suporders of $\calR_{sp}$ and class numbers: the nonreal
simple case}\label{subsec:supord-Rsp-simple-case}
Assume that $\pi$ is a supersingular Weil $p$-number of
dimension $2$ listed in (\ref{eq:2}). Only the case
$\pi=\sqrt{p}\zeta_{12}$ with $p\neq 3$ needs to be studied, as the
rest have already been covered in
Section~\ref{sec:sp.2}.
If $\pi=\sqrt{p}\zeta_{12}$, we have $K=\mathbb Q(\sqrt{-p}, \sqrt{-3})$,
and $\calR_{sp}=\mathbb Z[\sqrt{-p}, \zeta_6]$. Since the discriminants of
$\mathbb Q(\sqrt{-3})$ and $\mathbb Q(\sqrt{-p})$ are coprime, $O_K$ is the
compositum of $\mathbb Z[\zeta_6]$ and $O_{\mathbb Q(\sqrt{-p})}$. If $p=2$ or
$p\equiv 1\pmod{4}$, then $O_{\mathbb Q(\sqrt{-p})}=\mathbb Z[\sqrt{-p}]$, and
$\calR_{sp}$ is the maximal order in $K$. We assume that
$p\equiv 3\pmod{4}$ and $p\neq 3$ for the rest of this
subsection. Note that $2O_K\subseteq \calR_{sp}$, and
$\calR_{sp}/2O_K=\mathbb Z[\zeta_6]/(2)\simeq \mathbb F_4$, which embeds into
$O_K/2O_K\simeq \mathbb F_4\oplus \mathbb F_4$ diagonally. It follows that
$\calR_{sp}$ and $O_K$ are the only orders in $O_K$ containing
$\calR_{sp}$. By (\ref{eq:classNo-order}),
$h(\calR_{sp})=3h(O_K)/[O_K^\times : \calR_{sp}^\times]$. It remains
to calculate the index $[O_K^\times : \calR_{sp}^\times]$.
\begin{lem}
Let $p_1$ and $p_2$ be distinct primes with $p_1\equiv p_2 \equiv 3
\pmod{4}$, and $\epsilon$ be the fundamental unit of
$F=\mathbb{Q}(\sqrt{p_1p_2})$. Then $\sqrt{-\epsilon}\in
K=\mathbb{Q}(\sqrt{-p_1}, \sqrt{-p_2})$, and
$O_K^\times=\dangle{\sqrt{-\epsilon}}\times \boldsymbol \mu_K$, the direct
product of the free abelian group generated by $\sqrt{-\epsilon}$
and the group $\boldsymbol \mu_K$ of roots of unity in $K$. Moreover, if $\epsilon\in
\mathbb{Z}[\sqrt{p_1p_2}]$, then $\sqrt{-\epsilon}$ lies in the
$\mathbb{Z}$-module $\mathbb{Z}\sqrt{-p_1}+\mathbb{Z}\sqrt{-p_2} \subset O_K$; otherwise
$\sqrt{-\epsilon}\equiv (\sqrt{-p_1}+\sqrt{-p_2})/2
\pmod{\mathbb{Z}\sqrt{-p_1}+\mathbb{Z}\sqrt{-p_2}}$.
\end{lem}
\begin{proof}
By Dirichlet's Unit Theorem, the quotient group $O_K^\times/\boldsymbol \mu_K$ is a free abelian group
of rank 1 containing $O_F^\times/\{\pm 1\}\cong \dangle{-\epsilon}$ as a subgroup of finite
index. In fact, we have $[O_K^\times/\boldsymbol \mu_K: O_F^\times/\{\pm 1\}]\leq
2$ by \cite[Theorem 4.12]{Washington-cyclotomic} as $K$ is a
CM-field with maximal totally real subfield $F$.
Since both $p_i\equiv 3 \pmod{4}$, it follows from
\cite[(V.1.7)]{ANT-Frohlich-Taylor} that the norm $\Nm_{F/\mathbb{Q}}(\epsilon)=+1$.
By \cite[Lemma 3]{MR0441914}, $p_1\epsilon$ is a perfect square in $F^\times$.
Write $p_1\epsilon=(x+y\sqrt{p_1p_2})^2$ with $x,y\in \mathbb{Q}$. Then
\[\sqrt{-\epsilon}=\sqrt{p_1\epsilon}\cdot
\frac{-1}{\sqrt{-p_1}}=(x+y\sqrt{p_1p_2})\cdot
\frac{-1}{\sqrt{-p_1}}\in \mathbb{Q}\sqrt{-p_1}+
\mathbb{Q}\sqrt{-p_2}\subset K.\]
In particular, $[O_K^\times/\boldsymbol \mu_K: O_F^\times/\{\pm 1\}]\geq 2$. It follows that $[O_K^\times/\boldsymbol \mu_K: O_F^\times/\{\pm
1\}]=2$, and $O_K^\times/\boldsymbol \mu_K\cong \dangle{\sqrt{-\epsilon}}$. Hence
$O_K^\times =\dangle{\sqrt{-\epsilon}}\times \boldsymbol \mu_K$.
By our assumption on $p_i$, the prime $2$ is unramified
in $O_K$. One easily checks that the following statements are equivalent:
\begin{enumerate}
\item $\epsilon\in \mathbb{Z}[\sqrt{p_1p_2}]=\mathbb{Z}+2O_F$;
\item $\epsilon\equiv 1 \pmod{2O_F}$;
\item $\sqrt{-\epsilon}\equiv 1
\pmod{2O_K}$;
\item $\sqrt{-\epsilon}\in \mathbb{Z}+2O_K$.
\end{enumerate}
By Exercise~42(d) of \cite[Chapter~2]{MR0457396}, a $\mathbb{Z}$-basis of
$O_K$ is given by
\[\left\{1, \quad\frac{1+\sqrt{-p_1}}{2}, \quad
\frac{1+\sqrt{-p_2}}{2},\quad \frac{(1+\sqrt{-p_1})(1+\sqrt{-p_2})}{4}
\right\}.\]
It follows that
\begin{gather*}
O_K\cap (\mathbb{Q}\sqrt{-p_1}+
\mathbb{Q}\sqrt{-p_2})=\mathbb{Z}\sqrt{-p_1}+
\mathbb{Z}(\sqrt{-p_1}+\sqrt{-p_2})/2;\\
(\mathbb{Z}+2O_K)\cap (\mathbb{Q}\sqrt{-p_1}+
\mathbb{Q}\sqrt{-p_2})=\mathbb{Z}\sqrt{-p_1}+
\mathbb{Z}\sqrt{-p_2}.
\end{gather*}
Therefore, if $\epsilon\in \mathbb{Z}[\sqrt{p_1p_2}]$, then
$\sqrt{-\epsilon}\in \mathbb{Z}\sqrt{-p_1}+
\mathbb{Z}\sqrt{-p_2}$. Otherwise
$\sqrt{-\epsilon}$ lies in $\mathbb{Z}\sqrt{-p_1}+
\mathbb{Z}(\sqrt{-p_1}+\sqrt{-p_2})/2$ but not in $\mathbb{Z}\sqrt{-p_1}+
\mathbb{Z}\sqrt{-p_2}$. Hence $\sqrt{-\epsilon}\equiv (\sqrt{-p_1}+\sqrt{-p_2})/2
\pmod{\mathbb{Z}\sqrt{-p_1}+\mathbb{Z}\sqrt{-p_2}}$ in this case.
\end{proof}
We return to the assumption that $K=\mathbb Q(\sqrt{-p}, \sqrt{-3})$ with $p\equiv
3\pmod{4}$ and $p\neq 3$. Note that $\boldsymbol \mu_K=\dangle{\zeta_6}\subset
\calR_{sp}^\times$, and $\calR_{sp}\cap
(\mathbb Q\sqrt{-p}+\mathbb Q\sqrt{-3})=(\mathbb Z\sqrt{-p}+\mathbb Z\sqrt{-3})$. Let $\epsilon$ be the fundamental unit of
$F=\mathbb Q(\sqrt{3p})$. If $\epsilon\in \mathbb Z[\sqrt{3p}]$, then
$\sqrt{-\epsilon}\in \calR_{sp}$, and hence
$\calR_{sp}^\times=O_K^\times$. This holds in particular when
$p\equiv 3 \pmod{8}$ and $p\neq 3$ as remarked after (\ref{eq:varpi_d}).
Assume that $p\equiv 7\pmod{8}$ and
$\epsilon\not\in\mathbb Z[\sqrt{3p}]$. Then $(O_F/2O_F)^\times \simeq
\mathbb F_4^\times$ and $\epsilon^3\in \mathbb Z+2O_F=\mathbb Z[\sqrt{3p}]$. On the
other hand, $[(O_K/2O_K)^\times:
(\calR_{sp}/2O_K)^\times]=[(\mathbb F_4^\times)^2:\mathbb F_4^\times]=3$, so we have $\sqrt{-\epsilon}\not\in \calR_{sp}$ but
$(\sqrt{-\epsilon})^3\in \calR_{sp}$.
In summary, we find that
\[[O_K^\times: \calR_{sp}^\times]=[O_F^\times :\mathbb Z[\sqrt{3p}]^\times]=
\begin{cases}
1 &\quad \text{if } \epsilon \in \mathbb Z[\sqrt{3p}];\\
3 & \quad \text{otherwise.}
\end{cases}\]
Therefore, we have $h(\calR_{sp})=\varpi_{3p}h(O_K)$, where $\varpi_{3p}=3/[O_F^\times
:\mathbb Z[\sqrt{3p}]^\times]$ as defined in (\ref{eq:varpi_d}).
\section*{Acknowledgements}
J.~Xue is partially supported
by the 1000-plan program for young scholars of PRC. He thanks Academia
Sinica and the NCTS for their warm hospitality and great working
conditions. TC Yang and CF Yu
are partially supported by the grants MoST 100-2628-M-001-006-MY4,
103-2811-M-001-142, 104-2115-M-001-001MY3 and 104-2811-M-001-066.
\bibliographystyle{plain}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 3,379 |
\section{Introduction}
Higher-order networks \cite{giusti2016two,battiston2020networks,eliassirad,lambiotte} are attracting increasing attention as they are able to capture the many-body interactions of complex systems ranging from brain to social networks.
Simplicial complexes are higher-order networks that encode the network geometry and topology of real datasets. Using simplicial complexes allows the network scientist to formulate new mathematical frameworks for mining data \cite{otter2017roadmap,petri2013topological,massara2016network,sreejith2016forman,kartun2019beyond,katifori2} and for understanding these generalized network structures revealing the underlying deep physical mechanisms for emergent geometry \cite{wu,bianconi2017emergent,bianconi2016network,tadic1,tadic2} and for higher-order dynamics \cite{millan2020explosive,torres2020simplicial,reitz2020higher,
barbarossa2020topological,landry2020effect,skardal2019abrupt,skardal2019higher,
iacopini2019simplicial,taylor2015topological,lucas2020multiorder,zhang2020unified,skardal2020memory,lee,carletti2020dynamical,millan2018complex,mulas2020coupled,gambuzza2020master,millan2019synchronization}.
In particular, this very vibrant research activity is relevant in neuroscience to analyse real brain data and its profound relation to dynamics \cite{giusti2016two,severino2016role,sporns,giusti2015clique,reimann2017cliques,petri2013topological,tadic2} and in the study of biological transport networks \cite{katifori1,katifori2}.
\begin{figure*}[htb!]
\centering
\includegraphics[width=2.0\columnwidth]{figure1.pdf}
\caption{{\bf Schematic representation of the Kuramoto and the topological Kuramoto model.} Panel (a) shows a network in which nodes sustain a dynamical variable (a phase) whose synchronization is captured by the Kuramoto model. Panel (b) shows a simplicial complex in which not only nodes but also links sustain dynamical variables whose coupled synchronization dynamics is captured by the higher-order topological Kuramoto model.}
\label{fig:diagram}
\end{figure*}
In networks, dynamical processes are typically defined over signals associated to the nodes of the network. In particular, the Kuramoto model \cite{strogatz2000kuramoto,boccaletti2018synchronization,rodrigues2016kuramoto,
restrepo2005onset,ott2008low}
investigates the synchronization of phases associated to the nodes of the network. This scenario can change significantly in the case of simplicial complexes \cite{millan2020explosive,torres2020simplicial,barbarossa2020topological}.
In fact, simplicial complexes can sustain dynamical signals defined on simplices of different dimension, including nodes, links, triangles and so on, called {\em topological signals}. For instance, topological signals defined on links can represent fluxes of interest in neuroscience and in biological transportation networks.
The interest on topological signals is rapidly growing with new results related to signal processing \cite{torres2020simplicial,barbarossa2020topological} and higher-order topological synchronization \cite{millan2020explosive,lee,note}.
In particular, higher-order topological synchronization \cite{millan2020explosive} demonstrates that topological signals (phases) associated to higher dimensional simplices can undergo a synchronization phase transition. These results open a new uncharted territory for the investigation of higher-order synchronization.
Higher-order topological signals defined on simplices of different dimension can interact with one another in non-trivial ways.
For instance in neuroscience the activity of the cell body of a neuron can interact with synaptic activity which can be directly affected by gliomes in the presence of brain tumors \cite{araque2014gliotransmitters}.
In order to shed light on the possible phase transitions that can occur when topological signals defined on nodes and links interact, here we build on the mathematical framework of higher-order topological synchronization proposed in Ref.~\cite{millan2020explosive} and consider a synchronization model in which topological signals of different dimension are coupled. We focus in particular on the coupled synchronization of topological signals defined on nodes and links, but we note that the model can be easily extended to topological signals of higher dimension.
The reason why we focus on topological signals {defined on nodes and links} is three-fold. First of all we can have a better physical intuition of topological signals defined on nodes (traditionally studied by the Kuramoto model) and links (like fluxes) that is relevant in brain dynamics \cite{araque2014gliotransmitters,deville_brain} and biological transport networks \cite{katifori1,katifori2}. Secondly,
although the coupled synchronization dynamics of nodes and links can be considered as a special case of coupled synchronization dynamics of higher-order topological signals on a generic simplicial complex, this dynamics can be observed also on networks including only pairwise interactions. Indeed nodes and links are the simplices that remain unchanged if we reduce a simplicial complex to its network skeleton. Since currently there is more availability of network data than simplicial complex data, this fact implies that the coupled dynamics studied in this work has wide applicability as it can be tested on any network data and network model. Thirdly, defining the coupled dynamics of topological signals defined on nodes and links can open new perspectives in exploiting the properties of the line graph of a given network which is the network whose nodes corresponds to the links or the original network \cite{evans}.
{In this work, we show} that by adopting a global adaptive coupling of dynamics inspired by Refs.~\cite{d2019explosive,zhang2015explosive,dai2020discontinuous} the coupled synchronization dynamics of topological signals defined on nodes and links is explosive \cite{boccaletti2016explosive}, i.e., it occurs at a discontinuous phase transition in which the two topological signals of different dimension synchronize at the same time.
{We also illustrate} numerical evidence of this discontinuity on real connectomes and on simplicial complex models including the configuration model of simplicial {complexes} \cite{courtney2016generalized} and the non-equilibrium simplicial complex model called Network Geometry with Flavor \cite{bianconi2016network,bianconi2017emergent}.
We provide a comprehensive theory of this phenomenon on fully connected networks offering a complete analytical understanding of the observed transition. This approach can be extended to random networks treated within the annealed network approximation. The analytical results reveal that the investigated transition is discontinuous.
\section{Results}
\subsection{Higher-order topological Kuramoto model of topological signals of a given dimension }
Let us consider a simplicial complex $\mathcal{K}$ formed by $N_{[m]}$ simplices of dimension $m$, i.e., $N_{[0]}$ nodes, $N_{[1]}$ links, $N_{[2]}$ triangles, and so on.
In order to define the higher-order synchronization of topological signals we will make use of algebraic topology (see the Appendix for a brief introduction) and specifically we indicate with ${\bf B}_{[m]}$ the $m$-th incidence matrix representing the $m$-th boundary operator. \\ The higher-order Kuramoto model generalizes the classic Kuramoto model to treat synchronization of topological signals of higher-dimension.
The classic Kuramoto model describes the synchonization transition for phases
\begin{eqnarray}
\bm{\theta}=(\theta_1,\theta_2,\ldots \theta_{N_{[0]}})
\end{eqnarray}
associated to nodes, i.e., simplices of dimension $n=0$ (see Figure $\ref{fig:diagram}$).
The Kuramoto model is typically defined on a network but it can treat also synchronization of the phases associated to the nodes of a simplicial complex.
Each node $i$ {has associated} an internal frequency $\omega_{i}$ drawn from a given distribution, for instance a normal distribution $\omega_i\sim \mathcal{N}(\Omega_0,1/\tau_0)$. In absence of any coupling, i.e., in absence of pairwise interactions, every node oscillates at its own frequency. However in a network or in a simplicial complex skeleton the phases associated to the nodes follow the dynamical evolution dictated by the equation
\begin{eqnarray}
\dot{\bm \theta}=\bm \omega-\sigma {\bf B}_{[1]}\sin \left({\bf B}_{[1]}^{\top}\bm\theta\right),
\label{K0}
\end{eqnarray}
where here and in the following we use the notation $\sin({\bf x})$ to indicate the column vector where the sine function is taken element wise. Note that here we have chosen to write this system of equations in terms of the incidence matrix ${\bf B}_{[1]}$. However if we indicate with ${\bf a}$ the adjacency matrix of the network and with $a_{ij}$ its matrix elements, this system of equations is equivalent to
\begin{eqnarray}
\dot{\theta}_i=\omega_i+\sigma\sum_{j=1}^N a_{ij}\sin(\theta_j-\theta_i),
\end{eqnarray}
valid for every node $i$ of the network.
For coupling constant $\sigma=\sigma_c$ the Kuramoto model \cite{strogatz2000kuramoto,boccaletti2018synchronization,rodrigues2016kuramoto} displays a continuous phase transition above which the order parameter
\begin{eqnarray}
R_0&=&\frac{1}{N_{[0]}}\left|\sum_{i=1}^{N_{[0]}} e^{\mathbb{i}\theta_i}\right|
\label{op0}
\end{eqnarray}
is non-zero also in the limit $N_{[0]}\to \infty$.\\
The higher-order topological Kuramoto model \cite{millan2020explosive} describes synchronization of phases associated to the $n$ dimensional simplices of a simplicial complex.
Although the definition of the model applies directly to any value of $n$, here we consider specifically the case in which the higher-order Kuramoto model is defined on topological signals (phases) associated to the links
\begin{eqnarray}
\bm{\phi}=(\phi_{\ell_1},\phi_{\ell_2},\ldots \phi_{\ell_{N_{[1]}}}),
\end{eqnarray}
where $\phi_{\ell_r}$ indicates the phase
associated to the $r$-th link of the simplicial complex (see Figure $\ref{fig:diagram}$).
The higher order Kuramoto dynamics defined on simplices of dimension $n>0$ is the natural extension of the standard Kuramoto model defined by Eq.~(\ref{K0}).
Let us indicate with $\tilde{\bm\omega}$ the {internal} frequencies associated to the links of the simplicial complex, sampled for example from a normal distribution, $\tilde{\omega}_{\ell}\sim \mathcal{N}(\Omega_1,1/\tau_1)$. The higher-order topological Kuramoto model
is defined as
\begin{eqnarray}
\dot{\bm{\phi}}&=&\tilde{\bm{\omega}}-\sigma {\bf B}^{\top}_{[1]}\sin ({\bf B}_{[1]}\bm{\phi})-\sigma {\bf B}_{[2]}\sin ({\bf B}_{[2]}^{\top}\bm{\phi}).
\label{K1}
\end{eqnarray}
Once the synchronization dynamics is defined on higher-order topological signals of dimension $n$ (here taken to be $n=1$) an important question is whether this dynamics can be projected on $(n+1)$ and $(n-1)$ simplices.
Interestingly, algebraic topology provides a clear solution to this question. Indeed for $n=1$, when the dynamics describes the evolution of phases associated {to the links}, one can consider the projection $\bm\phi^{[-]}$ and $\bm\phi^{[+]}$ respectively on nodes and on triangles defined as
\begin{eqnarray}
\bm\phi^{[-]}={\bf B}_{[1]}\bm\phi,\nonumber \\
\bm\phi^{[+]}={\bf B}_{[2]}^{\top} \bm\phi.
\end{eqnarray}
Note that in this case ${\bf B}_{[1]}$ acts as a discrete divergence and ${\bf B}_{[2]}^{\top}$ acts as a discrete curl. Interestingly, since the incidence matrices satisfy ${\bf B}_{[1]}{\bf B}_{[{\text{\text{down}}}2]}={\bf 0}$ and ${\bf B}_{[2]}^{\top}{\bf B}_{[1]}^{\top}={\bf 0}$ (see Methods \ref{Ap0}) these two projected phases follow the uncoupled dynamics
\begin{eqnarray}
\dot{\bm \phi}^{[-]}&=&{\bf B}_{[1]}\tilde{\bm\omega}-\sigma {\bf L}_{[0]}\sin \bm\phi^{[-]},\nonumber \\
\dot{\bm \phi}^{[+]}&=&{\bf B}_{[2]}^{\top}\tilde{\bm\omega}-\sigma {\bf L}_{[2]}^{\text{\text{\text{down}}}}\sin \bm\phi^{[+]},\nonumber \\
\end{eqnarray}
where ${\bf L}_{[0]}={\bf B}_{[1]}{\bf B}_{[1]}^{\top}$ and ${\bf L}_{[2]}^{\text{\text{down}}}={\bf B}_{[2]}^{\top}{\bf B}_{[2]}$.
These two projected dynamics undergo a continuous synchronization transition at $\sigma_c=0$ \cite{millan2020explosive} with order parameters
\begin{eqnarray}
R_1^{\text{down}}&=&\frac{1}{N_{[0]}}\left|\sum_{i=1}^{N_{[0]}} e^{\mathbb{i} \phi_i^{[-]}}\right|,\nonumber \\
R_1^{\text{up}}&=&\frac{1}{N_{[2]}}\left|\sum_{i=1}^{N_{[2]}} e^{\mathbb{i} \phi_i^{[+]}}\right|.
\label{op1}
\end{eqnarray}
In Ref. \cite{millan2020explosive} an adaptive coupling between these two dynamics is considered formulating the explosive higher-order topological Kuramoto model in which the topological signal follows the set of coupled equations
\begin{eqnarray}
\dot{\bm{\phi}}&=&\tilde{\bm{\omega}}-\sigma R_1^{\text{up}} {\bf B}^{\top}_{[1]}\sin ({\bf B}_{[1]}\bm{\phi})\nonumber \\
&&-\sigma R_1^{\text{down}}{\bf B}_{[2]}\sin ({\bf B}_{[2]}^{\top}\bm{\phi}).
\end{eqnarray}
The projected dynamics on nodes and triangles are now coupled by the modulation of the coupling constant $\sigma$ with the order parameters $R_1^{\text{down}}$ and $R_1^{\text{up}}$, i.e.
the two projected phases follow the coupled dynamics
\begin{eqnarray}
\dot{\bm \phi}^{[-]}&=&{\bf B}_{[1]}\tilde{\bm\omega}-\sigma R^{\text{up}}_1{\bf L}_{[0]}\sin \bm\phi^{[-]},\nonumber \\
\dot{\bm \phi}^{[+]}&=&{\bf B}_{[2]}^{\top}\tilde{\bm\omega}-\sigma R_1^{\text{down}} {\bf L}_{[2]}^{\text{down}}\sin \bm\phi^{[+]}.\nonumber \\
\end{eqnarray}
{This} explosive higher-order topological Kuramoto model has been shown in Ref.~\cite{millan2020explosive} to lead to a discontinuous synchronization transition on different models of simplicial complexes and on clique complexes of real connectomes.
\subsection{Higher-order topological Kuramoto model of coupled topological signals of different dimension}
Until now, we have captured synchronization occurring only among topological signals of the same dimension. However, signals of different dimension can be coupled to each other in non-trivial ways.
In this work we will show how topological signals of different dimensions can be coupled together leading to an explosive synchronization transition. Specifically we focus on the coupling of the traditional Kuramoto model [Eq.(\ref{K0})] to a higher-order topological Kuramoto model defined for phases associated to the links [Eq.(\ref{K1})].
The coupling between these two dynamics is here performed considering the modulation of the coupling constant $\sigma$ with the global order parameters of the node dynamics [defined in Eq. (\ref{op0})] and the link dynamics [defined in Eq. (\ref{op1})].
{Specifically, we consider two models denoted as Model NL (nodes and links) and model NLT (nodes, links, and triangles).}
Model NL couples the dynamics of the phases of the nodes
$\bm{\theta}$ and of the links $\bm{\phi}$ according to the following dynamical equations
\begin{eqnarray}
\dot{\bm{\theta}}&=&\bm{\omega}-\sigma R_1^{\text{down}}{\bf B}_{[1]}\sin ({\bf B}^{\top}_{[1]}\bm{\theta}),\label{thc}\\
\dot{\bm{\phi}}&=&\tilde{\bm{\omega}}-\sigma R_0 {\bf B}^{\top}_{[1]}\sin ({\bf B}_{[1]}\bm{\phi})-\sigma {\bf B}_{[2]}\sin ({\bf B}_{[2]}^{\top}\bm{\phi}).
\label{model1}
\end{eqnarray}
The projected dynamics for $\bm\phi^{[-]}$ and $\bm\phi^{[+]}$ {then obey }
\begin{eqnarray}
\dot{\bm \phi}^{[-]}&=&{\bf B}_{[1]}\tilde{\bm\omega}-\sigma R_0{\bf L}_{[0]}\sin \bm\phi^{[-]},\label{phi-1} \\
\dot{\bm \phi}^{[+]}&=&{\bf B}_{[2]}^{\top}\tilde{\bm\omega}-\sigma {\bf L}_{[2]}^{\text{down}}\sin \bm\phi^{[+]}.\label{phi+1}
\end{eqnarray}
Therefore the projection on the nodes ${\bm\phi}^{[-]}$ of the phases $\bm\phi$ associated to the links [Eq.~(\ref{phi-1})] is coupled to the dynamics of the phases $\bm{\theta}$ [Eq.~(\ref{thc})] associated directly to nodes. However the projection on the triangles ${\bm\phi}^{[+]}$ of the phases $\bm\phi$ associated to the links is independent of ${\bm\phi}^{[-]}$ and of $\bm\theta$ as well.
Model NLT also describes the coupled dynamics of topological signals defined on nodes and links but the adaptive coupling captured by the model is different. In this case the dynamical equations are taken to be
\begin{eqnarray}
\dot{\bm{\theta}}&=&\bm{\omega}-\sigma R_1^{\text{down}}{\bf B}_{[1]}\sin ({\bf B}^{\top}_{[1]}\bm{\theta}),\label{thc2}\\
\dot{\bm{\phi}}&=&\tilde{\bm{\omega}}-\sigma R_0 R_1^{\text{up}} {\bf B}^{\top}_{[1]}\sin ({\bf B}_{[1]}\bm{\phi})\nonumber \\
&-&\sigma R_1^{\text{down}}{\bf B}_{[2]}\sin ({\bf B}_{[2]}^{\top}\bm{\phi}).
\label{model2}
\end{eqnarray}
For Model NLT the projected dynamics for $\bm\phi^{[-]}$ and for $\bm\phi^{[+]}$ obey
\begin{eqnarray}
\dot{\bm \phi}^{[-]}&=&{\bf B}_{[1]}\tilde{\bm\omega}-\sigma R_0R_1^{\text{up}}{\bf L}_{[0]}\sin \bm\phi^{[-]},\label{phi-2} \\
\dot{\bm \phi}^{[+]}&=&{\bf B}_{[2]}^{\top}\tilde{\bm\omega}-\sigma R_1^{\text{down}}{\bf L}_{[2]}^{\text{down}}\sin \bm\phi^{[+]}.\label{phi+2}
\end{eqnarray}
Therefore, as in Model NL, the dynamics of the projection ${\bm\phi}^{[-]}$ of the phases $\bm\phi$ associated to the links [Eq.~(\ref{phi-2})] is coupled to the dynamics of the phases $\bm{\theta}$ associated directly to nodes [Eq.~(\ref{thc2})] and vice versa. Moreover, the dynamics of the projection of the phases $\bm\phi$ on the triangles ${\bm\phi}^{[+]}$ [Eq.~(\ref{phi+2})] is now also coupled with the dynamics of ${\bm\phi}^{[-]}$ [Eq.~(\ref{phi-2})] and vice versa.
Here and in the following we will use the convenient notation (using the parameter $m$) to indicate both models NL and NLT with the same set of dynamical equations given by
\begin{eqnarray}
\dot{\bm{\theta}}&=&\bm{\omega}-\sigma R_1^{\text{down}}{\bf B}_{[1]}\sin ({\bf B}^{\top}_{[1]}\bm{\theta}),\label{thetai0}\\
\dot{\bm{\phi}}&=&\tilde{\bm{\omega}}-\sigma R_0 \left(R_1^{\text{up}}\right)^{m-1}{\bf B}^{\top}_{[1]}\sin ({\bf B}_{[1]}\bm{\phi})\nonumber \\
&&-\sigma \left(R_1^{\text{down}}\right)^{m-1} {\bf B}_{[2]}\sin ({\bf B}_{[2]}^{\top}\bm{\phi}),\label{phi0}
\end{eqnarray}
which reduce to Eqs.~(\ref{model1}) for $m=1$ and to Eqs.~(\ref{model2}) for $m=2$.
We make two relevant observations:
\begin{itemize}
\item
First, the proposed coupling between topological signals of different dimension can be easily extended to signals defined on higher-order simplices providing a very general scenario for coupled dynamical processes on {simplicial complexes.}
\item
Second, the considered coupled dynamics of topological signals defined on nodes and links can be also studied on networks with exclusively pairwise interactions where we assume that the number of simplices of dimension $n>1$ is zero.
Therefore in this specific case this topological dynamics can have important effects also on simple networks.
\end{itemize}
We have simulated {Model NL and Model NLT} on two main examples of simplicial complex models: the configuration model of simplicial complexes \cite{courtney2016generalized} and the Network Geometry with Flavor (NGF) \cite{bianconi2016network,bianconi2017emergent} (see Figure $\ref{fig:model}$). In the configuration model we have considered power-law distribution of the generalized degree with exponent $\gamma<3,$ and for the NGF model with have considered simplicial complexes of dimensions $d=3$ whose skeleton is a power-law network with exponent $\gamma=3$.
In both cases we observe an explosive synchronization of the topological signals associated to the nodes and to the links. On finite networks, the discontinuous transition emerge together with the hysteresis loop formed by the forward and backward synchronization transition. However the two models display a notable difference. In Model NL we observe a discontinuity for $R_0$ and $R_1^{\text{down}}$ at a non-zero coupling constant $\sigma=\sigma_c$, however $R_1^{\text{up}}$ follows an independent transition at zero coupling (see Figure $\ref{fig:model}$, panels in the second and fourth column). In Model NLT, on the contrary, all order parameters $R_0$, $R_1^{\text{down}}$, and $R_1^{\text{up}}$ display a discontinuous transition occurring for the same non zero value of the coupling constant $\sigma=\sigma_c$ (see Figure $\ref{fig:model}$ panels in the first and third column). This is a direct consequence of the fact that in Model NL the adaptive coupling leading to discontinuous phase transition only couples the phases $\bm\phi^{[-]}$ and $\bm\theta$, while for Model NLT the coupling involves also the phases $\bm\phi^{[+]}$.
Additionally we studied both Model NL and Model NTL on two real connectomes: the human connectome of Ref.~\cite{sapiens} and the c. elegans connectome from Ref.~\cite{celegans} (see Figure $\ref{fig:connectomes}$). Interestingly also for these real datasets we observe that in Model NL the explosive synchronization involves only the phases $\bm\theta$ and $\bm\phi^{[-]}$ while in Model NLT we observe that also $\bm\phi^{[+]}$ undergoes an explosive synchronization transition at the same value of the coupling constant $\sigma=\sigma_c$.
\begin{figure*}[htb!]
\centering
\includegraphics[width=2.0\columnwidth]{figure2.pdf}
\caption{{\bf The Higher-order topological synchronization models (Models NL and NLT) coupling nodes and links on simplicial complexes.} The order parameters $R_{0}$, $R_{1}^{\text{down}}$ and $R_1^{\text{up}}$ are plotted versus $\sigma$ for the higher-order topological synchronization Model NLT (panels (a)-(e)-(i) and (c)-(g)-(k)) and Model NL (panels (b)-(f)-(j) and (d)-(h)-(l)) defined over the Network Geometry with Flavor \cite{bianconi2016network} (panels (a)-(e)-(i) and (b)-(f)-(j)) and the configuration model of simplicial complexes \cite{courtney2016generalized} (panels (c)-(g)-(k) and (d)-(h)-(l)). The Network Geometry with Flavor on which we run the numerical results shown in (a) and (b) includes $N_{[0]}=500$ nodes and has flavor $s=-1$ and $d=3$. The configuration model of simplicial complexes on which we run the numerical results shown in (c) and (d) includes $N_{[0]}=500$ nodes and has generalized degree distribution which is power-law with exponent $\gamma=2.8.$ In both Model NL and in Model NLT we have set $\Omega_0=\Omega_1=2$ and $\tau_0=\tau_1=1$.}
\label{fig:model}
\end{figure*}
\begin{figure*}[htb!]
\centering
\includegraphics[width=2.0\columnwidth]{figure3.pdf}
\caption{{\bf The Higher-order topological synchronization models (Models NLT and NL) coupling nodes and links on real connectomes.} The order parameters $R_{0}$, $R_{1}^{\text{down}}$ and $R_1^{\text{up}}$ are plotted versus $\sigma$ on real connectomes. Panels (a)-(e)-(i) and (b)-(f)-(j) show the numerical results on the human connectome \cite{sapiens} for Model NLT and Model NL respectively. Panels (c)-(g)-(k) and (d)-(h)-(i) show the numerical results on the c. elegans connectome \cite{celegans} for Model NLT and Model NL respectively. In both Model NLT and in Model NL we have set $\Omega_0=\Omega_1=2$ and $\tau_0=\tau_1=1$. }
\label{fig:connectomes}
\end{figure*}
\section{Discussion}
\subsection{Theoretical solution of the NL model}
As mentioned earlier the higher-order topological Kuramoto model coupling the topological signals of nodes and links can be defined on simplicial complexes and on networks as well.
In this section we exploit this property of the dynamics to provide an analytical understanding of the synchronization transition on uncorrelated random networks.
It is well known that the Kuramoto model is challenging to study analytically. Indeed the full analytical understanding of the model is restricted to the fully connected case, while on a generic sparse network topology the analytical approximation needs to rely on some approximations. A powerful approximation is the annealed network approximation \cite{rodrigues2016kuramoto} which consists in approximating the adjacency matrix of the network with its expectation in a random uncorrelated network ensemble.
In order to unveil the fundamental theory that determines the coupled dynamics of topological signals described by the higher-order Kuramoto model here we combine the annealed approximation with the Ott-Antonsen method \cite{ott2008low}.
This approach is able to capture the coupled dynamics of topological signals defined on nodes and links. In particular the solution found to describe the dynamics of topological signals defined on the links is highly non trivial and it is not reducible to the equations valid for the standard Kuramoto model.
Conveniently, the calculations performed in the annealead approximation can be easily recasted in the exact calculation valid in the fully connected case previous a rescaling of some of the parameters.
The analysis of the fully connected network reveals that the discontinuous sychronization transition of the considered model is characterized by a non-trivial backward transition with a well defined large network limit. On the contrary the forward transition is highly dependent on the network size and vanishes in the large network limit, indicating that the incoherent state remains stable for every value of the coupling constant $\sigma$ in the large network limit. This implies that on a fully connected network the NL model does not display a closed hysteresis loop as it occurs also for the model proposed in Ref. \cite{skardal2019abrupt}.
This scenario is here shown to extend also to sparse networks with finite second moment of the degree distribution while scale-free networks display a well defined hysteresis loop in the large network limit.
\subsection{Annealed dynamics}
For the dynamics of the phases $\bm\theta$ associated to the nodes - Eq.~(\ref{thetai0}) - it is possible to proceed as in the traditional Kuramoto model \cite{ichinomiya2004frequency, lee2005synchronization,restrepo2005onset}. However the annealed approximation for the dynamics of the phases $\bm\phi$ defined in Eq. (\ref{phi0}) needs to be discussed in detail as it is not directly reducible to previous results.
To address this problem our aim is to directly define the annealed approximation for the dynamics of the projected variables $\bm\phi^{[-]}$ which, here and in the following are indicated as
\begin{eqnarray}
\bm \psi=\bm\phi^{[-]},
\label{projection0}
\end{eqnarray}
in order to simplify the notation. Moreover we will indicate with $N=N_{[0]}$ the number of nodes in the network or in the simplicial complex skeleton.
Here we focus on the NL Model defined on networks, i.e., we assume that there are no simplices of dimension two.
We provide an analytical understanding of the coupled dynamics of nodes and links in the NL Model by determining the equations that capture the dynamics in the annealed approximation and predict the value of the complex order parameters
\begin{eqnarray}
{R}_0e^{\mathbb{i}\Theta}&=&\frac{1}{N}\sum_{i=1}^Ne^{\mathbb{i}\theta_i}, \nonumber \\
{R}_1^{\text{down}}e^{\mathbb{i}\Psi}&=&\frac{1}{N}\sum_{i=1}^Ne^{\mathbb{i}\psi_i},
\label{order}
\end{eqnarray}
(with $R_0,R_1^{\text{down}},\Theta$ and $\Psi$ real) as a function of the coupling constant $\sigma$.
We notice that Eq.~(\ref{phi-1}), valid for Model NL, can be written as
\begin{eqnarray}
\dot {\bm{\psi}}&=&{\bf B}_{[1]}\tilde{\bm{\omega}}-\sigma R_0 {\bf L}_{[0]}\sin (\bm{\psi}).
\label{psi20}
\end{eqnarray}
This equation can be also written elementwise as
\begin{eqnarray}
\dot {\psi}_i=\hat{\omega}_i+\sigma R_0 \sum_{j=1}^N a_{ij}\left[\sin(\psi_j)-\sin(\psi_i)\right],
\label{psi30}
\end{eqnarray}
where the vector $\hat{\bm \omega}$ is given by
\begin{eqnarray}
\hat{\bf\omega}={\bf B}_{[1]}\tilde{\bm{\omega}}.
\end{eqnarray}
Let us now consider in detail these frequencies in the case in which the generic internal frequency $\tilde{\omega}_{\ell}$ of a link follows a Gaussian distribution, specifically in the case in which ${\tilde\omega}_{\ell}\sim \mathcal{N}(\Omega_1,1/\tau_1)$ for every link $\ell$.
Using the definition of the boundary operator on a link it is easy to show that the expectation of $\hat{\omega}_i$ is given by
\begin{eqnarray}
\avg{\hat{\omega}_i}=\left[\sum_{j<i}a_{ij}-\sum_{j>i}a_{ij}\right]\Omega_1.
\label{avgomega}
\end{eqnarray}
Given that each node has degree $k_i$, the covariance matrix ${\bf C}$ is given by the graph Laplacian ${\bf L}_{[0]}$ of the network, i.e.
\begin{eqnarray}
C_{ij}&=&\Avg{\hat{\omega}_i\hat{\omega}_j}_c=\sum_{\ell,\ell^{\prime}} \Avg{[{\bf B}_{[1]}\tilde{\bm\omega}]_i [{\bf B}_{[1]}\tilde{\bm\omega}]_j}_c\nonumber \\
&=&\frac{[{ L_{[0]}}]_{ij}}{\tau_1^2}=\frac{k_i\delta_{ij}-a_{ij}}{\tau_1^2},
\label{correlation}
\end{eqnarray}
where we have indicated with $\Avg{\ldots}_c$ the connected correlation.
Therefore the
variance of $\hat{\omega}$ in the annealed approximation is
\begin{eqnarray}
\Avg{\hat{\omega}_i^2}_c=\avg{\hat{\omega}_i^2}-\avg{\hat{\omega}_i}^2=\frac{k_i}{\tau_1^2}.
\end{eqnarray}
Moreover, the projected frequencies are actually correlated and for $i\neq j$ we have
\begin{eqnarray}
\Avg{\hat{\omega}_i\hat{\omega_j}}_c=\Avg{\hat{\omega}_i\hat{\omega}_j}-\Avg{\hat{\omega}_i}\Avg{\hat{\omega}_j}=-\frac{a_{ij}}{\tau_1^2}.
\end{eqnarray}
It follows that the frequencies $\hat{\bm\omega}$ are correlated Gaussian variables with average given by Eq. (\ref{avgomega}) and correlation matrix given by the graph Laplacian.
The fact that the frequencies $\hat{\omega}_i$ are correlated is an important feature of the dynamics of $\bm\psi$ and, with few exceptions (e.g.,~\cite{skardal2015frequency}), this feature has remained relatively unexplored in the case of the standard Kuramoto model.
Additionally let us note that the average of $\hat{\omega}$ over all the nodes of the network is zero. In fact
\begin{eqnarray}
\sum_{i=1}^N \hat \omega_i = {\bf 1}^T \hat {\bm \omega} = {\bf 1}^T {\bf B}_{[1]}{\bm \omega} = 0,
\label{u}
\end{eqnarray}
where with ${\bf 1}$ we indicate the $N$-dimensional column vector of elements $1_i=1$.
By using the symmetry of the adjacency matrix, i.e. the fact that $a_{ij} = a_{ji}$, Eq. (\ref{u}) implies that the sum of $\dot \psi_i$ over all the nodes of the network is zero, i.e.
\begin{eqnarray}
\sum_{i=1}^N \dot \psi_i &= \sum_{i=1}^N \hat \omega_i + \sigma R_0 \sum_{i,j} a_{ij}[\sin(\psi_j )- \sin(\psi_i)] = 0.\nonumber
\end{eqnarray}
We now consider the annealed approximation consisting in substituting the adjacency matrix element $a_{ij}$ with its expectation in an uncorrelated network ensemble
\begin{eqnarray}
a_{ij}\to \frac{k_ik_j}{\avg{k}N},
\label{annealed}
\end{eqnarray}
where $k_i$ indicates the degree of node $i$ and $\avg{k}$ is the average degree of the network. Note that the considered random networks can be both sparse \cite{anand2009entropy} or dense \cite{seyed2006scale} as long as they display the structural cutoff, i.e. $k_i\ll \sqrt{\avg{k}N}$ for every node $i$ of the network.
In the annealed approximation we can put
\begin{eqnarray}
\avg{\hat{\omega}_i}\simeq k_i\Omega_1 \left[1-2\sum_{j>i}\frac{k_j}{\avg{k}N}\right].
\label{homega_av0}
\end{eqnarray}
Also, in the annealed approximation the dynamical Eq.~(\ref{thetai0}) and Eq.~(\ref{psi20}) reduce to
\begin{eqnarray}
\dot{\bm \theta}&=&\bm\omega-\sigma R_1^{\text{down}}\hat{R}_0{\bf k}\cdot \sin(\bm{\theta}-\hat\Theta),\label{thetap} \\
\dot{\bm\psi}&=&\hat{\bm\omega}+\sigma R_0 \hat{R}_1^{\text{down}}{\bf k}\sin\hat\Psi-\sigma R_0 {\bf k}\odot \sin \bm\psi, \label{psip}
\end{eqnarray}
where $\odot$ indicates the Hadamard product (element by element multiplication) and
where two auxiliary complex order parameters are defined as
\begin{eqnarray}
\hat{R}_0e^{\mathbb{i}\hat\Theta}&=&\sum_{i=1}^N\frac{k_i}{\avg{k}N}e^{\mathbb{i}\theta_i}, \nonumber \\
\hat{R}_1^{\text{down}}e^{\mathbb{i}\hat\Psi}&=&\sum_{i=1}^N\frac{k_i}{\avg{k}N}e^{\mathbb{i}\psi_i},
\end{eqnarray}
with $\hat{R}_0,\hat\Theta,\hat{R}_1^{\text{down}}$ and $\hat\Psi$ real.
\subsection{The dynamics on a fully connected network}
On a fully connected network in which each node has degree $k_i=N-1$ the dynamics of the NL Model is well defined provided its parameter are properly rescaled.
In particular we require a standard rescaling of the coupling constant with the network size, given by
\begin{eqnarray}
\sigma\to \sigma/(N-1)
\end{eqnarray}
which guarantees that the interaction term in the dynamical equations has a finite contribution to the velocity of the phases.
The Model NL on fully connected networks requires also some specific model dependent rescalings associated to the dynamics on networks. Indeed in order to have a finite expectation $\avg{\hat{\omega}_i}$ of the projected frequencies $\hat{\omega}_i$ and a finite of the covariance matrix $\bf C$, [given by Eqs.~(\ref{avgomega}) and (\ref{correlation}), respectively] we require that on a fully connected network both $\Omega_1$ and $\tau_1$ are rescaled according to
\begin{eqnarray}
\Omega_1 &\to& \Omega_1/N,\nonumber \\
\tau_1 &\to &\tau_1 \sqrt{N-1}.
\end{eqnarray}
Considering these opportune rescalings and noticing that the order parameters obey $\hat{R}_0=R_0$, $\hat{R}_1^{\text{down}}=R_1^{\text{down}}$, $\Theta=\hat\Theta$, and $\Psi=\hat\Psi$, we obtain that Model NL dictated by Eqs. (\ref{thetap})-(\ref{psip}) can be rewritten here as
\begin{eqnarray}
\dot{\bm \theta}&=&\bm\omega-\sigma R_1^{\text{down}}{R}_0 \sin(\bm{\theta}-\Theta),\label{thetapf} \\
\dot{\bm\psi}&=&\hat{\bm\omega}+\sigma R_0 {R}_1^{\text{down}}\sin\Psi-\sigma R_0 \sin \bm\psi, \label{psipf}
\end{eqnarray}
with $R_0,R_1^{\text{down}}, \Theta$ and $\Psi$ given by Eq. (\ref{order})
and
\begin{eqnarray}
C_{ij}=\Avg{\hat{\omega}_i\hat{\omega}_j}_c=\delta_{ij}-\frac{1}{N-1}.
\end{eqnarray}
\subsection{Solution of the dynamical equations in the annealed approximation}
\subsubsection{General framework for obtaining the solution of the annealed dynamical equations}
In this section we will provide the analytic solutions for the order parameter of the higher-order topological synchronization studied within the annealed approximation, i.e., captured by Eqs.~(\ref{thetap}) and (\ref{psip}).
In particular first we will find an expression of the order parameters $R_0$ of the dynamics associated to the nodes {(Eq.~(\ref{thetap}))} and subsequently in the next paragraph we will derive the expression for the order parameter $R_1^{\text{down}}$ associated to the projection on the nodes of the topological signal defined on the links (Eq.~{\ref{psip})). By combining the two results it is finally possible to uncover the discontinuous nature of the transition.
\subsubsection{Dynamics of the phases of the nodes}
When we investigate Eq. (\ref{thetap}) we notice that this equation can be easily reduced to the equation for the standard Kuramoto model treated within the annealed approximation \cite{restrepo2005onset} if one performs a rescaling of the coupling constant $\sigma $ $R_0\to \sigma$. Therefore we can treat this model similarly to the known treatment of the standard Kuramoto model \cite{restrepo2005onset,boccaletti2018synchronization,rodrigues2016kuramoto}.
Specifically, starting from Eq.~(\ref{thetap}) and using a rescaling of the phases $\bm{\theta}$ according to
\begin{eqnarray}
{\theta}_i\to {\theta}_i-\Omega_0 t,
\end{eqnarray}
it is possible to show that we can set $\Theta=0$ and therefore Eq. (\ref{thetap}) reduces to the well-known annealed expression for the standard order Kuramoto model given by
\begin{eqnarray}
\dot{\bm \theta}&=&\bm\omega-\Omega_0 {\bf 1}-\sigma R_1^{\text{down}}\hat{R}_0{\bf k}\cdot \sin(\bm{\theta}).
\end{eqnarray}
Assuming that the system of {equations} reaches a steady state in which both $R_1^{\text{down}}$ and $\hat{R}_0$ become time independent, the order parameters of this system of equations in the coherent state $\hat{R}_0>0$ and $R_1^{\text{down}}>0$ can be found to obey \cite{ichinomiya2004frequency,restrepo2005onset,boccaletti2018synchronization,boccaletti2016explosive}
\begin{eqnarray}
&&\hat{R}_0=\sum_{i=1}^N\frac{k_i}{\avg{k}N}\int_{|\hat{c}_i|<1} d\omega g(\omega) \sqrt{1-\left(\frac{\omega-\Omega_0}{\sigma k_i \hat{R}_0R_1^{\text{down}}}\right)^2},\nonumber \\
&&R_0=\frac{1}{N}\sum_{i=1}^N \int_{|\hat{c}_i|<1} d\omega g(\omega) \sqrt{1-\left(\frac{\omega-\Omega_0}{\sigma k_i \hat{R}_0R_1^{\text{down}}}\right)^2},\label{R0s_th}
\end{eqnarray}
where $\hat{c}_i$ indicates
\begin{eqnarray}
\hat{c}_i=\frac{\omega-\Omega_0}{\sigma k_j \hat{R}_0R_1^{\text{down}}}.
\end{eqnarray}
and $g(\omega)$ is the Gaussian distribution with expectation $\Omega_0$ and standard deviation $1$.
\subsubsection{Dynamics of the phases of the links projected on the nodes}
In this paragraph we will derive the expression of the order parameters $R_1^{\text{down}}$ and $\hat{R}_1^{\text{down}}$ which, together with Eqs.~(\ref{R0s_th}), will provide the annealed solution of our model.
To start with we assume that the frequencies $\hat{\bm\omega}$ are known. In this case we can express the order parameters $R_1^{\text{down}}$ and $\hat{R}_1^{\text{down}}$ as a function of the
probability density function ${\rho}^{(i)}(\psi,t|\hat{\bm\omega})$ that node $i$ is associated to a projected phase of the link equal to $\psi$. Since in the annealed approximation $\psi_i$ has a dynamical evolution dictated by Eq.~(\ref{psip}) the probability density function obeys the continuity equation
\begin{eqnarray}
&&\partial_t {\rho}^{(i)}(\psi,t|{\hat{\bm\omega}})+\partial_{\psi}\left[{\rho}^{(i)}(\psi,t|{\hat{\bm\omega}}))v_{i}\right]=0\nonumber \\
\end{eqnarray}
with associated velocity $v_{i}$ given by
\begin{eqnarray}
v_{i}&=&\kappa_{i}-\sigma R_0 k_i\sin \psi_i,
\end{eqnarray}
where we have defined $\kappa_i$ as
\begin{eqnarray}
\kappa_{i}&=&\hat{\omega}_i+\sigma k_i R_0\hat{R}_1^{\text{down}}\sin\hat\Psi.
\end{eqnarray}
In this case the complex order parameters are given by
\begin{eqnarray}
\hat{R}_1^{\text{down}}e^{\mathbb{i}\Psi}&=&\sum_{i=1}^N\frac{k_i}{\avg{k}N}\int d\psi {\rho}^{(i)}(\psi,t|\hat{\bm\omega}) e^{\mathbb{i} \psi},\nonumber \\
{R}_1^{\text{down}}e^{\mathbb{i}\tilde{\Psi}}&=&\sum_{i=1}^N\frac{1}{N}\int d\psi {\rho}^{(i)}(\psi,t|\hat{\bm\omega}) e^{\mathbb{i} \psi}.
\label{R1_rho}
\end{eqnarray}
In order to solve the continuity equation we follow Ott-Antonsen \cite{ott2008low} and we express ${\rho}^{(i)}(\psi,t|\hat{\bm\omega})$ in the Fourier basis as
\begin{eqnarray}
{\rho}^{(i)}(\psi,t|\hat{\bm\omega})=\frac{1}{2\pi}\left\{1+\sum_{{m}=1}^{\infty}\hat{f}^{(i)}_{m}(\hat{{\omega}}_i,t) e^{\mathbb{i}{m} {{\psi}}}+c.c.\right\}.
\end{eqnarray}
Making the ansatz
\begin{eqnarray}
\hat{f}_{m}^{(i)}(\hat{\omega}_i,t)&=&[b_i(\hat{\omega}_i,t)]^m
\end{eqnarray}
we can derive the equation for the evolution of $b_i=b_i(\hat{\omega}_i,t)$ given by
\begin{eqnarray}
&&\partial_t b_i+\mathbb{i}b_i\kappa_{i}
+\sigma k_iR_0\frac{1}{2}(b^{2}_i-1)=0.
\label{dyn_0}
\end{eqnarray}
Since we showed before that the average value of $\dot \psi_i$ over nodes is zero, we look for non-rotating stationary solutions of Eq. (\ref{dyn_0}), $\partial_t b_i = 0$. As long as $R_0>0$ these stationary solutions are given by
\begin{eqnarray}
b_i&=&-\mathbb{i}d_i\pm \sqrt{1-d_i^2},
\end{eqnarray}
where $d_i$ is given by
\begin{eqnarray}
d_i=\frac{\hat{\omega}_i}{\sigma k_i R_0}+\hat{R}_1^{\text{down}}\sin\hat{\Psi}.
\end{eqnarray}
By inserting this expression into Eq.~(\ref{R1_rho}) we get the expression of the order parameters given the projected frequencies $\hat{\bm\omega}$, in the coherent phase in which $R_0>0$
\begin{eqnarray}
\hat{R}_1^{\text{down}}\cos\hat{\Psi}&=&
\sum_{i=1}^N\frac{k_i}{\avg{k}N}\sqrt{1-d_i^2}\theta(1-d_i^2),\nonumber \\
\hat{R}_1^{\text{down}}\sin\hat{\Psi}&=&\sum_{i=1}^N\frac{k_i}{\avg{k}N}\left\{\sqrt{d_i^2-1}\chi(d_i)+d_i\right\},\nonumber \\
{R}_1^{\text{down}}\cos\Psi&=&
\sum_{i=1}^N\frac{1}{N}\sqrt{1-d_i^2}\theta(1-d_i^2),\nonumber \\
{R}_1^{\text{down}}\sin\Psi&=&\sum_{i=1}^N\frac{1}{N}\left\{\sqrt{d_i^2-1}\chi(d_i)+d_i\right\},
\label{comp0}
\end{eqnarray}
where, indicating by $\theta(x)$ the Heaviside function, we have defined \begin{eqnarray}
\chi(d_i)=[-\theta(d_i-1)+\theta(-1-d_i)].
\end{eqnarray}
Finally, if the projected frequencies $\hat{\bm{\omega}}$ are not known we can average the result over the marginal frequency distribution of the projected frequency $\hat{\omega}_i$ given by $G_i(\hat{\bm\omega})$ getting
\begin{widetext}
\begin{eqnarray}
\hat{R}_1^{\text{down}}\cos\hat{\Psi}&=&
\sum_{i=1}^N\frac{k_i}{\avg{k}N}\int_{|d_i|\leq 1}d\hat{ \omega}_iG_i(\hat{ \omega}_i)\sqrt{1-\left(\frac{\hat{\omega}_i}{\sigma R_0k_i}+\hat{R}_1^{\text{down}}\sin \hat{\Psi}\right)^2}\nonumber ,\\
\hat{R}_1^{\text{down}}\sin\hat{\Psi}&=&-\sum_{j=0}^N\frac{k_i}{\avg{k}N}\int_{d_i>1}d\hat{\omega}_i G_i(\hat{ \omega}_i)\sqrt{\left(\frac{\hat{\omega}_i}{\sigma R_0k_i}+\hat{R}_1^{\text{down}}\sin \hat{\Psi}\right)^2-1}\nonumber \\
&&+\sum_{i=1}^N\frac{k_i}{\avg{k}N}\int_{d_i<-1}d\hat{ \omega}_iG_i(\hat{ \omega}_i)\sqrt{\left(\frac{\hat{\omega}_i}{\sigma R_0k_i}+\hat{R}_1^{\text{down}}\sin \Psi\right)^2-1}\nonumber \\
&&+\sum_{i=1}^N\frac{k_i}{\avg{k}N}\int_{-\infty}^{\infty}d\hat{ \omega}_iG_i(\hat{ \omega}_i) \left(\frac{\hat{\omega}_i}{\sigma R_0k_i}+\hat{R}_1^{\text{down}}\sin \Psi\right),\nonumber\\
R_1^{\text{down}}\cos \Psi&=&
\sum_{i=1}^N\frac{1}{N}\int_{|d_i|\leq 1}d\hat{ \omega}_iG_i(\hat{ \omega}_i)\sqrt{1-\left(\frac{\hat{\omega}_i}{\sigma R_0k_i}+\hat{R}_1^{\text{down}}\sin \hat{\Psi}\right)^2}\label{comp},
\end{eqnarray}
\end{widetext}
and an analogous equations for $R_1^{\text{down}} \sin(\Psi)$ (not shown).
We note that in the case of distributions $g(\omega)$ and $G_i(\hat \omega)$ that are symmetric around their means the above equations always admit the solution $\Psi = \hat \Psi = 0$. Such values of the phases are also confirmed by direct numerical integration of the NL model.
\begin{figure*}[htb]
\centering
\includegraphics[width=1.8\columnwidth]{figure4.pdf}
\caption{{\bf Comparison between the simulation results of the NL Model and its solution in the annealed approximation}
The order parameters $R_0$ and $R_1^{\text{down}}$ of the NL Model are shown as a function of $\sigma$ for a Poisson network with average degree $c=12$ and for an uncorrelated scale-free network with minimum degree $m=6$ and power-law exponent $\gamma=2.5$. Both networks have $N=1600$ nodes. The symbols indicate the simulation results for the forward (cyan diamonds) and the backward (green circles) synchronization transition. The solid lines indicate the analytical solution for the backward transition obtained by integrating Eq.~(\ref{comp0}).}
\label{fig:annealed}
\end{figure*}
These equations together with Eqs.~(\ref{R0s_th}) capture the steady-state behavior of the higher-order Kuramoto model coupling topological signals defined on nodes and links within the annealed approximation in the coherent synchronized phase.
Note that by derivation, these equations cannot capture the asynchronous phase which is instead always a trivial solution of the dynamical equations corresponding to $R_0=R_1^{\text{down}}=0$.
Finally we observe that for the NL Model as well as for the standard Kuramoto model on random networks, it is expected that the annealed approximation is more accurate for networks that are connected and are sufficiently dense.
To illustrate the applicability of the theoretical analysis, we consider two examples of connected networks with $N=1600$ nodes: a Poisson network with average degree $c=12$ and an uncorrelated scale-free network with minimum degree $m=6$ and power-law exponent $\gamma=2.5$ In Fig.~\ref{fig:annealed} we compare the values of $R_0$, $R_1^{\text{down}}$ obtained from direct numerical integration of Eqs.~(\ref{thetai0}) and (\ref{psi30}) and the steady state solutions obtained from the numerical solution of Eqs.~(\ref{comp0}).
The backward transition is fully captured by our theory, while the next paragraphs will clarify the theoretical expectations for the forward transition.
\subsection{Solution on the fully connected network}
The integration of Eq.~(\ref{comp}) requires the knowledge of the marginal distributions $G_i(\hat{\omega})$ which does not have in general a simple analytical expression.
However, in the fully connected networks with Gaussian distribution of the internal frequency of nodes and links this calculation simplifies significantly. Indeed, when the link frequencies are sampled from a Gaussian distribution with mean $\Omega_1/N$ and standard deviation $1/(\tau_1\sqrt{N-1})$, the marginal frequency distribution $G_i(\hat{\omega})$ of the internal frequency $\hat{\omega}_i$ of a node $i$ in a fully connected network is given by (see Methods for details)
\begin{eqnarray}
G_i(\hat{\omega})&=& \frac{\tau_1}{\sqrt{2\pi/\bar{c}}}\exp\left[-\tau_1^2\bar{c}\frac{(\hat{\omega}_i-\Avg{\hat{\omega}_i})^2}{2}\right],
\label{marginal}
\end{eqnarray}
where $\bar c=\frac{N}{N-1}$. By considering $\Omega_0=\Omega_1=\Avg{\hat{\omega}_i}=0,$ and performing a direct integration of Eqs.~(\ref{comp}) we obtain (see Methods section for details) the closed system of equations for $R_0$ and $R_1^{\text{down}}$
\begin{eqnarray}
&1 = \sigma R_1^{\text{down}} h\left(\sigma^2 R_0^2 (R_1^{\text{down}})^2\right),\nonumber \\
&R_1^{\text{down}} = \sigma R_0 \tau_1 \sqrt{\bar{c}} h\left(\sigma^2 \tau_1^2 R_0^2 \right),\label{sys}
\end{eqnarray}
where the scaling function $h(x)$ is given by
\begin{eqnarray}
h(x) = \sqrt{\frac{\pi }{2}} e^{-x/4} \left[I_0\left(\frac{x}{4}\right)+I_1\left(\frac{x}{4}\right)\right],
\end{eqnarray}
with $I_0$ and $I_1$ indicating the modified Bessel functions.
The numerical solution of Eqs.~(\ref{sys}) reveals the following picture: for low values of $\sigma$, only the incoherent solution $R_0 = R_1^{\text{down}} = 0$ exists. At a positive value of $\sigma$, two solutions of Eqs.~(\ref{sys}) appear at a bifurcation point, with the upper solution corresponding to a stable synchronized state and the lower solution to an unstable synchronized solution. For larger values of $\sigma$, the values of $R_0$ and $R_1^{\text{down}}$ corresponding to the upper solution approach one (full phase synchronization), while those for the lower solution approach zero asymptotically, thus indicating that the incoherent state never loses stability.
Indeed, it can be easily checked (see Methods for details) that for large $\sigma$ the unstable solution of Eqs. (\ref{sys}) has asymptotic behavior
\begin{eqnarray}
R_0&=&\sigma^{-2}J_0,\nonumber \\
R_1^{\text{down}}&=&{\sigma}^{-1}J_1,
\label{AsyFC}
\end{eqnarray}
with $J_0$ and $J_1$ constants given by
\begin{eqnarray}
J_0&=&\left[\frac{\pi}{2}\right]^{-2}\left[G(0)g(0)\right]^{-1},\\
J_1&=&\left[g(0)\frac{\pi}{2}\right]^{-1}.
\label{JFC}
\end{eqnarray}
Therefore the unstable branch approaches the trivial solution $R_0=R_1^{\text{down}}=0$ only asymptotically for $\sigma\to \infty$. This implies that the trivial solution remains stable for every possible value of $\sigma$ although as $\sigma$ increases it describes the stationary state of an increasingly smaller set of initial conditions.
\begin{figure*}[htb]
\centering
\includegraphics[width=2.0\columnwidth]{figure5.pdf}
\caption{{\bf The backward and the forward discontinuous phase transition on fully connected networks} The order parameters $R_0$ (circles) and $R_1^{\text{down}}$ (squares) are plotted as a function of the coupling constant $\sigma$ on a fully connected network. The solid and the dashed lines indicate the stable branch and the unstable branch predicted by Eqs.(\ref{sys}). Simulations (shown as data point) are here obtained by integrating numerically Eqs. (\ref{thetap}) and (\ref{psip}) for a fully connected network of $N=500$ (cyan circles), $N=1000$ (green squares), and $N=2000$ (purple diamonds) with $\Omega_0=\Omega_1=0$ and (rescaled) $\tau_0=\tau_1=1$. The backward transition is perfectly captured by the theoretical prediction and is affected by finite size effects very marginally. The forward transition is instead driven by stochastic fluctuations and moves to higher values of $\sigma$ as the network size increases.}
\label{fig:fully_connected}
\end{figure*}
This scenario is confirmed by numerical simulations (see Figure \ref{fig:fully_connected}) showing that
the backward transition is captured very well by our theory and does not display notable finite size effects.
The forward transition, instead, displays remarkable finite size effects. Indeed, as $\sigma$ increases, the system remains in the incoherent state until it explosively synchronizes at a positive value of $\sigma$ and reaches the stable synchronized branch.
However the incoherent state is stable in the limit $N \to \infty$, and this forward transition is the result of finite size fluctuations that push the system above the unstable branch, causing the observed explosive transition. This is consistent with the fact that for larger values of $N$, which have smaller finite size fluctuations, the system remains in the incoherent state for larger values of $\sigma$.
Therefore, while a closed hysteresis loop is not present in the NL model defined on fully connected networks, we observe fluctuation-driven hysteresis, in which finite-size fluctuations of the zero solution drive the system towards the synchronized solution, creating an effective hysteresis loop.
\subsection{Hysteresis on homogeneous and scale-free networks}
In this section we discuss how the scenario found for the fully connected network can be extended to random networks with given degree distribution.
We will start from the self-consistent Eqs.~(\ref{comp}) obtained within the annealed approximation model. These equations display a saddle point bifurcation with the emergence of two non-trivial solutions describing a stable and an unstable branch of these self-consistent equations. These solutions always exist in combination with the trivial solution $R_0=R_1^{\text{down}}=0$ describing the asynchronous state.
Two scenarios are possible: either the unstable branch converges to the trivial solution only in the limit $\sigma\to\infty$ or it converges to the trivial solution at a finite value of $\sigma$.
In the first case, the scenario is the same as the one observed for the fully connected network, and the trivial solution remains stable for any finite value of $\sigma $. In this case the forward transition is not obtained in the limit $N\to\infty$ and the transition observed on finite networks is only caused by finite size effects. In the second case the trivial solution loses its stability at a finite value of $\sigma$. Therefore the forward transition is not subjected to strong finite size effects and we expect to see a forward transition also in the $N\to\infty$ limit.
in order to determine which network topologies can sustain a non-trivial hysteresis loop we expand Eqs.~(\ref{comp}) for $0<R_0\ll1$, $0<\hat{R}_0\ll 1$, and $0<R_1^{\text{down}}\ll1$ under the hypothesis that the distributions $g(\omega)$ and $G_i(\hat{\omega}) $ are symmetric and unimodal.
Under these hypothesis it is easy to show that Eqs.~(\ref{comp}) predict an unstable solution in which $R_0$ and $R_1^{\text{down}}$ scale with $\sigma$ according to
\begin{eqnarray}
R_0&=&\sigma^{-2}J_0,\nonumber \\
R_1^{\text{down}}&=&{\sigma}^{-1}J_1,
\end{eqnarray}
with $J_0$ and $J_1$ constants given by
\begin{eqnarray}
J_0&=&\Avg{k}\left[\frac{\pi}{2}\frac{\Avg{k^2}}{\Avg{k}}\right]^{-2}\left[ g(\Omega_0)\frac{1}{N}\sum_i k_iG_i(\Avg{\hat{\omega}_i})\right]^{-1},\nonumber\\
J_1&=&\left[g(\Omega_0)\frac{\pi}{2}\frac{\Avg{k^2}}{\Avg{k}}\right]^{-1}.
\end{eqnarray}
As long as the network does not have vanishing $J_0$ and $J_1$ the unstable branch converges to the trivial solution $R_0=R_1^{\text{down}}$ only in the limit $\sigma\to\infty$.
This happens for instance for Gaussian distribution of the internal frequency of the links and converging second moment $\avg{k^2}$ of the degree distribution.
However, when the second moment diverges, i.e., the network is scale free with $\avg{k^2}\to \infty$ as $N\to\infty$, then $R_0$ and $R_1$ can converge to the trivial solution $R_0=R_1^{\text{down}}=0$ also for finite $\sigma$.
This analysis suggests that the scenario described for the fully connected network remains valid for sparse (connected) networks as long as the degree distribution does not have a diverging second moment, while a stable hysteresis loop can be observed for scale-free networks.
\section{Conclusions}
Until recently the synchronization phenomenon has been explored only in the context of topological signals associated to the nodes of a network. However, the growing interest in simplicial complexes opens the perspective of investigating synchronization of higher order topological signals, for instance associated to the links of the discrete networked structure. Here we uncover how topological signals associated to nodes and links can be coupled to one another giving rise to an explosive synchronization phenomenon involving both signals at the same time.
The model has been tested on real connectomes and on major examples of simplicial complexes (the configuration model \cite{courtney2016generalized} of simplicial complex and the Network Geometry with Flavor \cite{bianconi2016network}).
Moreover, we provide an analytical solution of this model that provides a theoretical understanding of the mechanism driving the emergence of this discontinuous phase transition and the mechanism responsible for the emergence of a closed hysteresis loop.
This work can be extended in different directions including the study of the de-synchronization dynamics of this coupled higher-order synchronization and the duality of this model with the same model defined on the line graph of the same network.
\section*{Acknowledgements}
This work is partially founded by SUPERSTRIPES Onlus. This research utilized Queen Mary's Apocrita HPC facility, supported by QMUL Research-IT.
http://doi.org/10.5281/zenodo.438045. G.B. acknowledge support from the Royal Society IEC\textbackslash NSFC \textbackslash 191147. J.J.T. acknowledges financial support from the Spanish Ministry of Science and Technology, and the Agencia Espa\~nola de Investigaci\'on (AEI) under grant FIS2017-84256-P (FEDER funds) and from the Consejer\'ia de Conocimiento, Investigaci\'on y Universidad, Junta de Andaluc\'ia and European Regional Development Fund, Refs.~A-FQM-175-UGR18 and SOMM17/6105/UGR.
\section*{Author contributions}
All authors have contributed in the design of the project, in the numerical implementations of the algorithm, the theoretical derivations and the writing of the manuscript.
\section*{Code Availability}
All codes are available upon request to the Authors.
\section*{Data Availability}
The connectome network dataset used in this study are freely available: the Homo sapiens dataset comes from Ref. \cite{sapiens} the C.elegans dataset comes from Ref. \cite{celegans}.
\section*{Competing interests}
The authors declare no competing interests.
\bibliographystyle{apsrev4-1}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 9,944 |
<a name="Communication-Styles"></a>
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<a name="Communication-Styles-1"></a>
<h3 class="section">16.2 Communication Styles</h3>
<p>The GNU C Library includes support for several different kinds of sockets,
each with different characteristics. This section describes the
supported socket types. The symbolic constants listed here are
defined in <samp>sys/socket.h</samp>.
<a name="index-sys_002fsocket_002eh"></a>
</p>
<dl>
<dt><a name="index-SOCK_005fSTREAM"></a>Macro: <em>int</em> <strong>SOCK_STREAM</strong></dt>
<dd><p>The <code>SOCK_STREAM</code> style is like a pipe (see <a href="Pipes-and-FIFOs.html#Pipes-and-FIFOs">Pipes and FIFOs</a>).
It operates over a connection with a particular remote socket and
transmits data reliably as a stream of bytes.
</p>
<p>Use of this style is covered in detail in <a href="Connections.html#Connections">Connections</a>.
</p></dd></dl>
<dl>
<dt><a name="index-SOCK_005fDGRAM"></a>Macro: <em>int</em> <strong>SOCK_DGRAM</strong></dt>
<dd><p>The <code>SOCK_DGRAM</code> style is used for sending
individually-addressed packets unreliably.
It is the diametrical opposite of <code>SOCK_STREAM</code>.
</p>
<p>Each time you write data to a socket of this kind, that data becomes
one packet. Since <code>SOCK_DGRAM</code> sockets do not have connections,
you must specify the recipient address with each packet.
</p>
<p>The only guarantee that the system makes about your requests to
transmit data is that it will try its best to deliver each packet you
send. It may succeed with the sixth packet after failing with the
fourth and fifth packets; the seventh packet may arrive before the
sixth, and may arrive a second time after the sixth.
</p>
<p>The typical use for <code>SOCK_DGRAM</code> is in situations where it is
acceptable to simply re-send a packet if no response is seen in a
reasonable amount of time.
</p>
<p>See <a href="Datagrams.html#Datagrams">Datagrams</a>, for detailed information about how to use datagram
sockets.
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| {
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\section{Introduction} \label{sec: Introduction}
Black holes are an essential and astrophysically relevant feature of the theory of General Relativity (GR), formulated by Einstein in the early 20th century. To date, theoretical predictions of Einstein's theory are in remarkable agreement with experimental observations~\cite{Berti:2015itd, LIGOScientific:2016aoc, EventHorizonTelescope:2019dse, LIGOScientific:2021sio}. Nevertheless, experiments leave room for deviations from GR and black holes provide a unique probe of the strong-field regime of gravity. In this context, two aspects of the black hole solutions in Einstein gravity are particularly intriguing.
First, stationary black holes in GR contain a curvature singularity~\cite{Hawking:1970zqf}. The respective unphysical divergence of curvature invariants results in the incompleteness of particle geodesics.\footnote{In the case of spinning black holes in GR, causality breaks down already before approaching the singularity, i.e., when crossing the inner Cauchy horizon.} These are signals that the theory is pushed beyond its regime of validity. At very short distances of the order of a Planck length, $l_{pl}\approx 10^{-35}$m, i.e.~at extremely high curvature scales, a theory of quantum gravity (QG) is expected to take over and provide a consistent microscopic description of spacetime.
Second, the stationary black hole solutions of GR are characterized by their event horizon. The event horizon is the portion of spacetime which is disconnected from future asymptotic infinity\footnote{In the stationary cases at hand the local definitions of event horizons, e.g.~as marginally trapped surfaces, agree with the global definition.}, in other words, from which no signal can escape. Event horizons are directly connected to recent observations including the quasinormal-mode ringdown of observed gravitational-wave signals~\cite{LIGOScientific:2016aoc} and a potential detection of the black-hole shadow, i.e., a characteristic central brightness depression in the intensity of very-long baseline interferometry~\cite{EventHorizonTelescope:2019dse}. Nevertheless, there is room in current experimental data for exotic compact objects as mimickers of black holes, see for instance gravastars~\cite{Mazur:2001fv,Visser:2003ge}, various versions of wormholes~\cite{Bronnikov:1973fh,Ellis:1973eft}, fuzzballs and firewalls~\cite{Susskind:1993ws,Mathur:2005zp,Almheiri:2012rt}, or, more recently, 2:2 holes~\cite{Holdom:2002xy,Holdom:2016nek,Holdom:2019bdv,Holdom:2019ouz,Ren:2019afg}. Potential observable signatures of horizon-like structures with a non-zero reflectivity include gravitational wave echoes associated with time-delayed signals following the main merger ringdown~\cite{Cardoso:2016rao,Abedi:2016hgu,Barcelo:2017lnx,Cardoso:2017njb,Oshita:2019sat}.
For astrophysical objects of high mass, the curvature at classical horizon scales is many orders of magnitude below the Planck scale. Therefore a common belief is that QG does not play a role at horizon scales and drastic modifications of gravity at the classical level are needed to obtain spacetimes which substantially deviate from the GR predictions at horizon scales. Such a viewpoint may be presumptuous, given that the question of what are the fundamental degrees of freedom and which of these are relevant at different energies is an open question in QG, see~\cite{Eichhorn:2018yfc,Ashtekar:2021kfp,Surya:2019ndm,Brennan:2017rbf} for reviews on prominent candidate theories.
Different approaches to QG may lead to distinct expectations about the presence of horizon-scale modifications. Hence, observational insights into the existence or absence of horizons provides one of the rare opportunities of QG phenomenology. More generally, investigating phenomenological consequences of different approaches to QG constitutes an important step towards understanding the physical significance of the building blocks of the theory.
In the following, we adopt the viewpoint of asymptotic safety~\cite{Weinberg:1976xy,Weinberg:1980gg,Reuter:1996cp} (cf.~\cite{Bonanno:2020bil} for a recent review) as providing a high-energy (ultraviolet) completion for gravity and matter based on quantum scale invariance. Formally, quantum scale invariance corresponds to approaching a fixed-point of the renormalization group (RG) flow in the theory space of couplings. At the RG fixed point, dimensionless couplings become scale-independent. By dimensionless couplings, we refer also to the dimensionless counterpart of dimensionful couplings, obtained by multiplying with an appropriate power of the RG-scale. As a result, dimensionful couplings, such as the Newton coupling, acquire a characteristic high-energy scale-dependence.
We investigate phenomenological implications of such a scale-dependence of couplings, by means of a mechanism known as {\it RG-improvement}. Essentially, RG-improvement consists in retaining the dependence of some of the couplings on the RG scale and identifying the latter with a characteristic energy scale of the physical system in consideration.
RG-improvement has been developed in the context of non-gravitational quantum-field theories~\cite{Coleman1973:rcssb} where it corresponds to a resummation of large logarithms and, for instance, can recover the quantum corrections to the Coulomb potential~\cite{Dittrich:2014eff}.
In the context of spacetime geometries, RG-improvement has been pioneered in~\cite{Bonanno:1998ye,Bonanno:2000ep} for the Schwarzschild geometry. Since then, RG improvement has been applied to black-hole physics~\cite{Bonanno:1998ye,Bonanno:2000ep,Bonanno:2006eu,Falls:2010he,Koch:2013owa,Kofinas:2015sna} and in cosmology~\cite{Bonanno:2001xi,Bonanno:2001hi,Reuter:2005kb,Hindmarsh:2011hx,Bonanno:2018gck,Platania:2019qvo,}, see also~\cite{Bonanno:2017pkg,Platania:2020lqb} for reviews. In RG-improvement of black-hole spacetimes, the RG-scale is typically identified with an appropriate notion of the local curvature scale and, as a result, could resolve classical curvature singularities~\cite{Bonanno:1998ye,Bonanno:2000ep,Torres:2014gta,Kofinas:2015sna,Torres:2017ygl,Adeifeoba:2018ydh}.
Potential phenomenological effects~\cite{Liu:2012ee, Eichhorn:2021iwq, Eichhorn:2021etc} face the challenge of overcoming the large separation between the Planck scale and the intrinsic scale of astrophysical black holes, see e.g.~\cite{Held:2019xde,Zhou:2020eth}. At the same time, the physical significance of the resulting regular black holes is currently challenged by a potential instability of the resulting inner Cauchy horizon~\cite{Carballo-Rubio:2018pmi,Bonanno:2020fgp}. In addition, there are indications that regular black holes which admit an asymptotic series expansion could be incompatible with a fundamental action principle~\cite{Knorr:2022kqp}.
We emphasize that QG phenomenology based on an RG-improvement of couplings does \emph{not} provide a unique prediction or strict first-principle derivation from asymptotic safety. First steps towards a first-principle derivation involve the physical RG running of couplings in terms of momentum-dependent correlation functions~\cite{Bosma:2019aiu}. The RG-improvement procedure is impacted by several physical assumptions.
First, RG-improvement depends inevitably on the specified scale-dependence of couplings.
As of now, only the scale dependence of the Newton coupling and the cosmological constant has been investigated: The fixed-point scaling of the Newton coupling leads to resolution of the classical curvature singularity~\cite{Bonanno:2000ep}. However, the fixed-point scaling of the cosmological constant is expected to reintroduce a curvature singularity~\cite{Koch:2013owa, Pawlowski:2018swz, Adeifeoba:2018ydh}.
Second, the scale-dependence of couplings can be retained
(i) at the level of the action~\cite{Reuter:2003ca,Reuter:2004nv},
(ii) at the level of the equations of motion~\cite{Bonanno:2001hi,Bonanno:2002zb,Babic:2004ev},
(iii) at the level of the metric of a classical solution~\cite{Bonanno:2000ep,Torres:2014gta,Kofinas:2015sna},
and, as more recently proposed, (iv) at the level of curvature invariants of a classical solution~\cite{Held:2021vwd}. In the cases (ii) and (iii), the RG-improvement is implemented in coordinate-dependent quantities and is thus additionally impacted by an unphysical dependence on the choice of coordinates~\cite{Held:2021vwd}.
Third, different scale identifications have been investigated. This includes a scale identification with curvature invariants, cf.~e.g.~\cite{Eichhorn:2021iwq, Held:2021vwd}, and a scale identification with geodesic distance to an asymptotic observer, cf.~e.g.~\cite{Bonanno:2000ep}.
Both essentially amount to an identification of the RG-scale with local curvature scales. As a result, classical GR remains a good approximation in regions of sub-Planckian local curvature. The present work highlights that this type of scale-identification is a physical assumption. Different physical assumptions can lead to qualitatively different results.
Insight into consistent choices of scale-identification might be gained from the Bianchi identities under the restrictive assumption of a separately conserved stress-energy tensor~\cite{Reuter:2003ca,Babic:2004ev,Domazet:2010bk}. The decoupling mechanism may provide another way of identifying the RG scale~\cite{Reuter:2003ca}.
\\
In the following, we implement the RG-improvement at the level of the action, and retain the scale-dependence of both the Newton coupling an the cosmological constant. Focusing on spherical symmetry, we investigate a scale-identification with local temperature instead of local curvature. The latter choice leads to qualitative differences in the results with regard to the existence of a horizon and therefore in particular with respect to the scale at which "QG effects" start to play a role.
Our choice of scale-identification is motivated by the field theory of a gravitating object surrounded by radiation in thermal equilibrium. The requirement of thermal equilibrium restricts the space of solutions to static spacetimes. For such spacetimes the RG-scale is identified with the local temperature of a stationary observer. The local temperature, introduced originally by Tolman and Ehrenfest~\cite{Tolman:1930etgr,Tolman:1930tes}, includes a blueshift-factor which diverges at the horizon of the classical black hole solutions.
The obtained spacetimes describe horizonless geometries which match the classical Schwarzschild spacetime extremely well down to a Planck distance away from the classical horizon. Therefore, if they provide a description of the spacetime realized in nature up to a finite energy scale, in particular up to or beyond classical horizon scales, the absence of a classical horizon structure may leave observational imprints on gravitational-wave signals or black-hole shadows.
The paper is organized as follows. In Sec.~\ref{subsec: RG-improvement idea}, we briefly review asymptotic safety as a motivation for including running couplings in the action. In Sec.~\ref{subsec: field equations}, we derive the general form of the modified field equations, whereas Sec.~\ref{sec: cutoff identification} is devoted to our choice of scale-identification. In Sec.~\ref{subsec: fixed-point analysis} and Sec.~\ref{subsec: numerical analysis}, spherically-symmetric solutions are studied analytically in the fixed-point regime and numerically at large distances, respectively. We finish with a discussion in Sec.~\ref{sec: discussion}.
\section{The modified Einstein-Hilbert theory}
\label{sec: Modified EH action}
\subsection{Running couplings in the classical action}
\label{subsec: RG-improvement idea}
This section reviews the origin and motivation behind an RG-improvement of couplings at the level of the classical action. Therefore, we introduce some of the ingredients of the asymptotic safety scenario~\cite{Weinberg:1976xy,Weinberg:1980gg,Reuter:1996cp} and the Wilsonian RG~\cite{Wilson:1971bg,Wilson:1971dh}.
The central object in the functional RG is the scale-dependent effective average action $\Gamma_k$~\cite{Wetterich:2001kra}, constructed formally from all operators which are compatible with the symmetries of the theory. In a gravitational context, these symmetries include diffeomorphism invariance and additional symmetries imposed on the matter sector. Starting from the bare classical action in the ultraviolet (UV), the effective average action at the scale $k$ is obtained by a momentum-shell integration of quantum fluctuations with momenta larger than the infrared (IR) cutoff $k$. In the limit $k\to 0$ the standard effective action is recovered. The scale-dependence of $\Gamma_k$ is governed by an exact functional RG flow equation~\cite{Wetterich:1992yh,Reuter:1996cp},
\begin{equation}\label{eq: flow equation}
k \partial_k \Gamma_k= \frac{1}{2} \text{STr}\qty[\qty(\Gamma_k ^{(2)}+\mathcal{R}_k)^{-1}k \partial_k \mathcal{R}_k],
\end{equation}
where $\Gamma_k ^{(2)}$ denotes the matrix of second functional derivatives of $\Gamma_k$ with respect to the fluctuation fields at fixed background. $\mathcal{R}_k$ is a regulator which suppresses IR modes, whereas its derivative in the flow equation results in the suppression for UV modes. $\text{STr}$ denotes a generalized functional trace. By construction, the main contribution to $\Gamma_k$ arises from momentum modes at the scale $k$. Although~\eqref{eq: flow equation} is an exact equation, finding solutions in the form of RG-trajectories requires approximation methods. For example, in a truncation of the theory space, the effective action is restricted to a finite sum of basis operators constructed from the fields and their derivatives. Local truncations of the effective action typically provide a good description for large values of $k$, whereas at low-momentum scales nonlocal terms become important and are difficult to handle~\cite{Capper:1973pv,Donoghue:1993eb}.
Here, the idea of RG-improvement comes into play. On the basis of the decoupling mechanism, a qualitative shortcut was proposed for the way from the UV to the IR~\cite{Reuter:2003ca}. The shortcut consists in including a spacetime-dependent cutoff $k=k(x)$ in the classical action, which has to be built from the properties of the classical physical system under consideration and therefore is expected to qualitatively capture the effect of some of the higher-order or nonlocal terms in the effective average action. The principle at work is that, at a certain decoupling scale, the physical parameters in $\Gamma_k$ may override the effect of the mathematical regulator appearing in the denominator of the trace in~\eqref{eq: flow equation}. As a consequence, $\Gamma_k$ at a finite scale below the cutoff scale does not deviate much from the standard effective action obtained in the limit $k\to 0$~\cite{Reuter:2003ca, Platania:2020lqb}. Identifying the decoupling scale might predict certain terms contained in the full effective action, but not in the original truncation. An example is given by the $\phi^4 \ln(\phi)$ term in the Coleman–Weinberg effective potential of massless $\phi^4$ theory~\cite{Coleman1973:rcssb}.
In light of the original motivation for RG-improvement inspired by asymptotic safety, the choice of cutoff should preserve the symmetries of the effective average action. In particular, imposing diffeomorphism invariance on the cutoff in an application of RG-improvement to gravity guarantees staying as close as possible to the theoretical framework of asymptotic safety. Nevertheless, the RG-improvement procedure does not provide a first-principle derivation from asymptotic safety even if such a symmetry requirement on $k$ is imposed. On the other hand, including characteristic information about the physical system is the key prerequisite for the interpretation and understanding of the phenomenological implications of the results obtained from RG-improvement.
Although the original motivation for RG-improvement suggests that the scale-dependence of couplings should be implemented at the level of the action, RG-improvements at the level of the classical field equations or classical solutions have been employed widely (see~\cite{Koch:2014cqa, Platania:2020lqb} for reviews). Diffeomorphism-invariant scale-identifications with local curvature invariants or measures of proper distance, but also single component entries of the stress-energy tensor, as the energy density, have been used to identify the RG-scale. More recently, it has been proposed to perform the RG-improvement at the level of curvature invariants in order to guarantee a manifestly coordinate-invariant procedure~\cite{Held:2021vwd}.
Including leading-order quantum effects via a coordinate-dependent cutoff at the level of the action was explored for example in~\cite{Babic:2004ev, Domazet:2010bk, Koch:2010nn} for cosmology and in~\cite{Contreras:2013hua,Koch:2015nva} in the context of black holes. A more general proposal is to promote the cutoff to a function of the yet unknown metric, which generally leads to modified theories of gravity with structurally different equations than those of classical Einstein gravity.
\\
In what follows, scale-dependent couplings are included at the level of the classical action with an RG scale-identification motivated by the field theory of a gravity-matter system in thermal equilibrium. The following two subsections are devoted to the derivation of the modified field equations for general and for a concrete scale-dependence of the gravitational couplings in the action.
\subsection{General form of the field equations}\label{subsec: field equations}
Our starting point is the Einstein-Hilbert action
\begin{equation}
\label{eq: modified EH action}
S = \int \dd[4]{x}\sqrt{-g}\frac{1}{16 \pi G(k)}\qty(R - 2\Lambda(k)),
\end{equation}
where the Newton coupling and cosmological constant are considered as functions of the IR cutoff scale $k$. If $k=k(x)$ is a a scalar function on spacetime, not depending on the the metric, the equations obtained from a variation of the action with respect to the metric are given by~\cite{Reuter:2003ca}
\begin{equation}
\label{eq: modified equations k metric-independent}
G_{\mu\nu} + \Lambda(k)g_{\mu\nu} = 8\pi G(k) T_{\mu\nu} + \Delta t_{\mu\nu},
\end{equation}
where $G_{\mu\nu}= R_{\mu\nu}-1/2\,R\,g_{\mu\nu}$ is the Einstein tensor and $T_{\mu\nu}$ is the energy-momentum tensor of the matter. The coordinate dependence of the Newton coupling introduces an additional effective energy-momentum tensor
\begin{equation}
\Delta t_{\mu\nu} = G(k) \Big(\nabla_\mu \nabla_\nu - g_{\mu\nu}\Box\Big){G(k)}^{-1}.
\end{equation}
In situations where the energy-momentum tensor $T_{\mu\nu}$ is separately conserved, the contracted Bianchi identities and the scalar nature of $k$ impose a self-consistency condition on the cutoff function~\cite{Reuter:2003ca,Babic:2004ev,Domazet:2010bk}. Allowed cutoffs and solutions to~\eqref{eq: modified equations k metric-dependent} have been investigated in cosmological settings in~\cite{Babic:2004ev, Domazet:2010bk, Koch:2010nn} and in the context of black holes, see~\cite{Contreras:2013hua,Koch:2015nva}. It should be noted that in modified theories of gravity the conservation of the matter energy-momentum tensor at the classical and quantum level is not guaranteed. In our case no matter in the form of an energy-momentum tensor is included in the gravitational action.
\\
Let us now come to a key difference in our setup compared to previous studies of RG-improvement at the level of the action.
In the reviewed derivation of the equations~\eqref{eq: modified equations k metric-independent}, the RG scale $k$ is regarded as independent of the metric. In contrast, when varying the action, we retain the explicit variation of the RG scale $k$ with respect to the metric.
For a general metric-dependent cutoff function $k=k(g_{\mu\nu})$ the equations for the metric field are modified in comparison to~\eqref{eq: modified equations k metric-independent} by an additional term depending on the variation of the cutoff function with respect to the metric,
\begin{equation}\label{eq: modified equations k metric-dependent}
G_{\mu\nu} + \Lambda(k)g_{\mu\nu} = \Delta t_{\mu\nu} + \frac{\delta k}{\delta g^{\mu\nu}}\qty(\frac{1}{G(k)}\partial_{k}G(k)\qty(R-2\Lambda(k))+2\partial _k \Lambda(k)).
\end{equation}
To proceed with the study of solutions to this equation, we need to specify the RG-scale dependence of $G(k)$ and $\Lambda(k)$ in Sec.~\ref{subsec: running couplings} and the scale-identification $k=k(g_{\mu\nu})$ in Sec.~\ref{sec: cutoff identification}.
\subsection{Concrete scale-dependence of the gravitational couplings}\label{subsec: running couplings}
In this section we specify the scale-dependence of the gravitational couplings from asymptotic safety. The scale-dependence of the dimensionless Newton coupling $g(k) = G(k)k^2$ and of the cosmological constant $\lambda(k)= \Lambda(k)k^{-2}$ are dictated by their beta functions which are defined as the scale-derivatives $\beta_{g} = k \partial_k g(k) $ and $\beta_{\lambda} = k \partial_k \lambda(k)$. These can be computed in a given truncation of the effective average action, using functional RG methods. At a fixed-point, corresponding to a zero of the set of beta functions, the dimensionless gravitational couplings become constants $g_*$ and $\lambda_*$. The fixed-point is called interacting or non-Gaussian if at least one of its components is non-zero. According to asymptotic safety, there exists a non-Gaussian fixed-point of the RG-flow in the UV where the theory becomes quantum scale-invariant. Provided that the fixed-point comes with a finite number of relevant directions, the theory is rendered UV-complete and predictive. Evidence for the so-called Reuter fixed-point~\cite{Reuter:1996cp,Souma:1999at} is accumulating, cf.~for example~\cite{Bonanno:2020bil} for references and critical reflections. Throughout this paper, we shall assume that $g_*$ and $\lambda_*$ are positive.
We specify to an explicit RG-scale dependence which connects the UV-scaling regime in the vicinity of the Reuter fixed point to the IR-scaling regime approaching (towards smaller $k$) the free Gaussian fixed point, cf.~\cite{Bonanno:2007wg, Gubitosi:2018gsl}. The underlying $\beta$-functions are derived in~\cite{Machado:2007ea}, see also App.~B in~\cite{Gubitosi:2018gsl}. An analytic expression which captures the two relevant scaling regimes can be derived by expanding the respective $\beta$-functions to second order in $g_k$. In terms of the dimensionful scale-dependent couplings $G(k)$ and $\Lambda(k)$, the resulting RG-scale dependence\footnote{Concretely, we refer to Eqs.~(4.3) and~(4.5) in~\cite{Gubitosi:2018gsl} with $k_0=0$ and $\Lambda(k=k_0)=0$. We set the effective dimensionful cosmological constant to zero at spatial infinity, as our analysis in Sec.~\ref{sec: spherically symmetric solutions} is focused on asymptotically flat spacetimes.} reads
\begin{align}
G(k) &= \frac{G_0}{\chi+(G_0/g_*)\;k^2}\;,
\notag\\[0.5em]
\Lambda(k) &= \frac{G_0\; (\lambda_*/g_*)\;k^4}{\chi + (G_0/g_*)\;k^2}\;.
\label{eq: scale dependence}
\end{align}
Here, we have introduced the fixed-point values $g_*$ and $\lambda_*$ and a fiducial dimensionless parameter $\chi$. The explicit values for $g_*$ and $\lambda_*$ which match \cite{Gubitosi:2018gsl} can be obtained by taking the respective limits of Eqs.~(4.3) and~(4.5) in~\cite{Gubitosi:2018gsl} and fixing $\chi=1$. For $\chi=0$, one recovers the UV scaling regime, i.e.,
\begin{align}
G(k) \sim k^{-2}\;,
\quad\quad\quad
\Lambda(k) \sim k^2.
\end{align}
For small $k^2\ll\chi$ (and $\chi=1$), the couplings scale towards the free IR fixed point, i.e.,
\begin{align}
G(k) \rightarrow G_0\;,
\quad\quad\quad
\Lambda(k) \sim k^4.
\end{align}
General Relativity with vanishing cosmological constant is recovered in the limit $\chi= 1$ and $\omega \equiv 1/g_* = 0$.
We introduce $\chi$, such that small $\chi>0$ allows us to perturb the fixed-point equations. We will make use of an expansion of the field equations around $\chi=0$ in our analysis of solutions to the field equations in the fixed-point regime, cf.~Sec.~\ref{subsec: fixed-point analysis}. In the following, in addition to $\hbar = c= 1$, the classical Newton constant is set to $G_0 = 1$, which implies that dimensionful quantities are measured in Planck units.
The $k$-dependence~\eqref{eq: scale dependence} illustrates the weakening of the dimensionful Newton coupling at high energies, whereby the transition scale to the quantum gravity regime is typically associated with the Planck scale. It was shown some time ago that such an anti-screening behaviour of the Newton coupling might lead to the resolution of classical singularities in the context of an RG-improvement~\cite{Bonanno:2000ep}.
In an RG-improvement of the classical Schwarzschild (Anti-) de Sitter solutions, however, the quadratic divergence of the cosmological constant in the UV reintroduces the curvature singularity~\cite{Koch:2013owa, Pawlowski:2018swz} at the center (see~\cite{Adeifeoba:2018ydh} for a detailed analysis of conditions for black hole singularity resolution).
\section{Scale-identification from thermal field theory}\label{sec: cutoff identification}
The IR cutoff $k$ in the definition of the effective average action $\Gamma_k$ is a mathematical parameter associated with the RG and the flow equations. Integrating out all the quantum fluctuations from the effective average action, the $k$-dependence cancels out, such that the full physics can be extracted from the field equations governed by the effective action $\Gamma_0$. In the context of RG-improvement, however, the integration is extended only down to finite $k$. Therefore, in physical applications of RG-improvement -- especially in the search of solutions to the field equations~\eqref{eq: modified equations k metric-dependent} -- the cutoff parameter must be related to a characteristic energy scale of the system. This procedure is referred to as scale-identification.
In highly symmetric frameworks, such as homogeneous and isotropic cosmology or spherically-symmetric black-hole spacetimes, the scale is typically identified with local curvature scales of the classical solutions.
In~\cite{Platania:2019kyx} an iterative RG-improvement was suggested based on a self-adjusting cutoff, which takes into account the backreaction effects due to corrections to the Einstein equations arising from a running Newton coupling.
More generally, RG-scale settings at the level of the action and equations were proposed in~\cite{Reuter:2003ca,Babic:2004ev,Domazet:2010bk,Koch:2010nn,Domazet:2012tw,Koch:2014joa} based on diffeomorphism invariance and the Bianchi identities. Alternatively, the decoupling mechanism in effective field theories may serve as a route towards a self-consistent scale-identification~\cite{Reuter:2003ca}.
