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package org.sagebionetworks.repo.web.service;
import static org.junit.Assert.assertEquals;
import static org.junit.Assert.assertTrue;
import static org.junit.Assert.fail;
import java.io.UnsupportedEncodingException;
import java.net.URLEncoder;
import org.junit.Test;
import org.sagebionetworks.repo.manager.search.SearchHelper;
/**
* @author deflaux
*
*/
public class SearchHelperTest {
/**
* @throws Exception
*/
@Test
public void testCleanUpFreeTextSearchQueries() throws Exception {
// with space separator
assertEquals(
"q=prostate+cancer&return-fields=name,id&facet=node_type,disease,species",
SearchHelper
.cleanUpSearchQueries("q=prostate cancer&return-fields=name,id&facet=node_type,disease,species"));
// with url encoded space
assertEquals(
"q=prostate+cancer&return-fields=name,id&facet=node_type,disease,species",
SearchHelper
.cleanUpSearchQueries("q=prostate+cancer&return-fields=name,id&facet=node_type,disease,species"));
}
/**
* @throws Exception
*/
@Test
public void testFreeTextSearchQueryEncodedTooManyTimes() throws Exception {
// Note that we are already skipping one level of encoding here because
// the spring stuff does the first decode, but these tests do not
// exercise that logic so the query below is only double-encoded to test
// the triple encoding case
try {
SearchHelper
.cleanUpSearchQueries("q%3Dprostate%252Bcancer");
fail("fail");
} catch (IllegalArgumentException e) {
assertTrue(e.getMessage()
.startsWith("Query is incorrectly encoded"));
}
}
@Test
public void testCleanUpSearchQueriesInvalidArgument() throws Exception {
try {
SearchHelper
.cleanUpSearchQueries("q=");
fail("fail");
} catch (UnsupportedEncodingException e) {
assertTrue(e.getMessage()
.startsWith("Query parameter is malformed"));
}
}
/**
* @throws Exception
*/
@Test
public void testCleanUpBooleanSearchQueries() throws Exception {
// just a free text query
assertEquals("q=prostate", SearchHelper
.cleanUpSearchQueries("q=prostate"));
// free text with other parameters
assertEquals(
"q=cancer&return-fields=name,id&facet=node_type,disease,species",
SearchHelper
.cleanUpSearchQueries("q=cancer&return-fields=name,id&facet=node_type,disease,species"));
// a simple boolean query
assertEquals("q.parser=structured&q=" + URLEncoder.encode("node_type:'dataset'", "UTF-8"),
SearchHelper.cleanUpSearchQueries("q.parser=structured&q=node_type:'dataset'"));
// boolean query embedded in front, middle, and end of query string
assertEquals(
"q=cancer&return-fields=name,id&facet=node_type,disease,species&bq="
+ URLEncoder.encode("node_type:'dataset'", "UTF-8"),
SearchHelper
.cleanUpSearchQueries("bq=node_type:'dataset'&q=cancer&return-fields=name,id&facet=node_type,disease,species"));
assertEquals(
"q=cancer&return-fields=name,id&facet=node_type,disease,species&bq="
+ URLEncoder.encode("node_type:'dataset'", "UTF-8"),
SearchHelper
.cleanUpSearchQueries("q=cancer&bq=node_type:'dataset'&return-fields=name,id&facet=node_type,disease,species"));
assertEquals(
"q=cancer&return-fields=name,id&facet=node_type,disease,species&bq="
+ URLEncoder.encode("node_type:'dataset'", "UTF-8"),
SearchHelper
.cleanUpSearchQueries("q=cancer&return-fields=name,id&facet=node_type,disease,species&bq=node_type:'dataset'"));
// a joined AND
assertEquals(
"q=cancer&return-fields=name,id&facet=node_type,disease,species&bq="
+ URLEncoder.encode(
"(and node_type:'dataset' num_samples:1000..)",
"UTF-8"),
SearchHelper
.cleanUpSearchQueries("q=cancer&bq=(and node_type:'dataset' num_samples:1000..)&return-fields=name,id&facet=node_type,disease,species"));
// a split AND
assertEquals(
"q=cancer&return-fields=name,id&facet=node_type,disease,species&bq="
+ URLEncoder.encode(
"(and node_type:'dataset' num_samples:1000..)",
"UTF-8"),
SearchHelper
.cleanUpSearchQueries("bq=node_type:'dataset'&bq=num_samples:1000..&q=cancer&return-fields=name,id&facet=node_type,disease,species"));
// OR query
assertEquals(
"return-fields=name,id&facet=node_type,disease,species&bq="
+ URLEncoder.encode(
"(or node_type:'layer' node_type:'dataset')",
"UTF-8"),
SearchHelper
.cleanUpSearchQueries("bq=(or node_type:'layer' node_type:'dataset')&return-fields=name,id&facet=node_type,disease,species"));
// nested query split
assertEquals(
"return-fields=name,id&facet=node_type,disease,species&bq="
+ URLEncoder
.encode(
"(and created_by:'nicole.deflaux@sagebase.org' (or node_type:'layer' node_type:'dataset'))",
"UTF-8"),
SearchHelper
.cleanUpSearchQueries("bq=created_by:'nicole.deflaux@sagebase.org'&return-fields=name,id&facet=node_type,disease,species&bq=(or node_type:'layer' node_type:'dataset')"));
// nested query joined
assertEquals(
"return-fields=name,id&facet=node_type,disease,species&bq="
+ URLEncoder
.encode(
"(and (or node_type:'layer' node_type:'dataset') created_by:'nicole.deflaux@sagebase.org')",
"UTF-8"),
SearchHelper
.cleanUpSearchQueries("bq=(and (or node_type:'layer' node_type:'dataset') created_by:'nicole.deflaux@sagebase.org')&return-fields=name,id&facet=node_type,disease,species"));
assertEquals(
"return-fields=name,id&facet=node_type,disease,species&bq="
+ URLEncoder
.encode(
"(and (or acl:'PUBLIC' acl:'AUTHENTICATED_USERS' acl:'nicole.deflaux@gmail.com') node_type:'dataset' created_by:'matt.furia@sagebase.org')",
"UTF-8"),
SearchHelper
.cleanUpSearchQueries("bq=(and (or acl:'PUBLIC' acl:'AUTHENTICATED_USERS' acl:'nicole.deflaux@gmail.com') node_type:'dataset' created_by:'matt.furia@sagebase.org')&return-fields=name,id&facet=node_type,disease,species"));
assertEquals(
"return-fields=name,id&facet=node_type,disease,species&bq="
+ URLEncoder
.encode(
"(and (or acl:'PUBLIC' acl:'AUTHENTICATED_USERS' acl:'nicole.deflaux@gmail.com') node_type:'dataset' created_by:'matt.furia@sagebase.org')",
"UTF-8"),
SearchHelper
.cleanUpSearchQueries("bq=(or acl:'PUBLIC' acl:'AUTHENTICATED_USERS' acl:'nicole.deflaux@gmail.com')&bq=(and node_type:'dataset' created_by:'matt.furia@sagebase.org')&return-fields=name,id&facet=node_type,disease,species"));
assertEquals(
"return-fields=name,id&facet=node_type,disease,species&bq="
+ URLEncoder
.encode(
"(and (or acl:'PUBLIC' acl:'AUTHENTICATED_USERS' acl:'nicole.deflaux@gmail.com') node_type:'dataset' created_by:'matt.furia@sagebase.org')",
"UTF-8"),
SearchHelper
.cleanUpSearchQueries("bq=(or acl:'PUBLIC' acl:'AUTHENTICATED_USERS' acl:'nicole.deflaux@gmail.com')&bq=node_type:'dataset'&bq=created_by:'matt.furia@sagebase.org'&return-fields=name,id&facet=node_type,disease,species"));
// fun characters inside values
assertEquals("q.parser=structured&q=some_key%3A%27one%26two%27&q=dave", SearchHelper
.cleanUpSearchQueries("q.parser=structured&q="+ URLEncoder.encode("some_key:'one&two'", "UTF-8") + "&q=" + URLEncoder.encode("dave", "UTF-8")));
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 8,147 |
\section{Introduction}\label{sec:introduction}}
An increasing deal of attention has recently been devoted to the
understanding and the analysis of collective social belief dynamics
over social networks~\cite{Colbaugh2010,Asur:2010,shi2013agreement,Baccelli}.
This interest has been stimulated by the growing awareness of the
fundamental role that social networks and media may play in the
formation and diffusion of opinions/beliefs. For example, it is
widely recognized that social media have played a pivotal role in
several recent political events, such as the ``Arabian Spring''
or the last US presidential campaign. Moreover, the availability of a
large amount of social data generated by users has attracted the
interest of companies and government agencies, which envision
opportunities for exploiting such data to get important real-time
insights on evolution of trends, tastes, and opinions in the society.
Several experimental approaches, based on sentiment analysis \cite{Pang2008}, have
been proposed for a timely analysis of social dynamics. Furthermore,
several analytic frameworks have been developed with the goal of understanding and predicting
dominant belief dynamics. These models aim at providing important
insights on the dynamics of social interactions, as well as possible explanatory
mechanisms for the emergence of strong collective opinions. Additionally, they
have also been used to devise possible efficient strategies to influence social
beliefs.
The existing models can be coarsely partitioned into two classes:
\begin{itemize}
\item Discrete models, in which a discrete variable is associated to
every individual corresponding to a node on a graph, and represents
the current belief/position of each individual, e.g., favorable,
contrary, neutral, with respect to the considered subject. This
body of work also includes studies such as~\cite{naming1,naming2}
where the naming game is used to model phenomena such as opinion
dynamics in a population of agents. The social interactions are
represented by the graph edges and the state of a node changes for
effect of the interactions with its neighbors, i.e., the state of a
node is a deterministic/stochastic function of the states of its
neighboring nodes. Several different mechanisms, such as the Voter
Model \cite{liggett1997}, Bootstrap-percolation \cite{Bootstrap},
and Linear Threshold Models \cite{kempe}, have been proposed. The
dynamics of the process terminates when the system reaches a
globally consistent configuration.
\item Continuous models, in which the opinion of individuals on a
particular subject is described by means of a continuous variable,
whose value is adapted as a result of social interactions with
individuals having different opinions \cite{F&J,Tempo,
Fagnani-Como,Garnier,9-Blondel,16-Weisbuch,17-Bhattacharyya,noi-TNSE}. Also
in this case, social interactions between individuals are typically
modeled by using (static or dynamic) graphs, which reflect the
structure of the society and describe how individuals
interact.
\end{itemize}
All of the above pieces of work have considered that
the beliefs of an individual depend on her social interaction and vary for
effect of pairwise ``attractive'' forces. In particular,
a sub-class of continuous models that have attained considerable
popularity, considers the so-called bounded confidence, according to
which interactions between individuals are effective only if their
beliefs are sufficiently close to each
other~\cite{7-Deffuant,8-Hegselmann,degroot1974reaching,Fagnani-Acemoglu,Fagnani-Como,Garnier,9-Blondel,16-Weisbuch,17-Bhattacharyya}.
Furthermore, all previous models represent the evolution of the individuals'
opinions about a specific subject, neglecting how beliefs on different,
yet correlated, topics may vary over time. This is essentially
equivalent to assume the evolution of opinions on different subjects
to be independent. Unfortunately, things are much more involved, and
opinions on correlated topics exhibit complex inter-dependencies.
Consider for example the following situation: A group of people
discuss about two correlated subjects, e.g., fish
(in general) and salmon (in particular) as a part of diet. A person disliking fish also dislikes
salmon. If the influence process changes the individual's attitude
toward fish, say promoting fish as a healthy part of a diet, then such
a person may change her food preferences in favor of salmon as
well.
So far, only few pieces of work
\cite{nedic,fortunato,li2013consensus,tempo-science,Tempo-MD} have
tackled the dynamics of opinions on multiple, correlated subjects. In
most of such pieces of work, opinions are represented as vector-valued
variables, evolving over a multidimensional space in which every axis
represents a different subject.
In particular, in \cite{nedic,fortunato,li2013consensus} the evolution of
opinions on different axes exhibit a weak dependency, i.e., two users interact only if
their Euclidean distance between opinions does not exceed a prefixed
threshold. The pieces of work in \cite{tempo-science, Tempo-MD}, instead,
propose a linear
multidimensional model that explicitly accounts
for the interdependence of opinions on various topics, and they provide conditions for
both stability and convergence \cite{Tempo-MD}.
{\em Our contribution and methodology.} In this paper, we move a step
forward with respect to the existing work. First, we generalize the model
in~\cite{tempo-science, Tempo-MD}, introducing a noise component,
which represents the individual endogenous process of opinion/belief
evolution. Second, we enhance the model by accounting for possible
{\em repulsive} interactions due to adversion between individuals. Using
such a model and adopting a mean field approach holding for large
population of users, we characterize the evolution of opinions on
correlated subjects through a multidimensional Fokker-Planck
equation. We derive ergodicity conditions (i.e., conditions for the
existence of a unique stationary solution), and, under mild
assumptions, we obtain a {\em closed-form} expression for the stationary
distribution of individuals' opinions. We remark that the stability
analysis in the presence of an adversarial individuals' attitude is much
less obvious than in the traditional models (such as \cite{noi-TNSE}) where only attractive
forces were considered. Finally, we provide novel, efficient
numerical techniques to analyze both the steady state and the
transient solution of the Fokker-Planck (FP) equation, and show interesting
effects of opinions dynamics and correlated topics in some relevant scenarios.
\subsection{Paper organization\label{sec:organization}}
The paper is organized as follows. In Sect.~\ref{sec:preliminaries}
we introduce the model and we derive the Fokker-Planck equation based on mean-field approach.
In Sect.~\ref{sec:FP} we develop a methodology for the solution of the FP equation. In Sect.~\ref{sec:stability} we analyse the stability of the system depending on the
different parameters. Sect.~\ref{sec:steady_state} presents an analysis of the steady-state regime. in Sect.~\ref{sec:summary} we summarize the mathematical tools we have used in the paper.
Numerical results are reported in Sect~\ref{sec:results}. Finally we draw some conclusions in Sect.~\ref{sec:conclusions}.
\subsection{Notation\label{sec:notation}}
Boldface uppercase and lowercase letters denote matrices ad column
vectors, respectively. ${\bf I}_n$ is the identity matrix of size $n$ and
the transpose of the generic matrix ${\bf A}$ is denoted by
${\bf A}^\mathsf{T}$. The notation ${\bf A} = \left\{a(i,j)\right\}$
is sometimes used to define a matrix ${\bf A}$ whose $(i,j)$-th element
is $a_{i,j}$. Similarly, ${\bf a} = {\rm cat}\left\{{\bf a}(i)\right\}$
indicates that the column vector ${\bf a}$ is obtained by concatenating
the column vectors ${\bf a}_i$. The Laplace transform of the function
$f(x)$ is denoted by $\hat{f}(s)$. Finally, the symbol $\otimes$
denotes the Kronecker product.
\section{System model}\label{sec:preliminaries}
Consider a set of agents $\mathcal U$, with cardinality $U$, with agent $i$
exhibiting personality $P_i\in \mathcal P$. The agent's personality accounts
for her interests and habits, e.g., the social networks
to which she has subscribed or the forums in which she participates.
We consider that agents have opinions on $N$ different topics and that
an opinion formed on one subject is influenced by the opinions on some
of the other subjects, i.e., topics are interdependent
\cite{Tempo-MD,Tempo45,Tempo46}. We define ${\bf C}$ as the coupling matrix, with
$c_{mn}$ expressing the entanglement of subject $m$ on subject $n$.
The opinions that agent $i\in \mathcal U$ has on the different subjects is
represented by a vector of size $N$, denoted by ${\bf x}_i(t) \in \mathcal X^N$,
which evolves over continuous time, $t\in \RR_+$.
We define the prejudice vector ${\bf u}(P_i)$ as the a-priori
$N$-dimensional belief that agent $i$ has on the different subjects;
also the prejudice depends on the agent's personality.
We represent through a graph the existence and the intensity of social
relationships between users, which depend on the personality of the
agents and on the similarity between the agents' beliefs. The actual
influence that agents exert on each other then depends on their
opportunity to interact, as well as on their sensitivity to others' beliefs.
As a result, the evolution of agent $i$'s belief over time can be
represented as:
\begin{equation}
{\bf x}_i(t\mathord{+}\partial t) = {\bf x}_i(t) +{\bf C} {\bf d}_{{\bf x},i}(t)
\label{eq:x_i1}
\end{equation}
where ${\bf x}_i(t)$ denotes the belief of agent $i$ on the $N$ topics at
the current time instant, ${\bf d}_{{\bf x},i}(t)$ accounts for the variation of agent $i$ opinions in the time interval $[t,t+\partial t]$, and
${\bf C}$ accounts for the influence of the opinion on one topic on the opinions on other topics. The quantity ${\bf d}_{{\bf x},i}(t)$ is given by
\begin{eqnarray}
\label{eq:deltax}
{\bf d}_{{\bf x},i} &=& \frac{1\mathord{-}\alpha(P_i)}{U-1}\sum_{\substack{j\in \mathcal U \\ j \neq i}} \underbrace{\zeta\left(P_i\mathord{,}P_j\right)\left[{\bf x}_j(t)\mathord{-} {\bf x}_i(t)\right] \partial t}_{(a)}\nonumber\\
&&\quad+ \underbrace{\alpha(P_i) \left[{\bf u}(P_i)\mathord{-}{\bf x}_i(t)\right] \partial t}_{(b)} + \underbrace{\sigma \partial {\bf w}_i(t)}_{(c)} \,.
\end{eqnarray}
The meaning of the terms on the right hand side of the above
expression is as follows.
\begin{itemize}
\item The first term $(a)$ represents the interaction of agent $i$ with
all other agents in $\mathcal U$. In particular,
\begin{itemize}
\item $\alpha(P_i)\in (0,1]$ indicates how insensitive $i$ is to
other agents' beliefs, which, as also discussed in
\cite{sociology-community}, plays an important role in opinion
dynamics. This parameter will also be referred to as the agent's
level of stubbornness. When $\alpha(P_i)\to 1$, the agent becomes
completely insensitive to others' beliefs (stubborn). Instead, as
$\alpha(P_i)$ decreases, the agent is more inclined to
accept others' beliefs and is less conditioned by her own
prejudice. For brevity, in the following we denote
$\bar{\alpha}(P_i) =1-\alpha(P_i)$;
\item $\zeta(P_i,P_j)$ represents the presence and the
strength of interactions between agents $i$ and $j$ (hereinafter
also referred to as mutual influence). It is a function of both
agents' personality and defines the structure of the social
graph~\cite{sociology-community}. Note that the interactions
between agents do not depend on the proximity of their opinions,
i.e., they are independent of ${\bf x}_j(t)\mathord{-} {\bf x}_i(t)$. Also,
whenever $\zeta(P_i,P_j) = 0$, the two agents do not influence
each other, i.e., they never interact. Finally, it is fair
to assume that each element of $\zeta(P_i,P_j)$ is upper bounded
by a constant and is continuous with respect to its first and
second arguments.
\end{itemize}
\item The second term $(b)$ represents the tendency of an agent to retain her
prejudice.
\item The third term $(c)$ accounts for the endogenous process of the belief
evolution within each agent. Such process is modeled as an
i.i.d. standard Brownian motion with zero drift and scale parameter $\sigma^2$~\cite{Baccelli}.
\end{itemize}
We remark that ${\bf x}_i(t+\partial t)$, i.e., the belief of agent $i$ at time $t+\partial t$,
depends on her personality $P_i$ and the current
agent's belief. In other words, the temporal evolution of agents' beliefs $
\{ {\bf x}_i(t),\; i\in \mathcal U\}$ is Markovian over ${\cal Y}^U$, where ${\cal Y} =\mathcal P \times \mathcal X^N$ is an
$(N+1)$-dimensional continuous space.
Furthermore, in the following we assume that $\alpha(P_i)$, $\zeta(P_i,P_j)$, and each element of ${\bf u}(P_i)$ are
continuous with respect to their arguments.
For ease of notation, we denote ${\bf Z}(P_1,P_2) = \zeta(P_i,P_j){\bf C}$;
thus, by replacing~\eqref{eq:deltax} in~\eqref{eq:x_i1}, the latter can be rewritten as
\begin{eqnarray}
\label{eq:x_i}
{\bf x}_i(t\mathord{+}\partial t) &\hspace{-2ex}\mathord{=}&\hspace{-2ex} {\bf x}_i(t)\mathord{+}\frac{1\mathord{-}\alpha(P_i)}{U-1}\hspace{-2ex}\sum_{\substack{j\in \mathcal U/\{i\}}}\hspace{-2ex}{\bf Z}\left(P_i\mathord{,}P_j\right)\left[{\bf x}_j(t)\mathord{-} {\bf x}_i(t)\right]\partial t \nonumber\\
&& +\alpha(P_i) {\bf C}\left[{\bf u}(P_i)\mathord{-}{\bf x}_i(t)\right] \partial t +\sigma{\bf C} \partial {\bf w}_i(t) \,.
\end{eqnarray}
\subsection{From the discrete to the continuous model}
We now extend the above model to the continuous case by using the mean-field theory.
We leverage on the procedure presented in~\cite{noi-TNSE} and apply it to the
multi-subject scenario. More in detail, we define the empirical
probability measure, $\rho^{(U)}(\partial p, \partial {\bf x}, t)$ over ${\cal Y}$ at
time $t$, as:
\begin{equation}
\rho^{(U)} (\partial p, \partial {\bf x}, t ) =\frac{1}{U}\sum_{i \in \mathcal U}
\delta_{(P_i,{\bf x}_i(t ))}(\partial p, \partial {\bf x})\,.
\label{eq:rho_U}
\end{equation}
In the above expression, $\delta_{(P_i,{\bf x}_i(t ))}(\partial p,\partial {\bf x})$ is
the Dirac measure centered at $(P_i,{\bf x}_i(t ))$, i.e.,
$\delta_{(P_i,{\bf x}_i(t ))}(\partial p,\partial {\bf x})$ represents the mass
probability associated with opinion ${\bf x}_i(t )$ of agent $i$, which
has personality $P_i$. Note that in~\eqref{eq:rho_U} agents are seen
as particles in the continuous space ${\cal Y}$, moving along the opinion
axis ${\bf x}$. As shown in \cite{noi-TNSE}, by applying the mean-field
theory~\cite{gartner1988mckean,dawson1983critical}, as $U\to \infty$,
$\rho^{(U)} (\partial p, \partial {\bf x}, t)$ converges in law to the asymptotic
distribution $\rho(p, {\bf x}, t)$, provided that
$\rho^{(U)} (\partial p, \partial {\bf x}, 0)$ converges in law to $\rho(p,{\bf x},
0)$. Moreover, $\rho(p, {\bf x}, t)$ can be obtained from the following
non-linear Fokker-Planck (FP)
equation~\cite{gartner1988mckean,dawson1983critical}:
\begin{align} \label{eq:fokker-planck}
\frac{\partial }{\partial t} \rho(p, {\bf x}, t) = &-\sum_{n=1}^N \frac{\partial }{\partial x_n} \left[ \mu_n(p, {\bf x}, t) \rho(p, {\bf x}, t) \right] \nonumber \\ & +
\frac{1}{2}\sum_{m,n=1}^N D_{mn} \frac{\partial^2 }{\partial x_m \partial x_n} \rho(p,{\bf x}, t)
\end{align}
where $D_{mn}$ is the $(m,n)$-th entry of the diffusion tensor ${\bf D} = \sigma^2{\bf C}\Cm^\mathsf{T}$, and $\mu_n(p, {\bf x}, t)$
is defined as the component of the instantaneous
speed along axis $x_n$ of a
generic agent whose personality and opinion at time $t$ are equal to $p$ and ${\bf x}$.
The instantaneous speed is given by:
\begin{align}
\label{eq:mu_x2}
\boldsymbol{\mu }(p,{\bf x} ,t) =& \;\bar{\alpha}(p) \int_{{\bf y}\in \mathcal X^N} \int_{\mathcal P} {\bf Z}(p,q) ({\bf y}-{\bf x}) \rho(q,{\bf y} , t) \partial^N {\bf y} \partial q \nonumber \\
&\quad+ \alpha(p) {\bf C} [{\bf u}(p) - {\bf x}] \nonumber \\
=& \;\bar{\alpha}(p) \left[ \boldsymbol{\gamma}(p,t) - \boldsymbol{\Gamma}(p) {\bf x} \right] + \alpha(p) {\bf C} [{\bf u}(p) - {\bf x}] \nonumber \\
= & \; -\boldsymbol{\Xi}(p) {\bf x} + \boldsymbol{\phi}(p,t)
\end{align}
where we defined
\begin{equation} \label{eq:Gammap}
\boldsymbol{\Gamma}(p) \mathord{\triangleq}\hspace{-0.5ex} \iint_{{\bf y},q} \hspace{-2ex}{\bf Z}(p,q) \rho(q,{\bf y},t) \partial^N {\bf y} \partial q \mathord{\stackrel{(a)}{=}}\hspace{-0.5ex} \int_q \hspace{-0.5ex}{\bf Z}(p,q) \rho_0(q) \partial q ,
\end{equation}
where in $(a)$ we wrote $\rho(q,{\bf y},t) =\rho({\bf y},t |q)\rho_t(q)$ and
exploited the fact that by definition $\int_{{\bf y}} \rho({\bf y},t |q)\partial^N
{\bf y} = 1 $. Since the distribution of the agents' personality at time
$t$, $\rho_t(q)$, does not depend on $t$,
we have: $\rho_t(q)=\rho_0(q)$. Furthermore,
\begin{equation} \label{eq:J1}
\boldsymbol{\gamma}(p, t) \triangleq \iint_{{\bf y},q} {\bf Z}(p,q) {\bf y} \rho(q,{\bf y} , t) \partial^N {\bf y} \partial q ,
\end{equation}
\begin{equation} \label{eq:Xi}
\boldsymbol{\Xi}(p) \triangleq \bar{\alpha}(p)\boldsymbol{\Gamma}(p) + \alpha(p) {\bf C} ,
\end{equation}
\begin{equation}
\boldsymbol{\phi}(p ,t) \triangleq \bar{\alpha}(p) \boldsymbol{\gamma}(p, t) + \alpha(p) {\bf C} {\bf u}(p) \label{eq:phi}
\end{equation}
and we considered a zero-drift Brownian motion process
${\bf w}(t)$.
In the following, we analyze the system dynamics by solving the above
FP equation for $\rho(p, {\bf x}, t)$ so as to obtain the
distribution of agents over ${\cal Y}$.
\section{Solution of the Fokker-Planck (FP) equation} \label{sec:FP}
In this section, we solve the $N$-dimensional FP equation, which
describes an $N$-dimensional Ornstein-Uhlenbeck (OU)\cite{risken1996fokker} random process.
Considering a general initial density $\rho(p,{\bf x}, 0) = \rho_0({\bf x}|p)
\rho_0(p)$, we obtain the solution of the FP equation $\rho(p,{\bf x}, t)$
as shown in Appendix~\ref{app:solution} in the Supplemental Material:
\begin{equation}
\rho(p,{\bf x}, t)
\mathord{=} \rho_0(p) \int_{{\bf y}}\hspace{-1ex} \mathcal G\left({\bf x}\mathord{,} {\bf m}(p\mathord{,}{\bf y}\mathord{,}t)\mathord{,} \boldsymbol{\Sigma} (p\mathord{,}t) \right)
\rho_0({\bf y}|p)\partial^N {\bf y} \label{eq:solution}
\end{equation}
where $\mathcal G\left({\bf x}, {\bf m}(p,{\bf y},t), \boldsymbol{\Sigma} (p ,t) \right)$ is the pdf of the Gaussian multivariate distribution with
covariance
\begin{equation}\label{eq:covariance}
\boldsymbol{\Sigma}(p ,t) \triangleq \int_0^{t} {\rm e}^{-\boldsymbol{\Xi}(p) \tau} {\bf D} {\rm e}^{-\boldsymbol{\Xi}(p)^\mathsf{T} \tau} \partial\tau
\end{equation}
and mean
\begin{eqnarray}
{\bf m}(p,{\bf y},t) &\hspace{-2ex}\triangleq&\hspace{-2ex}{\rm e}^{-\boldsymbol{\Xi}(p) t} \left[ {\bf y}+\int_0^{t} {\rm e}^{\boldsymbol{\Xi}(p) \tau} \boldsymbol{\phi}(p,\tau)\partial\tau \right] \nonumber\\
&\hspace{-2ex}\stackrel{(a)}{=}&\hspace{-2ex} {\rm e}^{-\boldsymbol{\Xi}(p) t} {\bf y} + \alpha(p)\boldsymbol{\Xi}^{-1}(p) \left({\bf I}_N\mathord{-}{\rm e}^{-\boldsymbol{\Xi}(p) t} \right) {\bf C} {\bf u}(p)\nonumber \\
&\hspace{-2ex} &+ \bar{\alpha}(p) \int_0^{t} {\rm e}^{-\boldsymbol{\Xi}(p) (t-\tau)} \boldsymbol{\gamma}(p,\tau) \partial\tau \label{eq:mv_x}
\end{eqnarray}
where in (a) we used the definition of $\boldsymbol{\phi}(p,\tau)$ provided in~\eqref{eq:phi}.
However, notice that ${\bf m}(p,{\bf y},t)$ in \eqref{eq:mv_x} is a function of
$\boldsymbol{\gamma}(p,t)$, which in turn is a function of $\rho(p,{\bf x}, t)$ (see
\eqref{eq:J1}). As such, we have to impose a self-consistency
condition; precisely, replacing~\eqref{eq:solution} in~\eqref{eq:J1}, we obtain
\begin{eqnarray}
\boldsymbol{\gamma}(p,t)
&\hspace{-2ex}=&\hspace{-2ex} \int_{{\bf y}} \int_q {\bf Z}(p,q) {\bf y} \rho(q,{\bf y} , t) \partial^N {\bf y} \partial q \nonumber\\
&\hspace{-2ex}=&\hspace{-2ex} \int_{{\bf y}} \int_q {\bf Z}(p,q) {\bf y} \rho_0(q) \int_{{\bf z}} \mathcal G\left({\bf y}, {\bf m}(q,{\bf z},t), \boldsymbol{\Sigma}(q ,t)\right) \nonumber\\
&\hspace{-2ex} &\hspace{-2ex}\qquad\cdot \rho_0({\bf z}|q)\partial^N {\bf z} \partial^N {\bf y} \partial q \nonumber\\
&\hspace{-2ex}=&\hspace{-2ex} \int_q {\bf Z}(p,q) \rho_0(q) \int_{{\bf z}} {\bf m}(q,{\bf z},t) \rho_0({\bf z}|q)\partial^N {\bf z} \partial q \nonumber\\
&\hspace{-2ex}=&\hspace{-2ex} \boldsymbol{\gamma}_0(p,t) + \boldsymbol{\gamma}_1(p,t) +\int_q {\bf Z}(p,q) \rho_0(q)\bar{\alpha}(q) \nonumber\\
&\hspace{-2ex}&\hspace{-2ex}\qquad\cdot\int_0^{t} {\rm e}^{-\boldsymbol{\Xi}(q) (t-\tau)} \boldsymbol{\gamma}(q,\tau)\partial\tau \partial q
\label{eq:fixed_point}
\end{eqnarray}
\noindent where we used~\eqref{eq:mv_x} and defined for brevity:
\begin{equation}
\boldsymbol{\gamma}_0(p,t) \triangleq \int_q {\bf Z}(p,q) \rho_0(q) {\rm e}^{-\boldsymbol{\Xi}(q) t} \int_{{\bf y}} {\bf y} \rho_0({\bf y}|q)\partial^N {\bf y} \partial q
\label{eq:gamma0}
\end{equation}
\begin{eqnarray}
\boldsymbol{\gamma}_1(p,t) & \triangleq & \int_q {\bf Z}(p,q) \rho_0(q) \alpha(q) \boldsymbol{\Xi}^{-1}(q)\nonumber\\
&&\qquad \cdot \left({\bf I}_N\mathord{-}{\rm e}^{-\boldsymbol{\Xi}(q) t} \right) {\bf C} {\bf u}(q) \partial q
\label{eq:gamma1}
\end{eqnarray}
Interestingly, \eqref{eq:fixed_point} is a linear Volterra equation of the
second kind \cite{Volterra}.
We can take over time the Laplace transform of \eqref{eq:fixed_point} and get
\begin{eqnarray} \label{eq:fixed_point_transform}
\widehat{\boldsymbol{\gamma}}(p,s) &\hspace{-1ex}=& \hspace{-1ex} \widehat{\boldsymbol{\gamma}}_0(p,s) + \widehat{\boldsymbol{\gamma}}_1(p,s) \nonumber \\
&\hspace{-3ex} & \hspace{-3ex} +\int_q\hspace{-0.5ex} {\bf Z}(p,q) \rho_0(q) \bar{\alpha}(q) {\bf X}^{-1}(s,q)\widehat{\boldsymbol{\gamma}}(q, s) \partial q \nonumber\\
\end{eqnarray}
where ${\bf X}(s,q) = s {\bf I}_N + \boldsymbol{\Xi}(q)$
\begin{equation}
\widehat{\boldsymbol{\gamma}}_0(p,s) = \hspace{-1ex} \int_q\hspace{-0.5ex} {\bf Z}(p,q) \rho_0(q) {\bf X}^{-1}(s,q)\hspace{-1ex} \int_{{\bf y}} {\bf y} \rho_0({\bf y}|q)\partial^N {\bf y} \partial q\label{eq:fixed_point_transform_0}
\end{equation}
\begin{equation}
\widehat{\boldsymbol{\gamma}}_1(p,s) =\hspace{-1ex} \int_q {\bf Z}(p,q) \rho_0(q) \alpha(q) s^{-1} {\bf X}^{-1}(s,q) {\bf C} {\bf u}(q) \partial q \label{eq:fixed_point_transform_1}
\end{equation}
Eq.\,\eqref{eq:fixed_point_transform} is a non-homogeneous integral equation,
whose solution is unique if and only if the associated homogeneous equation has no nonzero solutions. If the solution is unique, then it gives the solution of the FP through \eqref{eq:solution}.
\textbf{Remark}:
By restricting the definition of $\boldsymbol{\gamma}(p,t)$ over an arbitrary
compact domain $\mathcal {P}\times [0,T]$,
\eqref{eq:fixed_point} can be rewritten in an operational form as
$(\mathcal I-\mathcal A)[\boldsymbol{\gamma}(p,t)]=\boldsymbol{\gamma}_0(p,t) + \boldsymbol{\gamma}_1(p,t)$ where the
operator $\mathcal A[\boldsymbol{\gamma}(p,t)]=\int_q {\bf Z}(p,q) \rho_0(q)
\bar{\alpha}(q) \int_0^{t} {\rm e}^{-\boldsymbol{\Xi}(q) (t-\tau)} \boldsymbol{\gamma}(q,\tau)
\partial\tau \partial q $. Note that, as an immediate consequence of {the structure of Volterra equations over compact domains}, we have
$\|\mathcal A^n\| < 1$ for sufficiently large $n$ \cite[cap.\,2. pp.\,50--51]{Kolmogorov}. Thus,
$(\mathcal I-\mathcal A)[\cdot]$ is invertible (i.e., the solution is unique) and an expression for
$\boldsymbol{\gamma}(p,t)$ can be obtained as
$\boldsymbol{\gamma}(p,t)= (\mathcal I-\mathcal A)^{-1}[\left(\boldsymbol{\gamma}_0(p,t) + \boldsymbol{\gamma}_1(p,t)\right)]= \sum_n \mathcal A^n [\left(\boldsymbol{\gamma}_0(p,t) + \boldsymbol{\gamma}_1(p,t)\right)]$.
\begin{comment}
Now a sufficient condition for operator $ (\mathcal I-\mathcal A)^{-1})$ to be bounded in norm is that $\|\mathcal A\|<1$. (Indeed, under the assumption that $\|\mathcal A\|<1$ it holds:
$\|(I- \mathcal A^{-1}) \|<\frac{1}{1-\| \mathcal A\|}$.
Therefore a sufficient condition for the boundedness of $\gamma(p,t)$ is $\|\mathcal A\|_\infty<1$.
By direct inspection it can be easily shown that $\|\mathcal A\|_\infty<1$ whenever $Z(p,q)\ge 0$ and $\inf_p \alpha(p)>0$.
Indeed:
\[
\|\mathcal A\|_\infty\le \int_q \left| {{\bf Z}(p,q) \rho_0(q) (1-\alpha(q)) } \right| \int_0^{t} {\rm e}^{-\boldsymbol{\Xi}(q) (t-\tau)} d\tau \partial q\le \int_q \left| {{\bf Z}(p,q) \rho_0(q) (1-\alpha(q))\Gamma(q) \boldsymbol{\Xi}^{-1}(q)} \right| \partial q
\]
with
\[
\int_q \left| {{\bf Z}(p,q) \rho_0(q) (1-\alpha(q)) \boldsymbol{\Xi}^{-1}(q)} \right| \partial q= \int_q {{\bf Z}(p,q) \rho_0(q) (1-\alpha(q)) \boldsymbol{\Xi}^{-1}(q)} \partial q\le \sup_q \frac{ (1-\alpha(q)) \Gamma(q) }{\alpha(q) + (1-\alpha(q))\Gamma(q) }:=\beta<1
\]
Then over every compact domain $\mathcal {P}\times [0,T]$, we have
$\|\theta(p,t)\|_\infty< \| \frac{\boldsymbol{\gamma}_0(p,t) + \boldsymbol{\gamma}_1(p,t)}{\Gamma(p)} \|_\infty \frac{1}{1-\beta}$.
In other words, given that $\| \frac{\boldsymbol{\gamma}_0(p,t) + \boldsymbol{\gamma}_1(p,t)}{\Gamma(p)} \|_\infty$ is uniformly bounded with respect to $T$
also $\|\theta(p,t)\|_\infty$ is uniformly bounded with respect to $T$.
\end{comment}
Provided that the integral equation~\eqref{eq:fixed_point_transform} admits a unique solution,
in general it is still rather challenging to explicitly find it. In the
following, we will particularize our analysis to two cases in which it is
possible to find an explicit analytical expression for such as
solution.
\subsection{The case of $\zeta(p,q)$ in product form} \label{sec:easy}
We recall that ${\bf Z}(p,q)=\zeta(p,q){\bf C}$. If
$\zeta(p,q) = \zeta_1(p) \zeta_2(q)$, then from~\eqref{eq:J1} we have
\begin{eqnarray}
\boldsymbol{\gamma}(p, t)
&=& \zeta_1(p) \int_{{\bf y}} \int_q \zeta_2(q){\bf C}{\bf y} \rho(q,{\bf y} , t) \partial^N {\bf y} \partial q \,. \nonumber\\
&=& \zeta_1(p){\bf b}(t) \label{eq:gamma_delta}
\end{eqnarray}
Taking the Laplace transform of~\eqref{eq:gamma_delta}, we obtain
$\widehat{\boldsymbol{\gamma}}(p, s)=\zeta_1(p)\widehat{{\bf b}}(s)$. Using the
latter expression in~\eqref{eq:fixed_point_transform} and
recalling~\eqref{eq:fixed_point_transform_0} and
\eqref{eq:fixed_point_transform_1}, we get
\begin{eqnarray} \label{eq:product_fixed_point_transform}
\widehat{\boldsymbol{\gamma}}(p,s)
&=& \widehat{\boldsymbol{\gamma}}_0(p,s) + \widehat{\boldsymbol{\gamma}}_1(p,s) \nonumber\\
&&\hspace{-5ex} +\zeta_1(p)\int_q \zeta_2(q) {\bf C} \rho_0(q) \bar{\alpha}(q) {\bf X}^{-1}(s,q)\widehat{\boldsymbol{\gamma}}(q, s) \partial q \nonumber\\
&=& \zeta_1(p)\left(\widehat{{\bf b}}_0(s)+ \widehat{{\bf b}}_1(s)\right)\nonumber\\
&& \hspace{-5ex}+\zeta_1(p)\int_q\rho_0(q) \bar{\alpha}(q) \zeta_2(q) {\bf C} {\bf X}^{-1}(s,q)\zeta_1(q) \partial q \widehat{{\bf b}}(s) \nonumber\\
\end{eqnarray}
under the assumption that ${\bf X}(s,q) = s {\bf I}_N + \boldsymbol{\Xi}(q)$ is
invertible\footnote{Note that invertibility is granted for $\mathrm{Re}(s)>\sup_q \|\boldsymbol{\Xi}(q)\|$}. In~\eqref{eq:product_fixed_point_transform}
\begin{equation}
\widehat{{\bf b}}_0(s) = \int_q \zeta_2(q){\bf C}\rho_0(q) {\bf X}^{-1}(s,q)\hspace{-1ex} \int_{{\bf y}} {\bf y} \rho_0({\bf y}|q)\partial^N {\bf y} \partial q\label{eq:product_fixed_point_transform_0}
\end{equation}
\begin{equation}
\widehat{{\bf b}}_1(s) = \int_q \zeta_2(q){\bf C} \rho_0(q) \alpha(q) s^{-1}{\bf X}^{-1}(s,q) {\bf C} {\bf u}(q) \partial
q \label{eq:product_fixed_point_transform_1} \,.
\end{equation}
Note that we can also write
\begin{align} \label{eq:fixed_point_transform_sep}
\widehat{{\bf b}}(s) =& \left( \int_q \rho_0(q) \bar{\alpha}(q) \zeta_2(q) {\bf C} {\bf X}^{-1}(s,q) \zeta_1(q) \partial q \right) \widehat{{\bf b}}(s)\nonumber\\
&+\widehat{{\bf b}}_0(s) + \widehat{{\bf b}}_1(s)
\end{align}
from which, under the assumption that matrix
\[ {\bf T}(s)= {\bf I}_N -\int_q \rho_0(q) \bar{\alpha}(q) \zeta_2(q) {\bf C} {\bf X}^{-1}(s,q) \zeta_1(q) \partial q\]
is non singular, we obtain:
\[ \widehat{{\bf b}}(s) = {\bf T}(s)^{-1}\left( \widehat{{\bf b}}_0(s) + \widehat{{\bf b}}_1(s)\right)\,.\]
We now have an explicit solution (in the transformed domain) for
${\bf b}(t)$. Indeed, for $\mathrm{Re}(s)$ sufficiently large, matrix
${\bf T}(s)$ is non singular. Once
${\bf b}(t)$ is obtained, we can compute $\boldsymbol{\gamma}(p,t)$
through~\eqref{eq:gamma_delta}, then ${\bf m}(p,{\bf y},t)$
in~\eqref{eq:mv_x}, and, finally,
our opinion density $\rho(p,{\bf x},t)$ through~\eqref{eq:solution}.
\subsection{Discrete personality distribution}
Suppose that the personality distribution is discrete with $M$ probability masses; then we write the opinion distribution at $t=0$ as
\begin{equation}\label{eq:rho0}
\rho_0(p) = \sum_{i=1}^M r_i\delta(p-p_i).
\end{equation}
Consequently, the fixed-point equation \eqref{eq:fixed_point_transform} becomes
\begin{eqnarray} \label{eq:fixed_point_transform_disc}
\widehat{\boldsymbol{\gamma}}(p_i,s)&=& \sum_{k=1}^M {\bf Z}(p_i,p_k) r_k \bar{\alpha}(p_k) {\bf X}^{-1}(s,p_k)\widehat{\boldsymbol{\gamma}}(p_k,s)\nonumber\\
&&\qquad +\widehat{\boldsymbol{\gamma}}_{0}(p_i,s) + \widehat{\boldsymbol{\gamma}}_{1}(p_i,s)\,.
\end{eqnarray}
In~\eqref{eq:fixed_point_transform_disc}, we defined
\begin{equation} \label{eq:gamma0_disc}
\widehat{\boldsymbol{\gamma}}_0(p_i,s) =\sum_{k=1}^M {\bf Z}(p_i,p_k) r_k {\bf X}^{-1}(s,p_k){\bf x}_{0}(p_k)
\end{equation}
where ${\bf x}_{0,k}$ is the average opinion at time $t=0$ corresponding to personality $p_k$ and has been obtained by observing that the term $\rho_0(q)\int_{{\bf y}}{\bf y} \rho_0({\bf y}|q)\partial^N {\bf y}$ in~\eqref{eq:fixed_point_transform_0} is equal to ${\bf m}(q,{\bf x},0)={\bf x}_0(q)$.
Also, we defined:
\begin{equation} \label{eq:gamma1_disc}
\widehat{\boldsymbol{\gamma}}_1(p_i,s) =\sum_{k=1}^M \frac{{\bf Z}(p_i,p_k) r_k \alpha(p_k)}{s} {\bf X}^{-1}(s,p_k){\bf C} {\bf u}(p_k) \,.
\end{equation}
Next, recalling the notation defined in~Section~\ref{sec:notation} we define
$\underline{\widehat{\boldsymbol{\gamma}}}(s) = {\rm cat}\{\widehat{\boldsymbol{\gamma}}(p_i,s)\}$,
$\underline{\widehat{\boldsymbol{\gamma}}}_0(s)={\rm cat}\{\widehat{\boldsymbol{\gamma}}_0(p_i,s)\}$,
$\underline{\widehat{\boldsymbol{\gamma}}}_1(s) ={\rm cat}\{\widehat{\boldsymbol{\gamma}}_{1}(p_i,s)\}$,
$\underline{{\bf x}}_0 = {\rm cat}\{{\bf x}_0(q_i)\}$,
$\underline{{\bf u}} = {\rm cat}\{{\bf u}(q_i)\}$,
$\underline{\boldsymbol{\Xi}} = {\hbox{diag}} \left(\boldsymbol{\Xi}(p_1), \dots, \boldsymbol{\Xi}(p_M) \right)$,
${\bf P}_0 = {\hbox{diag}} \left(r_1, \dots, r_M \right) \otimes {\bf I}_N$, ${\bf P}_1 = {\hbox{diag}} \left(r_1 \alpha(p_1), \dots, r_M \alpha(p_M) \right) \otimes {\bf I}_N$,
${\bf P}_2 = {\hbox{diag}} \left(r_1 \bar{\alpha}(p_1), \dots, r_M\bar{\alpha}(p_M) \right) \otimes {\bf I}_N$, and
\begin{equation}
\underline{{\bf Z}} = \left\{{\bf Z}(p_i,p_j)\right\} = \left\{ \zeta(p_i,p_j)\right\}\otimes {\bf C}\,.
\end{equation}
Then, we can write \eqref{eq:fixed_point_transform_disc} as
\begin{equation} \label{eq:fixed_point_transform_mat}
\underline{\widehat{\boldsymbol{\gamma}}}(s) = \underline{\widehat{\boldsymbol{\gamma}}}_0(s) + \underline{\widehat{\boldsymbol{\gamma}}}_1(s) + \underline{{\bf Z}} \left(s {\bf I}_{MN} + \underline{\boldsymbol{\Xi}} \right)^{-1}{\bf P}_2 \underline{\widehat{\boldsymbol{\gamma}}}(s)
\end{equation}
while \eqref{eq:gamma0_disc} and \eqref{eq:gamma1_disc} become
\begin{equation} \label{eq:gamma0_mat}
\underline{\widehat{\boldsymbol{\gamma}}}_0(s) = \underline{{\bf Z}} \left(s {\bf I}_{MN} + \underline{\boldsymbol{\Xi}} \right)^{-1}{\bf P}_0 \underline{{\bf x}}_0
\end{equation}
and
\begin{equation} \label{eq:gamma1_mat}
\underline{\widehat{\boldsymbol{\gamma}}}_1(s) = \underline{{\bf Z}} \left(s {\bf I}_{MN} + \underline{\boldsymbol{\Xi}} \right)^{-1}s^{-1} {\bf P}_1 \left({\bf I}_M \otimes {\bf C} \right) \underline{{\bf u}}.
\end{equation}
Let $\underline{{\bf X}}(s)=s{\bf I}_{MN} + \underline{\boldsymbol{\Xi}}$. Then we can solve explicitly the fixed-point equation of \eqref{eq:fixed_point_transform_mat} (in the transform domain) as
\begin{eqnarray}
\underline{\widehat{\boldsymbol{\gamma}}}(s) &=& \left({\bf I}_{MN}\mathord{-} \underline{{\bf Z}} \underline{{\bf X}}(s)^{-1}{\bf P}_2 \right)^{-1} \left( \underline{\widehat{\boldsymbol{\gamma}}}_0(s) \mathord{+}\underline{\widehat{\boldsymbol{\gamma}}}_1(s) \right)\nonumber\\
& = & \left({\bf I}_{MN}\mathord{-} \underline{{\bf Z}} \underline{{\bf X}}(s)^{-1}{\bf P}_2 \right)^{-1} \underline{{\bf Z}} \underline{{\bf X}}(s)^{-1}\nonumber\\
&& \qquad \cdot\left( {\bf P}_0 \underline{{\bf x}}_0\mathord{+} s^{-1} {\bf P}_1 \left({\bf I}_M \otimes {\bf C} \right)\underline{{\bf u}}\right) \nonumber\\
& = & \left(\underline{{\bf X}}(s)\underline{{\bf Z}}^{-1} \mathord{-} {\bf P}_2 \right)^{-1} \left( {\bf P}_0 \underline{{\bf x}}_0 \mathord{+} s^{-1} {\bf P}_1 \left({\bf I}_M \otimes {\bf C} \right)\underline{{\bf u}}\right) \nonumber\\
& = & \underline{{\bf Z}} \left(\underline{{\bf X}}(s)\mathord{-} {\bf P}_2 \underline{{\bf Z}} \right)^{-1} \left( {\bf P}_0 \underline{{\bf x}}_0 \mathord{+} s^{-1} {\bf P}_1 \left({\bf I}_M \otimes {\bf C} \right)\underline{{\bf u}} \right)\,.\nonumber\\
\end{eqnarray}
Now, let
\begin{equation}\label{eq:Psim}
\boldsymbol{\Psi} = \underline{\boldsymbol{\Xi}} - {\bf P}_2 \underline{{\bf Z}} \,.
\end{equation}
Provided that $ \boldsymbol{\Psi}$ is invertible, the above solution can be inverse-transformed to get the solution in the time domain as
\begin{equation} \label{eq:gamma_t}
\underline{\boldsymbol{\gamma}}(t) = \underline{{\bf Z}} \left({\rm e}^{-\boldsymbol{\Psi} t} {\bf P}_0 \underline{{\bf x}}_0\mathord{+} \boldsymbol{\Psi}^{-1} \left({\bf I}_{MN}\mathord{-}{\rm e}^{-\boldsymbol{\Psi} t} \right) {\bf P}_1 \left({\bf I}_M \otimes {\bf C} \right) \underline{{\bf u}} \right).
\end{equation}
Then $\rho(p,{\bf x},t)$ can be obtained as described in the previous
section.
Finally, the following theorem complements the above result:
\begin{theorem}\label{theo1}
i) For any finite $t$ the Volterra equation
\eqref{eq:fixed_point} admits a unique
solution which is Lipschitz-continuous. ii) The solution $\gamma(p,t)$ of the Volterra
equation \eqref{eq:fixed_point} , under any distribution
$\rho_0(p)$, which is continuous at every point in $p$, is the
uniform limit of solutions $\gamma_n(p,t)$, obtained by replacing
distribution $\rho_0(p)$ with its discrete approximation $\rho_n(p)$
whose mesh-size is $\frac{1}{n}$.
\end{theorem}
The proof of this theorem is given following exactly the same lines of Appendix~\ref{app:C} (which proves a similar statement under steady state conditions).
\section{Stability analysis} \label{sec:stability}
Let us define the system as \emph{stable} if, for every initial
condition, the opinion distribution converges to the stationary
solution for $t \rightarrow \infty$.
In this section, we first focus on the case where the personality distribution is
discrete, then we generalize the result to the the case of continuous personality distributions.
\subsection{Discrete personality distribution}
Let us state the stability conditions as follows:
\begin{itemize}
\item $\underline{\boldsymbol{\Xi}}$ is Hurwitz-stable (i.e., all the eigenvalues of $\underline{\boldsymbol{\Xi}}$ have positive real part).
\item $\boldsymbol{\Psi}$ is Hurwitz-stable.
\end{itemize}
Indeed, recalling the expression of the covariance in~\eqref{eq:covariance}, its
value remains limited if the first condition is met, while, looking
at~\eqref{eq:mv_x} and~\eqref{eq:gamma_t}, the mean of the
distribution remains limited if both the above conditions are satisfied.
In particular, when the first condition is not met, there are some
personalities for which the opinion distribution scatters about along
some directions. We call this phenomenon \emph{type-I instability}.
Instead, if the first condition is met and the second one is not, for all
personalities the opinion covariance remains limited but there are some
personalities for which the mean opinion value drifts to
infinity. We will refer to this case as \emph{type-II instability}.
Below we elaborate on the conditions that are needed to ensure the system stability.
\subsubsection{Condition to avoid Type-I instability}
Let $\lambda_k({\bf A}) = \lambda^R_k({\bf A}) + j \lambda_k^I({\bf A})$ be the
$k$-th eigenvalue of matrix ${\bf A}$. Recall that $\boldsymbol{\Xi}(p_i)=\bar{\alpha}(p_i)\boldsymbol{\Gamma}(p_i)+\alpha(p_i){\bf C}$,
thus the necessary and sufficient condition to avoid type-I instability is
given by
\[ \min_k \lambda^R_k(\boldsymbol{\Xi}(p_i)) > 0, \quad \forall i\,.\]
Since $\boldsymbol{\Gamma}(p_i) = {\bf C} \sum_h \zeta(p_i,p_h) \rho_0(p_h)$, we can write
\[ \boldsymbol{\Xi}(p_i)=\left[\bar{\alpha}(p_i)\sum_h \zeta(p_i,p_h) \rho_0(p_h)+\alpha(p_i)\right]{\bf C}\,. \]
It follows that the stability condition becomes
\[ \left[\bar{\alpha}(p_i)\sum_h \zeta(p_i,p_h) \rho_0(p_h)+\alpha(p_i)\right] \min_k \lambda^R_k({\bf C}) > 0\,. \]
Assuming that $\bar{\alpha}(p_i)\sum_h \zeta(p_i,p_h) \rho_0(p_h)+\alpha(p_i)>0$ for every $i$, then
stability is ensured when
\[ \min_k \lambda^R_k({\bf C}) >0 \]
The above expression highlights
that if the opinion dynamics along every topic are stable, introducing
a Hurwitz stable matrix ${\bf C}$ preserves stability.
\subsubsection{Condition to avoid Type-II instability}
The necessary and sufficient condition to avoid type-II instability is
given by
\[ \min_k \lambda^R_k(\boldsymbol{\Psi}(p_i)) > 0, \quad \forall i\,.\]
Let us focus on the case where
$\alpha(p) \equiv \alpha$. By recalling~\eqref{eq:Psim}, we write
\begin{eqnarray}
\boldsymbol{\Psi} &=& \underline{\boldsymbol{\Xi}} - {\bf P}_2\underline{{\bf Z}} \nonumber\\
&=& \bar{\alpha}\underline{\boldsymbol{\Gamma}}+\alpha{\bf I}_{M}\otimes {\bf C} -\bar{\alpha}{\bf P}_0\underline{{\bf Z}} \nonumber\\
&=& \alpha {\bf I}_{M}\otimes {\bf C} + \bar{\alpha} \left( \underline{\boldsymbol{\Gamma}} - {\bf P}_0 \underline{{\bf Z}} \right) \label{eq:Psim2}
\end{eqnarray}
where
$\underline{\boldsymbol{\Gamma}} = {\hbox{diag}} \left(\boldsymbol{\Gamma}(p_1), \dots, \boldsymbol{\Gamma}(p_M)
\right)$, with $\boldsymbol{\Gamma}(p_i)$ obtained by
discretizing~\eqref{eq:Gammap}. Then
we define
$\underline{\boldsymbol{\Gamma}} - {\bf P}_0 \underline{{\bf Z}} \triangleq \boldsymbol{\Theta}
\otimes {\bf C}$, where $\boldsymbol{\Theta}$ is the $M \times M$ matrix whose
$(i,j)$-th entry is given by
\[
\theta_{ij} = \left\{
\begin{array}{rl}
\sum_{k \neq i} \zeta(p_i,p_k) r_k, & i=j \\
- \zeta(p_i,p_j) r_i , & i\neq j \,.
\end{array} \right.
\]
As a consequence, considering that $\lambda_{(j-1)N +i}(\boldsymbol{\Theta}\otimes
{\bf C})= \lambda_{i}(\boldsymbol{\Theta})\lambda_j({\bf C})$ and using~\eqref{eq:Psim2},
the condition for stability reads as follows:
\[
\Re\{\lambda_i(\boldsymbol{\Theta}) \lambda_j({\bf C})\} > -\frac{\alpha}{1-\alpha}
\Re\{\lambda_j({\bf C})\}\,\,\,\,\,\, \forall i,j \,.
\]
{\bf Remark.} In the scalar case ($N=1$), the following propositions
hold.
\begin{proposition} \label{prop:stab_1}
If $\zeta(p_i,p_j) \geq 0$ for all $i,j$, $\boldsymbol{\Psi}$ is Hurwitz-stable.
\end{proposition}
\noindent{\bf Proof:~} If $\zeta(p_i,p_j) \geq 0$ for all $i,j$, $\boldsymbol{\Psi}$ is a matrix whose off-diagonal elements are nonpositive and
we can apply Theorem 1 of \cite{Plemmons}. Precisely the Hurwitz stability of
$\boldsymbol{\Psi}$ (condition J29 in Theorem 1 of \cite{Plemmons}) is implied by
the fact that the row sums of $\boldsymbol{\Psi}$ are all positive (condition K35
applied with diagonal matrix ${\bf D}={\bf I}$). Since the row sums of $\boldsymbol{\Psi}$
are all equal to $\alpha$, the proposition follows.\hfill{\rule{2mm}{2mm}}
\begin{proposition} \label{prop:stab_2}
$\boldsymbol{\Psi}$ is Hurwitz-stable if $\min_{i,j}\zeta(p_i,p_j) < 0$ and, for every $i$
\begin{equation} \label{eq:cond_stab}
\sum_{i \neq j} \left( \zeta(p_i,p_j) r_j- |\zeta(p_i, p_j)| r_i
\right) > -\frac{\alpha}{1-\alpha} \,.
\end{equation}
\end{proposition}
\noindent{\bf Proof:~} If $\min_{i,j}\zeta(p_i,p_j) < 0$, the Hurwitz stability of $\boldsymbol{\Psi}$ (condition J29 in
Theorem 1 of \cite{Plemmons}) is implied by condition N39 applied with
diagonal matrix ${\bf D}={\bf I}$, which is expressed in
\eqref{eq:cond_stab}.
\hfill{\rule{2mm}{2mm}}
Note that the above propositions still hold in the case of continuous personalities.
We now present an example on the stability conditions for a simple case as described below.
\example{1}{Consider $N=1$ and that
there are $M$ personalities $p_i =\frac{(2i-1)}{M}-1$, $i=1,\dots,M$
($M$ even), with $r_i= \frac1{M}$ and
\[
\zeta(p_i,p_j) = \left\{
\begin{array}{rl}
\zeta_1, & p_i p_j > 0 \\
- \zeta_2 , & p_i p_j < 0
\end{array} \right.
\]
with $\zeta_1,\zeta_2$ being arbitrary positive values. Thus,
$\boldsymbol{\Xi}_i$ is a scalar equal to
$\alpha + \bar{\alpha} \frac{\zeta_1-\zeta_2}{2}$ for all
$i$. Moreover,
\[
\underline{{\bf Z}} = \left[
\begin{array}{cc}
\zeta_1 & - \zeta_2 \\
- \zeta_2 & \zeta_1
\end{array} \right] \otimes \mathbf{1}_{M/2}
\]
where $\mathbf{1}_{n}$ is a size-$n$ square matrix with all entries equal to 1. Thus,
\[
\lambda_{i}(\underline{{\bf Z}}) = \left\{
\begin{array}{rl}
0, & 1 \leq i \leq M-2 \\
M \frac{\zeta_1-\zeta_2}{2} , & i = M-1 \\
M \frac{\zeta_1+\zeta_2}{2} , & i = M \,.
\end{array} \right.
\]
Since $\boldsymbol{\Psi} = \left(\alpha + \bar{\alpha} \frac{\zeta_1-\zeta_2}{2}\right){\bf I}_M - \frac{\bar{\alpha}}{M} \underline{{\bf Z}}$, it follows that
\[
\lambda_{i}(\boldsymbol{\Psi}) = \left\{
\begin{array}{rl}
\alpha + \bar{\alpha} \frac{\zeta_1-\zeta_2}{2}, & 1 \leq i \leq M-2 \\
\alpha , & i = M-1 \\
\alpha - \bar{\alpha} \zeta_2 , & i = M \,.
\end{array} \right.
\]
We then have type-I instability if $\zeta_2 \geq \zeta_1 + 2 \frac{\alpha}{1-\alpha}$ and type-II
instability if
$ \frac{\alpha}{1-\alpha} \leq \zeta_2 < \zeta_1 + 2
\frac{\alpha}{1-\alpha}$. Conversely, the system is stable iff
$\zeta_2 < \frac{\alpha}{1-\alpha}$. It is easy to see that this is
equivalent to condition \eqref{eq:cond_stab}, which in this case is
necessary and sufficient. \hfill{\rule{2mm}{2mm}}
}
\subsection{Continuous personality distribution}
We consider the case of a continuous personality distribution as the
limit case of a family of discrete distributions with increasingly
small discretization steps, $\Delta p$.
Then similarly to what done in the previous section, we assume that
the following inequality holds:
\begin{align} \label{eq:stab_cond_cont}
& \lim_{\Delta p \to 0}\left(\sum_{j \neq i} \zeta(p_i,p_j) \rho_{0}(p_j) \Delta p - \sum_{j \neq i} |\zeta(p_i,p_j)| \rho_{0}(p_i) \Delta p\right) \nonumber\\
&\quad= \int \zeta(p,q) \rho_0(q) \partial q - \int |\zeta(p,q)| \partial q \rho_0(p) \nonumber\\
& \quad > -\frac{\alpha(p)}{1-\alpha(p)} \,.
\end{align}
We now prove that the above is the stability condition for
the continuous case, provided that some technical conditions, specified below, are met.
Suppose that \eqref{eq:stab_cond_cont} is true for our
continuous-personality system. Then, for $\Delta p$ sufficiently
small, \eqref{eq:cond_stab} is also satisfied for the discretized
system, so that the corresponding $\boldsymbol{\Psi}(\Delta p)$ is Hurwitz-stable. main-field
Next we assume that for $\Delta p$ sufficiently small:
\begin{itemize}
\item $\boldsymbol{\Psi}(\Delta p)$ is uniformly Hurwitz-stable, i.e., the minimum real part of its eigenvalues is bounded away from zero by an amount $\epsilon$ and
\item $\boldsymbol{\Psi}^{-1}(\Delta p)$ has a uniformly bounded norm, i.e., $\| \boldsymbol{\Psi}^{-1}(\Delta p)\| < K,$ $\forall \Delta p$ sufficiently small and $K<\infty$~\footnote{This last condition is implied by the previous one when $\boldsymbol{\Psi}(\Delta p)$ is diagonalizable. }.
\end{itemize}
It follows that, for $\Delta p$ sufficiently small, the fixed-point
solution in \eqref{eq:gamma_t} is uniformly bounded from
above. As $\Delta p \rightarrow 0$, the fixed-point solution for the
continuous-personality system is uniformly bounded from
above (by Theorem \ref{theo1} and norm continuity) for every finite $t$. Hence, the system
is stable.
\section{Steady-state analysis\label{sec:steady_state}}
Under previous stability/ergodicity conditions, it is interesting to
analyse the limiting solution for $t\to \infty$, which can be obtained
as solution of the associated steady state FP equation.
In the most general case, we can rewrite~\eqref{eq:fokker-planck} disregarding the dependence on $t$, as the following multi-dimensional FP equation:
\begin{eqnarray} \label{eq:fokker-planck-multi}
\sum_{n=1}^N \frac{\partial }{\partial x_n} \left( \mu_n(p, {\bf x}) \rho(p, {\bf x}) \right)\mathord{=}
\sum_{m,n=1}^N \frac{D_{mn}}{2} \frac{\partial^2\rho(p,{\bf x}) }{\partial x_m \partial x_n}
\end{eqnarray}
where $\rho(p,{\bf x}) =\lim_{t \to \infty}\rho(p,{\bf x},t) $ and,
according to the second line in~\eqref{eq:mu_x2}, $\boldsymbol{\mu }$ is given by
\begin{eqnarray} \label{eq:mu_steady_state}
\boldsymbol{\mu }(p,{\bf x})
&=& \bar{\alpha}(p) \left[ \boldsymbol{\gamma}(p) - \boldsymbol{\Gamma}(p) {\bf x} \right] + \alpha(p) {\bf C} [{\bf u}(p) - {\bf x}] \nonumber\\
&=& \bar{\alpha}(p){\bf C}( \boldsymbol{\beta}(p) - \eta(p) {\bf x}) + \alpha(p){\bf C} [{\bf u}(p) - {\bf x}] \nonumber\\
&=& -w(p) {\bf C}({\bf x}-{\bf f}(p)) \,.
\end{eqnarray}
In the above equation, we used the expressions of $\boldsymbol{\gamma}(p)$ and
$\boldsymbol{\Gamma}(p)$ in~\eqref{eq:J1} and~\eqref{eq:Gammap}, respectively, and
defined $\boldsymbol{\gamma}(p) = {\bf C} \boldsymbol{\beta}(p)$ and $\boldsymbol{\Gamma} = \eta(p){\bf C}$
where $\eta(p) = \int_q\zeta(p,q)\rho_0(q)\partial q$ and $\boldsymbol{\beta}(p) = \int_{\bf y}\int_q\zeta(p,q){\bf y} \rho(q,{\bf y})\partial^N{\bf y} \partial q$.
Moreover, we defined
$w(p) = \bar{\alpha}(p)\eta(p)+\alpha(p)$ and
${\bf f}(p) = \frac{\bar{\alpha}(p)\boldsymbol{\beta}(p)+\alpha(p){\bf u}(p)}{w(p)}$.
Interestingly, if vector $\boldsymbol{\beta}(p)$ is given, then $\boldsymbol{\mu }(p,{\bf x})$ does not
depend on $\rho(p,{\bf x})$ any longer, rather it depends only on
the personality $p$, the opinion vector ${\bf x}$, and the initial
distribution $\rho_0(p)$. Furthermore, observe that $\boldsymbol{\phi}(p)=\frac{\beta(p)}{\eta(p)}$ can
be expressed as the fixed point of the multidimensional Fredholm
equation:
\begin{eqnarray} \boldsymbol{\phi}(p) &=&
\underbrace{\int_q\frac{\zeta(p,q)\alpha(q){\bf u}(q) \rho_0(q)
}{\eta(p) w(q)}\partial q}_{{\bf h}(q)} \nonumber \\
&& \quad + \int_q\underbrace{\frac{\zeta(p,q)\bar{\alpha}(q)\eta(q)}{\eta(p)
w(q)}}_{\Phi(p,q)}\boldsymbol{\phi}(q) \rho_0(q)\partial q \nonumber \\
&=& {\bf h}(p) + \int_q\Phi(p,q)\boldsymbol{\phi}(q) \rho_0(q)\partial q \label{eq:beta_Fredholm-a}
\end{eqnarray}
(See Appendix~\ref{app:C} in the Supplemental Material for
details.)
From~\eqref{eq:mu_steady_state}, we observe that if $-w(p){\bf C}$ is
stable for every $p$, the stationary solution of the FP equation under
steady-state conditions is given by~\cite{risken1996fokker}:
\begin{equation}\label{eq:rho_W}
\rho(p,{\bf x}) = \frac{\rho_0(p)}{\sqrt{2 \pi |{\bf W}(p)|}} {\rm e}^{-1/2({\bf x}-{\bf f}(p))^\mathsf{T} {\bf W}(p)^{-1}({\bf x}-{\bf f}(p))}
\end{equation}
where ${\bf W}(p)$ is the unique solution of the Lyapunov equation ${\bf A}(p){\bf W}(p)+{\bf W}(p){\bf A}(p)^\mathsf{T}=-{\bf D}$ and can be expressed as:
\[
{\bf W}(p)=-\int_0^\infty {\rm e}^{{\bf A}(p)\tau}{\bf D} {\rm e}^{{\bf A}(p)^\mathsf{T}\tau}\partial
\tau \,.
\]
A more explicit expression of $\rho(p,{\bf x})$ can be obtained when ${\bf C}$ is symmetric.
Let us write $\boldsymbol{\mu }(p,{\bf x})$ as
\begin{equation} \label{eq:potential}
\boldsymbol{\mu }(p,{\bf x}) = -\nabla_{{\bf x}} \left[\underbrace{\frac{w(p)}{2}({\bf x}-{\bf f}(p))^\mathsf{T}{\bf C}({\bf x}-{\bf f}(p))}_{V(p,{\bf x})}\right]
\end{equation}
where $V(p,{\bf x})$ is a potential. If ${\bf D}$ is symmetric, it can be
written as ${\bf D} = {\bf Q}\boldsymbol{\Sigma}{\bf Q}^\mathsf{T}$ where ${\bf Q}$ is an orthogonal
matrix and $\boldsymbol{\Sigma}$ is diagonal. The multi-dimensional FP equation
in~\eqref{eq:fokker-planck-multi} can be rewritten as
\begin{eqnarray} \label{eq:fokker-planck-multi2}
-\nabla_{{\bf x}}^\mathsf{T} \left[\nabla_{{\bf x}}V(p,{\bf x}) \rho(p,{\bf x})\right] = \frac{1}{2}\mathsf{Tr}\left\{{\bf D} {\bf H}_{{\bf x}}(\rho(p,{\bf x}))\right\}
\end{eqnarray}
where ${\bf H}_{{\bf x}}(\rho(p,{\bf x}))$ is the Hessian matrix of $\rho(p,{\bf x})$
with respect to the variable ${\bf x}$. By defining
${\bf L} = {\bf Q}\boldsymbol{\Sigma}^{-1/2}$, we then have ${\bf L}^\mathsf{T}{\bf D}{\bf L}={\bf I}$. Now
consider a new system of coordinates, ${\bf y}$, such that ${\bf x} =
{\bf L}{\bf y}$. Then, for any twice differentiable function $f({\bf x})$, we have
$\nabla_{{\bf x}}f({\bf x}) = {\bf L} \nabla_{{\bf y}}f({\bf L}{\bf y})={\bf L}
\nabla_{{\bf y}}f_1({\bf y})$ and
${\bf H}_{{\bf x}}(f({\bf x})) = {\bf L} {\bf H}_{{\bf y}}(f({\bf L}{\bf y})){\bf L}^\mathsf{T} = {\bf L}
{\bf H}_{{\bf y}}(f_1({\bf y})){\bf L}^\mathsf{T}$. By replacing these expressions
in~\eqref{eq:fokker-planck-multi2}, we obtain
\begin{eqnarray}
\mathord{-}\nabla_{{\bf y}}^\mathsf{T}{\bf L}^\mathsf{T} \left[{\bf L} \nabla_{{\bf y}}V_1(p\mathord{,}{\bf y}) \rho_1(p\mathord{,}{\bf y})\right] = \frac{\mathsf{Tr}\left\{{\bf D} {\bf L} {\bf H}_{{\bf y}}(\rho_1(p\mathord{,}{\bf y})){\bf L}^\mathsf{T}\right\}}{2}\nonumber
\end{eqnarray}
which, after some algebra, reduces to
\begin{eqnarray} \label{eq:fokker-planck-multi32}
-\nabla_{{\bf y}}^\mathsf{T} \left[\boldsymbol{\Sigma}^{-1}\nabla_{{\bf y}}V_1(p,{\bf y})\rho_1(p,{\bf y})\right]
&=& \frac{1}{2}\mathsf{Tr}\left\{{\bf H}_{{\bf y}}(\rho_1(p,{\bf y}))\right\} \nonumber\\
&=& \frac{1}{2}\nabla^2_{{\bf y}}\rho_1(p,{\bf y}) \,.
\end{eqnarray}
In the above equation, $V_1(p,{\bf y})$ has the same structure as
$V(p,{\bf x})$ in~\eqref{eq:potential} and,
thus,~\eqref{eq:fokker-planck-multi32} is a FP equation in standard
form whose solution is given by the following
distribution:
\begin{equation}\label{gibbs}
\rho_1(p,{\bf y})=\frac{{\rm e}^{-2 V_1(p, {\bf y})}}{\int\int{\rm e}^{-2 V_1(q, {\bf z})} \mathrm{d}q \mathrm{d} {\bf z}}.
\end{equation}
Therefore the steady state solution of the FP equation can be
expressed as a Gibbs distribution~\eqref{gibbs} associated with a
potential $V_1(p,{\bf y})$.
{\bf Remark 1:} Note that the same expression can be obtained from the
transient solution for $t\rightarrow \infty$ under certain
conditions. Specifically, consider~\eqref{eq:fixed_point}-\eqref{eq:gamma1} and their Laplace
transforms~\eqref{eq:fixed_point_transform}-\eqref{eq:fixed_point_transform_1};
using the final-value theorem, we get
\begin{equation}
\lim_{t\rightarrow \infty} \boldsymbol{\gamma}(p,t) = \lim_{s \to 0} s
\widehat{\boldsymbol{\gamma}}(p,s) \,.
\end{equation}
Assuming that both $\boldsymbol{\Psi}$ and $\boldsymbol{\Xi}(p)$ are Hurwitz-stable for all $p$, then
$\lim_{t\rightarrow \infty} \boldsymbol{\gamma}_0(p,t) = \lim_{s \to 0} s \widehat{\boldsymbol{\gamma}}_0(p,s) = 0$ and
\begin{align}
&& \lim_{t\rightarrow \infty} \boldsymbol{\gamma}_1(p,t) = \lim_{s \to 0} s \widehat{\boldsymbol{\gamma}}_1(p,s) \nonumber \\
&=& \int_q {\bf Z}(p,q) \rho_0(q) \alpha(q)\boldsymbol{\Xi}^{-1}(q) {\bf C} {\bf u}(q) \partial q\,.
\end{align}
So doing, the stationary solution satisfies the integral equation
\begin{eqnarray}\label{eq:gammav_infinity}
\lim_{t\rightarrow \infty} \boldsymbol{\gamma}(p,t) &=& \lim_{t\rightarrow \infty} \boldsymbol{\gamma}_1(p,t) \nonumber \\
&&\hspace{-10ex} +\int_q {\bf Z}(p,q) \rho_0(q) \bar{\alpha}(q) \boldsymbol{\Xi}^{-1}(q) \lim_{t\rightarrow \infty} \boldsymbol{\gamma}(p,t) \partial q\,.
\end{eqnarray}
Note that the same observations made for
\eqref{eq:fixed_point_transform} hold also for the stationary
solution~\eqref{eq:gammav_infinity}.
{\bf Remark 2:} When $t\to \infty$, the expression of the
average in \eqref{eq:mv_x} becomes:
\begin{eqnarray}
&&\hspace{-6ex}{\bf m}(p,\cdot,\infty)\nonumber\\
&=&\hspace{-2ex} \lim_{t\to \infty} {\rm e}^{-\boldsymbol{\Xi}(p) t} {\bf y} + \alpha(p)\boldsymbol{\Xi}^{-1}(p) \left({\bf I}_N\mathord{-}{\rm e}^{-\boldsymbol{\Xi}(p) t} \right) {\bf C} {\bf u}(p)\nonumber\\
&& \qquad+ \bar{\alpha}(p) \int_0^{t} {\rm e}^{-\boldsymbol{\Xi}(p) (t-\tau)} \boldsymbol{\gamma}(p,\tau) \partial\tau \nonumber\\
&=&\hspace{-2ex} \alpha(p)\boldsymbol{\Xi}^{-1}(p){\bf C} {\bf u}(p)\nonumber\\
&& \qquad+\lim_{t\to \infty} \bar{\alpha}(p) \int_0^{\infty} u(t-\tau){\rm e}^{-\boldsymbol{\Xi}(p) (t-\tau)} \boldsymbol{\gamma}(p,\tau) \partial\tau \nonumber\\
&\stackrel{(a)}{=}&\hspace{-2ex} \alpha(p)\boldsymbol{\Xi}^{-1}(p){\bf C} {\bf u}(p)\nonumber\\
&& \qquad +\bar{\alpha}(p)\lim_{s\to 0} s {\cal L}\left\{u(t){\rm e}^{-\boldsymbol{\Xi}(p) t}\right\} {\cal L}\left\{\boldsymbol{\gamma}(p,t)\right\} \nonumber\\
&=&\hspace{-2ex} \alpha(p)\boldsymbol{\Xi}^{-1}(p){\bf C} {\bf u}(p)+ \bar{\alpha}(p)\boldsymbol{\Xi}^{-1}(p) \lim_{s\to 0} s\widehat{\boldsymbol{\gamma}}(p,s) \nonumber\\
&=&\hspace{-2ex} \boldsymbol{\Xi}^{-1}(p)\left[\alpha(p){\bf C}{\bf u}(p)+\bar{\alpha}(p)\boldsymbol{\gamma}(p,\infty)\right] \label{eq:m_infinity}
\end{eqnarray}
where in $(a)$ we applied the final value theorem and we noted that
the integral can be written as the convolution of two functions,
thus its Laplace transform is the product of the transforms of the aforementioned
functions.
{\bf Remark 3:} When $\rho_0(p)$ is discrete, letting
$t\to \infty$ in \eqref{eq:gamma_t}, we obtain an expression of the
steady state distribution as: $\underline{\boldsymbol{\gamma}}(\infty)= \underline{{\bf Z}} \boldsymbol{\Psi}^{-1} {\bf P}_1 {\bf C} \underline{{\bf u}}$.
At last, observe that the following result holds:
\begin{theorem} \label{theo2}
i) The Fredholm equation
defining $\phi(p)$ (given explicitly in \eqref{eq:beta_Fredholm-a}) admits a unique
solution which is Lipschitz-continuous. ii) The solution $\phi(p)$ of the Fredholm
equation \eqref{eq:beta_Fredholm-a}, under any distribution
$\rho_0(p)$, which is continuous at every point in $p$, is the
uniform limit of solutions $\phi_n(p)$, obtained by replacing
distribution $\rho_0(p)$ with its discrete approximation $\rho_n(p)$
whose mesh-size is $\frac{1}{n}$.
\end{theorem}
The proof is provided in Appendix \ref{app:C} in the Supplemental Material.
\section{Summary} \label{sec:summary}
For the sake of clarity, here we summarize the main steps and analytical tools that we used in our derivations.
\begin{itemize}
\item We first adopted the mean-field approach to model the opinion dynamic evolution through
a continuous distribution function whose expression can be obtained by solving a FP equation;
\item Then, by taking the Fourier transform of the FP equation and using the method of characteristics,
we rewrote it as a system of first-order partial-differential
equations;
\item Such a system was solved and the final solution
was obtained by the Fourier inverse transform;
\item The conditions ensuring the system stability were
derived for the matrices characterizing the system, by exploiting the
definition of Hurwitz stability;
\item Finally, we carried out the steady state analysis starting from the FP
equation and letting $t\to \infty$. We obtained an expression for
$\boldsymbol{\mu }(p,{\bf x})$ which is a function of quantities that can be
computed by solving a multidimensional Fredholm equation.
\end{itemize}
\section{Numerical results\label{sec:results}}
In this section, we show some numerical examples, which shed light on the impact of the model parameters on the stationary state of opinions as well as on their dynamics. In the following, we will always consider a uniform distribution of personalities in the range $[-1,1]$ and a two-dimensional opinion space, i.e., ${\bf x} = [x_1,x_2]^{^\mathsf{T}}$.
\subsection{Sensitivity of the stationary distribution on the model parameters}
In this first subsection, we evaluate the impact of the noise variance, the prejudice, and the coupling matrix ${\bf C}$, on the stationary distribution.
We first show the effect of noise variance $\sigma^2_n$ and the prejudice ${\bf u}$. To this purpose, we consider an asymmetric coupling matrix
\begin{equation} \label{eq:corr_mat_asym}
{\bf C} = \left[
\begin{array}{cc}
1 & \rho \\
\epsilon & 1
\end{array}
\right]
\end{equation}
where $\rho = 0.3$ and $\epsilon$ is a very small positive number
(namely, $10^{-10}$ while obtaining the results)\footnote{We have used
$\epsilon$ as an approximation to $0$, since setting $\epsilon = 0$
would yield a non-diagonalizable matrix ${\bf C}$.}. Such
model is well suited for a case where the reciprocal influence of the
opinions on two subjects is unidirectional (e.g., from subject 2 to subject
1): a possible example could be the appreciation of the government
action (subject 1), and the opinion on the right level of taxation
(subject 2), where subject 2 is more likely to affect subject 1 than
vice versa. We assume a constant level of stubbornness $\alpha(p)
= 0.01$, while the strength of opinion interaction is given by
\begin{equation} \label{eq:zeta_proximity}
\zeta(p,q) = \frac{1}{1+|p-q|^2}
\end{equation}
i.e., interactions are stronger between agents with similar personalities, a model which we will refer to as \emph{proximity} model.
Fig.\,\ref{fig:Noise_effect} shows the contour lines of the
stationary opinion distribution for different values of $\sigma_n^2$,
for two different prejudice scenarios. In the top row, the prejudice is given by
\begin{equation} \label{eq:prejudice_1}
{\bf u}(p) = \left\{
\begin{array}{cc}
[-1, 0]^\mathsf{T}, & p<0 \\
{[1, 0]^\mathsf{T}}, & p \geq 0
\end{array}
\right.
\end{equation}
while in the bottom it satisfies
\begin{equation} \label{eq:prejudice_2}
{\bf u}(p) = \left\{
\begin{array}{cc}
[0,-1]^\mathsf{T}, & p<0 \\
{[0,1]^\mathsf{T}}, & p \geq 0 \,.
\end{array}
\right.
\end{equation}
In both rows, we set $\sigma_n^2 = 10^{-3}$ for the left plot,
$\sigma_n^2 = 5 \times 10^{-3}$ for the center plot, and $\sigma_n^2 =
10^{-2}$ for the right plot. Observe that the stationary distribution features two peaks, corresponding to the two different prejudice points, with the same height and width. The width increases with increasing noise variance, for the highest noise variance, the peaks start to merge.
\begin{figure}[tb]
\centering
\includegraphics[width=1.0\columnwidth]{MD_noise_effect.eps}
\caption{Contour lines of stationary opinion distribution for
$\alpha = 0.01$ and $\zeta(p,q)$ as in \eqref{eq:zeta_proximity}. Plots (a), (b), (c): prejudice as in \eqref{eq:prejudice_1}. Plots (d), (e), (f): prejudice as in \eqref{eq:prejudice_2}. Plots (a), (d): $\sigma_n^2 = 10^{-3}$. Plots (b), (e): $\sigma_n^2 = 5 \times 10^{-3}$; plots (c), (f): $\sigma_n^2 = 10^{-2}$. \label{fig:Noise_effect}
\vspace{-3mm}
\end{figure}
Next, in Fig.\,\ref{fig:Corr_effect} we investigate the effect of topic correlation, as expressed by the coupling matrix ${\bf C}$. We use the same interaction strength as per \eqref{eq:zeta_proximity}, and the prejudice given in \eqref{eq:prejudice_2}. Also, as before, $\alpha(p) = 0.01$ and $\sigma_n^2 = 10^{-3}$. Finally, we consider the coupling matrix as in \eqref{eq:corr_mat_asym}, with $\rho \in \{-10,-5,0,5,10\}$.
For $\rho=0$ there is no interaction between the two opinion
components; changing $\rho$ does not have any effect on the mean of the stationary distribution for each personality (\eqref{eq:mv_x}), leading to peaks whose locations are essentially invariant with respect to $\rho$. Notice that the distribution of $x_2$ is independent of $\rho$. Moreover, due to the symmetric scenario, in all cases the stationary distribution shows a reflectional symmetry around the origin. Finally, changing $\rho$ into $-\rho$ has, in the considered scenario, the effect of reflecting the stationary distribution around the $x_2$-axis, or, in other words, of changing $x_1$ into $-x_1$. From the system point of view, the effect of different correlation values implies a larger share of agents that have strong positive opinions on both subjects (positive correlation value) or one strong positive and one strong negative opinion (negative correlation value). Going back to the previous practical example, agents which are in favor of a low level of taxation will judge more positively or more negatively the government action, depending on the correlation coefficient sign.
\begin{figure}[tb]
\centering
\includegraphics[width=1.0\columnwidth]{MD_corr_effect.eps}
\caption{Contour lines of stationary opinion distribution for
$\alpha = 0.01$, $\sigma_n^2 = 10^{-3}$, and $\zeta(p,q)$ as in \eqref{eq:zeta_proximity}. Coupling matrix given by \eqref{eq:corr_mat_asym}. From left to right: $\rho = -10, -5, 0 , 5, 10$. \label{fig:Corr_effect}
\vspace{-3mm}
\end{figure}
\subsection{Community-based scenario, the influence of \protect${\bf Z}$ and the insurgence of instability}
We now assess the impact of the interaction strength matrix ${\bf Z}$ on
the stability of opinions. We consider a different scenario, in which
there are $M$ personalities ($M$ even), all with the same stubbornness
level $\alpha = 0.01$, organized in two communities and with an
interaction similar to that considered in Example 1, but with $N=2$,
i.e., \begin{equation} \label{eq:zeta_community} \underline{{\bf Z}} = \left(\left[
\begin{array}{cc}
\zeta_1 & - \zeta_2 \\
- \zeta_2 & \zeta_1
\end{array} \right] \otimes \mathbf{1}_{M/2} \right) \otimes {\bf C}
\end{equation}
with $\zeta_1 = 1$ and $\zeta_2 $ used as a parameter. We consider
${\bf C}$ as given by \eqref{eq:corr_mat_asym} with $\rho = 1$, the
prejudice as given in \eqref{eq:prejudice_2}, and $\sigma_n^2
=10^{-3}$.
Also, $\rho_0(p)$ is given by~\eqref{eq:rho0} with $r_i=1/M$ for all $i$.
As a possible practical example of such a scenario, we can
think of two religious sects, say Bogumils and Cathars, which have
generally different views on two topics, represented by the two
opinions $x_1$ and $x_2$.
It is not difficult to show that the stability region boundaries are
the same as for Example 1, since ${\bf C}$ is Hurwitz-stable.
Furthermore, we can derive the asymptotic expression of the mean in~\eqref{eq:m_infinity}
by exploiting Remark 3 in Section~\ref{sec:steady_state} and writing from~\eqref{eq:Gammap} and~\eqref{eq:Xi}
$\underline{\boldsymbol{\Xi}}=\beta{\bf I}_M\otimes {\bf C}$, where $\beta=\bar{\alpha}\frac{1-\zeta_2}{2}+\alpha$.
If $\beta>0$, through some algebra and by applying the properties of the Kronecker product, we obtain
\begin{equation} \label{eq:asym_mean}
{\bf m}(p,\cdot,\infty) = \frac{\alpha}{\alpha - \overline{\alpha}
\zeta_2} {\bf u}(p). \end{equation}
Thus, if $\zeta_2 < \alpha/\overline{\alpha} \simeq 0.0101$, the system is
stable.
Fig. \ref{fig:Comm_stable} shows the stationary distribution
for increasing value of $\zeta_2$. As it can be seen, Bogumils and
Cathars merge because of the attractive forces for $\zeta_2 = -0.2$ or
lower, while, for $-0.1 < \zeta_2 < 0$, notwithstanding their
reciprocal attraction, they remain separated because of the effect of
the prejudice. For $ 0 < \zeta_2 < \alpha/\overline{\alpha}$, the two
communities repel each other, but this repulsion is not strong enough
to win the effect of prejudice, so stability is preserved while the
means grow larger and larger for
$\zeta_2 \uparrow \alpha/\overline{\alpha}$.
\begin{figure}[tb]
\centering
\includegraphics[width=1.0\columnwidth]{MD_comm_stable.eps}
\caption{Stationary opinion distribution for
$\alpha = 0.01$, $\sigma_n^2 = 10^{-3}$, ${\bf Z}$ as in \eqref{eq:zeta_community} ($\zeta_1 = 1$), for several values of $\zeta_2$ in the stability region. Coupling matrix given by \eqref{eq:corr_mat_asym} with $\rho = 1$. From left to right: $\zeta_2 = -0.3,-0.2,-0.1,0,0.004$. \label{fig:Comm_stable}
\vspace{-3mm}
\end{figure}
For $\zeta_2 > \alpha/\overline{\alpha}$, the system is not stable
anymore, and \eqref{eq:asym_mean} does not hold. In particular, when
$\zeta_2 < \zeta_1 + 2 \alpha/\overline{\alpha} = 1.0202$, the
intra-community attraction and the inter-community repulsion have such
a relative strength that the system experiences type-II instability:
the communities are preserved but their respective means tend to
diverge. In our example, the two religious sects are so enemy of each
other, to radicalize their views while retaining a strong identity
within themselves, giving rise to religious fanaticism.
We now look at the time evolution of opinions in the case where, for all personalities, opinions start deterministically from the origin. Fig.\,\ref{fig:Comm_unstable} shows the opinions at time instants $t = 5,10,15,20$, for $\zeta_2 = 0.1$. Notice that, because of the correlation, the two communities diverge along the bisector of the I-III quadrants.
\begin{figure}[tb]
\centering
\includegraphics[width=1.0\columnwidth]{MD_comm_unstable.eps}
\caption{Contour lines of opinion distribution for
$\alpha = 0.01$, $\sigma_n^2 = 10^{-3}$, ${\bf Z}$ as in \eqref{eq:zeta_community} ($\zeta_1 = 1$, $\zeta_2 = 0.1$). Coupling matrix given by \eqref{eq:corr_mat_asym} with $\rho = 1$. From left to right: $t = 5,10,15,20$. \label{fig:Comm_unstable}
\vspace{-3mm}
\end{figure}
Finally, for $\zeta_2 > \zeta_1 + 2 \alpha/\overline{\alpha}$, the
inter-community repulsion prevails on intra-community attraction, the
system experiences type-I instability, and the variance within the two
communities also grows to infinity. In other words, Bogumils and
Cathars dissolve themselves into heterogeneous crowds not having
definite views on the two topics of
interest. Fig.\,\ref{fig:Comm_unstable} shows the opinions at time
instants $t = 0.15,0.3,0.45,0.5$ for $\zeta_2 = 10$, again when the
opinions all start deterministically from the origin. The dynamic is
similar, but faster than in the previous case, and the communities
expand, so that, for $t$ sufficiently large, their boundaries disappear.
\begin{figure}[tb]
\centering
\includegraphics[width=1.0\columnwidth]{MD_comm_unstable_2.eps}
\caption{Contour lines of opinion distribution for
$\alpha = 0.01$, $\sigma_n^2 = 10^{-3}$, ${\bf Z}$ as in \eqref{eq:zeta_community} ($\zeta_1 = 1$, $\zeta_2 = 10$). Coupling matrix given by \eqref{eq:corr_mat_asym} with $\rho = 1$. From left to right: $t = 0.15,0.3,0.45,0.5$. \label{fig:Comm_unstable_2}
\vspace{-3mm}
\end{figure}
\subsubsection{Finite-network behavior}
In order to assess the validity of the mean-field approach, in
Fig. \ref{fig:Comm_simul} we show the opinion distribution behavior
for a finite set of $U = 500$ agents, by solving numerically
\eqref{eq:x_i}. In particular, we consider the same model as in
Figs. \ref{fig:Comm_stable}-\ref{fig:Comm_unstable} for
$\zeta_2 \in \{-0.1, 0, 0.004, 0.1\}$. The first half of the agents
are Bogumils, while the others are Cathars. For the first three values
of $\zeta_2$, which correspond to a stable system, we show the
stationary opinion distribution. For $\zeta_2 = 0.1$, we show the
distribution at time $t = 20$. As it can be seen, the results in
Figs. \ref{fig:Comm_stable}-\ref{fig:Comm_unstable} match those in Fig.
\ref{fig:Comm_simul}, indicating that the asymptotic analysis
well represents the opinion dynamics of relatively small populations.
\begin{figure}[tb]
\centering
\includegraphics[width=1.0\columnwidth]{MD_comm_simulations.eps}
\caption{Contour lines of simulated opinion distribution for
$U = 500$, $\alpha = 0.01$, $\sigma_n^2 = 10^{-3}$, ${\bf Z}$ as in \eqref{eq:zeta_community} ($\zeta_1 = 1$), where the first half of users belong to community 1, while the second half to community 2. Coupling matrix given by \eqref{eq:corr_mat_asym} with $\rho = 1$. From left to right: $\zeta_2 = -0.1, 0, 0.004, 0.1$. For the rightmost graph, $t=20$. \label{fig:Comm_simul}
\vspace{-3mm}
\end{figure}
\subsection{Rotational effects}
We now consider the case in which the dynamics is stable in
both dimensions, but the effect of the coupling matrix yields
instability (of Type I). Consider again the two-community scenario
given by \eqref{eq:zeta_community}, but with $\zeta_1 = 1$ and
$\zeta_2 = -0.1$. Note that, in the scalar case, this would be a stable scenario. However, let us now consider a coupling matrix given by
\begin{equation} \label{eq:corr_mat_complex}
{\bf C} = \left[
\begin{array}{cc}
0 & 1 \\
-1 & 0
\end{array}
\right]
\end{equation}
which is easily seen to have eigenvalues equal to $\pm j$. This is
quite an artificial scenario, since this coupling matrix has rather an
ad-hoc shape, to which it is difficult to associate any real-world situation. All the other parameters are the same as in the previous example, except for the prejudice, which is given by
\begin{equation} \label{eq:prejudice_22}
{\bf u}(p) = \left\{
\begin{array}{cc}
[0,-10], & p<0 \\
{[0,10]}, & p \geq 0 \,.
\end{array}
\right.
\end{equation}
It is easy to see that, for the $i$-th discrete personality, the covariance matrix in \eqref{eq:covariance} is given by:
\begin{equation}
\boldsymbol{\Sigma} \left( p_i,t \right) = \sigma^2_n t {\bf I}_2
\end{equation}
independently from $i$. Note that such covariance matrix does not
reach any finite limit for $t \rightarrow \infty$, hence the system is
unstable. Fig.\,\ref{fig:Comm_rotating} shows the temporal evolution of the opinion distribution, for $t = 1,10,20,30,40,50,60,70,80$. The peaks corresponding to the two communities widen along time, as expected, while their means undergo a rotation around the origin, at some instants (such as at $t = 60$) making the peaks temporarily merge.
\begin{figure*}[tb]
\centering
\includegraphics[width=1.0\textwidth]{MD_comm_rotating.eps}
\caption{Contour lines of opinion distribution for
$\alpha = 0.01$, $\sigma_n^2 = 10^{-3}$, ${\bf Z}$ as in \eqref{eq:zeta_community} ($\zeta_1 = 1$, $\zeta_2 = -0.1$). Coupling matrix given by \eqref{eq:corr_mat_complex}. From left to right: $t = 1,10,20,30,40,50,60,70,80$. \label{fig:Comm_rotating}
\vspace{-3mm}
\end{figure*}
\section{Conclusions\label{sec:conclusions}}
Several analytical models representing the dynamics of social opinions
have been proposed in the literature. By drawing on this body of work,
we developed a model that accounts for the individual endogenous
process of opinion evolution as well as for the possible adversarial
behavior of individuals. Importantly, our model also represents the interdependence
between opinions on different, yet correlated, subjects. Under
asymptotic conditions on the size of the individuals' population, we obtained the time
evolution of the opinions distribution as the solution of a
multidimensional Fokker-Planck equation. We then discussed the
stability conditions and derived the steady-state solution. Our
numerical results match the stability conditions we obtained and show
interesting phenomena in collective belief dynamics.
\bibliographystyle{IEEEtran}
| {
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Format Feature Film
Genre Horror | Thriller
Audience Teens | Adults
Rating PG-13
An apparently deranged woman arrives at a fracking outpost, she destroys the communications equipment and disables transportation. When the outpost crew detain her, she frantically rants "the sounds is coming to kill us" before being knocked out by one of the crew. Noah, the outpost's boss, decides to take 'Eagle-eye' to investigate the woman's base camp. It's been abandoned. The woman comes to suffering concussion and with no memory of recent events. The crew start to believe that there really is a noise tearing apart its victims, trying to bring forth an Alien Demon God. They decided to leave when help arrives but their numbers dwindle as they try different strategies to survive. When there are only two left they abandon the wait and try to make it out on their own.
We know more about Outer Space than the Deep Ocean.
Jump To: Director Writers Producer
Director Edward Lynden-Bell Edward Lynden-Bell is a New Zealand writer and Director known for running a prolific music video and short take New Zealand based production company and Directing The Last Great Snail Chase (2007), writing for Intrinsic Value films, the New Zealand Film Commission and for Simon Hunter's Edie (2017). Read More
Writers Edward Lynden-Bell Edward Lynden-Bell is a New Zealand writer and Director known for running a prolific music video and short take New Zealand based production company and Directing The Last Great Snail Chase (2007), writing for Intrinsic Value films, the New Zealand Film Commission and for Simon Hunter's Edie (2017). Read More
Sasha Whitehouse Sasha Whitehouse had a Diploma from New Zealand's Film and TV school before graduating with a BA in Film and TV from Melbourne's prestigious VCA School of Film and TV in 2005. He direct Read More
Producer Sacha Rodríguez A producer who is passionate about making films, Sacha Rodriguez joins Gabriel Beristain, and together, they lay the foundations for Vedado's filmed entertainment division, produce a feature length documentary and the feature film "Havana Kyrie" starring Franco Nero and Ron Perlman - and embark on development of a new slate of screen titles including the upcoming feature film "The House of Abraham Phillips", in collaboration with The Lan Film Company in Wales. Read More
+1 323 723 FILM
Unsolicited Project Submissions
© 2023 Matutino Films PTY. LTD.
Matutino FIlms does not accept or consider creative materials (including but not limited to text, graphics and audiovisual material) sent voluntarily. | {
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{"url":"https:\/\/artofproblemsolving.com\/wiki\/index.php?title=2011_AMC_8_Problems\/Problem_25&diff=prev&oldid=85453","text":"During AMC testing, the AoPS Wiki is in read-only mode. No edits can be made.\n\n# Difference between revisions of \"2011 AMC 8 Problems\/Problem 25\"\n\n## Problem\n\nA circle with radius $1$ is inscribed in a square and circumscribed about another square as shown. Which fraction is closest to the ratio of the circle's shaded area to the area between the two squares?\n\n$[asy] filldraw((-1,-1)--(-1,1)--(1,1)--(1,-1)--cycle,white,black); filldraw(Circle((0,0),1), mediumgray,black); filldraw((-1,0)--(0,1)--(1,0)--(0,-1)--cycle,white,black);[\/asy]$\n\n$\\textbf{(A)}\\ \\frac{1}2\\qquad\\textbf{(B)}\\ 1\\qquad\\textbf{(C)}\\ \\frac{3}2\\qquad\\textbf{(D)}\\ 2\\qquad\\textbf{(E)}\\ \\frac{5}2$\n\n## Solution 1\n\nThe area of the smaller square is the one half of the product of its diagonals. Note that the distance from a corner of the smaller square to the center is equivalent to the circle's radius so the diagonal is equal to the diameter: $2*2*1\/2=2.$\n\nThe circle's shaded area is the area of the smaller square subtracted from the area of the circle: $\\pi - 2.$\n\nIf you draw the diagonals of the smaller square, you will see that the larger square is split $4$ congruent half-shaded squares. The area between the squares is equal to the area of the smaller square: $2.$\n\nApproximating $\\pi$ to $3.14,$ the ratio of the circle's shaded area to the area between the two squares is about\n\n$$\\frac{\\pi-2}{2} \\approx \\frac{3.14-2}{2} = \\frac{1.14}{2} \\approx \\boxed{\\textbf{(A)}\\ \\frac12}$$\n\n## Solution 2\n\nFor the ratio of the circle's shaded area to the area between the squares to be $1,$ they would have to be approximately the same size. For any ratio larger than that, the circle's shaded area must be greater. However, we can clearly see that the circle's shaded area is part of the area between the squares, and is approximately $\\boxed{\\textbf{(A)}\\ \\frac12}$.\n\nNote that this solution is not rigorous, because we still should show that the ratio is less than $\\frac{3}{4}$.","date":"2022-01-23 18:30:31","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\": 13, \"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\": 0.8326128721237183, \"perplexity\": 233.84765069213333}, \"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\/1642320304309.5\/warc\/CC-MAIN-20220123172206-20220123202206-00089.warc.gz\"}"} | null | null |
Q: Can the ErrorProvider be queried to see if it has set any errors? I have this code to do some basic sanity checking before posting a record:
if (string.IsNullOrWhiteSpace(textBoxFirstName.Text))
{
errorProvider.SetError(textBoxFirstName, "Enter a first name");
}
if (string.IsNullOrWhiteSpace(textBoxLastName.Text))
{
errorProvider.SetError(textBoxLastName, "Enter a last name");
}
...but I want to then do something like this to exit the handler if either of those conditions has been met:
if (errorProvider.SetErrorCount > 0) then return;
...but I see no way to do that. I don't want to have to write an "OR" statement to see if either of the textBoxes I'm checking are empty and then short circuit the handler that way.
Is there a way to tell whether the errorProvider is "dirty" to avoid cluttery code?
A: Write a method and pass it the error message and the control. Have a counter variable and increase the counter within the method. Here is some pseudocode:
private int errorCount;
SetError(Control c, string message)
{
errorProvider.SetError(c, message);
errorCount++;
}
A: One option would be to use the GetError method off of the ErrorProvider.
// possibly use a backing field for all controls to evaluate
private readonly Control[] textBoxes = new[] { textBoxFirstName, textBoxLastName };
// helper property to evaluate the controls
private bool HasErrors
{
get { return textBoxes.Any(x => !string.IsNullOrEmpty(errorProvider.GetError(x)); }
}
| {
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Mechanical Engineer, R&D - San Jose, CA, USA
Title: Mechanical Engineer, R&D
Reporting to: Manager, Hardware Engineering
Location: San Jose, California
Venus Concept is committed to delivering innovative energy based and robotic technologies that disrupts the medical aesthetics industry. If you are interested in leveraging robotic technology and growing your skills in this area, consider joining our team. We are currently seeking a Mechanical Engineer, R&D who is highly organized and can ensure that our high standard of quality is maintained. Reporting to the Manager, Hardware Engineer, the Mechanical Engineer, R&D will be based in our San Jose office and will be working with a highly sophisticated medical device applications, encompassing many difference areas including but not limited to mechanical component design, mechanisms, sensors and actuators integration, robotics. We are looking for a self-starter who is comfortable in a dynamic working environment, working on innovative trend setting changes which will directly impact the medical aesthetic industry.
Plan, conceptualize, and create mechanical designs for new products, applications and features.
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Design, document and release parts, assemblies, and fixtures for novel medical system.
Work on iterate design concepts and mechanisms by rapid prototyping (machining, 3D printing).
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Break down concept and large/complex problems into achievable and measurable tasks.
Bachelor's degree in Mechanical Engineering or related field.
8+ years of relevant industry experience designing mechanical parts.
Medical device experience is an asset as is robotics experience.
Experience with SolidWorks design.
Experience with lathes, mills, 3d printers, and manual tools
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Robotics, mechatronics, or mechanisms experience is an asset.
Demonstrated ability to communicate effectively with both technical and non-technical individuals and to work in a fast paced, high quality, iterative projects.
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Venus Concept is an equal opportunity employer committed to diversity and inclusion. We welcome applications from all qualified individuals. Venus Concept is committed to accommodating persons with disabilities. Accommodations are available on request for qualified candidates during each stage of the recruitment process.
About Venus Concept
Venus Concept is an innovative global medical aesthetic technology leader with a broad product portfolio of minimally invasive and non-invasive medical aesthetic and hair restoration technologies and reach in over 60 countries and 19 direct markets. Venus Concept focuses its product sales strategy on a subscription-based business model in North America and in its well-established direct global markets. Venus Concept's product portfolio consists of aesthetic device platforms, including Venus Versa, Venus Legacy, Venus Velocity, Venus Fiore, Venus Freedom, Venus Viva, Venus Freeze Plus, Venus Glow, Venus Bliss, Venus Epileve and Venus Viva MD. Venus Concept's hair restoration systems include NeoGraft®, an automated hair restoration system that facilitates the harvesting of follicles during a FUE process and the ARTAS® and ARTAS iX® Robotic Hair Restoration systems, which harvest follicular units directly from the scalp and create recipient implant sites using proprietary algorithms. Venus Concept has been backed by leading healthcare industry growth equity investors including EW Healthcare Partners (formerly Essex Woodlands), HealthQuest Capital, Longitude Capital Management, and Aperture Venture Partners.
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Venus Versa™ is cleared by the FDA as a multi-application device intended to be used in aesthetic and cosmetic procedures. The SR515 and SR580 applicators are cleared by the FDA for the treatment of benign pigmented epidermal and cutaneous lesions and treatment of benign cutaneous vascular lesions. The HR650/HR650XL and HR690/HR690XL applicators are cleared by the FDA for the removal of unwanted hair and to effect stable long-term or permanent hair reduction for Fitzpatrick skin types I-IV. The AC Dual applicator is cleared by the FDA for the treatment of acne vulgaris. The DiamondPolar™ and OctiPolar™ applicators on the Venus Versa™ system are cleared by the FDA for non-invasive treatment of moderate to severe facial wrinkles and rhytides on females with Fitzpatrick skin types I-IV. The NanoFractional RF™ (Viva) applicator is cleared by the FDA for dermatological procedures requiring ablation and resurfacing of the skin.
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NeoGraft® is cleared by the FDA with indication for use in suction-assisted follicular extraction and re-implantation. NeoGraft® is an auto-graft system and can be used on both male and female patients.
Venus Legacy™ is cleared by the FDA for the non-invasive treatment of moderate to severe facial wrinkles and rhytides in females with Fitzpatrick skin types I-IV with the OctiPolar™ and DiamondPolar™ applicators, and temporary reduction in the appearance of cellulite with the 4D Body (LB2) and 4D Face (LF2) applicators.
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The Venus Viva™ MD is cleared by the FDA for the use in dermatological procedures requiring ablation and resurfacing of the skin, and the non-invasive treatment of moderate to severe facial wrinkles and rhytides for females in Fitzpatrick skin types I-IV with the DiamondPolar™ applicator.
Venus Freeze Plus™ is a non-invasive device cleared by the FDA for use in dermatologic and general surgical procedures for females for the treatment of moderate to severe facial wrinkles and rhytides in Fitzpatrick skin types I-IV, using the DiamondPolar™ and OctiPolar™ applicators.
Venus Heal™ is cleared by the FDA for the relief of minor muscle aches and pain, relief of muscle spasm, and temporary improvement of local blood circulation. The treatment of these conditions will lead to the treatment of certain soft tissue injuries and conditions.
Venus Glow™ is cleared by the FDA as a Class I motorized dermabrasion device. It provides a dermal rejuvenation treatment that works to open up and deep-clean pores. Venus Concept is the exclusive distributor for Venus Glow™.
Venus Epileve™ is cleared by the FDA for hair removal, permanent hair reduction (defined as the long-term stable reduction in the number of hairs
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Appu is a 2002 Indian Kannada-language romantic action comedy film directed by Puri Jagannadh. It stars Puneeth Rajkumar and Rakshita. The supporting cast features Avinash, Srinivasa Murthy and Sumithra. The film was produced by Puneeth's mother, Parvathamma under Poornima Enterprises, the production banner of the Rajkumar family. It marked the screen debut Puneeth and Rakshita in lead roles.
Up on theatrical release on 26 April 2002, the film was a success and completed a 200-day run in theatres. The lead actor Puneeth Rajkumar came to be known as "Appu" hereafter among the masses, colloquially.
The film was remade in Telugu in 2002 as Idiot (Directed by Puri Jagannadh), in Tamil in 2003 as Dum, in Bengali in 2006 as Hero and in Bangladeshi Bengali in 2008 as Priya Amar Priya.
Plot
Appu is a carefree guy, who spends time hanging with his friends, and is also the son of a head constable named Venkata Swamy. One night, Appu is thrashed by a rival gang at night, but was rescued by a girl named Suchitra. She pays his hospital bills and donates her blood. She is gone from the hospital by the time Appu regains consciousness. When Appu's friends inform him about the girl who rescued him, he starts to fall for her for her kindheartedness, though he did not see her. Suchi later turns out to be the daughter of the city commissioner Rajshekhar.
Appu meets Suchi in the college and expresses his love. When she does not agree, he teases her which leads to Suchi to complain about Appu to Rajshekhar, who takes Appu to the police station and severely beats him before being rescued by Venkatswamy and his superior SI Sudarshan. Even though Appu is beaten by Rajshekhar, he becomes more adamant to win over Suchi, and proposes to Suchi again in the college, where she asks him to jump from the building. When Appu is ready to do so, Suchi agrees to his love. However, Rajshekhar is not happy about their relationship and hires some goons to amputate Appu.
Suchi discovers this and runs to help him, but is met with an accident. Both of them get admitted to the same hospital where they unite there also. Rajshekhar arranges her marriage with another person, to which she openly opposes and tries to commit suicide. Appu arrives and rescues her, but Rajshekhar still wants to get her married to a man of his own choice, where he also engages goons to kill Appu. Appu finally escapes all the troubles and meets the DGP to help him to marry his love. The DGP finally suspends Rajshekhar and arranges Appu's marriage in the police station. Finally, Appu appears for civil services and is selected for IPS.
Cast
Production
After the success of Yuvaraja (2001), Puri Jagannadh was approached by the Rajkumar family to introduce their third son Puneeth Rajkumar to make his onscreen debut as the lead actor. Puri gladly accepted the opportunity. Rakshitha, daughter of cameraman B.C. Gowrishankar made her acting debut with this film, and she went on to play the same character in its Telugu and Tamil remakes.
Soundtrack
Gurukiran composed the film's background score and music for its soundtrack, with the lyrics written by Upendra, Sriranga and Hamsalekha. The soundtrack album consists of six tracks.
References
External links
Rediff article
2000s Kannada-language films
2002 films
Indian romantic action films
Kannada films remade in other languages
Films scored by Gurukiran
Films directed by Puri Jagannadh
2002 action comedy films
2002 romantic comedy films
Indian action comedy films
Indian romantic comedy films
Films shot in Bangalore
Films shot in Switzerland
2000s masala films
Films set in universities and colleges
2000s romantic action films | {
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as frances loom puts it… they are adding soul to your space, one rug at a time. this collection of antique rugs is curated by frances loom founder, kelly vittengl. with shipments coming in every thursday, you are guaranteed to find your perfect match. you can shop the collection and fall in love here. check out some of my favorite spaces with "soul" below. | {
"redpajama_set_name": "RedPajamaC4"
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<html>
<head>
<meta charset="utf-8">
<title>myPlugin Test Suite</title>
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<script src="../libs/jquery-loader.js"></script>
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<link rel="stylesheet" href="../libs/qunit/qunit.css" media="screen">
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<script src="../build/before.js"></script>
<script src="../build/myPlugin.js" data-cover></script>
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| {
"redpajama_set_name": "RedPajamaGithub"
} | 2,128 |
Schloss Vik (ältere Schreibweise Wik) ist ein Prachtbau in der schwedischen Gemeinde Uppsala in der historischen Provinz Uppland. Es liegt am See Mälaren ungefähr 10 km südwestlich von Uppsala.
Das Gebäude war ursprünglich eine Verteidigungsburg die um 1450 gebaut wurde. Im 17. Jahrhundert entstand daraus ein Schloss im französischen Stil, das heute als Byggnadsminne geschützt ist.
Schloss Vik gehört heute dem Provinziallandtag (Landsting) der Provinz Uppsala län und wird für Ausbildungen und Konferenzen genutzt. 1998 war das Schloss im Film Commander Hamilton als Wohnort der Hauptperson zu sehen.
Geschichte
Die erste urkundliche Erwähnung des Gutes Vik (Vihc) als Eigentum von Ramborg Israelsdotter (And), der Frau des Reichsrates Arvid Gustavsson (Sparre av Vik), erfolgte 1303. Danach gehörte der Hof dem Adelsgeschlecht Bielke, das die Burg errichten ließ. Laut dem Bischof und Chronisten Peder Swart belagerte Gustav Wasa 1521 "ein kleines Haus mit Namen Vik", das heute mit der Burg identifiziert wird. Die Besitzerwechsel erfolgten hauptsächlich durch Erbschaft und Heirat. Gut Vik gehörte unter anderem Gustaf Horn und den Adelsfamilien von Liewen und von Essen. 1912 wurde es erstmals verkauft.
Schloss Vik heute
Das Konferenzzentrum besteht aus 15 Sälen in unterschiedlichen Größen. Acht davon liegen im Schloss. In der näheren Umgebung befinden sich acht Hotels, die ganzjährig geöffnet haben. Dem Konferenzzentrum ist ein Aktivitätszentrum angeschlossen, das weitere Angebote für die Besucher bereithält.
Im Anschluss an das Schloss liegt ein weitläufiger Naturpark. Die Kieswege bilden einen Naturlehrpfad durch Laubwälder und Weideflächen. Im Sommer hat der nahe gelegene Badestrand mit Stegen und Grasflächen viele Besucher. Dort befindet sich auch ein Café, das von Juni bis August geöffnet ist.
Einzelnachweise
Weblinks
Schloss Vik
Eintrag beim Riksantikvarieämbetet
Schloss in Schweden
Bauwerk in der Gemeinde Uppsala
Byggnadsminne in Uppsala län
Schloss in Europa | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 6,104 |
Q: Create folder on OneDrive with API I am trying to create new folder on OneDrive through API, but I am getting Exception
Fatal error: Uncaught exception 'League\OAuth2\Client\Provider\Exception\IdentityProviderException' with message 'Must provide one of the following facets to create an item: Bundle, File, Folder, RemoteItem
This is with $a = ["name" => "Folder"];.
If I change it to this:
$a = ["name" => "Folder", "folder" => array()];
$this->provider->post('https://graph.microsoft.com/v1.0/me/drive/root/children',$a,$_SESSION['access_token']);
Then I get
Property folder in payload has a value that does not match schema
From the example page I am assuming that folder value should be empty array.
A: Correct format for the folder was
$folderParameters = ["name" => $folderName, "folder" => ["childCount" => '0']];
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 3,914 |
\section{Introduction}
\vspace{4 mm}
Parasites have evolved a diversity of sophisticated strategies to evade the host's immune response, among which antigenic variation is perhaps one of the most striking ones. This strategy consists in periodically changing a protective coat composed by an abundant and immunogenic protein. In this mechanism the parasites express only one variant antigenic protein copy from a large repertoire of silent genes. The mechanism allows transient immune evasion, since after changing the variable protein that is being expressed, an entirely new parasite population arises that is not recognized by the host's immune system that has developed an antibody response directed against the previous antigen. By repeating this cycle during the course of an infection, parasites are able to remain in the host for long periods of time.
Perhaps the most paradigmatic example is that of African trypanosomes (responsible of producing the sleeping sickness in humans), but antigenic variation is also observed in {\it Giardia llambia}, and the malaria agents belonging to the {\it Plasmodium} genus. Some viruses are also able to evade the immune response in a strategy similar to that just described. However in the latter systems antigenic diversity is generated by the introduction of point mutations in the gene encoding the antigen, rather than by switching the expressed gene. This implies some substantial differences in the dynamics since the new antigen is most likely somewhat similar to the previous one (parental) and perhaps is recognized (yet with lower affinity) by the same antibodies
Several models have been developed to study and predict the population dynamics of parasites and viruses during the course of an infection within a single host. Most of them are based on a system of coupled differential equations inspired on variations of the predator-prey models (see ref. \cite{Kosinski} to \cite{Blyuss}). In short, this approach consists in a set of differential equations describing the dynamical interaction between antigens and the host's immune system, in such a way that the outcome of one equation is a modulating parameter of the others, and including in some cases cross-reactive immune responses as well as other possible interactions. Stochasticity is incorporated ad hoc into the models by the emergence of new variants (which eventually are not recognized by immune system) at random times, usually driven by a Poisson process.
Very recently Gurarie et al. (see ref. \cite{Gurarie}) implemented a discrete time computer model for the case of malaria. This modeling approach, termed agent-based, consists in a set of coupled difference equations that describe the transition between successive iterations of the parasite population (i.e. parasite generations) and its interaction with the immune system. According to the authors the advantage of this approach is that, owing to its discrete nature, the aleatory components are incorporated more easily by adding random factors to the variables that represent the efficiency of immune system.
In spite of the existence of these models of antigenic variation, in our opinion it is worth re-addressing the problem from a different perspective. Here we present a model that tackles this topic from a microscopic point of view that consists in following the pathway and behavior of its individual elements through a multi-type branching process. Iwasa, Michor and Nowak (ref. \cite{Iwasa}) already used this methodology to study problems related to the ones presented here; however these authors focused into the evolutionary dynamics of viruses to escape antiviral therapy.
The model presented here has the following advantages: the role of each one of its parameters has a straightforward biological interpretation; its versatility easily permits the incorporation of increasing complexity and realism; the process can be studied backwards in time, like in population genetics' coalescent theory. Finally, its simplicity allowed us to obtain an analytical expression for the critical surface separating subcritical from supercritical regimes in the parameter space.
\section{The model}
Our model, a discrete-time non-independent multi-type branching process, assumes the existence of two types of {\em cells/infective particles} (viral particles, parasites, etc.) which are defined according to the host's immune system ability to recognize them; namely, sensitive (type-1) and resistant (type-2) cells. The model involves three parameters, $\delta, \mu, p \in [0,1]$, that are defined as follows:
\vspace{2 mm}
The population of cells proliferates by binary division and the offspring of sensitive cells die, independently, with probability $\delta$ (and consequently survive with probability $1-\delta$). Surviving cells may become resistant (i.e. start producing a new antigen variant) with probability $\mu$.
A newly arisen resistant cell creates a clan (or dynasty) of resistant cells in the following, recursive, way: at a given time the whole progeny of resistant cells divides into resistant cells, which remain as such with probability equal to $p$. In other words, $1-p$ is the probability that the immune system acquires the ability to recognize this particular clan, i.e. the clan bearing this specific variant antigen.
This means that for a given resistant cell appearing at generation $n$ we take a geometric random variable, $N$, of parameter $1-p$, and consider the whole dividing resistant clan until generation $n + N$, where resistant cells become sensitive.
Note that while the probabilities $\delta$ and $\mu$ are intrinsic properties of parasitic cells, $p$ measures the immune system capability to start recognizing a previously unidentified variant antigen (and consequently a clan).
Summarizing: parameter $\delta$ measures the efficiency of immune response against sensitive cells; parameter $\mu$ represents the rate at which new resistant variants appear; and parameter $p$ is related to the delay times spent by the immune systems to recognize a new variant.
\begin{figure}
\includegraphics[width=0.8 \textwidth]{branching_graph_2.pdf}
\caption{Green cells are sensitive. Red and blue cells represent clans of antigen variants not recognized by the immune system.}
\label{fig1}
\end{figure}
Figure \ref{fig1} illustrates a realization of the process: a sensitive (green) cell originates a resistant descendant clan (red) which in turn become sensitive (green) after three generations. At the bottom of the figure, the emergence of a new resistant variant (blue) is represented. Different clans of resistant cells, and sensitive cells, evolve independently.
We remark that this is not an standard multi-type branching process as for example those considered in ref. \cite{Kimmel}, in the sense that resistant cells in a given clan do not evolve independently: instead, their destiny is determined by immune system capacity, which does or does not recognize the whole population of cells carrying a specific variant antigen. The model could also be envisaged as a percolation process on the complete binary tree in presence of a random environment (the clans of resistant cells of random sizes).
\section{Extinction probability}
To compute the extinction/survival probabilities of the process --and thus obtaining the critical surface as a function of the parameters-- we introduce an additional multi-type branching process which, clearly, has the same extinction/survival probabilities as the antigenic variation model introduced before. This independent multi-type branching process with two types of cells is obtained from the antigenic variation model by {\em collapsing} to one generation each clan of resistant cells.
\subsection{Independent multi-type branching process}
Let consider two types of cells that evolve independently. The progeny of each cell is as follows:
\vspace{2 mm}
Type-1 cells give birth to:
\begin{itemize}
\item two type-1 cells with probability $(1-\mu)^2(1-\delta)^2$,
\item two type-2 cells with probability $\mu^2(1-\delta)^2$,
\item one type-1 cell and one type-2 cell with probability $2 \mu (1-\mu)(1-\delta)^2$,
\item one type-1 cell with probability $2 (1-\mu) \delta (1-\delta)$,
\item one type-2 cell with probability $2 \mu \delta (1-\delta)$,
\item no cell with probability $\delta^2$.
\end{itemize}
\vspace{2 mm}
Type-2 cells give birth to:
\begin{itemize}
\item $2^{N}$ type-1 cells with probability $p^{N-1}(1-p)$,
$N=1,2,3, \ldots$
\end{itemize}
\subsection{Extinction probability}
We get the equation for the extinction probability of this independent multi-type branching process --starting with one type-1 cell, one type-2 cell, or eventually any given initial configuration of cells-- following standard procedures that use probability generating functions (pgf) as given, for example, in ref. \cite{Kimmel}, \cite{Jagers}, \cite{Haccou}, \cite{Serra}.
For each $n=0,1,2, \ldots$ we denote $Z_1(n)$ (resp. $Z_2(n)$) the
number of type-1 (resp. type-2) cells present at generation $n$. In the
particular case $n=1$ we put $Z_1 = Z_1(1)$ (resp. $Z_2 = Z_2(1)$) to simplify the notation.
As it is well known, the distribution of $Z_1(n)$ and $Z_2(n)$ as much as the probability of extinction of the process could be computed from the pgf's of $Z_1$ and $Z_2$, which are:
\begin{eqnarray}
f_1(s,t) &=& \mathbb{E} \left[ s^{Z_1} \, t^{Z_2} | Z_1(0)
=1, \, Z_2(0) = 0 \right] \nonumber \\
f_2(s,t) &=& \mathbb{E} \left[ s^{Z_1} \, t^{Z_2} | Z_1(0) =0, \,
Z_2(0) = 1 \right], \nonumber
\end{eqnarray}
$s,t \in [0,1]$.
\vspace{4 mm}
We have:
\begin{eqnarray}
f_1(s,t) &=& \left\{ \delta + (1-\delta)[ \mu t + (1-\mu) s ] \right\}^2 \nonumber \\
f_2(s,t) &=& (1-p) \sum_{k=1}^{\infty} s^{2^k} p^{k-1}. \nonumber
\end{eqnarray}
Let $q_1$ (resp. $q_2$) denote the extinction probability for the
process starting with a single type-1 (resp. type-2) cell. The
following result is adapted from references \cite{Kimmel}, \cite{Jagers}, \cite{Haccou}:
\begin{theorem}
The probability of extinction of the process, $q=(q_1,q_2)$, is the solution of
equation
\begin{equation}
\label{extinction}
(f_1(s,t),f_2(s,t))=(s,t)
\end{equation}
that is closest to the origin in $[0,1]^2$.
\end{theorem}
\vspace{4 mm}
Note that $f_1(1,1)=f_2(1,1)=1$, so that $(1,1)$ is a solution
of (\ref{extinction}). Depending on the values of parameters
$\delta$, $\mu$ and $p$, it can happen: $i)$ $(1,1)$ is the only
solution of (\ref{extinction}) --and hence the extinction probability
is one--; $ii)$ there exists another solution of (\ref{extinction}) in $(0,1)^2$ --and non-extinction probability is positive.
\subsection{Critical surface}
In order to know which is the set of parameter values that make the multi-type branching process (and hence the antigenic variation model) become extinct with probability one, or survives forever with a positive probability, we look for the values $(s_0,t_0) \in [0,1]^2$ that satisfy:
\begin{eqnarray}
\label{extinction_2}
\left\{ \delta + (1-\delta)[ \mu t_0 + (1-\mu) s_0 ] \right\}^2 &=&
s_0 \nonumber \\
(1-p) \sum_{k=1}^{\infty} {s_0}^{2^k} p^{k-1} &=& t_0.
\end{eqnarray}
Note that $s_0 \neq 1$ (resp. $s_0 \neq 0$) if and only if $t_0 \neq 1$ (resp. $t_0 \neq 0$). From these equations we get that the probability of non-extinction is positive if and only if there
exists $s_0 \in (0,1)$ such that:
\begin{equation}
\label{theequation} \left( \delta + (1-\delta)\left[ \mu (1-p)
\sum_{k=1}^{\infty} {s_0}^{2^k} p^{k-1} + (1-\mu) s_0 \right]
\right)^2 = s_0. \nonumber
\end{equation}
Let introduce the function:
\begin{equation}
g(s) = \left( \delta + (1-\delta)\left[ \mu (1-p)
\sum_{k=1}^{\infty} {s}^{2^k} p^{k-1} + (1-\mu) s \right]
\right)^2, \nonumber
\end{equation}
$s \in [0,1]$. Note that, for $\delta, \mu, p \in (0,1)$, $g(.)$ is a strictly increasing and convex function with $g(0)= \delta^2$ and $g(1)=1$. So that, there exists $s_0 \in (0,1)$ satisfying $g(s_0) = s_0$ if and only if $g'(1) > 1$.
It holds:
\begin{equation}
g'(1)= 2(1-\delta)\left[ 2 \mu (1-p)
\sum_{k=0}^{\infty} (2p)^{k} + (1-\mu) \right]. \nonumber
\end{equation}
{\bf Case 1:} $p \geq \frac{1}{2}$. $g'(1) = \infty$, and so there exists $s_0 \in (0, 1)$
such that $g(s_0) = s_0$. The probability of non-extinction is positive; we say the process lies in the {\em supercritical region}.
\vspace{4 mm}
{\bf Case 2:} $p < \frac{1}{2}$. We have
\begin{equation}
g'(1) = \frac{2(1- \delta)}{1 - 2p}[1 + \mu - 2p]. \nonumber
\end{equation}
\vspace{4 mm}
{\bf Critical $\mu$:} For each pair $(\delta, p)$ we introduce the {\em critical value},
$\mu_c(\delta, p)$, defined by:
\begin{equation}
\frac{2(1- \delta)}{1 - 2p}[1 + \mu_c(\delta,p) - 2p] = 1, \nonumber
\end{equation}
$0 \leq \mu_c(\delta,p) \leq 1$.
It is clear that the probability of extinction is one --or strictly smaller than one-- depending on $\mu \leq \mu_c(\delta,p)$ --or $\mu > \mu_c(\delta,p)$.
We summarize the analysis in the following result:
\begin{theorem}
Let:
\begin{equation}
\label{critical}
\mu_c(\delta,p) = \left\{
\begin{array}{ll}
\min \left\{ 1, \max \left\{ 0, \frac{(1-2p)}{2(1-\delta)}(2 \delta -1) \right\} \right\} \;\;\; \mbox{if } p < \frac{1}{2} \\
0 \;\;\; \mbox{if } p \geq \frac{1}{2}
\end{array}
\right.
\nonumber
\end{equation}
It holds, for each $(\delta,p,\mu)$:
\begin{itemize}
\item[i)] The process is {\em subcritical} (i.e. the extinction probability is one)
if $\mu \leq \mu_c(\delta,p)$.
\item[ii)] The process is {\em supercritical} (i.e. the probability of non-extinction
is strictly positive) if $\mu > \mu_c(\delta,p)$.
\end{itemize}
\end{theorem}
\begin{figure}
\includegraphics[width=0.5\textwidth]{critical_surface_2.png}
\caption{Critical surface}
\label{fig2}
\end{figure}
{\bf Remark:} The critical surface does not depend on the initial condition of the process, provided that it starts with a finite number of type-1 and type-2 cells. That depends on the initial condition is the extinction probability in the superctitical region.
\vspace{2 mm}
Figure \ref{fig2} shows the critical surface in the non-trivial region: $\delta > \frac{1}{2}$, $p < \frac{1}{2}$.