\\
In what follows we introduce a metric-dependent cutoff function motivated by the Euclidean field theory of a finite-size gravity-matter system in thermal equilibrium. According to Hawking~\cite{Hawking:1975vcx}, a semi-classical black hole emits black-body radiation at a finite temperature. Putting the system consisting of the black hole and the radiation in an isolated box with finite volume, the requirement of thermal equilibrium implies that physical properties of the system cannot depend on time from the viewpoint of observers at rest with respect to the system. As a consequence, the spacetime must be stationary. Restricting to a system with a finite spatial volume $V$ in a static spacetime, the metric can be written as
\begin{equation}
\dd{s^2} = g_{00}(x^i)\dd{t^2} + h_{ij}\dd{x^i}\dd{x^j},
\end{equation}
where $g_{00}<0$ and $h_{ij}$ do not depend on the coordinate $t$. The latter is proportional to the proper time of stationary observers and can be fixed by identification with the proper time of a particular observer. Then the gravitational redshift factor between that observer and another one at $x^\mu$ is given by $\sqrt{-g_{00}(x^i)}$. The radiation is taken into account by considering a relativistic field with appropriate boundary conditions imposed at the walls of the box. In thermal equilibrium, the product of the local temperature of an infinitesimally small subsystem and the redshift factor of the gravitational field remains constant. This fact, also known as the Tolman-Ehrenfest effect~\cite{Tolman:1930tes}, expresses the dependence of the proper temperature of a local observer on the gravitational potential at the point where the measurement is made. In turn, if $T$ denotes the temperature of the black-body radiation measured by a stationary observer at infinity in the limit where $V\to \infty$, then $T/ \sqrt{-g_{00}(x^i)}$ defines the local temperature for an observer at the point $x^\mu$. On this basis a cutoff function is constructed as follows,
\begin{equation}\label{eq: cutoff identification}
k = \frac{T}{\sqrt{-g_{00}(x^i)}}.
\end{equation}
In other words, the momentum cutoff at a certain point in the confined spatial volume is identified with the local (blueshifted) value of the physical observable whose expectation value is $T$ for a stationary observer at infinity. For classical black holes, $T$ is to be identified with the characteristic temperature of the emitted black body spectrum. On the other hand, more general static geometries may not admit a straightforward interpretation of the parameter $T$. In what follows, we shall review the origin of $T$ classically and comment on its role in our situation.
For a classical black hole, the redshift factor goes to zero at the event horizon, where $g_{00} = 0$, and therefore the local temperature diverges. The cutoff as defined in~\eqref{eq: cutoff identification} therefore diverges at a finite distance away from the classical singularity. Let us note that previous studies of RG-improvement in spherical symmetry typically identify the RG cutoff-scale with local \emph{curvature} or \emph{proper distance} scales of the classical theory. Nevertheless, from the viewpoint of the thermal field theory of a static gravitational field interacting with its environment in an isolated region of space, it is the local \emph{temperature} which sets the physical energy scale.
To illustrate the origin of the parameter $T$ for classical black holes and its role in the definition of the cutoff, it is useful to revisit the central elements in the derivation of the Hawking temperature in the framework of Euclidean path integrals. Originally, Hawking's derivation of black hole radiance~\cite{Hawking:1975vcx} utilized methods from quantum field theory on curved background. The result, that a black hole radiates at finite temperature, suggests an equivalent treatment in terms of Euclidean field theory and thermal Green's functions~\cite{Gibbons:1976ue, Gibbons:1976pt, Hawking:1978jz}.
Let us consider a static spherically symmetric spacetime of the form
\begin{equation}\label{eq: metric with f(r)}
\dd{s^2} = -f(r)\dd{t^2} + g(r)^{-1}\dd{r^2} + r^2\dd{\Omega^2},
\end{equation}
where $\dd{\Omega^2}$ is the infinitesimal area element on the $S^2$. We assume that the spacetime has an event horizon at $r_h$ where $f(r_h)=g(r_h)=0$ and where the derivates of the metric functions do not vanish. A well-known example is the Schwarzschild spacetime with lapse function $f(r) = g(r) =1-2M/r$ and $r_h=2M$ which describes the exterior of a spherically symmetric object of radius $r_h$ and mass $M$ measured at infinity. The coordinate singularity at $r=r_h$ can be removed by a coordinate transformation to Eddington-Finkelstein coordinates. In the context of Euclidean path integrals for thermal systems, the positive definite Euclidean metric is defined by a complexification of the time coordinate, $t\to i\tau$. The canonical partition function for the gravitational field is consequently written as a sum over all smooth Riemannian geometries which are periodic in imaginary time, $\tau\to \tau + \beta$, with period $\beta = T^{-1}$ at infinity,
\begin{equation}
Z(\beta) = \int \dd{[g_{\mu\nu}]}e^{-S_E \qty[g_{\mu\nu}]},
\end{equation}
where $S_E$ is the Euclidean action. In particular, our scale identification~\eqref{eq: cutoff identification} retains the global diffeomorphism-invariance of the Euclidean action within the class of thermal Euclidean manifolds, i.e.~those with a timelike Killing vector and periodicity of $\tau \to \tau+\beta$. Within this class, $l= 1/k = \beta \sqrt{g_{00}(r)}$ has a covariant interpretation as the geometric circumference of the Euclidean manifold at radius $r$.
Taylor-expanding the metric functions in the near-horizon limit, i.e.~for small $(r-r_h)$, to first order the metric looks locally like Rindler space. After a coordinate transformation $(\tau, r)\to (\phi, \rho)$, where $\phi = \sqrt{\abs{f'(r_h)g'(r_h)}/4}\tau$ and $\rho^2 = 4 (r-r_h)/g'(r_h)$, the metric in a neighborhood of the horizon can be written as
\begin{equation}\label{eq: metric local horizon limit}
\dd{s^2} = \dd{\rho^2} + \rho^2\dd{\phi^2},
\end{equation}
where we have omitted the two extra dimensions corresponding to the $S^2$ in~\eqref{eq: metric with f(r)}. Smoothness of the metric~\eqref{eq: metric local horizon limit} requires the identification of $\phi$ as an angular variable with period $2\pi$ and restriction of the range of $r$ to $r\geq r_h$. The last condition arises from the fact that the Killing vector $\partial_\tau$ vanishes at $r=r_h$. Then the line element becomes that of a flat disc with radius $\rho$ and polar angle $\phi$. This in turn fixes the imaginary time period $\beta$ and thereby the temperature $T$ measured by a stationary observer at infinity,
\begin{equation}\label{eq: T}
T = \sqrt{\frac{\abs{f'(r_h)g'(r_h)}}{16 \pi^2}}.
\end{equation}
For a Schwarzschild black hole, $T = T_H = 1/8\pi M$ reproduces the Hawking temperature. The associated Euclidean manifold has topology $S^1 \cross \mathbb{R_{+}}$ and looks like the surface of a cigar with a smoothly closed end on the one side, while asymptotically approaching a cylinder at large $r$. With this picture in mind,~\eqref{eq: cutoff identification} identifies the characteristic length scale $l= 1/k$ with the radius of the cigar. In the UV limit $k\to \infty$, the point $r=r_h$ is reached. It is a fixed-point of the isometry $\tau \to \tau + \beta$ and represents the Euclidean continuation of the bifurcate Killing horizon in the Lorentzian black-hole solution~\cite{Wald:1995yp,Ross:2005sc}.
In our situation, the identification of the parameter $T$ is not straightforward. The dependence of the cutoff function~\eqref{eq: cutoff identification} on the $(0,0)$-component of the metric results in modified field equations whose solutions will in general differ from the classical GR solutions. In particular, it is a priori not clear whether these spacetimes exhibit a horizon. In a complete treatment one would aim to solve a variational problem for the metric components such that the Euclidean effective action is minimized. Nevertheless, in the context of RG-improvement introduced in Sec.~\ref{subsec: RG-improvement idea}, we expect the shortcut of introducing a local-temperature-dependent cutoff to capture qualitative features of nonlocal terms and radiative corrections contained in the full effective action for the thermal system. On these grounds, there are different ways of dealing with the parameter $T$. If the new spacetime is found to contain an event horizon with the necessary conditions for the derivation of~\eqref{eq: T}, then it would be consistent to identify $T$ with the temperature~\eqref{eq: T}. However, even in spherical symmetry, the full modified field equations can only be studied numerically. Therefore, in practice, as in other forms of RG-improvement, a classical input must be made at some stage of the procedure. The running couplings in~\eqref{eq: scale dependence} are defined such, that they approach $G=G_0\equiv1$ and $\Lambda = 0$ in the IR. Consequently, in the spherically symmetric case, the Schwarzschild solution at large distances can be used as a classical input. Then $T = T_H = 1/8\pi M$ provides a consistent identification for the numerical solution at large distances. Whereas we employ this identification in the numerical analysis of solutions in Sec.~\ref{subsec: numerical analysis}, we shall in general consider $T$ as a free parameter.
\section{Spherically symmetric solutions in different regimes}\label{sec: spherically symmetric solutions}
In the following, we limit the search for solutions to the modified Einstein-Hilbert theory discussed in Sec.~\ref{sec: Modified EH action} and Sec.~\ref{sec: cutoff identification} to static spherically symmetric spacetimes, written in the form
\begin{equation}\label{eq: metric ansatz}
\dd{s^2} = -A(r)\dd{t^2} + B(r)\dd{r^2} + r^2\dd{\Omega^2}.
\end{equation}
To derive the field equations governing the metric functions $A$ and $B$, we may follow two different routes: On the one hand, we can insert the ansatz~\eqref{eq: metric ansatz} into the general field equations~\eqref{eq: modified equations k metric-dependent}.\footnote{We note that the $(2,2)$ and $(3,3)$-components of the field equations~\eqref{eq: modified equations k metric-dependent} are redundant once $A(r)$ and $B(r)$ are governed by the $(0,0)$- and $(1,1)$-components of the field equations. Hence, there are only 2 nontrivial field equations, as for the derivation following~\eqref{eq: action variation}.} On the other hand, we can insert the ansatz~\eqref{eq: metric ansatz} into the modified Einstein-Hilbert action~\eqref{eq: modified EH action} and directly compute the variations
\begin{equation}
\label{eq: action variation}
\frac{1}{\sqrt{-g}}\frac{\delta S}{\delta A} = 0 \quad \text{and} \quad \frac{1}{\sqrt{-g}}\frac{\delta S}{\delta B} = 0.
\end{equation}
In both cases, we use the scale-dependent running couplings in~\eqref{eq: scale dependence} and the cutoff identification in~\eqref{eq: cutoff identification}. We confirm that both derivations result in the same field equations.
The cutoff function introduced in Sec.~\ref{sec: cutoff identification} depends only on the metric and not on its derivatives. Hence, the field equations remain of second order, as in GR.
For the ansatz in~\eqref{eq: metric ansatz}, the radial field equation can be solved algebraically to give $B(r)$ in terms of $A(r)$ and its derivatives, cf.~App.~\ref{app: differential equations} for details. Therefore, solutions are completely characterized by a second-order nonlinear ordinary differential equation for $A(r)$,
\begin{equation}
\label{eq: eom}
A'' + f(r,A, A')=0.
\end{equation}
The explicit form of $f(r,A,A')$ is given in App.~\ref{app: differential equations}.
\subsection{Exact solutions to the fixed-point equations}\label{subsec: fixed-point analysis}
\begin{figure}[t]
\centering
\includegraphics[width=0.55\textwidth]{img-G-L-running.pdf}
\caption{\label{fig: running couplings as function of k} Dimensionful Newton coupling (green) and cosmological constant (blue) as functions of the radial coordinate, dictated by the scale-dependence of the couplings according to~\eqref{eq: scale dependence} ($\chi = 1$) and inserting the power law solution $A \propto r^{\sqrt{3}-1}$ to the field equations in the fixed-point regime. We set $\omega = 1$, $\lambda_* = 1$ and $T = 1/8\pi M$ with $M=1$.}
\end{figure}
In this section, we shall study the differential equation for the $(0,0)$-component of the metric in the fixed-point regime, i.e.,~with the gravitational couplings running according to~\eqref{eq: scale dependence} with $\chi=0$. Setting $\chi = 0$ in the differential equation for the metric function $A$, cf.~App.~\ref{app: differential equations}, the equation becomes independent of the UV fixed-point value of the dimensionless Newton coupling. Consequently, the running of the cosmological constant is the only relevant ingredient for solutions to the fixed-point theory at high energies. In addition, the equation depends only on the parameter combination $\lambda_* T^2$, which can be traced back to the $k$-dependence of the dimensionful couplings at the fixed-point and our definition of the cutoff-function.
The momentum-cutoff $k\propto A^{-1/2}$ becomes large when the metric function $A$ is close to zero. A zero at a finite radius would correspond to a horizon. In contrast, we find that the fixed-point equations admit power-law solutions around $r=0$,
\begin{equation}\label{eq: power laws}
A(r) \propto r^\alpha.
\end{equation}
For a special choice of the proportionality constant in~\eqref{eq: power laws}, a parabolic solution is given by $A(r) = 2 \lambda_* T^2 r^2$. A trivial solution to the equations in the fixed-point regime is given by~\eqref{eq: power laws} with the power $\alpha = 0$, i.e.,~a constant solution for which the radius of the counterpart to the Euclidean cigar never shrinks to zero and therefore never reaches the fixed-point. Additionally, there are solutions with powers $\alpha = \pm \sqrt{3}-1$. The latter solution with the negative sign diverges at $r=0$ and is not appropriate to describe the UV regime $k\to \infty$. Therefore, except for the fine-tuned quadratic solution there is a distinct solution $A=a_0 r^{\alpha}$ with power-law exponent
\begin{equation}
\alpha = \sqrt{3}-1.
\end{equation}
Our numerical analysis in Sec.~\ref{subsec: numerical analysis} shows that this is the correct power law for $A$ at small $r$.
Inserting the power-law solution for $A$, together with the fixed-point scaling of $G$ and $\Lambda$, into the expression for $B$ given by App.~\eqref{eq: B rule}, leads to $B=0$. On the other hand, the numerical solutions presented in Sec.~\ref{subsec: numerical analysis} show that in a sufficiently small neighbourhood of the origin $B$ is given by a positive power of the radial coordinate. The leading non-zero term for $B$, representing an approximate solution to the fixed-point theory, can be obtained by perturbing the fixed-point equations. Concretely, we consider small $\chi >0$ corrections to the exact solution, $A(r) = a_0 r^{\sqrt{3}-1} + \chi a_1 r^{\alpha_1} $, and expand~\eqref{eq: general differential equation for A} around $\chi=0$ to lowest non-trivial order.
As a result, $B$ receives corrections proportional to $\chi^2$ with a leading $r$-dependence
\begin{equation}\label{eq: power laws B}
B(r) \propto r^{2\alpha}.
\end{equation}
The proportionality constant $b_1$ is fully fixed by $\chi^2$ times a factor depending on the proportionality constant $a_0$ in the expression for $A$, and on the free parameters $T$ and $\omega$, cf.~App~\ref{app: differential equations}.
\begin{figure}[t]
\centering
\includegraphics[width=0.6\textwidth]{img-A-analytical-numerical.pdf}
\caption{\label{fig: A numerical analytical} Numerical solution to the field equations (green) according to the interpolating scale-dependence~\eqref{eq: scale dependence} of the gravitational couplings ($\chi=1$) . For the numerical integration, Schwarzschild initial conditions are imposed at $r_i = 10M$, we set $\lambda_*=1$, $\omega = 1$ and $T = 1/8\pi M$ with $M=1$. The asymptotic solution in the fixed-point regime is given by $A\propto r^{\alpha}$ and receives $r^{2 \alpha}$-corrections in a finite neighborhood of $r=0$ (blue dashed). For comparison, the classical Schwarzschild solution is shown for $r/M>2$ (gray dashed).}
\end{figure}
The first observation is that the renormalization of the couplings in the action with a thermal scale setting function, leads to a finite metric at the origin. Fig.~\ref{fig: running couplings as function of k} shows the running Newton coupling,~\eqref{eq: scale dependence} with $\chi=1$, as a function of the radial coordinate for the critical solution. It vanishes at the origin, reflecting the anti-screening property of gravity in the UV, whereas at large distances from the origin the classical Newton constant is recovered. On the other hand, the effective cosmological constant becomes negligible in the IR, but diverges at the center.
Different from the classical Schwarzschild black hole, the fixed-point of infinite local temperature signalling the existence of a horizon in the thermal theory, is shifted towards the center. This result is non-trivial, in fact, a power law solution $A\propto (r-r_h)^\alpha$ at a horizon at finite $r=r_h$ could have been realized as well.
Put differently, we find that taking the Euclidean thermal Schwarzschild black bole as a guiding principle for the scale-identification, the outcome describes a spacetime where the fixed-point of the underlying isometry is now at the origin of the radial coordinate. In turn, the infinite temperature-blueshift is realized only in the limit of $r\to 0$, which coincides with the limit of approaching the UV fixed-point using measures such as the proper distance or local curvature of the classical spacetime. In the above sense, we are dealing with a spacetime for which various measures of energy such as the blueshift factor, the local curvature or inverse powers of the proper distance diverge at the same spatial point in the UV limit.
In the presence of a running cosmological constant, a curvature singularity is expected at the origin~\cite{Adeifeoba:2018ydh}. In fact, various curvature scalars such as the Ricci scalar, the Weyl scalar and the Kretschmann scalar diverge. As an example, the leading term of the Kretschmann scalar at small $r$ is given by
\begin{equation}\label{eq: Kretschmann}
R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}\propto r^{-4\sqrt{3}}.
\end{equation}
Compared to the scaling $R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}|_\text{Schwarzschild}\propto r^{-6}$, the classical singularity is strengthened.
To investigate the causal structure of the spacetime globally, in the following Sec.~\ref{subsec: numerical analysis}, we study numerical solutions away from the fixed-point.
\subsection{Numerical solutions at large distances}\label{subsec: numerical analysis}
\begin{figure}[t]
\centering
\includegraphics[width=.49\textwidth]{img-A-B-numerical.pdf}
\hfill
\includegraphics[width=.49\textwidth]{img-Vol-Kretschmann-Weyl-numerical.pdf}
\caption{\label{fig: numerical result} Numerical solution to the field equations obtained with scale-dependent gravitational couplings~\eqref{eq: scale dependence} ($\chi=1$) and scale-identification~\eqref{eq: cutoff identification}. Schwarzschild initial conditions are imposed at $r_i = 10M$, we set $\lambda_*=1$, $\omega = 1$ and $T = 1/8\pi M$ with $M=1$. Left: Metric function $A$ (green), $B$ (purple) and derivative $A'$ (red dashed). Right: Volume element (purple), Kretschmann scalar (green) and squared Weyl tensor (blue).}
\end{figure}
In the previous section, we derived analytical solutions to the field equations~\eqref{eq: modified equations k metric-dependent} in the fixed-point regime where $G$ and $\Lambda$ scale according to~\eqref{eq: scale dependence} with $\chi=0$. More generally, the scale-dependence of the couplings, interpolating between the UV and the IR, can be approximated via~\eqref{eq: scale dependence} with $\chi =1$. The effective cosmological constant at infinity is set to zero, allowing us to search for asymptotically flat solutions. To obtain numerical solutions away from the fixed-point, we transform to a system of two first-order differential equations and set $\chi=1$ in the field equations, cf.~App.~\ref{app: differential equations}. As was pointed out in Sec.~\ref{sec: cutoff identification}, a consistent identification and interpretation of the free parameter $T$ is not straightforward. For the numerical integration, $T$ is identified with the Hawking temperature of a Schwarzschild black hole, i.e.~$T= T_H = 1/8\pi M$. We make this choice, since the classical Schwarzschild spacetime is used to set initial conditions at large distances $r_{i}\gg r_h = 2M$. The parameters $\lambda_*$ and $\omega$, corresponding to the UV fixed-point value of the dimensionless cosmological constant and the inverse UV fixed-point value of the dimensionless Newton coupling, are of order unity in asymptotic safety. The result of the integration for an object of Planck mass is shown in Fig.~\ref{fig: numerical result}.
\begin{figure}[t]
\centering
\includegraphics[width=.49\textwidth]{img-A-numerical.pdf}
\hfill
\includegraphics[width=.49\textwidth]{img-B-numerical.pdf}
\caption{\label{fig: different masses} Metric functions $A$ (left) and $B$ (right) of for different masses. At small $r$ the solutions are given by power laws $A\propto a_0(M)r^{\alpha}$ and $B\propto b_1(M)r^{2\alpha}$ with $\alpha = \sqrt{3}-1$, $a_0(M) \propto M^{-\sqrt{3}-1}$ and $b_1(M)\propto M^{-2\sqrt{3}}$.}
\end{figure}
The function $A$ associated with the $(0,0)$-component of the metric matches the Schwarz-schild solution well down to a Planck distance away from the classical horizon, which can also be seen from Fig.~\ref{fig: A numerical analytical}. In the interior, deviations from GR are significant. In the intermediate transition region, the radial metric function $B$ and the derivative $A'$ are peaked. The vanishing second derivative $A''$ can be regarded as a phase transition between the Schwarzschild geometry and the new spacetime obtained from the modified field equations. Our numerical results show power-law behaviour at small $r$, close to the origin. In particular, they allow us to single-out the correct exponent from the allowed ones presented in Sec.~\ref{subsec: fixed-point analysis}. This power-law exponent is $\alpha=\sqrt{3}-1$ for $A$, which gives rise to $\beta = 2 \alpha$ as the asymptotic solution to the power-law behavior of $B$. The mass-dependence of the two proportionality constants in the power-law expressions can be approximated numerically by investigating solutions with different masses, as shown in~Fig.~\ref{fig: different masses}. We find $a_0(M) \propto M^{-\sqrt{3}-1}$ for the proportionality constant in the power law for $A$, and $b_1(M)\propto M^{-2\sqrt{3}}$ for the proportionality constant in the power law for $B$. With increasing mass, the transition between the classical and the interior region is sharpened. However, the metric functions remain smooth, in contrast to the classical Schwarzschild spacetime for which our choice of coordinates is singular at the horizon.
For astrophysical objects with mass of the order $M_\odot \sim 10^{38}$, deviations from a classical horizon would be tiny. Nonetheless, they could give rise to observable gravitational wave signatures in the form of late-time echoes~\cite{Abedi:2016hgu,Barcelo:2017lnx,Cardoso:2017njb}.
Different parameter values for $\lambda_*$ and $\omega$ impact the solutions as expected from~\eqref{eq: scale dependence}, i.e.~the limit $\omega = 0$ reproduces the Schwarzschild solution, whereas a finite $\omega > 0$ results in a horizonless spacetime. Deviations from the Schwarzschild geometry outside the classical horizon become larger as $\omega$ is increased. The result is similar if higher values for $\lambda_*$ are specified.
As anticipated from the analytical study in Sec.~\ref{subsec: fixed-point analysis}, the spacetimes exhibit a curvature singularity where the Kretschmann scalar scales as $\sim r^{-4\sqrt{3}}$ at the center. The curvature singularity is not hidden behind a classical event horizon as for the Schwarzschild black hole, but is instead naked and timelike. At this stage it is unclear whether such a singularity is physically relevant, keeping in mind that the nature of observables in quantum gravity in the UV is indeterminate.
\section{Discussion and future prospects}
\label{sec: discussion}
Inspired by Weinberg's asymptotic safety scenario for quantum gravity~\cite{Weinberg:1976xy,Weinberg:1980gg}, we derive modified field equations for general RG scale-dependent gravitational couplings in the Einstein-Hilbert action. In contrast to previous work, we include variations of the RG-scale $k$ with respect to the metric. Adopting the viewpoint of Euclidean field theory of an isolated system consisting of a static black hole in thermal equilibrium with Hawking radiation~\cite{Gibbons:1976ue, Gibbons:1976pt, Hawking:1978jz}, we construct a thermal scale-identification which associates $k$ with the local temperature~\cite{Tolman:1930etgr, Tolman:1930tes} as measured by a stationary observer.
We specify to an RG-scale dependence of the gravitational couplings which interpolates between the scaling regime of asymptotically safe gravity in the ultraviolet (UV), i.e., for large $k$, and the scaling regime approaching the free (Gaussian) fixed point in the infrared (IR), i.e., for small $k$. In the IR, the couplings scale like $G(k)\sim\text{const}$ and $\Lambda(k)\sim k^4$, implying that in this regime the field equations are equivalent to GR sourced by thermal radiation. Our approximation neglects a subsequent scaling-regime in the deep IR, in which the dimensionful cosmological constant freezes out to its observed value. Thereby, asymptotically flat spacetimes can be studied numerically. Assuming that at large radii the classical Schwarzschild spacetime is recovered, the free parameter in our scale-identification is identified with the Hawking temperature.
Our numerical solutions, obtained by imposing the Schwarzschild metric in the IR and integrating towards smaller distances, reproduce vacuum General Relativity in the exterior region remarkably well for large masses. Deviations of the quantum counterparts from classical black holes become significant only at a Planck distance away from the classical horizon and thereby could lead to observational signatures~\cite{Abedi:2016hgu,Barcelo:2017lnx,Cardoso:2017njb}.
\\
In the UV regime, we assume a scale-dependence of the gravitational couplings dictated by their canonical mass dimension, which is consistent with the prediction of a non-Gaussian UV fixed-point according to asymptotic safety. The fixed-point scaling of the dimensionful couplings, combined with the scale-identification via the local blueshift factor, results in modified field equations in the UV which admit analytical solutions in the form of power laws for the $(0,0)$-component of the metric. One of these power-law solutions with the power-law exponent $\alpha=\sqrt{3}-1$ is singled out by our numerical solutions that smoothly connects to a Schwarzschild exterior. As a consequence, the metric at the origin becomes scale-invariant with respect to a rescaling of the radial coordinate. An intriguing question is whether a scale-invariant black hole core can be obtained in other approaches to quantum gravity using the language of critical phenomena, e.g., in loop quantum gravity, string theory, or in a holographic set-up.
The scalar curvatures of our solutions diverge at the origin. The divergence in the Kretschmann scalar is stronger than for the Schwarzschild spacetime, cf.~Eq.~\eqref{eq: Kretschmann}. It is unclear, however, from a classical and quantum point of view, whether these types of singularities are physically relevant.
In classical theories of gravity, curvature singularities and the physically more relevant question of geodesic completeness are not equivalent, see e.g.~\cite{Bejarano:2017fgz}. Spacetimes with integrable singularities~\cite{Lukash:2011hd,Lukash:2013ts} may exist, for which the presence of a curvature singularity does not lead to the geodesic incompleteness of the spacetime.
In the context of destructive interference of singular spacetimes in the path integral as a selection-mechanism between candidates for a microscopic action~\cite{Lehners:2019ibe,Borissova:2020knn}, it was pointed out~\cite{Giacchini:2021pmr} that for many regular black hole geometries with regular Riemann invariants (i.e., invariants built solely from the Riemann tensor), some of the derivative invariants (i.e., invariants built from covariant derivatives and the Riemann tensor) can still diverge. Therefore, even at the classical level, the characterization of geometries based on only local curvature invariants is most likely insufficient to characterize the physical viability of the spacetimes.
At the quantum level, there are obstructions in defining local observables (e.g.,~\cite{Donnelly:2015hta}) related to the diffeomorphism-invariance of gravity. In particular, the divergence of the local curvature may not be meaningful in the deep UV, where the dynamics is described by quantum gravity.
The existence of a naked time-like singularity within a strong-gravity region interior to the classical horizon is reminiscent of the 2:2 hole solutions from quadratic gravity~\cite{Holdom:2002xy, Holdom:2016nek,Holdom:2019bdv}. These are characterized by the leading terms $A(r)\propto r^2$ and $B(r)\propto r^2$ in a series expansion around the origin. Quadratic curvature terms modify the behavior in the UV compared to classical GR with zero cosmological constant, whereas in the case of an RG-improved Einstein-Hilbert action a qualitatively similar effect arises as a consequence of the running of the cosmological constant. For the 2:2 solutions, thermodynamic quantities of a finite-energy wavepacket remain bounded at the origin~\cite{Holdom:2016nek,Holdom:2019bdv,Holdom:2019ouz,Ren:2019afg}. A similar outcome may be conjectured for the fate of wavepackets passing through our scale-invariant cores, which is subject to a better understanding of the dissipative properties of thermal quantum systems in the deep UV.
It should be pointed out that the qualitative similarity with solutions from quadratic gravity is in line with the original motivation for an RG-improvement of couplings. As reviewed in Sec.~\ref{subsec: RG-improvement idea}, through an RG-improvement of the classical action, it may be possible to qualitatively study the effect of higher-order and nonlocal terms in the effective action which were not taken into account in the original truncation. In our case, such terms would include operators quadratic in the curvature due to radiative corrections, and indeed qualitative similarity between our solutions to an RG-improved Einstein-Hilbert action and classical quadratic gravity solutions, is what we observe.
Extending our analysis to an RG-improved quadratic gravity action with the same choice of cutoff given by the local temperature, may reveal whether our result for the power-law exponent of the metric is only an artifact of the Einstein-Hilbert truncation of the action, or rather indicates a larger universality class. Yet another important question is the whether similar scale-invariant cores can also be found beyond spherical symmetry, e.g., for a stationary and axisymmetric spacetimes.
\\
Finally, we stress that quantum-gravity phenomenology based on an RG-improvement of couplings should not be viewed as a first-principle derivation from asymptotic safety. In fact, our results differ qualitatively from previous applications of RG-improvement, in that the deviations from GR become apparent already at the classical horizon. The relevance of quantum gravity effects at the horizon is usually debated in relation to the information loss problem~\cite{Mathur:2005zp,Almheiri:2012rt}.
We attribute this qualitative difference to the different physical assumptions underlying the scale-identification of the RG-scale $k$ with an energy scale in the classical theory: RG improvement in the literature, cf.~\cite{Bonanno:2000ep} and~\cite{Platania:2019kyx} for a recent review, is based on a scale identification tied to local curvature.
For astrophysically relevant black holes, the curvature at the event horizon is small compared to the Planck scale. As a result, modifications to astrophysically relevant black holes have been found to be small.
In contrast, the RG improvement in this paper is based on a scale-identification tied to local temperature. For any black hole, the local blueshifted temperature diverges at the horizon. As a result, modifications become significant at a Planck distance to the classical event horizon and result in a horizonless compact object.
We view this as a sign that the qualitative results of RG-improvement hinge on the physical assumptions made in the respective scale-identification. Differences compared to an RG-improvement with local curvature do not arise from considering only a subgroup of the full diffeomorphism group. Such a restriction is not strictly necessary. Qualitatively similar results, showing modifications at the classical horizon, can presumably be obtained from an RG-improvement at the level of the Schwarzschild solution with a horizon-detecting metric invariant given by the square of the covariant derivative of the Riemann tensor~\cite{Karlhede:1982fj}.
We conclude by emphasizing that, in a consistent treatment of asymptotic safety, physical quantum effects have to be calculated from the effective action, where all fluctuations have been integrated out.
\acknowledgments
We thank B.~Holdom and A.~Platania for invaluable discussions and comments. We also thank B.~Knorr for comments on the manuscript. A.~Held would like to thank the Perimeter Institute for Theoretical Physics for hospitality during the initial phase of this project.
The work leading to this publication was supported by the PRIME programme of the German Academic Exchange Service (DAAD) with funds from the German Federal Ministry of Education and Research (BMBF), by the Natural Sciences and Engineering Research Council of Canada, and by Perimeter Institute for Theoretical Physics. Research at Perimeter Institute is supported in part by the Government of Canada through the Department of Innovation, Science and Economic Development and by the Province of Ontario through the Ministry of Colleges and Universities.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 8,993 |
\section{Introduction}
The term boulder refers to particle sizes larger than $0.256 m$ \citep[][and
references therein]{Krumbein1937}. They can be found along many of the oceans'
coastlines. Boulders are thought to be good candidate deposits to improve
coastal hazard assessments because only coastal hazards, such as tsunami and
storms, carry enough energy to move these large particles. The problem, however,
is that tsunamis and storms are competing causative processes for boulder
transport on many coastlines, and that separating boulders moved during storm
from those moved by tsunami waves is important to avoid skewing the storm or
tsunami history along coastlines where both events can occur. Several
simplified methods
\citep[i.e,][]{not03,benetal10,bucetal11,nanetal11a} have been
put forward to calculate the wave amplitude of a "typical" storm or tsunami wave
needed to move a boulder of certain mass. What is a typical tsunami or storm
wave? It is impossible to answer this question quantitatively because the
characteristics of tsunami and storm waves vary greatly and are not only
controlled by the generation mechanism, but also by a complex interplay of water
depth and wave-wave interactions as the waves approach the shore, a process also
know as shoaling.
In order to take the temporal dimension of the interaction between a boulder and
a wave into account, \citet{weidip15} introduced the concept of the critical
angle of dislodgement that a boulder has to reach as it interacts with a storm
or tsunami wave. If the boulder does not reach or exceed the critical angle of
dislodgement, the boulder will not dislodge. In that case, \citet{weidip15}
argue that it is impossible to tell of the boulder moved. However, if the
boulder interacts with the wave long enough and the boulder reaches and exceeds
the critical angle of dislodgement, the boulder will dislodge and it can be
recognized in the field that the boulder moved. \citet{weidip15} related the
time it takes for the boulder to reach the critical position for dislodgement to
the half period of a monochromatic wave. The results of this study indicate that
the amplitudes of storm and tsunami waves are similar enough so that the
uncertainties involved in measuring the boulder mass and determining the
environmental parameters, such as slope and roughness in front the boulder, are
large enough to make it difficult if not impossible to distinguish between
boulders moved by tsunami or during storms where both causative processes are
agents of coastal change.
As mentioned earlier, the wave characteristics of storm and tsunamis wave are
also governed by water depth and other wave-related processes. In the past,
monochromatic wave were assumed to represent storm and tsunami waves reasonably
well. We argue that monochromatic waves are not a good model for storms and
tsunami waves when it comes to boulder transport. This is not only because
tsunami and storm wave have different frequencies, but also because they do not
exist using a full nonlinear system (for more details see below in sections
\ref{introrandom} and \ref{triads}) necessary to describe waves in the
nearshore area even in a simplified context. The closest approximation to
monochromatic waves is the so-called ``narrow spectrum'' that results into a
wave shape similar to Stokes waves. However, even this narrow spectrum will
undergo changes as the waves approach the shore.
For boulder transport in tsunamis, it should be acknowledged that a coupling of
boulder transport and dislodgement models with tsunami propagation and
inundation models has partly addressed the issues related to wave shoaling. For
more details about these models, we refer to \citet{nanetal11b}. Very little
work has been presented for boulder transport in storms. Most notably,
\citet{kenetal16} is one of the few if not the only scientific study that
considers boulder transport by shoaling storm waves.
The more advanced work for tsunami by \citet{nanetal11b} has benefited from
simple, yet ground-breaking work by \cite{not03}. Similar basic work does yet
exist for boulders moved by storm waves. With this contribution, we seek to
establish a basic understanding of boulders interacting with storm waves in the
nearshore area. For this endeavor, we couple the TRIADS model by \citet[][ and
references therein]{Sheremet2016TRIADS:Spectra} with boulder dislodgement model
by \citet[ hereafter referred to as \textbf{BoDiMo} for \textbf{Bo}ulder
\textbf{Di}slodgement \textbf{Mo}del]{weidip15}. Due to the characteristics of
the TRIADS model (see sections \ref{triads} and \ref{coupling}), the coupling
between TRIADS and BoDiMo constitutes an important step toward a new paradigm
for the use of deposits in hazard assessments integrated stochastic processes
provide a mathematically consistent framework.
\section{Theoretical Background}
\subsection{Waves as Random Processes}
\label{introrandom}
Ocean waves are a weakly nonlinear. Although the governing equations
are nonlinear, the non-linearity is small and the system is linear
in the leading order approximation. Therefore, in the leading order
the general solution can be represented as superposition of ``elementary"
solutions. This is the basic idea behind the Fourier representation. The
elementary solutions are sinusoids, or more general, complex exponentials
$e^{i\left[k(\omega_j)x-\omega_j t\right]}$. For example, in the
one-dimensional case, one formally writes the free surface elevation $\eta$ as
\begin{equation}
\eta(x,t)=\sum_{j=1} a_j(x,\omega)e^{i\left[k(\omega_j)x-\omega_j t\right]}.
\label{eq: wave}
\end{equation}
where the summation is carried over all angular frequencies $\omega_j$, and
$k(\omega_j)$ is the wave number, related to the frequency through the dispersion
relation. Equation \ref{eq: wave} is usually referred to as the Fourier
decomposition. Under certain quite general conditions this representation is
unique (in other words, the elementary functions provide a basis for the linear
solution space). The ``elementary" functions are also called modes, and are
identified by their angular frequency $\omega_j$. The coefficients $a_j$ are
complex, with the modulus $\left|a\right|$ proportional to the amplitude, and
$\theta=\arg a(\omega)$ the initial phase of mode $\omega$. In equation \ref{eq:
wave}, the summation should be regarded as a symbolic operation; for example,
for a continuum of modes, the sum should be replaced by integration.
Ocean waves are often described as random. This means that two wave measurements
$\eta_{1,2}(x,t)$ are not identical even if they represent what one would
describe intuitively the ``same ocean state" (for example two 10-min
measurements taken 20 min apart during a storm). Such measurements are usually
regarded as ``realizations" of the ``same ocean state". The fact that
$\eta_1\neq\eta _2$ implies that they have distinct sets of Fourier coefficients
coefficients, say $a_j(\omega)$. If the identity of the ``ocean state" is
defined by the set of all its realizations, it follows it is also completely
defined by the ensemble of all sets of Fourier coefficients of these
realizations. It can be shown that most of the statistical properties of
engineering interest that describe a given ``ocean state'' can be represented by
realizations that have the identical amplitudes $\left|a(\omega)\right|$, and
modal phases uniformly distributed in the interval $[0,\pi]$.
The Fourier representation \ref{eq: wave}, however, is not a solution of the
full nonlinear governing equations for waves. Because the system is
\emph{weakly} nonlinear, one can still use a Fourier representation, but in this
case the amplitudes cannot be constant, and therefore have to evolve in time.
Indeed, because the Fourier modes are solutions of the linear equation, the
Fourier decomposition \ref{eq: wave} yields a system of equations that describes
the evolution of modal amplitudes $a(\omega)$ through mutual (wave-wave)
interactions.
Wave-wave interactions have two important effects: 1) they transfer of energy
between Fourier modes, for example exciting modes that whose amplitude was
negligible initially; and 2) they generate weak correlations between modal
phases, which result in the deformation of the wave shape. These effects are
dominant in shoaling waves. For example, energy transfer toward low frequencies
excite infragravity waves, negligible in deep water but reaching heights of the
order of 0.5 m in the nearshore. Transfers of energy toward higher frequencies,
accompanied by strong phase correlations, play an important role in the wave
peaking and breaking process.
\subsection{The TRIADS Wave Model}
\label{triads}
The nonlinear shoaling evolution of waves in the nearshore area is simulated
using a uni-directional version of the TRIADS model
\citep{Davis14,Sheremet2016TRIADS:Spectra}, which integrates the directional,
hyperbolic equations describing the evolution directional triads proposed by
\citet{Agnon1997StochasticSpectra}. The formulation assumes the beach to be
cylindrical (laterally uniform) and mildly sloping in the cross shore direction
($h(x)$ with $x$ as the cross-shore direction). Waves are assumed to propagate
perpendicular to the shoreline. The free surface elevation $\eta(x,t)$ is
represented as a superposition Fourier modes (compare to Eq. \ref{eq: wave})
\begin{equation}
\eta(x,t) =\sum_{j=1}^{N}a_{j}(x,t)\
\exp\left[\theta_{j}(x,t)-\omega_{j}t\right]
\label{eq:FTeta}
\end{equation}
with complex amplitudes $a_{j}$ and phases $\theta_{j}(x,t)$.
Here, $N$ is the total number of Fourier modes, with a mode uniquely
defined by its radian frequency $\omega_{j}$ satisfying the linear
dispersion relation
\begin{equation}
\omega_{j}^{2}=gk_{j}\tanh k_{j}h;\quad
k_{j}=\frac{d\theta_{j}}{dx}.\label{eq: disp}
\end{equation}
Because we assume that the beach slopes mildly, the wave number $k_j$ varies
with the position at much lower rate that the phase. The evolution of the
amplitude $a_{j}$ is governed by the equation
\begin{eqnarray}
\frac{db_{j}}{dx} & = &
-i\sum_{p,q=1}^{N}W_{j,p,q}b_{p}b_{q}e^{-i\Delta_{j,p,q}\theta}\delta\left
(\Delta_{j,p,q}\omega\right)\nonumber
\\
& &
+2i\sum_{p,q=1}^{N}W_{j,-p,q}b_{-p}b_{q}e^{-i\Delta_{j,p,-q}\theta}\delta\left
(\Delta_{j,p,-q}\omega\right),
\label{eq:TRIADS}
\end{eqnarray}
where $b_{j}=a_{j}c_{j}^{\nicefrac{1}{2}}$, with $c_{J}$ the cross-shore
component of the modal group velocity, and $\Delta_{j,p,\pm
q}\xi=\xi_{j}-\xi_{p}\mp\xi_{q}$, with $\delta$ the Kronecker delta. The
interaction coefficient $W_{j,\pm p,q}$ depends on the frequencies and the
linear wave numbers (Eq. \ref{eq:TRIADS}) of the interacting modes $j$, $p$,
and $q$.
The model was run for plane beaches $h(x)=sx$, where $s$ denotes the constant
slope. Model wave significant wave height at the offshore boundary of the model
were specified using a JONSWAP spectrum \citep{hasetal73}. Assuming the
offshore boundary is far enough from the shoaling zone to allow for a linear
process representation, the complex modal amplitudes at the offshore boundary
can be written as \[
a_{j}^{\infty}=\sqrt{S_{j}\frac{\Delta\omega}{\pi}}\exp i\phi_{j},
\]
where $S_{j}=S\left(\omega_{j}\right)$ is the JONSWAP spectrum discretized
at frequencies $\omega_{j}$, and $0\le\phi_{j}\le2\pi$ are uniformly
distributed random initial phases. For a single set of initial phases $\left\{
\phi_{j}\right\} _{j=1,N}$, the numerical solution of equation
\eqref{eq:TRIADS} with boundary conditions $a_{j}^{\infty}$ corresponds to a
single realization of the shoaling of the JONSWAP spectrum. The wave spectrum is
retrieved from TRIADS simulations as function of water depth $h$:
\begin{equation}
S_{j}(h)=\frac{\pi}{\Delta\omega}\left\langle
\left|a_{j}(h)\right|^{2}\right\rangle ,\label{eq: S(h)}
\end{equation}
where the angular brackets denote the ensemble average. In this study, we
average over 100 realizations, i.e., over 100 simulations using different sets
of initial phases.
\subsection{Boulder Dislodgement Model}
The boulder dislodgement model is based on \citet{weidip15},
which employs the adapted version of the Newton's Second Law of Motion:
\begin{equation}
\label{EOMfinal}
\begin{split}
r\left({\frac{7}{5}}\rho_s + C_m \rho_f\right) V \theta_{tt}=& D
\sin{(\theta - \alpha)}
+[L+B]\cos{(\theta - \alpha)} \\
&- W \cos{(\theta)}
\end{split}
\end{equation}
in which $\rho_s$ and $\rho_f$ are the boulder and fluid densities, $D$,
$L$, $B$, and $W$ represent the drag and lift forces, the buoyancy, and
weight of the boulder. Parameter $\alpha$ denotes the slope on which the
boulder in questions is situated. The angle $\theta$ is the result
of the simplification of Newton's Second Law of Motion, which is based on
the assumption that the boulders are spherical and therefore has to rotate
out of its stable pocket. If the angle $\theta$ exceeds a critical angle,
the boulder dislodges. This critical angle of dislodgement, $\theta_c$ is
a function the slope angle $\alpha$ and the roughness elements in front
of the boulder. The governing equation, Eq.
\ref{EOMfinal}, is solved numerically employing an Adaptive Runge-Kutta
method \citep{caca90} with embedded integration formulas for the forth and
fifth-order terms \citep{feh69}. In order to unsure efficient and accurate
computations, the Python library {\tt odespy} by
\citet{Langtangen2013ThePackage} is utilized.
This model constitutes a significant improvement over previous models, because
it not only takes the magnitude of the forces into account but also their
duration. The duration is important because the amount of the time the sum of
the forces is larger than zero, which is the threshold of motion and the basis
criterion of previous models, might not be large enough for the boulder to reach
the critical angle of dislodgement. In that case, the boulder will move back
into its original position as soon as the resisting forces dominate the sum of
the forces, resulting in $\Sigma F <0$. \citet{weidip15} employed this model to
distinguish moved by tsunami and storm waves because, with out loss of
generality, the magnitude of the lift and drag forces are related to the wave
amplitude, but the duration is linked to their period (storm waves have periods
that are at least two orders of magnitude smaller than the period of tsunamis).
\subsection{Coupling between TRIADS and Boulder Dislodgement Model}
\label{coupling}
Because the drag and lift forces can be computed by their classic quadratic
dependency of the horizontal velocity, the coupling the wave and
boulder-dislodgement models reduces to a simple calculation of the
horizontal velocity associated with the nonlinear wave process described by
TRIADS:
\begin{equation}
u(x,z,t)=\sum_{J}^N\frac{gk_{j}}{\omega_{j}}a_{j}(x,t)\frac{\cosh
k_{j}(z+h)}{\sinh k_{j}h}\exp
i\left[\theta_{j}(x,t)-\omega_{j}t\right]\label{velo}
\end{equation}
where $z$ is the height above the bed where the velocity is calculated (top
of the boulder). Note that Eq. \eqref{velo} represents one realization of
the stochastic process of wave transformation in the nearshore; in this
study, one hundred different realizations were computed for each input
spectrum.
\subsection{Frequency of Boulder Dislodgement}
For the same geometric setup and initial spectrum, it can be expected that
not every realization will cause boulder dislodgement. In order to be able to
quantify how many of the realizations for the same geometric setup and
initial spectrum do, we introduce the frequency of dislodgement,
$D=N_D[N]^{-1}$, where $N$ is the total number of realizations ($N=100$),
and $N_D$ is the number of realizations for which boulder dislodgement
occurred.
\subsection{Parameter Study}
The parameter study comprises a total of about 5.6$\times
\textrm{10}^{\textrm{6}}$ runs of the coupled model, for three different
slopes, 16 different different initial wave characteristics (16 different
input spectra), 100 realizations using random relative phases, 20 different
roughness elements in front of the boulder, and 61 different boulder
masses.
\section{Results}
\subsection{The Rise of Infragravity Energy}
The nonlinear processes represented in the TRIADS model, specifically, the
second term in the right-hand side of Eq. \ref{eq:TRIADS}, transfer
energy from the peak of the frequency spectrum toward low frequencies, in
the range of 0.005 to 0.05 Hz. Waves in this frequency range, are called
"infragravity" waves, and are only produced during the shoaling process. For
discussion about the nonlinear shoaling process, we refer to
\citet{Herbers1994Infragravity-FrequencyWaves},
\citet{Herbers1995Infragravity-FrequencyWaves}, and
\citet{Sheremet2002ObservationsComponents}. Figure \ref{fig:spectra}a-c shows
the shoaling transformation over a 0.01 slope of a JONSWAP spectrum ($T_p= 8s$, $H_s=2m$).
\begin{figure}[!ht]
\centering
\noindent
\includegraphics[width=0.9\textwidth]{FigRiseInfraEngergy.png}
\caption{Spectra for in 10 m (a), 15 m (b), and 5 m (c) water depth. From 20 m
water depth (a) to 5 m water depth (c), the spectral density increases an
order of magnitude in the infragravity frequency band, $f_i$. The blue line in
(a) refers to input spectrum at the offshore boundary of the computational
domain on which the realizations are based. The red line in (a), (b) and (c)
represents the average over the hundred realizations, while the grey area
defines the envelope.}
\label{fig:spectra}
\end{figure}
The maximum spectral density in the infragravity band increases about an
order of magnitude as the waves travel from deeper into shallower water. In
this particular example, the ratio of infragravity energy to the total energy
\begin{equation}
\widetilde{E}(h)=\dfrac{\sum_{f_{j}<0.05}S_{j}(h)}{\sum_{j}S_{j}(h)}\label{eq:
IG content}
\end{equation}
(where $S_{j}$ is given by Eq. \ref{eq: S(h)}) increases from
$\widetilde{E}(20\ \text{m})=5.6\times10^{-4}$ to $\widetilde{E}(15\
\text{m})=2.3\times10^{-3}$, and $\widetilde{E}(5\
\text{m})=2.6\times10^{-2}$; the relative spectral content of infragravity
energy increases approximately 200 times from 20 m to 5 m water depth.
Figure \ref{fig:enfra} shows TRIADS simulations of the shoaling evolution
of the infragravity energy content $\widetilde{E}$ as a function of water depth
for all wave conditions and slopes examined. In general, the energy content
increases with increasing significant wave heights and increasing peak periods
at all water depths. The increase of the infragravity energy content is
stronger for smaller slopes, due to the increased spatial scale over which
nonlinear interaction is active.
\begin{figure}[!ht]
\centering
\noindent
\includegraphics[width=0.9\textwidth]{EnergyRatioSlopes1.png}
\caption{Energy ratio, $\widetilde{E}$, as a function of water depth. The
lines in each subplot represent the different slope, and the subplots represent
different wave conditions. }
\label{fig:enfra}
\end{figure}
Note that estimates of the infragravity energy content are based on spectral
quantities (i.e., ensemble-averaged values, Eq. \ref{eq: S(h)}, red line in
Fig. \ref{fig:spectra}). While the increase in the mean infragravity energy
content for $s=0.1$ is the smallest in our tests, it is possible that a small
number of realizations will exceed the mean increase corresponding to smallest
slope ($s=0.01$). Because individual realizations can exhibit significant
deviations from the mean, a significant number of realizations can cause
situations at which a boulder can be dislodged, while mean conditions will not
or vice versa. Therefore, it is necessary to consider individual realizations
to calculate the time series of the velocity that governs the dislodgement of
boulders.
\subsection{Boulder Dislodgement}
In order to find the realizations that for a given wave condition and slope are
able to dislodge boulders, time series of the horizontal velocity need the
calculated from the individual spectra. Figure \ref{fig:uts} depicts time series
of the horizontal velocity calculated with Eq. \ref{velo} for three of the one
hundred realizations. From the longer times in Fig. \ref{fig:uts}a-c, we can
see that the waves generally experience an increase from deep to shallower
water. Aside from the increase in significant wave height, we can also see that
the time series in deeper water has fewer spikes that are much larger than the
majority of wave crests. The number of these outliers increases as well from
deep to shallower water.
\begin{figure}[!ht]
\centering
\noindent\includegraphics[width=0.9\textwidth]{WaveveloReali.png}
\caption{Time series of the horizontal velocity inverted from the spectra for
the three randomly chosen realization for the $600 s$ (a-c) and from 50 to 100
s
(d-f). }
\label{fig:uts}
\end{figure}
The actual wave forms are shown in Fig. \ref{fig:uts}d-f in time series that
only cover 50 seconds instead of 600 seconds (Fig. \ref{fig:uts}a-c). In all
three plots, the superposition of different frequency components leads to
complicated velocity time series. We can discern an increase in significant wave
height from deeper to shallower water, but what can also be recognized is the
increasing asymmetry between the wave crest and trough, which is an effect of
the shoaling process. It is also important to note that the qualitative
difference between the individual time series increases significantly from deep
to shallower water depth. Therefore, a larger variability in the boulder
dislodgement frequency can be expected in shallower water. This is a direct
result of nonlinear processes acting on the wave during the shoaling process.
Figure \ref{fig:exee} shows the dislodgement frequency $D$ as a function of
boulder mass for a peak period of 16 seconds, 6 meters in significant wave
height, and a roughness of 0.5 of the boulder radius. As expected, we can see
that for smaller masses the number of realizations that are able to dislodge
boulders is larger than for bigger masses. For example, a dislodgement frequency
of larger than 95 is occurs for masses smaller than about $m_1=1.4$ tons; for $D
= 75$, the mass is $2.7 t$; for $D=50$, the mass is about $4.6 t$; and for $D =
25$, the mass is $8.1 t$.
\begin{figure}[!ht]
\centering
\noindent
\includegraphics[width=0.7\textwidth]{dislodgmentexeedance.png}
\caption{Frequency of dislodgement, $D$ as a function of boulder mass for a
peak wave period of $14 s$, a significant wave height of $6 m$, and roughness
of 0.5 the boulder radius. }
\label{fig:exee}
\end{figure}
Figure \ref{fig:probmap} depicts the frequency of dislodgement for
significant wave height, peak periods, slopes, a range of masses and
roughnesses. The roughness in all subplots varies from 0.1 to 1.0, and mass
varies from about $1 kg$ to about $ 40 t$. The different panes in the
subplots, marked with $\alpha_1$, $\alpha_2$ and $\alpha_3$, represent the
slopes $\alpha_1 = 0.01$, $\alpha_2 =0.05$, and $\alpha_3 = 0.1$. The
different rows indicate an increase of the significant wave height from $2
m$ to $8 m$, and the wave peak period increases from $8 s$ to $16 s$ in the
different columns. Employing a $\delta=0.5$ to look at the data, we see
that only the steepest slope ($\alpha_3$) for the condition $H_s=2m$,
$T_p=8s$ is able to have a frequency of dislodgement that is larger than
$D=50$. For a significant wave height of $H_s = 4m$, the mass at which
$D=50$ (assuming $\delta=0.5$) increases from about $4 kg$ for $T_p=8s$ to
about $100 kg$ for a peak period of $16 s$ independent of the slope. For
larger significant amplitudes ($H_s=6m$ and $H_s=8m$), differences for the
different slopes are significant. For example for $H_s=8m$ and $T_p=16s$,
the mass for $D=50$ and $\alpha_1$ is about $105 kg$, for $\alpha_2$ the
mass is about $900 kg$, and for $\alpha_3$ the mass is about $2000 kg$.
\begin{figure}
\centering
\noindent\includegraphics[width=\textwidth]{ProbMapAll.png}
\caption{Dislodgement frequency, $D$ as a function of boulder
mass for all wave conditions, roughnesses, and slopes.}
\label{fig:probmap}
\end{figure}
It is not only important to determine at which masses certain dislodgement
frequencies $D$ occur, but also over which mass range an increase from low to
high values of $D$ takes place. It should be noted that for the different wave
conditions this mass range over which the transition from low to high
values of $D$ occurs will take place in the single digit kilogram values to
several tons. To eliminate the bias introduced by the wide range of order
of magnitude, we define the log-scale difference $\xi$ with $\xi =
\log_{10}\left(m_{(\textrm{low} D)}[m_{(\textrm{high} D)}]^{-1}\right)$ in
which $ m_{(\textrm{low} D)}$ represents the mass with low and
$m_{(\textrm{high} D)}$ denotes the mass for a high value of $D$. An
example is shown in Fig. \ref{fig:exee} in which the log-scale difference
between $D = 5$ and $D = 95$ is calculated to be $\xi = 1.47$. Tab
\ref{tab:1} contains the log-scale differences for different wave
conditions. It is interesting to note that the log-scale difference more
than doubles for the different slope angles for larger significant wave
heights and longer peak periods and remains more or less constant for small
waves and shorter peak periods.
\begin{table}
\caption{Selected wave conditions and their respective log-scale differences,
$\xi$, for the
different slopes $\alpha_1$, $\alpha_2$, and $\alpha_3$} \centering
\begin{tabular}{ llc }
\toprule
Wave Condition & Slope & $\xi$ \\ \midrule
\multirow{3}{*}{6: $H_s=4m$,\,$T_s=8s$} & $\alpha_1$ & 1.27 \\
\cmidrule(r){2-3}
& $\alpha_2$ & 1.27 \\ \cmidrule(r){2-3}
& $\alpha_3$ & 1.27 \\ \midrule
\multirow{3}{*}{9: $H_s=4m$,\,$T_s=14s$} & $\alpha_1$ & 0.88 \\
\cmidrule(r){2-3}
& $\alpha_2$ & 0.98 \\ \cmidrule(r){2-3}
& $\alpha_3$ & 0.88 \\ \midrule
\multirow{3}{*}{12: $H_s=6m,\,T_s=12s$} & $\alpha_1$ & 0.98 \\
\cmidrule(r){2-3}
& $\alpha_2$ & 0.98 \\ \cmidrule(r){2-3}
& $\alpha_3$ & 1.08 \\ \midrule
\multirow{3}{*}{14: $H_s=6m,\,T_s=16s$} & $\alpha_1$ & 0.88 \\
\cmidrule(r){2-3}
& $\alpha_2$ & 1.08 \\ \cmidrule(r){2-3}
& $\alpha_3$ & 1.47\\ \midrule
\multirow{3}{*}{16: $H_s=8m,\,T_s=16s$} & $\alpha_1$ & 0.78 \\
\cmidrule(r){2-3}
& $\alpha_2$ & 1.57 \\ \cmidrule(r){2-3}
& $\alpha_3$ & 1.67\\
\bottomrule
\end{tabular}
\label{tab:1}
\end{table}
\section{Discussion}
As waves propagate from deeper into shallower water, wave-wave interaction
transfers energy toward lower and higher frequencies of the spectrum. The latter
causes a modification of the wave shape, for example, by increasing the skewness
and asymmetry of waves in shallower water (Fig. \ref{fig:uts}). Transferring
wave energy into higher frequencies results into the generation of infragravity
waves (Fig. \ref{fig:spectra}). While for all simulated wave conditions and
slopes, the increase in infragravity wave energy in shallower water is apparent,
the smallest slope exhibits the most significant increase (Fig.
\ref{fig:enfra}). This observation can be ascribed to fact that a milder slope
allows the waves to nonlinearly interact with each for longer and over a farther
distances. The generation of infragravity waves has profound consequences for
the individual realizations of the velocity time series needed in the boulder
dislodgement model (Fig. \ref{fig:uts}). As to whether a specific realization
can dislodge a boulder of certain mass depends on the specific wave-wave
interactions that developed within the time history of the wave propagation.
This fact results in the observation that from the same initial wave
characteristics one realization is and another realization is not capable of
dislodging a boulder of certain mass. How many realizations of a certain
initial wave characteristics are able to dislodge a boulder are collected in the
frequency of dislodgement. Figures \ref{fig:exee} and \ref{fig:probmap} show the
frequency of dislodgement depends on the magnitude of the initial wave
characteristics, mass and slope. Obviously, larger waves can dislodge heavier
boulders, but it also seems that a smaller slopes cause for heavier boulders to
be dislodged more easily that on steeper slopes, which seems to be linked to the
aforementioned more significant rise in infragravity energy. Another interesting
observation is that the log-scale difference between high and low number of the
frequency of dislodgement shows significant diversity for larger initial waves
and seems to be much larger for smaller slopes. A simple analysis of the data
presented in Fig. \ref{fig:probmap} reveals that the significant wave height and
offshore peak period are proportional to boulder mass and slope angle in a
nonlinear fashion, namely $H_s \propto (m^{2/3}, \alpha^{-2})$ and $T_p \propto
(m^{1/3}, \alpha^{-1})$. This is an interesting results because it goes beyond
of what the methods proposed by \citet{not03}, \citet{benetal10}, and
\citet{nanetal11a} are able to predict. More simulations with more offshore wave
conditions are needed to establish are robust analysis on how the significant
wave height and peak period are related to the roughness in front of the
boulder. However, it can be expected that the relationship between significant
wave height and peak period, and roughness is nonlinear as well.
\section{Conclusion}
In this contribution, we coupled the model TRIADS
\citep[][and references therein]{Sheremet2016TRIADS:Spectra} with the
boulder-dislodgement model from \citet{weidip15}. Because TRIADS is a nonlinear
wave model, it allows the transfer of wave energy across frequencies, which is
an important feature observed in coastal waves and was not considered in
previously published models of boulder dislodgement during storms. Furthermore,
TRIADS describes the evolution of directional triads \citep[as proposed
by][]{Agnon1997StochasticSpectra} based on one hundred different initial phases
of the same initial spectrum, making it possible to move from a simple framework
in which one particular wave is responsible for the dislodgement of one
particular boulder mass toward a ensemble approach that reflects the physical
and mathematical complexities more realistically. While this stochastic
framework is not fully developed in this contribution, we argue that the
definition of the frequency of dislodgement is a pivotal intermediate step.
The results of our parameter study match intuitively and quantitatively well
with previously published models. Our results also highlight the importance of
the environmental parameters, such as slope on which the boulder is resting and
the roughness elements in the direction of dislodgement, as long with the boulder
mass and characteristics of the waves. For more details on the influence of
roughness and slope, see \citet{not03} and \citet{weidip15}. The environmental
parameters are difficult, if not impossible, to observe in the field, but we
think that the frequency of dislodgement (and later the stochastic framework)
will help to, at least, qualitatively assess the uncertainty arising from this
shortcoming. Based on the wealth of information contained in Fig.
\ref{fig:probmap}, we argue that it is possible and necessary to derive a new
boulder dislodgement equation that not only includes boulder mass, roughness in
front of the boulder and slope angle, but also frequency of dislodgement.
Inverting both components of the wave characteristics is not trivial because
$H_s$ and $T_p$ are both unknowns and there are nonlinear relationships to
boulder mass and slope.
In summary, the theoretical consequences of our approach, i.e., the dislodgement
frequency and considering waves as a random process, allow us to extend our
thinking framework considerably toward a more realistic situation in which the
wave spectrum changes its shape depending on water depth and wave-wave
interaction and boulder dislodgement is governed not only by the amplitude of the
passing waves, but also how long sum of the forces is larger than zero. Through
our simulations, it becomes evident that a nonlinear treatment of the waves is
pivotal because the nonlinear deformation of the wave shape can generate forces
that can be both significantly stronger or weaker and act longer or shorter than
those generated by a linear wave with same spectral density distribution. Once
there is more information on how the peak period and significant wave height are
impacted by the roughness in front of boulder, the ways is paved to derive a new
formula for boulder dislodgement based on the frequency of dislodgement. However,
no matter the form this new formula will have, the nonlinear relationships
between the inverted values of offshore significant wave height and peak period,
and variables, such as mass, roughness and slope, the collected data in the
field, which are the basis for the inversion, need to be known much more
accurately. This is difficult to achieve, introducing, therefore, unwelcome
uncertainty. Yet such inversions are extremely important to estimate the hazard
coming from storms to improve mitigation efforts. We argue, that a stochastic
framework should be able to address the increased uncertainty. In the end, it
remains to be seen if a stochastic approach can truly achieve this. Our results,
however, indicate that a stochastic approach will be successful.
\newpage
\section*{References}
| {
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The folks at the DoubleTree by Hilton Boston – Downtown just finished renovating their hotel and are presenting a contest to celebrate the occasion. As contests go, it's pretty simple.
Called "The Ultimate Downtown Boston Experience Giveaway," users can enter by going to the hotel's Facebook page, "liking" it, citing their favorite Boston attraction, and explaining why. The winner will receive an overnight stay for two and breakfast at the hotel along with two, 2-day Go Boston cards. Pretty sweet, eh?
There are a lot of attractions that I love in Boston, and I find it difficult to narrow my favorite down to one. It probably depends on the time of year. Right about now, for example, I'm partial to the simple pleasures of visiting the Public Garden, enjoying the spring weather, seeing the flowers begin to emerge, and taking a timeless ride on the Swan Boats. The Go Card offers admission to the Swan Boats as well as a bunch of other Boston-area attractions.
How about you? What's your favorite place to go and thing to do in our fair city? Proclaim your choice at the DoubleTree Facebook page and you could win a complimentary visit. Don't dawdle. The contest ends April 30. | {
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Spy Novelist Silva's 'Secret Servant'
By Alan Cheuse
Writer Daniel Silva has a 10th spy thriller coming out next week. It's called "The Secret Servant."
And Alan Cheuse has a review.
ALAN CHEUSE: "The Secret Servant" features Silva's by now familiar hero, Israeli art restorer and special agent Gabriel Allon.
Scarcely having returned home from saving the Vatican from a missile attack by Islamic fundamentalists, Allon gets dispatched back to Europe to try and halt yet another insidious terror plot. This one pits Allon against the group of dedicated gunmen intent on kidnapping the daughter of the American ambassador to England and turning her into a weapon of hideous design.
This may sound thin and purely melodramatic, but it's not. In Allon and his cadre of fellow Israeli agents, Silva has created an authentic band of brothers. And he nearly does the same with the crew on the other side of the conflict, beginning with the father of the Islamist leader.
I won't give away too many of the details, which means, I suppose, that there is a kind of melodrama in the way the novel unfolds, moving from London and various European locations, to Tel Aviv and back again, with time ticking away in each chapter in the manner of Frederick Forsyth's classic, "Day of the Jackal."
But Silva's growing mastery of psychology and narrative suspense and the integration of serious research into the forward motion of the story is also quite evident.
The other thing you'll notice while reading this novel is how time disappears as you sink into the story, like a stone.
SIEGEL: The novel is "The Secret Servant" by Daniel Silva. Our reviewer, Alan Cheuse, teaches writing at George Mason University in Fairfax, Virginia. Transcript provided by NPR, Copyright NPR.
Alan Cheuse
Alan Cheuse died on July 31, 2015. He had been in a car accident in California earlier in the month. He was 75. Listen to NPR Special Correspondent Susan Stamburg's retrospective on his life and career. | {
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Q: Keep Android debugger alive even after device restart I am having trouble debugging some code in my DeviceBootReceiver (handles android.intent.action.BOOT_COMPLETED intent). I want to debug this bit, but how does one keep the debugger alive, when the device reboots? Is there any hack that anyone has come across for this?
What I want to do :
*
*Start debugging app via Android Studio
*Power down the device
*Power up the device
*Still be able to get the debugger attached to my app when it starts to handle android.intent.action.BOOT_COMPLETED intent
Any thoughts?
A: *
*Open your device "developer options" settings;
*Scroll down to "select debug app" and make sure your app is selected there;
*Check the option "wait for debugger".
This will make sure that when your app is executed, for example when it receives a BOOT_COMPLETED broadcast, the debugger gets attached first.
Hope it helps.
A: You can re-broadcast the intent by yourself via adb shell:
$ adb shell am broadcast -a android.intent.action.BOOT_COMPLETED
A: It takes about 30 seconds AFTER your device becomes available for interactions for android.intent.action.BOOT_COMPLETED to be broadcast. You have enough time to start up your application and enter debug mode manually
You might want to print a timestamp to your logcat when the android.intent.action.BOOT_COMPLETED intent is received so you have a better idea of when all this happens
| {
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Q: Paging of frequently changing data I'm developing a web application which display a list of let's say "threads". The list can be sorted by the amount of likes a thread has. There can be thousands of threads in one list.
The application needs to work in a scenario where the likes of a thread can change more than 10x in a second. The application furthermore is distributed over multiple servers.
I can't figure out an efficient way to enable paging for this sort of list. And I can't transmit the whole sorted list by likes to a user at once.
*
*As soon as an user would go to page 2 of this list, it likely changed and may contain threads already listed from page one
Solutions which don't work:
*
*Storing the seen threads on the client side (could be too many on mobile)
*Storing the seen threads on the Server side (too many users and threads)
*Snapshot the list in temp database table (it's too frequent changing data and it need to be actual)
(If it matters I'm using MongoDB+c#)
How would you solve this kind of problem?
A: This is typically handled using an OLAP cube. The idea here is that you add a natural time dimension. They may be too heavy for this application, but here's a summary in case someone else needs it.
OLAP cubes start with the fundamental concept of time. You have to know what time you care about to be able to make sense of the data.
You start off with a "Time" table:
Time {
timestamp long (PK)
created datetime
last_queried datetime
}
This basically tracks snapshots of your data. I've included a last_queried field. This should be updated with the current time any time a user asks for data based on this specific timestamp.
Now we can start talking about "Threads":
Threads {
id long (PK)
identifier long
last_modified datetime
title string
body string
score int
}
The id field is an auto-incrementing key; this is never exposed. identifier is the "unique" id for your thread. I say "unique" because there's no unique-ness constraint, and as far as the database is concerned it is not unique. Everything else in there is pretty standard... except... when you do writes you do not update this entry. In OLAP cubes you almost never modify data. Updates and inserts are explained at the end.
Now, how do we query this? You can't just directly query Threads. You need to include a star table:
ThreadStar {
timestamp long (FK -> Time.timestamp)
thread_id long (FK -> Threads.id)
thread_identifier long (matches Threads[thread_id].identifier)
(timestamp, thread_identifier should be unique)
}
This table gives you a mapping from what time it is to what the state of all of the threads are. Given a specific timestamp you can get the state of a Thread by doing:
SELECT Thread.*
FROM Thread
JOIN ThreadStar ON Thread.id = ThreadStar.thread_id
WHERE ThreadStar.timestamp = {timestamp}
AND Thread.identifier = {thread_identifier}
That's not too bad. How do we get a stream of threads? First we need to know what time it is. Basically you want to get the largest timestamp from Time and update Time.last_queried to the current time. You can throw a cache up in front of that that only updates every few seconds, or whatever you want. Once you have that you can get all threads:
SELECT Thread.*
FROM Thread
JOIN ThreadStar ON Thread.id = ThreadStar.thread_id
WHERE ThreadStar.timestamp = {timestamp}
ORDER BY Thread.score DESC
Nice. We've got a list of threads and the ordering is stable as the actual scores change. You can page through this at your leisure... kind of. Eventually data will be cleaned up and you'll lose your snapshot.
So this is great and all, but now you need to create or update a Thread. Creation and modification are almost identical. Both are handled with an INSERT, the only difference is whether you use an existing identifier or create a new one.
So now you've inserted a new Thread. You need to update ThreadStar. This is the crazy expensive part. Basically you make a copy of all of the ThreadStar entries with the most recent timestamp, except you update the thread_id for the Thread you just modified. That's a crazy amount of duplication. Fortunately it's pretty much only foreign keys, but still.
You also don't do DELETEs either; mark a row as deleted or just exclude it when you update ThreadStar.
Now you're humming along, but you've got crazy amounts of data growing. You'll probably want to clean it out, unless you've got a lot of storage budge, but even then things will start slowing down (aside: this will actually perform shockingly well, even with crazy amounts of data).
Cleanup is pretty straightforward. It's just a matter of some cascading deletes and scrubbing for orphaned data. Delete entries from Time whenever you want (e.g. it's not the latest entry and last_queried is null or older than whatever cutoff). Cascade those deletes to ThreadStar. Then find any Threads with an id that isn't in ThreadStar and scrub those.
This general mechanism also works if you have more nested data, but your queries get harder.
Final note: you'll find that your inserts get really slow because of the sheer amounts of data. Most places build this with appropriate constraints in development and testing environments, but then disable constraints in production!
Yeah. Make sure your tests are solid.
But at least you aren't sensitive to re-ordered data mid-paging.
A: Interesting question. Unless I'm misunderstanding you, and by all means let me know if I am, it sounds like the best solution would be to implement a system that, instead of page numbers, uses timestamps. It would be similar to what many of the main APIs already do. I know Tumblr even does this on the dashboard, where this is, of course, not an unreasonable case: there can be tons of posts added in a small amount of time at peak hours, depending on how many people the user follows.
So basically, your "next page" button could just link to /threads/threadindex/1407051000, which could translate to "all the threads that were created before 2014-08-02 17:30. That makes your query super easy to implement. Then, when you pull down all the next elements, you just look for anything that occurred before the last element on the page.
The downfall of this, of course, is that it's hard to know how many new elements have been added since the user started browsing, but you could always log the start time and know anything since then would be new. And it's also difficult for users to type in their own pages, but that's not a problem in most applications. You also need to store the timestamps for every record in your thread, but that's probably already being done, and if it's not then it's certainly not hard to implement. You'll be paying the cost of something like eight bytes extra per record, but that's better than having to store anything about "seen" posts.
It's also nice because, and again this might not apply to you, but a user could bookmark a page in the list, and it would last unchanged forever since it's not relative to anything else.
A: For constantly changing data such as likes I would use a two stage appraoch. For the frequently changing data I would use an in memory DB to keep up with the change rates and flush this peridically to the "real" db.
Once you have that the query for constantly chaning data is easy.
*
*Query the db.
*Query the in memory db.
*Merge the frequently changed data from the in memory db with the "slow" db data .
*Remember which results you already have displayed so pressing the next button will
not display an already dispalyed value twice because on different pages because its rank has changed.
If many people look at the same data it might help to cache the results of 3 in itself to reduce the load on the real db even further.
Your current architecture has no caching layers (the bigger the site the more things are cached). You will not get away with a simple DB and efficient queries against the db if things become too massive.
A: I would cache all 'thread' results on the server when the user first time hits the database. Then return the first page of data to the user and for each subsequent next page calls I'd return cached results.
To minimize memory usage you can cache only records ids and fetch whole data when user requests it.
Cache can be evicted each time user exits current page. If it isn't a ton of data I would stick to this solution because user won't get annoyed of data constantly changing.
| {
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2016-6-30 · With the sale, Newmont is offloading its Batu Hijau mine, Indonesia's the second-biggest copper and gold operation.
Alibaba.com offers 8,681 asic miner for sale products. About 12% of these are other pcb & pcba, 3% are multilayer pcb, and 2% are graphics cards. A wide variety of asic miner for sale options are available to you, such as free samples, paid samples.
2018-7-23 · Price – How much does the bitcoin miner cost? Cheap mining hardware will mine less bitcoins, Used Bitcoin Mining Hardware for Sale.
Modular to Sponsor, Exhibit, and Present at Indonesia Miner. Modular is a Gold Sponsor of this year's Indonesia Miner International Exhibition, taking place 11-12 April 2018 in Jakarta, Indonesia.
In May, 2016 the team successfully launched Baikal mini X11 algorithm miner. The miner's high cost-performance and stability have won high evaluation and recognition.
2018-2-27 · Several parties are interested in buying Freeport-McMoRan's cobalt project in the Democratic Republic of Congo but not at a price that would interest the miner and so it is not planning a sale, Freeport's chief executive officer said.
2018-3-27 · The sale of Kestrel, Indonesia — are going to burn more coal Rio decides to sell its interest in Grasberg to a state-backed Indonesian miner.
For Sale Project Has been mined by small scale miner over 20 years with continuous production since then. Exisitng 300,000 ton dump showing 2.5 g/t. | {
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Q: SugarCRM Home Page URL I am using sugarCRM Open Source Application for my own purpose.
When I click the Home link the above URL is like:
ajaxUILoc=index.php?module=Home&action=index&ParentTab=sales
How can I find out the flow of code using the above url. I have verified the sugarCRM developers guide regarding the url, but I cannot get the solution. How they call the functions or classes using like this.
Actually why I am asking is I want to modify the existing code to add some extra modules.
I am struggling for last one week without solution.
A: ajaxUILoc=index.php?module=Home&action=index&ParentTab=sales
That is saying you are in the sales Tab and on the 'Index' Page of the Home Module.
Firstly there is a modules directory in the root of the application inside which is a directory for each module, secondly for customisations there is a custom directory in the root of the application, where is generally all custom code is placed to make SugarCRM upgrade safe. There is a customCode variable you can add for placing custom code in the custom files.
Example:
'customCode'=>'{if $surrent_user->user_name==\'admin\'} .....some code...... {/if} {if $current_user->user_name!=\'admin\'} <input name="assigned_user_name" value="{$fields.assigned_user_name.value}" readonly="readonly"> {/if}',
The SugarCRM forums can be quite good http://www.sugarcrm.com/forums
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 6,978 |
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Physical simulation of extreme winds (e.g. tornadoes, downbursts) and other complex flow fields coupled with state-of-the-art visualization and measurement techniques. | {
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{"url":"https:\/\/mathinsight.org\/slides\/pure_time_differential_equation_graphical?bare=true","text":"### Graphical solution to pure-time differential equations\n\nSign of derivativeBehavior of function\nDerivative positiveFunction increasing\nDerivative negativeFunction decreasing\nDerivative zeroFunction constant\n How do we do this in general?How to determine $\\smash{f(x)}$ from graph of $\\smash{f'(x)}$? $f'(x)$ $f(x)$\n\nUse derivative facts to sketch solution of pure-time differential equation\n\n \\begin{align*} y'(t) &= 0.5t-3\\\\ y(0) &= 4 \\end{align*} (Initial condition)\n Sketch graph of $y'(t)$. When $t=0$, what is sign of $y'(t)$? Negative. $y(t)$ decreases at $t=0$. As $t \\uparrow$ , $y'(t) \\to 0$.$\\Rightarrow$ graph of $y(t)$ flattens. $y'(t)$ $y(t)$\n\n$y'(6)=0 \\Rightarrow$ graph of $y(t)$ is horizontal at $t=6$.\n\nFor $t>6$, $y'(t)>0 \\Rightarrow$ $y(t)$ increases.\nGraph of $y(t)$ represents solution.\n\nSecond example: Solve the pure-time differential equation\n\n ${}$ where $f(t)$ is plotted below.\n\nStep 1: Find $t$ where $f(t)=0$.\n\n$f(t)=0$ when $t=1, 3, 6$\n\nThe graph of $y(t)$ must be horizontal at these points.\n\nStep 2: Determine sign of $f(t)$ in between.\n\n\u2022 $f(t) \\lt 0$ for $t \\lt 1$\n\u2022 $f(t) \\gt 0$ for $1 \\lt t \\lt 3$\n\u2022 $f(t) \\lt 0$ for $3 \\lt t \\lt 6$\n\u2022 $f(t) \\gt 0$ for $t \\gt 6$\n\nStep 3: Start at initial condition.\n\nStep 4: Sketch solution that moves in the correct directions.\n\n(If change initial condition, graph just shifts up or down.)","date":"2022-01-29 10:37:26","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 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\": 0.9960650205612183, \"perplexity\": 2404.5772616415493}, \"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-2022-05\/segments\/1642320304883.8\/warc\/CC-MAIN-20220129092458-20220129122458-00687.warc.gz\"}"} | null | null |
Maley'sBhoys Latest – Bishop Says Police Incite Football Fans
The following is an article which featured in today's Scottish Sunday Times, written by Lorraine Davidson. I personally think it's a great article, so that's why I've featured it here.
Please forgive any spelling errors and I've typed this out as quickly as I can.
One of Scotland's most senior churchmen said football fans are being "unneccesarily provoked" by police as part of a crackdown on sectarianism.
Philip Tartaglia, the Catholic Bishop of Paisley, told how he was frisked by stewards under the glare of the police, filmed on video camera and made to feel "distinctly uncomfortable" when he attended a Celtic football match with two other priests in August.
Read more:http://maleysbhoys.com/index.cgi?board=celticchat&action=display&thread=420&page=1#ixzz1eGzfD3sN
celtic forum fans Maley's Bhoys | {
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\section{Introduction}
Image captioning is a popular research area as it combines the domains of Computer Vision and Natural Language Processing. Generating captions automatically is a difficult problem since it not only involves detecting the objects present in the image, but also involves expressing the semantic relationship between the corresponding objects in a natural language.
As the deep learning community matures, there have been several approaches for image captioning which have produced state-of-the-art models that produce sentences closer to natural language \cite{show-attend-tell}\cite{show-and-tell}. The different approaches usually follow the general encoder-decoder structure. The role of an encoder is to extract the semantic information from the image and is typically represented by Convolutional Neural Networks (CNN). The role of a decoder is to translate the encoded image features into natural language and this is usually represented by Recurrent Neural Network (RNN) \cite{show-attend-tell}. In particular, a special type of RNN is popular and is referred to as Long Short Term Memory (LSTM) which has the ability to handle long-term temporal dependencies better through the use of a memory cell. However, RNN is known to be sequential in the sense that it generates one word at a time and as such, the training time increases with increased sequence length of the caption. As such, the transformer architecture has been used in recent times in place of an RNN \cite{transformer-stacked-attention} for image captioning to avoid this sequential training problem.
Our key contribution in this paper is that we perform sensitivity analysis of several hyperparameters for both LSTM-based decoder and Transformer-based decoders by using Flickr8k dataset.
\subsection{Related Work}
After achieving a lot of success in Neural Machine Translation tasks, the encoder-decoder framework has been used several times for image captioning \cite{show-attend-tell}\cite{transformer-stacked-attention}\cite{show-and-tell}. \citet{show-and-tell} used CNN as the encoder for the image and then used Long Short Term Memory (LSTM) RNN as the decoder. In particular, they only used the encoded image representation for the first timestep for the LSTM. \citet{show-attend-tell} extended this approach by proposing a visual attention mechanism to attend to different parts of the images for every timestep during the caption generation using LSTM.
Beyond the recurrence-based decoders, \citet{transformer} established a new architecture for machine translation called Transformer which is purely based on attention mechanism. They also showed that a Transformer is superior is quality while taking significantly less time to train as it is parallelizable. Using Transformer for image captioning has achieved state-of-the-art results as shown by \citet{transformer-stacked-attention}.
\section{Approach}
In this section, we discuss the methodology used for our experiments. In particular, we used an encoder-decoder architecture for image captioning where the encoder consisted of a CNN and two different decoder networks were examined: LSTM and Transformer.
\subsection{Flickr8k Dataset}
The dataset used for experimentation is Flickr8K \cite{image-captioning-survey} which has 8,000 images in total. In particular, it is divided in to 6,000 training images, 1,000 validation images, and 1,000 test images. Furthermore, each of the images is associated with five reference captions annotated by humans. As such, our training set consists of 30,000 samples where each sample corresponds to one image and one caption.
\subsection{Encoder}
The encoder model we used was the ResNet CNN model. The CNN extracts the features of the image which are referred to as annotation vectors. These vectors form the hidden states of the encoder on which the attention mechanism is performed. We experimented with 3 different types of ResNet models, the ResNet18, ResNet50 and the ResNet101. The number following the model name indicates the number of layers. We removed the final pooling and softmax layer and extracted the features from the final convolutional layer. We obtain an output of size N x 14 x 14, where the value of N depends on the type of encoder used. This is then flattened to give us a 196-dimensional vector on which we perform attention.
\subsection{LSTM Decoder}
The LSTM network produces a caption by generating a word at every time-step. The output at a given time-step is conditioned on the current hidden state, a context vector which is obtained from the attention mechanism and all the previous hidden states. The initial hidden state and cell state of the LSTM is obtained by taking the average of the annotation vectors and passing it through different MLPs. At a given time-step, we perform attention to obtain a context vector which is then appended to the input word embedding. In particular, we follow the same soft attention training process that is described by \citet{show-attend-tell} for our experiments.
\subsubsection{Transformer Decoder}
The transformer model introduced by \citet{transformer} was a way forward for language modeling that did away with the recurrent nature of language modeling. It relied purely on self-attention. The transformer model is shown in Figure \ref{fig:transformer}.
\begin{figure}[htpb!]
\centering
\includegraphics[width=0.4\linewidth]{figures/tranformer.png}
\caption{Transformer architecture. We substitute the encoder with a CNN.}
\label{fig:transformer}
\end{figure}
We use the encoder as the CNN and adopt just the decoder of the transformer architecture. The transformer network relies on a series of computations known as scaled dot-product attention. The attention function is basically a mapping of queries(Q) and key-value pairs(K-V) to an output. The output is a weighted sum of the values, where the weights assigned to each value is based on a similarity function between the query and key. This is shown in Figure \ref{subfig:attention}. It is given by the following formula:
\begin{equation}
Attention(Q, K, V) = softmax(\frac{QK^T}{\sqrt{d_k}})V
\end{equation}
Here $\sqrt{d_k}$ is a scaling factor that prevents the absolute value of the dot product from blowing up.
The transformer employs another strategy known as Multi-Head self-attention. This ensures that the model learns a multi-modal representation of the input sentence. The model learns to attend to different representations of the same input. The idea is to project the input vectors to different sub-spaces followed by the self attention function in each subspace. The output of each subspace is concatenated and a linear layer projects the data back down to the original subspace. This is shown in Figure \ref{subfig:multihead}. The formula is given by:
\begin{align*}
MultiHead(Q, K, V) &= Concat(head_1, ..., head_h)W^O \\ where, head_i &= Attention(QW_i^Q, KW_i^K, VW_i^V)\\
\end{align*}
The decoder block can be broken into three main sub-blocks. The first is a masked multi-head self attention layer. This is a layer of self attention where the only difference is that each vector only attends to words that come before it. This is so that the model only uses past information to make judgements about the present and does not get a peek into the future. The second sub-layer is a layer of multi-head attention on the encoder hidden states. This is where the representation of the image is fed into the decoder layer. Finally, there is a feed forward layer which introduces some non-linearity to the model.
In between each sub-layer there is a residual connection which speeds up convergence and prevents the vanishing gradient problem. There are also dropout layers after each sub-layer to prevent over-fitting. Multiple such decoder blocks are stacked one on top of the other until the outputs of the last layer are passed through a softmax layer to obtain the output probabilities. Another technique for regularization we used was label-smoothing with $\epsilon=0.1$. It was noticed that the model had higher loss but the BLEU scores improved.
\begin{figure}[htpb!]
\centering
\begin{subfigure}[b]{0.4\linewidth}
\includegraphics[width=\linewidth]{figures/multihead.png}
\caption{Multi-head attention.}
\label{subfig:multihead}
\end{subfigure}
\begin{subfigure}[b]{0.4\linewidth}
\centering
\includegraphics[width=0.5\linewidth]{figures/attention.png}
\caption{Scaled dot-product attention function.}
\label{subfig:attention}
\end{subfigure}
\caption{Attention mechanisms used in a Transformer.}
\label{fig:multihead_&_attention}
\end{figure}
\subsection{Metrics}
We use BLEU \cite{bleu}, METEOR \cite{meteor}, CIDEr \cite{cider}, ROUGE \cite{rouge} to evaluate the quality of generated captions. BLEU \cite{bleu} measures similarity between a set of reference texts and the machine generated text through the use of n-grams. METEOR \cite{meteor} is based on explicit word to word matches through the use of corresponding word stems and synonyms. CIDEr \cite{cider} uses Term Frequency-Inverse Document Frequency (TF-IDF) weighting for each n-gram to measure similarity between reference texts and predicted text. ROUGE \cite{rouge} uses word pairs, n-grams, and word sequences to measure sentence similarity. Existing image captioning research use BLEU, METEOR, and ROUGE extensively. However, CIDEr has been found to be more correlated with human assessment \cite{image-captioning-survey}. As such, we decided to include CIDEr as well to get a better representation of caption quality.
\section{Experimental Analysis}
We performed several experiments to analyze the difference in using different encoder-decoder architectures.
\subsection{ResNet + LSTM}
Our baseline model was ResNet18 combined with an LSTM using a hidden vector size of 512. We do not finetune the encoder in the baseline. The word embedding size used in all LSTM experiments are of size 512. We conducted three different experiments: varying the encoder, fine-tuning the encoder, and varying the number of LSTM hidden units. All experiments were trained using the Adam optimizer with a learning rate of 0.0001 on Nvidia GeForce GTX 1080 GPU. In additon, the termination of training was determined by early stopping to obtain the best possible BLEU-4 scores.
For the first experiment, we used ResNet18, ResNet50, and ResNet101 models in order to examine the effect of improved image quality. It was hypothesized that using a larger CNN model would result in better caption generation. The summary of the experimental results is listed in Table \ref{tab:lstm-varying-encoder}. Moreover, Figure \ref{fig:exp-lstm-image-quality} shows that ResNet50 and ResNet101 perform much better than ResNet18 and hence, validates our original hypothesis. One key observation is that performance difference between RestNet50 and ResNet101 is minimal and ResNet101 actually gave a lower CIDEr score. As such, there is an indication that increasing image quality after some point may lead to saturation in terms of caption quality.
\begin{table}[htpb!]
\caption{Experimental results for varying encoder in CNN+LSTM architecture.}
\label{tab:lstm-varying-encoder}
\centering
\begin{tabular}{llllllll}
\toprule
Encoder & BLEU-1 & BLEU-2 & BLEU-3 & BLEU-4 & METEOR & ROUGE\_L & CIDEr \\
\midrule
ResNet18 & 58.57 & 40.37 & 27.39 & 18.32 & 19.99 & 44.94 & 47.71 \\
ResNet50 & 59.95 & 42.34 & 29.26 & 20.06 & 20.79 & 45.63 & 52.65 \\
ResNet101 & 60.73 & 43.06 & 29.55 & 20.17 & 20.46 & 46.10 & 51.01 \\
\bottomrule
\end{tabular}
\end{table}
\begin{figure}[htpb!]
\centering
\includegraphics[width=0.5\linewidth]{figures/image_quality_lstm.png}
\caption{Effect of varying encoders on image captioning for CNN+LSTM architecture.}
\label{fig:exp-lstm-image-quality}
\end{figure}
Next, we decided to fine-tune all three encoders discussed above and see it's effect on caption quality. It was hypothesized that fine-tuning would increase captioning quality as ResNet is trained on ImageNet while we are using Flickr8k. For each of the encoders, we compared the score of fine-tuned model with the corresponding model scores without fine-tuning. The summary of the results is listed in Table \ref{tab:lstm-finetuned}. Moreover, Figure \ref{fig:exp-lstm-finetuned} shows that fine-tuning is beneficial for all encoders as it outperforms the base models. One important observation is that the deeper models benefit more from fine-tuning as it can be clearly seen from the positive trend for CIDEr and METEOR scores. Additionally, we were able to outperform the metrics reported by \citet{show-attend-tell} through these experiments.
\begin{table}[htpb!]
\caption{Experimental results for fine-tuning last CNN layer in CNN+LSTM architecture. It shows the difference of scores by subtracting the base model scores from fine-tuned scores.}
\label{tab:lstm-finetuned}
\centering
\begin{tabular}{llllllll}
\toprule
Encoder & $\Delta$ BLEU-1 & $\Delta$ BLEU-2 & $\Delta$ BLEU-3 & $\Delta$ BLEU-4 & $\Delta$ METEOR & $\Delta$ ROUGE\_L & $\Delta$ CIDEr \\
\midrule
ResNet18 & 1.95 & 2.00 & 1.84 & 1.75 & 0.15 & 0.81 & 1.86 \\
ResNet50 & 2.74 & 2.75 & 2.60 & 2.06 & 0.63 & 2.18 & 2.48 \\
ResNet101 & 2.14 & 2.07 & 1.92 & 1.30 & 1.12 & 2.05 & 6.46 \\
\bottomrule
\end{tabular}
\end{table}
\begin{figure}[htpb!]
\centering
\includegraphics[width=0.5\linewidth]{figures/lstm-finetuning.png}
\caption{Effect of fine-tuning on image captioning for CNN+LSTM architecture. Positive difference indicates that fine-tuned model outperforms the base model without fine-tuning.}
\label{fig:exp-lstm-finetuned}
\end{figure}
Lastly, we looked at the effect of varying number of LSTM hidden units by experimenting with 256, 512, and 1024 units. The experiment was carried out by keeping the encoder the same and varying the LSTM units. This experiment was repeated for all encoders: ResNet18, ResNet50, and ResNet101. It was hypothesized that increasing the hidden units would increase captioning quality as the capacity to store information increases and hence, it would be able to model long term dependencies better. The results of this experiment is summarized in Table \ref{tab:lstm-varying-lstm-dimensions}. From Figure \ref{fig:exp-lstm-units}, we can see that the performance improves by a negligible amount as the number of units increase. We suspect this is due to over-fitting as we are using Flickr8k which is a small dataset compared to MSCOCO and Flickr30k.
\begin{table}[htpb!]
\caption{Experimental results for varying number of hidden units in an LSTM for the CNN+LSTM architecture.}
\label{tab:lstm-varying-lstm-dimensions}
\centering
\begin{tabular}{lllllllll}
\toprule
Encoder & LSTM Size & BLEU-1 & BLEU-2 & BLEU-3 & BLEU-4 & METEOR & ROUGE\_L & CIDEr \\
\midrule
ResNet18 & 256 & 58.90 & 41.04 & 28.07 & 19.04 & 19.93 & 44.84 & 48.31 \\
& 512 & 58.57 & 40.37 & 27.39 & 18.32 & 19.99 & 44.94 & 47.71 \\
& 1024 & 59.75 & 41.46 & 28.37 & 19.29 & 20.34 & 45.36 & 49.53 \\
\midrule
ResNet50 & 256 & 60.01 & 42.33 & 29.08 & 19.79 & 20.76 & 45.83 & 52.76 \\
& 512 & 59.95 & 42.34 & 29.26 & 20.06 & 20.79 & 45.63 & 52.65 \\
& 1024 & 60.68 & 42.98 & 29.96 & 20.80 & 20.62 & 45.90 & 51.88 \\
\midrule
ResNet101 & 256 & 60.51 & 42.42 & 29.28 & 19.92 & 20.60 & 46.03 & 53.52 \\
& 512 & 60.73 & 43.06 & 29.55 & 20.17 & 20.46 & 46.10 & 51.01 \\
& 1024 & 60.73 & 42.77 & 29.63 & 20.25 & 20.99 & 46.26 & 52.39 \\
\bottomrule
\end{tabular}
\end{table}
\begin{figure}[htpb!]
\centering
\begin{subfigure}[b]{0.45\linewidth}
\includegraphics[width=\linewidth]{figures/lstm-units-resnet18.png}
\caption{ResNet18}
\label{subfig:lstm-units-resnet18}
\end{subfigure}
\begin{subfigure}[b]{0.45\linewidth}
\includegraphics[width=\linewidth]{figures/lstm-units-resnet50.png}
\caption{ResNet50}
\label{subfig:lstm-units-resnet50}
\end{subfigure}
\begin{subfigure}[b]{0.45\linewidth}
\includegraphics[width=\linewidth]{figures/lstm-units-resnet101.png}
\caption{ResNet101}
\label{subfig:lstm-units-resnet101}
\end{subfigure}
\caption{Effect of varying number of LSTM hidden units keeping CNN encoder fixed.}
\label{fig:exp-lstm-units}
\end{figure}
\subsection{ResNet + Transformer}
Our baseline model for the ResNet-Transformer architecture was a ResNet18 model with 3 transformer layers. Each layer uses just a single head in the baseline. We conducted three experiments: varying the type of encoder model, varying the number of decoder layers, and varying the number of heads. Additionally we analyzed the effect of finetuning the encoder and seeing the performance of the model after finetuning. All experiments were trained using the Adam optimizer with a fixed learning rate of 0.00004 on a Geforece GTX 1080 Ti GPU. Termination of training was determined by early stopping to obtain the best possible BLEU-4 scores. The results are compiled in Tables 4-7.
For analyzing the different encoder models, we used ResNet18, ResNet50 and ResNet101. It was hypothesized that using a larger encoder model should improve the caption generation as the image representation would be better. Surprisngly, from Figure \ref{fig:exp-trans-image-quality}, we see that ResNet50 model performs best whereas the ResNet101 model gives the worst performance. This could be because the dataset size is very small. ResNet18 could be underfitting, whereas ResNet101 could be overfitting. The results are summarized in Table \ref{tab:trans-varying-encoder}.
\begin{figure}[!htpb]
\centering
\includegraphics[width=0.5\linewidth]{figures/image_quality_transformer.png}
\caption{Effect of varying encoders on image captioning for CNN+Transformer architecture.}
\label{fig:exp-trans-image-quality}
\end{figure}
\begin{table}[!htpb]
\caption{Experimental results for varying encoder in CNN+Transformer architecture.}
\label{tab:trans-varying-encoder}
\centering
\begin{tabular}{llllllll}
\toprule
Encoder & BLEU-1 & BLEU-2 & BLEU-3 & BLEU-4 & METEOR & ROUGE\_L & CIDEr \\
\midrule
ResNet18 & 59.08 & 40.35 & 26.87 & 17.60 & 18.55 & 42.62 & 43.36 \\
ResNet50 & 60.15 & 41.61 & 28.02 & 18.73 & 18.66 & 43.96 & 45.53 \\
ResNet101 & 59.46 & 40.89 & 26.90 & 17.37 & 18.01 & 42.44 & 42.10 \\
\bottomrule
\end{tabular}
\end{table}
For analyzing the effect of number of decoder layers and heads, we kept the type of encoder fixed as ResNet18. It was hypothesized that increasing the number of heads should improve the quality as the transformer gets to attend to information from different representation subspaces. The results are compiled in Table \ref{tab:trans-varying-heads}. From Figure \ref{fig:varying_heads}, we see that increasing the number of heads had either no effect or was even reducing the performance of the model. It is possible that our learning rate is not tuned perfectly, but we suspect that increasing the number of heads causes the model to overfit as our dataset size is very small compared to bigger datasets like MSCOCO or Flickr30k. Increasing dropout or other regularization techniques might help curb this effect but we have not experimented with this. Similar to changing the number of heads, it was hypothesized that increasing the number of layers should improve performance of caption generation as the transformer learns a better representation of the word as it gets deeper. From Figure \ref{fig:varying_layers}, we see that changing the number of layers also showed no significant or noticeable changes. At times the model performed better and at times worse. These results are compiled in Table \ref{tab:trans-varying-layers}.
\begin{table}[htpb!]