Numerically solving equations (\ref{extinction_2}) in the supercritical region we obtain the extinction probability $q_1=s_0$ (resp. $q_2=t_0$) for the process starting with one type-1 cell (resp. one type-2 cell). In Figure \ref{fig3} we show these extinction probabilities computed for $p=0.65$.
\begin{figure}
\includegraphics[width=0.5\textwidth]{probs_extincion_pfijo.png}
\caption{Extinction probabilities for the process starting with one type-1 cell (yellow) and one type-2 cell (red). $p=0.65$.}
\label{fig3}
\end{figure}
\section{Parasitemia Waves}
The model analyzed in the previous sections shows instability in the asymptotic limit $n \rightarrow \infty$, that means: the process becomes extinct (supposedly at an exponential or sub-exponential rate) or it explodes exponentially fast (see for example ref. \cite{Kimmel}). All states except $Z_1=Z_2=0$ are transient and consequently does not exist any kind of {\em oscillatory regime} as has been experimentally observed in mammals infected with parasites or virus exhibiting antigenic variation (see ref. \cite{Ross}, \cite{Barry_Turner_1991}). In this section we introduce some modifications to obtain waves of parasitemia at least with high probability for a reasonable number of generations (let say, $n$ of the order of hundreds or thousands).
\subsection{Random parameters}
To avoid fast extinction or explosion in the number of cells we consider parameters $\mu$ and $\delta$ as random variables (the distribution of which depends on the state of the process at every $n$).
In {\it Trypanosoma brucei} populations, several authors (\cite{Seed_Black}, \cite{Vassella}, \cite{Tyler}, \cite{Savill_Seed}) have proposed a mechanism that induces cell transformation from dividing to non-dividing forms in a density-dependent manner. This is a mechanism of self-regulation that prevents fast population explosions. Experimental results with immunosuppressed mice support this mechanism. Mathematical models involving a set of (deterministic, macroscopic) differential equations have been proposed to describe these concentration changes.
We follow an analogous approach by updating the parameter $\delta$ along the evolution of the (microscopic) process. On the other hand, when the density of cells is small enough, we randomly update the value of $\mu$ to avoid the fast extinction of the process with high probability.
\subsection{Varying $\delta$}
Here we consider parameter $\delta$ as a random variable updated with $n$. Fix $r_{\delta}>0$ and $0 \leq \delta_{min} \leq \delta_{max} \leq 1$. We denote $R(n)$ the number of cells recognized by the immune system (that is, type-1 cells) at generation $n$. Let $X_{\delta}(n)$ be a random variable with Beta distribution of parameters $\alpha = \frac{R(n)}{r_{\delta}}$, $\beta = 1$.
We put:
\[ \delta(n) = \delta_{min} + (\delta_{max} - \delta_{min}) X_{\delta}(n) , \,\, n=1,2,3,\ldots\]
and interpret $\delta_{min}$ as the background contribution of the immune system to the probability of death of sensitive cells; $\delta_{control}(n)=(\delta_{max} - \delta_{min}) X_{\delta}(n)$ takes into account the mechanism of self-control of parasite population at generation $n$: this mechanism regulates the rate of conversion of dividing cells into non-dividing forms.
\vspace{2 mm}
Figure \ref{fig4} shows the effect of $\delta_{control}$: the simulations correspond to realizations of Galton-Watson branching processes where cells at generation $n$ give birth to:
\begin{itemize}
\item two cells with probability $(1-\delta(n))^2$,
\item one cell with probability $2\delta(n)(1-\delta(n))$,
\item no cell with probability $\delta(n)^2$.
\end{itemize}
$\delta_{min} = 0.20$, $\delta_{max} = 0.55$ and $r_{\delta}= 10^3, 10^4, 10^5$. As it can be seen, as $n$ grows up the total number of cells, $Z(n)$, fluctuates around a value not too far away from $r_{\delta}$.
\begin{figure}
\vspace{-2 cm}
\includegraphics[width=0.6 \textwidth, angle=90]{autocontrol_final.pdf}
\vspace{-2 cm}
\caption{Self-control: simulations of three Galton-Watson processes. $\delta_{min}=0.20$; $\delta_{max}=0.55$. $r_{\delta}=10^3$ (green); $r_{\delta}=10^4$ (blue); $r_{\delta}=10^5$ (red).}
\label{fig4}
\end{figure}
\subsection{Varying $\mu$}
Rates of antigenic variation were measured experimentally in {\it Trypanosoma brucei} (see \cite{Barry_Turner_1991}, \cite{Turner_1997} and references therein). Values of parameter $\mu$ range from $10^{-3}$ to $10^{-5}-10^{-7}$ switches/cell/generation, depending on the conditions of infection (fly-transmitted or syringe-passaged). Low values of $\mu$ are comparable to mutation rates.
In our approach, we consider $\mu$ as a random variable updated with $n$:
\[ \mu(n) = \mu_{min} + (\mu_{max} - \mu_{min}) X_{\mu}(n), \,\, n=1,2,3,\ldots\]
where $0 \leq \mu_{min} \leq \mu_{max} \leq 1$ and for fixed $r_{\mu}>0$ $X_{\mu}(n)$ is a random variable with Beta distribution of parameters $\alpha = \frac{r_{\mu}}{R(n)}$, $\beta = 1$. As in the $\delta$ case, parameter $\alpha$ is updated with $n$. The effect of variable $\mu(n)$ is to favor the emergence of new variants when the population of cells is small enough.
\subsection{Simulations of parasitemia waves}
To visualize the emergence of parasitemia waves we implement simulations for the population of cells using the antigenic variation model with fixed $p$ and random variables $\delta(n)$ and $\mu(n)$ updated with $n$ as described before. Figure $5$ shows four typical realizations of the process for a given initial condition. Data are drawn every four generations. Black lines are used for the total population of cells, $Z(n)=R(n)+Q(n)$, and red lines for the population of resistant type-2 cells, $Q(n)$.
\vspace{2 mm}
Values of parameters in simulations:
\begin{itemize}
\item $p=0.65$
\item $\delta_{min}=0.60$, $\delta_{max}=0.95$
\item $\mu_{min}=0.004$, $\mu_{max}=0.01$
\item $r_{\delta}=10^4$, $r_{\mu}=10^2$
\item Initial conditions: $R(0)=5 \times 10^3$, $Q(0)=1$, $\delta_{initial}=0.60$, $\mu_{initial}=0.004$
\end{itemize}
We note that the vast majority of new variants have almost no effect at all. In figure (a) the process ended before $100$ generations, after a few parasitemia peaks at the beginning. Figure (b) shows several peaks of different heights. Figures (c) and (d) show extended regions with low concentration of parasites, after which the process emerges again. Because $p > \frac{1}{2}$ the process lies in the superctitical region for all times (whenever $Z(n)>0$). For each one of the independent processes starting with one type-1 cell or one type-2 cell at generation $n$, extinction probabilities computed from equations (\ref{extinction_2}) take values in the intervals: $s_0 \in (0.997, 1.000)$, $t_0 \in (0.935, 0.999)$; from those values it is possible to compute the total extinction probability for the process as a function of $n$.
The examples show that our model is able to reproduce the essential aspects of the dynamics of antigenic variation. New elements can be easily incorporated to the model thanks to its simplicity.
\begin{center}
\begin{table}[h]
\begin{tabular}{cc}
\includegraphics[height=40mm]{simulation1_10_13.pdf} &
\includegraphics[height=40mm]{simulation2_10_13.pdf} \\
FIG. 5: (a) & FIG. 5: (b) \\
\includegraphics[height=40mm]{simulation3_10_13.pdf} &
\includegraphics[height=40mm]{simulation4_10_13.pdf} \\
FIG. 5: (c) & FIG. 5: (d)
\end{tabular}
\end{table}
\end{center}
\section{Conclusions}
In this paper we introduce a new approach to model the dynamics of antigenic variation using a multi-type branching process. The model considers the following aspects: efficiency of immune response against sensitive cells; rate at which new resistant variants appear; and a random modelling for the delay times spent by the immune system to recognize a new variant.
We characterize analytically the subcritical and supercritical regions on the three-dimensional space of parameters. Simplicity and versatility are to be highlighted in our approach. By appropriately updating the set of parameters we introduce a density-dependent mechanism that accounts for qualitative behavior observed on in vivo experiments such as the characteristics peaks of parasitemia along an infection.
The model can be extended to include situations of higher complexity, as for instance: variable growth rates of the different antigenic variants (by adding random death parameters inside each clan of resistant cells), memory in the immune responses and delay times to recognize new antigen variants (by modifying the value of $p$ along the infection).
In our opinion multi-type branching processes have a great potential to model evolution of host-parasite interactions like the particular case of antigenic variation. The approach presented here opens a variety of possibilities for future work.
\vspace{4 mm}
\acknowledgements
This work has been partially supported by Agencia Nacional de Investigaci\'on e Innovaci\'on (ANII), Uruguay. We thank Enrique Lessa, Sebasti\'an Castro and Guillermo Lamolle for helpful discussions.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 304 |
\section{Introduction}
We consider the stationary nonlinear Schr\"{o}dinger equation
\begin{equation}
\label{problemNLSE}
\tag{\protect{$\mathcal{P}_\varepsilon$}}
\left\{
\begin{aligned}
- \varepsilon^2 \Delta u + Vu &= u^p, & & \text{in \(\mathbb{R}^N\)},\\
u(x) & \to 0 &&\text{as \(\abs{x} \to \infty\)},
\end{aligned}
\right.
\end{equation}
where \(\varepsilon > 0\) is a real parameter, \(N\geq 3\), \(\frac{N}{N-2} < p <
\frac{N+2}{N-2}\) and
\(V \in C(\mathbb{R}^N,\mathbb{R}^+)\).
In the semi-classical limit when \(\varepsilon\) is small, one expects quantum physics to be approximated by classical physics and thus the stationary solutions should concentrate around critical points of the potential.
A first way to construct such a family of solutions around a \emph{nondegenerate} critical point of the potential is the Lyapunov--Schmidt reduction
\cites{AmbrosettiBadialeCingolani1996,AmbrosettiBadialeCingolani1997,AmbrosettiMalchiodi2006,FloerWeinstein1986,Oh1988,Oh1988Errata,Oh1990}.
Solutions of \eqref{problemNLSE} can also be found by variational methods. The most natural method yields solutions concentrating around a \emph{global minimum} of the potential \(V\) \citelist{\cite{Rabinowitz1992}\cite{Wang1993}}. More elaborate critical constructions allow to construct solutions concentrating around \emph{strict local minima} \cites{delPinoFelmer1996,delPinoFelmer1998} and around \emph{strict local maxima} \cites{delPinoFelmer1997,delPinoFelmer2002}.
All the works mentioned above are concerned with \emph{subcritical frequency} case \(\inf_{\mathbb{R}^N} V > 0\).
In the \emph{critical frequency case} \(\inf_{\mathbb{R}^N} V = 0\), solutions concentrating around nondegenerate critical points \cite{AmbrosettiMalchiodiRuiz2006} and around local minima have been obtained \cites{AmbrosettiFelliMalchiodi2005,BonheureVanSchaftingen2008} provided that the potential \(V\) does not decay too fast at infinity.
In the case of local minima, the variational method has been adapted to construct solutions concentrating around a local minimum with a fast decay potential \(V\) --- including a compactly supported potential \citelist{\cite{MorozVanSchaftingen2009}\cite{MorozVanSchaftingen2010}\cite{YinZhang2009}\cite{BaDengPeng2010}}.
The goal of this work is to establish by a variational method the existence of solutions concentrating around local maxima for fast-decaying potentials. Since any potential that decays at infinity has a global maximum, this shows the existence of solutions for a quite general class of potentials. We also think that this problem is a good test of the robustness and flexibility of the variational methods for solutions concentrating around local maxima and of the penalization method for fast decay potentials.
\medbreak
Our main result is the following
\begin{theorem}
\label{theoremMainSmooth}
Let \(N \geq 1\), \(p > 1\) such that \( \frac{1}{p} > \frac{N - 2}{N + 2}\) and \(V \in C^N(\mathbb{R}^{N}, \mathbb{R}^+)\), \(V \not\equiv 0\) be a nonnegative potential.
If
\(
\lim_{\abs{x}\to\infty} V(x) = 0
\)
and either \(\frac{1}{p} < \frac{N - 2}{N}\) or \(\liminf_{\abs{x} \to \infty} V (x) \abs{x}^2 > 0 \), then, for \(\varepsilon>0\) small enough, the problem~\eqref{problemNLSE}
has a family of positive solution that concentrates around a global maximum of \(V\).
\end{theorem}
A typical new potential \(V\) for which this result applies is given by \(V(x) = \frac{1}{1+\abs{x}^4}\) for \(x \in \mathbb{R}^n\). The assumption on \(p\) is optimal, since when \(\frac{1}{p} \ge \frac{N-2}{N}\) and \(V\) is compactly supported, \eqref{problemNLSE} does not have any solution.
Indeed, such a solution would be positive and satisfy \(-\Delta u = u^p\) in \(\mathbb{R}^N \setminus \supp V\) and that would imply \(u = 0\) on \(\mathbb{R}^N \setminus \supp V\) \cite{BV}.
Theorem~\ref{theoremMainSmooth} follows from the following result:
\begin{theorem}\label{theoremMainLambda}
Let \(N \geq 1\), \(p > 1\) such that \( \frac{1}{p} > \frac{N - 2}{N + 2}\) and \(V \in C(\mathbb{R}^N,\mathbb{R}^+)\).
Assume that there exists a smooth bounded open set \(\Lambda \subset \mathbb{R}^N\) such that
\begin{equation*}
\sup_{\Lambda} V > \inf_{\Lambda} V = \sup_{\partial\Lambda} V
\end{equation*}
and
\begin{equation*}
\sup_{\Lambda} V^{\frac{p+1}{p-1}-\frac{N}{2}}
< 2 \inf_{\Lambda} V^{\frac{p+1}{p-1}-\frac{N}{2}}\:.
\end{equation*}
If either \(\frac{1}{p} < \frac{N - 2}{N}\) or \(\liminf_{\abs{x} \to \infty} V (x) \abs{x}^2 > 0 \), then for \(\varepsilon > 0\) small enough, problem~\eqref{problemNLSE} possesses a positive weak
solution \(u_{\varepsilon} \in H^1_\mathrm{loc}(\mathbb{R}^N)\) such that
\(u_{\varepsilon}\) achieves its maximum at \(x_{\varepsilon} \in \Lambda\),
\begin{equation*}
\liminf_{\varepsilon\to 0} u_{\varepsilon}(x_{\varepsilon}) > 0
\end{equation*}
and
\begin{equation*}
\liminf_{\varepsilon\to 0} \dist(x_\varepsilon, \mathbb{R}^N \setminus \Lambda) > 0\:.
\end{equation*}
\end{theorem}
The first assumption on \(V\) implies that
\[
\sup_{\partial\Lambda} V = \inf_{\partial\Lambda} V\:,
\]
that is, \(\partial \Lambda\) is a level line of \(V\).
Theorem~\ref{theoremMainSmooth} follows from Theorem~\ref{theoremMainLambda} by taking \(\Lambda_\delta = \{x \in \mathbb{R}^N \; \mid\; V(x) > \sup V - \delta \}\) for \(\delta > 0\). By Sard's lemma, the set \(\Lambda_\delta\) is smooth for almost every \(\delta > 0\). One applies then Theorem~\ref{theoremMainLambda} and a diagonal argument.
Our method of proof is based on the penalization method \citelist{\cite{delPinoFelmer1996}\cite{delPinoFelmer1997}} adapted to decaying potentials \citelist{\cite{BonheureVanSchaftingen2008}\cite{MorozVanSchaftingen2010}}.
However the decay of the potentials requires us to take some extra care at several steps, especially when lower bounds on the energy of solutions are needed.
\medbreak
This paper is organized as follows. We first introduce a penalized problem (section~\ref{sectionPenalization}) and recall some properties of the associated limiting problem (section~\ref{sectionLimiting}). We then study the asymptotic behaviour of families of critical points (section~\ref{sectAsymptoticsCritical}) and minimizers (section~\ref{sectAsymptoticsMinimizers}) of the energy functional associated to \eqref{problemNLSE}. This allows us to define a minimax level and prove the existence of a solution to the penalized problem in section~\ref{sectionMinimax}. Finally in section~\ref{sectionOriginalProblem} we use the asymptotics and some comparison argument to show that when \(\varepsilon\) is small, our solutions of the penalized problem solve the original problem. Whereas the proof is written for \(N \ge 3\), we highlight in section~\ref{sectionSlow} how the proof can be adapted to the case \(N \le 2\).
\section{The penalized problem}
\label{sectionPenalization}
Following M.\thinspace del Pino and P.\thinspace Felmer \cite{delPinoFelmer1997}, we introduce a penalized problem.
D.\thinspace Bonheure and J.\thinspace Van~Schaftingen \citelist{\cite{BonheureVanSchaftingen2008}\cite{BonheureVanSchaftingen2006}} have introduced a penalized problem for decaying potentials. The penalization for fast decay potentials is due to V.\thinspace Moroz and J.\thinspace Van~Schaftingen \citelist{\cite{MorozVanSchaftingen2009}\cite{MorozVanSchaftingen2010}}. It was used by D.\thinspace Bonheure together with the authors to study solutions concentrating around spheres \cite{DCBonheureVanSchaftingen2008}. Another penalized problem was defined by Yin Huicheng and Zhang Pingzheng \cite{YinZhang2009} (see also Fei Mingwen and Yin Huicheng \cite{FY} and Ba Na, Deng Yinbin and Peng Shuangjie \cite{BaDengPeng2010}).
\subsection{The penalization potential}
Recall that \(\Lambda\) is a bounded domain.
Let \(x_0 \in \Lambda\) and \(\rho > 0\) be such that
\(\overline{B(x_0,\rho)} \subset \Lambda\), and let
\(\chi_{\Lambda}\) denote the characteristic function of the set \(\Lambda\). For \(N \ge 3\), the penalization
potential \(H : \mathbb{R}^N \to \mathbb{R}\) is defined by
\begin{equation*}
H(x) := \bigl(1-\chi_{\Lambda}(x)\bigr) \frac{(N-2)^2}{4\abs{x-x_0}^2} \biggl(\frac{\log \frac{\rho}{\rho_0}}{\log \frac{\abs{x-x_0}}{\rho_0}
}\biggr)^{1+\beta}
\end{equation*}
for some fixed \(\beta > 0 \) and \(\rho_0 \in (0, \rho)\).
Let us recall that the operator \(-\Delta - H\) satisfies a positivity principle \cite{MorozVanSchaftingen2010}*{Lemma 3.1}.
\begin{lemma}
\label{lemmaPositivity}
For every \(u \in C^\infty_c (\mathbb{R}^N)\),
\begin{equation*}
\int_{\mathbb{R}^N} \bigl( \abs{\nabla u}^2 - H \abs{u}^2 \bigr) \geq 0\:.
\end{equation*}
\end{lemma}
\begin{proof}
Since \(N \ge 3\), this follows from the classical Hardy inequality since for every \(x \in \mathbb{R}^N \setminus B(x_0, \rho)\),
\[
H(x) \le \frac{(N-2)^2}{4\abs{x-x_0}^2}\:.\qedhere
\]
\end{proof}
\subsection{The penalized nonlinearity}
Fix \(\mu \in (0,1)\). The penalized nonlinearity \(g_{\varepsilon}: \mathbb{R}^N \times \mathbb{R} \rightarrow \mathbb{R}\) is defined for \(x \in \mathbb{R}^N\) and \(s \in \mathbb{R}\) by
\begin{equation*}
g_{\varepsilon}(x,s) := \chi_{\Lambda}(x) s_+{}^p + \bigl( 1-\chi_{\Lambda}(x) \bigr)
\min\bigl(\mu \bigl(\varepsilon^2 H(x) + V(x)\bigr), \abs{s}^{p-1} \bigr) s_+\:.
\end{equation*}
Also set \(G_{\varepsilon}(x,s) := \int_0^s g_{\varepsilon}(x,\sigma)\, d\sigma\). The function \(g_{\varepsilon}\) is a Carath\'eodory function with the following properties :
\begin{itemize}
\item[(\(g_1\))] \(g_{\varepsilon}(x,s) = o(s),\) as \(s \rightarrow 0\), uniformly on compact subsets of
\(\mathbb{R}^N\).
\item[(\(g_2\))] for every \(x \in \mathbb{R}^N\) and \(s \in \mathbb{R}\),
\[
g_{\varepsilon}(x,s) \le (s)_+^p,
\]
if moreover \(x \in \mathbb{R}^N \setminus \Lambda\), then
\[
g_{\varepsilon}(x,s) \leq \mu\bigl(\varepsilon^2 H(x) + V(x) \bigr)s_+
\]
\item[(\(g_3\))] for every \(s \in \mathbb{R}\), if \(x \in \Lambda\),
\[
(p+1) G_{\varepsilon}(x,s) \leq g_{\varepsilon}(x,s)s\:,
\]
and if \(x \not \in \Lambda\),
\[
2 G_{\varepsilon}(x,s) \leq g_{\varepsilon}(x,s)s \:,
\]
\item[(\(g_4\))] the function
\begin{equation*}
t \in (0, \infty) \mapsto \frac{g_{\varepsilon}(x, ts)s}{t}
\end{equation*}
is nondecreasing for all \(x \in \mathbb{R}^N\) and \(s \in \mathbb{R}\).
\end{itemize}
\subsection{The penalized functional}
The Hilbert space naturally associated to the linear part of our equation is the
weighted Sobolev space \(H^1_{V}(\mathbb{R}^N)\),
which is the closure of \(C^\infty_c(\mathbb{R}^n)\) under any of the equivalent norms
\begin{equation*}
\norm{u}_{\varepsilon}^2 := \int_{\mathbb{R}^N} \bigl( \varepsilon^2 \abs{\nabla u}^2 + V \abs{u}^2 \bigr)
\end{equation*}
defined for \(\varepsilon > 0\).
We look for a solution \(u \in H^1_V (\mathbb{R}^N)\) of the penalized equation
\begin{equation}
\label{problemPNLSE}
\tag{\protect{$\mathcal{Q}_\varepsilon$}}
- \varepsilon^2 \Delta u(x) + V(x)\,u(x) = g_{\varepsilon}\bigl(x,u(x)\bigr) \qquad \text{for \(x \in \mathbb{R}^N\)}.
\end{equation}
The associated functional is
\begin{equation*}
\mathcal{J}_{\varepsilon} : H^1_V \rightarrow \mathbb{R} : \mathcal{J}_{\varepsilon}(u) := \frac{1}{2} \int_{\mathbb{R}^N} \bigl( \varepsilon^2
\abs{\nabla u}^2 + V\abs{u}^2 \bigr) - \int_{\mathbb{R}^N} G_{\varepsilon}\bigl(x,u(x)\bigr)\: dx\;.
\end{equation*}
It is standard that \(\mathcal{J}_{\varepsilon}\) is well-defined and continuously differentiable and that its critical points are weak solutions of the penalized equation \eqref{problemPNLSE}.
\subsection{The Nehari manifold}
The Nehari manifold associated to the functional
\(\mathcal{J}_{\varepsilon}\) is defined by
\begin{equation*}
\mathcal{N}_{\varepsilon} := \bigl\{ u \in H^1_V(\mathbb{R}^N) \setminus \{0\}\; \mid\; \dualprod{ \mathcal{J}_{\varepsilon}'(u)}{u}=0
\bigr\}\;.
\end{equation*}
It is well-known that \(u \in H^1_V(\mathbb{R}^N)\setminus \{0\}\) is a critical point of \(\mathcal{J}_{\varepsilon}\) if and only if \(u \in \mathcal{N}_{\varepsilon}\) and \(u\) is a critical point of \(\mathcal{J}_{\varepsilon}\) restricted to \(\mathcal{N}_{\varepsilon}\).
We point out that \(\mathcal{N}_{\varepsilon}\) is bounded away from \(0\). We first have an integral estimate.
\begin{lemma}
\label{lemENehariEstimate1}
Let \(\varepsilon > 0\) and
\(
u \in \mathcal{N}_\varepsilon.
\)
Then
\[
\int_{\Lambda} (u)_+^{p+1}
\ge (1-\mu) \int_{\mathbb{R}^N} \varepsilon^2 \abs{\nabla u}^2 +V \abs{u}^2.
\]
\end{lemma}
\begin{proof}
By \((g_2)\), one has
\[
\begin{split}
\int_{\mathbb{R}^N} \bigl(\varepsilon^2 \abs{\nabla u}^2 +V \abs{u}^2\bigr)
&=\int_{\mathbb{R}^N} g_\varepsilon(x, u(x)) u(x)\:dx\:\\
&\le \int_{\Lambda} \abs{u}^{p+1} +\mu \int_{\mathbb{R}^N \setminus \Lambda} (V+\varepsilon^2 H)\abs{u}^2 .
\end{split}
\]
We deduce from Lemma~\ref{lemmaPositivity} that
\[
(1-\mu) \int_{\mathbb{R}^N} \varepsilon^2 \abs{\nabla u}^2 +V \abs{u}^2
\le \int_{\Lambda} \abs{u}^{p+1}.\qedhere
\]
\end{proof}
\begin{lemma}
\label{lemENehariEstimate2}
Let \(\varepsilon > 0\) and
\(
u \in \mathcal{N}_\varepsilon.
\)
Then
\[
\int_{\Lambda} \varepsilon^2 \abs{\nabla u}^2 +V \abs{u}^2
\ge c\varepsilon^{N}
\]
where \(c > 0\) is independent of \(\varepsilon\) and \(u\).
\end{lemma}
\begin{proof}
Since \(\inf_\Lambda V > 0\), the Sobolev and H\"older inequalities imply that
\[
\begin{split}
\int_{\Lambda} \abs{u}^{p+1} &\le C \Bigl( \int_{\mathbb{R}^N} \abs{\nabla u}^2 \Bigr)^{\frac{p-1}{4} N }
\Bigl( \int_{\Lambda} \abs{u}^2\Bigr)^{\frac{p+1}{2}-\frac{p-1}{4}N}\\
&\le \frac{C}{\varepsilon^{\frac{p-1}{2} N}} \Bigl( \int_{\Lambda} \varepsilon^2 \abs{\nabla u}^2 + V \abs{u}^2 \Bigr)^\frac{p+1}{2}.
\end{split}
\]
The conclusion follows from Lemma~\ref{lemENehariEstimate1}.
\end{proof}
We also have a uniform lower estimate on the maximum.
\begin{lemma}\label{lem:1}
Let \(\varepsilon > 0\) and
\(u \in \mathcal{N}_\varepsilon\). Then
\begin{equation*}
\sup_{\Lambda}\frac{u_+^{p-1}}{V} \ge 1\:.
\end{equation*}
\end{lemma}
This was proved for solutions of \eqref{problemPNLSE} by V.\thinspace Moroz and J.\thinspace Van Schaftingen \cite{MorozVanSchaftingen2010}*{Lemma 4.2} (see also~\cite{BonheureVanSchaftingen2008}*{Lemma 17}).
\begin{proof}
One has by \((g_2)\),
\[
\begin{split}
\int_{\mathbb{R}^N} \varepsilon^2 \abs{\nabla u}^2 +V \abs{u}^2 &\le \int_{\Lambda} (u)_+^{p+1}
+\mu \int_{\mathbb{R}^N \setminus \Lambda} (V + \varepsilon^2 H)\abs{u}^2 \\
&\le \sup_{\Lambda}\frac{u_+^{p-1}}{V} \int_{\Lambda} V \abs{u}^{2}
+\mu \int_{\mathbb{R}^N \setminus \Lambda} (V + \varepsilon^2 H)\abs{u}^2\:,
\end{split}
\]
and thus by Lemma~\ref{lemmaPositivity},
\[
\sup_{\Lambda}\frac{u_+^{p-1}}{V} \int_{\Lambda} V \abs{u}^{2} \ge \int_{\Lambda} V \abs{u}^{2}\:.
\]
By Lemma~\ref{lemENehariEstimate2}, \(\int_{\Lambda} V \abs{u}^{2} > 0\), and the conclusion follows.
\end{proof}
We also note the following coercivity estimate.
\begin{lemma}\label{lem:bounded}
For every \(\varepsilon > 0\) and \(u \in \mathcal{N}_{\varepsilon}\),
\[
\Bigl(\frac{1}{2}-\frac{1}{p+1}\Bigr)(1-\mu) \int_{\mathbb{R}^N} \bigl( \varepsilon^2 \abs{\nabla u}^2 + V
\abs{u}^2\bigr) \leq \mathcal{J}_{\varepsilon}(u)\:.
\]
\end{lemma}
\begin{proof}
Since \(u \in \mathcal{N}_\varepsilon\), one has
\begin{multline*}
\mathcal{J}_{\varepsilon}(u)=\Bigl(\frac{1}{2}-\frac{1}{p+1}\Bigr) \int_{\mathbb{R}^N} \bigl( \varepsilon^2 \abs{\nabla u}^2 + V \abs{u}^2 \bigr)\\
+ \frac{1}{p+1} \int_{\mathbb{R}^N} g_\varepsilon\bigl(x, u(x)\bigr) u(x) - (p+1) G_\varepsilon\bigl(x, u(x)\bigr)\,dx.
\end{multline*}
In view of \((g_3)\) and \((g_2)\),
\begin{multline*}
\frac{1}{p+1} \int_{\mathbb{R}^N} g_\varepsilon\bigl(x, u(x)\bigr) u(x) - {(p+1)} G_\varepsilon(x, u(x))\,dx\\
\geq -\frac{p-1}{p+1} \int_{\mathbb{R}^N \setminus \Lambda} G_\varepsilon\bigl(x, u(x)\bigr)\,dx \\
\geq - \Bigl(\frac{1}{2}-\frac{1}{p+1}\Bigr) \mu \int_{\mathbb{R}^N \setminus \Lambda} ( \varepsilon^2 H + V )\abs{u}^2.
\end{multline*}
Thanks to Lemma~\ref{lemmaPositivity}, we reach the conclusion.
\end{proof}
\subsection{The Palais-Smale condition}
For every \(\varepsilon > 0\), the functional \(\mathcal{J}_{\varepsilon}\) satisfies the Palais-Smale compactness condition:
\begin{lemma}\label{lem:PalaisSmale}
For every \(\varepsilon > 0\), if \( (u_n)_{n \in \mathbb{N}} \) is a sequence such that \((\mathcal{J}_{\varepsilon}(u_n))_{n \in \mathbb{N}}\) converges and \((\mathcal{J}_{\varepsilon}'(u_n))_{n \in \mathbb{N}} \) converges to \(0\) in \( (H^1_V(\mathbb{R}^N))' \), then, up to a subsequence, \( (u_n)_{n \in \mathbb{N}} \) converges in \(H^1_V(\mathbb{R}^N)\).
\end{lemma}
The proof of Lemma~\ref{lem:PalaisSmale} is a combination of the arguments for the penalization without \(H\) \cite{BonheureVanSchaftingen2008}*{Lemma 6} and without \(V\) \cite{MorozVanSchaftingen2010}*{Lemma 3.5} whose main lines originate in the proof for nondecaying potentials \cite{delPinoFelmer1996}*{Lemma 1.1}. It was already proved with the present penalization for the functional restricted to a subspace of symmetric functions \cite{DCBonheureVanSchaftingen2008}.
\subsection{Minimizers on the Nehari manifold}
\begin{proposition}\label{prop:Minimizer}
For every \(\varepsilon > 0\), there exists \(u \in \mathcal{N}_\varepsilon\) such that
\[
\mathcal{J}_{\varepsilon}(u) = \inf_{\mathcal{N}_\varepsilon}\mathcal{J}_{\varepsilon}\:.
\]
\end{proposition}
Proposition~\ref{prop:Minimizer} was proved for the penalization for nondecaying potentials \cite{delPinoFelmer1996}*{Lemma 2.1}, the penalization without \(H\) \cite{BonheureVanSchaftingen2008}*{Proposition 9} the penalization without \(V\) \cite{BonheureVanSchaftingen2008}*{Proposition 3.7} and the present penalization under symmetry constraints \cite{DCBonheureVanSchaftingen2008}.
\begin{proof}[Proof of Proposition~\ref{prop:Minimizer}]
The proof is standard: by \( (g_4)\) one has the equality \cite{Rabinowitz1992}*{Proposition 3.11}
\[
\inf_{\mathcal{N}_\varepsilon}\mathcal{J}_{\varepsilon}= \inf_{\substack{u \in H^1_V(\mathbb{R}^N) \\ u_+ \vert_\Lambda \ne 0}} \sup_{t > 0} \mathcal{J}_{\varepsilon}(tu)= \inf_{\substack{\gamma \in C([0, 1], H^1_V(\mathbb{R}^N)) \\ \gamma(0)=0 \\ \mathcal{J}_{\varepsilon}(\gamma(1)) < 0}} \sup_{t \in [0, 1]} \mathcal{J}_{\varepsilon}(\gamma(t))\:;
\]
by Lemmas \ref{lemENehariEstimate2} and \ref{lem:bounded}, \(\mathcal{J}_{\varepsilon}\) is bounded away from \(0\) on \(\mathcal{N}_\varepsilon\). Since \(\mathcal{J}_{\varepsilon} \) satisfies the Palais-Smale compactness condition by Lemma~\ref{lem:PalaisSmale}, the existence of \(u\) follows.
\end{proof}
\section{Limiting problems}
\label{sectionLimiting}
\subsection{The limit problem}
For \(\nu > 0\) let \(U_{\nu}\) be the unique positive solution of the problem
\begin{equation}\label{problemLimit}
\left\{
\begin{aligned}
-\Delta u + \nu u &= u^p & &\text{in \(\mathbb{R}^{N}\)}, \\
u &> 0, \\
u(0) &= \max_{\mathbb{R}^N} u.
\end{aligned}\right.
\end{equation}
The function \(U_{\nu}\) is radial around the origin \cite{Kwong}. The functional associated to \eqref{problemLimit}
is \(\mathcal{I}_{\nu} : H^1(\mathbb{R}^N) \to \mathbb{R}\) defined for \(u \in H^1(\mathbb{R}^N)\) by
\begin{equation*}
\mathcal{I}_{\nu}(u) := \frac{1}{2} \int_{\mathbb{R}^N} \bigl( \abs{\nabla u}^2 + \nu \abs{u}^2 \bigr) - \frac{1}{p+1} \int_{\mathbb{R}^N} u_+^{p+1}.
\end{equation*}
One has the variational characterization
\[
b_{\nu} := \inf_{\mathcal{M}_\nu} \mathcal{I}_\nu = \mathcal{I}_{\nu}(U_{\nu})
\]
where
\[
\mathcal{M}_\nu = \Bigl\lbrace u \in H^1(\mathbb{R}^N)\setminus \{0\} \; \mid\; \int_{\mathbb{R}^N} \bigl(\abs{\nabla u}^2 + \nu \abs{u}^2 \bigr) = \int_{\mathbb{R}^N} u_+^{p+1} \Bigr\rbrace\:.
\]
We also set
\begin{equation}\label{estim:by}
\mathcal{C}(y) := b_{V(y)} = \frac{S_{p+1}^r}{r} V(y)^{\frac{p+1}{p-1}-\frac{N}{2}},
\end{equation}
where \(\frac{1}{r}=\frac{1}{2}-\frac{1}{p+1}\) and
\begin{equation*}
S_{p+1}^2 := \inf \Bigl\lbrace \int_{\mathbb{R}^N} \bigl(\abs{\nabla u}^2 + \abs{u}^2\bigr) \; \mid\; u \in
H^1(\mathbb{R}^N)\ \text{and}\ \int_{\mathbb{R}^N} u_+^{p+1} = 1
\Bigr\rbrace\:.
\end{equation*}
We also recall the following classical result
\begin{lemma}
\label{lemmaConvergenceMinimizingLimiting}
Let \(\nu > 0\) and \( (v_n)_{n\in \mathbb{N}}\) be a sequence in \(\mathcal{M}_\nu \subset H^1(\mathbb{R}^N)\). If
\[\lim_{n \to \infty} \mathcal{I}_\nu(v_n) = b_{\nu}, \]
then there exists a sequence of points \((y_n)_{n\in \mathbb{N}}\) in \(\mathbb{R}^N\) such that \(v_n(\cdot -y_n) \to U_\nu\) in \(H^1(\mathbb{R}^N)\).
\end{lemma}
\begin{proof}
Let \(\mathcal{F}_\nu : H^1(\mathbb{R}^N) \to \mathbb{R}\) be defined for \(v \in H^1(\mathbb{R}^N)\) by
\[
\mathcal{F}_\nu (v) = \int_{\mathbb{R}^N} \abs{\nabla v}^2 + \nu \abs{v}^2 - \int_{\mathbb{R}^N} v_+^{p+1}
\]
By a standard application of the Ekeland variational principle on the manifold \(\mathcal{M}_\nu\) (see for
example \cite{MW}*{Theorem 4.1}), there exist sequences \((\tilde{v}_n)_{n \in \mathbb{N}} \subset \mathcal{M}_\nu\) and \((\lambda_n)_{n \in \mathbb{N}} \subset \mathbb{R}\) such that \(\mathcal{I}_{\nu}(\tilde{v}_n)\to b_{\nu}\),
\(\mathcal{I}_{\nu}'(\tilde{v}_n)+\lambda_n \mathcal{F}_\nu'(\tilde{v}_n)\to 0\) in \(H^{-1}(\mathbb{R}^N)\) and \(v_n - \tilde{v}_n \to 0\) in \(H^1(\mathbb{R}^N)\) as \(n \to \infty\).
The sequence \((\tilde{v}_n)_{n \in \mathbb{N}}\) is a Palais-Smale sequence for the unconstrained functional \(\mathcal{I}_{\nu}\), that is \(\mathcal{I}_{\nu}'(\tilde{v}_n) \to 0\).
Indeed one has
\begin{align*}
\lambda_n (p-1)\int_{\mathbb{R}^N} \abs{\nabla v_n}^2 + \abs{v_n}^2 &= -\lambda_n \dualprod{\mathcal{F}_\nu'(\tilde{v}_n)}{\Tilde{v}_n} \\
&= \dualprod{\mathcal{I}_{\nu}'(\tilde{v}_n)}{\tilde{v}_n}+ o(\norm{\Tilde{v}_n}_{H^1})=o(\norm{\Tilde{v}_n}_{H^1})\:,
\end{align*}
as \(n \to \infty\).
Since there exists a constant \(c>0\) such that \(\norm{v}_{H^1(\mathbb{R}^N)} \geq c\) for every \(v \in \mathcal{M}_\nu\) we deduce that
\(\lim_{n\to \infty} \lambda_n = 0\).
We compute that
\begin{equation*}
2 \mathcal{I}_{\nu}(\tilde{v}_n) - \dualprod{\mathcal{I}_{\nu}'(\tilde{v}_n)}{\tilde{v}_n} = \Bigl( 1 - \frac{2}{p+1} \Bigr) \int_{\mathbb{R}^N} (\tilde{v}_n)_+^{p+1}\: dx \to 2 b_{\nu}\:.
\end{equation*}
Hence,
\begin{equation*}
\liminf_{n \to \infty} \int_{\mathbb{R}^N} (\tilde{v}_n)_+^{p+1} > 0\:.
\end{equation*}
Since \((\tilde{v}_n)_{n \in \mathbb{N}}\) is bounded in \(H^1(\mathbb{R}^N)\), we deduce from \cite{Lions1984}*{Part 2, Lemma I.1} (see also \cite{Willem1996}*{Lemma 1.21}) that
\begin{equation*}
\int_{\mathbb{R}^N} (\tilde{v}_n)_+^{p + 1} \le C \Bigl(\int_{\mathbb{R}^N} \abs{\nabla \tilde{v}_n}^2 + \abs{\tilde{v}_n}^2 \Bigr)\sup_{z \in \mathbb{R}^N} \Bigl(\int_{B(z,1)} (\tilde{v}_n)^{p + 1} \Bigr)^\frac{p - 1}{p + 1}\:.
\end{equation*}
Consequently, there exists a sequence \((y_n)_{n \in \mathbb{N}} \subset \mathbb{R}^N\) such that, if we set \(\Bar{v}_n := \tilde{v}_n(\cdot-y_n)\), we have
\begin{equation}\label{ineqConvergenceMinimizingLimitingBalls}
\liminf_{n \to \infty} \int_{B(0,1)} (\Bar{v}_n)_+^{p + 1} > 0.
\end{equation}
Since \((\Bar{v}_n)_{n \in \mathbb{N}}\) is bounded in \(H^1(\mathbb{R}^N)\) and \(\frac{1}{p} > \frac{N - 2}{N}\), we can assume that \(\Bar{v}_n \rightharpoonup \Bar{v}\) in \(H^1(\mathbb{R}^N)\),
\(\Bar{v}_n \to
\Bar{v}\) in \(L^{p + 1}_{\text{loc}}(\mathbb{R}^N)\) and \(\Bar{v}_n \to \Bar{v}\) almost everywhere.
By \eqref{ineqConvergenceMinimizingLimitingBalls}, \(\Bar{v} \not\equiv 0\). For all \(v \in H^1(\mathbb{R}^N)\), we have \(\dualprod{\mathcal{I}_{\nu}'(\Bar{v}_n)}{v} \to 0\)
because \(\Bar{v}_n\) is a Palais-Smale sequence, and \(\dualprod{\mathcal{I}_{\nu}'(\Bar{v}_n)}{v} \to \dualprod{\mathcal{I}_{\nu}'(\Bar{v})}{v}\) because \(\Bar{v}_n \rightharpoonup \Bar{v}\). We
conclude that \(\dualprod{\mathcal{I}_{\nu}'(\Bar{v})}{v} = 0\) and so \(\Bar{v}\) is a solution of \eqref{problemLimit}. We compute that
\begin{equation*}
\frac{p-1}{p+1} \int_{\mathbb{R}^N} (\Bar{v}_n)_+^{p+1} = 2 \mathcal{I}_{\nu}(\Bar{v}_n) - \dualprod{\mathcal{I}_{\nu}'(\Bar{v}_n)}{\Bar{v}_n} \to 2 b_{\nu}
\end{equation*}
and
\begin{equation*}
\frac{p-1}{p+1} \int_{\mathbb{R}^N} (\Bar{v})_+^{p+1} = 2 \mathcal{I}_{\nu}(\Bar{v}) = 2 b_{\nu}.
\end{equation*}
Therefore \(\norm{(\Bar{v}_n)_+}_{L^{p+1}(\mathbb{R}^N)} \to \norm{(\Bar{v})_+}_{L^{p+1}(\mathbb{R}^N)}\). We infer that \(\Bar{v}_n \to \Bar{v}\) in \(L^{p+1}(\mathbb{R}^N)\). Finally we
can write
\begin{equation*}
\begin{split}
\norm{\Bar{v}_n-\Bar{v}}_{H^1(\mathbb{R}^N)}^2 &= \dualprod{\mathcal{I}_{\nu}'(\Bar{v}_n)-\mathcal{I}_{\nu}'(\Bar{v})}{\Bar{v}_n-\Bar{v}} \\ &\qquad+ \int_{\mathbb{R}^N} \left(
(\Bar{v}_n)_+^{p} - (\Bar{v})_+^{p} \right) \left( \Bar{v}_n-\Bar{v} \right)\:.
\end{split}
\end{equation*}
Since \(\mathcal{I}_{\nu}'(\Bar{v}) = 0\), \(\mathcal{I}_{\nu}'(\Bar{v}_n) \to 0\) as \(n \to \infty\) and the last term goes to \(0\) by H\"older's inequality, we conclude that
\(\Bar{v}_n \to \Bar{v}\) in \(H^1(\mathbb{R}^N)\). The conclusion follows.
\end{proof}
\subsection{Penalized limit problems}
The two following lemmas will provide information about the limit of sequences of rescaled solutions. The first lemma is due to M.\thinspace del Pino and P.\thinspace Felmer \cite{delPinoFelmer1998}*{Lemma 2.3}.
Let \(\mathbb{R}^N_+ := \left\lbrace x \in \mathbb{R}^N \; \mid\; x_N > 0 \right\rbrace\).
\begin{lemma}\label{lem:plimpenal1}
Let \(\nu \geq 0\) and \(\mu \in [0, \nu]\). If \(u \in H^1(\mathbb{R}^N)\) is a
solution of
\begin{equation*}
\left\{ \begin{aligned}
- \Delta u + \nu u &= u_+^p & & \text{in \(\mathbb{R}^N_+\)}, \\
- \Delta u + \nu u &= \min \bigl( \mu, \abs{u}^{p-1} \bigr)u_+ & &\text{in \(\mathbb{R}^N_-\)},
\end{aligned}\right.
\end{equation*}
then \(\abs{u}^{p-1} \leq \mu\) in \(\mathbb{R}^N_-\).
\end{lemma}
\begin{proof}
We follow the argument of M.\thinspace del Pino and P.\thinspace Felmer \cite{delPinoFelmer1998}*{Lemma 2.3}. By elliptic regularity, \(u \in H^2(\mathbb{R}^N) \cap C^1(\mathbb{R}^N)\).
Thus we can use \(\partial_N u\) as a test function in the equation.
Writing \(g(s) := \min ( \mu, \abs{s}^{p-1})s_+\) and \(G(s) := \int_0^s g(\sigma)\; d\sigma\), we obtain
\begin{equation*}
\frac{1}{2} \int_{\mathbb{R}^N} \partial_N \bigl( \abs{\nabla u}^2 + \nu \abs{u}^2 \bigr) = \frac{1}{p+1} \int_{\mathbb{R}^N_+}
\partial_N \bigl( u_+^{p+1} \bigr) + \int_{\mathbb{R}^N_-} \partial_N ( G \circ u)\:.
\end{equation*}
This reduces to
\begin{equation*}
\int_{\mathbb{R}^{N-1}} \Bigl( G\bigl( u(x', 0) \bigr) - \frac{1}{p+1} \bigl(u(x',0)\bigr)_+^{p+1} \Bigr)\; dx' = 0.
\end{equation*}
Since \(G(u) \leq \frac{u_+^{p+1}}{p+1}\) on \(\mathbb{R}^N\), we have \(G\left( u(x', 0) \right) =
\frac{1}{p+1} (u(x',0))_+^{p+1}\) for all \(x' \in \mathbb{R}^{N-1}\) and hence, for every \(x'\in \mathbb{R}^{N-1}\),
\(
u(x',0) \leq \mu^{\frac{1}{p-1}}\:.
\)
One has on \(\mathbb{R}^N_-\)
\[
-\Delta u + (\nu-\mu)u \le 0\:.
\]
Since \(\nu \ge \mu\), we deduce by the maximum principle that \(u \leq \mu^{\frac{1}{p-1}}\) in \(\mathbb{R}^N_-\).
\end{proof}
The second lemma is an application of the maximum principle.
\begin{lemma}\label{lem:plimpenal2}
Let \(\nu \geq 0\) and \(\mu \in [0, \nu]\). If \(u \in H^1(\mathbb{R}^N)\), \(u\geq 0\), is a solution of
\begin{equation*}
- \Delta u + \nu u = \min \bigl( \mu, u^{p-1} \bigr)u_+ \qquad
\text{in \(\mathbb{R}^N\)},
\end{equation*}
then \(u \equiv 0\).
\end{lemma}
\begin{proof}
If \(u\) is a solution, we have
\[
- \Delta u + (\nu - \mu) u \leq 0 \qquad \text{in \(\mathbb{R}^N\)}\:.
\]
Taking \(u\) as a test function, we obtain
\[
\int_{\mathbb{R}^N} \abs{\nabla u}^2 + (\nu - \mu) \abs{u}^2 \leq 0\:.
\]
Since \(\nu - \mu \geq 0\), this implies that \(u \equiv 0\).
\end{proof}
\section{Asymptotics of families of critical points}
\label{sectAsymptoticsCritical}
In this section we refine the
asymptotic analysis in \cite{BonheureVanSchaftingen2008}*{Section 5} in order to obtain an estimate of the energy of a critical point \(u_{\varepsilon}\) of
\(\mathcal{J}_{\varepsilon}\)
depending on the number and the location of its local maxima. The corresponding lower estimate was proved in \cite{BonheureVanSchaftingen2008}.
\subsection{Asymptotics on small balls}
The next lemma states that the sequences of rescaled solutions converge in \(C^1_{\text{loc}}(\mathbb{R}^N)\)
to a function in \(H^1(\mathbb{R}^N)\).
\begin{lemma}
\label{lemuepsrescaledcompact}
Let \((\varepsilon_n)_{n \in \mathbb{N}} \) be a sequence in \(\mathbb{R}^+\) such that
\(\varepsilon_n \to 0\) as \(n \to \infty\), let \((u_{n})_{n \in \mathbb{N}}\) be a sequence of solutions of \(\mathcal{Q}_{\varepsilon_n}\) such that
\[
\liminf_{n \to \infty} \varepsilon_n^{-N} \mathcal{J}_{\varepsilon_n}(u_n) < \infty
\]
and let \((x_n)_{n \in \mathbb{N}}\) be a sequence in \(\mathbb{R}^N\) such that \(x_n \to \Bar{x}\) as \(n \to \infty\).
Denote by \((v_n)_{n \in \mathbb{N}}\) the sequence
defined by \(v_n(x) = u_{\varepsilon_n}(x_n + \varepsilon_n x)\). If \(V(\Bar{x}) > 0\),
then there exists \(v \in H^1(\mathbb{R}^N)\) such that, up to a subsequence,
\(
v_n \to v
\)
in \(C^1_{\mathrm{loc}}(\mathbb{R}^{N})\)
as \(n \to \infty\).
\end{lemma}
This lemma was proved for minimal energy solutions in \cite{BonheureVanSchaftingen2008}*{Lemma 13}. We sketch here the argument in order to highlight that the proof only depends on the fact that \(u_n\) is a solution that satisfies an energy bound.
\begin{proof}
Take \(\varphi \in C^\infty_c(\mathbb{R}^N) \) such that \(\varphi \equiv 1 \) on \(B(0,1)\).
Set for \(R > 0\) and \(x \in \mathbb{R}^N\) \(\varphi_R(x)=\varphi(\frac{x}{R})\). The sequence
\( (\varphi_R v_n)_{n \in \mathbb{N}}\) is bounded in \(H^1(\mathbb{R}^N)\) for every \(R > 0\). By a diagonal argument, there exists \(v \in H^1_\mathrm{loc}(\mathbb{R}^N) \) such that \(v_n \to v\) weakly in \(H^1_\mathrm{loc}(\mathbb{R}^N)\) along a subsequence.
Now note that for every \(R > 0\),
\[
\int_{B(0,R)} \abs{\nabla v}^2 \leq \liminf_{n \to \infty} \int_{B(0,R)} \abs{\nabla v_n}^2 \leq \liminf_{n \to \infty} \int_{\mathbb{R}^N} \abs{\nabla v_n}^2
\]
and
\[
V(\Bar{x}) \int_{B(0,R)} \abs{v}^2 \leq \liminf_{n \to \infty} \int_{B(0,R)} V \abs{v_n}^2
\leq \liminf_{n \to \infty} \int_{\mathbb{R}^N} V \abs{v_n}^2\:,
\]
so that \(v \in H^1(\mathbb{R}^N)\).
The remainder follows from classical regularity and compactness results.
\end{proof}
\begin{lemma}
\label{lemsmallballs}
Let \((\varepsilon_n)_{n \in \mathbb{N}} \) be a sequence in \(\mathbb{R}^+\) such that
\(\varepsilon_n \to 0\) as \(n \to \infty\), let \((u_{n})_{n \in \mathbb{N}}\) be a sequence of solutions of \(\mathcal{Q}_{\varepsilon_n}\) such that
\[
\liminf_{n \to \infty} \varepsilon_n^{-N} \mathcal{J}_{\varepsilon_n}(u_n) < \infty
\]
and let \((x_n)_{n \in \mathbb{N}}\) be a sequence in \(\mathbb{R}^N\) such that \(x_n \to \Bar{x}\) as \(n \to \infty\). If \(V(\Bar{x}) > 0 \) and
\begin{equation*}
\liminf_{n\to\infty} u_{\varepsilon_n}(x_n) > 0\:,
\end{equation*}
then, \(\Bar{x} \in \Bar{\Lambda}\),
\[
\limsup_{n \to \infty} \frac{\dist(x_n, \Lambda)}{\varepsilon_n} < \infty\:,
\]
\[
\lim_{R\to \infty} \limsup_{n\to \infty} \Bigabs{\varepsilon_n^{-N} \int_{B(x_n, \varepsilon_n R)} \Bigl( \frac{1}{2}
\bigl( \varepsilon_n^2 \abs{\nabla u_{\varepsilon_n}}^2 + V \abs{u_{\varepsilon_n}}^2 \bigr) -
G_{\varepsilon_n}(.,u_{\varepsilon_n}) \Bigr) -\mathcal{C}(\Bar{x})} = 0\:,
\]
and
\[
\lim_{R\to \infty} \limsup_{n\to \infty} \varepsilon_n^{-N} \int_{B(x_n, 2\varepsilon_n R) \setminus B(x_n, \varepsilon_n R)}
\abs{\nabla u_{\varepsilon_n}}^2 + V \abs{u_{\varepsilon_n}}^2=0\:.
\]
\end{lemma}
\begin{proof}
Set \(v_n(x) := u_{\varepsilon_n}(x_n + \varepsilon_n x)\). By Lemma~\ref{lemuepsrescaledcompact} up to a subsequence, there exists a \(v \in H^1(\mathbb{R}^N)\) such that
\(v_n \to v\) in \(C^1_{\text{loc}}(\mathbb{R}^N)\). We have
\(v(0) = \lim_{n\to \infty} v_n(0) > 0\) so that \(v \not \equiv 0\).
Let us now prove by contradiction that
\begin{equation}
\label{limsupdistxneps}
\limsup_{n \to \infty} \frac{\dist(x_n, \Lambda)}{\varepsilon_n} < \infty\:.
\end{equation}
Up to a subsequence, we can assume that \(\lim_{n \to \infty} \dist(x_n, \Lambda)/\varepsilon_n = \infty\).
Then, since the sequence of characteristic functions \(\chi_n(x) :=
\chi_{\Lambda}(x_n + \varepsilon_n x)\) converges pointwise to \(0\), we have as \(n \to \infty\)
\[
\begin{split}
g_{\varepsilon_n}(x_n + \varepsilon_n \cdot, v_n) & = \min \bigl( \mu \bigl(\varepsilon_n^2 H(x_n + \varepsilon_n \cdot) v_n
+ V(x_n + \varepsilon_n \cdot) \bigr), \abs{v_n}^{p-1} \bigr)(v_n)_+ \\
&\qquad \to \min \bigl( \mu V(\Bar{x}) , \abs{v}^{p-1} \bigr)v_+\: ,
\end{split}
\]
in \(L^q_{\mathrm{loc}}(\mathbb{R}^N)\) for \(1 \le q < \frac{2N}{p(N-2)}\),
and thus \(v\) solves the limiting equation
\begin{equation*}
-\Delta v + V(\Bar{x}) v = \min \bigl(\mu V(\Bar{x}) , \abs{v}^{p-1} \bigr)v_+,
\qquad\text{in \(\mathbb{R}^N\)}.
\end{equation*}
By Lemma \ref{lem:plimpenal2}, \(v \equiv 0\), which is a contradiction. Thus \eqref{limsupdistxneps} holds.
Now, let us assume that
\begin{equation}
\label{limsupdistxnepsc}
\limsup_{n \to \infty} \frac{\dist(x_n, \mathbb{R}^N \setminus \Lambda)}{\varepsilon_n} = \infty\:.
\end{equation}
Since \(\chi_n(x)\) converges pointwise to \(1\), we have, up to a subsequence, for \(n\) large enough,
\begin{equation*}
g_{\varepsilon_n}(x_n + \varepsilon_n \cdot, v_n) = (v_n)_+^p \to v_+^p\:,
\end{equation*}
in \(L^q_{\mathrm{loc}}(\mathbb{R}^N)\) for \(1 \le q < \frac{2N}{p(N-2)}\).
Hence \(v\) solves the limiting equation
\begin{equation}\label{plim4}
- \Delta v + V(\Bar{x}) v = v_+^p \qquad \text{in }\ \mathbb{R}^N\:.
\end{equation}
If \eqref{limsupdistxneps} holds but \eqref{limsupdistxnepsc} does not, then
\[
\limsup_{n \to \infty} \frac{\dist(x_n, \partial \Lambda)}{\varepsilon_n} < \infty\:.
\]
Since \(\Lambda\) is smooth,
\(
\chi_n \to \chi_{E},
\)
almost everywhere as \(n \to \infty\), where \(E\) is a half-space.
By Lemma \ref{lem:plimpenal1}, \(v\) is again a solution of \eqref{plim4}.
In any case, \(v\) is thus a nontrivial solution of \eqref{plim4}. Now we claim that
\begin{multline}\label{eqlimhn}
\lim_{R\to \infty} \lim_{n\to \infty} \varepsilon_n^{-N} \frac{1}{2}\int_{B(x_n, \varepsilon_n R)}
\bigl( \varepsilon_n^2 \abs{\nabla u_{\varepsilon_n}}^2 + V \abs{u_{\varepsilon_n}}^2 \bigr) \\ = \frac{1}{2}\int_{\mathbb{R}^N} \abs{\nabla U_{V(\Bar{x})}}^2 + V(\Bar{x}) \abs{U_{V(\Bar{x})}}^2\:.
\end{multline}
For every \(R > 0\), the convergence of \(v_n\) to \(v\) in \(C^1_{\text{loc}}(\mathbb{R}^N)\) implies that
\begin{equation*}
\lim_{n\to \infty} \varepsilon_n^{-N} \frac{1}{2}\int_{B(x_n, \varepsilon_n R)}
\bigl( \varepsilon_n^2 \abs{\nabla u_{\varepsilon_n}}^2 + V \abs{u_{\varepsilon_n}}^2 \bigr) \\ = \frac{1}{2}\int_{B(0, R)} \abs{\nabla v}^2 + V(\Bar{x}) \abs{v}^2\:.
\end{equation*}
Since \(v \in H^1(\mathbb{R}^N)\), and \(v\) and \(U_{V(\Bar{x})}\) are equal up to a translation, we conclude that \eqref{eqlimhn} holds.
The argument for the other limit is similar.
\end{proof}
\subsection{Asymptotics outside small balls}
The solutions decay outside a neighborhood of \(\Lambda\):
\begin{lemma}\label{lemmaAsymptoticsOutsideLambda}
For every open set \(U\) such that \(\Bar{\Lambda} \subset U\), there exists \(C > 0\) such that for every \(\varepsilon > 0\), if \(u \in H^1_V (\mathbb{R}^N)\) is a solution of \(\mathcal{Q}_{\varepsilon}\),
\begin{equation*}
\int_{\mathbb{R}^N\setminus U} \bigl(\varepsilon^2 \abs{\nabla u}^2 + V \abs{u}^2 \bigr) \le C \varepsilon^2 \int_{\mathbb{R}^N} \bigl(\varepsilon^2 \abs{\nabla u}^2 + V \abs{u}^2 \bigr)\:.
\end{equation*}
\end{lemma}
\begin{proof}
Since \(V\) is continuous and \(\inf_{\Lambda} V > 0\), we can assume without loss of generality that \(\inf_{U} V > 0\).
Take \(\psi \in C^\infty_c(\mathbb{R}^N)\) such that \(\psi = 0\) on \(\Bar{\Lambda}\) and \(\psi=1\) on \(\mathbb{R}^N \setminus U\). By taking \(\psi^2 u_{n}\) as a test function in \eqref{problemPNLSE}, we obtain
\begin{multline*}
\int_{\mathbb{R}^N} \bigl(\varepsilon^2 \abs{\nabla (\psi u)}^2 + V \abs{\psi u}^2\bigr)
=\int_{\mathbb{R}^N} g_{\varepsilon} (x, u(x))\psi(x)^2 u(x) \: dx + \int_{\mathbb{R}^N} \varepsilon^2 \abs{\nabla \psi}^2 \abs{u}^2 \: .
\end{multline*}
Since \(\psi = 0\) in \(\Lambda\), we deduce from \( (g_2) \) and Lemma~\ref{lemmaPositivity} that
\begin{align*}
\int_{\mathbb{R}^N} g_{\varepsilon} (x, u(x))\psi (x)^2 u(x) \:dx &\leq \mu \int_{\mathbb{R}^N} (V+\varepsilon^2 H)\abs{\psi u}^2 \\
&\leq \mu \int_{\mathbb{R}^N} \bigl(\varepsilon^2 \abs{\nabla (\psi u)}^2 + V \abs{\psi u}^2\bigr)\: .
\end{align*}
Therefore, since \(\supp \nabla \psi \subset U \setminus \Bar{\Lambda}\) and \(\inf_{U} V > 0\), we have
\[
(1-\mu) \int_{\mathbb{R}^N} \bigl(\varepsilon^2 \abs{\nabla (\psi u)}^2 + V \abs{\psi u}^2\bigr)
\leq \int_{\mathbb{R}^N} \varepsilon^2 \abs{\nabla \psi}^2 \abs{u}^2 \leq C \varepsilon^2 \int_{U \setminus \Bar{\Lambda}} V \abs{u}^2\:.\qedhere
\]
\end{proof}
Now we have an estimate outside small balls.
\begin{lemma}\label{lemoutsmallballs}
Let \((\varepsilon_n)_{n \in \mathbb{N}} \) be a sequence in \(\mathbb{R}^+\) such that
\(\varepsilon_n \to 0\) as \(n \to \infty\), let \((u_{n})_{n \in \mathbb{N}}\) be a sequence of solutions of \((\mathcal{Q}_{\varepsilon_n})\) such that
\[
\limsup_{n \to \infty} \varepsilon_n^{-N} \mathcal{J}_{\varepsilon_n} (u_n) < \infty\: ,
\]
and let \((x^i_n)_{n \in \mathbb{N}} \subset \mathbb{R}^N\), \(1\leq i \leq M\), be sequences such that \(x^i_n \to \Bar{x}^i \in \mathbb{R}^N\) as \(n \to \infty\).
If for every \(i \in \{1, \dotsc, M\}\), \(V(\Bar{x}^i) > 0\) and
\begin{equation*}
\liminf_{n\to\infty} u_{n}(x^i_n) > 0\:,
\end{equation*}
then
\begin{equation*}
\liminf_{R\to \infty} \liminf_{n\to \infty} \varepsilon_n^{-N}\Bigl( \int_{\mathbb{R}^N\setminus \mathcal{B}_n(R)} \frac{1}{2}
\bigl( \varepsilon_n^2 \abs{\nabla u_{n}}^2 + V \abs{u_{n}}^2 \bigr) - G_{\varepsilon_n}(.,u_{n}) \Bigr) \geq 0\:,
\end{equation*}
where \(\mathcal{B}_n(R) := \bigcup_{i=1}^M B(x^i_n, \varepsilon_n R)\).
Furthermore, if
\begin{equation*}
\inf\bigl\{ \limsup_{n \to \infty} \norm{u_{n}}_{L^{\infty}(U \setminus \mathcal{B}_n(R))} \; \mid\; \text{\(R > 0\), \(U\) is open and \(\Bar{\Lambda} \subset U\)}\bigr\} = 0\:,
\end{equation*}
then
\begin{equation*}
\lim_{R\to \infty} \limsup_{n\to \infty} \varepsilon_n^{-N} \Bigabs{\int_{\mathbb{R}^N\setminus \mathcal{B}_n(R)} \frac{1}{2}
\bigl( \varepsilon_n^2 \abs{\nabla u_{n}}^2 + V \abs{u_{n}}^2 \bigr) - G_{\varepsilon_n}(.,u_{n})} = 0\:.
\end{equation*}
\end{lemma}
The first assertion was proved in \cite{BonheureVanSchaftingen2008}*{Lemma 15}.
\begin{proof}
First we claim that
\[
\lim_{R\to \infty} \limsup_{n\to \infty} \varepsilon_n^{-N} \Bigabs{\int_{\mathbb{R}^N\setminus \mathcal{B}_n(R)}
\bigl( \varepsilon_n^2 \abs{\nabla u_{n}}^2 + V \abs{u_{n}}^2 - g_{\varepsilon_n}(.,u_{n}) u_n \bigr)}= 0\: .
\]
This is proved in \cite{BonheureVanSchaftingen2008}*{Lemma 15} by taking a suitable family of test functions and using Lemma~\ref{lemuepsrescaledcompact}. We do not need to go to a subsequence since by Lemma~\ref{lemsmallballs},
\[
\lim_{R \to \infty} \limsup_{n \to \infty} \int_{B(x_n^i, \varepsilon_n R) \setminus B(x_n^i, \varepsilon_n R/2)} \varepsilon_n^2 \bigl(\abs{\nabla u_{n}}^2 + V \abs{u_{n}}^2\bigr) = 0 \: .
\]
The first assertion follows, as in \cite{BonheureVanSchaftingen2008}*{Lemma 15} from the inequality
\begin{multline*}
\frac{1}{2}\int_{\mathbb{R}^N\setminus \mathcal{B}_n(R)}
\bigl( \varepsilon_n^2 \abs{\nabla u_{n}}^2 + V \abs{u_{n}}^2 \bigr) - \frac{1}{2} \int_{\mathbb{R}^N\setminus \mathcal{B}_n(R)} g_{\varepsilon_n}(x,u_{n}(x)) u_n(x) \: dx \\
\le \frac{1}{2}\int_{\mathbb{R}^N\setminus \mathcal{B}_n(R)}
\bigl( \varepsilon_n^2 \abs{\nabla u_{n}}^2 + V \abs{u_{n}}^2 \bigr) - \int_{\mathbb{R}^N\setminus \mathcal{B}_n(R)} G_{\varepsilon_n}(x,u_{n}(x)) \:dx\:.
\end{multline*}
For the second assertion, we have by \((g_2)\) and \((g_3)\),
\begin{multline*}
\Bigabs{\int_{\mathbb{R}^N\setminus \mathcal{B}_n(R)} G_{\varepsilon_n}(x,u_{n}(x)) - \frac{1}{2} g_{\varepsilon_n}(x,u_{n}(x)) u_{n}(x)\: dx} \\
\le \int_{\mathbb{R}^N\setminus \mathcal{B}_n(R)} \frac{1}{2} g_{\varepsilon_n}(x,u_{n}(x)) u_{n}(x)\: dx\\
\leq \frac{1}{2} \int_{U \setminus \mathcal{B}_n(R)} (u_{n})_+^{p+1}
+\frac{\mu}{2} \int_{\mathbb{R}^N\setminus U} \bigl( \varepsilon_n^2 H + V\bigr) \abs{u_{n}}^2\:.
\end{multline*}
We compute that
\begin{equation*}
\begin{split}
\int_{U \setminus \mathcal{B}_n(R)} \abs{u_{n}}^{p+1}
&\leq C \norm{u_{n}}^{p-1}_{L^{\infty}(U \setminus \mathcal{B}_n(R))} \int_{U \setminus \mathcal{B}_n(R)} \abs{u_{n}}^{2} \\
&\le \frac{1}{(\inf_U V)^2} \norm{u_{n}}^{p-1}_{L^{\infty} (U\setminus \mathcal{B}_n(R))} \int_{\mathbb{R}^N} \bigl( \varepsilon_n^2 \abs{\nabla u_{n}}^2 + V \abs{u_{n}}^2 \bigr)\:.
\end{split}
\end{equation*}
In view of Lemma~\ref{lemmaAsymptoticsOutsideLambda}, for every \(U \supset \Lambda\) there exists \(C > 0\) such that we have
\begin{multline*}
\Bigabs{\int_{\mathbb{R}^N\setminus \mathcal{B}_n(R)} G_{\varepsilon_n}(x,u_{n}(x)) - \frac{1}{2} g_{\varepsilon_n}(x,u_{n}(x)) u_{n}(x)\: dx}\\
\le \frac{1}{2} \Bigl(\frac{1}{(\inf_U V)^2} \norm{u_{n}}^{p-1}_{L^{\infty} (U\setminus \mathcal{B}_n(R))} + C \mu \varepsilon_n^2\Bigr) \int_{\mathbb{R}^N} \bigl( \varepsilon_n^2 \abs{\nabla u_{n}}^2 + V \abs{u_{n}}^2 \bigr)\:.
\end{multline*}
We conclude by taking \(U \supset \Bar{\Lambda}\) small enough and \(R\) and \(n\) large enough, in view of the hypothesis and Lemma~\ref{lem:bounded}.
\end{proof}
\subsection{Conclusion}
We can now state and prove the main result of this section is
\begin{proposition}\label{estim:inf2}
Let \((\varepsilon_n)_{n \in \mathbb{N}} \) be a sequence in \(\mathbb{R}^+\) such that
\(\varepsilon_n \to 0\) as \(n \to \infty\), let \((u_{n})_{n \in \mathbb{N}}\) be a sequence of solutions of \(\mathcal{Q}_{\varepsilon_n}\) such that
\[
\limsup_{n \to \infty} \varepsilon_n^{-N} \mathcal{J}_{\varepsilon_n} (u_n) < \infty\: ,
\]
and let \((x^i_n)_{n \in \mathbb{N}} \subset \mathbb{R}^N\), \(1\leq i \leq M\), be sequences such that \(x^i_n \to \Bar{x}^i \in \mathbb{R}^N\) as \(n \to \infty\).
If for every \(i \in \{1, \dotsc, M\}\), \(V(\Bar{x}^i) > 0\) and
\begin{equation*}
\liminf_{n\to\infty} u_{n}(x^i_n) > 0\:,
\end{equation*}
and if for every \(i,j \in \{ 1, \dots, M \}\) such that \(i\neq j\),
\begin{equation*}
\lim_{n\to\infty} \frac{\abs{x^i_n-x^j_n}}{\varepsilon_n} = +\infty\:,
\end{equation*}
then for every \(i \in \{ 1, \dots, M \}\), \(\Bar{x}^i \in \Bar{\Lambda}\),
\begin{equation*}
\lim_{n\to\infty} \frac{\dist(x^i_n, \Lambda)}{\varepsilon_n} < +\infty\:,
\end{equation*}
and
\begin{equation*}
\liminf_{n\to\infty} \varepsilon_n^{-N} \mathcal{J}_{\varepsilon_n}(u_{\varepsilon_n}) \geq
\sum_{i=1}^M \mathcal{C}(\Bar{x}^i)\:.
\end{equation*}
Furthermore, if
\begin{equation*}
\inf\Bigl\{ \limsup_{n \to \infty} \norm{u_{n}}_{L^{\infty}(U \setminus \mathcal{B}_n(R))} \; \mid\; \text{\(R > 0\), \(U\) is open and \(\Bar{\Lambda} \subset U\)}\Bigr\} = 0\:,
\end{equation*}
then
\begin{equation*}
\lim_{n\to\infty}\varepsilon_n^{-N} \mathcal{J}_{\varepsilon_n}(u_{n})=
\sum_{i=1}^M \mathcal{C}(\Bar{x}^i)\:.
\end{equation*}
\end{proposition}
\begin{proof}
This follows from Lemmas~\ref{lemsmallballs} and \ref{lemoutsmallballs} (see \citelist{\cite{BonheureVanSchaftingen2008}*{Proposition 16}\cite{MorozVanSchaftingen2010}*{Lemma 4.3}} for the details).
\end{proof}
\section{Asymptotics of families of almost minimizers}
\label{sectAsymptoticsMinimizers}
\subsection{Families of minimizers}
Let us recall how the results of Section~\ref{sectAsymptoticsCritical} allow to study the asymptotics of \(\inf_{\mathcal{N}_\varepsilon} \mathcal{J}_\varepsilon\).
\begin{proposition}
\label{propAsymptotics}
If \(\inf_\Lambda V > 0\), then
\[
\lim_{\varepsilon \to 0} \varepsilon^{-N} \inf_{\mathcal{N}_\varepsilon} \mathcal{J}_\varepsilon
= \inf_\Lambda \mathcal{C}\:.
\]
\end{proposition}
This has been proved by M.\thinspace del Pino and P.\thinspace Felmer \cite{delPinoFelmer1996}*{(2.4) and Lemma 2.2} when \(\inf_{\mathbb{R}^N} V > 0 \), and has been extended to decaying potentials \citelist{\cite{BonheureVanSchaftingen2008}*{Lemma 12 and proof of Proposition 21}\cite{MorozVanSchaftingen2010}*{Lemma 2.2}}.
\begin{proof}[Sketch of the proof]
First one shows that for every \(x \in \Lambda\),
\[
\limsup_{\varepsilon \to 0} \varepsilon^{-N} \inf_{\mathcal{N}_\varepsilon} \mathcal{J}_\varepsilon
\le \mathcal{C}(x)\:
\]
by taking suitable multiples of cutoffs of \(U_{V(x)}(\frac{\cdot-x}{\varepsilon})\).
By Proposition~\ref{prop:Minimizer}, for every \(\varepsilon > 0\), there exists \(u_\varepsilon \in \mathcal{N}_\varepsilon\) such that \(\mathcal{J}_\varepsilon(u_\varepsilon) = \inf_{\mathcal{N}_\varepsilon} \mathcal{J}_\varepsilon\). By classical regularity theory, \(u_\varepsilon\) is continuous. Choose \(x_\varepsilon \in \Bar{\Lambda}\) such that \(u_\varepsilon(x_\varepsilon) = \sup_\Lambda u_\varepsilon\). By Lemma~\ref{lem:1}, \(\liminf_{\varepsilon \to 0} u_\varepsilon(x_\varepsilon) > 0\). By Proposition~\ref{estim:inf2},
\[
\liminf_{\varepsilon \to 0} \varepsilon^{-N} \mathcal{J}_\varepsilon(u_\varepsilon)
\ge \inf_\Lambda \mathcal{C}\:.\qedhere
\]
\end{proof}
\subsection{Decay of almost minimizers}
The next ingredient is a decay estimate that will allow to control the functional outside \(\Lambda\) in the proof of the strict inequality \eqref{estim}.
\begin{lemma}\label{lem:3}
Let \((\varepsilon_n)_{n \in \mathbb{N}} \subset \mathbb{R}^+_0\) be a sequence such that \(\varepsilon_n \to 0\) as \(n \to \infty\) and let \((u_n)_{n \in \mathbb{N}} \subset \mathcal{N}_{\varepsilon_n}\). If \(\inf_\Lambda V > 0\) and
\begin{equation*}
\limsup_{n\to\infty} \varepsilon_n^{-N} \mathcal{J}_{\varepsilon_n} (u_n) \leq \inf_{\Lambda} \mathcal{C},
\end{equation*}
then, for every open set \(U \subset \mathbb{R}^N\) such that \(\Bar{\Lambda} \subset U\),
\begin{equation*}
\lim_{n\to\infty} \varepsilon_n^{-N} \int_{\mathbb{R}^N\setminus U} \bigl( \varepsilon_n^2 \abs{\nabla u_n}^2 + V \abs{u_n}^2 \bigr)= 0\:.
\end{equation*}
\end{lemma}
This lemma is proved by M.\thinspace del Pino and P.\thinspace Felmer \cite{delPinoFelmer1997}*{(1.19)} when \(\inf_{\mathbb{R}^N} V > 0\).
The proof of Lemma~\ref{lem:3} relies on the following lemma
\begin{lemma}\label{lem:lowenergycoercivity}
Let \(U \subset \mathbb{R}^N\) be such that \(\Bar{\Lambda} \subset U\) and \(\inf_U V > 0\).
Let \(\psi \in C^1 (\mathbb{R}^N)\) and \(\varphi \in C^1 (\mathbb{R}^N)\) be such that \(\psi = 0\) on \(\Lambda\), \(\varphi = 0\) on \(\mathbb{R}^N \setminus U\), and \(\psi^2 + \varphi^2 = 1 \) on \(\mathbb{R}^N\).
There exists \(C > 0\) such that for every \(\varepsilon > 0\) and \(u \in H^1_V (\mathbb{R}^N)\),
\[
\mathcal{J}_{\varepsilon} ( \varphi u) +
(1-\mu) \int_{\mathbb{R}^N} \bigl(\varepsilon^2 \abs{\nabla (\psi u)}^2 + V \abs{\psi u}^2\bigr)
\le \mathcal{J}_{\varepsilon} (u) + C \varepsilon^2 \int_{U \setminus \Bar{\Lambda} } V \abs{u}^2\:.
\]
\end{lemma}
\begin{proof}
One has
\begin{multline*}
\mathcal{J}_{\varepsilon} (u) = \mathcal{J}_{\varepsilon} ( \varphi u) + \frac{1}{2}
\int_{\mathbb{R}^N} \bigl(\varepsilon^2 \abs{\nabla (\psi u)}^2 + V \abs{ \psi u}^2 \bigr)
-\frac{1}{2} \int_{U \setminus \Bar{\Lambda} } \varepsilon^2(\abs{\nabla \psi}^2 +\abs{\nabla \varphi}^2 ) \abs{u}^2\\
- \int_{\mathbb{R}^N} G_\varepsilon(x, u(x))-G_\varepsilon(x,\varphi(x) u(x))\:dx\:.
\end{multline*}
One has for every \(x \in \mathbb{R}^N \setminus \Lambda\), by \( (g_2) \),
\[
\begin{split}
G_\varepsilon(x, u(x))-G_\varepsilon(x,\varphi(x) u(x))&= \int_{\varphi(x) u(x)}^{u(x)} g_\varepsilon(x,\sigma)\:d\sigma\\
&\leq \mu \int_{\varphi(x) u(x)}^{u(x)} \bigl(V(x) + \varepsilon^2 H(x) \bigr) \sigma \:d\sigma\\[1.9mm]
&= \frac{\mu}{2}\bigl(V(x) + \varepsilon^2 H(x) \bigr) \abs{\psi(x)u(x)}^2,
\end{split}
\]
so that
\[
\int_{\mathbb{R}^N} G_\varepsilon(x, u(x))-G_\varepsilon(x,\varphi(x) u(x))\:dx \leq \frac{\mu}{2}\int_{\mathbb{R}^N}\bigl(V + \varepsilon^2 H \bigr) \abs{\psi u}^2 \:.
\]
On the other hand,
\[
\int_{U \setminus \Bar{\Lambda} } \varepsilon^2 (\abs{\nabla \psi}^2 +\abs{\nabla \varphi}^2 ) \abs{u}^2
\leq C\varepsilon^2 \int_{U \setminus \Bar{\Lambda} } V \abs{u}^2\:.
\]
The conclusion follows.
\end{proof}
\begin{proof}[Proof of Lemma~\ref{lem:3}]
Without loss of generality, assume that \(\inf_U V > 0\).
Let \(\psi \in C^1 (\mathbb{R}^N)\) and \(\varphi \in C^1 (\mathbb{R}^N)\) be such that \(\psi = 0\) on \(\Lambda\), \(\varphi = 0\) on \(\mathbb{R}^N \setminus U\), and \(\psi^2 + \varphi^2 = 1 \) on \(\mathbb{R}^N\).
Define \(t_n\) so that \(t_n \varphi u_n \in \mathcal{N}_{\varepsilon_n}\).
By Lemma~\ref{lem:bounded},
\[
\limsup_{n \to \infty} \varepsilon_n^{-N}\int_{\mathbb{R}^N} \varepsilon^2 \abs{\nabla u_n}^2 +V \abs{u_n}^2 < \infty\:,
\]
and thus
\[
\limsup_{n \to \infty} \varepsilon_n^{-N} \int_{\mathbb{R}^N} \varepsilon_n^2 \abs{\nabla (\varphi u_n)}^2 +V \abs{\varphi u_n}^2 < \infty\:.
\]
By the choice of \(t_n\),
\begin{align*}
t_n^2\int_{\mathbb{R}^N} \varepsilon_n^2 \abs{\nabla (\varphi u_n)}^2 +V \abs{\varphi u_n}^2 &= \int_{\mathbb{R}^N} g_{\varepsilon_n}(x, t_n \varphi(x) u_n(x)) t_n \varphi(x) u_n(x)\: dx \\
&\geq t_n^{p+1} \int_{\Lambda} \abs{u_n}^{p+1}\:.
\end{align*}
We infer from Lemmas~\ref{lemENehariEstimate1}~and~\ref{lemENehariEstimate2} that
\[
\liminf_{n \to \infty} \varepsilon_n^{-N} \int_{\Lambda} \abs{u_n}^{p+1} > 0\:.
\]
Therefore,
\(\limsup_{n \to \infty} t_n < \infty\)
and
\[
\limsup_{n \to \infty} \varepsilon_n^{-N} \int_{U \setminus \Bar{\Lambda} } V \abs{t_n u_n}^2 < \infty\:.
\]
By Lemma~\ref{lem:lowenergycoercivity},
\begin{multline*}
\limsup_{n \to \infty} \varepsilon_n^{-N}\mathcal{J}_{\varepsilon_n} (t_n u_n) \geq \liminf_{n \to \infty} \varepsilon_n^{-N}\mathcal{J}_{\varepsilon_n} (t_n \varphi u_n) \\
+ \limsup_{n \to \infty} \varepsilon_n^{-N} (1-\mu) t_n^2 \int_{\mathbb{R}^N} \bigl(\varepsilon_n^2 \abs{\nabla (\psi u_n)}^2 + V \abs{\psi u_n}^2\bigr)\:.
\end{multline*}
By assumption, we have
\[
\limsup_{n \to \infty} \varepsilon_n^{-N}\mathcal{J}_{\varepsilon_n} (t_n u_n) \le \limsup_{n \to \infty} \varepsilon_n^{-N}\mathcal{J}_{\varepsilon_n} (u_n) \le \inf_{\Lambda} \mathcal{C}\: ,
\]
and since \(t_n \varphi u_n \in \mathcal{N}_{\varepsilon_n}\), we deduce from Proposition~\ref{propAsymptotics} that
\[
\liminf_{n \to \infty} \varepsilon_n^{-N}\mathcal{J}_{\varepsilon_n} (t_n \varphi u_n) \ge \inf_{\Lambda} \mathcal{C}\: .
\]
Combining the last three inequalities, we obtain the conclusion.
\end{proof}
\subsection{Asymptotics of the barycenters}
As in \cite{delPinoFelmer1997}, we introduce a barycenter map in order to localize functions.
Let \(\psi \in C^1(\mathbb{R}^N)\) be such that \(\supp \psi\) is compact, \(\supp \psi \subset \{x \in \mathbb{R}^N \; \mid\; V(x) > 0\}\) and \(\psi=1\) on
a neighborhood of \(\Lambda\).
The barycenter of a function \(u \in L^2(\mathbb{R}^N)\) is defined by
\begin{equation*}
\beta(u) := \frac{\displaystyle \int_{\mathbb{R}^N} x \abs{\psi(x) u(x)}^2 \: dx}{\displaystyle \int_{\mathbb{R}^N} \abs{\psi(x)u(x)}^2\: dx}\:.
\end{equation*}
The map \(\beta\) is well-defined on the set \(\{ u \in H^1_V(\mathbb{R}^N) \; \mid\; \psi u \ne 0\}\), which contains \(\mathcal{N}_\varepsilon\) for each \(\varepsilon > 0\) by Lemma~\ref{lemENehariEstimate2}.
\begin{proposition}
\label{propositionAsymptoticsMoments}
Let \((\varepsilon_n)_{n\in \mathbb{N}} \subset \mathbb{R}^+_0\) be a sequence such that \(\varepsilon_n \to 0\) as \(n \to \infty\) and let \((u_n)_{n\in \mathbb{N}} \subset \mathcal{N}_{\varepsilon_n}\). If \(\inf_{\Lambda} V>0\) and
\begin{equation*}
\limsup_{n \to \infty} \varepsilon_n^{-N}\mathcal{J}_{\varepsilon_n} (u_n) \leq \inf_{\Lambda} \mathcal{C}\: ,
\end{equation*}
then
\begin{equation*}
\lim_{n\to\infty} V\bigl(\beta(u_n)\bigr) = \inf_{\Lambda} V\:.
\end{equation*}
\end{proposition}
\begin{proof}
Let \(t_n > 0\) be such that
\begin{equation*}
\int_{\mathbb{R}^N} \varepsilon_n^2 \abs{\nabla t_n \psi u_n}^2 + V_0 \abs{ t_n \psi u_n}^2
=\int_{\mathbb{R}^N} ( t_n \psi u_n)_+^{p+1}\:.
\end{equation*}
Define \(v_n : \mathbb{R}^N \to \mathbb{R}\) for \(y \in \mathbb{R}^N\) by
\[
v_n(y) := t_n \psi(\beta(u_n)+ \varepsilon_n y)u_n(\beta(u_n)+ \varepsilon_n y)\:.
\]
\medbreak
\paragraph{Claim 1}
\emph{The sequence \((t_n)_{n \in \mathbb{N}}\) is bounded.}
By Lemmas~\ref{lemENehariEstimate1} and \ref{lemENehariEstimate2},
\[
\liminf_{n \to \infty} \epsilon_n^{- N} \int_{\mathbb{R}^N} (\psi u_n)_+^{p+1}
\ge \liminf_{n \to \infty} \epsilon_n^{- N} \int_{\Lambda} (u_n)_+^{p+1}
> 0.
\]
Since
\[
\limsup_{n \to \infty} \int_{\mathbb{R}^N} \varepsilon_n^2 \abs{\nabla t_n \psi u_n}^2 + V_0 \abs{ t_n \psi u_n}^2
\]
by Lemma~\ref{lem:bounded}, the sequence \((t_n)_{n \in \mathbb{N}}\) is bounded.
\medbreak
\paragraph{Claim 2} \emph{One has }
\begin{equation*}
\limsup_{n \to \infty} \varepsilon_n^{-N} \mathcal{J}_{\varepsilon_n}(t_n \psi u_n) \leq \inf_{\Lambda} \mathcal{C}\:.
\end{equation*}
We can write
\begin{equation*}
\begin{split}
\mathcal{J}_{\varepsilon_n}(t_n \psi u_n)
&= \mathcal{J}_{\varepsilon_n}(t_n u_n) + \frac{\varepsilon_n^2}{2} t_n^2\int_{\mathbb{R}^N} \abs{\nabla (\psi u_n)}^2-\abs{\nabla u_n}^2\\
&\qquad + \frac{t_n^2}{2} \int_{\mathbb{R}^N} (\psi^2-1)V \abs{u_n}^2\\
&\qquad + \int_{\mathbb{R}^N} G_{\varepsilon_n}(x, t_n u_n(x)) - G_{\varepsilon_n}(x, t_n \psi(x) u_n(x)) \:dx
\end{split}
\end{equation*}
Now, in view of Lemma \ref{lem:3}, since \(\psi=1\) in a neighborhood of \(\Lambda\) and \((t_n)_{n \in \mathbb{N}}\) is bounded,
\begin{multline*}
\frac{\varepsilon_n^2}{2}t_n^2 \int_{\mathbb{R}^N} \abs{\nabla (\psi u_n)}^2-\abs{\nabla u_n}^2\\
=\frac{\varepsilon_n^2}{2} t_n^2\int_{\mathbb{R}^N} (\psi^2-1) \abs{\nabla u_n}^2
+ 2 \psi u_n \nabla \psi \cdot \nabla u_n + \abs{\nabla \psi }^2\abs{u_n}^2
=o(\varepsilon_n^N)\: ,
\end{multline*}
as \(n \to \infty\).
Lemma \ref{lem:3} also implies that
\[
\begin{split}
\int_{\mathbb{R}^N} (\psi^2-1)V \abs{u_n}^2
=o(\varepsilon_n^N)\:,
\end{split}
\]
as \(n \to \infty\).
Finally, since \(\psi = 1\) on a neighborhood \(U\) of \(\Lambda\), we deduce from \((g_2)\) that
\begin{multline*}
\int_{\mathbb{R}^N} G_{\varepsilon_n}\bigl(x, t_n u_n(x)\bigr) - G_{\varepsilon_n}(x, t_n \psi(x) u_n(x))\:dx\\
\le \frac{\mu}{2} t_n^2\int_{\mathbb{R}^N \setminus U} \bigl(\varepsilon_n^2 H + V\bigr) \abs{u_n}^2
=o(\varepsilon_n^N)\:,
\end{multline*}
as \(n \to \infty\).
where we have used Lemma \ref{lem:3} again.
It follows from the hypothesis that
\begin{equation*}
\limsup_{n \to \infty} \varepsilon_n^{-N} \mathcal{J}_{\varepsilon_n}(t_n u_n) \le \limsup_{n \to \infty} \varepsilon_n^{-N} \mathcal{J}_{\varepsilon_n}(u_n) \le \inf_{\Lambda} \mathcal{C}\:;
\end{equation*}
the claim follows.
\medbreak
\paragraph{Claim 3} \emph{There holds}
\begin{equation*}
\limsup_{n \to \infty} \mathcal{I}_{V_0}(v_n) \leq \inf_{\Lambda} \mathcal{C}\:,
\end{equation*}
\emph{where \(V_0=\inf_\Lambda V\).}
One computes that, using \((g_2)\),
\begin{equation*}
\begin{split}
\varepsilon_n^{N} \mathcal{I}_{V_0}(v_n)
&= \mathcal{J}_{\varepsilon_n}(t_n \psi u_n) + \frac{t_n^2}{2} \int_{\mathbb{R}^N} (V_0 - V)\psi^2 \abs{u_n}^2\\
&\qquad + \int_{\mathbb{R}^N} G_{\varepsilon_n}(x, t_n \psi(x) u_n(x)) - \frac{t_n^{p+1}}{p+1} (\psi(x) u_n(x))_+^{p+1} \:dx\\
&\leq \mathcal{J}_{\varepsilon_n}(t_n \psi u_n) + \frac{t_n^2}{2} \int_{\mathbb{R}^N} (V_0 - V)\psi^2 \abs{u_n}^2.
\end{split}
\end{equation*}
For \(\kappa \in (0, 1)\), define
\[
U_\kappa = \bigl\{ x \in \mathbb{R}^N \; \mid\; \bigl(V_0-V(x)\bigr)\psi^2(x) < \kappa V(x) \bigr\}\:.
\]
Since \(V\) is continuous, \(U_\kappa\) is open and \(\Bar{\Lambda} \subset U_\kappa\).
By Lemma \ref{lem:3},
\[
\begin{split}
\int_{\mathbb{R}^N} (V_0 -V)\psi^2 \abs{u_n}^2
&\le \kappa\int_{U_\kappa} V\abs{u_n}^2
+ \int_{\mathbb{R}^N \setminus U_\kappa} (V_0 -V)\psi^2 \abs{u_n}^2\\
&\le \kappa \int_{\mathbb{R}^N} V \abs{u_n}^2 + o(\varepsilon_n^N)\:,
\end{split}
\]
as \(n \to \infty\).
Since \(\kappa > 0 \) is arbitrary, and \((t_n)_{n \in \mathbb{N}}\) and \( (\norm{u_n}_\varepsilon)_{n \in \mathbb{N}} \) are bounded,
\[
\frac{t_n^2}{2} \int_{\mathbb{R}^N} (V_0-V) \psi^2 \abs{u_n}^2 \le o(\varepsilon_n^N)\:,
\]
as \(n \to \infty\).
The claim now follows from Claim 2.
\medbreak
\paragraph{Conclusion}
We know from Claim 3 that \((v_n)_{n \in \mathbb{N}}\) is a minimizing sequence of \(\mathcal{I}_{V_0}\) on its associated Nehari manifold
\(
\mathcal{M}_{V_0}.
\)
By Lemma~\ref{lemmaConvergenceMinimizingLimiting}, there exists a sequence of points \((y_n)_{n \in \mathbb{N}} \subset \mathbb{R}^N\) such that \(v_n(\cdot-y_n)\) converges in \(H^1(\mathbb{R}^N)\) to the positive solution
\(U_{V_0}\) of problem \eqref{problemLimit}.
Let \(x_n := \varepsilon_n y_n\).
Since
\[
\begin{split}
\beta(u_n)&=x_n + \frac{\displaystyle \int_{\mathbb{R}^N} (x-x_n) \abs{\psi(x) u_n(x)}^2 \: dx}{\displaystyle \int_{\mathbb{R}^N} \abs{\psi(x) u_n(x)}^2\: dx}\\
&=x_n + \frac{\displaystyle \varepsilon_n \int_{\mathbb{R}^N} y \abs{v_n(y-y_n)}^2 \: dy}{\displaystyle \int_{\mathbb{R}^N} \abs{v_n(y-y_n)}^2 \: dy}\:,
\end{split}
\]
we have \(\lim_{n \to \infty} \beta(u_n)-x_n = 0\).
Now, note that for \(n\) large enough, by Lemma~\ref{lemENehariEstimate2},
\[
\liminf_{n\to \infty} \varepsilon_n^{-N}\int_{\Lambda} \abs{u_n}^2 >0.
\]
Since \(v_n(\cdot-y_n) \to U_{V_0}\) in \(L^2(\mathbb{R}^N)\), we must have \(\dist(x_n, \Lambda) = O(\varepsilon_n)\) as \(n \to \infty\).
Let
\[
\Bar{V} := \limsup_{n \to \infty} V(x_n)= \limsup_{n \to \infty} V(\beta(u_n)).
\]
Since \(V\) is continuous on the compact set \(\supp \psi\), one has \(\lim_{n \to \infty} V(x_n+\varepsilon y) \geq V_0 \).
By Claim 2,
\[
\begin{split}
b_{V_0} &\geq \varepsilon_n^{-N} \mathcal{J}_{\varepsilon_n}(t_n \psi u_n) + o(1)\\
&\geq \frac{1}{2}\int_{\mathbb{R}^N} \abs{\nabla v_n}^2 + \frac{1}{2}\int_{\mathbb{R}^N} V(x_n+\varepsilon y) \abs{v_n}^2
- \frac{1}{p+1}\int_{\mathbb{R}^N} (v_n)^{p+1}_+ + o(1),
\end{split}
\]
as \(n \to \infty\),
and thus we obtain
\begin{multline*}
\frac{1}{2}\int_{\mathbb{R}^N} \abs{\nabla U_{V_0}}^2 + V_0 \abs{U_{V_0}}^2
- \frac{1}{p+1}\int_{\mathbb{R}^N} (U_{V_0})^{p+1}_+ = b_{V_0} \\
\geq \frac{1}{2}\int_{\mathbb{R}^N} \bigl(\abs{\nabla U_{V_0}}^2 + \Bar{V} \abs{U_{V_0}}^2\bigr)
- \frac{1}{p+1}\int_{\mathbb{R}^N} (U_{V_0})^{p+1}_+.
\end{multline*}
This implies that \(\Bar{V} \leq V_0\). The conclusion follows.
\end{proof}
\section{The minimax level}
\label{sectionMinimax}
\subsection{Definition of the minimax level}
Following M.\thinspace del Pino and P.\thinspace Felmer \cites{delPinoFelmer1997,delPinoFelmer2002}, we define a minimax
value for \(\mathcal{J}_{\varepsilon}\).
Let \(\eta \in C^{\infty}_{\mathrm{c}}(\mathbb{R}^+)\) be a cut-off function such that \(0 \leq \eta \leq 1\), \(\eta=1\) on a neighborhood of \(\Lambda\) and \(\supp \eta \subset \{x \in \mathbb{R}^N : V(x) > 0\}\).
We define \(w_{\varepsilon,y} \in C^\infty_{\mathrm{c}}(\mathbb{R}^N)\) by
\begin{equation}\label{wepsy}
w_{\varepsilon,y}(x) := t_{\varepsilon,y} \eta(x)
U_{V(y)}\Bigl(\frac{y-x}{\varepsilon}\Bigr),
\end{equation}
where \(t_{\varepsilon,y} > 0\) is such that \(w_{\varepsilon,y} \in \mathcal{N}_{\varepsilon}\).
Let \(\Lambda_\varepsilon \subset \Lambda\) be such that
\begin{equation}
\label{limsupLambdaepsilon}
\lim_{\varepsilon \to 0} \sup_{x \in \partial \Lambda_\varepsilon} \dist(x,\partial\Lambda) = 0
\end{equation}
and
\begin{equation}
\label{liminfLambdaepsilon}
\lim_{\varepsilon \to 0} \inf_{x \in \partial \Lambda_\varepsilon} \frac{\dist(x,\partial\Lambda)}{\varepsilon} = \infty\:.
\end{equation}
We define the family of paths
\begin{equation*}
\Gamma_{\varepsilon} := \left\lbrace \gamma \in C(\overline{\Lambda}_\varepsilon, \mathcal{N}_{\varepsilon}) \; \mid\; \text{for every \(y \in \partial\Lambda_\varepsilon\), } \gamma(y) = w_{\varepsilon,y} \right\rbrace
\end{equation*}
and the minimax value
\begin{equation}\label{ceps}
c_{\varepsilon} := \inf_{\gamma \in \Gamma_{\varepsilon}} \sup_{y \in \Lambda_\varepsilon} \mathcal{J}_{\varepsilon}( \gamma(y)
)\:.
\end{equation}
We want to apply the following theorem.
\begin{theorem}[General Minimax Principle \cite{Willem1996}*{Theorem 2.9}]
Let \(X\) be a Banach space. Let \(M_0\) be a closed subspace of the metric space \(M\) and $\Gamma_0 \subset
C(M_0,X)$. Define
\begin{equation*}
\Gamma := \left\lbrace \gamma \in C(M,X) \; \mid\; \gamma_{\vert M_0} \in \Gamma_0 \right\rbrace.
\end{equation*}
If \(\varphi \in C^1(X,\mathbb{R})\) satisfies
\begin{equation*}
\infty > c := \inf_{\gamma \in \Gamma} \sup_{z \in M} \varphi\bigl(\gamma(z)\bigr) > a := \sup_{\gamma_0 \in \Gamma_0}
\sup_{z \in M_0} \varphi\bigl(\gamma_0(z)\bigr)
\end{equation*}
and if \(\varphi\) satisfies the Palais-Smale condition at the level \(c\), then \(c\) is a critical value of \(\varphi\).
\end{theorem}
Since \(\mathcal{J}_{\varepsilon}\) satisfies the Palais-Smale condition (Lemma~\ref{lem:PalaisSmale}), we have
to show that for \(\varepsilon > 0\) small enough,
\begin{equation}\label{estim}
c_{\varepsilon} > \sup_{y \in \partial\Lambda_\varepsilon} \mathcal{J}_{\varepsilon}( w_{\varepsilon,y} ) =: a_{\varepsilon}\:.
\end{equation}
\subsection{Estimates on the levels}\label{sectionEstimatesLevels}
\subsubsection{Estimate of \(a_{\varepsilon}\)}
We begin with an estimate of \(a_{\varepsilon}\).
\begin{lemma}\label{lem:estimaeps}
We have
\begin{equation*}
\lim_{\varepsilon \to 0}\varepsilon^{-N} a_{\varepsilon} = \sup_{\partial\Lambda} \mathcal{C}\:,
\end{equation*}
\end{lemma}
\begin{proof}
By a standard computation, we find in view of \eqref{liminfLambdaepsilon}
\begin{equation*}
\mathcal{J}_{\varepsilon}( w_{\varepsilon,y} ) = \varepsilon^N \mathcal{I}_{V(y)}\left(U_{V(y)}\right) + o(\varepsilon^N)\:.
\end{equation*}
as \(\varepsilon \to 0\), uniformly in \(y \in \Lambda\).
Thus
\begin{equation*}
a_{\varepsilon} = \sup_{y \in \partial\Lambda_\varepsilon} \mathcal{J}_{\varepsilon}( w_{\varepsilon,y} ) = \varepsilon^N \sup_{y \in \partial\Lambda_\varepsilon}
b_{V(y)} + o(\varepsilon^N)\:.
\end{equation*}
The estimate follows from \eqref{estim:by}, \eqref{limsupLambdaepsilon} and the continuity of \(V\).
\end{proof}
\subsubsection{Upper estimate of the critical level \(c_{\varepsilon}\)}
The same method gives an upper estimate on \(c_{\varepsilon}\).
\begin{lemma}\label{estim:sup2}
We have
\begin{equation*}
\limsup_{\varepsilon \to 0}\varepsilon^{-N} c_{\varepsilon} \le \sup_{\Lambda} \mathcal{C}\:.
\end{equation*}
\end{lemma}
\begin{proof}
As a test path in \eqref{ceps}, we take \(w_{\varepsilon,y}\) defined by \eqref{wepsy} for every \(y \in \Lambda_\varepsilon\). We obtain the first
estimate after a straightforward computation in view of \eqref{liminfLambdaepsilon}.
\end{proof}
\subsubsection{Lower estimate of the critical level \(c_{\varepsilon}\)}
A more delicate construction gives a lower estimate of the critical level \(c_{\varepsilon}\).
\begin{lemma}
\label{lemmaEstimatecepsilon}
If
\[
\sup_{\Lambda} \mathcal{C} > \inf_{\Lambda} \mathcal{C},
\]
then
\[
\liminf_{\varepsilon \to 0} \varepsilon^{-N} c_{\varepsilon} > \inf_{\Lambda} \mathcal{C}.
\]
\end{lemma}
We do not know whether one has the natural stronger conclusion
\[
\liminf_{\varepsilon \to 0} \varepsilon^{-N} c_{\varepsilon} \geq \sup_{\Lambda} \mathcal{C}.
\]
\begin{lemma}\label{lem:barycenter}
Let \( x \in \Lambda\). There exists \(\varepsilon_0 > 0\) such that for every \(\varepsilon \in{} (0, \varepsilon_0)\) and every \(\gamma \in \Gamma_{\varepsilon}\), there exists \(z \in \Lambda_\varepsilon\) such that \(\beta (\gamma(z)) = x\).
\end{lemma}
\begin{proof}
For every \(z \in \partial \Lambda_\varepsilon\), one has in view of the definition of \(w_{\varepsilon, y}\),
\[
\beta(w_{\varepsilon, y} )=y + o(1),
\]
as \(\varepsilon \to 0\), uniformly in \(z\).
Let \(\gamma \in \Gamma_\varepsilon\).
Therefore, if \(\varepsilon\) is small enough, one has for every
\(y \in \partial \Lambda_\varepsilon\), by \eqref{limsupLambdaepsilon}, \(x \in \Lambda_{\varepsilon}\) for \(n\) large enough.
When \(\varepsilon\) is small enough, we have
\[
\sup_{y \in \partial \Lambda_{\varepsilon}} \abs{\beta(w_{\varepsilon, y} )-y} < \inf_{y \in \partial \Lambda_{\varepsilon}} \abs{y-x},
\]
therefore, by the properties of the topological degree, if \(x \in \Lambda_{\varepsilon}\) there exists \(z \in \Lambda_\varepsilon\) such that \(\beta (\gamma(z)) = x\).
\end{proof}
We follow the arguments of \cite{delPinoFelmer1997}. Heuristically, the idea is to show that a sequence of functions violating the strict
inequality \eqref{estim} cannot have enough energy to stay concentrated inside \(\Lambda\) and must thus concentrate around a
point of \(\partial \Lambda\). But this would in fact contradict the continuity of the paths in \(\Gamma_{\varepsilon}\).
\begin{proof}[Proof of Lemma~\ref{lemmaEstimatecepsilon}]
Assume by contradiction that there is a sequence \( (\varepsilon_n)_{n \in \mathbb{N}}\) such that \(\lim_{n \to \infty} \varepsilon_n = 0\) and \(\lim_{n \to \infty} \varepsilon_n^{-N} c_{\varepsilon_n} \le \inf_{\Lambda} \mathcal{C}\).
By definition of \(c_\varepsilon\), there exists \(\gamma_n \in \Gamma_{\varepsilon_n}\) such that
\[
\lim_{n \to \infty} \sup_{x \in \Lambda_{\varepsilon_n}} \varepsilon_n^{-N}\mathcal{J}_{\varepsilon_n}\bigl(\gamma_n(x)\bigr) \le \inf_{\Lambda} \mathcal{C}\:.
\]
Choose \(x\in \Lambda\) such that \(V(x) > \inf_{\Lambda} V\).
For each \(n \in \mathbb{N}\) large enough, let \(x_n\) be given by Lemma \ref{lem:barycenter} so that \(\beta(\gamma_n(x_n)) = x\). One has
\[
\limsup_{n \to \infty} \varepsilon_n^{-N}\mathcal{J}_{\varepsilon_n}\bigl(\gamma_n(x_n)\bigr) \leq \inf_{\Lambda} \mathcal{C}\:.
\]
Proposition~\ref{propositionAsymptoticsMoments} brings then a contradiction.
\end{proof}
\subsection{Existence of a solution}
We are now in a position to prove the strict inequality \eqref{estim}.
\begin{lemma}
\label{lemmaStrict}
If
\[
\sup_{\Lambda} \mathcal{C} > \inf_{\Lambda} \mathcal{C}= \sup_{\partial\Lambda} \mathcal{C}\:,
\]
then
\[
\liminf_{\varepsilon \to 0} \varepsilon^{-N} (c_{\varepsilon} - a_{\varepsilon})>0\:.
\]
\end{lemma}
\begin{proof}
This follows directly from Lemmas~\ref{lem:estimaeps} and \ref{lemmaEstimatecepsilon}.
\end{proof}
As a consequence of the General Minimax Principle, we have thus proved the following existence result for the penalized problem \eqref{problemPNLSE}.
\begin{proposition}
Let \(N\geq 3\), \(1 < p < \frac{N+2}{N-2}\) and let \(g_{\varepsilon}: \mathbb{R}^N \times \mathbb{R}^+ \rightarrow \mathbb{R}\) be a function satisfying assumptions \((g_1)\)-
\((g_4)\). For \(\varepsilon > 0\) small enough, there exists \(u_\varepsilon \in \mathcal{N}_\varepsilon\) such that \(\mathcal{J}_{\varepsilon}(u_\varepsilon)= c_{\varepsilon}\) and \(\mathcal{J}_{\varepsilon}'(u_\varepsilon)= 0\).
\end{proposition}
\begin{proof}
This follows from the general minimax principle (Theorem~\ref{lem:estimaeps}), the Palais-Smale condition coming from Lemma~\ref{lem:PalaisSmale} and the strict inequality of Lemma~\ref{lemmaStrict}.
\end{proof}
\section{Back to the original problem}
\label{sectionOriginalProblem}
\subsection{Asymptotics of solutions}
Thanks to the asymptotics of solutions of Section~\ref{sectAsymptoticsCritical} and the estimates on the critical level of Section~\ref{sectionEstimatesLevels}, we prove that the solution \(u_{\varepsilon}\) is single-peaked.
\begin{lemma}\label{lem:unifdecay}
Let \((u_{\varepsilon})_{\varepsilon>0}\) be a family such that for \(\varepsilon > 0\) small enough, \(\mathcal{J}_{\varepsilon}(u_\varepsilon)= c_{\varepsilon}\) and \(\mathcal{J}_{\varepsilon}'(u_\varepsilon)= 0\).
Let \((x_{\varepsilon})_{\varepsilon>0}\) in \(\Lambda\) be such that
\[
\liminf_{\varepsilon\to 0} u_{\varepsilon}(x_{\varepsilon}) > 0\:.
\]
If
\[
\sup_{\Lambda} \mathcal{C} < 2 \inf_{\Lambda} \mathcal{C}\: ,
\]
then for every \(U \subset \mathbb{R}^N\) such that \(\Bar{U}\) is compact and \(\inf_U V > 0\),
\[
\lim_{\substack{\varepsilon\to 0 \\ R \to \infty}} \norm{u_{\varepsilon}}_{L^{\infty}(U \setminus
B(x_{\varepsilon},\varepsilon R))} = 0\:.
\]
If moreover
\[
\sup_{\Lambda} \mathcal{C} > \inf_{\Lambda} \mathcal{C}= \sup_{\partial\Lambda} \mathcal{C}\:,
\]
then
\[
\liminf_{\varepsilon\to 0} d(x_{\varepsilon}, \mathbb{R}^N \setminus \Lambda) > 0\:.
\]
\end{lemma}
\begin{proof}
First we prove that
\[
\lim_{\substack{\varepsilon\to 0 \\ R \to \infty}} \norm{u_{\varepsilon}}_{L^{\infty}(U \setminus
B(x_{\varepsilon},\varepsilon R))} = 0\:.
\]
Assume by contradiction that there exist sequences \((\varepsilon_n)_{n\in \mathbb{N}}\) and
\((y_n)_{n\in \mathbb{N}}\) such that for every \(n \in \mathbb{N}\), \(y_n \in U\),
\begin{align*}
\lim_{n\to \infty} \varepsilon_n &= 0\:, & \liminf_{n\to \infty} u_{\varepsilon_n}(y_n) &> 0 &
&\text{and} & \lim_{n\to \infty} \frac{\abs{x_{\varepsilon_n}-y_n}}{\varepsilon_n} &= +\infty\:.
\end{align*}
Then, by Lemma~\ref{estim:sup2},
\begin{equation*}
\limsup_{n\to\infty} \varepsilon_n^{-N} \mathcal{J}_{\varepsilon_n}(u_{\varepsilon_n}) \le \sup_{\Lambda} \mathcal{C}\:,
\end{equation*}
while by Proposition~\ref{estim:inf2}
\begin{equation*}
\liminf_{n\to\infty} \varepsilon_n^{-N} \mathcal{J}_{\varepsilon_n}(u_{\varepsilon_n})
\geq \liminf_{n\to\infty} \bigl( \mathcal{C}(x_{\varepsilon_n})
+ \mathcal{C}(y_n) \bigr) \geq 2 \inf_{\Lambda} \mathcal{C}\: .
\end{equation*}
This is a contradiction with our assumption.
\medbreak
Now we turn to the second assertion. Assume by contradiction that there exists a sequence \((\varepsilon_n)_{n\in \mathbb{N}}\) such that
\(\varepsilon_n \to 0\) and \(\lim_{n\to \infty} \dist (x_{\varepsilon_n}, \mathbb{R}^N\setminus \Lambda) = 0\). Then, by the first assertion, the second part of Proposition~\ref{estim:inf2}
and Lemma~\ref{lemmaEstimatecepsilon},
\begin{equation*}
\inf_{\Lambda} \mathcal{C}
< \liminf_{n\to\infty} \varepsilon_n^{-N} \mathcal{J}_{\varepsilon_n}(u_{\varepsilon_n})
= \liminf_{n\to\infty} \mathcal{C}(x_{\varepsilon_n})
= \sup_{\partial\Lambda} \mathcal{C}\:.
\end{equation*}
But this contradicts our assumption.
\end{proof}
This allows now to show that \(u_\varepsilon\) is a subsolution for some second-order linear elliptic operator.
\begin{lemma}
\label{lemmaIneqSmallBalls}
Let \((u_{\varepsilon})_{\varepsilon>0}\) be a family of positive solutions of \eqref{problemPNLSE} at the level \(c_{\varepsilon}\)
and let \((x_{\varepsilon})_{\varepsilon>0} \subset \Lambda\) be such that
\[
\liminf_{\varepsilon\to 0} u_{\varepsilon}(x_{\varepsilon}) > 0.
\]
Then there exists \(\varepsilon_0 > 0\) and \(R>0\) such that for all \(\varepsilon \in (0,\varepsilon_0)\),
\[
-\varepsilon^2 (\Delta + \mu H) u_{\varepsilon} + (1-\mu) V u_{\varepsilon} \leq 0 \hspace{0.5cm} \text{in} \ \mathbb{R}^N\setminus
B(x_{\varepsilon},\varepsilon R).
\]
\end{lemma}
\begin{proof}
This follows from Lemma~\ref{lem:unifdecay}, see \cite{MorozVanSchaftingen2010}*{Lemma 5.1} for the details.
\end{proof}
We then have a comparison principle \cite{MorozVanSchaftingen2010}*{Lemma 3.2}.
\begin{lemma}
\label{lemmaComparison}
Let \(\Omega \subset \mathbb{R}^N\) be a domain with a smooth boundary.
Let $v \in H^1_V (\Omega)$ and \(w \in H^1_\mathrm{loc} (\Bar{\Omega})\) be such that \(w \ge 0\) in \(\Omega\).
If
\begin{equation*}
- \varepsilon^2 (\Delta + \mu H) v + (1 - \mu) V v
\le - \varepsilon^2 (\Delta + \mu H) w + (1 - \mu) V w, \qquad \text{weakly in \(\Omega\)}.
\end{equation*}
and \(v \leq w\) on \(\partial\Omega\),
then \(v \leq w\) in \(\Omega\).
\end{lemma}
\begin{proof}
Take \(\psi \in C^\infty_c(\mathbb{R}^N)\) such that \(\psi \equiv 1\) on \(B_1\) and \(\supp \psi \subset B_2\) and define \(\psi_n\in C^\infty_c(\mathbb{R}^N)\)
for \(n \in \mathbb{N}\) and \(x \in \mathbb{R}^N\) by \(\psi_n(x)=\psi(x/n)\).
Using \( \psi_n^2(v - w)_+ \in H^1_V(\Omega)\) as a test function in the inequation, one has
\begin{multline*}
\int_{\Omega} \bigl(\varepsilon^2 \abs{\nabla (\psi_n (v - w)_+)}^2 - \varepsilon^2 \mu H \abs{\psi_n (v - w)_+}^2 + (1 - \mu) V \abs{\psi_n (v -w)_+}^2 \bigr)\\
\le \varepsilon^2 \int_{\Omega} \abs{\nabla \psi_n}^2 (v - w)_+^2\:.
\end{multline*}
By Lemma~\ref{lemmaPositivity} on the one hand and by
definition of \(\psi_n\) and nonnegativity of \(w\) on the other hand, we have
\[
(1 - \mu) \int_{\Omega} \varepsilon^2 \abs{\nabla (\psi_n (v - w)_+)}^2 + V \abs{\psi_n (v - w)_+}^2
\le C \int_{B_{2 n} \setminus B_n} \frac{\abs{v(x)}^2}{\abs{x}^2}\,dx.
\]
By Lebesgue dominated convergence, we deduce that \( \psi_n (v - w)_+ \to 0\) strongly in \(H^1_V (\Omega)\) as \(n \to \infty\). Hence, \( (v - w)_+ = 0\).
\end{proof}
\subsection{Barrier functions and solution of the original problem}
Since \(u_\varepsilon\) is a subsolution for some second-order linear elliptic operator, we shall compare it with supersolutions of that operator. We first recall how suitable supersolutions can be constructed.
\subsubsection{The case of fast decaying potentials}
Indepentently of the decay rate of \(V\) we have
\begin{lemma}
\label{lemmaBarrierFast}
Let \((x_{\varepsilon})_{\varepsilon} \subset \Lambda\) be such that \(\liminf_{\varepsilon\to 0} d(x_{\varepsilon},\partial \Lambda) > 0\), let \(\mu \in (0,1)\) and let \(R>0\).
If \(N \ge 3\), then there exists \(\varepsilon_0 > 0\) and a family of functions \((W_{\varepsilon})_{0<\varepsilon<\varepsilon_0}\) in \(C^{1,1}(\mathbb{R}^N\setminus B(x_{\varepsilon},\varepsilon R))\)
such that, for \(\varepsilon \in (0,\varepsilon_0)\),
\begin{enumerate}[(i)]
\item \(W_{\varepsilon}\) satisfies the inequation
\[
-\varepsilon^2 (\Delta + \mu H) W_{\varepsilon} + (1-\mu) V W_{\varepsilon} \geq 0 \hspace{0.5cm} \text{in} \ \mathbb{R}^N\setminus
B(x_{\varepsilon},\varepsilon R),
\]
\item \(\nabla W_{\varepsilon} \in L^2(\mathbb{R}^N\setminus B(x_{\varepsilon},\varepsilon R))\),
\item \(W_{\varepsilon} = 1\) on \(\partial B(x_{\varepsilon},\varepsilon R)\),
\item there exist \(C, \lambda, \nu > 0\) such that for every \(x \in \mathbb{R}^N \setminus B(x_{\varepsilon},\varepsilon R)\),
\[
W_{\varepsilon}(x) \leq C \exp \left( -\frac{\lambda}{\varepsilon} \frac{\abs{x-x_{\varepsilon}}}{1+\abs{x-x_{\varepsilon}}} \right) \left( 1+\abs{x}^2 \right)^{-\frac{N-2}{2}}.
\]
\end{enumerate}
\end{lemma}
\begin{proof}
The arguments are the same as those of V.\thinspace Moroz and J.\thinspace Van Schaftingen \cite{MorozVanSchaftingen2010}*{Lemma 5.2}, since the penalization potential \(H\) is the same.
\end{proof}
The decay of \(u_\varepsilon\) is then similar to the decay of \(W_\varepsilon\).
\begin{proposition}
\label{propositionDecayFast}
Let \((u_{\varepsilon})_{\varepsilon>0}\) be a family of positive solutions of \eqref{problemPNLSE} at the level \(c_{\varepsilon}\)
and let \((x_{\varepsilon})_{\varepsilon>0} \subset \Lambda\) be such that
\[
\liminf_{\varepsilon\to 0} u_{\varepsilon}(x_{\varepsilon}) > 0\:.
\]
If
\(N \ge 3\)
then there exists \(C,\lambda > 0\) and \(\varepsilon_0 > 0\) and \(R>0\) such that for all \(\varepsilon \in (0,\varepsilon_0)\),
\begin{equation*}
u_{\varepsilon}(x) \leq C \exp \left( -\frac{\lambda}{\varepsilon} \frac{\abs{x-x_{\varepsilon}}}{1+\abs{x-x_{\varepsilon}}} \right)
\left( 1+\abs{x}^2 \right)^{-\frac{N-2}{2}},
\qquad x \in \mathbb{R}^N.
\end{equation*}
\end{proposition}
\begin{proof}
This is a consequence of Lemmas~\ref{lemmaIneqSmallBalls} and \ref{lemmaBarrierFast} together with the comparison principle (Lemma~\ref{lemmaComparison}).
\end{proof}
We can now go back to the original problem.
\begin{proposition}
\label{propOriginalFast}
Let \((u_{\varepsilon})_{\varepsilon>0}\) be a family of positive solutions of \eqref{problemPNLSE} at the level \(c_{\varepsilon}\).
If
\(
\frac{1}{p} < \frac{N-2}{N},
\)
then there exists \(\varepsilon_0 > 0\) such that, for every \(0 < \varepsilon < \varepsilon_0\), \(u_{\varepsilon}\) solves the original
problem \eqref{problemNLSE}.
\end{proposition}
\begin{proof}
The proof follows the lines of \cite{MorozVanSchaftingen2010}*{Proposition 5.4}. By Lemma \ref{lem:1},
there exists a family of points \((x_{\varepsilon})_{\varepsilon>0} \subset \Lambda\) such that
\[
\liminf_{\varepsilon\to 0} u_{\varepsilon}(x_{\varepsilon}) > 0.
\]
By Lemma \ref{lem:unifdecay}, \(d_0 := \inf d(x_{\varepsilon}, \partial \Lambda) > 0\). Hence, by
proposition~\ref{propositionDecayFast}, we have for \(\varepsilon > 0\) small enough and for \(x \in \mathbb{R}^N\setminus\Lambda\),
\begin{equation*}
\begin{split}
\left(u_{\varepsilon}(x)\right)^{p-1} &\leq \Bigl( C \exp \Bigl( -\frac{\lambda}{\varepsilon} \frac{\abs{x-x_{\varepsilon}}}{1+\abs{x-x_{\varepsilon}}} \Bigr)
\bigl( 1+\abs{x}^2 \bigr)^{-\frac{N-2}{2}} \Bigr)^{p-1} \\
&\leq \mu \varepsilon^2 \frac{(N-2)^2}{4\abs{x-x_0}^2} \Bigl(\frac{\log \frac{\rho}{\rho_0}}{\log \frac{\abs{x-x_0}}{\rho_0}}
\Bigr)^{1+\beta}
= \mu \varepsilon^2 H(x).
\end{split}
\end{equation*}
By definition of the penalized nonlinearity \(g_{\varepsilon}\), one has then
\begin{equation*}
g_{\varepsilon}\left( x, u_{\varepsilon}(x) \right) = \left( u_{\varepsilon}(x) \right)^p, \qquad x \in \mathbb{R}^N\setminus\Lambda,
\end{equation*}
and therefore \(u_{\varepsilon}\) solves the original problem \eqref{problemNLSE}.
\end{proof}
\subsubsection{The case of slow decaying potentials}
Now we assume that \(\liminf_{\abs{x} \to \infty} V(x) \abs{x}^2\). We first have a counterpart of Lemma~\ref{lemmaBarrierFast}.
\begin{lemma}
Let \((x_{\varepsilon})_{\varepsilon} \subset \Lambda\) be such that \(\liminf_{\varepsilon\to 0} d(x_{\varepsilon},\partial \Lambda) > 0\), let \(\mu \in (0,1)\) and let \(R>0\).
If
\[
\liminf_{\abs{x} \to \infty} V(x) \abs{x}^2 > 0,
\]
then, there exists \(\varepsilon_0 > 0\) and a family of functions \((W_{\varepsilon})_{0<\varepsilon<\varepsilon_0}\) in \(C^{1,1}(\mathbb{R}^N\setminus B(x_{\varepsilon},\varepsilon R))\)
such that, for \(\varepsilon \in (0,\varepsilon_0)\),
\begin{enumerate}[(i)]
\item \(W_{\varepsilon}\) satisfies the inequation
\[
-\varepsilon^2 (\Delta + \mu H) W_{\varepsilon} + (1-\mu) V W_{\varepsilon} \geq 0 \hspace{0.5cm} \text{in} \ \mathbb{R}^N\setminus
B(x_{\varepsilon},\varepsilon R),
\]
\item \(\nabla W_{\varepsilon} \in L^2(\mathbb{R}^N\setminus B(x_{\varepsilon},\varepsilon R))\),
\item \(W_{\varepsilon} = 1\) on \(\partial B(x_{\varepsilon},\varepsilon R)\),
\item there exist \(C, \lambda, \nu > 0\) such that for every \(x \in \mathbb{R}^N\setminus B(x_{\varepsilon},\varepsilon R)\),
\[
W_{\varepsilon}(x) \leq C \exp \Bigl( -\frac{\lambda}{\varepsilon} \frac{\abs{x-x_{\varepsilon}}}{1+\abs{x-x_{\varepsilon}}} \Bigr) \bigl( 1+\abs{x}^2 \bigr)^{-\frac{\nu}{\epsilon}}.
\]
\end{enumerate}
\end{lemma}
\begin{proof}
See the discussion after \cite{MorozVanSchaftingen2010}*{Theorem 7}.
\end{proof}
As a consequence we have
\begin{proposition}
\label{propositionDecaySlow}
Let \((u_{\varepsilon})_{\varepsilon>0}\) be a family of positive solutions of \eqref{problemPNLSE} at the level \(c_{\varepsilon}\)
and let \((x_{\varepsilon})_{\varepsilon>0} \subset \Lambda\) be such that
\[
\liminf_{\varepsilon\to 0} u_{\varepsilon}(x_{\varepsilon}) > 0\:.
\]
If
\[
\liminf_{\abs{x} \to \infty} V(x) \abs{x}^2 > 0,
\]
then there exists \(C,\lambda, \nu > 0\) and \(\varepsilon_0 > 0\) and \(R>0\) such that for all \(\varepsilon \in (0,\varepsilon_0)\),
\begin{equation*}
u_{\varepsilon}(x) \leq C \exp \left( -\frac{\lambda}{\varepsilon} \frac{\abs{x-x_{\varepsilon}}}{1+\abs{x-x_{\varepsilon}}} \right)
\bigl( 1+\abs{x}^2 \bigr)^{-\frac{\nu}{\epsilon}},
\qquad x \in \mathbb{R}^N.
\end{equation*}
\end{proposition}
This allows us to go back to our original problem.
\begin{proposition}
\label{propOriginalSlow}
Let \((u_{\varepsilon})_{\varepsilon>0}\) be a family of positive solutions of \eqref{problemPNLSE} at the level \(c_{\varepsilon}\).
If
\begin{equation*}
\liminf_{\abs{x} \to \infty} V(x) \abs{x}^2 > 0,
\end{equation*}
then there exists \(\varepsilon_0 > 0\) such that, for every \(0 < \varepsilon < \varepsilon_0\), \(u_{\varepsilon}\) solves the original
problem \eqref{problemNLSE}.
\end{proposition}
\begin{proof}
The proof begins as the proof of Proposition~\ref{propOriginalFast}. Applying
proposition~\ref{propositionDecaySlow}, we have for \(\varepsilon > 0\) small enough and for \(x \in \mathbb{R}^N\setminus\Lambda\),
\begin{equation*}
\begin{split}
\bigl(u_{\varepsilon}(x)\bigr)^{p-1} &\leq \Bigl( C \exp \Bigl( -\frac{\lambda}{\varepsilon} \frac{\abs{x-x_{\varepsilon}}}{1+\abs{x-x_{\varepsilon}}} \Bigr)
\bigl( 1+\abs{x}^2 \bigr)^{-\frac{\nu}{\epsilon}} \Bigr)^{p-1} \\
&\leq \mu \varepsilon^2 \frac{(N-2)^2}{4\abs{x-x_0}^2 \bigl(\log \frac{\abs{x-x_0}}{\rho_0}
\bigr)^{1+\beta}}
= \mu \varepsilon^2 H(x)\:,
\end{split}
\end{equation*}
and therefore \(u_{\varepsilon}\) solves the original problem \eqref{problemNLSE}.
\end{proof}
\subsubsection{Proof of the main theorem}
Finally we complete the proof of the main theorem.
\begin{proof}[Proof of Theorem \ref{theoremMainLambda}]
Let \(u_\varepsilon\) be the solution of the penalized problem \eqref{problemPNLSE} from Proposition~\ref{prop:Minimizer}. By Lemma \ref{lem:1}, there exists \((x_\varepsilon)_{\varepsilon > 0}\) such that
\[
\liminf_{\varepsilon \to 0} u_\varepsilon(x_\varepsilon) > 0.
\]
By Lemma \ref{lem:unifdecay},
\[
\liminf_{\varepsilon\to 0} d(x_{\varepsilon}, \mathbb{R}^N \setminus \Lambda) > 0\:.
\]
By Proposition~\ref{propOriginalFast} or \ref{propOriginalSlow},
\(u_\varepsilon\) solves \eqref{problemNLSE} for \(\varepsilon\) small enough.
\end{proof}
\section{The low-dimensional case}
\label{sectionSlow}
In the case \(N \le 2\), we do not have the Hardy inequality, but since one cannot have \(\frac{1}{p} < \frac{N-2}{N}\), we can use some information about the decay of \(V\).
We define the penalization
potential \(H : \mathbb{R}^N \to \mathbb{R}\) by
\begin{equation*}
H(x) := \bigl(1-\chi_{\Lambda}(x)\bigr) \frac{1}{\abs{x-x_0}^{2 + \beta}}
\end{equation*}
for some \(\beta > 0\).
In place of Lemma~\ref{lemmaPositivity}, we now have
\begin{lemma}
For every \(u \in C^{\infty}_c(\mathbb{R}^N)\),
\begin{equation*}
\int_{\mathbb{R}^N} \bigl( \abs{\nabla u}^2 - H \abs{u}^2 \bigr) \geq - C \int_{\mathbb{R}^N} V \abs{u}^2\:.
\end{equation*}
\end{lemma}
\begin{proof}
Since \(\liminf_{\abs{x} \to \infty} V(x) \abs{x}^2 > 0\), there exists \(R > 0\) such that if \(x \in R^N \setminus B_R\), \(H(x) \ge V(x)\). One has
\[
\int_{\mathbb{R}^N \setminus B_R} H \abs{u}^2 \leq \int_{\mathbb{R}^N \setminus B_R} V \abs{u}^2\:.
\]
Taking \(\varphi \in C^\infty (\mathbb{R}^N)\) such that \(\varphi \ge 0\), \(\supp \varphi \subset B_{2R}\) and \(\varphi = 1\) on \(B_R\), by the Sobolev inequality,
\[
\begin{split}
\int_{B_R} H \abs{u}^2 \le \int_{\mathbb{R}^N} H \abs{ \varphi u}^2 & \leq C \int_{\mathbb{R}^N} V \abs{(1-\varphi)u}^2 + \frac{1}{2}\int_{\mathbb{R}^N} \abs{\nabla (\varphi u) }^2 \\
&\le C \int_{\mathbb{R}^N} V \abs{(1-\varphi)u}^2 + \int_{\mathbb{R}^N} \bigl( \abs{\varphi}^2\abs{\nabla u}^2 + \abs{\nabla \varphi}^2 \abs{u}^2\bigr).
\end{split}
\]
from which the conclusion follows.
\end{proof}
The proof of the counterpart of the Palais-Smale condition (Lemma~\ref{lem:PalaisSmale}) for \(N \in \{1, 2\}\) relies on the condition \(\lim_{\abs{x} \to \infty} V(x) \abs{x}^2 > 0\).
The rest of the proof is the same up to minor modifications when Lemma~\ref{lemmaPositivity} is used. These modification of the argument works in fact also for \(N \ge 3\) when one assumes that \(\lim_{\abs{x} \to \infty} V(x) \abs{x}^2 > 0\).