\caption{Experimental results for varying number of heads keeping decoder layers fixed for the CNN+Transformer architecture. Encoder used is ResNet18.}
\label{tab:trans-varying-heads}
\centering
\begin{tabular}{lllllllll}
\toprule
Layers & Heads & BLEU-1 & BLEU-2 & BLEU-3 & BLEU-4 & METEOR & ROUGE\_L & CIDEr \\
\midrule
3 & 1 & 59.08 & 40.35 & 26.87 & 17.60 & 18.55 & 42.62 & 43.36 \\
& 2 & 60.10 & 41.31 & 27.22 & 17.73 & 17.92 & 42.92 & 41.57 \\
& 3 & 58.86 & 40.24 & 26.05 & 16.68 & 17.73 & 42.17 & 38.71 \\
\midrule
5 & 1 & 58.87 & 40.38 & 27.06 & 17.81 & 18.65 & 43.03 & 44.35 \\
& 2 & 57.25 & 39.04 & 25.62 & 16.40 & 17.91 & 41.90 & 40.47 \\
& 3 & 57.89 & 39.40 & 26.08 & 16.93 & 18.36 & 42.57 & 41.03 \\
\midrule
7 & 1 & 60.10 & 41.45 & 27.48 & 18.16 & 18.34 & 43.35 & 45.24 \\
& 2 & 59.68 & 41.14 & 27.48 & 17.91 & 17.90 & 42.91 & 42.12 \\
& 3 & 59.84 & 41.18 & 27.27 & 17.80 & 18.24 & 43.00 & 44.26 \\
\bottomrule
\end{tabular}
\end{table}
\begin{figure}[htpb!]
\centering
\begin{subfigure}[b]{0.45\linewidth}
\includegraphics[width=\linewidth]{figures/3_layers.png}
\caption{3 layers}
\label{subfig:3_layers}
\end{subfigure}
\begin{subfigure}[b]{0.45\linewidth}
\includegraphics[width=\linewidth]{figures/5_layers.png}
\caption{5 layers}
\label{subfig:5_layers}
\end{subfigure}
\begin{subfigure}[b]{0.45\linewidth}
\includegraphics[width=\linewidth]{figures/7_layers.png}
\caption{7 layers}
\label{subfig:7_layers}
\end{subfigure}
\caption{Effect of varying number of heads keeping number of decoder layers fixed.}
\label{fig:varying_heads}
\end{figure}
\begin{figure}[htpb!]
\centering
\begin{subfigure}[b]{0.45\linewidth}
\includegraphics[width=\linewidth]{figures/1_head.png}
\caption{1 head}
\label{subfig:1_head}
\end{subfigure}
\begin{subfigure}[b]{0.45\linewidth}
\includegraphics[width=\linewidth]{figures/2_heads.png}
\caption{2 heads}
\label{subfig:2_head}
\end{subfigure}
\begin{subfigure}[b]{0.45\linewidth}
\includegraphics[width=\linewidth]{figures/3_heads.png}
\caption{3 heads}
\label{subfig:3_head}
\end{subfigure}
\caption{Effect of varying number of decoder layers keeping number of heads fixed.}
\label{fig:varying_layers}
\end{figure}
\begin{table}[htpb!]
\caption{Experimental results for varying number of decoder layers keeping heads fixed for the CNN+Transformer architecture. Encoder used is ResNet18.}
\label{tab:trans-varying-layers}
\centering
\begin{tabular}{lllllllll}
\toprule
Heads & Layers & BLEU-1 & BLEU-2 & BLEU-3 & BLEU-4 & METEOR & ROUGE\_L & CIDEr \\
\midrule
1 & 3 & 59.08 & 40.35 & 26.87 & 17.60 & 18.55 & 42.62 & 43.36 \\
& 5 & 58.87 & 40.38 & 27.06 & 17.81 & 18.65 & 43.03 & 44.35 \\
& 7 & 60.10 & 41.45 & 27.48 & 18.16 & 18.34 & 43.35 & 45.24 \\
\midrule
2 & 3 & 60.10 & 41.31 & 27.22 & 17.73 & 17.92 & 42.92 & 41.57 \\
& 5 & 57.25 & 39.04 & 25.62 & 16.40 & 17.91 & 41.90 & 40.47 \\
& 7 & 59.68 & 41.14 & 27.48 & 17.91 & 17.90 & 42.91 & 42.12 \\
\midrule
3 & 3 & 58.86 & 40.24 & 26.05 & 16.68 & 17.73 & 42.17 & 38.71 \\
& 5 & 57.89 & 39.40 & 26.08 & 16.93 & 18.36 & 42.57 & 41.03 \\
& 7 & 59.84 & 41.18 & 27.27 & 17.80 & 18.24 & 43.00 & 44.26 \\
\bottomrule
\end{tabular}
\end{table}
Finally, the most notable and obvious difference is when fine-tuning. It was hypothesized that fine-tuning the encoder would improve the quality of the caption. The summary of the results is listed in Table \ref{tab:trans-finetuned}, and from Figure \ref{fig:exp-trans-finetuned}, it is clear that the fine-tuned encoder models consistently perform better than the encoder models which are not. This makes sense considering that the encoder models are trained on ImageNet whereas our dataset is Flickr8k.
\begin{figure}[htpb!]
\centering
\includegraphics[width=0.5\linewidth]{figures/fine_tuning_transformer.png}
\caption{Effect of fine-tuning on image captioning for CNN+Transformer architecture. Positive difference indicates that fine-tuned model outperforms the base model without fine-tuning.}
\label{fig:exp-trans-finetuned}
\end{figure}
\begin{table}[htpb!]
\caption{Experimental results for fine-tuning last CNN layer in CNN+Transformer architecture. It shows the difference of scores by subtracting the base model scores from fine-tuned scores.}
\label{tab:trans-finetuned}
\centering
\begin{tabular}{llllllll}
\toprule
Encoder & $\Delta$ BLEU-1 & $\Delta$ BLEU-2 & $\Delta$ BLEU-3 & $\Delta$ BLEU-4 & $\Delta$ METEOR & $\Delta$ ROUGE\_L & $\Delta$ CIDEr \\
\midrule
ResNet18 & 1.18 & 1.36 & 0.94 & 0.64 & 0.15 & 0.94 & 1.92 \\
ResNet50 & 0.20 & -0.34 & -1.05 & -1.52 & 0.24 & -0.50 & 0.68 \\
ResNet101 & 2.09 & 2.57 & 2.69 & 2.06 & 1.92 & 2.69 & 8.35 \\
\bottomrule
\end{tabular}
\end{table}
\section{Conclusions}
We have presented and compared two different architectures for image caption generation. A ResNet model is used as an image feature extractor. For decoding we experimented with LSTMs and Transformers. We performed a sensitivity analysis of various hyperparameters. Experiments show that fine-tuning the encoder model almost always improves the outcome of the decoder model. The LSTM model using ResNet50 and ResNet101 surpass our reference baseline, the Soft-Attention Model even without finetuning. This could be due to the higher representative capability of the ResNet model as compared to the VGG model used in the Soft-attention model. All the other models surpass the baseline upon finetuning, with the only exception being using Resnet50 and a Transformer decoder. We have shown the effect of tuning other hyperparameters such as the number of hidden units for LSTMs, the number of decoder layers and number of heads used in multi-head attention for Transformers. It was noticed that increasing the number of heads, layers or hidden vector size does not always improve results and may even result in reducing the output quality. We believe that this is due to model overfitting as our dataset size is very small. A natural continuation of this work would be to experiment with larger datasets such as Flickr30k or MSCOCO. Apart from that, it would also be interesting to experiment with changing the word embedding size used in the LSTM model, or even finding out the effect of using pre-trained word embeddings such as GloVe vectors. Finally, we could experiment with more complicated Transformer architectures such as those by \citet{transformer-stacked-attention}.
\subsubsection*{Acknowledgments}
The authors would like to thank Jimmy Ba for his thoughtful instruction throughout the Neural Networks course as well as providing helpful insights during our project consultation meetings.
\small
\bibliographystyle{plainnat}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 3,777 |
package com.qmx.wxmp.common.utils;
import java.util.concurrent.ThreadFactory;
public class DaemonThreadFactory implements ThreadFactory{
@Override
public Thread newThread(Runnable r) {
Thread t = new Thread(r);
t.setDaemon(true);
return t;
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 9,259 |
\section{Introduction}
Recently, with the advent of large pre-trained language models~\cite{devlin2018bert,brown2020language,smith2022using}, dense retrieval \cite{DBLP:conf/emnlp/KarpukhinOMLWEC20,DBLP:conf/iclr/XiongXLTLBAO21,DBLP:journals/corr/abs-2210-11773} has become a popular topic for information retrieval, which aims to identify relevant contents from a large collection of passages.
Despite the great success of dense retrieval on benchmarks like MS MACRO~\cite{DBLP:conf/nips/NguyenRSGTMD16}, the inference efficiency is still left as a problem.
Therefore, small and efficient models are welcomed in the practical scenarios of dense retrieval.
Considering the trade-off between efficiency and effectiveness, researchers resort to knowledge distillation (KD) \cite{DBLP:journals/corr/HintonVD15,sanh2019distilbert,jiao2019tinybert}, which aims to transfer knowledge from a strong and large teacher model to a small yet efficient student model.
Popular distillation methods involve response-based methods and feature-based methods.
Response-based methods \cite{DBLP:journals/corr/HintonVD15, kim2018paraphrasing, mirzadeh2020improved} train the student model to predict the output of the teacher model, which is simple yet effective in different tasks.
However, compared with feature-based methods, response-based methods ignore the intermediate signals of teachers, which are proven to be important in deep and thin neural network~\cite{DBLP:journals/corr/RomeroBKCGB14} and lead to lower upper limits than feature-based methods.
Feature-based methods mainly use the representations or attention maps of the intermediate layers as features, directly matching the features of the teacher and the student \cite{DBLP:journals/corr/RomeroBKCGB14,zagoruyko2016paying,DBLP:conf/iclr/ZagoruykoK17,passalis2018learning,xu2020feature,DBLP:journals/corr/abs-2112-04195,haidar2021rail}.
However, these methods suffer from two major weaknesses:
(1) In our preliminary study, feature-based methods do not work in dense retrieval, whose performances are even lower than response-based methods.
(2) Matching the features requires the teacher and the student to share the same tokenizer-vocabulary set or internal output size, which would somehow constrain the flexibility choices of the teacher model.
To fill the gap, we propose a liberal feature-based distillation method (LEAD) for dense retrieval.
Specifically, we use the internal \texttt{[CLS]} embedding vectors to calculate the similarity distribution of the passages as features in the mini-batch.
Then, we align the features between the teacher and the student, to accomplish the feature matching without tokenizer of architecture constraints.
We also randomly select layers from the model to participate in the feature matching, adaptively assigning weights to the selected layers based on their informativeness.
Besides, we jointly train the teacher model and student model to let them learn from each other.
The reason why we call our method \textbf{liberal} comes from two folds:
(1) LEAD supports various teacher architectures like dual encoder, ColBERT, cross encoder, and so on, only assuming that the teacher has a layer number no less than the student.
(2) LEAD uses the label distributions to calculate distillation loss, which is free of the constraints on vocabularies, tokenizers, or model architectures.
To evaluate the effectiveness of LEAD, we conduct experiments on several public benchmarks, such as MS MARCO \cite{DBLP:conf/nips/NguyenRSGTMD16} and TREC Deep Learning Track \cite{DBLP:journals/corr/abs-2003-07820,DBLP:journals/corr/abs-2102-07662}, where LEAD achieves better performances than traditional response-based and feature-based methods.
Furthermore, we also find that LEAD has two main advantages:
(1) LEAD is extendable to the broad sense ``layers'', which include Transformers layers and linear layers that can be appended after the models depending on the downstream tasks.
(2) LEAD is portable for the existing popular distillation pipeline for dense retrieval \cite{DBLP:conf/sigir/ZengZV22,lin2022prod}, which means it not only achieves superior performance but can also achieve further improvements with other effective pipelines.
\section{Related Work}
\subsection{Dense Retrieval}
Dense retrieval methods \cite{DBLP:conf/sigir/ZhanM0G0M21, lin2021batch, lin2022prod, lu2022ernie} have the potential of identifying semantically relevant but text-independent query-passage pairs.
The query and passage representations in dense retrieval models are usually obtained using dual encoders \cite{DBLP:conf/emnlp/KarpukhinOMLWEC20}.
ColBERT~\cite{DBLP:conf/sigir/KhattabZ20} pushes performance higher by introducing the late interaction while requiring higher computation and storage costs.
In spite of the model structure, recent work also focuses on utilizing training strategies to obtain better results, ranging from data-centric studies~\cite{DBLP:conf/acl/RenLQLZSWWW21, sciavolino2021simple} and negative sampling~\cite{gao2021your, DBLP:conf/iclr/XiongXLTLBAO21, DBLP:journals/corr/abs-2210-11773} to distillation~\cite{DBLP:journals/corr/abs-2010-02666, DBLP:conf/sigir/ZengZV22, DBLP:journals/corr/abs-2205-09153}.
Our work is in line with the distillation methods, where LEAD can be a better alternative to the widely-used response-based methods.
\subsection{Knowledge Distillation}
Knowledge distillation~\cite{DBLP:journals/corr/HintonVD15} aims to transfer knowledge from an effective teacher model to an efficient student model, which can be divided into two categories: response-based methods and feature-based methods.
The main idea of response-based methods~\cite{lin2022prod, DBLP:conf/sigir/ZengZV22} is to let student model mimic the final predictions of the teacher model~\cite{gou2021knowledge}, which is simple yet effective in different tasks.
However, these methods fail to take into account the intermediate signals of the teacher model, which turns out to be very important in very deep neural networks~\cite{romero2014fitnets}.
Feature-based methods are thought to be the extension of response-based methods and are especially useful in thinner and deeper networks~\cite{gou2021knowledge}, which focus on aligning the internal representations or attention scores.
\citet{DBLP:journals/corr/RomeroBKCGB14,xu2020feature,wang2020exclusivity,sun2019patient,haidar2021rail} directly match the internal representations of the teacher and the student, which is not suitable for cross encoders.
Besides, \citet{zagoruyko2016paying, passban2021alp, huang2017like,li2021virt} directly enclose the attention maps of the teacher and the student, which are constrained on used vocabularies and tokenizers.
Unlike distillation work on multiple teachers \cite{DBLP:journals/corr/abs-2205-09153,lin2022prod}, LEAD focuses on single teacher, taking advantage from both response-based and feature-based methods, enclosing the label distributions of intermediate layers, thus leaving no constraints on the vocabularies, tokenizers, or model architectures.
\section{Method}
We first introduce the task description of dense retrieval and then offer a unified view of retrieval models.
Based on the unified view, we elaborate the liberal feature-based distillation method.
\subsection{Task Description}
The target of dense retrieval is to retrieve passages based on learned distributed representations of queries and passages.
Formally, given a query set ${\mathbb{Q}}=\{ {\bm{q}}_1, {\bm{q}}_2, \dots, {\bm{q}}_{n} \}$ containing $n$ queries and a passage set ${\mathbb{P}}=\{ {\bm{p}}_1, {\bm{p}}_2, \dots, {\bm{p}}_{m} \}$ containing $m$ passages, dense retrieval aims to find the most relevant passages in corpus ${\mathbb{P}}$ for each query ${\bm{q}}_i$.
\subsection{Unified View of Retrieval Models}
\label{sec:view}
In the research line of dense retrieval, there are several popular models consisting of stacked neural networks, such as dual encoder \cite{DBLP:conf/emnlp/KarpukhinOMLWEC20}, ColBERT \cite{DBLP:conf/sigir/KhattabZ20}, and cross encoder \cite{DBLP:conf/naacl/QuDLLRZDWW21}.
Although these models have diverse architectures, they shall have common characteristics.
Therefore, we offer a unified view of these models in this section.
Consider dividing the above-mentioned models into two parallel aligned modules, $E_1$ and $E_2$, which are two piles of neural layers (including Transformers and linear layers) with the same layer numbers.
Given a model with $N$ layers, let's use $E_{1}^{i}$ and $E_{2}^{i}$ ($1 \leq i \leq N$) to denote the encoding of the first $i$ layers of $E_1$ and $E_2$, respectively.
The two major differences between those models are (1) the instantiations of $E_1$ and $E_2$ and (2) the similarity calculation function $f(\cdot)$.
To better elaborate the unified view, we show the following examples.
\textbf{Dual Encoder (DE)} \cite{DBLP:conf/emnlp/KarpukhinOMLWEC20} is the most widely used dense retrieval architecture, which encodes queries and passages into dense vectors separately, calculating the relevance score through the inner product.
For DE, $E_1$ is the query encoder and the $E_2$ is the passage encoder.
Both of them are Transformer encoders.
The similarity calculation function $f_{\textrm{DE}}(\cdot)$ is defined as:
\begin{equation}
\footnotesize
f_{\textrm{DE}}({\bm{q}}, {\bm{p}}) = E_{Q}({\bm{q}})^{T} \cdot E_{P}({\bm{p}})
\label{equ:descore}
\end{equation}
\textbf{ColBERT (CB)} \cite{DBLP:conf/sigir/KhattabZ20} can be viewed as a more expressive dual-encoder, which delays the interaction between query and passage after encoding.
The instantiation of $E_1$ and $E_2$ is the same as DE.
But the similarity calculation function $f_{\textrm{CB}}(\cdot)$ is defined as:
\begin{equation}
\footnotesize
f_{\textrm{CB}}({\bm{q}}, {\bm{p}}) = \sum_{1 \leq x \leq X} \mathop{\textrm{max}}\limits_{1 \leq y \leq Y} E_{Q}({\bm{q}})_x \cdot E_{P}({\bm{p}})_y
\label{equ:colscore}
\end{equation}
where $X$ and $Y$ denote the length of the query and passage token sequence, respectively.
Please note that, following \citet{DBLP:journals/corr/abs-2010-02666}, we remove the punctuation filter and the last linear layer of the encoders to focus on distillation.
\textbf{Cross Encoder (CE)} \cite{DBLP:conf/naacl/QuDLLRZDWW21} has strong abilities to capture the fine-grained relationships between queries and passages within the Transformer encoding.
Much different from DE and CB, for CE, $E_1$ is the query-passage pair encoder $E_{\textrm{CE}}$ and $E_2$ is the projection layer ${\bm{w}}$ after the Transformer encoder, which is used in a shared manner.
The similarity calculation function $f_{\textrm{CE}}(\cdot)$ is defined as:
\begin{equation}
\footnotesize
f_{\textrm{CE}}({\bm{q}}, {\bm{p}}) = {\bm{w}}^T \cdot E_{\textrm{CE}}([{\bm{q}};{\bm{p}}])
\label{equ:descore}
\end{equation}
where $[;]$ is the concatenation operation.
\begin{figure*}
\centering
\includegraphics[page=1,width=0.8\textwidth]{figures/framework7.pdf}
\caption{
The unified framework of LEAD, which supports various teacher architectures.
Assume the teacher and student model has $N$ and $M$ layers ($N \geq M$), respectively.
The \textit{Module1} and \textit{Module2} correspond to the $E_1$ and $E_2$ of the teacher model, respectively.
${\bm{t}}_i$ and ${\bm{s}}_i$ is the similarity distribution of the passages in the current batch, which is calculated with the $i$-th layer for the teacher and student model using $f(*)$ and $g(*)$, respectively.
$a_1, a_2, ..., a_K$ is the current random selection of teacher layers for layer-wise distillation.
$b_1, b_2, ..., b_K$ is the current random selection of student layers for layer-wise distillation.
The selected layers preserve the original partial order $1 \leq a_1 < a_2 < ... < a_K \leq N$ and $1 \leq b_1 < b_2 < ... < b_K \leq M$.}
\label{fig:framework}
\end{figure*}
\subsection{Distilling with Liberal Layer Feature}
Previous response-based distillation methods \cite{DBLP:conf/naacl/QuDLLRZDWW21,DBLP:journals/corr/abs-2010-11386,luan2021sparse} mainly distill knowledge from the output logits of the entire teacher model to the student model.
However, considering that different model layers try to capture the relationships between queries and passages at different granularity, we believe distilling knowledge from intermediate layers may be good for the student model.
Thus, based on the unified view, we propose a liberal feature-based distillation method (LEAD) based on layer-wise alignment, which is shown in \cref{fig:framework}.
\paragraph{Layer Feature Definition.}
In general, the backbone of our model is a pile of Transformer encoders, which abstracts the input information from low-level structure to high-level semantics.
For each layer, we calculate the score distribution of the input query-passage pairs in a minibatch, which is defined as the \textbf{layer feature}:
\begin{equation}
\footnotesize
{\bm{t}}_{i} = \mathop{\forall}_{{\bm{p}} \in {\mathbb{P}}^+ + {\mathbb{P}}^-} \frac{\exp(f({\bm{q}}, {\bm{p}}))}{\sum_{{\bm{p}}' \in {\mathbb{P}}^+ + {\mathbb{P}}^-}\exp(f({\bm{q}}, {\bm{p}}'))}
\label{equ:studentscorepd}
\end{equation}
\begin{equation}
\footnotesize
{\bm{s}}_{i} = \mathop{\forall}_{{\bm{p}} \in {\mathbb{P}}^+ + {\mathbb{P}}^-} \frac{\exp(g({\bm{q}}, {\bm{p}}))}{\sum_{{\bm{p}}' \in {\mathbb{P}}^+ + {\mathbb{P}}^-}\exp(g({\bm{q}}, {\bm{p}}'))}
\label{equ:studentscorepd}
\end{equation}
where ${\mathbb{P}}^+$ and ${\mathbb{P}}^-$ is the relevant and negative passage pool of ${\bm{q}}$, respectively;
$f(*),g(*)$ is the similarity score function of the student model and teacher model, respectively;
$i$ denotes the $i$-th layer.
\paragraph{Layer Selection.}
In each layer-wise alignment, we need to find a teacher layer for each student layer.
Without loss of the generality, we randomly select $K$ layers ($N \geq M \geq K$) from teacher model $A = \{a_1, a_2,..., a_K\} $ and $K$ layers from student model $B = \{b_1, b_2,..., b_K\}$, respectively, keeping their original partial orders.
Then, we distill knowledge from the $i$-th selected teacher layer $a_i$ to the $i$-th selected student layer $b_i$ by forcing the layer features ${\bm{t}}_{a_i}$ and ${\bm{s}}_{b_i}$ to be close:
\begin{equation}
\footnotesize
\mathcal{L}_{\textrm{lyr}}^{i}=D_{\mathrm{KL}}({\bm{t}}_{a_i}/\tau, {\bm{s}}_{b_i}/\tau)
\label{equ:ctdsoftloss}
\end{equation}
where $\tau$ is the temperature.
\paragraph{Layer Re-weighting.}
Although different teacher layers contain different information, some layers may contain noisy information which may be harmful to the model performance.
Thus, we propose a layer re-weighting technique by assigning different weights to involved teacher layers.
The weights are calculated based on the teacher layer features and the ground truth label distribution:
\begin{equation}
\footnotesize
w_i = \frac{\exp(-D_{\mathrm{KL}}({\bm{y}}, {\bm{t}}_{a_i},) / \tau)}{\sum_{{\bm{j}} = 1}^K \exp(-D_{\mathrm{KL}}({\bm{y}}, {\bm{t}}_{a_j}) / \tau)}
\end{equation}
where ${\bm{y}}$ is the ground truth one-hot label.
At last, the layer distillation loss is the weighted sum of the loss items of all the selected layers.
\begin{equation}
\footnotesize
\mathcal{L}_{\textrm{lyr}} = \sum_{i=1}^K w_i \times \mathcal{L}_{\textrm{lyr}}^{i}
\label{equ:layer_sum}
\end{equation}
\subsection{Joint Training of Teacher and Student}
Previous distillation methods for dense retrieval \cite{DBLP:journals/corr/abs-2010-02666, lin2020distilling, lin2022prod, DBLP:conf/sigir/ZengZV22} usually tune the student with a fixed teacher, while the supervision signals come from the differences in the output label distributions.
We think that the pre-learned teacher models that are converged on the training data could further benefit from distillation.
Therefore, we jointly train the teacher model and the student model, allowing them to learn from each other.
First, we adopt the bi-directional response-based loss as:
\begin{equation}
\footnotesize
\mathcal{L}_{\textrm{rep}}=D_{\mathrm{KL}}({\bm{t}}_{N}/\tau, {\bm{s}}_{M}/\tau) + D_{\mathrm{KL}}({\bm{s}}_{M}/\tau, {\bm{t}}_{N}/\tau)
\label{equ:ctdsoftloss}
\end{equation}
Second, we calculate the hard loss items by the output distributions and ground truth as:
\begin{equation}
\footnotesize
\mathcal{L}_{\textrm{tch}} = -\sum_{{\bm{p}}^+ \in {\mathbb{P}}^+} \log\frac{\exp(f({\bm{q}}, {\bm{p}}^+))}{\sum_{{\bm{p}} \in {\mathbb{P}}^+ + {\mathbb{P}}^-} \exp(f({\bm{q}}, {\bm{p}}))}
\label{equ:dtdhardloss}
\end{equation}
\begin{equation}
\footnotesize
\mathcal{L}_{\textrm{stu}} = -\sum_{{\bm{p}}^+ \in {\mathbb{P}}^+} \log\frac{\exp(g({\bm{q}}, {\bm{p}}^+))}{\sum_{{\bm{p}} \in {\mathbb{P}}^+ + {\mathbb{P}}^-} \exp(g({\bm{q}}, {\bm{p}}))}
\label{equ:dtdhardloss}
\end{equation}
As last, the total loss is the combination of the layer distillation loss, the response-based loss and the hard loss:
\begin{equation}
\footnotesize
\mathcal{L}_{\textrm{total}} = \mathcal{L}_{\textrm{lyr}} + \mathcal{L}_{\textrm{rep}} + \mathcal{L}_{\textrm{tch}} + \mathcal{L}_{\textrm{stu}}
\label{equ:layer_sum}
\end{equation}
\subsection{Discussion}
\paragraph{Extendibility.}
Since LEAD is based on the unified view of retrieval models presented in \cref{sec:view}, in which the models are abstracted as piles of neural layers, we think LEAD is extendable to different types of layers.
For example, LEAD also supports variants of DE and CB models, such as appending linear layers to the naive output of DE and CB, reducing the output dimensionalities, which supports building a smaller vector-based index for passages.
\paragraph{Portability.}
LEAD is a good alternative to the traditional response-based distillation method, which can also be applied in the state-of-the-art distillation pipelines for dense retrieval, such as curriculum learning \cite{DBLP:conf/sigir/ZengZV22} and learning with multiple teachers \cite{lin2022prod}.
\section{Experiments}
\subsection{Experimental Setting}
\paragraph{Datasets.}
Experiments are conducted on several popular retrieval datasets: MS MACRO Passage Ranking (\textsc{MS-Pas}) \cite{DBLP:conf/nips/NguyenRSGTMD16}, TREC 2019 DL Track (\textsc{TREC-Pas-19} and \textsc{TREC-Doc-19}) \cite{DBLP:journals/corr/abs-2003-07820}, MS MARCO Document Ranking (\textsc{MS-Doc}) \cite{DBLP:conf/nips/NguyenRSGTMD16} and TREC 2020 DL Track (\textsc{TREC-Pas-20} and \textsc{TREC-Doc-20}) \cite{DBLP:journals/corr/abs-2102-07662}.
The statistics are shown in \cref{sec:data}.
\paragraph{Metrics.}
Following previous works~\cite{DBLP:conf/sigir/ZhanM0G0M21,lin2022prod}, for \textsc{MS-Pas}, we report \textbf{MRR@10}, \textbf{MAP@1k} and \textbf{R@1k} on the dev set.
For \textsc{TREC-Pas-19} and \textsc{TREC-Pas-20}, we report \textbf{nDCG@10} and \textbf{MAP@1k}.
For \textsc{MS-Doc}, we report \textbf{MRR@10} and \textbf{R@100} on the dev set.
For \textsc{TREC-Doc-19} and \textsc{TREC-Doc-20}, we report \textbf{nDCG@10} and \textbf{R@100}.
\subsection{Baselines}
To verify the effectiveness of our methods, we compare LEAD with three groups of methods.
The first group contains sparse retrieval methods and popular dense retrieval methods without distillation, including
\textbf{BM25} \cite{DBLP:conf/sigir/Yang0L17},
\textbf{DeepCT} \cite{DBLP:conf/sigir/DaiC19},
\textbf{docT5query} \cite{DBLP:journals/corr/abs-1904-08375},
\textbf{ANCE} \cite{DBLP:conf/iclr/XiongXLTLBAO21}, \textbf{ME-BERT} \cite{luan2021sparse} and
\textbf{ADORE} \cite{DBLP:conf/sigir/ZhanM0G0M21}.
The second group consists of recent dense retrieval methods enhanced with distillation, including
\textbf{Margin-MSE} \cite{DBLP:journals/corr/abs-2010-02666},
\textbf{TCT-ColBERT} \cite{DBLP:journals/corr/abs-2010-11386}
and
\textbf{RocketQA v1} \cite{DBLP:conf/naacl/QuDLLRZDWW21}.
The third group contains different knowledge distillation strategies like response-based method (\textbf{RD}) \cite{DBLP:conf/naacl/QuDLLRZDWW21} and feature-based method (\textbf{FD}) \cite{DBLP:journals/corr/abs-2112-04195}.
Furthermore, we use \textbf{12-layer DE}, \textbf{12-layer CB} and \textbf{12-layer CE} as the teacher model and conduct knowledge distillation with a 6-layer DE student.
\subsection{Implementation Details}
\paragraph{Model Initialization.}
We use \textsc{luyu/co-condenser-marco} from huggingface\footnote{\url{https://huggingface.co/}} to initialize the 12-layer teacher models and \textsc{distilbert-base-uncased} for the 6-layer dual encoder student model.
\paragraph{Training Process.}
First, we use random or BM25 negatives to train a 12-layer DE and retrieve top-100 hard negatives.
Then, we train the teacher model and student model with the mined hard negatives as warming up.
Finally, we load the trained checkpoints for the teacher model and student model, conducting the knowledge distillation.
The experiments are conducted on 8 NVIDIA V100 GPUs.
Detailed hyper-parameters for reproducing our experiments are listed in \cref{sec:hyps}.
\paragraph{Metric Calculation.}
For DEs, we follow the previous work \cite{DBLP:conf/iclr/XiongXLTLBAO21,DBLP:conf/naacl/QuDLLRZDWW21} and compute the metrics using the \texttt{IndexFlatIP} of faiss\footnote{\url{https://faiss.ai/}}.
While for CB and CE teacher models, we follow \citet{DBLP:journals/corr/abs-2010-02666,lin2020distilling,lin2022prod} and rerank the top-1000 results and the top-100 results retrieved by the 12-layer DE teacher model for \textsc{MS-Pas} and \textsc{MS-Doc}, respectively.
\subsection{Main Results}
\paragraph{Comparing with existing methods.}
We compare LEAD with other state-of-the-art text retrieval methods on \textsc{MS-Pas}, \textsc{TREC-Pas-19}, \textsc{MS-Doc} and \textsc{TREC-Doc-19}.
We use ``LEAD'' to denote using 12-layer CB as teacher model.
The results are shown in \cref{tab:main_results_pas1} and \cref{tab:main_results_doc1}.
From the result, we can find that: (1) LEAD achieves the best performance on almost all the metrics on both \textsc{MS-Pas} and \textsc{MS-Doc}.
(2) Among the baselines, knowledge distillation methods include: TCT-ColBERT \cite{DBLP:journals/corr/abs-2010-11386} which uses CB as teacher model, RocketQA v1 \cite{DBLP:conf/naacl/QuDLLRZDWW21} which uses CE as teacher model, and Margin-MSE \cite{DBLP:journals/corr/abs-2010-02666} which uses the ensemble of multi CE as teachers.
The best setting of LEAD which uses CB as teacher model performs better than all of them.
\paragraph{Comparing with different KD strategies.}
To further compare with other knowledge distillation methods in a fair setting, we conduct experiments using different kinds of teacher models, such as DE, CB and CE.
The results on \textsc{MS-Pas}, \textsc{TREC-Pas-19}, \textsc{TREC-Pas-20}, \textsc{MS-Doc}, \textsc{TREC-Doc-19} and \textsc{TREC-Doc-20} are illustrated in \cref{tab:main_results2}.
As we can see from the table, it is obvious that we can get the following conclusions:
(1) Compared to the original 6-layer DE student, LEAD can further achieve improvements with various teacher models, especially for MRR@10, MAP and nDCG@10.
(2) When training with the same teacher model, generally speaking, RD is better than FD in all the comparisons, indicating that FD is not easy to work on distillation for dense retrieval.
Furthermore, LEAD outperforms other KD strategies, such as FD and RD, on all the datasets and metrics, showing the effectiveness of our method.
(3) Among the explored teacher models, 12-layer CB is the best for boosting a 6-layer DE student when using LEAD.
We guess the reason may be two folds.
First, CB and DE share a similar architecture while the late interaction brings CB better performances, which further leads to better distillation results.
Second, although the CE teacher outperforms the CB teacher, training with CB in RD and LEAD can benefit from the cross-batch negatives \cite{DBLP:conf/naacl/QuDLLRZDWW21}, which is limited in CE.
Therefore, we use 12-layer CB as the teacher model for LEAD in the rest experiments.
\subsection{Effect on Layer Selection}
\label{sec:layer_selection}
We explore the effects of two major factors in the layer selection of LEAD.
\input{tables/layer_selection1}
\paragraph{Study on Layer Number $K$.}
We tune the distillation layer number $K$ in the range of $\{1, 2, 3, 4, 5, 6\}$ on \textsc{MS-Pas} and \textsc{MS-Doc}, and observe the trends of student performances, which is shown in \cref{tab:layer_selection1}.
We can observe that the model performance gradually improves with the increase of $K$.
However, the model performance reaches the peak at $K=5$ on \textsc{MS-Pas} and $K=6$ on \textsc{MS-Doc}, respectively.
This is because, with more distillation layers, the student model can absorb more knowledge from the teacher model, which benefits model performance, but eventually the best layer number highly depends on the dataset characteristic.
Therefore, we use $K=5$ for \textsc{MS-Pas} and $K=6$ for \textsc{MS-Doc} in later experiments.
\input{tables/layer_selection2}
\paragraph{Study on Selection Strategies.}
In LEAD, we randomly select intermediate layers from teacher model and student model for knowledge distillation. We further investigate the performance of different layer selection strategies including Random, Last and Skip in \citet{sun2019patient,haidar2021rail}. Random refers to randomly selecting intermediate layers for knowledge distillation. Last refers to distilling information from the last layers. And Skip refers to distilling information across every $k$ layers.
More precisely, on \textsc{MS-Pas}, the layer choice for student model is \{2, 3, 4, 5, 6\}.
We set $k=2$ and the layer choice for teacher model is \{8, 9, 10, 11, 12\} in Last and \{1, 3, 5, 7, 9\} in Skip.
It is worth mentioning that the layer selection choice keeps dynamically changing during training in Random while remaining fixed in Last and Skip. The result of the selection strategies on \textsc{MS-Pas} is shown in \cref{tab:layer_selection2}.
From the result, we can find that our layer selection strategy Random is better than Last and Skip. Presumably, it might be due to the fact that Last and Skip only distill information from part of intermediate layers, which may lose information from unchosen layers. However, by distilling information from dynamically changing intermediate layers, Random captures more diverse distributions of richer semantics from low-level to high-level.
\subsection{Ablation Study}
\input{tables/ablation}
To show the effectiveness of each component, we conduct an ablation study on LEAD.
Since the layer selection has been discussed in \cref{sec:layer_selection}, we focus on the joint training and the layer re-weighting, whose results are shown in \cref{tab:ablation}.
First, to completely forbid the joint training, we directly remove the teacher's hard loss item $\mathcal{L}_{\textrm{tch}}$ and $D_{\mathrm{KL}}({\bm{s}}_{M}/\tau, {\bm{t}}_{N}/\tau)$ in $\mathcal{L}_{\textrm{rep}}$ from the total loss, freezing the teacher model during training.
It is clear that after removing the joint training from LEAD, the student's performances on nearly all the metrics decreased.
The reason may be that the information of the teacher model and the student model can complement each other.
Second, we replace the weight $w_i$ with a constant $1/K$ in \cref{equ:layer_sum} to show the effects of the layer re-weighting.
As we can see, removing the layer re-weighting causes a larger performance drop in \textsc{MS-Pas} compared with the joint training, but still outperforms response-based methods.
We think the reason may be that by assigning higher weights to more informative layers, the noises from uninformative layers can be largely reduced.
\subsection{Further Study}
\input{tables/extendability.tex}
\paragraph{Extendibility.}
To verify the extendibility of LEAD, we append a linear layer 768 $\rightarrow$ 768 after the student model and teacher model, and observe the performance of RD, FD and LEAD on \textsc{MS-Pas}, which is shown in \cref{tab:extend}.
We can find that after appending the linear layer, the performance of all distillation methods drops. However, our method LEAD+linear still achieves the best performance, which demonstrates the extendibility of our method.
\input{tables/plug_in.tex}
\paragraph{Portability.}
To verify the portability of LEAD, we follow PROD~\cite{lin2022prod} and choose three teacher models for continual distillation, which include DE, CB and CE.
PROD* refers to using RD in the distillation process, while PROD*+LEAD refers to using LEAD in the distillation process.
The results on \textsc{MS-Pas} are shown in \cref{tab:plug_in}.
From the result, we can find that:
(1) The performance of student model gradually improves in each step for both PROD* and PROD*+LEAD. However, the performance gain of PROD*+LEAD in each step is better than that of PROD*, which leads to better performance of the final student model. The reason may be that compared with RD, LEAD can fully utilize the information from the intermediate layers of the teacher model.
(2) Compared with LEAD which solely uses CB as teacher model, the result of PROD*+LEAD after continual distillation gets little improvement. This may be due to the fact that the multi-teachers used in the continual distillation share similar abilities.
Overall, the experiment results prove the feasibility of plugging LEAD into the existing distillation paradigm and further improving the model performance.
\input{tables/teacher.tex}
\paragraph{Joint training benefits teacher model.}
We propose to jointly train the teacher and student with the intuition that they can complement each other.
To verify it, we evaluate the performance of the teacher model before and after LEAD on \textsc{MS-Pas} and \textsc{MS-Doc}, which is shown in \cref{tab:teacher}.
We can find that the performance of teacher model can be greatly improved after LEAD on both datasets, which verifies that teacher model can learn from student model. Besides, as shown in \cref{tab:ablation}, when we remove joint training from LEAD, the model performance degrades severely. However, the performance of LEAD without joint training is still better than that of RD and FD.
\section{Conclusion}
In this paper, we propose a novel knowledge distillation method LEAD for dense retrieval.
To be specific, LEAD aligns the distribution of the student model and teacher model and pays more attention to the most informative layers using the layer re-weighting technique. LEAD is extendable to more linear layers and is easy to plug into the existing distillation pipelines.
Extensive experiments on six widely-used benchmarks show that LEAD can significantly improve the performance of the student model after distillation compared with response-based methods and feature-based methods.
We further carried out some hyper-parameter analysis and ablation experiments to prove the effectiveness of LEAD, which also inspires future research direction on the distillation of dense retrievers.
\section*{Acknowledgement}
\section{Hyper-parameters}
\label{sec:hyps}
\input{tables/hyper}
The detailed hyper-parameters are shown in \cref{tab:hyper}.
\section{Dataset Statictics}
\label{sec:data}
\input{tables/dataset_statistics}
The statistics are shown in \cref{tab:dataset}.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 7,976 |
Naseem Khan Achakzai is currently living in Lahore but born and raised in Quetta.
He Completed his education in O-levels from Wilderness School Quetta, A-Levels from City School Quetta, Bachelors in Sociology and Economics from the University of Balochistan, Masters in Public Administration from the University of Management and Technology Lahore, MSc in Poverty reduction from School of Oriental and African studies the University of London.
He was a Pashtoo Instructor at Kuch Khaas.
He was a Director Operations at Comprehensive Disaster Response Services (CDRS).
Founder and CEO of Hila a non-profitable, non-political and non-religious organization.
Chairman Pakistan Chapter at International Youth Council.
He was a consultant at Directorate General Monitoring & Evaluation P&D Department Govt of Punjab.
He was a Social Development Specialist at University of Management and Technology.
He is a director at the International Youth Summit.
He is a Founding Executive Director at CSRP.
Naseem Khan Achakzai is a source of Inspiration for all the young blood of Balochistan. He is living a successful life which others aspire.
But the fact is that he works very hard and dedicatedly acquire all the hurdles came into his way. He is showing the positive side of Balochistan in the world with his abilities. | {
"redpajama_set_name": "RedPajamaC4"
} | 4,977 |
Q: python - Check two different columns for distinct strings I am just trying to write a condition that checks two columns from my pandas df and return records only where plan = free and role = owner as shown below. BUT i get an error due to my 'and' condition below. Is there an easy solution here?
Table:
plan | role | [...]
free | owner | [...]
trial| guest | [...]
Python
free_admin = (df['plan'] == 'free' and df['role'] == 'owner')
df_try = df[free_admin]
df_try.head()
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 9,148 |
Die Tour der südafrikanischen Cricket-Nationalmannschaft in die West Indies in der Saison 2000/01 fand vom 4. März bis zum 16. Mai 2001 statt. Die internationale Cricket-Tour war Teil der internationalen Cricket-Saison 2000/01 und umfasste fünf Test Matches und sieben ODIs. Südafrika gewann die Testserie 2-1 und die ODI-Serie 5-2.
Vorgeschichte
Die West Indies bestritten zuvor ein Drei-Nationen-Turnier in Australien, Südafrika eine Tour gegen Sri Lanka.
Das letzte Aufeinandertreffen der beiden Mannschaften bei einer Tour fand in der Saison 1998/99 in Südafrika statt.
Stadien
Für die Tour wurden folgende Stadien als Austragungsorte vorgesehen und am 18. August 2000 bekanntgegeben.
Kader
Südafrika benannte seinen Test-Kader am 17. Februar 2001.
Die West Indies benannte ihren Test-Kader am 6. März und ihren ODI-Kader am 22. April 2001.
Tour Matches
Test Matches
Erster Test in Georgetown
Zweiter Test in Port of Spain
Dritter Test in Bridgetown
Vierter Test in St John's
Fünfter Test in Kingston
One-Day Internationals
Erstes ODI in Kingston
Zweites ODI in St John's
Drittes ODI in St. George's
Viertes ODI in St. George's
Fünftes ODI in Bridgetown
Sechstes ODI in Port of Spain
Siebtes ODI in Kingstown
Statistiken
Die folgenden Cricketstatistiken wurden bei dieser Tour erzielt.
Player of the Series
Als Player of the Series wurden die folgenden Spieler ausgezeichnet.
Player of the Match
Als Player of the Match wurden die folgenden Spieler ausgezeichnet.
Weblinks
Die Serie auf espncricinfo.com
Einzelnachweise
Internationale Cricket-Saison 2000/01
Tour der südafrikanischen Cricket-Nationalmannschaft in den West Indies
Cricket-Wettbewerb in Antigua und Barbuda
Cricket-Wettbewerb in Barbados
Cricket-Wettbewerb in Jamaika
Cricket-Wettbewerb in Grenada
Cricket-Wettbewerb in Guyana
Cricket-Wettbewerb in St. Vincent und den Grenadinen
Cricket-Wettbewerb in Trinidad und Tobago | {
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City deputy fire chief heading to Pa.
By ROBERT KOCH
Hour Staff Writer
Deputy Fire Chief Robert Talloni, a 28-year veteran of the Norwalk Fire Department, could be bound for Harrisburg, Pa., to head the fire department there.
"It sounds like it's moving forward," said Norwalk Fire Chief Denis McCarthy when asked about news reports from the Harrisburg area. "What the process is, because he's eligible for retirement, a request for retirement would have to be approved by the Fire Commission. He is eligible for a retirement pension. I have not received a request. There's been discussion internally. I think (Talloni) has spoken with members of his own shift."
Talloni could not be reached for comment Wednesday afternoon.
Michael Holmes, chief of staff for Harrisburg Mayor Linda Thompson, did not return several telephone calls to The Hour Newspapers on Wednesday afternoon.
WPTM FOX43 of York, Pa., reported Tuesday that "Talloni will be taking over as the Harrisburg Fire Chief." Holmes confirmed Monday that a new chief had been selected. Thompson will hold a press conference March 29 to introduce the new chief to the community, according to the York television station.
Harrisburg is the capital of Pennsylvania and has a population of roughly 49,000.
Talloni is one of five deputy fire chiefs employed by the Norwalk Fire Department. Each deputy chief works as shift commander and is responsible for all five engine companies, two ladder companies, the rescue trucks and all fire suppression activities during that shift, according to McCarthy.
"He is the incident commander for all of the calls," McCarthy said.
Talloni started working for the department in April 1982. He was promoted to inspector and later lieutenant in 1991, captain in 1994 and deputy chief in 2004, according to the city.
His salary in 2009 was $140,476. That included roughly $38,000 in overtime pay, according to the city Comptroller's Office.
If Talloni leaves the department, he is entitled per Fire Commission approval to a pension of 2.5 percent of his base salary multiplied by the numbers of years he was employed by the department. That would translate to an annual pension of about $69,298.
The position would be filled internally as set out in the collective-bargaining agreement between the city and the Norwalk Professional Firefighters Association Local 830, according to McCarthy.
"We haven't started any process," McCarthy said. "We'll wait until we receive official notification and request for retirement." | {
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{"url":"https:\/\/organicdesign.nz\/Category:Hardware","text":"Categoria:Hardware\n\nJump to: navigation, search\n\nPages in category \u2018Hardware\u2019\n\nThe following 41 pages are in this category, out of 41 total.\n\nMedia in category \u2018Hardware\u2019\n\nThe following 16 files are in this category, out of 16 total.","date":"2018-03-22 08:21:42","metadata":"{\"extraction_info\": {\"found_math\": false, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.9083003401756287, \"perplexity\": 6407.393474228475}, \"config\": {\"markdown_headings\": false, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": false}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2018-13\/segments\/1521257647782.95\/warc\/CC-MAIN-20180322073140-20180322093140-00214.warc.gz\"}"} | null | null |
Female Pelvic Core
On Demand Programs
Book First Session
Career & Internship
2 Weeks of Unlimited Training
Our origin, founders, and purpose.
Started as a Dream
Build into a Business
Influencing the Industry
Gymnazo started as a childhood dream. Growing up, Michael loved sports and strength training in high school, but was always curious about how to learn to fix injuries and help people get back to what they love. He moved to Fresno State to study Kinesiology. Desiring to live in a coastal community, he graduated college, packed up his room and moved to SLO weeks later. No house, no job, one friend and a strong vision.
After standing in Kennedy Club Fitness handing out business cards to generate enough clients to convince the manager to hire him as a Personal Trainer, Michael started to carve out his reputation in San Luis Obispo (SLO). Knowing that Kinesiology and a basic personal training certification wasn't nearly enough education to deliver the best service, he started seeking graduate level education.
After going to a seminar about Applied Functional Science (AFS), Michael was blown away by the impact Gary Gray was having in the physical therapy world. The scientific methodology empowers practitioners to step away from merely treating the symptoms and help them quickly pinpoint the cause and with greater results. Michael could envision how blending these restorative techniques with fitness performance would be a powerful hybrid. He enrolled in the Fellowship of Applied Functional Science and started to integrate restoration strategies and three dimensional movement into his workouts.
Starting with just 18 athletes, Michael began to build a small group training program which he eventually named Gymnazo. In 2010, Gymnazo was starting to grow. He started to wear the many hats of an entrepreneur and needed some help. He met Paden and initially sought to hire her to help him run his brand strategy. But after three "business dates" they started actually dating and collaborating on Gymnazo. It worked so well the same year they got married, they incorporated and moved out of Kennedy Club Fitness. Eventually, she left Collaboration Business Consulting to work full time at Gymnazo. Her passion to market and grow businesses into sustainable models allows Michael to focus on innovating new services and building a team of highly skilled coaches. Today, Michael and Paden enjoy collaborating and each focus on different aspects of the business.
Our founder, Michael, since 7th grade always knew he had a passion and calling on his life to fix broken bodies and get people back to enjoying their lives. A son of an entrepreneur, he witnessed the need for grit, ambition and vision when creating and sustaining a business. But everything he has pursued through education and training has been to make this dream into a reality. He is our innovator, our internal movement guru and the one who is deeply connected to the dream to blend restoration training with performance.
Paden Hughes
Paden, our CEO, grew up the oldest of five, growing up competing in soccer and speech and debate. At an early age found a passion for developing her friends and siblings to believe in their natural abilities and to be confident in their uniqueness. She Went to Cal Poly to study communication and business with a plan to go into market consulting like her father. Her career started while in college working for an international consulting firm. Several years later she found fulfillment in executive coaching and small business consulting at a local level. She is a problem solver and loves pushing those around her to achieve great outcomes. She uses these skills daily to set goals for Gymnazo and lead the team to achieve these. She is our business leader, the one who keeps the business on track, the team inspired and accountable to results.
Both owners share a deep belief that Gymnazo innovative methodology blended with a powerful customer experience advances the fitness industry to new heights. They are determined to bring this model to the masses.
The name Gymnazo is from the original Greek root word for Gymnasium, which in ancient times meant to
"exercise so as to discipline one's mind, body and soul."
Gymnazo was founded on the belief that true transformation goes beyond the physical. It is mental. It is soulful. | {
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Ayrton Fagundes (February 11, 1937 – March 14, 1994) was a Brazilian broadcast journalist.
Best known as an anchorman for prime time television newscasts such as Camera 10, at TV Difusora (now Band TV), Globo TV Southern subsidiary RBS, and TV Record. Ayrton Fagundes belongs to the first generation of Brazilian journalists that have introduced the nightly news programming and weekly magazines as a new genre.
A presenter, editor, and executive producer, he spent the last years of his career in the capital of Brazil, Brasilia, where he was appointed bureau-chief of the Brazilian financial news daily Jornal do Comercio, of Porto Alegre, writing a daily column in politics.
Later he was appointed chief-spokesperson for the Secretaria Especial de Informatica (SEI), at the Brazilian Ministry of Science and Technology, during the early years of the information technology industry in Brazil, where he worked with the new regulatory framework surrounding the nascent computer industry. He was interviewed by Alan Riding, of The New York Times, on several occasions, regarding Brazil's "prickly computer policies".
References
Brazilian journalists
1994 deaths
1937 births
20th-century journalists | {
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Ionuț Alin Pop (born 1 August 1997) is a Romanian footballer who plays as a goalkeeper for Liga II side CSM Slatina, on loan from Hermannstadt.
Club career
Pop started his career in local club Bihor Oradea, before he signed to Italian first tier A.S. Roma. He never played a match in the first team, he was loaned to third tier Fidelis Andria to the 2016–17 season. He made his professional debut on 14 September 2016 against Paganese, playing 90 minutes. On 7 July 2017, fellow third tier side Alessandria signed him for an undisclosed fee.
On 26 June 2019, Ionuț Pop signed a contract with Liga I side FC Hermannstadt.
International career
Pop made his debut in the Romania national under-19 football team on 17 November 2015 against Switzerland. He received a red card in the 82nd minute.
Career statistics
Club
Honours
US Alessandria Calcio 1912
Coppa Italia Serie C: 2017–18
References
Sources
1997 births
Living people
Sportspeople from Oradea
Romanian footballers
Romania youth international footballers
Association football goalkeepers
FC Bihor Oradea players
Serie A players
A.S. Roma players
Serie C players
S.S. Fidelis Andria 1928 players
U.S. Alessandria Calcio 1912 players
Liga I players
Liga II players
FC Hermannstadt players
CSM Slatina footballers
Romanian expatriate footballers
Romanian expatriate sportspeople in Italy
Expatriate footballers in Italy | {
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Q: How does a nine volt battery make a spark? With a nine volt battery, touching the two terminals together (or using a faulty terminal) will cause a spark roughly where I would want it to be.
How is this possible? Is it ionizing only a very small portion of air surrounding the wires when this happens and it is just more visible? I believe at an extremely small distance, ~300v is the breakdown point of air (often, for example according to Paschen's law) so I do not understand how the battery can do this.
A: As the contact is being broken, a connection is made through very small pieces of metal (microscopic features), which have enough current through them to vaporize, the ions of which then support a current through the air briefly.
While lower voltages do not, in general, jump a gap that is present
before the voltage is applied, interrupting an existing current flow
with a gap often produces a low-voltage spark or arc. As the contacts
are separated, a few small points of contact become the last to
separate. The current becomes constricted to these small hot spots,
causing them to become incandescent, so that they emit electrons
(through thermionic emission). Even a small 9 V battery can spark
noticeably by this mechanism in a darkened room. The ionized air and
metal vapour (from the contacts) form plasma, which temporarily
bridges the widening gap.
Also, when a flowing current is interrupted, it will cause inductive kickback, where the collapsing magnetic field causes an increase in voltage, to try to maintain the existing current. The voltage can increase enough to cause dielectric breakdown of air and allow current to flow through it.
Attempting to open an inductive circuit often forms an arc, since the inductance provides a high-voltage pulse whenever the current is interrupted
Wikipedia: High voltage § Sparks in air
I'm not sure if inductive kickback is strong enough with a 9 V battery to cause a spark by itself, but it would help current to flow after the plasma path has formed.
A: Back EMF only occurs with an inductive or capacitive circuit,you dont have this with resistive circuit. the spark is because at last instant of contact the metal vapourises as previously described.If the voltage is sufficicent over 20 volts, the spark can become an arc,and can reach a length of several inches,the current still flows,until the separation becomes too great.If the circuit is broken to an inductance,the back emf from the coil will intensify the arc, and will assist to maintain the arc. A flow of electric current is difficult stop,and this is the beuty of DC, (but can be a nuisance) With AC, there is no net flow of current,and this flow stops and starts,so arcing is not an issue with AC,hence switches are primitive.
A: To answer this question, you'll need to know Ohm's Law: V=IR, as well as inductance which "stores" current or rather, resists changes in current.
What this means is that once a wire connection is made across the battery terminals current starts flowing through the wire. The current 'I' is equal to V/R, which is the battery voltage (9V) divided by the the resistance of the wire and battery. Now, remember the inductance of the system is going to try and maintain that current. When you disconnect the wire, even for fractions of sections, the inductance tries to hold 'I' constant. The act of breaking the connection makes 'R' go from very low to very high. Now if 'I' is constant and 'R' approaches infinity, then 'V' must also approaches infinity to balance the V=IR equation. That's how you get the voltage high enough to ionize gas and spark or burn a very small amount of remaining metal contact. Of course the voltage doesn't hit infinity because once it goes high enough to arc then current flows and discharges the inductance.
Earlier in this thread someone mentioned that when the connection is made for the first time only through a few small pieces of metal which causes all the current to flow through and burn it. That's actually incorrect since the few pieces of metal have a very high resistance, which won't allow enough current though anyways. It's only when the connection is broken that the system inductance forces the current higher than the resistance alone would allow.
| {
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Q: Running C++ program multiple times I have a C++ program which I need to run it multiple times.
For example:-
Run ./addTwoNumbers 50 times.
What would be a good approach to solve this problem?
A: In POSIX shells,
for i in {1..50} ; do ./addTwoNumbers ; done
A: If this is code you are writing, take the number of times you want to "run" as an argument:
#include <stdio.h>
#include <stdlib.h>
int main(int argc, char* argv[]) {
int numTimes = 1;
if (argc > 1)
{
numtimes = atoi(argv[1]);
}
for (int i = 0; i < numTimes; i++)
{
// Your code goes here
}
}
(Note this doesn't do any sanity checking on the input, but it should point you in the right direction)
A: The way you were asking the question indicated that you had a finished binary. You want to run it as if it was from the command line. The forward slash, to me, is a clue that you are a Unix like operating system user. Well, that, and the fact that this post is tagged "Unix", which I just saw after writing the below. It should all be applicable.
The scheme of using the shell is probably the simplest one.
man bash tells you how to write a shell script. Actually we need to figure out what shell you are using. From the command line, type:
echo $SHELL
The response I get is
/bin/bash
Meaning that I am running bash. Whatever you get, copy down, you will need it later.
The absolutely lowest knowledge base is to simply create a file with any standard text editor and no suffix. Call it, simply (for example) run50.
The first line is a special line that tells the unix system to use bash to run the command:
#! /bin/bash
(or whatever you got from echo $SHELL).
Now, in the file, on the next line, type the complete path, from root, to the executable.
Type the command just as if you were typing it on the command line. You may put any arguments to your program there as well. Save your file.
Do you want to run the program, and wait for it to finish, then start the next copy? Or do you want to start it 50 times as fast as you can without waiting for it to finish? If the former, you are done, if the latter, end the line with &
That tells the shell to start the program and to go on.
Now duplicate that line 50 times. Copy and paste, it is there twice, select all, and then paste at the end, for 4 times, again for 8, again for 16, and again for 32. Now copy 18 more lines and paste those at the end and you are done. If you happen to copy the line that says #! /bin/bash don't worry about it, it is a comment to the shell.
Save the file.
From the command line, enter the following command:
chmod +x ./filenameofmyshellcommand
Where you will replace filenameofmyshellcommand with the name of the file you just created.
Finally run the command:
./filenameofmyshellcommand
And it should run the program 15 times.
If you are using bash, instead of duplicating the line 50 times, you can write a loop:
for ((i=1;i<=50;i++)) do
echo "Invocation $i"
/complete/path/to/your/command
done
I have included a message that tells you which run the command is on. If you are timing the program I would not recommend a "feelgood" message like this. You can end the line with & if you want the command to be started and the script to continue.
The double parenthesis are required for this syntax, and you have to pay your syntax.
for ((i=1;i<=50;i++)) do echo "invocation $i" & done
is an interesting thing to just enter from the command line, for fun. It will start the 50 echos disconnected from the command line, and they often come out in a different order than 1 to 50.
In Unix, there is a system() library call that will invoke a command more or less as if from the terminal. You can use that call from C++ or from perl or about a zillion other programs. But this is the simplest thing you can do, and you can time your program this way. It is the common approach in Unix for running one program or a sequence of programs, or for doing common tasks by running a series of system tools.
If youy are going to use Unix, you should know how to write a simple shell script.
A: int count=0;
int main()
{
beginning:
//do whatever you need to do;
int count++;
if (count<=50);
{
goto beginning;
}
return 0;
}
| {
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{"url":"https:\/\/techprpr.com\/2020\/08\/01\/page\/2\/","text":"## Server Bug Fix: OpenVPN connection issue on Azure VM\n\nI had many un succesfull attempts at installing openvpn access server on a VM but everytime the Issue is same. I connect to my VPN after installing but when I go to any website it says \u201cERR_NAME_NOT_RESOLVED\u201d. Please tell me what is the Issue and I had install openvpn via this link: https:\/\/medium.com\/@evgenijrenke\/configure-openvpn-access-server-on-azure-6e6120bacddf. T had even tried to flush my Windows DNS but of no use.\n\nTagged : \/ \/\n\n## Math Genius: Simplified tensor formula\n\nA is a fourth-order tensor, B is a second-order tensor, and C is a second-order tensor\n$$mathbf{A}:mathbf{B}cdotmathbf{C}=[quad ]:mathbf{B}$$\n$$mathbf{B}cdotmathbf{C}=[quad ]:mathbf{B}$$\n$$mathbf{B}cdotmathbf{C}=[quad ]:mathbf{C}$$\nWhat is the expression in brackets\uff1fCan you derive it step by step\uff1fthank you very much\uff01\n\nWrite the first equation in index notation (employing Einstein summation)\neqalign{ A_{ijkl}:B_{kp}C_{pl} &= left(A_{ijkl}C^T_{lp}right):B_{kp} \\ }\nThe term in parentheses is mystery tensor, i.e.\n$$T = Acdot C^T$$\nThe second and third equations can be written as\neqalign{ B_{ij}C_{jk} &= left(B_{ij}C_{jk};M_{lp}right):B_{lp} \\ B_{ij}C_{jk} &= left(B_{ij}C_{jk};N_{lp}right):C_{lp} \\ }\nSo any matrix $$M$$ which satisfies $$M:B=1$$ will solve the first equation, i.e.\n$$T=(Bcdot C)star M$$\nwhere $$(star)$$ denotes the dyadic product.\n\nSimilarly, any $$N$$ which satisfies $$N:C=1$$ solves the second equation, i.e.\n$$T=(Bcdot C)star N$$\nHere are some concrete examples. Let $$M$$ be a matrix in which all of the elements are zero except for $$M_{11}=B_{11}^{-1}$$. A second solution would take $$M_{12}=B_{12}^{-1}$$.\n\nAnother type of solution takes a random matrix $$R$$ and scales it to satisfy the constraint\n$$M=frac{R}{R:B}=frac{R}{{rm Tr}(RB^T)} quadimpliesquad M:B = 1$$\nObviously, there are an infinite number of solutions to the last two equations.\n\n## Server Bug Fix: Oracle VM and live storage migration (like storage vMotion on VMware), is it possible?\n\nI have to migrate VMs to new storage for a client. Hypervisor is Oracle VM, versions 3.3 & 3.4. Haven\u2019t access to the machines yet, I\u2019m searching for info for live storage migration on this platform, Oracle VM.\n\nIn Oracle docs, Oracle VM 3.4 docs this doesn\u2019t seem possible (only if the VM is stopped). In Oracle VM datasheet it mentions, Storage live VM migration, perform live migrations of running virtual machines that have virtual disks on local storage.\n\nSo I am confused. Does anybody have any experience of live storage migration on Oracle VM, versions 3.3 & 3.4?\n\nTagged : \/ \/\n\n## Ubuntu HowTo: Lenovo Touchpad issue 20.04\n\nI decided to swap from Windows to Ubuntu 20.04 (I\u2019m a newbie) and I found a problem with the touchpad.\n\n`uname -r` gives: `5.4.0-33-generic`\n\n`xinput` gives:\n\n``````\u23a1 Virtual core pointer id=2 [master pointer (3)]\n\u239c \u21b3 Virtual core XTEST pointer id=4 [slave pointer (2)]\n\u239c \u21b3 PS\/2 Logitech Wheel Mouse id=15 [slave pointer (2)]\n\u23a3 Virtual core keyboard id=3 [master keyboard (2)]\n\u21b3 Virtual core XTEST keyboard id=5 [slave keyboard (3)]\n\u21b3 Power Button id=6 [slave keyboard (3)]\n\u21b3 Video Bus id=7 [slave keyboard (3)]\n\u21b3 Video Bus id=8 [slave keyboard (3)]\n\u21b3 Power Button id=9 [slave keyboard (3)]\n\u21b3 Sleep Button id=10 [slave keyboard (3)]\n\u21b3 Lenovo EasyCamera: Lenovo EasyC id=12 [slave keyboard (3)]\n\u21b3 Ideapad extra buttons id=13 [slave keyboard (3)]\n\u21b3 AT Translated Set 2 keyboard id=14 [slave keyboard (3)]\n``````\n\nIt shows the touchpad as PS\/2 Logitech Wheel Mouse.\n\n``````xserver-xorg is already the newest version (1:7.7+19ubuntu14).\n``````\n\nWhat I\u2019ve tried:\n\nI installed first the binary:\n\n``````xserver-xorg-input-synaptics_1.9.1-1ubuntu1_amd64.deb\n``````\n\nNo problems, but the touchpad still not working.\n\nI\u2019d appreciate help to fix this issue; I know that it is known issue, and probably the solution is in front of me, but you can appreciate I\u2019m new to Ubuntu.\n\nI\u2019ve removed `xserver-xorg-input-synaptics`, unplugged the auxiliary mouse and restart the laptop.\n\nThe touchpad is recognised and the main functions working.\n\nThanks Pilot6 for the help.\n\nThe ONLY way I can get my trackpads to function properly on my Thinkpad t469 and t15 is to install xserver-xorg-input-synaptics\n\nTagged : \/ \/ \/ \/\n\n## Making Game: I need to get pip but don\u2019t have disutils\n\nI need to get disutils.util to unpackage things that I try to get using pip. The problem is I don\u2019t have pip and I don\u2019t have permission to use sudo or basically anything.\n\nI searched and searched but I couldn\u2019t find a solution that worked.\n\nI\u2019m on Linux the 18.04 LTS I have python 3.\n\nTagged : \/ \/\n\n## Linux HowTo: Adding a key with \u201cssh-add\u201d works but passing it with \u201c-i\u201d doesn\u2019t\n\nI have a case like follows:\n\n``````ssh -J\\${some-jump-host} -i some-key.pem [email\u00a0protected]\\${some-ip}\n``````\n\nwhich does not ask for the key password for `some-key.pem` but directly fail as it is not authenticated.\n\nHowever, if I do:\n\n``````ssh-add -K some-key.pem\nssh -J\\${some-jump-host} [email\u00a0protected]\\${some-ip}\n``````\n\nI have\n\n``````\\$ cat ~\/.ssh\/config\nInclude ~\/ssh_config\n\nHost *\nUseKeychain yes\nStrictHostKeyChecking no\n\nIdentityFile path-to-an-irrelevant-key\nIdentityFile path-to-some-other-irrelevant-key\nServerAliveInterval 60\nServerAliveCountMax 10\nStrictHostKeyChecking no\nUserKnownHostsFile=\/dev\/null\nForwardAgent yes\nForwardX11 yes\n``````\n\nin `~\/ssh_config` we just define some hosts and proxy jumps, and this:\n\n``````Host *\nPKCS11Provider \/Library\/OpenSC\/lib\/opensc-pkcs11.so\nForwardAgent yes\nStrictHostKeyChecking no\nUserKnownHostsFile=\/dev\/null\n``````\n\nso my question is, why ssh seem to be ignoring the key when it is passed with `-i`?\n\nNote:\nthis is the relevant output of the `-vvv` for the case with `-i`\n\n``````debug1: Will attempt key: some-key.pem explicit\ndebug2: pubkey_prepare: done\ndebug3: send packet: type 5\ndebug1: kex_input_ext_info: server-sig-algs=<rsa-sha2-256,rsa-sha2-512>\ndebug2: service_accept: ssh-userauth\ndebug3: send packet: type 50\ndebug3: input_userauth_banner\nAuthorized uses only. All activity may be monitored and reported.\ndebug1: Authentications that can continue: publickey\ndebug3: start over, passed a different list publickey\ndebug3: authmethod_lookup publickey\ndebug3: authmethod_is_enabled publickey\ndebug1: Next authentication method: publickey\n``````\n\nTagged : \/ \/ \/ \/\n\n## Math Genius: Meaning of the slash \u201c\/\u201d in \\$mathbb{Z}\/pmathbb{Z}\\$\n\nWhat the meaning of the slash \u201c\/\u201d in expression like this: $$mathbb{Z}\/pmathbb{Z}$$ ? I know that it is called \u201ca quotient ring\u201d, but quotient reminds division. It\u2019s an operator somewhat related to division?\n\nI\u2019m an engineer trying to fully understand the abstract algebra behind error correction codes. May you leave a suggestion of a book for non-mathematicians that explain this topic in an easy manner?\n\nIf \\$R\\$ is a ring and \\$J\\$ a two-sided ideal of \\$R\\$, the quotient ring \\$R\/J\\$ consists of the equivalence classes \\$x + J\\$ for \\$x in R\\$, where \\$x sim y\\$ if \\$x \u2013 y in J\\$.\nThis is a ring with operations \\$(x+J) + (y+J) = (x+y)+J\\$ and \\$(x+J)(y+J) = xy + J\\$.\n\nIn the case of \\$mathbb Z \/ pmathbb Z\\$, \\$pmathbb Z\\$ consists of the multiples of \\$p\\$ and the equivalence relation is congruence mod \\$p\\$. Thus \\$mathbb Z\/pmathbb Z\\$ consists of the congruence classes mod \\$p\\$.\n\nIn \\$Bbb Z\\$ ,we define the equivalence relation \\$R\\$ by \\$x ;R; y iff x-y\\$ is a multiple of \\$p \\$.\n\n\\$Bbb Z\/p Bbb Z \\$ is the set of equivalence classes.\n\n\\$\\$Bbb Z\/3Bbb Z={overline {0},overline {1},overline {2}}. \\$\\$\n\n\\$\\$overline {3}=overline {0}. \\$\\$\n\nRisking self-praise, I might recommend my own course notes (formerly a book published by a traditional publisher) \u201cCoding Notes\u201d at http:\/\/www.math.umn.edu\/~garrett\/coding\/CodingNotes.pdf There are also the rather telegraphic overheads for a course I taught many times on that subject using those notes\/book, at http:\/\/www.math.umn.edu\/~garrett\/coding\/\n\nThis course (and the book\/notes) was meant to be intelligible to people who\u2019d not studied any abstract algebra before, and, in particular, to engineering and computer science people, in addition to math majors in the relatively early part of their undergrad education.\n\nSo, in particular, these notes are very down-to-earth, and talk in a way precisely meant to be intelligible to engineering and computer science people\u2026 who may have a \u201cdifferent dialect\u201d in mathematics.\n\nSo, no, not abstract, yet mathematically accurate, and aimed at coding-theory issues. (Though not really high-end, and certainly no longer up-to-date.)\n\nThe generic notation for the quotient of a set \\$X\\$ by an equivalence relation \\$cal R\\$ is: \\$; X\/cal R\\$.\n\nAs the equivalence relation here can be defined as \\$;xcal R yiff x-yin pmathbf Z\\$, the ideal \\$pmathbf Z\\$ is taken as the name of the relation and used in the notation of the quotient set.\n\nTagged :\n\n## Code Bug Fix: Device Proximity LED for barcode\n\nThe fact is that old scanners do not read the generated 1D barcode from the phone screen. And I started looking for another option and came across Beaming Barcode Technology, and here.\n\nI wanted to get the `SDK` for `Android`, I did not find it anywhere. I tried to decompile the old `APK`, to no avail.\n\nAfter searching, I realized that they were bought by Samsung.\n\nBut for some reason they don\u2019t talk about it anywhere, there are no questions in `StackOverflow` about this technology, or about the implementation of a smartphone Proximity sensor to generate a barcode.\n\nQuestion:\n\nDoes this technology really work?\n\nIf so, how do you implement this yourself with a sensor?\n\nPerhaps the technology could be thought of as LoopPay(SamsungPay) in magnetic stripe cards.\nSamsung Paid Around \\$250 Million for LoopPay, Its Apple Pay Competitor\nSamsung Pay\n\nThe magnetic stripe card is highly compatible with the physical specifications regarding the reading method, and it was effective to some extent even with technologies such as LoopPay.\nHowever, there may be countries or regions that are technically possible but are not serviced due to differences in rules or business practices.\n\nOn the other hand, the basics of barcode scanners are to illuminate a barcode and convert the light that can be reflected and detected to be recognized, but there are various methods and shapes.\nMechanism of barcode scanning\n\nBy analogy with LoopPay, isn\u2019t it a method of simulating this reflected light with the light emission of the LED of the smartphone and making it look like it has read a barcode?\nThe sensor is the barcode scanner of the POS terminal, so the sensor may not be needed on the smartphone side.\n\nThat\u2019s why there are no questions or articles about proximity sensors or any sensors.\n\nHowever, as mentioned above, barcode scanners are not as limited in standards as magnetic stripe readers, and there are various methods and shapes of barcode scanners in the world.\n\nThe technology you asked may work, but if it works, it will have limited scope.\nIt is unlikely that it will spread in a practical range.\n\nTagged : \/ \/ \/\n\n## Server Bug Fix: Vyos: Redirect All Users Web Request to Webpage When Internet is Down\n\nWe are a small company and are currently using Vyos router for our internet connection.\nI would like to know if it is possible to redirect all the users web requests to a static web page whenever the internet goes down so that the users can know that there is a problem with the router itself which can make my work a lot easier.\nMy idea is something that is pretty similar to how plug and play routers function.\n\nI know how to make a webpage on my Vyos but i\u2019m not sure how to redirect all the web traffic to it when it can\u2019t reach the internet.\n\nTagged : \/ \/ \/ \/\n\n## Making Game: Is there a way of converting a hierarchical list into a table in Word?\n\nI have a list like:\n\n\u2022 Top Level 1\n\u2022 Sub Level 1\n\u2022 Sub Sub Level 1\n\u2022 Sub Sub Level 2\n\u2022 Sub Level 2\n\u2022 Sub Sub Level 3\n\n(but bigger)\n\nIs there a way I can convert the above into a table that looks like:\n\n```Top Level 1 | Sub Level 1 | Sub Sub Level 1\n| | Sub Sub Level 2\n| Sub Level 2 | Sub Sub Level 3\n```\n\nWord doesn\u2019t have this functionality in particular, I\u2019m afraid.\n\nWord does not do this.\n\nBut I do have a solution using Excel that you can paste back into Word.\n\nMy solution uses a simple search and replace\nand a native Text to Columns in a spreadsheet software.\n\n## Generic solution\n\nStep 1) Search and replace all the spaces\/ tabs that make your hierarchy with the bullet. (In Word or in Excel or anywhere or plain text editor).\nYou\u2019ll end up with something like this\n\n``````\u2219 item 1\n\u2219\u2219 item 2\n\u2219\u2219\u2219 item 3\n``````\n\nStep 3) Paste into a spreadsheet and use the Text to Columns feature and split in on `your bullet character`. Done!\n\n## My specific workflow using my apps of choice\n\nStep 1) Copy paste the list from Workflowy as a plain text into Vscode. This results in a markdown style list.\n\n``````- item 1\n- item 2\n- item 3\n``````\n\nStep 2) Search and replace every `two spaces` and replace them with `-`\n\n``````- item 1\n-- item 2\n--- item 3\n``````\n\nStep 3) Copy paste into OpenOffice\/ Excel. Use the Text to Columns feature and split in on `-`. Done! You get your table!\n\nWorkaround:\n\n1. In Word, go to New Number Format under Numbering.\n2. After the number, provide a distinct special\ncharacter.\n3. Do this for all hierarchical levels.\n4. Copy and paste the entire content to Excel or Google Sheet along\nwith the numbers and special characters. It will copy to a single\ncolumn.\n5. Write a formula using Find or Search to copy only the cells with the\nspecific Special Character for a specific hierarchy.\n6. Do this for all hierarchical levels.\n\nBulletsToTable.com\n\nI had the same problem \u2013 I make my notes as heirarchical lists in MS OneNote but sometimes want to show them as tables. This web app makes quick work of what you asked, just paste in your list!\n\nEdit: Demo GIF here to show it in action\n\nTagged :","date":"2020-08-15 01:43:59","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 19, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.35245391726493835, \"perplexity\": 5059.344464257649}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2020-34\/segments\/1596439740423.36\/warc\/CC-MAIN-20200815005453-20200815035453-00173.warc.gz\"}"} | null | null |
\section{Introduction}
Hairy surfaces submerged in fluid flow are ubiquitous in nature, from crustacean olfaction \cite{koehl2001lobster,koehl2001fluid}, suspension feeding \cite{conova1999role}, to nectar drinking \cite{nasto2018viscous}. While we can define a hairy surface in simple terms -- an array of filaments anchored at one end to a surface -- the utility and observed behavior of hairy surfaces is as varied as the organisms that employ them.
For example, the olfactory organ of decapod crustaceans consists of an array of sensilla, called aesthetascs, on the lateral flagellum of the antennule. Each aesthetasc contains olfactory receptor neurons. This system can be viewed as an array of rigid hairs attached to an antenna, which is then flicked through the water in order to sample odors.
The motion of marine organisms at micrometric scales is different from human scales. At small scales, the viscous drag on an organism vastly outweighs inertia. This phenomenon is indicated by the Reynolds number, $\Rey$, a dimensionless ratio of inertial to viscous stresses. However, flow past hairy surfaces is not limited to the low-$\Rey$ regime. Crustacean olfaction and suspension feeding both occur at $\Rey = O(1)$. Koehl \etal observed that crustaceans flick their antennae at different speeds, intentionally manipulating the Reynolds number $\Rey$ to achieve different states of flow. The two states observed are called \textit{rake} -- where the fluid inside the bed of hairs is stagnant, and \textit{sieve} -- where the fluid inside the bed of hairs travels opposite the motion of the antenna. Koehl \etal have conjectured that crustaceans use these different phases of flow to aid olfaction. On the slow down-stroke, the hair bed collects a stagnant packet of fluid, allowing chemicals to diffuse to the chemo-receptors on the hairs. Conversely, on the fast up-stroke, the hair bed releases the packet of fluid in order to sample a new packet.
Recent technological advances in fabrication have allowed engineers to implement hairy surfaces in devices. For example, a bed of flexible hairs anchored at an angle to the surface impedes flow in one direction \cite{alvarado2017nonlinear}, acting as a steady-state hydrodynamical diode. Robotic pollinators use hairs coated in a gel to transport pollen \cite{chechetka2017materially,amador2017sticky}.
Furthermore, passive and/or steady-state devices are highly valued for their simplicity and robustness. Hairy surfaces, then, are an ideal design element to study, for their ubiquity, range of exhibited behavior, and robustness.
Increasingly, inertia is used as a design element in microfluidic and hydrodynamic devices -- for particle filtering \cite{DiCarlo09,DiCarlo11}, flow cytometry \cite{DiCarlo10}, and entrapment of live cells in droplets for tissue printing \cite{edd2008controlled,amini2017inertial}.
While the equations of motion for fluids with $\Rey = O(1)$ are nonlinear and therefore difficult and costly to solve, there has been an emergence of asymptotic and numerical theories to enable rational design with inertia \cite{klotsa2009chain, hood15inertial, true2017hydrodynamics}. However, in this regime, results are sensitive to variations in the boundary conditions, and so far theories have been ad-hoc. Inertial flow over complicated and intricate surfaces is an open field for study.
To close this gap between biology and rational design, we investigate a bio-inspired model system of rigid hairs subject to inertial flow at $\Rey = O(1)$. In order to use inertia as a design element, similar to the olfaction mechanism in crustaceans, we need a theory for the flow phase based on experimental parameters such as hair length, diameter, and spacing length. Intuitively, the flow phase should be determined by the depth of the boundary layer on the hairs: rakes arise from large overlapping boundary layers and sieves arise from small boundary layers. We develop a quantitative theory for predicting the depth of the boundary layer, which can then be used to design a hair bed with desired flow phase.
\begin{figure}[t]
\centering
\includegraphics[scale = 1] {channel_bed_hairs_ver3.pdf}
\caption{ (A) Channel diagram. (B) The undistrubed channel flow is a rectangular Poisuille flow $\ubar$. (C) Hairs have diameter $d_h$, length $L_h$, and center-to-center spacing $\delta$. (D) The hairs are arranged in a $5 \times 5$ rectangular grid. }\label{fig:diagram}
\end{figure}
\section*{Discovery of deflection phase}\label{sec:exp_res}
To investigate the physical principles, we developed an experimental model system of rigid hairs immersed in water (Fig. \ref{fig:diagram}A-D) We mounted rigid steel rods on one wall of a rectangular channel and visualize the flow with tracer particles. Features of the flow are set by the smallest length scale in this system, which is the diameter of the hairs, $d_h = 1$mm. We define the characteristic velocity to be the maximum flow speed $U$ in an undisturbed channel. Then, for water with density $\rho$ and viscosity $\mu$, we define the Reynolds number to be: $\Rey = \rho U d_h/\mu$.
We measure the flow phase as a function of $\Rey$ and the separation lengths $\delta$ of the hair bed by measuring the magnitude of the flow velocity in the center of the bed. The channel had cross-section dimensions $40$mm by $62$mm, and the separation lengths $\delta$ varied from $2$mm to $10$mm.
In the rake phase, streamlines circumvent the hair bed (Fig. \ref{fig:phases} A). The velocity magnitude inside the bed is an order of magnitude less than the undisturbed channel flow. The streamlines appear to have time reversal symmetry -- a characteristic of creeping flows ($\Rey \ll 1$) where, if time and velocity are reversed, the flow will evolve along the same streamlines. The rake phase can be observed at $\Rey = 0.8$ for the hair bed with separation length $\delta = 2$mm (Fig. \ref{fig:phases}D). This is consistent with the observations and simulations for crustaceans \cite{koehl2001fluid,cheer1987paddles}.
The sieve phase is characterized by streamlines that fully penetrate the hair bed (Fig. \ref{fig:phases} C). The streamlines move predominantly in the $z-$direction, with a slight deflection in the $x-$direction, thereby breaking time-reversal symmetry. The magnitude of velocities inside the bed are similar to that of the undisturbed channel flow. The sieve phase can be observed at $\Rey = 19$ for the hair bed with separation length $\delta = 4$mm (Fig. \ref{fig:phases}F). This is consistent with the observations and simulations for crustaceans \cite{koehl2001fluid,cheer1987paddles}.
\begin{figure}[t]
\centering
\includegraphics[scale = 1]{phases_plot_ver6.pdf}
\caption{There are three phases of flow observed in experiment. (A) In the rake phase, numerical simulations show that streamlines circumvent the hair bed. (B) In the deflection phase, numerical simulations show that streamlines penetrate the hair bed but exit laterally. (C) In the sieve phase, numerical simulations show that the streamlines fully penetrate the hair bed. (D) Streamlines reconstructed from PIV measurements of the velocity show a rake at $\Rey = 0.8$ and $\delta=2$mm, black markers represent the centers of the cylinders. (E) Streamlines reconstructed from PIV measurements of the velocity show the deflection at $\Rey = 31$ and $\delta = 2$mm. (F) Streamlines reconstructed from PIV measurements of the velocity show the sieve at $\Rey = 19$ and $\delta = 4$mm.}\label{fig:phases}
\end{figure}
While two phases of flow have been observed in crustaceans -- rake and sieve -- in our geometry we observe a third transitional region of flow which we call the \textit{deflection} phase. Here, fluid penetrates into the hair bed, but is deflected laterally out of bed.
The deflection phase is characterized by streamlines with significant transverse displacement in the $x-$direction (Fig. \ref{fig:phases} B). The streamlines penetrate the hair bed, but are deflected transversely, exiting the bed before reaching the last row of hairs. We observe a circulation region in the wake of the hair bed. The velocity magnitude in the wake region is an order smaller than the undisturbed channel flow. Time reversal symmetry is clearly broken. The deflection phase can be observed at $\Rey = 31$ for the hair bed with separation length $\delta = 2$mm (Fig. \ref{fig:phases}E).
In order to design devices that can exploit these phases of flow, it would be useful to have a predictive theory that does not require a full numerical simulation. Given $\Rey$ and a hair bed separation length $\delta$, can we predict which phase the flow will exhibit? We conjecture that, to first order, the most significant predictor of the flow phase is the depth of the boundary layer on a single hair. In the next section, we develop a quantitative theory for boundary layer depth on a cylinder in a rectangular channel.
\section*{Boundary layer thickness around a single hair}\label{sec:1H}
A long slender body (like a hair or bristle) disturbs the surrounding flow due to the no-slip boundary condition on its surface. This gives rise to a region with a steep velocity gradient, called the boundary layer. The shape of the boundary layer depends on the geometry of the body and the Reynolds number of the flow: low-$\Rey$ flows have thick boundary layers and high-$\Rey$ flows have thin boundary layers.
For an array of hairs, we expect the flow to exhibit rake behavior when the boundary layers are so thick that they overlap in the region between the hairs (Fig. \ref{fig:gamma_contour_rc}A). Conversely, we expect the flow to exhibit sieve behavior when the boundary layers are too short to overlap (Fig. \ref{fig:gamma_contour_rc}E).
Therefore, in order to understand the flow around an array of rigid hairs, we first characterize the flow around a single hair. While the presence of neighboring hairs will affect the shape of the boundary layer non-linearly, these effects are higher order (since at $\Rey = O(1)$ the system is only weakly non-linear).
\begin{figure}[t]
\centering
\includegraphics[scale = 1] {gamma_rc_delta_plot_ver4.pdf}
\caption{ (A) Diagram of a single hair in a rectangular channel. (B) The boundary layer is the negative (blue) region of the disturbance flow $w'$ at $\Rey = 0.025$. (C) Plots of $\Gamma_p$ where $p = -0.1$ for various values of $\Rey$, here the black rectangle represents the hair. The critical radius $r_c$ as defined by equation (\ref{eq:rc}). (D) The critical radius $r_c$ as a function of $\Rey$ is fit well by a logistic function (\ref{eq:rc_pois}) and shown as a solid black line. }\label{fig:gamma_rc}
\end{figure}
Consider a single cylinder anchored to one wall in a rectangular channel (Fig. \ref{fig:gamma_rc}A). The flow $\bu$ around the hair is described by the steady-state Navier-Stokes equations (\ref{eq:NSE1})--(\ref{eq:NSE2}) (Full derivation in SI). We assume that the flow at the inlet is the Poiseuille flow $\ubar$ for a rectangular channel \cite{Papanastasiou99}. It is useful to consider the normalized disturbance flow: $\bu' = (\bu -\ubar)/U$.
Recall that $U$ is the characteristic velocity, which we determine to be the maximum velocity in an undisturbed channel. The resulting disturbance flow $\bu'$ has dimensionless units and therefore can be compared across $\Rey$.
The disturbance flow will take values between $-1 \le \bu' \le 0 + \epsilon$, where $\epsilon$ is a relative flow speedup. We observe speedup in this system due to the presence of the channel walls and conservation of mass. There is a large negative region of flow near the hair, which we call the boundary layer. In numerical simulations, we observe that at $\Rey = 0.025$ the boundary layer is thick and elliptical in shape (Fig. \ref{fig:gamma_rc}B). Whereas at a higher Reynolds number $\Rey = 2.5$, the boundary layer is thin and rectangular (Fig. \ref{fig:gamma_rc}C).
\begin{figure}[t]
\centering
\includegraphics[scale = 0.88] {gamma_contour_rc_ver3.pdf}
\caption{ The degree of overlap of the boundary layers determines the resulting flow phase. For all plots, the separation length is $\delta = 4mm$. The top row shows the disturbance flow $w'$ around two hairs, with the curve $\Gamma_p$ from a single hair super-imposed over each hair. The bottom row shows the absolute flow $w$ around the two hairs. At $\Rey = 0.1$, (A) the boundary layers overlap completely and (D) the total flow acts like a rake with stagnant flow in the gap between the hairs. At $\Rey = 2$, (B) the boundary layers partially overlap and (E) the total flow acts like a deflected flow with moderate speeds in the gap. And at $\Rey = 5$, (C) the boundary layers do not overlap, and (F) the total flow acts like a sieve, with large speeds in the gap.}\label{fig:gamma_contour_rc}
\end{figure}
Further, we restrict our evaluation to the plane $z=0$ and only consider the streamwise component of the disturbance flow (i.e. for $\bu' = (u',v',w')$, consider only $w'$). Then we can measure the level-set with value $p$ of the streamwise disturbance flow. We define the curve $\Gamma_p$ as follows:
\begin{equation}\label{eq:gamma}
\Gamma_p = \{ (x,y)\, \vert \,\, w'(x,y,0) = p \}\,.
\end{equation}
To determine the depth of the boundary layer, we consider the $p=-0.1$ level set. We define the critical radius $r_c$ as the x-component of the curve $\Gamma_p$ evaluated at the midway point of the channel $y = H/2$ (Fig. \ref{fig:gamma_rc}C).
\begin{equation}\label{eq:rc}
r_c = \Gamma_p \vert_{y = H/2} \, , \qquad p = -0.1\, .
\end{equation}
The critical radius is a decreasing function of $\Rey$ (Fig. \ref{fig:gamma_rc}D), and is well approximated by a logistic function in $\log_{10}{\Rey}$, specifically:
\begin{equation}\label{eq:rc_pois}
\frac{r_c}{L_h} = m + \frac{M}{\left(\frac{\log_{10}\Rey}{\log_{10}\Rey_*}\right)^{k}+1}
\end{equation}
Equation (\ref{eq:rc_pois}) has four fitting parameters: $m = 0.026$, $M = 0.175$, $\Rey_* = 1.2$, and $k=1.5$.
We can use this numerical fit of the critical radius to design hair beds to achieve either a rake, deflection, or sieve flow.
\section*{Designing arrays of rigid hairs}
We arrive at a simple design rule for predicting the phase of flow around a bed of rigid hairs. If the separation length $\delta < r_c$, then the boundary layers overlap in the gap between the hairs, which will result in a rake flow. Conversely, if $\delta > 2r_c$, then the boundary layers will not touch in the gap, which will result in a sieve flow. And intermediate $\delta$ will result in a deflection phase.
\begin{figure}[t]
\centering
\includegraphics[scale = 1]{exp_data_seive_v_rake_rc_plot_ver3.pdf}
\caption{(A) Phase diagram of the observed flow: rake (blue square), deflection (red triangle), and sieve (black circle). The critical radius (\ref{eq:rc_pois}) determines the separation between rake flow and deflected flow (thin dashed line), while twice the critical radius determines the separation between deflected flow and sieve flow (thick solid line).}\label{fig:phase_data_rc}
\end{figure}
We can further verify this design rule by modeling the flow around two cylinders in a channel with a separation length $\delta = 4$mm. We observe that the curve $\Gamma$ predicting the boundary layer on a single cylinder in a channel is consistent with the boundary layers on two cylinders in a channel (Fig. \ref{fig:gamma_contour_rc}A,C,E). This shows that hydrodynamic interactions between hairs are higher order effects and validates our decision to ignore them at first order.
At $\Rey = 0.1$, the critical radius for a single hair is $r_c = 0.2L_h = 5.9$mm. Here, $r_c > \delta = 4$mm, so the boundary layer from each hair overlaps, covering the gap between the hairs (Fig. \ref{fig:gamma_contour_rc}A). According to our design rule, this configuration should produce a rake flow. We observe that the absolute flow is nearly stagnant in the gap region (Fig. \ref{fig:gamma_contour_rc}B), consistent with rake flow.
At $\Rey = 2$, the critical radius is $r_c = 0.08L_h = 2.5$mm. Here $r_c$ is comparable to the separation length $\delta = 4$mm. The boundary layer from each hair overlaps in the gap region, however the disturbance flow $w'$ is only moderately negative in the gap (Fig. \ref{fig:gamma_contour_rc}C). According to our design rule, this configuration should produce a deflection flow. We observe that the absolute flow has moderate speed in the gap region (Fig. \ref{fig:gamma_contour_rc}D), consistent with deflection flow.
At $\Rey = 5$, the critical radius $r_c = 0.05L_h = 1.4$mm is much smaller than the separation length $\delta$. The boundary layers are distinct with no overlap (Fig. \ref{fig:gamma_contour_rc}E). According to our design rule, this configuration should produce a sieve flow. We observe that the absolute flow is close to unity in the gap region (Fig. \ref{fig:gamma_contour_rc}F), consistent with sieve flow.
This rule is consistent with our experimental data (Fig. \ref{fig:phase_data_rc}A). Equation (\ref{eq:rc_pois}) for $r_c$ predicts the separation between the rake flow and the deflection flow, while the equation for $2r_c$ predicts the separation between deflection flow and sieve flow. The rake and sieve phases of flow agree qualitatively with 2D numerical predictions \cite{cheer1987paddles,chow1989drag}.
\section*{Hairy surfaces in crustaceans}
We compare our design space with that employed by the hairy appendages found in crustaceans. The hairy surfaces serve one of two purposes: chemo-sensing (stomatopod and lobster) and suspension-feeding (mole crab and barnacle). For each crustacean, the relevant measurements are the length of the hair $L_h$, the hair diameter $d_h$, the cross-stream spacing length $\delta_x$, the stream-wise spacing length $\delta_z$, and the $\Rey$ (Table \ref{tab:biodata}).
\begin{figure}[t]
\centering
\includegraphics[scale = 0.8] {shear_gamma_rc_delta_plot_ver1.pdf}
\caption{ (A) Diagram of a single hair in a rectangular channel subject to a shear flow. (B) Plots of $\Gamma_p$ where $p = -0.1$ in a shear flow for various values of $\Rey$, here the black rectangle represents the hair. The critical radius $r_c$ as defined by equation (\ref{eq:rc}). (D) In a shear flow, the critical radius $r_c$ as a function of $\Rey$ is fit well by a logistic function (\ref{eq:rc_pois}) and shown as a solid black line. }\label{fig:shear_gamma_rc}
\end{figure}
Marine crustaceans use their sense of smell to detect food, detect information about their neighbors, and avoid predators \cite{mead1999stomatopod}. Most crustaceans, including stomatopods and lobsters, detect odors using chemosensory sensillae called aesthetascs -- stiff hair-like structures located on the antennules. On the stomatopod \textit{Gonodactylaceus mutatus}, the aesthetascs are arranged in 16 rows, with each row consisting of three aesthetascs (Fig. \ref{fig:crustaceans}A). The aesthetascs form a $50^{\circ}$ angle with the surface of the antennule \cite{mead2000stomatopod}. Similarly, the spiny lobster \textit{Panulinus argus} has aesthetascs arranged in an array of 15 rows, with each row consisting of 10 aesthetascs \cite{reidenbach2008antennule}. The aesthetascs are angled laterally to form a $32^{\circ}$ angle with the antennule \cite{goldman2001fluid} and angled streamwise so that the tips form a zig-zag pattern (Fig. \ref{fig:crustaceans}B). Furthermore, the entire array of aesthetascs are surrounded by much larger guard hairs.
\begin{figure}[t]
\centering
\includegraphics[scale = 1]{crustaceans_all_plot_ver4.pdf}
\caption{ (A) Stomatopod antennule, (B) lobster antennule, (C) barnacle cirra, and (D) mole crab setae. (E) The critical radius for a cylinder in an unbounded shear flow (\ref{eq:rc_pois}) determines the separation between rake flow and deflected flow (thin dashed line), while twice the critical radius determines the separation between deflected flow and sieve flow (thick solid line). Boxes on the plot indicate ranges of $\Rey$ and $\delta/L_h$ observed in lobsters and stomatopods. }\label{fig:crustaceans}
\end{figure}
Suspension-feeders use a diverse range of filamentous appendages to capture particles from the surrounding water. For example, the barnacle \textit{Balanus glandula} feeding structure is called a cirral fan, consisting of paired biramous appendages covered with setae. Barnacles use their cirri to directly capture food, filter food from the water column, and reject non-nutritious items \cite{geierman2009feeding}. Barnacles are able to switch between active feeding (beating their cirri) and passive feeding (holding their cirri extended), in response to the current speed \cite{trager1990barnacle}. Each segment of the cirra has a two groups of six setae, arranged in order of decreasing length, and separated by an $47^{\circ}$ angle. For the purpose of this study, we consider the length of the second longest seta and the spacing length between the first and second seta (Fig. \ref{fig:crustaceans}C). The mole crab \textit{Emerita talpoida} uses a pair of second antenna to collect food particles, each antenna composed of long plumose flagella. Each flagellum is covered with duplicate sets of four rows of setae, which in turn are covered in shorter setules \cite{conova1999role}. The exteriormost setae are covered in fine, cylindrical setules, while the second pair of setae are covered in shorter, flat setules (Fig. \ref{fig:crustaceans}D).
Each crustacean moves its hairy appendage through the water, and therefore only the anchoring wall effects (in contrast to channel walls in our experiment) are prevalent in the model. So, instead of measuring the boundary layer depth $r_c$ on a hair in Poiseuille flow, we model $r_c$ in a shear flow (Fig \ref{fig:shear_gamma_rc}A-C). The critical radius is still defined by Eq (\ref{eq:rc}), and can be fit by the logistic function in Eq (\ref{eq:rc_pois}). For shear flow, the fitting parameters are: $m = 0.058$, $M = 0.253$, $\Rey_* = 3$, and $k=1.5$. Note that the values of $m$, $M$, and $\Rey_*$ are significantly different than those for a cylinder in a rectangular channel. However, the design principle remains the same. If $\delta < r_c$, we predict a rake flow, if $\delta > 2r_c$ we predict sieve flow, and we predict deflection flow in the transition region.