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\section*{Acknowledgments}
The authors thank O. Dial, B. Halperin, V. Manucharyan, and J. Sau for helpful discussions. This work is supported by the Center for Integrated Quantum Materials (CIQM) under NSF award 1231319 (LSL and OS) and the U.S. DOE Office of Basic Energy Sciences, Division of Materials Sciences and Engineering under award DE-SC0001819 (PJH, MTA, AY). Nanofabrication was performed at the Harvard Center for Nanoscale Systems (CNS), a member of the National Nanotechnology Infrastructure Network (NNIN) supported by NSF award ECS-0335765. AA was supported by the Foundation for Fundamental Research on
Matter (FOM), the Netherlands Organization for Scientific Research
(NWO/OCW). ICF was supported by the European Research Council under
the European Union's Seventh Framework Programme (FP7/2007-2013) / ERC
Project MUNATOP, the US-Israel Binational Science Foundation, and the
Minerva Foundation for support.
\section*{Figure legends}
\textbf{Fig. 1.}${}\quad$\textbf{`Fiber-optic' modes and spatially resolved current imaging in a graphene Josephson junction}. \textbf{(A, B)} Guided edge modes induced by an intrinsic band bending near crystal boundary, for single-layer and bilayer graphene (schematic). Mode frequencies positioned outside the Dirac continuum ensure mode decoupling from the bulk states. Guided modes exist for any edge potential no matter how weak. In a single layer, mode velocity changes sign as the potential strength increases, see Eq.(5). In a bilayer, the modes occur in pairs [\textit{green and red curves}: dispersion for positive and negative potential strength, respectively].
\textbf{(C)} The
guided modes are manifested through peaks in
the density of current-carrying states at the crystal boundaries, prominent near charge neutrality
(\textit{red}: $n=0.05\times10^{11} cm^{-2}$; \textit{blue}: $n=2.5\times10^{11} cm^{-2}$). \textbf{(D)} Schematics of superconducting interferometry in a graphene Josephson junction,
which
is used to image the spatial structure of current-carrying states. A flux is threaded through the junction area to produce interference patterns, as current bias $V_{sd}$ is applied through
the superconducting electrodes and the voltage drop across the device is recorded. Carrier density $n$ is tuned by a gate voltage $V_{\rm b}$. \textbf{(E, F)}
The recorded interference pattern is of a single-slit Fraunhofer type at high carrier density, turning into a SQUID-like interference near neutrality
(colorscale is $dV/dI\,(\Omega)$ for device $BL1$). \textbf{(G, H)} Current flow, extracted from the interference data using Fourier techniques, is uniform at high carrier density and
peaks at the crystal edges for carrier density close to neutrality. \\
\textbf{Fig. 2.}${} \quad$\textbf{Gate-tunable evolution of edge and bulk current-carrying states in graphene}. \textbf{(A)} Edge-dominated
SQUID-like interference pattern at neutrality
in device $ML1$ ($n=2.38\times 10^9$ cm$^{-2}$; colorscale is $dV/dI (\Omega)$). \textbf{(B, C)} Real-space image of current flow confined to the boundaries over a range of densities near neutrality, shown alongside with the raw interference data
(corresponding to the white box in (D)).
\textbf{(D)} A real-space map of current flow as a function of electron concentration reveals coexistence of edge and bulk modes at intermediate densities. \textbf{(E)} Conventional Fraunhofer pattern for uniform current flow at high electron density ($n=7\times 10^{11}$ cm$^{-2}$). \textbf{(F)} Comparison of current amplitudes along the edge (red) and bulk (blue) from the plot in panel (C).
Current flow is edge-dominated near neutrality. Note that
minima for both contributions coincide in $n$, indicating that a positional
edge/bulk density offset is not present.\\
\textbf{Fig. 3.}${} \quad$\textbf{Boundary currents in bilayer graphene in the presence of broken crystal inversion symmetry}. \textbf{(A)} Spatially resolved supercurrent map in device $BL2$, in a normalized plot of $J(x)/J_{max}(x)$. Edge-dominated transport occurs near charge neutrality, while an increasing bulk contribution is tuned with carrier concentration. \textbf{(B)} Comparison of current amplitudes along the edge (red) and through the bulk (blue) from panel (A). Enhanced edge currents
are prominent at neutrality, whereas a uniformly distributed flow is recovered at high densities. The normal state conductance $G(e^2/h)$
{\it vs.} carrier density is also shown (black). \textbf{(C)} Measurement schematic for superconducting interferometry in a dual-gated bilayer graphene Josephson junction.
A dual-gated device consists of bilayer graphene flake on hBN with a suspended top gate, where application of voltages $V_{\rm t}$ and $V_{\rm b}$ on the top and back gates enables independent control of the transverse electric field $E$ and carrier density $n$. \textbf{(D)} Resistance map as a function of $V_{\rm b}$ and $V_{\rm t}$ for bilayer $BL4$. Enhanced resistance at high $E$ fields indicates the emergence of a gate-tunable insulating state due to broken crystal inversion symmetry. \textbf{(E)} Spatially-resolved boundary currents as a function of $E$ field. The vertical axis is a trace along the red path labeled in (B). \textbf{(F)} Sequence of Fraunhofer measurements at various locations on the current map in panel (E).\\
\textbf{Fig. 4.}${} \quad$\textbf{`Fiber-optics' theoretical model of transport in graphene}. \textbf{(A)} Real-space maps of measured current flow $J(x)$ in bilayer device $BL3$ at fixed carrier densities on the hole side, showing edge currents near the Dirac point and a continuous evolution towards
bulk flow. \textbf{(B)} Theoretical plot of spatially resolved density of states in bilayer graphene at fixed carrier densities for edge waveguide model. For the simulation, an effective delta function potential approximation is used with the best-fit value $\lambda = 0.5 $ eV$\cdot$nm (see SOM).
Band mass of bilayer graphene is taken $0.04 m_e$ where $m_e$ is electron mass.\\
\newpage
\textbf{Figure 1}
\begin{figure}[!h]
\includegraphics[width=160mm]{Fig1.png}
\end{figure}
\newpage
\textbf{Figure 2}
\begin{figure}[!h]
\includegraphics[width=160mm]{Fig2.PNG}
\end{figure}
\newpage
\textbf{Figure 3}
\begin{figure}[!h]
\includegraphics[width=160mm]{Fig3.PNG}
\end{figure}
\newpage
\textbf{Figure 4}
\begin{figure}[!h]
\includegraphics[width=120mm]{Fig4.PNG}
\end{figure}
\end{document}
\section{Modeling electronic guided modes}
\textbf{Materials and Methods}\\
\underline{Modeling electronic guided modes}\\
A full model of supercurrent-carrying states in our system should account, in principle, for a number of microscopic effects.
This includes, in particular, the microscopic details of transport through the NS interfaces, the realistic edge potential profile due to band bending near graphene edge, as well as the effects of disorder. Since treating all these issues simultaneously and on equal footing makes such a modeling a daunting task, here we resort to some simplifications. First, we will completely ignore the effects of induced superconductivity,
focusing on the normal metallic state of a pure graphene. Second, we consider a clean system and account for disorder scattering perturbatively at the end. Third, since states in a clean system, being delocalized, are capable of carrying supercurrent, we will focus on evaluating the density of states (DOS) taking it to reflect on the current-carrying capacity of the system. Of course, such an approach may be questioned for disordered systems in which some states are localized, and therefore can contribute to DOS but not to supercurrent. However, taking into account that in a clean system all states possess a roughly similar current-carrying capacity, we adopt this approximation on the merit of its simplicity.\\
Turning to the discussion of system geometry, we note two points. First, as discussed in the main text, the problem of guided states on a halfplane near the edge $x>x_0$ can be mapped onto a similar problem on a full plane by accounting for the states in valleys $K$ and $K'$ mixing at the edge. This mapping is particularly transparent for the armchair edge, where the boundary condition for the spinor wavefunctions in the two valleys is simply $\psi_{K}+\psi_{K'}=0$. In this case, one can see that the two-valley half-plane problem is mathematically equivalent to the problem posed on a full plane for particles in just one valley, provided the line potential for the latter problem is taken to be a sum of the original edge potential and its mirror-reflected double, $V(x>x_0)\,\to\,V(|x-x_0|)$. \\
Second, the states with the wavelengths larger than the edge potential width can be described by a delta function approximation. In that, a realistic microscopic potential $V(x)$ is replaced by a delta-function pseudopotential $\tilde V(x)=\lambda\delta(x-x_0)$, where $\lambda=\int V(x')dx'$ and $x_0$ is the edge position. For a system of width $w$ with two parallel edges positioned at $x_0=\pm w/2$ we therefore arrive at the model
\begin{equation} \label{model}
V(x) = \lambda \delta(x+w/2)+\lambda \delta(x-w/2),
\end{equation}
with $-\infty<x<\infty$. Carriers in this system are described by the massless Dirac Hamiltonian
\begin{equation} \label{hamiltonian}
H =H_0+V(x),\quad H_0=v\sigma_1 p_x+v\sigma_2 p_y
\end{equation}
with $v\approx 10^6{\rm m/s}$ the carrier velocity and $\sigma_{1,2}$ the pseudospin Pauli matrices.
As stated above, we will use spatially-resolved DOS for the problem (\ref{hamiltonian}) as a measure of current-carrying capacity of the system. In justification we note that an electron system carrying normal electric current can be understood in terms of changes in the occupancy of the states near the Fermi energy. As a result, the spatially-resolved current density will vary in the same manner as DOS
\begin{equation}
N(\mu,\vec r) = \frac{dn(\vec r)}{d\mu}
,\quad
n(\vec r)=\langle \psi^\dagger(\vec r)\psi(\vec r)\rangle
.
\end{equation}
Here $n$ is the total carrier density and $\mu$ is chemical potential.
Below we evaluate DOS as a function of position and energy, focusing on the characteristic features due to the guided modes. \\
Taking into account that typical wavelength values of relevant electronic states, $\lambda \sim 10$nm , are much smaller than the distance between edges $w\sim \,1{\rm \mu m}$, we can represent DOS
in the form
\begin{equation}
\label{full_density}
N(\mu,x) = N_0(\mu)+N_1(\mu,x-w/2)+N_1(\mu,x+w/2)
\end{equation}
where $N_0$ is the DOS of a uniform infinite system,
\begin{equation}
\label{n_bulk}
N_0(\varepsilon) = \frac{|\varepsilon|}{2\pi \hbar^2v^2}
\end{equation}
and $N_1$ is the contribution to DOS from a single delta-function line potential, placed at $x=0$. Below we derive an expression
\begin{equation}
\label{n_edge}
N_1(\varepsilon,x) = \frac{4\lambda}{\pi \hbar v}\,\Im\int\frac{dp}{2\pi}\frac{p^2 e^{-2\kappa_{\varepsilon,p}|x|/\hbar}}{\kappa_{\varepsilon,p}
\bigl[ 4\lambda\varepsilon+(4-\lambda^2)\hbar v\kappa_{\varepsilon,p}\bigl]}
,\quad
\kappa_{\varepsilon,p} = \sqrt{p^2-(\epsilon/\hbar v)^2}
\end{equation}
where the energy $\varepsilon$ is taken to have an infinitesimal positive imaginary part. In the final result for DOS $\varepsilon$ must be replaced by the chemical potential, $\varepsilon = \mu$.
The spatial dependence described by Eq.\eqref{n_edge} is shown on Fig. \ref{peaks}.\\
In our model, which is essentially non-interacting, the effects of screening can be included {\it ad hoc} by treating the potential strength in Eq.\eqref{model} as a function of carrier density.
Since the latter is parameterized by $\mu$, we will use a simple model
\begin{equation} \label{eq:screening}
\lambda\rightarrow\lambda'=\frac{\lambda}{1+(|\mu|/\mu_0)^\alpha}
\end{equation}
where the parameter $\mu_0$ depends on microscopic details. Comparing to the data indicates that a reasonably good fit can be achieved for $\alpha \approx 2$.\\
Modeling results are presented in \addOS{Fig.1(c) of the main text} for energies corresponding to carrier densities $n = 0.05\cdot 10^{11} \text{cm}^{-2}$ (red curve) and $n = 2.5\cdot10^{11} \text{cm}^{-2}$ (blue
curve), where we evaluated $n$ accounting for the spin and valley degeneracy in graphene. Potential strength is chosen to be
$\lambda = - 1.5\,\hbar v \approx 1$ eV$\cdot$nm and the screening parameter value is $\mu_0 = 0.2\sqrt{\pi\hbar^2v^2n_0} \approx 7\,\text{meV}$, where $n_0 = 10^{11} \text{cm}^{-2}$ is the corresponding scale for density. \\
The simulation for graphene bilayer, which was used to generate \addOS{Fig.1(b) and Fig. 4(b) of the main text}, was carried out using an effective delta function potential approximation, as above. Greens function expressed through the T-matrix was used to obtain mode dispersion and DOS in a manner similar to our treatment of modes in a single layer. For the delta function strength we used the best-fit value $\lambda = 0.5 $ eV$\cdot$nm (and no screening).
\underline{Microscopic derivation}\\
Here we consider long-wavelength modes for a potential line positioned at $x=0$. This problem is described by the Hamiltonian \eqref{hamiltonian} with $V(x) = \lambda \delta(x)$. For this problem we construct the Greens function which takes the full account of scattering by the potential.
As is well known, the discrete spectrum of the system (in our case, the guided modes) can be conveniently expressed through the poles of the electron Greens function. Likewise, the spatially-resolved DOS is expressed as the Greens function trace. The Greens function, in turn, can be straightforwardly evaluated using Dysons's equation and the T-matrix representation:
\begin{equation}\label{eq:G}
G = G_0+G_0 V G_0+G_0 V G_0 V G_0 + \dots
=G_0+G_0 T G_0
\end{equation}
where $G_0= (i\epsilon-H_0)^{-1}$.
Assuming that the
phase and amplitude of the electron wavefunction are given by a continuous function of $x$, we can express the quantity $T$ as
\begin{equation}\label{eq:T_general}
T(\epsilon, p_y) =\lambda\Bigl(1-\lambda\int\frac{dp_x}{2\pi\hbar}G_0(\vec p)\Bigl)^{-1}
\end{equation}
The continuity assumption should in practice be relaxed by a weaker assumption accounting for the phase jump of the wavefunction across the delta function potential at $x=0$ (see main text). Here, however, we will proceed with Eq.\eqref{eq:T_general} on the merit of its simplicity.
Evaluating the integral in Eq.\eqref{eq:T_general} gives
\begin{equation}\label{eq:T}
T(\epsilon, p_y) = \lambda\Bigl(1+\frac{\lambda }{2\hbar v}\left( i\tilde\epsilon+\sigma_1\tilde p\right)\Bigl)^{-1}
\end{equation}
where we defined
\begin{equation}
\tilde\epsilon=\frac{\epsilon}{\sqrt{\epsilon^2+v^2p_y^2}}\qquad \tilde p=\frac{\hbar v p_y}{\sqrt{\epsilon^2+\hbar^2v^2p_y^2}}
\end{equation}
Here $\varepsilon$ is the Matsubara frequency, with a suitable analytic continuation $i\varepsilon\to\varepsilon+i0$ to be performed at the end.\\
The T-matrix poles give the guided modes dispersion
\begin{equation}\label{eq:dispersion}
\epsilon=\pm \hbar u|p_y|,\quad u=v\frac{4 \hbar^2v^2-\lambda^2}{4 \hbar^2v^2+\lambda^2}
\end{equation}
where the sign is given by $\pm=\text{sign}\lambda$.
Since $|u|<v$, the energies $\epsilon=\pm u |p_y|$ are positioned, for each $p_y$ value, outside the Dirac continuum of the bulk states.
This expression behaves in a qualitatively similar way to the exact dispersion derived in the main text, Eq.(1) \addOS{[see Fig.1(a) of the main text]}. The guided modes described by Eq.\eqref{eq:dispersion} are quasi-1D states that propagate as plane waves in the $y$ direction along the $x=0$ line and
decay exponentially as evanescent waves in the transverse direction.\\
Spatially-resolved DOS can be evaluated as
\begin{equation}\label{eq:n(E)}
N(\epsilon,\vec r)=-\frac1{\pi}{\rm Im}\,{\rm Tr\,} G(\epsilon, \vec r,\vec r')_{\vec r=\vec r'}
\end{equation}
where the energy variable is analytically continued from positive imaginary to real values via $i\epsilon\to\epsilon+i0$ and a trace is taken over pseudospin variables.
To proceed with our calculation, we will need Greens function evaluated in a mixed position-momentum representation
\begin{align}
&G_0(\epsilon,p_y,x)=\int \frac{dp_x}{2\pi}e^{ip_x x}G_0(\epsilon,\vec p)
\\\nonumber
&= \frac{-i\tilde\epsilon-\sigma_2 \tilde p-i\sigma_1\text{sign}(x)
}{2\hbar v} \exp\bigl(-\kappa(i\varepsilon) |x|/\hbar\bigl)
\end{align}
where $\kappa(i\varepsilon)=\sqrt{(\epsilon/\hbar v)^2+p_y^2}$.
The trace of an equal-point Greens function in Eq.(\ref{eq:n(E)}) then could be evaluated from Eq.\eqref{eq:G} with the help of Eq.\eqref{eq:T}:
\begin{equation}\label{eq:TrG}
{\rm Tr\,} G(\epsilon,x'=x)=\sum_{p_y}\Biggl(
\frac{\tilde\epsilon}{i\hbar v}
+\frac{4\lambda \tilde p^2 e^{-2\kappa |x|/\hbar}}{\hbar v\left[\left( 2+i\lambda\tilde\epsilon\right)^2-\lambda^2\tilde p^2\right]}\Biggl)
\end{equation}
where the two terms represent contributions of $G_0$ and $G_0VG_0$, respectively. \\
As a warmup, we consider the first term of \eqref{eq:TrG}.
Introducing a UV cutoff $p_0 = \varepsilon_0/\hbar v$ we evaluate the sum over $p_y$ as
\begin{equation}
\int_{-p_0}^{p_0}\frac{dp_y}{2\pi\hbar}\frac{\epsilon}{\sqrt{\epsilon^2+\hbar^2v^2p_y^2}}=\frac{\epsilon}{\pi \hbar v}\ln\frac{\epsilon_0}{\epsilon}
.
\end{equation}
Performing analytic continuation $\epsilon\to \delta-i\epsilon$, we arrive at
\begin{equation}
N_0(\varepsilon) = - \frac{\epsilon}{\pi^2 \hbar^2v^2}\Im\ln\frac{\epsilon_0}{\delta-i\epsilon}
\end{equation}
where $\delta=+0$. Taking the imaginary part, we obtain the expression in Eq.\eqref{n_bulk}.\\
Next, we proceed to evaluate the second term in Eq.\eqref{eq:TrG}. Performing the same analytic continuation, we arrive at the result in Eq.\eqref{n_edge}. The expression in Eq.\eqref{n_edge} can be conveniently analyzed by dividing the integral into two parts, taken over the domains $|p_y|>\varepsilon/\hbar v$ and $|p_y|<\varepsilon/\hbar v$, respectively. The latter contribution is particularly simple because it is governed by the pole \eqref{eq:dispersion} and can be easily evaluated, giving
\begin{equation}
\label{guided}
N_{\rm g.w.}(\varepsilon,x)=\frac{2\epsilon\lambda }{\hbar^2vu(4-\lambda^2)}
e^{-2 \sqrt{(v/u)^2-1} |x| |\epsilon|/\hbar v}
\end{equation}
This contribution is solely due to the guided edge mode. As illustrated in the Fig \ref{peaks},
this term dominates the peak structure in DOS for guided waves. \\
We used the full expression in Eq.\eqref{n_edge} to produce the spatially-resolved DOS curves shown in \addOS{Fig.1(c) of the main text}. In that, we accounted for screening, as described in Eq.\eqref{eq:screening}. Because of screening, the peak structure is more prominent at low chemical potential, and is suppressed relatively to the bulk DOS at high chemical
potential values.\\
\underline{The effect of disorder}\\
Here we estimate the disorder scattering rate $\gamma(k)$ for guided modes [see Eq.(1) in the main text and accompanying discussion]. We will model edge roughness by a fluctuating delta function strength, treating the fluctuations as a gaussian white noise:
\begin{equation}\label{eq:V+dV}
V(x,y)=(\lambda +\delta\lambda(y))\,\delta(x)
,\quad
\langle \delta\lambda(y)\delta\lambda(y')\rangle =\alpha\delta(y-y')
.
\end{equation}
Writing the Greens function as a series in the potential $V+\delta V$, Eq.(\ref{eq:V+dV}), we have
\begin{equation}
G=G_0+G_0(V+\delta V)G_0+G_0(V+\delta V)G_0(V+\delta V)G_0+...
\end{equation}
Averaging the Greens function over disorder, we only need to account for the pair correlators $\langle \delta\lambda(y)\delta\lambda(y')\rangle$.
In a non-crossing approximation, we express the disorder-averaged Greens function through a suitable self-energy
\begin{equation}
\langle G\rangle = G_0+ G_0(V+\Sigma)G_0+ G_0(V+\Sigma)G_0(V+\Sigma)G_0+...
\end{equation}
where
\begin{equation}\label{eq:Sigma}
\Sigma(\epsilon)=\alpha
\int \frac{dp_x}{2\pi} G(\epsilon,p_y,x,x')_{x=x'=0}
\end{equation}
The quantity (\ref{eq:Sigma}) is complex-valued, with the imaginary part expressed through the density of states at $x=0$ as
\begin{equation}
{\rm Im}\,\Sigma(\epsilon)=-\pi \alpha N(\epsilon)_{x=0}
\end{equation}
The disorder scattering rate for the guided waves can now be found from the dispersion relation obtained from the T-matrix pole, Eg(\ref{eq:T_general}),
which is corrected by the presence of $\Sigma$ as follows
\begin{equation} \label{eq:dispersion_scatter}
1+(\lambda+\Sigma(i\epsilon))\frac{i\tilde\epsilon+\sigma_1\tilde p}{2\hbar v}=0
.
\end{equation}
Here we continue to use Matsubara notation, as in Eqs.(\ref{eq:T_general}),(\ref{eq:T}).
Since the density of states scales linearly with energy, $N(\epsilon)\sim |\epsilon|$, we can solve Eq.\eqref{eq:dispersion_scatter} in the long-wavelength limit treating $\Sigma(i\epsilon)$ as a perturbation. Writing $\epsilon=\epsilon_0(p_y)+\delta\epsilon$, where $\epsilon_0=u|p_y|$ is a solution for $\Sigma=0$, we linearize in $\delta\epsilon$ to obtain
\begin{equation}
\delta\varepsilon = -\frac1\lambda\Bigl(1-\frac{u^2}{v^2}\Bigl)\Sigma(i\epsilon_0)|p_y|
\end{equation}
After analytic continuation, we obtain
\begin{equation}
\gamma(p_y) = \frac{\pi\alpha}{|\lambda|}\Bigl(1-\frac{u^2}{v^2}\Bigl)|p_y|N\bigl(u|p_y|\bigl)_{x=0}
\end{equation}
Accounting for the linear scaling $N\sim|\epsilon|$, we find that the damping rate scales as a square of $p_y$,
\begin{equation}
\gamma(p_y) =\frac{\lambda}{\hbar^2 v (4-\lambda^2)}p_y^2
\end{equation}
at small $p_y$. A similar dependence, albeit with a different prefactor, is found at large $p_y$.
From this we conclude that the modes are undamped over lengthscales $\sim\lambda^2/\xi$, where $\lambda$ is a wavelength and $\xi$ is a disorder lengthscale. Taking realistic values $\lambda\approx 10-100\,{\rm nm}$ and $\xi\approx 0.1\,{\rm nm}$, we obtain an estimate for the guided mode mean free path in the $1-10\,{\rm \mu m}$ range. These large values can be traced to the weak confinement of the waves at small $p_y$. The weak confinement results in the mode wavefunction positioned mostly outside the confining potential, which reduces the impact of scattering. The mean free path rapidly grows with wavelength, in a direct analogy with guided optical waves in weakly guiding fiber designs, where weak confinement is employed to achieve exceptionally long mean free paths. \\
\underline{Josephson junctions: Device overview}\\
We analyze five graphene Josephson junctions on hBN with widths ranging from $W=800-1200$ nm and lengths ranging from $L=250-350$ nm (see Fig. 1d for a labeled device schematic). Listed in Table S1 are details on individual sample geometries. The small $L/W$ aspect ratios place these devices are in the narrow junction limit, where the the critical current $I_c$ can be approximated as a phase dependent summation over many parallel 1D current channels (Equation 2 in the main text). Electrical measurements are conducted using standard Lockin techniques in a Leiden Cryogenics Model Minikelvin 126-TOF dilution refrigerator with a base temperature of 10 mK, well below the critical temperature of Al. \\
Using a dry transfer method, graphene/hBN stacks are sequentially deposited on a 300 nm thermally grown SiO$_2$ layer, which covers a doped silicon substrate functioning as a global back gate. Graphene flakes are etched to the desired geometry using a 950 PMMA A4 polymer mask ($\sim 200$ nm thick; spun at 4000 rpm) followed by an RIE O2 plasma etch. Titanium/aluminum (Ti/Al) superconducting electrodes are defined on selected flakes using electron beam (ebeam) lithography on a 950 PMMA A4 resist mask, followed by thermal evaporation and liftoff in acetone. For the titanium adhesion layer, we evaporate 10 nm at a rate of 0.3 Angstrom/s. This is followed by an evaporation of a 70 nm aluminum layer at a rate of 0.5 Angstrom/s at pressures in the low to mid $10^{-7}$ Torr range. For dual-gated bilayers, suspended top gates are fabricated using a standard PMMA/MMA/PMMA trilayer resist method which leaves a 200 nm air gap between the top gate and graphene. After using ebeam lithography to define the gates, which employs position-dependent dosage, Cr/Au (3/425 nm) gates are deposited using thermal evaporation and liftoff in acetone. To remove processing residues and enhance quality, devices were current annealed in vacuum at dilution refrigerator temperatures. We note that edge currents were detected both in current-annealed and intrinsically high quality non-annealed devices; typically the appearance of edge currents coincided with the occurrence of Fabry-Perot interference in the ballistic transport regime. All five graphene Josephson junctions exhibit similar transport behavior. Additional data sets are provided in the Supplementary Figures.\\
\underline{Fourier method for extraction of supercurrent density distribution}\\
In a magnetic field $B$, the critical current $I_c(B)$ through a Josephson junction equals the magnitude of the complex Fourier transform of the current density distribution $J(x)$:
\begin{equation}
I_c(B)=|\mathcal{I}_c(B)|=\left| \int_{-\infty}^{\infty} J(x) \exp(2\pi i(L+l_{Al})Bx/\Phi_0) dx \right|
\end{equation}
where $x$ is the dimension along the width of the superconducting contacts (labeled in Fig. 1d), $L$ is the distance between contacts, $l_{Al}$ is the magnetic penetration length (due to a finite London penetration depth in the superconductor and flux focusing), and $\Phi_0=h/2e$ is the flux quantum. Relevant in the narrow junction limit where current is only a function of one coordinate, Equation (28) provides a simple and concise description of our system. We employ Fourier techniques introduced by Dynes and Fulton to extract the real space current density distribution from the magnetic interference pattern $I_c(B)$. By expressing the current density as $J(x)=J_{s}(x)+J_{a}(x)$, where $J_{s}(x)$ and $J_{a}(x)$ are the symmetric and antisymmetric components, the complex critical current can be rewritten as:
\begin{equation}
\mathcal{I}_c(B)= \int_{-\infty}^{\infty}
J_s(x)\cos(2\pi (L+l_{Al})Bx/\Phi_0) dx
+ i\int_{-\infty}^{\infty}
J_a(x)\sin(2\pi (L+l_{Al})Bx/\Phi_0) dx
\end{equation}
We calculate symmetric component of distribution, the relevant quantity for analyzing edge versus bulk behavior, as the antisymmetric component goes to zero in the middle of the sample. For symmetric solutions, $\mathcal{I}_c(B)$ is purely real. To reconstruct $\mathcal{I}_c(B)$ from the measured critical current, the sign of $I_c(B)$ is reversed for alternating lobes of the Fraunhofer interference patterns. The extracted supercurrent distribution is expressed as an inverse Fourier transform:
\begin{equation}
J_s(x)\approx\int_{-\infty}^{\infty} \mathcal{I}_c(B)
\exp(2\pi i(L+l_{Al})Bx/\Phi_0) dB
\end{equation}
Because $I_c(B)$ is only nonzero over a rectangular window dictated by the finite scan range $B_{min}<B<B_{max}$, distribution extracted numerically is given by the convolution of $J(x)$ with the sinc function. To reduce artifacts due the convolution, we employ a raised cosine filter to taper the window at the endpoints of the scan. Explicitly,
\begin{equation}
J_s(x)\approx\int_{B_{min}}^{B_{max}} \mathcal{I}_c(B)
\cos^n(\pi B /2 L_B)
\exp(2\pi i(L+l_{Al})Bx/\Phi_0) dB
\end{equation}
where $n=0.5-1$ and $L_B=(B_{max}-B_{min})/2$ is the magnetic field range of the scan.\\
\underline{Gaussian fits to extract edge state widths}\\
To extract a length scale for the width of the edge currents near the Dirac point, we fit the experimental supercurrent density distribution $J_c(x)$ to the Gaussian function
\begin{equation}
J_c^G(x)= b\left( \exp\left( \frac{-(x-a)^2}{c} \right) +\exp\left( \frac{-(x+a)^2}{c} \right) \right)
\end{equation}
where $a$ determines the spatial peak offset, $b$ determines peak height, and $c$ determines peak width. For the data in Fig. 1H, the fit parameters are $a=0.515$, $b=8.8$, and $c=0.017$. The effective edge current width, given by the Gaussian full width at half maximum $x_{FWHM}=2\sqrt{c\cdot\ln2}$, is 220 nm.\\
\underline{Edge versus bulk amplitudes}\\
To more quantitatively assess the evolution of edge and bulk currents with electronic carrier density $n$, we plot line cuts of the individual contributions (see Fig. 2f and 3b). These are given by:
\begin{equation}
J_c^{edge}(n)=\sum_{x_i=-x_W}^{-x_W+\epsilon_1}\frac{J_c(x_i,n)}{N_1} \quad \mathrm{and} \quad
J_c^{bulk}(n)=\sum_{x_i=-\epsilon_2}^{\epsilon_2}\frac{J_c(x_i,n)}{N_2}
\end{equation}
for a graphene flake whose full width spans from $-x_W$ to $x_W$. $J_c^{edge}(n)$ is the spatially averaged current amplitude over a small window of width $\epsilon_1$ from the edge of the flake. Similarly, $J_c^{bulk}(n)$ is the spatially averaged current amplitude over a strip of width $2\epsilon_2$ around the center of the flake. $N_1=\epsilon_1/x_{step}$ and $N_2=\epsilon_2/x_{step}$, where $x_{step}$ is the distance between data points (determined by the magnetic field range of the scan). For example, for the plots in Fig. 2F, $x_W=405$ nm, $\epsilon_1= 29$ nm, and $\epsilon_2=87$ nm.\\
Based on the edge versus bulk current profiles, one may infer whether edge doping is the dominant cause of edge currents in our devices. In the presence of edge doping, the edge versus bulk contributions should be reversed for opposite polarities of bulk carriers (for example, edge dominated behavior at high densities on the electron side and bulk dominated behavior at high densities on the hole side), which is not consistent with the data. Bulk-dominated or flat distributions appear at both high electron and hole doping fairly consistently. As a second test, one can track the edge versus bulk contributions through the Dirac point to detect an offset in gate voltage between the charge neutrality point at the edge versus in the bulk. We did not detect positional density offset substantial enough to account for the large edge currents in these devices (Fig. 2F).\\
\underline{Bayesian method for extraction of supercurrent density distribution}\\
The critical current as a function of the magnetic field, $I_c(B)$, is related to the current density through the junction, $J_c(x)$, as
\begin{equation}\label{eq:ij}
I_c(B) = \int_{-\frac{W}{2}}^{\frac{W}{2}} dx\,J_c(x) \exp \left( 2\pi ix LB/\Phi_0 \right),
\end{equation}
with $L$ and $W$ the length and width of the junction, and $\Phi_0=h/2e$ the superconducting flux quantum.\\
In the measured $|I_c(B)|$ all information about its complex phase is lost, making the problem of determining the current density not have a unique solution. Using the method of Dynes and Fulton (DF), a unique solution can be found under the assumption of a symmetric current distribution, $J_c(x)=J_c(-x)$. In practice however, disorder and inhomogeneities in the junction will lead to asymmetric current densities. Additionally, since experiments are performed over a finite range of magnetic fields, there is a cutoff in the current density resolution. Neither this finite resolution, nor experimental uncertainties are taken into account in the DF method, meaning it can only provide a qualitative estimate of $J_c(x)$.\\
To gain a more quantitative understanding, we instead ask what is the distribution of $J_c(x)$ which produces the same critical current $I_c(B)$. We answer this question by performing Bayesian inference to obtain the posterior distribution of the current density, given the measured critical current. In our case, Bayes' rule reads:
\begin{equation}\label{eq:bayes}
P ( J_c ; |I_c| ) = \frac{ P( |I_c| ;J_c ) P (J_c )}{ P (|I_c| )}.
\end{equation}
Here, $ P ( J_c ; |I_c| )$ is the posterior distribution of the current density, the quantity we want to calculate, while $ P (J_c )$ is its prior distribution. The likelihood function $ P( |I_c| ;J_c )$ indicates the compatibility of the measured critical current with a given current density:
\begin{equation}
P( |I_c| ;J_c ) = \exp \left[ -\frac{(|I_c| - |I_c^f|)^2 }{2\varepsilon^2} \right],
\end{equation}
where $I_c^f$ is the current obtained from $J_c$ by using Eq.~\eqref{eq:ij}, $I_c$ is the measured current, and $\varepsilon$ is the measurement error. The factor $P (|I_c| )$ is the same for all current densities, meaning it does not enter into determining their relative probabilities.\\
The experimental current profiles are extracted from scans of the differential resistance as a function of DC current bias and magnetic field, $dV/dI(I_{\rm DC}, B)$. Within the same scan, for some field values $dV/dI$ has a clear maximum, while for others it monotonically increases towards its normal state value. We extract the critical current as the value $I_{\rm DC}$ at which the differential resistance is $x \times {\rm max}\,dV/dI$, choosing a value of $x\lesssim 1$. This selects points close to the maxima at field values where they are well defined, and points close to where the differential resistance reaches its normal state value otherwise. The uncertainty is obtained in the same fashion, by choosing a slightly smaller cutoff.\\
We maximize the likelihood function using a Monte Carlo sampling algorithm\cite{pymc}.
To get a large resolution of the current density without a significant increase in the dimensionality of the sampling space, we expand $J_c(x)$ as
\begin{equation}\label{eq:jc}
J_c(x) = \sum_{n=0}^{N} A_n \cos(2\pi n x/L)
\end{equation}
and enforce $J_c(x)>0$ for all $x$. The $A_n$ coefficients determine the shape of the distribution, which in Eq.~\eqref{eq:jc} is assumed to be symmetric, $J_c(x)=J_c(-x)$. Using an asymmetric form would typically lead to a critical current which shows node lifting -- the minima of $I_c(B)$ have nonzero values. While this feature is present in the measured critical current, it can be accounted for by factors other than an asymmetric current distribution \cite{Heida1998},
such as relatively small aspect ratios ($\sim$5), and a non-sinusoidal current-phase relationship arising from a large junction transparency. Using a symmetric $J_c$ avoids this ambiguity, and has the additional advantage of providing a more direct comparison between our method and that of Dynes and Fulton.\\
The likelihood function is maximized by allowing the $A_n$ coefficients to vary at each Monte Carlo step. As $N$ is increased the posterior distribution of the current density widens, an indication of over-fitting. This increase in uncertainty serves as a criterion for choosing $N$, which for the typical dataset is between 4 and 8. The priors of $A_n$ are set to the uniform distribution $[-\max(I_c), \max(I_c)]$.\\
An example of our method is shown in Fig.~\ref{fig:jcic}, using $N=5$. The current density is peaked at the edges of the sample, a feature also recovered in the DF approach. The corresponding critical current is in good agreement with the measured one, with the exception of the regions close to the nodes. Fig.~\ref{fig:jcic} indicates that the supercurrent through the junction flows mainly along its edges. As a further test of the edge state contribution, we modify the functional form of the current density in Eq.~\eqref{eq:jc}, to explicitly allow for edge states. We add delta functions to the current density at the edges of the sample, $J_c(x) \rightarrow J_c(x) + d_L \delta(x+W/2) + d_R \delta (x-W/2)$, and estimate the contribution of edge states as the ratio of $d_L+d_R$ to the total current density $J_c^{\rm tot}$. As the carrier density approaches zero a significant fraction of the supercurrent is carried by the edge states, with $(d_L+d_R)/J_c^{\rm tot} \simeq 0.45$ (see Fig.~\ref{fig:deltas}).
\newpage
\textbf{Supplementary Figures}\\
\begin{figure*}[h!]
\begin{minipage}[h!]{0.49\linewidth}
\center{\includegraphics[width=0.9\linewidth]{02.png}}
\end{minipage}
\hfill
\begin{minipage}[h!]{0.49\linewidth}
\center{\includegraphics[width=0.9\linewidth]{01.png}}
\end{minipage}
\caption{}
\label{peaks}
\end{figure*}
\textbf{Fig. S1}.
The excess contribution to the spatially-resolved DOS near a line delta function potential, $\Delta N(\epsilon,x)=N(\epsilon,x)-N_0(\epsilon)$ vs. distance from the delta function. Subtracted is the bulk contribution $N_0$ given in Eq.(S5). The left panel shows the full excess contribution obtained from Eq.(S6), the right panel shows the contribution solely due to the guided modes, Eq.(S18). The two contributions are nearly identical, confirming that the peak in DOS can serve as a telltale of the guided modes.Parameter values used: $\lambda=-1.5\hbar v$, energies $\epsilon=\epsilon_0$, $0.5\epsilon_0$, $0.1\epsilon_0$, where $\epsilon_0=\pi\hbar \sqrt{\pi n_0}$, $n_0 = 10^{11}\,\text{cm}^{-2}$ (higher peaks correspond to higher energy $\epsilon$ values).
\newpage
\begin{center}
\begin{figure}[t]
\includegraphics[width=0.8\textwidth]{Fig2supp.PNG}
\caption{}
\end{figure}
\end{center}
\textbf{Fig. S2}.
\textbf{(A)} Edge-dominated
SQUID-like interference pattern at neutrality in device $ML1$ ($n=2.38\times 10^9$ cm$^{-2}$; colorscale is $dV/dI (\Omega)$). From Fig. 2A in main text. \textbf{(B)} Real-space image of current flow confined to the boundaries, from data in part (A). \textbf{(C)} Conventional Fraunhofer pattern for uniform current flow at high electron density ($n=7\times 10^{11}$ cm$^{-2}$). From Fig. 2E in main text. \textbf{(D)} Real-space image of current flow confined to the boundaries, from data in part (C).
\newpage
\begin{center}
\begin{figure}[t]
\includegraphics[width=1.0\textwidth]{Data1.png}
\caption{}
\end{figure}
\end{center}
\textbf{Fig. S3}.
\textbf{(A)} Sequence of Fraunhofer measurements in bilayer device $BL3$ for the current maps in panels (B) and (C), shown in plots of $dV/dI (\Omega)$ as a function of magnetic field $B$ (mT) and current bias $I_{DC}$ (nA). \textbf{(B)} Real space image of current flow $J(x)$ as a function of carrier density on the hole side, showing edge currents near the Dirac point and a continuous evolution of bulk flow. \textbf{(C)} Individual line cuts of $J(x)$ plotted from (B). This is the data set in Fig. 4A, plotted with a properly scaled vertical axis (supercurrent density, nA/$\mu$m).
\newpage
\begin{center}
\begin{figure*}[htb]
\includegraphics[width=0.44\textwidth]{b15_5_ix_sym.pdf}
\includegraphics[width=0.44\textwidth]{b15_5_ib_sym.pdf}
\caption{\label{fig:jcic}}
\end{figure*}
\end{center}
\textbf{Fig. S4}.
Bayesian estimation of the supercurrent distribution. Posterior distribution of the current density near the Dirac point in device \textit{BL3} (left panel), and corresponding critical current (right panel). The values of $I_c$ obtained from the posterior distribution (orange) are in good agreement with the measured critical current (blue).
\newpage
\begin{center}
\begin{figure}[t]
\includegraphics[width=0.44\textwidth]{deltas.pdf}
\caption{\label{fig:deltas}}
\end{figure}
\end{center}
\textbf{Fig. S5}.
Ratio of the supercurrent carried by the edge states as a function of carrier density in device \textit{BL3} over the density range of $n\sim -1$ to $-2.9 \times 10^{11}$ cm$^{-2}$. Each scan corresponds to a Fraunhofer pattern, with Fig.~\ref{fig:jcic} showing the 8$^{\rm th}$ scan. (Increasing scan number corresponds to decreasing carrier density.)
\newpage
\textbf{Supplementary Tables}\\
\begin{table}[h]
\begin{tabular}{|c|c|c|c|c|}
\hline
Device & L (nm) & W (nm) & Aspect ratio, L/W & Contact width (nm)\\
\hline \hline
BL1 & 250 & 1200 & 0.208 & 400\\
ML1 & 300 & 1200 & 0.25 & 300 \\
BL2 & 300 & 800 & 0.375 & 400 \\
BL3 & 350 & 1200 & 0.292 & 600\\
BL4 & 250 & 900 & 0.278 & 400\\
\hline
\end{tabular}
\end{table}
\textbf{Table S1}.
List of device dimensions for the graphene Josephson junctions studied in this work. $L$ and $W$ refer to junction length and width, respectively, as labeled in Fig. 1d of the main text. Contact width refers to the size of the superconducting Ti/Al electrodes in the direction perpendicular to W. BLx and MLx refer to bilayer and monolayer graphene devices, respectively.
\newpage
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 6,414 |
Q: XmlImport One Line Table Not Pulling (Using VBA) VBA is not fetching the headers of an XML table when the table only has one line. Is there a way to fix this? I'm using:
ActiveWorkbook.XmlImport url:=url, ImportMap:=Nothing, Overwrite:=True, Destination:=Worksheets("A").Range("A1")
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David Rene Gagner (born December 11, 1964) is a Canadian former professional ice hockey player and current Director of Player Development for the Orr Hockey Group player agency.
Playing career
Dave Gagner spent two full seasons with the OHL's Brantford Alexanders. In 1982-83 he registered 55 goals and 121 points in 70 games, catching the attention of NHL scouts. The same season he was named to the OHL Second All-Star Team.
He was selected in the first round of the 1983 NHL Entry Draft (12th overall) by the New York Rangers and bounced back and forth between New York and their American Hockey League farm team, the New Haven Nighthawks over the next several seasons. He remained with the Rangers' organization until 1987 but was never able to completely get over the hump and earn a full-time roster spot, largely due to his being considered too small for an NHL forward.
His career took off when he was traded to the Minnesota North Stars in 1987. Though initially spending time with Minnesota's farm team in Kalamazoo, Gagner broke out in 1989–90 with 40 goals and was in the NHL to stay. In his best season (1990–91), Gagner recorded 82 points in 73 games. He remained with the Stars when they moved to Dallas, until he was traded to Toronto on January 29, 1996. His stay in Toronto would last only 28 games before he was again traded, this time to Calgary, where he would spend the entire 1996–97 season, scoring a very respectable 27 goals and 60 points in 82 games.
In the summer of 1997 Gagner was on the move again, signing a free agent contract with the Florida Panthers. Gagner would spend a season and a half in Florida before being involved in a blockbuster trade on January 17, 1999. Gagner, along with Ed Jovanovski, Mike Brown, Kevin Weekes and Florida's 1st round choice (Nathan Smith) in the 2000 NHL Entry Draft were dealt to Vancouver in exchange for superstar Pavel Bure, Bret Hedican, Brad Ference and Vancouver's 3rd round choice (Robert Fried) 2000 NHL Entry Draft.
Gagner would finish the 1998–99 season with the Canucks before officially announcing his retirement on September 9, 1999, after 15 years in the NHL. He finished with 719 points in 946 NHL regular season games.
Post-playing career
In March, 2000, Gagner founded Custom Ice Inc., a company that manufactures permanent and portable ice skating rinks.
In 2001, Brad Grant sold the Milton Icehawks to an Oakville trio that consisted of ex-NHLer Gagner, Mario Forgione who owned the Mississauga IceDogs at the time and was an automotive parts manufacturing president, and wine distillery consultant Ken Chase.
In August, 2006, it was announced that Gagner would serve as assistant coach of the London Knights of the Ontario Hockey League; his son, Sam, also played for the team and currently plays for the NHL's Winnipeg Jets. His daughter, Jessica Gagner plays hockey for the Dartmouth Big Green women's ice hockey program. Dave spent the 2006–07 and 2007–08 seasons with the Knights, and also opened a training centre in London to work with young prospects.
In June, 2008, Gagner's former agent Mike Gillis, who had recently been named General Manager of the Vancouver Canucks, hired Gagner as the team's Director of Player Development. Gagner replaced Canucks' legend Stan Smyl, who had been named the team's new Director of Collegiate Scouting. On September 3, 2013, Gagner left the Canucks to work with the Orr Hockey Group player agency in the same capacity.
Career statistics
Regular season and playoffs
International
References
External links
Custom Ice Rinks
1964 births
Brantford Alexanders players
Calgary Flames players
Canadian ice hockey centres
Courmaosta HC players
Dallas Stars players
Florida Panthers players
Ice hockey people from Ontario
Ice hockey players at the 1984 Winter Olympics
Kalamazoo Wings (1974–2000) players
Living people
London Knights coaches
Minnesota North Stars players
National Hockey League All-Stars
National Hockey League first-round draft picks
New Haven Nighthawks players
New York Rangers draft picks
New York Rangers players
Olympic ice hockey players of Canada
Sportspeople from Chatham-Kent
Toronto Maple Leafs players
Vancouver Canucks executives
Vancouver Canucks players
Canadian expatriate ice hockey players in Italy
Canadian ice hockey coaches | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 7,108 |
using System;
using Newtonsoft.Json;
namespace Cedar.Foundation.WeChat.Entities.WeChat
{
public class GetRecordModel
{
/// <summary>
/// </summary>
[JsonProperty("account")]
public string Account { get; set; }
/// <summary>
/// </summary>
[JsonProperty("starttime")]
public DateTime StartTime { get; set; }
/// <summary>
/// </summary>
[JsonProperty("endtime")]
public DateTime EndTime { get; set; }
/// <summary>
/// </summary>
[JsonProperty("openid")]
public string OpenId { get; set; }
/// <summary>
/// </summary>
[JsonProperty("pagesize")]
public int PageSize { get; set; }
/// <summary>
/// </summary>
[JsonProperty("pageindex")]
public int PageIndex { get; set; }
}
} | {
"redpajama_set_name": "RedPajamaGithub"
} | 3,094 |
Stamatios Masouris () was a Greek athlete. He competed at the 1896 Summer Olympics in Athens. Masouris was one of 17 athletes to start the marathon race. He finished eighth of the nine athletes to have completed the race.
References
External links
Year of birth missing
Year of death missing
Greek male long-distance runners
Greek male marathon runners
Olympic athletes of Greece
Athletes (track and field) at the 1896 Summer Olympics
19th-century sportsmen
Place of birth missing
Place of death missing | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 9,701 |
Switch to: References
Citations of:
New Dog: Old Tricks
Journal of Applied Animal Welfare Science 5 (3):239-242 (2002)
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Export: Choose a format.. Formatted textPlain textBibTeXZoteroEndNoteReference Manager
Ethical Issues in Animal Cloning.Autumn Fiester - 2005 - Perspectives in Biology and Medicine 48 (3):328-343.details
The issue of human reproductive cloning has recently received a great deal attention in public discourse. Bioethicists, policy makers, and the media have been quick to identify the key ethical issues involved in human reproductive cloning and to argue, almost unanimously, for an international ban on such attempts. Meanwhile, scientists have proceeded with extensive research agendas in the cloning of animals. Despite this research, there has been little public discussion of the ethical issues raised by animal cloning projects. Polling data (...) show that the public is decidedly against the cloning of animals. To understand the public's reaction and fill the void of reasoned debate about the issue, we need to review the possible objections to animal cloning and assess the merits of the anti-animal cloning stance. Some objections to animal cloning (e.g., the impact of cloning on the population of unwanted animals) can be easily addressed, while others (e.g., the health of cloned animals) require more serious attention by the public and policy makers. (shrink)
Animal Experimentation in Applied Ethics
Cloning in Applied Ethics
Justifying a Presumption of Restraint in Animal Biotechnology Research.Autumn Fiester - 2008 - American Journal of Bioethics 8 (6):36 – 44.details
Articulating the public's widespread unease about animal biotechnology has not been easy, and the first attempts have not been able to provide an effective tool for navigating the moral permissibility of this research. Because these moral intuitions have been difficult to cash out, they have been belittled as representing nothing more than fear or confusion. But there are sound philosophical reasons supporting the public's opposition to animal biotechnology and these arguments justify a default position of resistance I call the Presumption (...) of Restraint . The Presumption of Restraint constitutes a justificatory process that sets out the criteria for permitting or rejecting individual biotechnology projects. This Presumption of Restraint can be overridden by compelling arguments that speak to a project's moral and scientific merit. This strategy creates a middle-of-the-road stance that can embrace particular projects, while rejecting others. The Presumption of Restraint can also serve as a model for assessing moral permissibility in other areas of technological innovation. (shrink)
Biomedical Ethics in Applied Ethics | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 1,981 |
myPortico
EmployerLink
ELCA Together
Employer Education
The Wholeness Wheel
Strengthening Those Who Serve
For over 200 years, Lutherans have been supporting the well-being of faith leaders and their families. Why? Because those leaders help others to live out their purpose. And that's not easy work.
Portico is the ELCA's modern-day benefits ministry. We offer quality, affordable benefits designed specifically to help our members strengthen their overall well-being, find resilience, and live into their calling to serve.
"The ELCA calls its people and ministries to engage publicly, working for lasting, positive change here and now, amidst all the world's complexities, tensions, and ambiguities."
Rev. Jeff Thiemann
Portico President & CEO
Guided by a Shared Philosophy
It's powerful when a community agrees on what's important and works together to meet shared needs.
For decades, Portico has partnered with the ELCA's churchwide organization and synods to understand and meet the evolving benefit needs of our community. A key product of this work is the living document called the ELCA Philosophy of Benefits.
ELCA organizations of all kinds rely on this document to maintain benefit programs that align with ELCA priorities. It affirms the:
Importance of rostered minister and lay employee health and well-being
Shared cost responsibility between those receiving benefits and those providing them
Financial value and equity gained by taking a unified approach across the ELCA
Historical context of today's philosophy
Download the ELCA Philosophy of Benefits PDF
Investing for the Sake of the World
The ELCA calls its people and institutions to encourage corporations to act in socially responsible ways and to invest in alignment with ELCA priorities.
In alignment with the ELCA's corporate social responsibility ministry, Portico has maintained a social purpose investing strategy for over 30 years — starting well before this type of investing became popular.
Always, our social purpose-related decisions and actions follow ELCA social teachings and policies.
Retirement plan members can feel proud to be answering this corporate social responsibility call in the following ways.
Across all our funds, Portico identifies companies we're invested in that do not demonstrate responsible, sustainable business practices — and advocates for change.
Through our social purpose funds, Portico seeks to deliver attractive returns over the long term while also investing in companies creating positive social impact and screening out those creating negative social impact.
Default Closed
1988: Well before socially responsible investing became popular, Portico introduced its first social purpose investment funds. Since then, our commitment has strengthened, our experience has deepened, and our social purpose fund options are broad enough to meet the needs of those wanting to be 100% social purpose fund investors.
1993: The ELCA social statement, Caring for Creation, was adopted. In the years following, Portico took multiple steps to strengthen our focus on the environment and continues to make this a priority.
2015: We approved and implemented the private prisons investment screen following the ELCA's 2013 adoption of its criminal justice social statement.
2017: We researched and made a five-year investment focused on community development in the Holy Land, as the ELCA developed its human rights screen.
2020: We made social purpose funds our default fund for new members.
As Shareholder Advocate
Via all our funds, we wield a collective shareholder voice to create change.
Shareholder advocacy is all about speaking up, as investors, to advocate for more responsible corporate behavior that increases shareholder value. Like all smart change strategies, it starts with dialogue.
First, We Talk
Together with other like-minded shareholders, we invite corporate leaders into conversations to offer them an alternative way to look at their bottom-line risks and competitive opportunities, and encourage them to act in the best interests of shareholders.
If That Fails, We File a Resolution
When dialog fails to prompt change, we file a resolution calling for a specific action that appears on proxy ballots to be voted on at the company's next annual meeting. If the company commits to take action, we withdraw the resolution before it comes up for a vote.
If That Fails, We Vote
When we don't get a favorable response, we vote our proxies in support of that resolution. Even votes of less than 50% can capture a company's attention and bring its leaders back into dialogue.
See Portico's Shareholder Resolution Outcomes
How We Hold Board of Directors Accountable
When board of directors come up for election and the company or board members are not acting in the best interest of shareholders, we vote against approval of that board member, and clarify our reason in a letter to company management. With many shareholders using their vote to voice their disapproval, companies pay attention. We sent 55 letters in 2020.
Example: We consider a company's executive compensation excessive. In response, we'll vote against the company's compensation strategy. We'll also vote against re-election of the responsible board members, and explain that vote in a letter to the company.
We'll likely take the same approach if:
A board director is nominated who is too closely tied to the company and not able to be an unbiased conduit between the board and corporate management.
A company only puts a portion of its board directors up for re-election each year rather than all of them.
Thanks to many shareholders like Portico expressing their opinions through voting over the years, it's become more common for large companies to practice good corporate governance. For this reason, Portico is now able to focus this kind of advocacy on more mid-size and smaller companies.
As Social Purpose Investor
Via social purpose funds, we invest to create positive impact and screen out companies creating negative impact.
Positive Investing
In social purpose funds, we're able to invest a portion of a fund's assets in companies demonstrating measurable social impact in alignment with ELCA social teachings and policies. We call it social impact first investing.
Fund guidelines allow for the possibility of somewhat higher projected risk and/or somewhat lower projected return on up to 10% of a fund's assets. Using this strategy, Portico can invest in companies who work to create strong financial returns while also tackling issues like affordable housing and threats to the environment.
Importantly, these companies must commit to measuring and reporting the social impact they create so that we, in turn, can share it with our social purpose fund investors.
In social purpose funds, we're able to screen out companies whose business practices conflict with these ELCA social criteria screens:
Political and civil human rights
Private prisons
In response to ELCA calls for action, Portico has, over the last five years, strengthened screening related to climate change and fossil fuel companies and implemented new screens related to private prisons and human rights.
Using the ELCA's current social criteria screens as a guide, Portico reviews thousands of potential companies on an ongoing basis and typically screens out about 10% of those it reviews.
We currently exclude about 740 companies from ELCA retirement plan investments in the social purpose funds. Guided by the ELCA Environment screen, we have screened out about 200 companies for being most egregious in terms of damage to the environment. These include:
Companies with a history of significant toxic spills and releases, energy and climate change issues, poor water management practices, and other waste management issues
Some of the largest fossil fuel-producing companies, including ExxonMobil, Royal Dutch Shell, Chevron, and BP
About 155 companies owning thermal coal, oil shale, and tar sands reserves, the most carbon-intensive (dirtiest) fossil fuels, accounting for about 82% of the emissions tracked by the Carbon Underground 200™'s top 100 coal companies and in total about 2/3 of the Carbon Underground 200™'s top 100 coal and top 100 oil and gas companies.
Plan members, visit myPortico for more about being a social purpose investor
See how our efforts are making a difference. For social purpose fund investors, the outcomes of positive social investing and screening are part of their return on investment.
Investing in the Future of Our Planet, May 2016
Shareholder Advocacy
Climate Change Resolutions Lead to Industry First, July 2017
Convincing Costco to Reduce Its Greenhouse Gas Emissions, Nov. 2015
Positive Social Investing
Impax Reports Positive Environmental Impact for 2019, Dec. 2020
Nuveen Green Bond Impact Results for 2018, Oct. 2020
Green Bond Builds Environmentally-Friendly Landscape, Nov. 2015
Companies Holding Oil Shale and Tar Sands Reserves Screened Out, Dec. 2018
Targeting Thermal Coal Users, March 2016
BUILDING STRONGER COMMUNITIES
IFF Investment Brings Positive Change to Communities, Feb. 2020
Bank of Palestine Investment Helps Strengthen Community, April 2019
Portico Implements New ELCA Human Rights Screen, April 2019
PROMOTING SUSTAINABLE BUSINESS PRACTICES
Why We Don't Invest in Private Prisons, Nov. 2015
Advocating for ELCA Priorities
Through the Church Alliance, Portico represents ELCA priorities on many fronts — even on Capitol Hill.
The Church Alliance is a multi-denominational advocacy organization. As an advocate for Portico and other church benefit organizations, it carries the combined voices of over 30 faith denominations into the national legislative and regulatory arena to promote legislation and legal actions of interest to faith organizations.
Portico's ongoing leadership in the Church Alliance amplifies our voice and has positioned us to help realize these recent positive outcomes.
Advocated for including churches, religious organizations, and their employees in the Cares Act which made significant assistance, including the PPP loan program, available to support ministry and church workers.
Successfully advocated, through passage of the 2020 Fiscal Year Spending Package, to repeal the 40% "Cadillac Tax" on health care coverage and the "Parking Lot Tax" on churches and other tax-exempt organizations.
Educated policymakers, in a closed White House meeting on the opioid crisis, about Portico's use of care coordinators to successfully steer members toward effective but less costly preventive and outpatient interventions.
Successfully defended the long-standing clergy housing allowance exclusion through years of protracted court action.
Advocated for legislation that now preserves the ability of certain church-affiliated organizations to participate in church-sponsored 403(b)(9) plans.
Championing Wellness
Healthy, resilient people shape healthy, resilient communities.
Our churches and organizations are stronger when God's people are good stewards of the lives God has given them. When they care for their well-being, they show up with strength, courage, vulnerability, compassion, and grace.
More than two decades ago, Portico began offering holistic health benefits. Since then, we've actively invited our members – and the entire ELCA community — to embrace a faith-based, whole-person approach to living well.
This call to live well invites people to:
Address the many dimensions of our well-being
Recognize that our health or ill-health in one dimension affects our well-being in other dimensions
Understand that one can find healing without necessarily finding a cure
Over the years, we've created a variety of cutting-edge inspirational campaigns and resources, each designed to help people live well financially, emotionally, physically, and spiritually.
Follow Portico's Facebook page for wellness resources and inspiration.
Creating Financially Healthy Leaders
Supporting the financial well-being of those who lead — from seminary to retirement — has much to do with creating and maintaining vital ministries.
Portico's Seminary Scholarship
More than ever, it's critical to ease financial barriers for those entering ministry. By investing in future ELCA leaders, we empower those whom God calls to serve as the church needs and the Holy Spirit leads.
Since 2000, the ELCA's Fund for Leaders has offered scholarships to ELCA seminary students pursuing rostered ministry, providing over $2.8 million in scholarships to 297 ELCA seminary students in the 2020-2021 academic year.
Portico contributes to the ELCA Fund for Leaders by providing one full-tuition scholarship — funded 100% through donations from Portico employees, trustees, advisers, and benefit partners. No funding comes from the benefit contributions paid to Portico by sponsoring employers or plan members.
Learn more about the ELCA Fund for Leaders
Resourceful Servants
The ELCA seeks to be a thriving church, spreading the gospel and deepening faith for all people. But it also recognizes that financial strain can keep leaders and congregations from thriving.
The ELCA's Resourceful Servants Initiative was created to promote financial wellness for ELCA congregations, rostered ministers, and seminarians. It is supported by a grant from the Lilly Endowment and a partnership between the ELCA churchwide organization, the Mission Investment Fund of the ELCA, the ELCA Federal Credit Union, the ELCA Foundation, Portico Benefit Services, and Lutheran Social Service (LSS) Financial Counseling.
Portico's role focuses on the financial well-being of rostered ministers. For many, financial stress stands in the way — due to a lack of emergency savings, a lack of retirement savings, or both.
Resourceful Servants offers two Savings Matching programs:
One encourages emergency savings through the ELCA Federal Credit Union
One encourages retirement savings through Portico
Portico also offers resources designed to help leaders shape their own financial wellness — recorded webinars, the online Retirement Planning Tool, and Portico Financial Planners.
Matching funds and learning resources are helping participants develop healthy savings habits and more rapidly grow their savings and retirement accounts.
Learn more about this ELCA initiative
ELCA Special Needs Retirement Fund
When a retired leader or churchworker experiences real financial hardship, the ELCA is there.
The ELCA Special Needs Retirement Fund was established in 1993 to provide financial assistance (monthly retirement income, and in some cases, health plan contributions) to eligible retired leaders and churchworkers and their surviving spouses living on a low income.
Portico co-administers this fund with the ELCA. It serves about 40 people a year, and was founded by generous donations from individuals and congregations, as well as support from the churchwide organization and Portico.
In general, eligibility requires:
Reaching full Social Security retirement age
Retiring with at least seven years of retirement plan participation
Being sponsored in the retirement plan on the date of your retirement
Total monthly income from all sources to be less than $2,602 if single or $3,522 if married
Assets valued less than $250,000, with no more than $75,000 of that in cash or investments
To learn more about this fund, eligibility requirements, and how to apply, please call Portico's Customer Care Center at 800.352.2876
© 2021 Portico Benefit Services
CONTACT US TERMS & CONDITIONS PRIVACY & LEGAL NOTICES
Neither Portico Benefit Services nor the funds it manages are subject to registration, regulation, or reporting under the Investment Company Act of 1940, the Securities Act of 1933, the Securities Exchange Act of 1940, the Employer Retirement Income Security Act of 1974, or state securities laws. Accordingly, members are not afforded the protections of the provisions of the laws and related regulations. Carefully consider the investment objectives, risks, charges, and expenses of any fund before investing in it. All funds are subject to risk and uncertainty. Past performance cannot be used to predict future performance. ELCA funds are not insured or guaranteed by the Federal Deposit Insurance Corporation or any other government agency. See the ELCA Master Institutional Retirement Plan Investment Fund Descriptions for more information about our funds. | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 1,662 |
Boules (Petanque) Oh, oh, French toilets!
The Squat Toilet Weblog tells about changes on this website and points at funny or useful websites about France. Apart from that, news from France. Not all of it, not allways the breaking news, but just some messages to give an idea of the country. | {
"redpajama_set_name": "RedPajamaC4"
} | 8,595 |
ACCEPTED
#### According to
Index Fungorum
#### Published in
Lilloa 22: 506 (1951)
#### Original name
Pholiotella blattariopsis Speg.
### Remarks
null | {
"redpajama_set_name": "RedPajamaGithub"
} | 3,160 |
Академическая гимназия:
Академическая гимназия — среднее учебное заведение, существовавшее при Петербургской академии наук.
Академическая гимназия имени Д. К. Фаддеева Санкт-Петербургского государственного университета — среднее учебное заведение.
Академическая гимназия — среднее и высшее учебное заведение, существовавшее в Данциге.
Львовская академическая гимназия — среднее учебное заведение при Львовском университете (ныне при национальном университете «Львовская политехника». | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 5,379 |
Q: Qt static linking problem: undefined reference to C++ std library I am using a Qt static library. I compiled the static version of Qt5.15.1. I wrote my program which links to the Qt static library.
When I compile it, it shows the error:
undefined reference to std::pmr::get_default_resource()
which is from the C++ std library memory_resource header.
The following is the minimal reproducible example:
OS: archlinux
c++ compiler: gcc10
c++ standard: c++17
main.cpp
////main.cpp
#include <QWidget>
#include <QApplication>
int main(int argc, char* argv)
{
QApplication app;
QWidget w;
w.show();
return app.exec();
}
CMakeLists.txt
////CMakeLists.txt
cmake_minimum_required(VERSION 3.0 FATAL_ERROR)
project(test LANGUAGES CXX)
set(CMAKE_AUTOMOC ON)
set(CMAKE_AUTORCC ON)
set(CMAKE_AUTOUIC ON)
find_package(Qt5 COMPONENTS Widgets REQUIRED)
include_directories(${Qt5Widgets_INCLUDE_DIRS)
add_executable(test ./main.cpp)
target_link_libraries(test Qt5::Widgets)
The linking error while compiling:
[ 3%] Automatic MOC and UIC for target w
[ 3%] Built target w_autogen
[ 50%] Built target w
[ 53%] Automatic MOC and UIC for target test
[ 53%] Built target test_autogen
[ 57%] Linking CXX executable ../bin/test
/usr/bin/ld: /opt/qt_5_15_1_static/lib/libQt5FontDatabaseSupport.a(qfontconfigdatabase.o): in function `QFontconfigDatabase::fallbacksForFamily(QString const&, QFont::Style, QFont::StyleHint, QChar::Script) const':
qfontconfigdatabase.cpp:(.text._ZNK19QFontconfigDatabase18fallbacksForFamilyERK7QStringN5QFont5StyleENS3_9StyleHintEN5QChar6ScriptE+0x1f3): undefined reference to `std::pmr::get_default_resource()'
/usr/bin/ld: qfontconfigdatabase.cpp:(.text._ZNK19QFontconfigDatabase18fallbacksForFamilyERK7QStringN5QFont5StyleENS3_9StyleHintEN5QChar6ScriptE+0x1fa): undefined reference to `vtable for std::pmr::monotonic_buffer_resource'
/usr/bin/ld: qfontconfigdatabase.cpp:(.text._ZNK19QFontconfigDatabase18fallbacksForFamilyERK7QStringN5QFont5StyleENS3_9StyleHintEN5QChar6ScriptE+0x443): undefined reference to `std::pmr::monotonic_buffer_resource::~monotonic_buffer_resource()'
/usr/bin/ld: /opt/qt_5_15_1_static/lib/libQt5Core.a(qstringlist.o): in function `QtPrivate::QStringList_removeDuplicates(QStringList*)':
qstringlist.cpp:(.text._ZN9QtPrivate28QStringList_removeDuplicatesEP11QStringList+0x3c): undefined reference to `std::pmr::get_default_resource()'
/usr/bin/ld: qstringlist.cpp:(.text._ZN9QtPrivate28QStringList_removeDuplicatesEP11QStringList+0x57): undefined reference to `vtable for std::pmr::monotonic_buffer_resource'
/usr/bin/ld: qstringlist.cpp:(.text._ZN9QtPrivate28QStringList_removeDuplicatesEP11QStringList+0x486): undefined reference to `std::pmr::monotonic_buffer_resource::~monotonic_buffer_resource()'
/usr/bin/ld: qstringlist.cpp:(.text._ZN9QtPrivate28QStringList_removeDuplicatesEP11QStringList+0x4c1): undefined reference to `std::pmr::monotonic_buffer_resource::~monotonic_buffer_resource()'
collect2: error: ld returned 1 exit status
make[2]: *** [CMakeFiles/test.dir/build.make:398: ../bin/test] Error 1
make[1]: *** [CMakeFiles/Makefile2:137: CMakeFiles/test.dir/all] Error 2
make: *** [Makefile:104: all] Error 2
The error shows the Qt static library cannot find the reference to memory_resource in the std library. How could this happen? I have built Qt static library without any problem.
The reason why I have to use static Qt library is that CUDA requires static linking.
update
I have enabled c++17 in CMakeLists.txt
In my project, it is a little different.
My project structure is as below:
.
|--include
|--src
| |--mainwindow
| | |--mainwindow.h
| | |--mainwindow.cpp
| | |--CMakeLists.txt
| |
| |--other
|
|--main.cpp
|--CMakeLists.txt
I add mainwindow as a static library, which inherits QWidget class. and main.cpp links mainwindow.
After I add cxx17 standard, the problem still occurs.
I am not sure whether it is the problem of mainwindow subdirectory.
update
I have updated my CMakeLists.txt
//CMakeLists.txt
project(my_proj LANGUAGES CXX C)
set(CMAKE_CXX_STANDARD 17)
set(CMAKE_CXX_STANDARD_REQUIRED TRUE)
set(CMAKE_CXX_EXTENSIONS OFF)
set(CMAKE_AUTOMOC ON)
set(CMAKE_AUTORCC ON)
set(CMAKE_AUTOUIC ON)
find_package(Qt5 ... ...)
...
...
However, it still doesn't works...
UPDATE
I have updated my CMakeLists.txt by adding
set(CMAKE_EXE_LINKER_FLAGS "${CMAKE_EXE_LINKER_FLAGS} -static-libstdc++ -static-libgcc -static")
It links targets to static stdlibc++ and stdlibc;
It works for linking stdlib.
Now I have another problem. Since my project is static linking, it need to link to 3rd party shared libs, for example, third_party_lib.so.
It shows another error:
/usr/bin/ld: attempted static link of dynamic object '/usr/local/lib/librealsense2.so.2.30.0'
How about link my static libs to a 3rd party shared library?
A: It's unclear whether or not C++17 was enabled when you built the test executable. At least, I don't see it in your CMake file. You can enable it for that CMake target using the following:
set_target_properties(test PROPERTIES
CXX_STANDARD 17
CXX_STANDARD_REQUIRED ON
CXX_EXTENSIONS OFF
)
The undefined references (such as std::pmr::get_default_resource) are C++17 features, so adding this to your CMake file may resolve the issue.
You could also try setting the C++17 standard globally for your entire CMake project using the following:
set(CMAKE_CXX_STANDARD 17)
set(CMAKE_CXX_STANDARD_REQUIRED ON)
set(CMAKE_CXX_EXTENSIONS OFF)
A: Another way to set c++17 besides explicitly setting its properties is to use target_compile_features
Like the following:
target_compile_features(test PUBLIC cxx_std_17)
Here is the list of features
A: I don't know the cause (didn't investigate deeply), but with C++17 you need to link your code with QtXML component.
CMakeLists.txt:
cmake_minimum_required(VERSION 3.0 FATAL_ERROR)
project(test LANGUAGES CXX)
set(CMAKE_AUTOMOC ON)
set(CMAKE_AUTORCC ON)
set(CMAKE_AUTOUIC ON)
find_package(Qt5 COMPONENTS Widgets Xml REQUIRED)
#include_directories(${Qt5Widgets_INCLUDE_DIRS)
add_executable(test ./main.cpp)
target_link_libraries(test Qt5::Widgets Qt5::Xml)
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 2,638 |
Q: MySQL checking if string exist and updating or creating i am trying to make an simple if exist check for MySQL via php and then i will create an String if not exist; or if an string exist, i will update and add to the value counter = counter + 1
But now i dont now why nothing happend :s
What i am doing wrong --- i had search for the problem but i didnt has find anything..
thanks ahead
heres the code
$rankset = '';
$steamid64 = "76561198070477917";
$localmap = "de_dolls";
mysql_connect("localhost", "######", "###############") or die(mysql_error());
mysql_select_db("server") or die(mysql_error());
$check1 = mysql_query("SELECT * FROM `darkrp_missingmap` WHERE map = '$localmap'") or die(mysql_error());
while($info1 = mysql_fetch_array( $check1 ))
{
if(mysql_num_rows($info1) == 0) {
print "existiert nicht wird erstellt...";
mysql_query("SELECT server INSERT INTO darkrp_missingmap (map, count) VALUES ('$localmap', '1')");
}
else {
print "existiert wird aufgestuft...";
mysql_query("SELECT server UPDATE darkrp_missingmap SET count = count + 1 WHERE map = '$localmap'");
}
}
A: mysql_num_rows() takes a resource object as a parameter which is your $check1 while you are passing an array $info1 to it.
You should be doing:
$check1 = mysql_query("SELECT * FROM `darkrp_missingmap` WHERE map = '$localmap'") or die(mysql_error());
if (mysql_num_rows($check1) === 0) {
#no records found
print "existiert nicht wird erstellt...";
#rest of your code
} else if (mysql_num_rows($check1) === FALSE) {
#query failed
} else {
#records found
print "existiert wird aufgestuft...";
#rest of your code
}
You should be careful about using ==0 because mysql_num_rows() returns FALSE on failure which can also be loosely compared to 0.
Note:
Please, don't use mysql_* functions in new code. They are no longer maintained and are officially deprecated. See the red box? Learn about prepared statements instead, and use PDO, or MySQLi - this article will help you decide which. If you choose PDO, here is a good tutorial.
A: You can try this, you can ref [this]
$check1 = mysql_query("SELECT * FROM `darkrp_missingmap` WHERE map = '$localmap'") or die(mysql_error());
$nums = mysql_num_rows($check1);
if($nums == 0) {
// do something
}
Or you can use this sql.
SELECT COUNT(*) AS count FROM `darkrp_missingmap` WHERE map = '$localmap'
And get this query's result count.
$result = mysql_query($sql);
$row = mysql_fetch_array($result);
$nums = $row[0]['count'];
A: i dont think you select and insert at one time and you dont need that either
try this
if(mysql_num_rows($info1) >0) {
print "existiert nicht wird erstellt...";
mysql_query("INSERT INTO darkrp_missingmap (map, count) VALUES ('$localmap', '1')");
} else {
print "existiert wird aufgestuft...";
mysql_query("UPDATE darkrp_missingmap SET count = count + 1 WHERE map = '$localmap'");
}
try this and tell
A: Ok i have found my trouble i have already select the db "server" and then i tried to make this
INSERT INTO darkrp_missingmap(map, count) VALUES ('zu_kotze', '100')
Here the working example:
$steamid64 = "76561198070477917";
$localmap = "de_dollsw";
mysql_connect("localhost", "####", "##########") or die(mysql_error());
mysql_select_db("server") or die(mysql_error());
//maplog
$check1 = mysql_query("SELECT * FROM `darkrp_missingmap` WHERE map = '$localmap'") or die(mysql_error());
$nums = mysql_num_rows($check1);
if($nums == 0) {
print "existiert nicht wird erstellt...";
mysql_query("INSERT INTO darkrp_missingmap (map, count) VALUES ('$localmap', '1')");
} else {
print "existiert wird aufgestuft...";
mysql_query("UPDATE darkrp_missingmap SET count = count + 1 WHERE map = '$localmap'");
}
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 3,353 |
For the chocolate donuts: Preheat the oven to 375 degrees F.
Spray a 12-cavity donut pan with cooking spray and set it aside.
Whisk together the flour, granulated sugar, cocoa, salt, baking powder and baking soda in a large bowl. Whisk together the coffee, buttermilk, oil, vanilla and egg in a separate medium bowl until combined. Add the wet ingredients to the dry ingredients and mix to combine.
Transfer the batter to a piping bag. Fill each cavity with batter until it comes about three-quarters of the way up the sides of the pan.
Bake until a toothpick inserted into a donut comes out clean, about 12 minutes. Cool the donuts in the pan for 5 minutes, then remove to a rack and cool completely.
For the coffee glaze: Mix together the confectioners' sugar, coffee and milk in a bowl until smooth. If it seems too thick, add more coffee little by little.
Dip each donut halfway into the glaze, then allow excess to drip off. Sprinkle the tops with hazelnuts and sprinkles. Let set, about 30 minutes. | {
"redpajama_set_name": "RedPajamaC4"
} | 8,640 |
{"url":"https:\/\/byjus.com\/physics\/free-forced-damped-oscillations\/","text":"# Free Forced And Damped Oscillations\n\nIn Physics, oscillation is a repetitive variation, typically in time. It is measured between two or more different states or about equilibrium or about a central value. Some familiar examples of oscillations include alternating current and simple pendulum. Some parameters governing oscillation are:\n\n\u2022 Period of oscillation\n\u2022 Oscillation frequency\n\u2022 Oscillation amplitude\n\nIn general, an oscillation is a back and forth movement in a regular rhythm. In this article, let us discuss oscillation in detail.\n\n## Definition of Oscillation\n\nOscillation is the regular variation in position or magnitude about a central point or about a mean position.\n\nThe commonly used unit for the number of oscillations per second is the Hertz. Oscillations in Physics are quantified using parameters such as \u2013 Frequency, Amplitude, and period. Each term is explained as follows-\n\n Terms Definitions Units Symbol Period Of Oscillation The time taken for one complete oscillation is called Period. Second T Oscillation Frequency The number of cycles\/oscillations per second is called the frequency of oscillation. Hertz f Oscillation amplitude The maximum displacement from the mean position is called the amplitude of oscillation. meter A\n\n## How is Oscillation Calculated?\n\nThe formula for parameters governing oscillations in Physics are listed below-\n\n\u2022 Period of oscillation: Time period of a simple pendulum can be mathematically expressed as\n $$\\begin{array}{l}T=2\\pi \\sqrt{\\frac{L}{g}}\\end{array}$$\n\nWhere,\nT is the Time Period of oscillation\nL is the length\ng is the acceleration due to gravity\n\n\u2022 Oscillation frequency: Mathematically expressed as \u2013\n $$\\begin{array}{l}f=\\frac{1}{T}\\end{array}$$\n\nWhere,\nT is the time period of oscillation.\n\n### Examples of Oscillation\n\nDid you know that the vibration of a guitar string is an example of oscillation? The motions of a playground swing, tuning forks are also examples of oscillatory motion. Since these are mechanical in nature, they are also called vibrations. The motion of alternating current (although electrical) is also an example of oscillatory motion.\n\nYou may also want to check out these topics given below!\n\n## Simple Harmonic Motion\n\nIt is the simplest form of oscillatory motion which all of us come across in our day-to-day life. It is an oscillatory motion in which retarding force is proportional to the amount of displacement of an object from an equilibrium position. Or in other words, the restoring force acts in the direction opposite to that of displacement of the object and is proportional to it.\n\nAgain, the motion of a simple pendulum is a perfect example of this, where if it is displaced in one direction, the restoring force acts in the opposite direction. Any simple harmonic motion can be categorized into three types of oscillation.\n\n## Different Types Of Oscillation\n\nThere are three main types of Simple Harmonic Motion\n\n\u2022 Damped Oscillation\n\u2022 Forced Oscillation\n\u2022 Free Oscillation\n\n### Free Oscillation\n\nThe free oscillation possesses constant amplitude and period without any external force to set the oscillation. Ideally, free oscillation does not undergo damping. But in all-natural systems damping is observed unless and until any constant external force is supplied to overcome damping. In such a system, the amplitude, frequency and energy all remain constant.\n\n### Damped Oscillation\n\nThe damping is a resistance offered to the oscillation. The oscillation that fades with time is called damped oscillation. Due to damping, the amplitude of oscillation reduces with time. Reduction in amplitude is a result of energy loss from the system in overcoming external forces like friction or air resistance and other resistive forces. Thus, with the decrease in amplitude, the energy of the system also keeps decreasing. There are two types of damping\n\n\u2022 Natural Damping\n\u2022 Artificial Damping\n\n### Forced Oscillation\n\nWhen a body oscillates by being influenced by an external periodic force, it is called forced oscillation. Here, the amplitude of oscillation, experiences damping but remains constant due to the external energy supplied to the system.\n\nFor example, when you push someone on a swing, you have to keep periodically pushing them so that the swing doesn\u2019t reduce.\n\nQ 1) Can a motion be oscillatory but not simple harmonic? Explain with valid reason.\n\nAns: Yes.\nConsider an example of a ball dropping from a height on a perfectly elastic surface. The type of motion involved here is oscillatory but not simple harmonic as restoring force F=mg is constant and not F\u221d\u2212x, which is a necessary condition for simple harmonic motion.\n\nQ 2) What is the basic condition for the motion of a particle to be a simple harmonic motion?\n\nAns: The restoring force acting on it is proportional to its displacement from the mean position. That is, F=\u2212kx.\n\nQ 3) What happens to the oscillation of the body in damped oscillation?\n\nAns: The oscillation of the body decreases with time.\n\nQ 4) What is free oscillation?\n\nAns: Free oscillation is the oscillation of the particle with a fundamental frequency under the influence of restoring force.\n\nQ 5) In free oscillation, what happens to the amplitude, frequency and energy of the oscillating body?\n\nAns: In free oscillation, the amplitude, frequency and energy of the oscillating body remain constant\/unchanged.\n\nQ 6) Why is the frequency of free oscillation called natural frequency?\n\nAns:\u00a0The frequency of free oscillation depends on the nature and the structure of the oscillating body.\n\nQ 7) Name a few damping forces\n\nAns: Some damping forces are \u2013 Hysteresis force, viscous force and frictional force.\n\nQ 8) In damping oscillation, why does the oscillation decrease exponentially?\n\nAns: In damping, oscillation decrease in the energy results in a decrease in amplitude of oscillation.\n\nQ 9) What is the damping force?\n\nAns: Force produces resistance to the oscillation.\n\nQ 10) What is forced oscillation?\n\nAns: When oscillation of a body is under the influence of external force the oscillation is called forced oscillation.\n\nQ 11) How does the damping effect amplitude in forced oscillation?\n\nAns: The forced oscillation does face the damping effect. But the supply of external force results in gaining the energy thereby maintaining constant amplitude.\n\nQ 12) What is resonance?\n\nAns: Oscillation of object with maximum amplitude, when the frequency of the applied force equal to the natural frequency of the object is called resonance.\n\nQ 13) When can resonance be observed?\n\nAns: When the frequency of the applied force is equal to the natural frequency of the object.\n\nQ 14) What is the nature of the frequency in forced oscillation?\n\nAns: In forced oscillation, the frequency of the damped oscillation is equal to the frequency of the applied external force.\n\nQ 15) Can a motion be periodic and not oscillatory?\n\nAns: Yes.\nIn the case of uniform circular motion, it is periodic but not oscillatory.\n\nHope you have understood the concept of Oscillation, what is oscillation, its definition, types of oscillation, oscillation examples, simple Harmonic motion, and its types like \u2013 Free oscillation, damped oscillation and forced oscillation along with formula, terms, symbol and SI units.\n\nTest Your Knowledge On Free Forced Damped Oscillations!\n\n1. Ashoka.m\n\nVery easy to learn concept\n\n2. M Alahi\n\nThanks a ton sir \ud83d\udda4","date":"2022-05-23 20:19:53","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 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\": 0.7762467265129089, \"perplexity\": 795.2069337194499}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2022-21\/segments\/1652662561747.42\/warc\/CC-MAIN-20220523194013-20220523224013-00079.warc.gz\"}"} | null | null |
\section{Introduction}
Nowadays the world is full of digital data, due to the large deployment of sensors, fast internet and more computational power to generate all such that data. This data is might be very useful to extract information and predict events, allowing us to control or profit from them. In order to achieve such goal, we need fast algorithms that are capable of finding features that could bring useful information. However, this is a non-trivial task, as data is very large and usual simple statistics are slow and inaccurate. Thus,
the term data mining appeared to describe the problem of finding useful information in large data sets by integrating methods from many fields, like machine learning, statistics and database systems, spatial or temporal data analysis, pattern recognition, image and signal processing.
In recent years many works have been done to use machine learning techniques in order to extract useful information from data. The application ranges over several fields, including pharmacokinetics, such as determining clusters of drug absorption~\cite{tom:vin:car:17,gue:asmc:pmat:18}, weather prediction, e.g.~find patterns in hurricanes trajectories in order to better forecast the location of a hurricane landfall~\cite{Lee2007}, among others. Currently, with the massive introduction of electronic health records, there is a huge opportunity for mining temporal data with the objective of improving the prediction tasks in the biomedical domain~\cite{Zhao2017}.
The main objective of this work is the analysis of multivariate time series, namely, the imputation of missing values. Our approach is to use Dynamic Bayesian Networks, which can represent in a compact way relations between random variables~\cite{Pearl:1988:PRI:52121}. As such, this work follows the work of \cite{monteiro2015polynomial} and \cite{sousa2018polynomial} by extending their models. The proposed method is implemented and assessed in synthetic and real data.
The real datasets are from \textit{UCI Machine Learning Repository} \cite{Dua:2017} and \textit{UCR Time Series Classification Archive} \cite{UCRArchive}.
\subsection*{Related work}
In this paper we only consider the most used methods that support categorical time series, namely the last observation carried forward (LOCF), mode, EM and Amelia. The simplest method is the LOCF, where the missing value is simply imputed as the previous observed value. The Mode imputes the missing value with the most recurrent value. In expectation maximisation (EM) the missing values are considered hidden variables, that are randomly initalized. Then, an EM procedure is performed to improve the distribution of these hidden variables in order to maximize the likelihood of the data. The idea is that the imputed value is the most probable value of the associated hidden variable.
A more complex method is the so called Amelia~\cite{honaker2011amelia}. This method employs a three-step approach to impute the missing values~\cite{rubin1976inference}, where the first step creates plausible values for the missing observations. These values are then used to impute the missing values. This process is repeated a number of times, from which it results from the creation of a number of "complete" datasets. In the second step, these datasets are then analysed using complete-data methods. Finally, the result of the analysis is then combined.
In this work we use an R package implementation of \textit{Amelia II: A program for missing data} as a state-of-the-art imputation method.
\section{Background}
\subsection{Bayesian Networks}
\label{subsec:bn}
Let $X$ be a discrete random variable that takes values over the finite set $\mathcal{X}$. Moreover, let $\boldsymbol X = (X_1, ..., X_n)$ be a $n$-dimensional random vector, where each $X_i$ takes values in $\mathcal{X}_i=\{ x_{i1}, \dots, x_{ir_i}\}$, where $r_i$ the number of values that $X_i$ can take.
A \gls{BN} is composed of a \gls{DAG} that encodes a joint probability distribution over a set of random variables \cite{Pearl:1988:PRI:52121}.
\begin{definition}
A $n$-dimensional \glsfirst{BN} is a triple $B = (\mathbf{X}, G, \Theta)$ where:
\begin{itemize}
\item $\mathbf{X}$ is an $n$-dimensional finite random vector where each random variable $X_i$ ranges over by a finite domain $D_i$. Henceforward, it is denoted the joint domain by $D = \prod^{n}_{i = 1} D_i$.
%
\item $G = (N,E)$ is a \gls{DAG} with nodes $N = \{X_1,\dots, X_n\}$ and edges $E$ representing direct dependencies between variables.
%
\item $\Theta$ encodes the parameters $\{\theta_{ijk}\}_{i \in \{1,\dots,n\}, j \in D_{\prod_{X_i}}, k \in D_i}$ of a network, given by:
\begin{equation*}
\theta_{ijk} = P_B\left(X_i=x_{ik} | \Pi _{X_i} = w_{ij}\right),
\end{equation*}
where $\Pi_{X_i}$ denotes the set of parents of the node $X_i$ in the \gls{DAG} $G$, $x_{ik}$ is the $k$-th values of $X_i$ and $w_{ij}$ is the $j$-th configuration of $\Pi_{X_i}$. Moreover, $q_i$ is the number of total parents configurations of $X_i$, $q_i = \prod_{X_j \in \Pi_{X_i}} r_j$.
\end{itemize}
\end{definition}
Intuitively, the network structure of a \gls{DAG} G encodes conditional independence assumptions, where each random variable $X_i$ is only dependent of the descendants nodes and independent of its nondecendants, given its parents. These independence assumption are used within the chain rule to provide a factorized joint distribution over $\mathbf{X}$, defined as:
\begin{equation}
P_B(X_1,\dots, X_n) = \prod_{i = 1}^{n} P_B(X_i\mid\Pi_{X_i}).
\end{equation}
Learning a \gls{BN} $B=\left(\mathbf{X},G,\Theta\right)$ reduces to the problem of finding the network structure $G$ and the parameters $\Theta$ that best fits the data $D$. When data $D$ is complete, i.e, there are no missing values and hidden variables, being $D = \{\mathbf{y}_1,\dots,\mathbf{y}_N\}$ given by a set of $N$ i.i.d. instances, usually score-based learning is employed. Therein, a scoring criterion $\phi$ is used to measure how well a candidate network $B$ describes $D$. . However, since learning \gls{BN} is a NP-hard problem, the search space of possible solutions is explored by restring the solution space (tree-like \cite{friedman1997bayesian} or C$\kappa$G-like network structures \cite{sousa2018polynomial} or heuristic methods (greedy-hill climber \cite{heckerman1995learning}). In both cases, learning can be stated as an optimization problem:
\begin{equation}
\maximize_{B \in \mathcal{B}_n}\ \phi(B, D),
\end{equation}
where $\mathcal{B}_n$ corresponds to the set of \glspl{BN} with $n$ variables being searched.
\subsection{Dynamic Bayesian Networks}
\label{subsec:dbn}
The \gls{DBN} extends the representation of \gls{BN} to temporal processes. For simplicity, assume the discretization of time in time slices $\{0,\dots, T\}$. Let $\mathbf{X}[t]=(X_1[t], \dots,X_n[t])$ be a random vector that denotes the values of the set of random variables $\mathbf{X}$ over the time $t$. Moreover, let $\mathbf{X}[t_1:t_2]$ denote the set of random vectors $\mathbf{X}[t]$ over $t_1 \leq t \leq t_2$. Finally, let the joint probability distribution over the trajectory of a stochastic process from $\mathbf{X}[0]$ to $\mathbf{X}[T]$, $P(\mathbf{X}[0],\dots,\mathbf{X}[T])$, abbreviated with $P(\mathbf{X}[0:T])$, be defined as:
\begin{equation}
P(\mathbf{X}[0:T]) = P(\mathbf{X}[0])\prod_{t=1}^{T-1}P(\mathbf{X}[t+1]\mid\mathbf{X}[0:t]).
\end{equation}
One common approach to ease computation is to assume that the process is \textit{Markovian} in $\mathbf{X}$.
\begin{definition}
A stochastic process is said to satisfy the $m$-th order \textit{Markov} assumption if, for all $t\geq 0$:
\begin{equation}
P(\mathbf{X}[t+1]\mid\mathbf{X}[0:t]) = P(\mathbf{X}[t+1]\mid\mathbf{X}[t-m+1:t]),
\end{equation}
where in this case $m$ is called the \textit{Markov} lag of the process.
\end{definition}
Another assumption that simplifies the computation of the joint probability distribution is to consider the process to be stationary. This assumption is usually adequate when number of time slices in the training data is small.
\begin{definition}
A Markovian process is said to be stationary if:
\begin{equation}
P(\mathbf{X}[t+1]\ |\ \mathbf{X}[t]) \text{ is equal for all time slices } t \in \{0, ..., T-1\}.
\end{equation}
\end{definition}
Rigorously, a \gls{DBN} is composed of two networks: an initial \gls{BN} that encodes dependencies among variables in the initial state $\mathbf{X}[0]$, and a transition network that explain how dependencies flow forward in time. These dependencies include the intra-slice (from $\mathbf{X}[t]$ to $\mathbf{X}[t+1]$) and inter-slide (among $\mathbf{X}[t+1]$) dependencies.
\begin{definition}
A stationary first-order Markov \gls{DBN} consists of:
\begin{itemize}
\item A prior network $B^0$, which specifies the distribution over the initial states $\mathbf{X}[0]$;
\item A transition network $B_{t}^{t+1}$ over the variables $\mathbf{X}[t] \cup \mathbf{X}[t + 1]$, for all $t$, that specifies the transition probability $P(\mathbf{X}[t+1]\mid \mathbf{X}[t])$.
\end{itemize}
\end{definition}
An example of a \gls{DBN} along with its unrolled network for three time-slices is depicted in \Cref{fig:dbn_un}.
\begin{figure}[h]
\centering
\begin{subfigure}[b]{0.45\linewidth}
\centering
\resizebox{0.9\linewidth}{!}{
\input{dbn}
}
\caption{Prior network (left) and transition network (right) defining a \gls{DBN}.}
\label{fig:dbn_norm}
\end{subfigure}
\begin{subfigure}[b]{0.45\linewidth}
\centering
\resizebox{0.9\linewidth}{!}{
\input{dbn_un}
}
\caption{The corresponding unrolled network for three time-slices.}
\label{fig:dbn_un}
\end{subfigure}
\caption{Example of a stationary first-order \textit{Markov} \gls{DBN}.}
\label{fig:dbn}
\end{figure}
In the case that all data is observed, there are three approaches for learning dynamic Bayesian networks. The first approach learn optimal \gls{DBN} using mutual information tests, but ignores intra-slice dependencies~\cite{vinh2011polynomial}. A polynomial-time algorithm that learns an
To address both inter and intra-slice connections, a polynomial-time algorithm was recently proposed~\cite{monteiro2015polynomial}. However, to keep the complexity low, the search space for the intra-slice connections is restricted to tree augmented networks, i.e, acyclic networks where each variable has only one parent from the same time slice, but can have a finite number of parents from the previous time slices. The resultant network is denoted by tDBN. More recently, the search space was extended exponentially~\cite{sou:car:18,sousa2018polynomial} to networks where the intra-slice network has in-degree at most $\kappa$ and is consistent with the \gls{BFS} order of the tDBN. The resultant network of this method is denoted by bcDBN.
\section{Structural EM}
\label{sec:learnfrommissing}
In many cases, the assumption that the training data are fully observed is simply unrealistic. Since this assumption is crucial for learning the structure and the parameters of a \gls{BN} some changes to the learning process need to be made.
When data $D$ is incomplete due to missing values or hidden variables, however, scoring functions are no longer decomposable. This shortcoming was addressed in~\cite{friedman1997bayesian}, where the \glsfirst{SEM} iterative method to learn a \gls{BN} with hidden variables and/or missing values, a brief description of this method can be seen in this section.
\subsection{Parameter Estimation}
The first learning task that will be considered is the parameter estimation task. As in the case of the complete data, the approach that will be used is the \gls{MLE}. So given a network structure $G$ and the form of the \glspl{CPD}, it is only necessary to compute the parameters $\Theta$ to define the distribution $P(X\ |\ \Theta)$. It is also given a data set $D$ that consists of $M$ partial instances of $X$, so it is needed to compute the values $\hat{\Theta}$ that maximize the log-likelihood function:
\begin{equation}
\hat{\Theta} = \argmax_{\Theta}\ \log L(\Theta:D).
\end{equation}
Unlike the complete data case, where sufficient statistics are collected for each \gls{CPD} allowing to compute the parameters that maximize the likelihood, in the case of missing data there is no access to full sufficient statistics. In order to have access to them, one can take a simple approach of filling the missing values arbitrarily. Some strategies to fill these missing values consist on choosing some default value or one according to some prior distribution. The problem with this approach is that the filled value will introduce bias in the learned parameters.
Another approach tries to solve two different problems at once, these problems being the learning of the parameters and imputation of the missing values.
To be noted that each of these tasks is very easy when the solution to the other is present.
The \gls{EM} algorithm starts by choosing some arbitrary starting point. This can be either a choice of parameters or some assignment to the missing values. Assuming that it begins with a parameter assignment, then the algorithms repeat two steps. The first step is to use the current parameters in order to complete the data, using probabilistic inference. The second step, consists in using the completed data as if it was observed and compute a new set of parameters. So, given a set of parameters $\theta^0$ and a partial instance, the posterior of all possible assignments to the missing value of that instance can be calculated. The \gls{EM} algorithm then uses this probabilistic completion of different data instances to estimate the expected value of the sufficient statistics. It is well known that each iteration of this method increases the log-likelihood (see for instance~\cite{Koller:2009:PGM:1795555}), and moreover, that this process converges to a local maximum of the likelihood function.
It is worthwhile to give a more detailed explanation of the \gls{EM} algorithm.
Assume the general \gls{BN} with table-\glspl{CPD} and an initial assignment for the parameters $\Theta^0$. Let $X$ be all the child variables, $W$ all the parent variables and $o$ the dataset composed by $M$ data instances. The algorithm iterates over the following steps.
\paragraph{Expectation (E-Step)}In this step the \gls{ESS} are computed, this is done by using the current parameters $\Theta^t$.
\begin{itemize}
\item For each family $X$, $W$ and for each data case $o[m]$, compute the joint probability $P(X,W\ |\ o[m], \Theta^t)$.
\item Compute the \gls{ESS} for each $x$, $w$,
\begin{equation}
\bar{M}_{\theta^t}[x,w] = \sum_{m} P(x,w\ |\ o[m], \theta^t).
\label{eq:ess}
\end{equation}
\end{itemize}
\paragraph{Maximization (M-Step)} Given the \gls{ESS}, it performs maximum likelihood estimation, with respect to them, in order to compute a new set of parameters,
\begin{equation}
\theta^{t+1}_{x\ |\ w} = \frac{\bar{M}_{\theta_t}[x,w]}{\bar{M}_{\theta_t}[w]}.
\end{equation}
In \Cref{al:ess} and \Cref{al:em} the E-Step and for the Parameter EM are given.
\begin{algorithm}
\caption{Compute the expected sufficient statistics}\label{al:ess}
\begin{algorithmic}[1]
\Procedure{Compute-ESS}{$G, \Theta, D$}
\For{each $i = 1, \cdots, n$}\Comment{Initialization of data structures}
\For{each $x_i, w_i\ \in Val(X_i, \Pi ^G_{X_i})$ }
\State $\bar{M_t}[x_i, w_i] \gets 0$
\EndFor
\EndFor
\For{each $m = 1, \cdots, M$}\Comment{Collect probabilities from all instances}
\State Run inference on $ \langle G, \Theta \rangle$ using evidence $o[m]$
\For{each $i = 1, \cdots, n$}
\For{each $x_i, w_i\ \in Val(X_i, \Pi ^G_{X_i})$ }
\State $\bar{M_t}[x_i, w_i] \gets \bar{M_t}[x_i, w_i] + P(x_i, w_i\ |\ o[m])$
\EndFor
\EndFor
\EndFor
\State \textbf{return} $\{\bar{M_t}[x_i, w_i] : \forall i = 1, ..., n,\ \forall x_i, w_i \in Val(X_i, \Pi ^G_{X_i})\}$
\EndProcedure
\end{algorithmic}
\end{algorithm}
\begin{algorithm}
\caption{Expectation-Maximization algorithm for Bayesian Network(using table-CPDs)}\label{al:em}
\begin{algorithmic}[1]
\Procedure{Expectation-Maximization}{$G, \Theta^0, D$}
\For{each $t = 0, \cdots,$ until convergence}
\State $\{\bar{M_t}[x_i, w_i]\} \gets$ Compute-ESS($G, \Theta^t, D$) \Comment{E Step}
\For{each $i = 1, \cdots, n$}\Comment{M Step}
\For{each $x_i, w_i\ \in Val(X_i, \Pi ^G_{X_i})$ }
\State $\theta^{t+1}_{x_i|w_i} \gets \frac{\bar{M_t}[x_i, w_i]}{\bar{M_t}[w_i]}$
\EndFor
\EndFor
\EndFor
\State \textbf{return} $\Theta^t$
\EndProcedure
\end{algorithmic}
\end{algorithm}
\subsection{Structure Learning}
The intuition behind Structural EM algorithm is the same that was applied to solve the problem of learning the parameters of a \gls{BN} when there is missing data. Like the parameter estimation, there is two main steps, the expectation, where a complete data set is generated, and a maximisation, where the network structure is learned. The main difference between the \glsfirst{SEM} and the parameter estimation is that the maximisation step, in the \gls{SEM}, besides learning the parameters, the network structure is also learned. Moreover, \cite{Koller:2009:PGM:1795555} state that by using the \gls{MDL} score it is guaranteed that, in each iteration, the learned structure is better than the one used in the previous iteration. From this statement, it results that the \gls{SEM} algorithm will monotonically improve the score.
The pseudo-code of the algorithm is given in \Cref{al:sem}.
\begin{algorithm}
\caption{Structural EM algorithm for Bayesian Networks}\label{al:sem}
\begin{algorithmic}[1]
\Procedure{Structural-EM}{$G^0, \Theta^0, D$}
\For{each $t = 0, \cdots,$ until convergence}
\State ${\Theta^{t}}' \gets$ Expectation-Maximization($G^t, \Theta^t, D$) \Comment{Optional parameter learning step}
\State $G^{t+1} \gets $ Structure-Learn($D_{G^t,{\Theta^{t}}'}^*$) \Comment{Run EM to generate the \gls{ESS} for $D_{G^t,{\Theta^{t}}'}^*$}
\State $\Theta^{t+1} \gets$ Estimate-Parameters($D_{G^t,{\Theta^{t}}'}^*, G^{t+1}$)
\EndFor
\State \textbf{return} $G^{t}, \Theta^{t}$
\EndProcedure
\end{algorithmic}
\end{algorithm}
\section{Proposed Method}
One common problem with multivariate time series is missing values. Mostly because many methods assume full data and are useless in this scenario. Thus, finding ways to work with missing values becomes crucial. One of the most used approaches to solve this problem is to drop the observations with missing values, however, when the dataset has few observations this approach can lead to enormous loss of information.
Another approach that one can take is to impute the missing values. In this approach, the missing values are ``filled'' using some method, like an interpolation. Since the focus of this thesis is the multivariate categorical time series, the most common methods for interpolation does not apply. So in order to impute the missing values, this work proposes a method that uses the \gls{SEM} algorithm, devised by \cite{friedman1998bayesian}, to learn the structure of the data with missing values. However, because the algorithm learns \glspl{BN}, it cannot model a time series, as such the algorithm was changed for the purpose of learning \glspl{DBN}. As before, the search space is restricted to tDBN \cite{monteiro2015polynomial} and bcDBN \cite{sousa2018polynomial}. The \gls{SEM} algorithm can be divided with two big steps the parameter learning and the structure learning, and because the dataset has missing values one step cannot be learned without the other. As such, this algorithm starts by generating a \gls{DBN} randomly. Then the ``true'' parameters of the fixed network can be learned. This is done in an iterative process where first the \gls{ESS} are computed and then the new set of parameters are computed. This is done until convergence. With the parameters learned the algorithm then learns a new structure and repeats this process until the convergence criterion is met.
Finally with the \gls{DBN} given by the \gls{SEM}, the imputation algorithm then generate again a new dataset without missing values, however instead of having all the possible combinations of values that could fill the missing values, this dataset fills the missing values with the combinations that maximizes the posterior probability.
In \cref{sec:learnfrommissing} can be seen a description of the \gls{SEM} algorithm which will be the base to develop the imputation algorithm.
\begin{algorithm}
\caption{Missing values imputation via a DBN}\label{al:imp}
\begin{algorithmic}[1]
\Procedure{Imputation-DBN}{$D$}
\State $G^0, \Theta^0 \leftarrow$ Generate a random \gls{DBN}
\State $G, \Theta \leftarrow$ Structural-EM($G^0, \Theta^0, D$)
\For{each observation in $D$}
\For{each transition in the observation}
\If{transition has missing values}
\State Generate all possible combinations of values for the missing values
\For{each new combination of values}
\State Calculate the posterior probability
\EndFor
\State Select the generated combination that maximizes the posterior probability and impute the missing values
\EndIf
\EndFor
\State Add the observation to $D'$
\EndFor
\State \textbf{return} $D'$
\EndProcedure
\end{algorithmic}
\end{algorithm}
\section{Experimental Results}
\label{sec:results}
\subsection{Simulated data}
\label{subsec:impsim}
To assess the merits of the \gls{SEM} algorithm as an imputation method, multivariate time series were randomly generated using generated \glspl{DBN}. Then various datasets were generated, where the characteristics like the number of observations and the number of variables were changed. Finally, the missing values were generated with respect to two parameters, the first is the percentage of subjects with missing values and the second is the percentage of missing values corresponding to a subject. With the purpose of comparing this imputation method with state of the art methods, first these methods were used to impute the datasets with missing values, then the number of errors between the original dataset, without missing values, and the imputed datasets were counted. To facilitate the visualization, the results of this experiments are grouped in \Cref{fig:pimperros}.
\begin{figure}[h]
\centering
\includegraphics[width=\linewidth]{Final_sim_error.png}
\caption{Imputation errors for multiple datasets~(color online). }
\label{fig:pimperros}
\end{figure}
When analysing the results presented in \Cref{fig:pimperros} it can be concluded that the imputation done by \gls{SEM} algorithm has fewer errors when comparing with other imputation methods like \gls{LOCF}, Mode and the Amelia \cite{honaker2011amelia}. This result was expected because the data were generated using \glspl{DBN}, however, it important to highlight that despite the \gls{SEM} algorithm has fewer errors, it has errors. One justification of this can be the fact that the imputation chosen by the algorithm is the one that maximizes the probability of the observation.
\subsection{Benchmark data}
\label{subsec:impreal}
In order to evaluate the performance of the \gls{SEM} algorithm as an imputation method it was used 10 datasets from \textit{UCI Machine Learning Repository} \cite{Dua:2017} and \textit{UCR Time Series Classification Archive} \cite{UCRArchive}. Moreover, because the implementation of the \gls{SEM} algorithm only works with categorical time series and most of these datasets are composed by real-valued time series a discretization of these time series must be done. The discretization was done using the \textit{SAX} algorithm~\cite{lin2007experiencing}, it is important to note that it was used only an alphabet size of four, a maximum size of the time series of one hundred time steps and an independently discretization of each dimension of multivariate time series .
With the resulting discretized datasets, in order to test the performance of the imputation methods, it was removed values. Once more, these values were removed with respect to two parameters, the percentage of observation with missing values and the percentage of missing values per missing observation.
Moreover, in order to use the \gls{SEM} algorithm some assumptions need to be made, these are the stationarity of the time series and the first-order \textit{Markov} assumption. Analysing the results, \Cref{fig:realerror}, it can be concluded that, in most datasets, the imputation done with \gls{SEM} algorithm has fewer errors than the other methods.
Given these results, a Wilcoxon signed ranks test was performed, in order to compare the \gls{SEM} algorithm with others imputation methods. As such, the results were grouped by methods and by the percentage of observations with missing values, \Cref{table:wil}. The use of a Wilcoxon signed ranks test is justified by the fact that is simple and a robust non-parametric test for statistical comparisons~\cite{demvsar2006statistical}.
\begin{figure}[h]
\centering
\includegraphics[width=\linewidth]{Final_error.png}
\caption{Imputation errors for real datasets.}
\label{fig:realerror}
\end{figure}
\begin{figure}[h]
\centering
\includegraphics[width=0.55\linewidth]{ArabicDigits_error.png}
\includegraphics[width=0.55\linewidth]{CharacterTrajectories_error.png}
\caption{Imputation errors for real datasets.}
\label{fig:realerror}
\end{figure}
\begin{table}[h]
\centering
\begingroup\catcode`"=9
\sisetup{round-mode=places,scientific-notation=true, round-precision=2}
\renewcommand{\arraystretch}{1.2}
\csvreader[head to column names,
tabular = @{}lcccc@{},
table head = \toprule \bfseries Method & 10 & 20 & 30 & 40 \\\midrule,
table foot = \hline]
{wilcoxon_test_new.csv}{
{\csvcolii & \csvcoliii & \csvcoliv & \csvcolv & \csvcolvi
\endgroup
\caption{Results from the Wilcoxon signed ranks test between \gls{SEM} algorithm and other methods.}
\label{table:wil}
\end{table}
When analysing results from \Cref{table:wil} it is easy to note that almost all p-values have a value below 0.05, which indicates that the null-hypothesis, that the two algorithms perform equally well, is discarded. However, it is important to note that the imputation errors of \gls{LOCF} are similar to the imputation errors of the \gls{SEM} algorithm which is a strange result. One explanation for this result can be the number of symbols used to discretize the time series, because since 4 symbols were used the discretized time series may not vary that much over time, which leads to a lower imputation error when using the \gls{LOCF}.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 3,411 |
The LBRH Writing Team
by Tom DeWolf | Aug 13, 2019 | Little Book of Racial Healing | 0 comments
The Little Book of Racial Healing, like the CTTT Approach shared within its pages, was a team effort. Below is more information about everyone.
Jodie Geddes, co-author. As a Jamaican native having grown up in Brooklyn, NY, Jodie uses her story as a catalyst for transforming systems. As of this writing, she serves as the Community Organizing Coordinator at Restorative Justice for Oakland Youth (RJOY) and leads RJOY's national Truth-telling, Racial Healing and Reparations project. Additionally, she is a trainer and facilitator of restorative processes for schools, in the justice system, and communities. A public speaker nationally on the subject of restorative justice, truth processes and reparation, Jodie is also a published poet and writer, with her work featured on the online platform For Harriet and Blavity.She earned an MA in conflict transformation from Eastern Mennonite University's Center for Justice and Peacebuilding. Jodie also serves as President of the Board of Managers for Coming to the Table.
Thomas Norman DeWolf, co-author. Tom is related to the largest slave-trading dynasty in U.S. history. His first book, Inheriting the Trade, is the story of his experiences in the making of the Emmy-nominated, PBS documentary, Traces of the Trade: A Story from the Deep North, in which he and nine distant cousins retrace the triangle slave trade route of their ancestors, and grappled with the present-day consequences of the legacy of slavery. He is co-author, with Sharon Morgan, of Gather at the Table: The Healing Journey of a Daughter of Slavery and a Son of the Slave Trade. Tom serves as Director for Coming to the Table and is a trained STAR Practitioner. More at his website.
Barb Toews, editor. Barb is an experienced practitioner, trainer, and educator in restorative justice, with a special interest in its application in correctional facilities. She is currently Assistant Professor in criminal justice at University of Washington Tacoma. She is co-founder of Designing Justice+Designing Spaces (DJ+DS), an initiative that explores the relationship between restorative justice and the architecture/design of justice spaces by engaging incarcerated individuals in a visionary design process. She has published numerous articles and book chapters on restorative justice in correctional settings and in relation to environmental design. Books include The Little Book of Restorative Justice for People in Prison and Critical Issues in Restorative Justice, co-edited with Howard Zehr. Prior to her current academic position, Barb held leadership positions in criminal justice non-profit organizations, including the Pennsylvania Prison Society and the Lancaster Area Victim Offender Reconciliation Program (now Advoz).
Lynda Davis, advance manuscript reader. Lynda is a descendant of enslavers. She has been a member of CTTT since February 2014, where she is a member of both the Reparations and Mindfulness/Meditation Working Groups. She's also a member of the Annapolis CTTT Local Affiliate Group. Lynda writes for the Slave Dwelling Project's blog, and for her local paper, The Capital Gazette.
Dionne Ford, advance manuscript reader. Dionne is co-editor of Slavery's Descendants: Shared Legacies of Race and Reconciliation (Rutgers, 2019) and author of Finding Josephine forthcoming from Putnam. Her work has won awards from the National Association of Black Journalists, the Newswomens' Club of New York, the Sustainable Arts Foundation, and the National Endowment for the Arts. Dionne served on the CTTT Board of Managers 2013-14, and served as Vice President.
Fabrice Guerrier, advance manuscript reader. Fabrice is a writer, poet, entrepreneur and former President of the Board of CTTT. He is the Founder and CEO of Syllble Studios, Inc., a collaborative storytelling startup organizing peer-to-peer online production houses to publish original fiction stories and books. He co-authored The Wall – a futuristic Scifi novella and Syllble: Collection of Collaboratively Written Short Stories. He is a 2018 Shafik Gabr Fellow, a Seth Godin AltMBA Alumni, a Senior Fellow at Humanity in Action and was named the 2016 PEN Haiti Fellow at the PEN American Center. He holds an MA in Conflict Transformation from Eastern Mennonite University and a BS in International Affairs and Leadership Studies from Florida State University. He is fluent in French, Haitian Creole and English. His writing has appeared in The PEN America Blog, Public Pool, Blavity and Moko: Caribbean Arts and Letters.
Sheri Bailey, sidebar story contributor. Sheri is a playwright, spoken word artist, and teacher, and NEA Individual Theatre Fellowship winner. She founded the Juneteenth Festival Co. in Virginia in 1997. Books, plays, screenplays, essays and stories she's written include All Kinds of Blue, Abolitionists' Museum, Dannie 'n Laurence, A Great & Dismal Swamp, 2 Harriets, Ben & Jefferson, Afrikan Kingz & Queenz, Southern Girls (co-written with Dura Curry), Passing, Mayhem @ the Red River Bar, Walking with a Panther, and Summers in Suffolk (1870, 1896, 1930, 1955, 1970, 1992). Sheri Bailey's body of work will be housed at the Huntington Library.
Sri Lalitambika Devi, sidebar story contributor. Sri is the author of several Indian scriptures in English translation, including the Bhagavad Gita and Sankaracharya's Atma Bodha. She is a founding editor of literary journal Lalitamba and teaches in the English Department at CUNY (Borough of Manhattan Community College).
Danita Rountree Green, sidebar story contributor. Author/ playwright Danita Rountree Green (Satiafa) is a Storyteller on a Mission, challenging the narratives on race, historical trauma and the legacy of slavery. Her titles include Broom Jumping: A Celebration of Love, Grandmother's Gift of Memories: An African American Family Keepsake, and The Love Locked Down Series. Co-founder of the Richmond, Virginia affiliate chapter of Coming to the Table. More at her website.
Kimberly James, sidebar story contributor. The Green Goddess, Kimberly James, is a professor, author, and world traveler. She has visited over 75 countries and every continent. Her travel memoir, Finding My Brave Space: A Black Girl's Tale of Wanderlust, was published in 2017.
Cassandra Lane, sidebar story contributor. Formerly a newspaper journalist and high school teacher, Cassandra Lane is an alum of VONA and AROHO. She received an MFA from Antioch University, and has published stories in The Bellingham Review, The Manifest-Station, Writers Resist, The Source, The Atlanta Journal Constitution, The Scream, Everything but the Burden, Ms. Aligned and more. Her story is part of The New York Times' recent "Conception" project—an animated visual/audio series exploring motherhood and choice. Her book-length project explores the generational impact of the lynching of her maternal great-grandfather. A Louisiana native, Cassandra lives with her family in Los Angeles and is the managing editor of L.A. Parent magazine.
Sharon Morgan, sidebar story contributor. Sharon Leslie Morgan, a writer and genealogist, is the founder of Our Black Ancestry. A pioneer in multicultural marketing, she is a founder of the National Black Public Relations Society. The Chicago native has lived abroad in the Caribbean, Europe and Africa. She is co-author, with Thomas DeWolf, of Gather at the Table: The Healing Journey of a Daughter of Slavery and a Son of the Slave Trade. Sharon's first book, My Daddy Is A Cool Dude, was published in 1975 by The Dial Press and nominated for a prestigious Caldecott Medal for children's literature. She is the co-author of Real Women Cook: Building Healthy Communities with Recipes that Stir the Soul, and her most recent book is Paris in a Pot: Living a Dream in the City of Light.
Leslie Stainton, sidebar story contributor. Leslie teaches first-year writing seminar to undergraduates at the University of Michigan, where she is Lecturer, Residential College. She is the author of two nonfiction books, Lorca: A Dream of Life and Staging Ground: An American Theater and Its Ghosts. Her writing has appeared in The Sun, The American Scholar, and Broad Street, among others. She is at work on a book about her slaveholding ancestors, the Scarletts of Georgia. She serves on the CTTT Board of Managers. | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 952 |
1803?-1897. Deposited in the Wall collection, Alexander Turnbull Library, New Zealand.
What is The Halfway House?
The Halfway House is an old name for Irish, Welsh and English accommodation houses or public inns. These former coaching stops are located at strategic travelling points, usually halfway between two destinations by horse and sometimes on a border between counties. The name came to New Zealand with the 1840 mass immigration of European people.
The first building known as the Halfway House in Glenside (then called "The Halfway") was built in c1840-1841 and occupied by the Wall family. It was built halfway between Wellington and Porirua. It no longer exists. However, there is a later built building known as the Halfway House in Glenside.
The most recent building known as the Halfway House in Glenside is located at 246 Middleton Road, Glenside, behind Twigland Gardeners World on the Glenside Reserve.
Wellington City Council own it. The existing building, built c1885 and known as the Halfway House, was privately owned until 1951. In 1951 Victoria College (now Victoria University) encroached onto the Wellington Town Belt. Wellington City Council purchased the the Halfway House and associated land with Crown funds paid in compensation for the encroachment. The house was rented out until about c1997 when the last tenant died. It has been unoccupied since then.
Since 1951, the Glenside Progressive Assn. Inc. has been concerned about the way the Wellington City Council has been managing the Glenside Reserve and Halfway House and has been raising these concerns with the Wellington City Council.
These concerns intensified during 2001-2011. This was because the house was not tenanted or being looked after. At the same time, the Wellington City Council was consulting on the Northern Growth Management Framework and the Assn. could see the potential for the Glenside Reserve and Halfway House to have a more prominent place in the community. During these years the Assn. made many submissions and representations to the City Council and Councillors in an effort to get the house renovated for public use.
Renovations commenced in 2012 and were completed in 2017.
The Glenside community vision is to work with Council in managing the future of the house, to develop the grounds as a Victorian farm garden using heritage plants and flowers, and to promote the house and grounds as an attractive and interesting place for visitors.
What is the heritage status of the Halfway House?
The Halfway House is listed on the Wellington District Plan as an historic site.
In 2010 the Historic Places Trust received a paper from the Glenside Progressive Assn. Inc. to have the building assessed for a Historic Category rating.
The Trust advised it was unable to progress the Halfway House nomination for registration due to its current resourcing levels and that the nomination will remain on file in the event circumstances change.
Information provided by the Association has been recorded in the Historic Paces Trust register database and will be kept a part of the Historic Places Trust records.
What is happening with the Halfway House?
In 2012 the Wellington City Council began restoration work on the Halfway House. There was a set-back in October 2015. Just days before renovations were due to be signed off, a crimped fire sprinkler pipe in the ceiling burst free and for three days water flooded the house. In the last week of February 2017, renovations recommenced. The house was officially opened on 29 October, 2017. Wellington City Council sought registrations of interest from people interested in leasing the house. It is currently occupied by the leaders of Challenge 2000, a youth development initiative. The downstairs front rooms remain available for community use and are managed by the Glenside Progressive Assn. Inc.
What is happening with the grounds?
The grounds are managed in partnership by the Halfway House Heritage Gardeners, who are creating an early settler Victorian farm garden, and the Wellington City Council.
What does Wellington City Council propose for the house?
Retain ownership of the building but find a suitable long-term use for it.
Carry out essential maintenance and work to make the house safe.
Lease the building for public good (recreation, leisure, community, cultural or education).
Develop a car-park (which has since been completed).
Recreate a garden in keeping with heritage and character and place historic interpretation panels.
Why is there a Halfway House in Glenside?
Glenside was known as The Halfway from c1840-1928. The name was given to the area because it was located on the main route halfway between Wellington and Porirua.
Early settlers Anthony and Susannah Wall settled in the area in the spring of 1841 and provided accommodation for travelers passing by. Their home is believed to have been located off present day Glenside Road and was known as the Halfway House. It was the first Halfway House in the locality.
After the Wall's left the area, history records six different proprietors operating a Halfway House in the Halfway (Glenside) up until c1900. By 1900 the railway line had opened taking travelers by steam train direct to towns, by-passing the former horse and coach route.
Where was the first Halfway House located?
The Wall family lived at The Halfway from 1841-1849. There are different opinions about the location of the Wall's house. The page 'Halfway House Location' on this website explains the details.
What do we know of the current Halfway House in Glenside?
Writers have different opinions about whether the Halfway House on the Glenside Reserve was originally used for a Halfway House business or was a private residence for Alexander Brown and his wife Margaret.
What people do agree is that Alexander "Sandy" Brown built it in about 1885 and that he was a licensee who operated a Halfway House in The Halfway, now called Glenside.
Brown's descendants say that Alexander operated a hostelry (public inn and accommodation) and way station (place for checking goods in transit) for stage coaches passing along the main road.
Map showing the Halfway House on the Porirua Road, north of Johnsonville. This was drawn when the Porirua Road was being upgraded, hence the reference to the "old" part of the road.
The existing c1885 Halfway House is located very close to where one of the Halfway House's was located in 1849. A map drawn by Thomas Henry Fitzgerald in 1849 for Captain A H Russell marks the location of a building called The Halfway House (see map above). Today this location is about where Twigland Gardeners World is, or perhaps it was behind Twigland's, where the ground rises at the base of the hill.
In 1849 the Wall's sold out and moved to Paremata, now known as Papakowhai. The Halfway House was then managed by John McKain. It is possible that Mr McKain relocated the operation of his business from the bottom of the Accommodation Road over to this 1849 site. (In the old days, it was the person who was licenced, not the premises). The owner of a private home opposite Twigland Gardener's World found many old horseshoes in her garden, suggesting that at a later time there may have been a blacksmith located opposite the Halfway House marked on Mr Fitzgerald's map. | {
"redpajama_set_name": "RedPajamaC4"
} | 8,789 |
Melanerpes herminieri е вид птица от семейство Picidae. Видът е почти застрашен от изчезване.
Разпространение
Видът е разпространен в Гваделупа.
Източници
Melanerpes | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 7,402 |
I gave this book 3.5 stars. It was fun and interesting. Not very surprising, but it still brings you on a great adventure with likeable characters.
The story was a bit slow in the beginning, there was nothing that really screamed out to me. I put this off for a few days, but I really wanted to get to the ARC of the second book soon so I decided to put it on my priority reads list. It mostly started out with me reading it for about 5 minutes every day when I would bring my dog to the park, but I did get very captivated by the story about 36% in.
The writing wasn't overly complicated to read. This novel didn't have anything complex that I really needed to have my full attention on the whole time I was reading, so I had no problems following along.
I wish the author was a bit more clear on the protagonist's age. I can't tell if she's in college or if she's a senior in a private school or something about to go to university. I know at the very beginning she was very worried about her GPA being lowered by Evan, her TA, but are there even TA's in high school? I never had any that's for sure. I'm really not sure about her age.
As for Evan, he completely grew on me. He started out as the hot yet Satan-esque TA in Kat's class who seemed to be out to ruin her life, but then when we spent more time with him he really wasn't bad at all. I honestly don't know how he went from being so awful to being actually sweet at times. He made a complete one eighty during the book and I really want to see more of him in the next book.
As for the plot itself, it was exciting and managed to capture my attention seamlessly. There is something about the mix of mystery and anything that's paranormal that just brings certain books to life. Now, I love real life mysteries and mysteries in TV shows and movies, but the mystery genre is one that I have a lot of problems with in books. I usually have a hard time getting into them if they aren't Agatha Christie or Stephen King, so the added magical element adds something to the plot that you can't find in a normal mystery novel.
This book also tugs at the heartstrings at times and that just gave more life to the story. I feel like when a book can make you feel those things then it is doing a good job. Books with sad scenes that can bring out your own emotions can't be bad books. Unless they are feelings of hatred towards the book, then it can be bad, but this wasn't one of those books.
Overall, I'm excited to read the next book and see what improvements that the author has made. I'm sure there will be a lot more world building and action to come. | {
"redpajama_set_name": "RedPajamaC4"
} | 9,384 |
\section{Introduction}
\label{secIntro}
Many analytical approaches to consideration of quantum magnetic systems with localized spins base on representations of spins via bosonic or fermionic operators allowing to turn from spin Hamiltonians to those describing ensembles of bosons or fermions. \cite{auer} At low temperature the form of the representation depends on the ground state of the spin system under discussion. The Holstein-Primakoff and the Dyson-Maleev representations are frequently used for magnetically ordered phases, the Jordan-Wigner transformation proved to be useful for $S=1/2$ spin chains, the bond-operator formalism \cite{18} was proposed for spin liquids with singlet ground state, etc.
We propose in the present paper a new representation of an integer spin $S$. This representation should be useful in describing the paramagnetic phase in which all spins are mainly in the quantum state with zero quantum number for projection onto a preferential direction. As an example of particular system of this type we choose a Heisenberg magnet with large single-ion easy-plane anisotropy which Hamiltonian has the form
\begin{eqnarray}
\label{ham}
{\cal H} = \half \sum_{i,j} J_{ij} \mathbf{S}_i \mathbf{S}_j + D \sum_i \left(S^z_i\right)^2,
\end{eqnarray}
where summations are taken over all sites of the lattice with arbitrary spatial dimension, $D>0$ and signs of exchange constants $J_{ij}$ are not important in the following. The first term in \eqref{ham} is supposed to be small enough compared to the second one so that the system to be in the paramagnetic phase at $T=0$. There is a quantum critical point (QCP) $D=D_{c}$ separating the paramagnetic phase at $D>D_{c}$ and a magnetically ordered or a spin-liquid one at $D<D_{c}$. The value of $D_{c}$ as well as the phase type at $D<D_{c}$ depend on the lattice spatial dimension and the exchange interaction \cite{add5, 6, wong}.
Quite a few compounds can be mentioned with the paramagnetic ground state of the considered type which are described by the Hamiltonian \eqref{ham}: $\rm CsFeBr_3$ \cite{add9,lind2,add18,add19}, $\rm CsFeCl_3$ \cite{lind2}, $\rm Sr_3NiPtO_6$ \cite{add1}, NENC \cite{add7,add16,add17}, NENP \cite{add13,add14,add15}, NBYC \cite{add10}, $\rm NiCl_2$-$\rm 4SC(NH_2)_2$ \cite{12,13,14,15,16,17,add8,chern} (apparently the most intensively studied compound of this type in recent years), $\rm(Ni(C_5H_5NO)_6)(NO_3)_2$ \cite{carlin} and $\rm NiSnCl_6\cdot6H_2O$ \cite{nisn1,nisn2}. All the mentioned materials have $S=1$ and all of them are quasi-1D magnets. It is probably the reason why the majority of the recent theoretical investigations of the model \eqref{ham} with large $D>0$ focus on weakly coupled or independent spin chains \cite{add2,add3,add4,add5,add6,add11,add12,add15,kolezh,mc1,mc2,1,4,5,6,7,8,9}. In particular, the elementary excitation spectrum has been derived before using a random phase approximation \cite{lind1,lind2}, the regular perturbation theory \cite{4}, a "generalized spin-wave approach" \cite{13,chern} and some other self-consistent procedures \cite{1,3,5,wang5}.
Treating the first term in Eq.~\eqref{ham} as a perturbation, using the proposed spin representation and the conventional diagram technique we derive below expressions for the elementary excitation spectrum and the ground state energy in the paramagnetic phase in the third order in the perturbation theory. For brevity, this approach is referred to hereafter as an expansion in terms of $J/D$ while one should bare in mind that constants $J_{ij}$ in Eq.~\eqref{ham} are assumed to be arbitrary. We also obtain these results in the special case of $S=1$ using simpler spin representations some of which were introduced before in Refs.~\cite{kolezh,3}. In the particular case of a spin chain with $S=1$ our expression for the spectrum coincides with that obtained in Ref.~\cite{4} only for this special case using the regular perturbation theory. Comparing our results with those of numerical calculations \cite{2,3} which were carried out for $S=1$, square lattice and antiferromagnetic exchange we show that our approach works better than other theoretical methods proposed so far \cite{1,lind1,5,chern} . In particular, our results are in very good agreement with the numerical ones \cite{2,3} not very close to the QCP. At $D\approx D_c$ only the spectrum of long-wavelength excitations is reproduced unsatisfactorily that is a consequence of strong fluctuations near the QCP.
We demonstrate that our approach is applicable to the intensively studied compound $\rm NiCl_2$-$\rm 4SC(NH_2)_2$ (DTN) \cite{12,13,14,15,16,17,add8,chern} described by the model \eqref{ham}. It is shown that our expression for the spectrum describes badly the spectrum obtained in the neutron experiment \cite{13} with the conventional set of parameters (values of exchange constants and $D$) attributed to DTN before. A new set of parameters is proposed for DTN using which we fit well the neutron data and reproduce values of critical fields found experimentally \cite{16}. It should be noted that in comparison with the conventional model used for DTN discussion we take into account also an exchange interaction between two magnetic sublattices of DTN \cite{prev} which should be taken into consideration as the recent ESR experiment \cite{12} demonstrates.
The rest of the present paper is organized as follows. In Sec.~\ref{spinrep} the new representation of an integer spin is proposed and details of the diagram technique based on this representation are discussed. In Sec. \ref{observ} expressions for the elementary excitation spectrum and the ground state energy are derived in the third order in $J/D$. In Sec.~\ref{disc} we compare our results with those obtained before using other approaches. We discuss also representations which are valid only for $S=1$ some of which are simpler than the general one. The expression for the spectrum derived in Sec.~\ref{observ} is applied for analysis of the DTN spectrum obtained in the neutron experiment \cite{13}. Sec.~\ref{conc} contains our conclusion.
\section{Representation of an integer spin and technique}
\label{spinrep}
The ground state of a system described by the Hamiltonian \eqref{ham} in the limit of $J/D \to 0$ is a direct product of states $| S^{z}_i = 0 \rangle$: $\Pi_i \otimes | S^z_i = 0 \rangle$. The lowest excited states can be constructed from the ground state by substituting $| S^z_i = \pm 1 \rangle$ for $| S^{z}_i = 0 \rangle$ at any $i$. The energy of such states is equal to $D$ and one leads to the doubly degenerate dispersionless elementary excitation spectrum
\begin{eqnarray}
\label{spec00}
\e_{0\pp} = D.
\end{eqnarray}
The exchange interaction gives rise to the spectrum dispersion. When it is small enough one can find expressions for the spectrum and other observables considering the exchange as a perturbation. In particular, the elementary excitation spectrum of the spin chain with $S=1$ and the exchange coupling between only nearest neighbors was calculated in Ref.~\cite{4} in the third order of the regular (non diagrammatic) perturbation theory. In contrast, our aim is to construct a spin representation which opens the door to calculations utilizing the diagram technique and which allows to obtain such expressions for observables easier. In particular, we recover below the result of Ref.~\cite{4} using the diagram technique.
We propose the following expressions for projections of an integer spin $S$:
\begin{eqnarray}
\label{sz}
S^z_i &=& n_{b,i} - n_{a,i}, \\
\label{s+}
S^+_i &=& S^x_i + i S^y_i = \bkr_i \sqrt{\dfrac{ ( S - n_{b,i} ) ( S+1 + n_{b,i}) }{ 1 + n_{b,i} }} +
\sqrt{\dfrac{ ( S - n_{a,i} ) ( S+1 + n_{a,i}) }{ 1 + n_{a,i} }} \cdot a_i,
\end{eqnarray}
where $a_i$ and $b_i$ are bosonic operators, $n_{a,i} = \akr_i a_i$, $n_{b,i} = \bkr_i b_i$ and
\begin{eqnarray}
\langle S^z_i = -(n+1) | \;\, \akr_i \;\, | S^z_i = -n \rangle &\;=& \sqrt{ n+1 }, \non \\
\langle S^z_i = n+1 | \;\, \bkr_i \;\, | S^z_i = n \rangle &\;=& \sqrt{ n+1 }, \non \\
a_i \; | S^z_i = 0 \rangle &\;=& 0, \non\\
b_i \; | S^z_i = 0 \rangle &\;=& 0, \non \\
n &\geq& 0. \non
\end{eqnarray}
Operators $\akr$ and $\bkr$ create excitations with $S^z=-1$ and $+1$, respectively. The subspace of physical states is constrained by the following conditions:
\begin{equation}
\label{consab}
n_{a,i} n_{b,i} = 0,
\end{equation}
\begin{equation}
\label{consaa}
\begin{array}{l}
n_{a,i} \leq S, \\
n_{b,i} \leq S.
\end{array}
\end{equation}
It can be readily verified that representation \eqref{sz}--\eqref{s+} reproduces the spin commutation relations on the physical subspace defined by Eqs.~\eqref{consab} and \eqref{consaa}.
Condition \eqref{consab} selects states having at any site only excitations of $a$- or $b$- type. This constraint can be satisfied by adding to the Hamiltonian the term describing an infinite repulsion between $a$ and $b$ particles
\begin{equation}
\label{u1}
{\cal H}_{U1} = \frac UN \sum_i \akr_i \bkr_i a_i b_i, \quad U \to + \infty.
\end{equation}
After this modification matrix elements of operators \eqref{sz}--\eqref{s+} become zero between states from physical and unphysical subspaces constrained by conditions \eqref{consaa}. It means that at zero (and most probably at low) temperature we can use Eqs.~\eqref{sz}--\eqref{s+} and the diagram technique forgetting about condition \eqref{consaa}, as it is done in a similar situation in the case of the Holstein-Primakoff representation. We prove this statement below for $S=1$ by performing calculations using Eqs.~\eqref{sz}--\eqref{s+}, taking into account constraint term \eqref{u1} and introducing to the Hamiltonian the additional term
\begin{eqnarray}
\label{u2}
{\cal H}_{U2} = \frac UN \sum_i \left( \akr_i \akr_i a_i a_i + \bkr_i \bkr_i b_i b_i \right), \quad U \to + \infty,
\end{eqnarray}
which explicitly selects states with no more than one $a$ or $b$ particles as condition \eqref{consaa} requires at $S=1$. It is worth noting that we could construct a spin representation similar to \eqref{sz}--\eqref{s+}, which matrix elements are zero on states from physical and unphysical subspaces so that it was not necessary to introduce term \eqref{u1} to the Hamiltonian. However such a representation would be very cumbersome. On the other hand term \eqref{u1} does not complicate calculations much at any integer $S$. That is why we use below Eqs.~\eqref{sz}--\eqref{s+} with constraint term \eqref{u1}.
At sufficiently small exchange interaction and low temperature we expect densities of $a$ and $b$ particles to be small. Therefore one can expand square roots in Eq.~\eqref{s+} into series and restrict oneself by the first terms of the resultant normally ordered expressions which have the form
\begin{eqnarray}
\label{s+series}
S^+_i &\approx& \bkr_i \left( c_1 - c_2 \; \bkr_i b_i \right) + \left( c_1 - c_2 \; \akr_i a_i \right) a_i, \\
\label{c1}
c_1 &=& \sqrt{ S(S+1) }, \\
\label{c2}
c_2 &=& \sqrt{ S(S+1) } - \sqrt{ \frac{(S-1)(S+2)}{2} } > 0.
\end{eqnarray}
Using Eqs.~\eqref{sz} and \eqref{s+series} and taking into account Eq.~\eqref{u1} we obtain from Eq.~\eqref{ham}
\begin{subequations}
\label{hamfur}
\begin{eqnarray}
\label{h2}
{\cal H} &=& \sum_\pp \e_{1\bf p} \left( \akr_\pp a_\pp + \bkr_\pp b_\pp \right) \\
\label{h2an}
&&{}+\sum_\pp \frac{ c_1^2 }{ 2 } J_\pp \left( \akr_\pp \bkr_{-\pp} + a_\pp b_{-\pp} \right) \\
\label{aaaa}
&&{} + \fracNN \sum_{\pp_1+\pp_2=\pp_3+\pp_4} \left\{ \left[ D + \half J_{3-1} - \frac{ c_1 c_2 }{ 2 } \left( J_1 + J_3 \right) \right] \left( \akr_1 \akr_2 a_3 a_4 + \bkr_1 \bkr_2 b_3 b_4 \right) + \left[ U - J_{3-1} \right] \akr_1 \bkr_2 a_3 b_4 \right\} \\
\label{abbb}
&&{} - \fracNN \sum_{\pp_1+\pp_2+\pp_3=\pp_4} \frac{ c_1 c_2 }{ 2 } J_1 \left( \bkr_1 \akr_2 \akr_3 a_4 + \akr_1 \bkr_2 \bkr_3 b_4 + \akr_4 a_3 a_2 b_1 + \bkr_4 b_3 b_2 a_1 \right),
\end{eqnarray}
\end{subequations}
where $ J_\pp = \sum_j J_{ij} e^{i \pp {\bf R}_{ij}} $, $N$ is the number of unit cells,
\beq
\label{spec0}
\e_{1\bf p} = \e_{0\bf p} + \frac{ c_1^2 }{ 2 } J_\pp = D + \frac{ c_1^2 }{ 2 } J_\pp,
\eeq
$\e_{0\bf p}$ is defined by Eq.~\eqref{spec00} and we omit indexes $\pp$ in Eqs.~\eqref{aaaa} and \eqref{abbb}. Note that we take into account only terms with no more than four operators in Eq.~\eqref{hamfur}. It can be shown that terms containing more than four operators which appear from higher order terms in the series expansion of square roots in Eq.~\eqref{s+} lead to contributions to the spectrum and to the ground state energy of the order of $(J/D)^4$ and higher. As our aim is to calculate these quantities only up to the third order in $J/D$, we can use Hamiltonian \eqref{hamfur}.
It is convenient to introduce the following Green's functions:
\begin{eqnarray}
\label{gdef}
G ( p ) &=& - i \langle a_p \akr_p \rangle = - i \langle b_p \bkr_p \rangle, \\
\label{fdef}
F ( p ) &=& - i \langle \bkr_{-p} \akr_p \rangle,
\end{eqnarray}
where $ p = (\w , \pp ) $ and $ a_p $ is the Fourier transform of $ a_\pp ( t ) $. Naturally, the equality $\langle a_p \akr_p \rangle = \langle b_p \bkr_p \rangle$ is satisfied. Dyson equations for these Green's functions have the form
\begin{eqnarray}
\label{dyson}
G ( p ) &=& G_0 ( p ) \left[ 1 + \Sig_p G ( p ) + \Pi_p F ( p ) \right], \\
F ( p ) &=& G_0 ( -p) \left[ \overline{\Pi}_p G ( p ) + \Sig_{-p} F ( p ) \right],\non
\end{eqnarray}
where $ G_0 ( p ) = ( \w - \e_{1\bf p}+ i\delta )^{-1} $, $\Sig_p$ and $\Pi_p$ are normal and anomalous self-energy parts, respectively. The solution of Eq.~\eqref{dyson} has the form
\begin{eqnarray}
\label{g}
G ( p ) &=& \frac{\w+ \e_{1\bf p} + \Sig_{-p}}{{\cal D}( p )}, \\
\label{f}
F ( p ) &=& - \frac{ \overline{\Pi}_p}{{\cal D}( p )}, \\
\label{den}
{\cal D}( p ) &=& \w^2 - \e^2_{1\bf p} - \e_{1\bf p} (\Sig_p + \Sig_{-p}) + \w ( \Sig_{-p} - \Sig_p ) - \Sig_p \Sig_{-p} + \left|\Pi_p\right|^2.
\end{eqnarray}
\section{Application of the approach}
\label{observ}
In this section we apply the method described above for calculation of the elementary excitation spectrum and the ground state energy.
\subsection{Elementary excitation spectrum}
\begin{figure}
\includegraphics[scale = 1.0]{Diagrams2nd}
\caption{(a) Diagrams for the normal self-energy part $\Sigma_p$ giving non-zero contributions of the second order in $J/D$. Solid and dashed lines stand for Green's functions $G(p)$ of $a$ and $b$ particles, respectively, defined by Eq.~\eqref{g}. Lines containing solid and dashed parts stand for anomalous Green's functions $F(p)$ defined by Eq.~\eqref{f}. Bare vertexes are defined by Eqs.~\eqref{aaaa} and \eqref{abbb}. Open and filled circles represent renormalized vertexes equations for which are presented in panel (b).}
\label{diag1}
\end{figure}
The elementary excitation spectrum $\e_{\pp}$ is defined by poles of Green's functions \eqref{g} and \eqref{f}:
\begin{eqnarray}
\label{speceq}
{\cal D}( \e_{\pp} , \pp ) = 0.
\end{eqnarray}
Let us consider diagrams for the normal self-energy part $\Sigma_p$ some of which are shown in Fig.~\ref{diag1}. If a diagram contains a contour that can be walked around while moving by arrows of functions $G_{0}(p)$, integrals over frequencies in such a diagram give zero. \cite{popov} In diagrams without such contours there are at least two vertexes \eqref{abbb} or at least one vertex \eqref{abbb} and one anomalous Green's functions $F(p)$. As the vertex \eqref{abbb} is of the order of $J/D$ and the numerator of $F(p)$ is $O(J/D)$ (see Eq.~\eqref{h2an}), the normal self-energy part is the value of the order of $(J/D)^{2}$. Thus, we obtain for $\Sigma_p$ in the first order in $J/D$
\begin{equation}
\label{sigma1}
\Sig_p^{(1)} = 0.
\end{equation}
Let us turn to diagrams for the anomalous self-energy parts $\Pi_{p}$. One of them is shown in Fig.~\ref{diag3}(b). The contribution to $\Pi_{p}$ of the first order in $J/D$ is given by term \eqref{h2an}. Diagrams for $\Pi_{p}$ with one vertex contain sums like $\sum_{\kk} J_{\pp+\kk} = 0$ in the first order in $J/D$. Therefore such diagrams are at least of the second order in $J/D$. A consideration of other diagrams similar to that presented above for $\Sigma_p$ shows that they give contributions of the second order in $J/D$ and higher. Then we have in the first order in $J/D$
\begin{equation}
\label{pi1}
\Pi^{(1)}_p = \frac{c_1^2}{2} J_\pp.
\end{equation}
\begin{figure}
\includegraphics[scale = 1.0]{Diagrams3rd}
\caption{(a) Diagrams for the normal self-energy part $\Sigma_{p}$ giving non-zero contributions of the third order in $J/D$. (b) The diagram for anomalous self-energy part $\Pi_{p}$ of the second order in $J/D$. Same notation as in Fig.~\ref{diag1}.}
\label{diag3}
\end{figure}
It is convenient to rewrite Eq.~\eqref{speceq} using Eq.~\eqref{den} in the form
\beq
\label{spec1}
\e_{\pp}^{2} &=& \left( \e_{1\pp} + \Sig_{(\e_{\pp}, \pp)} \right)^2 - \left|\Pi_{(\e_{\pp}, \pp)}\right|^2 + \left( \e_{\pp} - \e_{1\pp} - \Sig_{(\e_{\pp}, \pp)} \right) \left( \Sig_{(\e_{\pp}, \pp)} - \Sig_{(-\e_{\pp}, -\pp)}\right).
\eeq
It follows from the previous discussion that
\begin{eqnarray}
\e_{\pp} - \e_{1\pp} - \Sig_{(\e_{\pp}, \pp)} = O((J/D)^2),\\
\Sig_{(\e_{\pp}, \pp)} - \Sig_{(-\e_{\pp}, -\pp)} = O((J/D)^2). \non
\end{eqnarray}
Bearing in mind these equations and that our aim is to find the spectrum in the third order in $J/D$ we can write Eq.~\eqref{spec1} in the form
\beq
\label{spec2}
\e_{\pp}^{2} = \left( \e_{1\pp} + \Sig_{(\e_{\pp}, \pp)} \right)^{2}- \left|\Pi_{(\e_{\pp}, \pp)}\right|^2 + O((J/D)^4).
\eeq
As follows from this equation, the first corrections to $\e_{1\pp}$ are of the second order in $J/D$. Taking into account that the first corrections to self-energy parts are also of the second order in $J/D$, we can replace $(\e_{\pp}, \pp)$ by $(\e_{1\pp}, \pp)$ in Eq.~\eqref{spec2} and write down the final formula for the spectrum calculation up to the third order in $J/D$
\begin{eqnarray}
\label{spec3}
\e_\pp^2 &=& \left( \e_{1\pp} + \Sig_{(\e_{1\pp},\pp)} \right)^2 - \left|\Pi_{(\e_{1\pp}, \pp)}\right|^2. \end{eqnarray}
Let us find the spectrum in the second order in $J/D$. As is clear from Eq.~\eqref{spec3}, one has to use the anomalous self-energy part in the first order in $J/D$ for which we obtain Eq.~\eqref{pi1}. The normal self-energy part has to be found in the second order in $J/D$ in which diagrams presented in Fig.~\ref{diag1}(a) should be taken into account. In order to calculate these diagrams it is necessary to find vertexes denoted by open and filled circles in the zeroth order. As follows from the diagram analysis presented above, the zeroth and the first order contributions to the vertexes are presented by a series of ladder diagrams only (see Fig.~\ref{diag1}(b)). Diagrammatic equations in Fig.~\ref{diag1}(b) have the following explicit form in the zeroth order in $J/D$:
\begin{subequations}
\label{vert0}
\begin{eqnarray}
\Gamma_1^{(0)} &=& D - 2 \frac{ \Gamma_1^{(0)} D }{ 2D - \Omega}, \\
\Gamma_2^{(0)} &=& U - \frac{ \Gamma_2^{(0)} U }{ 2D - \Omega },
\end{eqnarray}
\end{subequations}
where $\Omega$ is the sum of incoming lines frequencies. The solution of Eq.~\eqref{vert0} at $ U \to +\infty $ has a simple form
\begin{subequations}
\label{vert1}
\begin{eqnarray}
\Gamma_1^{(0)} &=& \frac{ D \left( 2D - \Omega \right)}{ 4D - \Omega }, \\
\Gamma_2^{(0)} &=& 2D - \Omega.
\end{eqnarray}
\end{subequations}
After substitution of Eqs.~\eqref{vert1} to diagrams for self-energy parts and integration over frequencies $\Omega$ gives a difference of the kind $\epsilon_{1{\bf k}_1}-\epsilon_{1{\bf k}_2}$ with some momenta ${\bf k}_{1,2}$. Then $\Omega$ corresponds to values of the order of $J/D$ in diagrams for self-energy parts and it can be neglected in the vertex calculation in the zeroth order in $J/D$. As a result we obtain for vertexes from Eqs.~\eqref{vert1}
\begin{subequations}
\label{ver0res}
\begin{eqnarray}
\Gamma_1^{(0)} &=& D/2, \\
\Gamma_2^{(0)} &=& 2D.
\end{eqnarray}
\end{subequations}
One obtains for the normal self-energy part in the second order in $J/D$ as a result of calculation of diagrams shown in Fig.~\ref{diag1}(a) taking into account Eqs.~\eqref{ver0res}
\begin{equation}
\label{sigmas2}
\Sig^{(2)}_{( \e_{1\pp}, \pp )} = \frac{2c_1^4 + 4c_1 c_2 - c_1^2 c_2^2}{8D} \frac1N\sum_{\kk} J^{2}_{\kk}.
\end{equation}
Using Eqs.~\eqref{sigmas2}, \eqref{spec3} and \eqref{pi1} we have for the spectrum in the second order in $J/D$ (cf.\ Eq.~\eqref{spec0})
\begin{equation}
\label{specs2}
\e_{2\pp} = \e_{1\pp} + \Sig^{(2)}_{( \e_{1\pp}, \pp )} - \frac{1}{2\e_{0\pp}}\left|\Pi^{(1)}_{(\e_{1\pp}, \pp)}\right|^2 = D + \frac{c_1^2}{2} J_\pp + \frac{2c_1^4 + 4c_1 c_2 - c_1^2 c_2^2}{8D} \frac1N\sum_{\kk} J^{2}_{\kk} - \frac{ c_1^4 } {8} \frac{J_\pp^2} {D}.
\end{equation}
In order to calculate the spectrum in the third order in $J/D$ one has to find the normal and anomalous self-energy parts in the third and the second order, respectively. To find $\Sigma_{p}$ in the third order we have to take into account diagrams shown in Fig.~\ref{diag3}(a). Besides, one has to consider also diagrams presented in Fig.~\ref{diag1}(a) taking into account the first order corrections to the vertexes for which we have after solving equations in Fig.~\ref{diag1}(b)
\begin{eqnarray}
\Gamma_1^{(1)} &=& \frac12 D - \frac18 \Omega + \frac12 J_{3-1} - \frac{c_1 c_2}{4} (J_1 + J_3), \\
\label{gamma2}
\Gamma_2^{(1)} &=& 2D - \Omega - J_{3-1}.
\end{eqnarray}
The only diagram of the second order in $J/D$ for the anomalous self-energy part is shown in Fig.~\ref{diag3}(b). It should be found using Eq.~\eqref{gamma2}. As a result of straightforward calculation we obtain
\begin{eqnarray}
\label{sig3}
\Sigma^{(3)}_{( \e_{1\pp}, \pp )} &=& \frac{c_1^2(c_1 + c_2)^2}{16D^2}\frac{1}{N^2} \sum_{\kk,\qq} J_\qq J_\kk J_{\qq-\kk} - \frac{ 2c_1^4 + 4 c_1 c_2 - c_1^2 c_2^2 }{8D^2} J_\pp \frac{1}{N} \sum_\kk J^2_\kk
\non\\
&&{}
+
\frac{c_1^4 + 3 c_1^2 c_2^2 - 2 c_1^3 c_2 + c_1 c_2 }{8D^2} \frac{1}{N^2} \sum_{\kk,\qq} J_\pp J_\kk J_{\qq-\kk + \pp},\\
\label{pi2}
\Pi^{(2)}_{( \e_{1\pp}, \pp )} &=& \frac{c_1^2}{4D} \frac{1}{N} \sum_\kk J_\kk J_{\kk-\pp} + \frac{c_{1}^{4}}{4D} \frac1N \sum_{\kk} J^{2}_{\kk}.
\end{eqnarray}
Using Eqs.~\eqref{spec3}, \eqref{sigma1}, \eqref{pi1}, \eqref{sigmas2}, \eqref{pi2} and \eqref{sig3}, one has for the spectrum in the third order in $J/D$
\begin{eqnarray}
\label{specs3}
\e_{3\pp} &=& D + \frac{c_1^2}{2} J_\pp + \frac{ 2c_1^4 + 4 c_1 c_2 - c_1^2 c_2^2 }{8D} \frac{1}{N} \sum_\kk J^2_\kk - \frac{c_{1}^{4}}{8D} \frac1N J^{2}_{\pp} \non\\
&&{}+ \frac{c_1^2(c_1 + c_2)^2}{16D^2}\frac{1}{N^2} \sum_{\kk,\qq} J_\qq J_\kk J_{\qq-\kk} - \frac{ 4c_1^4 + 4 c_1 c_2 - c_1^2 c_2^2 }{8D^2} J_\pp \frac{1}{N} \sum_\kk J^2_\kk \non \\
&&{}+ \frac{c_1^4 + 3 c_1^2 c_2^2 - 2 c_1^3 c_2 + c_1 c_2 }{8D^2} \frac{1}{N^2} \sum_{\kk,\qq} J_\pp J_\kk J_{\qq-\kk + \pp} - \frac{c_1^4}{8D^2} J_\pp \frac{1}{N} \sum_\kk J_\kk J_{\kk-\pp} + \frac{c_1^6}{16D^2} J_\pp^3.
\end{eqnarray}
The spectrum $\e_{3\pp}$ has a minimum in some point $\pp=\pp_0$, in which it is separated from the ground state by a gap. For example, this minimum is located at $\pp$ equal to the antiferromagnetic vector $\pp_0$ in the simple square or the simple cubic lattices with antiferromagnetic exchange interaction between only nearest neighbors ($\pp_0=(\pi,\pi)$ and $(\pi,\pi,\pi)$, respectively). The gap decreases with decreasing $D$ and it vanishes at QCP $D=D_{c}$. Upon further decreasing of $D$ a ``condensation'' takes place of elementary excitations with momentum $\ppp$ which corresponds to appearance of long-range magnetic order. We show below by particular examples that at $D\agt D_c$ series in $J/D$ for some quantities converge slowly. It is a manifestation of strong fluctuations near QCP.
It is worth noting that $c_1,c_2\sim S$ at $S\gg1$ (see Eqs.~\eqref{c1} and \eqref{c2}). Therefore, as follows from Eq.~\eqref{specs3}, the expansion parameter is actually $S^2J/D$ (not $J/D$) at $S\gg1$. It means, in particular, that $D_c\sim S^2J$ in the case of antiferromagnetic exchange interaction between nearest neighbor spins on the simple square or the simple cubic lattices. This conclusion is consistent with the result of the spin wave analysis carried out in antiferromagnetic phase (see, e.g., Ref.~\cite{prev}). Thus, a very strong anisotropy or a very small exchange are required for the paramagnetic phase stability at $S\gg1$ and the paramagnetic phase is absent in the classical spin limit at any finite $J$ and $D$ as it must be.
\subsection{Ground state energy}
\begin{figure}
\includegraphics[scale = 1.0]{GSdiagrams}
\caption{ Diagrams giving non-zero contributions of the second and the third order in $J/D$ to the ground state energy. Same notation as in Fig.~\ref{diag1}.}
\label{gsfig}
\end{figure}
It is useful to calculate the ground state energy for the sake of comparison with numerical results. Diagrams giving non-zero contributions of the second and the third order in $J/D$ are shown in Fig.~\ref{gsfig}. The straightforward calculation of these diagrams leads to the following expression for the ground state energy:
\begin{equation}
\label{gs}
E_{gs} = -\frac{ c_1^4 }{ 8D } \frac1N \sum_\pp J^2_\pp - \frac{ c_1^4 }{ 16D^2 } \frac{1}{N^2} \sum_{\pp,\kk} J_\pp J_\kk J_{\pp+\kk}.
\end{equation}
Notice that the first non-zero term in $E_{gs}$ is of the second order in $J/D$.
\section{Discussion and comparison with previous results and experiment}
\label{disc}
Note that we do not specify the type of exchange interaction $J_{ij}$ in Eq.~\eqref{ham} and the lattice type and dimension while deriving Eq.~\eqref{specs3}. Then Eq.~\eqref{specs3} is applicable, in particular, for the spin chain with $S=1$ and with exchange interaction between nearest neighbors only. The spectrum in this special case was calculated before in the third order in $J/D$ in Ref.~\cite{4} using the regular perturbation theory. It is easy to verify that the result of Ref.~\cite{4} coincides with Eq.~\eqref{specs3} in this case.
\subsection{Other spin representations for $S=1$}
\label{secSEqualOne}
In the particular case of $S=1$ a number of simpler spin representations can be introduced. In one of them $S^{z}_{i}$ is given by Eq.~\eqref{sz} and $S^+_i$ has the form (cf.\ Eq.~\eqref{s+})
\begin{eqnarray}
\label{s+simple}
S^+_i = \sqrt{2} \left( \bkr_i + a_i \right).
\end{eqnarray}
In contrast to Eqs.~\eqref{sz}--\eqref{s+} the introduction of term \eqref{u2} into the Hamiltonian is necessary now because Eq.~\eqref{s+simple} has non-zero matrix elements between states from physical and unphysical subspaces. Representation \eqref{s+simple} is actually introduced in Ref.~\cite{kolezh}, where spin chains in magnetic field are discussed. We have made calculations of the spectrum and the ground state energy using this representation and recovered Eqs.~\eqref{specs3} and \eqref{gs} at $S=1$.
It should be noted that at $S=1$ one can make calculations without taking into account the constraint term \eqref{u2} by adding to Eq.~\eqref{s+simple} projector operators $1 - n_{a,i} - n_{b,i}$ as follows:
\begin{equation}
\label{s+no}
S^{+}_i = \sqrt2 \left[ (1 - n_{a,i} - n_{b,i}) b_i + \akr_i (1 - n_{a,i} - n_{b,i}) \right].
\end{equation}
In contrast to Eq.~\eqref{s+simple} this representation has zero matrix elements between states from physical and unphysical subspaces and there is no need to take into account Eq.~\eqref{u2} now. We have carried out calculations with Eq.~\eqref{s+no} and recovered Eqs.~\eqref{specs3} and \eqref{gs} at $S=1$ once again.
The following spin representation for $S=1$ is introduced in Ref.~\cite{3}:
\beq
\label{s+sq}
S^{+}_i = \sqrt2 \left[ \sqrt{1 - n_{a,i} - n_{b,i}}\; b_i + \akr_i \sqrt{1 - n_{a,i} - n_{b,i}} \right],
\eeq
which is equivalent to our one \eqref{s+no} on the physical subspace and which, therefore, should lead to the same results. But authors of Ref.~\cite{3} do not take advantage of the opportunity to find physical quantities in the form of series in terms of powers of $J/D$ which such representations provide. They expand roots in Eq.~\eqref{s+sq} and analyze the spectrum of the Hamiltonian in the harmonic approximation. They also find renormalization of this spectrum by taking into account the simplest diagrams and carrying out some self-consistent calculations for the square lattice and antiferromagnetic exchange. We compare below results obtained in Ref.~\cite{3} with our ones.
\subsection{ Other approaches to the spectrum calculation }
After substitution of the self-energy parts \eqref{sigma1} and \eqref{pi1} obtained in the first order in $J/D$ to the general expression \eqref{spec1} for the spectrum we recover Lindgard's formula \cite{lind1,lind2,11}
\begin{eqnarray}
\label{eqLin}
\e_{L\pp} = \sqrt{ D \left[ D + S(S+1) J_\pp \right] },
\end{eqnarray}
which was found in Refs.~\cite{lind1,lind2} within the random phase approximation. But as it is clear from the above discussion, Eq.~\eqref{eqLin} is correct only in the first order in $J/D$. The second and the third order terms in $J/D$ in Eq.~\eqref{eqLin} differ significantly from those in Eq.~\eqref{specs3}. Case study that is done below shows that Eq.~\eqref{eqLin} works quite badly when $J/D$ is not very small.
A "generalized spin-wave approach" (GSWA) is used in Refs.~\cite{13,chern} for the spectrum consideration in DTN. In the framework of this approach in which fluctuations are taken into account in a mean field fashion the following expression is obtained:
\begin{equation}
\label{gswa}
\e_\pp^{gswa}=\sqrt{\mu(\mu+2s^2 J_\pp)},
\end{equation}
where parameters $s$ and $\mu$ are determined as a result of self-consistent calculations using equations
\begin{eqnarray}
D &=& \mu\left( 1+\frac1N\sum_\pp\frac{J_\pp}{\e_\pp^{gswa}} \right),\\
\label{gswas}
s^2 &=& 2 - \frac1N\sum_\pp\frac{\mu+s^2J_\pp}{\e_\pp^{gswa}}.
\end{eqnarray}
It is seen that Eq.~\eqref{gswa} is a modification of Eq.~\eqref{eqLin} at $S=1$: $D$ and $J_\pp$ are renormalized. We show below by case study that Eqs.~\eqref{gswa}--\eqref{gswas} work much better than Eq.~\eqref{eqLin} but Eq.~\eqref{specs3} proves to be more precise and convenient because the numerical solving of integral equations and numerical calculations are not required.
It should be noted that an approach very similar to that leading to Eqs.~\eqref{gswa}--\eqref{gswas} is proposed in Refs.~\cite{1,wang5}. However the resultant equations in Refs.~\cite{1,wang5} are more cumbersome than Eqs.~\eqref{gswa}--\eqref{gswas} whereas they work slightly worse than Eqs.~\eqref{gswa}--\eqref{gswas} as our comparison shows with the numerical results of Refs.~\cite{2,3}. That is why we do not consider here in detail results of Refs.~\cite{1,wang5}.
\subsection{ Comparison with numerical results }
\label{secMC}
The elementary excitation spectrum is found in Ref.~\cite{2} by Monte-Carlo calculations for the square lattice and $S=1$. The exchange interaction is taken in Ref.~\cite{2} to be positive (antiferromagnetic) and equal for all nearest neighbors. Results of Refs.~\cite{2,3} together with the spectrum calculated using Eqs.~\eqref{specs3}, \eqref{gswa}--\eqref{gswas} and \eqref{eqLin} are shown in Fig.~\ref{figMC} for $D=6J$ and $D=10J$. It is seen from Fig.~\ref{figMC} that Eq.~\eqref{specs3} works well in the whole Brillouin zone when $D$ is not very close to the critical value $D_c^{2D}\approx5.65$ found numerically in Refs.~\cite{haas,2,wong}. The phase with the long-range magnetic order (antiferromagnetic phase) is stable at $D<D_c^{2D}$. Notice also that even at $D\agt D^{2D}_c$ Eq.~\eqref{specs3} describes well the spectrum of short-wavelength elementary excitations.
\begin{figure}
\includegraphics[scale=0.6]{D6}
\includegraphics[scale=0.6]{D10}
\caption{(Color online.) Elementary excitation spectrum $\e_\pp$ of the model \eqref{ham} on the square lattice with $S=1$ and antiferromagnetic exchange $J>0$ between nearest neighbors for (a) $D=6J$ and (b) $D=10J$. Monte-Carlo data of Refs.~\cite{2,3} are shown by points. Dash, dash-dotted and solid lines are drawn using Lindgard's formula \eqref{eqLin}, GSWA formula \eqref{gswa} and Eq.~\eqref{specs3} (the present paper result), respectively. The deviation of the spectrum \eqref{specs3} from the numerical data at $D=6J$ in the close vicinity of the minimum at $(\pi,\pi)$ is a consequence of the proximity to the QCP $D=D_c^{2D}\approx5.65$ separating the paramagnetic and the antiferromagnetic phases (see the main text).}
\label{figMC}
\end{figure}
The noticeable deviation of the long-wavelength excitation spectrum given by Eq.~\eqref{specs3} (with momenta ${\bf p}\approx\pp_0=(\pi,\pi)$) at $D\agt D^{2D}_c$ is a result of strong fluctuations near QCP which manifest themselves in a bad convergence of the series in terms of powers of $J/D$. It is illustrated by Fig.~\ref{figgap} which shows the dependence of the spectrum gap at $\pp=\ppp$ on $D$. It is seen that the second and the third order corrections to the gap are approximately equal to each other at $D\agt D^{2D}_c$ and the value of $D_{c}$ found from the relation $\e_{3\pp_0}=0$ is 12\% smaller than the value of $D_c^{2D}\approx5.65$ obtained numerically in Refs.~\cite{haas,2,wong}.
\begin{figure}
\includegraphics[scale=0.7]{gap-graph}
\caption{
(Color online.) The spectrum gap at $\pp=(\pi,\pi)$ is presented as a function of $D$ in the model \eqref{ham} with $S=1$ and antiferromagnetic exchange interaction $J$ between only nearest neighbors. Monte-Carlo data of Ref.~\cite{2,3} are shown by points, results by Hamer et al. \cite{3} are represented by dash line, dash-dotted and dotted lines are drawn using GSWA formula \eqref{gswa} and Lindgard's formula \eqref{eqLin}, respectively. Spectra $\e_{1\pp}$, $\e_{2\pp}$ and $\e_{3\pp}$ are also shown, which are found in the first, the second and the third orders in $J/D$ and which are given by Eqs.~\eqref{spec0}, \eqref{specs2} and \eqref{specs3}, respectively. The critical value of the anisotropy $D_c^{2D}\approx5.65$ \cite{haas,2} is marked, below which the phase with the antiferromagnetic long-range magnetic order is stable.}
\label{figgap}
\end{figure}
The result of the ground state energy calculation is shown in Fig.~\ref{figgs}, which demonstrates that Eq.~\eqref{gs} works well when $D$ is not very close to $D^{2D}_c$ and that series for $E_{gs}$ converges slowly at $D\agt D^{2D}_c$.
\begin{figure}
\includegraphics[scale=0.5]{ground}
\caption{(Color online.) The ground state energy of the model \eqref{ham} on the square lattice with $S=1$ and antiferromagnetic exchange between only nearest neighbors. Monte-Carlo data of Refs.~\cite{2,3} are shown by points, results by Hamer et al. \cite{3} are presented by dash line, the solid line is drawn using Eq.~\eqref{gs}, which is obtained in the third order in $J/D$.}
\label{figgs}
\end{figure}
Comparison of the results obtained within different approaches and presented in Figs.~\ref{figMC}--\ref{figgs} shows that formulas derived above using the method suggested in the present paper, on the whole, work better. At the same time it should be noted that despite its simplicity GSWA gives quite precise results.
\subsection{Application to $\rm NiCl_2$-$\rm 4SC(NH_2)_2$}
At the present time the most extensively studied compound described by the model \eqref{ham} and having the paramagnetic ground state is $\rm NiCl_2$--$\rm 4SC(NH_2)_2$ which is known as DTN \cite{12,13,14,15,16,17,chern,add8}. The magnetic subsystem of DTN consists of $\rm Ni$ ions with $S=1$ and the Lande factor $g=2.26$. Magnetic ions form a body-centered tetragonal lattice which can be viewed as two interpenetrating tetragonal sublattices. The exchange interaction between spins inside one sublattice is antiferromagnetic and strongly anisotropic: the exchange constant along the tetragonal hard axis ($z$ axis) is much larger than those along $x$ and $y$ axes. That is why DTN is considered as a quasi-1D compound. Hamiltonian \eqref{ham} with the following set of parameters is used for interpretation of the majority of experimental data:
\beq
\label{olddtnpar}
D &=& 8.9\, \rm{K}, \non \\
J_z &=& 2.2\, \rm{K},\\
J_{xy} &=& 0.18\, \rm{K}, \non
\eeq
where $J_{z}$ is the exchange constant along the chains and $J_{xy}$ is the exchange coupling constants between chains inside one tetragonal sublattice. Interaction between tetragonal sublattices is supposed to be negligibly small \cite{13} in the most of considerations.
DTN behavior attracts special attention near two QCPs in magnetic field $H$ applied along the hard $z$ axis. The first QCP $H=H_{c1}$ separates the paramagnetic and the canted antiferromagnetic phases and the second one separates the canted antiferromagnetic phase and the ferromagnetic one in which all spins are parallel to the field. Values of these critical fields are defined by the following {\it exact} relations (see, e.g., Refs.~\cite{13,chern}):
\begin{eqnarray}
\label{hc1}
H_{c1} &=& \e_{\pp_0}, \\
\label{hc2}
H_{c2} &=& 2 S J_{\bf 0} + V_{\bf0}+ D (2 S - 1),
\end{eqnarray}
where $\e_{\pp}$ is the spectrum at $H=0$ and $V$ is the exchange coupling between sublattices considered below which has been neglected so far. The following values of the critical fields are obtained experimentally in Ref.~\cite{16}:
\footnote{It should be noted that in some other papers (see, e.g., Refs.~\cite{12,14}) other values of the critical fields are reported: $H_{c1}=2.1$~T and $H_{c2}=12.6$~T. In the present paper we use values \eqref{hcdtn} because they were measured at extremely small temperature of 1~mK. Besides, our fitting of the neutron data of Ref.~\cite{13} by varying exchange coupling constants and $D$ while keeping the critical fields fixed gives better result with values \eqref{hcdtn}.
}
\begin{eqnarray}
\label{hcdtn}
\begin{array}{ll}
H_{c1}^{DTN} &= 2.05\,{\rm T}, \\
H_{c2}^{DTN} &= 12.175\, {\rm T}.
\end{array}
\end{eqnarray}
In the present paper we focus on analysis of the DTN elementary excitation spectrum at zero magnetic field, which is observed in neutron experiment \cite{13} and shown in Fig.~\ref{figolddtn}(a)--(c). Its current theoretical interpretation looks inconsistent. Eqs.~\eqref{gswa}--\eqref{gswas} are used for the spectrum analysis in Refs.~\cite{13,chern}. As is shown in Ref.~\cite{13}, Eqs.~\eqref{gswa}--\eqref{gswas} describe DTN spectrum very well with parameters $D=8.12$~K, $J_{xy}=0.17$~K and $J_z=1.74$~K which differ from those used in the literature now \eqref{olddtnpar} (see the dash line in Fig.~\ref{figolddtn}(a)--(c)). However in the subsequent paper \cite{14} these parameters were declined because the value of the critical field $H_{c2}$ found using Eq.~\eqref{hc2} with these parameters differs significantly from the experimentally obtained value. The set of parameters \eqref{olddtnpar} is proposed in Ref.~\cite{14}, which has been used up to the present. In particular, in the recent paper \cite{chern} Eqs.~\eqref{gswa}--\eqref{gswas} are used with the conventional parameters \eqref{olddtnpar} for the spectrum analysis in the paramagnetic phase near the antiferromagnetic vector $\pp_0$. However, as it is seen from Fig.~\ref{figolddtn}(a)--(c) (the dash-dotted line), GSWA with this set of parameters describes unsatisfactorily the spectrum of short-wavelength excitations.
\begin{figure}
\includegraphics[scale=0.55]{gXY-old}
\includegraphics[scale=0.55]{gZ-old}
\includegraphics[scale=0.55]{gXYZ-old}
\includegraphics[scale=0.55]{gXY-old-cor}
\includegraphics[scale=0.55]{gZ-old-cor}
\includegraphics[scale=0.55]{gXYZ-old-cor}
\caption{(Color online.) (a)--(c) Elementary excitation spectrum of DTN along three directions in the Brillouin zone at zero field. The data of the neutron experiment \cite{13} at $T=80$~mK are shown by points, dash and dash-dotted lines are drawn using Eqs.~\eqref{gswa}--\eqref{gswas} with parameters $D=8.12$~K, $J_{xy}=0.17$~K and $J_z=1.74$~K (which are proposed in Ref.~\cite{13}) and with the conventional set of parameters \eqref{olddtnpar}, respectively, solid lines are drawn using Eq.~\eqref{specs3} with the conventional parameters \eqref{olddtnpar}. (d)--(f) The spectrum of the first order in $J/D$ given by Eq.~\eqref{spec0} and the second and the third order corrections in $J/D$ to the spectrum found using Eqs.~\eqref{specs2} and \eqref{specs3} and the conventional set of parameters \eqref{olddtnpar}.}
\label{figolddtn}
\end{figure}
Because the body-centered tetragonal magnetic lattice of DTN is a Bravais lattice, all expressions obtained above are applicable to DTN. The spectrum obtained using Eq.~\eqref{specs3} with the conventional parameters \eqref{olddtnpar} for DTN is presented by the solid line in Fig.~\ref{figolddtn}(a)--(c). It is seen that the agreement with experimental data is poor. At the same time the method of the spectrum calculation proposed in the present paper looks suitable for DTN. It is illustrated by Fig.~\ref{figolddtn}(d)--(f), in which the second and the third order corrections in $J/D$ to the spectrum and the first order spectrum $\e_{1\pp}$ given by Eq.~\eqref{spec0} are presented. It is seen that the third order corrections are 3--5 times smaller than the second order ones in almost the whole Brillouin zone except for the vicinity of the antiferromagnetic vector, where they are almost equal but still remain much smaller than $\e_{1\pp}$.
Thus, the theoretical description of DTN needs revision that is indicated also by recent ESR-experimental data. As it is mentioned above, the interaction between spins from different DTN sublattices is usually ignored. But the data of the recent ESR experiment \cite{12} indicate that it should be taken into consideration: one of the spectrum branches has a gap in the canted antiferromagnetic phase at $H_{c1}<H<H_{c2}$ and the optical mode is slightly split. At the same time the model \eqref{ham} without the inter-sublattice interaction has the doubly degenerate spectrum (due to two equivalent magnetic sublattices).
In our previous paper \cite{prev} the spin-wave approach and the magnon Bose-condensation technique (near $H_{c2}$) are used for analysis of the canted antiferromagnetic phase. It is shown that the inter-sublattice interaction of the form
\begin{eqnarray}
\label{v}
{\cal H}_V = \sum_{i,j} V_{i,j} {\bf S}_i {\bf S}_j,
\end{eqnarray}
where index $i$ labels sites of one sublattice and $j$ labels sites of another sublattice nearest to $i$, leads to the effects observed in the ESR experiment (the gap in one of the spectrum branches and the optical mode splitting). Unfortunately the lack of experimental data near $H_{c2}$, large anisotropy and quasi-1D nature of DTN did not allow to make reliable quantitative predictions about the value of $V_{i,j}$. It is just shown in Ref.~\cite{prev} that if $V_{i,j} = V$, then $V\sim0.1$~K.
In the present paper we continue our discussion started in Ref.~\cite{prev} and fit the elementary excitation spectrum using the least square method by varying parameters $D$, $J_z$, $J_{xy}$ and $V$, while keeping the critical fields given by Eqs.~\eqref{hc1} and \eqref{hc2} to be equal to the experimentally obtained values \eqref{hcdtn}. As a result of this fit we obtain the following set of parameters that differs noticeably from the conventional one \eqref{olddtnpar}:
\begin{eqnarray}
D &=& 7.72 \; \rm{K}, \non \\
J_z &=& 1.86\; {\rm K}, \label{ourdtnpar} \\
J_{xy} &=& 0.2\; {\rm K}, \non\\
V &=& 0.1\; \rm{K}. \non
\end{eqnarray}
The spectrum obtained is presented in Fig.~\ref{figourdtn} that is in good agreement with the neutron data of Ref.~\cite{13}. The inter-sublattice interaction removes the double degeneracy of the spectrum in DTN. This splitting is zero in (a) and (b) panels of Fig.~\ref{figourdtn}, but it is clearly seen in panel (c). It should be noted that the upper branch of the spectrum in Fig.~\ref{figourdtn}(c) goes beyond the experimental error near $\pp={\bf 0}$. However, the branch splitting value $\delta\e_\pp=\e^{+}_\pp-\e^{-}_\pp$ is very small compared to $\e^{+}_\pp+\e^{-}_\pp$ and it appears to be given in DTN by a slowly convergent series in $J/D$: $\delta\e_\pp=(1.7-1.0+0.4)$~K, where the first, the second and the third terms stand for the values of corresponding corrections in $J/D$. Then, one has to take into account higher order terms in $J/D$ in order to find the small value of $\delta\e_\pp$ in DTN that is out of the scope of the present paper.
\begin{figure}
\includegraphics[scale=0.6]{gXY-our}
\includegraphics[scale=0.6]{gZ-our}
\includegraphics[scale=0.6]{gXYZ-our}
\caption{(Color online.) DTN spectrum along three directions in the Brillouin zone calculated using Eq.~\eqref{specs3} and parameters \eqref{ourdtnpar}. Neutron experimental data are taken from Ref.~\cite{13}.}
\label{figourdtn}
\end{figure}
Note also that the value of $V=0.1$~K found above agrees with the estimation $V\sim0.1$~K obtained as a result of our consideration \cite{prev} of the canted antiferromagnetic phase.
\section{Conclusion}
\label{conc}
We propose the new representation \eqref{sz}, \eqref{s+} for an integer spin $S$ via bosonic operators, which is useful in describing the paramagnetic phase and transitions to magnetically ordered phases in magnetic systems with large single-ion easy-plane anisotropy. Using this representation, the diagram technique and treating the exchange interaction as a perturbation we obtain Eq.~\eqref{specs3} for the elementary excitation spectrum of the model \eqref{ham} in the paramagnetic phase in the third order of the perturbation theory (that is referred to as an expansion in terms of $J/D$ for shot). Expression \eqref{gs} is also found for the ground state energy in the third order in $J/D$. Eq.~\eqref{specs3} coincides with that obtained in Ref.~\cite{4} in the special case of a spin chain with $S=1$ and the exchange interaction between nearest neighbors only. We recover Eq.~\eqref{specs3} also at $S=1$ using simpler spin representations \eqref{sz}, \eqref{s+simple} and \eqref{sz}, \eqref{s+no}.
Comparison with numerical results obtained in Refs.~\cite{2,3} for the square lattice, $S=1$ and the antiferromagnetic exchange between nearest neighbors only shows that Eqs.~\eqref{specs3} and \eqref{gs} work better than results of other analytical methods proposed so far. In particular, Eqs.~\eqref{specs3} and \eqref{gs} work very well when $D$ is not very close to the critical value $D_{c}$, below which the antiferromagnetic phase becomes stable \cite{2,3,wong} (see Figs.~\ref{figMC}--\ref{figgs}). At $D\agt D_c$ Eq.~\eqref{specs3} poorly describes only the spectrum of long-wavelength quasiparticles.
It is shown that Eq.~\eqref{specs3} is applicable for the spectrum analysis of the intensively studied compound $\rm NiCl_2$-$\rm 4SC(NH_2)_2$ (DTN), which is described by the model \eqref{ham} \cite{add8,chern,12,13,14,15,16,17}. We show that Eq.~\eqref{specs3} with the conventional set of parameters for DTN \cite{14} \eqref{olddtnpar} describes the experimentally obtained spectrum \cite{13} unsatisfactorily (see Fig.~\ref{figolddtn}). The new set of parameters \eqref{ourdtnpar} is proposed for DTN which provides a good description of the experimental spectrum (see Fig.~\ref{figourdtn}) and reproduces the experimentally obtained critical fields values \cite{16} \eqref{hcdtn}. In contrast to the conventional model proposed for DTN before we take into account also the exchange interaction \eqref{v} between DTN magnetic sublattices, which becomes apparent in the recent ESR experiment \cite{12}.
In the forthcoming paper we continue analysis of the model \eqref{ham} using representations \eqref{sz}, \eqref{s+}, \eqref{s+simple} and \eqref{s+no} and consider its behavior in the vicinity of QCP $H=H_{c1}$ at $T\ne0$.
\begin{acknowledgments}
We are thankful to Prof.~A.~I.~Smirnov for stimulating discussions. This work was supported by RF President (grant MK-329.2010.2), RFBR grant 09-02-00229, and Programs "Quantum Macrophysics", "Strongly correlated electrons in semiconductors, metals, superconductors and magnetic materials" and "Neutron Research of Solids".
\end{acknowledgments}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 376 |
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"redpajama_set_name": "RedPajamaCommonCrawl"
} | 6,649 |
{"url":"https:\/\/kisonecat.com\/blog\/cartan\/","text":"# Constructing a Lie group from a Lie algebra.\n\n##### November 30, 2006 mathematics\n\nCartan proved that every finite-dimensional real Lie algebra Error:LaTeX failed:\nThis is pdfTeX, Version 3.141592653-2.6-1.40.22 (TeX Live 2021\/nixos.org) (preloaded format=latex) restricted \\write18 enabled. entering extended mode (.\/working.tex LaTeX2e <2021-11-15> patch level 1 L3 programming layer <2022-02-24> (\/nix\/store\/pibhz89i08877bwjc13mmq7b3kaqkmhi-texlive-combined-full-2021-final\/s hare\/texmf\/tex\/latex\/base\/article.cls Document Class: article 2021\/10\/04 v1.4n Standard LaTeX document class (\/nix\/store\/pibhz89i08877bwjc13mmq7b3kaqkmhi-texlive-combined-full-2021-final\/s hare\/texmf\/tex\/latex\/base\/size12.clo)) (\/nix\/store\/pibhz89i08877bwjc13mmq7b3kaqkmhi-texlive-combined-full-2021-final\/s hare\/texmf\/tex\/latex\/preview\/preview.sty (\/nix\/store\/pibhz89i08877bwjc13mmq7b3kaqkmhi-texlive-combined-full-2021-final\/s hare\/texmf\/tex\/generic\/luatex85\/luatex85.sty) (\/nix\/store\/pibhz89i08877bwjc13mmq7b3kaqkmhi-texlive-combined-full-2021-final\/s hare\/texmf\/tex\/latex\/preview\/prtightpage.def)) (\/nix\/store\/pibhz89i08877bwjc13mmq7b3kaqkmhi-texlive-combined-full-2021-final\/s hare\/texmf\/tex\/latex\/amsmath\/amsmath.sty For additional information on amsmath, use the ?' option. (\/nix\/store\/pibhz89i08877bwjc13mmq7b3kaqkmhi-texlive-combined-full-2021-final\/s hare\/texmf\/tex\/latex\/amsmath\/amstext.sty (\/nix\/store\/pibhz89i08877bwjc13mmq7b3kaqkmhi-texlive-combined-full-2021-final\/s hare\/texmf\/tex\/latex\/amsmath\/amsgen.sty)) (\/nix\/store\/pibhz89i08877bwjc13mmq7b3kaqkmhi-texlive-combined-full-2021-final\/s hare\/texmf\/tex\/latex\/amsmath\/amsbsy.sty) (\/nix\/store\/pibhz89i08877bwjc13mmq7b3kaqkmhi-texlive-combined-full-2021-final\/s hare\/texmf\/tex\/latex\/amsmath\/amsopn.sty)) (\/nix\/store\/pibhz89i08877bwjc13mmq7b3kaqkmhi-texlive-combined-full-2021-final\/s hare\/texmf\/tex\/latex\/amsfonts\/amsfonts.sty) (\/nix\/store\/pibhz89i08877bwjc13mmq7b3kaqkmhi-texlive-combined-full-2021-final\/s hare\/texmf\/tex\/latex\/base\/fontenc.sty) (\/nix\/store\/pibhz89i08877bwjc13mmq7b3kaqkmhi-texlive-combined-full-2021-final\/s hare\/texmf\/tex\/latex\/lm\/lmodern.sty) (\/nix\/store\/pibhz89i08877bwjc13mmq7b3kaqkmhi-texlive-combined-full-2021-final\/s hare\/texmf\/tex\/latex\/lm\/t1lmr.fd) (\/nix\/store\/pibhz89i08877bwjc13mmq7b3kaqkmhi-texlive-combined-full-2021-final\/s hare\/texmf\/tex\/latex\/l3backend\/l3backend-dvips.def) No file working.aux. Preview: Fontsize 12pt (\/nix\/store\/pibhz89i08877bwjc13mmq7b3kaqkmhi-texlive-combined-full-2021-final\/s hare\/texmf\/tex\/latex\/lm\/ot1lmr.fd) (\/nix\/store\/pibhz89i08877bwjc13mmq7b3kaqkmhi-texlive-combined-full-2021-final\/s hare\/texmf\/tex\/latex\/lm\/omllmm.fd) (\/nix\/store\/pibhz89i08877bwjc13mmq7b3kaqkmhi-texlive-combined-full-2021-final\/s hare\/texmf\/tex\/latex\/lm\/omslmsy.fd) (\/nix\/store\/pibhz89i08877bwjc13mmq7b3kaqkmhi-texlive-combined-full-2021-final\/s hare\/texmf\/tex\/latex\/lm\/omxlmex.fd) (\/nix\/store\/pibhz89i08877bwjc13mmq7b3kaqkmhi-texlive-combined-full-2021-final\/s hare\/texmf\/tex\/latex\/amsfonts\/umsa.fd) (\/nix\/store\/pibhz89i08877bwjc13mmq7b3kaqkmhi-texlive-combined-full-2021-final\/s hare\/texmf\/tex\/latex\/amsfonts\/umsb.fd) ! Undefined control sequence. l.11 \\germ g Preview: Tightpage -32891 -32891 32891 32891 [1] (.\/working.aux) ) (see the transcript file for additional information) Output written on working.dvi (1 page, 1620 bytes). Transcript written on working.log. `\ncomes from a connected, simply-connected Lie group . I hadn\u2019t known the proof of this result (and apparently it is rather uglier than one might hope), but [1] gives a short proof of it, which I presented to the undergraduates in my Lie group seminar. I\u2019ll sketch the proof now.\n\nTheorem. For every Lie algebra , there is a simply-connected, connected Lie group having as its Lie algebra.\n\nFirst, if , then the exponential map gives , and we define . It turns out is a Lie group, and is its Lie algebra.\n\nIf has no center, then is injective, so we have realized as a Lie subalgebra of endomorphisms of a vector space, and by the above, there is a Lie group with as its Lie algebra. Taking its universal cover proves the theorem in this case.\n\nNow we induct on the dimension of the center . Let be a one-dimensional central subspace of , and construct a short exact sequence . But this central extension of by corresponds to a 2-cocycle .\n\nLemma. Let be the map which differentiates a (smooth!) -cocycle of the group cohomology of . The map is injective.\n\nConsequently, we can find with . Since , by induction there is a Lie group having as its Lie algebra. We build the central extension of by using the cocycle , namely, , where and the operation is . Since , it turns out that the Lie algebra corresponding to is . We finish the proof by taking the universal cover .\n\n[1] V.V. Gorbatsevich, Construction of a simply connected group with a given lie algebra, Uspekhi Mat. Nauk. 41 (1986) 177\u2013178.","date":"2023-03-31 23:14:59","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 1, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.94880211353302, \"perplexity\": 3559.4925536877718}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2023-14\/segments\/1679296949689.58\/warc\/CC-MAIN-20230331210803-20230401000803-00636.warc.gz\"}"} | null | null |
Styles ist der Familienname folgender Personen:
AJ Styles (* 1977), US-amerikanischer Wrestler
Brian Thomas Styles (1936–1993), britischer Botaniker
Darren Styles (* 1975), britischer DJ und Musikproduzent
Harry Styles (* 1994), britischer Popsänger der Boygroup One Direction
Gemma Styles (* 1990), ältere Schwester von Harry Styles
Joey Styles (* 1971), US-amerikanischer Wrestling-Kommentator
Stella Styles, deutsche Pornodarstellerin
Sya Styles († 2015), französischer DJ und Musikproduzent
William Styles (1874–1940), britischer Sportschütze
Styles ist ein Pseudonym von:
Styles P. (* 1974), US-amerikanischer Rapper
Sonstiges:
Styles Bluff, Felsenkliff im Kempland, Antarktika
Styles-Gletscher, Gletscher Mac-Robertson-Land, Antarktika
Styles Strait, Meerenge vor der Küste des Enderbylands, Antarktika
Siehe auch:
Stiles
Style | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 6,978 |
module MotionSpec
module Matcher
class HaveGeneric
def initialize(method_name, *args)
@method_name = method_name
@args = args
end
def matches?(subject)
subject.send("has_#{@method_name}?", *@args)
end
def fail!(subject, negated)
fail FailedExpectation.new(
FailMessageRenderer.message_for_have_generic(
negated, subject, @method_name, @args
)
)
end
end
end
end
| {
"redpajama_set_name": "RedPajamaGithub"
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-- To personalize your site experience and to allow us to deliver the type of content and product offerings in which you are most interested.
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"redpajama_set_name": "RedPajamaC4"
} | 8,671 |
"""Simple example that connects to a device with autodiscover."""
import asyncio
from pyatv import helpers
# Method that is dispatched by the asyncio event loop
async def print_what_is_playing(atv):
"""Print what is playing for the discovered device."""
playing = await atv.metadata.playing()
print("Currently playing:")
print(playing)
asyncio.get_event_loop().run_until_complete(helpers.auto_connect(print_what_is_playing))
| {
"redpajama_set_name": "RedPajamaGithub"
} | 965 |
Q: The correct form for Baltics or Baltic I am trying to figure out the difference between Baltics and Baltic words should there be any. Imagine I was to call a company and want to use the Baltic/s word in it. Which would fit the most? Let's imagine a fictional company called CityBee. If I want to emphasize on its location in the whole Baltic states, should this be:
*
*CityBee Baltics
*CityBee Baltic
A: Baltics is not a common expression. The GloWbE corpus has 145 instances,
against 3151 of Baltic. Of course you can call your company whatever you want.
| {
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} | 3,015 |
The Darwin Lagrangian (named after Charles Galton Darwin, grandson of the naturalist) describes the interaction to order between two charged particles in a vacuum and is given by
where the free particle Lagrangian is
and the interaction Lagrangian is
where the Coulomb interaction is
and the Darwin interaction is
Here and are the charges on particles 1 and 2 respectively, and are the masses of the particles, and are the velocities of the particles, is the speed of light, is the vector between the two particles, and is the unit vector in the direction of .
The free Lagrangian is the Taylor expansion of free Lagrangian of two relativistic particles to second order in v. The Darwin interaction term is due to one particle reacting to the magnetic field generated by the other particle. If higher-order terms in are retained, then the field degrees of freedom must be taken into account, and the interaction can no longer be taken to be instantaneous between the particles. In that case retardation effects must be accounted for.
Derivation in vacuum
The relativistic interaction Lagrangian for a particle with charge q interacting with an electromagnetic field is
where is the relativistic velocity of the particle. The first term on the right generates the Coulomb interaction. The second term generates the Darwin interaction.
The vector potential in the Coulomb gauge is described by (Gaussian units)
where the transverse current is the solenoidal current (see Helmholtz decomposition) generated by a second particle. The divergence of the transverse current is zero.
The current generated by the second particle is
which has a Fourier transform
The transverse component of the current is
It is easily verified that
which must be true if the divergence of the transverse current is zero. We see that is the component of the Fourier transformed current perpendicular to .
From the equation for the vector potential, the Fourier transform of the vector potential is
where we have kept only the lowest order term in .
The inverse Fourier transform of the vector potential is
where
(see ).
The Darwin interaction term in the Lagrangian is then
where again we kept only the lowest order term in .
Lagrangian equations of motion
The equation of motion for one of the particles is
where is the momentum of the particle.
Free particle
The equation of motion for a free particle neglecting interactions between the two particles is
Interacting particles
For interacting particles, the equation of motion becomes
Hamiltonian for two particles in a vacuum
The Darwin Hamiltonian for two particles in a vacuum is related to the Lagrangian by a Legendre transformation
The Hamiltonian becomes
Hamiltonian equations of motion
The Hamiltonian equations of motion are
and
which yield
and
Note that the quantum mechanical Breit equation originally used the Darwin Lagrangian with the Darwin Hamiltonian as its classical starting point though the Breit equation would be better vindicated by the Wheeler–Feynman absorber theory and better yet quantum electrodynamics.
See also
Static forces and virtual-particle exchange
Breit equation
Wheeler–Feynman absorber theory
References
Magnetostatics
Equations of physics | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 7,652 |
The Israeli Anti-Semitic Cartoons Contest (Hebrew: תחרות קריקטורות אנטישמיות ישראלית) was initiated by two Israeli artists in response to the Muhammad cartoons controversy and the subsequent "Holocaust Cartoon Competition" by the Iranian newspaper Hamshahri. Illustrator Amitai Sandy announced the contest on the website of his Tel Aviv-based graphic arts company on February 14, 2006, stating, "We'll show the world we can do the best, sharpest, most offensive Jew hating cartoons ever published! No Iranian will beat us on our home turf!"
The Jerusalem Post reported Sandy as saying that his intention was to challenge bigotry by using humour. According to Haaretz, within three days of the announcement of the contest Sandy was interviewed by more than thirty daily newspapers, as well as two television channels and a radio program broadcast on 450 local stations in the United States.
On April 6, the winner was announced on the contest homepage: "Fiddler on the Roof." It depicted a fiddler on the Brooklyn Bridge during the September 11, 2001 attacks on the World Trade Center.
Other common themes through the cartoons included world domination, the myth of Jews having horns, the Holocaust (and its denial), and the blood libel, all of which were familiar staples or topics of antisemitism. Many of those staples were canards as well.
References
External links
Israeli group announces anti-Semitic cartoons contest!
Israeli antisemitic cartoons contest, a web-archive of the cartoon gallery
Editorial cartooning awards
Jyllands-Posten Muhammad cartoons controversy
Israeli awards
Arts awards in Israel
Antisemitic works
Jewish comedy and humor
2006 in Israel
2006 in comics
he:פרשת קריקטורות מוחמד#התגובות לקריקטורות | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 4,503 |
Onésiphore Ernest Talbot (August 15, 1854 – May 6, 1934) was a Canadian politician.
Born in St-Arsène, Temiscouata County, Canada East, Talbot was educated at St. Michel and the Quebec Seminary. A farmer, he was a member of the Council of Agriculture in the Province of Quebec and was awarded Quebec's Ordre national du mérite agricole. He was a Lieutenant-Colonel with the 17th Regiment de Lévis and Bellechasse. He was elected to the House of Commons of Canada for Bellechasse in the 1896 federal election. A Liberal, he was re-elected in 1900, 1904, and 1908. He was defeated in 1911.
References
The Canadian Parliament; biographical sketches and photo-engravures of the senators and members of the House of Commons of Canada. Being the tenth Parliament, elected November 3, 1904
External links
1854 births
1934 deaths
Liberal Party of Canada MPs
Members of the House of Commons of Canada from Quebec | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 7,226 |
Ризеби () — коммуна в Германии, в земле Шлезвиг-Гольштейн.
Входит в состав района Рендсбург-Экернфёрде. Подчиняется управлению Шлай. Население составляет 2436 человек (на 31 декабря 2010 года). Занимает площадь 38,85 км². Официальный код — 01 0 58 137.
Примечания
Ссылки
Официальная страница
Города Шлезвиг-Гольштейна | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 5,438 |
<!doctype html>
<title>/resources/media/gamut/gamut-theora-bt470.ogv</title>
<script src=../framework.js></script>
<script>
// MANUALLY EDITED
// Format: Ogg
// Video: Theora
expectedDuration = 9.966666666;
expectedFrameRate = 30/1;
expectedHeight = 240;
expectedWidth = 320;
type = 'application/ogg; codecs=theora';
pixelDataTolerance = 1;
</script>
<pre>FAIL (script didn't run)</pre>
| {
"redpajama_set_name": "RedPajamaGithub"
} | 8,297 |
Membership in the Corporation shall be limited to persons and institutions interested in furthering the objects of the Corporation and shall consist of anyone whose application for admission as a Member has received the approval of the Board. Applications by prospective members shall be submitted in writing to the Secretary of the Corporation.
There shall be three classes of Membership in the Corporation, consisting of Regular Members, Institutional Members, and Honorary Life Members. All classes of membership have equal voting rights. Membership in the Corporation is not transferable.
policy-making, or have served the Corporation well over a significant period of time may, at the sole discretion of the Board, be appointed Honorary Life Members by a vote consisting of the majority of the Board held at a Directors' Meeting.
Any Honorary Life Membership may be revoked by the Board in its sole discretion by a vote consisting of the majority of the Board held at a Directors' Meeting.
Notwithstanding anything contained within these By-laws, Honorary Life Members shall not be required to pay membership dues or fees, nor shall their membership in the Corporation expire otherwise than upon the death of such Honorary Life Member.
No Institutional Member shall be an Honorary Life Member, but such Institutional Member's Nominee or any former such Nominees may be considered and approved for Honorary Life Membership in accordance with this Article 4.03.
The Board may by resolution create additional honorary positions of distinction in addition to those provided for in these By-laws. The creation of such positions must be ratified by a majority vote of Members at a General Meeting.
Membership in the Corporation for natural persons shall run annually from January 1 to December 31 and in the case of Institutions shall run annually from January 1 to December 31.
Membership fees or dues shall be such amounts as may be set by the Board by ordinary resolution, from time to time, and ratified at a General Meeting of Members. For greater clarity, the Board may set differential fees respecting separate classes of Members and respecting sub-classes of classes of Members, provided that any such differential fee structures are similarly ratified at a General Meeting of Members. Such fees are payable in Canadian funds.
Failure by a Member who is a natural person to pay dues for the current year by January 31 of that year, or, in the case of an Institutional Member, by January 31 of that year, shall result in the automatic termination of that Member's membership on that date.
Membership fees or dues shall not be refundable in whole or in part PROVIDED THAT the Board may in its sole discretion elect to waive this Condition in circumstances where the Board determines it to be appropriate.
Any Member may withdraw from the Corporation by delivering to the Corporation a written resignation and lodging a copy of same with the Secretary of the Corporation and such withdrawal shall be effective upon receipt by the Secretary of the Corporation of said resignation, or, if a time is specified in such resignation, then as at the time so specified, whichever is the later.
Any Member may be required to resign by a vote of TWO THIRDS (2/3) of the Members present at an Annual General Meeting, or by a vote consisting of a majority of Directors at a Directors' Meeting which is subsequently ratified at an Annual General Meeting or Special Meeting. | {
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{"url":"https:\/\/physics.stackexchange.com\/tags\/gyroscopes\/new","text":"# Tag Info\n\n0\n\nBefore giving the formula, note that the response of the gyroscope wheel will not be in opposite direction. For names of direction of rotation let me use the standard names of aircraft principle axes: roll, pitch and yaw. Take the spin axis of the gyro wheel as rolling, and apply a large torque that will pitch the gyro wheel fast. Then in response the gyro ...\n\n0\n\nFor a spinning flywheel, the external torque vector determines the rate of change of the angular momentum vector , L, which equals Iw. For a uniform disk, I, (The rotational inertia) = (\u00bd) m R^2, and the angular velocity vector, w, (in radians\/sec) is defined as being along the axis of rotation. (You may want to look up the associated \u201cright hand rules\u201d.) ...\n\n0\n\nThis answer elaborates on the earlier answer by use Krishna. As user Krishna describes: if the top is constructed in such a way that it can remain upright statically (which of course means that it needs a foot that is wide enough), then if if spins sufficiently close to vertical it will slow down to the statically upright position. But of course the ...\n\n0\n\nA top can spin in an upright position. If you manage to balance a top and let it spin, it will spin about its axis in an upright position. But, it will never fall down, because it is balanced. The friction would cause it to stop and it will be left standing after it loses its energy. But, if you spin it with a very slight (almost unnoticeable) initial angle,...\n\n0\n\nif you neglect the centrifugal and Coriolis torques this means that your system is rotate slowly , you get the equations of motion. $$\\ddot{\\varphi}= -{\\frac {\\sin \\left( \\psi \\right) T_{{\\vartheta }}}{J_{{\\vartheta }} \\cos \\left( \\vartheta \\right) }}+{\\frac {\\cos \\left( \\psi \\right) { \\it uu}_{{1}}}{\\cos \\left( \\vartheta \\right) J_{{\\varphi }}}}$$ $$... 1 Sudden removal of the precession inducing torque has the same effect as sudden onset of the precession inducing torque, but in opposite direction. In both cases the sudden change kicks in a nutation. Onset of precession: if the gyroscope wheel is spinning very fast then the nutation frequency is high, and the amplitude is small. In cases where the spin rate ... 3 You didn't miss anything. In your question you analyze the dynamics correctly at the instant t=0. If you use the same argument, after the application of the force F over the time interval dt you will see that$$-F.R = I_y\\dot \\omega_y \\implies \\omega_y = -F.R \\frac {dt}{I_y},$$and then from the Euler equations$$M_x = I_x\\dot \\omega_x - (I_y - I_z)\\...\n\nTop 50 recent answers are included","date":"2020-04-06 19:08:35","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\": 0.8119811415672302, \"perplexity\": 515.3942576903837}, \"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-2020-16\/segments\/1585371656216.67\/warc\/CC-MAIN-20200406164846-20200406195346-00397.warc.gz\"}"} | null | null |
Ayn Rand's rational modernism
Tags: Anne C. Heller, Atlas Shrugged, Ayn Rand, Captain Ahab, Hayek, Herman Melville, Jennifer Burns, modernism, The Fountainhead, von Mises
Russian Jew in her natural habitat
Is it not obvious why Ayn Rand continues to attract readers, followers, and bosom enemies? Her novels The Fountainhead (1943) and Atlas Shrugged (1957) have been read by millions of readers, despite the antagonism of critics from all colors in the political spectrum. Is it not obvious that major presses (Doubleday, Oxford UP) would publish "tell-all" biographies to discredit her major themes by harping on aspects of her private life that supposedly prove that she was ever the amoral Nietzschean Superman, a veritable Hitler to her finally disillusioned "Collective" of assimilated Jews (including Alan Greenspan!) that she gathered around her during the 1960s and on until her death? Why there is a veritable Ayn Rand industry out there, with a new documentary in the making, to be produced by conservatives, as I write this.
The subject of Ayn Rand and her reported mishegas is too large for a single blog; moreover her cultural significance is too great and my research too fragmentary to do justice to the problem. Consider this blog a first try at an explanation for her continued relevance and fascination.
These are the values she upholds in her two mega-meshugenah novels and in later public appearances: New York City and its heaven-assaulting skyscrapers, American exceptionalism as upward mobility, the gold standard, wealth-creation, laissez-faire capitalism, the puritan work-ethic, the irreplaceable good of heroic rugged individualists performing focused and intelligent labor, reason, empiricism, technology, abortion rights (in the first trimester), and materialist science–all of which are essential and intertwined elements of the modern world, a world of constant innovation and positive change furiously opposed by collectivists, New Dealers, and statists in general, the totalitarians whom she views as the source of incompetence, waste, disintegration, thuggery, and demoralization. Her affinity group includes Ludwig von Mises and Friedrich Hayek, although their ideas are not interchangeable with hers.
Unlike the irrationalism and tragic vision upheld by competing modernists, Rand's larger-than-life characters triumph over their reactionary opponents, overcoming obstacles that would intimidate the faint of heart. Impressionistic evidence in addition to remarkable book sales suggest that ordinary readers take courage from these adventures and stand more than a mite taller in the face of arbitrary authority. As I was reading both her novels (I haven't yet read Anthem) and the recent biographies by Anne C. Heller and Jennifer Burns,* I was reminded that her critics treated her as a twentieth-century Captain Ahab and distorted her messages in almost identical ways as have the Melville industry, for instance in deeming Herman Melville (another Romantic artist and "individualist" for whom Might did not make Right) as a synecdoche for Amerika, as a moral terrorist, as a lunatic, as personally destructive to his wife and children, and as a fatal influence upon the impressionable young. (And the anti-slavery Senator from Massachusetts Charles Sumner has been treated to similar attacks by organic conservatives, as I have shown elsewhere on this website. See https://clarespark.com/2009/10/05/charles-sumner-moderate-conservative-on-lifelong-learning/.)
Above all, Ayn Rand confronted in her most popular works the situation we face today as a Republican Party-controlled House of Representatives prepares to take its seats next week: Will the democratic republic/limited government envisioned by the Founders be at least partly reinstated either in the next two years or after the election of 2020? Or will "progressives" or, as Rand liked to call them, "looters and moochers," continue to hold sway? "Look not to the stars…." but finish the sentence yourselves.
*Someone once said that no one should write a biography until they are in at least late middle age. I would say over 60, and after having gone through at least some psychoanalytically oriented therapy. Rand's childhood and young adulthood was so filled with trauma, that had she survived it without scars and foibles, it would have been impossible. What I resent about the recent bios (Burns and Heller) is that they limn some of the reasons for trauma but do not draw the necessary conclusions. Hence the hatchet jobs that they may not have consciously intended. Still, they are writing from the p.o.v. of "moderation."
I read The Fountainhead in the summer of 1963. I had just turned 20. Dominique's behavior struck me as a little bit shocking, as it was not exactly in line with the mores I'd grown up with (the affair with Wynand, whom she didn't Love, but did, oddly, rather respect); but as to her sexuality, her hero-worship of her "Mr. Right," that struck me as obviously feminine (and all the more so with Dagny in Atlas).
It does seem to me that her love affair with the movies and her involvement with them would have had an effect on her. The relationship between the heroine and hero of these two novels was actually similar to that between the leads in various movies from the '30s throughout the '50s at least. Bogart-Bacall. Myrna Loy-William Powell (The Thin Man). Others too. The strong, capable woman and her hero, the even stronger and more capable man. There are other examples, but those two really stand out for me.
As for the Rape That Wasn't (and A.R. got that right — it was indeed "Rape by Engraved Invitation), that wasn't exactly a new notion either. I imagine there are other movie examples, but it's there for all to imagine in Gone with the Wind. Scarlett is horrified by Rhett's obvious sexual interest, but there's something delicious there too … of course the consummation isn't actually shown. That sort of stuff wasn't seen in "legitimate" movies until the '60s or so.
No doubt many of us here remember "Nancy Friday's" books of feminine sexual fantasies, which she claimed were common among women, and which often featured rape.
But it's one thing to work a very mild whiff of salaciousness into novels or to indulge it in imagination, and another thing entirely to experience rape-for-real. I imagine Miss R. would have decked anybody who tried it for real.
–Oh, another example of the same sort of thing, but this time explicit: Mike Russo and what's-her-name, the Mom, in Grace Metalious's Peyton Place. That one was close to a real rape, but just how close?
I didn't, and don't, see anything "kinky" in Miss R.'s novels.
By the way … I read We the Living after the other two, and I did think that as a "regular-fiction" book it was the best of the three. But also way too depressing to re-read, and I never have.
Comment by Julie near Chicago — November 25, 2018 @ 10:08 pm | Reply
Clare, elsewhere you have characterized Ayn Rand as a Romantic with Nietzschean tendencies. You've also pointed out that Ayn Rand's ideas are a product of her time, defined by her struggles with Soviet ideology, and best understood in that context.
In her forward to "The Fountainhead" and elsewhere, she specifically repudiates Nietzsche for his mystic view of man, for his "blood" and "will" theories. She rejects standard Romanticist ideas for much the same reason; in her lectures on aesthetics, she condemns overt emotional appeals in art. In her view, art should express the ideal, based on what one values, not wallow in base emotions or irrational ideas. (She is not arguing against emotion or beauty in art, but insists that the artist use his creativity to satisfy and inspire himself.) More than anything, Rand strove to be known as a liberating artist and a great philosopher on the order of Aristotle, not a self-glorifying crank nor an adherent of corrupt philosophies.
I suggest that the main appeal of Ayn Rand to today's readers is that she has a view of life so radically different from the progressive view that dominates political and social life, so refreshingly *human*, optimistic and man-loving. She does not bow to the Christian-oriented Conservative movement that continues to futilely attack progressives on a simplistic, "because God said so" basis. Conservatives are right on many social and fiscal issues — and embarrassingly wrong on others — but they cannot justify why "certain rights" and freedoms are "inalienable" and why the progressive "rights" to food, shelter, work, and education are arbitrary, "man-made," and morally wrong. Ayn Rand can, and *this* is what makes her exciting. Her system of ethics is profound and inspiring.
I can pick around the edges of her philosophy; I am troubled by aspects of her epistemology and ideas on concept formation; I find her attacks on mysticism and altruism a little too pat and simple; her defiant personality seems to have affected her "rationality" at times. But I can't find a major flaw or weakness in her system of philosophy that gives me cause to dismiss her statements or her idealism. Each essay and lecture is a serious challenge to my thinking and critical skills. By all accounts, Ayn Rand was a harsh critic, a bitch to be around. Anyone who was in her inner circle had their beliefs constantly challenged, and I doubt I would have lasted long in that group.
But that is the nature of a teacher who insists upon arriving at conclusions rationality instead of merely "believing" something because she or someone else said so. Ayn Rand left us the intellectual tools and language necessary to believe in ourselves, to continue to do our own research into matters of truth and reality, and to confront those who would oppress us.
Ayn Rand is a hero and a true Promethean.
Comment by Scott G Lloyd — December 21, 2012 @ 12:51 am | Reply
Scott G Lloyd: Where did I claim that Ayn Rand was Nietzschean? I couldn't find it in my blogs on her work. If you know of such a comparison, I will revise it, but so far it is not on the website.
Comment by clarelspark — June 2, 2015 @ 4:09 pm | Reply
I like this analysis very much. A much-needed antidote to the anti-American, anti-capitalist tone of things these days. Yesterday I saw a headline from Mother Jones magazine in which the word "free-market" was used as an adjective. Expecting a pleasant surprise–I haven't seen the magazine for a long time, maybe it had changed for the better–I clicked through to the story, and found to my horror that "free-market" was being used as a term of derision. Sigh.
Bosom enemies is a good term. The anti-Rand crowd is much more organized and vehement than I had realized–I wrote a negative review on Amazon of the book Ayn Rand Nation and I couldn't believe how much venom was directed toward me. Nobody wanted to engage on the substantive issues, they just applauded the author's demonization of Rand and set about demonizing me. Then I got a couple of emails from the other side with details about the commenters–one has a whole anti-Rand blog (yet didn't mention that in the comment, just indulged in name-calling, and trying to get me to take down the review–which made me laugh–I'm going to trash my writing because someone I never heard of says it doesn't meet his standards?!). Very creepy. Also very creepy is the anti-Jew sentiment in Ayn Rand Nation; one greatly hesitates to use the term self-loathing, but it definitely springs to mind when wading through Weiss's terrible mean-spirited book.
For example, here is a passage from my review with some quotes I found ugly:
Riffing on Jews and capitalism, Weiss tells us "The only capitalists I ever saw were overworked storekeepers, snarling gypsy-cab drivers, and smack dealers on 135th Street. She [Rand] saw a free, unregulated market as the defining institution of a free society. To me, a free, unregulated market was Benny the Goniff selling fruit from a stall in front of a butcher shop on Kingsbridge Road, screaming "Whoaaaa! We got melons here!" in a high-pitched Yiddish accent, sneaking rotten fruit into the bag and counting out ten when a dozen were ordered."
Yes, that really is a direct quote from the book (page 14). It gets worse. Weiss continues: "Benny's spirit drifted downtown to Wall Street. In place of Benny the Goniff as my archetypical capitalist was a new cast of characters. … Instead of red-faced Benny in his stained undershirt there was the esteemed electronic-trading advocate Bernie Madoff in his monogrammed underwear. Both blended together in my perceptions, small-time and big wheel."
(the rest is here: http://www.amazon.com/review/R2N4DH5IPRRSQX/ref=cm_cr_pr_perm?ie=UTF8&ASIN=0312590733&linkCode=&nodeID=&tag= )
As you say here, there's something significant here. An unpleasant part of the zeitgeist….
Comment by Sarah Rolph — December 8, 2012 @ 6:30 pm | Reply
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Pingback by Index to Ayn Rand blogs « YDS: The Clare Spark Blog — April 16, 2011 @ 2:35 pm | Reply
[Please read the second comment first.] I briefly took up the matter of the portrayal of sexuality here: https://clarespark.com/2011/01/04/railroading-ayn-randalissa-rosenbaumdagny-taggart/. It is worth adding to that blog as follows. First, Rand denied that there was any rape in any of her books, and publicly pronounced her horror at such a suggestion. She did state her belief in "man-worship" and in one of the introductions to The Fountainhead, said that Howard Roark was a condensation of the heroic male ideal. Her major novels should not be read as realistic or naturalistic, but as fantasies in the Symbolist tradition.
Having said that, her disapproval of "Women's Lib" should be contrasted with her insistence on abortion rights (especially in the first trimester, as later abortions were dangerous to the mother, she thought). As to her "man-worship" it is unrealistic to think that a woman of her generation, coming from the family that she did (the mother as both supportive and uncomfortable with motherhood) and the distant father who wished she would be an engineer, and then the horrors she experienced during the Leninist revolution and its aftermath, would not have contributed to her foibles and possibly too, her passion to fight collectivism as she saw it developing in the U.S., especially in the 1930s.
But more, we don't know enough about gender differences that are inherited to dismiss as unnatural or incorrect her search for the perfect male. I was raised that way, and would never have had an important relationship with a man whom I did not view (initially) as superior to me. It is possibly an evolutionary thing, and sociobiologists have commented upon it. To my knowledge, AR was not psychoanalyzed, so worked out her Oedipal drama in literature and philosophy. I don't think that one can understand her happy endings without a sip of Freud.
Comment by clarespark — January 7, 2011 @ 9:15 pm | Reply
Am I the only chick to have read Ayn Rand's novels for the richnography? Dominique's necklace, the jewels held together with invisible chains. Dagny's diamond bracelet, and the ruby pendant which was the only thing she wore while having at it with Hank. (Rand was a jewelry designer as well as a writer, and I think she invented the tennis bracelet in fiction before it existed in fact.) The fur coats, the fancy apartments. Better than Pat Booth's "Palm Beach".
And am I the only chick to notice that Dominique and Dagny are both kinky? Dominique glories in being raped,and she doesn't mind being the pampered prisoner in the tower. Dagny's bracelet give her the "most feminine of looks, the look of being chained," and she thrills at being Galt's servant girl.
Has anybody else noticed how the philosophy of the robber queen in Sade's "Justine" is but a step away from Rand's?
Comment by Normie — January 7, 2011 @ 5:17 pm | Reply
I don't deny that her sexuality is kinky. Her feminism is tied to laissez-faire capitalism. That is why I prefer We The Living to her later novels, for it is the most autobiographical of her major novels. There are mysteries unsolved by her biographers.
Comment by clarelspark — October 4, 2015 @ 7:13 pm | Reply
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\section{Introduction}
Let $(M^7,\varphi)$ be a $G_2$ manifold with the calibration 3-form
$\varphi$. If $\varphi$ restricts to be the volume form of an oriented
3-dimensional submanifold $Y^3$, then $Y$ is called an associative
submanifold of $M$. Associative submanifolds are very interesting
objects as they behave very similarly to holomorphic curves of
Calabi-Yau manifolds.
\vspace{.05in}
In \cite{as}, we studied the deformations of associative
submanifolds of $(M, \varphi)$ in order to construct Gromov-Witten
like invariants. One of our main observations was that oriented
$2$-plane fields on $M$ always exist by a theorem of Thomas
\cite{t}, and by using them one can split the tangent bundle
$T(M)={\bf E}\oplus {\bf V}$ as an orthogonal direct sum of an
associative $3$-plane bundle ${\bf E}$ and a complex $4$-plane
bundle $ {\bf V}$. This allows us to define ``complex associative
submanifolds'' of $M$, whose deformation equations may be reduced
to the Seiberg-Witten equations, and hence we can assign local
invariants to them, and assign various invariants to $(M,\varphi,
\Lambda)$, where $\Lambda$ is an oriented $2$-plane field on $M$.
It turns out that these Seiberg-Witten equations on the
submanifolds are restrictions of global equations on $M$.
\vspace{.05in}
In this paper, we explain how the geometric structures on $G_2$
manifolds with oriented $2$-plane fields $(M,\varphi, \Lambda )$
provide complex and symplectic structures to certain
6-dimensional subbundles of $T(M)$. When these bundles integrated we obtain a pair of Calabi-Yau manifolds whose complex and symplectic structures are remarkably related to each other.
We also study examples of Calabi-Yau manifolds which
fit nicely in our mirror set-up. Later, we do similar
constructions for $Spin(7)$ manifolds with oriented $3$-plane
fields. We then explain how these structures lead to the
definition of ``dual $G_2$ manifolds'' in a $Spin(7)$ manifold, with their own dual Calabi-Yau submanifolds.
\vspace{.07in}
{\em Acknowledgements:} We would like to thank R.Bryant and
S.Gukov for their valuable comments.
\vspace{.1in}
\section{Associative and Complex distributions of a $G_2$ manifold}
Let us go through quickly over the basic definitions about $G_2$
manifolds. The main references are the two foundational papers
\cite{hl} and \cite{b1}, as well as \cite{s}, \cite{b2},
\cite{bs}, and \cite{j}. We also need some properties introduced
in \cite{as}. Now let ${\mathbb O}={\mathbb H}\oplus l {\mathbb H}= {\mathbb R}^8$ be the octonions
which is an $8$ dimensional division algebra generated by $<1, i,
j, k, l, li ,lj, lk> $, and let $im {\mathbb O} ={\mathbb R}^7$ be the imaginary
octonions with the cross product operation $\times: {\mathbb R}^7\times
{\mathbb R}^7 \to {\mathbb R}^7$, defined by $u\times v=im(\bar{v}.u)$. The
exceptional Lie group $G_{2}$ is the linear automorphisms of $im
{\mathbb O}$ preserving this cross product operation, it can also be
defined in terms of the orthogonal $3$-frames in ${\mathbb R}^7$:
\begin{equation*}
G_{2} =\{ (u_{1},u_{2},u_{3})\in (im {\mathbb O})^3\;|\: \langle u_{i},u_{j} \rangle=\delta_{ij}, \;
\langle u_{1} \times u_{2},u_{3} \rangle =0 \; \}.
\end{equation*}
Another very useful definition popularized in \cite{b1} is the subgroup of $GL(7,{\mathbb R})$ which fixes a particular $3$-form
$\varphi_{0} \in \Omega^{3}({\mathbb R} ^{7})$. Denote
$e^{ijk}=dx^{i}\wedge dx^{j} \wedge dx^{k}\in \Omega^{3}({\mathbb R}^7)$,
then
$$ G_{2}=\{ A \in GL(7,{\mathbb R}) \; | \; A^{*} \varphi_{0} =\varphi_{0}\; \}. $$
\vspace{-0.15in}
\begin{equation}
\varphi _{0}
=e^{123}+e^{145}+e^{167}+e^{246}-e^{257}-e^{347}-e^{356}.
\end{equation}
\vspace{.05in}
{\Def A smooth $7$-manifold $M^7$ has a {\it $G_{2}$ structure} if
its tangent frame bundle reduces to a $G_{2}$ bundle.
Equivalently, $M^7$ has a {\it $G_{2}$ structure} if there is a
3-form $\varphi \in \Omega^{3}(M)$ such that at each $x\in M$
the pair $ (T_{x}(M), \varphi (x) )$ is isomorphic to $(T_{0}(
{\mathbb R}^{7}), \varphi_{0})$ (pointwise condition). We call $(M,\varphi)$ a manifold with $G_2$ structure.}
\vspace{.05in}
A $G_{2}$ structure $\varphi$ on $M^7$ gives an orientation
$\mu \in \Omega^{7}(M)$ on $M$, and $\mu$ determines a
metric $g=g_{\varphi }= \langle \;,\;\rangle$ on $M$, and a cross product
structure $\times$ on the tangent bundle of $M$ as follows: Let
$i_{v}=v\lrcorner $ be the interior product with a vector $v$, then
\begin{equation}
\langle u,v \rangle=[ i_{u}(\varphi ) \wedge i_{v}(\varphi )\wedge \varphi ]/6\mu .
\end{equation}
\begin{equation}
\varphi (u,v,w) = \langle u\times v,w \rangle .
\end{equation}
\vspace{.05in}
{\Def A manifold with $G_{2}$ structure $(M,\varphi)$ is called a
{\it $G_{2}$ manifold} if the holonomy group of the Levi-Civita
connection (of the metric $g_{\varphi }$) lies inside of $G_2 $.
Equivalently $(M,\varphi)$ is a $G_{2}$ manifold if $\varphi $ is
parallel with respect to the metric $g_{\varphi }$, that is
$\nabla_{g_{\varphi }}(\varphi)=0$; which is equivalent to
$d\varphi=0 $, $\;d(*_{g_{\varphi}}\varphi)=0$. Also equivalently, at each point
$x_{0}\in M$ there is a chart $(U,x_{0}) \to ({\mathbb R}^{7},0)$ on which
$\varphi $ equals to $\varphi_{0}$ up to second order term, i.e.
on the image of $U$
$\varphi (x)=\varphi_{0} + O(|x|^2)$}.
\vspace{.05in}
{\Rm One important class of $G_2$ manifolds are the ones obtained
from Calabi-Yau manifolds. Let $(X,\omega, \Omega)$ be a complex
3-dimensional Calabi-Yau manifold with K\"{a}hler form $\omega$
and a nowhere vanishing holomorphic 3-form $\Omega$, then
$X^6\times S^1$ has holonomy group $SU(3)\subset G_2$, hence is a
$G_2$ manifold. In this case $ \varphi$= Re $\Omega + \omega
\wedge dt$. Similarly, $X^6\times \mathbb{R}$ gives a noncompact
$G_2$ manifold.}
\vspace{.05in}
{\Def Let $(M, \varphi )$ be a $G_2$ manifold. A 4-dimensional
submanifold $X\subset M$ is called {\em coassociative } if
$\varphi|_X=0$. A 3-dimensional submanifold $Y\subset M$ is called
{\em associative} if $\varphi|_Y\equiv vol(Y)$; this condition is
equivalent to the condition $\chi|_Y\equiv 0$, where $\chi \in \Omega^{3}(M,
TM)$ is the tangent bundle valued 3-form defined by the
identity:}
\begin{equation}
\langle \chi (u,v,w) , z \rangle=*\varphi (u,v,w,z)
\end{equation}
The equivalence of these conditions follows from the `associator equality' of \cite{hl}
\begin{equation*}
\varphi (u,v,w)^2 + |\chi (u,v,w)|^2/4= |u\wedge v\wedge w|^2
\end{equation*}
Similar to the definition of $\chi$ one can define a tangent
bundle 2-form, which is just the cross product of $M$
(nevertheless viewing it as a $2$-form has its advantages).
\vspace{.05in}
{\Def Let $(M, \varphi )$ be a $G_2$ manifold. Then $\psi \in
\Omega^{2}(M, TM)$ is the tangent bundle valued 2-form defined by
the identity:}
\begin{equation}
\langle \psi (u,v) , w \rangle=\varphi (u,v,w)=\langle u\times v , w
\rangle
\end{equation}
\vspace{.05in}
Now we have two useful properties from \cite{as}, the first
property basically follows from definitions, the second property
fortunately applies when the first property fails to give anything
useful.
\vspace{.05in}
{\Lem (\cite{as}) To any $3$-dimensional submanifold $Y^3\subset
(M,\varphi)$, $\chi$ assigns a normal vector field, which
vanishes when $Y$ is associative.}
{\Lem(\cite{as}) To any associative manifold $Y^3\subset (M,\varphi)$ with a non-vanishing oriented $2$-plane field, $\chi$ defines a complex structure on its normal bundle (notice in particular that any coassociative submanifold $ X\subset M$ has an almost complex structure if its normal bundle has a non-vanishing section).}
\begin{proof}
Let $L\subset {\mathbb R}^7$ be an associative $3$-plane,
that is $\varphi_{0}|_{L}=vol(L)$. Then for every pair of
orthonormal vectors $\{u,v\}\subset L$, the form $\chi$ defines a
complex structure on the orthogonal $4$-plane $L^{\perp}$, as
follows: Define $j: L^{\perp} \to L^{\perp}$ by
\begin{equation}
j(X)=\chi(u,v,X)
\end{equation}
This is well defined i.e. $j(X)\in L^{\perp}$, because when $ w\in L$ we have:
$$ \langle \chi(u,v,X),w \rangle=*\varphi_{0}(u,v,X,w)=-*\varphi_{0}(u,v,w,X)=\langle \chi(u,v,w),X \rangle=0$$
\noindent Also $j^{2}(X)=j(\chi(u,v,X))=\chi(u,v,\chi(u,v,X))=-X$. We can check the last equality by taking an orthonormal basis $\{ X_{j}\}\subset L^{\perp}$ and calculating
\begin{eqnarray*}
\langle \chi(u,v,\chi(u,v,X_{i})),X_{j}\rangle &=&*\varphi_{0}(u,v,\chi(u,v,X_{i}),X_{j})=-*\varphi_{0}(u,v,X_{j},\chi(u,v,X_{i}))\\
&=&- \langle \chi(u,v,X_{j}),\chi(u,v,X_{i})\rangle =-\delta_{ij}
\end{eqnarray*}
The last equality holds since the map $j$ is orthogonal, and the
orthogonality can be seen by polarizing the associator equality, and by noticing $\varphi_{0}(u,v,X_i)=0$. Observe that the
map $j$ only depends on the oriented $2$-plane $\Lambda=<u,v>$ generated by $\{u,v\}$ (i.e. it only depends on the complex structure on
$\Lambda $). \end{proof}
{\Rm Notice that Lemma 1 gives an interesting flow on the $3$-dimensional submanifolds of $G_2$ manifolds
$f: Y \hookrightarrow (M,\varphi )$ (call $\chi$-flow), described by:
$$\frac{\partial}{\partial t} f=\chi(f_{*}vol \;(Y) )$$
For example, by \cite{bs} the total space of the spinor bundle $Q^7\to S^3$ (with ${\mathbb C}^2$ fibers) is a $G_2$ manifold, and the zero section $S^3\subset Q$ is an associative submanifold. We can imbed any homotopy $3$-sphere $\Sigma^{3} $ into $Q$ (homotopic to the zero-section). We conjecture that $\chi $- flow on $\Sigma\subset Q$, takes $\Sigma$ diffeomorphically onto the zero section. Note that, since any $S^3$ smoothly unknots in $S^7$ it is not possible to produce quick counterexamples by tying local knots; and an affirmative answer gives $\Sigma \cong S^3$. }
\begin{figure}[ht] \begin{center}
\includegraphics{E.eps} \caption{}
\end{center}
\end{figure}
Finally, we need some identities from \cite{b2} (also see \cite{k}) for $(M^7,\varphi)$, which follow from local calculations by using the definition (1). For
$\beta \in \Omega^{1}(M)$ we have:
\begin{equation}
|\;\varphi\wedge \beta \;|^2=4|\beta|^2, \; \;\mbox{and}\; \; |*\varphi\wedge \beta \;|^2=3|\beta|^2 , \\
\end{equation}
\begin{equation}
(\xi \lrcorner \; \varphi ) \wedge \varphi =2*(\xi \lrcorner \;\varphi )\;,\;\mbox{and}\; \;
*\;[ \; *( \beta \wedge *\varphi )\wedge * \varphi \;]=3\beta ,
\end{equation}
\begin{eqnarray}
\beta \times (\beta \times u)=-|\beta |^2u+\langle \beta,u\rangle \beta ,
\end{eqnarray}
where $*$ is the star operator. Let $\xi$ be a vector field on any Riemannian manifold $(M, g )$, and $\xi^{\#}\in \Omega^{1}(M)$ be its dual $1$-form, i.e. $\xi^{\#}(v)=\langle \xi,v\rangle$. Then for
$\alpha \in \Omega^{k}(M)$:
\begin{eqnarray}
*(\xi \lrcorner \; \alpha )&=&(-1)^{k+1}(\xi^{\#} \wedge *\alpha) .
\end{eqnarray}
\vspace{.05in}
\section{Mirror duality in $G_2$ manifolds}
\vspace{.05in}
On a local chart of a $G_2$ manifold $(M, \varphi )$, the form $\varphi $ coincides with the form $\varphi_{0} \in \Omega^{3} ({\mathbb R}^7)$ up to quadratic terms, we can express the corresponding tangent valued forms $\chi$ and $\psi$ in terms of $\varphi_{0}$ in local coordinates. More generally, if $e_1,...e_7$ is any local orthonormal frame and $e^1,..., e^7$ is the dual frame, from definitions we get:
\begin{equation*}
\begin{aligned}
\chi=&\;\;(e^{256}+e^{247}+e^{346}-e^{357})e_1\\
&+(-e^{156}-e^{147}-e^{345}-e^{367})e_2\\
&+(e^{157}-e^{146}+e^{245}+e^{267})e_3\\
&+(e^{127}+e^{136}-e^{235}-e^{567})e_4\\
&+(e^{126}-e^{137}+e^{234}+e^{467})e_5\\
&+(-e^{125}-e^{134}-e^{237}-e^{457})e_6\\
&+(-e^{124}+e^{135}+e^{236}+e^{456})e_7.\\
\end{aligned}
\end{equation*}
\vspace{.1in}
\begin{equation*}
\begin{aligned}
\psi=&\;\;(e^{23}+e^{45}+e^{67})e_1\\
&+(e^{46}-e^{57}-e^{13})e_2\\
&+(e^{12}-e^{47}-e^{56})e_3\\
&+(e^{37}-e^{15}-e^{26})e_4\\
&+(e^{14}+e^{27}+e^{36})e_5\\
&+(e^{24}-e^{17}-e^{35})e_6\\
&+(e^{16}-e^{25}-e^{34})e_7.\\
\end{aligned}
\end{equation*}
\vspace{.1in}
The forms $\chi$ and $\psi$ induce complex and symplectic
structures on certain subbundles of $T(M)$ as follows:
Let $\xi$ be a nonvanishing vector field of $M$. We can define a symplectic $\omega_{\xi}$ and a complex structure $J_{\xi}$ on the $6$-plane bundle $V_{\xi}:=\xi^{\perp}$ by
\vspace{.05in}
\begin{equation}
\omega_\xi=\langle \psi, \xi \rangle \;\;\;\mbox{and} \;\; J_{\xi}(X)=X\times \xi.
\end{equation}
Now we can define
\vspace{.05in}
\begin{equation}
\textup{Re}\; \Omega_{\xi} = \varphi|_{V_{\xi}}
\;\;\mbox{and}\;\;\; \textup{Im}\; \Omega_{\xi} = \langle \chi, \xi \rangle.
\end{equation}
\vspace{.1in}
\noindent In particular $\omega_{\xi}=\xi \lrcorner \; \varphi$, and $\textup{Im}\;\Omega_{\xi} =\xi \lrcorner \; *\varphi $. Call $\Omega_{\xi} = \textup{Re}\; \Omega_{\xi} +i \;\textup{Im}\; \Omega_{\xi}$. The reason for defining these is to pin down a Calabi-Yau like structure on any $G_2$ manifold. In case $(M, \varphi)=CY \times S^1$ these quantities are related to the ones in Remark 1. Notice that when $\xi \in {\bf E}$ then $J_{\xi}$ is an extension of $J$ of Lemma 2 from the $4$-dimensional bundle ${\bf V}$ to the $6$-dimensional bundle $V_{\xi}$.
\vspace{.1in}
By choosing different directions, i.e. different $\xi$, one can
find the corresponding complex and symplectic structures. In
particular we will get two different complex structures if we
choose $\xi$ in the associative subbundle ${\bf E}$ (where $\varphi$ restricts to be 1), or if we choose $\xi$ in the complementary subbundle ${\bf V}$, which we will call the coassociative subbundle. Note that $\varphi$ restricts to zero on the coassociative subbundle.
\vspace{.1in}
In local coordinates, it is a straightforward calculation that by choosing $\xi=e_i$ for any $i$, from equations (11) and (12), we can easily obtain the corresponding structures $\omega_{\xi}$, $J_{\xi}$,
$\Omega_{\xi}$.
For example, let us assume that $\{e_1,e_2,e_3\}$ is the local orthonormal basis for the associative bundle ${\bf E}$, and
$\{e_4,e_5,e_6,e_7\}$ is the local orthonormal basis for the coassociative bundle ${\bf V}$. Then if we choose $\xi=e_3=e_1\times e_2$
then we get $\omega_{\xi}= e^{12}-e^{47}-e^{56}$
and Im $\Omega_{\xi}=
e^{157}-e^{146}+e^{245}+e^{267}$. On the other hand, if we choose
$\xi=e_7$ then $\omega_{\xi}= e^{16}- e^{25}-e^{34}$
and Im $\Omega_{\xi}=
-e^{124}+e^{135}+e^{236}+e^{456}$ which will
give various symplectic and complex structures on the bundle $V_{\xi}$.
\vspace{.05in}
\subsection {A useful example}
$\;$
\vspace{.1in}
Let us take a Calabi-Yau
6-torus $\mathbb {T}^6=\mathbb{T}^3\times \mathbb{T}^3$, where $\{e_1,e_2,e_3\}$ is the basis for one $\mathbb{T}^3$ and $\{e_4,e_5,e_6\} $ the basis for the other (terms expressed with a slight abuse of notation). We can take the product $M=\mathbb {T}^6\times S^1$ as the corresponding $G_2$ manifold with the calibration 3-form
$\varphi=e^{123}+e^{145}+e^{167}+e^{246}-e^{257}-e^{347}-e^{356}
$, and with the decomposition $T(M)={\bf E}\oplus {\bf V}$, where ${\bf E}=\{e_1,e_2,e_3\}$ and ${\bf V}=\{e_4,e_5,e_6,e_7\}$. Now, if we choose $\xi=e_{7}$, then $V_{\xi}=<e_1,...,e_6>$ and the symplectic form is $\omega_{\xi}=
e^{16}-e^{25}-e^{34}$, and the complex structure is
\[
J_{\xi} =\left(
\begin{array}{ccc}
{\bf e_1} & \mapsto &-e_6 \\
{\bf e_2} & \mapsto & e_5 \\
{\bf e_3} & \mapsto & e_4
\end{array}
\right)
\]
\vspace{.05in}
\noindent and the complex valued
$(3,0)$ form is $\Omega _{\xi }=(e^1+ie^6)\wedge
(e^2-ie^5)\wedge(e^3-ie^4)$; note that this is just
$\Omega _{\xi }=(e^1-iJ_{\xi}(e^1))\wedge
(e^2-iJ_{\xi}(e^2))\wedge(e^3-iJ_{\xi}(e^3)) $.
\vspace{.05in}
On the other hand, if we choose $\xi '=e_{3}$ then
$V_{\xi '}=< e_1,..,\hat{e}_{3},..,e_7>$ and the symplectic form is
$\omega_{\xi '}= e^{12}-e^{47}-e^{56}$ and the complex structure is
\[
J_{\xi '} =\left(
\begin{array}{ccc}
{\bf e_1 } & \mapsto &- {\bf e_2 } \\
e_4 & \mapsto & e_7 \\
e_5 & \mapsto & e_6
\end{array}
\right)
\]
\vspace{.1in}
\noindent Also
$\Omega_{\xi '} =(e^1+ie^2)\wedge
(e^4-ie^7)\wedge(e^5-ie^6)$, as above this can be expressed more tidily as $\Omega_{\xi '}=(e^1-iJ_{\xi'}(e^1))\wedge
(e^4-iJ_{\xi'}(e^4))\wedge(e^5-iJ_{\xi'}(e^5))$.
In the expressions of $J$'s the basis of associative bundle ${\bf E}$ is indicated by bold face letters to indicate the differing complex structures on $\mathbb{T}^6$. To sum up: If we choose $\xi$ from the coassociative bundle ${\bf V}$ we get the complex structure which decomposes the 6-torus as $\mathbb{T}^3\times \mathbb{T}^3$. On the other hand if we choose $\xi$ from the associative bundle ${\bf E}$ then the induced complex structure on the $6$-torus corresponds to the decomposition as $\mathbb{T}^2\times \mathbb{T}^4$. This is the phenomenon known as ``mirror duality''. Here these two $SU(3)$ and $SU(2)$ structures are different but they come from the same $\varphi$ hence they are dual. These examples suggests the following definition of ``mirror duality'' in $G_2$ manifolds:
\vspace{.05in}
{\Def Two Calabi-Yau manifolds are mirror pairs of each other, if
their complex structures are induced from the same calibration
3-form in a $G_2$ manifold. Furthermore we call them strong
mirror pairs if their normal vector fields $\xi$ and $\xi'$ are
homotopic to each other through nonvanishing vector fields. }
\vspace{.05in}
{\Rm In the above example of $CY\times S^1$, where $CY=
\mathbb{T}^6$, the calibration form $\varphi=$Re $\Omega
+\omega\wedge dt $ gives Lagrangian tori fibration in $X_{\xi} $
and complex tori fibration in $X_{\xi'}$. They are different
manifestations of $\varphi $ residing on one higher dimensional
$G_2$ manifold $M^7$. In the next section this correspondence
will be made precise.}
\vspace{.1in}
In Section 4.2 we will discuss a more general notion of mirror
Calabi-Yau manifold pairs, when they sit in different $G_2$
manifolds, which are themselves mirror duals of each other in a
$Spin(7)$ manifold.
\subsection{General setting}
$\;$
\vspace{.1in}
Let $(M^7, \varphi , \Lambda)$ be a manifold with a $G_2$ structure and a non-vanishing oriented $2$-plane field. As suggested in \cite{as} we can view $(M^7, \varphi)$ as an analog of a symplectic manifold, and the $2$-plane field $\Lambda $ as an analog of a complex structure taming $\varphi$. This is because $\Lambda $ along with
$\varphi $ gives the associative/complex bundle splitting $T(M)={\bf E}_{\varphi,\Lambda}\oplus {\bf V}_{\varphi,\Lambda}$. Now, the next object is a choice of a non-vanishing unit vector field $\xi \in \Omega^{0}(M,TM)$, which gives a codimension one distribution $V_{\xi}:= \xi^{\perp}$ on $M$, which is equipped with the structures $(V_{\xi}, \omega_{\xi}, \Omega_{\xi}, J_{\xi})$ as given by (11) and (12).
\vspace{.05in}
Let ${\xi}^{\#}$ be the dual $1$-form of $\xi$. Let $e_{\xi^{\#}}$ and $i_{\xi}=\xi \lrcorner $ denote the exterior and interior product operations on differential forms. Clearly
$e_{\xi^{\#}}\circ i_{\xi }+i_{\xi }\circ e_{\xi^{\#}} =id $.
\begin{equation}
\varphi =e_{\xi^{\#}}\circ i_{\xi }(\varphi )+i_{\xi }\circ e_{\xi^{\#}}(\varphi )=\omega_{\xi}\wedge \xi^{\#}+Re \; \Omega_{\xi}.
\end{equation}
This is just the decomposing of the form $\varphi $ with respect to $\xi \oplus \xi^{\perp}$.
Recall that the condition that the distribution $V_{\xi}$ be integrable (the involutive condition which implies $\xi^{\perp}$ comes from a foliation) is given by:
\begin{equation}
d{\xi}^{\#}\wedge {\xi}^{\#}=0.
\end{equation}
Even when $V_{\xi}$ is not integrable, by \cite{th} it is homotopic to a foliation. Assume $X_{\xi }$ be a page of this foliation; for simplicity assume this 6-dimensional manifold is smooth.
\vspace{.05in}
It is clear from definitions that $J_{\xi}$ is an almost complex
structure on $X_{\xi}$. Also the $2$-form $\omega_{\xi }$ is
non-degenerate on $X_{\xi}$, because from (2) we can write
\begin{equation}
\omega_{\xi }^3=(\xi \lrcorner \;\varphi )^3=\xi \lrcorner \;
[\;(\xi \lrcorner \; \varphi ) \wedge (\xi \lrcorner \;\varphi )\wedge \varphi \;]=\xi \lrcorner \; (6 |\xi |^2 \mu )= 6 \mu_{\xi}
\end{equation}
where $\mu_{\xi}=\mu |_{V_{\xi}}$ is the induced orientation form on $V_{\xi}$.
\vspace{.05in}
{\Lem $J_{\xi}$ is compatible with $\omega_{\xi}$, and it is metric invariant.}
\proof Let $u,v\in V_{\xi}$
\begin{eqnarray*}
\omega_{\xi}(J_{\xi}(u), v )&=&\omega_{\xi} ( u\times \xi ,v)=
\langle \psi(u \times \xi,v),\xi\rangle =\varphi (u \times \xi,v, \xi) \;\;\;\;\;\;\;\;\;\mbox {by} \;(5)
\\
&=&-\varphi (\xi, \;\xi \times u,\;v)= -\langle \;\xi\times (\xi\times u), v \; \rangle \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\mbox {by} \;(3)\\
&=& -\langle \; - | \xi |^2 u + \langle \xi,u \rangle \xi, v \;\rangle = | \xi |^2 \langle u,v\rangle - \langle \xi,u \rangle \langle \xi,v\rangle \;\;\;\;\;\mbox {by} \;(9)\\
&=& \langle u,v\rangle .
\end{eqnarray*}
By plugging in $J_{\xi}(u)$, $J_{\xi}(v)$ for $u$, $v$:
$\langle J_{\xi}(u), J_{\xi}(v)\rangle =-\omega_{\xi}(u,J_{\xi}(v))=\langle u,v\rangle$
~\hfill$\square$
\vspace{.05in}
{\Lem $\Omega_{\xi} $ is a non-vanishing $(3,0)$ form.}
\proof By a local calculation as in Section 3.1 we see that
$\Omega_{\xi} $ is a $(3,0)$ form, and is non-vanishing bacause
$\Omega_{\xi} \wedge \overline{\Omega}_{\xi}=8i \;vol(X_{\xi})$,
i.e.
\begin{eqnarray*} \frac{1}{2i} \;\Omega_{\xi} \wedge \overline{\Omega}_{\xi}=Im\;\Omega_{\xi} \wedge Re\;\Omega_{\xi} &=& (\xi \lrcorner *\varphi)\wedge [\;\xi\lrcorner \;(\xi^{\#}\wedge \varphi)\;]\\
&=&-\xi \lrcorner \; [\;(\xi \lrcorner *\varphi)\wedge (\xi^{\#}\wedge \varphi)\;]\\
&=&\xi \lrcorner \;[ *(\xi^{\#}\wedge \varphi )\wedge ( \xi^{\#}\wedge \varphi ) \;]\;\;\;\;\;\;\;\mbox {by} \;(10)\\
&=& |\xi^{\#}\wedge \varphi |^2 \; \xi \lrcorner \; vol (M)\\
&=& 4|\xi^{\#}|^2 \; (*\xi^{\#}) =4 \;vol (X_{\xi} ). \;\;\;\;\;\;\;\;\mbox {by} \;(7)
\end{eqnarray*} ~\hfill$\square$
\vspace{.1in}
We can easily calculate $*Re \;\Omega_{\xi}=-Im \;
\Omega_{\xi}\wedge \xi^{\#}$ and $*Im \;\Omega_{\xi}=Re \;
\Omega_{\xi}\wedge \xi^{\#}$. In particular if $\star $ is the
star operator of $X_{\xi}$ (for example by (15) $\star
\omega_{\xi}=\omega_{\xi}^2$/2), then
\begin{equation}
\star Re \; \Omega_{\xi}=Im\; \Omega_{\xi}.
\end{equation}
\vspace{.05in}
Notice that $\omega_{\xi} $ is a symplectic structure on $X_{\xi}$
whenever $d \varphi =0$ and $\mathcal L_{\xi}(\varphi)|_{V_{\xi}} =0$,
where $\mathcal L_{\xi}$ denotes the Lie derivative along $\xi$. This is
because $\omega_{\xi} ={\xi} \lrcorner \;\varphi $ and:
$$ d\omega_{\xi}=\mathcal L_{\xi}(\varphi) - \xi \lrcorner \; d\varphi = \mathcal L_{\xi}(\varphi). $$
\vspace{.1in}
\noindent Also $d^{*}\varphi =0 \implies d^{\star}\omega_{\xi}=0 $, without any condition on the vector field $\xi$, since
\begin{equation}
*\varphi=\star \;\omega_{\xi}- Im \; \Omega_{\xi}\wedge \xi^{\#},
\end{equation}
\vspace{.05in}
\noindent and hence $d(\star \omega_{\xi})=d(*\varphi |_{X_{\xi}}
)=0$. Also $d\varphi=0 \implies $ $d
(Re\;\Omega_{\xi})=d(\varphi|_{X_{\xi}})=0$.
\vspace{.05in}
Furthermore, $d^{*}\varphi =0$ and
$\mathcal L_{\xi}(*\varphi)|_{V_{\xi}} =0 \implies $ $d( Im\;
\Omega_{\xi})=0 $; this is because $Im\;\Omega_{\xi} =
\xi\lrcorner \; (*\varphi)$, where $*$ is the star operator on
$(M, \varphi )$. Also, $J_{\xi}$ is integrable when $d\Omega=0$
(e.g. \cite{hi}). By using the following definition, we can sum up all the conclusions of the above discussion as Theorem 5 below.
\vspace{.05in}
{\Def $(X^6, \omega, \Omega, J)$ is called an {\it almost
Calabi-Yau manifold}, if $X$ is a Riemannian manifold with a
non-degenerate $2$-form $\omega$ (i.e. $\omega^3 =6vol (X)$) which
is co-closed, and $J$ is a metric invariant almost complex
structure which is compatible with $\omega$, and $\Omega $ is a
non-vanishing $(3,0)$ form with $Re\;\Omega $ closed.
Furthermore, when $\omega$ and $ Im\; \Omega$ are closed, we
call this a Calabi-Yau manifold.}
{\Thm Let $(M,\varphi)$ be a $G_2$ manifold, and $\xi $ be a unit
vector field which comes from a codimension one foliation on $M$,
then $(X_{\xi},\omega_{\xi}, \Omega_{\xi},J_{\xi})$ is an almost
Calabi-Yau manifold with $\varphi |_{X_{\xi}}= Re\; \Omega_{\xi} $
and $*\varphi |_{X_{\xi}}= \star \omega_{\xi} $. Furthermore, if
$\mathcal L_{\xi}(\varphi )|_{X_{\xi}}=0$ then $d\omega_{\xi}=0$, and if
$\mathcal L_{\xi}(*\varphi)|_{X_{\xi}}=0$ then $J_{\xi}$ is integrable; when both of these conditions are satisfied then
$(X_{\xi},\omega_{\xi}, \Omega_{\xi},J_{\xi})$ is a Calabi-Yau
manifold.}
{\Rm If $\xi$ and $\xi'$ are sections of ${\bf V}$ and ${\bf E}$ respectively, then from \cite{m} the condition
$\mathcal L_{\xi}(*\varphi)|_{X_{\xi } }=0$ (complex geometry of $X_{\xi }$)
implies that deforming associative submanifolds of $X_{\xi}$ along $\xi$ in $M$ keeps them associative; and
$\mathcal L_{\xi '}(\varphi )|_{X_{\xi ' }}=0$ (symplectic geometry of $X_{\xi '}$) implies that deforming coassociative submanifolds of $X_{\xi '}$ along $\xi '$ in $M$ keeps them coassociative (e.g. for an example see Example 1).}
\vspace{.1in}
Notice that both complex and symplectic structure of the
CY-manifold $X_{\xi}$ in Theorem 3 is determined by $\varphi$ when
they exist. Recall that (c.f. \cite{v}) elements $\Omega \in
H^{3,0}(X_{\xi},{\mathbb C})$ along with topology of $X_{\xi}$ (i.e. the
intersection form of $H^3(X_{\xi},{\mathbb Z})$) parametrize complex
structures on $X_{\xi}$ as follows: We compute the third betti
number $b_{3}(M)=2h^{2,1} +2 $ since
$$H^{3}(X_{\xi},{\mathbb C})=H^{3,0}\oplus H^{2,1} \oplus H^{1,2} \oplus
H^{0,3}=2({\mathbb C}\oplus H^{2,1}). $$
Let $\{ A^{i}, B_{j} \}$ be a symplectic basis of $H_{3}(X,{\mathbb Z})$,
$i=1,.., h^{2,1}+1$, then
\begin{equation}
X_{i}=\int_{A^{i}}\Omega
\end{equation}
give complex numbers which are local homegenous coordinates of the
moduli space of complex structures on $X_{\xi}$, which is a
$h^{2,1}$ dimensional space (there is an extra parameter here
since $\Omega$ is defined up to scale multiplication).
\vspace{.1in}
As we have seen in the example of the Section 3.1, the choice of
$\xi$ can give rise to quite different complex structures on
$X_{\xi}$ (e.g. $SU(2)$ and $SU(3)$ structures). For example,
assume $\xi \in \Omega^{0} (M, {\bf V})$ and $\xi'\in \Omega^{0}
(M, {\bf E})$ be unit vector fields, such that the the codimension
one plane fields $\xi^{\perp}$ and $\xi'^{\perp}$ come from
foliations. Let $X_{\xi }$ and $X_{\xi '}$ be pages of the
corresponding foliations. By our definition $X_{\xi }$ and
$X_{\xi' }$ are mirror duals of each other. Decomposition
$T(M)={\bf E}\oplus{\bf V}$ gives rise to splittings
$TX_{\xi}={\bf E}\oplus \bar{ {\bf E} }$, and $TX_{\xi'}={\bf
C}\oplus {\bf V} $, where $\bar{\bf E}= \xi^{\perp}( {\bf V}
)\subset {\bf V} $ is a $3$-dimensional subbundle, and ${\bf C}=(
\xi' )^{\perp} ({\bf E})\subset {\bf E} $ is a $2$-dimensional
subbundle. Furthermore, ${\bf E}$ is Lagrangian in $TX_{\xi}$ i.e.
$J_{\xi}({\bf E})=\bar{\bf E}$, and ${\bf C}$, ${\bf V}$ are
complex in $TX_{\xi'}$ i.e. $J_{\xi'}({\bf C})={\bf C}$ and
$J_{\xi'}({\bf V})={\bf V}$.
Also notice that, $Re\;\Omega_{\xi}$ is a
calibration form of ${\bf E}$ , and $\omega_{\xi}$ is a
calibration form of ${\bf C}$. In particular, $\langle
\Omega_{\xi}, {\bf E})=1$ and $\langle \omega_{\xi }\wedge \xi^{
\#}, {\bf E}\rangle=0$; and $\langle \Omega_{\xi'}, {\bf E})=0$
and $\langle \omega_{\xi' }\wedge (\xi')^{ \#}, {\bf
E}\rangle=1$.
\vspace{.1in}
If $X_{\xi}$ and $X_{\xi'}$ are strong duals of each other, we can
find a homotopy of non-vanishing unit vector fields $\xi_{t}$
($0\leq t \leq 1$) starting with $\xi\in {\bf V}$ ending with
$\xi'\in {\bf E}$. This gives a $7$-plane distribution
$\Xi=\xi^{\perp}_{t} \oplus \frac{\partial}{\partial t}$ on
$M\times [0,1]$ with integral submanifolds $X_{\xi} \times
[0,\epsilon)$ and $X_{\xi'} \times (1-\epsilon, 1]$ on a
neighborhood of the boundary. Then by \cite{th} and \cite{th1} we
can homotop $\Xi$ to a foliation extending the foliation on the
boundary (possibly by taking $\epsilon$ smaller). Let $Q^7\subset
M\times [0,1]$ be the smooth manifold given by this foliation,
with $\partial Q= X_{\xi} \cup X_{\xi'}$, where $X_{\xi}\subset
M\times \{0\}$ and $X_{\xi'}\subset M\times \{1\}$.
\vspace{.1in}
\begin{figure}[ht] \begin{center}
\includegraphics{f.eps} \caption{} \end{center}
\end{figure}
\vspace{.1in}
We can define $\Phi \in \Omega^{3}(M \times [0,1])$
with $\Phi |_{X_{\xi}}= \Omega_{\xi}$ and $\Phi |_{X_{\xi'}} =\xi
\lrcorner \star \omega_{\xi'} $
\begin{equation*}
\Phi =\Phi (\varphi, \Lambda, t )= \langle \omega_{\xi_{t}}\wedge \xi_{t}^{\#}, {\bf E} \rangle \; \xi_{t}'' \lrcorner \star \omega_{\xi_{t}}
+ \langle Re\; \Omega_{\xi_{t}}, {\bf E} \rangle \; \Omega_{\xi_{t}}
\end{equation*}
\noindent where $\xi''_{t}= J_{\xi\times \xi'}(\xi_{t})=\xi_{t}\times (\xi\times \xi' )$ (hence $\xi''_{0}=-\xi'$ and $\xi''_{1}=\xi$). This can be viewed as a correspondence between the complex structure of $X_{\xi}$ and the symplectic structure of $X_{\xi'}$.
In general, the manifold pairs $X_{\alpha }$ and $X_{\beta }$ (as constructed in Theorem 5) determine each others almost Calabi -Yau structures via $\varphi$ provided they are defined.
\vspace{.1in}
{\Prop Let $\{ \alpha, \beta \}$ be orthonormal vector fields
on $(M, \varphi) $. Then on $X_{\alpha}$ the following hold
\vspace{.05in}
\begin{itemize}
\item [(i)]$ Re\;\Omega_{\alpha }=\omega_{\beta} \wedge \beta^{\#} + Re\; \Omega_{\beta} $\\
\item [(ii)]$ Im\;\Omega_{\alpha }=\alpha \lrcorner \; (\star \omega_{\beta})-(\alpha \lrcorner \; Im\; \Omega_{\beta} )\wedge \beta^{\#} $\\
\item[(iii)] $\omega_{\alpha }= \alpha \lrcorner \;Re \; \Omega_{\beta } + (\alpha \lrcorner \; \omega_{\beta })\wedge \beta^{\#} $
\end{itemize} }
\proof Since $Re\;\Omega_{\alpha } =\varphi |_{X_{\alpha}}$ (i) follows.
Since $Im\; \Omega_{\alpha}=\alpha \lrcorner *\varphi$ following gives (ii)
\begin{eqnarray*}
\alpha \lrcorner \; (\star \omega_{\beta})&=& \alpha \lrcorner \; [\; \beta \lrcorner \;*(\beta \lrcorner \;\varphi )\;] \\
&=&\alpha \lrcorner \; \beta \lrcorner \; (\beta^{\#} \wedge * \varphi )\\
&=& \alpha \lrcorner *\varphi + \beta^{\#}\wedge (\alpha \lrcorner \;\beta \lrcorner *\varphi) \\
&=& \alpha \lrcorner *\varphi + (\alpha \lrcorner \; Im\; \Omega_{\beta} )\wedge \beta^{\#}
\end{eqnarray*}
(iii) follows from the following computation
$$\alpha \lrcorner \; Re\; \Omega_{\beta}= \alpha \lrcorner \;\beta \lrcorner \; (\beta ^{\#} \wedge \varphi)= \alpha \lrcorner \; \varphi +\beta^{\#}\wedge (\alpha \lrcorner \;\beta \lrcorner \;\varphi )= \alpha \lrcorner \; \varphi- (\alpha \lrcorner \; \omega_{\beta })\wedge \beta^{\#} $$
~\hfill$\square$
\vspace{.1in}
Notice that even though the identities of Proposition 6 hold only
after restricting the right hand side to $X_{\alpha}$, all the
individual terms are defined everywhere on $(M,\varphi)$. Also,
from the construction, $X_{\alpha }$ and $X_{\beta }$ inherit
vector fields $\beta$ and $\alpha$, respectively.
\vspace{.05in}
{\Cor Let $\{ \alpha, \beta \}$ be orthonormal vector fields on
$(M,\varphi)$. Then there are $A_{\alpha \beta} \in \Omega^{3}(M)$, and $W_{\alpha \beta}\in \Omega^{2}(M)$ satisfying
\begin{eqnarray*}
(a)\;\;\;\; \;\;\; \; \;\;\; \varphi |_{X_{\alpha}}= Re\;\Omega_{\alpha} &\mbox {and} & \varphi |_{X_{\beta}}=Re\; \Omega_{\beta}\\
(b)\;\;\;\; \;\; A_{\alpha \beta}|_{X_{\alpha}}= Im\;\Omega_{\alpha} &\mbox {and} &
A_{\alpha \beta}|_{X_{\beta}}= \alpha \lrcorner \; (\star \omega_{\beta})\\
(c) \hspace{.5in} W_{\alpha \beta}|_{X_{\alpha}}= \omega_{\alpha} &\mbox {and} &
W_{\alpha \beta}|_{X_{\beta}}= \alpha \lrcorner \;Re \; \Omega_{\beta }
\end{eqnarray*} }
For example, when $\varphi$ varies through metric preserving $G_2$ structures \cite{b2}, (hence fixing the orthogonal frame $\{\xi, \xi' \}$),
it induces variations of $\omega$ one side, and $\Omega$ on the
other side.
{\Rm By using Proposition 6, in the previous torus example of 3.1 one can show a natural correspondence between the groups $H^{2,1}(X_{\xi} )$ and $H^{1,1}(X_{\xi'} ) $. Even though ${\mathbb T}^7$ is a trivial example of a $G_2$ manifold, it is an important special case since the $G_2$ manifolds of Joyce are obtained by smoothing quotients of ${\mathbb T}^7$ by finite group actions. We believe this process turns the subtori $X_{\xi}$'s into Borcea-Voisin manifolds with a similar correspondence of their cohomology groups. }
\vspace{.1in}
For the discussion of the previous paragraph to work, we need a
non-vanishing vector field $\xi $ in $T(M) ={\bf E}\oplus {\bf
V}$, moving from ${\bf V}$ to ${\bf E}$. The bundle ${\bf E}$
always has a non-zero section, in fact it has a non-vanishing
orthonormal $3$-frame field; but ${\bf V}$ may not have a
non-zero section. Nevertheless the bundle ${\bf V}\to M$ does
have a non-vanishing section in the complement of a $3$-manifold
$Y\subset M $, which is a transverse self intersection of the zero
section. In \cite{as}, Seiberg-Witten equations of such
$3$-manifolds were related to associative deformations. So we can
use these partial sections and, as a consequence $X_{\xi}$ and
$X_{\xi '}$ may not be closed manifolds. The following is a
useful example:
\vspace{.05in}
{\Ex Let $X_1$, $X_2$ be two Calabi-Yau manifolds,
where $X_1$ is the cotangent bundle of $S^3$ and $X_2$ is the
$\mathcal{O}(-1)\oplus\mathcal{O}(-1)$ bundle of $S^2$. They are conjectured to be the mirror duals of each other by physicists (c.f. \cite{ma}). By using the approach of this paper, we identify them as 6-dimensional submanifolds of a $G_2$ manifold. Let's choose $M=\wedge ^2_+(S^4)$; this is a
$G_2$ manifold by Bryant-Salamon \cite{bs}.
\vspace{.05in}
Let $\pi: \wedge ^2_+ (S^4) \to S^4 $ be the bundle projection. The sphere bundle of $\pi$ (which is also $\overline{{{\mathbb C}{\mathbb P}}^3} $) is the so-called {\it twistor bundle}, let us denote it by $\pi_{1}: Z(S^4) \to S^4$. It is known that the normal bundle of each fiber $\pi_{1}^{-1}(p)\cong S^2$ in $Z(S^4)$ can be identified by
$\mathcal{O}(-1)\oplus\mathcal{O}(-1)$ \cite{s}. Now we take ${\bf E}$ to be the bundle of vertical tangent vectors of $\pi$, and
${\bf V}=\pi^{*} (T S^4)$, lifted by connection distribution. Let $\xi$ be the pull-back of the vector field on $S^4$ with two zeros (flowing from north pole ${\bf n}$ to south pole ${\bf s}$), and let $\xi '$ be the radial vector field of ${\bf E}$ . Clearly $X_{\xi }=T^{*}(S^3)$ and $X_{\xi' }=\mathcal{O}(-1)\oplus\mathcal{O}(-1)$.
\vspace{.05in}
Note that $\xi$ is non-vanishing in the complement of $\pi^{-1}\{{\bf n,s}\}$, whereas $\xi'$ is non-vanishing in the complement of the zero section of $\pi $. Clearly on the set where they are both defined, $\xi$ and $\xi'$ are homotopic through nonvanishing vector fields $\xi_{t}$. This would define a cobordism between the complements of the zero sections of the bundles $T^{*}(S^3)$ and $\mathcal{O}(-1)\oplus\mathcal{O}(-1)$, if the distributions $\xi_{t}^{\perp}$ were involutive.}
\begin{figure}[ht] \begin{center}
\includegraphics{D.eps} \caption{} \end{center}
\end{figure}
\vspace{.1in}
\vspace{.1in}
Here the change of complex structures $X_{\xi '} \leadsto X_{\xi }$ happens as follows. Let $S^{3}_{\lambda}\to S^2$ be the Hopf map with fibers consisting of circles of radius
$\lambda $, clearly $S^{3}_{\infty}=S^2\times {\mathbb R}$
$$({\mathbb C}^{2}- 0 )\times S^2 \to (S_{\lambda}^3 \times {\mathbb R})\times S^{2}\; \stackrel {\lambda \to \infty}{\longrightarrow} \;
S^3_{\infty}\times S^3_{\infty}$$
\noindent where the complex structure on $ S^3_{\infty}\times S^3_{\infty}$ is the obvious one, induced from exchanging the factors. In general if we allow the vector fields $\xi$ and $\xi '$ be homotopic through vector fields $\xi_{t}$ possibly with zeros, or the family $\xi_{t}^{\perp}$ not remain involutive the cobordism between $X_{\xi}$ and $X_{\xi'}$ will have singularities.