We observe that the hairy appendages for the chemo-sensors (stomatopods and lobsters) lie in the deflection and sieve region of the diagram (Fig. \ref{fig:crustaceans}E). Conversely, the hairy appendages for suspension-feeders (mole crabs and barnacles) lie on the rake region of the diagram. One exception is that the short setules on the mole crab lie in the sieve region of the graph. However, this may be explained by the fact that the aspect ratio of the short setules $\kappa = 1/10$ is significantly different from the aspect ratio of the other crustacean hairs and the model $\kappa = 1/30$.
\begin{table*}[t]
\begin{center}
\begin{tabular}{ lrrrrrr }
Crustacean & $L_h$ & $d_h$ & $\kappa$ & $\delta_s$ & $\delta_{\ell}$ & $\Rey$\\ \hline
Stomatopod (\textit{Gonodactylaceus mutatus} \cite{mead2000stomatopod}) & $516\mu$m & $20\mu$m & $1/26$ & $20\mu$m & $150\mu$m & 0.8 - 1.7 \\
Lobster (\textit{Panulirus argus} \cite{goldman2001fluid}) & $720\mu$m & $22\mu$m & $1/33$ & $23\mu$m & $140 \mu$m & 0.5 - 2 \\
Mole crab (\textit{Emerita talpoida} \cite{conova1999role}) & $200\mu$m & $4\mu$m & $1/50$ & - & $15\mu$m & 0.2 - 1.4 \\
Mole crab$^*$ (\textit{Emerita talpoida} \cite{conova1999role}) & $60\mu$m & $6\mu$m & $1/10$ & - & $30\mu$m & 0.3 - 2.1 \\
Barnacle (\textit{Balanus glandula} \cite{vo2018fluid}) & $363\mu$m & $10\mu$m & $1/36$ & - & $27\mu$m & 0.4 - 5.3 \\
\hline
\end{tabular}
\caption{Biological data for the dimensions of hairs on crustaceans. $^*$Mole crabs have different setae with longer setules (no star) and shorter setules (star). $^\dagger$Barnacles have different morphology depending on whether the surrounding water is stagnant (no dagger) or active (dagger).}\label{tab:biodata}
\end{center}
\end{table*}
\section*{Discussion}\label{sec:conc}
To summarize, we have shown that rectangular arrays of rigid hairs immersed in fluid flow exhibit separate phases of flow at different $\Rey$. Consistent with crustacean olfaction, we observe both the rake and sieve flow. However, we uncovered a new intermediary phase, which we call the \textit{deflection} phase, where fluid partially penetrates the bed and escapes perpendicularly to the direction of flow.
The deflection phase has not been discussed in the literature to the best of our knowledge. This phase may not be observed in crustaceans because they have hair beds with different lateral and streamwise spacing lengths. For example, the stomatopod has a streamwise spacing length that is much smaller than the lateral spacing length (Table \ref{tab:biodata}). This may prevent streamlines from escaping laterally from the hair bed. Alternatively, it also could be a desirable feature for sensing. Since stomatopod hairs have chemo-receptors, it might be desirable to have fluid guide particles directly to the hairs.
Here we have shown that the phase of flow can be predicted form the depth of the boundary layer on a single hair. We discovered a simple design principle based on the degree of overlap of the boundary layers in the gap between the hairs. The theory for the depth of the boundary layers in equation (\ref{eq:rc_pois}) is specific to the geometry in this experiment. However, we have shown that the design principle is also valid for shear flow, and correlates with various marine crustacean hairy appendages, despite their varying geometries.
In particular, the flow phases described here depend on the fact that the hairs are tall relative to the channel, suppressing the effect of fluid escaping over the top of the hairs. Further work is needed to examine this behavior. Additionally, we have only considered hairs anchored normal to the surface of the channel. In lobsters the chemo-sensory hairs are angled so that the tips of the hairs form a zig-zag formation. Other angles of the hairs could affect the onset of the phases of flow, perhaps even directing the deflection flow to one side of the channel. Finally, we have not considered any deformation of the hairs. Deformation and reconfiguration is ubiquitous in biology at low-$\Rey$ flows, and can be observed at intermediate-$\Rey$ in the sensory appendages of the crab \textit{Callinectes sapidus} \cite{gleeson1982morphological,reidenbach2008antennule}.
\section*{Materials and Methods}
Flow was observed in a 450 mm long channel fabricated from 1/4 inch acrylic sheets and joined with plastic cement (Scigrip). The channel cross-section dimensions were 62 mm $\times$ 40 mm ($W \times H$), respectively, with the shortest dimension identified as the depth ($y$) dimension (Fig. \ref{fig:diagram}A) and the longer dimension as the width dimension ($x$). The channel length was chosen to minimize the entry length effects in the hair bed (See SI). Fluid entered and exited the channel through 1/4 inch plastic barbed tube fittings (McMaster-Carr) placed 400mm apart (Fig. \ref{fig:diagram_experiment}). Six channels were constructed, one for each hair bed and one for empty channel flow.
The hair beds consisted of a $5 \times 5$ rectangular grid of cylinders perpendicular to the bottom surface of the channel (Fig. \ref{fig:diagram}A-D). Each cylinder was cut from 1 mm diameter steel rods (uxcell) to have an aspect ratio of $\kappa = 1/30$ comparable to aesthetacs observed in lobsters and stomatopods (Table \ref{tab:biodata}). The hair length $L_h = 30$mm was chosen to be close to the height $H$ in order to suppress the effect of fluid escaping over the hair bed. We constructed five hair beds and varied the center-to-center distance between the cylinders $\delta$ to be 2mm, 4mm, 6mm, 8mm, and 10mm. The cylinders were inserted into a laser-cut holes in an acrylic sheet and then secured to the bottom of each respective channel. The cylinders were coated with black spray paint (Rustoleum, flat black) to reduce light reflection.
The channel Reynolds number is defined as $\Rey_C = UH/\nu$, where $\nu = 1 \times 10^{-6} \mathrm{m}^2 \mathrm{s}^{-1}$ is the kinematic viscosity of deionized water at room temperature, $H = 40$mm is the short dimension of the channel, and $U$ is the maximum fluid velocity in the channel. The cylinder Reynolds number is defined at $\Rey = Ud_h/\nu$. Fifteen different total flow rates were used, including $Q$ = 20, 30, 40, 50, 60, 70 mL min$^{-1}$ and $Q$ = 2, 3, 5, 7, 9, 11, 13, 15, and 17.5 gph, corresponding to a range of channel Reynolds numbers $\Rey_C = 22.9 - 1260$ and cylinder Reynolds numbers $\Rey = 0.56 - 31 $.
Fluorescent red polyethylene microspheres (Cospheric) were used to visualize the flow. The particles had 25 $\mu$m diameter and density of $0.995$g/cc. The particles were dispersed at 0.004 volume fraction in a suspending fluid composed of deionized water and 0.001 (wt/vol) biocompatible surfactant (Tween 80). The particles were pumped into the channel in one of two ways. At low speeds, the particles were pumped at a controlled flow rate using a multichannel syringe pump (New Era NE-1600). At high speeds, the particles were pumped using a submersible 80gph water pump (Songjoy). The flow rate was measured at the output with an acrylic flowmeter (McMaster-Carr). The solutions were infused using 1/4 inch plastic tubing (McMaster-Carr).
The particles were chosen to be near-neutrally buoyant with a particle density of $0.995$g cm$^{-3}$. The particle density does not match the density of the suspending fluid (density 1.000g cm$^{-3}$). The sedimentation velocity can be determined by balancing buoyancy force with the drag force for a sphere. For this experiment, the sedimentation velocity is $1.6\mu$m/s, meaning that the particles sediment a distance between $0.02 - 2$ mm, depending on the $\Rey$.
At high speeds, the sedimentation can be ignored in this experiment, but sedimentation can be observed at lower speeds.
The fluorescent particles were illuminated by two 532nm 50mW green lasers fixed with a cylindrical lens attachment to flatten the laser line into a sheet (Laserland). The laser sheets were arranged perpendicularly to illuminate the plane $y=0.5H$ (Fig. \ref{fig:diagram}A-D). A support structure made of extruded aluminum was constructed to fix the location of the lasers.
Particle velocities were tracked by imaging using a camera (Phantom Miro M320S). At low speeds we used 24 fps with 40ms exposure time, and at high speeds we used 100 frames per second with 10ms exposure time. The camera was focused to the plane illuminated by the laser sheet
The effective pixel size ranged between $14\mu$m and $20\mu$m in each video.
\subsection*{Determining particle velocities}
Videography provided measurements of the $z$ and $x$ (stream-wise and lateral) velocities of particles located in the plane $y = 0.5H$ illuminated by the laser sheet. Since the particles particle fluoresce red light, we filter the color videos to retain only the red channel data. Then we use the particle image velocimetry (PIV) code MatPIV \cite{matpiv} to develop a vector field representing the displacements of all particles from one frame to the next.
\begin{figure}[t]
\centering
\includegraphics[scale = 1] {exp_diagram_piv_vel_ver2.pdf}
\caption{(A) The experimental setup. (B) Measurement of the normalized average streamwise velocity $\langle w \rangle / U$ can be used to determine whether or not the flow is a rake. Error bars represent standard error, while circles mark $\delta = 2$mm, squares mark $\delta = 4$mm, triangles mark $\delta = 6$mm, stars mark $\delta = 8$mm, and diamonds mark $\delta = 10$mm. (C) measurement of the average angle $\langle \theta \rangle$ distinguished between deflection and sieve flow.} \label{fig:diagram_experiment}
\end{figure}
Because the laser-sheet is located on one side of the channel, the intensity of the fluorescent particles declines as they move away from the light source. Additionally, there are significant shadows generated by the cylinders. In order to compensate, we measure the velocities of particles over five frames in order to sample enough particles moving through the shaded region of the channel.
To classify the flow phase, we restrict the domain to the space between two rows of hairs (i.e. $0 < x < \delta$ and $-2.5\delta < z <2.5\delta$. We compute the average stream-wise and transverse velocities ($w$ and $u$ respectively) over all of the particles in the starting frame that are located in $S_i$. To account for the sedimentation of particles in the $x$-direction, we subtracted off the sedimentation velocity $v_s = 1.7\mu$m/s off of the velocity $u$. The flow phase can be quantified by the average downstream velocity $\langle w \rangle$ inside the the hair bed (Fig. \ref{fig:diagram_experiment}B).
When $\langle w \rangle < 0.006 U$, then the flow inside the hair bed is essentially stagnant, representing the rake flow phase. Then at larger velocities, we determine whether the flow is a deflection or a sieve by calculating the average angle $\langle \theta \rangle$ (Fig. \ref{fig:diagram_experiment}C). At angles less than $35^{\circ}$ the flow is mostly stream-wise, which we categorize as sieve flow, and at larger angles the flow is deflecting laterally in the deflection phase.
\subsection*{Numerical Simulation}
We model the flow around a long cylinder attached at one end to a rectangular channel (Fig. \ref{fig:gamma_rc}A). Let the origin lie in the center of the channel. A cylinder with height $L_h = 30$mm and diameter $d_h = 1$mm is attached to the bottom of the channel at location $(x,y,z) = (0, -\frac{1}{2}H, 0)$. The long axis of the cylinder extends in the $y-$direction. The aspect ratio of the hair is $\kappa = d_h/L_h = 1/30$, similar to measurements of crustaceans (Tab. \ref{tab:biodata}).
Let $\bu$ be the flow around the hair and $p$ the corresponding pressure. Then the equations of motion are the steady state Navier-Stokes equations:
\begin{subequations}
\begin{align}
\mu \nabla^{2} \bu - \nabla p &= \rho \, \bu \cdot \nabla \bu \,, \label{eq:NSE1} \\
\nabla \cdot \bu & = 0 \,, \label{eq:NSE2}
\end{align}
\end{subequations}
Subject to the boundary conditions:
\begin{subequations}
\begin{align}
\bu & = 0 \mbox{ on cylinder } \sqrt{x^{2} + z^{2}} = \frac{d_h}{2}, \quad y \le L_h - \frac{H}{2}\,, \label{eq:NSE3} \\
\bu &= 0 \mbox{ on channel walls } x = \pm \frac{W}{2}, y = \pm \frac{H}{2} \,, \\
\bu &= \ubar \mbox{ as } z \to \pm \infty \,. \label{eq:NSE5}
\end{align}
\end{subequations}
We solve the equations (\ref{eq:NSE1})--(\ref{eq:NSE2}) and (\ref{eq:NSE3})--(\ref{eq:NSE5}) numerically using COMSOL Multiphysics (Cambridge, MA) with 188,159 elements. Accuracy of this model is evaluated in the SI.
Additionally, we model the flow around a cylinder attached to a wall in a shear flow. We define the shear to be $\bu_{\gamma} = (0,0,u_{\gamma})$, where: $u_{\gamma} = \gamma (y+.5H)$.
We define the Reynolds number as: $\Rey = \rho \gamma L_h d_h/\nu$.
Then the flow $\bu$ and corresponding pressure $p$ solve the steady-state Navier-Stokes equations (\ref{eq:NSE1})--(\ref{eq:NSE2}) with the following boundary conditions:
\begin{subequations}
\begin{align}
\bu &= 0 \mbox{ on bottom wall } y = - \frac{H}{2} \,, \label{eq:NSE3g} \\
\bu &= \gamma H \mbox{ on top wall } y = + \frac{H}{2}\,, \\
\bu &= \bu_{\gamma} \mbox{ as } z \to \pm \infty \,, \\
[-p \mathbf{I} + \mu(\nabla \bu &+ (\nabla \bu)^T]\cdot \mathbf{n} = 0 \mbox{ on } x = \pm \frac{W}{2} \,.\label{eq:NSE5g}
\end{align}
\end{subequations}
We solve these equations (\ref{eq:NSE1})--(\ref{eq:NSE2}) and (\ref{eq:NSE3g})--(\ref{eq:NSE5g}) numerically using COMSOL Multiphysics (Cambridge, MA) with 158,262 elements.
\section*{Acknowledgements}
K.H. was supported by National Science Foundation Grant DMS-1606487. We'd like to thank Anoop Rajappan for assistence in the fabrication of PIV setup.
\providecommand{\noopsort}[1]{}\providecommand{\singleletter}[1]{#1}%
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{"url":"https:\/\/science.sciencemag.org\/content\/324\/5933\/1394?ijkey=234ea81e409ea4460e0353b4a6183ecd602c5ca3&keytype2=tf_ipsecsha","text":"Policy ForumDRUG DISCOVERY\n\nRepurposing with a Difference\n\nSee allHide authors and affiliations\n\nScience\u00a0 12 Jun 2009:\nVol. 324, Issue 5933, pp. 1394-1395\nDOI: 10.1126\/science.1169920\n\nThere is widespread belief that current models of drug discovery and development need revamping and reinvention in order to make pharmaceutical research and development (R&D) more predictable, reliable, and less costly. We suggest a novel approach to this challenge that involves profound changes in the way postmarketing surveillance data are gathered and used. This approach capitalizes on recent advances in molecular medicine, human genomics, and information technology, as well as an increasingly sophisticated public eager for solutions to their unmet medical needs. Novel business models and imaginative legal and regulatory reforms will be critical to fulfill this promise and to maximize its impact.\n\nDespite enormous investments in basic science, technology development, and experimentation with new organizational and management structures, pharmaceutical product development still requires at least 10 to 15 years and costs between $500 million and$2 billion (1, 2). Furthermore, there is a widening productivity gap: Research and development spending continues to increase, yet the number of new therapeutic chemical and biological entities approved by the U.S. Food and Drug Administration (FDA) has been declining since the late 1990s (3). Overcoming these and other obstacles to increased productivity may require an overhaul of the R&D paradigm (4); some have called for a \u201cdisruptive\u201d transformation of the industry (5).\n\nOne response to the productivity gap is drug \u201crepurposing\u201d (6, 7) or \u201crepositioning\u201d (8)\u2014terms that refer to the identification and development of new uses for existing or abandoned pharmacotherapies. New uses of existing drugs cost much less to develop compared with de novo drug discovery and development (8). There are a number of remarkable examples of repurposed drugs whose additional indications were discovered serendipitously (8). For instance, buproprion (Wellbutrin) was originally developed to treat depression but found another use in smoking cessation (marketed as Zyban for this indication). Duloxetine (Cymbalta) was also developed to treat depression, but was hypothesized\u2014based on mechanism of action, not serendipity\u2014as a treatment for stress urinary incontinence. It was successfully developed and marketed for both indications.\n\nA very unusual case of repurposing (the rescue of an abandoned drug) is thalidomide. Originally developed as a treatment for morning sickness during pregnancy, its dangerous side effects became tragically evident in the late 1950s and early 1960s only after an epidemic of severe birth defects occurred in children exposed to the drug in utero (9). Thalidomide was withdrawn from the market but, several years later, was accidentally discovered to be uniquely effective in treating severe complications of leprosy. It is now marketed for this use under the trade name Thalomid. Twenty years later, Thalomid's use was granted a new method-of-use (MOU) patent (see below) as a treatment for a type of cancer (multiple myeloma).\n\nAnother form of repurposing is the off-label use of prescription medications to treat a condition other than that for which the drug was approved by the FDA (10). This is possible and common, because the FDA regulates how drugs are approved but not how medicine is practiced; physicians are free to prescribe approved drugs for any uses they see fit, provided they have exhausted \u201cstandard-of-care\u201d approaches and have reason to believe that the off-label use will be of clinical benefit. Because of our increasingly sophisticated understanding of human biology and the molecular pathways of disease, one would expect there to be increasing opportunities for expanding off-label use based on fully elucidated pathways and mechanisms of action, a situation that has been called a \u201cnew grammar of drug discovery\u201d (11). A classic example of this phenomenon is imatinib (Gleevec and Glivec), a drug originally developed to treat chronic myelogenous leukemia whose indications were expanded to other cancers on the basis of common underlying molecular pathways (11, 12).\n\nA more systematic way to explore the potential of existing drugs for repurposing would involve a new use of postmarketing surveillance information. Postmarketing safety monitoring (13) is regulated by the FDA in the United States and the European Medicines Agency in Europe (14, 15). The detection of potential adverse drug reactions has traditionally depended on voluntary and spontaneous reporting by individual patients and physicians, using the FDA's Adverse Events Reporting Systems (AERS). Also, pharmaceutical companies monitor the literature for case reports that may indicate a safety problem with their medicines. In addition, more proactive approaches, such as statistical data-mining of hospital records, are beginning to emerge (1619).\n\nAn increasingly important and influential resource is groups of patients who can access medical information on the Internet and see themselves as equal partners with\u2014if not the primary drivers of\u2014the medical profession in managing their health (20). Special online resources, such as Resounding Health, have recently been developed to serve this population. In a growing number of cases, patients or their relatives not only initiate, but also design and carry out, research programs that have, for example, advanced understanding and treatment of gastrointestinal stromal tumor, gastroesophageal reflux disease, autism, and the genetic disorder pseudoxanthoma elasticum (20). Most such efforts to date have been carried out as part of a \u201cgift economy,\u201d in which patients and their families volunteer time and effort to bypass what they consider the \u201clethal lag time\u201d of professional research processes and formalisms (20).\n\nSuch efforts are aided by the fact that consumers can now have their genomes typed for disease associations for as little as \\$399 and can share these data electronically with their families, friends, or self-defined networks of individuals (21). A notable recent case is that of Google cofounder Sergey Brin, whose commercial genome scan revealed a high risk of developing Parkinson's disease. Brin is personally funding a study of 10,000 patients through two nonprofit companies, including the Michael J. Fox Foundation for Parkinson's Research (22).\n\nDisease-oriented social networks, such as Genetic Alliance, PatientsLikeMe, and MyDaughtersDNA, have created online communities to advance the translation of research into new treatments. Google Health, Microsoft Health Vault, and Indivo (deployed by the Dossia consortium of employers) have created personally controlled, electronic health records for individuals or groups to share their medical conditions with health care providers, researchers, and others (23). This convergence of increasing consumer activism, along with access to genetic information services and sophisticated, advanced, and accessible information technologies, has created unprecedented opportunities to bring worldwide human resources and data to bear on biomedical research problems.\n\nWhereas the purpose of classical pharmacovigilance is to identify adverse side effects of drugs (13), the new kind of pharmacovigilance we envision aims to detect, assess, and understand beneficial drug side effects (or expanded drug indications) that may become apparent during their development or use. This \u201ctype 2\u201d pharmacovigilance could be carried out, for example, by professionals using data-mining methods to look for potential beneficial events in electronic health records (16). However, we also anticipate another approach, in which potential beneficial side effects of existing drugs are identified by online communities of drug consumers using social networking technologies in a process that has been called \u201ccrowdsourcing\u201d (24, 25). Potential beneficial side effects (or new indications) for existing drugs identified in this way could be assessed in a manner conceptually similar to the formal methods by which causality criteria are applied to adverse events (15). Following initial assessment, candidates would be prioritized for further investigation, including some form of clinical trials. Validation would be most straightforward for those phenomena that could be rationalized on the basis of known disease pathways or mechanisms of action. Potential new uses that are not consistent with known disease mechanisms might generate hypotheses that could lead to the discovery of new biological processes or disease pathways.\n\nDefinitive clinical trials for novel uses of existing drugs will remain costly, and pharmaceutical companies are reluctant to invest in such efforts without patent protection. New information about the uses of existing drugs may create intellectual property in the form of MOU patents as opposed to composition-of-matter (COM) patents. COM patents are generally considered to be more valuable than MOU patents, but this differential valuation may be changing because, as Eisenberg points out, \u201cDrugs are information-rich chemicals that in many respects are more akin to other information products (such as databases) than they are to other chemicals\u201d (26).\n\nClassical pharmacovigilance, which is exclusively focused on safety issues, can produce information that is of considerable social value for patients, physicians, and insurers at the expense of economic value to pharmaceutical companies (26). Thus, repurposed pharmacovigilance that is focused on beneficial new uses will need to be based on new business models [such as open-sourcing (27)] apart from the traditional, vertically integrated R&D enterprise (5). It would also be enabled by either patent reform by Congress or new doctrinal interpretations of current law by the FDA and the courts, as has already been suggested (28).\n\nOur proposal, to repurpose pharmacovigilance, outlines a new approach to drug and biomarker discovery and suggests ways of overcoming the inadequate incentives of current business models and regulatory regimes that contribute to the productivity gap in pharmaceutical R&D. This approach leverages the talents, motivations, and resources of individuals and groups whose unmet medical needs are the fundamental goals of developing new therapies.\n\nReferences and Notes\n\n1. M.S.B. is a former vice president of the Novartis Institutes for Biomedical Research and the founder of Resounding Health Inc. K.D.M. is a principal developer of Indivo, an open-source, personally controlled health record that has been deployed through Dossia, Inc. V.P.S. is a cofounder of GlobalCures Inc.\nView Abstract","date":"2021-04-14 11:33:50","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\": 0.18566830456256866, \"perplexity\": 4385.981097982529}, \"config\": {\"markdown_headings\": false, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2021-17\/segments\/1618038077810.20\/warc\/CC-MAIN-20210414095300-20210414125300-00383.warc.gz\"}"} | null | null |
require "spec_helper"
describe VIPS::PNGWriter do
before :all do
@image = simg 'wagon.v'
@writer = VIPS::PNGWriter.new @image
@path = tmp('wagon.png').to_s
end
it "should write to a png file" do
@writer.write @path
im = VIPS::Image.png @path
im.x_size.should == @image.x_size
im.y_size.should == @image.y_size
end
it "should write a png to memory", :vips_lib_version => "> 7.22" do
str = @writer.to_memory
if Spec::Helpers.match_vips_version(">= 7.34")
reader = VIPS::PNGReader.new(str)
im = reader.read_buffer
im.should match_image(@image)
end
end
it "should write a tiny png file to memory", :vips_lib_version => "> 7.22" do
im = VIPS::Image.black(10, 10, 1)
s = im.png.to_memory
if Spec::Helpers.match_vips_version(">= 7.34")
reader = VIPS::PNGReader.new(s)
im2 = reader.read_buffer
im2.should match_image(im)
end
end
it "should allow setting of the png compression" do
@writer.compression = 9
@writer.compression.should == 9
end
it "should raise an exception when trying to set an invalid compression" do
lambda{ @writer.compression = -2 }.should raise_error(ArgumentError)
lambda{ @writer.compression = "abc" }.should raise_error(ArgumentError)
lambda{ @writer.compression = 3333 }.should raise_error(ArgumentError)
end
it "should generate smaller memory images with higher compression settings" do
if Spec::Helpers.match_vips_version("> 7.22")
@writer.compression = 0
mempng = @writer.to_memory
@writer.compression = 9
mempng2 = @writer.to_memory
mempng2.size.should < mempng.size / 2
end
end
it "should write smaller images with lower quality settings" do
@writer.compression = 9
@writer.write(@path)
size1 = File.size @path
@writer.compression = 0
@writer.write(@path)
size2 = File.size @path
size1.should < size2 / 2
end
it "should write an interlaced png" do
@writer.interlace = true
@writer.write @path
im = VIPS::Image.png(@path)
im.should match_image(@image)
end
it "should write an interlaced png to memory" do
if Spec::Helpers.match_vips_version("> 7.22")
@writer.interlace = true
str = @writer.to_memory
end
end
it "should create a png writer" do
@writer.class.should == VIPS::PNGWriter
@writer.image.should == @image
end
it "should accept options on creation from an image" do
writer = @image.png(nil, :compression => 3, :interlace => true)
writer.compression.should == 3
writer.interlace.should == true
end
end
| {
"redpajama_set_name": "RedPajamaGithub"
} | 4,977 |
\section{Introduction}
The goal of this paper is simple enough. It is an attempt to
address the question:
\begin{itemize}
\item[] {\em What are the analogues of the classical set of badly approximable numbers within the multiplicative frameworks of Littlewood's conjecture and its mixed counterpart?}
\end{itemize}
\subsection{The classical setup and the set $\bad$}
A classical result of Dirichlet states that for any real number
$\alpha$ there exist infinitely many $q\in\NN$ such that
$$
q||q\alpha||<1 \ .
$$
Here and throughout $||\, . \, ||$ denotes the distance to the
nearest integer. In general the right hand side of the above
inequality cannot be replaced by an arbitrarily small constant.
Indeed a result of Jarn\'{\i}k \cite{Ja} and Besicovitch \cite{Best}
states that the set
$$
\bad:=\{\alpha\in \RR\;:\; \liminf_{q\to \infty} q||q \alpha ||>0\}
$$
of badly approximable numbers is of maximal Hausdorff dimension; i.e.
$$
\dim \bad=1.
$$
For details regarding Hausdorff dimension the reader is referred to \cite{falc}.
However, from a measure theoretic point of view the classical
theorem of Khintchine \cite{Kh24} enables us to improve on the
global statement (a statement true for all numbers) of Dirichlet by
a logarithm. In particular, for $\lambda\ge 0$ let
$$
\bad^{\lambda} := \{\alpha\in \RR\;:\; \liminf_{q\to\infty} q
\cdot (\log q)^\lambda \cdot ||q\alpha||>0\} \ .
$$
Then, Khintchine's theorem implies that
$$
\big|\bad^{\lambda} \big|=\left\{\begin{array}{ll}
0& \mbox{ if }\lambda\le 1\\[2ex]
{\mbox{\footnotesize FULL}}& \mbox{ if }\lambda> 1.
\end{array}\right.
$$
Here and throughout $|\, \cdot \, |$ denotes Lebesgue measure and
`{\footnotesize FULL}' means that the complement of the set under
consideration is of measure zero.
The upshot of the classical setup is that we are able to shave off a
logarithm from the measure theoretic `switch over' set $\bad^1$
before we precisely hit the set $\bad$. In addition, if we shave
off any more (i.e. $(\log q)^{1 + \epsilon}$ with $\epsilon > 0$
arbitrary) then the corresponding set becomes empty. This is a
theme which we claim reoccurs within the multiplicative framework
of Littlewood's conjecture and its mixed counterpart.
\subsection{The multiplicative setup and the set $\mad$}
A straightforward consequence of Dirichlet's classical result is
that for every $(\alpha,\beta) \in \RR^2$, there exist infinitely
many $q\in\NN$ such that
$$
q\cdot ||q\alpha||\cdot||q\beta||<1.
$$
Littlewood conjectured that the right hand side of the above
inequality can be replaced by an arbitrarily small constant.
\begin{Lconj}
For every $(\alpha,\beta) \in \RR^2$,
\begin{equation}\label{eq_LC}
\liminf_{q\to \infty}q\cdot ||q\alpha||\cdot ||q\beta||=0 \ .
\end{equation}
\end{Lconj}
\noindent Despite concerted efforts over the years this famous
conjecture remains open. For background and recent `progress'
concerning this fundamental problem see \cite{ELK,PVL} and
references therein.
A consequence of LC is that the set
$$
\{(\alpha,\beta)\in \RR^2\;:\; \liminf_{q\to\infty} q\cdot
||q\alpha||\cdot ||q\beta||>0\}
$$
is empty and therefore is not a candidate for the multiplicative
analogue of $\bad$. Regarding possible candidates, for $\lambda\ge
0$ let
$$
\mad^{\lambda}:=\{(\alpha,\beta)\in \RR^2\;:\; \liminf_{q\to\infty}
q \cdot (\log q )^\lambda \cdot ||q\alpha||\cdot ||q\beta||>0\}.
$$
From a measure theoretic point of view Gallagher's theorem \cite{Ga}
(the multiplicative analogue of Khintchine's theorem) implies that
$$
|\mad^\lambda|=\left\{\begin{array}{ll}
0& \mbox{ if }\lambda\le 2\\[2ex]
{\mbox{\footnotesize FULL}}& \mbox{ if }\lambda> 2.
\end{array}\right.
$$
Natural heuristic `volume' arguments
give evidence in favour of the following statement: for every
$(\alpha,\beta) \in \RR^2$ there exist infinitely many $q\in\NN$
such that
$$
q\cdot \log q \cdot ||q\alpha||\cdot||q\beta|| \ll 1 .
$$
The results of Peck \cite{Peck} and Pollington $\&$ Velani
\cite{PVL} give solid support to this statement which represents a
significant strengthening of Littlewood's conjecture and implies
that
\begin{itemize}
\item[] {\bf[L1]} \hspace*{1ex} $\badl^{\lambda} = \emptyset $ \ if \ $ \lambda < 1 $.
\end{itemize}
Moreover, we suspect that the heuristics are sharp and thus $\mad
:= \mad^{1} $ represents the natural analogue of $\bad$ within the
multiplicative setup. It is worth emphasizing that $\mad$ defined
in this manner is precisely the set we hit after shaving off a
logarithm from the measure theoretic `switch over' set $\mad^2$.
Note that this is in keeping with the classical setup. Furthermore,
we claim that the analogue of Jarn\'{\i}k-Besicovitch theorem is
true for $\mad$. In other words,
\begin{itemize}
\item[] {\bf[L2]} \hspace*{1ex} $ \dim \badl^{\lambda} = 2 $ \ if \ $ \lambda \ge 1 $.
\end{itemize}
\vspace*{2ex}
\noindent
Regarding {\bf [L1]}, notice that a counterexample to LC would imply that $\badl^{\lambda} $ is non-empty for any $\lambda \ge 0$. In principle, it should be easier to give a counterexample to {\bf[L1]}. To date all that is know is the remarkable result of Einsiedler, Katok $\&$ Lindenstrauss \cite{ELK} that states that
$ \dim \mad^0=0. $ The following would be a leap in the right
direction towards {\bf[L1]} and would represent a significant
strengthening of the Einsiedler-Katok-Lindenstrauss zero dimension
result.
\begin{itemize}
\item[] {\bf[L3]} \hspace*{1ex}
$\dim \mad^\lambda =0 \ \mbox{ if } \ \lambda<1.$
\end{itemize}
\noindent To the best of our knowledge, currently we do not even
know if $ \dim \badl^{\lambda} < 2 $ for strictly positive $
\lambda < 1 $.
\noindent
Regarding {\bf [L2]}, very little beyond the trivial is known. A simple consequence of the `{\mbox{\footnotesize FULL}}' statement above is that $ \dim \mad^\lambda = 2 $ if $\lambda > 2$. Recently, Bugeaud $\&$ Moshchevitin~\cite{BM} have shown that $\dim \mad^2=2$. Note that this is non-trivial since the set $ \mad^2$ is of measure zero. Surprisingly and somewhat embarrassingly we are unable to show that $ \mad^{2 - \epsilon} \neq \emptyset $ let alone
\begin{itemize}
\item[] {\bf[L4]} \hspace*{1ex}
$ \mad^\lambda \neq \emptyset \ \mbox{ if } \ 1 \le
\lambda<2.$
\end{itemize}
\noindent In other words, given our current state of knowledge, we
can not rule out the unlikely possibility that LC is actually true
with a $(\log q)^{2- \epsilon}$ term inserted in the left hand
side of~\eqref{eq_LC} -- see also \cite[Question 37]{aim}.
\vspace*{2ex}
In this paper, we are unable to directly contribute towards the
statements {\bf[L1]} -- {\bf[L4]}. However, we are able to make
a significant contribution towards establishing the analogue of
{\bf[L2]} within the framework of the mixed Littlewood conjecture.
Thus, if there is a genuine `dictionary' between the results
related to the two conjectures then indirectly our contribution
adds weight towards {\bf[L2]}.
\subsection{The mixed multiplicative setup and the set $\badl_{\DDD}$}
Recently, de Mathan $\&$ Teuli\'{e} in \cite{MT04} proposed the
following variant of Littlewood's conjecture. Let $\mathcal{D} $ be
a sequence $(d_k)_{k=1}^\infty$ of integers greater than or equal to
$2$ and let
$$
D_0 := 1 \quad {\rm and} \quad D_n := \prod_{k=1}^n d_k \ .
$$
For $q \in \ZZ$ set
$$
|q|_\DDD:=\inf\{D_n^{-1}\;:\; q\in D_n\ZZ\}.
$$
\begin{MLconj}
For every real number $\alpha$
\begin{equation}\label{eq_MLC}
\liminf_{q\to \infty}q\cdot |q|_\DDD\cdot ||q\alpha||=0 \ .
\end{equation}
\end{MLconj}
\noindent When $\mathcal{D}$ is the constant sequence equal to a
prime number $p$, the norm $|\, \cdot \, |_\mathcal{D}$ is the
usual $p$-adic norm $|\, \cdot \, |_p$. In this particular case,
there is a perfect dictionary between the current body of results
associated with ($p$-adic) MLC and LC. The following constitute the
main non-trivial entries.
\begin{itemize}
\item In \cite[Theorem 2.1]{MT04} de Mathan $\&$ Teuli\'{e} establish the analogue of Peck's cubic result.
\item In \cite[Section 1]{MT04} de Mathan $\&$ Teuli\'{e} observe that the ideas within \cite{PVL} establish the analogue of the Pollington-Velani full dimension result. Also see \cite[Theorem 4]{bdd}.
\item In \cite[Theorem 1]{BHV} Bugeaud, Haynes $\&$ Velani establish the analogue of Gallagher's measure theoretic result.
\item In \cite[Theorem 1.1]{EinsiedlerKleinbock} Einsiedler $\&$ Kleinbock establish the analogue of the Einsiedler-Katok-Lindenstrauss zero dimension result.
\item In \cite{BM} Bugeaud $\&$ Moshchevitin establish the analogue of their $\dim \mad^2=2$ result.
\end{itemize}
\noindent Moving away from the $p$-adic case, the results associated
with MLC in the first two items above are valid for any bounded
sequence $\DDD$. In all likelihood, this is also true for the other
three items. The biggest challenge of the three seems to lie in
generalising the ($p$-adic) result of Einsiedler $\&$ Kleinbock
to bounded sequences. We are pretty confident that the other two
items can be generalised to bounded $\DDD$ without too much trouble
but stress that we have not carried out the details\footnote{The
problem of generalizing the ($p$-adic) mixed result obtained in
\cite{BHV} to arbitrary sequences $\DDD$ is particularly
interesting since for unbounded $\DDD$ we suspect that the `volume'
sum is dependant on $\DDD$.}. The point being made here is that for
bounded $\DDD$ there is reasonably hard evidence in support of a
`LC--MLC' dictionary.
For $\lambda \ge 0$ let
\begin{equation}
\badl_{\DDD}^{\lambda} := \left\{\alpha\in \RR \;:\; \liminf_{q \to
\infty} \ q \cdot (\log q)^{\lambda}
\cdot
|q|_\DDD\cdot ||q\alpha||> 0 \right\}.
\end{equation}
\noindent For bounded $\DDD$, in view of the above discussion it
is natural to expect that the following statements correspond to the
entries {\bf[L1]} and {\bf[L2]} within the `LC--MLC' dictionary.
\begin{itemize}
\item[] {\bf[ML1]} \hspace*{1ex} $\badl_{\DDD}^{\lambda} = \emptyset $ if $ \lambda < 1 $.
\item[] {\bf[ML2]} \hspace*{1ex} $ \dim \badl_{\DDD}^{\lambda} = 1 $ if $ \lambda \ge 1 $.
\end{itemize}
\noindent In short, the upshot for bounded $\DDD$ is that
$\badl_{\DDD} := \badl_{\DDD}^{1} $ represents the natural analogue
of $\bad$ within the `mixed' multiplicative setup. The assumption
that $\DDD$ is bounded is absolutely necessary -- see Theorem
\ref{th_mainsvsv} below.
Obviously a counterexample to MLC would imply that
$\badl_{\DDD}^{\lambda} \neq \emptyset$ for any $\lambda
\ge 0$. In principle, it should be easier to give a
counterexample to {\bf[ML1]}. The Einsiedler--Kleinbock result
($\dim \badl_{\DDD}^{0} = 0$ within the $p$-adic case) represents
the current state of knowledge regarding {\bf[ML1]}. It would be
highly desirable to obtaining the following generalization.
\begin{itemize}
\item[] {\bf[ML3]} \hspace*{1ex} $\dim \badl_{\DDD}^{\lambda} = 0 $ if $ \lambda < 1 $.
\end{itemize}
\noindent As far as we are aware, it is not even known if $ \dim
\badl_{\DDD}^{\lambda} < 1 $ for strictly positive $ \lambda <
1 $.
The following contribution towards {\bf[ML2]} constitutes the main
result proved in this paper. In our opinion, up to powers of
logarithms it is best possible for bounded $\DDD$.
\begin{theorem}\label{th_main}
Let $\DDD$ be a sequence of integers greater than or equal to $2$.
Then the set of real numbers $\alpha$ such that
\begin{equation}\label{eq_main}
\liminf_{q\to\infty}\ q\cdot \log q\cdot \log\log q\cdot
|q|_\DDD\cdot ||q\alpha||>0.
\end{equation}
has Hausdorff dimension equal to 1.
\end{theorem}
A simple consequence of the theorem is the following statement.
\begin{corollary}\label{mainsv1}
Let $\DDD$ be a sequence of integers greater than or equal to $2$.
For $ \lambda > 1 $
$$
\dim \badl_{\DDD}^{\lambda} = 1 \, .
$$
\end{corollary}
\noindent Unfortunately, for bounded $\DDD$ we are unable to deal
with the case $\lambda =1$. In fact, we are unable to show that
\begin{itemize}
\item[] {\bf[ML4]} \hspace*{1ex}
$\badl_{\DDD}^{1} \neq \emptyset \, . $
\end{itemize}
\noindent However, for unbounded $\DDD$ we can do much better in
the following sense.
\begin{theorem}\label{th_main2}
Let $\DDD=\{2^{2^n}\}_{n\in\NN}$. Then the set of real numbers
$\alpha$ such that
\begin{equation}\label{eq_dunbound}
\liminf_{n\to\infty} q\cdot \log\log q\cdot \log\log\log q\cdot
|q|_\DDD\cdot ||q\alpha||>0
\end{equation}
has Hausdorff dimension equal to $1$.
\end{theorem}
A simple consequence of the theorem is the following statement.
\begin{corollary}\label{th_mainsvsv} There exist uncountably many unbounded sequences $\DDD$ of
integers greater than or equal to $2$ such that
\begin{equation}\label{eq_mainsvsv}
\dim \badl_{\DDD}^{\lambda} = 1 \quad \forall \ \lambda > 0
\ .
\end{equation}
\end{corollary}
\noindent The theorem shows that {\bf[ML1]} is not generally true
for unbounded $\DDD$. It also suggests that if there are
counterexamples to MLC then they may be easier to find among rapidly
increasing sequences.
Furthermore, for unbounded $\DDD$ it is not generally true that the natural analogue of $\bad$ within the `mixed' multiplicative setup is $\badl_{\DDD}^{1} $. This is yet an other reason to why we restrict the `LC--MLC' dictionary to bounded sequences. Indeed, we can deduce from the proof of Theorem~\ref{th_main2} that the analogue of $\bad$ for any given unbounded $\DDD$ is in fact dependant on the growth of~$\DDD$.
\section{Preliminaries}
To prove Theorems~\ref{th_main} and~\ref{th_main2} it will be
convenient to work with the `modified logarithm' function
$\log^*\;:\; \RR\to\RR$ defined as follows
$$
\log^*\! x:=\left\{\begin{array}{ll}1&\mbox{for
}x<e \\
\log x&\mbox{for }x\ge e \, .
\end{array}\right.
$$
This will guarantee that for small values of $x$ the function
$\log^*\!x$ is well defined.
\subsection{The basic strategy \label{bssv}}
Given a function $f\;:\;\NN\to \RR$ and a sequence $\DDD$ of
integers not smaller than 2, consider the set
\begin{equation}\label{def_badl}
\mad_\DDD(f):=\{\alpha\in \RR\;:\; \liminf_{q\to \infty} f(q)\cdot
q\cdot |q|_\DDD\cdot ||q\alpha||>0\}.
\end{equation}
By definition the set $\mad_\DDD(f)$ is a subset of $\RR$ and
therefore
$$\dim \mad_\DDD(f) \le 1.$$ Thus the proofs of Theorems~\ref{th_main} and~\ref{th_main2} are reduced to establishing the following respective statements.
\begin{proposition}\label{prop1sv}
Let $\DDD$ be a sequence of integers greater than or equal to $2$.
Then
\begin{equation}\label{eq_madf}
\dim\mad_\DDD(f)\ge 1\quad\mbox{with } \ f(q):=\log^*\!q\cdot\log^*\log q.
\end{equation}
\end{proposition}
\begin{proposition}\label{prop2sv}
Let $\DDD=\{2^{2^n}\}_{n\in\NN}$. Then
\begin{equation}\label{eq_madf2}
\dim\mad_\DDD(f)\ge 1\quad\mbox{with } \ f(q):=\log^*\log
q\cdot\log^*\log^*\log q.
\end{equation}
\end{proposition}
To establish the propositions we make use of the following
decomposition of $\mad_\DDD(f)$. For any constant $c>0$ define
$$
\badl_{\DDD}(f,c) := \left\{\alpha\in \RR \;:\; f(q)\cdot q\cdot
|q|_\DDD\cdot ||q\alpha||>c\;\ \forall \ q\in\NN\right\}.
$$
It is easily verified that $$ \badl_{\DDD}(f,c) \subset
\badl_\DDD(f)$$ and
$$\badl_\DDD(f)\, = \, \bigcup_{c > 0} \badl_{\DDD}(f,c)\ . $$
\noindent Geometrically, the set $\badl_{\DDD}(f,c) $ simply
consists of points on the real line that avoid all intervals
$$
\Delta(r/q):=\left[\frac{r}{q}-\frac{c}{f(q)q^2|q|_\DDD},\frac{r}{q}+\frac{c}{f(q)q^2|q|_\DDD}\right]
$$
centered at rational points $r/q $ with $q \ge 1$. Alternatively, points
on the real line that lie within any such interval are removed.
Given a rational $r/q$, let
\begin{equation}\label{def_height}
H(q):=q^2|q|_\DDD
\end{equation}
denote its {\em height}. Trivially, we have that
$$
|\Delta(r/q)|=\frac{2c}{f(q)H(q)}.
$$
\noindent In order to show that $\dim \badl_{\DDD}(f) \ge 1 $,
the idea is to construct a Cantor-type subset $\KK_{\DDD}(f,c) $
of $
\badl_{\DDD}(f,c) $ such that
$$
\dim \KK_{\DDD}(f,c) \ge 1
$$
for some small constant $c>0$. Hence, by construction we have that
$$
\dim\mad_\DDD(f) \, \ge \, \dim \badl_{\DDD}(f,c) \, \ge \, \dim
\KK_{\DDD}(f,c) \, \ge \, 1\, .
$$
Thus, the name of the game is to construct the `right type' of Cantor
set $\KK_{\DDD}(f,c)$. In short, the properties of the desired
set fall naturally within a general framework which we now
describe.
\subsection{A general Cantor framework \label{gcf}}
\noindent{\bf The parameters.} Let ${\rm I}$ be a closed interval
in $\RR$. Let $$\RRR:=(R_n) \quad {\rm with } \quad {n\in \ZZ_{\ge
0}}$$ be a sequence of natural numbers and $$\vr:=(r_{m,n}) \quad
{\rm with } \quad m,n\in \ZZ_{\ge 0} \ {\rm \ and \ } \ m\le n
$$ be a two parameter sequence of non-negative real numbers.
\vspace*{2ex}
\noindent{\bf The construction.} We start by subdividing the interval ${\rm I}$ into $R_0$
closed intervals $I_1$ of equal length and denote by $\II_1$ the collection of such intervals. Thus,
$$
\#\II_1 = R_0 \qquad {\rm and } \qquad |I_1| = R_0^{-1}\, |{\rm
I}| \ .
$$
Next, we remove at most $r_{0,0}$ intervals $I_1$ from $\II_1$ . Note that we do not specify which intervals should be
removed but just give an upper bound on the number of intervals to be removed. Denote by $\JJ_1$ the resulting
collection. Thus,
\begin{equation}\label{iona1}
\#\JJ_1 \ge \#\II_1 - r_{0,0} \, .
\end{equation}
For obvious reasons, intervals in $\JJ_1$ will be referred to as (level one) survivors. It will be convenient to define $\JJ_0 := \{J_0\} $ with $ J_0 :={\rm I} $.
\vspace*{1ex}
\noindent In general, for $n \ge 0$, given a collection
$\JJ_n$ we construct a nested collection $\JJ_{n+1}$ of closed intervals $J_{n+1}$ using the following two operations.
\begin{itemize}
\item{\em Splitting procedure.} We subdivide each interval $J_n\in \JJ_n$ into $R_n$ closed sub-intervals $I_{n+1}$ of equal length and denote by $\II_{n+1}$ the collection of such intervals. Thus,
$$
\#\II_{n+1} = R_n \times \#\JJ_n \qquad {\rm and } \qquad |I_{n+1}| = R_n^{-1}\, |J_n| \ .
$$
\item{\em Removing procedure.} For each interval $J_n\in \JJ_n$ we remove at most
$r_{n,n}$ intervals $I_{n+1} \in \II_{n+1} $ that lie within $
J_n$. Note that the number of intervals $I_{n+1}$ removed is allowed to vary amongst the intervals in $\JJ_n$. Let $\II_{n+1}^{n} \subseteq \II_{n+1} $ be the collection of intervals that remain. Next, for each interval $J_{n-1}\in \JJ_{n-1}$
we remove at most $r_{n-1,n}$ intervals $I_{n+1} \in \II_{n+1}^{n} $ that lie within $ J_{n-1}$. Let $\II_{n+1}^{n-1} \subseteq \II_{n+1}^{n} $ be the collection of intervals that remain. In general, for each interval $J_{n-k}\in \JJ_{n-k}$ $(1 \le k \le n)$
we remove at most $r_{n-k,n}$ intervals $I_{n+1} \in \II_{n+1}^{n-k+1} $ that lie within $J_{n-k}$. Also we let $\II_{n+1}^{n-k} \subseteq \II_{n+1}^{n-k+1} $ be the collection of intervals that remain.
In particular,
$\JJ_{n+1} := \II_{n+1}^{0} $
is the desired collection of (level $n+1$) survivors. Thus,
the total number of intervals $I_{n+1}$ removed during the removal procedure is at most
$
r_{n,n}\#\JJ_n+r_{n-1,n}\#\JJ_{n-1}+\ldots+r_{0,n}\#\JJ_0
$
and so
\begin{equation}\label{iona2}
\#\JJ_{n+1}\ge R_n\#\JJ_n-\sum_{k=0}^nr_{k,n}\#\JJ_k.
\end{equation}
\end{itemize}
\noindent Finally, having constructed the nested collections $\JJ_n$ of closed intervals we consider the limit set
$$
\KKK ({\rm I},\RRR,\vr) := \bigcap_{n=1}^\infty \bigcup_{J\in
\JJ_n} J.
$$
The set $\KKK({\rm I},\RRR,\vr)$ will be referred to as a {\em $({\rm I},\RRR,\vr)$ Cantor set.}
\vspace*{2ex}
\noindent{\em Remark.}
We stress that the triple $({\rm I},\RRR,\vr)$ does not
uniquely determine the set $\KKK ({\rm I},\RRR,\vr)$.
The point is that during the construction we only specify the maximum number of intervals rather than the specific intervals to be removed. Thus the triple $({\rm I},\RRR,\vr)$ gives rise to a family of $({\rm I},\RRR,\vr)$ Cantor sets that reflects the various available choices during the removing procedure.
\vspace*{1ex}
As an illustration of the general framework, it is easily seen that the standard middle third
Cantor set corresponds to a $({\rm I}, \RRR,\vr)$ Cantor set with
$$
{\rm I} := [0,1], \qquad \RRR= (3,3,3,\ldots)\quad\mbox{and}\quad
\vr=(r_{m,n})
$$
where
$$
r_{m,n}:=\left\{\begin{array}{ll}1&\mbox{if }m=n\\[1ex]
0&\mbox{otherwise}.
\end{array}\right.
$$
\vspace*{2ex}
\noindent{\bf The results.} By definition, if $\JJ_n$ is empty for some $n \in \NN$ then the corresponding set $ \KKK ({\rm I},\RRR,\vr)$ is obviously empty. On the other hand, by construction, each closed interval $J_n \in \JJ_n $ is contained in some closed interval $J_{n-1} \in \JJ_{n-1} $. Therefore
$$\KKK({\rm I}, \RRR,\vr) \neq \emptyset \qquad {\rm if } \qquad \# \JJ_n \ge 1 \quad \forall \ n \in \NN \, .$$
Our first result provides a natural condition that guarantees this cardinality hypothesis and therefore the non-empty statement.
\begin{theorem}\label{th_cantor1} Given $\KKK ({\rm I},\RRR,\vr) $,
let
\begin{equation}\label{def_t0}
t_0:=R_0-r_{0,0}
\end{equation}
and for $n \ge 1 $ let
\begin{equation}\label{def_tn}
t_n:=R_n-r_{n,n}-\sum_{k=1}^n \frac{r_{n-k,n}}{\prod_{i=1}^k
t_{n-i}} \, .
\end{equation}
Suppose that $t_n>0$ for all $n\in\ZZ_{\ge 0}$. Then
$$
\KKK ({\rm I},\RRR,\vr) \neq \emptyset \ .
$$
\end{theorem}
\vspace*{2ex}
The proof of Theorem~\ref{th_cantor1} is short and direct and there seems little point in delaying it.
\vspace*{1ex}
\noindent{\em Proof of Theorem~\ref{th_cantor1}.} We show that a consequence of the construction of $\KKK ({\rm I},\RRR,\vr) $ is that
\begin{equation}\label{ineq_jj}
\#\JJ_n\ge t_{n-1} \#\JJ_{n-1} \quad \forall \ n \in \NN \, .
\end{equation}
This together with the assumption that $t_n>0$ implies that
$
\#\JJ_n\ge \prod_{i=0}^{n-1} t_i \, \#\JJ_0 \, > \, 0
$
and thereby completes the proof of Theorem~\ref{th_cantor1}. To verify \eqref{ineq_jj} we use induction. In view of \eqref{iona1} and \eqref{def_t0} the statement is trivially true for $n=1$.
Now suppose that \eqref{ineq_jj} is true for all $1 \le k \le n$. In particular, for any such $k$ we have that
$$
\#\JJ_n\ge t_{n-1}\#\JJ_{n-1}\ge\ldots\ge \prod_{i=1}^{k}t_{n-i}\#
\JJ_{n-k}.
$$
\noindent Thus,
\begin{eqnarray*}
\#\JJ_{n+1} & \stackrel{\eqref{iona2}} \ge & R_n\#\JJ_n-\sum_{k=0}^nr_{n-k,n}\#\JJ_{n-k}
\\ &\ge &
R_n\#\JJ_n-r_{n,n}\#\JJ_n-\sum_{k=1}^n\frac{r_{n-k,n}\#\JJ_n}{\prod_{i=1}^k
t_{n-i}} \\
& \stackrel{\eqref{def_tn}} = & t_n\#\JJ_n.
\end{eqnarray*}
This completes the induction step and establishes \eqref{ineq_jj} as required.
\hspace*{\fill}$\boxtimes$
\vspace*{3ex}
Our next result enables us to estimate the Hausdorff dimension of
$\KKK({\rm I},\RRR,\vr)$. It is the key to establishing Propositions \ref{prop1sv} $\&$ \ref{prop2sv}.
\begin{theorem}\label{th_cantor2} Given $\KKK ({\rm I},\RRR,\vr) $, suppose that
$R_n\ge 4$ for all $n\in\ZZ_{\ge 0}$ and that
\begin{equation}\label{cond_th2}
\sum_{k=0}^n \left(r_{n-k,n}\prod_{i=1}^k
\left(\frac{4}{R_{n-i}}\right)\right)\le \frac{R_n}{4}.
\end{equation}
Then
$$
\dim \KKK ({\rm I},\RRR,\vr) \ge \liminf_{n\to \infty}(1-\log_{R_n} 2).
$$
\end{theorem}
\noindent Here we use the convention that the product term in \eqref{cond_th2} is one when $k=0$ and by definition $ \log_{R_n}\!2 := \log2/ \log R_n$. The proof of Theorem~\ref{th_cantor2} is rather involved and constitutes the main substance of \S\ref{Stalin} and \S\ref{Lenin}. To some extent the raw ideas required to establish Theorem~\ref{th_cantor2} can be found in \cite[\S7]{BPV} where a conjecture of W.M. Schmidt regarding the intersection of simultaneously badly approximable sets is proved. Nevertheless we stress that in this paper we develop a general Cantor type framework rather than address a specific problem.
As a consequence the key ideas of \cite{BPV} are foregrounded.
\vspace*{2ex}
\noindent {\em Remark.} Although Theorem~\ref{th_cantor2} is more than sufficient for the specific application we have in mind, we would like to point out that we have not attempted to establish the most general or best possible statement. For example, in the case $ R_n \to \infty$ as $n \to \infty$, the theorem together with the fact that $ \KKK ({\rm I},\RRR,\vr) \subset \RR$ implies that $\dim \KKK ({\rm I},\RRR,\vr) = 1 $.
However, we do not claim that condition \eqref{cond_th2} is optimal for establishing this full dimension result.
\vspace*{2ex}
In the final section of the paper, we show that the intersection of any finite number of sets $\KKK({\rm I},\RRR,\vr_i) $ is yet another $({\rm I},\RRR,\vr) $ Cantor set for some appropriately chosen $\vr$. We shall also see in \S\ref{FC} that this enables us to strengthen Theorem~\ref{th_main}.
\section{Proof of Proposition \ref{prop1sv} modulo Theorem~\ref{th_cantor2} \label{proofcone}}
Throughout, $f:\NN\to \RR : q \to f(q)=\log^*\!q\cdot\log^*\!\log q$
and $\DDD$ is a sequence of integers greater than or equal to
$2$.
Let
$$R > e^{12} $$
be an integer. Choose $c_1=c_1(R) > 0 $
sufficiently small so that
\begin{equation}\label{ineq_c1}
2e^2c_1\frac{\log R+2}{\log 2}R<1
\end{equation}
and let $c = c(R,c_1)>0$ be a constant such that
\begin{equation}\label{ineq_c}
c\left(\frac{64R^2(\log R+2)}{c_1\log 2}+\frac{16eR^2(\log
R+2)^2}{\log 2}\right)<1.
\end{equation}
With reference to \S\ref{bssv} we now describe the basic construction of the set $\KK_{\DDD}(f,c)$.
Let ${\rm I}$ be any interval of length $c_1$ contained within the unit interval $ [0,1]$. Denote by $ \JJ_0:=\{J_0\}$ where $J_0 := {\rm I}$.
The idea is to establish, by induction on $n$, the
existence of a collection $\JJ_n$ of closed intervals $J_n$ such
that $\JJ_n $ is nested in $\JJ_{n-1}$; that is, each interval
$J_n$ in $\JJ_n$ is contained in some interval $J_{n-1}$ in
$\JJ_{n-1}$. The length of an interval $J_n$ will be given by
$$ |J_n| \, := \, c_1 \, R^{-n}F^{-1}(n) \ ,
$$
where
$$
F(n):=\prod_{k=1}^n k \; [\log^*\! k]\;\ \mbox{ for } n\ge 1\quad\mbox{
and } \quad F(0):=1 \ \mbox{ for }n\le 0 .
$$
Moreover, each interval $J_n $ in $\JJ_n$ will satisfy the condition that
\begin{equation} \label{cond}
J_{n} \, \cap \, \Delta(r/q) \, = \, \emptyset \qquad \forall
\ \ r/q\in\QQ \ \ \mbox{with } \ H(q) < R^{n \!-\!1}F(n-1) \, .
\end{equation}
In particular, we put
$$\KK_{\DDD}(f,c) := \bigcap_{n=1}^\infty \bigcup_{J\in\JJ_n}J \ .
$$
By construction, condition \eqref{cond} ensures that
$$
\KK_{\DDD}(f,c) \subset \badl_{\DDD}(f,c) \ .
$$
Furthermore, with reference to \S\ref{gcf} it will be apparent from the construction of the collections $\JJ_n $ that
$ \KK_{\DDD}(f,c) $ is in fact a $({\rm I},\RRR,\vr)$ Cantor set with $\RRR=(R_n)$ given by
\begin{equation}\label{def_rn}
R_n:=R \, (n+1) \, [\log^*\!(n+1)]
\end{equation}
and $\vr=(r_{m,n})$ given by
\begin{equation}\label{def_rrnm}
r_{m,n} \, := \, \left\{\begin{array}{ll} 7 \, \log^2\!R\cdot
n^2(\log^*\!n)^2 &\mbox{
\ if }\; m=n-1\\[2ex]
0 &\mbox{ \ otherwise. }
\end{array}\right.
\end{equation}
\noindent By definition, note that for any
$ R > e^9$ we have that the
\begin{eqnarray*}
{\rm l.h.s. \ of \ } \eqref{cond_th2} & = & r_{n-1,n}\cdot\frac{4}{R_{n-1}} \, \le \, 7 \cdot 2^3 \cdot \frac{\log^2\!R \cdot n \, \log^*\!n }{R} \\[2ex]
&\le & \frac{ 7 \cdot 2^6 \log^2\!R}{R^2} \cdot \frac{R \, (n+1) \, [\log^*\!(n+1)] }{4} \\[2ex]
& \le & \frac{R_n }{4} \ = \ {\rm r.h.s. \ of \ } \eqref{cond_th2} \, .
\end{eqnarray*}
Since we are assuming that $ R > e^{12} $, it then follows via Theorem~\ref{th_cantor2} that
$$
\dim \KK_{\DDD}(f,c) \ge \liminf_{n\to \infty} (1-\log_{R_n}\!2)=1 \, .
$$
This completes the proof of Proposition \ref{prop1sv} modulo Theorem~\ref{th_cantor2} and the construction of the collections $\JJ_n $.
\subsection{Constructing the collections $\JJ_n$ \label{cone}}
For $n=0$, we trivially have that (\ref{cond}) is satisfied for the
interval $J_0 = {\rm I}$. The point is that there are no rationals satisfying
the height condition $H(q)<1$ since by definition $H(q) \ge 1$. For the same reason
(\ref{cond}) with $n=1$ is trivially satisfied for any interval
$J_1$ obtained by subdividing each $J_0$ in $\JJ_0$ into $R_0=R$
closed intervals of equal length $c_1 R^{-1} $. Denote by $\JJ_1 $
the resulting collection of intervals $J_1$ and note that
$
\# \JJ_1 = R \ .
$
In general, given $\JJ_n$ satisfying \eqref{cond} we wish to
construct a nested collection $\JJ_{n+1}$ of intervals $J_{n+1}$ for
which (\ref{cond}) is satisfied with $n$ replaced by $n+1$. By
definition, any interval $J_n$ in $\JJ_n$ avoids intervals
$\Delta(r/q)$ arising from rationals with height bounded above by the quantity
$R^{n-1}F(n-1)$. Since any `new' interval $J_{n+1}$ is to be nested
in some $J_n$, it is enough to show that $J_{n+1}$ avoids intervals
$\Delta(r/q)$ arising from rationals $r/q$ with height satisfying
\begin{equation}\label{zeq2}
R^{n-1}F(n-1)\le H(q)<R^nF(n) \ .
\end{equation}
Denote by $C(n)$ the collection of all rationals satisfying
this height condition. Formally
$$
C(n) := \left\{r/q \in \QQ \, : \, H(q) \ \ {\rm satisfies \
(\ref{zeq2}) \, } \right\} \
$$
and it is precisely this collection of rationals that comes into
play when attempting to construct $\JJ_{n+1}$ from $\JJ_{n}$. We now proceed
with the construction.
Assume that $n\ge 1$. We subdivide each $J_n$ in $\JJ_n$ into
$$
R_n\stackrel{\eqref{def_rn}}=R(n+1) \, [\log^*\!(n+1)]
$$
closed intervals $I_{n+1}$ of equal length $ c_1
R^{-(n+1)}F^{-1}(n+1)$ and denote by $\II_{n+1}$ the collection of
such intervals. Thus,
$$|I_{n+1}|=c_1 R^{-(n+1)}F^{-1}(n+1) $$
and
$$
\# \II_{n+1} \, = \, R(n+1) \ [\log^*\!(n+1)] \, \times \, \# \JJ_{n}
\, .
$$
It is obvious that the construction of $\II_{n+1}$ corresponds
to the splitting procedure associated with the construction of a $({\rm I},\RRR,\vr)$ Cantor set.
In view of the nested requirement, the collection $\JJ_{n+1}$ which
we are attempting to construct will be a sub-collection of
$\II_{n+1}$. In other words, the intervals $I_{n+1}$ represent
possible candidates for $J_{n+1}$. The goal now is simple --- it is
to remove those `bad' intervals $I_{n+1}$ from $\II_{n+1}$ for
which
\begin{equation}
I_{n+1} \, \cap \, \Delta(r/q) \, \neq \, \emptyset \ \ \mbox{
for some \ } r/q \in C(n) \ .
\end{equation}
The sought after collection $ \JJ_{n+1}$ consists precisely of those intervals that survive. Formally, for $n \ge 1 $ we let
$$
\JJ_{n+1}:=\{I_{n+1}\in\II_{n+1}\;:\;I_{n+1} \, \cap \,
\Delta(r/q)=\emptyset\ \mbox{ for any }r/q\in C(n)\}.
$$
For any interval $J_{n-1} \in \JJ_{n-1}$ and any integer $ R \ge e^{12}$, we claim that
\begin{eqnarray}
\#\{I_{n+1}\in \II_{n+1} \!\!\!\! &:&\!\!\!\! J_{n-1} \cap I_{n+1}\cap
\Delta(r/q)\neq\emptyset\ \mbox{ for some }\ r/q\in C(n)\} \nonumber \\[2ex]
&\le &
7 \, \log^2\!Rn^2(\log^*\!n)^2 \, . \label{ineq_jn1}
\end{eqnarray}
It then follows from the definition of $r_{m,n}$ that
$$
\#\{I_{n+1}\in\II_{n+1}\backslash\JJ_{n+1}\;:\; I_{n+1}\subset
J_{n-1}\}\le7\log^2\!R\cdot n^2(\log^*\!n)^2 \stackrel{\eqref{def_rrnm}}=r_{n-1,n}
$$
and therefore the act of removing `bad' intervals from $\II_{n+1}$ is exactly in keeping with the
removal procedure associated with the construction of a $({\rm I},\RRR,\vr)$ Cantor set. The goal now is to justify \eqref{ineq_jn1}.
\subsubsection{Counting removed intervals}\label{seq_counting}
\noindent{\em Stage 1.} Let $ r/q \in C(n)$. Then there exists a non-negative integer $k$ and an integer $\bar{q}$ such that
\begin{equation}\label{def_qk}
q=D_k\cdot \bar{q} \qquad\mbox{and } \quad q\not\in D_{k+1}\ZZ.
\end{equation}
Then, $$ H(q):=D_k\cdot \bar{q}^{2} \, . $$
Since all the terms $d_k$ of $\DDD$ are greater than or equal to two, we have that
\begin{equation}\label{ineq_ek}
D_k \le 2^k \, .
\end{equation}
Next, note that $q^2\ge H(q)\ge R^{n-1}F(n-1)$. Thus, for any $R > e^9$ it follows that
\begin{eqnarray}
f(q) & \ge & \mbox{$\frac12$} \, \log^*\! (R^{n-1}F(n-1)) \log^*\!\mbox{$\frac12$}\log
(R^{n-1}F(n-1)) \nonumber \\[2ex]
&\ge& \mbox{$\frac12$} \, n(\log^*\!n)^2 \, . \label{ineq_f}
\end{eqnarray}
To see this first observe that \eqref{ineq_f} for $n=1$ is clearly true. For $n \ge 2$, by Stirling formula we have that
$$
R^{n-1}F(n-1) \, \ge \, R^{n-1}(n-1)! \, > \, (8n)^n\quad\mbox{ for any } \ R > e^9 \, .
$$
Therefore the left hand side of \eqref{ineq_f} is bigger than
$$
\mbox{$\frac12$}n\log^*\!(8n)\cdot \log^*\! \left(\mbox{$\frac12$} n \log
(8n)\right) \; > \; \mbox{$\frac12$}n(\log^*\!n)^2.
$$
\vspace*{3ex}
\noindent{\em Stage 2.}
We subdivide the collection $C(n)$ of rationals into various `workable' sub-collections. In the first instance, for any integer $k \ge 0$, let $\cC(n,k) \subset \cC(n) $ denote the collection of rationals satisfying the additional condition \eqref{def_qk}. Formally,
\begin{equation}\label{def_cnk}
C(n,k):=\left\{r/q \in C(n)\;:\; q\mbox{ satisfies
\eqref{def_qk}}\right\}.
\end{equation}
For any $r/q\in C(n,k)$ we have that
$H(q)=D_k\cdot \bar{q}^{2}$ and thus in view of~\eqref{zeq2}
and~\eqref{ineq_ek} it follows that
\begin{eqnarray}
0 \, \le \, k &\le& [\log_2(R^n F(n)) ] \, < \, n\log_2R+n\log_2n+n\log_2\log^*\!n \nonumber \\[1ex]
& < & c_2 \,
n\log^*\! n, \label{ineq_k}
\end{eqnarray}
where $c_2:=(\log R+2)/\log 2$ is an absolute constant independent on
$n$. The upshot is that for fixed $n$ the number of (non-empty) collections $C(n,k)$
is at most $c_2n\log^*\! n$.
\noindent Next, for any integer $ l \ge 0$, let $\cC(n,k,l) \subset \cC(n,k)
$ denote the collection of rationals satisfying the
additional condition that
\begin{equation}\label{def_cnkl}
e^lR^{n-1}F(n-1) \, \le \,
H(q) \, < \, e^{l+1}R^{n-1}F(n-1) \, .
\end{equation}
Formally,
\begin{equation*}\label{def_cnklsv}
C(n,k,l):=\left\{r/q \in C(n,k)\;:\; q\mbox{ satisfies
\eqref{def_cnkl}}\right\}.
\end{equation*}
In view of \eqref{zeq2} we have that
\begin{equation*}
e^l<R n\log^*\!n
\end{equation*}
and thus it follows that
\begin{eqnarray}\label{ineq_l}
0 \ \le \ l & \le & [\log (Rn\log^*\! n)] \, < \, \log R + 2\log^*\! n \nonumber \\[1ex]
&<& c_3 \log^*\! n
\end{eqnarray}
where $c_3:=2+\log R$. The upshot is that for fixed $n$ and $k$ the number of (non-empty) collections $C(n,k,l)$
is at most $c_3 \log^*\! n$. Notice that within any collection $C(n,k,l)$ we
have extremely tight control on the height.
\vspace*{3ex}
\noindent{\em Stage 3.}
Fix an interval $J_{n-1}\in \JJ_{n-1}$. Recall, that our goal is establish \eqref{ineq_jn1}. This we will do by estimating the quantity
$$
\#\{I_{n+1}\in \II_{n+1}\;:\; J_{n-1} \cap I_{n+1}\cap
\Delta(r/q)\neq\emptyset\ \mbox{ for some }\ r/q\in C(n,k,l)\}
$$
and then summing over all possible values of $k$ and $l$. With this in mind,
consider a rational $r/q\in C(n,k,l)$ and assume that $R> e^9$. Then
\begin{eqnarray}
\#\{I_{n+1}\in \II_{n+1} \!\!\!\!\! & \!\!\!\!\! : \!\!\!\!\! \; & \!\!\!\!\! I_{n+1}\cap
\Delta(r/q)\neq\emptyset \} \ \le \
\displaystyle\frac{|\Delta(r/q)|}{|I_{n+1}|}+2 \nonumber \\[2ex] &=&
\displaystyle\frac{2cR^{n+1}F(n+1)}{c_1f(q)H(q)}+2 \nonumber \\[2ex] &\stackrel{\eqref{def_cnkl}}{\le} &
\frac{2c R^2 n(n+1) \; [\log^*\!n] \; [\log^*\!(n+1)]}{c_1 f(q) e^l}+2
\nonumber \\[2ex]
&\stackrel{\eqref{ineq_f}}{<} & \displaystyle\frac{8cR^2
(n+1)}{c_1e^l}+2 \, . \label{iona}
\end{eqnarray}
\noindent Next, consider two rationals $r_1/q_1$, $r_2/q_2 \in
C(n,k,l)$. By definition, there exit integers $ \bar{q}_1$, $\bar{q}_2 $ so that
$$
q_1=D_k\bar{q}_1\quad\mbox{and}\quad q_2=D_k\bar{q}_2.
$$
Thus $(q_1,q_2)\ge D_k$ and we have that
$$
\left|\frac{r_1}{q_1}-\frac{r_2}{q_2}\right| \ge
\frac{1}{D_k\bar{q}_1\bar{q}_2}=(H(q_1)H(q_2))^{-1/2} \; \stackrel{\eqref{def_cnkl}}> \; e^{-l-1}R^{-n+1}F^{-1}(n-1).
$$
\noindent It is easily verified that $2\, |\Delta(r/q)| $ is less than the right hand side of the above inequality -- this makes use of the fact that $4ec < 1$ which is true courtesy of \eqref{ineq_c}.
Therefore,
$$
\Delta(r_1/q_1) \cap \Delta(r_2/q_2) = \emptyset
$$
and it follows that
\begin{eqnarray}
\#\{ r/q\in C(n,k,l) \!\!\!\! &\!\!\!\! : \!\!\!\! \;& \!\!\!\! J_{n-1} \cap
\Delta(r/q)\neq\emptyset \}
\nonumber \\[2ex]
&\le&
2+\frac{|J_{n-1}|}{e^{-l-1}R^{-n+1}F^{-1}(n-1)}\ = \ 2+c_1e^{l+1}.
\label{ayesha}
\end{eqnarray}
\noindent The upshot of the cardinality estimates \eqref{iona} and \eqref{ayesha} is that
\begin{eqnarray*}
\#\{I_{n+1}\in \II_{n+1} \!\!\!\! &\!\!\!\! : \!\!\!\! \;& \!\!\!\! J_{n-1} \cap I_{n+1}\cap
\Delta(r/q)\neq\emptyset\ \mbox{ for some } r/q\in C(n,k,l)\}
\nonumber \\[2ex]
&\le&
\left(2+c_1e^{l+1}\right)\left(2+\frac{8cR^2(n+1)}{c_1e^l}\right) \nonumber \\[2ex]
&= & 4+ 2c_1e^{l+1}+\frac{16cR^2(n+1)}{c_1e^l}+8ecR^2(n+1) .
\label{ineq_1}
\end{eqnarray*}
By summing over $l$ satisfying \eqref{ineq_l} we find that
\begin{eqnarray*}
\#\{I_{n+1}\in \II_{n+1} \!\!\!\! &:& \!\!\!\! J_{n-1} \cap I_{n+1}\cap
\Delta(r/q)\neq\emptyset\ \mbox{ for some } r/q\in C(n,k)\}
\\[2ex]
&\le&\sum_{e^l<Rn\log^*\!n} \!\!\! 2c_1e^{l+1}+ \sum_{l=0}^{c_3\log^*\! n}
\frac{16cR^2(n+1)}{c_1e^l}+c_3\log^*\!n(4+8ec R^2 (n+1)) \\[2ex]
&\le& c_4 n
\log^*\! n
\end{eqnarray*}
where
$$c_4:= 2e^2c_1R+\frac{64c}{c_1}R^2+(\log R+2)(16e \, c \, R^2+4).$$
By summing over $k$ satisfying \eqref{ineq_k} we find that
\begin{eqnarray}
\#\{I_{n+1}\in \II_{n+1} \!\!\!\! &:& \!\!\!\! J_{n-1} \cap I_{n+1}\cap
\Delta(r/q)\neq\emptyset\ \mbox{ for some } r/q\in C(n)\} \nonumber \\[2ex]
&\le& c_2c_4 n^2(\log^*\!n)^2 \, . \label{prop_main}
\end{eqnarray}
In view of \eqref{ineq_c1} and \eqref{ineq_c}, for any $R > e^{12}$ the right hand side of
\eqref{prop_main} is bounded by
\begin{equation*}
\left(2+\frac{4(\log R+2)^2}{\log 2}\right)n^2(\log^*\!n)^2 \ < \ 7\log^2\!R\cdot n^2(\log^*\!n)^2 \, .
\end{equation*}
This establishes \eqref{ineq_jn1} as required.
\section{Proof of Proposition \ref{prop2sv} modulo Theorem~\ref{th_cantor2}}
The proof of Proposition \ref{prop2sv} follows the same structure and ideas as the proof of Proposition~\ref{prop1sv}. In view of this it is really only necessary to point out the key differences.
\vspace*{2ex}
Throughout, $f:\NN\to \RR : q \to f(q)=\log^*\!\log q\cdot \log^*\!\log^*\!\log q$ and $\DDD:=\{2^{2^n}\}_{n\in \NN}.
$
Note that by definition
\begin{equation}\label{ineq_dk}
D_k\ge 2^{2^k} \, .
\end{equation}
\noindent With $R $, $ c_1$ and $c$ as in the proof of Proposition~\ref{prop1sv}, the basic construction of
$$\KK_{\DDD}(f,c) := \bigcap_{n=1}^\infty \bigcup_{J\in\JJ_n}J \ \subset \ \badl_{\DDD}(f,c)$$
remains pretty much unchanged apart from the fact that the function $F$ is given by
$$
F(n):=\prod_{k=1}^n [\log^*\! k\cdot\log^*\!\log k] \;\ \mbox{ for }
n\ge 1\quad\mbox{ and } \quad F(0):=1 \ \mbox{ for }n\le 0 .
$$
Also, it becomes apparent from the construction of the collections $\JJ_n$ that
$ \KK_{\DDD}(f,c) $ is a $({\rm I},\RRR,\vr)$ Cantor set with $\RRR=(R_n)$ given by
\begin{equation*}\label{def_rnsv}
R_n:=R \, [\log^*\!(n+1)\log^*\!\log(n+1)]
\end{equation*}
and $\vr=(r_{m,n})$ given by
\begin{equation*}\label{def_rrnmsv}
r_{m,n} \, := \, \left\{\begin{array}{ll} 7 \, \log^2\!R(\log^*\!n)^2(\log^*\!\log n)^2 &\mbox{
\ if }\; m=n-1\\[2ex]
0 &\mbox{ \ otherwise. }
\end{array}\right.
\end{equation*}
Then, it is easily verified that \eqref{cond_th2} is valid for any $R > e^9$ and so Proposition~\ref{prop2sv} follows via Theorem~\ref{th_cantor2}.
\vspace*{2ex}
Regarding the construction of the collections $\JJ_n$, the induction procedure is precisely as in \S\ref{cone}. The upshot is that the proof of Proposition~\ref{prop2sv} reduces to establishing the following analogue of \eqref{ineq_jn1}. For any interval $J_{n-1} \in \JJ_{n-1}$ and any integer $ R > e^{12}$, we have that
\begin{eqnarray}
\#\{I_{n+1}\in \II_{n+1} \!\!\!\! &:&\!\!\!\! J_{n-1} \cap I_{n+1}\cap
\Delta(r/q)\neq\emptyset\ \mbox{ for some }\ r/q\in C(n)\} \nonumber \\[2ex]
&\le &
7 \, \log^2\!R(\log^*\!n)^2(\log^*\!\log n)^2 \, . \label{ineq_jn1sv}
\end{eqnarray}
This implies that act of removing `bad' intervals from $\II_{n+1}$ when constructing $\JJ_{n+1} $ from $\JJ_n$ is exactly in keeping with the
removal procedure associated with the construction of a $({\rm I},\RRR,\vr)$ Cantor set.
In order to establish \eqref{ineq_jn1sv} we follow the arguments set out in \S\ref{seq_counting}. For completeness and ease of comparison we briefly describe the analogue of the key estimates.
\vspace*{3ex}
\noindent{\em Stage 1.} The analogue of \eqref{ineq_f} is the statement that for any $R > e^4$
\begin{eqnarray}
f(q)&\ge &\log^*\! \mbox{$\frac12$} \log(R^{n-1}F(n-1))\cdot \log^*\!\log^*\! \mbox{$\frac12$}
\log(R^{n-1}F(n-1))\nonumber\\[2ex]
&\ge &\log^*\! n\log^*\!\log n.\label{ineq_ff}
\end{eqnarray}
This makes use of the fact that for $n \ge 2 $
$$
R^{n-1}F(n-1)\ge e^{2n} \quad \mbox{ for any } \ R >e^4.
$$
\vspace*{2ex}
\noindent{\em Stage 2.} In view of \eqref{ineq_dk}, it follows that the analogue of
\eqref{ineq_k} is that
\begin{equation}\label{ineq_kk}
0 \, \le \, k \, \le \, [\log_2\log_2(R^nF(n))] \, < \, \tilde{c}_2 \log^*\!n
\end{equation}
where
$$\tilde{c}_2:=\frac{1}{\log 2}\left(2+\log\frac{\log R+2}{\log 2}\right)<c_2 \, $$
Note that $ \tilde{c}_2 < c_2 $ is valid since $R \ge 6$. Next, in view of \eqref{zeq2} we have that
\begin{equation*}\label{ineq_ell}
e^l \, < \, R \log^*\!n\log^*\!\log n
\end{equation*}
and thus it follows that
\begin{equation}\label{ineq_ll}
0 \ \le \ l \ \le \ c_3\log^*\!\log n \, .
\end{equation}
\vspace*{2ex}
\noindent{\em Stage 3.} Fix an interval $J_{n-1}\in \JJ_{n-1}$.
Then \eqref{ayesha} remains unchanged and the analogue
of~\eqref{iona} is as follows. Consider a rational $r/q\in
C(n,k,l)$ and assume that $R> e^4$. Then
\begin{eqnarray*}
\#\{I_{n+1}\in \II_{n+1} \!\!\!\!\! & \!\!\!\!\! : \!\!\!\!\! \; & \!\!\!\!\! I_{n+1}\cap
\Delta(r/q)\neq\emptyset \} \ \le \
\displaystyle\frac{2cR^{n+1}F(n+1)}{c_1f(q)H(q)}+2 \nonumber \\[2ex] &\stackrel{\eqref{def_cnkl}}{\le} &\frac{2cR^2 \; [\log^*\!n\log^*\!\log
n] \; [\log^*\!(n+1)\log^*\!\log(n+1)]}{c_1f(q)e^l}+2
\\[2ex]
&\stackrel{\eqref{ineq_ff}}{<} &
\displaystyle\frac{8cR^2\log(n+1)\log^*\!\log(n+1)}{c_1e^l}+2 \, .
\end{eqnarray*}
\noindent The upshot is that
\begin{eqnarray*}
\#\{I_{n+1}\in \II_{n+1} \!\!\!\! &\!\!\!\! : \!\!\!\! \;& \!\!\!\! J_{n-1} \cap I_{n+1}\cap
\Delta(r/q)\neq\emptyset\ \mbox{ for some } r/q\in C(n,k,l)\}
\nonumber \\[2ex]
&\le&
\left(2+c_1e^{l+1}\right)\left(2+\frac{8cR^2\log^*\!(n+1)\log^*\!\log(n+1)}{c_1e^l}\right).
\end{eqnarray*}
By summing up over $l$ satisfying \eqref{ineq_ll} we find that
\begin{eqnarray*}
\#\{I_{n+1}\in \II_{n+1}\!\!\!\! &:& \!\!\!\! J_{n-1} \cap I_{n+1}\cap
\Delta(r/q)\neq\emptyset\ \mbox{ for some } r/q\in
C(n,k)\}\\[2ex]
&\le & \sum_{e^l<R\log^*\!n\log^*\log n} \!\!\!\!\!\!\!\!\!\!\!\! 2c_1e^{l+1} \ + \
\sum_{l=0}^{c_3\log^*\!\log n} \frac{16cR^2\log^*\!(n+1)\log^*\! \log
(n+1)}{c_1e^l}\\[2ex]
&& ~ \hspace*{6ex} + \ c_3\log^*\!\log n(4+8 e \, c \, R^2 \log^*\!(n+1)\log^*\!\log (n+1)) \\[2ex]
& \le & c_4
\log^*\!n (\log^*\!\log^*\! n)^2 \, .
\end{eqnarray*}
\noindent By summing up over $k$ satisfying \eqref{ineq_kk} we find that
\begin{eqnarray*}
\#\{I_{n+1}\in \II_{n+1}\!\!\!\!&:& \!\!\!\! J_{n-1} \cap I_{n+1}\cap
\Delta(r/q)\neq\emptyset\ \mbox{ for some } r/q\in C(n)\} \nonumber \\[2ex]
& \le & c_2c_4 (\log^*\!n)^2\cdot(\log^*\! \log n)^2 \label{prop_main2} .
\end{eqnarray*}
In view of \eqref{ineq_c1} and \eqref{ineq_c}, for any $R>e^{12}$ the right hand side of
the above inequality is bounded by
\begin{eqnarray*}
\left(2+\frac{4(\log R+2)^2}{\log
2}\right)(\log^*\!n)^2\cdot(\log^*\!\log n)^2
&< &7\log^2\!R\cdot (\log^*\!n)^2\cdot(\log^*\!\log n)^2 \, .
\end{eqnarray*}
This establishes \eqref{ineq_jn1sv} and thereby completes the proof of Proposition~\ref{prop2sv}.
\section{Preliminaries for Theorem~\ref{th_cantor2} \label{Stalin}}
The overall strategy is simple enough. We show that under the hypothesis of the theorem, any given set $\KK({\rm I},\RRR,\vr)$ contains a `local' subset $\LKK({\rm I},\RRR,\vs)$ satisfying the desired lower bound inequality for the Hausdorff dimension. A general and classical method
for obtaining a lower bound for the Hausdorff dimension of an
arbitrary set is the following mass distribution principle -- see
\cite[pg. 55]{falc}.
\begin{MDP}
Let $ \mu $ be a probability measure supported on a subset $X$ of $
\RR$. Suppose there are positive constants $a, s $ and $l_0$ such
that
\begin{equation}\label{mdp_eq1}
\mu ( B ) \le \, a \; |B|^s \; ,
\end{equation}
for any interval $B$ with length $|B|\le l_0$. Then, $\dim X \ge s
$.
\end{MDP}
\noindent As we shall soon see, the construction of the local set alluded to above is much simpler than that of $\KKK({\rm I},\RRR,\vr)$ and enables us to exploit the mass distribution principle.
\subsection{Local Cantor sets} A $({\rm I},\RRR,\vr)$ Cantor set $\KKK({\rm I},\RRR,\vr)$ is said to be local if $r_{m,n}=0$ whenever $m\neq n$. Furthermore, we write $\LKK({\rm I},\RRR,\vs)$ for $\KKK({\rm I},\RRR,\vr)$ where
$$
\vs:=(s_n)_{n\in \ZZ_{\ge 0}} \quad {\rm and } \quad
s_n:=r_{n,n}.
$$
The set $\LKK({\rm I},\RRR,\vs)$ will be referred to as a {\em $({\rm I},\RRR,\vs)$ local Cantor set.}
\vspace*{2ex}
In a nutshell, the removing procedure associated with the construction of a local Cantor set has no `memory' -- it depends only on the level under consideration. More formally, given the collection $\JJ_{n}$ of level $n$ survivors, the construction of $\JJ_{n+1} $ is completely independent of the previous level $k$ ($<n$) survivors. Indeed the construction is totally local within each interval $J_{n} \in \JJ_{n}$. It is this fact that is utilized when attempting to establish the following dimension result for the associated local Cantor set. Note that in view of Theorem~\ref{th_cantor1}, any local set $\LKK({\rm I},\RRR,\vs)$ is non-empty if
$
R_n-s_n>0 $ for all $ n\in \ZZ_{\ge 0}
$.
\begin{lemma}\label{lem_local}
Given $\LKK({\rm I},\RRR,\vs)$, suppose that
$$
t_n:=R_n-s_n > 0 \quad \forall \ n\in\ZZ_{\ge 0} \, .
$$
Furthermore, suppose there are positive constants $s$ and $n_0 $
such that for all $n>n_0$
\begin{equation}\label{ineq_lemloc}
R_n^s\le t_n \, .
\end{equation}
Then
$$
\dim \LKK({\rm I},\RRR,\vs) \ge s.
$$
\end{lemma}
\noindent{\em Proof. \, } We start by constructing a probability measure
$\mu$ supported on $$\LKK({\rm I},\RRR,\vs) := \bigcap_{n=1}^\infty \bigcup_{J\in
\JJ_n} J.$$ in the standard manner. For any $J_n \in
\JJ_n$, we attach a weight $\mu(J_n)$ defined recursively as
follows.
\vspace*{2ex}
\noindent For $n=0$, $$ \mu(J_0)\ := \frac{1}{\#\JJ_0}=1 \ $$ and
for $n\ge 1$,
\begin{equation}\label{beq4}
\mu(J_n) \, := \, \frac{\mu(J_{n-1})}{\# \{J\in \JJ_n\;:\;
J\subset J_{n-1}\}} \
\end{equation}
\vspace*{2ex}
\noindent where $J_{n-1} \in \JJ_{n-1}$ is the unique interval
such that $J_n\subset J_{n-1}$.
This procedure thus defines inductively a mass on any interval
appearing in the construction of $\LKK({\rm I},\RRR,\vs) $. In fact a lot more is true
--- $\mu$ can be further extended to all Borel subsets $F$ of
$\RR $ to determine $\mu(F) $ so that $\mu$ constructed as above
actually defines a measure supported on $\LKK({\rm I},\RRR,\vs) $. We now state this
formally.
\begin{itemize}
\item[] {\em Fact.} The probability measure $\mu$ constructed above is
supported on $\LKK({\rm I},\RRR,\vs) $ and for any Borel set $F$
\[
\mu(F):= \mu(F \cap\LKK({\rm I},\RRR,\vs) ) \; = \; \inf\;\sum_{J\in \cJ }\mu(J) \
.
\]
The infimum is over all coverings $\cJ$ of $F \cap \LKK({\rm I},\RRR,\vs) $ by
intervals $J\in \{\JJ_n: n \in \ZZ_{\ge 0} \}$.
\end{itemize}
\noindent For further details see \cite[Prop. 1.7]{falc}. It remains to show that $\mu$ satisfies \eqref{mdp_eq1}. Firstly, notice that for any interval $J_n
\in \JJ_n$ we have that
\begin{equation}\label{svdelta}
\mu(J_n) \stackrel{\eqref{beq4}}\le t_{n-1}^{-1}\ \mu(J_{n-1}) \le \prod_{i=0}^{n-1}
t_i^{-1}
\end{equation}
Next, let $\delta_n$ denote the length of a generic interval $J_n
\in \JJ_n$. In view of the splitting procedure associated with the construction of $\LKK({\rm I},\RRR,\vs)$, we find that
\begin{equation}\label{eq_delta}
\delta_n=|I|\cdot \prod_{i=0}^{n-1}R_i^{-1} \, .
\end{equation}
Consider an arbitrary interval $B\subset [0,1] $ with length $|B| <
\delta_{n_0}$. Then there exists an integer $ n\ge n_0 $ such that
\begin{equation} \label{fine}
\delta_{n+1} \, \le \, |B| \, < \, \delta_n \; .
\end{equation}
It follows that
\begin{eqnarray*}
\mu(B) &\le& \sum_{\substack{J \in \JJ_{n+1}:\\
\scriptstyle J \cap B \neq\emptyset }}
\!\! \mu(J) \ \ \stackrel{\eqref{svdelta}}\le \ \ \left\lceil
\frac{|B|}{\delta_{n+1}}\right\rceil
\prod_{i=0}^n t_i^{-1} \\[2ex]
& \stackrel{\eqref{eq_delta}}\le & 2 \,
\frac{|B|}{|I|}\prod_{i=0}^n \frac{R_i}{t_i} \ \ = \ \ 2 \,
\frac{|B|}{|I|}^{1-s} \ \ \prod_{i=0}^n
\frac{R_i}{t_i} \, \cdot \ |B|^s \\[2ex]
& \stackrel{\eqref{fine}}< & 2 \,
\frac{\delta_n}{|I|}^{\!\!\!\!1-s} \ \ \prod_{i=0}^n
\frac{R_i}{t_i} \, \cdot \ |B|^s \\[2ex]
& \stackrel{\eqref{eq_delta}}< & 2 |I|^{-s}\prod_{i=0}^n
\frac{R_i^{s}}{t_i} \ \cdot \
|B|^{s} \\[2ex]
& \stackrel{\eqref{ineq_lemloc}}{\le} & 2 \,
|I|^{-s}\prod_{i=0}^{n_0}\frac{R_i^{s}}{t_i} \ \cdot \ |B|^{s} \,
.
\end{eqnarray*}
In other words, \eqref{mdp_eq1} is valid with
$$a:=2 \,
|I|^{-s}\prod_{i=0}^{n_0}\frac{R_i^{s}}{t_i} \, $$ and on applying
the mass distribution principle we obtain the desired statement.
\newline\hspace*{\fill}$\boxtimes$
\vspace*{2ex}
In view of Lemma~\ref{lem_local}, the proof of Theorem~\ref{th_cantor2} reduces to establishing the following key statement.
\begin{proposition}\label{lem_subcantor}
Let $\KKK({\rm I},\RRR,\vr)$ be as in Theorem~\ref{th_cantor2}. Then
there exists a local Cantor type set $$\LKK({\rm I},\RRR,\vs) \subset
\KKK({\rm I},\RRR,\vr) $$ where
$$
\vs:=(s_n)_{n\in \ZZ_{\ge 0}} \quad { with } \quad s_n:=
\mbox{$\frac12$} \, R_n \, .
$$
\end{proposition}
\vspace*{2ex}
\noindent Indeed, by Proposition~\ref{lem_subcantor} we have that
$$
\dim \KKK({\rm I},\RRR,\vr)\ge \dim \LKK({\rm I},\RRR,\vs) .
$$
Now fix some positive $s<\liminf\limits_{n\to\infty}(1-\log_{R_n}\!2)$. Then, there
exists an integer $n_0$ such that
$$
s \, < \, 1-\log_{R_n}\!2 \quad {\rm \ for \ all } \quad n>n_0 \, .
$$
Also note that
$$
t_n=R_n-s_n =\frac{R_n}{2}
$$
and
$$
R_n^s<\frac{R_n}{2}= t_n \quad {\rm \ for \ all } \quad n>n_0
\, .
$$
Therefore, Lemma~\ref{lem_local} implies that
$$\dim \LKK({\rm I},\RRR,\vs) \ge s \, . $$ This inequality is true for any
$s<\liminf\limits_{n\to\infty}(1-\log_{R_n}\!2)$ and hence completes
the proof of Theorem~\ref{th_cantor2} modulo Proposition~\ref{lem_subcantor}.
\vspace*{2ex}
Before moving on to the proof of the proposition, it is useful to first
investigate the distribution of intervals within each collection $\JJ_n$ associated with $\KKK({\rm I},\RRR,\vr)$.
\subsection{The distribution of intervals within $\JJ_n$}
In this section, the set
$$
\KKK ({\rm I},\RRR,\vr) := \bigcap_{n=1}^\infty \bigcup_{J\in
\JJ_n} J
$$
and the sequence $\vs$ are as in Proposition~\ref{lem_subcantor}.
Let $\TTT_0:= \{ {\rm I}\}$. For $n \ge 1$,
let $\TTT_n$ denote a generic collection of intervals obtained from
$\TTT_{n-1}$ via the splitting and removing procedures
associated with a $({\rm I},\RRR,\RRR-\vs)$ local Cantor set. Here $\RRR-\vs$ is the sequence
$(R_n-s_n)$. Then, clearly
$$
\# \TTT_{n+1} \ \ge \ \# \TTT_{n} \times s_n \, \quad \forall \ n \in \ZZ_{\ge0} \, .
$$
Loosely speaking, the following result shows that the intervals
$J_n$ from $\JJ_n$ are ubiquitous within the interval ${\rm I}$.
\begin{lemma}\label{sl_lem1} For $R$ sufficiently large,
\begin{equation}\label{sl_lem_eq}
\TTT_n\cap \JJ_n \neq \emptyset \qquad \forall \ \ n \in \ZZ_{\ge0} \, .
\end{equation}
\end{lemma}
\noindent{\em Proof. \, } For an integer $n \ge 0$, let $h(n)$
denote the cardinality of the set $\TTT_n\cap \JJ_n$. Trivially,
$h(0)=1$ and lemma would follow on showing that
\begin{equation}\label{goodcount}
h(n+1)\ge \frac{R_n}{4} \, h(n) \, .
\end{equation}
\noindent for all $ n \in \ZZ_{\ge0}$. This we now do via induction. Consider the set $\TTT_n\cap \JJ_n$. By the construction of $\TTT_{n+1}$ and the splitting procedure associated with $\KKK({\rm I},\RRR,\vr)$, each
of the $h(n)$ intervals in $\TTT_n\cap \JJ_n$ gives rise to at least
$s_n$ intervals $I_{n+1}$ in $\TTT_{n+1}\cap \II_{n+1}$. By the
the removing procedure associated with $\KKK({\rm I},\RRR,\vr)$,
for each interval $J_k\in\TTT_k\cap\JJ_k$ ($ 0 \le k \le n$) we remove at
most $r_{k,n}$ intervals $I_{n+1}\in \TTT_{n+1}\cap \II_{n+1}$ that lie within
$J_k$. The upshot of this is that
\begin{equation}\label{ineq_hn}
h(n+1)\ge s_n h(n)-\sum_{k=0}^{n} r_{k,n}\, h(k).
\end{equation}
For $n=0$ this inequality is transformed to
$$
h(1)\ge \frac{R_0}{2}h(0)-r_{0,0}h(0)\ \stackrel{\eqref{cond_th2}}\ge\ \frac{R_0}{4}h(0)
$$
as required. Now assume
that \eqref{goodcount} is valid for all $ 1 \le m \le n$. In
particular, it means that
$$
h(m)\le \frac{4}{R_m} \, h(m+1)\le\ldots\le
\prod_{k=1}^{n-m}\frac{4}{R_{n-k}}\, h(n)
$$
which together with \eqref{ineq_hn} implies that
\begin{eqnarray*}
h(n+1) & \ge & \frac{R_n}{2}h(n)-\left(\sum_{k=0}^n
\left(r_{n-k,n}\prod_{i=1}^k\frac{4}{R_{n-i}}\right)\right)h(n) \\[2ex]
& \stackrel{\eqref{cond_th2}}\ge & \frac{R_n}{4} \, h(n) \, .
\end{eqnarray*}
This completes the induction step and thus establishes the desired inequality
\eqref{goodcount} for all $n \in \ZZ_{\ge 0}$.
\hspace*{\fill}$\boxtimes$
\section{Proof of Proposition \ref{lem_subcantor} \label{Lenin}}
By definition, the set $\KKK({\rm I},\RRR,\vr)$ is the intersection of closed intervals $J_n$ lying within nested collections $\JJ_n$.
For
each integer $n \ge 0$, the aim is to construct a nested collection $\LLL_n
\subseteq \JJ_n$ that complies with the construction of a $({\rm I},\RRR,\vs)$ local Cantor set. Then, it would follow that
$$
\bigcap_{n=0}^\infty \bigcup_{J\in
\LLL_n} J
$$
is precisely the desired set $\LKK({\rm I},\RRR,\vs)$.
\subsection{Construction the collection $\LLL_n$}
For any integer $n \ge 0$, the goal of this section is to construct the
desired nested collection $\LLL_n \subseteq \JJ_n$ alluded to above. This will involve constructing auxiliary collections
$\LLL_{m,n}$ and $\R_{m,n}$ for integers $m,n$ satisfying $0\le m\le
n$. For a fixed $n$, let
$$
\JJ_0 \, , \ \JJ_1 \, , \ \ldots, \ \JJ_n \;
$$
be the collections arising from the construction of $\KKK({\rm
I},\RRR,\vr)$. We will require $\LLL_{m,n}$ to satisfy the following
conditions.
\begin{itemize}
\item[\bf C1.] For any $0\le m\le n$, $\LLL_{m,n}\subseteq \JJ_m$.
\item[\bf C2.] For any $0\le m< n$, the collections $\LLL_{m,n}$ are nested; that is
$$\bigcup_{J\in \LLL_{m+1,n}}J \qquad \subset \quad \bigcup_{J\in \LLL_{m,n}}J.$$
\item[\bf C3.] For any $0\le m<n$ and $ J_m\in \LLL_{m,n}$, there are at
least $R_m-s_m$ intervals $J_{m+1}\in \LLL_{m+1,n}$ contained
within $J_m$; that is
\begin{equation*}
\# \{J_{m+1} \in \LLL_{m+1,n} \;:\; J_{m+1} \subset J_{m}\} \ \ge
\ R_m-s_m \ .
\end{equation*}
\end{itemize}
\noindent In addition, define $\R_{0,0} := \emptyset $ and for $n
\ge 1$
\begin{equation}\label{def_Rnn}
\R_{n,n}:=\left\{I_{n}\in \II_n\backslash \JJ_n\;:\; I_n\subset
J_{n-1}\mbox{ for some }J_{n-1}\in \LLL_{n-1,n-1}\right\} \ .
\end{equation}
Furthermore, for $ 0\le m< n $ define
\begin{equation}\label{def_rnm}
\R_{m,n}:=\R_{m,n-1}\cup \{J_m\in \LLL_{m,n-1}\;:\; \#\{J_{m+1}\in
\R_{m+1,n}\;:\; J_{m+1}\subset J_m\}\ge s_m\ \} \ .
\end{equation}
Loosely speaking and with reference to condition (C3), the
collections $\R_{m,n}$ are the `dumping ground' for those intervals
$J_m\in\LLL_{m,n-1}$ which do not contain enough sub-intervals
$J_{m+1}$. Note that for $n$ fixed, the collections $\R_{m,n}$ are
defined in descending order with respect to~$m$. In other words, we
start with $\R_{n,n}$ and finish with $\R_{0,n}$.
\vspace*{1ex}
The construction is as follows.
\vspace*{1ex}
\noindent{\em Stage 1.} Let $\LLL_{0,0}:=\JJ_0 $ and $
\R_{0,0}:=\emptyset$.
\vspace*{1ex}
\noindent {\em Stage 2.} Let $ 0 \le t \le n $. Suppose we have
constructed the desired collections
$$\LLL_{0,t}\subseteq \JJ_0, \ \LLL_{1,t}\subseteq \JJ_1,\ldots, \LLL_{t,t}\subseteq
\JJ_t$$ and
$$\R_{0,t},\ldots, \R_{t,t} \, . $$
We now construct the corresponding collections for $t=n+1$.
\noindent{\em Stage 3.} Define
$$
\LLL'_{n+1,n+1}:=\{J_{n+1}\in \JJ_{n+1}\;:\; J_{n+1}\subset
J_n\mbox{ for some }J_n\in \LLL_{n,n}\}
$$
and let $\R_{n+1,n+1}$ be given by \eqref{def_Rnn} with $n+1$
instead of $n$. Thus the collection $\LLL'_{n+1,n+1}$ consists of
`good' intervals from $\JJ_{n+1}$ that are contained within some
interval from $\LLL_{n,n}$. Our immediate task is to construct the
corresponding collections $ \LLL'_{u,n+1}$ for each $ 0 \le u \le
n$. These will be constructed together with the `complementary'
collections $\R_{u,n+1}$ in descending order with respect to $u$.
\vspace*{1ex}
\noindent{\em Stage 4.} With reference to Stage 3, suppose we have
constructed the collections $\LLL'_{u+1,n+1}$ and $\R_{u+1,n+1}$ for
some $0\le u\le n$. We now construct $\LLL'_{u,n+1}$ and
$\R_{u,n+1}$. Consider the collections $\LLL_{u,n}$ and $\R_{u,n}$.
Observe that some of the intervals $J_u$ from $\LLL_{u,n}$ may
contain less than $R_u-s_u$ sub-intervals from $\LLL'_{u+1,n+1}$ (or
in other words, at least $s_u$ intervals from $\R_{u+1,n+1}$). Such
intervals $J_u$ fail the counting condition (C3) for $\LLL_{u,n+1}$
and informally speaking are moved out of $\LLL_{u,n}$ and into
$\R_{u,n}$. The resulting sub-collections are $\LLL'_{u,n+1}$ and
$\R_{u,n+1}$ respectively. Formally,
$$
\LLL'_{u,n+1}:=\{J_u\in \LLL_{u,n}\;:\; \#\{J_{u+1}\in
\R_{u+1,n+1}\;:\; J_{u+1}\subset J_u\}< s_u\ \} \
$$
and $\R_{u,n+1}$ is given by \eqref{def_rnm} with $m$ replaced by $u$ and $
n$ replaced by $n+1$.
\vspace*{1ex}
\noindent{\em Stage 5.} By construction the collections
$\LLL'_{u,n+1}$ satisfy conditions (C1) and (C3). However, for
some $J_{u+1}\in \LLL'_{u+1,n+1}$ it may be the case that $J_{u+1}$
is not contained in any interval $J_u\in \LLL'_{u,n+1}$ and thus
the collections $\LLL'_{u,n+1}$ are not necessarily nested. The
point is that during Stage 4 above the interval $J_u\in \JJ_u$
containing $J_{u+1}$ may be `moved' into $\R_{u,n+1}$. In order to
guarantee the nested condition (C2) such intervals $J_{u+1}$ are
removed from $\LLL'_{u+1,n+1}$. The resulting sub-collection is the
required auxiliary collection $\LLL_{u+1,n+1}$. Note that
$\LLL_{u+1,n+1}$ is constructed via $\LLL'_{u+1,n+1}$ in ascending
order with respect to $u$. Formally,
$$
\LLL_{0,n+1}:=\LLL'_{0,n+1}
$$
and for $1\le u\le n+1$
$$
\LLL_{u,n+1}:=\{J_u\in \LLL'_{u,n+1}\;:\; J_u\subset J_{u-1} \mbox{
for some } J_{u-1}\in \LLL_{u-1,n+1}\} \, .
$$
With reference to Stage 2, this completes the induction step and
thereby the construction of the auxiliary collections.
For any integer $n \ge 0$, it remains to construct the sought
after collection $\LLL_n$ via the auxiliary collections
$\LLL_{m,n}$. Observe that since
$$
\LLL_{m,m}\supset \LLL_{m,m+1}\supset \LLL_{m,m+2}\supset \ldots
$$
and the cardinality of each collection $\LLL_{m,n}$ with $m\le n$
is finite, there exists some integer $N(m)$ such that
$$
\LLL_{m,n} \ = \ \LLL_{m,n'} \qquad \forall \quad n,n' \ge N(m)
\ .
$$
\noindent Now simply define $$ \LLL_n:=\LLL_{n,N(n)} \ . $$
\noindent Unfortunately, there remains one slight issue. The
collection $\LLL_n$ defined in this manner could be empty.
\vspace*{2ex}
The goal now is to show that $\LLL_{m,n}\neq \emptyset$ for any
$m\le n$. This clearly implies that $\LLL_n\neq\emptyset$ and
thereby completes the construction.
\subsection{The collection $\LLL_{m,n}$ is non-empty}
\begin{lemma}\label{lem_mnonempty}
For any $m,n\in \NN, m\le n$, the set $\LLL_{m,n}$ is nonempty.
\end{lemma}
\noindent{\em Proof.} Suppose the contrary:
$\LLL_{m,n}=\emptyset$ for some integers satisfying $ 0 \le m\le
n$. In view of the construction of $\LLL_{m,n}$ every interval in
$\LLL_{m-1,n}$ contains at least $R_{m-1}-s_{m-1}>0$ sub-intervals
from $\LLL_{m,n}$. Therefore each of the collections $\LLL_{m-1,n},
\LLL_{m-2,n}, \ldots, \LLL_{0,n}$ are also empty and it follows that
$\R_{0,n}=\JJ_0$.
Now consider the set $\R_{m,n}$. By the construction we have the
chain of nested sets
$$
\R_{m,n}\supseteq \R_{m,n-1}\supseteq\cdots\supseteq\R_{m,m}
$$
and in view of \eqref{def_Rnn} the elements of $\R_{m,m}$ are
intervals from $\II_m\backslash \JJ_m$. Consider any interval
$J_m\in \R_{m,n}\backslash \R_{m,m}$. Take $m < n_0 \le n $ such
that $J_m\in \R_{m,n_0}$ but $J_m\not\in \R_{m,n_0-1}$. Then $J_m$
was added to $\R_{m,n_0}$ on stage~4 of the construction. Hence
$I_m$ should have at least $s_m$ sub-intervals from $\R_{m+1,n_0}$
and therefore from $\R_{m+1,n}$. The upshot of this is the
following: for any interval $I_m$ from $\R_{m,n}$ either $I_m\in
\II_m\backslash \JJ_m$ or $I_m$ contains at least $s_m$
sub-intervals $I_{m+1}\in \R_{m+1,n}$.
Next we exploit Lemma \ref{sl_lem1}. Choose an interval $J_0$ from $
\R_{0,n} = \JJ_0$ and define $\TTT_0:=\{J_0\}$. For $ 0\le m<n$,
we define inductively nested collections
$$
\TTT_{m+1}:=\{I_{m+1}\in \TTT(I_m)\;:\; I_m\in \TTT_m\}
$$
with $\TTT(I_m)$ given by one of the following three scenarios.
\begin{itemize}
\item $I_m\in \R_{m,n}$ and $I_m$ contains at least $s_m$ sub-intervals
$I_{m+1}$ from $\R_{m+1,n}$. Let $\TTT(I_m)$ be the collection
consisting of these sub-intervals. Note that when $m=n-1$ we have
$\TTT(I_m)\subset \R_{n,n}\subset \II_n\backslash \JJ_n$. Therefore
$\TTT(I_{n-1})\cap \JJ_{n}=\emptyset$.
\item $I_m\in \R_{m,n}$ and $I_m$ contains strictly less than $s_m$ sub-intervals $I_{m+1}$ from
$\R_{m+1,n}$. Then the interval $I_m \in \II_m\backslash \JJ_m$ and
we subdivide $I_m$ into $R_m$ closed intervals $I_{m+1}$ of equal
length. Let $\TTT(I_m)$ be the collection consisting of all of
these sub-intervals. Note that $\TTT(I_m)\cap \JJ_{m+1}=~\emptyset$.
\item $I_m\not\in \R_{m,n}$. Then the interval $I_m$ does not
intersect any interval from $\JJ_m$ and we subdivide $I_m$ into
$R_m$ closed intervals $I_{m+1}$ of equal length. Let $\TTT(I_m)$
be any collection consisting of all such sub-intervals. Note that
$\TTT(I_m)\cap \JJ_{m+1}=~\emptyset$.
\end{itemize}
\noindent The upshot is that
$$
\# \TTT_{m+1} \ \ge \ \# \TTT_m \times s_m\qquad \forall \; 0<
m\le n
$$
and that
$$
\TTT_{n}\cap \JJ_{n} \, = \, \emptyset \; .
$$
However, in view of Lemma~\ref{sl_lem1} the latter is impossible and
therefore the starting premise that $\LLL_{m,n}=\emptyset$ is false.
This completes the proof of Lemma~\ref{lem_mnonempty} and therefore
Proposition~\ref{lem_subcantor}.
\newline\hspace*{\fill}$\boxtimes$
\vspace*{4ex}
\section{Intersecting Cantor sets \label{FC}}
With reference to \S\ref{gcf}, fix the interval ${\rm I}$ and the sequence $\RRR:= (R_n)$. Let $ k \in \NN$ and consider the two parameter sequences
$$\vr_i:=(r^{\!(i)}_{m,n}) \qquad 1\le i \le k \, .
$$
The following result shows that
the intersection of any finite number of $({\rm I},\RRR,\vr_i) $ Cantor sets is yet another $({\rm I},\RRR,\vr) $ Cantor set.
\begin{theorem}\label{th_icantor}
For each integer $1\le i \le k $, suppose we are given a set $\KKK
({\rm I},\RRR,\vr_i) $. Then
$$
\bigcap_{i=1}^{k} \KKK ({\rm I},\RRR,\vr_i)
$$
is a $({\rm I},\RRR,\vr) $ Cantor set
where
$$
\vr:=(r_{m,n}) \quad\mbox{with } \quad r_{m,n} :=\sum_{i=1}^k
r^{(i)}_{m,n} \, .
$$
\end{theorem}
\vspace*{2ex}
\noindent{\em Proof.\, }
Loosely speaking we need to show that there exists a $({\rm I},\RRR,\vr) $ Cantor set that simultaneously incorporates the splitting and removing procedures associated with the sets
$$\KKK({\rm I},\RRR,\vr_i) := \bigcap_{n=1}^\infty \bigcup_{J\in\JJ^{(i)}_n} J \qquad \quad (1\le i \le k) \, . $$
For each $n\in\ZZ_{\ge 0}$, consider the collection
$$
\JJ_n:=\bigcap_{i=1}^k \JJ_n^{(i)} \, .
$$
We claim that $\JJ_n$ complies with the construction of a $({\rm I},\RRR,\vr) $ Cantor set.
If true, then we are done since
$$
\KKK({\rm I},\RRR,\vr):=\bigcap_{n=1}^\infty \bigcup_{J\in\JJ_n} J = \bigcap_{n=1}^\infty \bigcap_{i=1}^k \bigcup_{J\in\JJ^{(i)}_n} J= \bigcap_{i=1}^k \bigcap_{n=1}^\infty \bigcup_{J\in\JJ^{(i)}_n}J:=\bigcap_{i=1}^{k} \KKK ({\rm I},\RRR,\vr_i) \, .
$$
Firstly note that the claim is true for $n=0$ since $\JJ_0:=\{{\rm I}\}$.
Now assume that the claim is true for some fixed $n\in\ZZ_{\ge 0}$. Consider an arbitrary interval
$J_n\in \JJ_n$. By definition, $J_n\in \JJ_n^{(i)}$ for each $i$.
By construction, every interval in $\JJ_n^{(i)}$ gives rise to $R_n$ intervals $I_{n+1}\in \II_{n+1}^{(i)}$. Thus, for each $J_n\in \JJ_n$ the
collection
$$\II_{n+1}:=\bigcap_{i=1}^k\II_{n+1}^{(i)}
$$
contains exactly $R_n$ intervals $I_{n+1}$ that lie within $J_n$. This coincides precisely with the splitting procedure associated with a $({\rm I},\RRR,\vr) $ Cantor set. We now turn our attention to the removing procedure. By construction, for each interval $J_n\in \JJ_{n}^{(i)}$ we remove at most $r_{n,n}^{(i)}$ intervals $I_{n+1}\in\II^{(i)}_{n+1}$ that lie within $J_n$.
Thus for any
$J_n\in\JJ_n$ there are at most
$$
r_{n,n} := \sum_{i=1}^k r_{n,n}^{(i)}
$$
intervals $I_{n+1}\subset J_n$ that are removed from $\II_{n+1}$.
In general, for each $0\le m \le n$ and each interval
$J_{m}\in \JJ_{m}$ there are at most
$$
r_{m,n}:=\sum_{i=1}^k r_{m,n}^{(i)}
$$
additional intervals $I_{n+1}\subset J_m$ that are removed from $ \II_{n+1}$. This coincides precisely with the removing procedure associated with a $({\rm I},\RRR,\vr) $ Cantor set.
The upshot is that $\JJ_{n+1}$ complies with the construction of a $({\rm I},\RRR,\vr) $ Cantor set. This completes the induction step and thereby completes the proof of Theorem~\ref{th_icantor}.
\newline\hspace*{\fill}$\boxtimes$
\vspace*{4ex}
\noindent{$\bullet $ \em An application.}
We now describe a simple application of Theorem~\ref{th_icantor} which enables us to deduce a non-trivial strengthening of Theorem~\ref{th_main}.
In the course of establishing Proposition~\ref{prop1sv} we show that the
set $\mad_\DDD(f)$ contains the Cantor-type set $\KKK({\rm I},\RRR,\vr)$
where $\RRR=(R_n)$ and $\vr=(r_{m,n})$ are given by~\eqref{def_rn} and
\eqref{def_rrnm} respectively; namely, for any fixed integer $R > e^{12}$
$$
R_n:=R \, (n+1) \, [\log^*\!(n+1)] \quad {\rm and } \quad
r_{m,n} \, := \, 7\, \log^2\!R\cdot
n^2(\log^*\!n)^2
$$
if $ m=n-1$ and zero otherwise. Note that although these quantities are dependent on the actual value of $R$ the statement that $\KKK({\rm I},\RRR,\vr) \subset \mad_\DDD(f)$ is not.
Now for each $1\le i \le k $, let $\DDD_i$ be a sequence of integers greater than or equal to two and let $f$ be as in Proposition~\ref{prop1sv}. Then, with $\RRR$ and $\vr$ as above, Theorem~\ref{th_icantor} implies that
$$
\bigcap_{i=1}^k \mad_{\DDD_i}(f)\supset
\KKK({\rm I},\RRR,k\vr) \qquad\mbox{ where } \quad k\, (r_{m,n}):=(kr_{m,n}).
$$
It is easily verified that for $ R > k \, e^9$
\begin{eqnarray*}
{\rm l.h.s. \ of \ } \eqref{cond_th2} & = & k \cdot r_{n-1,n}\cdot\frac{4}{R_{n-1}} \, \le \, k \cdot 7 \cdot 2^3 \cdot \frac{\log^2\!R \cdot n \, \log^*\!n }{R} \\[2ex]
&\le & \frac{ k \cdot 7 \cdot 2^6 \log^2\!R}{R^2} \cdot \frac{R \, (n+1) \, [\log^*\!(n+1)] }{4} \\[2ex]
& \le & \frac{R_n }{4} \ = \ {\rm r.h.s. \ of \ } \eqref{cond_th2} \, .
\end{eqnarray*}
Hence, for any fixed $ R > k \, e^{12} $, Theorem~\ref{th_cantor2} implies that
$$
\dim\left(\textstyle{\bigcap_{i=1}^k} \mad_{\DDD_i}(f)\right)\ge \liminf_{n\to
\infty}(1-\log_{R_n}\!2)=1.
$$
The complementary upper bound inequality for the dimension is trivial. Thus we have established the
following strengthening of Theorem~\ref{th_main}.
\begin{theorem}\label{thm1sv} For each $1\le i \le k $, let $\DDD_i$ be a sequence of integers greater than or equal to $2$ and let $f$ be as in Proposition~\ref{prop1sv}.
Then
\begin{equation*}
\dim\left(\bigcap_{i=1}^k \mad_{\DDD_i}(f)\right) =1 \, .
\end{equation*}
\end{theorem}
\vspace*{3ex}
\noindent{$\bullet$ \em What about other intersections?} There are two natural problems that arise in relation to Theorem~\ref{th_icantor}. Firstly, to generalise the statement so as to incorporate any finite number of sequences $\RRR_i:=(R_n^{(i)})$. Secondly, to establish the analogue of
Theorem~\ref{th_icantor} for countable intersections. This is more challenging than the first and in all likelihood will involve imposing extra conditions on the sequences $\RRR$ and $\vr$.
A direct consequence of the `correct' countable version of Theorem~\ref{th_icantor} would be the statement that
$$
\dim\left(\textstyle{\bigcap_{i=1}^{\infty}} \mad_{\DDD_i}(f)\right) = 1 \, .
$$
Note that establishing the countable analogue of Theorem~\ref{thm1sv} remains an open problem.
\vspace*{3ex}
\noindent{$\bullet$ \em A more general Cantor framework.} The Cantor framework of \S\ref{gcf} and indeed of this section is one-dimensional. Naturally it would be interesting to develop the analogous $n$--dimensional Cantor framework in which intervals are replaced by balls. Establishing the higher dimensional generalisation of Theorem~\ref{th_cantor2} and indeed Theorem~\ref{th_icantor} will almost certainly make use of standard covering arguments from geometric measure theory; for example, the `$5r$' and Besicovitch covering lemmas. Beyond higher dimensions, it would be highly desirable to develop an analogue of the framework of \S\ref{gcf} within the context of `reasonable' metric spaces -- such as a (locally) compact metric space equipped with an Ahlfors regular measure. A generalisation of this type would enhance the scope of potential applications.
\vspace*{8ex}
\noindent{\bf Acknowledgements.} SV would like to thank all those at Talbot Primary School who have made the dynamic duo most welcome! In particular a special thanks to Julia Alvin, David Young and Paulin Jacobson. Also many thanks to Victor Beresnevich and Maurice Dodson for various suggestions that have improved the clarity of the paper and to Andrew Pollington for many discussions over many many years regarding Littlewood.
\def\cprime{$'$}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 7,203 |
Petrus Ramus, Pierre de la Ramée, o Pedro Ramo (Cuts, Vermandois, 1515 - París, 26 de agosto de 1572), retórico, humanista y lógico francés, creador de la corriente antiaristotélica de pensamiento denominada en su honor ramismo.
Biografía
De noble familia originaria de la Picardía, pero empobrecida hasta el punto de que el padre debía trabajar como agricultor, a los ocho años huyó de casa y marchó a París; a fuerza de trabajar de día como criado y estudiar de noche en el Collège de Navarra, ingresó en la Sorbona cuando sólo contaba con doce años y llegó a maestro en Artes a los veintiuno con la tesis de que quaecumque ab Aristotele dicta essent commentitia esse (todo lo que había escrito Aristóteles no es más que falsedad). Independiente, adogmático e hipercrítico, se rebeló y protestó con energía contra el excesivo escolasticismo de unas universidades en las que Aristóteles era el modelo único y la base de toda investigación filosófica, rechazando en firme cualquier aristotelismo como pura labor de higiene. Preconizó a cambio una lógica viva y abierta.
Se estableció como profesor en el pequeño Colegio del Ave María en Le Mans, y retomó sus ideas antiaristotélicas en dos obras de 1543, las Dialecticae partitiones y sus Animadversiones in Dialecticam Aristotelis (Críticas a la dialéctica aristotélica), que fueron condenadas por la facultad de Teología de la Sorbona y disgustaron terriblemente a los académicos, dolidos por la fuerza de los palos que les sacudía, entre los que no era el menos flojo la elegancia y prestigio del sistema copernicano frente al torpe almagesto ptolemaico del Estagirita. Este aldabonazo repercutió en toda Europa y las discusiones en las universidades europeas provocaron al cabo la división general entre ramistas y antirramistas de los claustros, de forma que la Sorbona tuvo que recurrir al mismísimo rey Francisco I para que prohibiera sus obras con un edicto (1 de marzo de 1544) donde se afirma que sus ideas son temerarias, arrogantes e impudentes, es decir, desvergonzadas. Expulsado además de la universidad, fue acogido en el Colegio de Presles en 1545 hasta que el sucesor filoprotestante del rey Francisco, Enrique II, anuló tal expulsión en 1547; entonces los jesuitas, bien situados y poderosos en París, consiguieron poner las obras de Ramus en el Index librorum prohibitorum de libros prohibidos.
Sin embargo la política oficial era apoyar al Humanismo y se permitió que ocupara la cátedra de matemáticas del Collège de France (1551), gracias en parte a la protección del cardenal de Lorena. En este cargo se preocupó de introducir algunas mejoras en la enseñanza y prosiguió con su empeño de publicar nuevas gramáticas; si ya había publicado una Grammatica latina (1548) imprimió ahora una Grammatica Graeca (1560) y una Grammaire Française (1562) y en este último año a Carlos IX un plan para reformar la Universidad. Con todo, y dando de nuevo una muestra de la libertad e independencia de su juicio, tras el Coloquio de Poissy (1561) en que protestantes y católicos trataron de hallar un acuerdo, Ramus apoyó al calvinista Teodoro de Beza contra su protector el Cardenal de Lorena y abjuró de la fe católica para seguir la protestante. Esto le valió abandonar otra vez la cátedra universitaria y huir de París al año siguiente. Regresó en 1563 con la Paz de Ambroise y reanudó la enseñanza, si bien en 1567 tuvo que marcharse a causa de las guerras de religión otra vez. En 1568 está en Alemania, y en Suiza estuvo como profesor en Heidelberg, Ginebra y Lausana. La paz de Saint-Germain (1570) le llevó otra vez a París, a la cátedra y rectorado del Colegio de Presles, dentro de la universidad, a lo que se opusieron fuertemente los jesuitas desde su Colegio de Clermont. Allí, en Presles, le alcanzó la muerte en la famosa Matanza de San Bartolomé (1572), en la que miles de protestantes franceses fueron cruelmente perseguidos y asesinados por calles, plazas y casas. Su cuerpo fue arrastrado y arrojado al Sena.
Obra
Como humanista experimentó el influjo de Lorenzo Valla y de Rodolphus Agricola. Quería una lógica «viva» opuesta a la silogística aristotélica. Algunas de sus innovaciones tuvieron que ver también con mejoras en pequeños detalles, como la incorporación de la jota y de la uve para los valores consonánticos de la i y la u en la ortografía del latín. Fue además uno de los abuelos de la Ilustración en Francia y por su método de clasificación de las disciplinas según un orden lógico anuncia ya a Descartes.
En el terreno religioso, ejerció una influencia considerable sobre la Teología del pacto (Covenant Theology) en las iglesias congregacionalistas de Nueva Inglaterra. Los discípulos de Pierre de la Ramée, entre los que se encontraban los españoles Francisco Sánchez de las Brozas (1523-1600) y Pedro Núñez Vela (1522-1602), elaboraron una síntesis lógico-dialéctica en sustitución del aristotelismo de los escolásticos que también se introdujo entre los Platónicos de Cambridge.
Bibliografía del autor
Dialecticae partitiones (1543)
Dialecticae Institutiones (1543)
Aristotelicae Animadversiones (1543).
Brutinae questiones (1547)
Grammatica latina (1548)
Rhetoricae Distinctiones in Quintilianum (1549).
Dialectique (1555)
Arithmeticae libri III (1555)
Scholae grammaticae libri II (1559)
Grammatica Graeca (1560)
Grammaire Française (1562)
Scholarum physicarum libri VIII in totidem acroamaticos libros Aristotelis (1565)
Scholarum metaphysicarum libri XIV (1566)
Scholae in liberales artes, Basilea, (1569)
Defensio pro Aristotele adversus Jac. Schecium, Lausana, (1571)
Avertissement sur la réformation de l'université de Paris au Roi (1561)
Commentariolum de Religione Christiana libri IV, Frankfurt (1577).
Notas y referencias
Bibliografía
Nelly Bruyère, Méthode et dialectique dans l'oeuvre de La Ramée: Renaissance et Age classique, Paris, Vrin 1984
Desmaze, Charles. Petrus Ramus, professeur au Collège de France, sa vie, ses ecrits, sa mort (Paris, 1864).
Freedman, Joseph S. Philosophy and the Arts in Central Europe, 1500-1700: Teaching and Texts at Schools and Universities (Ashgate, 1999).
Graves, Frank Pierrepont. Peter Ramus and the Educational Reformation of the Sixteenth Century (Macmillan, 1912).
Høffding, Harald. History of Modern Philosophy (English translation, 1900), vol. i.185.
Howard Hotson, Commonplace Learning: Ramism and Its German Ramifications, 1543–1630 (Oxford: Oxford University Press, 2007).
Lobstein, Paul. Petrus Ramus als Theolog (Strassburg, 1878).
Miller, Perry. The New England Mind (Harvard University Press, 1939).
Milton, John. A Fuller Course in the Art of Logic Conformed to the Method of Peter Ramus (London, 1672). Ed. y trad. Walter J. Ong y Charles J. Ermatinger. Complete Prose Works of John Milton: v. 8. Ed. Maurice Kelley. New Haven: Yale UP, 1982. p. 206-407.
Ong, Walter J. (1982). Orality and literacy: The technologizing of the word. New York: Methuen. p. viii.
---.Ramus, Method, and the Decay of Dialogue: From the Art of Discourse to the Art of Reason (Harvard University Press, 1958; reissued with a new foreword by Adrian Johns, University of Chicago Press, 2004. ISBN 0-226-62976-7).
---. Ramus and Talon Inventory (Harvard University Press, 1958).
Owen, John. The Skeptics of the French Renaissance (London, 1893).
Pranti, K. "Uber P. Ramus" in Munchener Sitzungs berichte (1878).
Saisset, Émile. Les précurseurs de Descartes (Paris, 1862).
Sharratt, Peter. "The Present State of Studies on Ramus," Studi francesi 47-48 (1972) 201-13.
—. "Recent Work on Peter Ramus (1970–1986)," Rhetorica: A Journal of the History of Rhetoric 5 (1987): 7-58.
—. "Ramus 2000," Rhetorica: A Journal of the History of Rhetoric 18 (2000): 399-455.
Voigt. Uber den Ramismus der Universität Leipzig (Leipzig, 1888).
Waddington, Charles De Petri Rami vita, scriptis, philosophia (Paris, 1848).
Enlaces externos
Internet Archive author Petrus Ramus
'Ramism' entry in The Dictionary of the History of Ideas
Catholic Encyclopedia entry
Charles Waddington, Ramus (Pierre de la Ramée) sa vie, ses écrits et ses opinions (1855)
Index Librorum Prohibitorum
Matemáticos de Francia del siglo XVI
Retóricos de Francia
Filósofos de Francia del siglo XVI
Humanistas de Francia del siglo XVI
Escritores de Francia del siglo XVI
Alumnado de la Universidad de París
Miembros del Colegio de Francia
Protestantes de Francia
Fallecidos por puñalada
Fallecidos en París
Víctimas de la persecución religiosa | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 2,012 |
Q: What does the notation $\circ$ mean? I can't search it on google, because it doesn't support symbols like these.
It might have to do with binary operations, I think.
A: Since you mention "binary operation", I'll point out some of the more common meanings of $\circ$ when speaking of functions and/or binary operations.
The symbol "$\circ$" denotes the composition of functions, composition of relations, or the composition of permutations, and more generally, denotes a binary operation, sometimes described as a law of composition, where $f\circ g$ is the function or operation resulting from performing the function/operation $g$, followed by the function/operation of $f$.
When denoting function composition, for example, if we are given that $f(x)$ and $g(x)$ are functions of $x \in \mathbb R$ such that: $f: \mathbb R \to \mathbb R,\; g:\mathbb R \to \mathbb R$, then $\;f\circ g: \mathbb R \to \mathbb R$ can be expressed as $$(f\circ g)(x) = f(g(x))$$
A: Usually composition of functions as stated in the other answer, but also can mean Hadamard/Schur/entrywise matrix product.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 6,878 |
Q: how to include font awesome icons and how to bind in html this is my html code
<body ng-controller="myController" >
<div gridster="gridsterOpts">
<ul>
<li gridster-item="item" ng-repeat="item in Items">
<div my-widget data="item.obj" ng-class="item.classes"><!--ng-style="item.object" -->
</div>
</li>
</ul>
</div>
and my script goes here
var app=angular.module('myApp',['gridster'])
app.controller('myController',function($scope){
$scope.Items = [
{ sizeX: 2, sizeY: 1, row: 0, col: 0, obj: {data:3222,message:"TotalCost",classes:["six design","fa-motorcycle"]}}]
});
app.directive('myWidget',function(){
return{
restrict:"EA",
scope:{
data:'=',
title:'=',
label:'=',
details:'=',
message:'=',
},
templateUrl:'spare.html',
}
});
and my css:
six design{
text align:right;
float:left;
}
and spare html goes here
< span ng-controller="myController" >
<h5>{{data.message}}<h5>
<h3>{{data.details}}</h3>
<h3>{{data.label}}</h3>
<h2>${{data.data}}</h2>
<h4>{{data.title}}</h4>
</span>
and now i just need to include ng-class array i.e ="classes" in obj only and then how to bind that in html and i should get all the css which i had applied to them and i also i should render all my font icons
like `obj:{ data:3222,message:"TotalCost" ,classes:"fa fa-level-up"}`
and how to bind it in hmtl
A: You can add even an array to ng-class,
You have classess in item.obj so change ng-class="item.classes" to ng-class="item.obj.classes"
<div style="background: yellow;width: 50px;" ng-repeat="item in Items">
<span style="margin-left: 10px" ng-repeat="class in item.obj.classes" ng-class="class" class="fa"></span>
<div my-widget data="item.obj"></div>
</div>
Also never use classnames with spaces so change ,six design to six-design
so css will be,
.six-design{
text align:right;
float:left;
}
and the object changes to,
classes:["six-design","fa-motorcycle"]
The object is,
$scope.Items = [
{ sizeX: 2, sizeY: 1, row: 0, col: 0, obj:{data:8988643347,right:'',title:'Income',classes:["style1","fa fa-apple","fa fa-youtube-square","fa-motorcycle"]}},
{ sizeX: 2, sizeY: 1, row: 0, col: 0, obj: {data:65476756,right:'',title:"Population",classes:["fa fa-map-marker","style2"] }},
{ sizeX: 2, sizeY: 1, row: 0, col: 0, obj: {data:54564545,right:'',title:"Users",classes:["fa fa-youtube-square","style3"]}},
{ sizeX: 2, sizeY: 1, row: 0, col: 0, obj: {data:123.33,label:"Money",classes:["design1","design2"]}},
{ sizeX: 2, sizeY: 1, row: 0, col: 0, obj: {data:3222,details:"Amount",classes:["design3","fa fa-shopping-cart","fa fa-motorcycle"]}},
{ sizeX: 2, sizeY: 1, row: 0, col: 0, obj: {data:3222,message:"TotalCost",classes:["sixdesign","fa-motorcycle"]}}]
Here is a working plunker
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 4,505 |
\section{Introduction}
Research of novel materials is in the forefront of contemporary physics and chemistry due to the presence of fundamentally interesting phases and the broad range of applications. In particular low-dimensional nanomaterials, including fullerenes \cite{Kroto1985}, carbon nanotubes (CNT) \cite{Iijima1991}, graphene \cite{Novoselov2004}. and most recently black phosphorus \cite{Li2014,Liu2014,castellanos-gomez2015,ling2015,Liu2015,Ryder2016,gusmao2017}, attracted significant attention. Common to these materials is that the electronic properties can be finely tuned by charge transfer which led to e.g. the discovery of superconductivity \cite{Hebard1991}, spin-density waves \cite{Bommeli1995}, and a Mott transition \cite{Maeda2002} in fullerides, the bleaching of optical transitions \cite{Eklund1997} and the Tomonaga--Luttinger to Fermi liquid crossover \cite{Pichler2004} in single walled-carbon nanotubes (SWCNT). It is also common that alkali atom doped modifications of these materials are extremely air-sensitive thus conventional contact probing of the electronic properties is difficult. This difficulty is even more pronounced for black phosphorus (bP) for which the pristine, undoped material is highly air and moisture sensitive \cite{ziletti2015,AbellanJACS2017,ZhangJACS2018}.
In principle, contactless methods, such as infrared spectroscopy could provide the required information on the conductivity of such samples, {\color{black}especially when combined with ESR, NMR or Raman-spectroscopy}. The microwave frequency range arises as an automatic choice for these studies as the frequency is closer to DC. As a result, this type of measurements is more representative for the DC conductivity except for some exotic cases e.g. including heavy fermions \cite{Dressel2005,Dressel2013}. Microwave cavity perturbation \cite{Buravov1971,Klein1993} allows to determine \emph{relative} changes in the conductivity, $\sigma$, dielectric permittivity, $\epsilon_{\text{r}}$, and magnetic permeability, $\mu_{\text{r}}$, of air-sensitive nanomaterials: the samples are placed inside a microwave cavity and changes in the the quality factor, $Q$, and resonance frequency, $f$, allow to determine the relevant parameters. A disadvantage of the method is that absolute values of material parameters are difficult to attain. An important advantage of the method is that the appropriate choice of cavity resonance mode allows to disentangle the different properties, e.g. there are cavity modes which locally sustain a node in the electric or magnetic field.
To illustrate the versatility of this technique we present experiments on pristine and potassium intercalated black phosphorus (bP) showing the appearance of new metallic charge carriers. Afterwards the K$_3$C$_{60}$ samples are investigated near the superconducting phase transition, which is followed by \emph{in-situ} doping of single walled carbon nanotubes (SWCNTs), whose resistivity evolution is examined in real time. {\color{black}Furthermore the technique was used for many other materials previously, such as: superconductivity in Nb and Pb \cite{HolczerPRB}; insulating behavior of NH$_3$K$_3$C$_{60}$ systems \cite{MaedaPRL}; and many more \cite{DresselReview}.}
\section{Experimental}
In this section first we present briefly how the investigated samples are prepared and introduce the reader to microwave conductivity measurements.
\paragraph{Sample preparation}
Polycrystalline black phosphorus was purchased from smart-elements with purity of $99.998\%$. The crystals were ground inside an argon-filled glovebox (MBraun GmbH with $< 0.1$ ppm of O$_2$ and H$_2$O) and used as a crushed powder. Potassium intercalation of the material was carried out in a solid-state route: to the pulverized phosphorus the stoichiometric amount of potassium metal (Sigma-Aldrich, $99.95\%$) was added to obtain the required intercalation compound (in our case $1:8$ for KP$_8$). Afterwards, the mixture was heated up to $70~^{\circ}$C, where it was carefully mixed together. The detailed process of synthesis is described in a previous article \cite{Abellan2017}. For the experiments, $10.1$ and $10.3$ mg of bP and KP$_8$ was put into ESR grade quartz tubes, evacuated to high vacuum ($2 \times 10^{-6}$ mbar) and sealed under $20$ mbar of He ($6.0$ Helium), which allowed cryogenic measurements.
Single crystal and powder K$_3$C$_{60}$ samples were prepared by the conventional potassium intercalation method; the crystal sample was from the same batch as in Ref. \cite{NemesPRB2000}. The powder samples were further ground together with non-conducting SnO$_2$ powder to prevent conducting links between the grains. SQUID magnetometry attested that DC superconducting properties (such as the steepness of the superconducting transition) were unaffected by the mixing. Samples were sealed in quartz ampules under low pressure helium.
We used commercial SWCNTs prepared by the arc-discharge method. The material was obtained from Nanocarblab (Moscow, Russia) with a well known mean diameter of $d=1.4$ nm and variance of $\sigma_d=0.1$ nm. The diameter distribution can be estimated well by a Gaussian function. This batch is the same as we used previously during Raman \cite{SimonPRB2005}, NMR \cite{SimonPRL2005} and ESR \cite{SzirmaiPRB2017} characterization and peapod filling \cite{SimonPRL2006}. The material was purified with oxidation in air and various acid treatments \cite{SimonChemPhysLett2004}. Afterwards it was ground thoroughly to allow microwaves to penetrate the whole bulk, and to assist the intercalation of potassium. About $5$ mg of SWCNT material was then vacuum annealed at $500~^{\circ}$C for $1$ hour in an ESR quartz sample tube with a neck and inserted into an Ar filled glove-box without air exposure. To achieve potassium intercalation we followed the well developed path of two chamber vapor phase intercalation \cite{Dresselhaus1981} adopted to our setup. The K was placed above the nanotube powder, where the neck forbid the mixing of the materials, similarly to our previous work done on graphite \cite{FabianPRB2012}. Finally the sample was evacuated to vacuum without air exposure.
\paragraph{Microwave cavity perturbation technique}
Microwave properties were measured with the cavity perturbation method \cite{Klein1993,Donovan1993} as a function of temperature, $T$, and for the case of K$_3$C$_{60}$ samples various static magnetic fields, $B$, were also applied. The used copper cavity has an unloaded quality factor of $Q_0 \approx 10,000$ and a resonance frequency, $f_0 \approx 11.2$ GHz, whose temperature dependence is taken into account. The samples were placed in the node of the microwave electric field and maximum of the microwave magnetic field inside the TE011 cavity, which is the appropriate geometry to study minute changes in the conductivity \cite{KitanoPRL2002}. The alternating microwave magnetic field induces eddy currents in the sample, which causes a change in the microwave loss and shifts the resonator frequency. The $Q$ factor of the cavity is measured via rapid frequency sweeps near the resonance. A fit to the obtained resonance curve yields the position, $f$, and width, $\Gamma$, of the resonance. $Q$ is afterwards obtained from its definition $Q=f/\Gamma$. This value has to be corrected with the unloaded $Q$ factor of the cavity, thus the loss caused by the inserted sample is: $\mathrm{\Delta}\left(\frac{1}{2Q} \right) = \frac{1}{2Q}-\frac{1}{2Q_0}$ and the frequency shift is $\mathrm{\Delta}f/f_0 = (f-f_0)/f_0$. Physical properties such as the relative permittivity can be calculated in the following way:
\begin{equation}
\epsilon_{\text{r}}'-1 = - A \frac{V_{\text{s}}}{V_{\text{c}}}\times \mathrm{\Delta}f/f_0,
\end{equation}
and
\begin{equation}
\epsilon_{\text{r}}''=A \frac{V_{\text{s}}}{V_{\text{c}}} \times \mathrm{\Delta}\left(\frac{1}{2Q} \right),
\end{equation}
with $\epsilon_{\text{r}}=\epsilon_{\text{r}}' + \cplxi \epsilon_{\text{r}}''$. $V_{\text{s}}$ and $V_{\text{c}}$ are denoting the volume of the sample and the cavity, respectively. The $A$ constant can be calculated from the electric field in the cavity with, $E$, and without, $E_0$, the sample:
\begin{equation}
A\approx\frac{\int \dd V E_{0}^{\ast}E}{\int \dd V \left| E_0 \right|^2}.
\end{equation}
Calculation of the resistivity or conductivity from the measured parameters is summarized in the following section and in Refs \cite{Csosz2018,Gyure2018}.
For the black phosphorus and the fulleride samples, we applied a low temperature geometry as the interesting physics is expected to happen below room temperature. To achieve this, we placed our probehead into the cryostat of a superconducting magnet fabricated by Cryogenics Ltd. The temperature range of the setup can be varied between $3.3$ K up to $200$ K with magnetic field ranging from $0$ to $9$ T. The static magnetic field and the rf magnetic field are parallel in our geometry. The trapped flux of the solenoid is about $10-20$ mT or $100-200$ Oe. For the K$_3$C$_{60}$ samples zero field measurements were done in another cryostat without a magnet, where the temperature range is limited to $6.5$ K from below. The used low temperature setup is presented in details in Ref. \cite{KarsaPssb2012}.
In case of \emph{in-situ} measurements, high temperature is mandatory, for this a different setup is used with a very similar microwave cavity. The samples, which are sealed in a quartz capillary, are placed in an additional quartz insert, inside the cavity. Dry nitrogen gas flows through the insert, whose temperature can be varied between $100$ K and $1000$ K. The sample and the intercalant are placed in the same horizontal plane in our geometry to avoid molten potassium flowing inside the cavity. Outside of the insert the cavity is attached to a nitrogen purge line to remove the humidity and water, which would disturb the experiments due to their additional absorption. The temperature of the copper cavity is stabilized by water cooling to avoid thermal expansion. This setup could also serve to detect the opposite effect, e.g. defunctionalization or dedoping of materials, especially combined with e.g. Raman or TG-MS spectroscopy.
\section{Results}
We demonstrate the utility of microwave cavity measurements on the nowadays intensively studied low dimensional materials, such as black phosphorus ($2$D), single walled carbon nanotubes ($1$D) and fullerides ($0$D). In this work the conductivity of the materials was studied, however the method itself is not limited to this value, e.g. dielectric properties ($\epsilon_{\text{r}}$) can be examined. The first subsection is about black phosphorus and its alkali intercalated variant, namely KP$_8$. The second one concerns the superconducting phase transition in K$_3$C$_{60}$, emphasizing that the method is a great tool to study such materials, especially when it is combined with magnetic field. These samples were measured in the low temperature setup, in turn the method can also be combined with high temperature in a different geometry, enabling real time examination of vapor-phase intercalation. This is demonstrated in the third part on SWCNTs.
In microwave conductivity measurements, the sample morphology greatly affects the relation between the complex conductivity of the material, $\widetilde{\sigma}$, and the microwave parameters, the loss and shift. Two limiting cases are known: i) the sample is large and the field penetrates only into a limited distance from the surface. This approximates the measurement performed on the K$_3$C$_{60}$ single crystal. ii) The sample consists of small grains whose dimensions are comparable to the penetration depth, $\delta$. This approximates well the black phosphorus, the SWCNT and the powder K$_3$C$_{60}$ samples of divided small grains approximated with spherical particles \cite{Klein1993,Donovan1993,Csosz2018}.
In the first case, when the radio frequency field penetrates in the skin depth only, also referred as to the skin limit, the following equation holds between the microwave measurement parameters and the properties of the material:
\begin{equation}
\frac{\mathrm{\Delta}f}{f_0} - \cplxi \mathrm{\Delta} \left(\frac{1}{2Q} \right)=-\cplxi \nu Z_{\text{s}},
\end{equation}
where $Z_{\text{s}}$ is the surface impedance of the sample, related to the conductivity as $Z_\text{s} = \sqrt{\mu_\text{0}\omega/\cplxi\widetilde{\sigma}}$, $\widetilde{\sigma}=\sigma_1 + \cplxi \sigma_2$. {\color{black}Here $\sigma_1$ and $\sigma_2$ represents the real and the imaginary parts of the complex conductivity}. The dimensionless $\nu \ll 1$ parameter is the so-called resonator constant \cite{HolczerPRB1994} and it depends on the sample surface relative to that of the cavity. This is the most appropriate case for bulky samples, like single crystals.
In the second case, when the microwave field penetrates into the sample (known as the penetration limit), the cavity measurables depend differently on the sample parameters. It was shown for a sphere with a radius of $a$, that
\begin{gather}
\frac{\mathrm{\Delta}f}{f_0} - \cplxi \mathrm{\Delta} \left(\frac{1}{2Q} \right) = - \gamma \widetilde{\alpha}, \label{eq:pow} \\
\widetilde{\alpha} = \frac{1}{10} \left(a\widetilde{k}\right)^2,
\end{gather}
where $\widetilde{k}=\frac{\omega}{c} \sqrt{\cplxi \widetilde{\sigma}/\epsilon_0 \omega}$ is the complex wavenumber and $\gamma$ is a dimensionless sample volume dependent constant. This scenario is well applicable to finely ground powders. Further approximation can be made if the bulk conductivity and the particle sizes are known. In the case of bP and SWCNTs the $Q \sim \rho$ approximation holds.
\subsection{Tuning the electronic properties of black phosphorus}
Conductivity measurements performed on undoped and potassium doped black phosphorus are shown in Fig. \ref{fig:kp8}. within the temperature range of $3.3$ K and $180$ K. The data points are normalized to the value of the pristine material taken at $100$ K (as our method cannot provide absolute values, this is a necessary step). The normalization was done taking into account the mass differences and assuming that the grain distribution of the material is not changed significantly during the intercalation. It is also assumed that $Q\sim \rho$ still holds for the KP$_8$ sample, as the conductivity of the sample is probably not increased by more than $3$ orders of magnitude.
\begin{figure}[h!]
\includegraphics*[width=.92\linewidth]{fig1_bp_kp8}
\caption{Microwave conductivity measurement performed on pristine black phosphorus and its potassium intercalated derivative, KP$_8$. The pristine material exhibits a semiconducting behavior in the investigated temperature range, in contrast to the doped sample, that behaves in {\color{black}this} way below $49$ K and above $130$ K. The conductivity of this regime for both materials is dominated by the electrons excited thermally through the smaller gap. In KP$_8$ between $49$ and $130$ K a metallic regime is observed (noted with vertical dashed lines) yielding the presence of conduction electrons, similarly to our previous observations on the NaP$_6$ system \cite{Shiozawa2018}. We argue that the excess charges are present on the intercalated phosphorene sheets. Above $130$ K the resistivity is dominated again by the thermally excited electrons from the larger band gap. Please note that the resistivity values are normalized to the value of the pristine material taken at $100$ K. Furthermore the superconducting phase transition occurring at $3.8$ K \cite{Shiozawa2018,ZhangNatComm2017} is probably hindered by {\color{black}freezing} out of the thermal excitations as the volume fraction of the superconducting phase is low compared to the whole volume. {\color{black}Inset presents the observed data in an Arrhenius plot to emphasize the activated behavior and to make the metallic regime more pronounced. Dashed green lines are guides for the eye to demonstrate the two gaps present in the pristine material as observed in Ref. \cite{MarkusPssb2017}}.}
\label{fig:kp8}
\end{figure}
The pristine material behaves as a semiconductor with a smaller and a larger band gap {\color{black}(visualized in the inset of Fig. \ref{fig:kp8})}, in agreement with previous literature observations \cite{MarkusPssb2017,Narita1983,Baba1991_1}. KP$_8$ exhibits a similar role below $49$ K with a slightly different small gap. The conductivity of this regime is dominated by the thermally excited electrons through the small band gap. The major difference between the two materials is visible in the $49-130$ K temperature range, where the alkali intercalated material exhibits a metallic behavior. {\color{black}It is clear that the intercalated system cannot be understood with only $2$ band gaps.} Similarly to our previous findings in the sodium-phosphorus system, NaP$_6$, we assign this to the presence of conduction electrons, whose states are existing in the intercalated phosphorene sheets \cite{Shiozawa2018}. Above $130$ K the conductivity of KP$_8$ is again dominated by the thermally excited electrons, but originated from the larger gap. Here we would like to point out that the overall resistivity of the intercalated material is increased by about a factor of $2$ compared to the pristine material. This somewhat strange and unexpected observation can be explained taking into account that during our measurements the microwave absorption is the quantity, which is truly measured and black phosphorus itself is a surprisingly good microwave absorbent \cite{MarkusPssb2017}. Moreover during the intercalation process some of the P$-$P bonds are broken \cite{Abellan2017}, especially near the surface, which can result in the suppression of absorption and increase of the resistivity.
It was shown previously that alkali intercalated black phosphorus becomes superconductive at a universal transition temperature of ca. $3.8$ K \cite{Shiozawa2018,ZhangNatComm2017}. In our measurement this effect is hindered by {\color{black}freezing} out of the thermal excitations as the resistivity is increasing as the temperature is going to zero. This also means that the number of the {\color{black}conduction} electrons is low compared to the whole system, in agreement with the low superconducting volume fraction observed in SQUID measurements \cite{Shiozawa2018}.
\subsection{Superconductivity of K$_3$C$_{60}$}
Fig. \ref{fig:k3c60} shows the microwave cavity loss, $\mathrm{\Delta}(1/2Q)$ and cavity shift, $\mathrm{\Delta}f/f_0$ for a single crystal and a fine powder K$_3$C$_{60}$ sample as a function of temperature for a few magnetic field values. The microwave loss decreases rapidly below $T_{\text{c}}$ in zero magnetic field as expected for superconductors. The most important observation is that the microwave loss becomes significant for a magnetic field as small as $0.05$ T for the fine powder sample, whereas even $3$ T has small effect on the microwave absorption for the single crystal one. In fact, we observe a huge, about $3$ times larger, microwave absorption below $T_{\text{c}}$ than in the normal state.
\begin{figure}[h!]
\includegraphics*[width=\linewidth]{fig2_k3c60_v2}
\caption{Temperature dependent cavity loss, $\mathrm{\Delta}(1/2Q)$, and cavity frequency shift, $\mathrm{\Delta}f/f_0$ measurements for single crystal and powder K$_3$C$_{60}$ samples. The magnetic field was $0$ and $3$ T for the crystalline and $0$, $0.05$ and $0.5$ T for the powder sample. Note that the $Q$-factor changes significantly for the powder sample in contrast to the single crystal sample. Note the different scales for the $\mathrm{\Delta}f/f_0$ data. The obtained datasets can be fitted with theoretical curves calculated from the Coffey--Clem theory \cite{Csosz2018,CoffeyClemPRL1991,CoffeyClemPRB1992,CoffeyClemPRB19922,CoffeyClem19924,CoffeyClem1993}.}
\label{fig:k3c60}
\end{figure}
The fact that the enhanced microwave absorption occurs with the application of the magnetic field hints at a flux motion related phenomenon that is discussed in the framework of the Coffey--Clem (CC) theory \cite{CoffeyClemPRL1991,CoffeyClemPRB1992,CoffeyClemPRB19922,CoffeyClem19924,CoffeyClem1993}. The microwave absorption peak occurs above the irreversibility line, i.e. it is related to the physical behavior of the vortex-fluid state; for K$_3$C$_{60}$ $T_{\text{irrev}}(B = 0.1~\text{T}) \approx 15$ K and $T_{\text{irrev}}(B = 1~\text{T}) < 5$ K \cite{PrassidesBook}. Superconducting fullerides are type-II ($\lambda \gg \xi$) and have a short mean free path i.e. they can be described in the local electrodynamics limit, which simplifies the discussion \cite{GunnarsonRMP1997}. The phenomenological CC theory is based on a two-fluid model and considers the motion of vortices due to the exciting electromagnetic field in the presence of a viscous background (described by the viscous drag coefficient, $\eta$) and a restoring force (described by an effective pinning force constant, $\kappa_{\text{p}}$). The details of the theory and the exact calculation for the K$_3$C$_{60}$ described elsewhere \cite{Csosz2018}, here we only summarize the key points. The concept of the complex penetration depth, $\widetilde{\lambda}$ is introduced in the theory the following way:
\begin{equation}
\widetilde{\lambda}^2 = \frac{\lambda^2+(\cplxi/2)\widetilde{\delta}^2_{\text{eff}}}{1-2\cplxi \lambda^2/\widetilde{\delta}^2_{\text{nf}}},
\end{equation}
where $\widetilde{\delta}^2_{\text{nf}}$ is the skin depth in the normal fluid, $\lambda$ is the usual penetration depth and $\widetilde{\delta}^2_{\text{eff}}$ is the complex effective skin depth, which contains the effect of vortex motion. $\widetilde{\lambda}$ is linked to the conductivity by $\widetilde{\sigma} = \cplxi/\mu_0 \omega \widetilde{\lambda}^2$.
On the left panels of Fig. \ref{fig:k3c60} the application of the first case is shown, described above for the single crystal sample. We find that for both the calculation and experiment, the cavity loss parameter drops rapidly below $T_{\text{c}}$, although $\sigma_1/\sigma_{\text{n}}$, {\color{black}where $\sigma_{\text{n}}$ is the conductivity of the normal, Fermi liquid state,} is around unity due to the vortex motion. This effect is due to the development of a significant $\sigma_2/\sigma_{\text{n}} \sim 100$, which limits the penetration of microwaves into the sample and thus reduces the loss. This means that the microwave surface impedance measurement is less sensitive to provide information about $\sigma_1$ in the presence of vortex motion.
The right panels in Fig. \ref{fig:k3c60} show the measurements for the fine powder sample. {\color{black}The arisen peak alike structure can be interpreted as follows: due to the vortex motion, the magnetic field can penetrate into a greater volume, than without, which results a decrease in the weight of Dirac delta function present in the real part of the conductivity. The effect is proportional to the inverse of the penetration depth squared. The missing weight, due to sum rule, results a constant part up to cut in frequency. If the vortex motion is not negligible, and the frequency used during the measurement is below this cut it can happen that the real part of the conductivity is higher in the superconducting state, than in the normal, Fermi-liquid state. This results an enhancement in the microwave loss, measured by the applied microwave technique.} To obtain fits, the transport and magnetic parameters ($\rho_{\text{n}}=1/\sigma_\text{n},\xi_0,\lambda_0$) have to be fixed to the respective mean of literature values. {\color{black}Here $\xi_0$ and $\lambda_0$ denotes the coherence length and the penetration depth at zero temperature, respectively.} We assumed that the sample consists of small spheres with a uniform diameter of $a$. The zero magnetic field data depend only on $\gamma$ and $a$ when the other sample properties are fixed. A fit to the $B = 0$ data yields $\gamma = 5.5(2) \times 10^{-4}$, and $a = 6.2(2)$ {\textmu}m. We then proceed to fit the magnetic field dependent data with $\kappa_{\text{p}}$ as the only free parameter and we obtain $\kappa_{\text{p}} = 1.0(1) \times 10^{-3}$ N$/$m$^2$. The fits and the detailed calculations for the obtained data are presented elsewhere \cite{Csosz2018}.
\subsection{\emph{In-situ} intercalation of potassium into SWCNT bundles; transition from semiconducting to metallic}
The next advantage of the microwave conductivity measurements is that they are readily adaptable to make \emph{in-situ} measurements. We demonstrate this on an arc-discharge SWCNT sample in the high temperature setup. The intercalation takes place similarly like in the two chamber vapor-phase intercalation method: the intercalant is separated in space and the process is driven by the gradient in temperature and chemical potential \cite{Dresselhaus1981}. The difference is that the nanotube powder is now placed in the middle of the cavity, while the intercalant, in our case the potassium, is located outside. The hot nitrogen melts the potassium and sustains the vapor pressure required for the doping. In this geometry the resistivity change of the material inside is measured instantly. In Figure \ref{fig:k-swcnt}. a complete \emph{in-situ} measurement process is presented.
\begin{figure}[h!]
\includegraphics*[width=.85\linewidth]{fig3_k-swcnt}
\caption{\emph{In-situ} intercalation of arc discharge SWCNTs. Steps marked with {\color{black}red} color indicate measurement steps, while {\color{OliveGreen}green} parts show intercalation steps proceeded at $545$ K. In the {\color{black}first step} the material clearly shows a semiconducting behavior (resistivity decreases upon increasing temperature) as expected from a bundled branch of nanotubes, where $2/3$ of the tubes are non-metallic. In the {\color{OliveGreen}second step} the first intercalation takes place, which lasted for $16$ minutes. This is followed by the measurement denoted with {\color{black}III}, where the system was cooled from $545$ K down to $155$ K and back. On this part the SWCNTs show a completely metallic behavior (resistivity decreases upon decreasing temperature and increases upon increasing) proving that the intercalation took place. Further doping steps, {\color{OliveGreen}IV and VI} and measurement parts, {\color{black}V and VII} indicate the drop of the resistivity as the material is turning more and more metallic.}
\label{fig:k-swcnt}
\end{figure}
The process is carried out in the following steps: first the temperature of the steadily cooled sample was increased from $173$ K to $520$ K, this part is labeled as {\color{black}I} in Fig. \ref{fig:k-swcnt}. In this range the resistivity of the material is decreasing upon increasing temperature, clearly showing a semiconducting behavior with $\rho = \rho_0 \expe^{\Delta/T}$. This is expected as the used arc-discharge SWCNT bundle contains a mixture of $2/3$ non-metallic and $1/3$ metallic nanotubes. A fitted exponential to this regime yields a transport activation energy of $\Delta = 39(1)$ K in agreement with previous observations \cite{KarsaPssb2012,KaiserPRB1998}. At $520$ K the intercalation starts to take place resulting in a decrease in the resistivity. The temperature is then fixed at a value of $545$ K for $16$ minutes, this regime is noted as {\color{OliveGreen}II} in the figure. Cooling down the sample ({\color{black}III}) we observe a completely different behavior as before: the sample becomes metallic, as its resistivity decreases with lowering the temperature. Afterwards the system is heated up again and the doping process is continued. The conductivity of the sample is increased further in {\color{OliveGreen}IV} as the sample gets more doped. In the further measurement and intercalation steps the resistivity drops further and further ({\color{black}V}, {\color{OliveGreen}VI} and {\color{black}VII}) and the process can be continued until reaching the equilibrium stoichiometry of KC$_7$, which is a completely metallic state \cite{Pichler1999}.
\section{Summary}
We presented that physical properties of air-sensitive nanomaterials, especially in powder form can be investigated in the framework of microwave cavity measurements, which appeared to be very robust and versatile. Conductivity measurements made on black phosphorus, KP$_8$, two types of K$_3$C$_{60}$ samples and real time evolution of K-SWCNTs, {\color{black}potassium intercalated single-walled nanotubes} are demonstrated herein.
\section{Acknowledgement}
Work supported by the Hungarian National Research, Development and Innovation Office (NKFIH) Grant Nr. K119442, SNN118012 and 2017-1.2.1-NKP-2017-00001. The authors thank the European Research Council (ERC Advanced Grant 742145 B-PhosphoChem) for financial support. The research leading to these results was partially funded by the European Union Seventh Framework Programme under grant agreement No. 604391 Graphene Flagship. We also thank the Deutsche Forschungsgemeinschaft (DFG-SFB 953 "Synthetic Carbon Allotropes", Project A1), the Interdisciplinary Center for Molecular Materials (ICMM), and the Graduate School Molecular Science (GSMS) for financial support. G. A. thanks the FAU for the Emerging Talents Initiative (ETI) grant \#WS16-17\_Nat\_04, and support by the DFG and FLAG-ERA (AB694/2-1).
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El Club Atlético Boca Juniors es una entidad deportiva argentina con sede en el barrio de La Boca, Buenos Aires. Fue fundado en dicho barrio el 3 de abril de 1905 por seis vecinos adolescentes, hijos de italianos. El básquet de Boca Juniors es uno de los más importantes y exitosos de la Argentina, ya que tuvo gran protagonismo tanto en la era amateur y semi-amateur de las distintas ligas de básquet metropolitano de la ciudad de Buenos Aires, como en la Liga Nacional de Básquet, donde es uno de los máximos campeones, razón por la cual es considerado uno de los denominados grandes de la liga nacional de básquet.
Grandes jugadores y entrenadores vistieron la casaca azul y oro antes y/o después de representar a la selección nacional de básquet e incluso de jugar en las grandes ligas de Europa y la NBA.
Amateurismo
La sección de básquetbol del club es creada en 1929 y al año siguiente se afilia en la Federación Argentina de Básquetbol, la única entidad rectora del básquet metropolitano hasta entonces, donde competiría en divisiones menores. La historia grande del básquet surgiría recién con la creación de la nueva entidad rectora del básquetbol porteño, la Asociación de Básquet de Buenos Aires de organización semiamateur y donde se afiliarían la mayoría de los clubes de fútbol. Desde 1937 hasta 1974 el básquetbol de la ciudad y alrededores estaría dividido entre "los que no cobran" de la FABB, luego Asociación Porteña de Básquet desde 1954 y "los que cobran" de la ABA, aunque existirían varios matices para definir el amateurismo del básquet. Boca Juniors bajo la organización de la ABA compite los Torneos Apertura hasta mitad de año y los Campeonatos Oficiales en el segundo semestre y desde 1951 los Torneos Metropolitanos, que aglutinaban a los mejores equipos de la Asociación Buenos Aires y la Federación de Básquet (luego Asociación Porteña), a fin de año.
El equipo de las estrellas
En la década del '40, Boca Juniors era conocido como "el equipo de las estrellas". Boca ganó en esa etapa los Torneos Apertura de 1938 y 1939 y los Campeonatos Oficiales de la Asociación de Buenos Aires en 1940 y 1941. El plantel alistó a estos hombres, por orden alfabético: Pedro Aizcorbe, Daniel Anglés, Elías Bissio, Roberto Contini, Alberto Dayán, Víctor Di Vita, José Giuliano, Carlos Induni, Felipe Mattianich, Mario Mattioni, Pedro Rodríguez y Carlos Stroppiana.
La vuelta a la gloria
Ya en 1955, Boca tenía las pretensiones de restituir su prestigio en el básquet. El proyecto se inició con la contratación de los jugadores rosarinos Enrique Borda -en 1954- y Bernardo Schime y Rubén Petrilli (un año más tarde). En el plantel ya estaban Fazio, Alberto Noval y Egidio De Fornasari. La dirección técnica le fue confiada a Andrés "Naranjito" Rossi.
Entre 1956 y 1957 siguieron llegando grandes jugadores, como el santafesino José Olivera, José Novoa y Luis Pérez. Este proyecto de Rossi comenzó a dar sus frutos con el comienzo de una serie interminable de éxitos. El primero fue el Torneo Metropolitano en 1957, título que Boca repitió en 1959, luego de haber salido subcampeón en 1958. También logró el subcampeonato de la Asociación Buenos Aires en 1958 y 1959.
La triple-triple corona
Luego del paso de Rossi por la dirección técnica de Boca, arriba Abelardo Rafael Dasso, con quien se vivió el más importante ciclo para la historia del básquet de Boca Juniors. Junto a él, Boca salía al rectángulo con Enrique Borda, Jesús María Díaz, Miguel Carrizo, Bernardo Schime, Alberto Desimone, Luis Pérez, Egidio De Fornasari, Alberto Noval, Héctor Vázquez, Rubén Castelli, Juan Carlos Mazzini, Abel Rojas, Luis Torrás, Edgardo Molinari y Héctor Rosales. En 1961, 1962 y 1963, el Xeneize consigue la triple corona: sale campeón de la Asociación Buenos Aires, del Apertura y del Metropolitano. En dicho lapso, disputó 93 partidos, con un récord descomunal de 89 victorias.
Boca además se queda con el Torneo Apertura de 1964 y los campeonatos de la Asociación de Buenos Aires 1965, 1966 y 1967, siempre con la dirección técnica de Dasso y la guía dentro de la cancha del "Chino" Díaz.
Los últimos años de la ABA
Boca volvió a salir campeón del Metropolitano en 1969 y subcampeón del Torneo Oficial de la Asociación Buenos Aires. En 1970, el Xeneize gana una vez más el Oficial, al superar a Lanús en el estadio de River por 91 a 76, y se cobró revancha de la final perdida ante ese equipo el año anterior. Enrique Borda era el entrenador azul y oro, quien plantó un equipo base formado por Juan Carlos Mazzini, Néstor Delguy, Adalberto Gusso, Emilio Dumani, Juan Tito y Jesús Díaz.
Liga Nacional de Básquet
Comienzo oscuro
Boca Juniors participó de la primera edición de la Liga Nacional de Básquet en 1984, cuya edición se denominó Torneo de Transición, y fue ganada por San Andrés. Entre los diez equipos que jugaron, Boca finalizó en el octavo puesto, y así quedó relegado a una división menor. Recién en 1988 logró el ascenso a la máxima categoría, al consagrarse campeón del torneo organizado por la Confederación Argentina de Básquetbol (CABB). En la siguiente temporada, finalizó en el puesto 14 (entre 16 participantes) y volvió a descender. A pesar de perder las finales del torneo de 1990, organizado por CABB, el equipo Xeneize regresó a la elite del básquet nacional.
En la temporada 1990/91 Boca finalizó décimo, entre catorce participantes, y en la Liga 1991/92 culminó en la posición 13 (también entre catorce equipos), debiendo disputar un play off por el descenso ante River Plate, al cual derrotó y envió al Torneo Nacional de Ascenso (TNA). El conjunto de la ribera repitió el puesto 13 en la temporada 1992/93 pero, esta vez, con la participación de 16 equipos. En el torneo siguiente finalizó en el puesto 12, y en 1994/95 terminó en el cuarto puesto dirigido por León Najnudel.
Primer campeonato de LNB
Con la llegada de Julio Lamas como entrenador en la temporada 1995/96 comenzó un ciclo brillante. Luego del octavo lugar conseguido en esa Liga, se consagró, por primera vez, campeón de la Liga Nacional en Liga Nacional de Básquet 1996/97 tras vencer en las finales a Independiente de General Pico por 4-1. Sobre 58 partidos, ganó 42 (72,4%) y perdió 16.
En la temporada 1997/98 fue subcampeón tras perder las finales ante Atenas de Córdoba por 4-0. El equipo fue dirigido por Néstor García, quien en 1998/99 terminó en la quinta posición al ser derrotado por Obras en cuartos de final.
Dirigido por Fernando Duró, el equipo finalizó en el quinto lugar al ser derrotado por Libertad de Sunchales en cuartos de final. Boca se quedó con la primera edición de la Copa Argentina de Básquet en el 2002, luego de vencer en el cuadrangular final a Argentino de Junín, Gimnasia y Esgrima La Plata y Atenas de Córdoba.
En la Liga Nacional 2002/03 llegó al segundo puesto tras perder la final con Atenas.
Segundo campeonato de LNB
Sergio Hernández llegó a la dirección técnica en el 2003, y Boca se consagró campeón invicto de la Copa Argentina, ganándole a Atenas en el estadio Carlos Cerutti (en Córdoba). Pero el equipo Xeneize quería la Liga otra vez, y llegó la hazaña. Primero consigue el lugar en la tabla de posiciones logrando la localia en todas las series de play off que dispute. Vence a Belgrano de San Nicolás en cuartos de final y a Obras en semifinales. El comienzo de la serie final fue desalentador, perdía 2-0 la serie final de local frente a Gimnasia y Esgrima La Plata pero logró darla vuelta. El delirio boquense fue en el Polideportivo del club platense. La serie terminó 4-2. Sobre 57 encuentros, ganó 44 (77,2 %) y perdió 13 (uno por no presentación). Promedio de tantos: 95,8 a favor y 85,0 en contra.
Se consagró campeón de la edición 41 del Sudamericano de Clubes Campeones, lo que significó el primer título internacional jugado en Paraguay.
El equipo retuvo el título en 2005 en Rafaela y 2006 en Barquisimeto (Venezuela) logrando el tricampeonato, este último con Eduardo Cadillac como entrenador.
Tercer campeonato de LNB
Con Gabriel Piccato como entrenador, el equipo conquistó su tercera Liga Nacional temporada 2006/07 dando la sorpresa luego de un comienzo de temporada complicado donde el entrenador Eduardo Cadillac fue reemplazado por falta de resultados. El equipo se recuperó y logró el excelente tercer lugar en la tabla de posiciones de cara a los play offs donde derrota a Ben Hur en cuartos de final y a Libertad en semifinal. En la serie final venció 4-2 a Peñarol de Mar del Plata quitándole la localia luego de vencerlo en Mar del Plata y finalmente se consagra en la Bombonerita consiguiendo el tercer título de Liga A. Sobre 59 juegos, logró 35 triunfos (58,3 %) y 24 derrotas. Promedio de tantos: 79,3 a favor y 78,03 en contra.
Instalaciones
Estadio Luis Conde
Ubicado en Arzobispo Espinosa 600, en el barrio porteño de La Boca, el Estadio Luis Conde, conocido como «La Bombonerita», se inauguró el 29 de junio de 1996 y tiene una capacidad de 2400 espectadores. En esa fecha se realizó su bautismo con un enfrentamiento amistoso entre el local, y Obras Sanitarias, que terminó 85 a 74 para el dueño de casa.
Rivalidades
Boca Juniors carece de una fuerte rivalidad con algún equipo en particular. Se puede decir que normalmente ha tenido cierta enemistad con los otros equipos históricos de la Liga, tales como Atenas de Córdoba, Peñarol de Mar del Plata o inclusive Ferro, con los que ha tenido memorables partidos en instancias decisivas del campeonato. El Club Atlético River Plate ha sido su rival cuando han coincidido en la máxima categoría, aunque esto se deba más que nada a la muy fuerte competencia que estas dos instituciones tienen en el fútbol y que lógicamente, se transmite a los otros deportes. Sin embargo, al estar el club de Núñez generalmente desarrollando su disciplina entre el TNA o el Torneo Federal, este enfrentamiento ha perdido peso debido a los escasos enfrentamientos.
Desde su vuelta a la LNB en 2015, el principal rival de Boca es San Lorenzo, equipo con el que supo tener grandes enfrentamientos en la era amateur, donde ambos solían ser protagonistas. Son los únicos dos equipos de los denominados ''cinco grandes del fútbol argentino'' que compiten en la máxima categoría del deporte y son de los clubes más ganadores del Básquet, además de haberse enfrentado en numerosos encuentros decisivos.
Jugadores y cuerpo técnico
Plantel actual
Cuerpo técnico
Entrenador: Carlos Duro
Palmarés
Metropolitanos
Torneo Metropolitano (6): 1957, 1959, 1961, 1962, 1963, 1969.
Campeonato Oficial (7): 1961, 1962, 1963, 1965, 1966, 1967, 1970.
Torneo Apertura (6): 1938, 1939, 1961, 1962, 1963, 1964.
Nacionales
Liga Nacional de Básquet
Campeón (3): 1996-97, 2003-04 y 2006-07.
Subcampeón (3): 1997-98, 2002-03, 2004-05.
Campeonato Argentino de Clubes (1): 1963
Copa Argentina (5): 2002, 2003, 2004, 2005, 2006.
Torneo Top 4:
Campeón (1): 2004.
Subcampeón (2): 2002, 2003.
Copa Desafío
Subcampeón (1): 2007.
Liga Nacional B (1):
Campeón (1): 1988.
Internacionales
Campeonato Sudamericano de Clubes Campeones (3): 2004, 2005, 2006.
Entrenadores destacados
Abelardo Rafael Dasso
Alberto Finguer
Edgard Parizzia
León Najnudel
Julio Lamas
Rubén Magnano
Sergio Hernández
Gabriel Piccato
Néstor García
Fernando Duró
Pablo D'Angelo
Oscar "Huevo" Sánchez
Gonzalo Eugenio García
Jugadores destacados
Referencias
Enlaces externos
Sitio Oficial de la LNB
Facebook Oficial del Básquet de Boca
Twitter Oficial del Básquet de Boca
Secciones deportivas del Club Atlético Boca Juniors
Clubes de baloncesto de Argentina | {
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package org.keycloak.validation;
import org.keycloak.models.KeycloakSession;
public interface ValidationContext<T> {
enum Event {
CREATE,
UPDATE
}
Event getEvent();
KeycloakSession getSession();
T getObjectToValidate();
ValidationContext<T> addError(String message);
ValidationContext<T> addError(String fieldId, String message);
ValidationContext<T> addError(String fieldId, String message, String localizedMessageKey, Object... localizedMessageParams);
ValidationResult toResult();
}
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{"url":"http:\/\/www.alexdamour.com\/content\/variable_selection\/PQ_VS_slides.html","text":"# A Design-Based Perspective on Variable Selection\n\nAlex D'Amour\nNovember 25, 2014\n\n### Perspectives on Model Selection\n\nWhy?\n\n\u2022 Stealth regularization (AIC, CV, LASSO)\n\u2022 Legitimate regularization (nonparametric regression)\n\u2022 Infer \u201crelevant\u201d parameters (vaguely causal?)\n\nThis talk: Find a model that provides the best predictive performance for our given sample size. Note that predictive performance includes estimation uncertainty, bias, and residual variation.\n\nHoly grail: Eliminate high-dimensional nuisance without high-dimensional priors.\n\n### Perspectives on Model Selection\n\nIs the truth\u2026\n\n\u2022 Finite dimensional and sparse? (Donoho et al. Compressed sensing, fundamentally parametric.)\n\u2022 Infinite dimensional and dense? (Meng 2014, Everything is variation, fundamentally nonparametric.)\n\nThis talk: The latter. There may exist a sparse set of predictors, but no reason to believe that the predictors as collected define the proper basis for such sparsity.\n\nMore reading: Liu and Yang, 2009. \u201cParametric or nonparametric? A parametricness index for model selection.\u201d\n\n### Aside: What is truth?\n\nExample Let $$X_i$$ be a $$p$$-dimensional multivariate normal with covariance matrix $$\\Sigma$$ defined so that $$\\Sigma_{k,l} = \\rho^{|k-l|}$$, $$0 < \\rho < 1$$.\n\nConsider: $Y_i \\sim X_{i,2} - \\rho X_{i,1} + 0.2 X_{i,p} + \\mathcal N(0,3).$\n\n\u201cTrue\u201d\u201c model includes covariates $$(1,2,p)$$. But for any subset $$A \\subset \\{1,\\cdots ,p\\}$$, $Y_i | X_{i,A} \\sim \\beta_A^{\\top} X_{i,A}\\mathcal + N(0, \\sigma_A).$ because $$(Y_i,X_i)$$ are jointly multivariate normal.\n\n\"Truth\u201d only has special status because it has minimal residual variance.\n\n### Aside: What is truth?\n\nSimulation: $$N = 100$$, $$p=25$$, $$\\rho = 0.75$$.\n\nFor simplicity, consider only growing models $$A_k = \\{1, \\cdots, k\\}$$.\n\n### Perspectives on Model Selection\n\nIs it a\u2026\n\n\u2022 Inference problem? (LASSO)\n\u2022 Decision problem? (Carvalho)\n\nThis talk: Design problem!\n\n\u2022 Not a facet of the underlying system (so not inference).\n\u2022 Done beforehand to define the problem, incorporating inferential constraints (so not decision).\n\u2022 Which available conditional distribution can we reliably estimate?\n\n### Virtues of Design Perspective\n\nEstimands mean something.\n\n\u2022 $$\\beta_1$$ is only meaningful in context of the rest of $$A$$. Cross-model inference about $$\\beta_1$$ is awkward, requires meaningless symmetries (Berk et al).\n\nSeparation of selection inference.\n\n\u2022 Arguably post-selection inference impossible without separation (Leeb and Potscher).\n\u2022 Where distributions exist, require strong assumptions, difficult hypotheses (Lockhart et al).\n\n### Virtues of Design Perspective\n\nSeparation of selection and inference.\n\n\u2022 Garden of forking paths.\n[M]odels become stochastic in an opaque way when their selection is affected by human intervention based on post-hoc considerations such as \u201cin retrospect only one of these two variables should be in the model\u201d or \u201cit turns out the predictive benefit of this variable is too weak to warrant the cost of collecting it.\u201d (Berk et al 2013).\n\n### Seeds of Design Perspective\n\nWasserman's HARNESS\n\n\u2022 Response to Lockhart et al LASSO hypothesis testing paper.\n\u2022 Randomly split data.\n\u2022 Model selection with one half.\n\u2022 Conditional on selected model, standard inference on other half.\n\n### Seeds of a Design Perspective\n\nIssues with Data Splitting\n\nSome statisticians are uncomfortable with data-splitting. There are two common objections. The first is that the inferences are random: if we repeat the procedure we will get different answers. The second is that it is wasteful.(Wasserman in response to Lockhart et al.)\n\n### (Finally) A Contribution\n\nPrincipled Data Splitting\n\n\u2022 Can we use design principles to improve data-splitting techniques?\n\u2022 Splitting can be skewed to alleviate concerns.\n\u2022 Optimization can be exchanged with randomization to navigate optimality\/robustness tradeoff.\n\nKey idea: Inference is already conditional on $$X$$. \u201cSplitting on observables\u201d can be used to improve power, restrict randomization without biasing inference.\n\n### A Small Result\n\nAssume $$(Y_i, X_i)$$ multivariate normal, as before.\n\nProcedure:\n\n\u2022 Use a penalized log-likelihood information criterion to select a model (AIC, BIC, DIC, or other)\n\u2022 Compute predictive intervals for $$Y^{rep}$$ using selected model.\n\nLemma: Under the multivariate normal model, for fixed split sizes in the model selection set $$n_1$$ and the inference set $$n_2$$, the optimal (oracle) splitting policy maximizes the leverage of the points in the inference set with respect to the selected model.\n\n### A Small Result\n\nProof: Linear regression information criteria have the form $myIC = n_1\\log \\hat \\sigma_A^2 + 2g(p_A, n_1) + C,$ where $$g$$ is a function of model size and sample size, and $$C$$ is a constant shared by all models.\n\nBecause of multivariate normality, residuals for any set $$A$$ are mean-zero normal, so $\\hat \\sigma_A^2 \\sim \\sigma^2_A \\chi^2_{n-p},$ so all expectations of $$myIC$$ do not depend on $$X$$.\n\nMeanwhile, the predictive variance has the form: $Var(Y) = X_A^{rep}(X_A^{\\top}X_A)^{-1}X_A^{rep\\top} \\sigma^2_A$ with trace decreasing in the leverage of inference set.\n\n### Tip of the Iceberg\n\nAchieving the (leverage) oracle:\n\n\u2022 Sequential designs to maximize expected leverage for likely models.\n\nRelaxed assumptions:\n\n\u2022 Without MVN, model selection depends on $$X$$.\n\u2022 Selection\/inference tradeoffs need to be formulated.\n\n### Tip of the Iceberg\n\nEvaluation:\n\n\u2022 Oracle model recovery is not the goal.\n\u2022 $$p$$ is infinite for all $$N$$? Stochastic process perspective for finite samples.\n\nCross-pollenation:\n\n\u2022 Algorithms from CUR decompositions, algorithmic leveraging (Mohoney et al).\n\u2022 Leverage-based sampling from surveys.\n\u2022 Complementary model selection and inference methods.\n\n### Summary\n\nGoal:\n\n\u2022 Select model to minimize predictive risk given the current sample size.\n\u2022 Report valid inferences conditional on this model.\n\nDon't care if:\n\n\u2022 Different model at different sample sizes.\n\u2022 True\/false positives.\n\nAchievable by:\n\n\u2022 Separating model selection and inference.\n\u2022 Optimize using design principles.","date":"2018-07-21 21:17:54","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\": 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\": 0.6314051151275635, \"perplexity\": 5135.376343492939}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2018-30\/segments\/1531676592778.82\/warc\/CC-MAIN-20180721203722-20180721223722-00381.warc.gz\"}"} | null | null |
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{"url":"https:\/\/gmatclub.com\/forum\/if-20-men-or-24-women-or-40-boys-can-do-a-job-in-12-days-working-for-246626.html","text":"It is currently 20 Jan 2018, 23:06\n\n### GMAT Club Daily Prep\n\n#### Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.\n\nCustomized\nfor You\n\nwe will pick new questions that match your level based on your Timer History\n\nTrack\n\nevery week, we\u2019ll send you an estimated GMAT score based on your performance\n\nPractice\nPays\n\nwe will pick new questions that match your level based on your Timer History\n\n# Events & Promotions\n\n###### Events & Promotions in June\nOpen Detailed Calendar\n\n# If 20 men or 24 women or 40 boys can do a job in 12 days working for 8\n\nAuthor Message\nTAGS:\n\n### Hide Tags\n\nMath Expert\nJoined: 02 Sep 2009\nPosts: 43334\n\nKudos [?]: 139627 [4], given: 12794\n\nIf 20 men or 24 women or 40 boys can do a job in 12 days working for 8\u00a0[#permalink]\n\n### Show Tags\n\n09 Aug 2017, 00:12\n4\nKUDOS\nExpert's post\n14\nThis post was\nBOOKMARKED\n00:00\n\nDifficulty:\n\n75% (hard)\n\nQuestion Stats:\n\n62% (05:08) correct 38% (03:32) wrong based on 138 sessions\n\n### HideShow timer Statistics\n\nIf 20 men or 24 women or 40 boys can do a job in 12 days working for 8 hours a day, how many men working with 6 women and 2 boys would it take to do a job four times as big working for 5 hours a day for 12 days?\n\nA. 44\nB. 50\nC. 120\nD. 122\nE. 128\n[Reveal] Spoiler: OA\n\n_________________\n\nKudos [?]: 139627 [4], given: 12794\n\nMath Expert\nJoined: 02 Aug 2009\nPosts: 5536\n\nKudos [?]: 6443 [3], given: 122\n\nRe: If 20 men or 24 women or 40 boys can do a job in 12 days working for 8\u00a0[#permalink]\n\n### Show Tags\n\n09 Aug 2017, 00:23\n3\nKUDOS\nExpert's post\nBunuel wrote:\nIf 20 men or 24 women or 40 boys can do a job in 12 days working for 8 hours a day, how many men working with 6 women and 2 boys would it take to do a job four times as big working for 5 hours a day for 12 days?\n\nA. 44\nB. 50\nC. 120\nD. 122\nE. 128\n\nHi...\nLet's count in terms of men..\n1 w = 20\/24 m.....b = 1\/2 m..\nSo if 20 m can do a work in 12*8 H, 20*12*8 m will do it in 1 h..\n4 times the work will require 12*8*4*20 m..\nWorking 5*12 per man , $$\\frac{20*12*8*4}{5*12}=128m$$\nBut 6w and 2b are also there.. 6w=6*20\/24=5m....2b=2*1\/2=1m..\nSo remaining men required=128-5-1=122\n\nD\n_________________\n\nAbsolute modulus :http:\/\/gmatclub.com\/forum\/absolute-modulus-a-better-understanding-210849.html#p1622372\nCombination of similar and dissimilar things : http:\/\/gmatclub.com\/forum\/topic215915.html\n\nBANGALORE\/-\n\nKudos [?]: 6443 [3], given: 122\n\nManager\nJoined: 23 Jul 2015\nPosts: 158\n\nKudos [?]: 42 [3], given: 31\n\nConcentration: Strategy, Technology\nGMAT 1: 730 Q50 V40\nRe: If 20 men or 24 women or 40 boys can do a job in 12 days working for 8\u00a0[#permalink]\n\n### Show Tags\n\n09 Aug 2017, 00:41\n3\nKUDOS\n4\nThis post was\nBOOKMARKED\nBunuel wrote:\nIf 20 men or 24 women or 40 boys can do a job in 12 days working for 8 hours a day, how many men working with 6 women and 2 boys would it take to do a job four times as big working for 5 hours a day for 12 days?\n\nA. 44\nB. 50\nC. 120\nD. 122\nE. 128\n\n$$R_m = \\frac{1}{12* 8 * 20}$$\n$$R_w = \\frac{1}{12* 8 * 24}$$\n$$R_b = \\frac{1}{12* 8 * 40}$$\n\nLet no. of men needed = x\n\n$$(\\frac{x}{12* 8 * 20} + \\frac{6}{12*8*24}+\\frac{2}{12*8*40} ) * 5*12 = 4$$\n\n$$\\frac{6+x}{32} = 4$$\n\n$$x+6 = 128$$\n$$x = 122$$ (choice D)\n\nKudos [?]: 42 [3], given: 31\n\nPS Forum Moderator\nJoined: 25 Feb 2013\nPosts: 817\n\nKudos [?]: 382 [3], given: 42\n\nLocation: India\nGPA: 3.82\nIf 20 men or 24 women or 40 boys can do a job in 12 days working for 8\u00a0[#permalink]\n\n### Show Tags\n\n09 Aug 2017, 01:05\n3\nKUDOS\n2\nThis post was\nBOOKMARKED\nBunuel wrote:\nIf 20 men or 24 women or 40 boys can do a job in 12 days working for 8 hours a day, how many men working with 6 women and 2 boys would it take to do a job four times as big working for 5 hours a day for 12 days?\n\nA. 44\nB. 50\nC. 120\nD. 122\nE. 128\n\n20 men work for 12 days for 8 hours to do a job\nso number of men required to complete the work in 12 days working 5 hours = (20*8)\/5 =32\nas the work is 4 times as big as the previous one, so no of men required will be = 4*32 = 128\nNow we already have 6 women and 2 boys. we need to convert them to men to find the remaining no of men required\n\n24W=20M, or 6W=(20*6)\/24 = 5 Men\n\n40B=20M, or 2B = (20*2)\/40 = 1 Men\n\nSo number of men required = 128-(5+1) = 122\n\nOption D\n\nLast edited by niks18 on 26 Aug 2017, 07:56, edited 1 time in total.\n\nKudos [?]: 382 [3], given: 42\n\nManager\nJoined: 07 Jun 2017\nPosts: 109\n\nKudos [?]: 3 [0], given: 454\n\nRe: If 20 men or 24 women or 40 boys can do a job in 12 days working for 8\u00a0[#permalink]\n\n### Show Tags\n\n09 Aug 2017, 20:12\nniks18 wrote:\nBunuel wrote:\nIf 20 men or 24 women or 40 boys can do a job in 12 days working for 8 hours a day, how many men working with 6 women and 2 boys would it take to do a job four times as big working for 5 hours a day for 12 days?\n\nA. 44\nB. 50\nC. 120\nD. 122\nE. 128\n\n20 men work for 12 days for 8 hours to do a job\nso number of men required to complete the work in 12 days working 5 hours = (20*8)\/5 =32\nas the work is 4 times as big as the previous one, so no of men required will be = 4*32 = 128\nNo we already have 6 women and 2 boys. we need to convert them to men to find the remaining no of men required\n\n24W=20M, or 6W=(20*6)\/24 = 5 Men\n\n40B=20M, or 2B = (20*2)\/40 = 1 Men\n\nSo number of men required = 128-(5+1) = 122\n\nOption D\n\nThis solution is short and clear..\nI am trying to figure out how (20*8)\/5 =32..\nIs there a formula for this?\n\nKudos [?]: 3 [0], given: 454\n\nPS Forum Moderator\nJoined: 25 Feb 2013\nPosts: 817\n\nKudos [?]: 382 [0], given: 42\n\nLocation: India\nGPA: 3.82\nIf 20 men or 24 women or 40 boys can do a job in 12 days working for 8\u00a0[#permalink]\n\n### Show Tags\n\n09 Aug 2017, 20:19\npclawong wrote:\nniks18 wrote:\nBunuel wrote:\nIf 20 men or 24 women or 40 boys can do a job in 12 days working for 8 hours a day, how many men working with 6 women and 2 boys would it take to do a job four times as big working for 5 hours a day for 12 days?\n\nA. 44\nB. 50\nC. 120\nD. 122\nE. 128\n\n20 men work for 12 days for 8 hours to do a job\nso number of men required to complete the work in 12 days working 5 hours = (20*8)\/5 =32\nas the work is 4 times as big as the previous one, so no of men required will be = 4*32 = 128\nNo we already have 6 women and 2 boys. we need to convert them to men to find the remaining no of men required\n\n24W=20M, or 6W=(20*6)\/24 = 5 Men\n\n40B=20M, or 2B = (20*2)\/40 = 1 Men\n\nSo number of men required = 128-(5+1) = 122\n\nOption D\n\nThis solution is short and clear..\nI am trying to figure out how (20*8)\/5 =32..\nIs there a formula for this?\n\nHi pclawong\nas 12 days are constant for both the scenarios, let's not think about it.\n20 men take 8 hours to complete the job\nNo of men required if work is done only for 1 hour a day = 8*20 men(more men as hours worked are less)\nNo of men required if work is done for 5 hours a day = (8*20)\/5 (less men required now)\n\nKudos [?]: 382 [0], given: 42\n\nVP\nJoined: 26 Mar 2013\nPosts: 1375\n\nKudos [?]: 324 [0], given: 170\n\nRe: If 20 men or 24 women or 40 boys can do a job in 12 days working for 8\u00a0[#permalink]\n\n### Show Tags\n\n10 Aug 2017, 05:08\npclawong wrote:\nniks18 wrote:\nBunuel wrote:\nIf 20 men or 24 women or 40 boys can do a job in 12 days working for 8 hours a day, how many men working with 6 women and 2 boys would it take to do a job four times as big working for 5 hours a day for 12 days?\n\nA. 44\nB. 50\nC. 120\nD. 122\nE. 128\n\n20 men work for 12 days for 8 hours to do a job\nso number of men required to complete the work in 12 days working 5 hours = (20*8)\/5 =32\nas the work is 4 times as big as the previous one, so no of men required will be = 4*32 = 128\nNo we already have 6 women and 2 boys. we need to convert them to men to find the remaining no of men required\n\n24W=20M, or 6W=(20*6)\/24 = 5 Men\n\n40B=20M, or 2B = (20*2)\/40 = 1 Men\n\nSo number of men required = 128-(5+1) = 122\n\nOption D\n\nThis solution is short and clear..\nI am trying to figure out how (20*8)\/5 =32..\nIs there a formula for this?\n\nThe solution above stems from the basics\n\nWork = number of labor * Rate * Time\n\nBecause it is same work\n\nW1 = W2\n\nn1 * R1 * T1 = n2 * R2 * T2 ( in this case, R1 = R2, so cancel out as each man have same rate)\n\nn1 * T1 = n2 * T2\n\nFrom men's work numbers, apply in equation above\n\n20 * 8 * 12 = n2 * 5 * 12\n\nn2 = 20 * 8 \/ 5 ...............same result as described in previous post.\n\nI hope it helps\n\nKudos [?]: 324 [0], given: 170\n\nVeritas Prep GMAT Instructor\nJoined: 16 Oct 2010\nPosts: 7868\n\nKudos [?]: 18494 [2], given: 237\n\nLocation: Pune, India\nRe: If 20 men or 24 women or 40 boys can do a job in 12 days working for 8\u00a0[#permalink]\n\n### Show Tags\n\n10 Aug 2017, 09:26\n2\nKUDOS\nExpert's post\n2\nThis post was\nBOOKMARKED\nBunuel wrote:\nIf 20 men or 24 women or 40 boys can do a job in 12 days working for 8 hours a day, how many men working with 6 women and 2 boys would it take to do a job four times as big working for 5 hours a day for 12 days?\n\nA. 44\nB. 50\nC. 120\nD. 122\nE. 128\n\nTotal Number of men needed = 20 * 4 * (8\/5) = 128 (using the method discussed here: https:\/\/www.veritasprep.com\/blog\/2015\/1 ... made-easy\/)\n\nLet's see how many we already have. Convert all workforce to men.\n24 women = 20 men\n6 women = 5 men\n\n40 boys = 20 men\n2 boys = 1 man\n\nSo we already have 5 + 1 = 6 men.\nWe need another 128 - 6 = 122 men.\n_________________\n\nKarishma\nVeritas Prep | GMAT Instructor\nMy Blog\n\nGet started with Veritas Prep GMAT On Demand for \\$199\n\nVeritas Prep Reviews\n\nKudos [?]: 18494 [2], given: 237\n\nTarget Test Prep Representative\nStatus: Founder & CEO\nAffiliations: Target Test Prep\nJoined: 14 Oct 2015\nPosts: 2056\n\nKudos [?]: 1088 [0], given: 4\n\nLocation: United States (CA)\nRe: If 20 men or 24 women or 40 boys can do a job in 12 days working for 8\u00a0[#permalink]\n\n### Show Tags\n\n15 Aug 2017, 09:51\nBunuel wrote:\nIf 20 men or 24 women or 40 boys can do a job in 12 days working for 8 hours a day, how many men working with 6 women and 2 boys would it take to do a job four times as big working for 5 hours a day for 12 days?\n\nA. 44\nB. 50\nC. 120\nD. 122\nE. 128\n\nWe can let the job = 1, and we are given that the job can be completed in 12 x 8 = 96 hours by 20 men or 24 women or 40 boys. Let\u2019s determine the rate of 1 man per hour, which is (1\/20)\/96 = 1\/(20 x 96). Similarly, the rate of 1 woman per hour is (1\/24)\/96 = 1\/(24 x 96) and the rate of 1 boy per hour is (1\/40)\/96 = 1\/(40 x 96).\n\nNow we are given that another job is four times as big, and thus this new job = 4. We also are given that 6 women, 2 boys, and some number of men will be working on this job for 5 x 12 = 60 hours. We need to determine the number of men needed. We can let the number of men needed = n and create the following equation:\n\n6 * 1\/(24 x 96) * 60 + 2 * 1\/(40 x 96) * 60 + n * 1\/(20 x 96) * 60 = 4\n\n360\/(24 x 96) + 120\/(40 x 96) + 60n\/(20 x 96) = 4\n\nMultiplying the entire equation by 96, we have:\n\n360\/24 + 120\/40 + 60n\/20 = 384\n\n15 + 3 + 3n = 384\n\n3n = 366\n\nn = 122\n\n_________________\n\nScott Woodbury-Stewart\nFounder and CEO\n\nGMAT Quant Self-Study Course\n500+ lessons 3000+ practice problems 800+ HD solutions\n\nKudos [?]: 1088 [0], given: 4\n\nIntern\nJoined: 21 Sep 2016\nPosts: 28\n\nKudos [?]: 2 [0], given: 261\n\nRe: If 20 men or 24 women or 40 boys can do a job in 12 days working for 8\u00a0[#permalink]\n\n### Show Tags\n\n18 Sep 2017, 23:33\nBunuel wrote:\nIf 20 men or 24 women or 40 boys can do a job in 12 days working for 8 hours a day, how many men working with 6 women and 2 boys would it take to do a job four times as big working for 5 hours a day for 12 days?\n\nA. 44\nB. 50\nC. 120\nD. 122\nE. 128\n\nWe know the daily rate of:\n\nA) Man is 1\/(20*12*8) = 1\/1920\nB) Woman is 1\/2304\nC) Boy is 1\/3840\n\nSo, 1 man equals to 2 boys and 6 women produce as much as 5 men.\n\nSo, we want to know how many men should we add to the 6 men (6 women plus 2 boys).\n\nNow, we know that 20 men need to work 1920 hours to complete one job, working 8h\/day for 12 days.\n\nSince the second job is 4 times as big as the first one, it will need 1920*4 = 7680 hours from men.\n\nSo, we should divide that amount of hours per 60 hours (the amount of hours each person can work in the second job, as in, 5h\/day x 12 days).\n\nWe'll have: 7680\/60 = 128 men.\n\nSo, we need 128 men in total. But since we already have 6 women and 2 boys already working (or 6 men), there will be 128-122 men working alongside them.\n\nKudos [?]: 2 [0], given: 261\n\nRe: If 20 men or 24 women or 40 boys can do a job in 12 days working for 8 \u00a0 [#permalink] 18 Sep 2017, 23:33\nDisplay posts from previous: Sort by","date":"2018-01-21 07:06:14","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 1, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.4219811260700226, \"perplexity\": 1940.507707398205}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2018-05\/segments\/1516084890314.60\/warc\/CC-MAIN-20180121060717-20180121080717-00532.warc.gz\"}"} | null | null |
\section{Introduction}
Driven interacting many-particle systems \cite{Liggett} have been
of considerable interest in the past decades due to their rich
transport properties, especially in lower dimensions. The zero
range process (ZRP), a lattice gas model without any hardcore exclusions, is perhaps the simplest of them, which exhibit nontrivial static and dynamic properties in the steady state. The ZRP was introduced \cite{Spitzer70} as a mathematical model for
interacting diffusing particles and, since then, has found
applications in different branches of science \cite{ZRPRev,
ZRP_Evans}, such as in describing phase separation criterion in
driven lattice gases \cite{Criteria}, network re-wiring
\cite{Network}, statics and dynamics of extended objects
\cite{ExtendObj,Bijoy}, etc. Interestingly, the ZRP shows a
condensation transition for some specific choices of particle
hop rates for which, when the density becomes larger than a
critical density $\rho_c$, a macroscopic number of particles
accumulate on a single lattice site - representing
a classical real-space analogue of the Bose-Einstein condensation.