\vspace{.1in}
{\Rm If we apply the construction of Example 1 to the total space of the spinor bundle $Q\to S^3$ (see Remark 2), the two dual $6$-manifolds we get are $S^2\times {{\mathbb R}}^4$ and $S^3\times {\mathbb R}^3$.}
\vspace{.1in}
There is also a concept of mirror-dual $G_2$ manifolds in a Spin(7) manifold, hence we can talk about mirror dual CY manifolds coming from two different mirror dual $G_2$ submanifolds of a Spin(7) manifold. This is the the subject of the next section.
\vspace{.05in}
\section{ Mirror duality in $Spin(7)$ Manifolds}
\vspace{.1in}
Similar to Calabi-Yau case there is a notion of mirror duality
between $G_2$ manifolds \cite{ac}, \cite{av}, \cite{ gyz},
\cite{sv}. In this section we will give a definition of mirror
$G_2$ pairs, and an example which shows that associative and
co-associative geometries in mirror $G_2$ pairs are induced from
the same calibration $4$-form in a $Spin(7)$ manifold, and hence
these geometries dual to each other. Let us first recall the basic
definitions and properties of $Spin(7)$ geometries. The main
references in this subject are \cite{hl} and \cite{ti}.
{\Def An 8-dimensional Riemannian manifold $(N,\Psi)$ is called a
{\em Spin(7) manifold} if the holonomy group of its metric
connection lies in $Spin(7)\subset GL(8)$.}
\vspace{.05in}
Equivalently, a $ Spin(7) $ manifold is an $8$-dimensional
Riemannian manifold with a triple cross product $\times $ on its
tangent bundle, and a closed $4$-form $\Psi \in \Omega^{4}(N)$
with
$$ \Psi (u,v,w,z)=\langle u \times v \times w,z \rangle.$$
{\Def A 4-dimensional submanifold $X$ of a $Spin(7)$ manifold $(N,\Psi)$ is
called {\em Cayley} if $\Psi|_X\equiv vol(X)$. }
\vspace{.05in}
Analogous to the $G_2$ case, we introduce a tangent bundle valued
$3$-form, which is just the triple cross product of $N$.
{\Def Let $(N, \Psi )$ be a $Spin(7)$ manifold. Then $\Upsilon \in
\Omega^{3}(N, TN)$ is the tangent bundle valued 3-form defined by
the identity:}
\begin{equation*}
\langle \Upsilon (u,v,w) , z \rangle=\Psi (u,v,w,z)=\langle
u\times v\times w , z \rangle .
\end{equation*}
$Spin(7)$ manifolds can be constructed from $G_2$
manifolds. Let $(M,\varphi)$ be a $G_2$ manifold with a 3-form
$\varphi$, then $M\times S^1$ (or $M\times \mathbb{R}$) has holonomy group $G_2\subset Spin(7)$, hence is
a Spin(7) manifold. In this case $\Psi= \varphi\wedge dt +
*_{7} \varphi$, where $*_{7}$ is the star operator of $M^7$.
\vspace{.05in}
Now we will repeat a similar construction for a $Spin(7)$ manifold $(N, \Psi )$, which we did for $G_2$ manifolds. Here we make an assumption that $T(M)$ admits a nonvanishing $3$-frame field $\Lambda =\langle u,v,w \rangle$, then we decompose $T(M)={\bf K}\oplus{\bf D}$, where ${\bf K}=\langle u,v,w, u\times v\times w \rangle$ is the bundle of Cayley $4$-planes (where $\Psi$ restricts to be 1) and ${\bf D}$ is the complementary subbundle (note that this is also a bundle of Cayley $4$-planes since the form $\Psi$ is self dual). In the $G_2$ case, existence of an analogous decomposition of the tangent bundle followed from \cite{t} (in this case we can just restrict to a submanifold which a $3$-frame field exists).
On a chart in $N$ let $e_1,...e_8$ be an orthonormal frame and $e^1,...,e^8$ be the dual coframe, then the calibration
4-form is given as (c.f. \cite{hl}) \:
\begin{equation}
\begin{aligned}
\Psi=\;\; e^{1234}&+(e^{12}-e^{34})\wedge (e^{56}-e^{78})\\
&+(e^{13}+e^{24})\wedge (e^{57}+e^{68})\\
&+(e^{14}-e^{23})\wedge (e^{58}-e^{67})+e^{5678}\\
\end{aligned}
\end{equation}
\noindent which is a self dual $4$-form, and the corresponding tangent bundle valued 3-form is
\vspace{.1in}
\begin{equation*}
\begin{aligned}
\Upsilon=&\;\;(e^{234}+e^{256}-e^{278}+e^{357}+e^{368}+e^{458}-e^{467})e_1\\
&+(-e^{134}-e^{156}+e^{178}+e^{457}+e^{468}-e^{358}+e^{367})e_2\\
&+(e^{124}-e^{456}+e^{478}-e^{157}-e^{168}+e^{258}-e^{267})e_3\\
&+(-e^{123}+e^{356}-e^{378}-e^{257}-e^{268}-e^{158}+e^{167})e_4\\
&+(e^{126}-e^{346}+e^{137}+e^{247}+e^{148}-e^{238}+e^{678})e_5\\
&+(-e^{125}+e^{345}+e^{138}+e^{248}-e^{147}+e^{237}-e^{578})e_6\\
&+(-e^{128}+e^{348}-e^{135}-e^{245}+e^{146}-e^{236}+e^{568})e_7\\
&+(e^{127}-e^{347}-e^{136}-e^{246}-e^{145}+e^{235}-e^{567})e_8.\\
\end{aligned}
\end{equation*}
\vspace{.1in}
This time we show that the form $\Upsilon$ induce $G_2$ structures
on certain subbundles of $T(N)$. Let $\gamma$ be a nowhere
vanishing vector field of $N$. We define a $G_2$ structure
$\varphi_{\gamma}$ on the $7$-plane bundle
$V_{\gamma}:=\gamma^{\perp}$ by (where $*_{8}$ is the star operator on $N^8$)
\begin{equation}
\varphi_\gamma:=\langle \Upsilon, \gamma \rangle = \gamma \lrcorner \; \Psi= *_{8}(\Psi\wedge \gamma^{\#}).
\end{equation}
Assuming that $V_{\gamma}$ comes from a foliation, we let $M_{\gamma}$ be an integral submanifold of $V_{\gamma}$. We have $d\varphi_{\gamma}=0$, provided $\mathcal L_{\gamma } (\Psi )|_{V_{\gamma}} =0 $. On the other hand, we always have $d(\star\varphi_{\gamma})=0$ on $M_{\gamma}$. To see this, we use
$$ \Psi =\varphi_{\gamma}\wedge \gamma^{\#} +*_{7} \varphi_{\gamma}$$ where $*_{7} $ is the star operator on $M{\gamma}$, and use $d\Psi=0$ and the foliation condition $d \gamma^{\#}\wedge \gamma^{\#}=0$, and the identity $\theta|_{M_{\gamma}}= \gamma \lrcorner \; [\;\theta \wedge \gamma^{\#}\;]$ for forms $\theta $. In order to state the next theorem, we need a definition:
\vspace{.05in}
{\Def A manifold with $G_2$ structure $(M,\varphi )$ is called an almost $G_2$-manifold if $\varphi $ is co-closed.}
\vspace{.05in}
{\Thm Let $(N^8, \Psi )$ be a $Spin(7)$ manifold, and $\gamma $ be a unit vector field which comes from a foliation, then $(M_{\gamma},\varphi_{\gamma})$ is an almost $G_2$ manifold. Furthermore if $\mathcal L_{\gamma} (\Psi)|_{M_{\gamma}}=0$ then $(M_{\gamma},\varphi_{\gamma})$ is a $G_2$ manifold.}
\proof Follows by checking Definition 1 and by the discussion above.~\hfill$\square$
\vspace{.1in}
The following theorem says the induced $G_2$ structures on
$M_{\alpha}$, $M_{\beta }$ determine each other via $\Psi$; more
specifically $\varphi_{\alpha}$ and $\varphi_{\beta}$ are
restrictions of a global $3$-form of $N$.
\vspace{.05in}
{\Prop Let $(N, \Psi )$ be a $Spin(7)$ manifold, and $\{ \alpha,
\beta \}$ be an orthonormal vector fields on $N$.Then the
following holds on $M_{\alpha}$
$$ \varphi_{\alpha} = - \alpha \lrcorner \; (\varphi_{\beta} \wedge \beta^{\#} +*_{7}\;\varphi_{\beta})$$}
\proof The proof follows from the definitions, and by expressing
$\varphi_{\alpha}$ and $\varphi_{\beta}$ in terms of $\beta^{\#}$
and $\alpha^{\#}$ by the formula (13). ~\hfill$\square$
\vspace{.1in}
As in the $G_2$ case, by choosing different $\gamma $'s, one can
find various different $G_2$ manifolds $M_{\gamma}$ with
interesting structures. Most interestingly, we will get certain
``dual'' $M_{\gamma} $'s by choosing $\gamma $ in ${\bf K}$ or in
${\bf D}$. This will shed light on more general version of mirror
symmetry of Calabi-Yau manifolds. First we will discuss an
example.
\vspace{.1in}
\subsection {An example}
$\;$
\vspace{.1in}
Let $\mathbb
{T}^8=\mathbb{T}^4\times \mathbb{T}^4$ be the Spin(7) 8-torus, where
$\{e_1,e_2,e_3,e_4\}$ is the basis for the Cayley $\mathbb{T}^4$
and $\{e_5,e_6,e_7,e_8\} $ is the basis for the complementary
$\mathbb{T}^4$ . We can take
the corresponding calibration 4-form (20) above,
and take the decomposition $T(N)={\bf K}\oplus {\bf D}$,
where $\{e_1,e_2,e_3,e_4\}$ is the orthonormal basis for the
Cayley bundle ${\bf K}$, and $\{e_5,e_6,e_7,e_8\}$ is the local
orthonormal basis for the complementary bundle ${\bf D}$. Then if
we choose $\gamma=e_4=e_1\times e_2\times e_3$ then we get
$$\varphi_{\gamma}=
-e^{123}+e^{356}-e^{378}-e^{257}-e^{268}-e^{158}+e^{167}$$ On the
other hand, if we choose $\gamma'=e_5$ then we get
$$\varphi_{\gamma '}=e^{126}-e^{346}+e^{137}+e^{247}+e^{148}-e^{238}+e^{678}$$
which give different $G_2$ structures on the $7$ toris $M_{\gamma}$ and $M_{\gamma'}$.
\vspace{.1in}
Note that if we choose $\gamma$ from the Cayley bundle ${\bf K}$,
we get the $G_2$ structure on the 7-torus $M_{\gamma}$ which reduces the Cayley
4-torus $\mathbb{T}^4=\mathbb{T}^3 \times S^{1} $ (where $\gamma $ is tangent to $S^1$ direction) to an associative 3-torus $ \mathbb{T}^3 \subset M_{\gamma}$ with respect to this $G_2$
structure. On the other hand if we choose $\gamma '$ from the
complementary bundle ${\bf D}$, then the Cayley 4-torus $\mathbb{T}^4$ will be a coassociative submanifold of the $7$-torus $M_{\gamma'}$with the corresponding $G_2$ structure. Hence associative and coassociative geometries are dual to each other as they are induced from the same
calibration 4-form $\Psi$ on a Spin(7) manifold. This suggests the
following definition of the ``mirror duality'' for $G_2$
manifolds.
\vspace{.05in}
{\Def Two $7$-manifolds with $G_{2}$ structures are mirror pairs, if their $G_2$-structures are induced from the same calibration 4-form in a
$Spin(7)$ manifold. Furthermore they are strong duals if their normal vector fields are homotopic. }
\vspace{.05in}
{\Rm For example, by \cite{bs} the total space of an ${\mathbb R}^4 $ bundle over
$S^4$ has a $Spin(7)$ structure. By applying the process of Example 1, we obtain mirror pairs $M_{\gamma}$ and $M_{\gamma'}$ to be $S^3\times {\mathbb R}^4$ and ${\mathbb R}^4\times S^3$ with dual $G_2$ structures.}
\vspace{.1in}
\subsection{Dual Calabi-Yau's inside of Spin(7)}
$\;$
\vspace{.1in}
Let $(N^8, \Psi)$ be a $Spin(7)$ manifold, and let $\{\alpha, \beta\}$ be an orthonormal $2$-frame field in $N$, each coming from a foliation
Let $(M_{\alpha}, \varphi_{\alpha})$ and $(M_{\beta}, \varphi_{\beta})$ be the $G_2$ manifolds given by Theorem 8. Similarly by Theorem 5, the vector fields $\beta $ in $M_{\alpha }$, and $\alpha $ in $M_{\beta }$ give almost Calabi-Yau's $X_{\alpha \beta}\subset M_{\alpha}$ and $X_{\beta \alpha }\subset M_{\beta }$. Let us denote $X_{\alpha \beta}= (X_{\alpha\beta}, \omega_{\alpha \beta}, \Omega_{\alpha \beta}, J_{\alpha \beta})$ likewise
$X_{\beta \alpha }= (X_{\beta \alpha}, \omega_{\beta \alpha }, \Omega_{ \beta \alpha }, J_{\beta, \alpha})$. Then we have
\vspace{.1in}
{\Prop The following relations hold: \vspace{.05in}
\begin{itemize}
\item [(i)] $J_{\alpha \beta}(u)=u\times \beta\times \alpha $\\
\item [(ii)] $\omega_{\alpha \beta}=\beta \lrcorner \; \alpha \lrcorner \; \Psi$\\
\item [(iii)] $Re\; \Omega_{\alpha \beta}=\alpha \lrcorner \;
\Psi |_{X_{\alpha \beta} }$\\
\item [(iv)] $Im\; \Omega_{\alpha \beta}=\beta \lrcorner \;
\Psi |_{X_{\alpha \beta} }$
\end{itemize}}
\proof (i), (ii), and (iii) follow from definitions, and the formula $X\times Y= (X \lrcorner \; Y\lrcorner \;\varphi)^{\#}$.
\begin{eqnarray*}
Im\; \Omega_{\alpha \beta}&=& \beta\lrcorner *_{7} \varphi_{\alpha}=
\beta\lrcorner *_{7} (\alpha \lrcorner \; \Psi)\\
&=&\beta \lrcorner \;[\alpha \lrcorner \;*_{8}(\alpha \lrcorner \; \Psi)]\\
&=&\beta \lrcorner \; [\alpha \lrcorner \; (\alpha^{\#}\wedge \Psi )]\;\;\;\;\;\;\;\;\;\;\mbox{by}\;(10)\\
&=&\beta \lrcorner \; [\Psi- \alpha^{\#}\wedge (\alpha \lrcorner \; \Psi)] \\
&=&\beta \lrcorner \; \Psi -\alpha^{\#}\wedge (\beta \lrcorner \; \alpha \lrcorner \; \psi)] .
\end{eqnarray*}
Left hand side is already defined on $X_{\alpha \beta}$, by restricting to $X_{\alpha \beta}$ we get (iii). ~\hfill$\square$
\vspace{.05in}
{\Cor When $X_{\alpha, \beta}$ and $X_{\beta, \alpha}$ coincide, they are oppositely oriented manifolds and $\omega_{\alpha, \beta}=- \omega_{\beta, \alpha}$, and $Re\; \Omega_{\alpha \beta}=-Im\; \Omega_{\beta \alpha}$ (as forms on $X_{\alpha \beta}$).}
\vspace{.1in}
Now let $\{\alpha, \beta, \gamma\}$ be an orthonormal $3$-frame field in $(N^8, \Psi)$, and let
$(M_{\alpha}, \varphi_{\alpha})$, $(M_{\beta}, \varphi_{\beta})$, and $( M_{\gamma}, \varphi_{\gamma})$ be the corresponding almost $G_2$ manifolds. As before, the orthonormal vector fields $\{ \gamma, \beta \}$ in $M_{\alpha }$ and $\{\gamma, \alpha \}$ in $M_{\beta }$ give rise to corresponding almost Calabi-Yau's $X_{\alpha,\gamma}$, $X_{\alpha,\beta }$ in $M_{\alpha}$, and $X_{\beta,\gamma }$, $X_{\beta,\alpha }$ in $M_{\beta }$.
\vspace{.05in}
In this way $(N^8,\Psi )$ gives rise to $4$ Calabi-Yau descendents. By Corollary 11, $X_{\alpha \beta}$ and $X_{\beta \alpha}$ are different geometrically; they may not even be the same as smooth manifolds, but for simplicity we may consider it to be the same smooth manifold obtained from the triple intersection of the three $G_2$ manifolds.
\vspace{.05in}
In case we have a decomposition $T(N)={\bf K}\oplus {\bf D}$ of the tangent bundle of $(N^8,\Psi )$ by Cayley plus its orthogonal bundles(Section 4); we can choose our frame special and obtain interesting CY-manifolds. For example, if we choose $\alpha \in \Omega^{0}(M,{\bf K})$ and $\beta, \gamma \in \Omega^{0}(N,{\bf D})$ we get one set of complex structures, whose types are indicated by the first row of the following diagram. On the other hand, if we choose all $\{\alpha, \beta,\gamma \}$ lie entirely in ${\bf K}$ or ${\bf D }$ we get an other set of complex structures, as indicated by the second row of the diagram.
\vspace{.1in}
$\begin{array}{ccccccccc}
&&&&(N^8, \Psi )&&&&\\
&&&\swarrow &&\searrow&&&\\
&&(M_{\alpha},\varphi_{\alpha})&&&&(M_{\beta}, \varphi_{\beta})&&\\
&\swarrow&&\searrow&&\swarrow&&\searrow&\\
{\bf X}_{\alpha \gamma}&&&&X_{\alpha \beta}\;\;\;\;\;\;\;{\bf X}_{\beta \gamma} &&&&X_{\beta \alpha}\\
&&&&&&&&\\
SU(3)&&&&SU(3)\;\;\;\;\;\;\; SU(2) &&&& SU(3)\\
&&&&&&&&\\
SU(2)&&&&SU(2)\;\;\;\;\;\;\; SU(2) &&&& SU(2)\\
\end{array}$
\vspace{.1in}
Here all the corresponding symplectic and the holomorphic forms of the resulting Calabi-Yau's come from restriction of global forms induced by $\Psi $. The following gives relations between the complex/symplectic structures of these induced CY-manifolds; i.e. the structures $X_{\alpha \gamma}$, $X_{\beta \gamma}$ and $X_{\alpha \beta }$ satisfy a certain triality relation.
\vspace{.1in}
{\Prop We have the following relations;
\vspace{.05in}
\begin{itemize}
\item [(i)]$ Re\;\Omega_{\alpha \gamma}=\alpha \lrcorner (\; *_{6} \;\omega_{\beta \gamma }) +\omega_{\alpha \beta}\wedge\beta^{\#} $\\
\item [(ii)]$ Im\;\Omega_{\alpha \gamma}=\omega_{\beta\gamma }\wedge \beta^{\#} -\gamma \lrcorner \;*_{6}(\omega_{\alpha \beta} )$\\
\item[(iii)] $\omega_{\alpha \gamma }= \alpha \lrcorner \; Im\;\Omega_{\beta \gamma }+ (\gamma \lrcorner \;\omega_{\alpha \beta} )\wedge \beta^{\#} $
\end{itemize} }
\vspace{.05in}
\noindent First we need to prove a lemma;
\vspace{.05in}
{\Lem The following relations hold;
\begin{eqnarray*}
\alpha \lrcorner \; *_{6}(\omega_{\beta \gamma})&=& \alpha \lrcorner \;\Psi +\gamma^{\#}\wedge (\alpha \lrcorner \;\gamma \lrcorner \; \Psi) + \beta^{\#}\wedge (\alpha \lrcorner \;\beta \lrcorner \; \Psi)
-\gamma^{\#}\wedge \beta^{\#}\wedge (\alpha \lrcorner \; \gamma \lrcorner \; \beta \lrcorner \; \Psi). \\
Im\;\Omega_{\beta \gamma}&=&-\gamma \lrcorner \;\Psi - \beta^{\#}\wedge (\gamma \lrcorner \;\beta \lrcorner \;\Psi).\\
Re\; \Omega_{\alpha \gamma }&=& \alpha \lrcorner \; \Psi- \gamma^{\#}\wedge (\gamma \lrcorner \; \alpha \lrcorner \;\Psi).
\end{eqnarray*}
\proof
\begin{eqnarray*}
\alpha \lrcorner \; *_{6}(\omega_{\beta \gamma}) &=& \alpha \lrcorner \; [\gamma \lrcorner \; \beta \lrcorner \; *_{8}\;(\gamma \lrcorner \; \beta \lrcorner \Psi) ]\\
&=& - \alpha \lrcorner \;\gamma \lrcorner \; \beta \lrcorner \;(\gamma^{\#}\wedge \beta^{\#}\wedge \Psi) \\
&=&-\alpha \lrcorner \; \gamma \lrcorner \; [ -\gamma^{\#} \wedge \Psi + \gamma^{\#}\wedge \beta^{\#}\wedge (\beta \lrcorner \; \Psi) ] \\
&=& \alpha \lrcorner \;\Psi +\gamma^{\#}\wedge (\alpha \lrcorner \;\gamma \lrcorner \; \Psi) + \beta^{\#}\wedge (\alpha \lrcorner \;\beta \lrcorner \; \Psi) \\
& & -\gamma^{\#}\wedge \beta^{\#}\wedge (\alpha \lrcorner \; \gamma \lrcorner \; \beta \lrcorner \; \Psi) .
\end{eqnarray*}
\begin{eqnarray*}
Im\;\Omega_{\beta \gamma}&=& \gamma \lrcorner \; *_{7}\;(\beta \lrcorner \; \Psi) \\
&=& \gamma \lrcorner \; [\beta \lrcorner \; *_{8}\;(\beta \lrcorner \; \Psi) ]\\
&=& - \gamma \lrcorner \; \beta \lrcorner \; (\beta^{\#}\wedge \Psi) \\
&=& -\gamma \lrcorner \; \Psi - \beta ^{\#}\wedge (\gamma \lrcorner \; \beta \lrcorner \; \Psi).
\end{eqnarray*}
$$Re\; \Omega_{\alpha \gamma }=(\alpha \lrcorner \; \Psi ) |_{X_{\alpha \gamma}}= \gamma \lrcorner \; [ \gamma^{\#} \wedge (\alpha \lrcorner \; \Psi) ]= \alpha \lrcorner \; \Psi- \gamma^{\#}\wedge (\gamma \lrcorner \; \alpha \lrcorner \;\Psi).$$
~\hfill$\square$
\noindent {\it Proof of Proposition 12}. We calculate the
following by using Lemma 13:
\begin{eqnarray*}
\alpha \lrcorner \; *_{6}(\omega_{\beta \gamma})+\omega_{\alpha \beta}\wedge \beta^{\#}&=& \alpha \lrcorner \;\Psi +\gamma^{\#}\wedge (\alpha \lrcorner \;\gamma \lrcorner \; \Psi) + \beta^{\#}\wedge (\alpha \lrcorner \;\beta \lrcorner \; \Psi) \\
& & -\gamma^{\#}\wedge \beta^{\#}\wedge (\alpha \lrcorner \; \gamma \lrcorner \; \beta \lrcorner \; \Psi) +(\beta \lrcorner \;\alpha \lrcorner \;\Psi)\wedge \beta^{\#}\\
&= &\alpha \lrcorner \;\Psi - \gamma^{\#}\wedge (\gamma \lrcorner \;\alpha \lrcorner \; \Psi) -\gamma^{\#}\wedge \beta^{\#}\wedge (\alpha \lrcorner \; \gamma \lrcorner \; \beta \lrcorner \; \Psi).
\end{eqnarray*}
\vspace{.05in}
Since we are restricting to $X_{\alpha \gamma}$ we can throw away
terms containing $\gamma^{\#}$ and get (i). We prove (ii)
similarly:
\vspace{.05in}
\begin{eqnarray*}
Im\;\Omega_{\alpha \gamma }&=&(\gamma \lrcorner \; *_{7}\varphi_{\alpha} )=
\gamma \lrcorner \; [\;\alpha \lrcorner \; *_{8}\varphi_{\alpha} \; ]\\
&=& \gamma \lrcorner \; [\alpha \lrcorner \; *_{8} (\alpha \lrcorner \;\Psi )]= - \gamma \lrcorner \; [\alpha \lrcorner \; (\alpha^{\#} \wedge \;\Psi ) ) \\
&=&- \gamma \lrcorner \;\Psi +
\gamma \lrcorner \;[ \alpha^{\#}\wedge (\alpha \lrcorner \; \Psi) ]\\
&=& - \gamma \lrcorner \;\Psi -\alpha^{\#}\wedge( \gamma \lrcorner \;\alpha \lrcorner \; \Psi] .
\end{eqnarray*}
\begin{eqnarray*}
\omega_{\beta \gamma }\wedge \beta^{\#} - \gamma \lrcorner \;(*_{6}\;\omega_{\alpha \beta })&=& - \gamma \lrcorner \;(*_{6}\;\omega_{\alpha \beta })+ (\gamma \lrcorner \; \beta \lrcorner \; \Psi)\wedge \beta^{\#} \\
&=&-\gamma \lrcorner \;\Psi -\beta^{\#}\wedge (\gamma \lrcorner \;\beta \lrcorner \; \Psi) -\alpha^{\#}\wedge (\gamma \lrcorner \;\alpha \lrcorner \; \Psi) \\
& & +\beta^{\#}\wedge \alpha^{\#}\wedge (\gamma \lrcorner \; \beta \lrcorner \; \alpha \lrcorner \; \Psi) + (\gamma \lrcorner \; \beta \lrcorner \; \Psi)\wedge \beta^{\#}.
\end{eqnarray*}
\vspace{.1in}
\noindent Here, we used Lemma 13 with different indices
$(\alpha,\beta, \gamma) \mapsto (\gamma, \alpha, \beta)$, and
since we are restricting to $X_{\alpha \gamma}$ we threw away
terms containing $\alpha^{\#}$. Finally, (iii) follows by plugging
in Lemma 13 to definitions. ~\hfill$\square$
\vspace{.2in}
Following says that Calabi -Yau structures of $X_{\alpha \gamma}$ and $X_{\beta \gamma}$ determine each other via $\Psi$. Proposition 14 is basically a consequence of Proposition 6 and Corollary 11.
\vspace{.1in}
{\Prop We have the following relations
\vspace{.05in}
\begin{itemize}
\item [(i)]$ Re\;\Omega_{\alpha \gamma}=\alpha \lrcorner \;( *_{6}\;\omega_{\beta \gamma }) - (\alpha \lrcorner \; Re\;\Omega_{\beta \gamma}) \wedge \beta ^{\#}$\\
\item [(ii)]$ Im\;\Omega_{\alpha \gamma}=\omega_{\beta \gamma} \wedge \beta^{\#} +Im\; \Omega_{\beta \gamma}$\\
\item[(iii)] $\omega_{\alpha \gamma }= \alpha \lrcorner \;Im \; \Omega_{\beta \gamma} + (\alpha \lrcorner \; \omega_{\beta \gamma})\wedge \beta^{\#} $
\end{itemize} }
\proof All follow from the definitions and Lemma 11 (and by ignoring $\alpha^{\#}$ terms). ~\hfill$\square$
{\Rm After this paper was written we learned that the results similar to Theorem 5 already appeared in \cite{aw}, \cite{asa}, \cite{c}, \cite{cs} (where they use more restricted vector fields $\xi$), and we also found out that the idea of studying induced hypersurface structures, from manifolds with exceptional holonomy goes back to earlier works of Calabi and Gray. }
\vspace{.1in}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 971 |
\section{Introduction}
\IEEEPARstart{P}{roviding} efficient support for the Internet of Things (IoT) is one of the major objectives for the cellular wireless communication. Machine-to-Machine (M2M) communication is anticipated to support billions of Machine Type Communication (MTC) devices. In addition, the MTC devices are sporadically activated with short packets \cite{servicetype}. Therefore, the random access (RA) process for M2M communications in IoT is characterized by massive connection and sporadic transmission, as well as small-sized data packets.
Confronted with the characteristics described above, the conventional orthogonal multiple access (OMA) scheme becomes infeasible due to its low resource efficiency. To facilitate the sharing of uplink resources, different RA schemes were proposed and can be generally categorized into two types: grant-based RA \cite{ACB,seperation,dynamicone} and grant-free RA \cite{CRAN,2015ICC,ICASSP,AMPmassive,SBLAMP,LSAMPSBL,MPBSBL2019,MPBSBL}.
\subsection{Grant-Based Random Access}
In grant-based RA schemes, activated users contend for the RBs by transmitting a preamble sequence to the base station (BS), while a RB is assigned to the activated user, whose preamble sequence is received and accepted by the BS. One problem with the grant-based RA schemes is that the RB is wasted when more than one active device transmit the same preamble sequence. Some solutions were proposed to alleviate the RA congestion by reducing the collision probability, such as the Access Class Barring (ACB) scheme \cite{ACB}, delicate splitting of the RA preamble set \cite{seperation}, and automatic configuration of the RA parameters \cite{dynamicone}. However, the RB wastage cannot be fully avoided by grant-based RA schemes, which results in low resource efficiency. Furthermore, a handshaking process is always to recognize the contention winner, which undermines the uplink transmission efficiency of small data packets.
\subsection{Grant-Free Random Access}
\subsubsection{Compressed sensing-based grant-free RA schemes}
Compressed sensing (CS) algorithms employ pilot sequences to accomplish the user activity detection (UAD) and/or channel estimation (CE) problem. For example, the joint UAD and CE problem was addressed by a modified Bayesian compressed sensing algorithm \cite{CRAN} for the cloud radio access network (C-RAN). In addition, the powerful approximate message passing (AMP) algorithm was employed for the joint UAD and CE problem when the BS is equipped either with a single antenna \cite{2015ICC, ICASSP} or with multiple antennas \cite{AMPmassive}.
\subsubsection{Sparse Bayesian learning-based grant-free RA schemes}
The sparse Bayesian learning (SBL) algorithm considers the prior hyper-parameter of the sparse signal. The Expectation Maximization (EM) method was employed by the AMP-SBL algorithm \cite{SBLAMP} to update the sparse signal and the hyper-parameter. A least square (LS)-based AMP-SBL (LS-AMP-SBL) algorithm \cite{LSAMPSBL} was proposed to recover the sparse signal in three steps. Recently, a message-passing receiver design was proposed for the joint channel estimation and data decoding in uplink grant-free SCMA systems \cite{MPBSBL2019}. In addition, a message passing-based block sparse Bayesian learning (MP-BSBL) algorithm \cite{MPBSBL} was proposed for a grant-free NOMA system.
\begin{figure*}
\centering
\includegraphics[width=0.75\linewidth]{systemmodel.pdf}\vspace{-0.3cm}
\caption{LDS-OFDM based grant-free NORA system model. The joint UAD and CE problem is sloved by the proposed DNN-MP-BSBL algorithm, which processes the first $L_t$ received signals of the pilot sequences, while the MUD module is employed to process the remaining $L_c$ received signals to detect the data transmitted from active users.}\vspace{-0.3cm}
\label{SystemModel}
\end{figure*}
\subsection{Contributions}
In this paper, we consider a LDS-OFDM system, where devices perform grant-free RA once they are activated. A deep neural network-aided message passing-based block sparse Bayesian learning (DNN-MP-BSBL) algorithm is proposed to perform joint UAD and CE. The iterative message passing process of the MP-BSBL algorithm \cite{MPBSBL} is transferred from a factor graph to a neural network. Weights are imposed on the messages passing in the neural network and trained to minimize the estimation error.
The rest of this paper is organized as follows. The system model and the MP-BSBL algorithm are presented in Section \ref{ModelProblem}. The DNN structure for the DNN-MP-BSBL algorithm is illustrated in Section \ref{DNN_SBSL}, where the weighted message passing is explained in details. Simulation results are given in Section \ref{simulationSec} to verify the UAD and CE accuracy of the proposed DNN-MP-BSBL algorithm. Finally, Section \ref{conclusions} concludes this paper.
\section{Joint UAD and CE by MP-BSBL}\label{ModelProblem}
\subsection{System Model and Problem Formulation}
A LDS-OFDM system is considered in Fig. \ref{SystemModel}. There are $N$ sub-carriers and $K$ users. Each user is activated with probability $P_a$. For active user $k$, its information sequence $\mathbf{b}_k$ is encoded and mapped into a QAM sequence $\mathbf{c}_k\in\mathbb{C}^{L_c\times1}$ with length $L_c$. ZC sequences are adopted as pilot sequences. One unique sequence $\mathbf{p}_k$ with length $L_t$ is allocated for user $k$, and inserted into the transmitted sequence $\mathbf{x}_k$, i.e., $\mathbf{x}_k=[\mathbf{p}_k^T \ \mathbf{c}_k^T]^T$. Therefore, the length $L$ of $\mathbf{x}_k$ is $L=L_t+L_c$. The LDS spreader for user $k$ is characterized by $\mathbf{s}_k$, which is a sparse vector with length $N$ and $d_c$ non-zero elements. The LDS spreaders for all the $K$ users are characterized by a LDS spreading matrix $\mathbf{S}=[\mathbf{s}_1,...,\mathbf{s}_K]$. We consider a regular $\mathbf{S}$, i.e., the column degree $d_c$ and the row degree $d_r$ in $\mathbf{S}$ are constant. Each sub-carrier is shared by $d_r$ potential users, with $d_{r}=(K/N)d_{c}$. When multiple users are active on the same sub-carrier, the RA is conducted in a NOMA manner. After the OFDM de-modulator, the received matrix is
\begin{equation}\label{received}
\mathbf{Y}_{L\times N}=\mathbf{X}_{L\times K}\mathbf{H}_{K\times N}+\mathbf{W}_{N\times L},
\end{equation}
where the $(l,n)$-th entry of $\mathbf{Y}$ represents the $l$-th received symbol on the $n$-th sub-carrier, the $(l,k)$-th entry of $\mathbf{X}$ represents the $l$-th transmitted symbol of the $k$-th user, and $\mathbf{H}=[\alpha_1\mathbf{h}_1,...,\alpha_K\mathbf{h}_K]^T$ is a $K\times N$ row sparse channel matrix, which integrates the activity of the users. The activity indicator $\alpha_k=0$ if user $k$ is inactive. Otherwise, $\alpha_k=1$ and the $k$-th row $\mathbf{h}^T_k$ of $\mathbf{H}_{K\times N}$ represents the channel gain vector on $N$ sub-carriers for user $k$. The entries in the AWGN matrix $\mathbf{W}$ are assumed i.i.d with noise variance $\sigma_{w}^2$. $\mathbf{Y}$ can be decomposed as $\mathbf{Y}_{L\times N}=\left[(\mathbf{Y}^P_{L_t\times N})^T\ (\mathbf{Y}^c_{L_c\times N})^T \right]^T$, where $\mathbf{Y}^P$ and $\mathbf{Y}^c$ represent the received matrices w.r.t. the pilot sequences and the data sequences, respectively. We consider $\mathbf{Y}^P$ to solve the joint UAD and CE problem,
\begin{equation}\label{Pilotreceived}
\mathbf{Y}^P_{L_t\times N}=\mathbf{P}_{L_t\times K}\mathbf{H}_{K\times N}+\mathbf{W}_{N\times L_t},
\end{equation}
where $\mathbf{P}$ is assumed known to the receiver. Then we perform vectorization on the transpose of $\mathbf{Y}^P$ as in \cite{MPBSBL},
\begin{equation}\label{Vectorization}
\begin{split}
\mathbf{y}&=vec([{\mathbf{Y}^P}]^T)=(\mathbf{P}\otimes\mathbf{I}_N)vec(\mathbf{H}^T)+\mathbf{w}\\
&=\!\!\underbrace{\left[ {\begin{array}{*{20}{c}}
p_{1,1}\mathbf{I}_N&p_{1,2}\mathbf{I}_N&\cdots&p_{1,K}\mathbf{I}_N\\
p_{2,1}\mathbf{I}_N&p_{2,2}\mathbf{I}_N&\cdots&p_{2,K}\mathbf{I}_N\\
\vdots&\vdots&\ddots&\vdots\\
p_{L_t,1}\mathbf{I}_N&p_{L_t,2}\mathbf{I}_N&\cdots&p_{L_t,K}\mathbf{I}_N
\end{array}} \right]}_{\mathbf{P}_s}\!
\underbrace{\left[ {\begin{array}{*{20}{c}}
\alpha_1\mathbf{h}_1\\
\alpha_2\mathbf{h}_2\\
\vdots\\
\alpha_K\mathbf{h}_K
\end{array}} \right]}_{\mathbf{h}_s}\!\!+\mathbf{w}\\
&\overset{(a)}{=}\overline{\mathbf{P}}
\underbrace{\left[ {\begin{array}{*{20}{c}}
\alpha_1\mathbf{\overline{h}}_1\\
\alpha_2\mathbf{\overline{h}}_2\\
\vdots\\
\alpha_K\mathbf{\overline{h}}_K
\end{array}} \right]}_{\mathbf{\overline{h}}}+\mathbf{w},
\end{split}
\end{equation}
where $\mathbf{P}_s=\mathbf{P}\otimes\mathbf{I}_N$, $\mathbf{h}_s=vec(\mathbf{H}^T)$, and $\mathbf{h}_k$ is the channel gain vector of user $k$ on $N$ sub-carriers. According to the LDS spreading matrix $\mathbf{S}$, the transmitted symbols of each user are only spread onto $d_c$ sub-carriers. Therefore, $N-d_c$ elements in $\mathbf{h}_k$ are zero. We further simplify $\mathbf{h}_s$ by eliminating the zeros. Accordingly, the columns in $\mathbf{P}_s$ corresponding to the zeros in $\mathbf{h}_s$ are also removed. Finally, we obtain the simplified version of (\ref{Pilotreceived}) in equation ($a$) of (\ref{Vectorization}). According to (\ref{Vectorization}), the joint UAD and CE problem is equivalent to recovering $\overline{\mathbf{h}}$.
\subsection{MP-BSBL Algorithm \cite{MPBSBL}}
The recovery of $\mathbf{\overline{h}}$ can be addressed by the MP-BSBL algorithm \cite{MPBSBL}.
\begin{figure}
\centering
\includegraphics[width=0.87\linewidth]{factorgraph.pdf}\vspace{-0.3cm}
\caption{Factor graph for the message passing in the MP-BSBL algorithm \cite{MPBSBL}.}\vspace{-0.2cm}
\label{FactorGraph}
\end{figure}
For user $k$, the distribution of $\mathbf{\overline{h}}_k$ is assumed conditioned on a hyper-parameter ${\gamma}_k$, i.e., ${\mathbf{\overline{h}}_k}({\gamma}_k)\sim \mathcal{CN}(\mathbf{0},{\gamma}_k^{-1}\mathbf{I}_{d_c})$. The hyper-parameter ${\gamma}_k$ is assumed to follow a Gamma distribution. The noise precision $\lambda = 1/{\sigma_w^2}$ is unknown at the receiver but assumed with a priori probability $p(\lambda)$. With these assumptions above, the joint a posterior probability is factorized as follows
\begin{equation}\label{Factorize}
\begin{split}
p(\mathbf{\overline{h}},\gamma,\lambda|\mathbf{y}) &\propto p(\mathbf{y}|\mathbf{\overline{h}},\lambda)p(\mathbf{\overline{h}}|\mathbf{\gamma})p(\lambda)p(\gamma)\\
&=p(\lambda)\prod_{n=1}^{L_tN}p\left(y_n|\mathbf{\overline{h}},\lambda\right)\prod_{k=1}^{K}\prod_{d=1}^{d_c}p(\overline{h}_{k,d}|\gamma_k)p(\gamma_k),
\end{split}
\end{equation}
where $p(\lambda) \propto 1/\lambda$, $p(y_n|\mathbf{\overline{h}},\lambda)=\mathcal{CN}(y_n;\mathbf{\overline{p}^T_n}\mathbf{\overline{h}},\lambda^{-1})$, $\mathbf{\overline{p}^T_n}$ is the $n$-th row of matrix $\overline{\mathbf{P}}$ in (\ref{Vectorization}), $p(\overline{h}_{k,d}|\gamma_k)=\mathcal{CN}(\overline{h}_{k,d};0,\gamma_k^{-1})$, $p(\gamma_k)=Ga(\gamma_k;a_k,b_k)$, and $p(\gamma)=\prod_{k=1}^{d_c}p(\gamma_k)$, $\mathcal{CN}(x;\mu,\sigma^2)$ represents the complex Gaussian distribution probability density function (pdf) of $x$ with mean $\mu$ and variance $\sigma^2$, while $Ga(x;a,b)$ represents the Gamma distribution pdf of $x$ with parameters $a$ and $b$. The parameters $a$ and $b$ are usually assumed in the order of $10^{-4}$.
A factor graph is established for the MP-BSBL algorithm in Fig. \ref{FactorGraph}, where $f_{\overline{h}_{k,d}}(\overline{h}_{k,d},\gamma_k)$, $f_{\gamma_k}(\gamma_k)$ and $f_{y_n}(\mathbf{\overline{h}},\lambda)$ denote $p(\lambda)$, $p(\overline{h}_{k,d}|\gamma_k)$, $p(\gamma_k)$, and $p({y_n}|\mathbf{\overline{h}},\lambda)$. The extra variable $z_n=\mathbf{\overline{p}^T_n}\mathbf{\overline{h}}$ is introduced and the constraint $\delta(z_n-\mathbf{\overline{p}^T_n}\mathbf{\overline{h}})$ is represented by $f_{\delta_n}$. Then, $f_{y_n}$ is a function of $z_n$ and $\lambda$, i.e., $f_{y_n}(z_n,\lambda)=\mathcal{CN}(y_n;z_n,\lambda^{-1})$. The MP-BSBL algorithm performed on the factor graph in Fig. \ref{FactorGraph} is briefed as follows
Denote $l$ as the iteration index and $Q_{k,d}$ as the product of all the incoming messages from $\delta_{n'}$ to $\overline{h}_{k,d}$. Then, the variance $v_{Q_{k,d}}$ and mean $m_{Q_{k,d}}$ of $\overline{h}_{k,d}$ are,
\begin{equation}\label{vQkd}
v^{l}_{Q_{k,d}}\approx\left( \sum_{n'\in\mathcal{N}\left(\overline{h}_{k,d}\right)}\frac{\left|\overline{P}_{n',kd}\right|^2}{\left(\hat{\lambda}^{l-1}\right)^{-1}+v^{l-1}_{\delta_{n'} \to z_{n'}}} \right)^{-1}
\end{equation}
\begin{equation}\label{mQkd}
m^{l}_{Q_{k,d}}\approx v^{l}_{Q_{k,d}} \sum_{n'\in\mathcal{N}\left(\overline{h}_{k,d}\right)}\frac{\overline{P}_{n',kd}^{H}\left(y_{n'}-m^{l-1}_{\delta_{n'} \to z_{n'}}\right)}{\left(\hat{\lambda}^{l-1}\right)^{-1}+v^{l-1}_{\delta_{n'} \to z_{n'}}}+m_{\overline{h}_{k,d}}^{l-1}
\end{equation}
The variance $v^{l}_{\overline{h}_{k,d}}$ and mean $m^{l}_{\overline{h}_{k,d}}$ of $\overline{h}_{k,d}$ are updated as
\begin{equation}\label{vmhkd}
\begin{split}
v^{l}_{\overline{h}_{k,d}}&=\frac{1}{\left(v_{Q_{k,d}}^{l}\right)^{-1}+\hat{\gamma}_k^{l-1}}\\
m^{l}_{\overline{h}_{k,d}}&=\frac{m^{l}_{Q_{k,d}}}{1+v^{l}_{Q_{k,d}}\hat{\gamma}_k^{l-1}}
\end{split}
\end{equation}
The variance $v_{\delta_n \to z_n}$ and mean $m_{\delta_n \to z_n}$ from $\delta_n$ to $z_n$ are
\begin{equation}\label{delta2z}
\begin{split}
v^{l}_{\delta_n \to z_n}\! &\approx \!\sum_{ \left\{ i,j\right\} \in \mathcal{N}\left(f_{\delta_n}\right) }\left| \overline{P}_{n,ij}\right|^2v^{l}_{\overline{h}_{i,j}}\\
m^{l}_{\delta_n \to z_n}\! &\approx \!\sum_{ \left\{ i,j\right\} \in \mathcal{N}\left(f_{\delta_n}\right) }\overline{P}_{n,ij}m^{l}_{\overline{h}_{i,j}}\!-\!\frac{v^{l}_{\delta_n \to z_n}\left( y_n\!-\!m_{\delta_n \to z_n}^{l-1}\right)}{\left(\hat{\lambda}^{l-1}\right)^{-1}\!+\!v_{\delta_n \to z_n}^{l-1}}
\end{split}
\end{equation}
Then $\hat{\gamma}^{l}_k$ is updated by the MF message passing,
\begin{equation}\label{gamma}
\hat{\gamma}^{l}_k=\frac{a_k+d_c+1}{b_k+\sum\limits_{d=1}^{d_c}\left( \left|m^{l}_{\overline{h}_{k,d}}\right|^2+v^{l}_{\overline{h}_{k,d}}\right)}
\end{equation}
The variance $v^{l}_{z_n}$ and mean $m^{l}_{z_n}$ of $z_n$ are
\begin{equation}\label{zn}
\begin{split}
v^{l}_{z_n}&=\left(\hat{\lambda}^{l-1}+\left(v^{l}_{\delta_n \to z_n}\right)^{-1}\right)^{-1}\\
m^{l}_{z_n}&=v^{l}_{z_n}\left(y_n\hat{\lambda}^{l-1}+\frac{m^{l}_{\delta_n \to z_n}}{v^{l}_{\delta_n \to z_n}}\right)
\end{split}
\end{equation}
Then, $\hat{\lambda}^{l}$ is updated by MF message passing,
\begin{equation}\label{lambda}
\hat{\lambda}^{l}=\frac{L_tN}{\sum\limits_{n=1}^{L_tN}\left[ \left(m^{l}_{z_n}-y_n\right)^2+v^{l}_{z_n}\right]}
\end{equation}
If $\left({\hat{\gamma}^l_k}\right)^{-1}<\gamma_{th}$, user $k$ is detected as inactive. Otherwise, $\{m^l_{\overline{h}_{k,d}}, d=1,\ldots,d_c\}$ is the estimated channel gain.
\section{DNN-Aided MP-BSBL Algorithm}\label{DNN_SBSL}
\begin{figure*}
\centering
\includegraphics[width=0.82\linewidth]{dnnmpbsbl.pdf}\vspace{-0.35cm}
\caption{DNN for the weighted message passing in DNN-MP-BSBL algorithm with ($N=3, K=6, L_t=2, d_c=2$) and layer organization.}
\label{DNN-MP-BSBL}\vspace{-0.3cm}
\end{figure*}
\begin{table}
\renewcommand\arraystretch{1.2}
\caption{Input and Output for Each Layer of the DNN.}\vspace{-0.2cm}
\centering
\begin{tabular}{cccc}
\Xhline{1.2pt}
\!\!\!\!Index&\!\!\!\!\!\!\!\!\!\!\!\!\!\!Layer Input&\!\!Layer Output&\!\!\!\!Length\!\!\\
\hline
\!\!\!\!1&\!\!\!\!\!\!\!\!\!\!\!\!\!\!None&\!\!\!\!$\mathbf{y},\mathbf{I}^{l-1}_{\delta \!\to\! z}\!=\!\{\mathbf{v}^{l-1}_{\delta \!\to\! z},\mathbf{m}^{l-1}_{\delta \!\to\! z}\}$&\!\!\!\!$NL_t$\!\!\\
\!\!\!\!2&\!\!\!\!\!\!\!\!\!\!$\mathbf{y},\mathbf{I}^{l-1}_{\delta \!\to\! z},\hat{\lambda}^{l-1},\mathbf{P}$&\!\!\!\!$\mathbf{O}^l_{A_1}\!=\!\{\mathbf{O}^l_{A^v_1},\mathbf{O}^l_{A^m_1}\}$&\!\!\!\!$L_td_cK$\!\!\\
\!\!\!\!3&\!\!\!\!\!\!\!\!\!\!$\mathbf{m}^{l-1}_{\overline{h}},\mathbf{O}^l_{A_1}$&\!\!\!\!$\mathbf{I}^l_Q\!=\!\{\mathbf{v}^l_Q,\mathbf{m}^l_Q\}$&\!\!\!\!$d_cK$\!\!\\
\!\!\!\!4&\!\!\!\!\!\!\!\!\!\!$\mathbf{I}^l_Q,\hat{\gamma}^{l-1}$&\!\!\!\!$\mathbf{I}^l_{\overline{h}}\!=\!\{\mathbf{v}^l_{\overline{h}},\mathbf{m}^l_{\overline{h}}\}$&\!\!\!\!$d_cK$\!\!\\
\!\!\!\!5&\!\!\!\!\!\!\!\!\!\!$\mathbf{I}^l_{\overline{h}}$&\!\!\!\!$\hat{\mathbf{\gamma}}^l$&\!\!\!\!$K$\!\!\\
\!\!\!\!6&\!\!\!\!\!\!\!\!\!\!$\mathbf{P},\mathbf{I}^l_{\overline{h}}$&\!\!\!\!$\mathbf{O}^l_{A_2}\!=\!\{\mathbf{O}^l_{A^v_2},\mathbf{O}^l_{A^m_2}\}$&\!\!\!\!$L_td_cK$\!\!\\
\!\!\!\!7&\!\!\!\!\!\!\!\!\!\!$\mathbf{y},\mathbf{I}^{l-1}_{\delta \!\to\! z},\hat{\lambda}^{l-1},\mathbf{O}^l_{A_2}$&\!\!\!\!$\mathbf{I}^l_{\delta \!\to\! z}\!=\!\{\mathbf{v}^l_{\delta \!\to\! z},\mathbf{m}^l_{\delta \!\to\! z}\}$&\!\!\!\!$NL_t$\!\!\\
\!\!\!\!8&\!\!\!\!\!\!\!\!\!\!$\mathbf{y},\hat{\lambda}^{l-1},\mathbf{I}^l_{\delta \!\to\! z}$&\!\!\!\!$\mathbf{I}^l_{z}\!=\!\{\mathbf{v}^l_{z},\mathbf{m}^l_{z}\}$&\!\!\!\!$NL_t$\!\!\\
\!\!\!\!9&\!\!\!\!\!\!\!\!\!\!$\mathbf{I}^l_{z},\mathbf{y}$&\!\!\!\!$\hat{\lambda}^l$&\!\!\!\!$1$\!\!\\
\Xhline{1.2pt}
\end{tabular}
\label{LayerMeaning}
\end{table}
The factor graph in Fig. \ref{FactorGraph} is densely connected, which results in the correlation problem of the Gaussian messages \cite{GMPID,GMPCW,HCWTVT}. To address this problem, we propose a DNN-MP-BSBL algorithm to imposes weights on the Gaussian message update and the MF message update. To facilitate the training, the message passing is transferred from the factor graph to a DNN in Fig. \ref{DNN-MP-BSBL}(a). Each iteration of the MP-BSBL algorithm is now represented by one iteration block. Within each iteration block, one layer represents one particular message. Two auxiliary layers $A_1$ and $A_2$ are also added for illustration clarity. Therefore, as listed in Table \ref{LayerMeaning}, there are 9 layers in each iteration block. The layer organization is shown in Fig. \ref{DNN-MP-BSBL}(b). The weighted message passing is represented by a weighting matrix $\mathbf{W}$, whose $(i,j)$-th entry is non-zero if the $i$-th input node is connected to the $j$-th output node.
\subsection{Weighted Message Passing}
\noindent\underline{\textbf{Layer 1}}: Layer 1 is the input within one iteration block.
\noindent\underline{\textbf{Layer 2}}: Layer 2 is the auxiliary layer $A^l_1$ and the output $\mathbf{O}^l_{A_1}$ is derived as follows,
\begin{equation}\label{W_OAv}
\mathbf{O}^l_{A^v_1}=\frac{\mathbf{P}^2} {\frac{1}{\hat{\lambda}^{l-1}}\times \mathbf{W}^{l}_{\lambda\to A^v_1}+\mathbf{v}^{l-1}_{\delta \to z}\times\mathbf{W}^l_{v_\delta\to A^v_1}}
\end{equation}
\begin{equation}\label{W_OAm}
\mathbf{O}^l_{A^m_1}=\frac{\mathbf{P}^H.*\left( \mathbf{y}\times\mathbf{W}^{l}_{y\to A^m_1} \!-\!\mathbf{m}^{l-1}_{\delta\to z}\times\mathbf{W}^{l}_{{m_\delta}\to A^m_1} \right)} {\frac{1}{\hat{\lambda}^{l-1}}\times \mathbf{W}^{l}_{\lambda\to A^m_1}\!+\!\mathbf{v}^{l-1}_{\delta \to z}\times\mathbf{W}^l_{v_\delta\to A^m_1}}
\end{equation}
The fraction and $.*$ operations are performed elementwise while the $\times$ operation is the matrix multiplication.
\noindent\underline{\textbf{Layer 3}}: The output $\mathbf{I}^l_Q$ of Layer 3 is derived as follows,
\begin{equation}\label{W_Q}
\begin{split}
\mathbf{v}^l_Q&=\frac{1}{\mathbf{O}^l_{A^v_1}\times\mathbf{W}^{l}_{A^v_1\to v_Q}}\\
\mathbf{m}^l_Q&=\mathbf{v}^l_Q.*\left( \mathbf{O}^l_{A^m_1} \times \mathbf{W}^{l}_{A^m_1 \to m_Q} \right)\!+\!\mathbf{m}^{l-1}_{\overline{h}} \!\times \! \mathbf{W}^l_{\overline{h} \to Q}
\end{split}
\end{equation}
\noindent\underline{\textbf{Layer 4}}: The output $\mathbf{I}^l_{\overline{h}}$ of Layer 4 are derived as follows,
\begin{equation}\label{W_h}
\begin{split}
\mathbf{v}^l_{\overline{h}}&\!=\!\!\frac{1}{\frac{1}{\mathbf{v}_Q^l} + \left(\mathbf{\hat{\gamma}}^{l-1}\bigotimes \mathbf{1}_{d_c}\right) \times \mathbf{W}^l_{\gamma}}\\
\mathbf{m}^l_{\overline{h}}&\!=\!\!\frac{\mathbf{m}^l_Q}{\mathbf{1}_{d_c\!K}\!\!\times\!\! \mathbf{W}_{1\to\overline{h}} \!+\! \left(\mathbf{v}^l_Q .\!*(\mathbf{\hat{\gamma}}^{l-1}\bigotimes \mathbf{1}_{d_c})\right) \!\!\times\! \!\mathbf{W}^l_{v\gamma \to \overline{h}}}
\end{split}
\end{equation}
\noindent\underline{\textbf{Layer 5}}: The output $\hat{\gamma}^l=\{\hat{\gamma}^l_k, \forall k\}$ is derived as follows,
\begin{equation}\label{W_gamma}
\mathbf{\hat{\gamma}}^l=\frac{\mathbf{a}+d_c+1}{\mathbf{b}+|\mathbf{m}^l_{\overline{h}}|^2\times \mathbf{W}^l_{m_{\overline{h}}\to\gamma} + \mathbf{v}^l_{\overline{h}} \times \mathbf{W}^l_{v_{\overline{h}}\to\gamma}}
\end{equation}
\noindent\underline{\textbf{Layer 6}}: The output $\mathbf{O}^l_{A^v_2}$ and $\mathbf{O}^l_{A^m_2}$ of Layer 6 is derived as
\begin{equation}\label{OA2}
\begin{split}
\mathbf{O}^l_{A^v_2}&=\mathbf{P}^2.*\mathbf{v}^l_{\overline{h}}\\
\mathbf{O}^l_{A^m_2}&=\mathbf{P}.*\mathbf{m}^l_{\overline{h}}
\end{split}
\end{equation}
\begin{algorithm}[t!]\setstretch{1.0}
\caption{DNN-MP-BSBL algorithm}
\label{alg:MP-BSBL}
\KwIn{$\mathbf{y}$, $\mathbf{\overline{P}}$, $N_{it}$, $d_c$, weighting matrices}
\KwOut{$\hat{\mathbf{h}}_{\text{DNN}}$, the index set of active user IDX}
{\textbf{Initialize:}$\hat{\lambda}^0=10^3;\mathbf{\hat{\gamma}}^0=\mathbf{1};\mathbf{v}^0_{\delta\to z}=\mathbf{1}, \mathbf{m}^0_{\delta\to z}=\mathbf{0};\mathbf{m}^0_{\overline{h}}=\mathbf{0}$}
\For{$l=1:N_{it}$}{
1. Update $\mathbf{v}^l_Q$ and $\mathbf{m}^l_Q$ by (\ref{W_OAv}), (\ref{W_OAm}), and (\ref{W_Q}).
2. Update $\mathbf{v}^l_{\overline{h}}$ and $\mathbf{m}^l_{\overline{h}}$ by (\ref{W_h}).
3. Update $\mathbf{\hat{\gamma}}^l$ by (\ref{W_gamma}).
4. Update $\mathbf{v}^{l}_{\delta\to z}$ and $\mathbf{m}^{l}_{\delta\to z}$ by (\ref{OA2}) and (\ref{W_delta2z}).
5. Update $\mathbf{v}^l_z$ and $\mathbf{m}^l_z$ by (\ref{Wz}).
6. Update the noise precision $\hat{\lambda}^{l}$ by (\ref{W_lambda}).
}
\textbf{return:} IDX=find$\left(\{\left({\hat{\gamma}^l_k}\right)^{-1}\}>\gamma_{th}\right)$,
{\ \ \ \ $\hat{\mathbf{h}}_{\text{DNN}}=\{\mathbf{0}, k \notin \text{IDX}\} \bigcup \{m^l_{\overline{h}_{k,d}}, k \in \text{IDX}, d=1,\ldots,d_c\}$.}
\end{algorithm}
\noindent\underline{\textbf{Layer 7}}: The output $\mathbf{I}^l_{\delta\to z}$ of Layer 7 is
\begin{equation}\label{W_delta2z}
\begin{split}
\mathbf{v}^l_{\delta\to z}&\!=\!\mathbf{O}^l_{A^v_2} \times \mathbf{W}^l_{A_2^v \to v_\delta}\\
\mathbf{m}^l_{\delta\to z}&\!=\!\mathbf{O}^l_{A^m_2} \times \mathbf{W}^l_{A_2^m \to m_\delta}\\
&\!-\!\frac{\mathbf{v}_{\delta\to z}.*\left(\mathbf{y}\times \mathbf{W}^l_{y\to \delta}-\mathbf{m}^{l-1}_{\delta\to z}\times \mathbf{W}^l_{m_\delta \to m_\delta} \right)}{\left(\hat{\lambda}^{l-1}\right)^{-1}\times \mathbf{W}^l_{\lambda \to \delta}+\mathbf{v}^{l-1}_{\delta\to z}\times \mathbf{W}^{l}_{v_\delta \to m_\delta}}
\end{split}
\end{equation}
\noindent\underline{\textbf{Layer 8}}: The output $\mathbf{I}^l_{z}$ of Layer 8 is
\begin{equation}\label{Wz}
\begin{split}
\mathbf{v}^l_z&\!=\!\frac{1}{\hat{\lambda}^{l-1}\times\mathbf{W}^l_{\lambda\to z}+\frac{1}{\mathbf{v}^l_{\delta\to z}}\times\mathbf{W}_{{v_\delta}\to v_z}}\\
\mathbf{m}^l_z&\!=\!\mathbf{v}^l_z .\!*\!\left( (\mathbf{y}.*\hat{\lambda}^{l-1})\!\times\! \mathbf{W}^l_{y\lambda\to z} \!\!+\! \!\frac{\mathbf{m}^l_{\delta\to z}}{\mathbf{v}^l_{\delta\to z}}\!\times\! \mathbf{W}^l_{mv\to z}\right)
\end{split}
\end{equation}
\noindent\underline{\textbf{Layer 9}}: The output $\hat{\lambda}^l$ of Layer 9 is,
\begin{equation}\label{W_lambda}
\hat{\lambda}^l\!=\!\frac{L_tN}{\left(\mathbf{m}^l_z \!\times\! \mathbf{W}^l_{m_z\to\lambda}\!-\!\mathbf{y}\!\times\!\mathbf{W}^l_{y\to\lambda}\right)^2\!\times\! \mathbf{1}^T_{N L_t}\!+\!\mathbf{v}^l_z\!\times\! \mathbf{W}^l_{v_z\to\lambda}}
\end{equation}
\subsection{Summary of the Proposed DNN-MP-BSBL Algorithm}
Finally, the proposed DNN-MP-BSBL algorithm is summarized in Algorithm \ref{alg:MP-BSBL}. The mean square error (MSE) $\|\hat{\mathbf{h}}_\text{DNN}-\mathbf{h}\|_2^2$ is employed as the loss function for the training period, while the Normalized MSE (NMSE) $\|\hat{\mathbf{h}}_\text{DNN}-\mathbf{h}\|_2^2/\|\mathbf{h}\|_2^2$ is considered for the simulations in the testing period.
\section{Simulations}\label{simulationSec}
Parameters for the simulations are listed in Table \ref{SimulationParameters}. We consider a crowded NORA system with low-latency requirement, i.e., $N_{it}\leq20$. The NMSE performances of the LS-AMP-SBL esimator \cite{LSAMPSBL}, the BOMP estimator (with known active user number) \cite{MPBSBL} and the GA-MMSE estimator (with known user activity) are also considered. The NMSE performance of GA-MMSE estimator serves as the lower bound.
\begin{table}
\renewcommand\arraystretch{1.0}
\caption{Related Parameters for Simulations}\vspace{-0.1cm}
\centering
\begin{tabular}{ccc}
\Xhline{1.2pt}
Parameter&Symbol&Value\\
\hline
User number&$K$&110\\
Subcarrier number&$N$&8\\
Pilot length&$L_t$&11\\
Spreading factor&$d_c$&4\\
Activation probability for each user&$P_a$&0.1\\
UAD threshold&$\gamma_{th}$&0.1\\
Size of training set&&$10^5$\\
Size of test set&&$10^5$\\
Size of mini-batch&&200\\
Epoch number&&20\\
Learning rate&&$10^{-3}$\vspace{-0.1cm}\\
\Xhline{1.2pt}
\end{tabular}
\label{SimulationParameters}
\end{table}
The simulation results are shown in Fig. \ref{changeSNR}, in crowded NORA systems, both the MP-BSBL algorithm and the BOMP estimator diverge from the NMSE lower bound as SNR increases, and the LS-AMP-SBL algorithm fails to work even with 50 iterations. By contrast, the DNN-MP-BSBL algorithm could closely approach the lower bound within a wide range of SNR. Therefore, the DNN-MP-BSBL algorithm requires fewer iterations and provides better NMSE performance, indicating its advantages in crowded NORA system with low-latency requirement.
\begin{figure}
\centering
\includegraphics[width=0.77\linewidth]{changesnr.pdf}\vspace{-0.35cm}\\
\caption{NMSE performance with different SNR.}\vspace{-0.35cm}\label{changeSNR}
\end{figure}
\section{Conclusions}\label{conclusions}
A DNN-MP-BSBL algorithm was proposed in this paper for the joint UAD and CE problem in grant-free NORA systems. The iterative message passing process is transferred from a factor graph to a DNN, while weights are imposed on the messages and trained to improve the UAD and CE accuracy. Simulation results showed that the NMSE performance of the DNN-MP-BSBL algorithm could approach the lower bound in a feasible number of iterations, indicating its advantages for low-latency NORA systems.
\section*{Acknowledgement}
This work was supported by the NSFC under grant 61750110529.
\bibliographystyle{IEEEtran}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 7,748 |
Home Inspiration 20 Best Women's Beachwear Clothing Brands For 2023 | Panaprium
20 Best Women's Beachwear Clothing Brands For 2023
The best beachwear labels for women make affordable, flattering, luxury, designer bathing suits, bikinis, and beach clothing. They design iconic pieces focusing on wearability and sustainability to help you rock great styles with a clean conscience.
Many beachwear clothing brands offer sustainable, minimalist, and timeless swimwear for the contemporary woman. They craft fabulous beach aesthetic bikini sets, swimsuits, and kimonos from environmentally-friendly fabrics.
To get ready for your next sunny trip to the beach, check out our selection of the best women's beachwear clothing brands to discover and shop this season.
1. JADE Swim
Category: Swimwear, beachwear, bikinis, one-pieces
For: Women
From: NYC, New York, United States
Values: Recycled, luxury, minimalist, made in the USA
Prices: $70-$160
JADE Swim is a sustainable, black-owned beachwear brand founded by former fashion editor and stylist Brittany Kozerski. It aims to be different, innovative, unique, meaningful, luxurious, yet simple with a classic edge.
JADE Swim designs its bikinis and one-pieces with multifunctional versatility in mind. Its styles can be worn from day to night, from beach to city, as swimwear or ready-to-wear.
The luxury beachwear label stands for a minimalist aesthetic with innovative silhouettes and unique design details. It makes its swimsuits from regenerated nylon in Los Angeles, USA.
SHOP JADE SWIM
2. Casa Raki
Category: Basics, beachwear, swimwear, loungewear, resort wear
From: London, United Kingdom
Values: Organic, recycled, luxury
Casa Raki is a high fashion sustainable beachwear brand for travel lovers and designed in London with high-tech, organic and recycled fabrics. It creates a beautiful swimwear collection from recycled materials.
Casa Raki is a sustainable luxury beachwear brand focused on unique, responsible vacation wear for women. It only uses eco-friendly materials and processes while focusing on improving and extending garment life.
SHOP CASA RAKI
3. Hunza G
Category: Basics, beachwear, swimwear, accessories
Values: Recycled, luxury, made in the UK
Hunza G is a London-made clothing label making affordable, sustainable, designer beachwear. Creative Director Georgiana Huddart designs and produces swimsuits and ready-to-wear using stretchy and supportive fabrics.
Hunza G's signature fabric is knit and dyed locally, decreasing emissions and promoting a growing fashion production industry within the UK.
The swimwear and beachwear brand is shaping a new way of consuming fashion, reinventing how we view swimwear today. It designs each piece to be ethical, comfortable, and treasured.
SHOP HUNZA G
4. Evarae
Category: Basics, beachwear, swimwear, resort wear
Values: Organic, recycled, luxury, made in Italy
Prices: $330-$390
Evarae is a conscious luxury, sustainable swimwear and beachwear brand designed in London and made in Italy. It creates affordable, high-end, day to evening beach clothing inspired by warm weather and traveling around the globe.
Evarae is made with eco-friendly practices and the highest quality fabrics so that you don't have to choose between feeling good, style, and sustainability.
SHOP EVARAE
5. Sensi Graves Swim
Category: Basics, beachwear, swimwear, activewear, accessories
From: Hood River, Oregon, United States
Values: Recycled, made in the USA, give back
Prices: $20-$60
Sensi Graves Swim is an American-made swimwear brand that creates sustainable and stay-put beachwear from recycled fabrics in the USA. It partners with Hope Studio in Los Angeles and KAM in Salem for its production.
Sensi Graves Swim is committed to being transparent with its supply chain. It aims to become the most ethical and sustainable swimwear brand.
Sensi Graves Swim uses recycled hand tags, labels, and packaging and wraps its products in compostable bags without any plastic. It also donates a portion of its sales to environmental groups.
SHOP SENSI GRAVES SWIM
6. Peony
Category: Basics, swimwear, beachwear, resort wear, accessories
From: Burleigh Heads, Australia
Values: Organic, recycled, luxury, timeless
Peony is a luxury beachwear and resort wear brand from Australia, made with recycled and sustainable fabrics. Every peony piece is handmade in limited numbers.
Peony is committed to consciously creating beautiful swimwear from materials carefully chosen with intention and care. It believes good design is honest, innovative, durable, and sustainable.
SHOP PEONY
7. Stella McCartney
Category: Basics, denim, underwear, swimwear, beachwear, shoes, bags, accessories
For: Women, men, children
Stella McCartney is a British apparel designer selling sustainable beachwear and swimwear for men, women, and kids. She distributes her collections in 77 countries through 863 retailers.
Stella McCartney is committed to sustainability throughout its collections with a fair amount of circularity and material innovations. It aims to run a responsible, honest, and modern company in the UK.
Stella McCartney uses many sustainable materials, including organic cotton, regenerated cellulosic fabrics, and recycled plastics to create high-quality luxury clothing items.
She designs and manufactures premium sustainable apparel, footwear, and handbags through carefully selected suppliers in Europe.
SHOP STELLA MCCARTNEY
8. Girlfriend Collective
Category: Basics, swimwear, beachwear, knitwear, sportswear, loungewear, accessories
From: Seattle, Washington, United States
Values: Organic, recycled, inclusive
Girlfriend Collective makes size-inclusive, affordable, and sustainable swimwear and beachwear from recycled materials, such as recycled polyester from PET bottles and regenerated nylon.
Husband-and-wife duo Ellie and Quang Dinh launched the brand in 2016 in Seattle to create flattering, comfortable, and supportive sports clothing perfect for the gym, yoga, and leisure.
Girlfriend Collective aims to make luxurious, responsible sportswear and be as transparent as possible. It fosters a community of people who care about beautiful designs and the people making their clothes.
The sustainable label from Seattle makes eco-friendly and high-quality textiles from recycled materials in its facility in Taiwan. It offers affordable and eco-friendly bikini sets and one-pieces.
SHOP GIRLFRIEND COLLECTIVE
9. Vitamin A
Category: Basics, swimwear, beachwear, loungewear, accessories
From: Garden Grove, California, United States
Values: Organic, recycled, give back, made in the USA
Vitamin A produces luxury eco-friendly bikinis, swimsuits, and beachwear from ocean plastic. The sustainable beachwear brand is about feeling good, looking good, and doing good.
Vitamin A creates bikinis, bodysuits, and swimsuits designed to last while caring about fit and style details. Its pieces are flawless, supportive, and dependable in all the best ways.
The clothing label uses sustainable high-performance fabrics and eco-conscious textiles like organic cotton, linen, recycled cotton, and lyocell. It also gives back a portion of every sale to organizations that protect our oceans.
SHOP VITAMIN A
10. Wolven
Category: Basics, beachwear, swimwear, activewear, loungewear
For: Women, men
From: Los Angeles, California, United States
Values: Organic, recycled, give back
Wolven is a sustainable beachwear and swimwear brand that creates patterns inspired by nature to make sustainability sexy. It uses vibrant colors inspired by Indian art, celebration, and the roots of yoga.
Wolven believes that we should do whatever we can in our power to protect our home, the Earth. Beautiful fashion should not come at the expense of our planet.
Wolven makes swimsuits that help with athletic performance and confidence. It uses healthy, safe, upcycled, and recycled materials and gives a second life to plastic waste to keep plastic out of ecosystems.
Wolven offers stylish, flattering, responsible beachwear from plastic bottles and prevents them from entering our oceans. Additionally, it removes one pound of paste from the oceans for every purchase.
SHOP WOLVEN
11. Outerknown
Category: Basics, outerwear, beachwear, swimwear, loungewear, bags, accessories
Values: Organic, Fair Trade, recycled
Outerknown is a clothing brand making effortless, casual beach styles for women and men rooted in sustainability and transparency. It creates versatile beachwear essentials with recycled ocean plastic fibers.
Outerknown keeps strict sustainability rules and aims to raise the standard for sustainable design in the global fashion industry. It uses organic, upcycled, recycled, and regenerated materials to reimagine design and embrace circularity.
Accredited by the Fair Labor Association (FLA), Outerknown invests in the livelihoods of over 5K workers through Fair Trade USA. Three of its partners are Fair Trade Certified.
SHOP OUTERKNOWN
12. La Maison Simons
Category: Basics, denim, outerwear, knitwear, activewear, swimwear, loungewear, underwear, sleepwear, bags, shoes, accessories, jewelry
From: Quebec City, Canada
Values: Organic, recycled
La Maison Simons is a Canadian fashion retailer and creator of sustainable beachwear. It offers many ethical and affordable one-pieces and bikinis designed in Canada. It also makes some of its clothes locally.
The Canadian clothing brand creates swimwear that combine style and durability by adhering to high social and environmental standards. It uses sustainable fabrics made from recycled nylon.
SHOP LA MAISON SIMONS
13. prAna
Category: Basics, knitwear, outerwear, sportswear, denim, swimwear, loungewear, accessories
From: Carlsbad, California, United States
Values: Organic, recycled, Fair Trade
prAna is an outdoor brand that makes clothes for positive change to inspire new generations to thrive and stay active. Its premium lifestyle clothing includes an extensive collection of affordable, sustainable, and highly adjustable beachwear.
prAna offers ethical and cheap swimsuits for an idyllic day at the beach. It stocks a wide range of eco-friendly bikini bottoms, tops, tankinis, one-pieces, and more.
prAna uses Econyl regenerated nylon made from ocean waste in its women's swimwear. The material is infinitely recyclable and made in a closed-loop process to reduce carbon emissions and energy usage by 50% compared to virgin nylon yarn.
The outdoor label also uses sustainable materials such as organic cotton, hemp, recycled polyester, and cellulosic fibers like lyocell and modal made from renewable resources.
SHOP PRANA
14. Summersalt
Category: Basics, beachwear, swimwear, activewear, loungewear, sleepwear, knitwear
For: Women, children
From: St. Louis, Missouri, United States
Summersalt is a designer swimwear and beachwear brand making quality essentials for modern women. It offers a wide range of sustainable, affordable, and adjustable swimsuits for women and kids.
Summersalt creates luxury bathing suits designed to keep up with you before, during, and after pregnancy. Its pieces are comfortable, chic, and made of recycled materials to offer the best designer quality at the best price.
Summersalt makes flattering swimwear from a luxurious fabric that contains 78% recycled polyamide. It uses post-consumer materials like nylon waste from old fishing nets pulled from our oceans.
Summersalt is committed to numerous strategic initiatives for more diversity and inclusion. The brand is passionate about spreading love and kindness and learning how to be better every day.
SHOP SUMMERSALT
15. SELFISH swimwear
From: Montreal, Canada
Values: Recycled, local, timeless, made in Canada
SELFISH swimwear is an ethical and responsible beachwear brand that advocates body positivity. It makes each piece with care and love in its Montreal studio.
Naomie Caron is the designer and founder behind the label SELFISH swimwear. She always thought there was a better way of doing fashion and decided to make things differently.
SELFISH swimwear only uses recycled fabrics. It creates beachwear that flatters women's curves of every age and helps you dress confidently with elegant and comfortable styles.
SHOP SELFISH SWIMWEAR
16. Everlane
Category: Basics, swimwear, beachwear, denim, knitwear, outerwear, activewear, underwear, bags, shoes, accessories
From: San Francisco, California, United States
Everlane is an ethical fashion brand creating modern and beautiful essentials, at the best factories, without traditional markups. It's a great destination to buy high-quality, affordable, and sustainable beachwear.
The clothing retailer is headquartered in San Francisco, California, and sells primarily online. It uses sustainable materials and recycled fabrics such as regenerated nylon to create versatile, flattering swimsuits.
The apparel brand offers ethically made, designed-to-last beachwear and swimwear, including bikini sets and one-pieces for women. It believes in exceptional quality to make a difference.
Everlane is widely known for its radical transparency. It educates consumers on its supply chain, factories, employees, and the price breakdown of each product.
SHOP EVERLANE
17. NA-KD
Category: Basics, beachwear, swimwear, denim, outerwear, knitwear, underwear, nightwear, accessories
From: Gothenburg, Sweden
NA-KD is a top next-generation fashion marketplace, a mobile-based online shopping sanctuary for fashion addicts. It offers affordable, stylish, and sustainable beachwear for women.
NA-KD sells many different clothing labels at various price points and designs a range of flattering, stylish, versatile swimwear and beachwear under its brand.
The NA-KD brand is one of the fastest-growing on social media. To launch new collections, it regularly partners with influencers such as Andrea Hedenstedt, Anika Teller, Astrid Olsen, Camille Botten, Dilara Avci, Pamela Reif, Kristin Sundberg, and more.
SHOP NA-KD
18. Londre Bodywear
Category: Basics, swimwear, beachwear, loungewear, underwear, accessories
From: Vancouver, Canada
Values: Recycled, inclusive, give back, made in Canada
Londre Bodywear is a sustainable beachwear brand founded by Ainsley Rose and Hannah Todd. After working together on a photoshoot in Sayulita, Mexico, they launched the most flattering, high-quality swimwear with the lowest possible impact.
The beachwear label uses recycled materials, including plastic bottles from the streets and beaches of Taiwan, to offer affordable, sustainable, and fully adjustable swimwear for all sizes with added support and enough coverage.
Londre Bodywear uses business as a force for good, raises awareness, and funds women's health and environmental initiatives such as Amazon Watch and the Yellow Hammer Fund.
SHOP LONDRE BODYWEAR
19. Stay Wild Swim
Category: Basics, swimwear, accessories
Stay Wild Swim is a premium designer beachwear brand making sustainable swimwear, bikinis, bodysuits, and one-pieces from regenerated ocean plastic. It aims to flatter every woman and turn threats into threads.
Stay Wild Swim designs and makes its eco-friendly swimsuits in London from recycled materials such as Econyl and ensures that its production is as sustainable and ethical as possible.
The brand promotes conscious consumption and encourages its customers to buy less but buy better. Its sustainable beachwear is beautiful and made to last so that you don't have to shop for new pieces every season.
SHOP STAY WILD SWIM
20. August Society
Category: Basics, swimwear, activewear, bags, accessories
Values: Recycled, timeless
August Society is a sustainable beachwear brand from Singapore that creates high-quality eco-friendly swimsuits for women, men, and kids. It ethically makes all pieces in Bali from regenerated waste plastic and recycled plastic bottles.
August Society designs beautiful, comfortable, sustainable beachwear with care for functionality, quality, and the environment. It selects durable, top-quality fabrics that will last for years.
SHOP AUGUST SOCIETY
fashion luxury shopping tips sustainability travel | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 9,447 |
Maine ›
Central Maine Hoops Winter Basketball League ›
Maine Hoops presents
Central Maine Hoops Winter Basketball League Payments
Saco/Portland, Maine
There's an additional 3% + $3.00 convenience fee.
- Event Fee
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I understand any payments are non refundable unless granted permission from Director.
No Refunds unless extenuating circumstances. Tournament Credit may be given.
All event related questions should be directed to Lenny Holmes at mainehoops@gmail.com or call 2077499492. | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 4,720 |
Q: Firebase Rules for a social app I am trying to extend a social android aap, currently its rules are public, its data structure is something like this,
{
"posts" : {
"postId1" : {
"authorId" : "abcd",
},
"postId2" : {
"authorId" : "abcd",
},
"postId3" : {
"authorId2" : "wxyz",
},
"postId4" : {
"authorId2" : "wxyz",
}
}
}
I want to allow an authenticated user to create and delete his own post in "posts" node
I tried this,
{
"rules": {
".read":"auth.uid != null",
".write":false,
"questions": {
"$uid": {
".write": "$uid === auth.uid"
}
}
}}
But this does not allow a user to create a post although a user can edit or delete his pre-existing post in "posts" node, it seems that there is no write permission within the "posts" node.
But if i allow write permission for "posts" then due to cascading rules, every authenticated user can access other's data. How can I achieve my desired functionality?
A: First please see firebase-bolt for writing rules for real time database: https://github.com/firebase/bolt
This tool makes it easy to write rules.
And for your rules here is a sample which will allow only the author to update post:
{
"rules": {
".read": "auth.uid != null",
"posts": {
"$postId": {
".write": "data.val() == null && newData.child('uid').val() == auth.uid || data.val() != null && newData.val() != null && data.child('uid').val() == auth.uid || data.val() != null && newData.val() == null && newData.child('uid').val() == auth.uid"
},
".read": "auth.uid != null"
}
}
}
and the firebase bolt equivalent is below :
path / {
read() {auth.uid != null}
}
path /posts {
read() {auth.uid != null}
/{postId}{
create() { this.uid == auth.uid}
delete() { this.uid == auth.uid}
update() { prior(this.uid) == auth.uid}
}
}
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 5,688 |
Theoson-Jordan Siebatcheu Pefok, mais conhecido como Jordan Pefok (Washington, D.C., 26 de abril de 1996) é um futebolista americano que atua como atacante. Atualmente, joga pelo .
Carreira
Reims
Pefok iniciou no juvenil do Stade de Reims, tendo ingressado no clube aos 7 anos. Ele fez sua estreia na Ligue 1 em 31 de janeiro de 2015, contra o Toulouse, substituindo Alexi Peuget após 67 minutos em uma derrota por 1–0. Em 9 de agosto de 2015, Pefok marcou seu primeiro gol na Ligue 1 em apenas sua segunda partida contra o Girondins de Bordeaux. Assinou seu primeiro contrato profissional em setembro de 2015 se comprometendo a um contrato de três anos com Reims.
Young Boys
Em 13 de setembro de 2020, foi emprestado ao time suíço Young Boys. Depois de marcar 15 gols e quatro assistências em todas as competições durante o empréstimo, o Young Boys exerceu a opção de compra, a partir de 1º de julho de 2021.
Em 14 de setembro de 2021, Pefok marcou o gol da vitória no último minuto na partida de abertura da Liga dos Campeões da UEFA de 2021–22, uma vitória por 2–1 contra o Manchester United.
Nasceu em Washington, D.C. filho de pais camaroneses e cresceu na França. Como resultado, ele foi inicialmente elegível para jogar internacionalmente pela França, Camarões ou Estados Unidos. Ele é um ex-jogador da seleção sub-21 da França.
Foi convocado pela primeira vez para a Seleção Francesa de Futebol Sub-21 para dois amistosos em junho de 2017. Ele marcou contra a Albânia em sua estreia e também jogou contra os sub-20 de Camarões.
A Federação de Futebol dos Estados Unidos perguntou sobre Pefok, e ele foi convidado pela seleção americana para uma partida contra a França em junho de 2018, mas recusou a convocação, citando sua transferência de Reims para Rennes, enquanto deixava a porta aberta para possível inclusão na seleção dos Estados Unidos no futuro. Em 10 de março de 2021, ele anunciou que se comprometeu a jogar pelos Estados Unidos.
Fez sua estreia pelos Estados Unidos em 25 de março de 2021, em uma vitória de 4–1 sobre a Jamaica. Em 3 de junho de 2021, marcou o gol da vitória aos 89 minutos em uma vitória por 1–0 sobre Honduras na semifinal da Liga das Nações da CONCACAF de 2019–20, após entrar no jogo como substituto.
Nome
Jordan usa Pefok, sobrenome de sua mãe, nas costas de sua camisa dos Estados Unidos e do Young Boys. Desde agosto de 2021, a Seleção de Futebol dos Estados Unidos tem se referido a ele como Jordan Pefok em comunicações públicas.
Títulos
Reims
Ligue 2: 2017–18
Rennes
Copa da França de Futebol: 2018–19
Young Boys
Campeonato Suíço de Futebol: 2020–21
Estados Unidos
Liga das Nações da CONCACAF: 2019–20
Ligações externas
Perfil em ESPN.com.br
Nascidos em 1996
Pessoas vivas
Futebolistas da França
Futebolistas dos Estados Unidos
Franceses de ascendência camaronesa
Estado-unidenses expatriados na Suíça
Estado-unidenses expatriados na França
Futebolistas do Stade Rennais Football Club
Futebolistas de La Berrichonne de Châteauroux
Futebolistas do Stade de Reims
Futebolistas do BSC Young Boys
Futebolistas do 1. FC Union Berlin
Jogadores da Seleção Estadunidense de Futebol | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 96 |
<?php
namespace App\Bot\Facebook;
/**
* Class Bootstrap
* @package App\Bot\Facebook
*/
class Bootstrap
{
/**
* @var \Botomatic\Engine\Core\Entities\Session
*/
protected $session;
/**
* Bootstrap constructor.
*
* @param \Botomatic\Engine\Core\Entities\Session $session
*/
public function __construct(\Botomatic\Engine\Core\Entities\Session $session)
{
$this->session = $session;
$this->resolveLocalisation();
}
/**
* Localisation driver is set in .env file, the default locale is the user's Facebook locale, to overwrite
* this update code below
*/
protected function resolveLocalisation()
{
\Botomatic\Engine\Facebook\Localization\Localizator\Factory::build($this->session->getUser()->getLocale());
}
} | {
"redpajama_set_name": "RedPajamaGithub"
} | 6,425 |
Making it possible
In my own words
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\section{Introduction}
In an infinite-horizon, investment-consumption problem the goal is to maximise the discounted expected utility of consumption, where maximisation takes place over investment strategies in a financial market and nonnegative consumption streams (constrained such that the resulting wealth process is nonnegative for all time). Merton~\cite{merton1969lifetime} solved this problem for a constant parameter financial market and constant relative risk aversion (CRRA) utility. In this article we build on the prior work of the authors~\cite{herdegen2021infinite} and analyse the investment-consumption problem for
Epstein--Zin stochastic differential utility (EZ-SDU). EZ-SDU is a generalisation of time-additive CRRA utility that allows a disentanglement of the agent's risk aversion parameter from their temporal variance aversion preferences; see, for example, Duffie and Epstein \cite{duffie1992stochastic},
Schroder and Skiadas \cite{schroder1999optimal} and Herdegen, Hobson and Jerome \cite{herdegen2021infinite}.
For SDU the utility process is given in implicit form as the solution of a backward stochastic differential equation (BSDE), and before we can attempt to optimise over consumption streams, we must first decide how to associate a utility process to a given consumption stream.
The issue is complicated by the fact that since we work over the infinite horizon there is no terminal condition for the BSDE. For some parameter combinations Herdegen et al.~\cite{herdegen2021infinite} show that there exists a unique utility process for every consumption stream (perhaps taking values in $\overline{\mathds{R}}$). However, in the parameter combinations studied in this paper, uniqueness fails. Thus, to make progress, we must first decide which utility process to associate to a given consumption stream; only then can we attempt to optimise over investment-consumption pairs.
The goals of the paper are as follows: first, to illustrate non-uniqueness using constant proportional strategies as an exemplar, showing that non-uniqueness is often the norm, and not an aberration; second, to introduce the economically motivated concept of a {\em proper} solution and argue that it gives the `true' utility process to associate to a consumption stream; third, to prove that for a wide class of consumption streams there exists a (unique) proper utility process; and fourth, to show that if we restrict attention to consumption streams for which there exists a unique proper utility process, we can prove a verification argument for the investment-consumption problem and the natural candidate optimal strategy is optimal.
EZ-SDU is parametrised by two coefficients $R$ and $S$ taking values in $(0,1)\cup(1,\infty)$, corresponding to the agent's risk aversion and their elasticity of intertemporal complementarity (the reciprocal of which is the better known as elasticity of intertemporal substitution). The parameter $\theta\coloneqq\frac{1-R}{1-S}$ is critical. First, it is argued in \cite{herdegen2021infinite} that $\theta>0$ is necessary for the Epstein--Zin SDU equation to have a meaningful solution over the infinite horizon. Second, the mathematics of the problem is vastly different depending on whether $\theta \in (0,1)$ or $\theta \in (1,\infty)$. (The boundary case of $\theta = 1$, or equivalently $R=S$, is CRRA utility.)
The parameter combinations leading to $\theta\in(0,1]$ are studied in \cite{herdegen2021infinite} over the infinite horizon and by Kraft, Seifried and Steffensen \cite{kraft2013consumption}, Seiferling and Seifried \cite{seiferling2016epstein}, Kraft and Seifried \cite{kraft2017optimal} and Matoussi and Xing \cite{matoussi2018convex} over the finite time horizon. In all of these papers, the case $\theta>1$ is avoided due to mathematical difficulties. One (highly desirable) property of EZ-SDU when $\theta\in(0,1]$ is that all evaluable consumption streams are uniquely evaluable -- if there exists a solution to the EZ-SDU equation, then it is necessarily unique. This follows from a comparison theorem (see, for example, \cite[Theorem 9.8]{herdegen2021infinite}) which shows that a \textit{subsolution} to the EZ-SDU equation always lies below a \textit{supersolution}; since a solution is both a subsolution and a supersolution, applying the comparison theorem twice to two such solutions yields uniqueness.
This paper deals with the case $\theta>1$. In this case, the requirements of the comparison theorem in \cite{herdegen2021infinite} are not met.
It is tempting to hope that this is a technical issue and that by being smarter it is possible to extend the comparison theorem to the case $\theta>1$, thus resolving issues of uniqueness. However, this is not the case -- the problem with $\theta>1$ is fundamentally different to the problem with $\theta <1$. When $\theta>1$, it is not just that the comparison theorem fails but rather that non-uniqueness is endemic to the problem. (Indeed the only right-continuous consumption stream with a unique utility process is the zero process.) Note that the same issue of non-uniqueness arises in finite time horizon EZ-SDU as well, unless a nonzero bequest function is added at the terminal time. The main goals of this paper are to illuminate \textit{how} non-uniqueness occurs and to provide an economically motivated criterion for selecting the proper solution, before finally solving the investment-consumption problem.
We begin by studying the utility process associated to constant proportional investment-consumption strategies. In this case, an explicit, time-homogeneous utility process is provided in \cite[Proposition 5.4]{herdegen2021infinite} which is valid in the case $\theta>1$.
However, we show in Section~\ref{sec:constant proportional} that for a constant proportional strategy this solution is not the only solution and there exists an infinite family of (equally explicit, but time-inhomogeneous) utility processes.
It is clear that to be able to formulate Merton's optimal investment-consumption problem for EZ-SDU, there must be a rule which assigns a particular utility process to each consumption stream over which we maximise. Various candidates for this assignation rule are plausible. Perhaps the most obvious choice is the \textit{maximal} utility process. The rationale behind this would be that the agent gets to choose which utility process they associate to a given consumption stream and so they naturally choose the best one. However, when $R>1$ the maximal utility process associated to any consumption stream is the zero process, rendering the problem degenerate. An alternative choice might be the ``game theoretic'' or minimax version of the Merton problem, where the agent maximises over the worst utility process associated to each consumption stream. However, when $R<1$ the minimal utility process associated to any consumption stream is the zero process, again rendering the problem degenerate.
Instead, one of the key contributions of the paper is to introduce the notion of a {\em proper} solution.
It will follow from the discussion of constant proportional investment-consumption strategies in Section~\ref{ssec:family of utility processes} below that if $C$ is the consumption stream which arises from a constant proportional strategy (and if $\mathbb{E}[\int_0^\infty C_s^{1-R} ds ] < \infty$), then for each $T \in [0,\infty]$ there exists a utility process associated to $C$ which is nonzero for $t<T$ and zero for $t \geq T$. (In particular, zero is a solution, and corresponds to $T=0$). Economically, this may be interpreted as saying that the amount consumed after time $T$ has no effect on the agent's utility. This may be considered to be undesirable, and motivates the definition of a proper solution:
a solution $V=(V_t)_{t\geq0}$ (of the defining equation for an EZ-SDU utility process, see \eqref{eq:Epstein--Zin SDU} below) is proper if $\mathbb{E}\big[\int_t^\infty C_s^{1-R} \,\mathrm{d} s~\big|~\mathcal{F}_t\big]>0$ implies that $(1-R)V_t>0$..
The notion of a proper solution is based on the economic idea that strictly positive bounded consumption should imply a nonzero utility. We also introduce other notions of a solution of a more mathematical nature, namely the concepts of an extremal solution and a CRRA-order solution.
The main results of the paper for the case $\theta>1$ are as follows (more precise statements follow as Theorems~\ref{thm:proper solution rc}, \ref{thm:O^1-R_+ subset UP}, \ref{thm:three solutions coincide} and \ref{thm:verification} respectively).
\begin{mainresult}\label{mr:A}
For a very wide class of consumption streams $C$, there exists a proper utility process $V$ associated to $C$.
\end{mainresult}
\begin{mainresult}\label{mr:B}
For a wide class of consumption streams, the proper utility process is unique.
\end{mainresult}
\begin{mainresult}\label{mr:C}
For a wide class of consumption streams, the three solution concepts agree.
\end{mainresult}
\begin{mainresult}\label{mr:D}
In the constant parameter financial model, if we maximise over attainable consumption streams which have an associated unique proper utility process, then the investment-consumption problem is solved by a constant proportional investment-consumption strategy {\rm(}whose parameters may be identified in terms of the parameters of the EZ-SDU and the financial market{\rm\thinspace)}.
\end{mainresult}
These results can be compared with those of Melnyk, Muhle-Karbe and Seifried~\cite{melnyk2020lifetime}, where we restrict the comparison to the case $\theta>1$ -- the subject of this paper. The main focus of \cite{melnyk2020lifetime} is to understand the impact of transaction costs on the investment-consumption problem under EZ-SDU, but the frictionless case is also covered and \cite{melnyk2020lifetime} presents some of the most complete results in the current literature. Melnyk et al. do not prove any existence results and instead choose to optimise over the abstract class of consumption streams for which a solution exists.
When $\theta>1$, they further restrict to consumption streams $C=(C_t)_{t\geq0}$ whose EZ-SDU utility process $V=(V_t)_{t\geq0}$ satisfies $(1-R)V_t \geq C_t^{1-R}$ for all $t\geq0$ as well as a transversality condition $\lim_{t\to\infty}e^{-\gamma t}\EX{|V_t|}=0$ for a discount factor $\gamma$. It is unclear exactly how large this class is, but there are many consumption streams which we show to have a unique proper solution and which do not lie in this class. In particular, the bound on $V$ and the transversality condition together rule out the candidate optimal strategy for some parameter combinations, forcing Melnyk et al. to impose additional restrictions on the parameter values \cite[Assumption 3.3]{melnyk2020lifetime} . Finally, their approach only works when $R>1$. A more thorough comparison of our results with those in \cite{melnyk2020lifetime} is provided in Section~\ref{ssec:comparison to MMKS}.
The main results of this paper may also be compared with the prior results of the authors for the case $\theta<1$. In that setting, in \cite{herdegen2021infinite} the authors show
\begin{mainresultHHJ}[Herdegen et al., Theorem 10.8] For every consumption stream $C$ there exists a unique utility process $V$ associated to $C$, if we allow the utility process to take values in $[-\infty,\infty]$.
\end{mainresultHHJ}
\begin{mainresultHHJ}[Herdegen et al., Theorem 11.1]
In the constant parameter financial model, if we maximise over attainable consumption streams, then the investment-consumption problem is solved by a constant proportional investment-consumption strategy {\rm(}whose parameters may be identified in terms of the parameters of the EZ-SDU and the financial market{\rm\thinspace)}.
\end{mainresultHHJ}
In particular, the results of this paper are less complete than those of \cite{herdegen2021infinite}: we do not show that there exists a utility process for {\em every} consumption stream, but only a wide class; nor when we consider the investment-consumption problem can we maximise over {\em every} consumption stream, again we can only optimise over a wide class.\footnote{Note however, that even in the context of the classical Merton problem for additive utility, many authors restrict attention to a subclass of consumption streams with nice integrability properties, see the paper by Herdegen, Hobson and Jerome~\cite{herdegen2020elementary}
for a discussion. In this sense the results of \cite{herdegen2021infinite} are unusual in their completeness.} But this lack of completeness must be set against the additional complexity of the problem we consider here -- the non-uniqueness of the utility process is an unavoidable and major issue.
The remainder of the paper is structured as follows. In Section~\ref{sec:constant proportional}, we introduce EZ-SDU and Merton's investment-consumption problem under EZ-SDU. We show that we may associate to each constant proportional investment-consumption strategy a family of EZ-SDU utility processes indexed by $T\in[0,\infty]$, each of which corresponds to ignoring consumption from time $T$ onwards. In Section~\ref{sec:three solutions}, we introduce the notion of a \textit{proper solution}, an \textit{extremal solution}, and a \textit{CRRA-order solution}. The main results are restated precisely in Section~\ref{sec:maintheorems}.
In Section~\ref{sec:comparison}, we introduce subsolutions and supersolutions to the EZ-SDU equation, which, roughly speaking, differ from solutions by replacing the equality in the EZ-SDU equation with an inequality. We then prove a comparison theorem which provides a sufficient criterion under which subsolutions are dominated from above by supersolutions.
Section~\ref{sec:O-solutions} is dedicated to proving the existence of a \textit{CRRA-order solution} via analysis of a transformation of the original problem to one where it is possible to guarantee existence via a contraction mapping and a fixed point argument.
Here, a CRRA-order solution, for a consumption stream $C$, is a utility process $V$ such that there exists constants $k,K \in (0,\infty)$ such that for all $t$,
$k \mathbb{E}\big[\int_t^\infty \frac{C_s^{1-R}}{1-R} \,\mathrm{d} s~\big|~\mathcal{F}_t\big] \leq V_t \leq K \mathbb{E}\big[\int_t^\infty \frac{C_s^{1-R}}{1-R} \,\mathrm{d} s~\big|~\mathcal{F}_t\big]$. In Section~\ref{sec:extremal solution}, we prove existence and uniqueness of the extremal solution for a class of consumption streams.
Section~\ref{sec:proper solution} considers existence and uniqueness of proper solutions. Furthermore, we show that all three solution concepts agree for consumption streams that are bounded above and below by constant multiples of constant proportional strategies.
In Section~\ref{sec:verification} we verify that the candidate solution proposed in Section~\ref{sec:constant proportional} is optimal over the class of attainable consumption streams to which we may assign a unique proper solution, thus completing the study of the infinite-horizon investment-consumption problem for EZ-SDU.
Appendix \ref{sec:proofs existence of proper} contains some very technical results on the existence of proper solutions.
\section{Epstein--Zin stochastic differential utility and the Merton investment-consumption problem}\label{sec:constant proportional}
Throughout this paper, we work on a filtered probability space $(\Omega, {\mathcal F}, (\mathcal{F}_t)_{t \geq 0}, \P)$
such that $(\mathcal{F}_t)_{t\geq0}$ is complete and continuous\footnote{This is slightly stronger than the \textit{right-continuity} assumed in the \textit{usual conditions}. However, it is a necessary assumption for the existence arguments in Appendix \ref{sec:proofs existence of proper} to go through. Whilst we do not believe that left-continuity of the filtration is a necessary assumption for existence of a proper solution to hold in general, it is a pragmatic and convenient assumption, and it does cover the main application of a Brownian filtration and a constant parameter Black-Scholes-Merton financial market.} and $\mathcal{F}_0$ is $\P$-trivial. Let $\mathscr{P}$ denote the set of progressively measurable processes, and let $\mathscr{P}_+$ and $\mathscr{P}_{++}$ be the restrictions of $\mathscr{P}$ to processes that take nonnegative and positive values, respectively. Moreover, denote by $\mathscr{S}$ the set of all semimartingales. We identify processes in $\mathscr{P}$ or $\mathscr{S}$ that agree up to indistinguishability.
\subsection{Stochastic differential utility}
To understand Epstein--Zin stochastic differential utility (EZ-SDU), it is beneficial to introduce stochastic differential utility (SDU) in its more general form. We will contrast SDU with \textit{time-additive utility}. We consider infinite-horizon (lifetime) stochastic differential utility.
Time-additive (expected) utility is characterised by a utility function $U:\mathds{R}_+\times\mathds{R}_+\to\mathds{V}$, where $\mathds{V} \subseteq \ol{\mathds{R}}$ is the domain of the utility function. The utility of a consumption stream $C=(C_t)_{t \geq 0}$ is given by $\EX{\int_0^\infty U(t,C_t)\,\mathrm{d} t}$ (provided this expectation is well-defined) and the \textit{utility process} $V=(V_t)_{t \geq 0}$ -- which measures the utility starting from a given time -- is given by $V_t=\cEX[t]{\int_t^\infty U(s,C_s)\,\mathrm{d} s}$. Under SDU, the utility function $U$ is generalised to become an \textit{aggregator} $g:\mathds{R}_+\times\mathds{R}_+\times\mathds{V}\to\mathds{V}$. The SDU process $V^C=(V^C_t)_{t\geq0}$ associated to a consumption stream $C$ and an aggregator $g$ is then the solution to
\begin{equation}\label{eq:stochastic differential utility aggregator g}
V_t = \cEX[t]{\int_t^\infty g(s,C_s,V_s)ds}, \quad t\geq0.
\end{equation}
This creates a feedback effect in which the value at time $t$ may depend in a non-linear way on the value at future times and permits the modelling of a much wider range of preferences. However, in addition to issues about whether the conditional expectation is well-defined, there are new issues concerning the existence and uniqueness of solutions to \eqref{eq:stochastic differential utility aggregator g} which are not present for additive utilities.
\begin{defn}\label{defn:integrable set I(g,C)}
An \textit{aggregator} is a function $g:[0,\infty)\times \mathds{R}_+ \times \mathds{V} \to \mathds{V}$.
For $C\in\mathscr{P}_+$, define $\mathds{I}(g,C)\coloneqq\left\{V\in\mathscr{P}:~ \mathbb{E}\int_0^\infty \left| g(s,C_s,V_s)\right| \,\mathrm{d} s < \infty\right\}$. Further, let $\mathds{U}\mathds{I}(g,C)$ be the set of elements of $\mathds{I}(g,C)$ which are uniformly integrable. Then $V \in \mathds{I}(g,C)$ is a \textit{utility process} associated to the pair $(g,C)$ if it has c\`adl\`ag paths and satisfies \eqref{eq:stochastic differential utility aggregator g} for all $t \in [0,\infty)$.
\end{defn}
\begin{rem}\label{rem:utility process UI}
It can be easily shown that all utility processes are uniformly integrable (see \cite[Remark 3.2]{herdegen2021infinite}).
\end{rem}
\subsection{Epstein--Zin stochastic differential utility}\label{ssec:EZ-SDU}
Epstein--Zin stochastic differential utility (see \cite{duffie1992stochastic,herdegen2021infinite,kraft2014stochastic,schroder1999optimal}) is parameterised by $R,S\in(0,1)\cup(1,\infty)$ and, with $\mathds{V}=(1-R)\mathds{R}_+$, has time-homogeneous aggregator $f_{EZ}:\mathds{R}_+\times\mathds{V}\to\mathds{V}$ defined as
\begin{equation}\label{eq:Epstein--Zin aggregator}
f_{EZ}(c,v) = \frac{c^{1-S}}{{1-S}}\left((1-R)v\right)^\frac{S-R}{1-R}.
\end{equation}
It is convenient to introduce the parameters $\theta=\frac{1-R}{1-S}$ and $\rho=\frac{S-R}{1-R}=\frac{\theta-1}{\theta}$ so that $f_{EZ}(c,v) = \frac{c^{1-S}}{{1-S}}\left((1-R)v\right)^\rho$. Note that when $S=R$ the aggregator reduces to the discounted CRRA utility function $U(c)=\frac{c^{1-R}}{1-R}$. This case corresponds to $\theta=1$ and $\rho=0$.
\begin{rem}
(a) In \cite{herdegen2021infinite}, the authors begin by defining the (time-inhomogeneous) Epstein--Zin aggregator $g_{EZ}:\mathds{R}_+ \times \mathds{R}_+\times\mathds{V}\to\mathds{V}$, via
\begin{equation}\label{eq:Epstein--Zin aggregator previous paper}
g_{EZ}(t,c,v) = b e^{-\delta t}\frac{c^{1-S}}{{1-S}}\left( (1-R)v\right)^\frac{S-R}{1-R}.
\end{equation}
In addition to \eqref{eq:Epstein--Zin aggregator}, this aggregator incorporates a multiplier $b$ and a discount factor $e^{-\delta t}$. However, a simple scaling argument shows that it is possible to set $b=1$ without loss of generality and the authors show in \cite[Section~5.2]{herdegen2021infinite} that the discount factor can be absorbed into the financial market by a change of num\'eraire. Hence, we may also assume $\delta=0$ without loss of generality yielding the aggregator in \eqref{eq:Epstein--Zin aggregator}.
(b) One of the advantages of the aggregator $f_{EZ}$ (or $g_{EZ}$) is that it is one-signed and hence the conditional expectation $\cEX[t]{\int_t^\infty g(s,C_s,V_s)ds}$ is always well defined in $[-\infty,0]$ or $[0,\infty]$. Other authors (e.g. \cite[Appendix C]{duffie1992stochastic} or \cite{melnyk2020lifetime}) have used a slightly different aggregator $g_{EZ}^\Delta$, but in general existence of a utility process for $g^\Delta_{EZ}$ implies existence of a utility process for $f_{EZ}$ and not vice versa. A fuller discussion of this issue can be found in \cite[Section~5.2]{herdegen2021infinite}.
\end{rem}
\begin{defn}\label{def:SDU}
A process $V^C=V=(V_t)_{t\geq0}$ is an EZ-SDU utility process associated to consumption $C$ if $\int_0^\infty\frac{C_s^{1-S}}{{1-S}}\left( (1-R)V_s\right)^\rho \,\mathrm{d} s \in L^1$ and if, for each $t \geq 0$ it solves
\begin{equation}\label{eq:Epstein--Zin SDU}
V_t = \mathbb{E}\left[\left.\int_t^\infty\frac{C_s^{1-S}}{{1-S}}\left( (1-R)V_s\right)^\rho \,\mathrm{d} s\right| \mathcal{F}_t \right].
\end{equation}
\end{defn}
Implicit in the form of $f_{EZ}$ is the fact that in order to define $((1-R)V)^\rho$ we must have $\sgn(V) = \sgn(1-R)$ and hence that $\mathds{V} \subseteq (1-R)\overline{\mathds{R}}_+$. Then, for there to be equality in \eqref{eq:Epstein--Zin SDU} we must have at least that the signs of the left-hand and right-hand sides of \eqref{eq:Epstein--Zin SDU} agree, i.e. $\sgn(V) = \sgn(1-S)$. This forces $\sgn(1-R)= \sgn(1-S)$ or equivalently $\theta>0$. Some authors have considered the case $\theta<0$, but as argued in \cite[Section~7.3]{herdegen2021infinite}, the solutions they find are akin to bubbles, and are not economically justified. For the rest of the paper we will assume $\theta>0$.
The case $0<\theta<1$ was considered in \cite{herdegen2021infinite} and when $\theta=1$, EZ-SDU reduces to the widely-studied CRRA utility.
In this paper, we consider parameter combinations leading to $\theta>1$; either we have $R<S<1$ or $1<S<R$.
\begin{sass}
The parameters $R$ and $S$ are such that $\theta\coloneqq\frac{1-R}{1-S}>1$.
\end{sass}
\subsection{The financial market and attainable consumption streams}\label{ssec:financial market}
In this paper, we shall consider a frictionless Black--Scholes--Merton financial market. The market consists of a risk free asset with interest rate $r\in\mathds{R}$, whose price process $S^0=(S^0_t)_{t\geq0}$ is given by $S^0_t = S^0_0 e^{r t}$, and a risky asset given by geometric Brownian motion with drift $\mu\in\mathds{R}$, volatility $\sigma>0$ and initial value $S_0 = s_0>0$. Explicitly, the price process $S=(S_t)_{t \geq 0}$ of the risky asset is given by $S_t = s_0\exp(\sigma B_t + (\mu-\frac{1}{2}\sigma^2))$, for a Brownian motion $B=(B_t)_{t\geq0}$.
At each time $t\geq0$, the agent chooses to consume at a rate $C_t\in\mathds{R}_+$ per unit time, and to invest a proportion $\Pi_t\in\mathds{R}$ of their wealth into the risky asset. The proportion that they invest in the risk free asset $S^0$ at time $t$ is then given by $1-\Pi_t$. It follows that the wealth process of the agent $X=(X_t)_{t\geq0}$ satisfies the SDE
\begin{equation}\label{eqn:wealth process}
\,\mathrm{d} X_t = X_t \Pi_t \sigma \,\mathrm{d} B_t + \left( X_t (r + \Pi_t (\mu - r)) - C_t\right)\mathrm{d} t,
\end{equation}
subject to the initial condition $X_0=x$, where $x>0$ is the agent's initial wealth.
\begin{defn}
Given $x>0$ an \emph{admissible investment-consumption strategy} is a pair $(\Pi,C)= (\Pi_t,C_t)_{t \geq 0}$ of progressively measurable processes, where $\Pi$ is real-valued and $C$ is nonnegative, such that the SDE \eqref{eqn:wealth process} has a unique strong solution $X^{x, \Pi, C}$ that is $\as{\P}$ nonnegative. We denote the set of admissible investment-consumption strategies for $x > 0$ by $\mathscr{A}(x)$.
\end{defn}
Since the value associated to a strategy only depends upon consumption, and not upon the amount invested in each of the assets, we introduce the following definition:
\begin{defn}
A consumption stream $C \in\mathscr{P}_+$ is called \emph{attainable} for initial wealth $x > 0$ if there exists a progressively measurable process $\Pi = (\Pi_t)_{t\geq0}$ such that $(\Pi, C)$ is an admissible investment-consumption strategy. Denote the set of attainable consumption streams for $x > 0$ by $\mathscr{C}(x)$.
\end{defn}
\subsection{The Merton investment-consumption problem for Epstein--Zin stochastic differential utility}
The Merton investment-consumption problem combines elements from Sections \ref{ssec:EZ-SDU} and \ref{ssec:financial market} to consider the problem facing an agent with preferences governed by EZ-SDU who may choose an admissible investment-consumption strategy so as to maximise their subjective utility. Therefore, the goal of the Merton investment-consumption problem is to find
\begin{equation}\label{eq:Control problem first formulation}
V^*(x)=\sup_{C\in\mathscr{C}(x)}V^C_0,
\end{equation}
where $V^C=V=(V_t)_{t\geq0}$ solves \eqref{eq:Epstein--Zin SDU}.
However, \eqref{eq:Epstein--Zin SDU} may not possess a unique solution, so the problem in \eqref{eq:Control problem first formulation} is not well formulated until we have decided which utility process to assign to each and every consumption we want to optimise over. Section~\ref{ssec:family of utility processes} illustrates what can go wrong by providing a family of utility processes associated to a simple constant proportional investment-consumption strategy. We will then explain in Section~\ref{sec:three solutions} how to select the economically meaningful utility process from the many available and how this impacts the control problem \eqref{eq:Control problem first formulation}. We reformulate the control problem in Section~\ref{sec:verification}.
\subsection{An explicit utility process associated to constant proportional strategies}
From the scaling properties of the problem it is to be expected that the optimal strategy is to invest a constant proportion of wealth in the risky asset, and to consume a constant proportion of wealth. Consider, therefore, the investment-consumption strategy $\Pi \equiv \pi\in\mathds{R}$ and $C \equiv \xi X$ for $\xi\in\mathds{R}_+$. Fixing a proportional strategy determined by $(\pi,\xi)$, and using It\^o's lemma and the dynamics of $X$ given in \eqref{eqn:wealth process}, we find
\begin{equation}
X_t^{1-R} = \ x^{1-R} \exp\left(\pi \sigma (1-R) B_t + (1-R)\left({r} + (\mu-r)\pi - \xi - \frac{\pi^2\sigma^2}{2}\right)t\right).
\end{equation}
In particular, $\cEX[t]{X_s^{1-R}} = X_t^{1-R} e^{-H(\pi,\xi)(s-t)}$, wher
\begin{equation}
\label{eq:H(pi,xi)>0}
H(\pi, \xi) =(R-1)\left({r} + \lambda (\mu-r)\pi - \xi - \frac{\pi^2\sigma^2}{2}R\right).
\end{equation}
Let $\eta = \frac{S-1}{S} \left(r +\frac{\lambda^2}{2R}\right)$ and define $\hat{V}:\mathds{R}_+\to\mathds{R}$ by
\begin{equation} \label{eq:hat V candidate}
\hat{V}(x) = \eta^{-\theta S }\frac{x^{1-R}}{1-R}.
\end{equation}
Then, the following proposition holds.
\begin{prop}[{[}\cite{herdegen2021infinite},~Proposition 5.4{]}]\label{prop:derivecandidate}
Define $D= \{ (\pi, \xi)\in\mathds{R}\times\mathds{R}_{++} : H(\pi, \xi)>0 \}$. Consider constant proportional strategies with parameters $(\pi,\xi)\in D$. Suppose $\theta>0$ and $\eta > 0$. Then,
\begin{enumerate}[{\rm(}i{\rm)}]
\item For $(\pi,\xi)\in D$, one EZ-SDU utility process $V=(V_t)_{t\geq0}$ associated to the strategy $(\pi,\xi X)$ is given by
\begin{equation}\label{eq:valfungenstrat2}
V_t = h(\pi,\xi)X_t^{1-R}, \qquad \text{where}\qquad h(\pi,\xi) = \frac{\xi^{1-R}}{1-R} \left( \frac{\theta }{H(\pi,\xi)} \right)^\theta.
\end{equation}
\item The global maximum of $h(\pi,\xi)$ over the set $D$ is attained at $(\pi, \xi) = (\frac{\lambda}{\sigma R}, \eta)$ and the maximum is $\frac{\eta^{-\theta S}}{1-R}$.
\item Suppose we only consider constant proportional strategies, and suppose that to each such strategy we associate the utility process given in \eqref{eq:valfungenstrat2}. Then the optimal strategy is $(\hat{\pi},\hat{\xi}) = (\frac{\lambda}{\sigma R}, \eta)$ and satisfies $V^{\hat{\pi},\hat{\xi}X}_0 = \eta^{-\theta S }\frac{x^{1-R}}{1-R}= \hat{V}(x)$, where $x$ denotes initial wealth.
\end{enumerate}
\end{prop}
In particular, Proposition \ref{prop:derivecandidate}(i) gives us one explicit utility process associated to the constant strategy $(\pi,\xi X)$ . However, the next section shows that it is far from being the unique utility process. Proposition \ref{prop:derivecandidate}(iii) gives us a candidate optimal strategy.
\begin{rem}\label{rem:constantproportionalsatisfiesselforder}
For future reference note that, provided $H(\pi,\xi)>0$, if $C$ is the constant proportional consumption stream associated with parameters $(\pi,\xi)$, then $C^{1-R}$ is a geometric Brownian motion and $\cEX[t]{\int_t^\infty C_s^{1-R} ds} = \frac{1}{H(\pi,\xi)}C_t^{1-R}$, for all $t\geq0$.
\end{rem}
\subsection{A family of utility processes indexed by absorption time}\label{ssec:family of utility processes}
In this section we show that when $\theta>1$ for each proportional consumption stream $(\pi,\xi X)$, for $\pi\in\mathds{R}$, $\xi\in\mathds{R}_{++}$ such that $H(\pi,\xi)>0$, there exists a family of utility processes, parametrised by the first time they hit zero (and are absorbed).
We postulate a time-dependent form of the utility process $V=(V_t)_{t \geq 0}$ given by $V_t = \frac{A(t)\xi^{1-R}}{1-R}X_t^{1-R}$ for a nonnegative process $A=(A(t))_{t\geq0}$.
Finding a solution associated to the constant proportional investment-consumption strategy with parameters $(\pi,\xi)$ then becomes that of solving
\begin{equation}\label{eq:proportional pi xi integral eq absorption time T}
\frac{A(t)\xi^{1-R}}{1-R}X_t^{1-R} = V_t
= \cEX[t]{\int_t^\infty \frac{\xi^{1-S}}{1-S} X_s^{1-S} \left(A(s) \xi^{1-R}X_s^{1-R}\right)^\rho \,\mathrm{d} s }
\end{equation}
Using the expression for $\mathbb{E}[X_s^{1-R}|\mathcal{F}_t]$, we find that $A=(A(t))_{t\geq0}$ solves the integral equation
$ A(t)e^{-H(\pi,\xi)t} = \theta\int_t^\infty e^{-H(\pi,\xi)s} A(s)^\rho \,\mathrm{d} s $
and taking derivatives with respect to $t$ shows that $A$ solves the
the ODE
\begin{equation}\label{eq:ODE}
A'(t) = H(\pi,\xi)A(t) - \theta A(t)^\rho.
\end{equation}
Note that one solution is the constant solution $A(t)= \big(\frac{\theta}{H(\pi,\xi)}\big)^\theta$. More generally,
the ODE~\eqref{eq:ODE} is separable and can be solved to give
\begin{equation}\label{eq:Adef}
A(t) = \left( \frac{\theta - \left(\theta - H(\pi,\xi)A(0)^{1/\theta}\right) e^{\frac{H(\pi,\xi)}{\theta}t}}{H(\pi,\xi)} \right)^\theta
\end{equation}
If we assume that $A(0)<\big(\frac{\theta}{H(\pi,\xi)}\big)^\theta$, then $A$ hits zero at $t = T\coloneqq\frac{\theta}{H(\pi,\xi)} \ln \big( \frac{\theta}{\theta - H(\pi,\xi) A(0)^{1/\theta}} \big)$.
Since $(A(t))_{t\geq T}\equiv0$ is a solution on $[T,\infty)$ we can define a family of solutions to \eqref{eq:ODE} and hence to \eqref{eq:Epstein--Zin SDU}, indexed by $A(0)$ such that
\[ A(t) = \left\{ \begin{array}{lll} \left( \frac{\theta - \left(\theta - H(\pi,\xi)A(0)^{1/\theta}\right) e^{\frac{H(\pi,\xi)}{\theta}t}}{H(\pi,\xi)} \right)^\theta & \; &t < T = \frac{\theta}{H(\pi,\xi)} \ln \left( \frac{\theta}{\theta - H(\pi,\xi) A(0)^{1/\theta}} \right) \\
0 & & t \geq T \end{array} \right. \]
(Note that if $A(0)> \big(\frac{\theta}{H(\pi,\xi)}\big)^{\theta}$ then $A$ diverges to $\infty$, but this is not consistent with the fact that $\mathbb{E}[V_t] \rightarrow 0$.)
Alternatively, the family of solutions can be thought of as indexed by $T$ where $T = \inf\{t\geq0:V_t=0\}$. Effectively, for the solution indexed by $T$, consumption after $T$ does not yield any utility, and the utility process is zero thereafter.
It is hard to argue, when considering the infinite time horizon, that this represents an economically meaningful reduction of the problem. Hence, intuitively the ``correct'' utility process should correspond to $T=\infty$ and be given by \eqref{eq:valfungenstrat2}.
\begin{rem}
This issue also arises when trying to evaluate finite-horizon EZ-SDU. A variant of the Merton problem for finite-horizon EZ-SDU and $\theta>1$ is considered by \cite{matoussi2018convex,schroder1999optimal,seiferling2016epstein,xing2017consumption}, among others.
In \cite{matoussi2018convex,seiferling2016epstein,xing2017consumption} the issue of uniqueness is finessed by incorporating a strictly positive bequest function $U_\epsilon(C_T) = \epsilon\frac{C_T^{1-R}}{1-R}$ with $\epsilon>0$ at the finite time horizon $T$ (and either restricting to consumption streams that are strictly positive or only considering the case $R>1$), so that the EZ-SDU equation in this case becomes
\begin{equation}\label{eq:finite horizon with bequest}
V_t = \mathbb{E}\left[\left.\int_t^T\frac{C_s^{1-S}}{{1-S}}\left( (1-R)V_s\right)^\rho \,\mathrm{d} s + U_\epsilon(C_T)\right| \mathcal{F}_t \right], \quad \text{for all }0\leq t\leq T.
\end{equation}
This is not a viable approach in the infinite-horizon case, as a bequest ``at infinity'' has no meaning. In \cite{schroder1999optimal}, the authors claim that EZ-SDU utility processes are unique by finding a solution to \eqref{eq:finite horizon with bequest}, letting $\epsilon\searrow0$, and then claiming that the limiting process -- which is an EZ-SDU utility process for \eqref{eq:finite horizon with bequest} with zero bequest -- is the unique EZ-SDU utility process. As we have seen in this section, this is not the case.
\end{rem}
The approach taken in this paper is to embrace the multiple utility processes and distinguish the \textit{proper} utility process (which we introduce in the next section) from other utility processes. The aim of the proper utility process is to rule out solutions which
ignore the utility gained from consumption from some finite time onwards.
\section{Three solution concepts and three spaces of consumption streams}\label{sec:three solutions}
\subsection{Solution concepts}\label{ssec:solutionconcepts}
The following definition of a \textit{proper} solution to the EZ-SDU equation is motivated by the arguments of the previous section.
\begin{defn}\label{def:proper}
Let $C\in\mathscr{P}_+$ and suppose that $V=(V_t)_{t\geq0}$ is a solution to \eqref{eq:Epstein--Zin SDU}. Then $V$ is called a \textit{proper solution} if $(1-R)V_t>0$ on $\{\cEX[t]{\int_t^\infty C^{1-R}_s \,\mathrm{d} s } > 0\}$ for all $t\geq 0$
up to null sets.
\end{defn}
The notion of a proper solution immediately excludes the time-inhomogeneous utility processes found in Section~\ref{ssec:family of utility processes}.
\begin{rem}
Note that $\cEX[t]{\int_t^\infty C^{1-R}_s \,\mathrm{d} s } > 0$ on $\{(1-R)V_t>0\}$ up to null sets for all $t \geq 0$. Indeed, seeking a contradiction, suppose there are $t \geq 0$ and $D\in\mathcal{F}_t$ with $\P[D] > 0$ such that $(1-R)V_t>0$ on $D$ but ${\mathbf{1}_D \cEX[t]{\int_t^\infty C_s^{1-R} \,\mathrm{d} s} =0}$ $\as{\P}$ Then, $\EX{\mathbf{1}_D\int_t^\infty C_s^{1-S} \,\mathrm{d} s} = 0$ and hence $C=0$ for $\P \otimes \mathrm{d} t$ almost every $(\omega, t)$ in $D \times [t,\infty)$. Consequently, $\mathbb{E}[\mathbf{1}_D V_t] = \EX{\mathbf{1}_D \int_t^\infty f_{EZ}(C_s,V_s) \,\mathrm{d} s} = 0$, and we arrive at a contradiction.
\end{rem}
We also define the \textit{extremal} solution. This corresponds to either the maximal solution or the minimal solution -- depending on the sign of $\mathds{V}=(1-R)\mathds{R}_+$. Such solutions frequently appear in BSDE theory; see e.g. \cite{drapeau2013minimal,drapeau2016dual,peng1999monotonic}.
\begin{defn}\label{def:extremal solution}
Let $C\in\mathscr{P}_+$ and suppose that $V=(V_t)_{t\geq0}$ is a solution to \eqref{eq:Epstein--Zin SDU}.
$V$ is an \textit{extremal} solution if $(1-R)V\geq(1-R)Y$ for any other solution $Y=(Y_t)_{t\geq0}$.
\end{defn}
\begin{rem}\label{rem:maximal solution proper}
If there exists a proper solution, then extremal solutions are proper. This follows from Definitions \ref{def:proper} and \ref{def:extremal solution}; if $Y$ is a proper solution and $V$ is an extremal solution, then for each $t \geq 0$, up to null sets,
\begin{equation}
(1-R)V_t\geq(1-R)Y_t > 0 \text{ on } \left\{\cEX[t]{\int_t^\infty C^{1-R}_s \,\mathrm{d} s } > 0 \right\}.
\end{equation}
\end{rem}
The third solution concept that will be useful is a CRRA-order solution.
\begin{defn}\label{def:O equivalence relation}
Suppose that $X=(X_t)_{t\geq0}$ and $Y=(Y_t)_{t\geq0}$ are nonnegative progressive processes. We say that $X$ has the same order as $Y$ (and write $X \stackrel{\mathds{O}}{=} Y$, noting that $\stackrel{\mathds{O}}{=}$ is an equivalence relation) if there exist constants $k,K\in(0,\infty)$ such that
\begin{equation}\label{eq:O(Y) condition}
0\leq k Y \leq X \leq K Y.
\end{equation}
Denote the set of progressive processes with the same order as $Y$ by $\mathds{O}(Y)\subseteq\mathscr{P}_+$. Further, denote by $\mathds{O}(Y; k,K)$ the set of processes such that \eqref{eq:O(Y) condition} holds for a pre-specified $k$ and $K$.
\end{defn}
For $X \in \mathscr{P}_+$, define $J^{X} = (J^{X})_{t\geq0}$ by $J^{X}_t = \cEX[t]{\int_t^{\infty} X_s ds}$.
\begin{defn}
Let $C\in\mathscr{P}_+$ and suppose that $V=(V_t)_{t\geq0}$ is a solution to \eqref{eq:Epstein--Zin SDU}. We say that $V$ is a CRRA-order solution if $(1-R)V \stackrel{\mathds{O}}{=} J^{C^{1-R}}$.
\end{defn}
\begin{rem}
If $V$ is a CRRA-order solution, then $V$ is a proper solution.
\end{rem}
\subsection{Spaces of consumption streams}
Define $\mathds{B}\mathds{C}^{\pi,\xi}$ to be the class of consumption streams that are the same order as a constant proportional strategy with parameters $(\pi,\xi)$,
\( \mathds{B}\mathds{C}^{\pi,\xi} \coloneqq \left\{C\in\mathscr{P}_{++}:~ C \stackrel{\mathds{O}}{=} \xi X^{\pi,\xi X} \right\}
\),
and set $\mathds{B}\mathds{C} = \cup_{\pi,\xi : H(\pi,\xi)>0} \mathds{B}\mathds{C}^{\pi,\xi}$. Then, $\mathds{B}\mathds{C}$ is the set of consumption streams which are the same order as a constant proportional strategy with a finite utility process. Here, $X^{\pi,\xi X}$ is an abbreviation of $X^{1,\pi,\xi X}$ and will be used in cases, where the initial wealth is not important.
By Remark~\ref{rem:constantproportionalsatisfiesselforder}, if $C=\xi X^{\pi,\xi X}$ is a constant proportional investment-consumption strategy with $H(\pi,\xi)>0$ then $C^{1-R} \stackrel{\mathds{O}}{=} J^{C^{1-R}}$.
Let $\mathds{S}\mathds{O}$ be the class of positive progressively measurable processes of \textit{self-order}, i.e. such that $X$ is of the same order as $J^X$. More precisely, define
\begin{equation}
\mathds{S}\mathds{O} \coloneqq \left\{X\in\mathscr{P}_{++}:~ \mathbb{E} \left[ \int_0^\infty X_t \,\mathrm{d} t\right] < \infty \mbox{ and } X \stackrel{\mathds{O}}{=} J^X\right\}.
\end{equation}
Furthermore, define $\mathds{S}\mathds{O}(k,K) = \left\{X\in\mathscr{P}_{++}:~ \mathbb{E} \left[ \int_0^\infty X_t \,\mathrm{d} t\right] < \infty \mbox{ and } k J^X \leq X \leq K J^X\right\}$.
On some occasions we need a slightly stronger condition. Define
$$\mathds{S}\mathds{O}_\nu \coloneqq \left\{X\in\mathds{S}\mathds{O}:~ \big(e^{\nu t} X_t\big)_{t \geq 0} \in \mathds{S}\mathds{O} \right\}\quad \text{for }\nu\geq0, \qquad \text{and} \qquad \mathds{S}\mathds{O}_+ = \cup_{\nu > 0} \mathds{S}\mathds{O}_\nu.$$
For arbitrary $\alpha\in\mathds{R}$, let $\mathds{S}\mathds{O}^\alpha$, $\mathds{S}\mathds{O}^\alpha_\nu$ and $\mathds{S}\mathds{O}^\alpha_+$ be the sets of processes $X$ such that $X^\alpha$ is in $\mathds{S}\mathds{O}$, $\mathds{S}\mathds{O}_\nu$ and $\mathds{S}\mathds{O}_+$ respectively. In particular, for a pair $(\pi,\xi)$ such that $H(\pi,\xi)>0$ we have that $C = \xi X^{x,\pi,\xi X} \in \mathds{S} \mathds{O}^{1-R}$.
\begin{rem}\label{rem:YinSO implies KYinSO}
If $Y\in\mathds{S}\mathds{O}$, then $K Y\in\mathds{S}\mathds{O}$ for any $K>0$.
\end{rem}
The following two lemmas provide some set inclusions.
\begin{lemma}\label{lem:SO^mu subset SO^nu}
For $0 \leq \mu \leq \nu$, $\mathds{S}\mathds{O}_\mu \supseteq \mathds{S}\mathds{O}_\nu$.
\end{lemma}
\begin{proof}
Suppose that $X\in\mathds{S}\mathds{O}_\nu$ and $\mu < \nu$. Then, there exists $k>0$ such that
\begin{equation}\label{eq:SO_nu decreasing k}
X_t \geq k \cEX[t]{\int_t^\infty e^{\nu(s-t)}X_s \,\mathrm{d} s}\geq k \cEX[t]{\int_t^\infty e^{\mu(s-t)} X_s\,\mathrm{d} s}.