The ZRP has been generalized to multi-species models
\cite{Mult},the misanthrope process \cite{Beyond_ZRP}, urn models
\cite{Urn},the inclusion process \cite{Inclusion} and inhomogeneous hop rates \cite{Inhomo}, etc.
In the ZRP, the particles hop stochastically to one of the nearest neighbours with a rate that depends only on the number of particles on the departure site. As a consequence, the ZRP has a
factorized steady state (FSS), which is amenable to exact analytic
studies. However, when the hop rate depends on the neighbouring
sites, the steady state does not factorize in general
\cite{PFSS_Evans, EvansNew}. In such situations, one may naturally expect a cluster-factorized steady state (CFSS), a straightforward generalization of the factorized steady state (FSS), where the steady
state weight is a product of cluster-weight functions (see Eq.
(\ref{eq:CFSS})) of several variables, i.e., the occupation numbers
at {\it two} or more consecutive sites.
In this paper, we study a class of nonequilibrium lattice models where particles hop in a particular direction, say from a site to its right nearest neighbour, where hop rates not only depend on the occupation of the departure site but also
on the occupation of all of its neighbours within a range $R$;
hereafter, we refer to this process as the finite range process
(FRP). We demonstrate that, in one spatial dimension, one can have a CFSS for various specific choices of hop rates; what we mean by the CFSS here is that the steady state
probability weight can be written as a product of functions of
$R+1$ variables, each of them being an occupation number in the
cluster of $R$ consecutive sites.
A special case of the CFSS with $R=1$, called the pair-factorized
steady state (PFSS), was recently proposed and studied in \cite{PFSS_Evans} where it was shown that, for a particular class of PFSS,
the system can also undergo a condensation transition. Later, the PFSS has been found in continuous mass-transfer models \cite{PFSS_Mass, Bertin}, in systems with open boundaries \cite{PFSS_Open} and in random graphs
\cite{PFSS_Graph}, etc. However, non-trivial spatial structure, which is not present in a FSS, has not been explored before.
We show that, for a broad class of systems having a CFSS with any $R$, there exists a finite dimensional transfer-matrix representation of the steady state. Being finite dimensional, these matrices are quite convenient to manipulate and help in exact calculations of spatial correlation functions of any order. Moreover, we propose a sufficient criterion for the hop rates that can give rise to condensation transition in FRP in general. Surprisingly, we find that a small perturbation to an FSS could destroy condensation transition, if any.
The paper is organized as follows. In Sec. \ref{sec:model},
we discuss the model and its steady state in general for the $(R+1)$-
cluster. In Sec. \ref{sec:2clust}, we formulate a transfer
matrix method to calculate the correlation functions for $2$-
cluster and then, in Sec. \ref{sec:gen}, we generalize the
matrix formulation to $R>1$; continuous mass transfer models are
also discussed in this section. Some useful applications of the FRP
in the context of steady state thermodynamics for systems with
short-ranged correlations, is studied in Sec. \ref{sec:app}.
Finally, Sec. \ref{sec:conclude} provides conclusions, followed
by open issues and discussions.
\section{Model\label{sec:model}}
The model (see Fig. $1$) is defined on a one dimensional periodic lattice with
sites labeled by $i=1,2, \dots L.$ Each site $i$ has a non-
negative integer variable $n_i$ representing the number of
particles at that site (for a vacant site $n_i=0$). Particle from
any randomly chosen site $i$ can hop to one of its nearest
neighbours, say the right neighbour, with a rate that depends on
the number of particles at all the sites which are
within a range $R$ with respect to the departure site:
\begin{eqnarray}
(\dots, n_{i-1}, n_i, n_{i+1},\dots )&&\longrightarrow (\dots, n_{i-1}, n_i-1, n_{i+1}+1,\dots) \cr
{\rm with} ~{\mathbf{rate }}&&~ u(n_{i-R}, \dots, n_i, \dots, n_{i+R}).~~~
\end{eqnarray}
Clearly the total number of particles $N= \sum_i n_i$, or the density $\rho= N/L$, is conserved by this dynamics.
\begin{figure}[h]
\centering \includegraphics[width=8 cm]{cartoon.eps}
\caption{(Color online) In the one dimensional finite range process
(FRP) a particle hop from a site $i$ to its right nearest neighbour with a
rate that depends on occupation of site $i$ (here $n_i=3$) and
all its neighbours within a range $R$. The lattice model, for a certain
hop rate, can have an $(R+1)$-cluster-factorized steady state.}
\label{fig:dyn}
\end{figure}
\\
For $R=0,$ this model is identical to the zero range process (ZRP)
\cite{ZRP_Evans} with hop rate $u(n_i),$ an exactly solvable non-
equilibrium model that evolves to a factorized steady state (FSS)
\be
P(\{n_i\}) \propto \prod_{i=1}^L f(n_i) \delta( \sum_i n_i -N), \label{eq:FSS}
\ee
with $f(n) = \prod_{m=1}^n u(m)^{-1}.$
The process considered in this paper is a generalized version
of the ZRP and hereafter we refer to it as finite range process
(FRP).
For $R>0,$ the steady state of FRP in general cannot have an
FSS as there are nonzero spatial correlations; however, there can
be exceptions in specific cases. We provide explicit proof, in
the Appendix, that, for $R=1,$ the factorized steady state can be
achieved only for two cases - when the hop rate is $u(k,m,n)
= v(m)$ or when $u(k,m,n) = w(m,n).$ The first case is the
ZRP and the second one, where hop rate depends on both number
of particles in both departure and arrival sites, is known as the
misanthrope process (MP; see \cite{Beyond_ZRP} for a review).
Since a FRP with $R>1$ includes $R=1$ as a special case, one
expects that, except for the ZRP and the MP, there cannot be a
factorized steady state (FSS) for these classes of systems.
For the FRP, we first try whether a $(R+1)-$ cluster-factorized
form,
\begin{equation}
P(\{n_i\}) \propto \prod_{i=1}^L g(n_i, n_{i+1},\dots n_{i+R}) \delta( \sum_i n_i -N) \label{eq:CFSS}
\end{equation}
with cluster-weight function $g$ of $R+1$ occupation variables,
can be a steady state weight for Master equation
\bea
\frac{d}{dt} P(\{n_i\}) &=& \sum_{i=1}^L u(n_{i-R},\dots,n_i,\dots,n_{i+R}) P(\{ n_i\}) \cr
& - & \sum_{i=1}^Lu(n_{i-R},\dots,n_i+1,n_{i+1}-1,..,n_{i+R}) \cr
&& ~~~~~~~ \times P(\dots,n_{i-1}+1,n_{i}-1,\dots).
\label{eq:master}
\eea
Now, one can verify that a cluster-factorized form of steady state, as in Eq. (\ref{eq:CFSS}), is indeed possible
when the hop rate at site $i$ satisfies the following condition
\begin{eqnarray}
u(n_{i-R}, \dots, n_{i}, \dots,n_{i+R}) ~~~~~~~~~~~~~~~~~~\cr
~~~ ~~~= \prod_{k=0}^R \frac{g(\bar n_{i-R+k},\bar n_{i-R+1+k}, \dots, \bar n_{i+k})}{ g(n_{i-R+k},n_{i-R+1+k}, \dots, n_{i+k})}, \label{eq:Z20}
\label{eq:gen_hop_rate}
\end{eqnarray}
where $ \bar n_{j}=n_{j}-\delta_{ji} $.
A simple way to prove this is to construct a pair-wise
balance $-$ for every hop that takes configuration $C \to
C'$ there is a suitable and unique configuration $C''$ such that
$P(C) W(C\to C') = P(C'') W(C''\to C).$ For any configuration
$C= \{\dots, n_{i-1}, n_i,n_{i+1},..\}$, a particle hopping from
site $i$ can be balanced by taking $C'' = \{\dots,
n_{i-1}+1, n_i-1,n_{i+1},..\}$ with hopping from $i-1.$ Equation
(\ref{eq:gen_hop_rate}) is important as it says that any
desired cluster-factorized state can be obtained in FRP by a
choosing a suitable $R$-range hop rate $u(n_{i-R}, \dots, n_{i},
\dots,u_{i+R}).$
In the rest of the paper, we discuss various features of the
cluster-factorized steady state and their applications.
\section{2-clusters : Pair factorized steady state (PFSS) \label{sec:2clust}}
Let us start with $R=1$, for which the steady state is
factorized as product of $2$-site clusters, commonly known as the
pair-factorized steady state (PFSS). In this case, particles
hop from a site $i$ to $i+1$ with rate $u(n_{i-1}, n_i,n_{i+1})$ that depends
on the occupation of departure site and its neighbours. To have a
pair-factorized steady state of the form
\begin{equation}
P(\{n_i\}) = \frac{1}{ Z_{L,N}} \prod_{i=1}^L g(n_i, n_{i+1}) \delta( \sum_i n_i -N) \label{eq:PFSS}
\end{equation}
with a canonical partition function
\bea
Z_{L,N} = \sum_{\{n_i\}} \prod_{i=1}^L g(n_i, n_{i+1}) \delta \left( \sum_i n_i -N \right), \nonumber
\eea
the hop rate must satisfy Eq. (\ref{eq:Z20}) with $R=1,$
\begin{equation}
u(n_{i-1}, n_{i}, n_{i+1}) = \frac{g(n_{i-1}, n_{i}-1)}{g(n_{i-1}, n_{i}) } \frac{g(n_{i}-1, n_{i+1})}{g(n_{i}, n_{i+1})}. \label{eq:Z21}
\end{equation}
Unlike the FSS, the PFSS inherently generates spatial correlations
and, like the FSS, it can lead to real-space condensation for
certain hop rate \cite{PFSS_Evans}. This study has been later generalized
on arbitrary graphs \cite{PFSS_Graph}, open boundaries
\cite{PFSS_Open} and for studying mass transport processes and
condensation transition therein for discrete (particle) as well as
continuous mass \cite{PFSS_Mass}, etc. None of these studies, however,
attempted to calculate the spatial correlations in these
systems. In fact, the presence of spatial correlations can
change the nature of transitions by creating spatially extended condensates
with or without tunable shapes \cite{PFSS_Shape}.
To calculate spatial correlation functions we use the transfer
matrix formulation which is possible for a large class of systems having a CFSS.
For the purpose of illustration we mainly discuss this approach elaborately for
the PFSS. Since the PFSS with any arbitrary cluster-weight function $g(n_i,n_{i+1})$ can be
obtained from a suitable hop rate $u(n_{i-1}, n_i,n_{i+1})$ [as in Eq. (\ref{eq:Z21})],
we rather focus on the functional form of $g(n_i,n_{i+1})$, not
on the hop rate. In fact, any arbitrary function $g(n_{i},n_{i+1})$
is an element
of the infinite dimensional matrix
\be
G =\sum_{n=0}^\infty \sum_{n'=0}^\infty g(n,n') |n\ra \la n'| \label{eq:G}
\ee
where $\{|n\ra\}$ are the standard infinite dimensional basis vectors
which satisfy a completeness relation $\la n|n'\ra = \delta_{nn'}$.
Then, in the grand canonical ensemble (GCE), where a fugacity $z$ controls the density $\rho$, the partition sum
can be written as
\be
\cZ_L(z) = \sum_{N=0}^\infty Z_{L,N} z^N =Tr[T^L]
\ee
where the transfer matrix $T$ has element $\langle n |T|n' \rangle = z^{(n+n')/2} g(n,n')$. In the thermodynamic limit $Z_L(z)\simeq \lambda_{max}^L$ (when $\lambda_{max},$ the largest eigenvalue
of $T$ is non-degenerate).
Once we know the grand partition sum, we can calculate various observables; for example, all the moments for occupation number
$n$ at a site,
\bea
\langle n^{k} \rangle = \frac{1}{L}\frac{1}{\cZ_L(z)} \left(z\frac{d}
{dz} \right)^{k} \cZ_L(z).
\eea
For $k=1$, we get density of the system $\rho= \langle n \rangle
=\frac{1}{L}\frac{d}{dz} \ln \cZ_L(z);$ by inverting this density-fugacity relation, one can express other observables as a function of $\rho.$
This matrix formulation is quite general and works for any
form of weight function $g(n_{i},n_{i+1})$; however managing infinite dimensional matrices is a challenging task.
In the following, we show that, for a large class of weight functions, one can have a finite dimensional representation which, in some cases, can even be extended to $R>1.$
Let us consider a weight function which has the following form
\be
g(n_i,n_i+1)= \sum_{\kappa=0}^K a_\kappa (n_i) b_\kappa (n_{i+1})\label{eq:g_gen},
\ee
where $a_\kappa(n), b_\kappa(n)$ are arbitrary functions, not necessarily analytic.
It is evident that $g(n_i,n_{i+1})$ can be
written as an inner product of two $(K+1)$-dimensional vectors,
\bea
g(n_i,n_{i+1}) = \langle\alpha(n_{i})| \beta(n_{i+1})\rangle,
\eea
where
\bea
\langle \alpha(n)|& =& ( a_0(n), a_1(n),\dots, a_K(n))\cr
\langle \beta(n)| &=& ( b_0(n), b_1(n),\dots, b_K(n)). \label{eq:ab}
\eea
Then the partition sum in grand canonical ensemble
is $\cZ_L(z) = Tr[T(z)^L] $ with
\be T (z) = \sum_{n=0}^{\infty} z^n |\beta(n)\rangle \langle\alpha(n)|
\label{eq:Z}
\ee
a $(K+1)$-dimensional matrix. Now the partition sum and the stationary
correlation functions can be calculated easily.
To illustrate this, let us consider a simple example by setting
$K=1,$ $b_0(.) =1 =a_1(.),$ and renaming functions $a_0(.), b_1(.)$ as $f_0(.), f_1(.)$ respectively.
The weight function is now,
\be
g(n_i,n_{i+1})= f_0(n_i) + f_1(n_{i+1}) \label{eq:Z1}.
\ee
which we refer to as {\it sum-form.} This particular choice, {\it i.e.,} a pair-factorized steady state with a weight function in sum-form, does not lead to condensation transition, which we discuss later in Sec. \ref{sec:IV.B}. Also, in Sec. \ref{sec:condensation}, we consider a general case of Eq. (\ref{eq:g_gen}), which gives condensation transition, and we develop a possible criterion for the transition.
For any functional form of $f_0(n)$ and $f_1(n)$
we always have an infinite dimensional representation given by Eq. (\ref{eq:G}). However, interestingly in this case, we can do away with the infinite dimensional representation and get a simple $2$-dimensional representation by taking,
\bea
\langle\alpha(n) |= ( f_0(n),1) ~~{ \rm and }~~ \langle\beta(n) |= (1,f_1(n)).
\eea
The partition sum in GCE is then ${\cal Z} = Tr[T(z)^L],$ where
\be
T(z)=\sum_{n=0}^\infty z^n
\left( \begin{array}{cc}
f_0(n) & 1 \\
f_0(n) f_1(n) & f_1(n) \\
\end{array}
\right). \label{eq:Z2}
\ee
To see how the spatial correlation functions can be obtained,
let us take a specific form of the functions $f_0(.)$ and $f_1(.),$
\be
g(n_i,n_{i+1})=\frac{\bar q}{ (n_i+1)^\nu} + \frac{q}{ (n_{i+1}+1)^\nu}, \label{eq:Z5}
\ee
where parameters $\nu$ and $0 \leq q \leq 1$ tune the hop
rate of particles and $\bar q= 1-q,$ corresponding to
$f_0(n)/{\bar q} = f_1(n)/q = (n+1)^{-\nu}.$ In this case, the
desired hop rate, for which the PFSS with weight-function as in Eq.
\ref{eq:Z5} is realized, is given by
\bea
&&u(n_{i-1},n_i,n_{i+1})=\left( 1+\frac{1}{n_i} \right)^{2\nu} \left[ \frac{\bar q n_{i}^\nu +q(n_{i-1}+1)^\nu}{\bar q (n_{i}+1)^\nu +q(n_{i-1}+1)^\nu} \right] \cr
&&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\times \left[ \frac{\bar q(n_{i+1}+1)^\nu +q n_{i}^\nu}{\bar q (n_{i+1}+1)^\nu +q(n_{i}+1)^\nu}\right].\nonumber
\eea
In the extreme limits $q=0$ and $q=1,$ the model reduces to zero range process (details will be discussed in Sec. \ref{sec:app}).
The transfer matrix , following Eq. (\ref{eq:Z2}), becomes
\be
T(z)=\frac{1}{z}\left( \begin{array}{cc}
\bar qLi_{\nu}(z) &q\bar q \frac{z }{1-z} \\
Li_{2\nu}(z) & q Li_{\nu}(z) \\
\end{array}
\right) \label{eq:Z7}
\ee
where $Li_{\nu}(z)$ are the Polylog functions. The eigenvalues
of $T$ are
\be
\lambda_\pm=\frac{Li_{\nu}(z)}{2z}\left(1\pm
\sqrt{(q- \bar q)^2 +\frac{4q\bar q z Li_{2\nu}(z)}{(1-z) Li_{\nu}(z)^2}}\right). \label{eq:Z8}
\ee
The partition function $ \cZ_L (z) = \lambda_+^L + \lambda_-^L$
in the thermodynamic limit $(L\rightarrow\infty)$ becomes
$\cZ_L (z) \simeq \lambda_+^L$ and thus the density
\be
\rho(z)=z \frac{d}{dz} \mathrm{ln} \lambda_+. \label{eq:Z9}
\ee
Throughout the paper, we calculate observables only in the
thermodynamic limit. Let us consider $q=\frac{1}{2}$ and $\nu
=-1$ (results for different $q$ and $\nu$ are discussed in
Sec. \ref{sec:app}); here $\lambda_\pm = \frac{1}{2}{(1\pm \sqrt{1+z})}/{
(1-z)^2}$ and the density is
\be
\rho=\frac{2}{1-z}-\frac{1}{2 \sqrt{1+z}}-\frac{3}{2}.
\ee
Now we proceed to calculate the correlation functions, first the
two-point correlation function and later the higher order. The two point
correlation function is defined by
\bea
C(r)=\langle n_i n_{i+r} \rangle - \langle n_i \rangle \langle n_{i+r} \rangle. \label{eq:corfor}
\eea
For $r>0$ we have
\bea
C(r)&=& \frac{Tr [T'T^{r-1}T'T^{L-r-1}]}{ Tr[T^L]} -\rho^2 \label{eq:2corr}
\eea
where $T'= dT/d(\ln z).$
For $q=\frac{1}{2}$ and $\nu=-1$,
we get
\be
C(r) = \rho^2 \frac{z(3+z)^2}{4(1+z)(1-z)^2} e^{-r/\xi}
\ee
with $\xi^{-1}= |\ln \frac{\lambda_-}{\lambda_+}| = |\ln
\frac{1-\sqrt{1+z}}{1+\sqrt{1-z}}|$ being the inverse correlation
length. The correlation function for $r=0$ is nothing but the
variance $\sigma^2(\rho)$ of single-site occupation variable
$n_i$, i.e.,
\bea
C(0)\equiv \sigma^2 (\rho)= \langle n^2_i \rangle - \langle n_i \rangle^{2}= \frac{Tr [T''T^{L-1}]}{ Tr[T^L]} -\rho^2
\eea
where $T''=d^2T/d(\ln z)^2$
and, again for $q=\frac{1}{2}, \nu=-1$,
\be
C(0) = \frac{z}{4(1-z)^2} \left[ \frac{z^2+14z+17}{(1+z)}-\frac{8}{\sqrt{1+z}} \right] .
\ee
Now, we turn our attention to higher order correlation functions.
The $3$-point correlation function, for example, is defined as
\bea
{\cal C}(r_1,r_2)&=&\langle n_i n_{i+r_1} n_{i+r_1+r_2}
\rangle - \langle n_i \rangle \langle n_{i+r_1} \rangle \langle n_{i+r_1+r_2} \rangle\nonumber
\eea
which, in terms of transfer matrix, can be evaluated from the
expression
\bea
{\cal C}(r_1,r_2)= \frac{Tr [T'T^{r_1-1} T'T^{r_2-1} T'T^{L-r-1}]}{ Tr[T^L]} -\rho^3.
\eea
We find that the three-point correlation function can be written in
terms of the two-point correlation functions as
\be
{\cal C} (r_1,r_2)=\rho \left[ C(r_1)+C(r_2) - B(z) C(r_1)C(r_2) \right]
\ee
where $B(z)$ also depend on the parameters $q$ and $\nu$; for
$q=\frac{1}{2}$ and $\nu=-1$, we get $B(z) =1 + 8z(1+z) /(3+z)^2.$
In a similar way, one can calculate all the higher order
correlation functions exactly.
To conclude, when the weight function $g(n_i,n_{i+1})$ is a sum of two
functions as in Eq. (\ref{eq:Z1}), the correlation length
$\xi = |\ln\frac{ \lambda_- }{ \lambda _+}|$ remains finite
at any density as $\lambda_- < \lambda_+$ for any choice of $q$ and
$\nu.$
\section{Generalizations\label{sec:gen}}
\subsection{3-clusters and general (R+1)-clusters}
In this section we consider some specific models of FRP
with $R>1$ which give rise to $(R+1)$-cluster-factorized steady state. Corresponding partition function in the
grand canonical ensemble would require contraction of a
tensor product which is usually a hard task \cite{Tensor}.
Our aim here would be to obtain, if possible, a matrix formulation that can accommodate some cluster-
factorized steady states for any $R>1$. For $R=2$ we have a $3$-site cluster factorized steady state,
\bea
P(\{ n_i\}) = \prod_{i=1}^L g(n_i, n_{i+1}, n_{i+2}) \delta\left(\sum_{i=1}^L n_i -N\right).\nonumber
\eea
For illustration we consider a cluster-weight function,
\be
g(n_{i},n_{i+1},\dots n_{i+R})=\sum_{\kappa =0}^R f_\kappa(n_{i + \kappa}) \label{eq:Z3}
\ee
which is a simple generalization of the sum-form given in Eq. (\ref{eq:Z1}).
We will now show that a grand partition function of a
finite range process which has a $(R+1)$-cluster-factorized steady state with a
weight function given by Eq. (\ref{eq:Z3}) can be written
as $\cZ_L(z) =Tr[T^L]$ where $z$ is the fugacity and $T$ is a
$2^{R}$-dimensional transfer matrix. Since we intend to obtain
the transfer matrix for iteratively, let us rewrite the
transfer matrix given by Eq. (\ref{eq:Z2}) for $R=1$ in a
convenient form,
\be
T_1(z)=\sum_{n=0}^\infty z^n {\bf F}_1(n); ~
{\bf F}_1(n)=
\left( \begin{array}{cc}
f_0(n) & 1 \\
f_0(n) f_1(n) & f_1(n) \\
\end{array}
\right) \label{eq:F1}
\ee
In a similar way, we extend to $R>1$ and write
$T_{R}=\sum_{n=0}^\infty z^n {\bf F}_R(n)$ where the $2^{R}$-
dimensional matrix can be written as
\be
{\bf F}_{R}= \left( \begin{array}{cc}
{\bf F}_{R-1} & A_{R-1}{\bf F}_{R-1} \\
f_R {\bf F}_{R-1} & f_R A_{R-1} {\bf F}_{R-1} \\
\end{array}\right), \label{eq:FR}
\ee
using a constant matrix
$$A_{R}=\left( \begin{array}{cc}
0 & 0 \\
I_{2^{R-1}} & 0 \\
\end{array}
\right),
$$ \\
where $I_{2^{R-1}}$ is a $2^{R-1}$-dimensional identity matrix.
For $R=0$, we take $A_0 =1.$ Since $R=0$ corresponds to the ZRP
which has a factorized steady state, we have ${\bf F}_0(n)=f_0(n),$
which is a scalar. Clearly ${\bf F}_1$ in Eq. (\ref{eq:F1})
satisfy Eq. (\ref{eq:FR}). A little more algebra would show
that the transfer matrix for $R=2$ is
\be
T_2=\sum_{n} z^n \left( \begin{array}{cc}
{\bf F}_1 & A_1 {\bf F}_1 \\
f_2{\bf F}_1 & f_2 A_1 {\bf F}_1 \\
\end{array}
\right)=\sum_{n} z^n {\bf F}_2(n). \label{eq:Z4}
\ee
From the transfer matrix, one can, in principle, calculate the
expectation value of any desired observable. We will not
discuss further the finite range process $R>1$; the finite
dimensional transfer matrix is expected to generate spatial
correlations which was absent in the ZRP. We discuss some of the
models in details which undergo condensation transitions
(see Sec. \ref{sec:app}).
\subsection{ Continuous mass model }
Until now, we have studied CFSS on a one dimensional lattice
with each site having a discrete variable, called the
occupation variables or number of particles. The model and the
matrix formulation can be extended, without any particular
difficulty, to systems with continuous mass $m$. As an example,
let us consider
\be
g(m_{i},m_{i+1},m_{i+2})=m_i+m_{i+1}+m_{i+2}.
\ee
A $3$-cluster-factorized steady state with the above weight-function
can be obtained when $\epsilon$ amount of mass is transferred
from site $i$ to $i+1$ with the rate
\bea
&& u(m_{i-2},m_{i-1},m_i,m_{i+1},m_{i+2}) \cr
&&~~~~= \prod_{k=0}^2 \left[1- g(m_{i-2+k},m_{i-1+k},m_{i+k} )^{-1} \right].
\eea
For small $\epsilon$, the model is equivalent to a discrete
model where mass is measured in units of $\epsilon.$ In fact, the
residual mass (actual mass modulo $\epsilon$) at any site does not
change during evolution. The residual masses, each being smaller than a pre-defined
value $\epsilon$ which can be made arbitrary small,
does not contribute to the asymptotic form of the hop rate.
Thus we would obtain a transfer matrix $T_2$ discussed in the
previous section, with $f_{0,1,2}(m) = m,$ but the sum
$\sum_m$ will now be replaced by an integral $\int dm.$
Defining, a chemical potential $\mu$ (where $z=e^{\mu}$), we
get the transfer matrix, as in Eq. (\ref{eq:Z4}),
\be
T(\mu)=\frac{1}{\mu^2} \left( \begin{array}{cccc}
1 & \mu & 0 & 0 \\
\frac{2}{\mu} & 1 & 1 & \mu \\
\frac{2}{\mu} & 1 & 0 & 0 \\
\frac{6}{\mu^2} & \frac{2}{\mu} & \frac{2}{\mu} & 1 \\
\end{array}
\right).
\ee
This matrix has eigenvalues $\frac{1}{\mu^2}\{ \lambda, \lambda_1
e^{\pm i \theta}, \lambda_2\},$ where $\lambda$ (the
largest eigenvalue), $\lambda_1$, $\theta$ and $\lambda_2$ are
independent of $\mu$, and their approximate numerical values are
$\lambda \approx 3.86841$, $\lambda_1 \approx 1.10465,$ $\theta
\approx 1.87254$ and $\lambda_2 \approx -0.21184.$ In the
thermodynamic limit, the partition function is $\cZ_L = \left(
\lambda/\mu^2 \right)^L,$ and density $\rho=-{2}/{\mu}.$ The
two-point correlation function for $r>0$ is
\bea
C(r)&=&\langle m_i m_{i+r} \rangle -\rho^2 \cr
&=&\rho^2\left[c_{2} \left(\frac{\lambda_2}{\lambda}\right)^{r}+2 c_{1}\left(\frac{\lambda_1}{\lambda}\right)^{r} \cos(r \theta+\alpha) \right]
\eea
where, $c_1=0.3380 $, $c_2= -0.0375$ and $\alpha=0.1804.$ And, for $r=0$, the correlation (actually $\sigma^2$) is
$C(0)=\langle m^2 \rangle -\rho^2= 0.6704 \rho^2.
\section{Applications \label{sec:app}}
\subsection{Condensation transition \label{sec:condensation}}
One important feature in these simple one dimensional models is
that they can exhibit condensation transition at a finite
density when one or more parameters in the rate functions are
tuned. To demonstrate the possibility of a condensation
transition in the CFSS, for any $R$, we consider the weight of
$(R+1)$-cluster to be,
\be
g(n_i,n_{i+1},...,n_{i+R})= \frac{\left[q+ \sum_{j=0}^{R} n_{i+j} \right]^K}{(n_i+1)^\nu}, \label{eq:g_condensation}
\ee
where $K, \nu$ and $q$ are positive and $K$ is an integer.
This steady state
weight can be generated from a hop rate given by Eq.
(\ref{eq:gen_hop_rate}),
\be
u=\left( 1+\frac{1}{n_i} \right)^\nu \left[ \prod_{k=0}^R \left(1-\frac{1}{q+\sum_{j=0}^{R} n_{i-j+k}} \right) \right]^K . \label{eq:uRcondensation}
\ee
{\it Case with $R= 1$ (PFSS). $-$} We first consider $R=1.$ It is easy to see that for any $K,$ the weight function Eq. (\ref{eq:g_condensation})
can be expressed as Eq. (\ref{eq:g_gen}) with suitable choice of $a_\kappa(n)$ and $b_\kappa(n)$ where $\kappa$ varies from $0$ to $K,$ leading to a $(K+1)$ dimensional transition matrix. We further set the parameters $K=1=q;$ this gives rise to a PFSS, as in Eq. (\ref{eq:PFSS}), with $g(n_i,,n_{i+1}) = (n_i+n_{i+1}+1)/(n_{i}+1)^\nu$, which can be realized when a particle hops out from a site $i$ (to the right neighbour), having $n_i>0$ particles, with the following rate
\be
u=\left(1+\frac{1}{n_i}\right)^\nu \frac{n_i +n_{i-1}}{ 1+n_i +n_{i-1} } \frac{n_i +n_{i+1}}{ 1+n_i +n_{i+1} } \label{eq:Z101}.
\ee
For this case, we can obtain exact results following the
matrix formulation developed here. First we write $g(m,n)=
\langle \alpha(m)| \beta(n) \rangle$ where $ \langle \alpha(m) |
=\left( (m+1)^{-\nu}, (m+1)^{1-\nu} \right)$,
$ \langle\beta(n) |=\left(n,1 \right)$. Thus the grand
partition function can be written as $Z(z)= Tr(T^L)$ with
\bea
T = \sum_{n=0}^\infty |\beta(n) \rangle \langle \alpha(n)| z^n = \frac{1}{z}
\left( \begin{array}{cc}
Li_{\nu-1}(z) & Li_{\nu-2}(z) \\
Li_{\nu}(z)& Li_{\nu-1}(z)
\end{array}
\right)
\nonumber
\eea
The eigenvalues of $T$ are
\bea
\lambda_\pm(z) = \frac{1}{z} \left( Li_{\nu-1}(z) \pm\sqrt{ Li_{\nu}(z) Li_{\nu-2}(z)}\right), \nonumber
\eea
which leads to the density-fugacity relation $\rho(z)= z \lambda_+'(z)/\lambda_+(z)$ and
the critical density $\rho_c =\lim\limits_{z\to1} \rho(z).$
It turns out that for $\nu\le 4,$ $\rho_c$ diverges
$-$ indicating a fluid phase for any density. For $\nu>4$ we get,
\bea
\rho_c&=& \frac{\xi_1(\nu-1) - 2\xi_2(\nu)+\xi_3(\nu)}{2 \xi_2(\nu)+ 2 \zeta(\nu-1)\sqrt{\xi_2(\nu)} }
+\frac{\zeta(\nu-2)- \zeta(\nu-1) }{ \sqrt{\xi_2(\nu)}+ \zeta(\nu-1) }\nonumber
\eea
where $\xi_k(\nu) = \zeta(\nu) \zeta(\nu-k)$ and $\zeta(\nu)$
are Riemann zeta functions. Thus, for $\nu>4$ we have a
condensate when density exceeds this critical value. Unlike
the ZRP, where particles at different sites are not correlated,
here we have non-vanishing correlation that extends up to a length scale
$\xi(z)= |\ln\frac{ \lambda_-(z)}{\lambda_+(z)}|^{-1}$ which is finite throughout.
{\it Case with $R\ge 2$ (CFSS). $-$}
It is straightforward to extend the matrix formalism to $R>1$ when
$K=1.$ First, let us take $\nu =0.$ In this case, the weight function $g$ takes a sum-form
given by Eq. (\ref{eq:Z3}), for which we have already constructed a general transfer-matrix. For $\nu>0,$ the dimension of the transfer matrix remains the
same as in $\nu=0;$ it is only that each element of ${\bf F}_R$ in
Eq. (\ref{eq:Z4}) will be multiplied by an extra factor
$(n_i+1)^{-\nu}.$ We omit the exact analytic expressions of the density-fugacity relation and the critical density - the calculations
are straightforward but the expressions are very long. Only the numerical
values of critical density are tabulated in Table \ref{tbl} for different parameters.
\begin{table}
\caption{ Critical density $\rho_c$ for $K=1.$ }
\begin{tabular}{cccc} \hline \hline
$~~~~~~$& $~~~q=1~~~$ & $~~~q=1~~~$ & $~~~q=2~~~$ \cr
$~~~\nu~~~$& $~~~R=1~~~$ & $~~~R=2~~~$ & $~~~R=1~~~$ \cr
\hline
\hline
5 & 0.3254 & $ \infty $ & 0.1591 \cr
6 & 0.1054 &0.2773 & 0.0544 \cr
7 & 0.0429 &0.0981 & 0.0228 \cr
\hline
\end{tabular}
\label{tbl}
\end{table}
{\it Criterion for condensation transition.$-$}
For the ZRP, it is well known that, provided the hop rate $u_0(n)$ has an asymptotic form
\be
u_0(n) =1+ \frac{b}{ n^{\sigma}} + \dots \label{eq:criteria}
\ee
condensation occurs at a finite density, when $\sigma <1,$ or
when $\sigma =1$ but $b>2.$ This criterion can be extended to
any other system (without any constraint on occupation number) when
the steady state has a factorized form (\ref{eq:FSS}); one needs
to consider and effective rate function $u_0(n) \equiv
f(n-1)/f(n)$ and find its asymptotic form. This criterion
determines whether a model can undergo a condensation
transition and helps in understanding phase coexistence
in hardcore lattice gas models \cite{Criteria, Bijoy}.
Such a criterion for cluster-factorized steady state would be very
useful. At present, we do not have a general criterion, but
the examples studied above suggest a sufficient condition
for CFSS to have condensation. If the
rate function can be expanded as
$$
u(n_{i-R}, .., n_{i+R}) = \sum_{\nu=0}^{\infty} \frac{B_\nu (n_{i-R}, .., n_{i-1},n_{i+1} .., n_{i+R})}{ n_i^\nu},
$$
the condensation transition can occur when both the conditions
\bea
(i) ~&& {\rm both } \: B_0 \:{\rm and } \:B_1 \: {\rm are ~ constant} \cr
~(ii)&& B_1/B_0 >2 \label{eq:pfss_criteria}
\eea
are satisfied. This is only a simple generalization of the
criterion of condensation in the ZRP. Effectively, $B_1/B_0$ plays
the role of $b$ in Eq. (\ref{eq:criteria}). As the hop
rate in Eq. (\ref{eq:uRcondensation}) can be expanded as
\bea
u(\dots,n_{i-1}n_i,n_{i+1}\dots ) = 1+ \frac{\nu - K(R+1)}{n_i} + {\cal O} (\frac{1}{n_i^2}), \nonumber
\eea
and thus $B_0=1$ and $B_2= \nu - K(R+1),$ the criterion
correctly predicts the condensation which occurs only when
$\nu>K(R+1)+2.$ This is same as the usual condensation
criterion in the ZRP if we treat $b\equiv B_2/B_0.$
In this particular case, we have also checked that moments
$\langle n^k \rangle$ as a function of $z$, in leading order,
are the same as that in the ZRP with corresponding $b$ (see Eq.
(\ref{eq:criteria}) ). This criterion, however, cannot be
applied to some of the following cases studied recently, such
as, the misanthrope process \cite{Beyond_ZRP} and the PFSS
\cite{PFSS_Evans}. For the first case, $B_0$ and $B_1$ are not
constants and, for the later case, hop rates are not analytic
functions. A criterion of condensation, which can
apply to a cluster-factorized steady state in general is
desirable and remains a challenge.
\begin{figure}[h]
\centering \includegraphics[width=8.2 cm]{compare.eps}
\caption{ (Color online) Particle distribution in FRP with weight function (\ref{eq:g_condensation})
after $t=10^6$ MCS, starting from a random distribution of particles. Density is $\rho=\rho_c+0.01.$
The critical density for $R=0$ is $\rho_c=0.01925$; the same for $R>0$ are taken from
table \ref{tbl}. Clearly, for all cases, the condensate is localized to a single site.
The condensate size is written beside the condensate site.
}
\label{fig:single_site}
\end{figure}
We end this section with the following remark. The
condensation transition here is different from that
obtained for PFFS by Evans {\it et. al.} \cite{PFSS_Evans}.
There, one observes an extended condensate where both the size and the spatial extent of condensate scales with system size as $\sqrt L.$ This indicates that the transition is associated with a diverging
spatial correlation length. Whereas for the PFSS (and the CFSS)
studied here, the correlation length remains finite throughout and the transition is
characterized by a diverging mass fluctuation, as in the ZRP.
The condensate is also localized to a single site
(see Fig. \ref{fig:single_site}). A detailed comparison of
nature of condensate would be reported elsewhere \cite{ComingSoon}.
\subsection{ Pair factorized state with weight function in sum-form\label{sec:IV.B}}
In this section, we first show that a pair factorized steady state with weight function $g(n_i, n_{i+1}) = f_0(n_i) + f_1(n_{i+1})$,
which we refer to as {\it sum-form,} cannot give rise to
condensation. Then, we demonstrate this considering a
perturbation to a ZRP that converts the existing
factorized steady state of the ZRP to a PFSS with weight function in the sum-form. For the PFSS with weight function in the sum-form, the transfer matrix $T(z)$ is given by Eq. (\ref{eq:Z2}).
The largest eigenvalue of the matrix
$\lambda_+ = \frac{1}{2} ( T_{11} + T_{22}+ \sqrt { (T_{11}+ T_{22})^2 - 4 {\cal D}},$ where
${\cal D}$ is the determinant of $T$ can be used in Eq. (\ref{eq:Z9}) to get the density
$\rho(z).$ With some straightforward algebraic manipulations, one can show that the maximum density at
$z=z_c=1$ is,
\bea
\rho_c =\lim_{z\to 1} \rho(z) = \lim_{z\to 1}\frac{1}{2} \left[ \frac{1}{T_{21}} \frac{d T_{21}}{dz~} + \frac{1}{1-z}\right].\nonumber
\eea
Clearly $\rho_c$ diverges independent of the first term, leading to a conclusion that there can not be a condensation transition at any finite density. Thus, a PFSS {\it cannot} have condensation transition if the weight function has a sum-form.
To illustrate this, we consider a simple zero range process
with weight function $f(n) = 1/(n+1)^\nu$ or hop rate $u(n)= f(n-1)/f(n) =(n+1)^\nu/ n^\nu $
and add a perturbative term get a new weight function
\be
g(n_i, n_{i+1}) = (1-q) f(n_i) + q f(n_{i+1})
\ee
which depends on occupation of two consecutive sites. Here $ 0\le q\le 1 , \bar q = 1-q $ and we choose $f(n) = 1/(n+1)^\nu.$
A pair-factorized state, as in
Eq. (\ref{eq:PFSS}), with the above weight function occurs
when particles hop rate is
\bea
u(n_{i-1},n_i,n_{i+1}) =\frac{ \bar qf(n_{i-1}) + q f(n_{i}-1)}{\bar q f(n_{i-1}) + q f(n_{i})} \cr
\times \frac{ \bar q f(n_i-1) + q f(n_{i+1})}{ \bar qf(n_i) + q f(n_{i+1})}. \nonumber
\eea
For both $q=0$ and $q=1$ we have a factorized steady state, as in Eq. (\ref{eq:FSS}), which corresponds to
the ZRP with particle hop rate
\be
u(n) = \frac{f(n-1)}{f(n)}= \left( 1+\frac{1}{n} \right) ^\nu \simeq 1+\frac{\nu}{n} + {\cal O} (\frac{1}{n^2}).
\ee
Thus we expect a condensation transition for $q=0,1$ when
$\nu>2$ and the density is larger than a
critical value $\rho_c.$ In this case the $\cZ(z) = F(z) ^L$
(the transfer matrix $T(z)$ reduces to a scalar), where
$F(z) = \sum_{n=0}^\infty h(n) z^n = Li_\nu(z).$ The density is
$\rho = z \frac {d}{dz} F(z)$ and thus the critical density for
$q=0,1$ is
\be
\rho_c = \lim_{z\to 1} \rho(z)=\left\{ \begin{array}{ll}
\infty & \mathrm{for} \: \nu\leq 2, \\
\frac{\zeta(\nu-1)}{\zeta(\nu)}-1 & \mathrm{for} \: \nu >2. \\
\end{array}\right.
\ee
\begin{figure}[h]
\centering \includegraphics[width=8.2 cm]{phase.eps}
\caption{ (Color online) Small perturbation to the ZRP:
For small $q= 10^{-2},10^{-3}$ or $10^{-4}$, density $\rho(z)$
diverges when $z\to1.$ However for $q=0$ or for $q=1,$
$\rho_c = \rho(1) =\frac{\zeta(\nu-1)}{\zeta(\nu)}-1$ is finite,
leading to a condensation transition when $\rho> \rho_c.$ Inset
shows the phase diagram for $\nu=3.$ }
\label{fig:rho_c}
\end{figure}
The phase-diagram of the condensation transition in the $\rho$-$\nu$
plane is shown in Fig. \ref{fig:rho_c}. The critical line
$\rho_c$ separates the condensate phase from the fluid phase.
For a general $0<q<1$, we need to calculate the density using Eqs.
(\ref{eq:Z7}), (\ref{eq:Z8}), and (\ref{eq:Z9}),
\bea
\rho(z) =\frac{1}{a(a-(1-z)Li_{\nu}(z))} [Li_{\nu-1}(z)(\bar q^2(1-z)^2Li_{\nu}(z)-\cr (1-z)a)+2\bar qqz(Li_{2\nu}(z) +(1-z)Li_{2\nu-1}(z))]-1 \cr
{\rm where }~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\cr
a(q,z)=\sqrt{(q-\bar q)^2(1-z)^2Li_{\nu}^{2}(z)+4q\bar q z(1-z)Li_{2\nu}(z)} . \nn
\eea
In the limit $z\to 1,$ $\rho(z)$ diverges for all $\nu>0$ and
thus the condensation transition is destroyed.
It is somewhat surprising why for any non-zero $q$ however small, the condensation transition is destroyed.
It seems that this perturbation, which takes the factorized steady state of the ZRP to a pair-factorized steady
state, forces the condensation to disappear. One could understand
this following the criterion (\ref{eq:pfss_criteria}). For $\nu=3$, the rate for general $q$ has an asymptotic form ({\it i.e.} when $n_{i} \to \infty$)
)\bea
u(n_{i-1},n_i,n_{i+1})=1+3 \frac{q^2(1+n_{i-1})^3+\bar q^2 (1+n_{i+1})^3}{q \bar q n_{i}^4 } . \nn
\eea
Thus, here $B_1/B_0=0$ and therefore we should not expect
condensation for this hop rate. It can be shown easily that for any $\nu \geq 1$
the asymptotic form of the hop rate does not satisfy condition $(ii)$ of ansatz (\ref{eq:pfss_criteria}) ruling out the
possibility of a condensation transition.
\subsection{Subsystem mass distribution}
It was argued in recent works \cite{Bertin, gamma, zero-th} that,
for systems with short-ranged interaction, irrespective of whether they
are in equilibrium or not, one could obtain a state function
which plays the role of a free energy function. It was shown
in \cite{gamma, zero-th}
that the steady state distribution $P_v(m)$ of mass in
subsystems of volume $v \gg \xi$ can be determined from the
functional dependence of the scaled variance $\sigma^2 (\rho) = (\langle m^2 \rangle - \langle m \rangle^2)/v$, in the limit of large $v$, on the mass density $\rho$. When $\sigma^2(\rho) \propto \rho^2$ is a
quadratic function of density $\rho$, the subsystem mass
distribution can be characterized through a gamma distribution,
i.e., $P_v(m) \propto m^{\eta-1} \exp (\mu m)$, where $\mu = -
\eta/ \rho $ is an equilibrium-like chemical potential. The
exponent $\eta$ however depends on the details of the model and
it can be calculated from the knowledge of two point-correlation
function only. The matrix formulation developed here for the CFSS
can thus help in determination of $\eta.$
To illustrate this, let us consider a continuous finite range
process with $R=1,$ and calculate explicitly the variance of
the subsystem mass. Consider the following homogeneous
weight function for a pair-factorized steady state,
\be
g(m_i,m_{i+1}) = m_{i}^\delta + c~ m_{i}^\gamma m_{i+1}^{\delta-\gamma}
\ee
The grand partition sum is ${\cal Z}= Tr[T^L]$ where the transfer matrix $T(\mu)$ ($\mu=\ln(z)$ is the corresponding chemical potential) is given below
\begin{equation}
T(\mu)=\frac{1}{\mu^{1+\delta}} \left( \begin{array}{cc}
\Gamma(\delta+1) & c \frac{\Gamma(\gamma+1)}{\mu^{\gamma-\delta}} \\
\frac{\Gamma(2 \delta-\gamma+1)}{\mu^{ \delta-\gamma}} & c \Gamma(\delta+1) \\
\end{array}
\right) ,
\end{equation}
where $\Gamma(.)$ are Gamma functions.
Eigenvalues of $T(\mu)$ are $\lambda_{\pm}=\Lambda_{\pm}(\delta,\gamma,c)/ \mu^{1+\delta}$ where
\bea
&&2 \Lambda_{\pm}(\delta,\gamma,c)= (1+c) \Gamma(\delta+1)\cr
&& ~~~~~~~~\pm
\sqrt{(\delta+1)^{2}(1-c)^{2}+4c\Gamma(2 \delta-\gamma+1)\Gamma(\gamma+1)} \nn
\eea
and the particle density is
\be
\rho=\frac{\partial}{\partial
\mu}\mathrm{ln}\lambda_{+}=-\frac{\delta+1}{\mu}, \label{rho-mu}
\ee
implying a fluctuation-response (FR) relation
\be
\frac{d \rho}{d \mu} = \sigma^2(\rho), \label{FR}
\ee
analogous to the fluctuation-dissipation theorem in equilibrium.
Now, as shown below, one can check the above FR relation by explicitly calculating both sides of Eq. (\ref{FR}). The r.h.s of Eq. (\ref{FR}) can be calculated by integrating two-point correlation function $\sigma^2(\rho)=\sum_{r=-\infty}^{r=\infty} C(r)$, using Eq. (\ref{eq:2corr}),
\be
C(r)= \langle n_i n_{i+r}\rangle -\rho^2= \rho^2\left[ A(r)-1 \right]
\ee
where, for $r>0,$
\bea
A(r)= 1+\left( \frac{\Lambda_-}{\Lambda_+} \right)^r \frac{ (\delta-\gamma)^2/(\delta-1)^2 }{1-\frac{\Gamma(\delta+1)^2}{\Gamma(2\delta-\gamma+1)\Gamma(\gamma+1)}}
\nonumber
\eea
and
\bea
A(0)= \frac{\delta +2}{\delta +1}+\frac{2c}{\Lambda_+} \frac{(\delta-\gamma)^2}{(\delta +1)^2}
\frac{\Gamma(2\delta-\gamma+1)\Gamma(\gamma+1)}{2\Lambda_+-(1+c)\Gamma(\delta +1)}. \nn
\eea
In this system, the {\it gap} $(\lambda_+ - \lambda_-)$ between
the two eigenvalues is nonzero and the correlation length $\xi=|
\ln \frac{\Lambda_-}{\Lambda_+}|^{-1}$ is finite.
Therefore, following Ref. \cite{gamma}, the subsystem mass distribution $P_v(m)$, for $v \gg \xi$, is a gamma distribution
where the exponent $\eta$ can be written, using Eq. (\ref{eq:Z9}), as
\bea
\eta^{-1}&=&\sum_{r=-\infty}^{\infty} (A(r)-1),
= \frac{1}{\delta+1}.
\eea
Note that the exponent $\eta$ depends only on the homogeneity exponent $\delta$ but neither on $\gamma$ nor on $c.$
The left-hand side, the compressibility $d\rho/d\mu$, of Eq. (\ref{FR}) gives the same $\eta= \rho^2 (\frac{d\rho}{d\mu})^{-1} = \delta+1$, by differentiating the expression $\rho= -(\delta+1)/\mu$ in Eq. (\ref{rho-mu}) with respect to $\mu$; this is a proof that the fluctuation-response relation indeed holds here and also is consistent with the additivity property proposed earlier for these systems \cite{Bertin, gamma}.
In principle, the single-site mass distribution (for $v=1$) can
be calculated straightforwardly from the moments, but the exact
closed form expression is hard to obtain. In this regard, this
formulation \cite{gamma, Arghya} for obtaining the subsystem mass
distribution from the two-point correlation function is quite useful
in obtaining the macroscopic behaviour of the system.
\section{Summary \label{sec:conclude}}
We have introduced a class of nonequilibrium finite range processes (FRP) where particles on a one dimensional periodic lattice can hop in a particular direction, from one site to one of its nearest neighbours, with a rate that depends on the occupation of all the sites within a range $R$ starting from the
departure site. We show that, for certain specific functional forms
of the hop rates, the FRP has a cluster-factorized steady state (CFSS), i.e., the steady state probability of a microstate can be written as a product
of cluster-weight functions $g$ having $(R+1)$ arguments ~-
~the occupation numbers of $(R+1)$ consecutive sites. The model with $R=0$ reduces to the familiar zero range process (ZRP), which has factorized steady state.
The CFSS with $R=1$ reduces to the pair-factorized steady state (PFSS) and its steady state can always be represented by an infinite-dimensional transfer matrix. However, for the CFSS with $R>1$, a matrix formulation is {\it not} guaranteed.
In this work, we show that, for a large class of systems having CFSS with $R>0$, there exists a {\it finite} dimensional matrix representation. Being finite dimensional, these matrices are easy to manipulate and thus help in exactly calculating the $n$-point correlation functions for any $n$. The two-point correlation function ($n=2$) can be utilized to characterize the subsystem mass distribution in these nonequilibrium systems in terms of a nonequilibrium chemical potential and a free-energy function, which are obtained through a fluctuation-response relation \cite{Bertin, gamma} - analogous to the equilibrium fluctuation-dissipation theorem.
Even though the transfer-matrix is finite dimensional, the CFSS can undergo a condensation phase transition. We obtain a sufficient condition for the condensation transition for a particular class of hop rates in the FRP in general. The nature of the condensation transition studied in this paper are however different from those studied in systems having a PFSS \cite{PFSS_Evans}; the condensate here remains localized, as in the ZRP, in contrast to the extended condensate observed in \cite{PFSS_Evans, PFSS_Mass, PFSS_Graph}. Moreover, the condensation transition studied here occurs solely due to the diverging mass fluctuations at certain critical density $\rho_c,$ not due to a diverging correlation length; in fact, the correlation length remains finite throughout.
We should mention that it is {\it always} possible to
construct a hopping dynamics of FRP so that it evolves to a
desired steady state which is cluster-factorized. However,
for a given hop rate, there is no simple way {\it to check} if
it can give rise to a CFSS. In our opinion, the FRP is a very general
class of models as it includes the Ising model, Potts model,
misanthrope process, urn models, symmetric and asymmetric
exclusion processes on a ring and many other models
(one of them, of course, zero range process). More
importantly, the method developed here, could help in
finding the exact steady state structure in models even when
the interactions extend beyond two sites.
\section*{Appendix}
In this Appendix, we provide an argument that FRP can have a factorized steady state only for $R=0$ (namely the ZRP) and
for some specific misanthrope process (special cases of $R=1$). For any $R>1,$ however, one cannot have a factorized
steady state in general.
First we consider $R=1$ and show that, in this case, the hop rate reduces to those in the ZRP or in the misanthrope process, when one demands a factorized steady state. One can construct a general proof in a similar way,
that condition of FSS would reduce the hop rate of
FRP with $R>1$ to the ZRP or the misanthrope process. In Appendix B, we provide a proof of the above
for the hop rates which can be written in a product form.
\section*{Appendix A : NO FSS FOR $R=1$}
In this section, we show that, for $R=1$, one cannot have
a factorized steady state for the general hop rate
$u(n_{i-1},n_i,n_{i+1})$.
The Master equation for FRP for general $R>0$ is
\bea
\frac{d}{dt} P(\{n_i\}) = \sum_{i=1}^L F(n_{i-R},\dots,n_{i},\dots,n_{i+R}), \label{eq:master}
\eea
where
\bea
&&F(n_{i-R},..,n_{i},..,n_{i+R})= u(n_{i-R},..,n_i,..,n_{i+R}) P(\{ n_i\})\cr
&&~~~~~~~~~~~~~~~~-u(n_{i-R},..,n_i+1,n_{i+1}-1,..,n_{i+R}) \cr
&&~~~~~~~~~~~~~~~~~~~~~~~\times P(..,n_i+1,n_{i+1}-1,..). \label{eq:F}
\eea
In the steady state, right hand side of Eq. (\ref{eq:master}) must vanish, which can happen if
\bea
&&F(n_{i-R},\dots,n_{i},\dots,n_{i+R})=h(n_{i-R},\dots,n_i,\dots,n_{i+R-1})\cr
&&~~~~~~~~~~~~-h(n_{i-R+1},\dots,n_{i},\dots,n_{i+R}) \label{eq:h}
\eea
for some arbitrary function $h$ of $2R$ arguments. Note, that the above cancellation scheme
is only a sufficient condition.
Now let us consider $R=1,$ and demand that the steady state has a factorized form given by Eq. (\ref{eq:FSS}). Then
\bea
u(n_{i-1},n_i+1,n_{i+1}-1)\frac{f(n_i+1)}{f(n_i)}\frac{f(n_{i+1}-1)}{f(n_{i+1})}\cr
- u(n_{i-1},n_i,n_{i+1})=h(n_i,n_{i+1})- h(n_{i-1},n_i) \label{eq:R_1}
\eea
where $h$ is an arbitrary function, yet to be determined.
Since the hop rate $u(n_{i-1},n_i,n_{i+1})=0 $ when $n_i=0$ and we must have a boundary condition $f(m<0)=0,$
we can use specific values of $n_i$s in Eq. (\ref{eq:R_1}) to find recursion relation for $h$.
For $n_i=0=n_{i+1}$ equation (\ref{eq:R_1}) in $h(n_{i-1},0)=h(0,0).$ Again putting $n_{i+1}=0=n_{i-1}$ we get
$h(0,n_i)-h(0,0)=u(0,n_i,0).$
These two conditions leaves Eq. (\ref{eq:R_1}) for $n_i=0$ as
\bea
u(n_{i-1},1,n_{i+1}-1)\frac{f(1)}{f(0)}\frac{f(n_{i+1}-1)}{f(n_{i+1})}=u(0,n_{i+1},0). \nonumber
\eea
Clearly, in order to be consistent, $u(n_{i-1},1,n_{i+1})$ must be independent of $n_{i-1}$. For convenience,
without any loss of generality, lets set $u(n_{i-1},1,n_{i+1})=u(0,1,n_{i+1})$.
Thus, to have the factorized steady state for $R=1,$ the hop rate $u(n_{i-1},n_i,n_{i+1})$ must satisfy
\bea
u(n_{i-1},n_i+1,n_{i+1}-1) \frac{u(0,1,n_i)}{u(0,n_i+1,0)} \frac{u(0,n_{i+1},0)}{u(0,1,n_{i+1}-1)} \cr -u(n_{i-1},n_i,n_{i+1})=u(n_{i},n_{i+1},0)-u(n_{i-1},n_i,0). \label{eq:rate_constraint} \nn
\eea
Now if we take $n_i=1$ and use $u(n_{i-1},1,n_{i+1})=u(0,1,n_{i+1})$ in the above equation to rearrange the terms, we have
\bea
u(n_{i-1},2,n_{i+1}-1) \frac{u(0,1,1)}{u(0,2,0)} \frac{u(0,n_{i+1},0)}{u(0,1,n_{i+1}-1)}\cr -u(0,1,n_{i+1})= u(1,n_{i+1},0)- u(0,1,0), \nonumber
\eea
which implies that $u(n_{i-1},2,n_{i+1})$ must be independent of $n_{i-1}$. A similar recursion would result
that $u(n_{i-1},n_i,n_{i+1})$ must be independent of $n_{i-1}.$ This again reflects the fact that a factorized steady state is possible
for $R=1$ only when hop rate is $u=u(n_i,n_{i+1})$ {i.e.} the process is a \textit{misanthrope} process.
\section*{Appendix B : NO FSS FOR HOP RATE HAVING PRODUCT FORM }
In this section, we show that the FRP, for any $R>0$, cannot have a FSS when the hop rate has the following product form,
\be
u(n_{i-R}, \dots, n_i, \dots, n_{i+R})=\prod_{j=-R}^R v_{j}(n_{i+j}) \label{eq:product_u}.
\ee
The Master equation along with a demand of a factorized steady state of the
form (\ref{eq:FSS}), and then Eqs. (\ref{eq:F}) and (\ref{eq:h}) together, implies
\bea
&&v_{-R}\dots v_{-1} v_2 \dots v_R G(n_i,n_{i+1})=h(n_{i-R},\dots,n_{i+R-1})\cr
&&~~~~~~~~~~~~-h(n_{i-R+1},\dots,n_{i+R}) \label{eq:vG}
\eea
where $v_k \equiv v_k (n_{i+k})$ and
\bea
G(n_i,n_{i+1})=&-&v_0(n_i+1)v_1(n_{i+1})\frac{f(n_i+1)f(n_{i+1}-1)}{f(n_i)f(n_{i+1})}\cr
&+& v_0(n_i)v_1(n_{i+1}).\nonumber
\eea
Now differentiating both sides of Eq. (\ref{eq:vG}) with respect to $n_{i-R}$ and $n_{i+R}$, we have
\bea
\frac{\partial v_{-R}}{\partial n_{i-R}}\frac{\partial v_{R}}{\partial n_{i+R}} v_{-R+1}\dots v_{-1}v_2\dots v_{R-1}G(n_i,n_{i+1}) =0 .
\nonumber
\eea
This implies that, either $v_{-R}(n_{i-R})$ or $v_{R}(n_{i+R})$ must be a constant,
because the other solution $f(n)=1/v_0(n)=1/v_1(n)$ cannot be accepted as it means $v_1(0)=v_0(0)=0$, {\it i.e.}, a
particle cannot be transferred to a vacant neighbouring site. So, let us proceed with $v_{-R}=$constant (say $1$).
Then Eq. (\ref{eq:h}) reads as
\bea
&&v_{1-R}.. v_{-1} v_2..v_R G(n_i,n_{i+1})=h(n_{i-R},..,n_i,..,n_{i+R-1})\cr
&&~~~~~~~~~~~~~~~-h(n_{i-R+1},..,n_{i},..,n_{i+R}). \nn
\eea
Clearly for this equation to be valid its right hand side must be independent of $n_{1-R}$ and that in turn leads to
$h(x_1,x_2,\dots,x_k)=h(x_2,\dots,x_k)$.
This way we can eliminate one variable at each step until finally we reach to
\bea
v(n_{i-R}, \dots, n_i, \dots, n_{i+R})=v(n_i,n_{i+1})=v_0(n_i)v_1(n_{i+1}),\cr
{~\rm and ~} v_0(n_i+1)v_1(n_{i+1}-1) \frac{f(n_i+1)}{f(n_i)}\frac{f(n_{i+1}-1)}{f(n_{i+1})}\cr
-v_0(n_i)v_1(n_{i+1})=h(n_{i+1})-h(n_i).\nn
\eea
This is exactly the criterion for having a factorized steady state in misanthrope process with a hop rate that has
a product form $u(n_i, n_{i+1}) = v_0(n_i) v_1(n_{i+1}) $ \cite{Beyond_ZRP}.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 5,304 |
{"url":"https:\/\/www.compadre.org\/portal\/..\/picup\/exercises\/exercise.cfm?A=FreqExtraction","text":"+\nFrequency Extraction via Test Functions\n\nDeveloped by Peter Bryant - Published July 27, 2022\n\nThis exercise set walks students through a conceptual approach to computationally extract the frequency content from oscillatory signals. The purpose is to give students an intuitive understanding of projecting signals onto basis functions, such as is done when computing Fourier (co)sine series coefficients, and also to gain intuition into what is meant by the \"frequency content of a signal.\" Students create and plot signals composed of sinusoidal functions and investigate different metrics for determining correlation with test functions. They reach a result similar to a finite sum approximating the Fourier (co)sine transform, and use what they have computed to extract frequencies from an unknown signal. The end product is a computational tool, created by the student, that can be used to extract frequency information from measurements made in the laboratory.\nSubject Areas Experimental \/ Labs and Waves & Optics First Year Spreadsheet Students who complete the exercises will be able to: - explain how frequency content can be extracted computationally from an oscillatory signal (Exercises 1 - 4) - extract frequency content from an unknown signal (Exercise 5) - reconstruct an unknown signal from extracted frequencies (Exercise 5) - explain the physical irrelevance of the magnitudes of the discrete sums approximating the Fourier (co)sine transform (Exercise 7) 90 min\n\nThese exercises are not tied to a specific programming language. Example implementations are provided under the Code tab, but the Exercises can be implemented in whatever platform you wish to use (e.g., Excel, Python, MATLAB, etc.).\n\n**Exercise 1: Sinusoidal signals and test functions** 1. Plot the sum of at least two sine functions with with differing frequencies and amplitudes. For now, don't include phase angles. Your sum will be the \"signal\" function, representing a hypothetical measurement in time of an oscillatory quantity. Plot over enough time that you can see many cycles but not so much that you can't make out the features of the signal. Note that, in a spreadsheet, an approximation to $\\pi$ can often be entered into formulas as PI(). 1. Overlay a \"test\" function on your plot. The test function should be a sine function with arbitrary amplitude and with a frequency that you will vary. As a start, enter any frequency you'd like. *Suggestion for spreadsheets:* Reference the test function's frequency to a cell so that you can change its frequency simply by changing the value in the cell. **Exercise 2: A notion of correlation** Vary the frequency of the test function and observe your plot. 1. Is there a noticeable difference between the cases when - your test function has a frequency equal to one of the frequencies from the signal, or - your test function does not have a frequency from the signal? 2. If you were given a signal function composed of sine functions with unknown frequencies, do you think you could determine its unknown frequencies visually by varying the frequency of a test function? **Exercise 3: A better metric for correlation?** At each value of time multiply your signal by your test function and plot the product as a function of time on a new axis. Again vary the frequency of the test function and observe your new plot. 1. Is there a noticeable difference between the cases when - your test function has a frequency equal to one of the frequencies from the signal, or - your test function does not have a frequency from the signal? Pay particular attention to the maximum and minimum values of the plot of the product. 2. If you were given a signal function composed of sine functions with unknown frequencies, do you think you could determine its unknown frequencies by varying the frequency of a test function and looking at a plot of the product? **Exercise 4: Amplify any asymmetry in the product** 1. Compute the *sum* of the product over all time values. Now vary the frequency of the test function and observe the *magnitude* of the sum. Is there a noticeable difference between the cases when - your test function has a frequency equal to one of the frequencies from the signal, or - your test function does not have a frequency from the signal? 2. Explain the difference you've observed. Why is the magnitude of the sum noticeably different when the test function has a frequency close to the frequencies in the signal? **Exercise 5: Frequency content in an unknown signal** 1. Use what you have learned or developed to extract frequency content from an unknown signal, which will be provided to you. This means you must discover what frequencies are significantly present in the signal, and your answer should be a list of the frequencies you find. 3. Create a plot of a sum of sine functions with the frequencies you have extracted from the signal, and compare it with the signal itself. Adjust the amplitudes of the different sine functions in your sum until your function resembles the signal. Then change the amplitude of the term with the smallest frequency to zero and observe how your sum of sine functions changes. Does it still resemble the signal? **Exercise 6: Automate the frequency scan** Rather than changing the frequency of the test function by hand, 1. create a range of test function frequencies, 2. compute the sum of the product of functions for each frequency, and 3. create a plot of the magnitude of the sum with the varying test frequencies. What can you say about the peaks of this plot? **Exercise 7: Phase** Until now we have not considered phase. In general, however, oscillatory signals won't be well-approximated by sums of in-phase sine functions (sine functions all having zero or identical phase). To investigate the effect of differing phases, begin with your results from **Exercise 6** and use a new signal function for which each separate sine function in the sum has its own phase angle. *Suggestion for spreadsheets:* Reference the different phases in your formula to different cells so that you can change the phase angles simply by changing the values in the cells. 1. When you vary the phases of the sine functions in your signal, how are the major peaks in your plot affected? 2. When your signal function is composed of sine functions with arbitrary phases, are you still able to extract or identify its frequencies? **Exercise 8 (optional): Peak height** Argue that the absolute heights of the peaks on your plot from **Exercise 6** are not physically relevant.","date":"2022-11-27 05:46:57","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\": 0.5925500988960266, \"perplexity\": 538.3500859937916}, \"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\/1669446710192.90\/warc\/CC-MAIN-20221127041342-20221127071342-00831.warc.gz\"}"} | null | null |
9 Tabletop RPG character creation tips!
By admin November 26, 2022 November 25, 2022 Advice, TTRPG
So you're at the start of a new adventure, traveler? You found a party, and you're ready to create your shiny new character. But where do you begin? What are things you need to consider when bringing your character to life? We have 9 character creation tips to help you on your journey!
What is a TTRPG character?
Have you been thrown into the void by friends who want to start playing tabletop role-playing games? And you're not quite sure what you got yourself into? Let us explain!
A tabletop role-playing game is a form of a role-playing game, usually played in person, where the players play by talking to each other and rolling dice. The Game Master, also known as GM, runs the game and offers a story and scenario in a set world, which can be fantasy, sci-fi, or even steampunk-themed. Other players play a character in this setting and decide what to do in that given scenario.
Step 1: Take your time
Your character doesn't have to be perfect before the first session, and doesn't need to be set in stone. Why? Well, true magic happens when you play! You discover so much more about your character when you interact with other characters and the story. So don't make it too difficult for yourself. Take your time with character creation, so you have a character you want to play.
Step 2: Don't be scared of tropes and clichés
It might sound cliché, but you can play whoever you want. Do you want to be a fighter who settles scores with their fists? Or perhaps a bard who could end up in a fabulous yelling contest (interpret that how you will)? There's nothing wrong with cliches. They exist because they have worked pretty well in the past. Cliches are familiar to everyone, and it might be easier to put yourself in your new role.
Step 3: Think of themes or concepts
What would your character do if they were about to lose their family? Or what if they have traveled with a circus for their whole life? Or do you have a certain aesthetic in mind, like a dreamlike forest? Thinking of a theme, concept, or mood surrounding your character can help shape them.
Step 4: Use art and music to your advantage
Did you see character art that sparked an idea? Great! Did you hear a song that completely embodied the vibe and story of your character? Amazing! Art and music are a source of inspiration that you can use to your advantage. Maybe you have a favorite album that would be the perfect backstory, or a Pinterest board has the exact aesthetic you want. It's a great starting point for character creation. Creating a playlist or mood board can help flesh out an existing character idea as well.
Step 5: Know your abilities and features
Before the first session, it's wise to familiarize yourself with the character sheet. You don't need to know the whole rulebook, but having a character sheet can help progress the game and help you shape the intricacies of your character. Just make sure you have your character sheet ready when you sit down to play, and have read it once or twice.
Step 6: Communicate with your GM
You can always ask your GM for help when creating your character. Besides getting all your abilities and features straight, they can also help place the character in the campaign. This might help form an idea of your character and where they come from.
Step 7: Communicate with your fellow players
Nothing more fun than brainstorming with your fellow players and joining backstories! You can talk with each other about some ideas and maybe link some parts of your backstories. For example, your character collects rare seashells. Another character is a sailor who sells seashells alongside the fish they catch. You already know each other as buyer and seller, and it's a fun addition to both of your backstories.
Step 8: Play the type of character you like
It doesn't matter if you base your character on your favorite fictional (or nonfictional) character. If you want to escape reality as a female Indiana Jones, go for it! It will motivate you and spark inspiration. You can start with an example, and who knows, your character could end up completely different. But you have a starting point.
Step 9: Use tools to your advantage
Do you like to put your character in the hands of fate? Consult character creation tables online or in books. If you have no inspiration, these tools are super useful to roll up a random character.
Do you feel ready to create a character? Perfect! If you want to know how to make an Æther Void character, keep an eye out for an upcoming article. Can't wait? Check out our one-shot adventures! We've included pre-made characters, so you can experience what Æther Void characters have to offer.
What is your number one tip for creating a TTRPG character?
Also read: Reintroducing Original One-Shot Characters
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\section{Introduction}
As the volume and complexity of data grows, statistical data integration has gained increasing attention as it can lead to discoveries which are not evident in analyses of a single data set. We study a specific data-integration problem where we seek to leverage common samples measured across multiple diverse sets of features that are of different types (e.g., continuous, count-valued, categorical, skewed continuous and etc.). This type of data is often called mixed, multi-view data \citep{hall1997introduction,acar2011all,lock2013joint,tang2018integrated,baker2019feature}.
While many techniques have been developed to analyze each individual data type separately, there are currently few methods that can directly analyze mixed multi-view data jointly. Yet, such data is common in many areas such as electronic health records, integrative genomics, multi-modal imaging, remote sensing, national security, online advertising, and environmental studies. For example in genomics, scientists often study gene regulation by exploring only gene expression data, but other data types, such as short RNA expression and DNA methylation, are all part of the same gene regulatory system.
Joint analysis of such data can give scientists a more holistic view of the problem they study. But, this presents a major challenge as each individual data type is high-dimensional (i.e., a larger number of features than samples) with many uninformative features. Further, each data view contains different data types: expression of genes or short RNAs measured via sequencing is typically count-valued or zero-inflated plus skewed continuous data whereas DNA methylation data is typically proportion-valued. In this paper, we seek to leverage multiple sources of mixed data to better cluster the common samples as well as select relevant features that distinguish the inherent group structure.
We propose a convex formulation which integrates mixed types of data with different data-specific losses, clusters common samples with a joint fusion penalty and selects informative features that separate groups. Due to the convex formulation, our methods enjoy strong statistical, mathematical and empirical properties. We make several methodological contributions. First, we consider employing different types of losses for better handling non-Gaussian data with Generalized Convex Clustering Optimization (Gecco), which replaces Euclidean distances in convex clustering with more general convex losses. We show that for different losses, Gecco's fusion penalty forms different types of centroids which we call loss-specific centers. To integrate mixed multi-view data and perform clustering, we incorporate different convex distances, losses, or divergences for each of the different data views with a joint convex fusion penalty that leads to common groups; this gives rise to Integrative Generalized Convex Clustering (iGecco). Further, when dealing with high-dimensional data, practitioners seek interpretability by identifying important features which can separate the groups. To facilitate feature selection in Gecco and iGecco, we develop an adaptive shifted group-lasso penalty that selects features by shrinking them towards their loss-specific centers, leading to Gecco+ and iGecco+ which performs clustering and variable selection simultaneously.
To solve our methods in a computationally efficient manner, we develop a new general multi-block ADMM algorithm using sub-problem approximations, and make an optimization contribution by proving that this new class of algorithms converge to the global solution.
\subsection{Related Literature}
Our goal is to develop a unified, convex formulation of integrative clustering with feature selection based on increasingly popular convex clustering methods.
\cite{pelckmans2005convex, lindsten2011just, hocking2011clusterpath} proposed convex clustering which uses a fusion penalty to achieve agglomerative clustering like hierarchical clustering. This convex formulation guarantees a global optimal solution, enjoys strong statistical and mathematical theoretical properties, and often demonstrates superior empirical performance to competing approaches. Specifically, in literature, \citet{pelckmans2005convex,chi2017convex} showed it yields stable solutions to small perturbations on the data or tuning parameters; \cite{radchenko2017convex} studied statistical consistency; \cite{tan2015statistical} established its link to hierarchical clustering as well as prediction consistency; and many others have studied other appealing theoretical properties \citep{zhu2014convex,sui2018convex,chi2019recovering}. Despite these advantages, convex clustering has not yet gained widespread popularity due to its intensive computation. Recently, some proposed fast and efficient algorithms to solve convex clustering and estimate its regularization paths \citep{chi2015splitting,weylandt2019dynamic}. Meanwhile, convex clustering has been extended to biclustering \citep{chi2017convex} and many other applications \citep{chi2018provable,choi2019regularized}.
One potential drawback to convex clustering however, is that thus far, it has only been studied employing Euclidean distances between data points and their corresponding cluster centers. As is well known, the Euclidean metric suffers from poor performance with data that is highly non-Gaussian such as binary, count-valued, skewed data, or with data that has outliers. While \cite{wang2016robust} studied robust convex clustering and \cite{sui2018convex} investigated convex clustering with metric learning, there has not been a general investigation of convex clustering for non-Gaussian data and data integration on mixed data has not been studied. But, many others have proposed clustering methods for non-Gaussian data in other contexts. One approach is to perform standard clustering procedures on transformed data \citep{anders2010differential,bullard2010evaluation,marioni2008rna,robinson2010scaling}. But, choosing an appropriate transformation that retains the original cluster signal is a challenging problem. Another popular approach is to use hierarchical clustering with specified distance metrics for non-Gaussian data \citep{choi2010survey,fowlkes1983method}. Closely related to this, \cite{banerjee2005clustering} studied different clustering algorithms utilizing a large class of loss functions via Bregman divergences. Yet, the proposed methods are all extensions of existing clustering approaches and hence inherit both good and bad properties of those approaches. There has also been work on model-based clustering, which assumes that data are generated by a finite mixture model; for example \citet{banfield1993model,si2013model} propose such a model for the Poisson and negative binomial distributions. Still these methods have a non-convex formulation and local solutions like all model-based clustering methods.
We propose to adopt the method similar to \cite{banerjee2005clustering} and study convex clustering using different loss functions; hence our method inherits the desirable properties of convex clustering and handles non-Gaussian data as well. More importantly, there is currently no literature on data integration using convex clustering and we achieve this by integrating different types of general convex losses with a joint fusion penalty.
Integrative clustering, however, has been well-studied in the literature. The most popular approach is to use latent variables to capture the inherent structure of multiple types of data. This achieves a joint dimension reduction and then clustering is performed on the joint latent variables \citep{shen2009integrative,shen2012integrative,shen2013sparse,mo2013pattern,mo2017fully,meng2015mocluster}. Similar in nature to the latent variables approach, matrix factorization methods assume that the data has an intrinsic low-dimensional representation, with the dimension often corresponding to the number of clusters \citep{lock2013joint,hellton2016integrative,zhang2012discovery,chalise2017integrative,zhang2011novel,yang2015non}. There are a few major drawbacks of latent variable or dimension reduction approaches, however. First it is often hard to directly interpret latent factors or low-dimensional projections. Second, many of these approaches are based on non-convex formulations yielding local solutions. And third, choosing the rank of factors or projections is known to be very challenging in practice and will often impact resulting clustering solutions. Another approach to integrative clustering is clustering of clusters (COC) which performs cluster analysis on every single data set and then integrates the primary clustering results into final group assignments using consensus clustering \citep{hoadley2014multiplatform,lock2013bayesian,kirk2012bayesian,savage2013identifying,wang2014similarity}.
This, however, has several potential limitations as each individual data set might not have enough signal to discern joint clusters or the individual cluster assignments are too disparate to reach a meaningful consensus.
Finally, others have proposed to use distance-based clustering for mixed types of data by first defining an appropriate distance metric for mixed data (for example, the Gower distance by \citealp{gower1971general}) and then applying an existing distance-based clustering algorithm such as hierarchical clustering \citep{ahmad2007k,ji2012fuzzy}. Consequently, this method inherits both good and bad properties of distance-based clustering approaches. Notice that all of these approaches are either two-step approaches or are algorithmic or non-convex problems that yield local solutions. In practice, such approaches often lead to unreliable and unstable results.
Clustering is known to perform poorly for high-dimensional data as most techniques are highly sensitive to uninformative features. One common approach is to reduce the dimensionality of the data via PCA, NMF, or t-SNE before clustering \citep{ghosh2002mixture,bernardo2003bayesian, tamayo2007metagene}. A major limitation of such approaches is that the resulting clusters are not directly interpretable in terms of feature importance. To address this, several have proposed sparse clustering for high-dimensional data. This performs clustering and feature selection simultaneously by iteratively applying clustering techniques to subsets of features selected via regularization \citep{witten2010framework,sun2012regularized,chang2014sparse}. The approach, however, is non-convex and is highly susceptible to poor local solutions. Others have proposed penalized model-based clustering that selects features \citep{raftery2006variable,wang2008variable,pan2007penalized}. Still, these methods inherit several disadvantages of model-based clustering approaches. Moreover, sparse integrative clustering is relatively under-studied. \citet{shen2013sparse,mo2013pattern} extended iCluster using a penalized latent variable approach to jointly model multiple omics data types. They induce sparsity on the latent variable coefficients via regularization. As feature selection is performed on the latent variables, however, this is less interpretable in terms of selecting features directly responsible for distinguishing clusters.
Recently, and most closely related to our own work, \citet{wang2018sparse} proposed sparse convex clustering which adds a group-lasso penalty term on the cluster centers to shrink them towards zero, thus selecting relevant features.
This penalty, however, is only appropriate for Euclidean distances when the data is centered; otherwise, the penalty term shrinks towards the incorrect cluster centers. For feature selection using different distances and losses, we propose an adaptive shifted group-lasso penalty that will select features by shrinking them towards their appropriate centroid.
\section{Integrative Generalized Convex Clustering with Feature Selection}
In this section, we introduce our new methods, beginning with the Gecco and iGecco and then show how to achieve feature selection via regularization. We also discuss some practical considerations for applying our methods and develop an adaptive version of our approaches.
\subsection{Generalized Convex Clustering Optimization (Gecco)}
In many applications, we seek to cluster data that is non-Gaussian. In the literature, most do this using different distance metrics other than Euclidean distances \citep{choi2010survey,fowlkes1983method,de2004clustering}. Some use losses based on exponential family or deviances closely related to Bregman divergences \citep{banerjee2005clustering}.
To account for different types of losses for non-Gaussian data, we propose to replace the Euclidean distances in convex clustering with more general convex losses; this gives rise to Generalized Convex Clustering Optimization (Gecco).
$$
\minimize_{\mathbf U} \hspace{2mm}\sum_{i=1}^n \boldsymbol\ell(\mathbf X_{i.},\mathbf U_{i.}) + \gamma \sum_{i < i'} w_{ii'} \| \mathbf U_{i.} - \mathbf U_{i'.} \|_q
$$
Here, our data $\mathbf X$ is an $n \times p$ matrix consisting of $n$ observations and $p$ features; $\mathbf U$ is an $n \times p$ centroid matrix with the $i^{th}$ row, $\mathbf U_{i.}$, the cluster centroid attached to point $\mathbf X_{i.}$. The general loss $\boldsymbol\ell(\mathbf X_{i.},\mathbf U_{i.}) $ refers to a general loss metric that measures dissimilarity between the data point $\mathbf X_{i.}$ and assigned centroids $\mathbf U_{i.}$.
$\|\cdot\|_q$ is the $\ell_q$-norm of a vector and usually $q \in \{1,2,\infty\}$ is considered \citep{hocking2011clusterpath}. Here we prefer using the $\ell_2$-norm in the fusion penalty $(q=2)$ as it encourages the entire rows of similar observations to be fused together simultaneously and is also rotation-invariant; but one could use $\ell_1$ or $\ell_{\infty}$ norms as well. $\gamma$ is a positive tuning constant and $w_{ij}$ is a nonnegative weight.
When $\gamma$ equals zero, each data point occupies a unique cluster. As $\gamma$ increases, the fusion penalty encourages some of the rows of the cluster center $\mathbf U$ to be exactly fused, forming clusters. When $\gamma$ becomes sufficiently large, all centroids eventually coalesce to a single cluster centroid, which we define as the loss-specific center associated with $\boldsymbol\ell(\cdot)$. Hence $\gamma$ regulates both the cluster assignment and number of clusters, providing a family of clustering solutions. The weight $w_{ij}$ should be specified by the user in advance and is not a tuning parameter; we discuss choices of weights for various convex losses in Section \ref{practical}.
Going beyond Euclidean distances, we propose to employ convex distance metrics as well as deviances associated with exponential family distributions and Bregman divergences, which are always convex. Interestingly, we show that each of these possible loss functions shrink the cluster centers, $\mathbf U$, to different loss-specific centers, instead of the mean-based centroid as in convex clustering with Euclidean distances. For example, one may want to use least absolute deviations ($\ell_{1}$-norm or Manhattan distances) for skewed data or for data with outliers; with this loss, we show that all observations fuse to the median when $\gamma$ is sufficiently large. We emphasize loss-specific centers here as they will be important in feature selection in the next section. For completeness, we list common distances and deviance-based losses, as well as their loss-specific centers $\tilde x_j$ respectively in Table~\ref{loss-table}. (See Appendix~\ref{centroidcal} for all calculations associated with loss-specific centers, and we provide a formal proof when studying the properties of our approaches in Section \ref{property}.)
\begin{table}[h]
\vskip 0.15in
\begin{center}
\begin{small}
\scalebox{0.78}{
\begin{tabular}{ |l|l|l|l| }
\hline
\textbf{Data Type} & \textbf{Loss Type} & \textbf{Loss Function} & \textbf{Loss-specific Center} $\tilde \bx$ \\ \hline
\multirow{1}{*}{Continuous} & Euclidean ($\ell_2$) & $ \frac{1}{2} \|\bx_i - \bu_i\|_2^2$ & $\bar \bx$ \\ \hline
\multirow{6}{*}{Skewed Continuous} & Manhattan ($\ell_1$) & $\sum_{j=1} |x_{ij} - u_{ij}|$ & median($\bx$) \\
& Minkowski ($\ell_q$) & $\sqrt[q]{\sum_{j=1} |x_{ij} - u_{ij}|^q}$ & no closed form \\
& Mahalanobis (weighted $\ell_2$) & $(\bx_i-\bu_i)^T \textbf C^{-1}(\bx_i-\bu_i)$ & no closed form \\
&Chebychev ($\ell_{\infty}$) & $\max_j \{|x_{ij} - u_{ij}|\}$ &no closed form \\
&Canberra (weighted $\ell_1$) & $\sum_{j=1} \frac{|x_{ij} - u_{ij}|}{|x_{ij}| + |u_{ij}|}$ & no closed form\\
\hline
\multirow{4}{*}{Binary} & Bernoulli log-likelihood & $ - x_{ij} u_{ij} + \log(1+e^{u_{ij}})$ & $\text{logit}(\bar \bx)$ \\
& Binomial Deviance & $ - x_{ij} \log{u_{ij}} - (1-x_{ij})\log({1-u_{ij}})$ & $\bar \bx$ \\
& Hinge Loss & $\max(0,1-u_{ij} x_{ij})$ & mode$(\bx)$\\ & KL divergence & $-x_{ij} \log_2 u_{ij}$ & no closed form \\
&Hamming ($\ell_0$) &$ \sum_j \#(x_{ij} \neq u_{ij})/n$ & mode ($\bx$) \\
\hline
\multirow{6}{*}{Count} & Poisson log-likelihood & $ -x_{ij} u_{ij} + \exp(u_{ij})$ & $\log (\bar \bx)$ \\
& Poisson Deviance & $ -x_{ij} \log u_{ij} + u_{ij}$ & $\bar \bx$ \\
& Negative Binomial log-likelihood & $-x_{ij} u_{ij} + (x_{ij} + \frac{1}{\alpha} ) \log (\frac{1}{\alpha} + e^{u_{ij}} ) $ &$\log (\bar \bx)$ \\
& Negative Binomial Deviance & $ x_{ij} \log(\frac{x_{ij}}{u_{ij}}) - (x_{ij} + \frac{1}{\alpha}) \log (\frac{1+\alpha x_{ij}}{1+\alpha u_{ij} } ) $ & $\bar \bx$ \\
& Manhattan ($\ell_1$) & $\sum_{j=1} |x_{ij} - u_{ij}|$ & median($\bx$) \\
& Canberra (weighted $\ell_1$) & $\sum_{j=1} \frac{|x_{ij} - u_{ij}|}{|x_{ij}| + |u_{ij}|}$ & no closed form\\
\hline
\multirow{2}{*}{Categorical} & Multinomial log-likelihood & $ \big\{ \sum_{k=1}^{K} -x_{ijk} u_{ijk} + \log(\sum_{k=1}^K e^{u_{ijk}}) \big \} $ & $\text{mlogit}(\bar \bx)$ \\
& Multinomial Deviance & $ \big\{ \sum_{k=1}^{K} -x_{ijk} \log(u_{ijk}) \big \}$, $\sum \limits_{k=1}^K u_{ijk} = 1$ & $\bar \bx$ \\
\hline
\end{tabular}
}
\end{small}
\caption{Different losses and their loss-specific centers. We provide all calculations associated with loss-specific centers in Appendix~\ref{centroidcal}. Note the Gecco problem with Hamming or Canberra distances is not convex. Though we discuss general convex losses in this paper, we list those non-convex losses for reference. For multinomial log-likelihood and multinomial deviance, we change Gecco formulation slightly to accommodate three indices; we provide a detailed formulation in Appendix~\ref{multin}.}
\label{loss-table}
\end{center}
\vskip -0.1in
\end{table}
\subsection{Integrative Generalized Convex Clustering (iGecco)} \label{igecco}
In data integration problems, we observe data from multiple sources and would like to get a holistic understanding of the problem by analyzing all the data simultaneously. In our framework, we integrate mixed multi-view data and perform clustering by employing different convex losses for each of the different data views with a joint convex fusion penalty that leads to common groups. Hence we propose Integrative Generalized Convex Clustering (iGecco) which can be formulated as follows:
\begin{align*}
\minimize_{\mathbf U^{(k)}} \hspace{2mm} \sum_{k=1}^K \pi_k \boldsymbol\ell_k(\mathbf X^{(k)},\mathbf U^{(k)}) + \gamma \sum_{i < j} w_{ii'} \sqrt{ \sum_{k=1}^K \| \mathbf U_{i.}^{(k)} - \mathbf U_{i'.}^{(k)} \|^2 }
\end{align*}
Here, we have $K$ data sources. The $k^{th}$ data-view $\mathbf X^{(k)}$ is an $n \times p_k$ matrix consisting of $n$ observations and $p_k$ features; $\mathbf U^{(k)}$ is also an $n \times p_k$ matrix and the $i^{th}$ row, $\mathbf U_{i.}^{(k)}$, is the cluster center associated with the point $\mathbf X_{i.}^{(k)}$. And, $\boldsymbol\ell_k(\mathbf X_{i.}^{(k)},\mathbf U_{i.}^{(k)}) $ is the loss function associated with the $k^{th}$ data-view. Each loss function is weighted by $\pi_k$, which is fixed by the user in advance. We have found that setting $\pi_k$ to be inversely proportional to the null deviance evaluated at the loss-specific center, i.e., $\pi_k = \frac{1}{\boldsymbol\ell_k(\mathbf X^{(k)},\tilde \mathbf X^{(k)})}$, performs well in practice. Note that $\tilde \mathbf X = \begin{pmatrix} \tilde \mathbf X^{(1)} \cdots \tilde \mathbf X^{(K)} \end{pmatrix}$ where each $j^{th}$ column of $\tilde \mathbf X^{(k)}$ denotes the loss-specific center $\tilde x_j^{(k)}$. We employ this loss function weighting scheme to ensure equal scaling across data sets of different types. Finally, notice that we employ a joint group-lasso penalty on all of the $\mathbf U^{(k)}$'s; this incorporates information from each of the data sources and enforces the same group structure amongst the shared observations. We study this further and prove these properties in Section \ref{property}.
\subsection{Feature Selection: Gecco+ and iGecco+}
In high dimensions, it is important to perform feature selection both for clustering purity and interpretability.
Recently, \citet{wang2018sparse} proposed sparse convex clustering by imposing a group-lasso-type penalty on the cluster centers which achieves feature selection by shrinking noise features towards zero. This penalty, however, is only appropriate for Euclidean distances when the data is centered; otherwise, the penalty term shrinks towards the incorrect cluster centers. For example, the median is the cluster center with the $\ell_1$ or Manhattan distances. Thus, to select features in this scenario, we need to shrink them towards the median, and we should enforce ``sparsity" with respect to the median and not the origin. To address this, we propose adding a shifted group-lasso-type penalty which forces cluster center $\mathbf U_{\cdot j}$ to shrink toward the appropriate loss-specific center $\tilde x_j$ for each feature. Both the cluster fusion penalty and this new shifted-group-lasso-type feature selection penalty will shrink towards the same loss-specific center.
To facilitate feature selection with the adaptive shifted group-lasso penalty for one data type, our Generalized Convex Clustering Optimization with Feature Selection (Gecco+) is formulated as follows:
\begin{align*}
\minimize_{\mathbf U} \hspace{2mm} &\sum_{i=1}^n \boldsymbol\ell(\mathbf X_{i.},\mathbf U_{i.}) + \gamma \sum_{ i < i'}^n w_{ii'} \| \mathbf U_{i.} - \mathbf U_{i'.} \|_2 \\
&+ \alpha \sum_{j=1}^p \zeta_j \| \mathbf U_{.j} - \tilde x_j \cdot \textbf 1_n \|_2
\end{align*}
Again, $\mathbf U$ is an $n \times p$ matrix and $\tilde x_j$ is the loss-specific center for the $j^{th}$ feature introduced in Table~\ref{loss-table}. The tuning parameter $\alpha$ controls the number of informative features and the feature weight $\zeta_j$ is a user input which plays an important role to adaptively penalize the features. (We discuss choices of $\zeta_j$ in Section \ref{adaptive} when we introduce the adaptive version of our method.) When $\alpha$ is small, all features are selected and contribute to defining the cluster centers. When $\alpha$ grows sufficiently large, all features coalesce at the same value, the loss-specific center $\tilde{x}_j$, and hence no features are selected and contribute towards determining the clusters. Another way of interpreting this is that the fusion penalty exactly fuses some of the rows of the cluster center $\mathbf U$, hence determining groups of rows. On the other hand, the shifted group-lasso penalty shrinks whole columns of $\mathbf U$ towards their loss-specific centers, thereby essentially removing the effect of uninformative features. Selected features are then columns of $\mathbf U$ that were not shrunken to their loss-specific centers, $\mathbf U_{.j} \neq \tilde x_j \cdot \textbf 1_n$. These selected features, then, exhibit differences across the clusters determined by the fusion penalty. Clearly, sparse convex clustering of \citet{wang2018sparse} is a special case of Gecco+ using Euclidean distances with centered data. Our approach using both a row and column penalty is also reminiscent of convex biclustering \citep{chi2017convex} which uses fusion penalties on both the rows and columns to achieve checker-board-like biclusters.
Building upon integrative generalized convex clustering in Section \ref{igecco} and our proposed feature selection penalty above, our Integrative Generalized Convex Clustering Optimization with Feature Selection (iGecco+) is formulated as follows:
\begin{align}
\minimize_{\mathbf U^{(k)}} \hspace{2mm} &\sum_{k=1}^K \pi_k \boldsymbol\ell_k(\mathbf X^{(k)},\mathbf U^{(k)}) + \gamma \sum_{i < i'} w_{ii'} \sqrt{ \sum_{k=1}^K \| \mathbf U_{i.}^{(k)} - \mathbf U_{i'.} ^{(k)} \|^2 } \nonumber \\
& + \alpha \sum_{k=1}^K \sum_{j=1}^{p_k} \zeta_j^{(k)} \| \mathbf U_{.j}^{(k)} - \tilde x_j^{(k)} \cdot \textbf 1_n \|_2 \label{eq:1}
\end{align}
Again, $\mathbf U^{(k)}$ is an $n \times p_k$ matrix and $\tilde x_j^{(k)}$ is the loss-specific center for the $j^{th}$ feature for $k^{th}$ data type. By construction, iGecco+ directly clusters mixed multi-view data and selects features from each data view simultaneously. Similarly, adaptive choice of $\zeta_j^{(k)}$ gives rise to adaptive iGecco+ which will be discussed in Section \ref{adaptive}. Detailed discussions on practical choices of tuning parameters and weights can be also found in Section \ref{practical}.
\subsection{Properties}\label{property}
In this section, we develop some properties of our methods, highlighting several advantages of our convex formulation. The corresponding proofs can be found in Section~\ref{propprove} of the Appendix.
Define the objective function in \eqref{eq:1} as $F_{\gamma,\alpha} (\mathbf U)$ where $\mathbf U = \begin{pmatrix} \mathbf U^{(1)} \cdots \mathbf U^{(K)} \end{pmatrix}$. Then due to convexity, we have the following:
\begin{restatable}{proposition}{prop1}
\label{theorem:diff1}(Global solution)
If $\boldsymbol\ell_k$ is convex for all $k$, then any minimizer of $F_{\gamma,\alpha}(\mathbf U)$, $\mathbf U^*$, is a global minimizer. If $\boldsymbol\ell_k$ is strictly convex for all $k$, then $\mathbf U^*$ is unique.
\end{restatable}
\begin{restatable}{proposition}{prop2}
\label{theorem:diff2}(Continuity with respect to data and input parameters)
The global minimizer $\mathbf U^{*}_{\bw, \mathbf \pi, \bzeta,\mathbf X}(\gamma, \alpha)$ of iGecco+ exists and depends continuously on the data, $\mathbf X$, tuning parameters $\gamma$ and $\alpha$, the weight matrix $\bw$, the loss weight $\pi_k$, and the feature weight $\zeta_j^{(k)}$.
\end{restatable}
\begin{restatable}{proposition}{prop3}
\label{theorem:diff3}(Loss-specific center)
Define $\tilde \mathbf X = \begin{pmatrix} \tilde \mathbf X^{(1)} \cdots \tilde \mathbf X^{(K)} \end{pmatrix}$ where each $j^{th}$ column of $\tilde \mathbf X^{(k)}$ equals the loss-specific center $\tilde x_j^{(k)}$. Suppose each observation corresponds to a node in a graph with an edge between nodes $i$ and $j$ whenever $w_{ij}>0$. If this graph is fully connected, then $F_{\gamma,\alpha}(\mathbf U)$ is minimized by the loss-specific center $\tilde \mathbf X$ when $\gamma$ is sufficiently large or $\alpha$ is sufficiently large.
\end{restatable}
\textbf{Remark:} As Gecco, Gecco+ and iGecco are special cases of iGecco+, it is easy to show that all of our properties hold for these methods as well.
These properties illustrate some important advantages of our convex clustering approaches. Specifically, many other widely used clustering methods are known to suffer from poor local solutions, but any minimizer of our problem will achieve a global solution. Additionally, we show that iGecco+ is continuous with respect to the data, tuning parameters, and other input parameters. Together, these two properties are very important in practice and illustrate that the global solution of our method remains stable to small perturbations in the data and input parameters. Stability is a desirable property in practice as one would question the validity of a clustering result that can change dramatically with small changes to the data or parameters.
Importantly, most popular clustering methods such as k-means, hierarchical clustering, model-based clustering, or low-rank based clustering, do not enjoy these same stability properties.
Finally in Proposition \ref{theorem:diff3}, we verify that when the tuning parameters are sufficiently large, full fusion of all observations to the loss-specific centers is achieved. Hence, our methods indeed behave as intended, achieving joint clustering of observations. We illustrate this property in Figure \ref{viz-path-all} where we apply Gecco+ to the authors data set (described fully in Section \ref{Simulation}). Here, we illustrate how our solution, $\hat \mathbf U({\gamma,\alpha})$, changes as a function of $\gamma$ and $\alpha$. This so-called ``cluster solution path" begins with each observation as its own cluster center when $\gamma$ is small and stops when all observations are fused to the loss-specific center when $\gamma$ is sufficiently large. In between, we see that observations are fusing together as $\gamma$ increases. Similarly, when $\alpha$ is small, all features are selected and as $\alpha$ increases, some of the features get fused to their loss-specific center.
\begin{figure}[ht]
\centering
\includegraphics[scale = 0.75]{author_combined_transpose4.pdf}
\caption{Regularization path of Gecco+ solutions $\hat \mathbf U({\gamma,\alpha})$ for authors data. From left to right, we increase the parameter for fusion penalty $\gamma$. From top to bottom, we increase the parameter for feature penalty $\alpha$. The interpretation of regularization path is discussed in more detail in Section \ref{property}.}
\label{viz-path-all}
\end{figure}
\subsection{Practical Considerations and Adaptive iGecco+} \label{practical}
In this section, we discuss some practical considerations for applying our method to real data. We discuss choosing user-specific inputs such as weights as well as how to select tuning parameters. In doing so, we introduce an adaptive version of our method as well.
\subsubsection{Choice of Weights and Tuning Parameters}
In practice, a good choice of fusion weights ($w_{ij}$) has been shown to enhance both computational efficiency and clustering quality of convex clustering \citep{chi2015splitting}.
It has been empirically demonstrated that using weights inversely proportional to the distances yields superior clustering performance; this approach is widely adopted in practice. Further, setting many of the weights to zero helps reduce computation cost. Considering these two, the most common weights choice for convex clustering is to use $K$-nearest-neighbors method with a Gaussian kernel. Specifically, the weight between the sample pair ($i,j$) is set as $w_{ij} = I_{ij}^k \exp(- \phi d(\mathbf X_{i.} , \mathbf X_{j.}))$, where $I_{ij}^k$ equals 1 if observation $j$ is among observation $i$'s $K$ nearest neighbors or vice versa, and 0 otherwise. However, this choice of weights based on Euclidean distances may not work well for non-Gaussian data in Gecco(+) or for mixed data in iGecco(+). To account for different data types and better measure the similarity between observations, we still adopt $K$-nearest-neighbors method with an exponential kernel, but further extend this by employing appropriate distance metrics for specific data types in the exponential kernel.
In particular, for weights in Gecco and Gecco+, we suggest using the same distance functions or deviances in the loss function of Gecco and Gecco+. For weights in iGecco and iGecco+, the challenge is that we need to employ a distance metric which measures mixed types of data. In this case, the Gower distance, which is a distance metric used to measure the dissimilarity of two observations measured in different data types \citep{gower1971general}, can address our problem. To be specific, the Gower distance between observation $i$ and $i'$ overall can be defined as $d(\mathbf X_{i.},\mathbf X_{i'.}) = \sum_{k=1}^K \sum_{j=1}^{p_k} d_{ii'j}^{(k)} \big/ \sum_{k=1}^K p_k$ where $d_{ii'j}^{(k)} = \frac{ | \mathbf X_{ij}^{(k)} - \mathbf X_{i'j}^{(k)} |}{R_j^{(k)}}$ refers to the Gower distance between observation $i$ and $i'$ for feature $j$ in data view $k$ and $ R_j^{(k)} = \max_{i , i'} | \mathbf X_{ij}^{(k)} - \mathbf X_{i'j}^{(k)} |$ is the range of feature $j$ in data view $k$. In the literature, Gower distance has been commonly used as distance metrics for clustering mixed types of data \citep{wangchamhan2017efficient,hummel2017clustering,akay2018clustering} and shown to yield superior performance than other distance metrics \citep{ali2013k,dos2015categorical}.
Alternatively, we also propose and explore using stochastic neighbor embedding weights based on symmetrized conditional probabilities \citep{maaten2008visualizing}. These have been shown to yield superior performance in high-dimensions and if there are potential outliers. Specifically, the symmetrized conditional probabilities are defined as $p_{ij} = \frac{p_{j|i} + p_{i|j}}{2n}$, where $p_{j|i} = \frac{\exp(-\phi d(\mathbf X_{i.} , \mathbf X_{j.}))}{\sum_{k \neq i} \exp(-\phi d(\mathbf X_{i.} , \mathbf X_{k.}))}$.
We propose to use the weights $w_{ij} = I_{ij}^k \cdot p_{ij}$ where $I_{ij}^k$ still equals 1 if observation $j$ is among observation $i$'s $K$ nearest neighbors or vice versa, and 0 otherwise.
Again, we suggest using distance metrics appropriate for specific data types or the Gower distance for mixed data. In empirical studies, we experimented with both weight choices and found that stochastic neighbor embedding weights tended to work better in high-dimensional settings and if there are outliers. Hence, we recommend these and employed them in our empirical investigations in Section \ref{Simulation} and \ref{realdata}.
Estimating the number of clusters in a data set is always challenging. Current literature for tuning parameter selection mainly focuses on stability selection or consensus clustering \citep{wang2010consistent,fang2012selection} and hold-out validation \citep{chi2017convex}. In this paper, we adopt hold-out validation approach for tuning parameter selection and we follow closely the approach described in \cite{chi2017convex}; we have found that this performs well in empirical studies.
For the choice of the feature selection tuning parameter, $\alpha$, we find that clustering result is fairly robust to choices of $\alpha$. Hence, we suggest using only a few possibilities for $\alpha$ and to choose the combination of $\alpha$ and $\gamma$ which jointly minimizes hold-out error or within-cluster deviance. In many cases, we know the number of clusters a priori (or have an idea of an appropriate range for the number of clusters) and we can directly choose $\alpha$ which minimizes the hold-out error or within cluster deviance for that number of clusters.
\subsubsection{Adaptive Gecco+ and iGecco+ to Weight Features}\label{adaptive}
Finally, we consider how to specify the feature weights, $\zeta_j$ used in the shifted group-lasso penalty. While employing these weights are not strictly necessary, we have found, as did \citet{wang2018sparse}, that like the fusion weights, well-specified $\zeta_j$'s can both improve performance and speed computation. But unlike the fusion weights where we can utilize the pairwise distances, we don't have prior information on which features may be relevant in clustering. Thus, we propose to use an adaptive scheme that first fits the iGecco+ with no feature weights and uses this initial estimate to define feature importance for use in weights. This is similar to many adaptive approaches in the literature \citep{zou2006adaptive,wang2018sparse}.
Our adaptive iGecco+ approach is given in Algorithm~\ref{alg:adaptive-feature}; this applies to adaptive Gecco+ as a special case as well. We assume that the number of clusters (or a range of the number of clusters) is known a priori.
We begin by fitting iGecco+ with $\alpha = 1$ and uniform feature weights $\zeta_j^{(k)}=1$. We then find the $\gamma$ which gives the desired number of clusters, yielding the initial estimate, $\hat \mathbf U^{(k)}$. (Alternatively, we can use hold-out validation to select $\gamma$.) Next, we use this initial estimate to adaptively weight features by proposing the following weights: $\zeta_j^{(k)} = 1/\| \hat \mathbf U_{.j}^{(k)} - \tilde x_j^{(k)} \cdot \textbf 1_n \|_2$. These weights place a large penalty on noise features as $\| \hat \mathbf U_{.j}^{(k)} - \tilde x_j^{(k)} \cdot \textbf 1_n \|_2$ is close to zero in this case.
We also notice that noise features impact the distances used in the fusion weights as well. Hence, we suggest updating the distances adaptively by using the selected features to better measure the similarities between observations. To this end, we propose a new scheme to compute weighted Gower distances. First, we scale the features within each data view so that informative features in different data views contribute equally and on the same scale. Then, we employ the inverse of $\pi_k$, i.e., the null deviance, to weight the distances from different data types, resulting in an aggregated and weighted Gower distance, $\hat d(\mathbf X_{i.},\mathbf X_{i'.})$ as further detailed in Algorithm \ref{alg:adaptive-feature}.
Note that if the clustering signal from one particular data type is weak and there are few informative features for this data type, then our weighting scheme will down-weight this entire data type in the weighted Gower distance. In practice, our adaptive iGecco+ scheme works well as evidenced in our empirical investigations in the next sections.
\begin{algorithm}[H]
\caption{Adaptive iGecco+}
\label{alg:adaptive-feature}
\begin{algorithmic}
\STATE {1. Fit iGecco+ with $\alpha = 1$ and a sequence of $\gamma$}
\STATE {2. Find $\gamma$ which gives desired number of clusters ; Get the estimate $\hat \mathbf U^{(k)}$}\\
\STATE {3. Update the feature weights $\hat \zeta_j^{(k)} = \frac{1}{1+\|\hat \mathbf U_{.j}^{(k)} - \tilde x_j^{(k)} \cdot \textbf 1_n \|_2 }$ and fusion weights $\hat w_{ij} = I_{ij}^k \cdot \exp(-\phi \hat d(\mathbf X_{i.},\mathbf X_{i'.}))$ where $\hat d(\mathbf X_{i.},\mathbf X_{i'.}) = \sum_{k=1}^K \sum_{j=1}^{p_k}
\frac{\|\mathbf U_{.j}^{(k)} - \tilde x_j^{(k)} \cdot \textbf 1_n \|_2}{\max_j \|\mathbf U_{.j}^{(k)} - \tilde x_j^{(k)} \cdot \textbf 1_n \|_2} \cdot \frac{1}{\pi_k} \cdot d_{ii'j}^{(k)}$.}
\STATE {4. Fit adaptive iGecco+ with $\hat \bzeta$ and $\tilde \bw$; }
\end{algorithmic}
\end{algorithm}
\section{iGecco+ Algorithm}
In this section, we introduce our algorithm to solve iGecco+, which can be easily extended to Gecco, Gecco+ and iGecco. We first propose a simple, but rather slow ADMM algorithm as a baseline approach. To save computation cost, we further develop a new multi-block ADMM-type procedure using inexact one-step approximation of the sub-problems. Our algorithm is novel from optimization perspective as we extend the multi-block ADMM to higher number of blocks and combine it with the inexact sub-problem solve ADMM literature, which often results in major computational savings.
\subsection{Full ADMM to Solve iGecco+ (Naive Algorithm)}
Given the shifted group-lasso and fusion penalties along with general losses, developing an optimization routine for iGecco+ method is less straight-forward than convex clustering or sparse convex clustering. In this section, we propose a simple ADMM algorithm to solve iGecco+ as a baseline algorithm and point out its drawbacks.
The most common approach to solve problems with more than two non-smooth functions is via multi-block ADMM \citep{lin2015global,deng2017parallel}, which decomposes the original problem into several smaller sub-problems and solves them in parallel at each iteration. \cite{chen2016direct} established a sufficient condition for the convergence of three-block ADMM. We develop a multi-block ADMM approach to fit our problem for certain types of losses and prove its convergence.
We first recast iGecco+ problem \eqref{eq:1} as the equivalent constrained optimization problem:
\begin{align*}
&\minimize_{\mathbf U^{(k)},\mathbf V} \hspace{5mm} \sum_{k=1}^K \pi_k \boldsymbol\ell_k(\mathbf X^{(k)},\mathbf U^{(k)}) + \gamma \underbrace{ \bigg(\sum_{l \in \mathcal E} w_l \|\mathbf V_{l.}\|_2\bigg)}_{P_1(\mathbf V;\bw)}+ \alpha \sum_{k=1}^K \sum_{j=1}^{p_k} \zeta_j^{(k)} \| \mathbf U_{.j}^{(k)} - \tilde x_j^{(k)} \cdot \textbf 1_n \|_2 \\
& \subto \hspace{5mm} \mathbf D \begin{bmatrix} \mathbf U^{(1)} & \cdots & \mathbf U^{(K)} \end{bmatrix} - \mathbf V = \mathbf 0
\end{align*}
Recently, \cite{weylandt2019dynamic} derived the ADMM for convex clustering in matrix form and we adopt similar approach. We index a centroid pair by $l = (l_1, l_2)$ with $l_1 < l_2$, define the set of edges over the non-zero weights $\mathcal E = \{l = (l_1,l_2) : w_l > 0\}$, and introduce a new variable $\mathbf V = \begin{bmatrix} \mathbf V^{(1)} & \cdots & \mathbf V^{(K)} \end{bmatrix} \in \mathbb R^{|\mathcal E | \times \sum {p_k}}$ where $\mathbf V^{(k)}_{l.} = \mathbf U^{(k)}_{l_1.} - \mathbf U^{(k)}_{l_2.} $ to account for the difference between the two centroids. Hence $\mathbf V^{(k)}$ is a matrix containing the pairwise differences between connected rows of $\mathbf U^{(k)}$ and the constraint is equivalent to $\mathbf D \mathbf U^{(k)} - \mathbf V^{(k)} = \mathbf 0$ for all $k$; $\mathbf D \in \mathbb R^{ |\mathcal E | \times n} $ is the directed difference matrix corresponding to the non-zero fusion weights. It is clear the $\mathbf V$ sub-problem has closed-form solution for each iteration. We give general-form multi-block ADMM (Algorithm~\ref{alg:full-admm}) to solve iGecco+. Here $\text{prox}_{h(\cdot)} (\bx) = \argmin_{\bz} \frac{1}{2} \|\bx - \bz\|_2^2 + h(\bz)$ is the proximal mapping of function $h$.
\begin{algorithm}[h]
\caption{General Multi-block Algorithm for iGecco+}
\label{alg:full-admm}
\begin{algorithmic}
\WHILE{not converged}
\FOR{all $k = 1,\cdots,K$}
\STATE $\mathbf U^{ (k) } =
\argmin \limits_{\mathbf U} \hspace{1mm} \pi_k \boldsymbol\ell_k(\mathbf X^{(k)},\mathbf U) + \frac{\rho}{2} \| \mathbf D \mathbf U - \mathbf V^{(k)} + \bLambda^{(k)} \|_F^2 + \alpha \sum_{j=1}^{p_k} \zeta_j^{(k)} \| \mathbf U_{.j} - \tilde x_j^{(k)} \cdot \textbf 1_n \|_2 $
\ENDFOR
\STATE $\mathbf V = \text{prox}_{\gamma /\rho P_1(\cdot; \bw )} (\begin{bmatrix} \mathbf D \mathbf U^{(1)} + \bLambda^{(1)} & \cdots & \mathbf D \mathbf U^{(K)} + \bLambda^{(K)} \end{bmatrix} )$
\STATE $\bLambda^{(k)} = \bLambda^{(k)} + ( \mathbf D \mathbf U^{(k)} - \mathbf V^{(k)} ) $ \hspace{5mm} for all $k$
\ENDWHILE
\end{algorithmic}
\end{algorithm}
Notice that there is no closed-form solution for the $\mathbf U^{(k)}$ sub-problem for general losses. Typically we need to apply an inner optimization routine to solve the $\mathbf U^{(k)}$ sub-problem until full convergence. In the next section, we seek to speed up this algorithm by using $\mathbf U^{(k)}$ sub-problem approximations. But, first we propose two different approaches to fully solve the $\mathbf U^{(k)}$ sub-problem based on specific loss types and then use these to develop a one-step update to solve the sub-problem approximately with guaranteed convergence.
\subsection{iGecco+ Algorithm}
We have introduced Algorithm \ref{alg:full-admm}, a simple baseline ADMM approach to solve iGecco+. In this section, we consider different ways to solve the $\mathbf U^{(k)}$ sub-problem in Algorithm \ref{alg:full-admm}. First, based on specific loss types (differentiable and non-differentiable), we propose two different algorithms to solve the $\mathbf U^{(k)}$ sub-problem to full convergence. These approaches, however, are rather slow for general losses as there is no closed-form solution which results in nested iterative updates. To address this and in connection with current literature on variants of ADMM with sub-problem approximations, we propose iGecco+ algorithm, a multi-block ADMM which solves the sub-problems approximately by taking a single one-step update. We prove convergence of this general class of algorithms, a novel result in the optimization literature
\subsubsection{Differentiable Case}
When the loss $\boldsymbol\ell_k$ is differentiable, we consider solving the $\mathbf U^{(k)}$ sub-problem with proximal gradient descent, which is often used when the objective function can be decomposed into a differentiable and a non-differentiable function. While there are many other possible optimization routines to solve the $\mathbf U^{(k)}$ sub-problem, we choose proximal gradient descent as there is existing literature proving convergence of ADMM algorithms with approximately solved sub-problems using proximal gradient descent \citep{liu2013linearized,lu2016fast}. We will discuss in detail how to approximately solve the sub-problem by taking a one-step approximation in Section \ref{inexact}. Based upon this, we propose Algorithm~\ref{alg:full-diff}, which solves the $\mathbf U^{(k)}$ sub-problem by running full iterative proximal gradient descent to convergence. Here $P_2(\tilde \mathbf U^{(k)};\bzeta^{(k)}) = \sum_{j=1}^{p_k} \zeta_j^{(k)} \|\tilde \mathbf U^{(k)}_{.j}\|_2$.
\begin{algorithm}[h]
\caption{$\mathbf U^{(k)}$ sub-problem for differentiable loss $\boldsymbol\ell_{k}$ (Proximal gradient):}
\label{alg:full-diff}
\begin{algorithmic}
\WHILE{not converged}
\STATE $\mathbf U^{(k)} = \prox_{s_{k} \cdot \alpha P_2(\cdot;\bzeta^{(k)})} \begin{footnotesize} \big( \mathbf U^{(k)} - \tilde \mathbf X^{(k)} - s_{k} \cdot [
\pi_k \nabla \boldsymbol\ell_{k}(\mathbf X^{(k)},\mathbf U^{(k)} ) + \rho \mathbf D^T (\mathbf D \mathbf U^{(k)} - \mathbf V^{(k)} + \bLambda^{(k)} )] \big) + \tilde \mathbf X^{(k)} \end{footnotesize} $
\ENDWHILE
\end{algorithmic}
\end{algorithm}
In Algorithm~\ref{alg:full-diff} and typically in general (proximal gradient) descent algorithms, we need to choose an appropriate step size $s_k$ to ensure convergence. Usually we employ a fixed step size by computing the Lipschitz constant as in the squared error loss case; but in our method, it is hard to compute the Lipschitz constant
for most of our general losses. Instead, we suggest using backtracking line search procedure proposed by \cite{beck2009gradient,parikh2014proximal}, which is a common way to determine step size with guaranteed convergence in optimization.
Further, we find decomposing the $\mathbf U^{(k)}$ sub-problem to $p_k$ separate $\mathbf U_{.j}^{(k)}$ sub-problems brings several advantages such as (i) better convergence property (than updating $\mathbf U^{(k)}$'s all together) due to adaptive step size for each $\mathbf U_{.j}^{(k)}$ sub-problem and (ii) less computation cost by solving each in parallel. Hence, in this case, we propose to use proximal gradient for each separate $\mathbf U_{.j}^{(k)}$ sub-problem. To achieve this, we assume that the loss is elementwise, which is satisfied by every deviance-based loss. Last, as mentioned, there are many other possible ways to solve the $\mathbf U^{(k)}$ sub-problem than proximal gradient, such as ADMM. We find that when the loss is squared Euclidean distances or the hessian of the loss can be upper bounded by a fixed matrix, this method saves more computation.
We provide all implementation details discussed above in Section~\ref{diffdetail} of the Appendix.
\subsubsection{Non-differentiable Case}
When the loss $\boldsymbol\ell_k$ is non-differentiable, we can no longer adopt the proximal gradient method to solve the $\mathbf U^{(k)}$ sub-problem as the objective is now a sum of more than one separable non-smooth function. To address this, as mentioned, we can use multi-block ADMM; in this case, we introduce new blocks for the non-smooth functions and hence develop a full three-block ADMM approach to fit our problem.
To augment the non-differentiable term, we assume that our loss function can be written as $\boldsymbol\ell_k(\mathbf X^{(k)},\mathbf U^{(k)}) = f_k(g_k(\mathbf X^{(k)},\mathbf U^{(k)}))$ where $f_k$ is convex but non-differentiable and $g_k$ is affine. This condition is satisfied by all distance-based losses with $g_k(\mathbf X^{(k)},\mathbf U^{(k)}) = \mathbf X^{(k)} - \mathbf U^{(k)}$; for example, for Manhattan distances, we have $f_k(\mathbf Z) = \sum_{j=1}^p \|\bz_j\|_1 = \|\text{vec}(\mathbf Z)\|_1$, and $g_k(\mathbf X,\mathbf U) = \mathbf X -\mathbf U$.
The benefit of doing this is that now the $\mathbf U^{(k)}$ sub-problem has closed-form solution. Particularly, we can rewrite the $\mathbf U^{(k)}$ sub-problem as
\begin{align*}
&\minimize_{\mathbf U^{(k)},\mathbf V} \hspace{5mm} \sum_{k=1}^K \pi_k f_k(\mathbf Z^{(k)} ) + \frac{\rho}{2} \| \mathbf D \mathbf U - \mathbf V^{(k)} + \bLambda^{(k)} \|_F^2 + \alpha \sum_{k=1}^K \bigg( \underbrace{ \sum_{j=1}^{p_k} \zeta_j^{(k)} \| \br_j^{(k)} \|_2}_{P_2(\mathbf R^{(k)};\bzeta^{(k)})} \bigg) \\
& \subto \hspace{5mm} \mathbf X^{(k)} - \mathbf U^{(k)} = \mathbf Z^{(k)} , \hspace{5mm} \mathbf U^{(k)} - \tilde \mathbf X^{(k)} = \mathbf R^{(k)}
\end{align*}
where $ \tilde \mathbf X^{(k)}$ is an $n \times p_k$ matrix with $j^{th}$ columns equal to scalar $\tilde x_j^{(k)}$.
It is clear that we can use multi-block ADMM to solve the problem above and each primal variable has simple update with closed-form solution. We propose Algorithm~\ref{alg:full-non-diff}, a full, iterative multi-block ADMM, to solve the $\mathbf U^{(k)}$ sub-problem when the loss is a non-differentiable distance-based function. Algorithm~\ref{alg:full-non-diff} applies to iGecco+ with various distances such as Manhattan, Minkowski and Chebychev distances and details are given in Section~\ref{nondiffdetail} of the Appendix.
\begin{algorithm}[h]
\caption{$\mathbf U^{(k)}$ sub-problem for non-differentiable distance-based loss $\boldsymbol\ell_{k}$ (Multi-block ADMM):}
\label{alg:full-non-diff}
\begin{algorithmic}
\STATE {\bfseries Precompute:} Difference matrix $\mathbf D$, $\mathbf M = (\mathbf D^T \mathbf D + 2 \mathbf I)^{-1}$.
\WHILE{not converged}
\STATE $\mathbf U^{(k)} =
\mathbf M(\mathbf D^T (\mathbf V^{(k)} - \bLambda^{(k)} ) + \tilde \mathbf X^{(k)} + \mathbf R^{(k)} - \mathbf N^{(k)} + \mathbf X^{(k)} - \mathbf Z^{(k)} + \mathbf \Psi^{(k)} )$
\STATE $\mathbf Z^{(k)} = \text{prox}_{\pi_k f_k/\rho} (\mathbf X^{(k)} - \mathbf U^{(k)} + \mathbf \Psi^{(k)} )$
\STATE $\mathbf R^{(k)} = \text{prox}_{\alpha /\rho P_2(\cdot; \bzeta^{(k)} )} (\mathbf U^{(k)} - \tilde \mathbf X^{(k)} + \mathbf N^{(k)} ) $
\STATE $\mathbf \Psi^{(k)} = \mathbf \Psi^{(k)} + (\mathbf X^{(k)} - \mathbf U^{(k)} - \mathbf Z^{(k)}) $
\STATE $\mathbf N^{(k)} = \mathbf N^{(k)} + ( \mathbf U^{(k)} - \tilde \mathbf X^{(k)} - \mathbf R^{(k)} ) $
\ENDWHILE
\end{algorithmic}
\end{algorithm}
\subsubsection{iGecco+ Algorithm: Fast ADMM with Inexact One-step Approximation to the Sub-problem}\label{inexact}
Notice that for both Algorithm~\ref{alg:full-diff} and \ref{alg:full-non-diff}, we need to run them iteratively to full convergence in order to solve the $\mathbf U^{(k)}$ sub-problem for each iteration, which is dramatically slow in practice. To address this in literature, many have proposed variants of ADMM with guaranteed convergence that find an inexact, one-step, approximate solution to the sub-problem (without fully solving it); these include the generalized ADMM \citep{deng2016global}, proximal ADMM \citep{shefi2014rate,banert2016fixing} and proximal linearized ADMM \citep{liu2013linearized,lu2016fast}. Thus, we propose to solve the $\mathbf U^{(k)}$ sub-problem approximately by taking a single one-step update of the algorithm for both types of losses and prove convergence. For the differentiable loss case, we propose to apply the proximal linearized ADMM approach while for the non-differentiable case, we show that taking a one-step update of Algorithm~\ref{alg:full-non-diff} is equivalent to applying a four-block ADMM to the original problem and we provide a sufficient condition for the convergence of four-block ADMM. Our algorithm, to the best of our knowledge, is the first to incorporate higher-order multi-block ADMM and inexact ADMM with a one-step update to solve sub-problems for general loss functions.
When the loss is differentiable,
as mentioned in Algorithm~\ref{alg:full-diff}, one can use full iterative proximal gradient to solve the $\mathbf U_{.j}^{(k)}$ sub-problem, which however, is computationally burden-some.
To avoid this, many proposed variants of ADMM which find approximate solutions to the sub-problems. Specifically, closely related to our problem here, \citet{liu2013linearized,lu2016fast} proposed proximal linearized ADMM which solves the sub-problems efficiently by linearizing the differentiable part and then applying proximal gradient due to the non-differentiable part. We find their approach fits into our problem and hence develop a proximal linearized 2-block ADMM to solve iGecco+ when the loss $\boldsymbol\ell_k$ is differentiable and gradient is Lipschitz continuous. It can be shown that applying proximal linearized 2-block ADMM to Algorithm~\ref{alg:full-admm} is equivalent to taking
a one-step update of Algorithm~\ref{alg:full-diff} along with $\mathbf V$ and $\bLambda$ update in Algorithm~\ref{alg:full-admm}.
In this way, we avoid running full iterative proximal gradient updates to convergence for the $\mathbf U^{(k)}$ sub-problem as in Algorithm~\ref{alg:full-diff} and hence save computation cost.
When the loss is non-differentiable, we still seek to take an one-step update to solve the $\mathbf U^{(k)}$ sub-problem. We achieve this by noticing that
taking a one-step update of Algorithm~\ref{alg:full-non-diff} along with $\mathbf V$ and $\bLambda$ update in Algorithm~\ref{alg:full-admm} is equivalent to applying multi-block ADMM to the original iGecco+ problem recast as follows (for non-differentiable distance-based loss):
\begin{align*}
&\minimize_{\mathbf U^{(k)},\mathbf V} \hspace{5mm} \sum_{k=1}^K \pi_k f_k(\mathbf Z^{(k)} ) + \gamma \underbrace{ \bigg(\sum_{l \in \mathcal E} w_l \|\mathbf V_{l.}\|_2\bigg)}_{P_1(\mathbf V;\bw)}+ \alpha \sum_{k=1}^K \bigg( \underbrace{ \sum_{j=1}^{p_k} \zeta_j^{(k)} \| \br_j^{(k)} \|_2}_{P_2(\mathbf R^{(k)};\bzeta^{(k)})} \bigg) \\
& \subto \hspace{5mm} \mathbf X^{(k)} - \mathbf U^{(k)} = \mathbf Z^{(k)} , \hspace{5mm} \mathbf D \begin{bmatrix} \mathbf U^{(1)} & \cdots & \mathbf U^{(K)} \end{bmatrix} - \mathbf V = \mathbf 0, \hspace{5mm} \mathbf U^{(k)} - \tilde \mathbf X^{(k)} = \mathbf R^{(k)}
\end{align*}
Typically, general higher-order multi-block ADMM algorithms do not always converge, even for convex functions \citep{chen2016direct}. We prove convergence of our algorithm and establish a novel convergence result by casting the iGecco+ with non-differentiable losses as a four-block ADMM, proposing a sufficent condition for convergence of higher-order multi-block ADMMs, and finally showing that our problem satisfies this condition. (Details are given in the proof of Theorem \ref{theorem:inexact-full} in Appendix~\ref{algcovproof}.) Therefore, taking a one-step update of Algorithm~\ref{alg:full-non-diff} converges for iGecco+ with non-differentiable losses.
So far, we have proposed inexact-solve one-step update approach for both differentiable loss and non-differentiable loss case. For mixed type of losses, we combine those two algorithms and this gives Algorithm~\ref{alg:inexact-admm}, a multi-block ADMM algorithm with inexact one-step approximation to the $\mathbf U^{(k)}$ sub-problem to solve iGecco+. We also establish the following convergence result.
\begin{algorithm}[h]
\caption{iGecco+ Algorithm}
\label{alg:inexact-admm}
\begin{algorithmic}
\WHILE{not converged}
\FOR{all $k = 1,\cdots,K$}
\STATE Update $\mathbf U^{(k)}$:
\IF{$\boldsymbol\ell_k$ is differentiable}
\STATE Take a one-step update of Algorithm~\ref{alg:full-diff}
\ELSIF{$\boldsymbol\ell_k$ is non-differentiable}
\STATE Take a one-step update of Algorithm~\ref{alg:full-non-diff}
\ENDIF
\ENDFOR
\STATE $\mathbf V = \text{prox}_{\gamma /\rho P_1(\cdot; \bw )} (\begin{bmatrix} \mathbf D \mathbf U^{(1)} + \bLambda^{(1)} & \cdots & \mathbf D \mathbf U^{(K)} + \bLambda^{(K)} \end{bmatrix} )$
\STATE $\bLambda^{(k)} = \bLambda^{(k)} + ( \mathbf D \mathbf U^{(k)} - \mathbf V^{(k)} ) $ \hspace{5mm} for all $k$
\ENDWHILE
\end{algorithmic}
\end{algorithm}
\begin{restatable}{theorem}{inexact-full}
\label{theorem:inexact-full}(iGecco+ convergence)
If $\boldsymbol\ell_k$ is convex for all $k$, Algorithm~\ref{alg:inexact-admm} converges to a global solution. In addition, if each $\boldsymbol\ell_k$ is strictly convex, it converges to the unique global solution.
\end{restatable}
\textbf{Remark:}
Our corresponding Theorem \ref{theorem:inexact-full} establishes a novel convergence result as it is the first to show the convergence of four-block or higher ADMM using approximate sub-problems for both differentiable and non-differentiable losses.
It is easy to see that Algorithm \ref{alg:inexact-admm} can be applied to solve other Gecco-related methods as special cases. When $K=1$, Algorithm~\ref{alg:inexact-admm} gives the algorithm to solve Gecco+. When $\alpha= 0$, Algorithm~\ref{alg:inexact-admm} gives the algorithm to solve iGecco+. When $K=1$ and $\alpha = 0$, Algorithm~\ref{alg:inexact-admm} gives the algorithm to solve Gecco.
To conclude this section, we compare the convergence results of both full ADMM and inexact ADMM with one-step update in the sub-problem to solve Gecco+ ($n=120$ and $p=210$) in Figure \ref{conv_plot}. The left plots show the number of iterations needed to yield optimization convergence while the right plots show computation time. We see that Algorithm \ref{alg:inexact-admm} (one-step update to solve the sub-problem) saves much more computational time than Algorithm \ref{alg:full-admm} (full updates of the sub-problem). It should be pointed out that though Algorithm \ref{alg:inexact-admm} takes more iterations to converge due to inexact approximation for each iteration, we still reduce computation time dramatically as the computation time per iteration is much less than the full-solve approach.
\begin{figure}[ht]
\vskip 0.2in
\begin{center}
\centerline{\includegraphics[scale = 0.67]{PLL_conv.png}}
\centerline{\includegraphics[scale = 0.67]{Man_conv.png}}
\caption{Comparisons of full ADMM and one-step inexact ADMM algorithm to solve Gecco+ with Poisson log-likelihood (top panel, differentiable loss) and Gecco+ with Manhattan distances (bottom panel, non-differentiable loss). Left plots show the number of iterations needed to converge while right plots show computation time. Algorithm with one-step update to solve the sub-problem saves much more computational time.}
\label{conv_plot}
\end{center}
\vskip -0.2in
\end{figure}
\section{Simulation Studies}\label{Simulation}
In this section, we first evaluate performance of Gecco+ against existing methods on non-Gaussian data. Next we compare iGecco+ with other methods on mixed multi-view data.
\subsection{Non-Gaussian Data} \label{Gecco+_sim}
In this subsection, we evaluate the performance of Gecco and (adaptive) Gecco+ by comparing it with k-means, hierarchical clustering and sparse convex clustering. For simplicity, we have the following naming convention for all methods: loss type name + Gecco(+). For example, Poisson Deviance Gecco+ refers to Generalized Convex Clustering with Feature Selection using Poisson deviance.
Sparse CC refers to sparse convex clustering using Euclidean distances. We measure the accuracy of clustering results using adjusted Rand index \citep{hubert1985comparing}.
The adjusted Rand index is the corrected-for-chance version of the Rand index, which is used to measure the agreement between the estimated clustering assignment and the true group label. A larger adjusted Rand index implies a better clustering result. For all methods we consider, we assume oracle number of clusters for fair comparisons. For our Gecco+, we choose the largest $\alpha$ which minimizes within cluster variance or hold-out error.
Each simulated data set is comprised of $n=120$ observations with 3 clusters. Each cluster has an equal number of observations. Only the first 10 features are informative while the rest are noise. We consider the following simulation scenarios.
\begin{itemize}
\item S1: Spherical data with outliers
The first 10 informative features in each group are generated from a Gaussian distribution with different $\mu_k$'s for each class. Specifically, the first 10 features are generated from $N(\mu_k, \mathbf I_{10})$ where $\mu_1= (-2.5\cdot \mathbf 1_{5}^T, \mathbf 0_{5}^T )^T$, $\mu_2= (\mathbf 0_{5}^T , 2.5\cdot \mathbf 1_{5}^T)^T$, $\mu_3= (2.5\cdot \mathbf 1_{5}^T,\mathbf 0_{5}^T)^T$. The outliers in each class are generated from a Gaussian distribution with the same mean centroid $\mu_k$ but with higher variance, i.e, $N(\mu_k, 5 \cdot \mathbf I_{10})$. The remaining noise features are generated from $N(0,1)$.
In the first setting (S1A), the number of noise features ranges in $25,50,75,\cdots$ up to 225 with the proportion of the number of outliers fixed ( = 5\%). We also consider the setting when the variance of noise features increases with number of features fixed $p=200$ and number of outliers fixed (S1B) and high-dimensional setting where $p$ ranges from $250,500,750$ to 1000 (S1C).
\item S2: Non-spherical data with three half moons
Here we consider the standard simulated data of three interlocking half moons as suggested by \cite{chi2015splitting} and \cite{wang2018sparse}. The first 10 features are informative in which each pair makes up two-dimensional three interlocking half moons. We randomly select 5\% of the observations in each group and make them outliers. The remaining noise features are generated from $N(0,1)$.
The number of noise features ranges from $25,50,75,\cdots$ up to 225. In both S1 and S2, we compare Manhattan Gecco+ with other existing methods.
\item S3: Count-valued data
The first 10 informative features in each group are generated from a Poisson distribution with different $\mu_k$'s $(i=1,2,3)$ for each class. Specifically, $\mu_1 = 1 \cdot \mathbf 1_{10}$, $\mu_2 = 4 \cdot \mathbf 1_{10}$, $\mu_3 = 7 \cdot \mathbf 1_{10}$. The remaining noise features are generated from a Poisson distribution with the same $\mu$'s which are randomly generated integers from 1 to 10. The number of noise features ranges from $25,50,75,\cdots$ up to 225.
\end{itemize}
\begin{figure}[ht]
\vskip 0.2in
\begin{center}
\centerline{\includegraphics[width=\textwidth,height = 11cm]{Rplot_v0827_bw3.pdf}}
\caption{Simulation results of non-Gaussian data: (S1A) We increase number of noise features for spherical data with outliers; (S2) We increase number of noise features for non-spherical data with outliers; (S3) We increase number of noise features for count-valued data; (S1B) We increase noise level for spherical data with outliers; (S1C) We further increase number of noise features for spherical data with outliers in high dimensions. The adaptive Gecco+ outperforms existing methods in high dimensions.}
\label{sim_plot}
\end{center}
\vskip -0.2in
\end{figure}
We summarize simulation results in Figure~\ref{sim_plot}.
We find that for spherical data with outliers, adaptive Manhattan Gecco+ performs the best in high dimensions. Manhattan Gecco performs well in low dimensions but poorly as number of noisy features increases. Manhattan Gecco+ performs well as the dimension increases, but adaptive Manhattan Gecco+ outperforms the former as it adaptively penalizes the features, meaning that noisy features quickly get zeroed out in the clustering path and that only the informative features perform important roles in clustering. We see that, without adaptive methods, we do not achieve the full benefit of performing feature selection. As we perform adaptive Gecco+, we show vast improvement in clustering purity as the number of noise features grows where regular Gecco performs poorly. Sparse convex clustering performs the worst as it tends to pick outliers as singleton classes. Our simulation results also show that adaptive Manhattan Gecco+ works well for non-spherical data by selecting the correct features. For count data, all three adaptive Gecco+ methods perform better than k-means, hierarchical clustering and sparse convex clustering. We should point out that there are several linkage options for hierarchical clustering. For visualization purposes, we only show the linkage with the best and worst performance instead of all the linkages. Also we use the appropriate data-specific distance metrics in hierarchical clustering. Table~\ref{vs-accuracy-F} shows the variable selection accuracy of sparse convex clustering and adaptive Gecco+ in terms of F$_1$ score. In all scenarios, we fix $p=225$. We see that adaptive Gecco+ selects the correct features, whereas sparse convex clustering performs poorly.
\begin{table}[ht]
\vskip 0.15in
\begin{center}
\begin{tabular}{l c c c}
\toprule
\multirow{1}{*}{Method} &
\multicolumn{1}{c}{Scenario 1 (A)} &
\multicolumn{1}{c}{Scenario 2} &
\multicolumn{1}{c}{Scenario 3} \\
\midrule
Sparse Convex Clustering & 0.37 (3.1e-2) & 0.25 (2.4e-2) & 0.14 (7.2e-3) \\
Adaptive Gecco+ & 0.97 (1.9e-2) & 0.99 (1.0e-2) & 0.81 (8.0e-2)\\
\bottomrule
\end{tabular}
\caption{Comparisons of F$_1$ score for adaptive Gecco+ and sparse convex clustering}
\label{vs-accuracy-F}
\end{center}
\vskip -0.1in
\end{table}
\subsection{Multi-View Data}
In this subsection, we evaluate the performance of iGecco and (adaptive) iGecco+ on mixed multi-view data by comparing it with hierarchical clustering, iCluster+ \citep{mo2013pattern} and Bayesian Consensus Clustering \citep{lock2013bayesian}. Again, we measure the accuracy of clustering results using the adjusted Rand index \citep{hubert1985comparing}.
As before, each simulated data set is comprised of $n=120$ observations with 3 clusters. Each cluster has an equal number of observations. Only the first 10 features are informative while the rest are noise. We have three data views consisting of continuous data, count-valued data and binary/proportion-valued data. We investigate different scenarios with different dimensions for each data view and consider the following simulation scenarios:
\begin{itemize}
\item S1: Spherical data with $p_1 = p_2 = p_3 = 10$
\item S2: Three half-moon data with $p_1 = p_2 = p_3 = 10$
\item S3: Spherical data with $p_1 = 200$, $p_2= 100$, $p_3 = 50$
\item S4: Three half-moon data with $p_1 = 200$, $p_2= 100$, $p_3 = 50$
\item S5: Spherical data with $p_1 = 50$, $p_2= 200$, $p_3 = 100$
\item S6: Three half-moon data with $p_1 = 50$, $p_2= 200$, $p_3 = 100$
\end{itemize}
We employ a similar simulation setup as in Section \ref{Gecco+_sim} to generate each data view. The difference is that here for informative features, we increase the within-cluster variance for Gaussian data and decrease difference of cluster mean centroids $\mu_k$'s for binary and count data so that there is overlap between different clusters. Specifically, for spherical cases, Gaussian data is generated from $N(\mu_k, 3 \cdot \mathbf I_{10})$; count data is generated from Poisson with different $\mu_k$'s ($\mu_1 = 2$, $\mu_2 = 4$, $\mu_3 = 6$, etc); binary data is generated from Bernoulli with different $\mu_k$'s ($\mu_1 = 0.5$, $\mu_2 = 0.2$, $\mu_3 = 0.8$, etc). For half-moon cases, continuous data is simulated with larger noise and the count and proportion-valued data is generated via a copula transform. In this manner, we have created a challenging simulation scenario where accurate clustering results cannot be achieved by considering only a single data-view.
Again, for fair comparisons across methods, we assume the oracle number of clusters.
When applying iGecco methods, we employ Euclidean distances for continuous data, Manhattan distances for count-valued data and Bernoulli log-likelihood for binary or proportion-valued data. We use the latter two losses as they perform well compared with counterpart losses in Gecco+ and demonstrate faster computation speed. Again, we choose the largest $\alpha$ that minimizes within-cluster deviance.
Simulation results in Table~\ref{iGecco_rand} and Table~\ref{iGecco+_rand} show that our methods perform better than existing methods. In low dimensions, iGecco performs comparably with iCluster and Bayesian Consensus Clustering for spherical data. For non-spherical data, iGecco performs much better. For high dimensions, iGecco+ performs better than iGecco while adaptive iGecco+ performs the best as it achieves the full benefit of feature selection.
\begin{table}[ht]
\vskip 0.15in
\begin{center}
\begin{tabular}{ |l|c|c|}
\hline
Method & Scenario 1 & Scenario 2 \\ \hline
\multirow{1}{*}{Hclust: $\mathbf X_1$} & 0.35 (2.9e-2) & 0.54 (1.3e-2) \\
\multirow{1}{*}{Hclust: $\mathbf X_2$} & 0.53 (4.6e-2) & 0.61 (4.0e-2) \\
\multirow{1}{*}{Hclust: $\mathbf X_3$} & 0.52 (2.2e-2) & 0.70 (3.0e-2) \\
\multirow{1}{*}{Hclust: $[\mathbf X_1 \mathbf X_2 \mathbf X_3]$ - Euclidean} & 0.68 (4.7e-2) & 0.66 (4.4e-2) \\
\multirow{1}{*}{Hclust: $[\mathbf X_1 \mathbf X_2 \mathbf X_3]$ - Gower } & 0.86 (1.5e-2) & 0.84 (4.0e-2) \\
\multirow{1}{*}{iCluster+ with $\lambda = 0$} & 0.90 (1.6e-2) & 0.70 (8.0e-3) \\
\multirow{1}{*}{Bayesian Consensus Clustering} & \textbf{0.95 (1.2e-2)} & 0.63 (1.0e-2)\\
\multirow{1}{*}{iGecco} & \textbf{0.93 (4.7e-3)} & \textbf{1.00 (0.0e-0)} \\ \hline
\end{tabular}
\caption{Comparisons of adjusted Rand index for mixed multi-view data}
\label{iGecco_rand}
\end{center}
\vskip -0.1in
\end{table}
\begin{table}[ht]
\vskip 0.15in
\begin{center}
\begin{small}
\scalebox{0.9}{
\begin{tabular}{ |l|c|c|c|c|}
\hline
Method & Scenario 3 & Scenario 4 & Scenario 5 & Scenario 6 \\ \hline
\multirow{1}{*}{Hclust: $\mathbf X_1$} & 0.57 (1.8e-2) & 0.57 (1.4e-2) & 0.44 (2.4e-2)& 0.49 (1.7e-2)\\
\multirow{1}{*}{Hclust: $\mathbf X_2$} & 0.22 (1.9e-2) & 0.20 (1.8e-2) & 0.51 (1.7e-2)& 0.51 (2.6e-2) \\
\multirow{1}{*}{Hclust: $\mathbf X_3$} & 0.28 (1.1e-2) & 0.25 (2.6e-2) & 0.57 (2.7e-2)& 0.48 (3.3e-2)\\
\multirow{1}{*}{Hclust: $[\mathbf X_1 \mathbf X_2 \mathbf X_3]$ - Euclidean} & 0.72 (2.2e-2) & 0.43 (1.9e-2) & 0.53 (1.9e-2)& 0.56 (2.3e-2) \\
\multirow{1}{*}{Hclust: $[\mathbf X_1 \mathbf X_2 \mathbf X_3]$ - Gower} & 0.78 (1.0e-2) & 0.41 (3.0e-2) & 0.58 (4.2e-2) & 0.64 (2.6e-2) \\
\multirow{1}{*}{iCluster+} & 0.61 (2.5e-2) & 0.74 (2.8e-2) & 0.62 (1.7e-2) & 0.61 (1.4e-2) \\
\multirow{1}{*}{Bayesian Consensus Clustering} & 0.47 (1.1e-1) & 0.53 (1.0e-2) & 0.60 (1.0e-2) & 0.63 (1.1e-2) \\
\multirow{1}{*}{iGecco} & 0.14 (8.7e-2) & 0.13 (8.2e-2) & 0.45 (2.9e-2) & 0.42 (4.4e-2) \\
\multirow{1}{*}{iGecco+} & 0.37 (6.7e-2) & 0.37 (5.6e-2) & 0.48 (2.8e-2) & 0.48 (4.7e-2) \\
\multirow{1}{*}{Adaptive iGecco+} & \textbf{0.91 (6.1e-3)} & \textbf{0.92 (8.3e-3)} & \textbf{0.96 (2.4e-2)} & \textbf{0.94 (4.3e-2)} \\ \hline
\end{tabular}
}
\end{small}
\caption{Comparisons of adjusted Rand index for high-dimensional mixed multi-view data}
\label{iGecco+_rand}
\end{center}
\vskip -0.1in
\end{table}
Also we show the variable selection results in Table~\ref{iGecco+_vsa} and compare our method to that of iClusterPlus. Our adaptive iGecco+ outperforms iClusterPlus for all scenarios.
\begin{table}[ht]
\vskip 0.15in
\begin{center}
\resizebox{\linewidth}{!}{%
\begin{tabular}{lcccccccc}
\toprule
\multirow{2}{*}{} &
\multicolumn{2}{c}{Overall} &
\multicolumn{2}{c}{Gaussian} &
\multicolumn{2}{c}{Count} &
\multicolumn{2}{c}{Binary} \\
& {iCluster+} & {A iGecco+} & {iCluster+} & {A iGecco+} & {iCluster+} & {A iGecco+} & {iCluster+} & {A iGecco+} \\
\midrule
S3 & 0.88 (1.1e-2) & \textbf{0.96 (4.5e-3)} & 0.96 (9.2e-3) & 1.00 (0.0e-0) & 0.81 (1.3e-2) & 0.89 (5.7e-3) & 0.87 (1.3e-2) & 0.98 (7.8e-3) \\
S4 & 0.93 (1.5e-2) & \textbf{0.99 (4.5e-3)} & 0.97 (2.0e-2) & 0.99 (7.0e-3) & 0.93 (1.5e-2) & 1.00 (4.8e-3) & 0.89 (2.2e-2) & 0.99 (6.3e-3) \\
S5 & 0.95 (3.0e-2) & \textbf{1.00 (3.3e-3)} & 0.95 (3.3e-2) & 1.00 (0.0e-0) & 0.93 (3.6e-2) & 0.99 (1.0e-2) & 0.96 (2.2e-2) & 1.00 (0.0e-0)\\
S6 & 0.93 (3.1e-2) & \textbf{1.00 (1.6e-3)} & 0.95 (3.3e-2) & 1.00 (0.0e-0) & 0.88 (4.4e-2) & 1.00 (0.0e-0) & 0.95 (2.5e-2) & 1.00 (0.0e-0) \\
\bottomrule
\end{tabular}
}
\caption{Comparisons of F$_1$ score for adaptive iGecco+ and iClusterPlus}
\label{iGecco+_vsa}
\end{center}
\vskip -0.1in
\end{table}
\section{Real Data Examples}\label{realdata}
In this section, we apply our methods to various real data sets and evaluate our methods against existing ones. We first evaluate the performance of Gecco+ for several real data sets and investigate the features selected by various Gecco+ methods.
\subsection{Authors Data} \label{author}
The authors data set consists of word counts from $n = 841$ chapters written by four famous English-language authors (Austen, London, Shakespeare, and Milton). Each class contains an unbalanced number of observations with 69 features. The features are common ``stop words" like ``a", ``be" and ``the" which are typically removed before text mining analysis. We use Gecco+ not only to cluster book chapters and compare the clustering assignment with true labels of authors, but also to identify which key words help distinguish the authors. We choose tuning parameters using hold-out validation.
\begin{table}[ht]
\vskip 0.05in
\centering
\begin{tabular}{|l|c|}
\hline
Method & Adjusted Rand Index \\\hline
K-means & 0.73\\
Hierarchical Clustering & 0.73\\
Sparse Convex Clustering & 0.50 \\
Manhattan Gecco+ & 0.96 \\
Poisson LL Gecco+ & 0.96 \\
Poisson Deviance Gecco+ & 0.96 \\
\hline
\end{tabular}
\caption{Adjusted Rand index of different methods for authors data set}
\label{author-rand}
\vskip -0.1in
\end{table}
In Table~\ref{author-rand}, we compare Gecco+ with existing methods in terms of clustering quality. For hierarchical clustering, we only show the linkage with the best performance (in this whole section). Our method outperforms k-means and the best hierarchical clustering method. Secondly, we look at the word texts selected by Gecco+.
As shown in Table~\ref{author-feature}, Jane Austen tended to use the word ``her" more frequently than the other authors; this is expected as the subjects of her novels are typically females. The word ``was" seems to separate Shakespeare and Jack London well. Shakespeare preferred to use present tense more while Jack London preferred to use past tense more. To summarize, our Gecco+ not only has superior clustering performance but also selects interpretable features.
\begin{table}[ht]
\vskip 0.05in
\begin{center}
\begin{tabular}{lcccr}
\toprule
Method & Features \\
\midrule
Manhattan Gecco+ & \begin{tabular}{@{}c@{}} ``be" ,``had" ,``her", \\ ``the" ,``to", ``was" \end{tabular} \\
Poisson LL Gecco+ & ``an" , ``her" , ``our", ``your" \\
Poisson Deviance Gecco+ & \begin{tabular}{@{}c@{}} ``an", ``be" , ``had", ``her", \\ ``is", ``my" , ``the", ``was" \end{tabular} \\
\bottomrule
\end{tabular}
\caption{Features selected by different Gecco+ methods for authors data set}
\label{author-feature}
\end{center}
\vskip -0.1in
\end{table}
\subsection{TCGA Breast Cancer Data}
The TCGA data set consists of log-transformed Level III RPKM gene expression levels for 445 breast-cancer patients with 353 features from The Cancer Genome Atlas Network \citep{cancer2012comprehensive}. Five PAM50 breast cancer subtypes are included, i.e, Basal-like, Luminal A, Luminal B, HER2-enriched, and Normal-like. Only 353 genes out of 50,000 with somatic mutations from COSMIC \citep{forbes2010cosmic} are retained. The data is Level III TCGA BRCA RNA-Sequencing gene expression data that have already been pre-processed according to the following steps:
i) reads normalized by RPKM, and ii) corrected for overdispersion by a log-transformation. We remove 7 patients, who belong to the normal-like group and the number of subjects $n$ becomes 438. We also combine Luminal A with Luminal B as they are often considered one aggregate group \citep{choi2014identification}.
\begin{table}[ht]
\centering
\begin{tabular}{|l|c|}
\hline
Method & Adjusted Rand Index \\\hline
K-means & 0.44\\
Hierarchical Clustering & 0.26\\
Sparse Convex Clustering & 0.01 \\
Manhattan Gecco+ & 0.76 \\
Poisson LL Gecco+ & 0.72 \\
Poisson Deviance Gecco+ & 0.72 \\
\hline
\end{tabular}
\caption{Adjusted Rand index of different methods for TCGA data set}
\label{TCGA-rand}
\end{table}
From Table~\ref{TCGA-rand}, our method outperforms k-means and the best hierarchical clustering method. Next, we look at the genes selected by Gecco+ in Table~\ref{TCGA-feature}. FOXA1 is known to be a key gene that characterizes luminal subtypes in DNA microarray analyses \citep{badve2007foxa1}. GATA binding protein 3 (GATA3) is a transcriptional activator highly expressed by the luminal epithelial cells in the breast \citep{mehra2005identification}. ERBB2 is known to be associated with HER2 subtype and has been well studied in breast cancer \citep{harari2000molecular}. Hence our Gecco+ not only outperforms existing methods but also selects genes which are relevant to biology and have been implicated in previous scientific studies.
\begin{table}[ht]
\vskip 0.05in
\begin{center}
\begin{tabular}{lcccr}
\toprule
Method & Features \\
\midrule
Manhattan Gecco+ & \begin{tabular}{@{}c@{}}``BCL2" , ``ERBB2" ,``GATA3" \\ ``HMGA1", ``IL6ST" \end{tabular} \\
Poisson LL Gecco+ & ``ERBB2" ``FOXA1" ``GATA3" \\
Poisson Deviance Gecco+ & \begin{tabular}{@{}c@{}} ``ERBB2" , ``FOXA1", ``GATA3" \\ ``RET", ``SLC34A2"\end{tabular}
\\
\bottomrule
\end{tabular}
\caption{Features selected by different Gecco+ methods for TCGA data set}
\label{TCGA-feature}
\end{center}
\vskip -0.1in
\end{table}
Next we evaluate the performance of iGecco+ for mixed multi-view data sets and investigate the features selected by iGecco+.
\subsection{Multi-omics Data}\label{omics}
One promising application for integrative clustering for multi-view data lies in integrative cancer genomics. Biologists seek to integrate data from multiple platforms of high-throughput genomic data to gain a more thorough understanding of disease mechanisms and detect cancer subtypes. In this case study, we seek to integrate four different types of genomic data to study how epigenetics and short RNAs influence the gene regulatory system in breast cancer.
We use the data set from \cite{cancer2012comprehensive}. \cite{lock2013bayesian} analyzed this data set using integrative methods and we followed the same data pre-processing procedure: i) filter out genes in expression data whose standard deviation is less than 1.5, ii) take square root of methylation data, and iii) take log of miRNA data.
We end up with a data set of 348 tumor samples including:
\begin{itemize}
\item RNAseq gene expression (GE) data for 645 genes,
\item DNA methylation (ME) data for 574 probes,
\item miRNA expression (miRNA) data for 423 miRNAs,
\item Reverse phase protein array (RPPA) data for 171 proteins.
\end{itemize}
The data set contains samples used on each platform with associated subtype calls from each technology platform as well as integrated cluster labels from biologists. We use the integrated labels from biologists as true label. Also we merged the subtypes 3 and 4 in the integrated labels as those two subtypes correspond to Luminal A and Luminal B respectively from the predicted label given by gene expression data (PAM50 mRNA).
Figure \ref{fig:hist} in Appendix~\ref{genohist} gives the distribution of data from different platforms. For our iGecco+ methods, we use Euclidean distances for gene expression data and protein data as the distributions of those two data sets appear gaussian; binomial deviances for Methylation data as the value is between $[0,1]$; Manhattan distances for miRNA data as the data is highly-skewed. We compare our adaptive iGecco+ with other existing methods. From Table \ref{real-data-multi}, we see that our method outperforms all the existing methods.
\begin{table}[ht]
\centering
\begin{tabular}{|l |c|}
\hline
Method & Adjusted Rand Index \\\hline
Hclust: $\mathbf X_1$ GE & 0.51 \\
Hclust: $\mathbf X_2$ Meth & 0.39 \\
Hclust: $\mathbf X_3$ miRNA & 0.21 \\
Hclust: $\mathbf X_4$ Protein & 0.24 \\
Hclust: $[\mathbf X_1 \mathbf X_2 \mathbf X_3 \mathbf X_4]$ - Euclidean & 0.51 \\
Hclust: $[\mathbf X_1 \mathbf X_2 \mathbf X_3 \mathbf X_4]$ - Gower & 0.40 \\
iCluster+ & 0.36 \\
Bayesian Consensus Clustering & 0.35\\
Adaptive iGecco+ & \textbf{0.71} \\
\hline
\end{tabular}
\caption{Adjusted Rand index of different methods for multi-omics TCGA data set}
\label{real-data-multi}
\end{table}
We also investigate the features selected by adaptive iGecco+, shown in Table \ref{TCGA-feature-igecco}, and find that our method is validated as most are known in the breast cancer literature. For example, FOXA1 is known to segregate the luminal subtypes from the others \citep{badve2007foxa1}, and AGR3 is a known biomarker for breast cancer prognosis and early breast cancer detection from blood \citep{garczyk2015agr3}. Several well-known miRNAs were selected including MIR-135b, which is upregulated in breast cancer and promotes cell growth \citep{hua2016mir} as well as MIR-190 which suppresses breast cancer metastasis \citep{yu2018mir}. Several known proteins were also selected including ERalpha, which is overexpressed in early
stages of breast cancer \citep{hayashi2003expression} and GATA3 which plays an integral role in breast luminal cell differentiation and breast cancer progression \citep{cimino2013gata3}.
\begin{table}[ht]
\vskip 0.05in
\begin{center}
\begin{tabular}{lcccr}
\toprule
Data view & Features \\
\midrule
Gene Expression & \begin{tabular}{@{}c@{}}``AGR3", ``FOXA1", ``AGR2", ``ROPN1", \\ ``ROPN1B", ``ESR1", ``C1orf64", ``ART3",``FSIP1" \end{tabular} \\
miRNA & \begin{tabular}{@{}c@{}}``hsa-mir-135b", ``hsa-mir-190b", ``hsa-mir-577",
``hsa-mir-934" \end{tabular} \\
Methylation & \begin{tabular}{@{}c@{}} ``cg08047457", ``cg08097882", ``cg00117172", ``cg12265829" \end{tabular} \\
Protein & ``ER.alpha", ``GATA3", ``AR", ``Cyclin\_E1" \\
\bottomrule
\end{tabular}
\caption{Features selected by adaptive iGecco+ methods for multi-omics TCGA data set}
\label{TCGA-feature-igecco}
\end{center}
\vskip -0.1in
\end{table}
We also visualize resulting clusters from adaptive iGecco+. In Figure \ref{viz-omics}, we see that there is a clear separation between groups and adaptive iGecco+ identifies meaningful subtypes.
\begin{figure}[ht]
\centering
\includegraphics[width=\textwidth,height = 7.8cm]{igecco_viz3.pdf}
\caption{Cluster heatmap of multi-omics TCGA data with row orders determined by cluster assignments from iGecco+. The left bar refers to the integrated cluster labels from biologists. The black bars at the bottom of each data view correspond to the selected features. Our adaptive iGecco+ identifies meaningful subtypes.}
\label{viz-omics}
\end{figure}
\section{Discussion}
In this paper, we develop a convex formulation of integrative clustering for high-dimensional mixed multi-view data. We propose a unified, elegant methodological solution to two critical issues for clustering and data integration: (i) dealing with mixed types of data and (ii) selecting interpretable features in high-dimensional settings.
Specifically, we show that clustering for mixed, multi-vew data can be achieved using different data specific convex losses with a joint fusion penalty.
We also introduce a shifted group-lasso penalty that shrinks noise features to their loss-specific centers, hence selecting features that play important roles in separating groups. In addition, we make an optimization contribution by proposing and proving the convergence of a new general multi-block ADMM algorithm with sub-problem approximations that efficiently solves our problem. Empirical studies show that iGecco+ outperforms existing clustering methods and selects interpretable features in separating clusters.
This paper focuses on the methodological development for integrative clustering and feature selection, but there are many possible avenues for future research related to this work. For example, we expect in future work to be able to show that our methods inherit the strong theoretical properties of other convex clustering approaches such as clustering consistency and prediction consistency. An important problem in practice is choosing which loss function is appropriate for a given data set. While this is beyond the scope of this paper, an interesting direction for future research would be to learn the appropriate convex loss function in a data-driven manner. Additionally, many have shown block missing structure is common in mixed data \citep{yu2019optimal,xiang2013multi}. A potentially interesting direction for future work would be to develop an extension of iGecco+ that can appropriately handle block-missing multi-view data. Additionally, \citet{weylandt2019dynamic} developed a fast algorithm to compute the entire convex clustering solution path and used this to visualize the results via a dendrogram and pathwise plot. In future work, we expect that algorithmic regularization path approaches can also be applied to our method to be able to represent our solution as a dendrogram and employ other dynamic visualizations. Finally, while we develop an efficient multi-block ADMM algorithm, there may be further room to speed up computation of iGecco+, potentially by using distributed optimization approaches.
In this paper, we demonstrate that our method can be applied to integrative genomics, yet it can be applied to other fields such as multi-modal imaging, national security, online advertising, and environmental studies where practitioners aim to find meaningful clusters and features at the same time. In conclusion, we introduce a principled, unified approach to a challenging problem that demonstrates strong empirical performance and opens many directions for future research.
\section*{Acknowledgements}
The authors would like to thank Michael Weylandt and Tianyi Yao for helpful discussions. GA and MW also acknowledge support from NSF DMS-1554821, NSF NeuroNex-1707400, and NSF DMS-1264058.
\newpage
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 36 |
\section{Introduction}
In this paper we will consider the normalized Ricci flow equation
on a closed smooth $n$-dimensional manifold $M$,
\begin{equation}\label{0}
\left\{ \begin{array}{ll}
\frac{\partial}{\partial t}g=-2Ric+\frac{2r}{n}g, \\
g(0)=g_{0},
\end{array} \right.
\end{equation}
where $Ric$ and $r$ denote the Ricci tensor and average scalar
curvature $\frac{\int_{M}Rdv}{\int_{M}dv}$ of the Riemannian
metric $g$ respectively. Following Hamilton \cite{H}, a solution
to this equation is called {\it non-singular}, if the flow exists
for all time $t\geq0$ and the Riemannian curvature tensor
satisfies $|Rm|\leq C<\infty$ uniformly for some constant $C$
independent of $t$.
In \cite{H}, Hamilton classified $3$-dimensional non-singular
solutions. In particular, he proved that the underlying 3-manifold
is geometrizable in the sense of Thurston. Hamilton's theorem is
of great importance to understand the long-time behavior of
solutions to the Ricci flow, even to the Ricci flow with surgery
which was used by Perelman in \cite{P1} as a technique to study
the global property of the Ricci flow solution, modulo modifying
the singular points in space time (cf. for example \cite{CZ} or
\cite{KL} for detailed discussion.)
Our main result is the following generalization of Hamilton's
results to higher dimensions.
\begin{theorem}\label{001}
Any non-singular solution to the normalized Ricci flow equation
(\ref{0}) on a closed manifold $M$ does one and only one of the
following things:
\begin{itemize}\label{00}
\item[(1.1.1)] the solution collapses;
\item[(1.1.2)] the solution converges along a subsequence to a
shrinking Ricci soliton solution on $M$;
\item[(1.1.3)] the solution converges along a subsequence to a
Ricci flat metric solution on $M$;
\item[(1.1.4)] the solution converges along a subsequence to a
negative Einstein metric solution on $M$;
\item[(1.1.5)] the solution converges along a subsequence of times and space points to a complete non-compact
negative Einstein metric solution in the pointed Gromov-Hausdoff
sense.
\end{itemize}
\end{theorem}
A solution $(M,g(t))$ to equation (\ref{0}) is called collapse if
the maximum of the local injectivity radius
$\text{inj}(x,g(t)),x\in M,$ tends to zero along some subsequence
of times $t_{k}\rightarrow\infty$.
In dimension 3, by a result of Ivey \cite{I}, every closed Ricci
soliton is Einstein. So the convergence result coincides with that
of Hamilton's. For case (1.1.5), Hamilton proved that the
$3$-manifold splits into pieces of hyperbolic manifolds and
residual graph manifolds. Each hyperbolic piece has finite volume
and finite cusps at infinity, every cusp contains an
incompressible torus $T$, $i.e.$, $\pi_{1}(T)$ injects into
$\pi_{1}(M)$, with constant mean curvature and small area. In
general, by Cheeger-Gromov [3], the Riemannian manifold $(M,g(t))$
for large $t$ admits a thick-thin decomposition
$M=M^{\varepsilon}\bigcup M_{\varepsilon}$ for small
$\varepsilon$, where
$$M^{\varepsilon}=\{x\in M|\text{Vol}(B(x,1,g(t)))\geq\varepsilon\},$$
$$M_{\varepsilon}=\{x\in M|\text{Vol}(B(x,1,g(t)))\leq\varepsilon\}.$$
By (1.1.5) it is easy to see that the thick part is recognized as
the negative Einstein pieces, while the thin part has an
$F$-structure which partially collapses when we take the limit
(cf. \cite[Thm. 1.3]{A}). In 4-dimension we will prove the thin
part is indeed volume collapsed (cf. Theorem 1.5).
The following is a generalization of the celebrated Hitchin-Thorpe
inequality for non-singular solutions to the Ricci flow on closed
$4$-manifolds.
\vskip 3mm
\begin{theorem} \label{002}
Let $M$ be a closed oriented 4-manifold $M$, and
$\{g(t)\},t\in[0,\infty)$, be a non-singular solution to
(\ref{0}).
Then $M$ satisfies one of the following
\noindent (1.2.1) $M$ admits a shrinking Ricci solition;
\noindent (1.2.2) $M$ admits a positive rank $F$-structure;
\noindent (1.2.3) the Hitchin-Thorpe type inequality holds
$$ 2\chi (M)\ge 3|\tau(M)|$$
where $\chi (M)$ (resp. $\tau(M)$) is the Euler characteristic
(resp. signature) of $M$.
\end{theorem}
By [8] a closed shrinking Ricci soliton has finite fundamental
group. Thus, a $4$-dimensional closed shrinking Ricci soliton has
Euler characteristic at least $2$. On the other hand, manifold
with positive rank $F$-structure has vanishing Euler
characteristic. Therefore Theorem 1.2 implies readily that
\vskip 3mm
\begin{corollary}\label{003}
If a closed 4-manifold $M$ admits a non-singular solution, then
$\chi(M)\geq0$.
\end{corollary}
The above corollary gives a topological obstruction to the
existence of non-singular solutions on a closed $4$-manifold.
To state a strengthened result of Theorem 1.2 we need to introduce
Perelman's $\lambda$-functional (cf. \cite{P1}\cite{KL}\cite{FZ}).
For a smooth function $f\in C^\infty(M)$ on a Riemannian
$n$-manifold with a Riemannian metric $g$, let
\begin{equation}\mathcal{F}(g,f)=\int_{M} (R_g+|\nabla f|^{2})e^{-f}
dvol_{g},\end{equation} where $R_g$ is the scalar curvature of
$g$.
The Perelman's
$\lambda$-functional is defined by
\begin{equation}\lambda_{M}(g)=\inf _f \{\mathcal{F}(g,f)|\int_{M}e^{-f}
dvol_{g}=1\}.\end{equation} Note that $\lambda_{M}(g)$ is the lowest eigenvalue of the
operator $-4\triangle+R_{g}$. Let \begin{equation}\overline{\lambda}_{M}(g)
=\lambda_{M}(g)\text{Vol}_{g}(M)^{\frac{2}{n}}\end{equation}
which is invariant up to rescale the metric. Perelman \cite{P1}
has established the monotonicity property of $\overline{\lambda}
_M(g_t)$ along the Ricci flow $g_t$, namely, the function is
non-decreasing along the Ricci flow $g_t$ whenever
$\overline{\lambda} _M(g_t)\leq 0$. Therefore, it is interesting
to study the upper bound of $\overline{\lambda}_{M}(g)$. This
leads to define a diffeomorphism invariant
$\overline{\lambda}_{M} $ of $M$ due to Perelman (cf. \cite{P2}
\cite{KL})
by \begin{equation}\overline{\lambda}_{M}=
\sup\limits_{g\in \mathcal{M}} \overline{\lambda}_{M}(g),
\end{equation} where $\mathcal{M}$ is the set of Riemannian
metrics on $M$. As we pointed out in \cite{FZ} $\overline{\lambda}
_M$ is not a homeomorphism invariant. Observe that
$\overline{\lambda} _M=0$ if $M$ admits a volume collapsing with
bounded scalar curvature but does not admit any metric with
positive scalar curvature (cf. \cite{KL}).
A upper bound in \cite{FZ} for the invariant
$\overline{\lambda}_M$ was obtained by using the Seiberg-Witten
theory, whenever $\overline{\lambda}_M< 0$ and $M$ has a
non-trivial monopole class.
Now we are ready to state
\begin{theorem}\label{004}
Let $M$ be a closed oriented 4-manifold $M$ with
$\overline{\lambda}_{M}< 0$, and $\{g(t)\},t\in[0,\infty)$, be a
solution to (\ref{0}). If $|R(g(t))|<C$ where $C$ is a constant
independent of $t$, then
$$2\chi(M)-3|\tau(M)|\geq \frac{1}{96\pi^{2}}\overline{\lambda}_{M}^{2},$$
In particular, if $\{g(t)\},t\in[0,\infty)$ is a non-singular
solution to (\ref{0}), then the solution $(M,g(t))$ does not
collapse.
\end{theorem}
The above theorem combined with the Seiberg-Witten theory implies
that
\vskip 3mm
\begin{corollary} \label{005}
Let $M$ be a closed symplectic 4-manifold $M$ with
$\overline{\lambda}_{M} < 0$, and $\{g(t)\},t\in[0,\infty)$, be a
solution to (\ref{0}) with $|R(g(t))|<C$ where $C$ is a constant
independent of $t$.
If $b^{+}_{2}(M):=\text{dim}H^2_+(M;\Bbb R)
>1$, then $$\chi(M) \geq 3\tau(M).$$
\end{corollary}
In dimension $4$ Theorem (1.1.5) may be improved as follows:
\vskip 3mm
\begin{theorem} \label{006}
Let $(M,g(t)),t\in[0,\infty)$ be a non-singular
solution to (\ref{0}) on a closed oriented 4-manifold $M$ with
$\overline{\lambda}_{M} < 0$. Then, for any $\delta >0$, there is a
time $T\gg 1$, and a compact 4-submanifold $M^{\varepsilon}$ with
boundary in $M$, $M^{\varepsilon}\subset M$, such that
\begin{itemize}\label{00}
\item[(1.5.1)] $\text{Vol}(M-M^{\varepsilon}, g(T))< \delta$, and $M-M^{\varepsilon}$ admits an F-structure of positive rank.
\item[(1.5.2)] The
components of $\partial M^{\varepsilon}$ are graph 3-manifolds. \item[(1.5.3)] $M^{\varepsilon}$ admits
an Einstein metric with negative scalar curvature $g_{\infty}$ which is close to
$g(T)|_{M^{\varepsilon}}$ in the $C^{\infty}$-sense.
\end{itemize}
\end{theorem}
In view of Theorem 1.6, it is natural to wonder what kind of
information can be obtained on the thick part $M^\varepsilon$ in
the above theorem. Applying the Seiberg-Witten theory we obtain
the following result which provides a partial answer for certain
symplectic manifolds.
\vskip 3mm
\begin{theorem}\label{006} Let $(M,g(t))$ and $(M^{\varepsilon},g_{\infty})$ be the same as in Theorem 1.6. If $M$
admits a symplectic structure satisfying that $b^{+}_{2}(M)
>1$ and $\chi(M)=3 \tau (M)$, then $g_{\infty}$ is
a complex hyperbolic metric.
\end{theorem}
\vskip 3mm
We conclude this section by posing the following
\begin{conjecture}
Theorem (1.2.3) may be replaced by the following
Hitchin-Thorpe-Gromov-Kotschik type inequality
$$2\chi (M)-3|\tau (M)|\ge \frac 1{1296\pi ^2}\|M\|$$
where $\|M\|$ is a simplicial volume of $M$.
\end{conjecture}
The organization of the paper is as follows: In $\S$2 we give a proof of
Theorem 1.1. In $\S$3 we are concerned with $4$-dimensional
non-singular solutions, and we will prove Theorem 1.2, Theorem 1.4
and Corollary 1.5. In $\S 4$ we will prove Theorem 1.6. Finally,
we will prove Theorem 1.7 in $\S 5$.
\vskip 10mm
\section{Proof of Theorem 1.1}
In \cite{H2}, Hamilton introduced the (unnormalized) Ricci flow
equation
\begin{equation}\label{101}
\left\{ \begin{array}{ll} \frac{\partial}{\partial t}g=-2Ric, \\
g(0)=g_{0},
\end{array} \right.
\end{equation}
and its normalized equation (\ref{0}). By \cite{H2} the equation
(\ref{0}) is just a change of equation (\ref{101}) via rescalings
in space and a reparametrization in time, such that the volumes of
the Riemannian metrics are preserved to be constant. In this
paper, we will always assume the volumes of the metrics equal $1$,
whenever we consider the solutions to the normalized Ricci flow.
Especially we have $\text{Vol}(g_{0})=1$.
As in \cite{P1}, one also can define a scale invariant version of
$\mathcal{F}$ functional, the so called $\mathcal{W}$ functional.
For each smooth $f$ and constant $\tau>0$, let
\begin{equation}
\mathcal{W}(g,f,\tau)=\int_{M}[\tau(R+|\nabla
f|^{2})+f-n](4\pi\tau)^{-n/2}e^{-f}dv,\nonumber
\end{equation}
and then set
\begin{equation}
\mu(g,\tau)=\inf\{\mathcal{W}(g,f,\tau)|\int_{M}(4\pi\tau)^{-n/2}e^{-f}dv=1\},\nonumber
\end{equation}
\begin{equation}
\nu(g)=\inf\limits_{\tau>0}\mu(g,\tau).\nonumber
\end{equation}
By a result
of Rothaus \cite{R}, for each $\tau>0$, there exists a smooth
minimizer $\phi$ such that $\mu(g,\tau)=\mathcal{W}(g,\phi,\tau)$.
By Claim 3.1 of \cite{P1}, if $\lambda(g)>0$, then the infimum in
the definition of $\nu$ is attained by some $\tau>0$ and
$\nu(g)\leq0$. By \cite{P1} the $\lambda$ functional and the $\nu$
functional are non-decreasing along the Ricci flow, which plays a
central role in our proof (cf. Lemma \ref{105} and Lemma \ref{106}
for proofs of these facts.)
We start with the definition of convergence of solutions to the
equation (\ref{0}) or (\ref{101}). We assume that all the
Riemannian metrics taking into account are complete.
\begin{definition}[\cite{H1}]
Let $(M_{k},g_{k}(t),p_{k}), p_{k}\in M, t\in(A,B)$ with
$-\infty\leq A<0$ and $0<B\leq\infty$, be a sequence of marked
solutions to the Ricci flow equation (\ref{0}) or (\ref{101}). We
say that $(M_{k},g_{k}(t),p_{k})$ converges to another solution
$(M_{\infty},g_{\infty}(t),p_{\infty}),t\in(A,B),$ to (\ref{0}) or
(\ref{101}) respectively, if there is a sequence of increasing
open subsets $U_{k}\subset M_{\infty}$ containing $p_{\infty}$,
$i.e.$, $U_{k}\subset U_{k+1}$ for each $k$, and a sequence of
diffeomorphisms $F_{k}:U_{k}\to V_{k}\subset M_{k}$ mapping
$p_{\infty}$ to $p_{k}$, such that the pull-backed metrics
$F_{k}^{*}g_{k}(t)$ converge to $g_{\infty}(t)$ on every compact
subset of $M_{\infty}\times(A,B)$ uniformly together with all
their derivatives.
\end{definition}
In \cite{H1}, Hamilton proved his famous compactness theorem for
solutions to the unnormalized Ricci flow equation (\ref{101}),
under the assumption: (i) the local injectivity radii of the
metrics $g_{k}(0)$ at $p_{k}$
are uniformly bounded below;
(ii) the supremum norm of Riemannian curvature tensors are
uniformly bounded above on any compact time interval. For a local
version of this theorem, see \cite{CZ} for example. We remark that
the compactness theorem of Hamilton remains valid for the
normalized Ricci flow solutions under the same assumption. This
can be checked easily from the proof of the theorem.
Recall that $(M,g(t))$ is a {\it Ricci solition} solution to the
Ricci flow, if $g(t)$ is obtained from $g(0)$ via changes by
diffeomorphisms and rescalings. At each time the solution
satisfies $$Ric+\mathcal{L}_{X}g=cg$$ for some vector field $X$
and some constant $c$. The Ricci soliton is said to be shrinking,
steady or expanding according to $c>0,c=0,c<0$ respectively. Note
that if the manifold is closed, then $c=r$, which can be seen by
taking the trace of above equation and then integrating it over
$M$, where $r$ is the average scalar curvature of $g$.
In \cite{H}, Hamilton considered metrics with positive scalar
curvature. Our next proposition deals with the solutions so that
the $\lambda$-functional is positive.
\vskip 3mm
\begin{proposition}\label{102}
Let $(M,g(t)),t\in[0,\infty),$ be a non-singular solution to
(\ref{0}) on a closed $n$-manifold $M$ with $\lambda(g(0))>0$.
Then for any sequence of times $t_{k}\rightarrow\infty$, there is
a subsequence $t_{k_{i}}$ such that $(M,g(t_{k_{i}}+t))$ converges
to a shrinking Ricci soliton solution.
\end{proposition}
\begin{proof}
If we denote by $(M,\bar{g}(t))$ the corresponding unnormalized
Ricci flow solution, then by Proposition 1.2 of \cite{P1},
$$\frac{d}{dt}\lambda(\bar{g}(t))\geq\frac{2}{n}\lambda(\bar{g}(t))^{2}.$$
Thus the unnormalized Ricci flow must extinct in finite time and
$\lambda(\bar{g}(t))>0$ remains hold. Since $g(t)$ are just
rescalings of $\bar{g}(t)$, $\lambda(g(t))>0$ is preserved. Then
Perelman's no local collapsing theorem \cite[Chap. 4]{P1} tells us
that there exist $\kappa,\rho>0$ such that each metric ball
$B=B_{g(t)}(p,r)\subset M$, with radius $r\leq\rho$ and
$\sup_{x\in B}|Rm|(x,t)\leq r^{-2}$, has volume
$\text{Vol}_{g(t)}(B)\geq\kappa r^{n}$. By the definition of
non-singular solution, there is a constant $C>0$ such that
$|Rm|(x,t)\leq C$ for all $(x,t)\in M\times[0,\infty)$. By
replacing $\rho$ by a smaller constant, we may assume $\rho\leq
C^{-\frac{1}{2}}$. Thus $|Rm|(x,t)\leq\rho^{-2}$ always hold and
so $Vol(B_{g(t)}(x,\rho))\geq\kappa\rho^{n}$ for all $(x,t)$. This
implies that the diameters of $g(t)$ are bounded above uniformly,
since we can fill only $\frac{1}{\kappa\rho^{n}}$ disjoint balls
of radius $\rho$ in $M$ at each time $t$. By a result of Cheeger,
Gromov and Taylor \cite{CGT}, the injectivity radii of $g(t)$ are
bounded below uniformly for any $t$. Then Hamilton's compactness
theorem yields the convergence result.
Let $(M,g_{k}(t)=g(t_{k}+t))$, $t_{k}\rightarrow\infty$, be such a
sequence which converges to a limit solution to the normalized
Ricci flow equation
$(M_{\infty},g_{\infty}(t)),t\in(-\infty,\infty)$. By the uniform
boundedness of diameters of $g(t)$, $M_{\infty}=M$ and the
convergence is smooth on any compact time interval. Next we will
show that $(M,g_{\infty}(t))$ is a shrinking Ricci soliton.
Now for each
time $t\in(-\infty,\infty)$, $\nu(g(t))$ is achieved by some
positive number $\tau(t)$, $i.e.$,
$\nu(g(t))=\mu(g(t),\tau(t))\leq0$ holds. Further, by Lemma
\ref{106} below, $\nu(g(t))$ increases. By the smooth convergence
of $(M,g_{k}(t))$ for any $t$, we have
$$\nu(g_{\infty}(t))=\lim\limits_{k\rightarrow\infty}\nu(g(t_{k}+t))
=\lim\limits_{k\rightarrow\infty}\nu(t_{k}+t)
=\lim\limits_{t\rightarrow\infty}\nu(t),$$ is a constant
independent of $t$. We consider two
subcases for the limit solution. \\
Case 1: $\lambda(g_{\infty}(t_{0}))>0$ for some $t_{0}\geq0$. In
this case, $\nu(g_{\infty}(t))$ is attainable for $t$ around
$t_{0}$ and using Lemma \ref{106} again, one sees that
$(M,g_{\infty}(t))$ is really a Ricci soliton. By definition of
$\lambda$, the average scalar curvature
$r(g_{\infty}(t_{0}))\geq\lambda(g_{\infty}(t_{0}))>0$, so the
Ricci
soliton is a shrinking one.\\
Case 2: $\lambda(g_{\infty}(t))\equiv0$. In this case, the
monotonicity of the $\lambda$ functional implies that
$(M,g_{\infty}(t))$ is Einstein. See Lemma \ref{105} below. The
scalar curvature is zero because it equals
$\lambda(g_{\infty}(t))$. We will exclude this exception by
showing that $\nu(g_{\infty}(t))=-\infty$, which is a
contradiction since
$\nu(g_{\infty}(t))=\lim_{t\rightarrow\infty}\nu(g(t))\geq\nu(g(0))$
by the increasing of $\nu(g(t))$ along the Ricci flow. Hence the
proposition from the following claim.
\end{proof}
\begin{claim}
Let $(N,g)$ be a closed Riemannian manifold with average scalar
curvature $r\leq0$. Then we have
$\lim\limits_{\tau\rightarrow\infty}\mu(g,\tau)=-\infty$.
\end{claim}
\begin{proof}[Proof of the claim]
For any smooth function $f$, set $u=(4\pi\tau)^{-n/2}e^{-f}$; then
by the definition of the $\mathcal{W}$ functional,
$$\mathcal{W}(g,f,\tau)=\int_{M}[\tau(Ru+\frac{|\nabla u|^{2}}{u})-u\ln u]dv-\frac{n}{2}\ln(4\pi\tau)-n.$$
Choosing $u=1$ and substituting it into the functional, we have
\begin{eqnarray}
\mu(g,\tau)+n&\leq&\tau
r-\frac{n}{2}\ln(4\pi\tau)\leq-\frac{n}{2}\ln(4\pi\tau)\rightarrow-\infty\nonumber
\end{eqnarray}
as $\tau\rightarrow\infty$. This ends the proof of the claim.
\end{proof}
From the proof of above proposition, we have an immediate
corollary.
\begin{corollary}
If $(M,g)$ is a closed Riemannian manifold with $\lambda(g)>0$,
then the normalized Ricci flow solution, with $g$ as initial
metric, will never converge to a Ricci flat metric.
\end{corollary}
By a result of Fern\'{a}ndez-L\'{o}pez and Garc\'{i}a-R\'{i}o
\cite{FG}, any shrinking Ricci soliton has finite fundamental
group, so we have
\begin{corollary}
Let $(M,g)$ be a closed Riemannian manifold with $\lambda(g)>0$.
If the normalized Ricci flow solution on $M$ with $g$ as initial
metric is non-singular, then $\pi_{1}(M)$ is finite.
\end{corollary}
For a Riemannian metric $g$ on a manifold $M$, denote by
$\mbox{inj}(x,g)$ the injectivity radius of the metric $g$ at $x$.
We say a solution to the normalized Ricci flow collapses if there
exists a sequence of times $t_{k}\rightarrow T$ such that
$\sup_{x\in M}\mbox{inj}(x,g(t_{k}))\rightarrow0$, where $T$ is
the maximal existence time for the solution, which may be finite
or infinite. If a non-singular solution to the normalized Ricc
flow equation doesn't collapse, then by Hamilton's compactness
theorem, for each sequence of times $t_{k}\rightarrow\infty$,
there exists a subsequence $t_{k_{i}}$ and a sequence of points
$p_{i}\in M$ such that $(M,g(t_{k_{i}}+t),p_{i})$ converges to
another normalized Ricci flow solution
$(M_{\infty},g_{\infty}(t),p_{\infty})$. Denote by
$r_{\infty}(t)=\lim_{i\rightarrow\infty}r(g(t_{k_{i}}+t))$ the
limit constant in the normalized Ricci flow equation, then we have
$$\frac{\partial}{\partial t}g_{\infty}(t)=-2Ric_{\infty}(t)+\frac{2}{n}r_{\infty}(t)g_{\infty}(t),$$
where $Ric_{\infty}(t)$ is the Ricci tensor of $g_{\infty}(t)$.
Note that $r_{\infty}(t)$ may not equal to the average scalar
curvature $r(g_{\infty}(t))$.
Denote by $\breve{R}(g)=\min_{x\in M}R(x)$ the minimum of the
scalar curvature of a given metric $g$.
\vskip 2mm
\begin{proposition}\label{103}
Let $(M,g(t)),t\in[0,\infty)$, be a non-singular solution to
(\ref{0}) on a closed $n$-manifold $M$. Assume that
$\breve{R}(g(t))\leq-c<0$ uniformly for some constant $c>0$
independent of $t$. If the solution doesn't collapse, then there
exists a sequence of points $p_{k}\in M$ and a sequence of times
$t_{k}\rightarrow\infty$ such that the solutions
$\{(M,g(t_{k}+t),p_{k})\}_{k=1}^{\infty}$ converge to an Einstein
metric solution to (\ref{0}), whose scalar curvature is negative.
\end{proposition}
For proving this proposition, we need the following lemma:
\begin{lemma}\label{107} Let $(M,g(t)),t\in[0,\infty)$ be a solution to
(\ref{0}) on a closed $n$-manifold $M$. Assume that
$\breve{R}(g(t))\leq-c<0$ uniformly for some constant $c>0$
independent of $t$. If $|R(g(t))|\leq C$, where $C$ is a constant
independent of $t$, then
$$\int_{0}^{\infty}(r(g(t))-\breve{R}(g(t)))dt<\infty, \ \ \ {\rm and} \ \
\
$$ $$\int_{0}^{\infty}\int_{M}|R(g(t))-r(g(t))|dvdt<\infty.$$
\end{lemma}
\begin{proof}
We follow the proof given in Section 7 of \cite{H} by Hamilton.
Consider the evolution equation of $R$
\begin{eqnarray}
\frac{\partial}{\partial t}R=\triangle
R+2|Ric\textordmasculine|^{2}+\frac{2}{n}R(R-r),\nonumber
\end{eqnarray}
where $Ric\textordmasculine$ denotes the traceless part of Ricci
tensor. By maximal principle,
$\frac{d}{dt}\breve{R}\geq\frac{2}{n}\breve{R}(\breve{R}-r)$ and
so $\breve{R}(g(t))$ increases whenever it is negative. By
assumption
$\frac{d}{dt}\breve{R}\geq\frac{2}{n}\breve{R}(\breve{R}-r)\geq\frac{2c}{n}(r-\breve{R})$,
which implies that
$$\int_{0}^{\infty}(r-\breve{R})dt<\infty.$$
Come back to the original solution $(M,g(t))$. We have
$$\int_{M}|R-r|dv\leq\int_{M}(R-\breve{R})dv+\int_{M}(r-\breve{R})dv=2\int_{M}(r-\breve{R})dv=2(r-\breve{R}).$$
Thus
$$\int_{0}^{\infty}\int_{M}|R-r|dvdt<\infty.$$
\end{proof}
\begin{proof}[Proof of Proposition \ref{103}]
Assume that
$\lim_{t\rightarrow\infty}\breve{R}(t)=-\delta\leq-c<0$. Note that
by assumption, there exists a constant $C>0$ such that
$|Rm|(x,t)\leq C$ uniformly for all $(x,t)\in M\times[0,\infty)$.
First we will show that $\lim_{t\rightarrow\infty}r(t)$ exists and
equals to $-\delta$. For this, consider the evolution equation
$$\frac{d}{dt}(r-\breve{R})\leq\int_{M}(2|Ric|^{2}-\frac{2}{n}Rr)dv-\frac{2}{n}\breve{R}(\breve{R}-r)\leq D,$$
where $D\geq1$ is a constant depending only on $C$ and $n$. We
claim that for any $0<\epsilon<1$, there is $T$ such that
$r(t)-\breve{R}(t)<\epsilon$ whenever $t>T$. Otherwise, there will
be a sequence of times $t_{k}\rightarrow\infty$ satisfying
$t_{k+1}\geq t_{k}+1$ and $r(t_{k})-\breve{R}(t_{k})\geq\epsilon$.
From the above equation, we obtain that
$r(t)-\breve{R}(t)\geq\frac{\epsilon}{2}$ whenever
$t\in[t_{k}-\epsilon/D,t_{k}]$. Thus
$$\int_{0}^{\infty}(r-\breve{R})dt\geq\sum_{k=1}^{\infty}\int_{t_{k}-\epsilon/D}^{t_{k}}(r-\breve{R}(t))dt
\geq\sum_{k=1}^{\infty}\int_{t_{k}-\epsilon/D}^{t_{k}}\frac{\epsilon}{2}dt=\infty,$$
which contradicts the Lemma \ref{107}. Hence
$\lim_{t\rightarrow\infty}r(t)=-\delta$ and consequently
$r_{\infty}(t)=-\delta$ on any limit solution as we mentioned in
the paragraphs before Proposition \ref{103}.
Considering the time interval $[t_{0},t_{0}+1],\forall
t_{0}\in{\mathbb{R}},$ on the limit solution, we have
$\int_{t_{0}}^{t_{0}+1}\int_{M_{\infty}}|R(g_{\infty}(t))-r_{\infty}(t)|dvdt=0$.
Hence $R(g_{\infty}(t))\equiv r_{\infty}(t)=-\delta$. Then by the
evolution of scalar curvature we have
$Ric\textordmasculine(g_{\infty}(t))=0$, $i.e.$,
$(M_{\infty},g_{\infty}(t))$ is Einstein for all $t$. The scalar
curvature
$R_{\infty}(t)=\lim_{t\rightarrow\infty}\breve{R}(t)=-\delta<0$.
\end{proof}
\begin{remark}
It is remarkable that the conclusion in Proposition \ref{103}
remains valid if we replace the boundedness of the Riemannian
curvature tensor by the boundedness of the Ricci tensor. This can
be seen from the process of proving the Proposition \ref{103}.
Also note that in both of these special cases, the limit constant
$r_{\infty}(t)$ equal to $r(g_{\infty}(t))$, the average scalar
curvature of the limit metrics.
\end{remark}
By now for a solution to the normalized Ricci flow equation, if
$\lambda(g(t))>0$ for some time $t$, then we can use Proposition
\ref{102}; while if $\breve{R}(g(t))\leq-c<0$ holds for all $t$,
then we can use Proposition \ref{103}. As for the remaining case,
we have the following proposition similar as the zero sectional
curvature limit case considered by Hamilton \cite{H}.
\begin{proposition}\label{104}
Let $(M,g(t)),t\in[0,\infty)$, be a non-singular solution to
(\ref{0}) on a closed $n$-manifold $M$. Assume that
$\breve{R}(g(t))\leq0$, $\lambda(g(t))\leq0$ and
$\breve{R}(g(t))\nearrow0$ as $t\rightarrow\infty$. If the
solution doesn't collapse, then there exists a sequence of times
$t_{k}\rightarrow\infty$ such that the solutions
$\{(M,g(t_{k}+t))\}_{k=1}^{\infty}$ converge to a Ricci flat
metric solution to (\ref{0}).
\end{proposition}
\begin{proof}
We divide the proof into several subcases.
Case 1: The unnormalized Ricci flow extincts in finite time.
Perelman's no local collapsing theorem shows that the limit
manifold is closed and so $M_{\infty}=M$. Now
$\lambda(g(t))\nearrow0$ implies that $\lambda(g_{\infty}(t))=0$
for all $t$. By Lemma \ref{105}, $(M,g_{\infty}(t))$ is a Ricci
flat solution.
Case 2: The unnormalized Ricci flow exists for all time
$t\in[0,\infty)$ and there is a sequence of times
$t_{k}\rightarrow\infty$ such that the average scalar curvature
$r(t_{k})$ of $g(t_{k})$ converges to zero. In this case, we use a
modified version of the proof used in Section 6 of \cite{H} by
Hamilton. Denote by $R_{\infty}$ and $Ric_{\infty}$ the scalar
curvature and Ricci tensor of the limit solution respectively. By
assumption,
\begin{equation}
\int_{M}(R(t_{k})-\breve{R}(t_{k}))dv_{g(t_{k})}=r(t_{k})-\breve{R}(t_{k})\rightarrow0,\nonumber
\end{equation}
as $k\rightarrow\infty$. Note that
$R(t_{k})-\breve{R}(t_{k})\geq0$. By taking the limit, one obtains
$\int_{M_{\infty}}R_{\infty}(0)dv_{g_{\infty}}(0)=0$. But
$R_{\infty}(0)\geq0$ over $M_{\infty}$ since
$\lim_{k\rightarrow\infty}\breve{R}(t_{k})=0$, so
$R_{\infty}(0)\equiv0$. Note that $r_{\infty}(0)=0$, since
$r(t_{k})\rightarrow0$. Consider the evolving equation of the
scalar curvature on the limit solution
\begin{equation}\nonumber
\frac{\partial}{\partial t}R_{\infty}(t)=\triangle
R_{\infty}(t)+2|Ric_{\infty}(t)|^{2}-\frac{2}{n}r_{\infty}(t)R_{\infty}(t),t\in(-\infty,\infty).
\end{equation}
It follows from the strong maximal principle that
$R_{\infty}\equiv0$ and $Ric_{\infty}\equiv0$ over
$M_{\infty}\times(-\infty,\infty)$, $i.e.$,
$(M_{\infty},g_{\infty}(t))$ is a Ricci flat solution. Now the
volume of the limit manifold is less than or equals to 1. By a
result of Yau \cite{SY}, $M_{\infty}$ is compact. So
$M_{\infty}=M$ and the convergence is smooth.
Case 3: The unnormalized Ricci flow, say $(M,\bar{g}(t))$, exists
for all time $t\in[0,\infty)$ and the average scalar curvature of
normalized Ricci flow $r(t)\geq\delta$ uniformly for some constant
$\delta>0$. We want to show this case will never happen. Denote by
$\overline{V}(t)=\text{Vol}(\bar{g}(t))$ the volume of
$\bar{g}(t)$. Since $r(g)\text{Vol}(g)^{\frac{2}{n}}$ is scale
invariant, we have
$\bar{r}(t)\geq\delta\overline{V}^{\frac{-2}{n}}(t)$. By the
evolving equation
\begin{equation}\nonumber
\frac{d}{dt}\overline{V}=\int_{M}-R(\bar{g})dv_{\bar{g}}=-\bar{r}\overline{V}\leq-\delta\overline{V}^{\frac{n-2}{n}},
\end{equation}
we obtain $\overline{V}^{\frac{2}{n}}(t)\leq1-\frac{2}{n}\delta t$
for all $t$, which contradicts with the assumption that the
unnormalized Ricci flow solution exists for all time. The desired
result follows.
\end{proof}
\begin{remark}
In fact, by a refined argument using the monotonicity of
Perelman's $\mu$ functional along the Ricci flow, Case 1 of
Proposition \ref{104} also can be excluded. So the only phenomena
is that of Case 2, under the assumption in the proposition.
\end{remark}
Summing up the results of Proposition \ref{102}, \ref{103} and
\ref{104}, we finish the proof of Theorem \ref{001}.