\end{equation}
Furthermore, since $X\in\mathds{S}\mathds{O}$,
\begin{equation*}
X_t \leq K \cEX[t]{\int_t^\infty X_s\,\mathrm{d} s} \leq K \cEX[t]{\int_t^\infty e^{\mu(s-t)} X_s\,\mathrm{d} s}.\qedhere
\end{equation*}
\end{proof}
\begin{rem}\label{rem:bounds increasing in nu}
For each $\nu\geq0$, let $(k(\nu),K(\nu))$ be the tightest interval such that $(e^{\nu t}X_t)_{t\geq0}\!\in\!\mathds{S}\mathds{O}(k(\nu),K(\nu))$. Then, $k(\nu)$ is decreasing in $\nu$ by \eqref{eq:SO_nu decreasing k}. Furthermore, since $X\in\mathds{S}\mathds{O}$, $k(\nu)\leq k(0)<\infty$. By a symmetric argument, one can show that $K(\nu)$ is decreasing in $\nu$ as well.
\end{rem}
\begin{lemma}
$\mathds{B}\mathds{C} \subseteq \mathds{S}\mathds{O}^{1-R}_+ \subseteq \mathds{S}\mathds{O}^{1-R}$.
\end{lemma}
\begin{proof}
Take $C\in\mathds{B}\mathds{C}$. Then, there exists $\pi\in\mathds{R}$ and $\xi\in\mathds{R}_{++}$ and $C^\dagger=\xi X^{\pi,\xi X}$ such that ${C\stackrel{\mathds{O}}{=}C^\dagger}$. Note that $Z^{(\nu)}$ defined by $Z^{(\nu)}_t=e^{\nu t}(C^\dagger_t)^{1-R}$ is a geometric Brownian motion such that $\mathbb{E}\big[Z^{(\nu)}_t\big\vert\mathcal{F}_s\big]=Z^{(\nu)}_s e^{(\nu-H(\pi,\xi))(s-t)}$ for $s\geq t$. Let $\nu<H(\pi,\xi)$ and $Z= Z^{(\nu)}$. Then, $Z\stackrel{\mathds{O}}{=}J^{Z}$ by Remark \ref{rem:constantproportionalsatisfiesselforder}.
Define $Y$ by $Y_t=e^{\nu t}C^{1-R}$. Then, $Y\stackrel{\mathds{O}}{=}Z$ since $C\stackrel{\mathds{O}}{=}C^\dagger$. By integrating from time $T$ onwards and taking conditional expectations $J^Y\stackrel{\mathds{O}}{=}J^Z$. Since $\stackrel{\mathds{O}}{=}$ is an equivalence relation, combining these gives $Y\stackrel{\mathds{O}}{=}Z\stackrel{\mathds{O}}{=}J^Z\stackrel{\mathds{O}}{=}J^Y$ and $Y\in\mathds{S}\mathds{O}$. Thus, $C\in\mathds{S}\mathds{O}_\nu^{1-R}\subseteq\mathds{S}\mathds{O}_+^{1-R}$.
\end{proof}
\section{Main theorems}\label{sec:maintheorems}
The four theorems in this section correspond to the four main results presented in the introduction. Proofs of these and the associated propositions, are given in later sections.
The first proposition states that all consumption streams $C\in\mathscr{P}_+$ with $C^{1-R}\in\mathds{S}\mathds{O}$ have a CRRA-order solution associated to them. It is proven in \cite[Theorem 8.6]{herdegen2021infinite} and explicit bounds on $J^{C^{1-R}}((1-R)V)^{-1}$ are derived in Section \ref{sec:O-solutions}.
\begin{prop}\label{prop:O solution for C^1-R in SO}
Suppose that $C^{1-R}\in\mathds{S}\mathds{O}$. Then, there exists a unique CRRA-order solution to \eqref{eq:Epstein--Zin SDU}.
\end{prop}
The next result says that we may relax the lower bound on the consumption streams in Proposition~\ref{prop:O solution for C^1-R in SO} and still find a solution $V$. In particular, we may evaluate consumption streams such that $C^{1-R}$ is bounded above by a process in $\mathds{S}\mathds{O}$. We find an extremal solution in these cases.
\begin{prop}\label{prop:extremal solution for C^1-R < hatC^1-R}
For each $C\in\mathscr{P}_+$ such that $C^{1-R}\leq Y\in\mathds{S}\mathds{O}_+$, there exists a unique extremal solution $V^C$ to \eqref{eq:Epstein--Zin SDU}. Furthermore, $V^C$ is increasing in $C$.
\end{prop}
We now turn to \textit{proper solutions}. The following result shows that we may find a proper solution associated to a large class of consumption streams.
\begin{thm}\label{thm:proper solution rc}
Suppose that $C\in\mathscr{P}_+$ is a right-continuous consumption stream such that ${C^{1-R}\leq Y\in\mathds{S}\mathds{O}_+}$. Then, there exists a proper solution to \eqref{eq:Epstein--Zin SDU}.
\end{thm}
Proper solutions are an economically meaningful concept that allow us to choose from the many solutions to the EZ-SDU equation and Theorem \ref{thm:proper solution rc} provides a large class of consumption streams which have proper solutions. However, we have not yet discussed their uniqueness. If the property of being proper does not provide a criteria for selecting a unique solution, then it does not help to overcome the issues of non-uniqueness intrinsic to EZ-SDU when $\theta>1$. The following definition is therefore of great importance.
\begin{defn}
We say that $C\in\mathscr{P}_+$ is \textit{uniquely proper} if there exists a unique proper solution to \eqref{eq:Epstein--Zin SDU}.
Let $\mathds{U}\mathds{P}$ denote the set of uniquely proper consumption streams.
\end{defn}
\begin{thm}\label{thm:O^1-R_+ subset UP}
$\mathds{S}\mathds{O}^{1-R}_+ \subseteq\mathds{U}\mathds{P}$.
\end{thm}
For consumption streams in $\mathds{S}\mathds{O}^+$, all three solution concepts coincide.
\begin{thm}\label{thm:three solutions coincide}
Suppose that $C\in\mathds{S}\mathds{O}^{1-R}_+$. Then, the following three solutions to \eqref{eq:Epstein--Zin SDU} all coincide:
\begin{enumerate}
\item the CRRA-order solution associated to $C$ given in Proposition \ref{prop:O solution for C^1-R in SO};
\item the unique extremal solution;
\item the unique proper solution.
\end{enumerate}
\end{thm}
Finally, in Section~\ref{sec:verification}, we prove a verification theorem that shows that the candidate optimal proportional investment-consumption strategy given in Proposition \ref{prop:derivecandidate} is optimal over all attainable and uniquely proper right-continuous consumption streams.
\begin{defn}Let $\mathds{U}\mathds{P}^*$ be the restriction of $\mathds{U}\mathds{P}$ to the right-continuous processes.
\end{defn}
\begin{thm}[Verification Theorem]\label{thm:verification}
Let initial wealth be $x>0$. Suppose that $\theta> 1$ and $\eta>0$. If $V^C$ is the unique proper utility process associated to $C\in\mathds{U}\mathds{P}^*$ and $\hat{V}(x)$ is the candidate optimal utility given in \eqref{eq:hat V candidate}, then
\begin{equation}
\sup_{C \in \mathscr{C}(x)\cap\mathds{U}\mathds{P}^*} V^C_0 = V^{\hat{C}}_0 = \hat{V}(x)
\end{equation}
and the optimal investment-consumption strategy is given by $(\hat{\Pi},\hat{C}) = (\hat{\pi},\hat{\xi} X^{x,\hat{\pi}, \hat{\xi}X})$ where $\hat{\pi}\coloneqq \frac{\mu-r}{\sigma^2 R}$ and $\hat{\xi}\coloneqq\eta = \frac{S-1}{S} \big(r +\frac{\lambda^2}{2R}\big)$.
\end{thm}
It is interesting to contrast the results of this paper with those of Herdegen et al.~\cite{herdegen2021infinite} and Melnyk et al.~\cite{melnyk2020lifetime}.
\subsection{A comparison with Herdegen et al}
The goals of \cite{herdegen2021infinite} are two-fold; first to discuss how best to formulate the infinite-horizon investment-consumption problem, and second to solve the problem when $0<\theta<1$. The discussion in Part~I of \cite{herdegen2021infinite} motivates the set-up of the problem we use here, including the form of the EZ-aggregator $f_{EZ}$.
In Part~II of \cite{herdegen2021infinite}, the authors consider existence and uniqueness of the utility process and the investment-consumption problem in the case $0<\theta<1$.
The fundamental existence result \cite[Theorem 10.8]{herdegen2021infinite} is reused in this paper to give existence for $\theta>1$ when the consumption stream satisfies a self-order condition. Thereafter, the methodology and approach of \cite{herdegen2021infinite} and this paper differ considerably. In \cite{herdegen2021infinite}, the authors prove a comparison lemma (\cite[Theorem 9.8]{herdegen2021infinite}) whose use is threefold. First, it gives a simple proof of uniqueness of the solution to the equation for the utility process. Second, it can be used to prove a monotonicity result which can be used to generalise the existence result from the consumption streams satisfying the self-order condition to all consumption streams. Third, the comparison theorem plays a crucial role in the verification lemma.
When $\theta>1$, the hypotheses of the comparison lemma in \cite{herdegen2021infinite} are not satisfied. Indeed, the result cannot be true when $\theta>1$ as then uniqueness would follow -- and we have seen that uniqueness fails in general. Hence, we are required to introduce a new concept to identify the economically relevant solution. We call these proper solutions. Our goal is then to prove existence and uniqueness of such solutions. This requires a new and fundamentally different comparison lemma and new ideas to prove existence and uniqueness.
To prove existence of a proper solution, we first consider the case of a filtration which is constant (with respect to time) except at a finite set of time-points. In that setting, and for a class of particularly simple consumption streams, we give an explicit bound on the utility process associated to a proper solution. This bound can then be extended to a continuous filtration and a wide class of consumption streams, thus showing that there exists a proper solution in this setting.
Uniqueness of a proper solution also needs a completely new argument, and involves showing that given a pair of proper solutions, the ratio of the pair must either stay constant at unity or explode. The verification lemma now follows in a similar spirit to \cite{herdegen2021infinite} and relies on a perturbation argument first given in \cite{herdegen2020elementary}. However, some of the arguments need to be modified since we are discussing proper solutions. In particular, we need to prove an important result which says that we may approximate $C\in\mathds{U}\mathds{P}$ by a sequence $(C^n)_{n\in\mathds{N}}$ of consumption streams with associated proper solutions.
\subsection{A comparison with Melnyk et al}\label{ssec:comparison to MMKS}
It is also interesting to compare the results and techniques of this paper with those of the recent and wide-ranging paper by Melnyk et al.~\cite{melnyk2020lifetime}. The ultimate focus of \cite{melnyk2020lifetime} is to consider the investment-consumption problem for EZ-SDU with transaction costs. However, they first consider the frictionless case. They use a slightly different set-up, and a reduced set of parameter combinations -- in particular, they require that $R>1$.
The main result in \cite{melnyk2020lifetime} regarding the optimal investment-consumption strategy (Theorem 3.4) involves calculating the optimal strategy in a space $\mathscr{A}^0$ \cite[Definition 3.1]{melnyk2020lifetime}. For a strategy to belong to this class, the wealth and investment process must satisfy some integrability conditions, and there must exist a unique utility process. (Although uniqueness of the utility process is considered via a comparison lemma, which is different in both statement and proof to those in this paper and \cite{herdegen2021infinite}, existence is not considered in \cite{melnyk2020lifetime}). Finally, for the case $\theta>1$ results are proved under three extra conditions: the first is that consumption streams $C=(C_t)_{t\geq0}$ must be strictly positive; the second \cite[Equation 5]{melnyk2020lifetime} is that the associated utility process $V=(V_t)_{t\geq0}$ satisfies $(1-R)V_t \geq C_t^{1-R}$ for $t\geq0$; the third \cite[Equation 3]{melnyk2020lifetime} is that the utility process satisfies a transversality condition $\lim_{t\to\infty}e^{- \gamma t}\EX{|V_t|}=0$ for an appropriate $\gamma \in \mathds{R}$. The second and third of these assumptions are rather ad-hoc conditions but form a crucial element of the proof of the comparison theorem. Combining the first and second of the trio of assumptions ensures that $(1-R)V_t>0$ for all $t\geq0$. This has the effect of identifying the proper solution within the class of utility processes; however, these extra assumptions rule out many consumption streams for which there exists a proper utility process. Furthermore, for certain parameter combinations the second and third assumptions even rule out the candidate optimal strategy. As a consequence, Melynk et al. are forced to require additional restrictions on the parameters \cite[Assumption 3]{melnyk2020lifetime} beyond those which are necessary to ensure that the problem is well-posed.
\section{Subsolutions and supersolutions}\label{sec:comparison}
This section introduces subsolutions and supersolutions and then proves that, under certain conditions, subsolutions are bounded above by supersolutions. This important result will be used in all of the remaining sections.
We recall from \cite{herdegen2021infinite} the definition of an a \textit{aggregator random field} (a generalisation of the aggregator that is allowed to depend on the state of the world $\omega\in\Omega$) as well as the definition of a sub- and supersolution.
\begin{defn}[[\citealp{herdegen2021infinite},~Definition 9.1{]}]
An \textit{aggregator random-field} $g:[0,\infty) \times \Omega \times \mathds{R}_+ \times \mathds{V} \to \mathds{V}$ is a product measurable mapping such that
$g(\cdot,\omega,\cdot,\cdot)$ is an aggregator for fixed $\omega\in\Omega$, and for progressively-measurable processes $C=(C_t)_{t\geq0}$ and $V=(V_t)_{t\geq0}$, the process $(g(t,\omega,C_t(\omega),V_t(\omega)))_{t\geq0}$ is progressively-measurable.
\end{defn}
We need the more general notion of an aggregator random fields since it permits us to stochastically perturb our Epstein--Zin aggregator.
\begin{defn}[[\citealp{herdegen2021infinite},~Definition 9.3{]}]
Let $C \in \mathscr{P}_+$ and $g$ be an aggregator random field such that $g$ takes only one sign, i.e. $\mathds{V}\subseteq \ol \mathds{R}_+$ or $\mathds{V}\subseteq \ol \mathds{R}_-$. A $\mathds{V}$-valued l\`ad optional process $V$ is called
\begin{itemize}
\item a \textit{subsolution} for the pair $(g,C)$ if $\limsup_{t\to\infty}~ \EX{V_{t+}} \leq 0$ and for all bounded stopping times $\sigma\leq\tau$,
\begin{equation}\label{eq:subsolution equation}
V_\sigma ~\leq~ \cEX[\sigma]{V_{\tau{+}} + \int_\sigma^{\tau} g(s,\omega, C_s,V_s)\,\mathrm{d} s}.
\end{equation}
\item a \textit{supersolution} for the pair $(g,C)$ if $\liminf_{t\to\infty}~ \EX{V_{t+}} \geq 0$ and for all bounded stopping times $\sigma\leq\tau$,
\begin{equation}\label{eq:supersolution equation}
V_\sigma ~\geq~ \cEX[\sigma]{V_{\tau{+}} + \int_\sigma^{\tau} g(s,\omega, C_s,V_s)\,\mathrm{d} s}.
\end{equation}
\item a \emph{solution} for the pair $(g,C)$ if it is both a subsolution and a supersolution and ${V\in\mathds{I}(g,C)}$.
\end{itemize}
\end{defn}
\begin{rem}\label{rem:solutions are utility processes} By taking the limit as $\tau\to\infty$ and using the transversality condition, it is clear that a solution $V$ for the pair $(g,C)$ satisfies \eqref{eq:stochastic differential utility aggregator g}. We then choose a c\`adl\`ag version of $V$ so that $V$ is a utility process associated to $(g,C)$. This implies in particular that if $V$ is a solution for the pair $(g,C)$, then $V_\sigma = \cEX[\sigma]{\int_\sigma^\tau g(s,\omega, C_s,V_s)\,\mathrm{d} s +V_{\tau}}$ for all bounded stopping times $\sigma\leq\tau$ since $V_{\tau} = V_{\tau+}$.
\end{rem}
\subsection{Comparison of subsolutions and supersolutions}
It is shown in \cite[Theorem 9.8]{herdegen2021infinite} that when $g$ is decreasing in its last argument, if $V^1$ is a subsolution and $V^2$ is a supersolution (both associated to $g$ and some $C\in\mathscr{P}_+$) and either $V^1$ or $V^2$ is in $\mathds{U}\mathds{I}(g,C)$, then $V^1_\tau\leq V^2_\tau$ for all finite stopping times $\tau$. However, when $\theta>1$, the Epstein--Zin aggregator $f_{EZ}$ is increasing in its last argument so this theorem does not apply.
The next proposition shows that we may weaken the condition that the aggregator has a negative derivative with respect to its last argument, and instead assume that it has a derivative which is bounded above by some positive decreasing exponential.
We introduce the following condition on a pair $(V^1,V^2)$ of stochastic processes which will be a requirement for our comparison theorems.
\begin{condition}\label{cond:V^1,V^2 condition}
Let $g:[0,\infty) \times \Omega \times \mathds{R}_+ \times \mathds{V} \to \mathds{V}$ be an aggregator random field and $C\in\mathscr{P}_+$. The pair $(V^1,V^2)$ satisfies Condition~\ref{cond:V^1,V^2 condition} for the pair $(g,C)$ if either $V^1$ or $V^2$ is in $\mathds{U}\mathds{I}(g,C)$ and $(V^1-V^2)^+$ is $L^1$ bounded.
\end{condition}
\begin{rem}\label{rem:UI stronger than condition A}
Note that for a subsolution $V^1$ and a supersolution $V^2$ associated to the pair $(g,C)$, a sufficient condition for the pair $(V^1,V^2)$ to satisfy Condition \ref{cond:V^1,V^2 condition} for the pair $(g,C)$ is that $V^1,V^2\in\mathds{U}\mathds{I}(g,C)$. This is because $\EX{(V^1-V^2)^+} \leq |V^1_0|+|V^2_0| + \EX{\int_0^\infty |g(s,\omega, C_s,V^1_s)|\,\mathrm{d} s}+\EX{\int_0^\infty |g(s,\omega, C_s,V^2_s)|\,\mathrm{d} s} < \infty$. However, Condition \ref{cond:V^1,V^2 condition} is more general.
\end{rem}
\begin{prop}[Comparison Theorem]\label{prop:comparison}
Let $g:[0,\infty) \times \Omega \times \mathds{R}_+ \times \mathds{V} \to \mathds{V}$ be an aggregator random field that is concave and nondecreasing in its last argument. Let $C\in\mathscr{P}_+$. Suppose that $V^1$ is a subsolution and $V^2$ is a nonzero supersolution for the pair $(g,C)$ and that the pair $(V^1,V^2)$ satisfies Condition~\ref{cond:V^1,V^2 condition} for the pair $(g,C)$. Suppose further that for $\P$-a.e.~$\omega$, $g_v(t,\omega,C_t(\omega),V^2_t(\omega)) \leq \kappa e^{-\nu t}$ for all $t \geq 0$ for some $\kappa,\nu>0$. Here, we interpret $g_v$ to be the right-derivative of $g$ with respect to $v$, which exists everywhere for $v \neq 0$ by the concavity assumption. Then, $V^1_\sigma \leq V^2_\sigma$ $\as{\P}$ for all finite stopping times $\sigma$.
\end{prop}
\begin{proof}
Seeking a contradiction, suppose there is a finite stopping time $\sigma$ such that ${\P[V^1_{\sigma} > V^2_{\sigma}] > 0}$. By replacing $\sigma$ with $\sigma \wedge T$ for $T$ sufficiently large, we may assume without loss of generality that $\sigma$ is bounded. Set $A := \{V^1_\sigma > V^2_{\sigma} \}$. Since $V^1$ and $V^2$ are l\`ad, we may define the right-continuous processes $(V^1_{t+})_{t \geq 0}$ and $(V^2_{t+})_{t \geq 0}$. Further, define the stopping time ${\tau} := \inf\{t \geq \sigma: V^1_{t+} \leq V^2_{t+}\}$. Then $(V^1_{\tau+} - V^2_{\tau+})\mathbf{1}_{\{\tau < \infty\}}\leq 0$ by the right-continuity of $(V^1_{t+})_{t \geq 0}$ and $(V^2_{t+})_{t \geq 0}$.
First, we show that $\P[A \cap \{\sigma < \tau\}] > 0$. Indeed, otherwise if $\mathbf{1}_{\{\sigma = \tau\} \cap A} = \mathbf{1}_A$ $\as{\P}$, the definition of sub- and supersolutions yields
\begin{eqnarray}
\label{eq:pf:prop:comparision}\qquad~~
\mathbf{1}_A \left(V^1_\sigma - V^2_\sigma \right) \leq \cEX[\sigma]{\mathbf{1}_A \left(V^1_{\sigma+}- V^2_{\sigma+} \right)} = \cEX[\sigma]{\mathbf{1}_A \left(V^1_{\tau+}- V^2_{\tau+} \right)\mathbf{1}_{\{\tau < \infty\}} } \leq 0,
\end{eqnarray}
and we arrive at a contradiction.
Next, by the definition of sub- and supersolutions and by Jensen's inequality for the convex function $f(x)=x^+ = \max\{x,0\}$, for $t\leq T$, letting $B_t = A\cap\{\sigma\leq t<\tau\}$
\begin{align}
\lefteqn{\mathbf{1}_{B_t}\left(V^1_t - V^2_t \right)^+}
\\
&\leq \cEX[t]{\mathbf{1}_{B_t}\!\left(V^1_{(T\wedge\tau)+}- V^2_{(T\wedge\tau)+} + \int_t^{T\wedge\tau}\left[g(s,\omega,C_s,V^1_{s})-g(s,\omega,C_s,V^2_{s})\right]\,\mathrm{d} s \right)^+}
\\
&\leq\cEX[t]{\mathbf{1}_{B_t}\!\big(V^1_{(T\wedge\tau)+}- V^2_{(T\wedge\tau)+}\big)^+\! + \mathbf{1}_{B_t}\!\!\int_t^{T\wedge\tau}\!\!\!\left(g(s,\omega,C_s,V^1_{s})-g(s,\omega,C_s,V^2_{s})\right)^+\!\!\,\mathrm{d} s}
\end{align}
where the right hand side is well-defined since either $V^1$ or $V^2$ is in $\mathds{U}\mathds{I}(g,C)$. Taking expectations yields
\begin{align}
\lefteqn{\EX{\mathbf{1}_{B_t}\left(V^1_t - V^2_t \right)^+}}
\\ \label{eq:proof:comparison:1}
~~~~&~~\leq\EX{\mathbf{1}_{B_t}\!\big(V^1_{(T\wedge\tau)+}- V^2_{(T\wedge\tau)+}\big)^+\! + \mathbf{1}_{B_t}\!\!\int_t^{T\wedge\tau}\!\!\!\left(g(s,\omega,C_s,V^1_{s})-g(s,\omega,C_s,V^2_{s})\right)^+\!\!\,\mathrm{d} s}
\end{align}
Taking the $\limsup$ as $T \to \infty$ and using the fact that $\mathbf{1}_{B_t} \mathbf{1}_{\tau\leq T}\left(V^1_{\tau+}- V^2_{\tau+}\right)^+=0~\as{\P}$ for all $T\geq0$ and the transversality condition of sub- and supersolutions gives
\begin{eqnarray}
\lefteqn{\limsup_{T\to\infty}\EX{\mathbf{1}_{B_t}\left(V^1_{(T\wedge\tau)+}- V^2_{(T\wedge\tau)+}\right)^+}}
\\
&=&~ \limsup_{T\to\infty}\EX{\mathbf{1}_{B_t}\mathbf{1}_{T<\tau}\left(V^1_{T+}- V^2_{T+}\right)^+} + \limsup_{T\to\infty}\EX{\mathbf{1}_{B_t} \mathbf{1}_{\tau\leq T}\left(V^1_{\tau+}- V^2_{\tau+}\right)^+}
\\
&\leq&~ \limsup_{T\to\infty}\EX{\left(V^1_{T+}- V^2_{T+}\right)^+}
~\leq~ \limsup_{T\to\infty}\EX{(V^1_{T+})^+ +(V^2_{T+})^-} ~\leq~ 0,
\end{eqnarray}
where the last inequality follows since either $\mathds{V}=\mathds{R}_+$ (and $(V^2_{T+})^-=0$) along with the transversality condition for supersolutions, or $\mathds{V}=\mathds{R}_-$ (and $(V^1_{T+})^+=0$) along with the transversality condition for supersolutions.
Hence, by taking the $\limsup$ as $T\to\infty$ and using the positivity of the integrand, \eqref{eq:proof:comparison:1} becomes
\begin{eqnarray*}
\lefteqn{\EX{\mathbf{1}_{B_t}\left(V^1_t - V^2_t \right)^+}}
\\
&\leq&~\EX{\mathbf{1}_{B_t}\int_t^\tau\left(g(s,\omega,C_s,V^1_{s})-g(s,\omega,C_s,V^2_{s})\right)^+\,\mathrm{d} s},
\\
&\leq&\EX{\int_t^\infty \mathbf{1}_{B_s}g_v(s,\omega,C_s,V^2_s)\left(V^1_s - V^2_s \right)^+ \,\mathrm{d} s}
\\
&\leq&~\EX{\int_t^\infty \kappa e^{-\nu s} \mathbf{1}_{B_s} \left(V^1_s - V^2_s \right)^+ \,\mathrm{d} s},
\end{eqnarray*}
where in the middle line we have used that $g$ is concave and nondecreasing in its last argument and $V^2_s \neq 0$.
If $\Gamma(t) \coloneqq\EX{\mathbf{1}_{B_t}\left(V^1_t - V^2_t \right)^+}$, then $\Gamma = (\Gamma(t))_{t\geq0}$ is a nonnegative process such that $\Gamma(t) \leq \int_t^\infty \kappa e^{-\nu s} \Gamma(s) \,\mathrm{d} s$. Note that ${\Gamma(t)\leq \EX{(V^1_t-V^2_t)^+}\leq \gamma}$ for some $\gamma>0$, by the $L^1$-boundedness of $(V^1-V^2)^+$. Therefore, since $\int_0^\infty \kappa e^{-\nu t}\,\mathrm{d} t = \frac{\kappa}{\nu}$ and
$\int_0^\infty \kappa e^{-\nu s} \Gamma(s) \,\mathrm{d} s\leq \frac{\gamma\kappa}{\nu}$, we can apply Gr\"onwall's inequality for Borel functions (\cite[Theorem 2.5]{herdegen2017minimal} with $y(t)=\Gamma(-t)$ and $\mu(A)=\int_{A\cap\mathds{R}_-} \kappa e^{\nu t}\,\mathrm{d} t$) to conclude that $\Gamma(t)=0$ for all $t > 0$.
Note that $\mathbf{1}_{B_t}\left(V^1_t - V^2_t\right)^+ \geq 0$ for each $t \geq 0$. Hence, by Fatou's Lemma,
\begin{equation}\label{eq:pf:prop:comparison3}
0\leq\EX{\mathbf{1}_{B_t}\left(V^1_{t+} - V^2_{t+} \right)^+} \leq \liminf_{s\downdownarrows t}\EX{\mathbf{1}_{B_s}\left(V^1_s - V^2_s \right)^+}=0.
\end{equation}
Furthermore, since $\mathbf{1}_{B_t}\left(V^1_{t+} - V^2_{t+}\right) = \mathbf{1}_{B_t}\left(V^1_{t+} - V^2_{t+}\right)^+ \geq 0$ for each $t \geq 0$ by the definition of $\tau$, it follows from \eqref{eq:pf:prop:comparison3} that $P_t = \mathbf{1}_{B_t}\left(V^1_{t+} - V^2_{t+}\right) =0$ $\as{\P}$ for all $t\geq0$.
Since $(V^1_{t+})_{t \geq 0}$, $(V^2_{t+})_{t \geq 0}$ and $\mathbf{1}_{B_t}$ are right-continuous, $P=(P_t)_{t\geq0}$ is right continuous and is therefore indistinguishable from zero. In particular, $\mathbf{1}_A\mathbf{1}_{\sigma < \tau}\left(V^1_{\sigma+} - V^2_{\sigma+} \right) = \mathbf{1}_{B_\sigma}\left(V^1_{\sigma+} - V^2_{\sigma+} \right) = 0$ $\as{\P}$ But then the definition of sub-and supersolutions implies that
\begin{eqnarray}
\mathbf{1}_A\mathbf{1}_{\sigma < \tau} \left(V^1_\sigma - V^2_\sigma \right) \leq \cEX[\sigma]{\mathbf{1}_A \mathbf{1}_{\sigma < \tau}\left(V^1_{\sigma+}- V^2_{\sigma+} \right)} =0,
\end{eqnarray}
and we arrive at a contradiction.
\end{proof}
The following corollary will be used in the Verification Theorem later.
\begin{cor}\label{cor:comparison C^1-R < K (1-R)V}
Let $f_{EZ}$ be the Epstein--Zin aggregator and suppose that $R<1$. Let $C\in\mathscr{P}_+$. Suppose that $V^1$ is a subsolution and $V^2$ is a positive supersolution for the pair $(f_{EZ},C)$ and that the pair $(V^1,V^2)$ satisfies Condition~\ref{cond:V^1,V^2 condition} for the pair $(f_{EZ},C)$. Suppose further that $C_t^{1-R}\leq K e^{-\gamma t}(1-R)V^2_t$ for some $K,\gamma>0$ and all $t\geq0$. Then, $V^1_\sigma \leq V^2_\sigma$ for all finite stopping times $\sigma\geq0$.
\end{cor}
\begin{proof}
Taking derivatives of $f_{EZ}$ with respect to its second argument gives for $v > 0$
\begin{align}
\label{eq:f_EZ first derivative}
\dfrac{\partial f_{EZ}}{\partial v}(c, v) =&~ (\theta-1)c^{1-S}((1-R)v)^{-\frac{1}{\theta}} \geq 0,
\\
\dfrac{\partial^2 f_{EZ}}{\partial v^2}(c, v) =&~ -\rho (1-R)c^{1-S}((1-R)v)^{-(1+\frac{1}{\theta})} \leq 0.
\end{align}
Hence, using \eqref{eq:f_EZ first derivative}, $\frac{\partial f_{EZ}}{\partial v}(C_s, V^2_s) \leq (\theta-1)K^{\frac{1}{\theta}} e^{-\frac{\gamma}{\theta}t}$ and the conditions of Proposition \ref{prop:comparison} are met with $\kappa=(\theta-1)K^\frac{1}{\theta}$ and $\nu=\frac{\gamma}{\theta}$.
\end{proof}
\begin{rem}
Note that the utility process associated to constant proportional strategies is given in \eqref{eq:valfungenstrat2} by $h(\pi,\xi) X_t^{1-R}$. Since $C_t = \xi X$, the conditions of Corollary \ref{cor:comparison C^1-R < K (1-R)V} are not met, and we cannot use it to give a uniqueness result (even over constant proportional strategies). We instead use Corollary \ref{cor:comparison C^1-R < K (1-R)V} in the proof of the Verification Theorem (Theorem \ref{thm:verification}), in which we perturb the candidate value function beforehand.
\end{rem}
\section{The CRRA-order solution}\label{sec:O-solutions}
This section summarises \cite[Theorem B.2]{herdegen2021infinite} which gives conditions under which a unique CRRA-order solution can be shown to exist. Under such conditions, explicit bounds on the associated utility process are provided. We begin by introducing a simplifying change of coordinates.
Define the nonnegative processes $W = (1-R)V$ and $U = U(C) = \theta C^{1-S}$ and the aggregator
\begin{equation}\label{eq:aggregator h_EZ}
h_{EZ}(u,w) = u w^\rho.
\end{equation}
Further define $J = J^{U^\theta}$ by
\begin{equation}\label{eq:J^Utheta}
J_t = \cEX[t]{\int_t^\infty U_s^\theta \,\mathrm{d} s}, \quad \text{for all }t\geq0.
\end{equation}
Note that $V \in \mathds{I}(f_{EZ},C)$ if and only if $ W \in \mathds{I}(h_{EZ}, U(C))$. Hence, $V^C$ is a utility process associated to consumption stream $C$ with aggregator $f_{EZ}$ if and only if $W=W^{U(C)}$ is a utility process associated to consumption stream $U(C)$ with aggregator $h_{EZ}$. Furthermore, $V$ is a CRRA-order solution if and only if $W$ is solution such that $W\stackrel{\mathds{O}}{=}J$ and $V$ is an extremal solution if and only if $W$ is a \textit{maximal} solution. Finally, $V=(V_t)_{t\geq0}$ is a proper solution associated to $(f_{EZ},C)$ if and only if $W=(W_t)_{t\geq0}$ is a solution associated to $(h_{EZ},U)$ such that $W_t >0 $ on $\{ \cEX[t]{\int_t^\infty U_s^\theta \,\mathrm{d} s} >0 \} $ up to null sets for all $t\geq0$. In a slight abuse of the definition, we then also refer to $W$ as being proper.
We will work in this coordinate system until Section~\ref{sec:verification} and prove existence and uniqueness results in these coordinates; translation to the original coordinate system is immediate.
Define the operator $F_U:\mathds{I}(h_{EZ},U)\to \mathscr{P}_+$ by
\begin{equation}\label{eq:fixed point operator F}
F_{U}(W)_t = \cEX[t]{\int_t^\infty U_s W_s^\rho \,\mathrm{d} s},\quad \text{for all }t\geq0;
\end{equation}
we always choose a c\`adl\`ag version for the right-hand side of \eqref{eq:fixed point operator F}. Note that $W$ is a solution associated to $(h_{EZ},U)$ if and only if it is a fixed point of the operator $F_U$. To show existence of a \textit{maximal} solution associated to $(h_{EZ},U)$ in Section~\ref{sec:extremal solution}, it will instead be beneficial to prove existence of a fixed point to a more general (perturbed) operator.
\begin{prop}[[\citealp{herdegen2021infinite},~Theorem B.2{]}]\label{prop:existence of a solution, bounded and perturbed} Let $\epsilon\geq0$ and let $U\stackrel{\mathds{O}}{=}\Lambda$ for $\Lambda\in\mathds{S}\mathds{O}^\theta$. Let $J=(J_t)_{t\geq0}$ be defined by \eqref{eq:J^Utheta}. Then, $F^{\epsilon}_{U,\Lambda}:\mathds{I}(h_{EZ},U)\to \mathscr{P}_+$, defined by
\begin{equation}\label{eq:perturbed fixed point operator}
F^\epsilon_{U,\Lambda}(W)_t = \cEX[t]{\int_t^\infty U_s W_s^\rho + \epsilon \Lambda_s^\theta \,\mathrm{d} s},
\end{equation}
has a fixed point $W\in\mathds{I}(h_{EZ},U)$. It is the unique fixed point such that $W\stackrel{\mathds{O}}{=}J$.
\end{prop}
\begin{rem}
Often this theorem will be applied with $\Lambda=U$. In that case, if $U^\theta \in \mathds{S}\mathds{O}$ then there exists a fixed point $W$ of the operator $F_U=F^0_{U,U}$ and $W \stackrel{\mathds{O}}{=} J \stackrel{\mathds{O}}{=} U^\theta$.
\end{rem}
The following corollary to Proposition \ref{prop:comparison} shows that there is a natural ordering for subsolutions and supersolutions to \eqref{eq:perturbed fixed point operator} for different values of $\epsilon$ and $U$. In particular, since a solution is both a subsolution and a supersolution, it follows that for $\epsilon>0$, the solution found in Proposition \ref{prop:existence of a solution, bounded and perturbed} is the unique solution, and not just the unique solution of order $J$.
\begin{cor}\label{cor:comparison W,U positive eps} Fix $\nu>0$. Suppose $\Lambda\in\mathds{S}\mathds{O}_\nu^\theta$ and define the perturbed aggregator
\begin{equation}\label{eq:perturbed aggregator}
h^{\epsilon,\nu,\Lambda}_{EZ}(t,\omega,u,w)=u w^\rho + \epsilon e^{\nu t}\Lambda_t^\theta(\omega), \quad \text{for }\epsilon>0.
\end{equation}
Fix $\epsilon_2>0$ and $0\leq\epsilon_1\leq\epsilon_2$. Let $U^1,U^2\in\mathscr{P}_+$ satisfy $U^1\leq U^2\leq \Lambda$. Suppose that $W^1$ is a subsolution for the pair $(h_{EZ}^{\epsilon_1,\nu,\Lambda},U^1)$ and $W^2$ is a supersolution for the pair $(h_{EZ}^{\epsilon_2,\nu,\Lambda},U^2)$ such that $W^1,W^2\in [0,\infty)$ and such that $(W^1,W^2)$ satisfies Condition~\ref{cond:V^1,V^2 condition} for the pair $(h_{EZ}^{\epsilon_1,\nu,\Lambda},U^1)$.
Then, $W^1_\sigma \leq W^2_\sigma$ for all finite stopping times $\sigma\geq 0$.
\end{cor}
\begin{proof}
First note that $W^2$ is a supersolution for $(h_{EZ}^{\epsilon_1,\nu,\Lambda},U^1)$, since
\begin{equation}
W^2_\sigma\geq\cEX[t]{\int_\sigma^\tau h_{EZ}^{\epsilon_2,\nu,\Lambda}(U^2_s,W^2_s)\,\mathrm{d} s + W^2_{\tau+}}\geq\cEX[t]{\int_\sigma^\tau h_{EZ}^{\epsilon_1,\nu,\Lambda}(U^1_s,W^2_s)\,\mathrm{d} s + W^2_{\tau+}}.
\end{equation}
As $\Lambda^\theta \in \mathds{S}\mathds{O}_\nu$, there exists $K_\Lambda$ such that $W^2_t \geq \cEX[t]{\int_t^\infty \epsilon_2 \mathrm{e}^{\nu s}\Lambda_s^\theta\,\mathrm{d} s} \geq \frac{\epsilon_2}{K_\Lambda}e^{\nu t}\Lambda_t^\theta$. Therefore, since $U^1\leq \Lambda$ and $W^2 > 0$,
\begin{equation}
\dfrac{\partial h_{EZ}^{\epsilon_1,\nu,\Lambda}}{\partial w}\left(t,\omega,U^1_t,W^2_t\right) = \rho U^1_t(W^2_t)^{- \frac{1}{\theta}} \leq\rho\left(\tfrac{K_\Lambda}{\epsilon_2}\right)^\frac{1}{\theta} e^{-\frac{\nu}{\theta} t}.
\end{equation}
Furthermore, $\frac{\partial^2 h_{EZ}^{\epsilon_1,\nu,\Lambda}}{\partial w^2}(u,w)=-\frac{\theta-1}{\theta^2}u w^{-(1+\frac{1}{\theta})}\leq 0$ for $w > 0$, so that $h_{EZ}^{\epsilon_1,\nu,\Lambda}$ is concave.
Since $(W^1,W^2)$ satisfies Condition~\ref{cond:V^1,V^2 condition} for the pair $(h_{EZ}^{\epsilon_1,\nu,\Lambda},U^1)$, the assumptions of Proposition \ref{prop:comparison} are met, and the conclusion follows.
\end{proof}
The following Corollary gives explicit bounds $\hat{k},\hat{K}>0$ such that the fixed point $W$ found in Proposition \ref{prop:existence of a solution, bounded and perturbed} satisfies $\hat{k}J \leq W \leq\hat{K}J$.
\begin{cor}\label{cor:perturbed existence explicit bounds}
Let $\epsilon\geq0$ and suppose that $U^\theta\in\mathds{S}\mathds{O}\big(k,K\big)$. For $\epsilon>0$, suppose that $A$ and $B$ solve
\begin{equation}\label{eq:A and B equation}
A=K^{-1} (A^\rho + \epsilon) \qquad B=k^{-1} (B^\rho + \epsilon)
\end{equation}
and, if $\epsilon=0$, set $A = K^{-\theta}$ and $B = k^{-\theta}$ {\rm(}the positive solution to \eqref{eq:A and B equation}{\rm)}.
Then, the fixed point $W$ of $F^\epsilon_{U,U}$ found in Proposition \ref{prop:existence of a solution, bounded and perturbed} is in $\mathds{O}(J; kA,KB)$.
\end{cor}
\begin{proof}
We first show that $F^\epsilon_{U,U}$ maps from $\mathds{O}(U^\theta;A,B)$ to itself. We only prove the upper bound as the lower bound is symmetric. Suppose that $W\leq BU^\theta$. Then,
\begin{align*}
F^\epsilon_{U,U}(W)_t =&~\cEX[t]{\int_t^\infty U_s W_s^\rho + \epsilon U_s^\theta \,\mathrm{d} s}
\\
\leq&~ \cEX[t]{\int_t^\infty U_s {B^\rho}U_s^{\theta\rho} + \epsilon U_s^\theta\,\mathrm{d} s}
\\
=&~ \left( B^\rho + \epsilon\right)\cEX[t]{\int_t^\infty U_s^\theta \,\mathrm{d} s}
\\
=&~ \left( B^\rho + \epsilon\right) J_t \leq \frac{1}{k} \left( B^\rho + \epsilon\right) U^\theta_t = B U^\theta_t.
\end{align*}
The proof of Proposition \ref{prop:existence of a solution, bounded and perturbed} given in \cite[Theorem B.2]{herdegen2021infinite} first shows that $F^\epsilon_{U,U}: \mathds{O}(U^\theta)\to\mathds{O}(U^\theta)$ is a contraction mapping and then uses Banach's fixed point theorem. Hence, if we choose an initial process $W^0\in\mathds{O}(U^\theta;A,B)$, then repeated application of $F^\epsilon_{U,U}$ yields a fixed point $W^*\in\mathds{O}(U^\theta;A,B)$. Since the fixed point $W$ found in Proposition \ref{prop:existence of a solution, bounded and perturbed} is unique in the class $\mathds{O}(U^\theta)$, $W=W^*\in\mathds{O}(U^\theta;A,B)$. Finally, since $U^\theta\in\mathds{S}\mathds{O}(k,K)$, $\mathds{O}(U^\theta;A,B)\subseteq \mathds{O}(J;kA,KB)$.
\end{proof}
\section{The extremal solution}\label{sec:extremal solution}
In this section, we investigate the \textit{extremal} solution, from Definition \ref{def:extremal solution}. Recall that in the transformed coordinates the extremal solution is the maximal solution.
\begin{prop}\label{prop:maximal solution unique}
If the maximal solution exists, it is unique.
\end{prop}
\begin{proof}
Suppose for contradiction that there are two maximal solutions $W^1$ and $W^2$. Then, for all $t\geq0$, $W^1_t\geq W^2_t$ since $W^1$ is maximal in the class of solutions and $W^1_t\geq W^2_t$ since $W^1$ is maximal also. Thus, $W^1_t=W^2_t$ for all $t\geq0$. Since both $W^1$ and $W^2$ are c\`adl\`ag, they are indistinguishable.
\end{proof}
We now turn to existence of a maximal solution associated to $(h_{EZ},U)$.
\begin{prop}\label{prop:maximal existence U<K Lambda}
Suppose that $U\in\mathscr{P}_+$ satisfies $U\leq\Lambda$ for $\Lambda\in\mathds{S}\mathds{O}^\theta_+$. Then, there exists a unique maximal solution associated to $(h_{EZ},U)$. It is also maximal in the class of $L^1$-bounded subsolutions.
\end{prop}
\begin{proof}
Since $\Lambda\in\mathds{S}\mathds{O}_+^\theta$, there exists $\nu>0$ such that $\Lambda\in\mathds{S}\mathds{O}_{\nu\theta}^\theta$. For such $\nu$, let $\Lambda^{(\nu)}_t = e^{\nu t}\Lambda_t$.
Then, $\Lambda^{(\nu)}=(\Lambda^{(\nu)}_t)_{t\geq0}\in\mathds{S}\mathds{O}^\theta$. For each $n\in\mathds{N}$, let $U^n \coloneqq \max\left\{U,\frac{1}{n}\Lambda^{(\nu)}\right\}$. Then, $U^n\stackrel{\mathds{O}}{=}\Lambda^{(\nu)}$ as $U^n\leq \Lambda\leq \Lambda^{(\nu)}$. Let $(\epsilon_n)_{n\in\mathds{N}}$ be a positive-valued sequence such that $\epsilon_n\searrow0$.
By Proposition \ref{prop:existence of a solution, bounded and perturbed}, for each $n\in\mathds{N}$, there exists a solution $W^n$ associated to $(h_{EZ}^{\epsilon_n,\nu\theta,\Lambda},U^n)$, where $h_{EZ}^{\epsilon_n,\nu\theta,\Lambda}$ is defined in \eqref{eq:perturbed aggregator}. Furthermore, $W^n$ is decreasing in $n$ by Corollary \ref{cor:comparison W,U positive eps} and $U^n (W^n)^\rho$ is dominated by $U^1 (W^1)^\rho$. Hence, by the Dominated Convergence Theorem, we find that ${W \coloneqq \lim_{n\to\infty}W^n}$ satisfies
\begin{equation}
W_t = \lim_{n\to\infty}\cEX[t]{\int_t^\infty U^n_s (W^n_s)^\rho + \epsilon_n e^{\nu\theta s}\Lambda_s^\theta \,\mathrm{d} s}=
\cEX[t]{\int_t^\infty U_s W_s^\rho\,\mathrm{d} s},
\end{equation}
so that $W\in\mathds{I}(h_{EZ},U)$ is a solution associated to $(h_{EZ},U)$.
Suppose that $W'\in\mathds{I}(h_{EZ}, U)$ is a solution (or an $L^1$-bounded subsolution) associated to $(h_{EZ},U)$. Then, the pair $(W',W^n)$ satisfies Condition \ref{cond:V^1,V^2 condition} for the pair $(h_{EZ},U)$, since $W^n\in\mathds{U}\mathds{I}(h_{EZ}^{\epsilon_n,\nu\theta,\Lambda},U^n)\subset\mathds{U}\mathds{I}(h_{EZ},U)$ and $(W'-W^n)^+\leq W'$, where $W'$ is $L^1$-bounded. Hence, $W^n \geq W'$ for each $n\in\mathds{N}$ by Corollary \ref{cor:comparison W,U positive eps} and $W \geq W'$ is a maximal ($L^1$-bounded sub-) solution. Uniqueness in the class of maximal solutions follows from Proposition \ref{prop:maximal solution unique}.
\end{proof}
\textit{Proposition \ref{prop:extremal solution for C^1-R < hatC^1-R}} is a direct result of the Proposition \ref{prop:maximal existence U<K Lambda} and the following comparison result for maximal solutions.
\begin{prop}\label{prop:maximal solutions increasing in U}
Let $h_{EZ}$ be the aggregator defined in \eqref{eq:aggregator h_EZ} and suppose that $U^1,U^2\in\mathscr{P}_+$ satisfy $U^1\leq U^2 \leq \Lambda\in\mathds{S}\mathds{O}^\theta_+$. If $W^1$ and $W^2$ are the maximal solutions associated to $h_{EZ}$ and consumption $U^1$ and $U^2$ respectively, then $W^1_t\leq W^2_t$ for all $t\geq0$.
\end{prop}
\begin{proof}
Let $\nu$ be such that $\Lambda\in\mathds{S}\mathds{O}_{\nu\theta}^\theta$. Define $\Lambda^{(\nu)}=\big(\Lambda^{(\nu)}_t\big)_{t\geq0}$ by $\Lambda^{(\nu)}_t = e^{\nu t}\Lambda_t$ and for $n\in\mathds{N}$ and $i\in\{1,2\}$ define $U^{i,n}=\max\big\{U^i,\frac{1}{n}\Lambda^{(\nu)}\big\}$ and ${\epsilon_n = \frac{1}{n}}$. Then, by Proposition \ref{prop:existence of a solution, bounded and perturbed}, there exists a solution $W^{i,n}$ associated to $(h_{EZ}^{\epsilon_n,\nu\theta,\Lambda},U^{i,n})$. Furthermore, for all $n\in\mathds{N}$ and $t\geq0$, $W^{1,n} \leq W^{2,n}$ by Corollary \ref{cor:comparison W,U positive eps}.
As in Proposition \ref{prop:maximal existence U<K Lambda}, the unique maximal solution associated to $U^i$ is given by $W^i \coloneqq \lim_{n \to \infty}W^{i,n}$ for $i\in\{1,2\}$. Thus, $W^1_t=\lim_{n \to \infty}W^{1,n}_t \leq \lim_{n \to \infty}W^{2,n}_t= W^2_t$ for all $t\geq0$.
\end{proof}
We may also deduce the following Corollary to Proposition \ref{prop:maximal existence U<K Lambda}.
\begin{cor}\label{cor:extremal is maximal subsolution}
Let $C\in\mathscr{P}_+$ be such that $C^{1-R}\leq Y\in\mathds{S}\mathds{O}_+$. Then, the extremal solution associated to $(f_{EZ},C)$ is the maximal $L^1$-bounded subsolution when $R<1$ and the minimal $L^1$-bounded supersolution when $R<1$.
\end{cor}
The final result of this section shows that the solution found by fixed point argument in Proposition \ref{prop:existence of a solution, bounded and perturbed} is the unique maximal solution.
\begin{prop}\label{prop:fixed point solution is maximal solution for O(Lambda)}
Let $h_{EZ}$ be the aggregator defined in \eqref{eq:aggregator h_EZ}. Suppose that $U\in\mathds{S}\mathds{O}^\theta_+$. Then the solution associated to $(h_{EZ},U)$ found in Proposition \ref{prop:existence of a solution, bounded and perturbed} is the maximal solution.
\end{prop}
\begin{proof}
Fix $\theta>1$. Since $U\in\mathds{S}\mathds{O}^\theta_+$, there exists $\hat{\nu}>0$ such that $U\in\mathds{S}\mathds{O}_{\hat{\nu}\theta}^\theta$. By Lemma \ref{lem:SO^mu subset SO^nu}, it further follows that $U\in\mathds{S}\mathds{O}_{\nu\theta}^\theta$ for $\nu\leq\hat{\nu}$. For each $\nu\leq\hat{\nu}$, define $U^{(\nu)}_t = e^{-\nu t}U_t$ and $J^{(\nu)}=(J^{(\nu)}_t)_{t\geq0}$ by $J^{(\nu)}_t = \cEX[t]{\int_t^\infty e^{\nu s} U^\theta_s \,\mathrm{d} s}$. We can then find $k(\nu),K(\nu)$ such that $\left(U^{(\nu)}\right)^\theta\in\mathds{S}\mathds{O}(k(\nu),K(\nu))$. By Remark \ref{rem:bounds increasing in nu}, we may choose $k(\nu),K(\nu)$ such that $0<k(\hat{\nu})\leq\lim_{\nu\to0}k(\nu)=k(0)\eqqcolon k<\infty$ and $0<K(\hat{\nu})\leq\lim_{\nu\to0}K(\nu)=K(0)\eqqcolon K<\infty$, where both limits are decreasing in $\nu$.
For each $\epsilon>0$ and $0<\nu\leq\hat{\nu}$, there exists a solution $W^{\epsilon,\nu}$ associated to $(h_{EZ}^{\epsilon,\nu\theta,\Lambda},U^{(\nu)})$ by Proposition \ref{prop:existence of a solution, bounded and perturbed}.
Furthermore, by Corollary \ref{cor:perturbed existence explicit bounds}, $W^{\epsilon,\nu}_t\leq K(\nu) B^{\epsilon,\nu} J^{(\nu)}_t$ where $B=B^{\epsilon,\nu}$ solves $B=k(\nu)^{-1}(B^\rho + \epsilon)$. By Proposition \ref{prop:maximal existence U<K Lambda}, the unique maximal solution associated to $U$ is given by $W\coloneqq\lim_{\epsilon\to0}W^{\epsilon,\nu}$. Therefore, since $\lim_{\epsilon\to0}B^{\epsilon,\nu}=B^{0,\nu}=k(\nu)^{-\theta}$, $W_t \leq K(\nu)k(\nu)^{-\theta}J^{(\nu)}$ for all $\nu\leq \hat{\nu}$. Taking the limit as $\nu\searrow0$, gives $W_t\leq K k^{-\theta}J^{(0)}\eqqcolon K k^{-\theta}J$.
Similarly, by maximality of $W$ and the lower bound found in Corollary \ref{cor:perturbed existence explicit bounds}, $W\geq kK^{-\theta} J$. Hence, the maximal solution is in $\mathds{O}(J)$. Since the solution found in Proposition \ref{prop:existence of a solution, bounded and perturbed} is unique in $\mathds{O}(J)$, it is equal to the maximal solution.
\end{proof}
\section{The proper solution}\label{sec:proper solution}
We first focus on existence of proper solutions and then turn to uniqueness.
\subsection{Existence of proper solutions}
The goal of this section is to prove Theorem \ref{thm:proper solution rc}. To this end, we first prove that there exists a proper solution $W$ associated to the aggregator $h_{EZ}$ and consumption stream $U$ given by a discounted indicator function of a stochastic interval, i.e. $U_t = e^{-\gamma t} \mathbf{1}_{\{\sigma \leq t < \tau\}}$
for $\sigma$ and $\tau$ stopping times such that $\sigma \leq \tau$. Since any right-continuous consumption stream can locally be approximated from below by (a scaled version of) these processes, we can show that there exists a proper solution associated to right-continuous processes.
\begin{prop}\label{prop:proper solution discounted 1_[sigma,tau)}
Let $\gamma>0$ and $\sigma$ and $\tau$ be stopping times such that $\sigma\leq\tau$. Let ${U=(U_t)_{t\geq0}}$ be given by $U_t:=e^{-\gamma t}\mathbf{1}_{\{\sigma\leq t <\tau\}}$. Then, there exists a proper solution $W=(W_t)_{t\geq0}$ associated to $U$, for which $W_t \geq \big(\frac{1}{\gamma\theta}\cEX[t]{e^{-\gamma (t\vee\sigma)}- e^{-\gamma(t\vee\tau)}}\big)^\theta$.
\end{prop}
The proof of Proposition \ref{prop:proper solution discounted 1_[sigma,tau)} is long and technical and therefore relegated to the appendix.
To prove Theorem \ref{thm:proper solution rc}, we introduce two technical lemmas.
\begin{lemma}\label{lemma:utility process associated to 1_A U}
Suppose that $W$ is a utility process associated to $(h_{EZ},U)$ and suppose that there exists $t_0\geq0$ such that $U_t=0$ for $t<t_0$. If $A\in\mathcal{F}_{t_0}$, and $\widetilde{U}=(\widetilde{U}_t)_{t\geq0}$ is given by $\widetilde{U}_t = \cEX[t]{\mathbf{1}_A}U_t$ then $\widetilde{W}_t = \cEX[t]{\mathbf{1}_AW_{t\vee t_0}}$ is a utility process associated to $(h_{EZ},(\widetilde{U}_t)_{t\geq0})$.
\end{lemma}
\begin{proof}
Suppose that $t\geq t_0$. Then, since $W$ is a utility process associated to $U$,
\begin{align}
\widetilde{W}_t = \mathbf{1}_AW_t = \mathbf{1}_A^{1+\rho}W_t
= \cEX[t]{\int_t^\infty \mathbf{1}_AU_s (\mathbf{1}_AW_s)^\rho\,\mathrm{d} s}
= \cEX[t]{\int_t^\infty \widetilde{U}_s \widetilde{W}_s^\rho\,\mathrm{d} s}.
\end{align}
Conversely, suppose that $t<t_0$. Then, since $\widetilde{U}_s=0$ for $s<t_0$ and both $\widetilde{W}_{t_0} = \mathbf{1}_A W_{t_0}$ and $\widetilde{W}_{t_0} = \cEX[t_0]{\int_{t_0}^\infty \widetilde{U}_s \widetilde{W}_s^\rho\,\mathrm{d} s}$,
\begin{equation*}
\widetilde{W}_t = \cEX[t]{\mathbf{1}_A W_{t_0}}=
\cEX[t]{ \int_{t_0}^\infty \widetilde{U}_s \widetilde{W}_s^\rho\,\mathrm{d} s} =\cEX[t]{\int_t^\infty \widetilde{U}_s \widetilde{W}_s^\rho\,\mathrm{d} s}.\qedhere
\end{equation*}
\end{proof}
\begin{lemma}\label{lemma:J^U,theta>0 implies U_T>eps for T>t}
Suppose $U\in\mathscr{P}_+$ is right-continuous. Fix $t \geq 0$ and let
${A_t=\{J^{U^\theta}_t > 0\}}$ and $B_t=\bigcup_{\substack{T\geq t\\T\in\mathds{Q}}}\bigcup_{\substack{\epsilon>0\\ \epsilon\in\mathds{Q}}} \left\{\cEX[t]{\mathbf{1}_{\{U_T\geq\epsilon\}}}>0\right\}$. Then, $\P(A_t\setminus B_t)=0$.
\end{lemma}
\begin{proof}
Seeking a contradiction, suppose $C_t\coloneqq A_t\setminus B_t$ has positive measure. Then $\mathbb{E}[\mathbf{1}_{C_t} J^U_t]>0$ by the definition of $A_t$. Moreover, for each \textit{rational} $T \geq t$, the definition of $B_t$ gives $\cEX[t]{\mathbf{1}_{C_t} U_T}\leq\cEX[t]{\mathbf{1}_{B_t^c} U_T}=0$ which yields $\mathbf{1}_{C_t} U_T = 0$ $\as{\P}$~ Since $U$ is right-continuous, $\mathbf{1}_{C_t} U_T=0$ for all $T\geq t$ $\as{\P}$~ Taking expectations yields $\mathbb{E}[\mathbf{1}_{C_t} J^U_t] = \EX{\int_t^\infty \mathbf{1}_{C_s} U_s \,\mathrm{d} s}=0$ and we arrive at a contradiction.
\end{proof}
We may now prove Theorem \ref{thm:proper solution rc}. To show that $W$ is proper, we must show that if $A_t=\{J^{U^\theta}_t > 0\}$ and $C_t =\{W_t>0\}$, then $\P(A_t\setminus C_t)=0$. By Lemma \ref{lemma:J^U,theta>0 implies U_T>eps for T>t}, since $A_t\subseteq B_t$ up to null sets, we may instead prove that $\P(B_t\setminus C_t)=0$.
\begin{proof}[Proof of Theorem \ref{thm:proper solution rc}]\label{proof:thm:proper solution rc process}
Since $C^{1-R}\leq Y$ for $Y\in\mathds{S}\mathds{O}_+$, $U\coloneqq\theta C^{1-S}\leq\theta Y^{\frac{1}{\theta}}$. As ${Y\in\mathds{S}\mathds{O}_+}$ and by Remark \ref{rem:YinSO implies KYinSO}, $\theta Y^{\frac{1}{\theta}}\in\mathds{S}\mathds{O}^\theta_+$.
Therefore, by Proposition \ref{prop:maximal existence U<K Lambda} there exists a maximal solution $W$ associated to $(h_{EZ},U)$. We now show that $W$ is proper.
Fix $t^* \geq 0$. Set ${A_{t^*}:=\{J^{U^\theta}_{t^*} > 0\}}$ and $C_{t^*} =\{W_{t^*}>0\}$. By Lemma~\ref{lemma:J^U,theta>0 implies U_T>eps for T>t}, it suffices to show that $\P(B^\epsilon_T \setminus C_{t^*}) = 0$ for all rational $T \geq {t^*}, \epsilon > 0$, where $B^\epsilon_T=\{\cEX[{t^*}]{\mathbf{1}_{\{U_T\geq\epsilon\}}}>0\}$. So, fix rational $T\geq {t^*}$ and $\epsilon > 0$. Define the stopping time $\tau_\epsilon=\inf\{t\geq T: U_t\leq \frac{\epsilon}{2}\}$ and note that $\{\tau_\epsilon > T\}$ on $\{U_T \geq\epsilon\}$ by right-continuity of $U$. Define the process $\widetilde{U} = (\widetilde{U}_t)_{t \geq 0}$ by
${\widetilde{U}_t := \frac{\epsilon}{2} e^{-\gamma t}\cEX[t]{\mathbf{1}_{\{U_T\geq\epsilon\}}} \mathbf{1}_{\{t\in[T,\tau_\epsilon)\}}}$. Then $\widetilde{U}$ is dominated by $U$. Moreover, Proposition \ref{prop:proper solution discounted 1_[sigma,tau)} for $\hat U_t = e^{-\gamma t} \mathbf{1}_{\{T \leq t < \tau_\epsilon\}}$ with corresponding solution $\hat W$, Lemma \ref{lemma:utility process associated to 1_A U} for $t_0=T$ and $A = \{ \hat U_T \geq \epsilon \}$ and Jensen's inequality show that there exists a solution $\widetilde{W}$ associated to $\widetilde{U}=(\widetilde{U}_t)_{t\geq0}$ such that for all $t \geq 0$
\begin{align}
\widetilde{W}_t \geq&~\cEX[t]{\mathbf{1}_{\{U_T\geq\epsilon\}}\left(\frac{\epsilon}{2\gamma\theta}\cEX[t]{(e^{-\gamma (t\vee T)}- e^{-\gamma(t\vee\tau_\epsilon)})}\right)^\theta}
\\
\geq&~\left(\frac{\epsilon}{2\gamma\theta}\cEX[t]{\mathbf{1}_{\{U_T\geq\epsilon\}}}\cEX[t]{(e^{-\gamma (t\vee T)}- e^{-\gamma(t\vee\tau_\epsilon)})}\right)^\theta.
\end{align}
Since $\{\tau_\epsilon > T\}$ on $\{U_T \geq\epsilon\}$, it follows that $\tilde W_{t^*} > 0$ on $\{U_T \geq\epsilon\}$. Now the claim follows from the fact that $W_{t^*}\geq\widetilde{W}_{t^*}$ by Proposition \ref{prop:maximal solutions increasing in U}.
\end{proof}
\subsection{Uniqueness of proper solutions}
We now turn to uniqueness of proper solutions. The aim of this section will be to prove Theorem \ref{thm:O^1-R_+ subset UP} and then, as a corollary, Theorem \ref{thm:three solutions coincide}. The following two lemmas will be useful.
\begin{lemma}\label{lem:J = Me^A}
Fix $X \in \mathds{S}\mathds{O}$, and let $J^X=(J^X_t)_{t\geq0}$ be defined by $J^X_t = \cEX[t]{\int_t^\infty X_s ds}$. Then, there exists a martingale $M=(M_t)_{t\geq0}$ such that $J^X_t = M_t e^{- \int_0^t (X_s/J^X_s) ds }$.
\end{lemma}
\begin{proof}
Let $N=(N_t)_{t\geq0}$ be the uniformly integrable martingale given by $N_t = \cEX[t]{\int_0^\infty X_s \,\mathrm{d} s}$. Then $J^X_t = N_t - \int_0^t X_s \,\mathrm{d} s$. Define the increasing process $A$ by $A_t = \int_0^t (X_s/J^X_s) \,\mathrm{d} s$. Since $X \in \mathds{S}\mathds{O}$ we have $0< X_t \leq K J^X_t$ for some $K$ and hence $0 < A_t \leq Kt$.
Define $M$ via $M_t = e^{A_t} J^X_t$. Then $dM_t = e^{A_t} dJ^X_t + X_t e^{A_t} dt = e^{A_t} dN_t$ and all that remains to show is that the local martingale $M$ is a martingale. Since $A$ is increasing and $A_t \leq Kt$, we have $\mathbb{E}[\lVert e^{A}\rVert_T|N_T|]\leq\mathbb{E}[(e^{KT}-1)|N_T|]<\infty$ for $T\geq0$, where $\lVert e^{A}\rVert_T$ is the total variation of $e^A=(e^{A_t})_{t\geq0}$ at time $T$. Hence, $M$ is a martingale by \cite[Lemma A.1]{herdegen2019sensitivity}.
\end{proof}
\begin{lemma}\label{lemma:X>1 finite time}
Let $\alpha>0$, $\beta\in(0,1)$ and let $G=(G_t)_{t\geq0}$ be a c\`adl\`ag submartingale. Suppose that $X=(X_t)_{t\geq0}$ is is a right-continuous process such that $X_0=\beta$ and $X_t\leq1$ for all $t\geq0$. Define $\tau=\inf\{t\geq0:~X_t=1\}$ and suppose that
\begin{equation}\label{eq:Xdynam}
\,\mathrm{d} X_t \geq \alpha X_t \,\mathrm{d} t + \,\mathrm{d} G_t, \quad\text{for all }t<\tau.
\end{equation}
Then, there exists $\nu \in (0,1)$ and $T \in (0,\infty)$ such that $\P(\tau<T)>\nu$.
\end{lemma}
\begin{proof}
First note that $X$ is a (local) submartingale bounded above by 1 and so converges almost surely to an $\mathcal{F}_\infty$-measurable random variable $X_\infty\leq1$ by the Martingale Convergence Theorem.
Fix $\xi\in(0,\beta)$. Let $\sigma=\inf\{t\geq0:~X_t\notin(\xi,1)\}\leq\tau$. From the dynamics of $X$ given in \eqref{eq:Xdynam},
$$X_{t\wedge\sigma} \geq X_0 +\int_0^{t\wedge\sigma} \alpha X_s \,\mathrm{d} s + G_{t\wedge\sigma}-G_0 \geq \beta+\alpha\xi (t\wedge\sigma) + G_{t\wedge\sigma}-G_0.$$
Then, using that $G$ is a c\`adl\`ag submartingale and the Optional Sampling Theorem,
\begin{equation}\label{eq:stoppped X inequality}
\EX{X_{t\wedge\sigma}}\geq \beta+\alpha\xi \EX{t\wedge\sigma}.
\end{equation}
Since $X\leq1$, taking the $\limsup$ and using the Reverse Fatou's Lemma on the left hand side of \eqref{eq:stoppped X inequality} and the Monotone Convergence Theorem on the right hand side gives
\begin{equation}
1\geq\EX{X_\sigma}=\EX{\mathbf{1}_{\sigma<\infty}X_\sigma}+\EX{\mathbf{1}_{\sigma=\infty}X_\infty}\geq \beta+\alpha\xi \EX{\sigma}\geq \beta.
\end{equation}
Therefore, $\mathbb{E}[\sigma] \leq \frac{1-\beta}{\alpha \xi}$ and $\P(\sigma=\infty)=0
. Consequently, since $X$ is right-continuous, $X_\sigma\in(-\infty,\xi]\cup\{1\}$ $\as{\P}$ and
\begin{equation}
1 - (1-\xi)\P(X_\sigma\leq\xi)=\P(X_\sigma=1)+\xi\P(X_\sigma\leq\xi)\geq\EX{X_\sigma}\geq \beta.
\end{equation}
In particular, $\P(X_\sigma\leq\xi)\leq \frac{1-\beta}{1-\xi}$ and $\P(\sigma=\tau)=\P(X_\sigma=1)\geq 1- \frac{1-\beta}{1-\xi} = \frac{\beta-\xi}{1-\xi}$. Furthermore,
\begin{align}
\P(\sigma\geq T; \sigma=\tau)\leq \EX{\frac{\sigma}{T}\mathbf{1}_{\sigma\geq T}\mathbf{1}_{\sigma=\tau}}\leq \frac{1}{T}\EX{\sigma}\leq \frac{1-\beta}{\alpha\xi T}, \quad \text{for all }T\geq0,
\shortintertext{and}
\P(\tau<T)\geq\P(\sigma<T;\sigma=\tau)=\P(\sigma=\tau)-\P(\sigma\geq T;\sigma=\tau)\geq\frac{\beta-\xi}{1-\xi} - \frac{1-\beta}{\alpha\xi T}.
\end{align}
Choose $\nu = \frac{1}{2}\left(\frac{\beta-\xi}{1-\xi}\right)$ and $T = \frac{1-\beta}{\alpha\xi \nu}$. Then, $\P(\tau<T)\geq \nu$.
\end{proof}
We may now prove Theorem \ref{thm:O^1-R_+ subset UP} in the $(U,W)$ coordinates.
\begin{proof}[Proof of Theorem \ref{thm:O^1-R_+ subset UP}]
Fix $U \in\mathds{S}\mathds{O}^\theta_+$ and recall $J=(J_t)_{t\geq0}$ is given by $J_t = \cEX[t]{\int_t^\infty U^\theta_s \,\mathrm{d} s}$. Suppose that $U^\theta\in \mathds{S}\mathds{O}(k,K)$ and let $W\in\mathds{O}(J)$ be the solution associated to $(h_{EZ},U)$ found in Proposition \ref{prop:existence of a solution, bounded and perturbed} (after setting $\epsilon = 0$ and $U = \Lambda$). By Proposition \ref{prop:fixed point solution is maximal solution for O(Lambda)}, $W$ is the unique maximal solution, and as we saw in the proof of Theorem~\ref{thm:proper solution rc} it is also proper. We now prove uniqueness.
For contradiction, assume that there exists a proper solution $Y=(Y_t)_{t\geq0}$ such that $Y\neq W$. Since $W$ is maximal, $Y\leq W$. Then, since $W$ is unique in the class $\mathds{O}(J)$, it follows that $Y\stackrel{\mathds{O}}{\neq}J$. Hence, there exists $t\geq0$, $B\in\mathcal{F}_t$ and $\epsilon>0$ such that $\P(B)>\epsilon$ and $Y_t<kK^{-\theta} J_t$ on $B$. For ease of exposition, assume that $t=0$, $B=\Omega$ and $Y_0=y=(1-\epsilon)kK^{-\theta}J_0$. The general case is similar.
Define $Z=(Z_t)_{t\geq0}$ by $Z_t=Y_t(kK^{-\theta}J_t)^{-1}$ and note that $Z$ is c\`adl\`ag by Remark \ref{rem:solutions are utility processes} and $Z_0=(1-\epsilon)$. Since $kK^{-\theta}JZ \equiv Y$ is a utility process and since $U^\theta\geq kJ$, for all $t\leq T$,
\begin{align}
kK^{-\theta}J_t Z_t = Y_t =~ &\cEX[t]{\int_t^T U_s Y_s^\rho \,\mathrm{d} s+ Y_T}
\\
\geq~&\cEX[t]{\int_t^T (kJ_s)^\frac{1}{\theta} (kK^{-\theta}J_sZ_s)^\rho \,\mathrm{d} s+kK^{-\theta}J_T Z_T}.
\intertext{By Lemma \ref{lem:J = Me^A}, we find that $J_t = M_t e^{-A_t}$ for $A_t = \int_0^t (U_s^\theta/J_s) ds$. Hence, dividing by $kK^{-\theta}M_t$ and collecting terms gives}
e^{-A_t}Z_t \geq~ \frac{1}{M_t}&\cEX[t]{\int_t^T K M_s e^{-A_s}Z_s^\rho \,\mathrm{d} s+M_T e^{-A_T} Z_T}.
\end{align}
Define $\widetilde{Z}_t=e^{-A_t}Z_t$ and consider an equivalent measure $\widetilde{\P}$ defined by $\frac{\mathrm{d}\widetilde{\P}}{\mathrm{d} \P}\big|_{\mathcal{F}_t} = M_t$, so that
\begin{equation}\label{eq:Zeq first}
\widetilde{Z}_t \geq \cEXt[t]{\int_t^T K e^{-(1-\rho)A_s} \widetilde{Z}_s^\rho \,\mathrm{d} s +\widetilde{Z}_T}, \quad \text{for all }t\leq T.
\end{equation}
If we define $O$ by $O_t = \widetilde{Z}_t + \int_0^t K e^{-(1-\rho)A_s} \widetilde{Z}_s^\rho \,\mathrm{d} s$, then $O$ is a c\`adl\`ag supermartingale. By the Doob--Meyer decomposition, $O$ can therefore be decomposed as $O = N + P$, where $N$ is a local martingale and $P$ is a decreasing process such that $P_0=0$, both of which are c\`adl\`ag. In particular, rearranging gives
\begin{equation}\label{eq:subsolution as subsolution to BSDE?}
\,\mathrm{d}\widetilde{Z}_t = - K e^{-(1-\rho)A_t}\widetilde{Z}_t^\rho \,\mathrm{d} t + \,\mathrm{d} N_t -\,\mathrm{d} P_t.
\end{equation}
Let $\sigma\coloneqq \{t\geq0: Z_t\notin(0,1)\}$, $\hat{Z}_t=Z_{t\wedge{\sigma}}$, $\hat{N}_t = \int_0^{t\wedge\sigma}e^{A_s}\,\mathrm{d} N_s$ and $\hat{P}_t=\int_0^{t\wedge \sigma} e^{A_s}\,\mathrm{d} P_s$. Then, applying the product rule to $\hat{Z}_t=e^{A_t}\widetilde{Z}_t$ up to $t\leq \sigma$, and noting that $\frac{\,\mathrm{d} A_t}{\,\mathrm{d} t} = \frac{U_t^\theta}{J_t}\leq K$,
\[ \,\mathrm{d} \hat{Z}_t = \hat{Z}_t\,\mathrm{d} A_t - K\hat{Z}_t^\rho \,\mathrm{d} t+ \,\mathrm{d} \hat{N}_t - \,\mathrm{d} \hat{P}_t
\leq K(\hat{Z}_t - \hat{Z}_t^\rho)\,\mathrm{d} t + \,\mathrm{d} \hat{N}_t \leq \,\mathrm{d} \hat{N}_t.\]
Since $\hat{N}_t \geq \hat{Z}_t - \hat{Z}_0 \geq - \hat{Z}_0$, $\hat{N}$ is a supermartingale.
Let $X_t=1-\hat{Z}_t^{1-\rho}\leq 1$. Then, $X=(X_t)_{t\geq0}$ is c\`adl\`ag and, for $t<\sigma$,
\begin{align}
\,\mathrm{d} X_t =&~ -\!{(1-\rho)}\hat{Z}_t^{-\rho}\,\mathrm{d} \hat{Z}_t + \frac{\rho(1-\rho)}{2}\hat{Z}_t^{-(\rho+1)}\,\mathrm{d} \langle Z\rangle_t
\\
=&~ -\!{(1-\rho)}(\hat{Z}_t^{1-\rho}\,\mathrm{d} A_t -K\,\mathrm{d} t) + \,\mathrm{d} L_t + \,\mathrm{d} Q_t
\\
\geq&~ K(1-\rho)X_t \,\mathrm{d} t + \,\mathrm{d} L_t + \,\mathrm{d} Q_t \geq \,\mathrm{d} L_t
\end{align}
where
\begin{equation}
L_t \coloneqq -{(1-\rho)}\!\!\int_0^t \!\hat{Z}^{-\rho}_s\,\mathrm{d} \hat{N}_s \quad \text{and} \quad Q_t\coloneqq(1-\rho)\!\!\int_0^t\!\hat{Z}^{-\rho}_s\,\mathrm{d} \hat{P}_s + \int_0^t \frac{\rho(1-\rho)}{2}\hat{Z}^{-(\rho+1)}\,\mathrm{d} \langle Z\rangle_s.
\end{equation}
Since $L_t \leq X_t - X_0 \leq 1$, $L$ is a (continuous) submartingale. Hence, $G \coloneqq L+Q$ is a continuous submartingale. The result that $X$ explodes to $1$ in finite time with positive probability follows from Lemma \ref{lemma:X>1 finite time}. This implies that $Z$ hits zero in finite time and, consequently, that $Y$ is not proper.
\end{proof}
\begin{proof}[Proof of Theorem \ref{thm:three solutions coincide}]
Let $W$ be the unique solution such that $W\stackrel{\mathds{O}}{=}J$ given by Proposition \ref{prop:O solution for C^1-R in SO} ($V=\frac{W}{1-R}$ is the CRRA-order solution). Since $J>0$, $W$ is proper and uniqueness in the class of proper solutions follows from Theorem \ref{thm:O^1-R_+ subset UP}. Finally, $W$ is the maximal solution by Proposition \ref{prop:fixed point solution is maximal solution for O(Lambda)}.
\end{proof}
\section{Verification of the candidate optimal strategy}\label{sec:verification}
This section aims to prove that the candidate optimal strategy given in Proposition \ref{prop:derivecandidate} is indeed optimal. We will roughly follow the approach detailed in \cite{herdegen2021infinite} which goes as follows: first, show that if $\hat{X}=X^{\hat{\Pi},\hat{C}}$ is the wealth process under the candidate optimal strategy, then $\hat{V}(X+\epsilon\hat{X})$ is a supersolution for $(f_{EZ},C)$ (The reasons that we perturb the input of $\hat{V}=\hat{V}(X)$ by the optimal wealth process are to ensure that we can apply It\^o's lemma to $\hat{V}$, to make sure that the local martingale part of $\hat{V}$ is a supermartingale, and to ensure that $\hat{V}$ satisfies the supersolution transversality condition. This is explained in more detail in \cite{herdegen2020elementary} and \cite[Section~11]{herdegen2021infinite}); next, use a version of the Comparison Theorem (Corollary \ref{cor:comparison C^1-R < K (1-R)V}) for sub- and supersolutions to conclude that $\hat V(x(1+\epsilon)) \geq V^C_0$; finally, letting $\epsilon\searrow0$ gives $\hat V(x) \geq V^C_0$. Optimality follows since $V^{\hat{C}}_0 = \hat V (x)$ by Proposition \ref{prop:derivecandidate}.
However, the approach is not quite this simple for two main reasons. The most pressing reason is that when $\theta>1$ the utility process $V^C$ fails to be unique. The optimisation therefore takes place over the attainable and right-continuous consumption streams for which there exists a unique proper solution to the EZ-SDU equation. As we have argued in Section~\ref{sec:proper solution}, the proper solution is the economically meaningful solution, and we may only consider the consumption streams that have a unique proper solution associated to them. The assumption of right-continuity, which is necessary for the proof, is not overly restrictive.
The next issue is that the hypotheses of the relevant comparison theorem (Corollary \ref{cor:comparison C^1-R < K (1-R)V}) are not satisfied, even for right-continuous consumption streams with a unique proper solution. To overcome this issue, one must approximate an arbitrary consumption stream in $\mathds{U}\mathds{P}^*$ by a series of consumption streams satisfying the conditions and then take limits. The requirement of right-continuity ensures that we may choose right-continuous approximating consumption streams, which then have an associated proper solution $V^n$ by Theorem \ref{thm:proper solution rc}. Since the limiting process $V=\lim_{n\to\infty}V^n$ is a proper solution associated to $C$, it must agree with the unique proper solution $V^C$.
To prove Theorem \ref{thm:verification}, we will use the following lemma, which is proved as an intermediate part of \cite[Theorem 11.1]{herdegen2021infinite}. The result in \cite{herdegen2021infinite} is written for the case $0<\theta<1$, but it is not difficult to check that the argument extends to the case $\theta > 1$.
\begin{lemma}\label{lem:V hat (X+eps Y) a supersolution}
Let $\epsilon>0$ and let $\hat{X}=X^{\hat{\Pi},\hat{C}}$ denote the wealth process under our candidate optimal strategy. Fix $(\Pi,C)$ and let $X = X^{\Pi,C}$ denote the corresponding wealth process. Then, $\hat{V}(X + \epsilon \hat{X})$ is a supersolution for the pair $(f_{EZ},C+\eta\epsilon \hat{X})$.
\end{lemma}
We may then prove Theorem \ref{thm:verification}. Note that $\hat{V}(\hat{X})$ is a solution for the pair $(f_{EZ}, \eta \hat{X})$ and, by scaling, $\hat{V}( \epsilon \hat{X})$ is a solution for the pair $(f_{EZ},\eta\epsilon\hat{X})$. We expect that $\hat{V}(X^{\Pi,C})$ is a supersolution for $(f_{EZ},C)$ but, when $R>1$, the transversality condition might not hold. Furthermore, the conditions required for Proposition \ref{prop:comparison} to hold may be impossible to verify. However, as we show in the proof below, by considering the perturbed problem, the transversality condition is guaranteed and the comparison theorem can be applied.
\begin{proof}[Proof of Theorem \ref{thm:verification}]
It follows from Proposition \ref{prop:derivecandidate} that $\hat{V}(\hat{X})$ is a utility process associated to candidate optimal strategy $(\hat{\Pi}, \hat{C})$. Since $\hat{V}(\hat{X})$ is a CRRA-order solution to \eqref{eq:Epstein--Zin SDU}, it is the unique proper solution to \eqref{eq:Epstein--Zin SDU} by Theorem \ref{thm:three solutions coincide}. Hence, $V^{\hat{C}}_0 = \hat{V}(x)$. It therefore only remains to show that $V^C_0\leq\hat{V}(x)$ for all $C \in \mathscr{C}(x)\cap\mathds{U}\mathds{P}^*$. Fix an arbitrary $C \in \mathscr{C}(x)\cap\mathds{U}\mathds{P}^*$ and let $\Pi=(\Pi_t)_{t\geq0}$ be an associated investment process.
We first prove the result when $R>S>1$. In this case $V^C$ is the minimal solution (since ${C\in\mathds{U}\mathds{P}}$ and the minimal solution is proper). By Lemma \ref{lem:V hat (X+eps Y) a supersolution}, for each $\epsilon>0$, $\hat{V}(X^{C,\Pi} + \epsilon \hat{X})$ is a supersolution associated to $C^\epsilon=C+\eta\epsilon\hat{X}$.
Since $(C^\epsilon)^{1-S}\leq( \eta\epsilon)^{1-S}\hat{X}^{1-S}$, there exists an extremal solution $V^{C^\epsilon}$ associated to $C^\epsilon$ by Proposition \ref{prop:extremal solution for C^1-R < hatC^1-R} which is increasing in $\epsilon$ by Proposition \ref{prop:maximal solutions increasing in U}. It is the minimal supersolution by Corollary \ref{cor:extremal is maximal subsolution}. Hence, by minimality, $V^{C^\epsilon}_t\leq\hat{V}(X^{C,\Pi}_t + \epsilon \hat{X}_t)<0$ for all $t\geq0$ and $V^{C^\epsilon}$ is proper.
Let $V^*=\lim_{\epsilon\to0}V^{C^\epsilon}$. Then, $V^*_0\leq\hat{V}(x)$. Consequently, since $f_{EZ}$ is increasing in both arguments, and $C^\epsilon$ and $V^\epsilon$ are increasing in $\epsilon$, $f_{EZ}(C^\epsilon,V^\epsilon)$ is increasing in~$\epsilon$, and applying the Monotone Convergence Theorem for conditional expectations yields
\begin{equation*}
V^*_t = \lim_{\epsilon\to0}V^{C^\epsilon}_t = \lim_{\epsilon\to0}\cEX[t]{\int_t^\infty f_{EZ}(C^\epsilon_s,V^{C^\epsilon}_s) \,\mathrm{d} s} = \cEX[t]{\int_t^\infty f_{EZ}(C_s,V^*_s) \,\mathrm{d} s}.
\end{equation*}
Therefore, $V^*$ is a solution associated to $(f_{EZ},C)$. Since $V^*_t = \lim_{\epsilon\to0}V^{C^\epsilon}_t<0$ for all $t\geq0$, $V^*$ is proper. It therefore agrees with the unique proper solution $V^C$ so that $V^C_0\leq\hat{V}(x)$.\\
We now prove the result when $R<S<1$. Fix an arbitrary $C\in\mathscr{C}(x)\cap\mathds{U}\mathds{P}^*$ with associated investment process $\Pi=(\Pi_t)_{t\geq0}$.
Let $0<\zeta<\eta\frac{S}{1-S}$ and define $\widetilde{X}_t=e^{\zeta t}X^{C,\Pi}_t$, $Y_t = e^{\frac{\zeta}{S}t}\hat{X}_t$ and $\widetilde{C}_t=e^{\zeta t}C_t$ for $t\geq0$. Note that if ${r_\zeta=r+\zeta}$ and $\mu_\zeta=\mu+\zeta$, then
\begin{eqnarray*}
\,\mathrm{d} \widetilde{X}_t &=& \widetilde{X}_t \Pi_t \sigma \,\mathrm{d} B_t + \left( \widetilde{X}_t (r_\zeta + \Pi_t (\mu_\zeta- r_\zeta)) - \widetilde{C}_t\right)\mathrm{d} t
\end{eqnarray*}
We may think of $\widetilde{X}=(\widetilde{X}_t)_{t\geq0}$ as being the wealth process associated to the strategy $(\Pi,\widetilde{C}=(\widetilde{C}_t)_{t\geq0})$ in a more favourable financial market with risk-free rate $r_\zeta$, drift of the risky asset $\mu_\zeta$, and well-posedness parameter $\eta_\zeta = -\frac{1-S}{S}(r_\zeta + \frac{\lambda^2}{2R}) = \eta - \frac{1-S}{S}\zeta \in (0,\eta)$. The volatility is unchanged.
Furthermore, since
\begin{equation}
\frac{dY_t}{Y_t } = \frac{\lambda}{R} \,\mathrm{d} B_t + \left(\left(r + \frac{\lambda^2}{R} - \eta\right) + \frac{\zeta}{S}\right)\,\mathrm{d} t = \frac{\lambda}{R} \,\mathrm{d} B_t + \left( r_\zeta + \frac{\lambda^2}{R} - \eta_\zeta \right)\,\mathrm{d} t,
\end{equation}
$Y=(Y_t)_{t\geq0}$ is the wealth process under the optimal strategy in the new financial market with parameters $r_\zeta$ and $\mu_\zeta$. Define $\hat{V}^\zeta(x) = \eta_\zeta^{-\theta S }\frac{x^{1-R}}{1-R}$. Then, $\hat{V}^\zeta(\widetilde{X}+\epsilon Y)$ is a supersolution for $(f_{EZ},\widetilde{C}+\eta \epsilon Y)$ by Lemma \ref{lem:V hat (X+eps Y) a supersolution} and then also for $(f_{EZ},C)$ since $\widetilde{C}+\eta\epsilon Y\geq C$.
Let $C^n\coloneqq C\wedge n\hat{X}$. Then, there exists an extremal solution $V^{C^n}$ associated to $C^n$ by Proposition \ref{prop:extremal solution for C^1-R < hatC^1-R} which is monotonously increasing in $n$ by Proposition \ref{prop:maximal solutions increasing in U}. Also, $\hat{V}^\zeta = \hat{V}^\zeta(\widetilde{X}+\epsilon Y)$ is a supersolution for $(f_{EZ},C^n)$ since $C\geq C^n$. Furthermore, since $C$ is right-continuous, $C^n$ is right-continuous. The extremal solution $V^{C^n}$ associated to $C^n$ is therefore proper by Theorem \ref{thm:proper solution rc} and Remark \ref{rem:maximal solution proper}.
Next, using that $(C^n)^{1-R}\leq n^{1-R}\hat{X}^{1-R}$, we obtain
\begin{align}
(1-R)\hat{V}^\zeta(\widetilde{X}+\epsilon Y) &\geq \eta_\zeta^{-\theta S }(\epsilon Y)^{1-R} = \eta_\zeta^{-\theta S }\epsilon^{1-R} e^{\frac{\zeta(1-R)}{S}t}\hat{X}^{1-R} \\
&\geq \frac{\eta_\zeta^{-\theta S }\epsilon^{1-R} }{n^{1-R}}e^{\frac{\zeta(1-R)}{S}t} (C^n)^{1-R}.
\end{align}
Furthermore,
$V^{C^n}\in\mathds{U}\mathds{I}(f_{EZ},C^n)$ by Remark \ref{rem:utility process UI} and $$\EX{(V^{C^n}_t-\hat{V}^\zeta_t)^+}\leq \EX{V^{C^n}_t}=\EX{\int_t^\infty f_{EZ}(C^n_s,V^{C^n}_s)\,\mathrm{d} s}\leq \EX{\int_0^\infty f_{EZ}(C^n_s,V^{C^n}_s)\,\mathrm{d} s}<\infty.$$ Hence, the conditions of Corollary \ref{cor:comparison C^1-R < K (1-R)V} are met and $\hat{V}^\zeta_t \geq V^{C^n}_t$ for all $t\geq0$. In particular, $\hat{V}^\zeta_0 \geq V^{C^n}_0$. If $V^*=\lim_{n\to\infty}V^{C^n}$ is the monotone limit, then
\begin{equation}
V^*_t = \lim_{n\to\infty}V^{C^n}_t = \lim_{n\to\infty}\cEX[t]{\int_t^\infty f_{EZ}(C^n_s, V^{C^n}_s)\,\mathrm{d} s}=\cEX[t]{\int_t^\infty f_{EZ}(C_s, V^*_s)\,\mathrm{d} s}.
\end{equation}
Hence, $V^*$ is a solution. It is a proper solution since for each $t \geq 0$, $V^*_t > 0$ if $V^{C^n}_t>0$ for some $n$ and $V^{C^n}_t>0$ on $\{J^{(C^n)^{1-R}} > 0\} = \{J^{C^{1-R}} > 0\}$ up to null sets by the fact that each $V^{C^n}$ is proper and $\hat X$ is strictly positive. Therefore, $V^*$ must agree with the unique proper solution $V^C$ associated to $C$. In particular, $\hat{V}^\zeta_0=\hat{V}^\zeta(x(1+\epsilon))\geq \lim_{n\to\infty} V^{C^n}_0 =V^*_0 = V^C_0$. Finally, taking $\zeta,\epsilon\searrow0$ gives $\hat{V}(x)\geq V^C_0$.
\end{proof}
\bibliographystyle{plain}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 2,487 |
Q: Criteria api two table using foreign keys and where clause I have 3 entities where id are a primary keys and building_id are foreign keys and
I'm trying to create an SQL query which would be something like this
simplified SQL of what I want to accomplish
select
t.id as id,
t1.building_id as building_id,
t2.building_id as building_id
from
my_db.building t,
my_db.owner t1,
my_db.janitor t2
where
t.id = t1.building_id
and t.id = t2.building_id
and t.name = ?1
I want to generate the same SQL using criteria api. Here the same means that it uses the same where clause to join foreign keys with primary keys. I don't want any joins, unions, etc.
But, for the entities below when I use
CriteriaBuilder cb = em.getCriteriaBuilder();
CriteriaQuery<Tuple> cq = cb.createTupleQuery();
Root<Building> buildingRoot = cq.from(Building.class);
Root<Janitor> janitorRoot = cq.from(Janitor.class);
Root<Owner> ownerRoot = cq.from(Owner.class);
cq.multiselect(buildingRoot.get(Building_.id),
janitorRoot.get(Janitor_.buildingId),
ownerRoot.get(Owner_.buildingId))
.where(cb.and(
cb.equal(buildingRoot.get(Building_.id), ownerRoot.get(Owner_.buildingId)),
cb.equal(buildingRoot.get(Building_.id), requestProcedureRoot.get(janitor_.requestId))
cb.equal(buildingRoot.get(Building_.name, buildingName)))
I get a generated query and it has cross joins in it
select
request0_.id as col_0_0_,
request0_.building_id as col_1_0_,
requestpro1_.building_id as col_2_0_
from
my_db.building building0_
cross join my_db.owner owner0_
cross join my_db.janitor janitor0_
where
building0_.name = ?
and building0_.id = owner0_.building_id
entities
public class Building {
@Id
@Column(name="id")
private Long id;
@Column(name="name")
private String name;
@OneToMany(fetch = FetchType.LAZY, mappedBy = "building")
private List<Owner> owners;
@OneToMany(fetch = FetchType.LAZY, mappedBy = "building")
private List<Janitor> janitors;
}
public class Owner {
@Id
@Column(name="id")
Long id;
@Column(name = "building_id")
Long buildingId;
@ManyToOne
@JoinColumn(name = "building_id")
Building building;
}
@Entity
public class Janitor {
@Id
@Column(name="id")
private Long id;
@Column(name = "building_id")
private Long buildingId;
@ManyToOne
@JoinColumn(name = "building_id")
private Building building;
}
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 1,253 |
Aravind Swamy Looks Handsome As Former Tamil Nadu Cm
A recent picture of Aravind Swamy is doing rounds on social media where he looks exactly like Tamil legendary actor MG Ramachandran
Written By Aravind Peesapati | Updated: November 14, 2019 17:49 IST
Popular actor Aravind Swamy who recently played a crucial role in Mani Ratnam's directorial 'Chekka Chivantha Vaanam' is now going to play a key role in 'Thalaivi'. The news has come out long back and the actor will soon take part in the upcoming shooting schedule.
Aravind Swamy is playing the role of legendary Tamil actor and former Tamil Nadu chief minister MG Ramachandran in 'Thalaivi' movie. Starring Kangana Ranaut, 'Thalaivi' is the much-awaited biopic on veteran actress and former Tamil Nadu Chief Minister J Jayalalitha. Recently, a picture of Aravinda Swamy in MGR's look got released on the internet and has been grabbing the attention. Aravinda Swamy is looking handsome and is just like MGR in the picture.
AL Vijay is helming this project. The movie is going to get released in 2020 in Hindi, Tamil, and Telugu languages. Vishnu Induri and Shailesh Singh are funding this project under Vibri Media Works banner. A Hollywood makeup artist is on board to work on the look of Kangana in the movie. | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 2,133 |
{"url":"https:\/\/brilliant.org\/discussions\/thread\/erdosmoser-equation-proof-part-1-setting-the\/","text":"# Erd\u0151s\u2013Moser Equation Proof (Part $1$ - Setting the Boundaries and Definition)\n\nIn number theory, the Erd\u0151s\u2013Moser equation is defined as\n\n$1^k + 2^k + ... + m^k = (m + 1)^k$\n\nOf course, a well-known trivial solution is\n\n$1^1 + 2^1 = 3^1$\n\nHowever, Paul Erd\u0151s conjectured that no further solutions exist to the Erd\u0151s\u2013Moser equation other than\n\n$1^1 + 2^1 = 3^1$\n\nSo, now what?\n\nWell, a solution to find $m$ is\n\n$m^k - (m + 1)^k = - 1 - 2^k$\n\nsince\n\n$1^k = 1$\n\nLeo Moser also posed certain constraints on the solutions:\n\n$1$. $\\frac{2}{k} = x$, where $x$ is a positive number and that there is no other solution for $m < 10^{1,000,000}$, which Leo Moser himself proved in $1953$.\n\n$2$. $k + 2 < m < 2k$, which was shown in $1966$.\n\n$3$. $lcm (1, 2, ..., 200)$ divides $k$ and that any prime factor of $m + 1$ must be irregular and $> 10,000$, which was shown in $1994$.\n\n$4$. In $1999$, Moser's method was extended to show that $m > 1,485 \\times 10^{9,321,155}$ .\n\n$5$. In $2002$, it was shown that $200 < p < 1000$ must divide $k$, where $p$ is all the primes in between.\n\n$6$. In $2009$, it was shown that $\\frac{2k}{2m - 3}$ must be a convergent of $In (2)$ - large-scale computation was used to show that $m > 2.7139 \\times 10^{1,667,658,416}$.\n\nNote by Yajat Shamji\n5\u00a0months, 3\u00a0weeks ago\n\nThis discussion board is a place to discuss our Daily Challenges and the math and science related to those challenges. Explanations are more than just a solution \u2014 they should explain the steps and thinking strategies that you used to obtain the solution. Comments should further the discussion of math and science.\n\nWhen posting on Brilliant:\n\n\u2022 Use the emojis to react to an explanation, whether you're congratulating a job well done , or just really confused .\n\u2022 Ask specific questions about the challenge or the steps in somebody's explanation. Well-posed questions can add a lot to the discussion, but posting \"I don't understand!\" doesn't help anyone.\n\u2022 Try to contribute something new to the discussion, whether it is an extension, generalization or other idea related to the challenge.\n\u2022 Stay on topic \u2014 we're all here to learn more about math and science, not to hear about your favorite get-rich-quick scheme or current world events.\n\nMarkdownAppears as\n*italics* or _italics_ italics\n**bold** or __bold__ bold\n- bulleted- list\n\u2022 bulleted\n\u2022 list\n1. numbered2. list\n1. numbered\n2. list\nNote: you must add a full line of space before and after lists for them to show up correctly\nparagraph 1paragraph 2\n\nparagraph 1\n\nparagraph 2\n\n[example link](https:\/\/brilliant.org)example link\n> This is a quote\nThis is a quote\n # I indented these lines\n# 4 spaces, and now they show\n# up as a code block.\n\nprint \"hello world\"\n# I indented these lines\n# 4 spaces, and now they show\n# up as a code block.\n\nprint \"hello world\"\nMathAppears as\nRemember to wrap math in $$ ... $$ or $ ... $ to ensure proper formatting.\n2 \\times 3 $2 \\times 3$\n2^{34} $2^{34}$\na_{i-1} $a_{i-1}$\n\\frac{2}{3} $\\frac{2}{3}$\n\\sqrt{2} $\\sqrt{2}$\n\\sum_{i=1}^3 $\\sum_{i=1}^3$\n\\sin \\theta $\\sin \\theta$\n\\boxed{123} $\\boxed{123}$\n\n## Comments\n\nThere are no comments in this discussion.\n\n\u00d7\n\nProblem Loading...\n\nNote Loading...\n\nSet Loading...","date":"2021-07-31 10:29:16","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 42, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 1, \"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\": 0.9750093817710876, \"perplexity\": 2018.1282106622573}, \"config\": {\"markdown_headings\": true, \"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-2021-31\/segments\/1627046154085.58\/warc\/CC-MAIN-20210731074335-20210731104335-00034.warc.gz\"}"} | null | null |
Colin McFadden, Author at Why Should I Visit?
I love to travel because it gives me hope. Whether it's a small town in rural America or a city halfway around the world, I believe in the value of exploring, learning, and reflecting. In my non-travel life, I develop software and technology to enhance learning and research in higher education.
You know what goes well with meat? Meat. | {
"redpajama_set_name": "RedPajamaC4"
} | 5,137 |
Склад збірної Швеції на чемпіонаті Європи 2008
Докладніше…Швеція
Швеція 2008 | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 4,264 |
\section{Introduction}
\label{sec1:level1}
As a quantum-mechanical analogs of classical random walks, quantum walks have become increasingly popular in recent years, and have played a fundamental and important role in quantum computing. Owing to quantum superpositions and interference effects, quantum walks have been effectively used to simulate quantum phenomena \cite{bracken2007free}, and realize universal quantum computation \cite{childs2009universal,lovett2010universal}, as well as develop extensively quantum algorithms \cite{venegas2012quantum}. A wide variety of discrete quantum walk models have been successively proposed. The first quantization model of a classical random walk, which is the coined discrete-time model and which is performed on a line, was proposed by Aharonov \textit{et al}. \cite{aharonov1993quantum} in the early 1990s. Aharonov later studied its generalization for regular graphs in Ref. \cite{aharonov2001quantum}. Szegedy \cite{szegedy2004quantum} proposed a quantum walks model that quantizes the random walks, and its evolution operator is driven by two reflection operators on a bipartite graph. Moreover, in discrete models, the most-studied topology on which quantum walks are performed and their properties studied are a restricted family of graphs, including line \cite{nayak2000quantum,portugal2015one}, cycle\cite{bednarska2003quantum,melnikov2016quantum}, hypercube\cite{moore2002quantum,potovcek2009optimized}, and general graphs\cite{chakraborty2016spatial,wong2015faster,krovi2016quantum}. Indeed, most of the existing quantum walk algorithms are superior to their classical counterparts at executing certain computational tasks, e.g., element distinctness \cite{ambainis2007quantum,belovs2012learning}, triangle finding \cite{ magniez2007quantum,lee2013improved}, verifying matrix products \cite{buhrman2006quantum}, searching for a marked element \cite{shenvi2003quantum,krovi2016quantum},quantized Google's PageRank\cite{paparo2012google} and graph isomorphism \cite{douglas2008classical,gamble2010two,berry2011two,wang2015graph}.
In mass scenarios, a graph-based representation is incomplete, since graph edges can only represent pairwise relations between nodes. However, hypergraphs are a natural extension of graphs that allow modeling of higher-order relations in data. Because the mode of representation is even nearer to the human visual grouping system, hypergraphs are more available and effective than graphs for solving many problems in several applications. Owing to Zhou's random walks on hypergraphs for studying spectral clustering and semi-supervised ranking\cite{zhou2007learning}, hypergraphs have made recent headlines in computer vision \cite{ yu2014high,huang2017effect}, information retrieval\cite{hotho2006information,yu2014click,zhu2017unsupervised}, database design \cite{jodlowiec2016semantics}, and categorical data clustering \cite{ochs2012higher}. Many interesting and promising findings were covered in random walks on hypergraphs, and quantum walks provide a method to explore all possible paths in a parallel way, due to constructive quantum interference along the paths. Therefore, paying attention to quantum walks on hypergraphs is a natural choice.
In this paper, we focus on discrete-time quantum walks on regular uniform hypergraphs. By analyzing the mathematical formalism of hypergraphs and three existing discrete-time quantum walks\cite{portugal2016establishing}(coined quantum walks, Szegedy's quantum walks, and staggered quantum walks), we find that discrete-time quantum walks on regular uniform hypergraphs can be transformed into Szegedy's quantum walks on bipartite graphs that are used to model the original hypergraphs. Furthermore, the mapping is one to one. That is, we can study Szegedy's quantum walks on bipartite graphs instead of the corresponding quantum walks on regular uniform hypergraphs. In Ref. \cite{szegedy2004quantum}, Szegedy proved that his schema brings about a quadratic speed-up. Hence, we construct a model for quantum walks on bipartite graphs of regular uniform hypergraphs with Szegedy's quantum walks. In the model, the evolution operator of an extended Szegedy's walks depends directly on the transition probability matrix of the Markov chain associated with the hypergraphs.
In more detail, we first introduce the classical random walks on hypergraphs, in order to get the vertex-edge transition matrix and the edge-vertex transition matrix. We then define a bipartite graph that is used to model the original hypergraph. Lastly, we construct quantum operators on the bipartite graph using extended Szegedy's quantum walks, which is the quantum analogue of a classical Markov chain. In this work, we deal with the case that the cardinalities of the two disjoint sets can be different from each other in the bipartite graph. In addition, we deliver a slightly different version of the spectral properties of the transition matrix, which is the essence of the quantum walks. As a result, our work generalizes quantum walks on regular uniform hypergraphs by extending the classical Markov chain, due to Szegedy's quantum walks.
The paper is organized as follows. In Sec. II, we provide basic definitions for random walks on hypergraphs. In Sec. III, we construct a method for quantizing Markov chain to create discrete-time quantum walks on regular uniform hypergraphs. In Sec. IV, we analyze the eigen-decomposition of the operator. In Sec. V, we present conclusions and outlook on possible future directions.
\section{Review of random walks on hypergraphs}
\label{sec2:level1}
We start by defining some standard definitions of a hypergraph that will be used throughout this paper. We then briefly describe random walks on hypergraphs.
\subsection{Notations}
Let $HG=(V,E)$ denote a hypergraph, where $V$ is the vertex set of the hypergraph and $E\subset {{2}^{V}}\backslash \{\{\}\}$ is the set of hyperedges. $n=\left| V \right|$ is used to denote the number of vertices in the hypergraph and $m=\left| E \right|$ the number of hyperedges. Let $V=\{{{v}_{1}},{{v}_{2}},\cdots ,{{v}_{n}}\}$ and $E=\{{{e}_{1}},{{e}_{2}},\cdots ,{{e}_{m}}\}$. Given a hypergraph, define its incidence matrix $H\in {{R}^{n\times m}}$ as follows:
\begin{equation}
\begin{aligned}
h(i,j)=\left\{ \begin{matrix}
1 & \begin{matrix}
if & {{v}_{i}}\in {{e}_{j}} \\
\end{matrix} \\
0 & \begin{matrix}
if & {{v}_{i}}\notin {{e}_{j}} \\
\end{matrix} \\
\end{matrix} \right\}.
\end{aligned}
\label{eq:ClassificationModel}
\end{equation}
Note that the sum of the entries in any column is the degree of the corresponding edge. Similarly, the sum of the entries in a particular row is the degree of the corresponding vertex. Then, the vertex and hyperedge degrees are defined as follows:
\begin{equation}
\begin{aligned}
d(v)=\sum\limits_{e\in E}{h(v,e)=}\left| E(v) \right|
\end{aligned},
\label{eq:ClassificationModel}
\end{equation}
\begin{equation}
\begin{aligned}
E(v)=\{e\in E:v\in e\}
\end{aligned},
\label{eq:ClassificationModel}
\end{equation}
\begin{equation}
\begin{aligned}
\delta (e)=\sum\limits_{v\in V}{h(v,e)=}\left| e \right|
\end{aligned},
\label{eq:ClassificationModel}
\end{equation}
where $E(v)$ is the set of hyperedges incident to $v$. Let ${{D}_{v}}$ and ${{D}_{e}}$ denote the diagonal matrices of the degrees of the vertices and edges, respectively. A hypergraph is $d-regular$ if all its vertices have the same degree. Also, a hypergraph is $k-uniform$ if all its hyperedges have the same cardinality. In this paper, we will restrict our reach to quantum walks on $d-regular$ and $k-uniform$ hypergraphs from now on, denoting them as $H{{G}_{k,d}}$.
\subsection{Random walks on hypergraphs}
\label{sec2.1:level2}
A random walk on a hypergraph $HG=(V,E)$ is a Markov chain on the state space $V$ with its transition matrix $P$. The particle can move from vertex ${{v}_{i}}$ to vertex ${{v}_{j}}$ if there is a hyperedge containing both vertices. According to Ref. \cite{zhou2007learning}, a random walk on a hypergraph is seen as a two-step process. First, the particle chooses a hyperedge $e$ incident with the current vertex $v$. Then, the particle picks a destination vertex $u$ within the chosen hyperedge satisfying the following: $v,u\in e$. Therefore,the probability of moving from vertex ${{v}_{i}}$to ${{v}_{j}}$is:
\begin{equation}
\begin{aligned}
{{P}_{ij}}=P({{v}_{i}},{{v}_{j}})=\sum\limits_{k=1}^{m}{\frac{{{h}_{ik}}{{h}_{jk}}}{d({{v}_{i}})\delta ({{e}_{k}})}}=\frac{1}{d({{v}_{i}})}\sum\limits_{k=1}^{m}{\frac{{{h}_{ik}}{{h}_{jk}}}{\delta ({{e}_{k}})}}
\end{aligned},
\label{eq:ClassificationModel}
\end{equation}
or, more accurately, the equation can be written as
\begin{equation}
\begin{aligned}
P=\sum\limits_{\begin{smallmatrix}
e\in E, \\
\{v,u\}\subseteq e
\end{smallmatrix}}{\frac{1}{d(v)\delta (e)}}=\frac{1}{d(v)}\sum\limits_{\begin{smallmatrix}
e\in E, \\
\{v,u\}\subseteq e
\end{smallmatrix}}{\frac{1}{\delta (e)}}
\end{aligned}.
\label{eq:ClassificationModel}
\end{equation}
Alternately, a random walk on a hypergraph can be seen as a Markov chain on the hyperedges. At each step, the particle randomly chooses a hyperedge from the set of neighbors of the current hyperedge through the chosen vertex from the current hyperedge. Let the state space of the chain be $E$ and the transition matrix $Q$. The probability of moving form ${{e}_{i}}$ to ${{e}_{j}}$ is
\begin{equation}
\begin{aligned}
{{Q}_{ij}}=Q({{e}_{i}},{{e}_{j}})=\sum\limits_{k=1}^{n}{\frac{{{h}_{ki}}{{h}_{kj}}}{\delta ({{e}_{i}})d({{v}_{k}})}}=\frac{1}{\delta ({{e}_{i}})}\sum\limits_{k=1}^{n}{\frac{{{h}_{ki}}{{h}_{kj}}}{d({{v}_{k}})}}
\end{aligned},
\label{eq:ClassificationModel}
\end{equation}
or, alternatively,
\begin{equation}
\begin{aligned}
Q=\sum\limits_{\begin{smallmatrix}
v\in V, \\
v\in e\bigcap f
\end{smallmatrix}}{\frac{1}{\delta (e)d(v)}}=\frac{1}{\delta (e)}\sum\limits_{\begin{smallmatrix}
v\in V, \\
v\in e\bigcap f
\end{smallmatrix}}{\frac{1}{d(v)}}
\end{aligned}.
\label{eq:ClassificationModel}
\end{equation}
Let ${{P}_{VE}}$ denote the vertex-edge transition matrix
\begin{equation}
\begin{aligned}
{{P}_{VE}}=D_{v}^{-1}H
\end{aligned}
\label{eq:ClassificationModel}
\end{equation}
and ${{P}_{EV}}$ the edge-vertex transition matrix
\begin{equation}
\begin{aligned}
{{P}_{EV}}=D_{e}^{-1}{{H}^{T}}
\end{aligned}
\label{eq:ClassificationModel}
\end{equation}
with transition probability
\begin{equation}
\begin{aligned}
\sum\limits_{e\in E}{{{p}_{ve}}=1},\forall v\in V,
\end{aligned}
\label{eq:ClassificationModel}
\end{equation}
\begin{equation}
\begin{aligned}
\sum\limits_{v\in V}{{{p}_{ev}}=1},\forall e\in E.
\end{aligned}
\label{eq:ClassificationModel}
\end{equation}
Naturally, we can indicate $P$ and $Q$ in matrix form, respectively, as
\begin{equation}
\begin{aligned}
P=D_{v}^{-1}HD_{e}^{-1}{{H}^{T}}={{P}_{VE}}{{P}_{EV}},
\end{aligned}
\label{eq:ClassificationModel}
\end{equation}
\begin{equation}
\begin{aligned}
Q=D_{e}^{-1}{{H}^{T}}D_{v}^{-1}H={{P}_{EV}}{{P}_{VE}}.
\end{aligned}
\label{eq:ClassificationModel}
\end{equation}
\section{Quantum walks on hypergraphs}
\label{sec3:level1}
In this section, we design quantum walks on regular uniform hypergraphs by means of Szegedy's quantum walks. We first convert the hypergraph into its associated bipartite graph, which can be used to model the hypergraph. We then define quantum operators on the bipartite graph using Szegedy's quantum walks, which are a quantization of random walks.
\subsection{Bipartite graphs model of the hypergraphs}
\label{sec3.1.1:level3}
A hypergraph $HG$ can be represented usefully by a bipartite graph $BG$ as follows: the vertices $V$ and the edges $E$ of the hypergraph are the partitions of $BG$, and $\left( {{v}_{i}},{{e}_{j}} \right)$ are connected with an edge if and only if vertex ${{v}_{i}}$ is contained in edge ${{e}_{j}}$ in $HG$ . Formally, $B(H)=G(V\bigcup E,{{E}_{B}})$ and $\left( {{v}_{i}},{{e}_{j}} \right)\in {{E}_{B}}$ iff ${{h}_{ij}}=1$. The biadjacency matrix describing $B(H)$ is the following $\left( n+m \right)\times \left( n+m \right)$ matrix:
\begin{equation}
\begin{aligned}
{{A}_{B}}=\left( \begin{matrix}0 & H \\{{H}^{T}} & 0 \\\end{matrix} \right)
\end{aligned},
\label{eq:ClassificationModel}
\end{equation}
where $H$ with elements (1) is the incidence matrix of $HG$. Under this correspondence, the biadjacency matrices of bipartite graphs are exactly the incidence matrices of the corresponding hypergraphs. A similar reinterpretation of adjacency matrices may be used to show a one-to-one correspondence between regular uniform hypergraphs and bipartite graphs. That is, discrete-time quantum walks on regular uniform hypergraphs can be transformed into quantum walks on bipartite graphs that are used to model the original hypergraphs.
The transformation process is outlined in detail below. If there is a hyperedge ${{e}_{k}}$ containing both vertices ${{v}_{i}}$ and ${{v}_{j}}$ in the original hypergraph $HG=(V,E)$, convert it into two edges $({{v}_{i}},{{e}_{k}})$ and $({{e}_{k}},{{v}_{j}})$ in the bipartite graph. As a concrete example, we consider a $3-uniform$ and $2-regular$ hypergraph with the vertexes set $V=\{{{v}_{1}},{{v}_{2}},{{v}_{3}},{{v}_{4}},{{v}_{5}},{{v}_{6}}\}$ and the set of hyperedges $E=\{{{e}_{1}},{{e}_{2}},{{e}_{3}},{{e}_{4}}\}$. Then, a bipartite graph $B{{G}_{6,4}}$ with partite sets $V=\{{{v}_{1}},{{v}_{2}},{{v}_{3}},{{v}_{4}},{{v}_{5}},{{v}_{6}}\}$ and $E=\{{{e}_{1}},{{e}_{2}},{{e}_{3}},{{e}_{4}}\}$ can represent the hypergraph $H{{G}_{3,2}}$, which is depicted in Fig.1.
\begin{figure}
\centerline{\includegraphics [width=0.65\textwidth] {fig1.png}}
\caption{Example of a hypergraph with six vertexes and four hyperedges, and its associated bipartite graph.}
\label{fig1:}
\end{figure}
\textbf{Theorem 1 } Let $HG=(V,E)$ be a hypergraph, and we have
\begin{equation}
\begin{aligned}
\sum\limits_{v\in V}{d(v)}=\sum\limits_{e\in E}{d(e)}
\end{aligned}.
\label{eq:ClassificationModel}
\end{equation}
\textbf{Proof:} Let $B(H)=G(V\bigcup E,{{E}_{B}})$ be the incidence graph of $HG=(V,E)$. We sum the degrees in the part $E$ and in the part $V$ in $B(H)$. Since the sums of the degrees in these two parts are equal, we obtain the result.
In particular, if the hypergraph is $d-regular$ and $k-uniform$, we obtain $nd=mk$.
\subsection{Szegedy quantum walks on the bipartite graphs}
\label{sec3.2:level2}
Since we have transformed the hypergraph $H{{G}_{k,d}}$ into its bipartite graph $B{{G}_{n,m}}$, we now describe Szegedy quantum walks that take place on the obtained bipartite graph $B{{G}_{n,m}}$ by extending the class of possible Markov chains. The quantum walks on the hypergraph $HG$ start by considering an associated Hilbert space that is a linear subspace of the vector ${{H}^{{{n}^{2}}m}}=H_{v}^{n}\otimes H_{e}^{m}\otimes H_{v}^{n}$, where $n=\left| V \right|$, $m=\left| E \right|$. The computational basis of ${{H}^{{{n}^{2}}m}}$ is $\left\{ \left| {{v}_{i}},e,{{v}_{j}} \right\rangle :e\in E,{{v}_{i}},{{v}_{j}}\in V,{{v}_{i}},{{v}_{j}}\in e \right\}$. In addition, quantum walks on the bipartite graph $B{{G}_{n,m}}$ with biadjacent matrix (15) have an associated Hilbert space ${{H}_{A}}=H_{v}^{n}\otimes H_{e}^{m}$ and ${{H}_{B}}=H_{e}^{m}\otimes H_{v}^{n}$.
To identify quantum analogues of Markov chains - that is, the classical random walks with probability matrices (9) and (10) with entries of (11) and (12) - we define the vertex-edge transition operators: $A:{{H}^{n}}\to {{H}^{nd}}$ and edge-vertex transition operators: $B:{{H}^{m}}\to {{H}^{mk}}$ as follows:
\begin{equation}
\begin{aligned}
A=\sum\limits_{v\in V}{\left| {{\alpha }_{v}} \right\rangle \left\langle v \right|}
\end{aligned},
\label{eq:ClassificationModel}
\end{equation}
\begin{equation}
\begin{aligned}
B=\sum\limits_{e\in E}{\left| {{\beta }_{e}} \right\rangle \left\langle e \right|}
\end{aligned},
\label{eq:ClassificationModel}
\end{equation}
where
\begin{equation}
\begin{aligned}
\left| {{\alpha }_{v}} \right\rangle =\left| v \right\rangle \otimes \left( \sum\limits_{e\in E}{\sqrt{{{p}_{ve}}}\left| e \right\rangle } \right)
\end{aligned},
\label{eq:ClassificationModel}
\end{equation}
\begin{equation}
\begin{aligned}
\left| {{\beta }_{e}} \right\rangle =\left( \sum\limits_{v\in V}{\sqrt{{{p}_{ev}}}\left| e \right\rangle } \right)\otimes \left| v \right\rangle
\end{aligned}.
\label{eq:ClassificationModel}
\end{equation}
The transition operators are defined on the Hilbert space ${{H}_{A}}$ and ${{H}_{B}}$ separately, where the computational basis of ${{H}^{nd}}$ is $\left\{ \left| v,e \right\rangle :v\in V,e\in E \right\}$ and the computational basis of ${{H}^{mk}}$ is $\left\{ \left| e,v \right\rangle :e\in E,v\in V \right\}$. The states $\left| {{\alpha }_{v}} \right\rangle$ and $\left| {{\beta }_{e}} \right\rangle$ as superpositions that start from vertex $v$ to hyperedge $e$ and from hyperedge $e$ to vertex $v$, respectively. Obviously, the dimensions of $A$ and $B$ are $nd\times n$ and $mk\times m$, respectively. Note that $nd=mk$ from theorem 1. Using (19) and (20) along with (11) and (12), we obtain the following properties:
\begin{equation}
\begin{aligned}
\langle {{\alpha }_{v}} \left| {{\alpha }_{v'}} \right\rangle ={{\delta }_{vv'}}
\end{aligned},
\label{eq:ClassificationModel}
\end{equation}
\begin{equation}
\begin{aligned}
\langle {{\beta }_{e}} \left| {{\beta }_{e'}} \right\rangle ={{\delta }_{ee'}}
\end{aligned},
\label{eq:ClassificationModel}
\end{equation}
as well as
\begin{equation}
\begin{aligned}
{{A}^{T}}A ={{I}_{n}}
\end{aligned},
\label{}
\end{equation}
\begin{equation}
\begin{aligned}
{{B}^{T}}B ={{I}_{m}}
\end{aligned}.
\label{}
\end{equation}
One can easily verify that $\left| {{\alpha }_{v}} \right\rangle$ and $\left| {{\beta }_{e}} \right\rangle$ are unit vectors due to the stochasticity of ${{P}_{VE}}$ and ${{P}_{EV}}$. Distinctly, these equations imply that the action of $A$ preserves the norm of the vectors. The same is true regarding $B$.
We now immediately define the projectors ${{\Pi }_{A}}$ and ${{\Pi }_{B}}$ as follows:
\begin{equation}
\begin{aligned}
{{\Pi }_{A}}=A{{A}^{T}}=\sum\limits_{v\in V}{\left| {{\alpha }_{v}} \right\rangle \left\langle {{\alpha }_{v}} \right|}
\end{aligned},
\label{eq:ClassificationModel}
\end{equation}
\begin{equation}
\begin{aligned}
{{\Pi }_{B}}=B{{B}^{T}}=\sum\limits_{e\in E}{\left| {{\beta }_{e}} \right\rangle \left\langle {{\beta }_{e}} \right|}
\end{aligned}.
\label{eq:ClassificationModel}
\end{equation}
Using Eqs.(25) and (26), it is easy to see that ${{\Pi }_{A}}$ projects onto subspace ${{H}_{A}}$ spanned by $\left\{ \left| {{\alpha }_{v}} \right\rangle :v\in V \right\}$, and ${{\Pi }_{B}}$ projects onto subspace ${{H}_{B}}$ spanned by $\left\{ \left| {{\beta }_{e}} \right\rangle :e\in E \right\}$.
After obtaining the projectors, we can define the associated reflection operators, which are
\begin{equation}
\begin{aligned}
{{R}_{A}}=2{{\Pi }_{A}}-{{I}_{nd}}
\end{aligned},
\label{eq:ClassificationModel}
\end{equation}
\begin{equation}
\begin{aligned}
{{R}_{B}}=2{{\Pi }_{B}}-{{I}_{mk}}
\end{aligned},
\label{eq:ClassificationModel}
\end{equation}
where ${{I}_{nd}}={{I}_{mk}}$ is the identity operator. ${{R}_{A}}$ is the reflection though the line generated by $\left| {{\alpha }_{v}} \right\rangle$, and ${{R}_{B}}$ is the reflection though the line generated by $\left| {{\beta }_{e}} \right\rangle$. Note that the reflection operators ${{R}_{A}}$ and ${{R}_{B}}$ are unitary and Hermitian. With all the information, a single step of the quantum walks is given by the unitary evolution operator
\begin{equation}
\begin{aligned}
W={{R}_{B}}{{R}_{A}}
\end{aligned}
\label{eq:ClassificationModel}
\end{equation}
based on the transition matrix $P$. In the bipartite graph, an application of $W$ corresponds to two quantum steps of the walk from $v$ to $e$ and from $e$ to $v$. At time $t$, the whole operator of the quantum walks is ${{W}^{t}}$.
\section{Spectral analysis of quantum walks on hypergraphs}
In many classical algorithms, the eigen-spectrum of the transition matrix $P$ plays a critical role in the analysis of Markov chains. In a similar way, we now proceed to study the quantitative spectrum of the quantum walks unitary operator $W$.
Szegedy proved a spectral theorem for quantum walks, $W=re{{f}_{2}}re{{f}_{1}}$, in Ref. \cite{szegedy2004quantum}. In this section, we deliver a slightly different version in that the cardinality of set $X$ may be different from the cardinality of set $Y$ in the bipartite graph. In order to analyze the spectrum, we need to study the spectral properties of an $n\times m$ matrix $D$, which indeed establishes a relation between the classical Markov chains and the quantum walks. This matrix is defined as follows:
\textbf{(Discriminant Matrix) } The discriminant matrix for $W$ is
\begin{equation}
\begin{aligned}
{{D}_{nm }}={{A}^{T}}B
\end{aligned}.
\label{eq:ClassificationModel}
\end{equation}
Herein, suppose that $n\ge m$. Also, it follows from the definition that $D=\sqrt{{{P}_{VE}}\circ {{P}_{EV}}}$ with entries
\begin{equation}
\begin{aligned}
{{D}_{ve}}=\sqrt{{{p}_{ve}}{{p}_{ev}}},\forall v\in V,\forall e\in E
\end{aligned}.
\label{eq:ClassificationModel}
\end{equation}
Suppose that the discriminant matrix $D$ has the singular value decomposition $D=U\Sigma {{V}^{T}}=\sum\nolimits_{i}{{{\sigma }_{i}}}{{\mu }_{i}}\nu _{i}^{T}$. The left singular vectors $\left| {{\mu }_{k}} \right\rangle $ satisfy
\begin{equation}
\begin{aligned}
D\left| {{\nu }_{k}} \right\rangle ={{\sigma }_{k}}\left| {{\mu }_{k}} \right\rangle
\end{aligned}
\label{eq:ClassificationModel}
\end{equation}
and the right singular vectors
\begin{equation}
\begin{aligned}
\left\langle {{\mu }_{k}} \right|D=\left\langle {{\nu }_{k}} \right|{{\sigma }_{k}}
\end{aligned}
\label{eq:ClassificationModel}
\end{equation}
with ${{\sigma }_{k}}$ the singular value.
\textbf{Theorem 2 } For any ${{\sigma }_{k}}$ the singular value of $D$, $0\le {{\sigma }_{k}}\le 1$.
\textbf{Proof: } First, let $Dk={{\sigma }_{k}}k$.
Then we obtain
\begin{equation}
\begin{aligned}
{{\left| {{\sigma }_{k}} \right|}^{2}}{{\left\| k \right\|}^{2}}& ={{\left\| Dk \right\|}^{2}} \\
& =\left\langle Dk,Dk \right\rangle \\
& =\left\langle {{A}^{T}}Bk,{{A}^{T}}Bk \right\rangle \\
& =\left\langle Bk,A{{A}^{T}}Bk \right\rangle \\
& \le \left\langle Bk,Bk \right\rangle \\
& =\left\langle k,{{B}^{T}}Bk \right\rangle \\
& =\left\langle k,k \right\rangle ={{\left\| k \right\|}^{2}}.
\end{aligned}
\label{}
\end{equation}
Thus, $\left| {{\sigma }_{k}} \right|\le 1$. Since $\left\langle k,{{D}^{T}}Dk \right\rangle \ge 0$ for all $k$, we have $0\le {{\sigma }_{k}}$. Therefore, $0\le {{\sigma }_{k}}\le 1$.
Observing theorem 2 , we can write the singular value ${{\sigma }_{k}}$ as $\cos {{\theta }_{k}}$, where ${{\theta }_{k}}$ is the principal angle between subspace ${{H}_{A}}$ and ${{H}_{B}}$. In the early literature \cite{bjorck1973numerical}, Bj{\"o}rck and Golub deducted the relationship between the singular value decomposition and the principal angle ${{\theta }_{k}}$ between subspace ${{H}_{A}}$ and ${{H}_{B}}$. That is, $\cos ({{\theta }_{k}})={{\sigma }_{k}}$.
In the remainder of this section, we will explore the eigen-decomposition of the operator $W$, which can be calculated from the singular value decomposition of $D$.
Using $A$ to left-multiply (32) and $B$ to left-multiply (33), We have
\begin{equation}
\begin{aligned}
AD\left| {{\nu }_{k}} \right\rangle = \left( A{{A}^{T}} \right)B\left| {{\nu }_{k}} \right\rangle ={{\sigma }_{k}}A\left| {{\mu }_{k}} \right\rangle
\end{aligned},
\label{eq:ClassificationModel}
\end{equation}
\begin{equation}
\begin{aligned}
B{{D}^{T}}\left| {{\mu }_{k}} \right\rangle =\left( B{{B}^{T}} \right)A\left| {{\mu }_{k}} \right\rangle ={{\sigma }_{k}}B\left| {{\nu }_{k}} \right\rangle
\end{aligned}.
\label{eq:ClassificationModel}
\end{equation}
As we know, the action of $A$ and $B$ preserve the norm of the vectors, and $\left| {{\nu }_{k}} \right\rangle $ and $\left| {{\mu }_{k}} \right\rangle$ are unit vectors, so $A\left| {{\mu }_{k}} \right\rangle$ and $B\left| {{\nu }_{k}} \right\rangle$ also are unit vectors. Further, (35) and (36) imply that ${{\Pi }_{A}}$ and ${{\Pi }_{B}}$ have a symmetric action on $A\left| {{\mu }_{k}} \right\rangle$ and $B\left| {{\nu }_{k}} \right\rangle$. Therefore, we can conclude that the subspace $span\{A\left| {{\mu }_{k}} \right\rangle ,B\left| {{\nu }_{k}} \right\rangle \}$ is invariant under the action of ${{\Pi }_{A}}$ and ${{\Pi }_{B}}$.
We then have
\begin{equation}
\begin{aligned}
WA\left| {{\mu }_{k}} \right\rangle & ={{R}_{B}}{{R}_{A}}A\left| {{\mu }_{k}} \right\rangle \\
& ={{R}_{B}}A\left| {{\mu }_{k}} \right\rangle \\
& =2B{{B}^{T}}A\left| {{\mu }_{k}} \right\rangle -A\left| {{\mu }_{k}} \right\rangle \\
& =2{{\sigma }_{k}}B\left| {{\nu }_{k}} \right\rangle -A\left| {{\mu }_{k}} \right\rangle
\end{aligned}
\label{}
\end{equation}
and
\begin{equation}
\begin{aligned}
WB\left| {{\nu }_{k}} \right\rangle& ={{R}_{B}}{{R}_{A}}B\left| {{\nu }_{k}} \right\rangle \\
& ={{R}_{B}}(2A{{A}^{T}}-I)B\left| {{\nu }_{k}} \right\rangle \\
& ={{R}_{B}}(2A{{A}^{T}}B\left| {{\nu }_{k}} \right\rangle )-B\left| {{\nu }_{k}} \right\rangle \\
& ={{R}_{B}}(2{{\sigma }_{k}}A\left| {{\mu }_{k}} \right\rangle )-B\left| {{\nu }_{k}} \right\rangle \\
& =2{{\sigma }_{k}}(2{{\sigma }_{k}}B\left| {{\nu }_{k}} \right\rangle -A\left| {{\mu }_{k}} \right\rangle )-B\left| {{\nu }_{k}} \right\rangle \\
& =\left( 4\sigma _{k}^{2}-1 \right)B\left| {{\nu }_{k}} \right\rangle -2{{\sigma }_{k}}A\left| {{\mu }_{k}} \right\rangle.
\end{aligned}
\label{}
\end{equation}
Hence, we can conclude that the subspace $span\{A\left| {{\mu }_{k}} \right\rangle ,B\left| {{\nu }_{k}} \right\rangle \}$ is invariant under the action $W$. This, in turn, helps us characterize the eigenvalues of $W$ using the singular values of $D$.
Suppose that
\begin{equation}
\begin{aligned}
W\left| k \right\rangle ={{\lambda }_{k}}\left| k \right\rangle
\end{aligned}
\label{}
\end{equation}
and
\begin{equation}
\begin{aligned}
\left| k \right\rangle =aA\left| {{\mu }_{k}} \right\rangle +bB\left| {{\nu }_{k}} \right\rangle
\end{aligned}.
\label{}
\end{equation}
Simply plugging (40) into formulas (39), we obtain the following equation:
\begin{equation}
\begin{aligned}
W\left| k \right\rangle ={{\lambda }_{k}}aA\left| {{\mu }_{k}} \right\rangle +{{\lambda }_{k}}bB\left| {{\nu }_{k}} \right\rangle
\end{aligned}.
\label{}
\end{equation}
Then, left-multiplying (40) by $W$, we have
\begin{equation}
\begin{aligned}
W\left| k \right\rangle & =W(aA\left| {{\mu }_{k}} \right\rangle +bB\left| {{\nu }_{k}} \right\rangle ) \\
& =aWA\left| {{\mu }_{k}} \right\rangle +bWB\left| {{\nu }_{k}} \right\rangle \\
& =-(a+2b{{\sigma }_{k}})A\left| {{\mu }_{k}} \right\rangle +[2a\sigma _{k}^{2}+b(4{{\sigma }_{k}}-1)]B\left| {{\nu }_{k}} \right\rangle.
\end{aligned}
\label{}
\end{equation}
Comparing formulas (41) and (42), we can obtain the following equations:
\begin{equation}
\begin{aligned}
{{\lambda }_{k}}a=-(a+2b{{\sigma }_{k}})
\end{aligned},
\label{}
\end{equation}
\begin{equation}
\begin{aligned}
{{\lambda }_{k}}b=2a\sigma _{k}^{2}+b(4{{\sigma }_{k}}-1)
\end{aligned}.
\label{}
\end{equation}
Concerning unit vectors $A\left| {{\mu }_{k}} \right\rangle$ and $B\left| {{\nu }_{k}} \right\rangle$, we consider two cases with respect to non-collinearity and collinearity, as follows.
Case 1. First, we consider that $A\left| {{\mu }_{k}} \right\rangle$ and $B\left| {{\nu }_{k}} \right\rangle$ are linearly independent.
Using ${{\sigma }_{k}}=\cos {{\theta }_{k}}$, we obtain
\begin{equation}
\begin{aligned}
{{\lambda }_{k}}={{e}^{\pm 2i{{\theta }_{k}}}}
\end{aligned}
\label{}
\end{equation}
through a series of algebraic operations. Furthermore, we have the corresponding eigenvectors
\begin{equation}
\begin{aligned}
\left| k \right\rangle =\frac{A\left| {{\mu }_{k}} \right\rangle -{{e}^{\pm i{{\theta }_{k}}}}B\left| {{\nu }_{k}} \right\rangle }{\sqrt{2}\sin {{\theta }_{k}}}
\end{aligned}.
\label{}
\end{equation}
Case 2. Then, we consider that $A\left| {{\mu }_{k}} \right\rangle$ and $B\left| {{\nu }_{k}} \right\rangle$ are collinear. However, since $A\left| {{\mu }_{k}} \right\rangle$ is invariant under the action of ${{\Pi }_{A}}$, $B\left| {{\nu }_{k}} \right\rangle$ also is; and vice versa, since $B\left| {{\nu }_{k}} \right\rangle$ is invariant under ${{\Pi }_{B}}$, and $A\left| {{\mu }_{k}} \right\rangle $ also is. Therefore, $A\left| {{\mu }_{k}} \right\rangle$ and $B\left| {{\nu }_{k}} \right\rangle$ are invariant under the action of $W$, and $A\left| {{\mu }_{k}} \right\rangle$ are eigenvectors of $W$ with eigenvalue 1.
Now, we turn to the dimensionality of the spaces. We learned earlier that $nd$ is the dimension of edge Hilbert space about the bipartite graph $B{{G}_{n,m}}$, and the discriminant matrix $D$ has $m$ singular values, only some of which are non-zero. Space ${{H}_{A}}$ spanned by $\left\{ \left| {{\alpha }_{v}} \right\rangle :v\in V \right\}$, and space ${{H}_{B}}$ spanned by $\left\{ \left| {{\beta }_{e}} \right\rangle :e\in E \right\}$, are $n-dimension$ and $m-dimension$ subspaces of ${{H}^{nd}}$, respectively. Let ${{H}_{AB}}$ be the space spanned by $\left\{ \left| {{\alpha }_{v}} \right\rangle :v\in V \right\}$ and $\left\{ \left| {{\beta }_{e}} \right\rangle :e\in E \right\}$. Then, the dimension of ${{H}_{AB}}$ is $m+n$, when $A\left| {{\mu }_{k}} \right\rangle$ and $B\left| {{\nu }_{k}} \right\rangle$ are linearly independent. On the other hand, the dimension of ${{H}_{AB}}$ is $n-m$. Therefore, the operator $W$ has $nd-(m+n)$ eigenvalues 1 and $n-m$ eigenvalues -1 in the one-dimensional subspaces invariant, and $2m$ eigenvalues in the two-dimensional subspaces. Table 1 gives the eigenvalues of $W$ and the singular values of $D$ up to now.
\begin{table}[tbp]
\centering
\caption{Eigenvalues of $W$ obtained from the singular values of $D$, and angles ${{\theta }_{k}}$ obtained from the formula ${{\sigma }_{k}}=\cos {{\theta }_{k}}$, where $k=1,2,\cdots ,m$.}
\begin{tabular}{|c|c|c|}
\hline
Number of the eigenvalue &Eigenvalue of $W$ &Singular values of $D$ \\
\hline
$2m$ &${{\lambda }_{k}}={{e}^{\pm 2i{{\theta }_{k}}}}$ &${{\sigma }_{k}}=\cos {{\theta }_{k}}$ \\
$nd-(m+n)$ &1 &1 \\
$n-m$ &0 &-1\\
\hline
\end{tabular}
\end{table}
As a consequence, we obtain the following theorem:
\textbf{Theorem 3 } Let $W$ be the unitary evolution operator on $B{{G}_{n,m}}$ . Suppose that $n\ge m$. Then $W$ has $nd-(m+n)$ eigenvalues 1 and $n-m$ eigenvalues -1 in the one-dimensional invariant subspaces, and $2m$ eigenvalues in the two-dimensional subspaces. The $2m$ eigenvalues are ${{\lambda }_{k}}={{e}^{\pm 2i{{\theta }_{k}}}}$($0<{{\theta }_{k}}<\frac{\pi }{2}$) where ($k=1,2,\cdots ,m$ ) and the eigenvectors $\left| k \right\rangle =\frac{A\left| {{\mu }_{k}} \right\rangle -{{e}^{\pm i{{\theta }_{k}}}}B\left| {{\nu }_{k}} \right\rangle }{\sqrt{2}\sin {{\theta }_{k}}}$.
\section{Conclusions and outlook}
\label{sec7:level1}
Quantum walks are one of the elementary techniques of developing quantum algorithms. The development of successful quantum walks on graphs-based algorithms have boosted such areas as element distinctness, searching for a marked element, and graph isomorphism. In addition, the utility of walking on hypergraphs has been probed deeply in several contexts, including natural language parsing, social networks database design, or image segmentation, and so on. Therefore, we put our attention on quantum walks on hypergraphs considering its promising power of inherent parallel computation.
In this paper, we developed a new schema for discrete-time quantum walks on regular uniform hypergraphs using extended Szegedy's walks that naturally quantize classical random walks and yield quadratic speed-up compared to the hitting time of classical random walks. We found the one-to-one correspondence between regular uniform hypergraphs and bipartite graphs. Through the correspondence, we convert the regular uniform hypergraph into its associated bipartite graph on which extended Szegedy's walks take place. In addition, we dealt with the case that the cardinality of the two disjoint sets may be different from each other in the bipartite graphs. Furthermore, we delivered spectral properties of the transition matrix, which is the essence of quantum walks, and which has prepared for followup studies.
Our work presents a model for quantum walks on regular uniform hypergraphs, and the model opens the door to quantum walks on hypergraphs. We hope our model can inspire more fruitful results in quantum walks on hypergraphs. Our model provides the foundation for building up quantum algorithms on the strength of quantum walks on hypergraphs. Moreover, the algorithms of quantum walks on hypergraphs will be useful in quantum computation such as quantum walks search, quantized Google's PageRank, and quantum machine learning, based on hypergraphs.
Based on the preliminary research presented here, the following areas require further investigation:
1 Since the quantum walk evolutions are unitary, the probability of finding an element will oscillate through time. Therefore, the hitting time must be close to the time where the probability seems to peak for the very first time. One can calculate analytically the hitting time and the probability of finding a set of marked vertices on the hypergraphs using Szegedy' s quantum hitting time.
2 We have defined discrete-time quantum walks on regular uniform hypergraphs, while the condition can be relaxed. If the hypergraph is any one of several types, the questions then become a) how to generalize the quantum walks from regular uniform hypergraphs to any hypergraphs, and b) how to define quantum walks on hypergraphs?
\begin{acknowledgments}
I would like to thank Juan Xu , Yuan Su and Iwao Sato for helpful discussions. This work was supported by the Funding of National Natural Science Foundation of China (Grant No. 61571226,61701229), and the Natural Science Foundation of Jiangsu Province, China (Grant No. BK20170802).
\end{acknowledgments}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 6,978 |
Africa Journey
Sharing Wisdom About our Africa Journey
About the Africa Journey Fellowship
The Myth of the Blank Canvas
Africa inspires me to feel unlimited hope for life change. When I first met Rwandan orphans who had bright eyes, wide smiles showing perfect white teeth, and surprising plans for the future, I was jolted into a swirling dance of ideas, hope and action. I've lived this energizing journey for more than ten years now. It's a thrilling but difficult place. Along my journey, I've come to discover an odd behavior in myself, and others like me. I call it the myth of the blank canvas.
We look at the African orphan (or widow, or poor person) as a blank canvas, ready for us to splash brilliantly colored paint that will fill out the emptiness of their life. By blank canvas, I mean we forget to imagine what or who came before us. We don't realize the values, priorities and relationships that constitute the deeply ingrained identity of the person we want to help.
In 2007, I had the good luck of meeting a group of business entrepreneurs at Bourbon Coffee in Kigali. A friend connected me because he thought I would benefit from collaborating with ambitious American businessmen working in Rwanda. I sat within the circle of dreamers and thinkers, all of whom had become successful and wealthy in America. They talked of founding a high quality university in Kigali, with just the right educational programs that would fill the deficit of skills in Rwanda, as well as starting business enterprises to employ the students. The plan was enormous, and sounded perfect in every way. They expected their plan to be one of the most pivotal events in Rwanda's development. Then, in the midst of the euphoria, one of the guys said, "It's so exciting to be here in this moment today. I feel like I am eavesdropping, as the founding fathers discuss the plans that will establish the country."
Founding fathers? Who do we think we are? At least one of these men saw Rwanda as a blank canvas, ready for their big splashes of paint, that would make everything beautiful. I have come to see this is really how many of us westerners think. Why do we disregard what is already? When we want to help someone, why do we not honor and learn about the other relationships in that person's life? Is it because we see the past as being too broken to be significant?
I've helped many Rwandans over the years, and looking back I can see that in early days, I did not adequately regard the identities of those I helped. I viewed the youth of Rwanda as enormous opportunities for growth and life change, and thought they would be open to any idea I had for helping. In fact, I believed so strongly in the wisdom of my own ideas, I declined to follow advice from older Rwandans. Of course, pain resulted.
As time passed, I realized that my simplistic view was a myth, especially when I saw other Americans behaving as if nothing and no one had come before them.
My husband and I brought a group of Rwandan singers on music tours for three years in row. As we toured around the West Coast, we noticed that host families seemed very intent on imparting their knowledge and interests onto the guys. They treated the guys like molds ready to be imprinted. One person gave a violin and insisted that the Rwandan youth learn it. People gave them books, CDs, art, gadgets and hobby tools, and insisted that they adopt it all. The guys would hear all sorts of strong advice that was wholly impractical for their situations. This pattern was ridiculously common to the extent that my husband and I joked about it. But the jokes pricked me into awareness of my own self-absorbed perspective.
One of the most emotional moments of realizing the myth came a few years back when I asked my friend Angie if she would open her home to host Eric, one of the young musicians I had helped for many years, to be able to attend school in America. Angie had been to Rwanda and had started to think about adoption, so her heart was ready and open to welcome a Rwandan youth into her home. Several months after Eric joined her family, Angie wrote a wonderful, heartfelt blog about her adoption of Eric, and what it meant to her to have Eric like a son in her family. Angie is a great friend (and gave me permission to tell this story), but her blog struck me oddly and made me feel empty. The blog made me feel as if Angie wasn't considering the long journey Eric had travelled to get to her home, or the other families, including my own, who had sacrificed to bring him that far.
When we don't know what is already there, or who has come before, we are not learning the identity of the person we are trying to love. We are loving our ideas of how to change a person, more than loving the person.
There was certainly nothing negative about Angie's high commitment to parent Eric, just as there is nothing negative about our desires to help the poor, orphaned and widowed. Angie and her family remain one of the most important, life giving relationships to Eric. But this kind of parenting is very different, because while Eric is an orphan, he came to her home with a huge world of relationships, experiences and identity; a beautiful and complex picture. The best outcome for Angie, me and others like us will come after a long time of patient listening and learning how to best fit into a complex story.
Many short-term mission and humanitarian projects fail. I believe failure is partly due to the speed at which the idea came together. We don't take time to listen and learn how to best fit into what is already there. Jean Hatzfield, author of the "The Antelope Strategy," translated the word Muzungu as "usurper." I felt shocked to read such strongly negative translation of the word for white person. But as I reflect, I realize that it's true that our ambitious world changing plans often usurp what was already in process, either in one person's life or in a community. Wouldn't it be better to weave our efforts to help into what already is in process, than to launch a whole new initiative?[1]
Why do we try to make people into copies of ourselves, instead of first understanding the person we are trying to help? I don't think it's because we are bad people. I believe it's because we all deeply yearn for significance. We want to imprint our identity on needy people, because we want to feel valuable, smart and capable of changing the world. But our desire for significance cannot be gratified by helping others. Our need for significance will only be filled when we live in eternity in the presence of God.
By Serena Morones
[1] A major thesis of the book, When Helping Hurts by Brian Fikkert and Steve Corbett, is that poverty alleviation projects should build upon the existing assets in a community. Start with the strengths of a community and slowly build from there. Bringing in outside technology and resources doesn't have long term impact.
Posted by smorones on February 14, 2011 in Posts from the Group
Posts from the Group (11)
Recommended Reading Lists (1) | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 5,226 |
In electric power transmission, a CIM Profile is a subset model of the CIM UML model. These profiles are designated as parts documents in the IEC 61970 standard by working group 14. Each profile is itself a self-contained model which can be used for generating specific artifacts, such as CIM RDF or XML Schema.
Profile Groups
A CIM Profile Group (e.g., 61970-456 Steady-state Solution Profile Group) is a logical grouping of CIM Profiles. In general, each Parts document encompasses an entire Profile group which has one or more profiles in it.
Standards
IEC 61970-452
Equipment Profile
IEC 61970-453
Schematics Layout Profile
IEC 61970-456
Analog Measurements Profile
Discrete Measurements Profile
State Variable Profile
Topology Profile
See also
IEC 61970
CIM
External links
CIM EA - A tool written for Enterprise Architect which can manipulate and create profiles.
CIM Tool - A tool written for eclipse that can manipulate and create profiles.
EA schema composer - A tool part of Enterprise Architect which can manipulate and create profiles.
IEC standards
Electric power
Smart grid | {
"redpajama_set_name": "RedPajamaWikipedia"
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{"url":"http:\/\/eprints.iisc.ernet.in\/9780\/","text":"# Mechanism of $BaTiO_3$, formation through gel-to crystallite conversions\n\nKutty, TRN and Padmini, P (1995) Mechanism of $BaTiO_3$, formation through gel-to crystallite conversions. In: Materials Chemistry and Physics, 39 (3). pp. 200-208.\n\n PDF Mechanism_of.pdf Restricted to Registered users only Download (1063Kb) | Request a copy\n\n## Abstract\n\nThe wet chemical synthesis of $BaTiO_3$, through gel-to-crystallite conversions involves the reaction of coarse $TiO_2.xH_2O$ (10<x<120) gels, free of anionic contaminants, with $Ba(OH)_2$, solutions under refluxing conditions at $T\\leq 100^o C$, giving rise to nanosized crystallites. The paper focuses on the mechanism of formation of $BaTi0_3$. The reaction kinetics were monitored for various temperatures and concentrations and were found to be strongly dependent on $Ba(OH)_2$, concentration. Two regions of concentration could be detected: below 0.15 M $Ba(OH)_2$, polytitanates are formed, whereas at higher concentrations, i.e., greater than 0.15 M, perovskite phase is stabilised. Analyses of the kinetic data were carried out using various kinetic models used for heterogeneous reactions. Under higher concentrations of $Ba(OH)_2$ a bulk diffusional process dominates, accompanied by the collapse of the gel and a large decrease in volume. The present results also indicate the general features of gel-to-crystallite conversions, involving instability of the metal hydroxide gel brought about by the upset of ionic pressure in the gel, as a result of faster migration of $Ba^{2+}$ ions through the solvent cavities within the gel framework. It is proposed that with increasing pH within the gel, de-olation of the bridging groups such as Ti-OH-Ti and Ti-O-Ti takes place, followed by oxolation, leading to the formation of corner-sharing $TiO_6$ octahedra that are charge-compensated by $Ba^{2+}$ ions.\n\nItem Type: Journal Article Copyright of this article belongs to Elsevier. Metal hydroxide gels;Perovskites;Diffusion;Liesegang rings Division of Chemical Sciences > Materials Research Centre 15 Mar 2007 19 Sep 2010 04:35 http:\/\/eprints.iisc.ernet.in\/id\/eprint\/9780","date":"2016-09-28 10:30:19","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 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\": 0.6584880948066711, \"perplexity\": 5046.32308226471}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 20, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2016-40\/segments\/1474738661349.6\/warc\/CC-MAIN-20160924173741-00059-ip-10-143-35-109.ec2.internal.warc.gz\"}"} | null | null |
\section{Introduction}
Barnsley, Harding, Vince, and Viswanathan introduced a notion of Weierstrass Fourier Series to approximate rough functions (see \cite{1}).\\
Key idea of constructing Weierstrass Fourier Series is:
\\1. Construct a linear operator on $L^p(\mathbb{R})$;
\\2. Using this linear operator, transform the classical Fourier basis to a new basis;
\\3. By Gram-Schmidt process, get an orthonormal basis;
\\4. Using this orthonormal basis, do what we can do in classical Fourier Analysis.\\
In this paper, we are going to extend their theory by the following steps:
\\1. Following the key idea in their paper, establish a new expression of Weierstrass Fourier Series, which is more suitable for subsequent discussions in this paper.
\\2. Deduce the notion of Discrete Weierstrass Fourier Transform.
\\3. Provide some numerical examples to test this transform.
\section{Weierstrass Fourier Series}
In classical Fourier Analysis, $\{e^{2\pi ikx}\}_{k\in \mathbb{Z}}$ is a complete orthonormal basis for $L^2 ([0,1])$, and we can approximate functions with Fourier Series. Barnsley, Harding, Vince, and Viswanathan constructed another orthonormal basis for $L^2 ([0,1])$, and established a theory for approximating functions with Weierstrass Fourier Series. A key result in their paper is listed below (see Theorem \ref{dwft 1}).
\begin{theorem}[Barnsley et al.) (see \cite{1}, Corollary 3.1]\label{dwft 1}
Assume that $p \in [1, \infty]$, $a, b\in \mathbb{R}$, $b\ne 0$, and $|a|\ne |b|^{\frac{1}{p}}$. For any $g\in L^p (\mathbb{R})$, there is a unique solution $f\in L^p (\mathbb{R})$ to the equation
\begin{equation}\label{dwft 1 eq 1}
f(x)-af(bx)=g(x),
\end{equation}
and the solution is given by the following series, which are absolutely convergent in $L^p(\mathbb{R})$:
$$f(x)=\left\{\begin{array}{ll}
\sum_{m=0}^{\infty}\,a^{m}\,g(b^{m}x) &\textrm{if $|a|<|b|^{\frac{1}{p}}$}\\
\\
-\sum_{m=1}^{\infty}\,\left (\frac{1}{a})^{m}\,g(\frac{x}{b^m} \right ) & \textrm{if $|a|>|b|^{\frac{1}{p}}$}.
\end{array}\right.$$
\end{theorem}
The following corollary is a direct conclusion from this theorem.
\begin{corollary}\label{dwft 2}
Assume that $a \in [0,1)$ and $b=2$. If $g\in L^\infty (\mathbb{R})$, then (\ref{dwft 1 eq 1}) has a unique solution $f\in L^\infty (\mathbb{R})$, and the solution is given by the following series, which are absolutely convergent in $L^\infty(\mathbb{R})$:
\begin{equation}
f(x)=\sum_{m=0}^{\infty}\,a^{m}\,g(2^{m}x).
\end{equation}
Furthermore, $f\in L^2 ([0,1])$.
\end{corollary}
\begin{proof}
Let $p=\infty$ in Theorem \ref{dwft 1}, and the first part of the corollary is proved. As $f\in L^\infty (\mathbb{R})$, $f\in L^\infty ([0,1])$. So, $f\in L^2 ([0,1])$.
\end{proof}
In paper \cite{1}, Barnsley et al. introduced Weierstrass Fourier Series by substituting $g(x)=\sin kx$ and $\cos kx$. However, another form shown below is more suitable for subsequent discussions in this paper.
In (\ref{dwft 1 eq 1}), assume that $a \in [0,1)$ and $b=2$. For each $k\in \mathbb{Z}$, let $g(x)=e_k(x)=e^{2\pi ikx}$. By Corollary \ref{dwft 2}, there exists a unique solution $f_k(x)=\sum_{m=0}^{\infty}\,a^{m}\,e_k(2^{m}x)\in L^2 ([0,1])$. By normalizing $\{f_k\}$, a normalized basis $\{\hat{e}_k\}$ for $L^2 ([0,1])$ is established:
$$\hat{e}_k=\left\{\begin{array}{ll}
1 &\textrm{if $k=0$}\\
\\
\sqrt{1-a^2}\sum_{m=0}^{\infty}a^m e^{2\pi ik\cdot 2^m x} &\textrm{if $k\ne 0$}.
\end{array}\right.$$
By Gram-Schmidt process, an orthonormal basis $\{\tilde{e}_k\}$ for $L^2 ([0,1])$ is obtained:
$$\tilde{e}_k=
\left\{\begin{array}{lll}
1 &&\textrm{if $k=0$}\\
\\
\hat{e}_k &=\sqrt{1-a^2}\sum_{m=0}^{\infty}a^m e^{2\pi ik\cdot 2^m x} &\textrm{if $k$ is odd}\\
\\
\frac{\hat{e}_k-a\hat{e}_{k/2}}{\sqrt{1-a^2}} &=(1-a^2)\sum_{m=0}^{\infty}a^m e^{2\pi ik\cdot 2^m x}-ae^{\pi ikx} &\textrm{if $k$ is even and $k\ne 0$}.
\end{array}\right.$$
Notice that when $a=0$, $\tilde{e}_k=e_k$ for all $k\in \mathbb{Z}$. To sum up:
\begin{theorem}\label{dwft 3}
Given $a\in [0,1)$, the set of functions $\{\tilde{e}_k\}_{k \in \mathbb{Z}}$ is an orthonormal basis for $L^2 ([0,1])$.
\end{theorem}
Weierstrass Fourier coefficients for a function $f$ can be calculated as follows:
$$\tilde{\alpha}_k=\langle f,\tilde{e}_k \rangle =
\left\{\begin{array}{lll}
\alpha_0 &&\textrm{if $k=0$}\\
\\
\langle f,\hat{e}_k \rangle &=\sqrt{1-a^2}\sum_{m=0}^{\infty}a^m \alpha_{k\cdot 2^m} &\textrm{if $k$ is odd}\\
\\
\frac{\langle f,\hat{e}_k \rangle-a\langle f,\hat{e}_{k/2} \rangle}{\sqrt{1-a^2}} &=(1-a^2)\sum_{m=0}^{\infty}a^m \alpha_{k\cdot 2^m}-a\alpha_{k/2} &\textrm{if $k$ is even and $k\ne 0$},
\end{array}\right.$$
where $\alpha_k$ is the $k^{th}$ Fourier coefficient. Notice that when $a=0$, $\tilde{\alpha}_k=\alpha_k$ for all $k\in \mathbb{Z}$. Now, we can approximate any function in $L^2 ([0,1])$ using Weierstrass Fourier Series.
\begin{theorem}\label{dwft 4}
With the notations above, given $a\in [0,1)$, if $f\in L^2 ([0,1])$ has Fourier expansion
$$f(x)=\sum_{k=-\infty}^{\infty}\alpha_k e_k(x),$$
then it has Weierstrass Fourier expansion
$$f(x)=\sum_{k=-\infty}^{\infty}\tilde{\alpha}_k \tilde{e}_k(x).$$
In particular, when $a=0$, two expansions are the same.
\end{theorem}
\begin{proposition}\label{dwft 5}
With the notations above, given $a\in [0,1)$, if $f\in L^2 ([0,1])$ is a real function, then the following statements hold:\\
(1) $\overline{\tilde{\alpha}_k}=\tilde{\alpha}_{-k}$ for all $k\in \mathbb{Z}$;\\
(2) $\overline{\tilde{e}_k}=\tilde{e}_{-k}$ for all $k\in \mathbb{Z}$;\\
(3) $\sum_{k=-n}^n \tilde{\alpha}_k \tilde{e}_k(x)$ is real and $\sum_{k=-n}^n \tilde{\alpha}_k \tilde{e}_k(x)=\alpha_0 +2Re(\sum_{k=1}^n \tilde{\alpha}_k \tilde{e}_k(x))$ for all $n\in \mathbb{Z_+}$.
\end{proposition}
\section{Discrete Weierstrass Fourier Transform}
With the notations above, given $a\in [0,1)$, for any fixed $n\in \mathbb{N}_+$, let $A$ be an $n\times n$ matrix such that
\begin{equation}
A_{ij}=\tilde{e}_j(\frac{i}{n}), {0\le i,j\le n-1}.
\end{equation}
Given $n$ data points $b_0, b_1, b_2, \cdots, b_{n-1}\in \mathbb{C}$, let $b=(b_0, b_1, b_2, \cdots, b_{n-1})^T$. \textbf{Discrete Weierstrass Fourier Transform}(DWFT) is defined as the linear operator on $\mathbb{C}^n$: $dwft(b)=A^{-1}\,b$. \textbf{Inverse Discrete Weierstrass Fourier Transform}(IDWFT) is defined as the linear operator on $\mathbb{C}^n$: $idwft(b)=A\,b$.
\begin{theorem}\label{dwft 6}
Given $a\in [0,1)$, $dwft$ and $idwft$ definded above are inverse linear operators on $\mathbb{C}^n$. In particular, when $a=0$, $dwft$ is classical Discrete Fourier Transform and $idwft$ is classical Inverse Discrete Fourier Transform.
\end{theorem}
Like DFT, DWFT is useful for data compression. Fix $n\in \mathbb{N}_+$. Let $k\in \mathbb{N}_+$ and $1\le k\le \frac{n+1}{2}$ if $n$ is odd, $1\le k\le \frac{n+2}{2}$ if $n$ is even. Assume
\begin{equation}\label{dwft 6 eq 1}
b=(b_0, b_1, b_2, \cdots, b_{n-1})^T
\end{equation}
is a set of data. Do DWFT on $b$ and get
\begin{equation}\label{dwft 6 eq 2}
c=dwft(b)=(c_0, c_1, c_2, \cdots, c_{n-1})^T.
\end{equation}
Change $c_k, c_{k+1}, \cdots, c_{n-k}$ to $0$ and get
\begin{equation}\label{dwft 6 eq 3}
c'(k)=(c_0, c_1, \cdots, c_{k-1}, 0, \cdots, 0, c_{n-k+1}, \cdots, c_{n-1})^T.
\end{equation}
Do IDWFT on $c'(k)$ and get
\begin{equation}\label{dwft 6 eq 4}
b'(k)=idwft(c'(k))=(b_0', b_1', b_2', \cdots, b_{n-1}')^T.
\end{equation}
When $b=(b_0, b_1, b_2, \cdots, b_{n-1})^T$ is real, $b'(k)$ we got above is not necessarily real. However, we always want to get real approximation for real data. So, if the data are real, a last procedure that discarding the imaginary part of $b'(k)$ should be taken, say
\begin{equation}\label{dwft 6 eq 5}
b''(k)=Re(b'(k)).
\end{equation}
Say $b''(k)$ is the approximation of $b$ using $k$ terms using DWFT. Nevertheless, the imaginary part of $b'(k)$ is usually very small compared to its real part.
Now, if $b''(k)$ is very closed to $b$ even if $k$ is very small compared to $n$, we can store $c'(k)$, $a$, and $n$ instead of data $b$. Then we can calculate matrix $A$ as well as $b''(k)$ when we need and use $b''(k)$ as an approximation of $b$. In this way, plenty of space is saved.
If DWFT and IDWFT are replaced by DFT and IDFT, then the approximation using DFT is obtained.
By Theorem \ref{dwft 6}, DWFT is a generalization of DFT. Since we can choose a suitable $a$ for each set of data, DWFT will never be worse than DFT in approximation.
\section{Experiments}
In this section, some data sets are approximated using DFT and DWFT. With the notations in (\ref{dwft 6 eq 1}), (\ref{dwft 6 eq 2}), (\ref{dwft 6 eq 3}), (\ref{dwft 6 eq 4}) and (\ref{dwft 6 eq 5}), given data vector $b$ and $a\in [0,1)$, \textbf{error vector} for $k$ terms is defined as
\begin{equation}\label{error vector}
\vec{E}_k=b''(k)-b,
\end{equation}
and \textbf{error function} for $f$ is defined as the function
\begin{equation}\label{error function}
\mu (k)=||\vec{E}_k||_2,
\end{equation}
which maps $k$ to the 2-norm of $\vec{E}_k$, for $k\in \{1, 2, \dots, \frac{n}{2}+1\}$. In particular, $\mu (\frac{n}{2}+1)=0$ by Theorem \ref{dwft 6}.
\subsection{Data from real functions}
In this part, the number of data $n=1024$ for all examples, and each data vector $b=(f(0), f(\frac{1}{n}), f(\frac{2}{n}), \dots, f(\frac{n-1}{n}))^T$ for some real-valued function $f$ on $[0,1]$.
\subsubsection{Linear function}
\begin{center}
$f(x)=x-0.5$ and $a=0.5$\\
\scalebox{0.5}{\includegraphics{figure1_1_1.png}}
\end{center}
Figure 1.1.1 The figure of the error function for $f(x)=x-0.5$ with $a=0.5$. The blue curve represents approximation using DFT, and the green curve represents approximation using DWFT. DWFT is always better.
\begin{center}\scalebox{0.5}{\includegraphics{figure1_1_2.png}}\end{center}
Figure 1.1.2 Approximation of $f(x)=x-0.5$ using $10$ terms. The red curve represents the original data, the blue curve represents approximation values using DFT, and the green curve represents approximation values using DWFT. See also Figure 1.1.3 and Figure 1.1.4.
\begin{center}\scalebox{0.5}{\includegraphics{figure1_1_3.png}}\end{center}
\begin{center}
Figure 1.1.3 Approximation of $f(x)=x-0.5$ using $50$ terms.
\end{center}
\begin{center}\scalebox{0.5}{\includegraphics{figure1_1_4.png}}\end{center}
\begin{center}
Figure 1.1.4 Approximation of $f(x)=x-0.5$ using $300$ terms.
\end{center}
\subsubsection{Triangular function with low frequency and high frequency}
\begin{center}
$f(x)=\sin x+0.01\sin 105x$ and $a=0.5$\\
\scalebox{0.5}{\includegraphics{figure1_2_1.png}}
\end{center}
Figure 1.2.1 The figure of the error function for $f(x)=\sin x+0.01\sin 105x$ with $a=0.5$. The blue curve represents approximation using DFT, and the green curve represents approximation using DWFT. DWFT is better at the first $510$ terms.
\begin{center}\scalebox{0.5}{\includegraphics{figure1_2_2.png}}\end{center}
Figure 1.2.2 Approximation of $f(x)=\sin x+0.01\sin 105x$ using $10$ terms. The red curve represents the original data, the blue curve represents approximation values using DFT, and the green curve represents approximation values using DWFT. See also Figure 1.2.3 and Figure 1.2.4.
\begin{center}\scalebox{0.5}{\includegraphics{figure1_2_3.png}}\end{center}
Figure 1.2.3 Approximation of $f(x)=\sin x+0.01\sin 105x$ using $50$ terms.
\begin{center}\scalebox{0.5}{\includegraphics{figure1_2_4.png}}\end{center}
Figure 1.2.4 Approximation of $f(x)=\sin x+0.01\sin 105x$ using $300$ terms.
\subsubsection{Discontinuous function}
\begin{center}
$f(x)=
\left\{\begin{array}{ll}
0 & x\in [0,\frac{1}{2}]\\
1 & x\in (\frac{1}{2},1]
\end{array}\right.$ and $a=0.5$\\
\scalebox{0.5}{\includegraphics{figure1_3_1.png}}
\end{center}
Figure 1.3.1 The figure of the error function for $f(x)=
\left\{\begin{array}{ll}
0 & x\in [0,\frac{1}{2}]\\
1 & x\in (\frac{1}{2},1]
\end{array}\right.$
with $a=0.5$. The blue curve represents approximation using DFT, and the green curve represents approximation using DWFT. DWFT is not better.
\begin{center}\scalebox{0.5}{\includegraphics{figure1_3_2.png}}\end{center}
Figure 1.3.2 Approximation of $f(x)=
\left\{\begin{array}{ll}
0 & x\in [0,\frac{1}{2}]\\
1 & x\in (\frac{1}{2},1]
\end{array}\right.$
using $10$ terms. The red curve represents the original data, the blue curve represents approximation values using DFT, and the green curve represents approximation values using DWFT. See also Figure 1.3.3 and Figure 1.3.4.
\begin{center}\scalebox{0.5}{\includegraphics{figure1_3_3.png}}\end{center}
Figure 1.3.3 Approximation of $f(x)=
\left\{\begin{array}{ll}
0 & x\in [0,\frac{1}{2}]\\
1 & x\in (\frac{1}{2},1]
\end{array}\right.$
using $50$ terms.
\begin{center}\scalebox{0.5}{\includegraphics{figure1_3_4.png}}\end{center}
Figure 1.3.4 Approximation of $f(x)=
\left\{\begin{array}{ll}
0 & x\in [0,\frac{1}{2}]\\
1 & x\in (\frac{1}{2},1]
\end{array}\right.$
using $300$ terms.
\subsubsection{Rough function}
\begin{center}
$f(x)=\sum_{k=0}^{\infty}0.42^k\cos (\pi \cdot 2^k x)$ and $a=0.42$\\
\scalebox{0.5}{\includegraphics{figure1_4_1.png}}
\end{center}
Figure 1.4.1 The figure of the error function for $f(x)=\sum_{k=0}^{\infty}0.42^k\cos (\pi \cdot 2^k x)$ with $a=0.42$. The blue curve represents approximation using DFT, and the green curve represents approximation using DWFT. DWFT is better at the first $454$ terms.
\begin{center}\scalebox{0.5}{\includegraphics{figure1_4_2.png}}\end{center}
Figure 1.4.2 Approximation of $f(x)=\sum_{k=0}^{\infty}0.42^k\cos (\pi \cdot 2^k x)$ using $10$ terms. The red curve represents the original data, the blue curve represents approximation values using DFT, and the green curve represents approximation values using DWFT. See also Figure 1.4.3 and Figure 1.4.4.
\begin{center}\scalebox{0.5}{\includegraphics{figure1_4_3.png}}\end{center}
Figure 1.4.3 Approximation of $f(x)=\sum_{k=0}^{\infty}0.42^k\cos (\pi \cdot 2^k x)$ using $50$ terms.
\begin{center}\scalebox{0.5}{\includegraphics{figure1_4_4.png}}\end{center}
Figure 1.4.4 Approximation of $f(x)=\sum_{k=0}^{\infty}0.42^k\cos (\pi \cdot 2^k x)$ using $300$ terms.
\subsection{Discrete data}
In this part, data are practical data from some websites. Given data vector $b$ and $a\in [0,1)$, error vector and error function for the data are defined in (\ref{error vector}) and (\ref{error function}).
\subsubsection{Stock price of Commonwealth Bank of Australia}
Daily open prices of Commonwealth Bank of Australia from Sep 30, 2010. Totally 1024 data points. Source from https://au.finance.yahoo.com/.
\begin{center}\scalebox{0.5}{\includegraphics{figure2_1_1.png}}\end{center}
Figure 2.1.1 The figure of the error function for 1024 daily open prices of Commonwealth Bank of Australia with $a=0.3$. The blue curve represents approximation using DFT, and the green curve represents approximation using DWFT. DWFT is better at the first $88$ terms.
\begin{center}\scalebox{0.5}{\includegraphics{figure2_1_2.png}}\end{center}
Figure 2.1.2 Approximation of 1024 daily open prices of Commonwealth Bank of Australia using $10$ terms. The red curve represents the original data, the blue curve represents approximation values using DFT, and the green curve represents approximation values using DWFT. See also Figure 2.1.3 and Figure 2.1.4.
\begin{center}\scalebox{0.5}{\includegraphics{figure2_1_3.png}}\end{center}
Figure 2.1.3 Approximation of 1024 daily open prices of Commonwealth Bank of Australia using $50$ terms.
\begin{center}\scalebox{0.5}{\includegraphics{figure2_1_4.png}}\end{center}
Figure 2.1.4 Approximation of 1024 daily open prices of Commonwealth Bank of Australia using $300$ terms.
\subsubsection{Water level of Alameda in California}
Water level of Alameda in California per hour from May 29, 2014. Totally 1024 data points. Source from http://tidesandcurrents.noaa.gov/.
\begin{center}\scalebox{0.5}{\includegraphics{figure2_2_1.png}}\end{center}
Figure 2.2.1 The figure of the error function for 1024 water levels of Alameda in California per hour from May 29, 2014 with $a=0.3$. The blue curve represents approximation using DFT, and the green curve represents approximation using DWFT. DWFT is not better.
\begin{center}\scalebox{0.5}{\includegraphics{figure2_2_2.png}}\end{center}
Figure 2.2.2 Approximation of 1024 water levels of Alameda in California using $10$ terms. The red curve represents the original data, the blue curve represents approximation values using DFT, and the green curve represents approximation values using DWFT. See also Figure 2.2.3 and Figure 2.2.4.
\begin{center}\scalebox{0.5}{\includegraphics{figure2_2_3.png}}\end{center}
Figure 2.2.3 Approximation of 1024 water levels of Alameda in California using $50$ terms.
\begin{center}\scalebox{0.5}{\includegraphics{figure2_2_4.png}}\end{center}
Figure 2.2.4 Approximation of 1024 water levels of Alameda in California using $300$ terms.
\subsection{Comments}
\subsubsection{}
Figure 1.1.2 and Figure 1.1.3 in Section 4.1.1 coincide with Figure 1, Figure 2, Figure 3 and Figure 4 of \cite{1}, which means that the discrete algorithm in this paper gives the same result as the continuous algorithm when the number of the data is much bigger than the terms of approximation.
\subsubsection{}
In the figures above we can see that at some of the data points, DWFT approximates better, while at some other data points DFT is better. For example, in Figure 1.1.3, at the data points near 0.0 or 1.0, DWFT is better while at the data points near 0.5, it is significantly worse, but overall, it approximates the data better in the norm defined above.
\subsubsection{}
We can also see a characteristic that approximation values using DWFT shakes heavily at the points near $\frac{1}{2}$, less heavily at the points near $\frac{1}{4},\frac{3}{4}$, and less heavily at the points near $\frac{1}{8},\frac{3}{8},\frac{5}{8},\frac{7}{8}$, $\cdots$
\subsubsection{}
In the above examples, DWFT is better than DFT for some smooth continuous functions. For some other smooth functions such as $f(x)=x^2$ with $a=0.5$ and $f(x)=e^x$ with $a=0.5$, DWFT is also better than DFT. But for $f(x)=x(x-1)$ with $a=0.5$, DWFT behaves very bad.
\subsubsection{}
For triangle functions with low frequency and high frequency such as the example in Section 4.1.2, $f(x)=\sin x+0.01\cos 100x+0.01\cos 200x$ with $a=0.5$ and $f(x)=10\sin 0.1x-\cos x+0.01\sin 100x+0.02\cos 200x$ with $a=0.5$, DWFT is better than DFT.
However, make sure that the low frequencies are low enough with coefficients big enough compared to the high frequencies, and the low frequencies are not counteracted each other. For instance, DWFT doesn't work better than DFT for the function $f(x)=10\sin 0.1x-\sin x+0.01\sin 100x+0.02\cos 200x$ with $a=0.5$ because the low frequency terms ``$10\sin 0.1x$'' and ``$-\sin x$'' are counteracted each other.
\subsubsection{}
For discontinuous functions, DWFT doesn't seem better than DFT. As we can see in Section 4.1.3, approximation values using DWFT for discontinuous function shake heavily.
\subsubsection{}
As we can see in Section 4.1.4, DWFT has some advantages in approximating self-similar rough functions.
\subsubsection{}
As we can see in Section 4.2, DWFT seems not good for approximating practical data, because practical data are not perfectly self-similar. A small shake in the data can induce shakes everywhere in the approximation values using DWFT. In this case, the approximation values using DWFT look much rougher than the data.
\subsubsection{}
If some data set can be approximated better using DWFT, we have a better way to compress the data. However, the calculation of DWFT is much more complicated than DFT. We might wish to find a fast way to calculate DWFT.
| {
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A. James Hudspeth is the F.M. Kirby Professor at Rockefeller University in New York City, where he is director of the F.M. Kirby Center for Sensory Neuroscience. His laboratory studies the physiological basis of hearing.
Early life and education
As a teenager, James Hudspeth spent his summers working as a technician in the lab of neurophysiologist Peter Kellaway at Baylor College of Medicine. Hudspeth was expelled from high school for mixing dangerous chemicals and other mischief.
Hudspeth graduated from Harvard College in 1967, and received his master's degree from Harvard University in 1968. He enrolled in a graduate program in neurobiology to avoid being drafted into the military, but a year later the policy was changed, requiring him to enter medical school for exemption. He studied under Nobel prize winners Torsten Wiesel and David Hubel. He completed both programs and received his PhD in 1973 and MD in 1974, both from Harvard University.
He began a postdoctoral fellowship with Åke Flock at the Karolinska Institute, but left early without much success to return to Harvard Medical School.
Career
Following his postdoctoral training, Hudspeth was a professor at Caltech from 1975 to 1983. He then moved to the UCSF School of Medicine where he was a professor from 1983 to 1989. He directed the neuroscience program at University of Texas Southwestern Medical Center from 1989 until 1995, when the department was closed. In 1995, he was recruited to the Rockefeller University.
Hudspeth has been an HHMI investigator since 1993.
Research
Hudspeth's research is focused on sensorineural hearing loss, and the deterioration of the hair cells, the sensory cells of the cochlea. Hudspeth's bold interpretation of the data obtained in his careful experimental research combined with biophysical modelling lead him to propose for the first time that the sense of hearing depends on a channel that is opened by mechanical force: The hair cells located in the inner ear perceive sound when their apical end -consisting of a bundle of filaments- bends in response to the movement caused by this sound. The activated hair cell rapidly fills with calcium entering from the outside of the cell, which in turn activates the release of neurotransmitters that start a signal to the brain. Hudspeth proposed the existence of a "gating spring" opened by direct mechanical force that would open an hypothetical channel responsible for the entry of calcium ions. The hypothesis was based on the following evidence: 1) Part of the energy needed to bend the filament bundle was mysteriously lost, but could be explained if it was used to opening this gating spring, 2) The entry of calcium ions was microseconds long, this is so fast that only direct opening -without a cascade of chemical reactions- could account for it and 3) Hudspeth tested a model analogue to the opening of a door with a string attached to the door knob and demonstrated that a similar process was taking place when the filaments of the hair cell moved. Furthermore, microscopic evidence showed the existence of such a string-like structure tethering the tip of one filament to the side of and adjacent filament that could be the elusive gating spring; this string—called the tip link—would tense if the filament bundle was bend and then open the channel. Although the precise identity of the proteins forming the tip link and the mechanosensitive channel is still controversial 30 years later. Hudspeth's hypothesis was correct and fundamental for the understanding of the sense of hearing.
Noted publications
Holton T & A.J. Hudspeth A Micromechanical contribution to cochlear tuning and tonotopic organization. Science (1983); 222 (4623): 508–510
D.P. Corey, A.J. Hudspeth Kinetics of the receptor current in bullfrog saccular hair cells. J. Neurosci., 3 (1983): 962-976
Rosenblatt KP, Sun ZP, Heller S, A.J. Hudspeth Distribution of Ca2+-activated K+ channel isoforms along the tonotopic gradient of the chicken's cochlea. Neuron (1997): 19(5): 1061–1075 (note: this research was continued several years later taking advantage of newly available technology)
A.J. Hudspeth How hearing happens. NEURON (1997): 19(5): 947-950
Lopez-Schier H, Starr CJ, Kappler JA, Kollmar R, A.J. Hudspeth Directional cell migration establishes the axes of planar polarity in the posterior lateral-line organ of the zebrafish. Dev CELL (2004): 7(3):401–412
Chan DK, A.J. Hudspeth Ca2+ current-driven nonlinear amplification by the mammalian cochlea in vitro. Nature Neuro (2005): 8(2):149–155
Kozlov AS, Risler T, A.J. Hudspeth Coherent motion of stereocilia assures the concerted gating of hair-cell transduction channels. Nature Neuro (2007): 10(1):87–92
Kozlov AS, Baumgart J, Risler T, Versteegh CP, A.J. Hudspeth Forces between clustered stereocilia minimize friction in the ear on a subnanometre scale. Nature. (2011): 474 (7351):376–9
Fisher JA, Nin F, Reichenbach T, Uthaiah RC, A.J. Hudspeth The spatial pattern of cochlear amplification Neuron (2012): 76(5):989–9
Awards
1985 W. Alden Spencer Award
1991 K.S. Cole Award, Biophysical Society
1994 Charles A. Dana Award
1996 Rosenstiel Award
2002 Award of Merit, Association for Research in OtolaryngologyForces between clustered stereocilia minimize friction in the ear on a subnanometre scale. Kozlov AS, Baumgart J, Risler T, Versteegh CP, A.J. Hudspeth. Nature. 2011 May 22;474(7351):376-9. doi: 10.1038/nature10073.
2003 Ralph W. Gerard Prize, Society for Neuroscience
2010 Guyot Prize, University of Groningen
Elected member of the National Academy of Sciences and the American Academy of Arts and Sciences
2015 Elected member of the American Philosophical Society
2018 Kavli Prize in Neuroscience (shared with Christine Petit and Robert Fettiplace)
2020 Louisa Gross Horwitz Prize (shared with Christine Petit and Robert Fettiplace).
References
Living people
Year of birth missing (living people)
Harvard Medical School alumni
Rockefeller University faculty
Members of the United States National Academy of Sciences
California Institute of Technology faculty
Kavli Prize laureates in Neuroscience | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 8,480 |
\section{Introduction}
The standard model of fundamental particles and interactions has been very successful in describing the observed phenomena, but it is incomplete. First of all, the experimental evidences of neutrino oscillations caused by nonzero small neutrino masses and flavor mixing require new physics beyond the standard model \cite{neutrino}. Additionally, the cosmological challenges of particle physics such as inflation, dark matter, and baryon asymmetry also acquire the standard model extension \cite{pdg2018}. Hence, it is worthwhile to look for a theory that addresses all these puzzles.
The standard model actually contains a hidden/accident symmetry $U(1)_{B-L}$. If one includes, e.g., three right-handed neutrinos it behaves as a gauge symmetry free from all the anomalies. The resulting theory based on $SU(3)_C\otimes SU(2)_L\otimes U(1)_Y\otimes U(1)_{B-L}$ can provide consistent neutrino masses via induced seesaw mechanism \cite{seesaw}. This theory also generates suitable baryon asymmetry converted from the leptogenesis resulting from the seesaw mechanism \cite{leptog}. However, within this framework, it is not naturally to understand dark matter. Indeed, a matter parity can be induced as residual gauge symmetry due to the $U(1)_{B-L}$ breaking. However, the theory does not contain any odd field responsible for dark matter candidate. Let us note that the Majoron associated with $B-L$ breaking is actually eaten by the new neutral gauge boson, which should rapidly decay into quarks and leptons. The $B-L$ Higgs field and right-handed neutrinos are also unstable, since they decay to ordinary particles.
In this work, we discuss a class of models based upon gauge symmetry, $SU(3)_C\otimes SU(P)_L\otimes U(1)_X\otimes U(1)_{N}$, called 3-$P$-1-1, for $P=3,4$. Here, $SU(2)_L$ is extended to $SU(P)_L$ which offers a natural solution for the question of generation number \cite{331}. It is easily verified that the electric charge $Q$ and baryon-minus-lepton charge $B-L$ neither commute nor close algebraically with $SU(P)_L$ \cite{3311,dong2015}. Hence, the two Abelian factors $U(1)_{X,N}$ are resulted from algebraic closure condition, in which the new charges $X$ and $N$ are related to $Q$ and $B-L$ via the Cartan generators of $SU(P)_L$, respectively. Besides the answer of generation number, the model manifestly accommodates dark matter which is unified with normal matter to form $SU(P)_L$ multiplets. This is a consequence of the noncommutative $B-L$ symmetry and matter partiy as a residual gauge symmetry. Such dark fields have ``wrong'' $B-L$ charge in comparison to the standard model definition, that is old under the matter parity, providing dark matter candidates. They may be a fermion, scalar or gauge boson. The abundance of dark matter observed today can either be thermally produced as a WIMP or results from a standard leptogenesis similarly to the baryon asymmetry \cite{a3311}. Therefore, in the second case both the dark and normal matter asymmetries are produced due to the CP-violating decay of the lightest right-handed neutrino. In a scenario, the $U(1)_{N}$ breaking field successfully inflates the early universe, and its decay reheats the universe producing such right-handed neutrinos, as desirable \cite{a3311}.
The 3-3-1-1 model has been extensively investigated in the literature \cite{3311,dong2015,a3311}, but the 3-4-1-1 model has not considered yet. In this work, we construct the 3-4-1-1 model with general fermion and scalar contents, obtain the matter parity, and interpret dark matter candidates. Since the theory contains two $U(1)$ factors, the kinetic mixing between the corresponding gauge bosons is not avoidable \cite{kineticmixing}. Therefore, we diagonalize the gauge boson sector when including the kinetic mixing term. The effect of the kinetic mixing is present in the $\rho$ parameter and the coupling of $Z$ with fermions, which can alter the electroweak precision test. It significantly modifies the neutral meson mixings and rare meson decays. The last aim of this work is to probe the new physics of the model at the LHC. This work also revisits the kinetic mixing effect in the 3-3-1-1 model, which was previously studied \cite{dong2016}.
The rest of this work is organized as follows. In Sec. \ref{model}, we introduce the model and show dark matter. In Sec. \ref{gauge}, we diagonalize the gauge sector. In Sec. \ref{pheno}, we examine the $\rho$ parameter, mixing parameters, and the $Z$ couplings. In Sec. \ref{pheno1}, we investigate the FCNCs. The search for the new physics is presented in Sec. \ref{lhc}. In Sec. \ref{3311}, the kinetic mixing effect in a previous study is revisited. Finally, we conclude this work in Sec. \ref{con}.
\section{\label{model} The model}
In this section we propose the 3-4-1-1 model, while the 3-3-1-1 model \cite{dong2015,dong2016} was well established and skipped.
\subsection{Gauge symmetry}
As stated, the gauge symmetry is given by
\begin{equation} SU(3)_C\otimes SU(4)_L\otimes U(1)_X\otimes U(1)_N.\end{equation}
The electric and baryon-minus-lepton charges are embedded as
\begin{eqnarray} && Q=T_3+\beta T_8 + \gamma T_{15}+X,\\
&& B-L= b T_8 +c T_{15}+N,\end{eqnarray} where $T_i$ ($i=1,2,3,...,15$), $X$, and $N$ are $SU(4)_L$, $U(1)_X$, and $U(1)_N$ charges, respectively.
Nontrivial commutation relations are obtained by
\begin{eqnarray} && [Q,T_1\pm i T_2]=\pm(T_1\pm i T_2),\nonumber \\ &&[Q,T_4\pm i T_5]=\mp q(T_4\pm i T_5),\nonumber \\
&& [Q,T_6\pm i T_7]=\mp(1+q)(T_6\pm i T_7),\nonumber \\
&& [Q,T_9\pm i T_{10}]=\mp p(T_9\pm i T_{10}),\nonumber \\
&& [Q,T_{11}\pm i T_{12}]=\mp(1+p)(T_{11}\pm i T_{12}),\nonumber \\
&& [Q,T_{13}\pm i T_{14}]=\mp(p-q)(T_{13}\pm i T_{14}),\nonumber \\
&& [B-L,T_4\pm i T_5]=\mp (1+n) (T_4\pm i T_5),\nonumber \\
&& [B-L,T_6\pm i T_7]=\mp(1+n)(T_6\pm i T_7),\nonumber \\
&& [B-L,T_9\pm i T_{10}]=\mp (1+m)(T_9\pm i T_{10}),\nonumber \\
&& [B-L,T_{11}\pm i T_{12}]=\mp(1+m)(T_{11}\pm i T_{12}),\nonumber \\
&& [B-L,T_{13}\pm i T_{14}]=\mp(m-n)(T_{13}\pm i T_{14}),\end{eqnarray} where we define the basic electric charges as $q=-(1+\sqrt{3}\beta)/2$ and $p=-(1+\sqrt{6}\gamma-q)/3$ and the basic baryon-minus-lepton charges as $n=-(2+\sqrt{3}b)/2$ and $m=-(2+\sqrt{6}c-n)/3$. Hence, ($q,p$) and ($n,m$) will determine the $Q$ and $B-L$ charges of new particles, respectively.
\subsection{Particle presentation}
The fermions transform under the 3-4-1-1 gauge symmetry as
\begin{eqnarray} && \psi_{aL}\equiv \left(\begin{array}{c}\nu\\ e \\ E^{q,n}\\ F^{p,m}\end{array}\right)_{aL}\sim \left(1,4,\frac{p+q-1}{4},\frac{m+n-2}{4}\right), \\
&& Q_{\alpha L}\equiv \left(\begin{array}{c}d \\ -u \\ J^{-q-1/3,-n-2/3}\\ K^{-p-1/3,-m-2/3} \end{array}\right)_{\alpha L}\sim \left(3,4^*,-\frac{p+q+1/3}{4},-\frac{m+n+2/3}{4}\right),\\
&& Q_{3 L}\equiv \left(\begin{array}{c}u \\ d \\ J^{q+2/3,n+4/3}\\ K^{p+2/3,m+4/3}\end{array}\right)_{3L}\sim \left(3,4,\frac{p+q+5/3}{4},\frac{m+n+10/3}{4}\right),\\
&& \nu_{aR}\sim \left(1,1,0,-1\right),\hspace*{0.5cm} e_{aR}\sim \left(1,1,-1,-1\right),\\
&& E_{aR}\sim \left(1,1,q,n\right), \hspace*{0.5cm} F_{aR}\sim \left(1,1,p,m\right),\\
&& u_{a R}\sim \left(3,1,2/3,1/3\right),\hspace*{0.5cm} d_{aR}\sim \left(3,1,-1/3,1/3\right),\\
&& J_{\alpha R}\sim \left(3,1,-q-1/3,-n-2/3\right),\hspace*{0.5cm} K_{\alpha R}\sim \left(3,1,-p-1/3,-m-2/3\right), \\
&&J_{3R}\sim \left(3,1,q+2/3,n+4/3\right),\hspace*{0.5cm} K_{3R}\sim \left(3,1,p+2/3,m+4/3\right),\end{eqnarray}
where $a=1, 2, 3$ and $\alpha =1, 2$ denote generation indices. Additionally, $\nu_{R}, E, F, J$, and $K$ are new fields, included to complete the representations. This fermion content is independent of all the anomalies (cf. Appendix \ref{appa}).
In order for gauge symmetry breaking and mass generation, we introduce the scalar content,
\begin{eqnarray} \eta &=&
\left(
\begin{array}{l}
\eta^{0,0}_1\\
\eta^{-1,0}_2\\
\eta^{q,n+1}_3\\
\eta^{p,m+1}_4
\end{array}\right)\sim \left(1,4,\frac{p+q-1}{4},\frac{m+n+2}{4}\right),\\
\rho &=&
\left(
\begin{array}{l}
\rho^{1,0}_1\\
\rho^{0,0}_2\\
\rho^{q+1,n+1}_3\\
\rho^{p+1,m+1}_4
\end{array}\right)\sim \left(1,4,\frac{p+q+3}{4},\frac{m+n+2}{4}\right),\\
\chi &=&
\left(
\begin{array}{l}
\chi^{-q,-n-1}_1\\
\chi^{-q-1,-n-1}_2\\
\chi^{0,0}_3\\
\chi^{p-q,m-n}_4
\end{array}\right)\sim\left(1,4,\frac{p-3q-1}{4},\frac{m-3n-2}{4}\right), \\
\Xi &=&
\left(
\begin{array}{l}
\Xi^{-p,-m-1}_1\\
\Xi^{-p-1,-m-1}_2\\
\Xi^{-p+q,-m+n}_3\\
\Xi^{0,0}_4
\end{array}\right)\sim\left(1,4,\frac{-3p+q-1}{4},\frac{-3m+n-2}{4}\right),\\
\phi &\sim& (1,1,0,2),\end{eqnarray}
where the superscipts stand for ($Q,B-L$) respectively, while the subscripts indicate $SU(4)_L$ components. The scalars obtain such quantum numbers, provided that they couple left-handed fermions to corresponding right-handed counterparts, except that $\phi$ couples to $\nu_R\nu_R$ (see below).
\subsection{Total Lagrangian}
The total Lagrangian has the form,
\begin{eqnarray}
\mathcal{L}= \mathcal{L}_{\mathrm{kinetic}} + \mathcal{L}_{\mathrm{Yukawa}}-V,
\end{eqnarray} where the first part combines kinetic terms and gauge interactions, given by
\begin{eqnarray} \mathcal{L}_{\mathrm{kinetic}} &=& \sum_F \bar{F}i\gamma^\mu D_\mu F+\sum_S (D^\mu S)^\dagger (D_\mu S)\nonumber \\
&& -\frac 1 4 G_{r \mu\nu} G_r^{\mu\nu}-\frac 1 4 A_{i \mu\nu} A_i^{\mu\nu}-\frac 1 4 B_{\mu\nu} B^{\mu\nu}-\frac 1 4 C_{\mu\nu} C^{\mu\nu}-\frac{\delta}{2} B_{\mu\nu} C^{\mu\nu}.\end{eqnarray} The covariant derivative is
\begin{equation} D_\mu = \partial_\mu + i g_s T_r G_{r\mu}+ i g T_i A_{i \mu}+ i g_X X B_\mu + i g_N N C_\mu,\end{equation} and we denote the coupling constants $(g_s,g,g_X,g_N)$, generators $(T_r,T_i,X,N)$, and gauge bosons $(G_r, A_i, B,C)$ corresponding to the 3-4-1-1 subgroups, respectively. Above, $F$ and $S$ run over the fermion and scalar multiplets, while the parameter $\delta$ is dimensionless, called kinetic mixing.\footnote{This kinetic mixing term is always presented due to the gauge invariance and cannot be removed by rescaling the corresponding fields. Even if its tree-level value vanishes, it can be radiatively induced \cite{kineticmixing}.}
The second and last parts are the Yukawa interactions and scalar potential, given respectively by
\begin{eqnarray} \mathcal{L}_{\mathrm{Yukawa}}&=&h^\nu_{ab}\bar{\psi}_{aL}\eta\nu_{bR}+ h^e_{ab}\bar{\psi}_{aL}\rho e_{bR} + h^E_{ab}\bar{\psi}_{aL}\chi E_{bR}+ h^{F}_{ab}\bar{\psi}_{aL}\Xi F_{bR}+h'^\nu_{ab}\bar{\nu}^c_{aR}\nu_{bR}\phi\nonumber \\
&& + h^J_{33}\bar{Q}_{3L}\chi J_{3R}+ h^{K}_{33}\bar{Q}_{3L}\Xi K_{3R} + h^J_{\alpha \beta}\bar{Q}_{\alpha L} \chi^* J_{\beta R}+ h^{K}_{\alpha \beta}\bar{Q}_{\alpha L} \Xi^* K_{\beta R} \nonumber \\
&&+ h^u_{3a} \bar{Q}_{3L}\eta u_{aR}+h^u_{\alpha a } \bar{Q}_{\alpha L}\rho^* u_{aR} +h^d_{3a}\bar{Q}_{3L}\rho d_{aR} + h^d_{\alpha a} \bar{Q}_{\alpha L}\eta^* d_{aR} +H.c.,\\
V &=& \mu^2_1\eta^\dagger \eta + \mu^2_2 \rho^\dagger \rho + \mu^2_3 \chi^\dagger \chi + \mu^2_4 \Xi^\dagger \Xi + \mu^2_5 \phi^\dagger \phi +\lambda_1 (\eta^\dagger \eta)^2 + \lambda_2 (\rho^\dagger \rho)^2 \nonumber \\
&&+ \lambda_3 (\chi^\dagger \chi)^2 + \lambda_4 (\Xi^\dagger \Xi)^2+ \lambda_5 (\phi^\dagger \phi)^2 + \lambda_6 (\eta^\dagger \eta)(\rho^\dagger \rho) +\lambda_7 (\eta^\dagger \eta)(\chi^\dagger \chi)\nonumber \\
&&+\lambda_8 (\eta^\dagger \eta)(\Xi^\dagger \Xi)+\lambda_{9} (\eta^\dagger\eta)(\phi^\dagger \phi)+\lambda_{10} (\rho^\dagger \rho)(\chi^\dagger \chi) +\lambda_{11} (\rho^\dagger \rho)(\Xi^\dagger \Xi) \nonumber \\
&& +\lambda_{12}(\rho^\dagger \rho)(\phi^\dagger \phi)+\lambda_{13}(\chi^\dagger \chi)(\Xi^\dagger \Xi)+\lambda_{14}(\chi^\dagger \chi)(\phi^\dagger \phi) +\lambda_{15}(\Xi^\dagger \Xi)(\phi^\dagger \phi) \nonumber \\
&&+\lambda_{16} (\eta^\dagger \rho)(\rho^\dagger \eta) +\lambda_{17} (\eta^\dagger \chi)(\chi^\dagger \eta)+\lambda_{18} (\eta^\dagger \Xi)(\Xi^\dagger \eta)+\lambda_{19} (\rho^\dagger \chi)(\chi^\dagger \rho)\nonumber \\
&& +\lambda_{20} (\rho^\dagger \Xi)(\Xi^\dagger \rho)+\lambda_{21} (\chi^\dagger \Xi)(\Xi^\dagger \chi)+ (\lambda\eta \rho \chi \Xi+H.c.),
\end{eqnarray} where the Yukawa ($h$'s) and scalar ($\lambda$'s) couplings are dimensionless, while the $\mu$'s parameters have the mass dimension.
\subsection{Matter parity}
Since $Q$ is conserved, only the neutral components $\eta_1, \rho_2, \chi_3, \Xi_4$, and $\phi$ develop vacuum expectation values (VEVs),
\begin{eqnarray} \langle \eta \rangle &=& \frac{1}{\sqrt{2}}\left(
\begin{array}{c}
u \\
0\\
0\\
0
\end{array}\right),\hspace*{0.5cm}
\langle \rho\rangle =
\frac{1}{\sqrt{2}} \left(
\begin{array}{c}
0\\
v \\
0\\
0
\end{array}\right),\hspace*{0.5cm}
\langle \chi\rangle =
\frac{1}{\sqrt{2}} \left(
\begin{array}{c}
0\\
0\\
w\\
0
\end{array}\right),\hspace*{0.5cm}
\langle \Xi\rangle =
\frac{1}{\sqrt{2}} \left(
\begin{array}{c}
0\\
0\\
0\\
V
\end{array}\right),\hspace*{0.5cm}
\langle \phi\rangle = \frac{1}{\sqrt{2}} \Lambda.\end{eqnarray}
The VEVs $V, w, u, v$ break the 3-4-1-1 symmetry to $SU(3)_C\otimes U(1)_Q\otimes U(1)_{B-L}$, while the VEV $\Lambda$ breaks $B-L$ to the matter parity, $U(1)_{B-L}\rightarrow P$, where \begin{equation} P=(-1)^{3(B-L)+2s}=(-1)^{3(bT_8+cT_{15}+N)+2s}\end{equation} is multiplied by the spin parity $(-1)^{2s}$ as conserved by the Lorentz symmetry, similar to the 3-3-1-1 model \cite{3311,dong2015}.\footnote{This kind of the matter parity is also recognized in the class of the left-right extensions \cite{3331}.}
Because $w, V, \Lambda$ provide the masses of new particles, whereas $u,v$ do so for the ordinary particles, we assume $u,v\ll w, V, \Lambda$, to keep a consistency with the standard model.
The matter parity $P$ divides particles into two types: \begin{enumerate} \item Normal particles according to $P=1$ (even): $\nu$, $e$, $u$, $d$, $\eta_{1,2}$, $\rho_{1,2}$, $\chi_3$, $\Xi_4$, $\phi$, $\gamma$, $W$, $Z_{1,2,3,4}$, which include the standard model particles. \item Wrong particles according to $P=P^\pm_n\equiv (-1)^{\pm(3n+1)}$ or $P=P^\pm_m\equiv (-1)^{\pm(3m+1)}$: $E$, $F$, $J$, $K$, $\eta_{3,4}$, $\rho_{3,4}$, $\chi_{1,2,4}$, $\Xi_{1,2,3}$, $W_{13}^{\mp q,\mp (n+1)}$, $W_{14}^{\mp p,\mp (m+1)}$, $W_{23}^{\mp (q+1),\mp (n+1)}$, $W_{24}^{\mp (p+1),\mp (m+1)}$, where the $W$'s fields are non-Hermitian gauge bosons which couple to the mentioned weight-raising/lowering operators. The remainders $\chi_4$, $\Xi_3$, and $W_{34}^{\pm (q-p),\pm (n-m)}$ have $P=P^+_n P^-_m$ or conjugated. \end{enumerate}
Generally, the wrong fields transform nontrivially under the matter parity for $n,m\neq (2k-1)/3$ and $n-m\neq 2k/3$ for every $k$ integer. However, an alternative case is that both $P_{n,m}=-1$ are odd, i.e. $n,m=2k/3$. In this case all the wrong fields are odd, except that $\chi_4$, $\Xi_3$, and $W_{34}$ are even which belong to the first type of normal particles.
\subsection{Dark matter}
It is easily to prove that the wrong particles always couple in pairs or self-interacted due to the matter parity conservation, which is analogous to superparticles in supersymmetry \cite{dong2015} (see also Dong and Huong in \cite{3331}). Hence, the lightest wrong particle (LWP) is stabilized, responsible for dark matter.
Since the candidate must be color and electrically neutral, we have several dark matter models: (i) $q=0$ including $E^0$, $\eta^0_3$, $\chi^0_1$, $W^0_{13}$; (ii) $p=0$ including $F^0$, $\eta^0_4$, $\Xi^0_1$, $W^0_{14}$; (iii) $q=-1$ consisting of $\rho^0_3$, $\chi^0_2$, $W^0_{23}$; (iv) $p=-1$ consisting of $\rho^0_4$, $\chi^0_2$, $W^0_{24}$. In each case, the remaining basic electric charge is left arbitrary.
The specific dark matter models that combine above cases are
\begin{enumerate}
\item $q=p=-1$: The candidate is a scalar combination of $\rho_{3,4}$, $\chi_2$, and $\Xi_2$, or a gauge boson combination of $W_{23}$ and $W_{24}$.
\item $q=-1, p=0$: The candidate is a fermion combination of $F_{1,2,3}$, a scalar combination of $\eta_4$, $\rho_3$, $\chi_2$, and $\Xi_1$, or a gauge boson combination of $W_{14}$ and $W_{23}$.
\item $q=0, p=-1$: The candidate is a fermion combination of $E_{1,2,3}$, a scalar combination of $\eta_3$, $\rho_4$, $\chi_1$, and $\Xi_2$, or a gauge boson combination of $W_{13}$ and $W_{24}$.
\item $q=p=0$: The candidate is a fermion combination of $E_{1,2,3}$ and $F_{1,2,3}$, a scalar combination of $\eta_{3,4}$, $\chi_1$, and $\Xi_1$, or a gauge boson combination of $W_{13}$ and $W_{14}$.
\end{enumerate}
The last model is for $p=q\neq 0,-1$. The candidate includes a scalar combination of $\chi^0_4$ and $\Xi^0_3$, or a vector $W^0_{34}$.
\subsection{Fermion mass}
When the scalars develop VEVs, the fermions gain masses and we write Dirac masses as $-\bar{f}_L m_f f_R+H.c.$ and Majorana masses as $-\frac 1 2 \bar{f}^c_{L,R} m^{L,R}_f f_{L,R}+H.c.$
The mass matrices of new fermions $E_a$, $F_a$, $J_a$, and $K_a$ are given by \begin{eqnarray} && [m_E]_{ab}=-h^E_{ab}\frac{w}{\sqrt{2}},\hspace*{0.5cm} [m_{F}]_{ab}=-h^{F}_{ab}\frac{V}{\sqrt{2}},\\
&& [m_{J}]_{33}=-h^{J}_{33}\frac{w}{\sqrt{2}},\hspace*{0.5cm} [m_{J}]_{\alpha\beta}=-h^J_{\alpha\beta}\frac{w}{\sqrt{2}},\\ &&
[m_{K}]_{33}=-h^{K}_{33}\frac{V}{\sqrt{2}},\hspace*{0.5cm}
[m_{K}]_{\alpha\beta}=-h^{K}_{\alpha\beta}\frac{V}{\sqrt{2}},\end{eqnarray} which all have masses at $w$, $V$ scale.
The mass matrices of charged-leptons and quarks $e_a$, $u_a$ and $d_a$ are obtained as \begin{eqnarray} && [m_e]_{ab}=-h^e_{ab}\frac{v}{\sqrt{2}},\nonumber \\
&& [m_u]_{3 a}=-h^u_{3 a}\frac{u}{\sqrt{2}},\hspace*{0.5cm} [m_u]_{\alpha a}=h^u_{\alpha a}\frac{v}{\sqrt{2}},\nonumber \\
&& [m_d]_{3 a}=-h^d_{3 a}\frac{v}{\sqrt{2}},\hspace*{0.5cm} [m_d]_{\alpha a}=-h^d_{\alpha a}\frac{u}{\sqrt{2}},\end{eqnarray} which provide appropriate masses at $u,v$ scale.
For the neutrinos, $\nu_{aL,R}$, the Dirac and Majorana masses are $[m_\nu]_{ab}=-h^\nu_{ab}\frac{u}{\sqrt{2}}$ and $[m^R_\nu]_{ab}=-\sqrt{2}h'^\nu_{ab}\Lambda$, respectively. Since $u\ll \Lambda$, the observed neutrinos $(\sim \nu_{aL})$ achieve masses via the type I seesaw mechanism, \begin{equation} m^L_\nu \simeq - m_\nu (m^R_\nu)^{-1} (m_\nu)^T\sim u^2/\Lambda,\end{equation} which is small, as expected. The sterile neutrinos $(\sim \nu_{aR})$ obtain large masses, such as $m^R_\nu$.
\section{\label{gauge}Kinetic mixing}
\subsection{Canonical basis}
Let us write down the kinetic terms of the two $U(1)$ gauge fields as
\begin{equation} \mathcal{L}_{\mathrm{kinetic}}\supset -\frac 1 4 B^2_{\mu\nu}-\frac 1 4 C^2_{\mu\nu}-\frac{\delta}{2} B_{\mu\nu}C^{\mu\nu}= -\frac 1 4 (B_{\mu\nu}+\delta C_{\mu\nu})^2-\frac 1 4 (1-\delta^2) C^2_{\mu\nu},\end{equation} where $B_{\mu\nu}=\partial_\mu B_\nu-\partial_\nu B_\mu$ and $C_{\mu\nu}=\partial_\mu C_\nu-\partial_\nu C_\mu$ are the corresponding field strength tensors.
Because of the kinetic mixing term ($\delta$), the two gauge bosons $B_\mu$ and $C_\mu$ are generally not orthonormalized. We change to the canonical basis by a nonunitary transformation $(B_\mu,C_\mu)\rightarrow (B'_\mu,C'_\mu)$, where
\begin{equation} B'=B+\delta C,\hspace*{0.5cm} C'=\sqrt{1-\delta^2}C.\end{equation}
We substitute $B,C$ in terms of $B',C'$ into the covariant derivative. It becomes
\begin{equation}
D_\mu \supset ig_X X B_\mu+ig_N N C_\mu = i g_X X B'_\mu+\frac{i}{\sqrt{1-\delta^2}}(g_N N-g_X X\delta) C'_\mu,
\end{equation} which is given in terms of the orthonormalized (canonical) fields $(B'_\mu,C'_\mu)$.
\subsection{Gauge boson mass}
The 3-4-1-1 symmetry breaking leads to mixings among $A_{3}$, $A_8$, $A_{15}$, $B'$, and $C'$. Their mass Lagrangian arises from $\sum_S (D_\mu\langle S\rangle)^\dagger (D^\mu\langle S\rangle)$, such that
\begin{eqnarray}
\mathcal{L}_{\mathrm{mass}}^{\mathrm{neutral}} =
\fr1 2\left(A_3 \ A_8 \ A_{15} \ B' \ C' \right) M^2 \left(A_3 \ A_8 \ A_{15} \ B' \ C' \right)^T, \end{eqnarray}
where the mass matrix $M^2=\{m^2_{ij}\}$ is symmetric, possessing the elements,
\begin{eqnarray}
m^2_{11}&= & \frac {g^2} 4 (u^2 + v^2),\hspace*{0.5cm} m^2_{12}= \frac {g^2} {4\sqrt3} (u^2-v^2),\hspace*{0.5cm} m^2_{13} = \frac {g^2} {4\sqrt6} (u^2-v^2),\nonumber \\
m^2_{14} &=& -\frac{g^2t_X}{4\sqrt6}[\beta_1u^2+(2\sqrt6-\beta_1)v^2], \nonumber \\
m^2_{15}&=& \frac{g^2}{4\sqrt{6(1-\delta^2)}}\{[\delta\beta_1t_X-(\sqrt2b+ c )t_N] u^2 +[\delta (2\sqrt6-\beta_1)t_X+(\sqrt2b+c )t_N] v^2 \} ,\nonumber \\
m^2_{22}&=&\frac {g^2} {12}(u^2+v^2+4w^2),\hspace*{0.5cm} m^2_{23}=\frac {g^2} {12\sqrt2}(u^2+v^2-2w^2),\nonumber \\
m^2_{24}&=& -\frac{g^2t_X}{12\sqrt2}[\beta_1u^2-(2\sqrt6-\beta_1)v^2+2(2\sqrt2\beta-\gamma) w^2], \nonumber \\
m^2_{25}&=& \frac{g^2} { 12\sqrt{2(1-\delta^2)}}\{[\delta \beta_1t_X-(\sqrt2b+c )t_N] u^2-[\delta (2\sqrt6-\beta_1)t_X+(\sqrt2b+c )t_N] v^2\nonumber \\
&& +2[\delta(2\sqrt2\beta-\gamma) t_X-(2\sqrt2 b-c) t_N]w^2\},\nonumber \\
m^2_{33}&=&\frac {g^2} {24}(u^2+v^2+w^2+9V^2),\hspace*{0.5cm} m^2_{34}= -\frac{g^2t_X}{24}[\beta_1 u^2-(2\sqrt6-\beta_1) v^2 -(2\sqrt2\beta-\gamma)w^2+9\gamma V^2],\nonumber \\
m^2_{35}&=& \frac{g^2} {24\sqrt{1-\delta^2}}\{[\delta\beta_1t_X-(\sqrt2b+c )t_N] u^2-[\delta (2\sqrt6-\beta_1)t_X+(\sqrt2b+c )t_N] v^2\nonumber \\
&&-[\delta(2\sqrt2\beta-\gamma) t_X-(2\sqrt2 b-c) t_N]w^2+9 (\delta\gamma t_X-ct_N)V^2\},\nonumber \\
m^2_{44}&=& \frac{g^2t^2_X} {24}[\beta_1^2u^2+(2\sqrt6-\beta_1)^2 v^2+(2\sqrt2\beta-\gamma)^2w^2+9 \gamma^2V^2],\nonumber \\
m^2_{45}&=&-\frac{g^2t_X} {24\sqrt{1-\delta^2}}\{[\delta\beta_1t_X-(\sqrt2 b+c)t_N] \beta_1u^2+[\delta (2\sqrt6-\beta_1)t_X+(\sqrt2 b+c)t_N] (2\sqrt6-\beta_1)v^2\nonumber \\
&&+[\delta (2\sqrt2\beta-\gamma )t_X-(2\sqrt2 b-c)t_N](2\sqrt2\beta- \gamma)w^2+9\gamma (\delta\gamma t_X-ct_N) V^2\},\nonumber \\
m^2_{55}&=& \frac{g^2} {24(1-\delta^2)}\{[\delta \beta_1t_X-(\sqrt2 b+c )t_N]^2 u^2+[\delta (2\sqrt6-\beta_1)t_X+(\sqrt2 b+c )t_N]^2v^2\nonumber \\
&&+[\delta (2\sqrt2 \beta-\gamma)t_X-(2\sqrt2 b-c )t_N]^2w^2+9(\delta\gamma t_X-ct_N)^2 V^2+96t_N^2\Lambda^2\},\nonumber
\end{eqnarray} where we have defined $t_X=g_X/g$, $t_N=g_N/g$, and $\beta_1=\sqrt6+\sqrt2\beta+\gamma$.
The mass matrix always provides a zero eigenvalue with corresponding eigenstate (photon field),
\begin{equation}
A = s_W A_3 + c_W \left(\beta t_W A_8 +\gamma t_W A_{15} +\frac {t_W}{t_X} B'\right),\label{amu}\end{equation} where $s_W=e/g=t_X/\sqrt{1+(1+\beta^2+\gamma^2)t_X^2}$ is the sine of the Weinberg's angle \cite{donglong}. Since the field in parentheses of (\ref{amu}) is properly the hypercharge field coupled to $Y=Q-T_3$, we define the standard model $Z$ as
\begin{eqnarray}
Z=c_W A_3 -s_W \left(\beta t_W A_8 +\gamma t_W A_{15} +\frac {t_W}{t_X} B'\right).\end{eqnarray}
The new neutral gauge bosons, called $Z'_2, Z'_3$, orthogonal to the hypercharge field take the forms,
\begin{eqnarray}
Z'_2 &=& \frac{1}{\sqrt{1-\beta^2 t_W^2}} \left[(1-\beta^2 t_W^2) A_8-\beta\gamma t_W^2 A_{15}-\frac{\beta t_W^2}{t_X} B'\right], \\
Z'_3 &=& \frac{1}{\sqrt{1+\gamma^2t_X^2}}\left(A_{15}-
\gamma t_X B'\right).
\end{eqnarray}
At this stage, $C'$ is always orthogonal to $A, Z, Z'_2, Z'_3$.
Let us change to the new basis $A, Z, Z'_2, Z'_3$, and $C'$, such that $(A_3\, A_8\, A_{15}\, B'\, C')^T=U_1(A\, Z\, Z'_2\, Z'_3\, C')^T$, where
\begin{eqnarray}
U_1= \left(%
\begin{array}{ccccc}
s_W&c_W&0&0&0\\
\beta s_W&-\beta s_Wt_W&\sqrt{1-\beta^2t_W^2}&0&0\\
\gamma s_W&-\gamma s_Wt_W&-\frac{\beta\gamma t_W^2}{\sqrt{1-\beta^2t^2_W}}&\frac {1}{\sqrt{1+\gamma^2t_X^2}}&0\\
\frac{s_W}{t_X}&-\frac{s_Wt_W}{t_X}&-\frac{\beta t_W^2}{t_X\sqrt{1-\beta^2 t_W^2}}&-\frac{\gamma t_X}{\sqrt{1+\gamma^2t_X^2}}&0\\0&0&0&0&1
\end{array} \right).
\end{eqnarray}
The mass matrix $M^2$ is correspondingly changed to
\begin{eqnarray}
M'^{2} =U_1^T M^2 U_1 =\left(%
\begin{array}{ccc}
0& 0 \\
0& M_s^{\prime 2}
\end{array} \right), \hspace*{0.5cm} M'^2_s \equiv
\left(
\begin{array}{cccc}
m^2_{Z} & m^2_{ZZ'_2}& m^2_{ZZ'_3} & m^2_{ZC'}\\
m^2_{ZZ'_2} & m^2_{Z'_2} & m^2_{Z'_2Z'_3} & m^2_{Z'_2C'}\\
m^2_{ZZ'_3} & m^2_{Z'_2Z'_3} & m^2_{Z'_3} & m^2_{Z'_3C'}\\
m^2_{ZC'} & m^2_{Z'_2C'} & m^2_{Z'_3C'}& m^2_{C'}
\end{array}\right).
\end{eqnarray} where
\begin{eqnarray}
m^2_Z&=&\frac {g^2}{4c^2_W} (u^2 + v^2),\hspace*{0.5cm} m^2_{ZZ'_2} =\frac{g^2\sqrt{1-(\beta^2+\gamma^2)t_W^2}}{4\sqrt3 c_W\sqrt{1+\gamma^2 t^2_X}}\{\beta_2u^2+(2\sqrt3 \beta t_X^2-\beta_2)v^2\},\nonumber \\
m^2_{ZZ'_3}& =&\frac{g^2}{4\sqrt6 c_W\sqrt{1+\gamma^2 t^2_X}}\{(1+\gamma\beta_1t_X^2)u^2+[\gamma(2\sqrt6-\beta_1)t_X^2-1]v^2\},\nonumber \\
m^2_{ZC'}&=& \frac{g^2}{4\sqrt6 c_W \sqrt{1-\delta^2}}\{[\delta\beta_1t_X-(\sqrt2 b+c)t_N]u^2+[\delta(2\sqrt6-\beta_1)t_X+(\sqrt2 b+c)t_N]v^2\},\nonumber \\
m^2_{Z'_2}&= & \frac{g^2[1-(\beta^2+\gamma^2)t_W^2]}{12(1+\gamma^2t_X^2)}\{\beta_2^2u^2+(2\sqrt3\beta t_X^2-\beta_2)^2v^2+4[1+(\beta^2+\gamma^2)t_X^2]^2w^2\} ,\nonumber \\
m^2_{Z'_2Z'_3}&= & \frac{g^2\sqrt{1-(\beta^2+\gamma^2)t_W^2}}{12\sqrt2(1+\gamma^2t_X^2)}\{(1+\gamma\beta_1t_X^2)\beta_2u^2+[\gamma(2\sqrt6-\beta_1)t_X^2-1](2\sqrt3\beta t_X^2-\beta_2)v^2\nonumber \\
&&+2[\gamma(2\sqrt2 \beta-\gamma)t_X^2-1][1+(\beta^2+\gamma^2)t_X^2]w^2\},\nonumber \\
m^2_{Z'_2C'}&= & \frac{g^2\sqrt{1-(\beta^2+\gamma^2)t_W^2}}{12\sqrt2\sqrt{(1-\delta^2)(1+\gamma^2t_X^2)}}\{[\delta\beta_1t_X-(\sqrt2 b+c)t_N]\beta_2u^2+[\delta(2\sqrt6-\beta_1)t_X+(\sqrt2 b+c)t_N]\nonumber \\
&&\times(2\sqrt3\beta t_X^2-\beta_2)v^2+2[\delta(2\sqrt2\beta-\gamma)t_X-(2\sqrt2 b-c)t_N][1+(\beta^2+\gamma^2)t_X^2]w^2\},\nonumber \\
m^2_{Z'_3}&= &\frac{g^2}{24(1+\gamma^2t_X^2)}\{(1+\gamma\beta_1t_X^2)^2u^2+[\gamma(2\sqrt6-\beta_1)t_X^2-1]^2v^2+[\gamma(2\sqrt2 \beta-\gamma)t_X^2-1]^2w^2\nonumber \\
&&+9(1+\gamma^2t_X^2)^2V^2\},\nonumber \\
m^2_{Z'_3C'}&= &\frac{g^2}{24\sqrt{(1-\delta^2)(1+\gamma^2t_X^2)}}\{[\delta\beta_1t_X-(\sqrt2 b+c)t_N](1+\gamma\beta_1t_X^2)u^2+[\delta(2\sqrt6-\beta_1)t_X+(\sqrt2 b+c)t_N]\nonumber \\
&&\times[\gamma(2\sqrt6-\beta_1)t_X^2-1]v^2+[\delta(2\sqrt2 \beta-\gamma)t_X-(2\sqrt2 b-c)t_N][\gamma(2\sqrt2 \beta-\gamma)t_X^2-1]w^2\nonumber \\
&&+9(\delta\gamma t_X-ct_N)(1+\gamma^2t_X^2)V^2\},\nonumber \\
m^2_{C'}&= & m^2_{55}, \nonumber
\end{eqnarray} where we have defined $\beta_2=1+(\sqrt3 \beta+\beta^2+\gamma^2)t_X^2$.
Since $u, v \ll w, V, \Lambda$, the first row and first column of $M'^2_s$ consist of the elements much smaller than those of the remaining entries. We diagonalize $M'^2_s$ using the seesaw formula \cite{seesaw} that separates $Z$ from the heavy fields, given by
\begin{eqnarray}
\left(Z \, Z'_2 \, Z'_3 \, C' \right)^T = U_2 \left(Z_1 \, \mathcal{Z}'_2 \, \mathcal{Z}'_3 \, \mathcal{C}' \right)^T, \hspace*{0.5cm} M''^2=U^{T}_2M'^2_sU_2= \left(\begin{array}{cc}
m^{2}_{Z_{1}} &0 \\
0&M''^2_{s}\end{array}\right),\label{tran2}
\end{eqnarray}
where $Z_1$ is physical as decoupled, while $\mathcal{Z}'_2$, $\mathcal{Z}'_3$ and $\mathcal{C}'$ mix via $M''^2_s$, such that
\begin{eqnarray}
U_2 &\simeq & \left(\begin{array}{cccc}
1 & \epsilon_1 & \epsilon_2 & \epsilon_3 \\
-\epsilon_1 & 1 & 0 & 0\\
-\epsilon_2 & 0 & 1 & 0 \\
-\epsilon_3 & 0 & 0 & 1 \\
\end{array}
\right), \hspace*{0.5cm} M''^2_s \simeq
\left(
\begin{array}{ccc}
m^2_{Z'_2} & m^2_{Z'_2Z'_3} & m^2_{Z'_2C'}\\
m^2_{Z'_2Z'_3} & m^2_{Z'_3} & m^2_{Z'_3C'}\\
m^2_{Z'_2C'} & m^2_{Z'_3C'}& m^2_{C'}
\end{array}\right),\label{tranu2}\\
m^{2}_{Z_{1}}&\simeq & m^2_Z - \epsilon_1 m^2_{ZZ'_2} - \epsilon_2 m^2_{ZZ'_3} - \epsilon_3 m^2_{ZC'}.
\end{eqnarray}
We further separate $\epsilon_{1,2,3}\equiv\epsilon_{1,2,3}^0+\epsilon_{1,2,3}^\delta$, where $\epsilon^0_{1,2,3}$ are the mixing of $Z$ with $Z'_2, Z'_3,$ and $C'$ due to the symmetry breaking, whereas $\epsilon^{\delta}_{1,2,3}$ determine those mixings due to the kinetic mixing,
\begin{eqnarray}
\epsilon_1^0&=&\frac{1}{4c_W\sqrt{1+\gamma^2 t_X^2}[1+(\beta^2+\gamma^2)t_X^2]^{3/2}}\left\{\frac{\sqrt3(1+\gamma^2 t_X^2)[\beta_2u^2+(2\sqrt3 \beta t_X^2-\beta_2)v^2]}{w^2}\right.\nonumber \\
&&+\frac{(\beta+2\sqrt2\gamma)[1+\gamma(\gamma-2\sqrt2\beta)t_X^2]t_X^2(u^2+v^2)+\sqrt3[1-\gamma(2\sqrt2\beta-\gamma)t_X^2][1+(\beta^2+\gamma^2)t_X^2](u^2-v^2)}{3V^2}\nonumber \\
&&\left.+\frac{(b\beta+c\gamma)[b-\gamma(c\beta-b\gamma)t_X^2]t_X^2(u^2+v^2)}{4\Lambda^2}\right\},\\
\epsilon_2^0&=&\frac{1}{c_W\sqrt{1+\gamma^2 t_X^2}[1+(\beta^2+\gamma^2)t_X^2]}\left\{\frac{(\beta+2\sqrt2\gamma)t_X^2(u^2+v^2)+\sqrt3[1+(\beta^2+\gamma^2)t_X^2](u^2-v^2)}{3\sqrt2V^2}\right.\nonumber \\
&&\left.+\frac{c(b\beta+c\gamma)t_X^2(u^2+v^2)}{16\Lambda^2}\right\},\\
\epsilon_3^0&=&\frac{(b\beta+c\gamma)t_X^2(u^2+v^2)}{16c_W[1+(\beta^2+\gamma^2)t_X^2]t_N\Lambda^2},\\
\epsilon_1^\delta&=&\frac{\delta\{[b(1+\gamma^2 t_X^2)-\beta(b\beta+2c\gamma)t_X^2]t_N-\delta\beta t_X\}t_X(u^2+v^2)}{16c_W\sqrt{1+\gamma^2 t_X^2}[1+(\beta^2+\gamma^2)t_X^2]^{3/2}t_N^2\Lambda^2},\\
\epsilon_2^\delta&=&\frac{\delta[(ct_N-\delta\gamma t_X)-\gamma(b\beta+c\gamma)t_X^2t_N]t_X(u^2+v^2)}{16 c_W\sqrt{1+\gamma^2 t_X^2}[1+(\beta^2+\gamma^2)t_X^2]t_N^2\Lambda^2},\\
\epsilon_3^\delta&=&\frac{\delta t_X(u^2+v^2)}{16c_W[1+(\beta^2+\gamma^2)t_X^2]t_N\Lambda^2}\left\{\frac{\sqrt{1-\delta^2}}{t_N}-\frac{\delta(b\beta+c\gamma)t_X}{1+\sqrt{1-\delta^2}}\right\}.
\end{eqnarray}
Because $\epsilon_{1,2,3}\sim (u^2, v^2)/ (w^2, V^2, \Lambda^2)$, the mixings are very small.
Next, the symmetry breaking is done through three possible ways, corresponding to the assumptions: $w, V\ll \Lambda$, $w\ll V, \Lambda$, or $w, \Lambda\ll V$. Let us consider the first case, $w, V\ll \Lambda$. We have the element $m^2_{C'}$ much larger than the remainders. The mass matrix $M''^2_s$ can be diagonalized by using the seesaw formula, which yields
\begin{eqnarray}
(\mathcal{Z}'_2\, \mathcal{Z}'_3\, \mathcal{C}')^T=U_3 (\mathcal{Z}_2\,\mathcal{Z}_3\,Z_4)^T, \hspace*{0.5cm} M'''^2 =U_3^{T}M''^2_sU_3= \left(\begin{array}{cc}
M^2_{2\times 2} &0 \\
0&m^{2}_{Z_{4}}\end{array}\right),
\end{eqnarray}
where $Z_4$ is physical as decoupled, while $\mathcal{Z}_2,\mathcal{Z}_3$ mix via $M^2_{2\times 2}$. We obtain
\begin{eqnarray}
U_3 &\simeq & \left(\begin{array}{ccc}
1 & 0& \zeta _1 \\
0 & 1 & \zeta_2 \\
-\zeta_1 & -\zeta_2 & 1\\
\end{array}
\right), \hspace*{0.5cm} M^2_{2\times 2} =
\left(
\begin{array}{cc}
m^2_{11} & m^2_{12} \\
m^2_{12} & m^2_{22}
\end{array}\right), \hspace*{0.5cm} m^{2}_{Z_{4}}\simeq m^2_{C'},
\end{eqnarray}
where $\zeta_{1,2}\equiv \zeta_{1,2}^0+\zeta_{1,2}^\delta$,
\begin{eqnarray}
\zeta_1^0&=&-\frac{(2\sqrt2 b-c) w^2}{24\sqrt2 \sqrt{1-\beta^2t^2_W}t_N \Lambda^2},\\
\zeta_2^0&=&-\frac{9V^2(1+\gamma^2t_X^2)c-w^2[1-\gamma(2\sqrt2 \beta-\gamma)t_X^2](2\sqrt2 b-c)}{96\sqrt{1+\gamma^2t_X^2}t_N\Lambda^2},\\
\zeta_1^\delta&=&\frac{\delta w^2[(1-\delta^2+\sqrt{1-\delta^2})(2\sqrt2 \beta-\gamma)t_X+\delta(2\sqrt2 b-c)t_N]}{24\sqrt2\sqrt{1-\beta^2t_W^2}(1+\sqrt{1-\delta^2})t_N^2\Lambda^2},\\
\zeta_2^\delta&=&\frac{\delta}{96\sqrt{1+\gamma^2t_X^2}(1+\sqrt{1-\delta^2})t_N^2\Lambda^2}\left\{9V^2(1+\gamma^2t_X^2)[\gamma(1-\delta^2+\sqrt{1-\delta^2})t_X+\delta c t_N]\right.\\
&&\left.-w^2[1-\gamma(2\sqrt2 \beta-\gamma)t_X^2][(1-\delta^2+\sqrt{1-\delta^2})(2\sqrt2 \beta-\gamma)t_X+\delta(2\sqrt2 b-c)t_N] \right\},
\end{eqnarray}
which are very small, and
\begin{eqnarray}
m^2_{11}&\simeq&m^2_{Z'_2} - \zeta_1 m^2_{Z'_2C'}\simeq m^2_{Z'_2},\\
m^2_{12}&\simeq&m^2_{Z'_2Z'_3}- \zeta_1 m^2_{Z'_3C'}\simeq m^2_{Z'_2Z'_3},\\
m^2_{22}&\simeq&m^2_{Z'_3}- \zeta_2 m^2_{Z'_3C'}\simeq m^2_{Z'_3}.
\end{eqnarray}
Last, we diagonalize $M^2_{2\times 2}$ to yield two remaining physical gauge bosons,
\begin{eqnarray}
Z_2 = c_\varphi \mathcal{Z}_2 - s_\varphi \mathcal{Z}_3, \hspace*{0.5cm} Z_3 = s_\varphi \mathcal{Z}_2+ c_\varphi \mathcal{Z}_3.
\end{eqnarray}
The $\mathcal{Z}_2-\mathcal{Z}_3$ mixing angle and $Z_2, Z_3$ masses are given by
\begin{eqnarray}
t_{2\varphi }&\simeq& \frac{4\sqrt2 w^2[1-\gamma(2\sqrt2 \beta-\gamma)t_X^2]\sqrt{1+(\beta^2+\gamma^2)t_X^2}}{w^2[7-\gamma^2(2\sqrt2 \beta-\gamma)^2t_X^4+(8\beta^2+4\sqrt2 \beta\gamma+6\gamma^2)t_X^2]-9V^2(1+\gamma^2t_X^2)^2},\\
m^2_ {Z_2,Z_3}&=& \frac 1 2 [m^2_{11} + m^2_{22} \mp \sqrt{(m^2_{11} - m^2_{22})^2 + 4 m^4_{12}}].
\end{eqnarray}
Now we consider two other cases, $w\ll V, \Lambda$ and $w,\Lambda\ll V$. Because $m^2_{Z'_2}$, $ m^2_{Z'_2Z'_3}$, $ m^2_{Z'_2C'}\ll $ $ m^2_{Z'_3}$, $ m^2_{Z'_3C'}$, $ m^2_{C'}$, the mass matrix $M''^2_s$ can be diagonalized, obeying
\begin{eqnarray}
(\mathcal{Z}'_2\, \mathcal{Z}'_3\, \mathcal{C}')^T=U_3' (Z_2\,\mathcal{Z}_3\,\mathcal{C})^T, \hspace*{0.5cm} M'''^2 =U_3'^{T}M''^2_sU_3'= \left(\begin{array}{cc}
m^{2}_{Z_{2}} &0 \\
0&M'^2_{2\times 2}\end{array}\right),
\end{eqnarray}
where $Z_2$ is physical as decoupled, while $\mathcal{Z}_3$ and $\mathcal{C}$ mix via $M'^2_{2\times 2}$, and
\begin{eqnarray}
U_3' &\simeq & \left(\begin{array}{ccc}
1 & \mathcal{E}_1 & \mathcal{E}_2 \\
-\mathcal{E}_1 & 1 & 0 \\
-\mathcal{E}_2 & 0 & 1\\
\end{array}
\right), \hspace*{0.5cm} M'^2_{2\times 2} \simeq
\left(
\begin{array}{cc}
m^2_{Z'_3} & m^2_{Z'_3C'}\\
m^2_{Z'_3C'}& m^2_{C'}
\end{array}\right),\\
m^{2}_{Z_{2}}&\simeq & m^2_{Z'_2} - \mathcal{E}_1 m^2_{Z'_2Z'_3} -\mathcal{E}_2 m^2_{Z'_2C'}.
\end{eqnarray}
Further for the case $w\ll V, \Lambda$, we achieve $\mathcal{E}_{1,2}\equiv\mathcal{E}^0_{1,2}+\mathcal{E}^\delta_{1,2}$, where
\begin{eqnarray}
\mathcal{E}_1^0&=&\frac{\sqrt{1+(\beta^2+\gamma^2)t_X^2}w^2}{3(1+\gamma^2t_X^2)^2}\left\{\frac{4[\gamma(2\sqrt2\beta-\gamma)t_X^2-1]}{3\sqrt2V^2}-\frac{[b(1+\gamma^2t_X^2)-c\beta\gamma t_X^2]c}{4\Lambda^2}\right\},\\
\mathcal{E}_2^0&=&\frac{\sqrt{1+(\beta^2+\gamma^2)t_X^2}[\gamma(c\beta-b\gamma)t_X^2-b]w^2}{12(1+\gamma^2t_X^2)^{3/2}t_N\Lambda^2},\\
\mathcal{E}_1^\delta&=&\frac{\delta\sqrt{1+(\beta^2+\gamma^2)t_X^2}t_X[b\gamma(1+\gamma^2t_X^2)t_N+c\beta(1-\gamma^2t_X^2) t_N-\delta\beta\gamma t_X]w^2}{12(1+\gamma^2t_X^2)^2t_N^2\Lambda^2},\\
\mathcal{E}_2^\delta&=&\frac{\delta\sqrt{1+(\beta^2+\gamma^2)t_X^2}w^2}{12(1+\gamma^2t_X^2)^{3/2}t_N\Lambda^2}\left\{\frac{\delta[b+\gamma(b\gamma-c\beta)t_X^2]}{1+\sqrt{1-\delta^2}}+\frac{\sqrt{1-\delta^2}\beta t_X}{t_N}\right\},
\end{eqnarray}
which are very small. Otherwise, for the case $w, \Lambda \ll V$, we have
\begin{eqnarray}
\mathcal{E}_1&=&-\frac{\sqrt{1+(\beta^2+\gamma^2)t_X^2}(\delta\gamma t_X-ct_N)[\beta(\delta+c\gamma t_Xt_N)t_X-b(1+\gamma^2t_X^2)t_N]w^2}{[\beta(\delta+c\gamma t_Xt_N)t_X-b(1+\gamma^2t_X^2)t_N]^2w^2+12(1+\gamma^2t_X^2)^2t_N^2\Lambda^2},\\
\mathcal{E}_2&=&\frac{\sqrt{(1-\delta^2)(1+\gamma^2t_X^2)[1+(\beta^2+\gamma^2)t_X^2]}[\beta(\delta+c\gamma t_Xt_N)t_X-b(1+\gamma^2t_X^2)t_N]w^2}{[\beta(\delta+c\gamma t_Xt_N)t_X-b(1+\gamma^2t_X^2)t_N]^2w^2+12(1+\gamma^2t_X^2)^2t_N^2\Lambda^2},
\end{eqnarray}
which may be large.
We diagonalize the mass matrix $M'^2_{2\times 2}$ to get two remaining physical gauge bosons, such that
\begin{eqnarray}
Z_3 = c_\xi \mathcal{Z}_3 - s_\xi \mathcal{C}, \hspace*{0.5cm} Z_4 = s_\xi \mathcal{Z}_3+ c_\xi \mathcal{C}.
\end{eqnarray}
The $\mathcal{Z}_3-\mathcal{C}$ mixing angle for the case $w\ll V,\Lambda$ is given by
\begin{eqnarray}
t_{2\xi}\simeq \frac{6\sqrt{1-\delta^2}\sqrt{1+\gamma^2t_X^2}(\delta\gamma t_X-ct_N)V^2}{3[(\delta\gamma t_X-ct_N)^2-(1-\delta^2)(1+\gamma^2t_X^2)]V^2+32t_N^2\Lambda^2},\label{mixingangle}
\end{eqnarray}
which may be large. For the case $w,\Lambda\ll V$, the $\mathcal{Z}_3-\mathcal{C}$ mixing angle is defined similarly to \eqref{mixingangle}, but the term associated to $\Lambda$ should be omitted. In particular, all the two cases imply $\xi =0$ when $\delta=ct_N/\gamma t_X$, the condition by which the kinetic mixing and symmetry breaking effects cancels out. Besides, the $Z_3, Z_4$ masses are given by
\begin{eqnarray}
m^2_ {Z_3,Z_4}= \fr1 2 [m^2_ {Z'_3} +m^2_{C'} \mp \sqrt{(m^2_ {Z'_3} - m^2_{C'})^2 + 4m^4_ {Z'_3 C'} }].
\end{eqnarray}
In summary, the original fields are related to the mass eigenstates by $(A_3\, A_8\, A_{15}\, B\, C)^T=U(A\, Z_1\, Z_2\,Z_3\,Z_4)^T$. For the first case, $w,V\ll\Lambda$, we have $U= U_\delta U_1U_2U_3U_\varphi\simeq U_\delta U_1U_2U_\varphi$. For the second case, $w\ll V, \Lambda$, we obtain $U= U_\delta U_1U_2U'_3U_\xi\simeq U_\delta U_1U_2U_\xi$. For the last case, $w, \Lambda\ll V$, the mixing matrix is $U= U_\delta U_1U_2U'_3U_\xi$. Here we define
\begin{eqnarray}
U_\delta= \left(%
\begin{array}{ccccc}
1&0&0&0&0\\
0&1&0&0&0\\
0&0&1&0&0\\
0&0&0&1&-\frac{\delta}{\sqrt{1-\delta^2}}\\
0&0&0&0&\frac{1}{\sqrt{1-\delta^2}}
\end{array} \right), \hspace*{0.5cm}
U_\varphi= \left(%
\begin{array}{ccccc}
1&0&0&0&0\\
0&1&0&0&0\\
0&0&c_\varphi&s_\varphi&0\\
0&0&-s_\varphi&c_\varphi&0\\
0&0&0&0&1
\end{array} \right), \hspace*{0.5cm} U_\xi= \left(%
\begin{array}{ccccc}
1&0&0&0&0\\
0&1&0&0&0\\
0&0&1&0&0\\
0&0&0&c_\xi&s_\xi\\
0&0&0&-s_\xi&c_\xi
\end{array} \right).
\end{eqnarray}
The fields $A, Z_1$ are identical to the standard model, whereas $Z_2, Z_3$ and $Z_4$ are new, heavy gauge bosons. The mixings of the standard model gauge bosons with the new gauge bosons are very small, while the mixing within the new gauge bosons may be large.
\section{\label{pheno}Electroweak precision test}
\subsection{$\rho$ parameter}
The new physics that contributes to the $\rho$-parameter starts from the tree-level. This is caused by the mixing of the $Z$ boson with the new neutral gauge bosons. We evaluate
\begin{eqnarray} \Delta \rho &=&\frac{m_W^2}{c_W^2m_{Z_1}^2}-1=\frac{m_Z^2}{m^2_Z - \epsilon_1 m^2_{ZZ'_2} - \epsilon_2 m^2_{ZZ'_3} - \epsilon_3 m^2_{ZC'}}-1\simeq\frac{\epsilon_1 m^2_{ZZ'_2} + \epsilon_2 m^2_{ZZ'_3} + \epsilon_3 m^2_{ZC'}}{m_Z^2}\nonumber \\
&\equiv& (\Delta \rho)^0+(\Delta \rho)^\delta,
\end{eqnarray}
where
\begin{eqnarray}
(\Delta \rho)^0&\simeq&\frac{1}{4[1+(\beta^2+\gamma^2)t_X^2]^2}\left\{\frac{[\beta_2u^2+(2\sqrt3\beta t_X^2-\beta_2)v^2]^2}{(u^2+v^2)w^2}\right.\nonumber \\
&&\left.+\frac{\{(\beta+2\sqrt2\gamma)t_X^2(u^2+v^2)+\sqrt3[1+(\beta^2+\gamma^2)t_X^2](u^2-v^2)\}^2}{3(u^2+v^2)V^2}+\frac{(b\beta+c\gamma)^2t_X^4(u^2+v^2)}{4\Lambda^2}\right\},\\
(\Delta \rho)^\delta&\simeq&\frac{\delta[\delta+2(b\beta+c\gamma)t_Xt_N]t_X^2(u^2+v^2)}{16[1+(\beta^2+\gamma^2)t_X^2]^2t_N^2\Lambda^2}.
\end{eqnarray} This tree-level contribution is appropriately suppressed due to $u,v\ll w, V,\Lambda$. The $\rho$ deviation may receive one-loop corrections by non-degenerate vector multiplets, such as $(W_{13},W_{23})$ and $(W_{14},W_{24},W_{34})$, similar to the 3-3-1 model~\cite{sturho}. However, this source can be neglected if the new gauge bosons are heavy at TeV. In this analysis, we consider only the tree-level contribution.
Let us note that $\beta^2+\gamma^2<1/s^2_W-1\simeq 3.329$, which fixes $|\gamma|,|\beta|<1.82456$. The condition $q=-(1+\sqrt3\beta)/2$ leads to $-2.08012<q<1.08012$. Considering $q$ to be integer implies $q=-2,-1,0,$ and $1$. When $q=-2$, i.e. $\beta=\sqrt3$, we obtain $|\gamma|<0.57359$. The condition $p=-(1+\sqrt6\gamma-q)/3$ provides $-1.46833<p<-0.53167$, thus $p=-1$, i.e. $\gamma=0$, given that $p$ is integer. When $q=-1$, i.e. $\beta=1/\sqrt3$, we get $p=-2, -1, 0$, thus $\gamma=4/\sqrt6,1/\sqrt6, -\sqrt2/\sqrt3$, respectively. When $q=0$, i.e. $\beta=-1/\sqrt3$, we gain $p=-1, 0, 1$, thus $\gamma=\sqrt2/\sqrt3,-1/\sqrt6, -4/\sqrt6$, respectively. When $q=1$, i.e. $\beta=-\sqrt3$, we have $p=0$, thus $\gamma=0$.
However, we are interested in the four models for dark matter, such that $q=p=-1$ (or $\beta=1/\sqrt3,\gamma=1/\sqrt6$), $q=-1,p=0$ (or $\beta=1/\sqrt3,\gamma=-\sqrt2/\sqrt3$), $q=0,p=-1$ (or $\beta=-1/\sqrt3,\gamma=\sqrt2/\sqrt3$), and $q=p=0$ (or $\beta=-1/\sqrt3,\gamma=-1/\sqrt6$).\footnote{In these cases, the Landau pole is high enough, such that the new physics is viable \cite{landau}.} Besides, we take $n=m=0$, thus $b=-2/\sqrt3$, $c=-\sqrt2/\sqrt3$, for brevity. This case implies that the wrong particles are old. On the other hand, the $W$ mass, $m_W^2=\frac{g^2}{4}(u^2+v^2)$, implies $u^2+v^2=(246\ \mathrm{GeV})^2$. We will take $u$ in the range $(0,246)\ \mathrm{GeV}$, while $v$ is related to $u$.
The $\rho$ deviation is given from the global fit by $0.0002 < \Delta \rho < 0.00058$ \cite{pdg2018}.
For the cases, $w, V\ll\Lambda$ and $w\ll V, \Lambda$, $\Delta\rho$ is independent of $\delta$. Additionally, in the latter case ($w\ll V, \Lambda$), $\Delta\rho$ is independent of $\gamma$. However, in the case $w,\Lambda\ll V$, all the parameters contribute to $\Delta\rho$, except for $V$. Without loss of generality, we impose $V=2w$ for the case $w, V\ll\Lambda$ while $\Lambda=2w$ for the case $w,\Lambda\ll V$. Besides, we put $t_N=0.5$.
In Fig. \ref{DT1}, we make a contour of $\Delta\rho$ as the function of ($u,w$) concerning the first case of VEV arrangement. Here, the panels arranging from left to right correspond to the four dark matter models such as ($\beta=1/\sqrt3, \gamma=1/\sqrt6$), ($\beta=1/\sqrt3, \gamma=-\sqrt2/\sqrt3$), ($\beta=-1/\sqrt3, \gamma=\sqrt2/\sqrt3$), and ($\beta=-1/\sqrt3, \gamma=-1/\sqrt6$), respectively.
\begin{figure}[!h]
\begin{center}
\includegraphics[scale=0.31]{DT411.pdf}
\includegraphics[scale=0.31]{DT412.pdf}
\includegraphics[scale=0.31]{DT413.pdf}
\includegraphics[scale=0.31]{DT414.pdf}
\caption[]{\label{DT1} The $(u,w)$ regime that is bounded by the $\rho$ parameter for $w=0.5V\ll\Lambda$, where the panels from left to right correspond to the four dark matter models.}
\end{center}
\end{figure}
In Fig. \ref{DT2}, we make a contour of $\Delta\rho$ as the function of ($u,w$) for the second case of VEV arrangement. Here, we have only two viable cases, the left panel for $\beta=1/\sqrt3$ and the right panel for $\beta=-1/\sqrt3$.
\begin{figure}[!h]
\begin{center}
\includegraphics[scale=0.35]{DT421.pdf}
\includegraphics[scale=0.35]{DT422.pdf}
\caption[]{\label{DT2} The $(u,w)$ regime that is bounded by the $\rho$ parameter for $w\ll V, \Lambda$, where the left and right panels correspond to $\beta=1/\sqrt3$ and $\beta=-1/\sqrt3$.}
\end{center}
\end{figure}
The third case depending on the kinetic mixing parameter is given in Figs. \ref{DT3}, \ref{DT4}, \ref{DT5}, and \ref{DT6} according to the dark matter models ($\beta=1/\sqrt3, \gamma=1/\sqrt6$), ($\beta=1/\sqrt3, \gamma=-\sqrt2/\sqrt3$), ($\beta=-1/\sqrt3, \gamma=\sqrt2/\sqrt3$), and ($\beta=-1/\sqrt3, \gamma=-1/\sqrt6$), respectively. It is clear that the new physics scale bound is increased, when $|\delta|$ increases. The effect of $\delta$ is strong, when $u$ reaches values near $145$ GeV for the first dark matter model. By contrast, when $u$ approaches $0$ or $246$ GeV, the effect is negligible. In summary, the kinetic mixing effect is important when the new physics is considered.
\begin{figure}[!h]
\begin{center}
\includegraphics[scale=0.31]{DT4311.pdf}
\includegraphics[scale=0.31]{DT4312.pdf}
\includegraphics[scale=0.31]{DT4313.pdf}
\includegraphics[scale=0.31]{DT4314.pdf}
\caption[]{\label{DT3} The $(u,w)$ regime that is bounded by the $\rho$ parameter for ($\beta=1/\sqrt3, \gamma=1/\sqrt6, b=-2/\sqrt3$, $c=-\sqrt2/\sqrt3$) and $w=0.5\Lambda\ll V$, where the panels from left to right correspond to $\delta=-0.9,\ 0,\ 0.3$, and $0.9$.}
\end{center}
\end{figure}
\begin{figure}[!h]
\begin{center}
\includegraphics[scale=0.35]{DT4321.pdf}
\includegraphics[scale=0.35]{DT4322.pdf}
\includegraphics[scale=0.35]{DT4323.pdf}
\caption[]{\label{DT4} The $(u,w)$ regime that is bounded by the $\rho$ parameter for ($\beta=1/\sqrt3, \gamma=-\sqrt2/\sqrt3, b=-2/\sqrt3$, $c=-\sqrt2/\sqrt3$) and $w=0.5\Lambda\ll V$, where the panels from left to right are for $\delta=-0.9,\ 0$, and $0.9$, respectively.}
\end{center}
\end{figure}
\begin{figure}[!h]
\begin{center}
\includegraphics[scale=0.35]{DT4331.pdf}
\includegraphics[scale=0.35]{DT4332.pdf}
\includegraphics[scale=0.35]{DT4333.pdf}
\caption[]{\label{DT5} The $(u,w)$ regime that is bounded by the $\rho$ parameter for ($\beta=-1/\sqrt3, \gamma=\sqrt2/\sqrt3, b=-2/\sqrt3$, $c=-\sqrt2/\sqrt3$) and $w=0.5\Lambda\ll V$, where the panels from left to right are for $\delta=-0.9,\ 0$, and $0.9$, respectively.}
\end{center}
\end{figure}
\begin{figure}[!h]
\begin{center}
\includegraphics[scale=0.31]{DT4341.pdf}
\includegraphics[scale=0.31]{DT4342.pdf}
\includegraphics[scale=0.31]{DT4343.pdf}
\includegraphics[scale=0.31]{DT4344.pdf}
\caption[]{\label{DT6} The $(u,w)$ regime that is bounded by the $\rho$ parameter for ($\beta=-1/\sqrt3, \gamma=-1/\sqrt6, b=-2/\sqrt3$, $c=-\sqrt2/\sqrt3$) and $w=0.5\Lambda\ll V$, where the panels ordered correspond to $\delta=-0.9,\ -0.3,\ 0$, and $0.9$, respectively.}
\end{center}
\end{figure}
\subsection{$Z_1 \bar{f}f $ couplings}
As stated, the considering model has the mixing of the $Z$ boson with the new neutral gauge bosons. From \eqref{tran2} and \eqref{tranu2}, we get $Z=Z_1+\epsilon_1\mathcal{Z}'_2+\epsilon_2\mathcal{Z}'_3+\epsilon_3\mathcal{C}'$, $Z'_2=-\epsilon_1Z_1+\mathcal{Z}'_2$, $Z'_3=-\epsilon_2Z_1+\mathcal{Z}'_3$, and $C'=-\epsilon_3Z_1+\mathcal{C}'$. Hence, the couplings of $Z_1$ to fermions are modified by the mixing parameters $\epsilon_{1,2,3}$. Fitting the standard model precision test, the room for the mixing parameters is only $10^{-3}$ order. Hence, we impose the bound $|\epsilon_{1,2,3}|=10^{-3}$.
It is observed that in the first case ($w, V\ll \Lambda$), $\epsilon_3=0$ while $\epsilon_{1,2}$ are independent of $\delta, \Lambda$. In the second case ($w\ll V,\Lambda$), $\epsilon_{2,3}=0$ while $\epsilon_1$ is independent of $\delta, V, \Lambda$. In the last case ($w,\Lambda\ll V$), all the parameters contribute to $\epsilon_{1,2,3}$, except for $V$. Hence, we consider only the sensitivity of the new physics scales in terms of the kinetic mixing parameter for the last case. Since the effect of kinetic mixing does not depend on the $u,v$ relation, we impose $u=v=246/\sqrt2$ GeV and use also the previous inputs. The results are given in Fig. \ref{DT7}. It indicates that the new physics regime changes when $\delta$ varies.
\begin{figure}[!h]
\begin{center}
\includegraphics[scale=0.31]{DT4431.pdf}
\includegraphics[scale=0.31]{DT4432.pdf}
\includegraphics[scale=0.31]{DT4433.pdf}
\includegraphics[scale=0.31]{DT4434.pdf}
\caption[]{\label{DT7} The bounds on new physics scales as the functions of $\delta$ for $|\epsilon_{1,2,3}|=10^{-3}$, where the red, blue, and black lines correspond to $\epsilon_1$, $\epsilon_2$, $\epsilon_3$ for the four kinds of dark matter models ($\beta=1/\sqrt3, \gamma=1/\sqrt6$), ($\beta=1/\sqrt3, \gamma=-\sqrt2/\sqrt3$), ($\beta=-1/\sqrt3, \gamma=\sqrt2/\sqrt3$), and ($\beta=-1/\sqrt3, \gamma=-1/\sqrt6$), respectively.}
\end{center}
\end{figure}
\section{\label{pheno1} FCNCs}
Because the fermion generations transform differently under the gauge symmetry $SU(4)_L\otimes U(1)_X\otimes U(1)_N$, the tree-level FCNCs are present. Indeed, the neutral currents arise from
\begin{eqnarray}
\mathcal{L}_{\mathrm{NC}}&=& -g\bar{F}\gamma^\mu [T_3A_{3\mu}+T_8A_{8\mu}+T_{15}A_{15\mu}+t_XXB_{\mu}+t_NNC_\mu] F \nonumber \\
&=&-g\bar{F}\gamma^\mu [T_3A_{3\mu}+T_8A_{8\mu}+T_{15}A_{15\mu}+t_X(Q-T_3-\beta T_8-\gamma T_{15})B_{\mu}+t_N(B-L-bT_8-cT_{15})C_\mu] F.
\end{eqnarray}
It is clear that the leptons and exotic quarks do not flavor-change. Furthermore, the terms of $T_3$, $Q$, and $B-L$ also conserve flavors. Hence, the FCNCs couple only the ordinary quarks to $T_{8,15}$, such that
\begin{eqnarray}
\mathcal{L}_{\mathrm{NC}}\supset-g[\bar{q}_L\gamma^\mu T_8^q q_L(A_{8\mu}-\beta t_X B_\mu-bt_N C_\mu)+\bar{q}_L\gamma^\mu T_{15}^q q_L(A_{15\mu}-\gamma t_X B_\mu-ct_N C_\mu)],
\end{eqnarray}
where $q$ is denoted either $q=(u_1, u_2, u_3)$ or $q=(d_1, d_2, d_3)$, $T_8^q=\frac{1}{2\sqrt3}\text{diag}(-1,-1,1)$, and $T_{15}^q=\frac{1}{2\sqrt6}\text{diag}(-1,-1,1)$. Changing to the mass basis, $q_{L,R}=V_{qL,qR}q'_{L,R}$ where either $q'=(u,c,t)$ or $q'=d,s,b$, and $(A_3\, A_8\, A_{15}\, B\, C)^T=U(A\, Z_1\, Z_2\, Z_3\, Z_4)$, this yields
\begin{eqnarray}
\mathcal{L}_{\mathrm{FCNC}}=-\bar{q}'_{iL}\gamma^\mu q'_{jL}(V^*_{qL})_{3i}(V_{qL})_{3j}(g_0A_\mu+g_1Z_{1\mu}+g_2Z_{2\mu}+g_3Z_{3\mu}+g_4Z_{4\mu})\hspace*{0.5cm} (i\neq j).
\end{eqnarray}
It is noted that the photon always conserves flavors, $g_0=0$. In the first case ($w,V\ll \Lambda$), the couplings $g_{1,2,3,4}$ are
\begin{eqnarray}
g_1&=&-\frac{g}{\sqrt6}\left[\frac{\sqrt2}{\sqrt{1-\beta^2t_W^2}}\epsilon_1+\frac{1+\gamma (\sqrt2 \beta+\gamma)t_X^2}{\sqrt{1+\gamma^2t_X^2}}\epsilon_2+\frac{\delta(\sqrt2 \beta+\gamma)t_X-(\sqrt2 b+c)t_N}{\sqrt{1-\delta^2}}\epsilon_3\right],\label{g1}\\
g_2&=&\frac{g}{\sqrt6}\left[\frac{\sqrt2}{\sqrt{1-\beta^2t_W^2}}c_\varphi-\frac{1+\gamma(\sqrt2 \beta+\gamma)t_X^2}{\sqrt{1+\gamma^2t_X^2}}s_\varphi\right],\\
g_3&=&g_2(c_\varphi\rightarrow s_\varphi, s_\varphi\rightarrow -c_\varphi),\\
g_4&=&\frac{g}{\sqrt6}\frac{\delta(\sqrt2 \beta+\gamma)t_X-(\sqrt2 b+c)t_N}{\sqrt{1-\delta^2}}.
\end{eqnarray}
In the third case ($w, \Lambda\ll V$), the coupling $g_1$ is identical to (\ref{g1}), while
\begin{eqnarray}
g_2&=&-\frac{g}{\sqrt6}\left[\frac{\sqrt2}{\sqrt{1-\beta^2t_W^2}}+\frac{1+\gamma (\sqrt2 \beta+\gamma)t_X^2}{\sqrt{1+\gamma^2t_X^2}}\mathcal{E}_1+\frac{\delta(\sqrt2 \beta+\gamma)t_X-(\sqrt2 b+c)t_N}{\sqrt{1-\delta^2}}\mathcal{E}_2\right],\\
g_3&=&\frac{g}{\sqrt6}\left\{c_\xi\left[\frac{\sqrt2}{\sqrt{1-\beta^2t_W^2}}\mathcal{E}_1+\frac{1+\gamma (\sqrt2 \beta+\gamma)t_X^2}{\sqrt{1+\gamma^2t_X^2}}\right]-s_\xi\left[\frac{\sqrt2}{\sqrt{1-\beta^2t_W^2}}\mathcal{E}_2+\frac{\delta(\sqrt2 \beta+\gamma)t_X-(\sqrt2 b+c)t_N}{\sqrt{1-\delta^2}}\right]\right\},\\
g_4&=&g_3(c_\xi\rightarrow s_\xi, s_\xi\rightarrow -c_\xi).
\end{eqnarray}
In the second case ($w\ll V, \Lambda$), the couplings can be obtained from those in the third case by $\mathcal{E}_{1,2}\rightarrow 0$.
The contribution of the new physics to the meson mixing is given after integrating $Z_{1,2,3,4}$ out,
\begin{eqnarray}
\mathcal{L}^{\mathrm{eff}}_{\mathrm{FCNC}}&=&(\bar{q}'_{iL}\gamma^\mu q'_{jL})^2 [(V^*_{qL})_{3i}(V_{qL})_{3j}]^2\left(\frac{g^2_1}{m^2_{Z_1}} + \frac{g^2_2}{m^2_{Z_2}}+\frac{g^2_3}{m^2_{Z_3}}+\frac{g^2_4}{m^2_{Z_4}}\right)\nonumber \\
&\simeq&(\bar{q}'_{iL}\gamma^\mu q'_{jL})^2 [(V^*_{qL})_{3i}(V_{qL})_{3j}]^2\left(\frac{g^2_2}{m^2_{Z_2}}+\frac{g^2_3}{m^2_{Z_3}}+\frac{g^2_4}{m^2_{Z_4}}\right),
\end{eqnarray}
where the $Z_1$ contribution is small and omitted.
The strongest bound comes from $B^0_s-\bar{B}^0_s$ mixing, implying \cite{pdg2018}
\begin{eqnarray}
[(V^*_{dL})_{32}(V_{dL})_{33}]^2\left(\frac{g^2_2}{m^2_{Z_2}}+\frac{g^2_3}{m^2_{Z_3}}+\frac{g^2_4}{m^2_{Z_4}}\right)<\frac{1}{(100 \text{ TeV})^2}.
\end{eqnarray}
We assume the sector of up quarks to be flavor diagonal, i.e. $V_{\text{CKM}}\equiv V^\dagger_{uL}V_{dL} =V_{dL}$. We have $|(V^*_{dL})_{32}(V_{dL})_{33}|\simeq 3.9\times 10^{-2}$ \cite{pdg2018}, which leads to
\begin{eqnarray}
\sqrt{\frac{g^2_2}{m^2_{Z_2}}+\frac{g^2_3}{m^2_{Z_3}}+\frac{g^2_4}{m^2_{Z_4}}}<\frac{1}{3.9 \text{ TeV}}.\label{constraint}
\end{eqnarray}
Our remark is that since $u,v\ll w, V, \Lambda$, the l.h.s of (\ref{constraint}) depends only on the new physics scales, not on the weak scales.
In the first case ($w, V\ll \Lambda$), the $Z_4$ contribution is negligible. The l.h.s of (\ref{constraint}) is independent of $\delta$. The other inputs given previously are used, implying the bound for $w> 4.36$ TeV for all the four dark matter models.
In the second case ($w\ll V, \Lambda$), the $Z_{3,4}$ contributions are negligible. The l.h.s of (\ref{constraint}) is independent of $\beta$, $\gamma$, and $\delta$. The bound yields $w> 3.9$ TeV for all the four models.
In the third case ($w, \Lambda \ll V$), since the mixing angles $\mathcal{E}_{1,2}$ are finite, the l.h.s of (\ref{constraint}) depends on $\beta$, $\gamma$, and $\delta$, and is depicted in Fig. \ref{FCNC}. The figure yields that the new physics regime changes when $\delta$ varies. Furthermore, those bounds are obviously lower than that given by the two case above.
\begin{figure}[!h]
\begin{center}
\includegraphics[scale=0.31]{FCNC1.pdf}
\includegraphics[scale=0.31]{FCNC2.pdf}
\includegraphics[scale=0.31]{FCNC3.pdf}
\includegraphics[scale=0.31]{FCNC4.pdf}
\caption[]{\label{FCNC} The bounds on the new physics scales as functions of $\delta$ from the FCNCs for $w=0.5 \Lambda\ll V$, where the panels left to right are for the four dark matter models, respectively.}
\end{center}
\end{figure}
\section{\label{lhc} Collider bounds}
Since the new neutral gauge bosons couple to leptons and quarks, they contribute to the Drell-Yan and dijet processes at colliders.
The LEPII searches for $e^+e^-\to \mu^+\mu^-$ happen similarly to the case of the 3-3-1-1 model, where all the new gauge bosons $Z_{2,3,4}$ mediate the process. Assuming that all the new physics scales are the same order, they are bounded in the TeV scale \cite{3311}.
The LHC searches for dijet and dilepton final states can be studied. Using the above condition, the new physics scales are also in TeV, similarly to \cite{331p}.
\section{\label{3311} The 3-3-1-1 model revisited}
The 3-3-1-1 model is based upon the gauge symmetry $SU(3)_C\otimes SU(3)_L\otimes U(1)_X\otimes U(1)_N$. Thus it contains four neutral gauge bosons $A_{3,8}$, $B$, and $C$ according to the last three gauge groups, in which $B,C$ has a kinetic mixing term, $-(\delta/2)B_{\mu\nu}C^{\mu\nu}$. The kinetic mixing effect in the 3-3-1-1 model was explicitly studied in \cite{dong2016}. Here we present only new results beyond the previous investigation.
Changing to the canonical basis, $A_{3}$, $A_8$, $B'$, and $C'$, the corresponding mass matrix $M^2=\{m^2_{ij}\}$ is given by
\begin{eqnarray}
m^2_{11}&= & \frac {g^2} 4 (u^2 + v^2),\hspace*{0.5cm} m^2_{12}= \frac {g^2} {4\sqrt3} (u^2-v^2),\hspace*{0.5cm} m^2_{13} = -\frac{g^2t_X}{4\sqrt3}[(\sqrt3+\beta)u^2+(\sqrt3-\beta)v^2], \nonumber \\
m^2_{14}&=& \frac{g^2}{4\sqrt{3(1-\delta^2)}}\{[\delta (\sqrt3+\beta)t_X-b t_N] u^2 +[\delta (\sqrt3-\beta)t_X+b t_N] v^2 \} ,\nonumber \\
m^2_{22}&=&\frac {g^2} {12}(u^2+v^2+4w^2),\hspace*{0.5cm}
m^2_{23}= -\frac{g^2t_X}{12}[(\sqrt3+\beta)u^2-(\sqrt3-\beta)v^2+4\beta w^2],
\nonumber \\
m^2_{24}&=& \frac{g^2} { 12 \sqrt{1-\delta^2}}\{[\delta (\sqrt3+\beta)t_X-b t_N] u^2-[\delta (\sqrt3-\beta)t_X+b t_N] v^2 +4(\delta\beta t_X-b t_N)w^2\},\nonumber \\
m^2_{33} &=& \frac{g^2t^2_X}{12}[(\sqrt3+\beta)^2u^2 +(\sqrt3-\beta)^2v^2 + 4\beta^2w^2],\nonumber \\
m^2_{34}&=& \frac{-g^2 t_X } { 12\sqrt{1-\delta^2}}\{ (\sqrt3+\beta)[\delta (\sqrt3+\beta)t_X-b t_N] u^2+ (\sqrt3-\beta)[\delta (\sqrt3-\beta)t_X+b t_N] v^2+4\beta \left(\delta\beta t_X-b t_N\right)w^2\}, \nonumber \\
m^2_{44}&=& \frac{g^2} { 12(1-\delta^2)}\{[\delta (\sqrt3+\beta)t_X - b t_N]^2 u^2+[\delta (\sqrt3-\beta) t_X+ b t_N]^2 v^2 +4(\delta\beta t_X-b t_N)^2w^2+48t^2_N\Lambda^2\}.\nonumber
\end{eqnarray} This result is similar to that in \cite{dong2016}, except for the last element, $m^2_{44}$, that differs in the coefficient of $\Lambda^2$. Note that $t_X=g_X/g$, $t_N=g_N/g$, $\beta$, $b$, $u$, $v$, and $w$ are those parameters belonging to the 3-3-1-1 model and in this case we have $s_W=e/g=t_X/\sqrt{1+(1+\beta^2)t_X^2}$.