To conclude this section, let us prove two lemmas used previously,
which are basically due to Perelman. These are basic facts in the
study of Ricci flow solutions.
\vskip 3mm
\begin{lemma}[Perelman]\label{105}
Let $(M,g(t)),t\in[0,T),$ be a solution to the normalized Ricci
flow equation (\ref{0}) on a closed $n$-manifold $M$. Then
$\lambda(g(t))$ increases whenever $\lambda(g(t))\leq0$, and the
increasing is strict unless $g(t)$ is Einstein.
\end{lemma}
\begin{proof}
Consider the coupled equation
\begin{equation}
\left\{ \begin{array}{ll}
\frac{\partial}{\partial t}g=-2Ric+\frac{2r}{n}g, \\
\frac{\partial}{\partial t}f=-\triangle f+|\nabla f|^{2}-R+r.
\end{array} \right.\nonumber
\end{equation}
Under this evolving equation, we have
\begin{eqnarray}
\frac{d}{dt}\mathcal{F}(g(t),f(t))&=&\int_{M}[2|Ric+\nabla^{2}f|^{2}-\frac{2}{n}r(R+\triangle
f)]e^{-f}dv.\nonumber
\end{eqnarray}
Let $\bar{f}$ be the eigenfunction of $-4\triangle_{g(t)}+R(g(t))$
. Denote by $\lambda(t)=\lambda(g(t))$, then
\begin{eqnarray}
\frac{d}{dt}\lambda(t)&=&\int_{M}[2|Ric+\nabla^{2}\bar{f}|^{2}-\frac{2}{n}r(R+\triangle
\bar{f})]e^{-\bar{f}}dv\nonumber\\
&\geq&\int_{M}[\frac{2}{n}(R+\triangle\bar{f})^{2}-\frac{2}{n}r(R+\triangle
\bar{f})]e^{-\bar{f}}dv\nonumber\\
&\geq&\frac{2}{n}(\int_{M}(R+\triangle\bar{f})e^{-\bar{f}}dv)^{2}-
\frac{2r}{n}\int_{M}(R+\triangle\bar{f})e^{-\bar{f}}dv\nonumber\\
&=&\frac{2}{n}(\int_{M}(R+|\nabla\bar{f}|^{2})e^{-\bar{f}}dv)^{2}-
\frac{2r}{n}\int_{M}(R+|\nabla\bar{f}|^{2})e^{-\bar{f}}dv\nonumber\\
&=&\frac{2}{n}\lambda(t)(\lambda(t)-r).\nonumber
\end{eqnarray}
Now $\lambda(t)\leq r(t)$ and the monotonicity when
$\lambda(t)\leq0$ follows.
If $\frac{d}{dt}\lambda(t)=0$, then the equalities in the above
estimate hold. So we have $$R+\triangle\bar{f}=const.$$ and
$$Ric+\nabla^{2}\bar{f}=\frac{1}{n}(R+\bar{f})g.$$ On the other hand,
$e^{-\bar{f}/2}$ is the only eigenfunction of $-4\triangle+R$, so
$$2\triangle\bar{f}-|\nabla\bar{f}|^{2}+R=\lambda(t).$$ Thus
$\triangle\bar{f}-|\nabla\bar{f}|^{2}=const.,$ which equals zero
since
$\int_{M}(\triangle\bar{f}-|\nabla\bar{f}|^{2})e^{-\bar{f}}dv=0$.
By the maximal principle $f=const.$ over $M$. Hence $R=const.$ and
$Ric=\frac{R}{n}g$. This proves the lemma.
\end{proof}
\begin{lemma}[Perelman]\label{106}
Let $(M,g(t)),t\in[0,T),$ be a solution to the Ricci flow equation
(\ref{0}) or (\ref{101}). If $\lambda(g(t))>0$ for all $t$, then
$\nu(g(t))$ increases. Furthermore, the increasing is strict
unless the solution is a shrinking Ricci soliton.
\end{lemma}
\begin{proof}
We only need to prove the unnormalized case, since the $\nu$
functional is invariant up to rescalings and diffeomorphism
transformations, $i.e.$, $\nu(\alpha\phi^{*}g)=\nu(g)$ for any
constant $\alpha>0$ and diffeomorphism $\phi$ of $M$. Now for any
times $t_{1}$ and $t_{2}$ such that $0\leq t_{1}<t_{2}< T$, choose
$\tau_{0}>0$ and $f_{0}\in C^{\infty}$ satisfying
$\nu(g(t_{2}))=\mu(g(t_{2}),\tau_{0})=\mathcal{W}(g(t_{2}),f_{0},\tau_{0})$.
Solving the equation \cite[Equ. (3.3)]{P1}
\begin{equation}
\left\{ \begin{array}{ll}
\frac{\partial}{\partial t}g=-2Ric, \\
\frac{\partial}{\partial t}f=-\triangle f+|\nabla
f|^{2}-R+\frac{2n}{\tau},\\
\frac{\partial}{\partial t}\tau=-1,\\
f(t_{2})=f_{0},\tau(t_{2})=\tau_{0},
\end{array} \right.\nonumber
\end{equation}
and computing directly \cite{KL}, under this evolving equation, we
have
\begin{equation}\nonumber
\frac{d}{dt}\mathcal{W}=\int_{M}2\tau|Ric+\nabla^{2}f-\frac{1}{2\tau}g|^{2}(4\pi\tau)^{-n/2}e^{-f}dv\geq0.
\end{equation}
Hence by the definitions of $\mu$ and $\nu$, we have
\begin{eqnarray}
\nu(t_{1})&\leq&\mu(g(t_{1}),\tau_{0}+t_{2}-t_{1})\nonumber\\
&\leq&\mathcal{W}(g(t_{1}),f(t_{1}),\tau_{0}+t_{2}-t_{1})\nonumber\\
&\leq&\mathcal{W}(g(t_{2}),f(t_{2}),\tau_{0})=\nu(t_{2}).\nonumber
\end{eqnarray}
The equality doesn't hold unless the integrand equals zero,
$i.e.$, it is a shrinking Ricci soliton. On the other hand, $\nu$
remains obviously constant on a Ricci soliton by the invariance of
this functional under changes by rescalings and diffeomorphism
transformations.
\end{proof}
\vskip 10mm
\section{4-dimensional non-singular solutions}
From now on we are concerned with Ricci flow on $4$-manifolds. We
will continue to use the same notations and conventions in $\S 2$.
We assume all closed manifolds have constant volume $1$.
\begin{lemma} Let $(M,g(t)),t\in[0,\infty)$ be a solution to
(\ref{0}) on a closed $4$-manifold $M$. If $|R(g(t))|\leq C$ and
$ \breve{R}(g(t))\leq -c< 0$, where $C$ and $c$ are constants
independent of $t$, then
$$\int_{0}^{\infty}\int_{M}|Ric\textordmasculine (g(t))|^{2}dvdt<\infty.$$
\end{lemma}
\begin{proof} Note that, for $t\in [0, \infty)$,
$$|R(g(t))|< C, \ \ \ {\rm and} \ \ \ \breve{R}(g(t))\leq -c< 0,$$ where $C$ and $c$ are constants independent of $t$. By
Lemma 2.7, we have
$$\int_{0}^{\infty}\int_{M}|R-r|dv dt<\infty.$$ From the equation
$$\frac{\partial}{\partial t}R=\triangle R
+2|Ric\textordmasculine|^{2}+\frac{2}{4}R(R-r),$$ we obtain
\begin{eqnarray}
\int_{0}^{\infty}\int_{M} 2|Ric\textordmasculine|^{2}dv
dt&=&\int_{0}^{\infty}\int_{M}\frac{\partial}{\partial t}R dv dt-
\frac{1}{2}\int_{0}^{\infty}\int_{M} R(R-r)dv dt\nonumber\\
&=&\int_{0}^{\infty}\frac{\partial}{\partial t}r dt+
\frac{1}{2}\int_{0}^{\infty}\int_{M} R(R-r)dv dt\nonumber\\
&\leq&\lim_{t\longrightarrow \infty
}\sup|r(g(t))-r_{0}|+\frac{C}{2}\int_{0}^{\infty}\int_{M}|R-r|dv dt\nonumber\\
&\leq& 2C+\frac{C}{2}\int_{0}^{\infty}\int_{M}|R-r|dv
dt<\infty.\nonumber
\end{eqnarray}
\end{proof}
Observe that $\breve{R}(g(t))\leq \lambda _M=\overline{\lambda}
_M$. Theorem 1.4 follows immediately from
\begin{lemma}
Let $M$ be a closed oriented 4-manifold $M$ and let
$\{g(t)\},t\in[0,\infty)$, be a solution to (\ref{0}). If
$|R(g(t))|<C$ and $\breve{R}(g(t))\leq
-c< 0$,
where $C$ and $c$ are constants independent of $t$, then
$$2\chi(M)\geq 3|\tau(M)|.$$
Furthermore, if $\overline{\lambda}_{M}< 0$, then
$$2\chi(M)- 3|\tau(M)|\geq \frac{1}{96\pi^{2}}\overline{\lambda}_{M}^{2}.$$
and any non-singular solution $\{g(t)\},t\in[0,\infty)$ does not
collapse.
\end{lemma}
\begin{proof}
From Lemma 3.1 we have
$$\int^{m+1}_{m}\int_{M} |Ric\textordmasculine(g(t))|^{2}dvdt \longrightarrow 0,$$ when
$m\longrightarrow \infty$.
By the Chern-Gauss-Bonnet formula and the Hirzebruch signature
theorem, for any metric $g$ on $M$,
$$\chi(M)=
\frac{1}{8\pi^{2}}\int_{M}(\frac{R(g)^{2}}{24}+|W^{+}(g)|^{2}+|W^{-}(g)|^{2}-\frac{1}{2}
|Ric\textordmasculine(g)|^{2})dv, \ \ \ \ { \rm and}$$
$$\tau(M)=\frac{1}{12\pi^{2}}\int_{M}(|W^{+}(g)|^{2}-|W^{-}(g)|^{2})dv,$$
where $W^{+}(g)$ and $W^{-}(g)$ are the self-dual and
anti-self-dual Weyl tensors respectively (cf. [1]). Thus
\begin{eqnarray*} 2\chi(M)-3|\tau(M)|& \geq &
\liminf\limits_{m\longrightarrow \infty }
\frac{1}{4\pi^{2}}\int^{m+1}_{m}\int_{M}(\frac{1}{24}
R(g(t))^{2}-\frac{1}{2}|Ric\textordmasculine(g(t))|^{2})dvdt
\\ & = & \liminf\limits_{m\longrightarrow \infty }
\frac{1}{4\pi^{2}}\int^{m+1}_{m}\int_{M}\frac{1}{24}R(g(t))^{2}
dvdt \ge 0.\end{eqnarray*} This proves the first inequality.
Observe that $\breve{R}(g(t))\leq \lambda _M=\overline{\lambda}
_M$. Note that
$$\int^{m+1}_{m}\int_{M}R(g(t))^{2}dvdt\geq
(\int^{m+1}_{m}r(g(t))dt)^{2}=(\int^{m+1}_{m}(r-\breve{R})dt+\int^{m+1}_{m}\breve{R}dt)^{2}.$$
From Lemma 2.7, we have
$$\int_{0}^{\infty}(r-\breve{R})dt<\infty.$$ Thus $$\lim\limits_{m\longrightarrow \infty
}\int^{m+1}_{m}(r-\breve{R})dt=0,$$ as $r-\breve{R}\geq 0$. By
taking $m\gg 1$ so that $|\int^{m+1}_{m}(r-\breve{R})dt|\ll 1$,
and
$$(\int^{m+1}_{m}(r-\breve{R})dt+\int^{m+1}_{m}\breve{R}dt)^{2}\geq
(\int^{m+1}_{m}(r-\breve{R})dt+\overline{\lambda}_{M})^{2}.$$
Thus we obtain $$2\chi(M)-3|\tau(M)|\geq \liminf\limits_{m\longrightarrow \infty }
\frac{1}{96\pi^{2}}(\int^{m+1}_{m}(r-\breve{R})dt+\overline{\lambda}_{M})^{2}=
\frac{1}{96\pi^{2}}\overline{\lambda}_{M}^{2}.$$
The second inequality clearly implies that $\chi (M)>0$ whenever
$\overline{\lambda }_M<0$. On the other hand, by Cheeger-Gromov's
collapsing theorem (c.f. [2] [3]) it holds that $\chi(M)=0$ if
$M$ collapse with bounded sectional curvature. This implies the
desired result.
\end{proof}
\begin{proof}[Proof of Theorem \ref{002}] By Theorem 1.1 and the Hitchin-Thorpe inequality for
Einstein manifold it remains only to verify the inequality $$2\chi
(M)\ge 3|\tau (M)|$$ in the case (1.1.5). By $\S 2$ we know that
this only happens when $\breve{R}(g(t))\leq -c< 0$ and
$\lambda(g(t))\leq 0$, where $c$ is a constant independent of $t$.
By Lemma 3.2 the desired result follows.
\end{proof}
\begin{proof}[Proof of Corollary \ref{003}] If $M$ admits an $F$-structure of
positive rank, then $\chi(M)=0$ (c.f. [2] [3]). If $M$ admits a
shrinking soliton, by [8] the fundamental group $\pi _1(M)$ is
finite, and so $b_{1}(M)=b_{3}(M)=0$ by the Poincar\`e duality.
Hence $\chi(M)\geq 2$. By Theorem 1.2 the desired result follows.
\end{proof}
\begin{proof}[Proof of Corollary \ref{005}]By Theorem 1 in
\cite{T}, the Spin$^{c}$ structure induced by a compatible almost complex
structure on $(M, \omega)$ has Seiberg-Witten invariant equal to $\pm
1$. Thus, by Corollary 4.4 in \cite{Le}, we have
$$\int_{M}R(g(t))^{2}dv\geq 32\pi^{2}(2\chi(M)+3\tau(M)).$$
From the proof of Lemma 3.2, we know that \begin{eqnarray*}
2\chi(M)-3|\tau(M)| & \geq & \liminf\limits_{m\longrightarrow
\infty }
\frac{1}{4\pi^{2}}\int^{m+1}_{m}\int_{M}\frac{1}{24}R(g(t))^{2}
dvdt
\\ & \geq & \frac{1}{3}(2\chi(M)+3\tau(M)).\end{eqnarray*} Hence $$\chi(M) \geq 3\tau(M).$$
\end{proof}
\vskip 8mm
\section{Proof of Theorem 1.6}
\vskip 3mm
Let $(M,g(t)),t\in[0,\infty)$ be a non-singular
solution to (\ref{0}) on a closed oriented 4-manifold $M$ with
$\overline{\lambda}_{M} < 0$. Since $|Rm(g(t))|<C$, there is a
constant $\varepsilon
>0$ depending only on $C$ such that, for any $t$,
$M_{t, \varepsilon}=\{x\in M: \text{Vol}(B_{x}(1), g(t))<
\varepsilon\}$ admits an
F-structure of positive rank, and the Euler number $\chi(M_{t, \varepsilon})=0$ (cf. \cite{CG1} \cite{CG2} \cite{A}).
By Theorem 1.4, $\chi(M)\geq
\frac{1}{96\pi^{2}}\overline{\lambda}_{M}^{2}> 0$. Hence, for any $t$, there
is an $x\in M$ such that $\text{Vol}(B_{x}(1), g(t))\geq \varepsilon$.
By Lemma 2.7 and Lemma 3.1, we have $$\int_{0}^{\infty}\int_{M} 2|Ric\textordmasculine|^{2}dv
dt< \infty, \ \ \ \int_{0}^{\infty}(r-\breve{R})dt<\infty \ \ \ \
{\rm and}$$ $$ \ \ \ \int_{0}^{\infty}\int_{M}|R-r|dvdt<\infty.$$
Thus we may choose a sequence of times $\{t_{k}\}$ so that
$t_{k}\longrightarrow
\infty$ and \begin{equation}\nu(k)=\int_{M} 2|Ric\textordmasculine(g(t_{k}))|_{k}^{2}dv_{k}\longrightarrow 0,
\ \ \ \ |r(g(t_{k}))-\breve{R}(g(t_{k}))|\longrightarrow 0 \end{equation} \begin{equation} \ \ \
\mu(k)=\int_{M}|R(g(t_{k}))-r(g(t_{k}))|dv_{k}\longrightarrow 0
\end{equation} as $k\longrightarrow \infty$.
We start the proof by proving three lemmas.
\begin{lemma} There is a sequence points $\{x_{j,k}\in M\}$, $j=1, \cdots, m$,
satisfying that, for any $j$, $(M,g(t_{k}), x_{j, k})$
$C^{\infty}$-converges to a complete Einstein manifold
$(M_{j,\infty},g_{j,\infty},x_{j,\infty})$, i.e. there are
embeddings $F_{j,k, \rho }: B_{x_{j,\infty}}(\rho)\longrightarrow M$
such that, for any $\rho
> 0$, $F_{j,k,\rho}^{*}g(t_{k})$ $C^{\infty}$-converges to
$g_{j,\infty}$ on $B_{x_{j,\infty}}(\rho)\subset M_{j,\infty}$, and
$F_{j,k,\rho}( x_{j, \infty})=x_{j,k}$. Furthermore, all of the
manifolds $M_{j,\infty}$ are distinct, and, for any $\rho> 0$ and
$k\gg 1$, $\{F_{j,k,\rho}(B_{x_{j,\infty}}(\rho))\}$ are disjoint.
\end{lemma}
\begin{proof} For each $g(t_k)$, choose a
maximal number of disjoint unit balls $B_{x_{j,k}}(1)\subset (M, g(t_k))$
so that $\text{Vol}(B_{x_{j,k}}(1), g(t_{k}))\geq
\varepsilon$. Since $\text{Vol}(M, g(t))\equiv 1$, there is a uniform bound on
the number of the balls. Therefore, by passing to a subsequence if necessary, we may assume that
the maximal numbers of the disjoint unit balls for all $g(t_{k})$
are the same, saying $\ell$.
For every $1\le j\le \ell$, by Hamilton's compactness
theorem \cite{H1}, $(M,g(t_{k}+t), x_{j, k})$
$C^{\infty}$-converges to another normalized Ricci flow solution
$(M_{j,\infty},g_{j,\infty}(t),x_{j,\infty})$ after passing to a
subsequence. By Proposition \ref{103},
$(M_{j,\infty},g_{j,\infty}(t),x_{j,\infty})$ is a complete
Einstein manifold with negative scalar curvature for every $t$.
This proves that $(M,g(t_{k}), x_{j, k})$ $C^{\infty}$-converges
to a complete Einstein manifold
$(M_{j,\infty},g_{j,\infty},x_{j,\infty})$, where
$g_{j,\infty}\equiv g_{j,\infty}(0)$, i.e. there are
embeddings $F_{j,k, \rho }: B_{x_{j,\infty}}(\rho)\longrightarrow
M$ such that, for any $\rho > 0$, $F_{j,k,\rho}^{*}g(t_{k})$
$C^{\infty}$-converges to $g_{j,\infty}$ on
$B_{x_{j,\infty}}(\rho)\subset M_{j,\infty}$, and $F_{j,k,\rho}(
x_{j, \infty})=x_{j,k}$. Note that it is not necessary that all of
the manifolds $M_{j,\infty}$ are distinct. Let $\{M_{j,\infty}\}$,
$j=1, \cdots, m\le \ell $, be the resulting collection of distinct
manifolds. It is easy to see that, for any $\rho> 0$ and $k\gg 1$,
$\{F_{j,k,\rho}(B_{x_{j,\infty}}(\rho))\}$ are disjoint, and $\sum
\text{Vol}(M_{j,\infty}, g_{j,\infty})\leq 1$.
\end{proof}
\vskip 2mm
\noindent{\it Remark}: It is easy to see that two limit manifolds
$M_{j_1, \infty}$ and $M_{j_2, \infty}$ are the same if and only
if the distance $\text{dist}_{g(t_k)}(x_{j_1, k},x_{j_2, k})$ is
uniformly bounded above, independent of $k$.
\vskip 2mm
By the above, $(M_{j,\infty},g_{j,\infty},x_{j,\infty})$ satisfies
$|Rm(g_{j,\infty})|<C$ and $\text{Vol}(M_{j,\infty},
g_{j,\infty})\leq 1$. By \cite{CG3} there is a good
chopping
of $M_{j,\infty}$, i.e. an exhaustion, $ \{U_{j,i}\}$, where every $ U_{j,i}$
is a compact $4$-submanifold with boundary $\partial U_{j,i}$,
$\bigcup_{i=1}^{\infty} U_{j,i}=M_{j,\infty}$ such that $$\cdots
\subset U_{j,i}\subset B_{x_{j,\infty}}(i) \subset
U_{j,i+1}\subset B_{x_{j,\infty}}(i+1) \subset U_{j,i+2} \cdots
\subset
M_{j,\infty}$$
satisfying that $ | \rm II \it (\partial U_{j,i}) |<\Lambda$ for
all $i$
and $\rm Vol\it ( \partial U_{j,i}, g_{j,\infty}|_{\partial
U_{j,i}}) \longrightarrow 0$ as $i
\longrightarrow \infty$, where $\rm II \it (\partial U_{j,i})$ is the
second fundamental form of $\partial U_{j,i}$, and $\Lambda$ is
a constant independent of $i$. By Lemma 4.1,
we have \begin{equation}\lim_{k\longrightarrow \infty}|\text{Vol}(U_{j,i}, F_{j,k,i+1}^{*}g(t_{k}))-
\text{Vol}(U_{j,i}, g_{j,\infty})|=0,\end{equation}
\begin{equation}\lim_{i\longrightarrow \infty}|\text{Vol}(U_{j,i},
g_{j,\infty})- \text{Vol}(M_{j,\infty}, g_{j,\infty})|=0, \ \ \ {\rm
and} \ \ \ \lim_{i\longrightarrow \infty}\text{Vol}(\partial
U_{j,i}, g_{j,\infty}|_{\partial U_{j,i}})=0.\end{equation}
\begin{lemma} Let $M_{k,i}=\coprod_{j=1}^{m}F_{j,k,i+1}(U_{j,i})\subset (M, g(t_k))$. Then
$M_{k,i}$ is a $4$-submanifold of $(M, g(t_k))$ with boundary
$\partial M_{k,i}\cong \coprod_{j=1}^{m}\partial U_{j,i}$, and every boundary component
is a graph 3-manifolds for $i\gg 1$.
\end{lemma}
\begin{proof} Since the
second fundamental forms $\rm II \it (\partial U_{j,i}) $ of $\partial
U_{j,i}$ have a uniform bound, the sectional
curvatures of $\partial U_{j,i}$ have a uniform bound, i.e. $|\rm Riem| _{\partial
U_{j,i}}< \Lambda'$. Because $\rm dim \partial U_{j,i}=3$ and $\lim_{i\longrightarrow \infty }\rm Vol\it ( \partial U_{j,i}, g_{j,\infty}|_{\partial
U_{j,i}})=0$, the components
of $\partial U_{j,i}$ are graph 3-manifolds for $i\gg 1$ (cf. \cite{CG3}). By
Lemma 4.1, $F_{j,k,i+1}(U_{j,i})$ are disjoint. Hence
$M_{k,i}$ is a $4$-submanifold of $M$ with boundary $\partial M_{k,i} =\coprod_{j=1}^{m}\partial
U_{j,i}$. The desired result follows.
\end{proof}
\begin{lemma} $$\lim_{k\longrightarrow
\infty}|\sum_{j=1}^{m}\rm {Vol}(U_{j,i}, F_{j,k,i+1}^{*}g(t_{k}))-
\rm {Vol}(M, g(t_{k}))|\leq \overline{C}(\sum_{j=1}^{m} \rm Vol(\partial U_{j,i},
g_{j,\infty}|_{\partial U_{j,i}}))^{\frac{1}{2}},$$
$$\ \
\ {\rm and} \ \ \ \sum_{j=1}^{m} \rm Vol(M_{j,\infty},
g_{j,\infty})= 1$$
where $\overline C$ is a constant independent of the indices.
\end{lemma}
\begin{proof} We first claim that there is an $i_0>0$ such that, for any $i>i_0$, there
is a $k_0$ satisfying that, for any
$k> k_0$, $\rm Vol(B_{y}(1), g(t_{k}))\leq \varepsilon$ for all $y\in
M- M_{k,i}$.
We may choose an $i_0\gg 1$ such that, for all
$y_\infty \in \bigcup_{j=1}^{m}(M_{j,\infty}-
B_{x_{j,\infty}}(i_0-2))$, $\rm Vol(B_{y_\infty}(1), g_{j,\infty})\leq
\frac{1}{2}\varepsilon$. If the claim is false, for any fixed $i> i_0$, there is a
subsequence of times $\{t_{k_{s}}\}$, and a sequence of points
$\{y_{k_{s}}\}$ such that $y_{k_{s}}\in M- M_{k,i}$, and
\begin{equation} \rm Vol(B_{y_{k_{s}}}(1), g(t_{k_{s}}))>
\varepsilon
\end{equation}
Observe that the distance $\rm dist_{g(t_{k_{s}})}(y_{k_{s}},x_{j,
k_{s}})\longrightarrow \infty$ as $k_s\to \infty$ for all $1\le
j\le m$. Otherwise, assuming $\rm
dist_{g(t_{k_{s}})}(y_{k_{s}},x_{j, k_{s}})< {\rho }$ for some $j$
and $\rho >0 $, we get that $F_{j,k_{s},\rho
}^{-1}(y_{k_{s}})\longrightarrow y_{\infty}\in
B_{x_{j,\infty}}(\rho )- B_{x_{j,\infty}}(i-1)$, and so
\begin{equation}
\rm Vol(B_{y_{k_{s}}}(1), g(t_{k_{s}}))\longrightarrow \rm
Vol(B_{y_{\infty}}(1), g_{j,\infty})\le \frac 12 \varepsilon
\end{equation}
when $k_{s}\longrightarrow \infty$, since $F_{j,k_{s},
\rho}^{*}g(t_{k_{s}}) $ $C^{\infty}$-converges to $g_{j,\infty}$.
This contradicts to (11).
On the other hand, $(M,g(t_{k_{s}}), y_{ k_{s}})$
$C^{\infty}$-converges to a complete Einstein manifold
$(M_{\infty},g_{\infty},y_{\infty})$, and $M_{\infty}$ is distinct
from everyone of $M_{j,\infty}$ for $1\le j \le \ell$ (cf. the
remark after Lemma 4.1). This violates the choice of maximality of
$m$. The claim follows.
By \cite{CG3} and the above claim, for any $i>i_0$, there is a
$k_0$ such that, for any $k>k_0$, $ M-M_{k,i}$ admits an
F-structure of positive rank, and $\chi(M- M_{k,i})=0$. By
Chern-Gauss-Bonnet theorem, \begin{eqnarray*}0=\chi(M-
M_{k,i})& =&
\frac{1}{8\pi^{2}}\int_{M-
M_{k,i}}(\frac{R(g(t_{k}))^{2}}{24}+|W^{\pm}(g(t_{k}))|_{k}^{2}-\frac{1}{2}
|Ric\textordmasculine(g(t_{k}))|_{k}^{2})dv_{k}\\
& & +\int_{\partial
(M- M_{k,i})}P_{k}(\rm II\it), \end{eqnarray*} where $P_{k} (\rm
II\it) $ is a polynomial of the second
fundamental form $\rm II\it(\partial
(M- M_{k,i}))$ of $\partial (M- M_{k,i})$ and its sectional
curvature. Hence
$$ \int_{M- M_{k,i}}\frac{R(g(t_{k}))^{2}}{24}dv_{k}\leq
\int_{M}\frac{1}{2} |Ric\textordmasculine(g(t_{k}))|_{k}^{2}dv_{k}
- 8\pi^{2}\int_{\partial (M- M_{k,i})}P_{k}(\rm II\it).$$
Since $F_{j,k,i+1}^{*}g(t_{k})$ $C^{\infty}$-converges to
$g_{j,\infty}$ on $B_{x_{j,\infty}}(i+1)\subset M_{j,\infty}$, we
have $|P_{k}(\rm II\it)|< \overline{C}$, $k\gg 1$, where
$\overline{C}$ is a constant depending only on the bounds $\Lambda$
of the
second fundamental forms $\rm II \it (\partial
U_{j,i})$ of $\partial U_{j,i}$, and
$$\upsilon(k): =|\rm Vol(\partial (M- M_{k,i}), g(t_{k})|_{\partial
(M- M_{k,i})})-\sum_{j=1}^{m} \rm Vol(\partial U_{j,i},
g_{j,\infty}|_{\partial U_{j,i}})|\longrightarrow 0$$ if
$k\longrightarrow \infty$. Hence
$$\int_{M- M_{k,i}}\frac{R(g(t_{k}))^{2}}{24}dv_{k}\leq
\frac{\nu(k)}{4}+\overline{C}(\sum_{j=1}^{m} \rm Vol(\partial
U_{j,i}, g_{j,\infty}|_{\partial U_{j,i}})+\upsilon(k))$$ where
$\nu (k)$ is defined in equation (7).
Clearly, \begin{eqnarray*}& & |r(g(t_{k}))|\rm Vol(M-
M_{k,i},g(t_{k}))-\int_{M- M_{k,i}}|R(g(t_{k}))|dv_{k} \\
& \leq & \int_{M- M_{k,i}}|R(g(t_{k}))-r(g(t_{k}))|dv_{k}\leq
\mu(k), \ \ \
\end{eqnarray*}
where $\mu (k)$ is as in (8).
\begin{eqnarray*}\int_{M- M_{k,i}}|R(g(t_{k}))|dv_{k} & \leq &
(\int_{M- M_{k,i}}R(g(t_{k}))^{2}dv_{k})^{\frac{1}{2}}\rm Vol(M-
M_{k,i},g(t_{k}))^{\frac{1}{2}}\\ & \leq & (\int_{M-
M_{k,i}}R(g(t_{k}))^{2}dv_{k})^{\frac{1}{2}}.\end{eqnarray*} Thus
\begin{eqnarray*}\rm Vol(M-
M_{k,i},g(t_{k}))& \leq &
\frac{2}{|r_{\infty}|}(\mu(k)+(6\nu(k)\\
& & +\overline{C}(\sum_{j=1}^{m} \rm Vol(\partial U_{j,i},
g_{j,\infty}|_{\partial U_{j,i}})+\upsilon(k)))^{\frac{1}{2}}),
\end{eqnarray*}
where we used the fact that $r(g(t_{k}))\longrightarrow
r_{\infty}<0$ if $k\longrightarrow \infty$ (cf. the proof of
Proposition 2.6.) Therefore,
\begin{eqnarray*}| \sum_{j=1}^{m}\rm Vol(U_{j,i},g_{j,\infty})-1|
& =& \lim_{k\longrightarrow \infty}|\rm Vol(M_{k,i}, g(t_{k}))-
1| \\ & \leq &\overline{C}(\sum_{j=1}^{m} \rm Vol(\partial U_{j,i},
g_{j,\infty}|_{\partial U_{j,i}}))^{\frac{1}{2}}.\end{eqnarray*}
By letting $i\longrightarrow \infty$, we get the second equality
in the lemma.
\end{proof}
\begin{proof}[Proof of Theorem 1.6] By Lemma 4.3, for any $\delta >0$, we can choose $i\gg 1$ and $k\gg 1$ such
that $\rm Vol(M- M_{k,i},g(t_{k}))<\delta $. Let $T=t_{k}$ and
$M^{\varepsilon}=M_{k,i}$. The desired results follows by Lemmas
4.1, 4.2 and 4.3.
\end{proof}
\vskip 8mm
\section{proof of Theorem 1.7}
\vskip 8mm
Let us first recall some facts about Seiberg-Witten equations,
which will
be used to prove Theorem 1.7 (See \cite{Le} for details).
Let $(M, g)$ be a compact oriented Riemannian $4$-manifold
with a $\rm Spin^{c}$ structure $\mathfrak{c}$. Let
$b^{+}_{2}(M)$ denote the dimension of the space of self-dual
harmonic $2$-forms in $M$. Let $S^{\pm}_{\mathfrak{c}}$ denote the
$\rm Spin^{c}$-bundles associated to $\mathfrak{c}$, and let $L$
be the determinant line bundle of $\mathfrak{c}$. There is a
well-defined Dirac operator
$$\mathcal{D}_{A}:
\Gamma(S^{+}_{\mathfrak{c}})\longrightarrow
\Gamma(S^{-}_{\mathfrak{c}})$$ Let $c:
\wedge^{*}T^{*}M \longrightarrow {\rm End}(S^{+}_{\mathfrak{c}}\oplus
S^{-}_{\mathfrak{c}})$ denote the Clifford multiplication on the
$\rm{Spin}^c$-bundles, and, for any $\phi\in \Gamma(S^{\pm})$, let
$$q(\phi)=\overline{\phi}\otimes\phi-\frac{1}{2}|\phi|^{2}{\rm id}.$$
The Seiberg-Witten equations read
$$\begin{array}{ccc}\mathcal{D}_{A}\phi=0 \\
c(F^{+}_{A})=q(\phi)
\end{array} $$
where the unknowns are a hermitian connection $A$ on $L$ and a
section $\phi\in \Gamma(S^{+}_{\mathfrak{c}})$, and $F^{+}_{A}$ is
the self-dual part of the curvature of $A$. A resolution of the
Seiberg-Witten equations is called {\it reducible} if $\phi\equiv
0$; otherwise, it is called {\it irreducible}.
Let $(M,g(t)),t\in[0,\infty)$ be a non-singular
solution to (\ref{0}) on a closed oriented 4-manifold $M$ with
$\overline{\lambda}_{M} < 0$. We will continue to assume the
volume of $(M, g(t))$ is $1$. Let $\breve{R}(g(t))$ denote the
minimum of the scalar curvature of $g(t)$. Recall that
$\breve{R}(g(t))\le \overline{\lambda}_{M}$. Assume that $M$
admits a symplectic structure $\omega$ satisfying that $b^{+}_{2}(M)
>1$ and $\chi(M)=3 \tau (M)$. Let $t_{k}$, $x_{j,k}$, $U_{j,i}$, $M_{j,\infty}$ and $F_{j,k,\rho}$
be the same as in Section 4. By Theorem 1 in
\cite{T}, the Spin$^{c}$ structure induced by a compatible almost complex
structure $J$ on $(M, \omega)$ has Seiberg-Witten invariant equal to $\pm
1$. Hence, for any $k$, there is an irreducible solution $(\phi_{k}, A_{k})$
to the Seiberg-Witten equations (cf. \cite{Le}). For the sake of
simplicity we will use $|\cdot |_k$ to denote the norm with
respect to the metric $g(t_k)$.
\begin{lemma} $$8\int_{M}|F^{+}_{A_{k}}|_{k}^{2}dv_{k} \geq
\int_{M}R(g(t_{k}))^{2}dv_{k}-\int_{M}48\pi^{2}
|Ric\textordmasculine(g(t_{k}))|_{k}^{2}dv_{k},$$
$$\ \ \ {\rm and} \ \ \ \lim_{k\longrightarrow \infty}\int_{M}|\nabla^{k} F^{+}_{A_{k}}|_{k}^{2}dv_{k}=0.$$
\end{lemma}
\begin{proof}The Bochner formula implies that
$$0=\frac{1}{2}\Delta_{k}
|\phi_{k}|_{k}^{2}+|\nabla^{A_{k}}\phi_{k}|_{k}^{2}+\frac{R(g(t_{k}))}{4}|\phi_{k}|_{k}^{2}+\frac{1}{4}|\phi_{k}|_{k}^{4},$$
$$4\int_{M}|\nabla^{A_{k}}\phi_{k}|_{k}^{2}dv_{k}=-\int_{M}(R(g(t_{k}))|\phi_{k}|_{k}^{2}+|\phi_{k}|_{k}^{4})dv_{k}.$$
From the estimate $|\phi_{k}|_{k}^{2}\leq - \breve{R}(g(t_{k}))$
(cf. \cite{Le}) and $-R(g(t_k))\le -\breve{R}(g(t_{k}))<0$ we get
that
$$4\int_{M}|\nabla^{A_{k}}\phi_{k}|_{k}^{2}dv_{k} \leq \int _M \breve{R}(g(t_{k}))^{2}dv_k-\int_{M}|\phi_{k}|_{k}^{4}dv_{k}.$$
By the second equation of the Seiberg-Witten equations and
$\chi(M)=3\tau(M)$,
\begin{eqnarray*}\int_{M}|\phi_{k}|_{k}^{4}dv_{k} & =&
8\int_{M}|F^{+}_{A_{k}}|_{k}^{2}dv_{k} \\ &\geq &
32\pi^{2}[c_{1}^{+}]^{2}[M] \geq 32\pi^{2}[c_{1}]^{2}[M] \\ &=&
32\pi^{2}(2\chi(M)+3\tau(M))\\ &=& 96 \pi^{2}(2\chi(M)-3\tau(M)) \\
&\geq & \int_{M}R(g(t_{k}))^{2}dv_{k}-\int_{M}48\pi^{2}
|Ric\textordmasculine(g(t_{k}))|_{k}^{2}dv_{k} ,
\end{eqnarray*}
where the last inequality follows by the Chern-Gauss-Bonnet
formula and Hirzebruch's signature formula (cf. section 4). Thus
\begin{eqnarray} 8\int_{M}|F^{+}_{A_{k}}|_{k}^{2}dv_{k} \geq
\int_{M}R(g(t_{k}))^{2}dv_{k}-\int_{M}48\pi^{2}
|Ric\textordmasculine(g(t_{k}))|_{k}^{2}dv_{k},\end{eqnarray}
where $c_{1}^{+}$ is the self-dual part of the harmonic form
representing the first Chern class $c_{1}$ of $M$. Hence, by (7)
(8), we have
\begin{eqnarray*}4\int_{M}|\nabla^{A_{k}}\phi_{k}|_{k}^{2}dv_{k} & \leq &
\int_{M}(\breve{R}(g(t_{k}))^{2}-R(g(t_{k}))^{2})dv_{k}+\int_{M}48\pi^{2}
|Ric\textordmasculine(g(t_{k}))|_{k}^{2}dv_{k} \\ & \leq &
C|\breve{R}(g(t_{k}))-r(g(t_{k}))|+\int_{M}48\pi^{2}
|Ric\textordmasculine(g(t_{k}))|_{k}^{2}dv_{k} \longrightarrow 0,
\end{eqnarray*}$k\longrightarrow 0$, where $C$ is a constant
independent of $k$. By the second one of the Seiberg-Witten
equations again (cf.
\cite{Le}), $$|\nabla^{k} F^{+}_{A_{k}}|_{k}^{2}\leq
\frac{1}{2}|\phi_{k}|_{k}^{2}|\nabla^{A_{k}} \phi_{k}|_{k}^{2},$$ where $\nabla^{k}$ is the connection induced by
Levi-civita connection. Hence
$$\int_{M}|\nabla^{k} F^{+}_{A_{k}}|_{k}^{2}dv_{k}\leq \frac{1}{2}|\breve{R}(g(t_{k}))|\int_{M}|
\nabla^{A_{k}}\phi_{k}|_{k}^{2}dv_{k}
\longrightarrow 0,$$ when $k\longrightarrow \infty$.
\end{proof}
Regard $F^{+}_{A_{k}}$ as self-dual 2-forms of $g'(t_{k})$ on $
U_{j,i}\subset M_{j,\infty}$, where
$g'(t_{k})=F_{j,k,i+1}^{*}g(t_{k})$. Since
$|F^{+}_{A_{k}}|_{k}^{2}=\frac{1}{8}|\phi_{k}|_{k}^{4}\leq
\frac{1}{8}\breve{R}(g(t_{k}))^{2}\leq C$, where $C$ is a constant
independent of $k$, $F^{+}_{A_{k}}\in L_{1}^{2}(g'(t_{k}))$, and $$\|F^{+}_{A_{k}}\|_{L_{1}^{2}(g'(t_{k}))}\leq C',$$
where $C'$ is a constant independent of $k$. Note that $\|\cdot
\|_{L_{1}^{2}(g_{j,\infty})}\leq 2 \|\cdot
\|_{L_{1}^{2}(g'(t_{k}))}$ for $k\gg 1$ since $g'(t_{k})$
$C^{\infty}$-converges to
$g_{j,\infty}$ on $ U_{j,i}$. Thus, by passing to a subsequence,
$F^{+}_{A_{k}}$ $L_{1}^{2}$-converges to a 2-form $\Omega_{j}\in
L_{1}^{2}(g_{j,\infty})$, which is a self-dual 2-form of
$g_{j,\infty}$.
\begin{lemma} For any $j$, $\Omega_{j}$ is a smooth self-dual 2-form on
$U_{j,i}-
\partial U_{j,i}$ such that $\nabla^{\infty} \Omega_{j}\equiv 0$,
and $| \Omega_{j}|_{\infty}\equiv {\rm cont.}\neq 0$, where
$\nabla^{\infty}$ is the connection induced by the
Levi-civita connection of $g_{j,\infty}$.
\end{lemma}
\begin{proof}
Note that
$$0 \leq \int_{U_{j,i}}|\nabla^{\infty} \Omega_{j}|_{\infty}^{2}dv_{\infty}=\lim_{k\longrightarrow \infty}
\int_{U_{j,i}}|\nabla^{\infty}
F^{+}_{A_{k}}|_{\infty}^{2}dv_{\infty} \leq \lim_{k\longrightarrow
\infty} \int_{M}|\nabla^{k} F^{+}_{A_{k}}|_{k}^{2}dv_{k}=0.$$ It
is easy to see that $\Omega_{j}$ is a weak solution of the
elliptic equation $(d+d^{*})\Omega_{j}=0$ on $U_{j,i}$. By
elliptic equation theory, $\Omega_{j}$ is a smooth self-dual
2-form on $U_{j,i}-
\partial U_{j,i}$, and $\nabla^{\infty} \Omega_{j}\equiv 0$.
Now we claim that, for any $j$ and $i\gg 1$,
$\int_{U_{j,i}}|\Omega_{j}|_{\infty}^{2}dv_{\infty}\neq 0$. If it is
not true, there is a $j_{1}$ such that
$\int_{U_{j_{1},i}}|\Omega_{j_{1}}|_{\infty}^{2}dv_{\infty}\equiv
0$. Note that, by the results in Section 4,
\begin{eqnarray*}\int_{U_{j,i}}|\Omega_{j}|_{\infty}^{2}dv_{\infty}=\lim_{k\longrightarrow
\infty} \int_{U_{j,i}}| F^{+}_{A_{k}}|_{\infty}^{2}dv_{\infty}&=&
\lim_{k\longrightarrow \infty} \int_{U_{j,i}}|
F^{+}_{A_{k}}|_{k}^{2}dv_{k}\\
&\leq &\frac{1}{8}\lim_{k\longrightarrow \infty}
\breve{R}(g(t_{k}))^{2}\rm Vol(U_{j,i}, g'(t_{k}))\\ &=&
\frac{1}{8}r_{\infty}^{2}\rm Vol(U_{j,i}, g_{j,\infty})
\end{eqnarray*}
$$|\int_{M}(R(g(t_{k}))^{2}-r_{\infty}^{2})dv_{k}|\leq
C\int_{M}(|R(g(t_{k})-r(g(t_{k}))|+|r_{\infty}-r(g(t_{k}))|)dv_{k}
\longrightarrow 0,$$ and \begin{eqnarray*}
\lim_{k\longrightarrow \infty}|\int_{M}|F^{+}_{A_{k}}|_{k}^{2}dv_{k}
-\sum_{j=1}^{m}
\int_{U_{ji}}|F^{+}_{A_{k}}|_{k}^{2}dv_{k}|&=&\lim_{k\longrightarrow \infty}\int_{M- M_{k,i}}|F^{+}_{A_{k}}|_{k}^{2}dv_{k}
\\& \leq & C \lim_{k\longrightarrow \infty}|\sum_{j=1}^{m}\rm Vol(U_{j,i},
g'(t_{k}))\\ & &
-
\rm Vol(M, g(t_{k}))|\\ & \leq &\overline{C}(\sum_{j=1}^{m} \rm Vol(\partial U_{j,i},
g_{j,\infty}|_{\partial U_{j,i}}))^{\frac{1}{2}}, \end{eqnarray*} if
$k\longrightarrow \infty$, where $C$ and $\overline{C}$ are
constants independent of $k$. By Lemma 5.1, we obtain
\begin{eqnarray*}r_{\infty}^{2}\sum_{j\neq j_{1}}\rm Vol(U_{j,i}, g_{j,\infty})&
\geq &
\sum_{j=1}^{m}\int_{U_{j,i}}8|\Omega_{j}|_{\infty}^{2}dv_{\infty}=\lim_{k\longrightarrow
\infty}\int_{M_{k,i}}8|F^{+}_{A_{k}}|_{k}^{2}dv_{k}\\ & \geq &
\lim_{k\longrightarrow
\infty}\int_{M}8|F^{+}_{A_{k}}|_{k}^{2}dv_{k}-\overline{C}(\sum_{j=1}^{m}
\rm Vol(\partial U_{j,i}, g_{j,\infty}|_{\partial
U_{j,i}}))^{\frac{1}{2}}
\\ & \geq& \lim_{k\longrightarrow \infty}\int_{M}(R(g(t_{k}))^{2}dv_{k}-\overline{C}(\sum_{j=1}^{m} \rm
Vol(\partial U_{j,i},
g_{j,\infty}|_{\partial U_{j,i}}))^{\frac{1}{2}}
\\ & =& r_{\infty}^{2}-\overline{C}(\sum_{j=1}^{m} \rm Vol(\partial U_{j,i},
g_{j,\infty}|_{\partial U_{j,i}}))^{\frac{1}{2}}.
\end{eqnarray*} Note that, for $i\gg 1$, $$1\gg \overline{C}(\sum_{j=1}^{m} \rm Vol(\partial U_{j,i},
g_{j,\infty}|_{\partial U_{j,i}}))^{\frac{1}{2}}\geq
r_{\infty}^{2}\rm Vol(U_{j_{1},i}, g_{j_{1},\infty}).$$ A
contradiction. Thus, for any $j$,
$\int_{U_{j,i}}|\Omega_{j}|_{\infty}^{2}dv_{\infty}\neq 0$. Thus
we obtain that $\nabla^{\infty} \Omega_{j}\equiv 0$, $|
\Omega_{j}|_{\infty}\equiv {\rm cont.}\neq 0$.
\end{proof}
\begin{proof}[Proof of Theorem 1.7]
Since all $\Omega_{j}$, $1\le j\le m$, are self-dual 2-forms, and $\nabla^{\infty} \Omega_{j}\equiv 0$, $|
\Omega_{j}|_{\infty}\equiv {\rm cont.}\neq 0$. Hence, on any
$U_{j,i}$, $g_{j, \infty}$ is a K\"{a}hler metric with K\"{a}hler
form $\sqrt{2}\frac{\Omega_{j}}{|\Omega_{j}|}$. By Lemma 4.1,
$g_{j, \infty}$ is a K\"{a}hler-Einstein metric on
$M_{j,\infty}$.
Now, by the Chern-Gauss-Bonett theorem and the Hirzebruch theorem,
\begin{eqnarray*} 0=\chi(M)-3\tau(M) & \geq & \liminf_{k\longrightarrow
\infty}\frac{1}{2\pi^{2}}( \int_{M}|W^{-}(g(t_{k}))|_{k}^{2}dv_{k}
-\frac{1}{4} \int_{M}|Ric\textordmasculine(g(t_{k}))|_{k}^{2}dv_{k})\\
&= & \liminf_{k\longrightarrow \infty}\frac{1}{2\pi^{2}}
\int_{M}|W^{-}(g(t_{k}))|_{k}^{2}dv_{k} \geq 0,
\end{eqnarray*} where $W^{-}$ is the anti-self-dual Weyl tensor.
Thus $$\liminf_{k\longrightarrow \infty}
\int_{M}|W^{-}(g(t_{k}))|_{k}^{2}dv_{k} = 0.$$ Therefore, for any
$j$,
$$0\leq \int_{M_{j,\infty}}|W^{-}(g_{j,\infty})|_{\infty}^{2}dv_{\infty}\leq \liminf_{k\longrightarrow \infty}
\int_{M}|W^{-}(g(t_{k}))|_{k}^{2}dv_{k} = 0.$$ Hence
$g_{j,\infty}$ is a K\"{a}hler-Einstein metric with
$W^{-}(g_{j,\infty})\equiv 0$. This implies that $g_{j,\infty}$ is
a complex hyperbolic metric by the proof of Theorem 4.5 in
\cite{Le}. The desired result follows.
\end{proof}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 8,696 |
Old alliances in the age of "America First"
By Karl-Theodor zu Guttenberg and Ulf Gartzke
America's political pendulum has swung back once again – and this time in a "huge" and unparalleled way. GOP Congressman and Donald Trump supporter Tom Cole put it succinctly when he commented on the 45th president's inauguration: "It really is a leap into the dark. And I think that's true for the country and that's true for Trump." It is also true for Europe and the rest of the world.…
Europe reacts to "America First"
By Jean-Marie Guéhenno
It is unclear whether US President Donald Trump is aware of the history behind the expression "America First," the term he uses to describe his foreign policy vision. The catchphrase was first used just before World War II by isolationists who opposed any American engagement in the mounting European crisis. The echo of that dark period has relevance for today. At that time, the structures that had been put in …
We can – and must – build on the unity of the transatlantic alliance
By Sigmar Gabriel
The paradigm of a world in crisis is dominating the international debate. "Post-truth?", "Post- West?", "Post-order?" are questions certain to be raised at the Munich Security Conference. In the echo chambers of the "every nation for itself" ideologists, it sounds as if we are already on the slippery slope towards a Hobbesian system in which the European Union, NATO, the merits of rules-based order and even long-standing friendships and partnerships … | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 6,237 |
Caproic acid is a short chain fatty acid used in manufacture of lubricants, surfactants, plasticizers among, others. Caproic acid is a white crystalline solid or oily colorless liquid with a characteristic odor. The other names for caproic acid is hexanoic acid. Caproic acid is used as a synthetic lubricant and refrigeration lubricant in production of metal working fluids to enhance rust resistance, cutting capabilities and grinding capabilities of fluids; in production of specialty soaps; as a chain terminator in production of polyester plasticizer among, others. Caproic acid is widely used in lotions, conditioners and creams in personal care industry. Caproic acid is intensely toxic and harmful and causes mild skin irritation when exposed in large quantities to the skin. In surfactants, caproic acid finds applications in shower gels, laundry detergents, liquid soaps and fabric softeners. Caproic acid degrades in the environment as it has the property of biodegradability.
Global caproic acid market size is expected to be positively influenced by the rising food and beverage industry. Rising synthetic and refrigeration lubricants demand for metal working fluids will drive the global caproic acid market growth. Global caproic acid market is observing growth owing to the increasing surfactant based industries like the textiles, soaps and detergents, personal care and cosmetics, oil & gas among, others.
Population demographics and lifestyle dynamics will continue to drive the personal care and cosmetics demand, which in turn will propel the pharmaceutical industry. In addition, the growing demand of fatty acids as growth inhibitors in the animal-feed additives industry is anticipated to boost the global caproic acid market during the forecast period.
Owing to the wide-area flexibility in feedstock availability and consumption of fatty acids in the emerging economies, the caproic acid market is witnessing growth. Their extensive and effective use in manufacturing lubricant additives and fire resistant hydraulic fluids is creating opportunities for global caproic acid market.
The use of sustainable chemicals has increased owing to the concerns regarding the effect of petrochemicals on the environment. A shift towards bio-based raw materials is observed owing to the trend to decrease carbon footprint and to reduce complete dependence on petrochemicals. The prohibition on variable cost parameters and key raw ingredients is expected to rise. The increasing cost of production and extensive research and development programs to meet the quality of products is a major constraint for the caproic acid market. As well as, the adoption of stringent regulatory guidelines is anticipated to hamper the prospective growth of caproic acid market.
The global caproic acid market is segmented into Asia Pacific, Europe, North America, Middle East and Africa and Latin America. The growth of polymer processing industries in North America and Europe is anticipated to steer the growth of caproic acid market. Asia-Pacific is projected to dominate the caproic acid market owing to the increasing demand for personal care products. India and China are anticipated to be the prominent consumer of caproic acid in the Asia-pacific region owing to the presence of established end-user industries in the countries of these emerging economy.
Some of the key players identified across the value chain of the global caproic acid market are P&G Chemicals, Emery Oleochemicals, KLK OLEO, Ecogreen Oleochemicals, Pacific Oleochemicals Sdn Bhd, Oleon NV, Ecogreen Oleochemicals, Timur OleoChemicals, Mosselman s.a, among others. | {
"redpajama_set_name": "RedPajamaC4"
} | 5,406 |
Sonny Burgess - Legendary Pacers Live In Spain LP
Sonny Burgess - Legendary Pacers Live In Spain LP. Sonny Burgess was one of the earliest of the original Sun Records rock'n'rollers. He signed with Sam Phillips in 1956 and produced a series of wild, raw 45s which cemented his reputation forever. Sonny refused all offers to perform in Europe until 1984, when he played a festival in Weymouth and stunned fans with his fabulous guitar playing and raw vocals. This LP shows Sonny raw, wild and in his prime on this stellar sounding live recording mixed and mastered by Sonny Burgess with the help of Mike Mariconda from the Raunch Hands. THIS LP IS LIMITED TO 1000 COPIES WITH TOP NOTCH GATEFOLD PACKAGING AND PRESSED ON SNAZZY CREAMY YELLOW WAX! Import from Spain. | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 2,523 |
<?php
/**
* Created by IntelliJ IDEA.
* User: KryDos
* Date: 22/03/16
* Time: 21:31
*/
namespace PHPNotifier;
use PHPNotifier\interfaces\RWInterface;
class PHPNotifier
{
/**
* This constant has partial name of
* Reader/Writer class, that will be
* used in the library
*/
const FILE_METHOD = 'File';
/** @var RWInterface $rw*/
protected $rw;
public function __construct($method, $store)
{
$this->rw = new RWFactory($method, $store);
}
/**
* puts task inside a store
*
* @param Task $task
*/
protected function schedule(Task $task)
{
$this->rw->getWriter()->createDb();
$this->rw->getWriter()->write($task);
}
/**
* set exact time when command have to be executed
*
* @param mixed $when
* @param string $command
* @param array $params
*/
public function scheduleTaskAtTime($when, $command, array $params = [])
{
if ($when instanceof \DateTime) {
$task = new Task($when->getTimestamp(), $command, $params);
} elseif (is_numeric($when)) {
$task = new Task($when, $command, $params);
} elseif(is_string($when) && ($timestamp = strtotime($when))) {
$task = new Task($timestamp, $command, $params);
} else {
throw new \InvalidArgumentException('time argument is not supported. Only integer, DateTime or valid date string are allowed');
}
$this->schedule($task);
}
/**
* execute task after $run_after seconds
*
* @param integer $run_after
* @param string $command
* @param array $params
*/
public function scheduleTaskIn($run_after, $command, array $params = [])
{
$task = new Task((time() + $run_after), $command, $params);
$this->schedule($task);
}
/**
* @return interfaces\ReaderInterface
*/
public function getReader()
{
return $this->rw->getReader();
}
/**
* @return interfaces\WriterInterface
*/
public function getWriter()
{
return $this->rw->getWriter();
}
} | {
"redpajama_set_name": "RedPajamaGithub"
} | 5,321 |
from sys import maxsize
class Group:
def __init__(self, name=None, header=None, footer=None, id=None):
self.name = name
self.header = header
self.footer = footer
self.id = id
def __repr__(self):
return "%s:%s%s:%s" % (self.id, self.name, self.header, self.footer)
def __eq__(self, other):
return (self.id is None or other.id is None or self.id == other.id) and self.name == other.name
def id_or_max(self):
if self.id:
return int(self.id)
else:
return maxsize
class Contact:
def __init__(self, firstname=None, lastname=None, company=None, homepage=None, id=None, all_emails_from_home_page=None,
all_phones_from_home_page=None, homephone=None, mobilephone=None, workphone=None, secondaryphone=None,
address=None, email=None, email2=None, email3=None):
self.lastname = lastname
self.firstname = firstname
self.company = company
self.homepage = homepage
self.address = address
self.homephone = homephone
self.mobilephone = mobilephone
self.workphone = workphone
self.secondaryphone = secondaryphone
self.id = id
self.email = email
self.email2 = email2
self.email3= email3
self.all_emails_from_home_page = all_emails_from_home_page
self.all_phones_from_home_page = all_phones_from_home_page
def __repr__(self):
return "%s:%s:%s:%s:%s:%s:%s" % (self.firstname, self.lastname,self.company,
self.address, self.email, self.homepage, self.id)
def __eq__(self, other):
return (self.id is None or other.id is None or self.id == other.id) and self.firstname == other.firstname
def id_or_max(self):
if self.id:
return int(self.id)
else:
return maxsize
| {
"redpajama_set_name": "RedPajamaGithub"
} | 6,485 |
{"url":"https:\/\/phys.libretexts.org\/Bookshelves\/University_Physics\/Book%3A_Introductory_Physics_-_Building_Models_to_Describe_Our_World_(Martin_Neary_Rinaldo_and_Woodman)\/26%3A_Calculus\/26.01%3A_Functions_of_Real_Numbers","text":"$$\\require{cancel}$$\n\n# 26.1: Functions of Real Numbers\n\nIn calculus, we work with functions and their properties, rather than with variables as we do in algebra. We are usually concerned with describing functions in terms of their slope, the area (or volumes) that they enclose, their curvature, their roots (when they have a value of zero) and their continuity. The functions that we will examine are a mapping from one or more independent real numbers to one real number. By convention, we will use $$x,y,z$$ to indicate independent variables, and $$f()$$ and $$g()$$, to denote functions. For example, if we say: \\begin{aligned} f(x) &= x^2\\\\ \\therefore f(2) &= 4\\end{aligned} we mean that $$f(x)$$ is a function that can be evaluated for any real number, $$x$$, and the result of evaluating the function is to square the number $$x$$. In the second line, we evaluated the function with $$x=2$$. Similarly, we can have a function, $$g(x,y)$$ of multiple variables: \\begin{aligned} g(x,y)&=x^2+2y^2\\\\ \\therefore g(2,3)&=22\\end{aligned}\n\nWe can easily visualize a function of 1 variable by plotting it, as in Figure A2.1.1.\n\nPlotting a function of 2 variables is a little trickier, since we need to do it in three dimensions (one axis for $$x$$, one axis for $$y$$, and one axis for $$g(x,y)$$). Figure A2.1.2 shows an example of plotting a function of 2 variables.\n\nUnfortunately, it becomes difficult to visualize functions of more than 2 variables, although one can usually look at projections of those functions to try and visualize some of the features (for example, contour maps are 2D projections of 3D surfaces, as shown in the $$xy$$ plane of Figure A1.1.2). When you encounter a function, it is good practice to try and visualize it if you can. For example, ask yourself the following questions:\n\n\u2022 Does the function have one or more maxima and\/or minima?\n\u2022 Does the function cross zero?\n\u2022 Is the function continuous everywhere?\n\u2022 Is the function always defined for any value of the independent variables?","date":"2022-01-27 18:02:41","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\": 0.9673076272010803, \"perplexity\": 292.7855858480294}, \"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-2022-05\/segments\/1642320305277.88\/warc\/CC-MAIN-20220127163150-20220127193150-00696.warc.gz\"}"} | null | null |
There are an assortment of ways which can be utilized to make your bathroom look shining. You can make rich structures or incorporate a few hues yet it is never finished until you incorporate a mirror plan which brings the best watch out of your bathroom. In this way, it is essential you look at on our assortment of styles we have accommodated you at whatever point you have to finish your bathroom.
As it is known, two mirrors for the most part give a more innovative and upscale look than when you utilize one mirror. It has been appeared to have a ton of points of interest while contrasted and different sorts as they enable numerous individuals to use in the meantime. This enables them to spare time as there is no time squandered when on the pausing.
Hence, it helps numerous couples when they are getting ready to go to different obligations and it is likewise agreeable when you are checking your looks in the mirror with somebody. Hence, it is prescribed that on the off chance that you need to get such an involvement in your bathroom, you should attempt this thought.
It is extremely simple to guarantee that you influence your bathroom to have a style and present day. It will require little exertion to make your bathroom look shimmering and overly cool when you put roundabout mirrors which will change the presence of your bathroom from having those normally utilized mirrors with sharp edges.
This shows innovativeness and in the meantime, it makes a novel look in your bathroom which will leave everybody charmed in the wake of utilizing the bathroom. Such a bathroom has an inviting impact as you will feel to visit your bathroom regularly. | {
"redpajama_set_name": "RedPajamaC4"
} | 850 |
Q: Is it possible to have dynamically generated query names in ColdFusion? What I am trying to do is
<cfloop array="#LOCAL.someArray" index="LOCAL.aString">
<cfset LOCAL.queryName = "uniqueQueryName_" & LOCAL.aString />
<cfquery name="#LOCAL.queryName#" datasource="db" cachedwithin="#CreateTimeSpan(1,0,0,0)#">
SELECT count(*) AS c FROM someTable
</cfquery>
<cfdump var="#LOCAL.queryName#" />
</cfloop>
is this possible, or is there a better way to do it?
Edit
This works with <cfloop query="LOCAL.queryName"> but not when I try to do <cfset ArrayAppend(LOCAL.returnArray, LOCAL.queryName.c) />
A: There is no need to use evaluate() to do this, and one shouldn't (so I've down-voted that answer, sorry).
All you need to do is use associative array notation:
<cfdump var="#local[qname]#">
If one wants to access a column of that query, it's:
#local[qname][columnName]#
And for a specific cell:
#local[qname][columnName][rowNumber]#
There are very very very few situations in which evaluate() is the correct answer to anything. One cannot rely on the Adobe docs because - unfortunately - an awful lot of it was not written by very experienced ColdFusion developers.
A: You can dump the query, and I imagine also access it by doing something like this:
<cfloop list="q1,q2,q3" index="qname">
<cfquery name="#qname#" datasource="dsn">
SELECT * from some_table;
</cfquery>
<cfdump var="#Evaluate('#qname#')#" />
</cfloop>
The Evaluate function is what allows you to do what you want.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 7,672 |
{"url":"https:\/\/www.physicsforums.com\/threads\/nano-fusion-micro-fusion-fusion-learning-source.822526\/","text":"Nano Fusion? Micro Fusion? Fusion Learning Source?\n\nSupaVillain\n\nWhen experimenting with fusion, why do we always go so big and make extremely expensive reactors that take years to create and even construct facilities for? I've seen some failed attempts at making fusion happen in carbon nanotubes, failing in the sense that the carbon nanotubes are just completely demolished. It makes more sense to me (I'm new to this stuff) to make small reactors that could fit in your hand or smaller to have far many more experiments conducted,had the same amount of money that's put into these massive reactors been put into a large quantity of smaller projects.\n\nAlso, what's a good way to learn about the parts related to the operating, testing, and computing inside of fusion devices? Is there any place online that has tons of data directly derived from fusion devices that I can view or is all of this stuff really not published for the public eye?\n\nIf I were to build a fusion reactor, what could I do to make the biggest difference possible in the world's understanding of fusion?\n\nRelated Nuclear Engineering News on Phys.org\n\nDrakkith\n\nStaff Emeritus\n2018 Award\nWhen experimenting with fusion, why do we always go so big and make extremely expensive reactors that take years to create and even construct facilities for? I've seen some failed attempts at making fusion happen in carbon nanotubes, failing in the sense that the carbon nanotubes are just completely demolished. It makes more sense to me (I'm new to this stuff) to make small reactors that could fit in your hand or smaller to have far many more experiments conducted,had the same amount of money that's put into these massive reactors been put into a large quantity of smaller projects.\nI'm no expert, but I don't think we can't make reactors that small. Regardless of the type of reactor, the things you need, like electromagnets, fuel injectors, etc, have a limit to how small they can be before you start running into problems. In addition, for magnetic confinement reactors, the ions and electrons spiral around because of the magnetic field, so your reactor needs to be larger than the spiral diameter.\n\nI'm sure there are plenty of other reasons too, but those are the only ones I can think of at the moment.\n\nSupaVillain\n\nYes, this is what I was assuming was the reason, I'd love to know the exact parts that are limiting size reduction in these reactors. I mean look at the new one at Lockheed Martin's Skunk Works, they're at least able to lower the reactor and its supporting system's size down to one semi truckload. Projects trying to achieve ignition might be able to be completed with even smaller fuel microcapsules and cheaper smaller lasers.\n\nDrakkith\n\nStaff Emeritus\n2018 Award\nI'd love to know the exact parts that are limiting size reduction in these reactors.\nPretty much every part is contributing.\n\nSupaVillain\n\nI'm realizing that we probably can't scale down the amount of energy that could cause fusion with thermonuclear, because it will require a certain amount of energy to make fusion possible, and that amount of required energy requires \"big\" devices.\n\nDrakkith\n\nStaff Emeritus\n2018 Award\nNot really. The amount of energy depends on the amount of plasma. A smaller volume means less plasma and less energy needed. However, various scaling laws apply and it turns out that really, really big reactors are more efficient than small reactors. The bigger the reactor, the more output power you get per input power. That's why ITER and similar reactors are multi-ton behemoths.\n\nDrakkith\n\nStaff Emeritus\n2018 Award\nNote that it is very easy to perform fusion. High school students have done it using home made electrostatic fusion reactors. The hard part is getting more energy out of fusion than you put into it. That's the part that we've been chasing for 70 years or so.\n\ne.bar.goum\n\nWhen experimenting with fusion, why do we always go so big and make extremely expensive reactors that take years to create and even construct facilities for? I've seen some failed attempts at making fusion happen in carbon nanotubes, failing in the sense that the carbon nanotubes are just completely demolished. It makes more sense to me (I'm new to this stuff) to make small reactors that could fit in your hand or smaller to have far many more experiments conducted,had the same amount of money that's put into these massive reactors been put into a large quantity of smaller projects.\n\nAlso, what's a good way to learn about the parts related to the operating, testing, and computing inside of fusion devices? Is there any place online that has tons of data directly derived from fusion devices that I can view or is all of this stuff really not published for the public eye?\n\nIf I were to build a fusion reactor, what could I do to make the biggest difference possible in the world's understanding of fusion?\nThree key parameters of a plasma reactors scale with radius: The ratio of plasma pressure to magnetic pressure $\\beta$, the product of collision frequency with thermal transit time $\\nu^*$, and the ratio of the lamor radius to the radius of the toroid $\\rho^*$ in the following ways:\n\u03b2 ~ nTB\u22122\n\u03bd* ~ nT\u22122R\n\u03c1* ~ T 1\/2B\u22121R\u22121\nSo, if you want to keep those constant, and reduce $R$,\nn ~ R\u22122\nT ~ R\u22121\/2\nB ~ R\u22125\/4\nSo reducing the radius means that you've got to increase the number density, the temperature and the magnetic field a lot. The magnetic field is a bit of a killer. Anything bigger than 10T or so is a problem. So, for lots of power, and long confinement times, you need big devices.\n\nAlso: It's not true that all fusion experiments are big and extremely expensive. Well, not on the scale of ITER. Even modest universities in countries without tonnes of research money can have fusion devices, and contribute important information to achieving fusion power. You don't have to have a power-producing fusion reactor to understand the physics processes at work! Here's a list of worldwide fusion experiments: https:\/\/en.wikipedia.org\/wiki\/List_of_fusion_experiments\n\nFurther, valuable information about fusion power can be done on plasma machines (that don't do fusion at all) - for instance, you can understand what plasmas will do to the materials on the walls of ITER.\n\nSupaVillain\n\nI know this about fusion, being \"easy\" to perform, but knowing that big is more efficient than small, why can't we just come up with a goal of efficiency with these smaller reactors, that would translate to the real efficiency we want with bigger reactors, after scaling everything up?\n\nDrakkith\n\nStaff Emeritus\n2018 Award\nI know this about fusion, being \"easy\" to perform, but knowing that big is more efficient than small, why can't we just come up with a goal of efficiency with these smaller reactors, that would translate to the real efficiency we want with bigger reactors, after scaling everything up?\nBecause that's like starting out your hobby as a mountain climber by climbing Mount Everest instead of a 50 ft cliff. You try to do the easiest stuff first and then, using what you've learned, move onto the hard stuff.\n\nSupaVillain\n\nFurther, valuable information about fusion power can be done on plasma machines (that don't do fusion at all) - for instance, you can understand what plasmas will do to the materials on the walls of ITER.\nI could tell, without knowing much on the subject, that even producing more data on plasmas and other things involved could help fusion along its way in the end. I sometimes think that discoveries made from the LHC or really any other projects going on in Universities will end up giving us the answers that we need to achieve efficiency in fusion, rather than directly working with it.\n\ne.bar.goum\n\nI could tell, without knowing much on the subject, that even producing more data on plasmas and other things involved could help fusion along its way in the end. I sometimes think that discoveries made from the LHC or really any other projects going on in Universities will end up giving us the answers that we need to achieve efficiency in fusion, rather than directly working with it.\nThe LHC really won't tell you anything about conditions in plasma machines. But that is exactly what does go on at many universities (see my wiki link, for starters) - plasma physicists and materials physicists are often heavily involved in fusion research, using machines that produce plasmas or smaller fusion machines.\n\nSupaVillain\n\nBecause that's like starting out your hobby as a mountain climber by climbing Mount Everest instead of a 50 ft cliff. You try to do the easiest stuff first and then, using what you've learned, move onto the hard stuff.\nI'm not making any sense, sorry. I meant, why can't we just work on the smaller scale and try to achieve an efficiency on that scale that \"should\" translate to the efficiency we dream of on the big scale, once we finally scale up to it?\n\nSupaVillain\n\nplasma physicists and materials physicists are often heavily involved in fusion research, using machines that produce plasmas or smaller fusion machines.\nThat's really what I want to hear, see I was starting out building a vacuum system for thin film deposition but then I realized fusion is just some deuterium and a grid away from the setup I already am finishing. I started looking into fusion and it inspired the crap outta me. I'd love to put my system to more use like what these researchers you speak of are doing.\n\nDrakkith\n\nStaff Emeritus\n2018 Award\nI could tell, without knowing much on the subject, that even producing more data on plasmas and other things involved could help fusion along its way in the end.\nNot if they are too small. If you want to understand how plasma behaves in a magnetic field, you're going to get very, very different results from a reactor that's 3-inches across compared to a reactor that's 30 feet across. That's not to say a 3-inch reactor would be useless. On the contrary, I'm sure you could get a lot of data out of such a device. But since we HAVE to make really big reactors first, before scaling down, we need to know how plasma behaves at much larger scales in addition to the smaller scales. Plasma instability at large scales almost certainly behaves differently than at small scales.\n\nAlso, note that we've had small-scale reactors for decades. We haven't had large-scale reactors similar to ITER until recently. Not ones that incorporate everything we've learned to date into their design and operation at least.\n\nI sometimes think that discoveries made from the LHC or really any other projects going on in Universities will end up giving us the answers that we need to achieve efficiency in fusion, rather than directly working with it.\nI don't see any reason to believe that. The LHC is VERY different from a fusion reactor. They don't even study the same effects.\n\nI'm not making any sense, sorry. I meant, why can't we just work on the smaller scale and try to achieve an efficiency on that scale that \"should\" translate to the efficiency we dream of on the big scale, once we finally scale up to it?\nWhy would we work on something that is orders of magnitude more difficult than fusion power already is? It makes little sense to do the hard stuff before you can do the easy stuff.\n\nSupaVillain\n\nOkay I see what you're saying Drakkith, makes sense to me now.\n\nruss_watters\n\nMentor\nI'm not making any sense, sorry. I meant, why can't we just work on the smaller scale and try to achieve an efficiency on that scale that \"should\" translate to the efficiency we dream of on the big scale, once we finally scale up to it?\nI'm not sure you read the answer you got. The answer is: because it is harder to make it small than big. And since they currently can't make fusion work at all, researchers are doing everything they can to make it easier!\n\nmfb\n\nMentor\nI'm not making any sense, sorry. I meant, why can't we just work on the smaller scale and try to achieve an efficiency on that scale that \"should\" translate to the efficiency we dream of on the big scale, once we finally scale up to it?\nThat is exactly what is happening, and ITER is the scaled up and improved version of previous reactors that should - based on that extrapolation - be able to get more power out than it needs for heating. Some issues are unique to larger machines, however, so you cannot test everything with smaller devices.\n\nSupaVillain\n\nSo the ITER is expected to achieve the fusion we want to have or is it just supposed to be used for further testing and hopefully eventually the fusion we want?\n\nmfb\n\nMentor\nITER won't be practical as a power plant - they don't even plan to generate electricity at all. The goal is to get about 500 MW of fusion power with 50 MW heating power. That is still too small for a power plant, but the main purpose is research.\nThe next larger reactor, DEMO, is supposed to be a demonstration power plant delivering power to the grid, and giving a reliable estimate of costs of future reactors.\n\nDrakkith\n\nStaff Emeritus\n2018 Award\nSo the ITER is expected to achieve the fusion we want to have or is it just supposed to be used for further testing and hopefully eventually the fusion we want?\nBoth. ITER is expected to be the first reactor to hit breakeven and generate more power than it takes to run it. It's also a test platform to figure out how to make large reactors better and more efficient so we can use them for electrical power. ITER will not be producing electrical power at all. The output power will solely be a measure of the amount of heat produced per input power.\n\nBengey\n\nTwo questions:\n1. When blasting fuel pellets with lasers, wouldn't it be easier to use really small pellets, on the micro scale?\n2. Instead of magnetic or inertial confinement, could we try mechanical confinement? I mean put some deuterium in a vice, like a nutcracker, a few atoms at a time. I believe we have the technology to manipulate small quantities. Just need to overcome the Coulomb force.\n\ne.bar.goum\n\nTwo questions:\n1. When blasting fuel pellets with lasers, wouldn't it be easier to use really small pellets, on the micro scale?\n2. Instead of magnetic or inertial confinement, could we try mechanical confinement? I mean put some deuterium in a vice, like a nutcracker, a few atoms at a time. I believe we have the technology to manipulate small quantities. Just need to overcome the Coulomb force.\n1. NIF uses 2mm diameter pellets. I'm not sure I see any benefit for anything smaller - why do you think it would be easier to use smaller pellets? And there would also be some downsides (less fuel, harder to manufacture, harder to align)\n\n2. Fusion occurs on femtometer scales. There exists no mechanical device that can force nuclei sufficiently close together.\n\nBengey\n\n1. I'm guessing that smaller pellets would give instabilities less time to develop. Alignment issues yes, but that's just engineering.\n2. The business end of a pair of scissors goes down to zero spacing, no? The challenge is whether a practical device could be designed, but again that's just engineering. First question is whether there's any theoretical barrier. Maybe no device could be smooth enough and captured atoms would hide in crevices? But crystals have smooth enough surfaces I would think.\n\ne.bar.goum\n\n1. I'm guessing that smaller pellets would give instabilities less time to develop. Alignment issues yes, but that's just engineering.\n2. The business end of a pair of scissors goes down to zero spacing, no? The challenge is whether a practical device could be designed, but again that's just engineering. First question is whether there's any theoretical barrier. Maybe no device could be smooth enough and captured atoms would hide in crevices? But crystals have smooth enough surfaces I would think.\nScissors certainly don't go to zero spacing. Sub millimetre if you're lucky, but that's 10^12 times bigger than a nucleus. I'm not sure you understand the scales at play here. Atoms are on the order of angstroms in size -- 10^5 times the order of the size of nuclei. 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Education and Crime Research Paper
View sample criminology research paper on education and crime. Browse other research paper examples for more inspiration. If you need a thorough research paper written according to all the academic standards, you can always turn to our experienced writers for help. This is how your paper can get an A! Feel free to contact our writing service for professional assistance. We offer high-quality assignments for reasonable rates.
In modern societies, an individual's life trajectory—including an individual's involvement in criminal activity—has become increasingly determined by his or her educational experiences. Over the past few centuries, schools have in many ways come to challenge families as the primary site for childhood socialization. The expanding role of formal education in the lives of youth has many causes. Economic production has become more dependent on cognitive skills taught in schools. Work has become typically set off from home life, limiting parents' ability to monitor and train children informally. Increasing female labor participation rates in recent decades have accelerated this trend, with over two thirds of mothers with children under age eighteen now currently employed. At the same time that work responsibilities have increasingly separated parents from their children, public education has been expanded to command greater portions of a youth's time. At the beginning of the nineteenth century only about ten percent of U.S. individuals age fourteen to seventeen attended high school; by the end of the century, only about ten percent of young adults failed to complete high school. As recently as in the 1940s, less than ten percent of individuals attained a bachelor's degree; by the end of the century, almost one-third of young adults were expected to attain such degrees. Not only have the number of years an individual is involved in a formal education system increased, but the amount of time per year has also dramatically expanded. The length of the school day has grown and the days in an academic school year have roughly doubled over the past century.
Research has clearly demonstrated how an individual's educational outcomes structure a wide range of adult life-course outcomes. Given the prominent role of education in an individual's life, educational experience has both significant direct and indirect effects on criminality. Over the past decade, educational experience has come to mediate the influence of social background on occupational destinations. By the end of the twentieth century, educational attainment had come to replace social origins as the primary determinant of occupational status, earnings, and even one's choice of marital partners. It is not surprising, therefore, that educational attainment plays a prominent role in explaining who is likely to commit criminal acts or subsequently to become incarcerated. Individuals who are incarcerated are less likely to have had previous success either in labor or marriage markets: about half of jail and prison inmates have never been married, close to half were unemployed prior to incarceration, and more than half had been living in poverty. More direct effects of educational experience are apparent when one examines the educational characteristics of those who are incarcerated. Only about 28 percent of incarcerated individuals in state and federal prisons have successfully graduated from high school (U.S. Department of Justice).
Schools play such a critical role in adult lifecourse outcomes because they affect individuals through several important social mechanisms. Schools are responsible for the socialization of youth. Schools work to train individuals for different roles in society and thus determine the selection of individuals for the allocation of scarce resources. Schools also structure an individual's interpersonal interactions and associations. The criminological significance of these distinct educational functions will first be explored and then connected to the relationship between crime and variation in educational performance and the structure of schooling. Lastly, conclusions and implications about the relationship between education and crime will be identified.
Mechanisms Producing Education-Crime Associations
As youth increasingly spend time in educational (rather than family) settings, the role of schools in the socialization of children and adolescents increases. Schools provide the context where much of the drama of the maturation process now unfolds. Children and particularly adolescents struggle—often in interaction with school authority—to define themselves as individuals with distinct identities. Identity formation involves challenges in many social psychological domains, including moral development. Educational psychologists have long argued that a critical stage in the process of moral development occurs during adolescence. Youths struggle to create their own definitions of right and wrong, as well as their own place in such a moral order (see Gilligan; Kohlberg).
Émile Durkheim, one of the founding influences on modern sociology, devoted a significant portion of his writings to how schools contribute to this socialization process. In Moral Education: A Study in the Theory and Application of the Sociology of Education (1903), Durkheim argued that schools confront individual students as the embodiment of society's moral authority. Youths learn in schools to respect society's moral authority if the rules they confront do not appear arbitrary, unenforceable, or unjust. Durkheim argued that discipline is needed in education ''to teach the child to rein in his desires, to set limits on his appetites of all kinds, to limit and, through limitation, to define the goals of his activity'' (p. 43). Essential to Durkheim's conception of the role of school discipline in the socialization of youth is his attention to the Hobbesian problem of order. The philosopher Thomas Hobbes argued that since individuals are governed by passions and desires, the threat of sanctions from a greater authority was necessary to constrain individual actions and promote social order. Durkheim countered that the strength of external sanctions was ultimately dependent on individuals internalizing these restrictions as normative rules. Durkheim argued that schools provide social settings whereby individuals are able to develop attachments to and integration with a larger societal moral order.
Durkheim's insights were most effectively introduced into contemporary criminological research by Travis Hirschi. Following Durkheim's insights, Hirschi was instrumental in developing criminological control theory, which has argued that individuals are subject to greater likelihood of criminal involvement when they have less attachment and integration with conventional authority. Since control theory owes its intellectual origins to earlier explorations of the role of schools in moral development, it is not surprising that—given the dramatic expansion of the role of schools in the lives of youth—much of the contemporary research from this perspective has emphasized the relationship between educational experience and criminality. Hirschi in later work with Michael Gottfredson argued that schools in fact were in many respects better situated than families to control and properly socialize youth. School personnel were argued to have a greater ability than family members to monitor, assess, and sanction youth misbehavior. School personnel were also claimed to have a greater incentive and need to control youthful behavior because of the large concentration of children and adolescents in close proximity to each other. Regardless of whether it has in any way replaced family-based socialization, involvement in schooling also serves an important role in the socialization of individuals. Schools provide youth with forms of attachment to conventional activities and thus increase an individual's ability to resist the temptations of criminal behavior.
While socialization of youth is one of the primary mechanisms whereby a causal relationship develops between educational experience and crime, the role of the education system in training, selection, and allocation is also critical. Sociologists Max Weber and Pitrim Sorokin, writing in the first third of the twentieth century, highlighted the fact that schools not only were responsible for training individuals for specific occupational tasks, but more importantly schools also served as closure mechanisms preventing individuals from gaining access to lucrative subsequent occupational positions. A second primary function of schools is thus ''to sort and sieve'' students for either success or failure. Schools directly determine through grades and promotions which students will have access to privileged advanced training leading to coveted occupational positions in a society and which will instead face the greatest risk of economic hardship.
Criminologists have argued that since schools are involved in selection and the allocation of scarce resources, they are sites where individuals confront obstacles to their aspirations for upward social mobility. Social scientists such as Richard Cloward, Lloyd Ohlin, and Arthur Stinchombe have developed strain theories of delinquency that link criminal behavior to blocked and frustrated status attainment. To the extent that schools produce resistance and misbehavior associated with institutional barriers to adult occupational success, a second mechanism underlying an association between crime and education is identified.
In addition to socialization and selection, schools also function to structure patterns of individual interpersonal interactions and associations. Social scientists, such as George Simmel and George Herbert Mead, argued early in the twentieth century that interpersonal interactions and associations were critical dimensions of how individuals came to understand and act in society. Criminologists have applied these insights by focusing on two processes. First, researchers such as Edwin Sutherland argued that delinquency could result from patterns of differential association. Since schools can structure youth interaction through a variety of mechanisms, the likelihood of youth misbehavior could be increased or dampened through such a structuring process. Second, schools provide settings where individual interactions occur. Researchers have argued that personnel within formal institutions often engage in a labeling process. Students are argued to have negative labels applied to them, which carry social stigmas. Since this research tradition assumes that individual meanings are the product of the dynamics of social interactions, often students will accept the negative labels assigned to them by authority figures. Rather than labels being easily rejected by students as being erroneous, they instead are argued to often become self-fulfilling prophecies.
Crime and Educational Performance
Given the multiple mechanisms whereby schools can influence adult life-course outcomes, it is not surprising that researchers repeatedly and consistently have demonstrated that educational performance and commitment are both negatively associated with adolescent delinquency, adult criminality, and incarceration. The more education an individual has the lower the risk of both criminal behavior and penal sanction. The higher the score on standardized cognitive tests, which partially reflect school learning, the lower the risk of criminality. High grade point averages and positive student attitudes toward school also have repeatedly been demonstrated to reduce the likelihood of adolescent delinquency and presumably adult criminality. Youth records of school sanction for student misbehavior, such as expulsion and suspension, are also clearly associated with adult criminality (Laub and Sampson; Gottfredson and Hirschi; Wilson and Herrnstein). These patterns are consistent with various criminological theoretical expectations discussed above. Students who are successful in terms of test score, grade point average, and years of education, are: defined as ''bright'' and ''good'' (labeling theory); have generally high degrees of attachment to conventional school activities (control theory); face easier success in pursuit of their ambitions (strain theory); and often are segregated off from students who are disruptive (differential association).
Several important research efforts have documented the relationship between school performance and crime. In 1950, Sheldon and Eleanor Glueck published an influential study of delinquency that documented the early onset of delinquent behaviors. Nearly half the delinquent youth had identifiable behavior problems before entering the fourth grade. Individuals who demonstrate early onset of serious identifiable misbehavior are likely to have entered school predisposed to failure as a result of the absence of early childhood family socialization. Even for these students, however, it is likely that schools can serve to either reinforce or dampen their preexisting tendencies for misbehavior. In 1969, Travis Hirschi published a seminal study of delinquency that focused much greater attention on educational behavior than did the earlier study by the Gluecks. Hirschi surveyed over five thousand junior and senior high school students in the San Francisco Bay area. He found systematic evidence that school performance and attachment (as measured by cognitive test scores, grades, and attitudes toward school) each had significant effects on the number of self-reported delinquent acts. Hirschi attributed this pattern of results to variation in the extent to which students formed positive attachments to school authority and activities. In the early 1990s, criminologists John Laub and Robert Sampson extended Hirschi's work, demonstrating that school attitudes and performance (as measured by grades) affect delinquency rates.
Variation in The Structure of Schooling and Crime
Years of educational attainment, cognitive test score, student grades, and attitudes toward school, however, are only a small part of how schools structure adolescent experience. Educational research demonstrates that other school factors—such as curriculum, resources, and school peer climates—also strongly influence a student's life chances. While numerous studies have examined the overall effect of schooling on deviance and crime, much of the existing criminological research has largely ignored the actual character of schooling. Criminological research has only begun to provide a more pedagogically sensitive examination of an adolescent's involvement with educational institutions. Such an examination requires a more complete elaboration and specification of the high school context that serves to diminish or increase the probability of criminality. Educational research has begun to inform criminological investigation by focusing on the role of vocational education, educational resources, and peer climates in affecting the incidence of delinquency, crime, and incarceration.
Vocational programs were instituted and expanded in high schools based on proponents' claims that occupational course work would reduce unemployment, crime, and deviant behavior in young adults. Criminological research has suggested mixed evidence on whether these programs have actually served to reduce individual propensity for criminal behavior. Because vocational education can function to segregate lowachieving students in particular courses either within a school or actually in a separate school within a larger district, many criminologists are skeptical that any positive effects of the programs can emerge. Setting vocational students off from academic students could lead to detrimental patterns of differential association or the labeling of vocational students as ''less able'' or as ''youthful troublemakers.''
It is important to note, however, that such negative effects are conditional on the actual structure of how vocational programs are organized. In many European countries such as Germany, for example, vocational programs and adolescent apprenticeships are an integral part of a socially validated educational system. In these settings, there is neither great stigma nor profound social segregation associated with these programs. In the United States, many schools in recent years have attempted to adopt an academy model for their vocational programs, where vocational education is integrated into both academic course work and the world of work: in these programs significant stigma or segregation is less likely. In 1971, Ahlstrom and Havighurst published what became a prominent skeptical evaluation of the role of vocational education in reducing the prevalence of delinquency. Ahlstrom and Havighurst investigated a specialized vocational work-study program designed for four hundred inner-city, maladjusted youth. The program was shown to have little effect on crime rates during student teen years.
Vocational education, however, has been demonstrated to have positive effects on student reports of satisfaction with school and positive perceptions of their teachers. Positive adolescent work experience is also related to psychological feelings of mastery, internal control, and self-competence. Given the significance of these factors in predicting criminality, it is likely that under certain circumstances vocational education can significantly discourage criminality. Recent criminological research has demonstrated that vocational education course work significantly reduces the likelihood of adult incarceration, if the course work occurs in an educational setting that does not concentrate and segregate high proportions of economically disadvantaged youth (Arum and Beattie).
Few criminological studies have attempted to estimate the effects of educational resources on individual delinquency and propensity for criminal behavior. One exception is Gary Gottfredson and Denise Gottfredson's Victimization in Schools (1985). The Gottfredsons argue that rates of student and teacher victimization in schools are a product of a range of school characteristics, including school resources, peer composition, and vocational curricular emphasis. Educational resources are likely important in that they can allow schools to reduce class size and thus increase a student's opportunities for learning from, and relating to, their teachers—that is, their likelihood of attachment to conventional activities. Educational resources can also be used to ensure greater monitoring of youth.
Educational resources likely affect a school's ability to influence positively an individual's life course, since schools with greater resources are better able to provide more positive enriched educational experiences for adolescents (such as costly vocational education programs). Recent noncriminological research has identified a clear pattern of the effects of educational resources on a range of socioeconomic outcomes including growth in test scores, increased years of educational attainment, and higher lifetime earnings. These socioeconomic outcomes have all been related to individual criminality and incarceration risk. It is therefore not surprising that high school student-teacher ratios have also been demonstrated to affect adult incarceration risk (Arum and Beattie).
Peer Climates
Peer climates can affect criminality in a number of ways, including differential association and altering social norms for acceptable behavior. Peer climates emerge in school as a product of both ecological and institutional factors. While peer climates are partly a reflection of peer composition, they are also structured by institutional factors. School practices in general and school disciplinary practices in particular define the parameters in which specific peer climates emerge and flourish. In the United States, significant variation in disciplinary practices exist: many public schools still practice corporal punishment, while in other schools often little is done to control student misbehavior and gang activity.
Peer composition has been demonstrated to be clearly associated with delinquency and subsequent incarceration in a large number of studies. Peer climates characterized by higher dropout rates and students of lower socioeconomic origins provide settings that make conventional school attachment more difficult. Research by James Coleman has emphasized, however, that schools have a role in structuring the manner in which peer climates exist. Work by Émile Durkheim also suggests the importance of school disciplinary practices in the socialization of youth. Punishment is necessary, according to Durkheim, because it unequivocally communicates that a normative rule has been broken.
Challenges to school disciplinary practices, regardless of whether they are from external environmental or internal organizational sources, would be particularly unsettling to the normative order of the school. Conservatives argue that due to administrative and legal challenges to school authority, students no longer view school rules as inviolate (Toby). At a practical level, school discipline works to generate student compliance and academically focused peer cultures. Peer climates have long been associated with student academic performance. In recent work, Coleman and his colleagues have argued that private schools outperform public schools in part because they are able to maintain stricter disciplinary climates with lower rates of student absenteeism, vandalism, drug use, and disobedience. Sociologists have also found that rates of misbehavior during the senior year are lower in schools that have higher rates of disciplining of sophomore students (Diprete et al.). Misbehaving students also have lower levels of educational achievement as measured by change in grades and test scores. Conservatives claim that without proper order and discipline, schools are unable to function properly and effective socialization is impossible.
Progressive educators, however, have countered that as traditional authoritarian disciplinary practices are eliminated from public schools, students will be less alienated from their educational environments, and more likely to remain in school and apply themselves to their studies. Support for this is suggested by the fact that the use of strict disciplinary practices, such as corporal punishment, leads to lower educational achievement and higher rates of delinquency. Researchers also argue that these school practices can lead to the formation of oppositional peer groups that resist formal education.
Conclusions and Implications
Criminologists who believe that propensity for adult criminality is established in early childhood attempt to dismiss empirical research that identifies significant school effects on delinquency and crime. These critics argue that selection bias accounts for education-crime associations. That is, some criminologists will argue that both educational and criminal trajectories are set at a very early preschool age. By the time that children enter school, the argument goes, families (or genetics) have already produced ''bad kids.'' Individuals fail in school because they lack social control: failure in school thus reflects individuallevel socialization problems that underlie criminal propensity; poor educational performance itself therefore does not produce criminal behavior. While some criminologists might still argue this position, it is fundamentally inconsistent with the larger social scientific research community's understanding of the role of education in life course development. At least since the late 1960s, social scientists have recognized that educational experience has come to mediate the relationship between social origins and adult lifecourse outcomes. While poorly socialized youth certainly are less likely to do well in terms of educational attainment, schools—if properly structured—can successfully counter these tendencies. Schools are institutions that can serve as ''turning points'' in individual lives. As the criminologists John Laub and Robert Sampson have argued: ''despite the connection between childhood events and experiences in adulthood, turning points can modify life trajectories—they can 'redirect paths.'''
Since schools play a critical role in determining the likelihood of delinquency, crime, and incarceration, policymakers historically have turned to educational reform to address social problems associated with adolescent delinquency and adult criminality. The last two decades of the twentieth century, however, were exceptional in U.S. history in terms of both educational and criminological policy. In unprecedented ways, policymakers have relied on incapacitation by the penal system to address the crime problem in society. Concurrently, educational policy has lost its focus on designing programs to integrate and socialize economically disadvantaged youths to become productive members of society. Instead, educational policymakers have become fixated on the narrow task of improving school performance and efficiency in terms of measurable student gains on cognitive standardized tests. While prison rolls have more than doubled in the last two decades of the twentieth century, high school vocational education enrollments have plummeted as the programs have been dismantled due to their high cost. While the penal system has demanded an increasing portion of local, state, and federal finances, educational budgets have struggled just to keep up with inflation and demographic growth in school age populations. While government officials increasingly threaten to sanction schools for the lack of student progress on cognitive tests, schools as institutions have become legally constrained from applying disciplinary sanctions to maintain peer climates conducive to learning and socialization. How policy reformers reconcile these tensions and contradictions in educational and social policy will determine the character of the educationcrime relationship in the future.
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ARUM, RICHARD, and BEATTIE, IRENE. ''High School Experience and the Risk of Adult Incarceration.'' Criminology 37, 3 (1999): 515– 538.
CLOWARD, RICHARD, and OHLIN, LLOYD. Delinquency and Opportunity. New York: Free Press, 1960.
COLEMAN, JAMES, and HOFFER, THOMAS. Public and Private High Schools: The Impact of Communities. New York: Basic Books, 1987.
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The fastest-growing source markets for Ponant are Australia, France and the United States, according to Herve Bellaiche, executive vice president of sales. The company has recently seen a significant expansion in the U.S. market with the purchase of assets from Travel Dynamics. The office has grown from a few workers in Miami to a staff of 30 and a Manhattan address.
An Australia office overseeing Asia-Pacific that opened in late 2013 is also starting to pay dividends, while France remains the company's number one market with loads of market penetration potential, said Bellaiche.
The arrival of a fourth new ship this past spring in the Lyrial marks more luxury expedition tonnage, as Bellaiche works to grow his brand globally.
The French brand may also be the best financed of the smaller luxury lines with the ideally sized five-star expedition ships occupying a niche with little competition.
Now included in the ticket price are all beverages, moving the product to a more all-inclusive experience.
"We are going to do more and more expedition cruises, that is clearly the plan," noted Bellaiche.
And they will, with double Northwest Passage sailings scheduled for 2015 before three of the four new vessels will head for Antarctica for the entire winter season.
Next year will see the company's first call in New York as well and provide an opportunity to show off tonnage in the U.S. market after a brief visit in 2014.
"Our deployment has been more of an evolution," said Bellaiche.
"We consider ourselves different from (the other luxury lines), we do things they don't. I don't see a clear competitor," Bellaiche continued.
Driving strong sales has been the decision to extend the booking window.
"Right now (in early June), every cruise is bookable through April 2017," Bellaiche noted. | {
"redpajama_set_name": "RedPajamaC4"
} | 9,610 |
Bethlehem ist eine Stadt in der Gemeinde Dihlabeng, Distrikt Thabo Mofutsanyana, Provinz Freistaat in Südafrika. Sie hat 16.236 Einwohner (Stand: 2011; Volkszählung). Die umliegenden Townships Bohlokong, Thorisong, Vuka und Old Location hatten im selben Jahr zusammen 60.431 Einwohner. Bethlehem liegt in 1706 Meter Höhe am nördlichen Fuß der Rooiberge, einem Vorgebirge der Drakensberge. Bethlehem ist ein landwirtschaftliches Zentrum (Weizenanbau, Schafhaltung).
Die Stadt wurde im Jahr 1864 von strenggläubigen Voortrekkern gegründet. Benannt ist sie nach Bethlehem, dessen hebräischer Name Beit Lechem בית לחם "Haus des Brotes" bedeutet. Der durch die Stadt fließende Fluss wurde passend dazu Jordan genannt.
Bethlehem liegt an der Bahnstrecke von Bloemfontein nach Durban und wird im Güter- und Personenverkehr bedient. Eine weitere, nur noch im Güterverkehr betriebene Strecke führt nordwärts Richtung Johannesburg.
Sehenswürdigkeiten
Pretoriuskloof Nature Reserve
am Ufer des Jordan, zu sehen sind viele Vogelarten
Wolhuterskop Nature Reserve
4 km entfernt, 800 ha groß; zu sehen sind Antilopen, Gazellen, Vögel, 15 km Wanderwege
National Hot Air Balloon Championship
jedes Jahr im Mai/Juni
Persönlichkeiten
Jimmy Kruger (1917–1987), Politiker, Minister für Justiz, Polizei und Strafvollzug
Einzelnachweise
Ort in der Provinz Freistaat
Dihlabeng
Ort in Afrika | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 3,202 |
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