Changing to the electroweak basis, $(A_3\, A_8 \, B' \, C')^T =U_1 (A \, Z \, Z' \, C')^T$, where $C'$ is orthogonal to $A = s_W A_3 + c_W \left(\beta t_W A_8 +\sqrt{1-\beta^2 t^2_W}B'\right)$,
$Z=c_W A_3 - s_W \left(\beta t_W A_8 +\sqrt{1-\beta^2 t^2_W}B'\right)$, and
$Z' =\sqrt{1-\beta^2 t^2_W}\left(A_8-
\beta t_W B'\right)$, thus
\begin{eqnarray}
U_1 = \left (\begin{array}{cccc}
s_W &c_W&0& 0\\
\beta s_W &-\beta s_W t_W &\sqrt{1-\beta^2t^2_W}&0\\
c_W\sqrt{1-\beta^2t^2_W}&- s_W\sqrt{1-\beta^2 t^2_W}&- \beta t_W&0\\
0&0&0&1
\end{array} \right ), \label{neutral2}
\end{eqnarray}
the mass matrix $M^2$ changes to
\begin{eqnarray} M'^2 = U^T_1 M^2 U_1=
\left(
\begin{array}{cc}
0 & 0 \\
0 & M'^2_{s}\end{array} \right),\hspace*{0.5cm} M'^2_s \equiv
\left(
\begin{array}{ccc}
m^2_{Z} & m^2_{ZZ'} & m^2_{ZC'}\\
m^2_{ZZ'} & m^2_{Z'} & m^2_{Z'C'}\\
m^2_{ZC'} & m^2_{Z'C'} & m^2_{C'}
\end{array}\right),
\end{eqnarray} which has the elements as given in \cite{dong2016}, in which $m^2_{C'}= m^2_{44}$.
The light state $Z$ can be separated by using the seesaw approximation,
\begin{eqnarray}
\left(Z \, Z' \, C' \right)^T = U_2 \left(Z_1 \, \mathcal{Z}' \, \mathcal{C}'\right)^T, \hspace*{0.5cm} M''^2=U^{T}_2M'^2_sU_2= \left(\begin{array}{cc}
m^{2}_{Z_{1}} & 0\\
0&M^2_{2\times 2}\end{array}\right),
\end{eqnarray}
where
\begin{eqnarray}
&& U_2 \simeq \left(\begin{array}{ccc}
1 & \epsilon_1 & \epsilon_2 \\
-\epsilon_1 & 1 &0 \\
-\epsilon_2 &0 & 1 \\
\end{array}
\right), \hspace*{0.5cm} M^2_{2\times 2}\simeq \left(\begin{array}{cc}
m^2_{Z'}&m^2_{Z'C'}\\
m^2_{Z'C'}&m^2_{C'} \end{array}\right). \\
&& m^2_{Z_1}\simeq m^2_{Z}- \epsilon_1m^2_{ZZ'}-\epsilon_2m^2_{ZC'}.
\end{eqnarray}
We separate $\epsilon_{1,2}\equiv\epsilon_{1,2}^0+\epsilon_{1,2}^\delta$, where $\epsilon^0_{1,2}$ are the mixing parameters due to the symmetry breaking \cite{3311,dong2015}, while $\epsilon^{\delta}_{1,2}$ determine the kinetic mixing effect,
\begin{eqnarray}
\epsilon_1^0&=&\frac{\sqrt{1-\beta^2t_W^2}}{4c_W}\left\{\frac{\sqrt3[u^2-v^2+\sqrt3\beta t_W^2(u^2+v^2)]}{w^2}+\frac{b^2\beta t_W^2(u^2+v^2)}{4\Lambda^2}\right\},\\
\epsilon_2^0&=&\frac{b\beta t_W^2(u^2+v^2)}{16c_Wt_N\Lambda^2},\\
\epsilon_1^\delta&=&\frac{\delta t_W[b(1-2\beta^2t_W^2)t_N-\delta\beta t_W\sqrt{1-\beta^2t_W^2}](u^2+v^2)}{16c_Wt_N^2\Lambda^2},\\\epsilon_2^\delta&=&\frac{\delta t_W(u^2+v^2)}{16c_Wt_N\Lambda^2}\left(\frac{\sqrt{1-\delta^2}\sqrt{1-\beta^2t_W^2}}{t_N}-\frac{\delta b \beta t_W}{1+\sqrt{1-\delta^2}}\right),
\end{eqnarray}
where $\epsilon_{1,2}^\delta$ differ from those in \cite{dong2016}.
We diagonalize $M^2_{2\times 2}$ to obtain mass eigenstates,
\begin{eqnarray}
Z_2 = c_\xi \mathcal{Z}' - s_\xi \mathcal{C}', \hspace*{0.5cm} Z_3 = s_\xi \mathcal{Z}'+ c_\xi \mathcal{C}',
\end{eqnarray} in which the $\mathcal{Z}'-\mathcal{C}'$ mixing angle and masses are
\begin{eqnarray}
t_{2\xi}&\simeq& \frac{2\sqrt{1-\delta^2}(\delta\beta t_W-bt_N \sqrt{1-\beta^2t_W^2})w^2}{[(\delta\beta t_W-bt_N\sqrt{1-\beta^2t_W^2})^2-(1-\delta^2)]w^2+12(1-\beta^2t_W^2)t_N^2\Lambda^2},\\
m^2_ {Z_2,Z_3}&=& \fr1 2 [m^2_ {Z'} +m^2_{C'} \mp \sqrt{(m^2_ {Z'} - m^2_{C'})^2 + 4m^4_ {Z' C'} }].
\end{eqnarray}
Generally, $\xi$ is finite if $w\sim \Lambda$. The kinetic mixing and symmetry breaking effects cancel out if $\delta=bt_N/\beta t_X$, which takes place between $\delta$ and $b/\beta$---the embedding coefficients of $T_8$. Whereas, in the 3-4-1-1 model, it happens between $\delta$ and $c/\gamma$---the embedding coefficients of $T_{15}$.
Hence, the gauge states are connected to the physical states by $(A_3\, A_8\, B\, C)^T=U_\delta U_1U_2U_\xi (A\, Z_1\, Z_2\,Z_3)^T$, where
\begin{eqnarray}
U_\delta= \left(%
\begin{array}{cccc}
1&0&0&0\\
0&1&0&0\\
0&0&1&-\frac{\delta}{\sqrt{1-\delta^2}}\\
0&0&0&\frac{1}{\sqrt{1-\delta^2}}
\end{array} \right), \hspace*{0.5cm}
U_\xi= \left(%
\begin{array}{cccc}
1&0&0&0\\
0&1&0&0\\
0&0&c_\xi&s_\xi\\
0&0&-s_\xi&c_\xi
\end{array} \right).
\end{eqnarray}
The $\rho$ deviation starts from the tree-level contribution,
\begin{eqnarray}
(\Delta\rho)_{\mathrm{tree}} &=& \frac{m^2_W}{c^2_Wm^2_{Z_1}}-1=\frac{m^2_Z}{m^2_Z-\epsilon_1m^2_{ZZ'}-\epsilon_2 m^2_{ZC'}}-1\simeq \frac{\epsilon_1m^2_{ZZ'}+\epsilon_2 m^2_{ZC'}}{m^2_Z}\equiv (\Delta \rho)_{\mathrm{tree}}^0+(\Delta \rho)_{\mathrm{tree}}^\delta,
\end{eqnarray}
where
\begin{eqnarray}
(\Delta \rho)_{\mathrm{tree}}^0&\simeq&\frac{[u^2-v^2+\sqrt3\beta t_W^2(u^2+v^2)]^2}{4(u^2+v^2)w^2}+\frac{b^2\beta^2t_W^4(u^2+v^2)}{16\Lambda^2},\\
(\Delta \rho)_{\mathrm{tree}}^\delta&\simeq&\frac{\delta\sqrt{1-\beta^2t_W^2}(\delta\sqrt{1-\beta^2t_W^2}+2b\beta t_Wt_N)t_W^2(u^2+v^2)}{16t_N^2\Lambda^2}.\label{rhodelta}
\end{eqnarray}
In this computation, we also include one-loop contributions by the gauge vector doublet $(X,Y)$, as supplied in \cite{dong2016}.
If $\Lambda\gg w$, $\Delta\rho$ does not depend on $\Lambda$, $t_N$, $b$, and $\delta$. If $\Lambda \sim w$, all the parameters modify $\Delta\rho$. Comparing to \cite{dong2016}, the difference is only expressions related to $\delta$. Hence, the first case is not investigated in this work. To finalize the result, we use the parameter values similar to those in \cite{dong2016}, namely $\Lambda=2w$, $t_N=0.5$, $n=0$ (thus $b=-2/\sqrt{3}$), and $q=-1,0, 1$ (thus $\beta=1/\sqrt{3},-1/\sqrt{3},-\sqrt{3}$, respectively).
We make a contour of $\Delta \rho$ as the function of $(u,w)$, as depicted in Figs. \ref{rho3311r}, \ref{rho3311s}, and \ref{rho3311m} for $\beta=-1/\sqrt{3}$, $\beta=1/\sqrt{3}$, and $\beta=-\sqrt{3}$, respectively. The effect of $\delta$ is quite similar to the 3-4-1-1 model and obviously different from \cite{dong2016}.
\begin{figure}[!h]
\begin{center}
\includegraphics[scale=0.31]{DT311}
\includegraphics[scale=0.31]{DT312}
\includegraphics[scale=0.31]{DT313}
\includegraphics[scale=0.31]{DT314}
\caption[]{\label{rho3311r} The $(u,w)$ regime constrained by $\Delta\rho$ for $\beta=-1/\sqrt{3}$, $b=-2/\sqrt{3}$, $t_N=0.5$, and $\Lambda=2w$, where the panels correspond to $\delta=-0.9,\ -0.2$, and $0.9$.}
\end{center}
\end{figure}
\begin{figure}[!h]
\begin{center}
\includegraphics[scale=0.31]{DT321}
\includegraphics[scale=0.31]{DT322}
\includegraphics[scale=0.31]{DT323}
\includegraphics[scale=0.31]{DT324}
\caption[]{\label{rho3311s} The $(u,w)$ regime constrained by $\Delta\rho$ for $\beta=1/\sqrt{3}$, $b=-2/\sqrt{3}$, $t_N=0.5$, and $\Lambda=2w$, where the panels correspond to $\delta=-0.9,\ 0.2$, and $0.9$.}
\end{center}
\end{figure}
\begin{figure}[!h]
\begin{center}
\includegraphics[scale=0.35]{DT331}
\includegraphics[scale=0.35]{DT332}
\includegraphics[scale=0.35]{DT333}
\caption[]{\label{rho3311m} The $(u,w)$ regime that is bounded by the $\rho$ parameter for $\beta=-\sqrt{3}$, $b=-2/\sqrt{3}$, $t_N=0.5$, and $\Lambda=2w$, where the panels, ordering from left to right, correspond to $\delta=-0.9,\ 0$, and $0.9$, respectively. In this case, the Landau pole, which is roundly $w=5$ TeV, is imposed.}
\end{center}
\end{figure}
The new physics contribution is safe, given that $|\epsilon_{1,2}|=10^{-3}$. Without loss of generality, we impose $u=v=246/\sqrt{2}$ as well as the given values of $\Lambda = 2w, t_N, \beta, b$ are used. In Fig. \ref{mixing}, $\epsilon_{1,2}$ are contoured as the functions of ($w,\delta$) for $\beta = -1/\sqrt3$, $\beta = 1/\sqrt3$, and $\beta = -\sqrt3$. It is clear that the new physics regime significantly changes when $\delta$ varies, in contradiction to \cite{dong2016}.
\begin{figure}[!h]
\begin{center}
\includegraphics[scale=0.35]{DT3mix1}
\includegraphics[scale=0.35]{DT3mix2}
\includegraphics[scale=0.35]{DT3mix3}
\caption[]{\label{mixing} The bounds on the new physics scales as functions of $\delta$, contour by $|\epsilon_{1,2}|=10^{-3}$, for the three kinds of the models $\beta=-1/\sqrt{3}$, $\beta=1/\sqrt{3}$, and $\beta=-\sqrt{3}$, respectively.}
\end{center}
\end{figure}
The meson mixing is described via the effective interaction \cite{dong2016}
\begin{eqnarray}
\mathcal{L}^{\mathrm{eff}}_{\mathrm{FCNC}}=(\bar{q}'_{iL}\gamma^\mu q'_{jL})^2 [(V^*_{qL})_{3i}(V_{qL})_{3j}]^2\left(\frac{g^2_2}{m^2_{Z_2}}+\frac{g^2_3}{m^2_{Z_3}}\right),
\end{eqnarray}
where
\begin{eqnarray}
g_2&=&\frac{g}{\sqrt3}\left(\frac{1}{\sqrt{1-\beta^2t_W^2}}c_\xi+\frac{bt_N-\delta\beta t_X}{\sqrt{1-\delta^2}}s_\xi\right),\\
g_3&=&g_2(c_\xi\rightarrow s_\xi, s_\xi\rightarrow -c_\xi).
\end{eqnarray}
The $B^0_s-\bar{B}^0_s$ bound leads to \cite{dong2016}
\begin{eqnarray}
\sqrt{\frac{g_2^2}{m_{Z_2}^2}+\frac{g_3^2}{m_{Z_3}^2}}<\frac{1}{3.9\text{ TeV}}.\label{FCNC3311}
\end{eqnarray}
When $w\ll \Lambda$, the above bound translates to $w>3.9$ TeV, independent of $\beta$, $b$, $g$, $g_X$, $g_N$, and $\delta$. When $w\sim \Lambda$, using the existing values of parameters, the bound for both scales is similar to the previous case, which is quite in agreement with the conclusion in \cite{dong2016}.
\section{\label{con} Conclusion}
We have proved that the 3-4-1-1 model provides dark matter candidates naturally, besides supplying small neutrino masses via the seesaw mechanism induced by the gauge symmetry breaking.
The kinetic mixing effects are evaluated, yielding the new physics scales at TeV scale, in agreement with the collision bound. The kinetic mixing and symmetry breaking effects are canceled out only in the new gauge sector and differs between the 3-4-1-1 and 3-3-1-1 models.
Similar to the 3-3-1-1 model \cite{a3311}, the 3-4-1-1 model can address the question of cosmic inflation as well as asymmetric dark and normal matter, which attracts much attention.
\section*{Acknowledgments}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 7,237 |
Q: Hyperledger: The required parameter 'sequence' is empty. Rerun the command with --sequence flag I'm trying out the NFT Auction repo at https://github.com/hyperledger-labs/nft-auction
I get the error
Error: The required parameter 'sequence' is empty. Rerun the command with --sequence flag
Finished vendoring Go dependencies
Skipping Chaincode packaging and installing...
Using organization org1
Installed chaincodes on peer:
Query installed successful on peer0.org1 on channel
Using organization org1
Error: The required parameter 'sequence' is empty. Rerun the command with --sequence flag
Usage:
peer lifecycle chaincode approveformyorg [flags]
Flags:
--channel-config-policy string The endorsement policy associated to this chaincode specified as a channel config policy reference
-C, --channelID string The channel on which this command should be executed
--collections-config string The fully qualified path to the collection JSON file including the file name
--connectionProfile string The fully qualified path to the connection profile that provides the necessary connection information for the network. Note: currently only supported for providing peer connection information
-E, --endorsement-plugin string The name of the endorsement plugin to be used for this chaincode
-h, --help help for approveformyorg
--init-required Whether the chaincode requires invoking 'init'
-n, --name string Name of the chaincode
--package-id string The identifier of the chaincode install package
--peerAddresses stringArray The addresses of the peers to connect to
--sequence int The sequence number of the chaincode definition for the channel
--signature-policy string The endorsement policy associated to this chaincode specified as a signature policy
--tlsRootCertFiles stringArray If TLS is enabled, the paths to the TLS root cert files of the peers to connect to. The order and number of certs specified should match the --peerAddresses flag
-V, --validation-plugin string The name of the validation plugin to be used for this chaincode
-v, --version string Version of the chaincode
--waitForEvent Whether to wait for the event from each peer's deliver filtered service signifying that the transaction has been committed successfully (default true)
--waitForEventTimeout duration Time to wait for the event from each peer's deliver filtered service signifying that the 'invoke' transaction has been committed successfully (default 30s)
Global Flags:
--cafile string Path to file containing PEM-encoded trusted certificate(s) for the ordering endpoint
--certfile string Path to file containing PEM-encoded X509 public key to use for mutual TLS communication with the orderer endpoint
--clientauth Use mutual TLS when communicating with the orderer endpoint
--connTimeout duration Timeout for client to connect (default 3s)
--keyfile string Path to file containing PEM-encoded private key to use for mutual TLS communication with the orderer endpoint
-o, --orderer string Ordering service endpoint
--ordererTLSHostnameOverride string The hostname override to use when validating the TLS connection to the orderer
--tls Use TLS when communicating with the orderer endpoint
--tlsHandshakeTimeShift duration The amount of time to shift backwards for certificate expiration checks during TLS handshakes with the orderer endpoint
Chaincode definition approved on peer0.org1 on channel 'defaultchannel' failed
Deploying chaincode failed
Tried looking at the Issues tab of the repo
A: You must pass a --sequence number to the command. The first time the chaincode is deployed it should be set to '1', and then increment from there.
For example:
peer lifecycle chaincode approveformyorg -o localhost:7050 --ordererTLSHostnameOverride orderer.example.com --channelID mychannel --name basic --version 1.0 --package-id $CC_PACKAGE_ID --sequence 1 --tls --cafile "${PWD}/organizations/ordererOrganizations/example.com/orderers/orderer.example.com/msp/tlscacerts/tlsca.example.com-cert.pem"
See the full example at https://hyperledger-fabric.readthedocs.io/en/latest/deploy_chaincode.html#approve-a-chaincode-definition.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 7,610 |
EXHIBITIONSLONG ISLAND
The Long Island Biennial Opens at The Heckscher Museum
August 4, 2018 by Pat Rogers Contemporary Art, EXHIBITIONS, Huntington & Dix Hills, LONG ISLAND, MUSEUMS, NY METRO
The 2018 "Long Island Biennial," a juried exhibition featuring art from contemporary artists across Suffolk and Nassau Counties, opens at The Heckscher Museum of Art this weekend after receiving a record-breaking submission of 351 artists, vying for inclusion at the Huntington's museum's prestigious curated exhibition. Curators selected 52 artists based in Montauk to New Hyde Part, NY. Of the 52 accepted, 38 artists are first-time exhibitors, reported The Heckscher.
The jurors are Christine Berry, Berry Campbell Gallery, New York; Robert Carter, Professor of Art, Nassau Community College, State University of New York; and Bobbi Coller, Independent Art Historian and Curator.
"The art world needs as many venues as possible for new artists; this is so important and very much appreciated," said juror Robert Carter. "The artist entries were surprising in how they varied in media use and subject matter – touching on nature, social issues and more. And, each juror brought their own unique perspective to the judging."
"Horizon with Thread" by Bastienne Schmidt, 2018. Pigment and string on raw canvas, 24 x 24 x 1 1/2 inches. Courtesy of The Heckscher Museum of Art.
As in past editions, artists from the East End are represented in the Open Call juried exhibition. They include Peter Beston (East Quogue); Donna Corvi (Montauk); Alex Ferrone (Peconic); Christa Maiwald (East Hampton); Michael McLaughlin (Riverhead); Bastienne Schmidt (Bridgehampton); Susan Christine Tango (Westhampton Beach); Dan W. Weldon (Sag Harbor); Frank Wimberley (Sag Harbor) and Amy Worth (Greenport).
"Abrasha in Port-au-Prince" by Peter Beston, 2018. Oil on canvas, 41 x 41 inches. Courtesy of The Heckscher Museum of Art.
Long Island artists accepted are Alexander Adell, Nicholas Alberti, Beth Atkinson, Mario Bakalov, Arthur Bernstein, Jane Breskin Zalben, Stuart Burton, John Cino, Riccarda De Eccher, Pat Detullio, Peter Dicke, Richard Gardiner, Raymond Germann, Naomi Grossman, Jan Guarino, Qin Han, Lenore Ann Hanson, Elizabeth Heaton, Stanley Horowitz, Holly Hunt, Warren Infield, Marc Josloff, Lita Kelmenson, Roshanak Keyghobadi, Anthony Klinger-Cooley, Rachelle Krieger, Joyce Kubat, Christopher Lauto, Craig E. Marcin, Paul Mele, Margaret Minardi, Kasmira Mohanty, Min Myar, Christopher Harold Parrott, Inna Pashina, Laura Powers-Swiggett, Anna Prikazchikova, George Schulman, Alisa Shea, Bill Shillalies, Susan Christine Tango, Pamela Waldroup and Tmima Z.
"Incubation" by Anthony Klinger-Cooley, 2018. Colored pencil, watercolor, 38 x 26 inches. Courtesy of The Heckscher Museum of Art.
Art featured in the Long Island Biennial can be viewed online by clicking here. In addition, the museum presents artwork by all the artist applications in an online gallery that can be viewed by clicking here. The artists featured in the exhibition will gather on September 16, 2018 from 1 to 3 p.m. for a Gallery Talk at The Heckscher. Museum members are admitted free; admission is $5 for non-members. The exhibition remains on view through November 11, 2018.
"Blue Angel" by George Schulman, 2018. Oil paint, graphite, fabric, gold and palladium leafing on wood, 33 x 33 inches. Courtesy of The Heckscher Museum of Art.
Inaugurated in 2010 by The Heckscher Museum of Art, the "Long Island Biennial" is designed to offer the opportunity for professional artists on Long Island to be part of the Open Call juried exhibition. The show also offers a snapshot of some of the art that's being made on Long Island at the time of the biennial.
BASIC FACTS: The Long Island Biennial is on view August 4 to November 11, 2018 at The Heckscher Museum of Art. An Artist Talk with the Artists of the Long Island Biennial will be held September 16, 2018 from 1 to 3 p.m. The Heckscher Museum is located at 2 Prime Avenue, Huntington, NY 11743. www.heckscher.org.
To discover all the exhibitions currently on view at the Long Island museum, visit www.heckscher.org.
Keith Sonnier's Art Featured in Two Museums in The Hamptons
NYC Museum Highlights – August 2018
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"redpajama_set_name": "RedPajamaCommonCrawl"
} | 5,081 |
Prices of construction materials fall
Laxman Kafle
By Laxman Kafle
Kathmandu, Feb. 9: Prices of construction materials -- iron, sand and pebbles -- have come down of late. With the decline in the price of iron billets in the international market, iron has also become cheaper in Nepal, said the traders.
Its price has decreased by Rs. 10-12 per kg in the local market since the end of 2020. The market price of a kilogram of iron bar has now fallen to Rs. 82 from Rs. 100 per kilogram about a month ago.
Pradeep Kumar Shrestha, managing director of Panchakanya Group, said that the price of steel has also come down in Nepali market along with the fall in the price of raw materials required for iron production in the Indian market.
He added that the price of iron reached above Rs. 96 per kilogram in the local market a few weeks ago due to skyrocketing price of the billets and that the demand for iron could not increase as expected as the construction work was not gathering steam.
People, expecting a further decline in iron price, are in a 'wait and watch' position, so they are reluctant to start constructing houses, he said.
Rabi Singh, president of the Federation of Construction Association of Nepal, said that contractors have breathed a sigh of relief following the reduction in price of construction materials.
The price of sand and pebbles have also declined due to their unhindered supply as the government has eased their ban. The price of tipper-full sand, which used to be around Rs. 35,000 until recently, has now dropped to Rs. 26,000.
He informed that the price of pebbles had also gone down to Rs. 22,000 from Rs. 28,000 and that the construction works had also gathered momentum.
"The government extending the deadline of contract agreement for ongoing projects has come as a huge relief to the contractors. It deadline for construction works, including road blacktopping and construction of bridges, must be extended at the earliest," he told The Rising Nepal.
If the government delayed extending deadline for these projects, the construction of these projects would again be affected, he added.
Traders have said that the price of cement, which had gone up by Rs. 20 per sack (50 kilogram) from December last year, has come down slightly.
According to Dhruba Thapa, president of the Cement Manufactures' Association, the demand for cement in the market had not increased as expected.
He added that the market was getting better even though the coronavirus had affected the demand for cement and that cement's price had almost remained constant over the last six months.
The current average price of Portland Pozzolana Cement (PPC) is Rs. 600 per sack, while that of Ordinary Portland Cement (OPC) is Rs. 700.
Finance Minister getting flexible towards business...
Nepal Strides Towards Robust Growth
Cooking gas hold-up leaves consumers choking
Govt assures smooth supply of daily goods
Demand for electricity declines due to lockdown
Demand for electricity declines NEA urges clients...
Valley retailers seek passes to run their shops for...
Poultry business facing losses above Rs 220 million...
No shortage of rice even in remote areas: FMTCL | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 7,090 |
Skewered! With Nohm Serious Yakitori Lands in SLC
February 21, 2020 by Ted Scheffler Leave a Comment
Last spring during a visit to New York City, my wife and I enjoyed a superb meal in America's only Michelin-starred yakitori restaurant, called Torishin. And since that time I've been tugging on the shirtsleeves of every restaurateur and chef I know, encouraging someone to open a yakitori eatery here in the Beehive State. Yakitori is an up and coming food trend that hasn't quite made its way to Utah yet … or has it?
Yakitori in Japanese, roughly speaking, refers to grilled chicken. Yaki means to grill or burn and tori means bird – typically chicken. Many Japanese restaurants offer some yakitori items on their menus, though not many are seriously dedicated to the art of yakitori. But now, thanks to a new restaurant called Nohm, that's changing. Hallelujah.
Nohm is a pan-Asian bistro that is the creation of a very talented young chef named David Chon. It's in the location that was previously home to Meditrina, but Chon has given the restaurant a total makeover and now the space is sleek, lively and modern, with dark brown and grey hues that serve as a reminder that this is largely a charcoal-powered kitchen.
The menu features raw, sashimi-style fish and seafood, along with cooked shareable items like deep-fried chicken meatballs with marinated egg yolk called agemono ($7), or soondae, which is squid stuffed with veggies, glass noodles and tofu, and others. In a nutshell, the cuisine here reminds me somewhat of David Chang's popular Momofuku restaurants in its creative combination of simplicity and culinary surprises.
Blue-fin Tuna
On the simplicity side is something like a plate of raw blue-fin tuna ($13), served simply with freshly grated wasabi and ginger. With a dish like this, the fish – tuna in this case – must be absolutely superb, because it's not hiding under a sauce, or even seared and crusted. And that's precisely what this blue-fin tuna was: a generous serving of superb sashimi tuna.
Grilled Spanish Mackerel
If you prefer your fish cooked, I can highly recommend Chef Chon's Spanish Mackerel ($10). It's a tender, delicious scored and grilled skin-on fillet of Spanish mackerel (yakimono) with saikyo miso – a lovely, golden miso that originated in and around Kyoto, served with Japanese scarlet scallions. Another shareable cooked menu item that I really love is Nohm's pan-fried Asian chive pancake ($7) – a dangerously addictive dish, indeed.
But where I think David Chon's Nohm really shines is in its respect for the art of yakitori and its expertise in preparing it. One key – according to yakitori experts I've talked to – is in the use of top-notch, ridiculously expensive Japanese charcoal called binchō-tan. It's made from Japanese oak and I'm told that what makes binchō-tan so special isn't that it burns at an extremely high temperature – which some people think – but rather that it is a very pure, high-quality charcoal product that burns very evenly.
To pair with Nohm's eclectic food selections there's a tempting selection of quality hot, slightly chilled, and cold sakes available, along with local and imported beers and a handful of red, white and sparkling wines available by the glass. But if you're looking to celebrate with something fancy, wine-wise, better to BYOB. BTW, we got a kick out of the restroom signage, which says simply: "Thinking Room."
Chicken Oysters, Thigh & Gizzard
I mentioned that yakitori traditionally refers to cooking chicken over a grill or charcoal. However, the skewered-and-grilled choices at Nohm go beyond chicken – as they did at the aforementioned Torishin – and include Wagyu beef, marinated pork, asparagus, enoki, baby tomatoes, shimofuri mushrooms, and others. Skewers at Nohm run $3 to $8, with most of them in the $3 to $4 range.
Gluten-free Yakitori
Other chicken parts available include wings, tenderloin, heart, neck and tail. We loved all of the chicken choices we made, yakitorily speaking, although I do wish that Nohm offered the chicken kidneys and tender belly skin that we loved so much at Torishin. That said, the menu at Nohm changes very frequently, so it's entirely possible that what was available last week when we dined there may be different this week.
As with the fish served at Nohm, the chicken is the highest-quality Mary's organic, free-range chicken, from head-to-chicken feet. Some of the yakitori is either marinated in soy-based sauce or is drizzled with sauce containing soy, and my wife applauded the effort the kitchen made to prepare chicken yakitori for her that was gluten-free.
Stir-fried Spicy Rice Cake
And speaking of gluten-free foods, Korean rice cakes have become very popular at our house since my wife avoids gluten. Nohm serves a big dish of spicy – no, make that incendiary – rice cakes bathed in fiery gochujang sauce and topped with a sort of fried noodle dumpling ($8). It's a dish that really hits the spot in cold weather. Fans of Korean flavors should also try the braised pork shank with Napa cabbage kimchi ($12).
Simply put, I am thrilled that SLC has yet another new, eclectic, economical eatery in which to dine. And I'm even more thrilled that now, with Nohm open, us yakitori yahoos can get skewered on demand.
Culinary quote of the week:
"Food should be fun." — Thomas Keller
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Filed Under: Bites, Restaurant reviews, Utah Bites Tagged With: restaurant review, slc restaurants
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The Ups and Downs of Mormon Media | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 7,237 |
\section{Introduction.}
Quadrupole collectivity is one of the most relevant features of nuclear
structure \cite{Bo-M,rs}. In this context, the theoretical understanding
of the evolution of the nuclear shapes, and the
structural changes associated with it, represent
an active research field
\cite{review,review-Bender,Heenen-nature,Werner,Robledo-1,Robledo-2,Rayner-1}.
From the experimental side, low-lying
spectroscopy is one of the most
powerful sources of information about nuclear shapes and/or shape
transitions since one can establish signatures correlating the
excitation energies with the deformation properties
\cite{Julin,draco,davidson,wu96,podolyak,pfun,caamano}.
Nowhere, however, the evolution of nuclear shapes
is more documented and challenging than
around the proton shell closure $Z=82$. For example, the
neutron deficient Lead isotopes with neutron number
$N \approx 104$ display three $0^{+}$ states within
1 MeV excitation energy \cite{Andreyev-1}. The
very rich and challenging variety of nuclear
shapes also extends to the neighboring Hg and Po isotopes
\cite{Julin}. In
particular, it has been demonstrated
\cite{delta2p-1} that in the case of the Lead isotopes
the
decreasing
trend observed in the
binding energy difference
${\delta}_{2p}(Z,N)=E(Z-2,N)-2E(Z,N)+E(Z+2,N)$
for decreasing mass number $A$, can already
be described quantitatively by mean field models
in terms of deformed ground states of Hg and Po nuclei
while the inclusion of the quadrupole correlation energy
\cite{delta2p-2} brings the calculations even closer to
experiment. From the experimental point of view
\cite{Julin}, the neutron deficient Mercury isotopes
exhibit deformed ground
states while the situation is more involved in
the case of Po nuclei (see, for
example \cite{Grahn}, and references
therein).
A considerable effort, has also
been devoted to
characterize Pt nuclei \cite{davidson,wu96,King-exp,Cederwall-exp,
Seweryniak-exp,Kondev-exp,Blanc,Joss-exp,Soramel-exp,Popescu-exp,
Xu-exp}. In this case, several deformation regimes
have been suggested. Previous theoretical
investigations \cite{Sauvage,Ansari,Krieger,bengtsson,Moller-1995,Moller-2008}
have found triaxial and oblate
ground state shapes
for the heaviest Pt isotopes
while for the light ones a
prolate deformed regime
is predicted.
From the experimental point of view
\cite{podolyak,caamano,CDROM}
the energy ratio $E_{4^{+}}/E_{2^{+}}$
is almost 2.5
for Pt nuclei with neutron number
$110 \le N \le 118$ already pointing
to $\gamma$ soft shapes.
The role played by the $\gamma$ degree of freedom
in Pt isotopes has also been stressed
by the comparison
of experimental and theoretical results
performed in Ref. \cite{Hilberath}
which shows, that good agreement can
be obtained if triaxiality is taken into account.
Further down, a transition
to a vibrational regime is suggested for $^{168-172}$Pt
by both the experimental data and their theoretical interpretation
\cite{Cederwall-exp}.
The shape evolution
provided by the mean field
framework \cite{rs}, based on the most recent
incarnations of the Gogny interaction \cite{gogny}, in the
isotopic chain
$^{166-204}$Pt is considered in the present
study as a representative sample
of nuclei close to the $Z=82$ proton shell closure
for which prolate, triaxial, oblate and spherical
ground state shapes are found.
Nuclear shapes
around the proton magic number $Z=82$ have been studied using
a wide variety of theoretical
models. Low-lying minima
in both Pb and Hg isotopes, have been predicted
within the framework of the Strutinsky method
\cite{Pashkevich,Be-Na,Na-other}. From a mean field perspective, the
coexistence between different nuclear shapes
in Pt, Hg and Pb nuclei, has been considered
within the relativistic mean field approximation
\cite{Yoshida-1,Yoshida-2}. Deformed ground states
were predicted in the case of Pb isotopes as well as superdeformed
ground states in Hg isotopes at variance with
experiment. In order to cure this problem, a new
parametrization of the relativistic mean field Lagrangian, called
NLSC, was introduced in Ref. \cite{NLSC}. More recently,
a new set called NL3$^{*}$ has been proposed in Ref. \cite{NL3-star}
providing an improved description of the ground state properties of many nuclei
like Pb isotopes. Studies based on Skyrme-like
models have been reported
\cite{Valor,Meyer,Smirnova,Sarri-Moreno,Sarri-Moreno-other} while
the shape coexistence in
$^{182-192}$Pb was analyzed in Ref. \cite{rayner-PRL} using
the Hartree-Fock-Bogoliubov (HFB) approach based on the
parametrization D1S \cite{gogny-d1s}
of the Gogny interaction \cite{gogny}. From a beyond mean
field perspective, symmetry projected configuration mixing, based
on both Skyrme \cite{Duguet-1,Duguet-2} and Gogny
\cite{Rayner-Pb}
energy density
functionals, has been successfully employed in this region
of the nuclear chart establishing a firm ground to support
the experimental evidences
for rotational bands in the neutron deficient Pb isotopes
built, on coexisting low-lying $0^{+}$ states.
On the other hand, evidences for $\gamma$ vibrations and shape
evolution have also been considered in the nuclei
$^{184-190}$Hg \cite{delaroche94}, where a five-dimension collective
Hamiltonian was built with the help of constrained Gogny-D1S
HFB calculations.
Just below the proton magic number $Z=82$, the nuclei with
$A=170-200$ are particularly interesting because, small islands of
oblate deformation might be favored energetically. Transitions from
prolate to oblate shapes, as the number of neutrons
increases, have been predicted
in this mass region, using collective models
\cite{wu96}, phenomenological Woods-Saxon or Nilsson potentials
\cite{bengtsson,naza90,wheldon}, relativistic mean field
\cite{naik,fossion,Sharma-Ring} as well as non-relativistic
deformed Hartree-Fock (HF) \cite{stevenson,Robledo-2}
and HFB calculations \cite{Robledo-2,ours2}. Signatures
for a transition from prolate to oblate
ground states, as the number of neutrons increases
from $N=110$ to $N=122$, have been found in Hf, W and Os isotopes. In
particular, such a transition was found \cite{Robledo-2} to happen at
$N=116-118$. Subsequently, the evolution of the ground state shapes
along the triaxial landscape of several isotopes of
Yb, Hf, W, Os and Pt has been studied in Ref. \cite{ours2} within
the framework of the mean field approximation based on both
Skyrme and Gogny interactions. This region is also an active research field
within the Interacting Boson Model (IBM)
\cite{Barret-1,pairs-IBM-1,pairs-IBM-2,pairs-IBM-3,pairs-IBM-4,pairs-IBM-5,pairs-IBM-6,gramos,morales}.
Taking into account that
around the neutron mid-shell N=104 examples of
coexisting configurations
have been found \cite{Julin}, it is very interesting to study
the propagation of
the nuclear shapes in the Pt isotopic chain
and the way it can be correlated with
the details in the underlying single-particle levels as functions
of the deformation parameters. For such a task, the
mean field approximation appears as a first tool
incorporating, important correlations within the concept of spontaneous
symmetry breaking and allowing a description of the evolution
of shell structure and deformation all over the
nuclear chart
(see, for example, \cite{review-Bender,Robledo-1,Robledo-2,Rayner-1,ours2,mf-1}
and references therein).
\begin{figure*}
\includegraphics[width=0.98\textwidth]{figure1}
\caption{Potential energy curves for the isotopes
$^{166-204}$Pt as functions of the axial
quadrupole moment $Q_{20}$ calculated with the
parametrizations D1S (continuous line), D1N (dashed line)
and D1M (dotted line)
of the Gogny interaction.}
\label{fig_axial}
\end{figure*}
In the present work, our
study will be performed within the
selfconsistent HFB
framework based on the
Gogny interaction \cite{gogny}. In addition to D1S \cite{gogny-d1s}, which is
still the most standard and thoroughly tested parametrization, we also
considered the two most recent parameter sets of the Gogny interaction, i.e.,
D1N \cite{gogny-d1n} and D1M \cite{gogny-d1m}. To the best of our
knowledge, the results
to be discussed later on in this paper, are the first systematic
mean field study reported in the
literature, using
both D1N and D1M parametrizations, to describe
(mean field)
ground state properties of Pt nuclei.
The selected
isotopes $^{166-204}$Pt cover almost the whole
shell
(i.e., $N=88-126$) and
display a range of ground state
shapes wide enough so as to be considered a
very challenging testing ground
for a comparison of the (mean field) ground state
properties predicted with the new parametrizations
D1N and D1M against the
standard D1S Gogny functional. Our calculations also
included the isotopes $^{160-164}$Pt which turn
out to be spherical and will
not be further discussed in the present study.
The paper is organized as follows. In Sec. \ref{formalism} we present a
brief description of the theoretical framework
used. The
results of our study are discussed
in Sec. \ref{results}. There, we discuss, in Sec. \ref{results-PECPES}, our
Gogny-HFB
calculations, for which axial symmetry is preserved as selfconsistent
symmetry, used to construct Potential Energy Curves (PECs). In a
second step we discuss our study of the triaxial
landscape, providing Potential Energy Surfaces (PESs), by
constraining on both $\beta$ and $\gamma$
quadrupole deformations. In our axial and triaxial HFB
calculations we considered at the same time, the three parameter sets
D1S, D1N and D1M. The interaction Gogny-D1S, taken as a reference
in the present study, is already considered as a global force able to describe reasonably well
low energy experimental data all over the nuclear chart
(see, for example, \cite{rayner-PRL,gogny,gogny-d1s,
gogny-other-1,gogny-other-2,
gogny-other-3,gogny-other-4,gogny-other-5,
gogny-other-6,Robledo-1,Rayner-Pb,bertsch,
peru,gogny-other-7,gogny-other-8,hilaire,delaroche10} and references therein).
This is also likely to be the situation with the new Gogny interactions D1N
\cite{gogny-d1n,ours2} and D1M \cite{gogny-d1n}
but still further explorations are required. Therefore, we consider
all these interactions/functionals in the present study
to asses to which extent the fine details of our mean field
predictions for the nuclei $^{166-204}$Pt are independent
of the particular version of the Gogny force employed.
After discussing the mean field
systematics of deformation for the considered nuclei, we
turn our attention, in Sec. \ref{results-SPE}, to the
underlying single-particle properties as functions of both
deformation (axial and triaxial) and mass number. This is
relevant if one keeps in mind that, from a mean field
perspective, shape changes arise when the deformed single-particle levels
are energetically favored to a different degree in open shell nuclei
(Jahn-Teller effect \cite{JTE-1}). We also consider the
behaviour with neutron number of the spherical shell occupancies corresponding
to the ground state of the different Pt isotopes in order to
shed some light into the phenomena involved in the different deformation regimes.
Finally, Sec. \ref{conclusions} contains the concluding remarks and work perspectives.
\begin{figure*}
\includegraphics[width=0.98\textwidth]{figure2}
\caption{(Color online) $Q- \gamma$ planes computed
with the Gogny-D1S force for the isotopes $^{166-204}$Pt.
The range of $Q$ values has been reduced to focus
on the interval around the minima. The contour lines
extend from the minimum up to 2 MeV in steps of 0.25 MeV.
Blue (black) contours are the three lowest, green (light gray)
ones the next three and red (dark gray) contours
correspond to the three with higher energies.
For each color the full line corresponds to the lower energy,
dashed to the next contour and dotted to the higher energy.
The minimum of the
triaxial landscape can be identified by the small ellipse
surrounding it.
}
\label{fig_d1s}
\end{figure*}
\section{Theoretical Framework.}
\label{formalism}
In order to compute both PECs and PESs we have used the
(constrained) HFB method
together with the parametrizations D1S, D1N
and D1M
of the
Gogny interaction. The solution of the HFB
equations, leading to the vacuum $| \Phi_{\rm HFB} \rangle$, was
based on the so called gradient method
\cite{gradient,ours2} to locate the minima. The kinetic energy of the center of
mass motion has been subtracted from the Routhian to be minimized in
order to ensure
that the center of mass is kept at rest. The exchange
Coulomb energy was considered in the Slater approximation and
we neglected the contribution
of the Coulomb interaction to the pairing field.
The HFB quasiparticle operators have been expanded
in an Harmonic Oscillator (HO) basis containing enough
number of shells (i.e., N= 13 major shells) to grant
convergence for all values of the mass quadrupole operator
and for all the nuclei
studied. Energy contour plots will be shown in the $(Q,\gamma)$
plane \cite{ours2} (instead of $(\beta,\gamma)$)
with
\begin{equation}
\label{Q20}
Q_{20} = \frac{1}{2} \langle \Phi_{\rm HFB} | 2z^{2} -
x^{2} - y^{2} | \Phi_{\rm HFB} \rangle
\end{equation}
\begin{equation}
\label{Q22}
Q_{22} = \frac{\sqrt{3}}{2} \langle \Phi_{\rm HFB} | x^{2} - y^{2} | \Phi_{\rm HFB}
\rangle
\end{equation}
\begin{equation}
\label{Q0}
Q = \sqrt{ Q_{20}^{2}+ Q_{22}^{2}}
\end{equation}
\begin{equation}
\label{gamma}
\tan \gamma = \frac{Q_{22}}{Q_{20}}
\end{equation}
\begin{figure*}
\includegraphics[width=0.98\textwidth]{figure3}
\caption{(Color online) The same as Fig. \ref{fig_d1s}
but for the Gogny-D1N force.
}
\label{fig_d1n}
\end{figure*}
Other interesting pieces of information coming from the mean field
are the single particle energies (SPEs) for protons and neutrons. In our
calculations, with the Gogny interaction, we are solving the HFB equations
and therefore the only quantities that can be properly defined are the
quasiparticle energies. However, in order to have the more
usual, Nilsson-like, diagrams we have chosen to plot the eigenvalues
of the Routhian \cite{rs}
$h = t+ \Gamma - {\lambda}_{20}Q_{20} - {\lambda}_{22}Q_{22}$, with
$t$ being the kinetic energy operator, $\Gamma$ the Hartree-Fock
field. The term ${\lambda}_{20}Q_{20} +{\lambda}_{22}Q_{22}$
contains the
Lagrange multipliers used to enforce the corresponding
constraints.
We have first performed calculations
restricted to axially symmetric shapes and in a second step
triaxiality is included in our mean field analysis. In the first
case, obviously, the term ${\lambda}_{22}Q_{22}$
is missing. In addition, the usual mean field constraints
on both neutron and proton average numbers are
taken into account. Parity and time-reversal are selfconsistent
symmetries in our axial calculations whereas
parity and simplex are the ones imposed in the
triaxial case \cite{ours2}.
\section{Discussion of the results.}
\label{results}
In this section, we discuss the results of the present study. The
systematics of deformation obtained for the isotopes $^{166-204}$Pt
is described in
Sec. \ref{results-PECPES}. Single particle properties
are considered in Sec. \ref{results-SPE}.
\subsection{Mean field systematics of deformation for $^{166-204}$Pt.}
\label{results-PECPES}
The PECs obtained for the isotopes $^{166-204}$Pt with our
constrained Gogny-HFB calculations
preserving axial symmetry, are shown
in Fig. \ref{fig_axial} as functions of the quadrupole
moment $Q_{20}$. Both prolate ($Q_{20} > 0$) and oblate
($Q_{20} < 0$) sides are displayed. The prolate side
is equivalent to the triaxial results, to be discussed later on,
with $Q=Q_{20}$ and $\gamma = 0^\circ$ whereas the oblate
side is equivalent to $Q=|Q_{20}|$ and $\gamma = 60^\circ$.
As we can see, the
interactions D1N and D1M provide PECs which are
extremely similar to the
ones obtained with Gogny-D1S.
The deformations of the oblate and prolate minima are
practically independent of the force.
The axial quadrupole
moment $Q_{20}$ corresponding to the absolute minima of the
PECs increases until $A\approx 180$ ($N\approx 102$)
when it reaches the value $Q_{20} \approx 10$ b.
A sudden prolate to oblate shape change occurs around $A=188$ for
all the Gogny
interactions considered in the present study.
Beyond $A=190$, absolute oblate minima are obtained
with quadrupole moments decreasing until
the spherical shape is reached for $^{204}$Pt ($N=126$). The opposite
situation occurs with the
secondary minima. On the other hand, the depth of both
prolate and oblate wells (as compared
to the spherical maximum) increases with increasing
$N$ up to $A=182$ (roughly mid-shell) and it decreases from
there on being
the decrease more pronounced for the prolate wells. This, as explained
in Ref. \cite{ours2} can be understood first as a consequence of
the filling of down slopping levels coming from the high $j$, unique
parity $i_{13/2}$ neutron orbital that
would explain the increase of the depth and then, at
mid shell, the filling of the up slopping
levels that would lead to the decrease of the height of the wells.
\begin{figure*}
\includegraphics[width=0.98\textwidth]{figure4}
\caption{(Color online) The same as Fig. \ref{fig_d1s}
but for the Gogny-D1M force.
}
\label{fig_d1m}
\end{figure*}
Slightly lower spherical barrier
heights (i.e., the difference between the energy of the
absolute minimum of the PEC and the energy of the spherical configuration) are
predicted by the new Gogny forces D1N and D1M. Such a
sensitivity of the spherical barriers with
respect to details of the effective interactions
used has already
been found in previous studies \cite{Robledo-2,Rayner-1,Tajima,Sarri-Moreno}.
In our calculations, the largest and smallest values of the
total pairing energies corresponding
to the spherical configurations are obtained with the sets
D1M and D1S, respectively. The
set D1N provides pairing energies in between.
This already reflects the different pairing content
of the considered Gogny functionals but we postpone
a discussion on this point for later on.
\begin{figure*}
\includegraphics[width=0.89\textwidth]{figure5}
\caption{Mean field excitation energies $\Delta E$ computed with the
Gogny interaction D1S (continuous line), D1N (dashed line)
and D1M (dotted line) are
displayed as functions of the deformation parameter $\gamma$ for fixed
values of the quadrupole moment $Q$ corresponding
to the lowest minima of the axially symmetric calculations
(see Fig. \ref{fig_axial}).
Results for $^{204}$Pt
are not included due to the presence of a spherical ground state.}
\label{evol-min-gamma}
\end{figure*}
The absolute values
$| \Delta E_{p-o} |$
of the energy differences
between the prolate and oblate
minima of the PECs exhibit two bumps, the
first with a maximum at $A=178$
(2.19, 1.97 and 2.11 MeV for D1S, D1N and
D1M) corresponds to
isotopes with a prolate ground state
and the second one at $A=198$
(1.16, 0.95 and 0.99 MeV for D1S, D1N and
D1M) corresponds to isotopes
with an oblate ground state. The nucleus $^{188}$Pt, separating
the two regions,
has almost degenerate
prolate and oblate minima with
$| \Delta E_{p-o} |$ values of
45, 111 and 71 keV for D1S, D1N and
D1M, respectively.
\begin{figure*}
\includegraphics[width=0.98\textwidth]{figure6}
\caption{Ground state triaxial coordinates
($Q,\gamma$) (left panel), neutron and proton
pairing energies (middle panel) and Thouless-Valatin
moments of inertia (right panel) are plotted
as functions of the mass number A. Calculations have
been performed with the Gogny interactions D1S, D1N and
D1M.
}
\label{figevolQ}
\end{figure*}
The previous axially symmetric results, agree well with the ones
in Ref. \cite{Robledo-2} using the parametrization SLy4 \cite{sly4}
of the Skyrme interaction in the particle-hole channel plus a
zero range and density dependent pairing
interaction \cite{Terasaki} (with strength
$g= 1000$ MeV fm$^{3}$ for both protons and neutrons)
and with previous Skyrme-HF+BCS calculations with the parameter set
SIII \cite{Sauvage}.
They also agree well with the results of the axial
calculations reported in Ref. \cite{Sharma-Ring} using the parametrizations
NL1 and NL2 of the relativistic mean field Lagrangian and
with axial macroscopic-microscopic calculations reported in Ref. \cite{Moller-1995}.
However, in our axial calculations,
prolate and oblate minima lie
quite close in energy ($| \Delta E_{p-o} | \le 2.2 $ MeV).
Thus, a
$\gamma$-path connecting them would be possible
and triaxiality could play a role, as will be discussed
below, converting some of the axially symmetric minima
into saddle points. Therefore, in a second step, we have also explored
the triaxial landscape and construct PESs
for all the considered nuclei.
The PESs obtained with the Gogny sets D1S, D1N and
D1M
are shown in Figs. \ref{fig_d1s}, \ref{fig_d1n} and \ref{fig_d1m}
in the form of $Q-\gamma$ planes.
In order to simplify the presentation, the range
of Q values plotted
is reduced to
0 b $\le Q \le 11$ b and the contour lines are
also severely reduced by considering contours every
250 keV up to 2 MeV higher than the energy
of the minimum. We can see, the spherical structure
in $^{204}$Pt while $^{202-198}$Pt
exhibit oblate
($\gamma =60^\circ$)
ground states with $Q \approx$ 3-4.5 b. An
{\it{island of triaxiality}}, centered around the nucleus
$^{188}$Pt,
is clearly visible from these figures. The ground
state triaxial coordinates
$(Q$,$\gamma)$ within such an {\it{island of triaxiality}}
evolve from
$(Q \approx $5 b, $\gamma \approx 50^\circ-58^\circ)$ in $^{196}$Pt
to $(Q \approx $9 b, $\gamma \approx 10^\circ)$ in
$^{184}$Pt. The depth of these triaxial
ground states, compared with the axial ones, is very small
(see below). In the case of $^{182-178}$Pt, our
calculations predict
prolate ($\gamma = 0^\circ)$
ground states with
deformations $Q \approx 10 $ b.
Again, for $^{176-172}$Pt
very shallow
triaxial minima are predicted. In the case of
$^{172}$Pt, for example, we find
$(Q \approx $5 b, $\gamma \approx 20^\circ)$.
The isotopes $^{170-166}$Pt display prolate ground states
with $Q \approx 3-4.5$ b.
\begin{figure*}
\includegraphics[width=0.98\textwidth]{figure7}
\caption{(Color online) Upper panels: Proton (left panel)
and neutron (right panel) SPEs
for the nucleus $^{180}$Pt
as functions of the axial quadrupole moment $Q_{20}$.
The Fermi levels
are also plotted with a thick (red) dotted line.
The results correspond to the force Gogny D1S.
Solid (dashed) lines are used for positive (negative) parity states.
With
increasing $K=1/2,3/2,5/2, \dots$ values color labels
are black, red, green, blue, dark-blue, brown, dark-green, etc.
Lower panel: the same as above but for neutron SPEs
of the nuclei $^{188}$Pt (left panel) and
$^{202}$Pt (right panel). The spherical quantum numbers at $Q_{20}=0$
are given for a number of orbitals close to the Fermi level.
}
\label{speq20}
\end{figure*}
\begin{figure*}
\includegraphics[width=0.95\textwidth]{figure8}
\caption{(Color online) Proton
SPEs are plotted for the nucleus $^{188}$Pt. In the left panel, the SPEs are plotted
as a function of the axially symmetric quadrupole moment $Q_{20}$
from $Q_{20}=0$ up to $Q_{20}=7.8$ b. In the middle part the triaxial
proton SPEs are plotted as functions of the deformation
parameter $\gamma$ and for $Q=7.8$ b. Finally, on the right panel, the axially symmetric
SPEs are plotted
as a function of $Q_{20}$ from $Q_{20}=-7.8$ b up to $Q_{20}=0$. The proton
Fermi level is depicted as a thick (red) dotted line. Some relevant $K$ values
are also included. In the axial plots, at $Q_{20}=0$ (i.e. sphericity) we
have the $d_{5/2}$ at $\epsilon \approx -9.5$ MeV, at an energy of
around -7.5 MeV we have the $d_{3/2}$ and at $\epsilon \approx -7.2$ MeV we have
the $h_{11/2}$ orbital. Also at $Q_{20}=0$ the $Z=82$
shell gap is clearly visible.
Solid (dashed) lines are used for
positive (negative) parity states. For more details, see the main text.
}
\label{speAXTRI-proton}
\end{figure*}
In order to obtain a more quantitative understanding of the
PESs we have plotted in Fig. \ref{evol-min-gamma}, the mean field
energies corresponding
to the lowest axial minima $Q=Q_{20}$
of the PECs in Fig. \ref{fig_axial}
as functions of the deformation parameter $\gamma$. We
observe that only one
of the two axial minima
remains in most of the cases. The nuclei
$^{202-198}$Pt exhibit oblate $(\gamma =60^\circ)$
absolute minima and the
prolate (axial) solutions become saddle points with excitation
energies $\Delta E \le 2 $ MeV. On the other hand, $^{196-188}$Pt are
rather $\gamma$-soft with triaxial minima almost
degenerate with the (axially symmetric) prolate and oblate solutions.
In the case of $^{188}$Pt, for example, we
obtain $|\Delta E|_{triaxial-saddle} \le 0.9 $ MeV.
Still inside the {\it{island of triaxiality}}, the oblate
configurations in both $^{186,184}$Pt already show the tendency
to increase their excitation energies. A similar trend
for the oblate solutions
is
observed within the mass range $182 \le A \le 174$.
Oblate and prolate configurations for the nuclei
$^{172-166}$Pt are quite close
($0.3 \le \Delta E \le 1.2$ MeV) and
softly linked along the $\gamma$ direction.
A detailed account of the evolution of
the ground
state triaxial coordinates $(Q,\gamma)$ as functions of the mass
number $A$ is presented in the left panel of
Fig. \ref{figevolQ} for the sake of completeness. The
striking similarity of the ground state deformations
obtained with
the Gogny interactions D1S, D1N and D1M becomes evident from this plot.
We observe the emergence of weakly oblate ground
states for the isotopes $^{202-198}$Pt. On the other hand, the
sudden shape
transition observed in the framework of the axially symmetric
HFB calculations is now
replaced by a smooth shape change through
the {\it{island of triaxiality}}
represented, in our case, by the isotopes
$^{184-196}$Pt. A prolate
deformed regime is predicted for the isotopes
$^{178-182}$Pt. We find
that, the trend
of shape changes predicted by our calculations
in the considered Pt isotopes
agrees well with the ones
obtained in
Refs. \cite{Ansari,Krieger}
and the conclusions
extracted from the combination of total routhian surface
calculations plus quasiparticle random phase approximation
(TRS+QRPA) and IBM models in Ref. \cite{Cederwall-exp}.
The general trend in our calculations is also
consistent with results obtained in the framework of the
Strutinsky approach \cite{bengtsson,Moller-2008}.
Further down in neutron number, the TRS results
\cite{review,Ce-other}
predict a rapid change to a triaxial shape
also found in our calculations around $^{172}$Pt.
Finally, our PESs for $^{170-166}$Pt
exhibit features that could
be interpreted as the onset of a more pronounced
vibrational character for these nuclei \cite{Cederwall-exp}.
\begin{figure*}
\includegraphics[width=0.95\textwidth]{figure9}
\caption{(Color online) The same as Fig. \ref{speAXTRI-proton} but
for neutrons. In the axial plots, at $Q_{20}=0$ (i.e. sphericity) we
have the $h_{9/2}$ and $f_{7/2}$ at $\epsilon \approx -10.5$ MeV, at an energy of
around -8 MeV we have the $i_{13/2}$ and at $\epsilon \approx -7.5$ MeV we have the
$f_{5/2}$ and $p_{3/2}$ orbitals. Also at $Q_{20}=0$ the $N=126$
shell gap is clearly visible.
}
\label{speAXTRI-neutron}
\end{figure*}
The rather involved behavior of the neutron and
proton pairing
energies $E_{pp}$ (with opposite sign)
versus mass number
for the Pt isotopes is shown
in the middle panel
of Fig. \ref{figevolQ}. As expected for a pure $N=126$
shell closure, neutron pairing collapses for the
nucleus $^{204}$Pt. We
observe that proton pairing shows a non constant behavior
in spite of having constant proton number ($Z=78$)
that comes from selfconsistency effects. On the other hand, neutron
pairing energies are lower inside
the region between $A=174$ and $A=186$ that is precisely where the
strong prolate deformation develops.
The lowering of pairing energies is a consequence of the lowering of the level density that is
needed (Jahn-Teller effect) to induce the deformed minima. For other
values of $A$, the neutron level
density around the Fermi level is higher and, as a consequence, pairing
correlations are stronger.
Concerning different values of the neutron and proton pairing,
for the three Gogny functionals the pairing energies follow
the same isotopic trend
and the only relevant change is in the absolute value
that tends to be slightly lower for D1S. At this
point it is worth to remember that the value of
the pairing energy shown is
related to the amount of pairing correlations
present in the system but it is by far not certain that
the correlation is linear, in other words, the
different values of $E_{pp}$ for different interactions
do not necessarily imply the same quantitative
behavior for pairing correlations.
Finally, the Thouless-Valatin moments of inertia
$J^{(1)}= 3/\left(E_{2^{+}}- E_{0^{+}}\right)$
of the first 2$^{+}$ states
in $^{166-204}$Pt are
plotted in the right panel of Fig. \ref{figevolQ}
as functions of the mass number
$A$. They are compared with the experimental values
extracted
from the available systematics for the excitation energies
of the
first 2$^{+}$ states
in $^{166-204}$Pt (see, for example, \cite{Cederwall-exp}).
The energies needed for the computation of
$J^{(1)}$ have been obtained using the
selfconsistent cranking approximation introducing
the usual time reversal breaking
term $-\omega J_{x}$ and the subsidiary
condition $\langle J_{x} \rangle= \sqrt{I(I+1)}$
\cite{rs,ours2,gogny-other-6}. The Thouless-Valatin
moments of inertia strongly depend on pairing
and therefore a comparison of the results obtained
with the three Gogny functionals considered in this
work, which exhibit different pairing contents, can also give
a hint on the quality of their predictions. As can be seen, the results
follow the same isotopic trend irrespective of the Gogny force
with the tendency to be the largest
for Gogny-D1S and the smallest in the case
of Gogny-D1M. Nevertheless, the differences in the predicted
values can still be attached to the uncertainties in the
effective interactions and we observe how the
selfconsistent cranking results tend to overestimate
the experimental values.
This defect, well known already, is a direct consequence of too
low pairing at the mean field level and its solution would
require an improved treatment of pairing correlations. There
are many mechanisms beyond mean field that modify the amount
of pairing correlations in a given nuclear system, but there are two
particularly important that tend to increse correlations. One is
the restoration of the particle number symmetry broken by the
HFB method and the other is shape fluctuations around the HFB minimum.
The latter is connected with the fact that the HFB minimum corresponds to
a low level density region of the single particle spectrum (see next
subsection)
and therefore its amount of pairing correlations is far lower than
the one of the neighbouring configurations. Taking into account
the first mechanism would involve particle number projection, whereas
the second can only be treated in the scope of the GCM with
the quadrupole moment as generating coordinate or in the Bohr Hamiltonian
method \cite{delaroche10}. Clearly, both methods are out of the scope of
the present study.
On the other hand, we observe certain correlation between
the evolution with the number of neutrons of the quadrupole
moments $Q$ and the moments of inertia $J^{(1)}$, as it is apparent
by looking at the left and right panels of Fig. \ref{figevolQ}.
This correlation is not so evident for the $\gamma$ degree of freedom.
The results discussed in this Section indicate, that the new
interactions/functionals D1N and
D1M provide
the same quality of mean field ground state
predictions for the considered Pt isotopes as compared
with the Gogny-D1S
force taken as a reference in our calculations.
This coincidence give us confidence
on the robustness of our mean field
predictions with respect
to a change in the particular parametrization
of the Gogny interaction used. The
agreement is also rather good with the
mean field picture obtained in Refs. \cite{Robledo-2,ours2} within the HF+BCS framework
based on the Skyrme parametrization SLy4
in the particle hole
channel plus a zero range and density dependent pairing
(with strength
$g= 1000$ MeV fm$^{3}$ for both protons and neutrons). Both
Gogny-D1S and Skyrme-SLy4 represent well reputed interactions
whose reasonable predictive power has already been
thoroughly tested all over the nuclear chart
and
it is very satisfying to observe how the
new parametrizations D1N and D1M, in spite
of the relaxation of some of the original
constraints in their fitting protocols and more
oriented to reproducing nuclear
masses \cite{gogny-d1n,gogny-d1s}, still follow
very closely the fine details predicted
with Gogny-D1S, Skyrme-SLy4 and
other theoretical models
\cite{Robledo-2,ours2,bengtsson,Krieger,Ansari,Sharma-Ring,Sauvage,Cederwall-exp,Ce-other}
for an isotopic chain
with such a challenging shape evolution.
Let us remark that we are perfectly aware of the fact that
the (static) mean field picture
described above, should also be
extended to a dynamical treatment of the relevant
degrees of freedom. This becomes clear from the topology
of the PESs indicating that, in order
to access a quantitative comparison of the energy
spectra and reduced transition probabilities with the considered
Gogny interactions, the dynamical interplay between the
zero point motion associated with the restoration
of broken symmetries (mainly, angular momentum
and particle number) and fluctuations in the collective
parameters $(\beta,\gamma)$ should be taken into account. For such
a cumbersome and computer power demanding extension the
Gaussian Overlap Approximation (GOA) appears as a first
suitable choice \cite{bertsch,Libert-1,GOA-other-1}.
For a very recent and excellent pedagogical review, the
reader is also referred to \cite{Prochniak}. Work
along these lines is in progress and will be reported
elsewhere.
\begin{figure*}
\includegraphics[width=0.98\textwidth]{figure10}
\caption{Spherical occupancies in the
proton (left panel) and neutron (right panel)
ground state wave functions for the isotopes
$^{166-204}$Pt. For details, see the main text.
}
\label{occupations}
\end{figure*}
\subsection{Single particle properties}
\label{results-SPE}
In this section, we pay attention to single particle
properties of the considered isotopes. To this end, we first
show in Fig. \ref{speq20}
SPE
plots, as functions of the
axial quadrupole moment
$Q_{20}$. The isotopes $^{180}$Pt, $^{188}$Pt and
$^{202}$Pt are taken as illustrative examples. The SPEs
correspond to our Gogny-D1S calculations. For
other nuclei
and/or Gogny interactions the results
are quite similar.
The energy levels in Fig. \ref{speq20}
gather together around the spherical configuration $Q_{20}$=0
forming the spherical shell model orbitals $nlj$. Due to axial
and time-reversal symmetries SPE levels, tagged by the
$K$ quantum number corresponding to the third component of
the angular momentum in the intrinsic frame, are doubly
degenerate. Positive and negative parity states
are plotted with full and dashed lines, respectively.
The proton ${\lambda}_{Z}$ and neutron ${\lambda}_{N}$
Fermi levels
are also shown
with a thick (red) dotted line.
As it is well known, atomic nuclei "avoid" regions
with high single particle level densities (Jahn-Teller effect)
and therefore the plots of SPEs versus deformation
help us to identify regions where
energy gaps favor the
appearance of deformed minima
\cite{HW}.
We also look at the onset of deformation in the
considered Pt nuclei using the Federman-Pittel (FP)
criteria \cite{pittel}. The origin of nuclear deformation
is certainly a much more complicated phenomenon involving
different mechanisms (see, for
example, \cite{Werner,rayner-PRL,Naza-def,Mottelson}
and references therein for detailed discussions
on this issue). Even though the FP argument
may be
incomplete, it is
certainly playing a role in the onset of nuclear deformation
and we resort to it in the present
mean field study
as a way to establish a very qualitative and
simplified overall picture of
shape changes in terms of the evolution of the underlying
SPEs. For recent studies using
similar ideas, see \cite{ours2,Fossion-def}. In order
to incorporate the terminology of spherical orbitals
to discuss deformed configurations, we assign to a given
deformed single particle orbital the label of the
spherical orbit from which it originates at $Q_{20}=0$
\cite{ours2}.
In Fig. \ref{speq20}, we can see at zero
deformation the spherical proton shells
$3s_{1/2}$, $1h_{11/2}$, $2d_{3/2}$
and $2d_{5/2}$ below the Fermi level ${\lambda}_{Z}$ and the
$1h_{9/2}$ level above it.
The relative position of
${\lambda}_{Z}$, with respect to the SPEs, is
quite stable. The main effect of the different neutron numbers
appears in the scale of Fermi energies ${\lambda}_{Z}$
ranging from around -9 MeV
in the case of $^{204}$Pt to values close
to zero in the case of the neutron deficient isotope $^{166}$Pt.
Therefore, in Fig. \ref{speq20} we have only plotted the
proton SPEs for the isotope $^{180}$Pt.
The
closeness to the proton magic number $Z=82$ makes
Pt isotopes ($Z=78$) to display a tendency to be spherical or
slightly oblate
as it can be seen in Fig. \ref{speq20}
from the huge spherical gap
of $\sim$ 6
MeV
and the large gap
of $\sim$ 3
MeV
on the oblate side centered around $Q_{20}=-5$ b.
For neutrons the relevant spherical orbitals
shown in Fig. \ref{speq20} are
$1h_{9/2}$, $2f_{7/2}$, $1i_{13/2}$, $2f_{5/2}$,
$3p_{3/2}$, $3p_{1/2}$
and $1g_{9/2}$. Notice also that in the case
of $^{204}$Pt
the tendency to be spherical is reinforced by its magic neutron
number $N=126$ and the final result is a spherical nucleus.
In our calculations, the $N=126$ spherical shell gap
is observed to change with mass number
from $\sim$ 5.5 MeV in $^{180}$Pt to $\sim$ 4.5 MeV in $^{202}$Pt
going through $\sim$ 5 MeV for $^{188}$Pt.
Between $^{202}$Pt
and $^{196}$Pt, ${\lambda}_{N}$
crosses a region on the
oblate side where an energy gap of $\sim$ 3 MeV (i.e. half the size
of the spherical gap) is found. This occurs at
around $Q_{20}=-5$ b, which overlaps perfectly with the gap
on the oblate proton sector already mentioned above.
On the other
hand, as can be seen from Fig. \ref{speq20}, the
$1i_{13/2}$ gets more and more occupied for increasing
neutron number $N$ and at a certain point (i.e., beyond $^{194}$Pt)
its role is transferred to the $2f_{5/2}$ and
$3p_{3/2}$ orbitals. According to FP only the neutron
$2f_{5/2}$ can interact with the proton
$2d_{3/2}$
(i.e., $n_{p}=n_{n}$, $l_{p}=l_{n}-1$)
but since the $l$ values are in this case
low we should not expect a very strong interaction among them.
This, qualitatively
explains the appearance of the, very soft and close
to spherical, oblate minima displayed
in Fig. \ref{fig_axial}
within the mass range $196 \le A \le 202$.
A similar mechanism leads
to very shallow and weakly prolate
secondary minima in $^{196,198}$Pt.
Between $^{194}$Pt and $^{184}$Pt the neutron Fermi level
on the
prolate sector
crosses a region with energy gaps of $\sim$ 3 MeV and between
$Q_{20}=3$ b and 10 b.
From the neutron SPE plot of $^{188}$Pt, we observe
how within this mass
range, the most prominent role
is played by the $1i_{13/2}$
which, according to the FP criteria, interacts optimally
with the proton $1h_{11/2}$ to favor a
prolate shape. On the oblate side, the interaction
between the relevant orbitals, leading to
the corresponding minima, takes place around
$Q_{20} =-5$ b. The final net
effect of these driving forces is the
appearance of prolate and oblate minima
that become saddle points inside the
{\it{island of triaxiality}}
(see
also the
discussion
below).
Below $^{182}$Pt, our axially symmetric
calculations predict strong prolate deformations.
In this case there is a strong gap on the neutron sector
centered at $Q_{20}$= 10 b, as it can be seen
from the neutron SPE plot of $^{180}$Pt.
The most active deformed orbitals are the
ones coming from the neutron $1i_{13/2}$
and $2f_{7/2}$. They interact
with the deformed proton orbitals coming from the
$1h_{11/2}$ and $2d_{5/2}$
to produce strongly prolate deformed shapes.
The secondary oblate minima predicted for
these nuclei, can be associated with
the proton and neutron energy gaps observed
around $Q_{20}$= -5 b. Further down in neutron
number, the neutron Fermi level ${\lambda}_{N}$ explores
regions lower in energy resulting
in less pronounced prolate minima located
around $Q_{20}$= 4 b in
the neutron deficient isotopes
$^{166,168}$Pt.
Let us now turn our attention to the origin of triaxiality
and for this, proton and neutron SPE plots
are shown in Figs. \ref{speAXTRI-proton} and
\ref{speAXTRI-neutron}, respectively, as functions of the
triaxial deformation parameter $\gamma$
\cite{Heenen-nature,ours2} for the nucleus $^{188}$Pt.
We consider $Q=$7.8 b which corresponds to a rather large
region in the $Q-\gamma$ plane near the triaxial minimum (located
at $Q=$6.2 b, see left panel of Fig.\ref{figevolQ}) where the PES is very flat
in the two coordinates (see Fig. \ref{fig_d1s}). The value of $Q=$7.8 b also
corresponds to the position of the axial prolate minimum (see
Fig. \ref{fig_axial}).
These plots allow us to identify
the $K$ values of the triaxial SPEs at the prolate
($\gamma = 0^\circ$) and oblate ($\gamma = 60^\circ)$ limits
and the change of the $K$ contents observed in most of the levels
as $\gamma$ evolves. Typical examples in this context, are the
negative parity K=1/2 proton level with SPE $\epsilon \approx $ -3 MeV
at $Q$=7.8 b and $\gamma = 0^\circ$ which transforms into a
$K=9/2$ level at $\gamma = 60^\circ$ and the
positive parity $K=13/2$ neutron level
with SPE $\epsilon \approx $ -4.2 MeV at $Q$=7.8 b and $\gamma = 0^\circ$
which transforms into a
$K=1/2$ level at $\gamma = 60^\circ$. The rather
low level density below the proton
Fermi level for $\gamma$ between $0^\circ$ and $30^\circ$ favors
the flateness of the energy curve as a function of $\gamma$ helping thereby
the development of the triaxial minimum in $^{188}$Pt around
$\gamma = 30^\circ$. From this plot we also conclude that
the isotopes with two protons less (Os isotopes) will be more prone to
triaxiallity as discussed in Ref \cite{ours2}. Concerning the
behaviour of the neutron SPEs with $\gamma$ depicted on Fig.
\ref{speAXTRI-neutron} we observe a region of low level density
in the interval of $\gamma$ between $0^\circ$ and $20^\circ$ that
would favor the development of triaxiallity. From there on the level
density increases and therefore, in order to develop a triaxial
minimum, the system is forced to change its $Q$ deformation
to lower values (see Fig. \ref{fig_d1s} for the nucleus under
consideration, $^{188}$Pt). The removal of two or four neutrons
makes the level density in the range of $\gamma$ between
$0^\circ$ and $20^\circ$
even lower than in the case of $^{188}$Pt explaining the $\gamma$
deformed minima observed in Fig. \ref{evol-min-gamma} for the nuclei
$^{184,186}$Pt. On the other hand, the addition of two or four extra
neutrons leads to a decrease of the level density in the interval
between $\gamma \approx 30^\circ$ and
$\gamma = 60^\circ$
favoring
the appeareance of triaxial minima in $^{190,192}$Pt and flat curves
in $^{194,196}$Pt, as observed in Fig. \ref{evol-min-gamma}.
In the spirit of the shell model, it is also interesting
to compute the spherical occupancies
$\nu (lj,Q,\gamma)$ of the different $lj$ orbitals in the
(usually) deformed ground states
$| \Phi_{\rm HFB}(Q,\gamma) \rangle$
of the Pt isotopes studied in this paper. They are given by
\begin{small}
\begin{eqnarray} \label{eq-occup}
\nu (lj,Q,\gamma) =
\sum_{n} \sum_{m} \langle \Phi_{\rm HFB}(Q,\gamma) | c^+_{nljm} c_{nljm} |
\Phi_{\rm HFB}(Q,\gamma) \rangle
\end{eqnarray}
\end{small}
where $c^+_{nljm}$ and $c_{nljm}$ are
the creation and annihilation operators of spherical harmonic oscillator orbits
characterized by the
quantum numbers $n$, $l$, $j$, $m$. The sum in $m$ is introduced to make the
quantity (\ref{eq-occup}) invariant under changes in
orientation (and therefore to represent a genuine
"spherical" quantity). The sum in the radial quantum
number $n$ does not allow
to pin down which specific HO orbital is occupied but, on the other
hand, allows us to get rid
of the uncertainties associated to the fact that the radial wave function of
the nuclear orbitals is close but not exactly the one of the HO.
The proton spherical occupancies in the ground state wave functions
of all the Pt isotopes considered are shown on the left
panel of Fig. \ref{occupations}. As the number of protons
remains constant along the isotopic chain one could expect a flat
behavior of all the occupancies. However, we observe in that figure
how the occupancies of the different orbitals can be classified in two
different regimes, namely the weak deformation regime including $^{166-172}$Pt
and $^{188-204}$Pt and the strong deformation
regime including the nuclei
$^{174-186}$Pt. It is noteworthy that in each of the regimes the proton
occupancies remain rather constant irrespective of the $\gamma$ deformation and
even the specific value of the $Q$ deformation parameter. The strong
deformation regime differs from the weak deformation one in that the
$d_{5/2}$, $d_{3/2}$ and $h_{11/2}$ orbitals loose occupancy in favor
of the $f_{7/2}$, $f_{5/2}$ and $h_{9/2}$. This rearrangement of the occupancies
is mainly due to the smearing out of the Fermi surface as a consequence of the
increasing proton pairing correlations (see Fig. \ref{figevolQ}) and
to a lesser extent to the quadrupole
interaction among orbits that can transfer particles from one orbit to another.
Given the shift of occupancies from the $d_{5/2}$, $d_{3/2}$ and $h_{11/2}$
to the $f_{7/2}$, $f_{5/2}$ and $h_{9/2}$ orbitals
we can interpret the well
deformed ground state of $^{174-186}$Pt as a multiparticle-multihole
excitation out of a reference spherical ground state.
The number of
particles exchanged in this kind of spherical shell model language is
in between two to four according to the results shown in the left panel of
Fig. \ref{occupations}.
In the case of neutrons, shown on the right
panel of Fig. \ref{occupations}, as neutron number increases we are occupying orbitals
belonging to the $N=5$ negative parity major shell and the positive parity
intruder $i_{13/2}$. As a consequence of deformation, the spherical orbitals
are mixed up and therefore placing two particles in a given deformed orbital
by no means implies placing two particles in an spherical orbit. The two particles
will be distributed among all the components of the deformed orbital when expressed
in the spherical basis. As a consequence, the behavior of the spherical
occupancies with neutron number is more or less linear in the whole interval
and for all the orbitals involved. The only noticeable deviation from this trend
takes place when entering the strong deformation regime where the $f_{7/2}$,
$h_{9/2}$ and $h_{11/2}$ orbitals loose particles in favor of the
high $j$ orbitals $i_{13/2}$, $i_{11/2}$, $j_{13/2}$ and $j_{15/2}$. Also the
$g_{9/2}$ orbital gets more particles (through the coupling to the $N=6$ orbital).
This change in occupancies can be mostly attributed to the quenching of neutron
pairing correlations in the ground state of the strongly deformed isotopes $^{174-186}$Pt.
When the weak deformation regime is entered at $A=188$ a much smoother behavior
of the spherical occupancies is recovered. This, together with the smooth behavior
of proton occupancies is quite unexpected as in the weak deformation regime
the different isotopes have a variety of ground state deformations ranging from
prolate to triaxial to oblate. The conclusion is that the occupancies
are more sensitive to the magnitude of the deformation $Q$ than to the
$\gamma$ degree of freedom.
The proton levels closest to the Fermi level are the $s_{1/2}$, $h_{11/2}$, $d_{3/2}$
and $d_{5/2}$ and therefore are the ones expected to strongly interact with the
neutron spin orbit partner according to the FP mechanism. The more effective orbitals
are those with high $j$ and therefore the proton $h_{11/2}$ is expected to strongly
interact with neutrons $h_{9/2}$ which is empty at the beginning of the chain and gets
steadily occupied as more neutrons are added. Also a strong interaction is expected with
the neutron $i_{13/2}$ orbital that shares with the $h_{9/2}$ one the occupation
pattern as a function of $N$. It is also noteworthy to point out how proton $h_{9/2}$, that
should be empty according to the spherical shell model, gets some occupancy for the
well deformed nuclei $^{174-186}$Pt as a consequence of the smearing out of the
Fermi surface, that allows those orbitals to interact via FP with
the neutrons $h_{11/2}$ and $g_{9/2}$. It could be argued that the proton $h_{11/2}$
orbital is loosing two particles in the strong deformation regime and this could imply
a strong impact in its interaction with its neutron spin-orbit partner. This would
be true if the $h_{11/2}$ orbital were occupied by a small amount of neutrons (of the
order of two) as this would imply completely emptying out the orbtial, but it has to
be bear in mind that there are ten neutrons at the begining of the region under consideration.
\section{Conclusions}
\label{conclusions}
In this paper we have studied the evolution of the ground state
nuclear shapes in a series
of Pt isotopes ranging from $N=88\, (A=166)$ up
to $N=126\, (A=204)$, covering practically one complete major shell.
The study has been performed within
the selfconsistent HFB approach based on the
D1S \cite{gogny-d1s} and the recent D1N \cite{gogny-d1n} and D1M
\cite{gogny-d1m} parametrizations of the Gogny interaction
and we have included, in addition to the axially symmetric limit, the
triaxial degrees of freedom $\beta$ and
$\gamma$.
From the analysis of the axially symmetric limit, we conclude that a
sudden prolate to oblate shape change occurs around $N=110\, (A=188)$
for all the Gogny parametrizations considered. This result is also in
agreement with those obtained either with Skyrme forces or from
relativistic mean field calculations
\cite{Robledo-2,Sharma-Ring,ours2,Sauvage}. On the other hand, when
triaxiality is taken into account, the picture that
finally emerges is that of smooth transitions between
the different shape regimes. We find that the absolute minimum of the
PESs for the Pt isotopes evolves from prolate shapes
with increasing values of their quadrupole moments $Q$ in the lighter
isotopes $A=166-182$ (with the exception of $A=172-176$, which
exhibit a tendency to triaxiality), to triaxial $\gamma$-soft in the
intermediate isotopes with $A=184-196$, and to oblate shapes in the
most neutron-rich isotopes $A=198-202$. Finally, the isotope
$^{204}$Pt becomes spherical. By analyzing
PECs and the $Q-\gamma$ landscapes, we observe
that the (axial) prolate and oblate minima, well separated
by high energy barriers in the $\beta$ degree of freedom, are
softly linked along the $\gamma$ direction. Indeed, most
of the secondary axial minima become saddle
points when the $\gamma$ degree of freedom is included
in the analysis. Pairing energies and Thouless-Valatin
moments of inertia have also been analyzed
as functions of the number of neutrons, finding some correlation
with the evolution of the quadrupole deformation $Q$. Such a
correlation, on the
other hand, is not so evident in the case of the $\gamma$ angle.
We consider the similarity between the mean field
ground state properties predicted for the considered
Pt chain with the most recent versions of the Gogny interaction
(i.e., D1S, D1N and D1M)
employed
in the present study as very positive.
On one hand, the results give us confidence
concerning the robustness of the predictions
against the details of the particular effective Gogny interaction
used. The robustness is reinforced when
the trend of our calculations is compared with the ones obtained
with Skyrme forces \cite{Robledo-2,ours2} and other theoretical approaches
\cite{review,bengtsson,Cederwall-exp,Ce-other}, which are
again very similar. On the other hand, our results also
point to the fact that the new incarnations D1N and D1M
of the Gogny interaction, based on fitting protocols
more in the direction of astrophysical
applications \cite{gogny-d1n,gogny-d1m}, essentially
keep the predictive power of the Gogny-D1S force \cite{gogny-d1s} already
considered as a (standard) global force.
We have analyzed the proton and neutron SPEs as functions
of both axial and triaxial deformation parameters in some
illustrative examples. The analysis has been
done in terms of the density of levels around the Fermi surface
(Jahn-Teller effect \cite{JTE-1})
and the Federman-Pittel mechanism \cite{pittel}. Our discussion has
also been illustrated with the calculation of
the spherical occupancies
in the ground state wave functions of the considered Pt nuclei. As
a result, we obtain a qualitative
understanding of the emergence of deformed configurations
with
two main ingredients which are, the
energy gaps that appear at different deformations in the SPEs of
neutrons
when the Fermi level ${\lambda}_{N}$ crosses different regions, and
the special role of the overlap between the proton $ 1h_{11/2}$ and the
neutron $ 1i_{13/2}$ orbitals.
The study of the low-lying excitation spectra and transition rates
in conjunction with shape transitions, would require to extend the
present mean field approach to take into account
correlations related to restoration of broken symmetries
and fluctuations of collective variables. The restoration of
broken symmetries would imply, among others, triaxial projections to
restore rotational symmetry and this is a very difficult and delicated
issue with realistic forces
\cite{Bender.08,Duguet-kernels-1,Duguet-kernels-2,Duguet-kernels-3}.
These difficulties
lead to consider instead of the exact projection some kind of approximation
to it that could eventually end up in a kind of collective Bohr
Hamiltonian
\cite{bertsch,Libert-1,GOA-other-1,Prochniak}
with deformation dependent parameters.
Upon completion of this work, a preprint has appeared
\cite{delaroche10} dealing with the calculation
of $2^+$ excitation energies in the framework of the 5-D Bohr hamiltonian
method with parameters extracted from a microscopic mean field calculation
with the Gogny D1S force. As a consequence of the number of nuclei considered
(around two thousand) in that calculation, their analysis of the mean field
results is not as exahustive as ours.
\begin{acknowledgments}
This work was partly supported by MEC
(Spain) under Contract FPA2007-66069, MICINN (Spain) under Contract
FIS2008-01301, the Consolider-Ingenio
2010 program CPAN (Spain) under Contract CSD2007-00042 and Junta de
Andaluc\'ia (Spain)
under Contract P07-FQM-02962. The first steps of this work were
undertaken in the framework of the FIDIPRO program (Academy of
Finland and University of Jyv\"askyl\"a) and one of us (R.R)
thanks Profs. J. Dobaczewski, J. \"Aysto, R. Julin and the experimental
teams of the University of Jyv\"askyl\"a (Finland) for
warm hospitality and encouraging discussions. We thank
Prof. S. Goriely (Universit\'e Libre de Bruxelles, Belgium)
for making the parametrization Gogny-D1M available to us
prior to publication and also for valuable
suggestions. Valuable
suggestions from Prof. K. Heyde are also acknowledged.
\end{acknowledgments}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 259 |
\section{INTRODUCTION}
Heavy flavour production in $e^{\pm}p$ collisions at HERA provides a good testing
ground of perturbative Quantum Chromodynamics (pQCD) as the high quark mass
provides a hard scale. Furthermore, other hard scales such as $Q^2$, the virtuality of
the exchanged boson, or $p_t$, the transverse momentum of the heavy
quark, allow resummation techniques to be tested.
Measurements of production rate and kinematic
properties of heavy quark bound states such as charmonium also give direct access
to the non-perturbative part of the production process.
New theoretical models can also be tested by searches for exotic
bound states like the $D^{*}p$ resonance.\par
Different kinematic variables are used to describe the $ep$ interaction
at HERA: $Q^2$, the Bjorken scaling variable, $x$, and the inelasticity, $y$.
Until 1997 HERA ran at a centre-of-mass energy of
$\sqrt{s}=300\,\text{GeV}$. This energy was increased to
$\sqrt{s}=320\,\text{GeV}$ for data taken from 1998 onwards.
Due to improvements of the detector and accelerator during a break in the data
taking, the available dataset is split into two periods. In the HERA\,I period
from 1996 until 2000 about $130\,\text{pb}^{-1}$ and between 2003
and 2007 of HERA\,II about $400\,\text{pb}^{-1}$ per experiment were collected.
The kinematic range of the analysed data can be separated in the following
two regimes: photoproduction ($\gamma p$), where the exchanged photon in the
process is almost real, and deep inelastic scattering (DIS), where the
exchanged photon is virtual. Experimentally,
$\gamma p$ is defined by the scattered electron not being in
the acceptance region of the detector, corresponding to a cut $Q^2 \lesssim 1\,\text{GeV}^2$.
\section{THEORY}
There are different approaches for the calculation of heavy
flavour production in next-to-leading order perturbative QCD. The
massive approach assumes no initial charm or beauty in the
proton (or photon). Heavy flavours are only generated dynamically from
the gluon distribution. This approach is particularly valid if the heavy quark
mass is of the order of other hard scales like $Q^2$ or the transverse
momentum $p_t$ of the heavy quarks.
In the massless approach, the leading-logarithmic and next-to-leading
logarithmic terms for example in $\alpha_s\log Q^2/m^2_{c,b}$
are resummed. This approach assumes the heavy quark to be massless and
is therefore only valid if other quantities like $Q^2$ or $p_t$
provide the dominant scale. Models using the $k_{T}$
factorisation approach~\cite{kt} are based on non-collinear parton dynamics
based on the CCFM~\cite{cascade} evolution equations.
In heavy quark bound states like charmonium, only the production of the
$c\bar{c}$ pair can be described in pQCD, whereas the formation of the $J/\psi$
bound state which occurs at long distances has to be described by
phenomenological models. In the framework of non-relativistic QCD (NRQCD)~\cite{nrqcd}
so-called colour singlet (CS) and colour octet (CO) states coexist.
In the spectroscopy of heavy quark bound states containing only one heavy
quark, Heavy Quark Effective Theory (HQET) is used to predict the
production and decay properties.
\section{BEAUTY IN PHOTOPRODUCTION}
\begin{wrapfigure}[10]{r}{0.38\textwidth}
\vspace{-35pt}
\begin{center}
\includegraphics[bb = 30 0 510 370,clip=true,width=60mm]{Fig01.eps}
\end{center}
\vspace{-20pt}
\caption{Cross sections for beauty production as a function of $p^{b}_{T}$
from various decay channels.}
\label{beauty}
\end{wrapfigure}
An important test of pQCD is provided by open heavy flavour production. In Figure
\ref{beauty} the measurements of the beauty cross section in photoproduction
of both the H1 and the ZEUS collaborations are shown as a function of the $b$
quark transverse momentum, $p_{T}$. The analyses used different datasets, techniques and decay channels
providing independent measurements cross-checking each other.
The different measurements agree well with each other and are in reasonable
agreement with the NLO prediction from FMNR\cite{FMNR}.
They cover a wide range in $p_{T}^{b}$ giving a consistent picture of $b$
quark photoproduction.
\vspace{0.5cm}
\section{INELASTIC $\mathbf{J/\psi}$ PRODUCTION}
At HERA, the charmonium state $J/\psi$ is produced predominantly by the boson gluon fusion (BGF) process.
The theoretical models which are used to compare with the measurements follow
either the DGLAP evolution or use the $k_{T}$ factorisation approach.
After the heavy quark pair ($c\bar{c}$) is produced at short distances,
the formation of the $J/\psi$ bound state is described by non-perturbative
long distance matrix elements (LDME). In the NRQCD models, contributions
from both CS and CO states are predicted; one aim of the measurements is the
extraction of their relative contributions.
\begin{wrapfigure}[13]{r}{0.38\textwidth}
\begin{center}
\vspace{-20pt}
\includegraphics[bb = 0 0 520 370, clip=true,width=0.36\textwidth]{Fig02.eps}
\end{center}
\vspace{-20pt}
\caption{Differential cross section as a function of $p_{T,\psi}^{2}$ in the
($\gamma p$) regime}
\label{jpsi}
\end{wrapfigure}
The H1 collaboration measured inelastic $J/\psi$ production
with decays to $\mu^+\mu^-$ in the photoproduction region using $\sim
\unit{166}{\text{pb}^{-1}}$ from the 2006-2007 data, complementing their previous
measurement in the DIS region~\cite{h1jpsi}.
In Figure ~\ref{jpsi} the differential cross section as a function of
$p_{T}^{2}$ is shown and compared with Monte Carlo predictions and the CS
calculations at leading order (LO) and at next-to-leading order (NLO).
The scaled CASCADE~\cite{cascade} Monte Carlo, which uses the $k_{T}$ factorisation in the
colour changing flavour mode, is able to reproduce the slope better than the
scaled EPJPSI~\cite{epjpsi} Monte Carlo, following the DGLAP evolution.
While the CS LO calculation is not able to describe the data, there is a good
agreement with the NLO calculation within the large normalisation uncertainties.
As the CS provides a generally good description of the data when using
the $k_{T}$ factorisation or calculations at higher orders, no significant
colour octet contribution is required, although there are still
large normalisation uncertainties on the prediction.
In order to reduce the effect of the normalisation uncertainty,
polarisation measurements are an important testing ground for the NRQCD
predictions. Normalised quantities can be measured using helicity parameters
to characterise the decay angular distributions. A measurement of the $J/\psi$
helicity~\cite{zeusjps} has been performed by the ZEUS collaboration using
the complete HERA statistics ($\sim \unit{470}{\text{pb}^{-1}}$).
The angular distributions of the $J/\psi\rightarrow l^{+}l^{-}$ decay can be
parametrised as $\frac{d^{2}\sigma}{d\Omega d y}\propto 1 + \lambda(y)\cos^2{\Theta} + \mu(y)\sin{2\Theta}\cos{\phi} +\frac{1}{2}
\nu(z)\sin{^2\Theta}\cos{^2 \phi}$ ~\cite{beneke}.
The following integrated helicity formulae are used, depending on the
chosen reference frame:\\[0.2cm]
\begin{minipage}{0.06\textwidth}
~\\
\end{minipage}
\begin{minipage}{0.39\textwidth}
\begin{tabular}{l l}
$\frac{1}{\sigma}\frac{d^{2}\sigma}{d \cos{\Theta}d z}$ &$\propto 1 +\lambda(z)\cos^2{\Theta}$,\\
$\frac{1}{\sigma}\frac{d^{2}\sigma}{d \phi d z}$ & $\propto 1 + \frac{\lambda(z)}{3}+\frac{\nu(z)}{3}\cos{^2\phi}$,\\[0.2cm]
\end{tabular}
\end{minipage}
\par\noindent where $\theta$ is the angle between the $\mu^{+}$ vector in the $J/\psi$ rest
frame and the z axis, and $\phi$ is the azimuthal angle in the x-y plane of
the $\mu^{+}$ vector in the $J/\psi$ rest frame.\\
\begin{wrapfigure}[15]{r}{0.38\textwidth}
\vspace{-20pt}
\begin{center}
\includegraphics[bb = 0 0 530 530, clip=true,width=0.33\textwidth]{Fig03.eps}
\end{center}
\vspace{-20pt}
\caption{Helicity parameter, $\nu$, as a function of the inelasticity, $z$}
\label{jpsihel}
\end{wrapfigure}
Figure~\ref{jpsihel} shows the measurement of the helicity parameter $\nu$ as
a function of the inelasticity $z$.
In addition to the NRQCD predictions two different predictions following the
$k_{t}$ factorisation approach are compared with the measurement.
The LO NRQCD predictions (BKV)~\cite{beneke} do not describe the dependency of this
parameter well, whereas the prediction including the CO contribution seems to be
favoured within the large uncertainties.
The other two predictions (Baranov)~\cite{baranov}, predict a small negative
polarisation and are always below the measurement.
In this scheme where only CS contributions have been taken into account two
different parametrisations of un-integrated gluon distributions have been used.
The CS model with $k_{T}$ factorisation gives predictions that are more similar to
CS+CO compared to the CS only prediction.
To distinguish between the different theoretical contributions the large
variation between the calculations has to be understood and it would be good
if the theoretical predictions could be improved and made available at NLO.
\vspace{-0.3cm}
\section{SPECTROSCOPY}
\vspace{-0.2cm}
\begin{wrapfigure}[15]{r}{0.38\textwidth}
\vspace{-30pt}
\begin{center}
\includegraphics[bb = 0 0 422 506, clip=true,width=0.36\textwidth]{Fig04.eps}
\end{center}
\vspace{-20pt}
\caption{$M(D^{*+}\pi_{a})$ and $M(D^{+}\pi_{a})$ distributions for the
$D_{1}(2420)^{0}$ and the $D_{2}^{*}(2460)^{0}$ candidates.}
\label{spect}
\end{wrapfigure}
The large charm production cross section at HERA permits measurements to be
made of excited charm and charm-strange mesons.
The production of excited charm and charm-strange mesons is observed by the
ZEUS collaboration using the full HERA\,I dataset corresponding to an
integrated luminosity
of $\sim \unit{126}{\text{pb}^{-1}}$ ~\cite{zeusmeson:08}.
The following decay channels were investigated:\\
\begin{minipage}{0.06\textwidth}
~\\
\end{minipage}
\begin{minipage}{0.39\textwidth}
\begin{tabular}{l l}
$D_{1}(2420)^{0}$ & $\rightarrow D^{*\pm}\pi^{\mp}$\\
$D_{2}^{*}(2460)^{0}$ & $\rightarrow D^{*\pm}\pi^{\mp}$\\
& $\rightarrow D^{\pm}\pi^{\mp}$\\
$D_{S1}(2536)^{0}$ & $\rightarrow D^{*+}K^{0}_{s}$\\
& $\rightarrow D^{*0}K^{+}_{s}$\\
\end{tabular}
\end{minipage}
\par\noindent Figure~\ref{spect} shows the $M(D^{*+}\pi_{a})$ and $M(D^{+}\pi_{a})$
distributions for the charm meson candidates reconstructed in the given
decay channels.
The measured masses of the observed mesons have been found to be in reasonable
agreement with the world average values.
In addition to the masses also the widths, helicity and the relative branching
fractions could be extracted.
The measured $D_{1}^{0}$ width is $\Gamma(D^{0}_{1})=53.2 \pm 7.2(\text{stat.})^{+3.3}_{-4.9}(\text{syst.})$ MeV
which is above the world average value of $20.4 \pm 1.7$ MeV.
A larger S-wave admixture at ZEUS with respect to that in measurements with
restricted phase space could explain the differences as already a small S-wave
admixture could have sizeable contributions to the $D_{1}^{0}$ width.
The measured value for the helicity parameter of $h(D_{1}^{0}) =
5.9^{+3.0}_{-1.7}(\text{stat.})^{+2.4}_{-1.0}(\text{syst.})$ has to be compared with the
prediction of the HQET for a pure S-wave ($h=0$) and a pure D-wave ($h=3$).
For the charm-strange meson $D_{s1}^{+}$ the measured parameter is $h(D_{s1}^{+}) =
-0.74^{+0.23}_{-0.17}(\text{stat.})^{+0.06}_{-0.05}(\text{syst.})$ which is inconsistent
with the prediction for a pure D-wave and more than two standard deviations
away from the prediction for a pure S-wave.
So the measurement suggests a significant contribution of both D-wave and S-wave
amplitudes to the $D_{s1}(2536)^{+}\rightarrow D^{*+}K_{S}^{0}$ decay.
In addition, a search for the radially excited charm meson
$D^{*'}(2640)^{\pm}$has been made, which was reported by DELPHI as a narrow resonance in the final
state $D^{*\pm}\pi^{+}\pi^{-}$ at 2637 MeV.
In the inspected mass range no signal was observed and according to the
expected mass and width the following limit at ($95 \%$ confidence level) was extracted:
$f(c\rightarrow D^{*'+})\cdot\mathcal{B}_{D^{*'+}\rightarrow D^{*+}\pi^{+}\pi^{-}}<0.4 \% $.
\section{D*p RESONANCE}
\begin{wrapfigure}[21]{r}{0.4\textwidth}
\vspace{-30pt}
\begin{center}
\includegraphics[bb = 0 0 550 390, clip=true,width=0.38\textwidth]{Fig05.eps}
\includegraphics[bb = 0 0 550 390, clip=true,width=0.38\textwidth]{Fig06.eps}
\end{center}
\vspace{-20pt}
\caption{Distribution of $M(D^{*}p$ in the HERA\,I (bottom) and
HERA\,II (top) data samples.}
\label{pent}
\end{wrapfigure}
The H1 collaboration observed a narrow signal in the $D^{*}p$ channel at $M(D^{*}p)= 3099\pm 3(\text{stat.})\pm 5(\text{syst.})$
MeV in the HERA\,I dataset~\cite{h1pent1:08}.
This signal was interpreted as an anti-charm baryon with a minimal constituent
quark composition of $uudd\bar{c}$ with a relative contribution of
$\frac{N(D^{*}p)}{N(D^{*})}\sim 1 \%$ in the identified $D^{*}$ meson sample.
In contrast to this, no evidence was found by the ZEUS~\cite{zeuspent:08} collaboration.
Using the full HERA dataset H1 does not confirm the observation.
The data of 2004 to 2007 correspond to an integrated luminosity of
$\unit{348}{\,\text{pb}^{-1}}$ and thus increase the available statistics by a
factor of $\sim 4$.
In contrast to the previous analysis of the $D^{*}p$ final state no dE/dx
requirements were used for particle identification. A
detailed description of the event selection can be found in~\cite{h1pent2:08}.
The mass spectrum was reconstructed both for the HERA\,II dataset as well as for
the HERA\,I dataset using the new selection criteria.
Figure~\ref{pent} shows the reconstructed invariant mass for the $D^{*}p$
candidates separately for the two datasets. In the reanalysed HERA\,I dataset the previously
reported signal is reproduced. Under the assumption of the width from the
HERA\,I measurement there is no excess in the HERA\,II data visible.
The upper limit at a 95 \% confidence limit was
calculated to be $\frac{N(D^{*}p)}{N(D^{*})}\sim 0.1 \%$.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 274 |
<!DOCTYPE html>
<html>
<head>
<meta charset='utf-8'>
<meta http-equiv="X-UA-Compatible" content="chrome=1">
<link rel="shortcut icon" href="https://raw.githubusercontent.com/jordanwesthoff/jordanwesthoff.github.io/master/cerberus/images/cerberus_favicon.ico">
<link rel="stylesheet" type="text/css" href="http://jordanwesthoff.github.io/cerberus/stylesheets/stylesheet.css" media="screen">
<link rel="stylesheet" type="text/css" href="http://jordanwesthoff.github.io/cerberus/stylesheets/pygment_trac.css" media="screen">
<link rel="stylesheet" type="text/css" href="http://jordanwesthoff.github.io/cerberus/stylesheets/print.css" media="print">
<title>Cerberus Cluster Statistics</title>
</head>
<body>
<header>
<div class="container">
<h1>Spring Progress Timeline</h1>
<h2>(Still) Hosted via rSync of GitHub Repo (Dynamic Stats)</h2>
<section id="downloads">
<a href="https://github.com/jordanwesthoff/senior_thesis" class="btn btn-github"><span class="icon"></span>Go To Repo</a>
<a href="http://intruder.student.rit.edu/cerberus" class="btn btn-cerberus "><span class="icon"></span>Go To Home</a>
<a href="http://intruder.student.rit.edu/cerberus/pages" class="btn btn-cerberus "><span class="icon"></span>Pages Index</a>
</section>
</div>
</header>
<div class="container">
<section id="main_content">
<h3>
<a id="cluster-stats" class="anchor" href="#cluster-stats" aria-hidden="true"><span class="octicon octicon-link"></span></a>About Jordan Westhoff</h3>
<p>
<hr />
<img alt="portrait" src="https://raw.githubusercontent.com/jordanwesthoff/jordanwesthoff.github.io/master/cerberus/images/portrait.jpg" />
Hey there,
<br>
My name is Jordan Westhoff and I am an enthusiastic worker who places value on efficiency,
professional delivery and integrity. I have a diverse knowledge of digital video, networking and
computational programming with an express interest in how they can collide to develop advances in networking,
cinema and imaging science technology.
<br>
<br>
My work at the Rochester Institute of Technology encompasses quite a diverse range of subjects,
from film and digital film productions to radiometry and IT work that involves High Performance Computing and
4K video technologies. In addition to all of this, I am working on my own Senior Thesis project, <a href="http://intruder.student.rit.edu/cerberus">the Cerberus cluster</a>,
in order to complete my degree and graduate this spring. I have posted several interesting links below that are relevant,
go ahead and give them a look!
</p>
<br>
<br>
<br>
<h3>
<a id="author" class="anchor" href="#author" aria-hidden="true"><span class="octicon octicon-link"></span></a>GitHub</h3>
<p>Project and code by <a href="https://github.com/jordanwesthoff" class="user-mention">@jordanwesthoff</a>. Click on the profile for my GitHub page! </p>
<pre><code>"You are not like Cerberus, three gentlemen at once, are you?"
- Richard Brinsley Sheridan: Act IV, sc. ii.
</pre></code>
</section>
</div>
</body>
</html>
| {
"redpajama_set_name": "RedPajamaGithub"
} | 9,299 |
Database Log Plugin for CakePHP
===============================
[](LICENSE.txt)
[](https://travis-ci.org/burzum/cakephp-database-log)
[](https://coveralls.io/r/burzum/cakephp-database-log)
Requirements
------------
* PHP 5.4+
* CakePHP 3.0
Documentation
-------------
See the [installation guide](docs/Installation.md) for setting the plugin up.
But please check [the official documentation](http://book.cakephp.org/3.0/en/core-libraries/logging.html) on how to use logging adapters in detail.
Support
-------
For bugs and feature requests, please use the [issues](https://github.com/burzum/cakephp-database-log/issues) section of this repository.
Contributing
------------
To contribute to this plugin please follow a few basic rules.
* Pull requests must be send to the ```develop``` branch except hotfixes.
* Contributions must follow the PSR2 coding standard.
* [Unit tests](http://book.cakephp.org/3.0/en/development/testing.html) are required.
License
-------
Copyright 2012 - 2015, Florian Krämer
Licensed under The MIT License
Redistributions of files must retain the above copyright notice.
| {
"redpajama_set_name": "RedPajamaGithub"
} | 4,528 |
Demetrio "Mino" Carta (Genoa, 6 September 1933) is an Italian-born Brazilian journalist, publisher and writer. Carta helped in the creation of Veja, Istoé and CartaCapital, three of the four leading newsmagazines currently published in Brazil.
Biography
Carta arrived in São Paulo, São Paulo with his family after the War in 1946, when he "still wore short pants". He was probably 12 or 13 years old at that time. He recalls São Paulo as a "quiet and orderly" town with "human measures".
In 1951, Carta did a vestibular exam and was admitted at the University of São Paulo's traditional Law School of Largo São Francisco. His enrollment records state that he was born on September 6, 1933. He attended the classes of the first years, but quit and ended up never graduating from higher education.
In 1960 he started his career in journalism by helping to found Editora Abril automobile magazine Quatro Rodas. In 1966, he introduced new journalism in Brazil by founding São Paulo-based newspaper Jornal da Tarde. Two years later, he helped Victor Civita of Abril to found Veja, which currently is the leading newsmagazine in the country, with a circulation of over a million copies per edition. Unsatisfied with the result, he helped in the foundation of Istoé in 1976. Yet not completely satisfied with the result, he founded CartaCapital in 1994. On the new magazine, he and other columnists emphatically criticize neoliberal and neocon politics that are recently defended by Veja.
Of all the publications Carta helped create during his life, only one, the defunct Jornal da República, failed to succeed. The 1970s newspaper had a large deficit on its budget.
Disappointed with the position of Luiz Inácio Lula da Silva in the Cesare Battisti case, Carta decided to retire from his blog and his column on CartaCapital.
Published books
In 2000, Carta released his first novel, titled O Castelo de Âmbar (The Castle of Amber), in which he narrates a semi-biographical story. The main character, Mercúcio Parla, may be his alter-ego; Parla narrates what he considers to be the promiscuous relationship between politicians, journalists and media thanes during almost half a century in the recent History of Brazil. Written as a fictional story, some connections may be done with the reality, such as the character's home land "Ausônia" being Italy and his house on "Rua Áurea" being Rua Augusta.
In 2003, Carta published his second novel A Sombra do Silêncio (The Shadow of Silence), a follow-up to O Castelo de Âmbar. In this book, the main character finds himself on the Rua Áurea with Cuore Mio, "the most laughing girl in the neighbourhood", starting what is described by the author as the "only and authentic love of their lives".
Awards
2003: Comunique-se Award for Best Businessperson of a News Vehicle – CartaCapital
2005: Comunique-se Award for Best Businessperson of a News Vehicle – CartaCapital
2006: Award of the Association of Foreign Press Correspondents in Brazil for Most Prestigious Journalist of the Year
2007: Comunique-se Award for Best Businessperson of a News Vehicle – CartaCapital
Notes
References
External links
Mino's blog
CartaCapital official website
Year of birth missing (living people)
Living people
Brazilian journalists
Brazilian columnists
Italian emigrants to Brazil
University of São Paulo alumni
Brazilian male writers
Brazilian newspaper founders
Brazilian magazine founders | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 4,472 |
/* global QUnit */
sap.ui.define([
"sap/base/util/LoaderExtensions",
"jquery.sap.dom",
"sap/ui/qunit/QUnitUtils" // implicit dependency, implements jQuery#_sapTest_dataEvents
], function (LoaderExtensions, jQuery /*, qutils */) {
"use strict";
return LoaderExtensions.loadResource("static/jquery.sap.dom.html", {
dataType: "html",
async: true
}).then(function (sHTML) {
document.body.innerHTML += sHTML;
}).then(function() {
QUnit.test("domById", function (assert) {
assert.ok(jQuery.sap.domById('control1'), "jQuery.sap.domById('control1') may not be null");
assert.equal(jQuery.sap.domById('contro10'), null, "jQuery.sap.domById('control10') should be null");
});
QUnit.test("byId - escaping", function (assert) {
var id = ".A:B::C..D";
var element = document.createElement("div");
element.setAttribute("id", id);
document.body.appendChild(element);
var $element = jQuery.sap.byId(id);
assert.ok($element instanceof jQuery, "Is jQuery object");
assert.equal($element.get(0), element, "Element found");
assert.equal($element.attr("id"), id, "Element id found");
document.body.removeChild(element);
});
QUnit.test("byId - context", function (assert) {
var id = ".A:B::C..D";
var element = document.createElement("div");
element.setAttribute("id", id);
var parentElement = document.createElement("div");
parentElement.appendChild(element);
document.body.appendChild(parentElement);
var wrongParentElement = document.createElement("div");
document.body.appendChild(wrongParentElement);
var $element1 = jQuery.sap.byId(id, parentElement);
assert.ok($element1 instanceof jQuery, "Is jQuery object");
assert.equal($element1.get(0), element, "Element found");
assert.equal($element1.attr("id"), id, "Element id found");
var $element2 = jQuery.sap.byId(id, wrongParentElement);
assert.ok($element2 instanceof jQuery, "Is jQuery object");
assert.equal($element2.get(0), null, "No element found");
document.body.removeChild(parentElement);
document.body.removeChild(wrongParentElement);
});
QUnit.test("byId - no input", function (assert) {
var $element1 = jQuery.sap.byId();
assert.ok($element1 instanceof jQuery, "Is jQuery object");
assert.equal($element1.get(0), null, "No element found");
var $element2 = jQuery.sap.byId("");
assert.ok($element2 instanceof jQuery, "Is jQuery object");
assert.equal($element2.get(0), null, "No element found");
var $element3 = jQuery.sap.byId(null);
assert.ok($element3 instanceof jQuery, "Is jQuery object");
assert.equal($element3.get(0), null, "No element found");
});
QUnit.test("byId - invalid input (special characters)", function (assert) {
// The following chars are special characters but won't be escaped
// (see comment in jQuery.sap.byId)
assert.throws(function () {
jQuery.sap.byId("[foo]");
},
/Syntax error, unrecognized expression/,
"Should thrown an syntax error");
assert.throws(function () {
jQuery.sap.byId("a!b(c)");
},
/Syntax error, unrecognized expression/,
"Should thrown an syntax error");
});
QUnit.test("focus", function (assert) {
assert.expect(1);
var invisibleDiv = jQuery.sap.domById("invisibleDiv");
jQuery.sap.focus(invisibleDiv);
assert.ok(true, "jQuery.sap.focus() may not have caused an error for invisible elements"); // would never be reached in case of error
});
QUnit.test("it should convert px values to rem", function (assert) {
assert.strictEqual(jQuery.sap.pxToRem("0px"), 0);
assert.strictEqual(jQuery.sap.pxToRem("0"), 0);
assert.strictEqual(jQuery.sap.pxToRem(0), 0);
assert.strictEqual(jQuery.sap.pxToRem("16px"), 1);
assert.strictEqual(jQuery.sap.pxToRem("32"), 2);
assert.strictEqual(jQuery.sap.pxToRem(64), 4);
});
QUnit.test("it should convert rem values to px", function (assert) {
assert.strictEqual(jQuery.sap.remToPx("0rem"), 0);
assert.strictEqual(jQuery.sap.remToPx("0"), 0);
assert.strictEqual(jQuery.sap.remToPx(0), 0);
assert.strictEqual(jQuery.sap.remToPx("1rem"), 16);
assert.strictEqual(jQuery.sap.remToPx("2"), 32);
assert.strictEqual(jQuery.sap.remToPx(4), 64);
});
QUnit.test("SetGetCursorPos", function (assert) {
jQuery('#textinput').cursorPos(4);
assert.equal(jQuery('#textinput').cursorPos(), 4, "wrong cursor position after setting and getting");
});
QUnit.test("SelectText", function (assert) {
jQuery('#testsel').selectText(2, 5);
var start = jQuery('#testsel').get(0).selectionStart;
var end = jQuery('#testsel').get(0).selectionEnd;
assert.equal(start, 2);
assert.equal(end, 5);
jQuery('#testsel').selectText(-2, 50);
var start = jQuery('#testsel').get(0).selectionStart;
var end = jQuery('#testsel').get(0).selectionEnd;
assert.equal(start, 0);
assert.equal(end, jQuery('#testsel').get(0).value.length);
jQuery('#testsel').selectText(2, null);
var start = jQuery('#testsel').get(0).selectionStart;
var end = jQuery('#testsel').get(0).selectionEnd;
assert.equal(start, 0);
assert.equal(end, 0);
});
QUnit.test("selectText()/getSelectedText(): The input element's type 'number' does not support selection", function (assert) {
// arrange
var $DomRef = jQuery("<input>", {
type: "number",
value: 1000000
});
jQuery(document.body).append($DomRef);
// act
var oReturnValue = $DomRef.selectText(0, 6);
$DomRef.getSelectedText();
// assertions
assert.strictEqual(oReturnValue, $DomRef, "No exception is thrown");
// cleanup
$DomRef.remove();
});
QUnit.test("selectText() should return this to allow method chaining", function (assert) {
// arrange
var $Empty = jQuery();
// act
var $This = $Empty.selectText(0, 6);
// assert
assert.strictEqual($This, $Empty);
});
QUnit.test("getSelectedText() should return an empty string when the jQuery collection is empty", function (assert) {
// assert
assert.strictEqual(jQuery().getSelectedText(), "");
});
function fnGetSelectedTextCase(mSettings) {
QUnit.test("getSelectedText()", function (assert) {
// arrange
var $DomRef = mSettings.input;
jQuery(document.body).append($DomRef);
$DomRef.selectText(mSettings.selectionStart, mSettings.selectionEnd);
// assertions
assert.strictEqual($DomRef.getSelectedText(), mSettings.output);
// cleanup
$DomRef.remove();
});
}
fnGetSelectedTextCase({
input: jQuery("<input>", {
value: "Hello World!"
}),
output: "World",
selectionStart: 6,
selectionEnd: 11
});
fnGetSelectedTextCase({
input: jQuery("<input>", {
type: "url",
value: "www.sap.com"
}),
output: "www",
selectionStart: 0,
selectionEnd: 3
});
fnGetSelectedTextCase({
input: jQuery("<input>", {
type: "search",
value: "Hello World!"
}),
output: "World",
selectionStart: 6,
selectionEnd: 11
});
fnGetSelectedTextCase({
input: jQuery("<input>", {
type: "tel",
value: "7818523054"
}),
output: "852",
selectionStart: 3,
selectionEnd: 6
});
fnGetSelectedTextCase({
input: jQuery("<input>", {
type: "password",
value: "Hello World!"
}),
output: "World",
selectionStart: 6,
selectionEnd: 11
});
QUnit.test("OuterHTML", function (assert) {
function getExpected(bReversAttributeOrder) {
function att(n, v) {
return " " + n + "=\"" + v + "\"";
}
var sRes = "<DIV";
sRes += bReversAttributeOrder ? att("ID", "CONTROL3") : att("TABINDEX", "0");
sRes += bReversAttributeOrder ? att("TABINDEX", "0") : att("ID", "CONTROL3");
sRes += ">CONTROL 3</DIV>";
return sRes;
}
var sOuterHTML = jQuery("#control3").outerHTML().toUpperCase(); //Uppercase needed for cross browser consistency (Firefox returns lowercase tags)
assert.ok(sOuterHTML === getExpected(false) || sOuterHTML === getExpected(true), "outerHTML is wrong, Current: '" + sOuterHTML + "'");
});
QUnit.test("ContainsOrEquals", function (assert) {
var oDomRef3 = jQuery.sap.domById('control3');
var oDomRef1 = jQuery.sap.domById('control1');
assert.ok(jQuery.sap.containsOrEquals(oDomRef1, oDomRef3), "jQuery.sap.containsOrEquals(oDomRef1,oDomRef3) control3 is contained in control1");
assert.ok(jQuery.sap.containsOrEquals(document.body, oDomRef1), "jQuery.sap.containsOrEquals(document.body,oDomRef1) control1 is contained in body");
assert.ok(jQuery.sap.containsOrEquals(oDomRef3, oDomRef3), "jQuery.sap.containsOrEquals(oDomRef3,oDomRef3) control3 is contained in control3");
assert.ok(jQuery.sap.containsOrEquals(document.body, document.body), "jQuery.sap.containsOrEquals(document.body,document.body) body is contained in body");
// text nodes are no longer supported! ok(jQuery.sap.containsOrEquals(oDomRef3,oDomRef3.firstChild),"jQuery.sap.containsOrEquals(control3,control3.firstChild) control3s text node is contained in control3");
assert.ok(!jQuery.sap.containsOrEquals(oDomRef3, document.body), "jQuery.sap.containsOrEquals(control3,document.body) body is not contained in control3");
assert.ok(!jQuery.sap.containsOrEquals(oDomRef3, oDomRef1), "jQuery.sap.containsOrEquals(control3,control1) control1 is not contianed in control3");
});
QUnit.test("Rect", function (assert) {
var oRect = jQuery('#PositionedSpan').rect();
assert.equal(oRect.left, 10, "jQuery('#PositionedSpan').rect() left is 10");
assert.equal(oRect.top, 10, "jQuery('#PositionedSpan').rect() top is 10");
assert.equal(oRect.width, 200, "jQuery('#PositionedSpan').rect() width is 200");
assert.equal(oRect.height, 100, "jQuery('#PositionedSpan').rect() height is 100");
});
QUnit.test("HasTabIndex", function (assert) {
assert.ok(!jQuery("#control2").hasTabIndex(), "control2 does actually not have a tabindex");
assert.ok(jQuery("#control3").hasTabIndex(), "control3 does actually have a tabindex");
});
QUnit.module("focusable dom refs");
QUnit.test("firstFocusableDomRef, simple container", function (assert) {
var oFocusableItem = jQuery('#control1').firstFocusableDomRef();
var oDomRef = jQuery.sap.domById('control3');
assert.ok(oFocusableItem, "jQuery(oDomRef).firstFocusableDomRef() not found");
assert.equal(oFocusableItem, oDomRef, "jQuery(oDomRef).firstFocusableDomRef() is not control3");
});
QUnit.test("lastFocusableDomRef, simple container", function (assert) {
var oFocusableItem = jQuery('#control1').lastFocusableDomRef();
var oDomRef = jQuery.sap.domById('control6');
assert.ok(oFocusableItem, "jQuery(oDomRef).firstFocusableDomRef() not found");
assert.equal(oDomRef, oFocusableItem, "jQuery(oDomRef).firstFocusableDomRef() is not control6");
});
QUnit.test("firstFocusableDomRef, complex container", function (assert) {
var oFocusableItem = jQuery('#container1').firstFocusableDomRef();
assert.ok(oFocusableItem, "a focusable item should be found in container1");
assert.equal(oFocusableItem, jQuery.sap.domById('container1-item2-button'), "it should be the button in item2");
});
QUnit.test("lastFocusableDomRef, complex container", function (assert) {
var oFocusableItem = jQuery('#container1').lastFocusableDomRef();
assert.ok(oFocusableItem, "a focusable item should be found in container1");
assert.equal(oFocusableItem, jQuery.sap.domById('container1-item2-button'), "it should be the button in item2");
});
QUnit.test("firstFocusableDomRef, hidden ancestor", function (assert) {
var oFocusableItem = jQuery('#container2').firstFocusableDomRef();
assert.ok(!oFocusableItem, "no focusable item should be found in the hidden container2");
});
QUnit.test("lastFocusableDomRef, simple container", function (assert) {
var oFocusableItem = jQuery('#container2').lastFocusableDomRef();
assert.ok(!oFocusableItem, "no focusable item should be found in the hidden container2");
});
QUnit.module("Others");
QUnit.test("ParentByAttribute", function (assert) {
var oDomRef = jQuery.sap.domById('control2');
var oParentWithAttribute = jQuery(oDomRef).parentByAttribute("data-sap-ui-test");
assert.notEqual(oParentWithAttribute, null, "jQuery(oDomRef).parentByAttribute('data-sap-ui-test') is null)");
assert.equal(oParentWithAttribute.getAttribute('data-sap-ui-test'), "true", "jQuery(oDomRef).parentByAttribute('data-sap-ui-test').getAttribute('data-sap-ui-test') should be 'true'");
var oParentWithAttributeFalse = jQuery(oDomRef).parentByAttribute("data-sap-ui-test", "false");
assert.ok(!oParentWithAttributeFalse, "jQuery(oDomRef).parentByAttribute('data-sap-ui-test', 'false') should be undefined");
var oParentWithAttributeTrue = jQuery(oDomRef).parentByAttribute("data-sap-ui-test", "true");
assert.ok(oParentWithAttributeTrue, "jQuery(oDomRef).parentByAttribute('data-sap-ui-test', 'true') should be not null");
});
QUnit.test("OwnerWindow", function (assert) {
assert.expect(0);
// tests not easily possible, as there is no defined, stable environment with different windows
});
QUnit.test("Focus", function (assert) { // actually tests jQuery itself
var oDomRef = jQuery.sap.domById('control3');
jQuery(oDomRef).trigger("focus");
assert.equal(document.activeElement, oDomRef, "jQuery().trigger(\"focus\") failed");
});
QUnit.test("ScrollbarSize", function (assert) {
var size = jQuery.sap.scrollbarSize(true);
assert.ok(size, "No size for scroll bar returned");
assert.ok(typeof size.width === "number", "No width for scroll bar returned");
assert.ok(typeof size.height === "number", "No height for scroll bar returned");
var size = jQuery.sap.scrollbarSize(null, true);
assert.ok(size, "No size for scroll bar returned");
assert.ok(typeof size.width === "number", "No width for scroll bar returned");
assert.ok(typeof size.height === "number", "No height for scroll bar returned");
var size = jQuery.sap.scrollbarSize("someclass", true);
assert.ok(size, "No size for scroll bar returned");
assert.ok(typeof size.width === "number", "No width for scroll bar returned");
assert.ok(typeof size.height === "number", "No height for scroll bar returned");
var size = jQuery.sap.scrollbarSize("someclass");
assert.ok(size, "No size for scroll bar returned");
assert.ok(typeof size.width === "number", "No width for scroll bar returned");
assert.ok(typeof size.height === "number", "No height for scroll bar returned");
});
QUnit.test("disableSelection", function (assert) {
// arrange
var oDomRef = document.createElement("div");
oDomRef.innerHTML = "text";
document.body.appendChild(oDomRef);
// act
jQuery(oDomRef).disableSelection();
// arrange
var sEvent = "onselectstart" in document.createElement("div") ? "selectstart" : "mousedown";
var oSelectstarListener = jQuery._data(oDomRef, "events")[sEvent];
// assertions
assert.strictEqual(oSelectstarListener.length, 1);
// cleanup
document.body.removeChild(oDomRef);
});
QUnit.test("enableSelection", function (assert) {
// arrange
var oDomRef = document.createElement("div");
oDomRef.innerHTML = "text";
document.body.appendChild(oDomRef);
// act
jQuery(oDomRef).disableSelection();
jQuery(oDomRef).enableSelection();
// assertions
assert.strictEqual(jQuery._data(oDomRef, "events"), undefined);
// cleanup
document.body.removeChild(oDomRef);
});
QUnit.test("jQuery aria extensions", function (assert) {
var $El = jQuery("<div>Aria Test</div>");
jQuery(document.body).append($El);
$El.addAriaLabelledBy("test1");
assert.strictEqual($El.attr("aria-labelledby"), "test1", "aria-labelledby attribute is added to the DOM");
$El.addAriaDescribedBy("test1");
assert.strictEqual($El.attr("aria-describedby"), "test1", "aria-describedby attribute is added to the DOM");
$El.addAriaLabelledBy("test1");
assert.strictEqual($El.attr("aria-labelledby"), "test1", "already existing aria-labelledby is ignored");
$El.addAriaDescribedBy("test1");
assert.strictEqual($El.attr("aria-describedby"), "test1", "already existing aria-describedby is ignored");
$El.addAriaLabelledBy("test");
assert.strictEqual($El.attr("aria-labelledby"), "test1 test", "new aria-labelledby attribute is added with space concating");
$El.addAriaDescribedBy("test");
assert.strictEqual($El.attr("aria-describedby"), "test1 test", "new aria-describedby attribute is added with space concating");
$El.removeAriaLabelledBy("t");
assert.strictEqual($El.attr("aria-labelledby"), "test1 test", "non-existing aria-labelledby is ignored");
$El.removeAriaDescribedBy("t");
assert.strictEqual($El.attr("aria-describedby"), "test1 test", "non-existing aria-describedby is ignored");
$El.removeAriaLabelledBy("test1");
assert.strictEqual($El.attr("aria-labelledby"), "test", "existing aria-labelledby is removed from DOM");
$El.removeAriaDescribedBy("test1");
assert.strictEqual($El.attr("aria-describedby"), "test", "existing aria-describedby is removed from DOM");
$El.removeAriaLabelledBy("test");
assert.strictEqual($El.attr("aria-labelledby"), undefined, "last existing aria-labelledby is removed the attribute");
$El.removeAriaDescribedBy("test");
assert.strictEqual($El.attr("aria-describedby"), undefined, "last existing aria-describedby is removed the attribute");
$El.removeAriaLabelledBy("test");
assert.strictEqual($El.attr("aria-labelledby"), undefined, "non-existing aria-labelledby is ignored");
$El.removeAriaDescribedBy("test");
assert.strictEqual($El.attr("aria-describedby"), undefined, "non-existing aria-describedby is ignored");
$El.remove();
});
QUnit.module("Selector", {
beforeEach: function () {
this.$Image = jQuery('' +
'<img id="image" data-sap-ui="image" src="jquery/SAPLogo.gif" class="sapUiImg" usemap="#Map1" alt="Alternative image text for Image2" tabindex="0" style="width:300px;height:200px;">'
);
this.$ImageMap = jQuery('' +
'<map tabindex="-1" id="imgMap1" data-sap-ui="imgMap1" name="Map1">' +
'<area id="area11" data-sap-ui="area11" style="display: inline;" aria-describedby="imgMap1-Descr" shape="rect" coords="1,1,100,100" href="#" alt="Text on Alt1" tabindex="-1">' +
'<area id="area12" data-sap-ui="area12" style="display: inline;" aria-describedby="imgMap1-Descr" shape="rect" coords="101,1,200,100" href="#" alt="Text on Alt2" tabindex="-1">' +
'<area id="area13" data-sap-ui="area13" style="display: inline;" aria-describedby="imgMap1-Descr" shape="rect" coords="201,1,300,100" href="#" alt="Text on Alt3" tabindex="-1">' +
'<area id="area14" data-sap-ui="area14" style="display: inline;" aria-describedby="imgMap1-Descr" shape="rect" coords="1,101,100,200" href="#" alt="Text on Alt4" tabindex="0">' +
'<area id="area15" data-sap-ui="area15" style="display: inline;" aria-describedby="imgMap1-Descr" shape="rect" coords="101,101,200,200" href="#" alt="Text on Alt5" tabindex="-1"></map>' +
'</map>'
);
// let the reference have parent to replace
jQuery(document.body).append(this.$Image);
jQuery(document.body).append(this.$ImageMap);
sap.ui.getCore().applyChanges();
},
afterEach: function () {
this.$Image.remove();
this.$ImageMap.remove();
}
});
QUnit.test("Using Quotes for selector ':sapFocusable' on ImageMap", function (assert) {
var $Area11 = jQuery("#area11");
// this caused an issue while introducing jQuery 2.2.2 in the ImageMap
// visual test page
var bFocusable = $Area11.is(":sapFocusable");
assert.ok(bFocusable, "Map is :sapFocusable");
});
QUnit.module("rtl vertical scrolling", {
beforeEach: function () {
this.container = document.createElement("div");
this.$container = jQuery(this.container);
this.container.style = "direction:rtl;overflow-x:scroll;overflow-y:hidden;width:100px;height:20px";
this.element = document.createElement("div");
this.element.style = "width:200px;height:10px;background:linear-gradient(to right,red,yellow);";
this.container.appendChild(this.element);
document.body.appendChild(this.container);
},
afterEach: function () {
document.body.removeChild(this.container);
}
});
QUnit.test("scrollLeftRTL - get", function (assert) {
var iScrollLeft = this.$container.scrollLeftRTL();
assert.strictEqual(iScrollLeft, 100, "initial scroll position is set to 100");
});
QUnit.test("scrollLeftRTL - set", function (assert) {
this.$container.scrollLeftRTL(100);
assert.strictEqual(this.$container.scrollLeftRTL(), 100, "scroll left for 100px");
this.$container.scrollLeftRTL(0);
assert.strictEqual(this.$container.scrollLeftRTL(), 0, "scroll back for 100px");
this.$container.scrollLeftRTL(101);
assert.strictEqual(this.$container.scrollLeftRTL(), 100, "scroll left exceeding width");
});
QUnit.test("scrollRightRTL", function (assert) {
var iScrollRight = jQuery.fn.scrollRightRTL.call(this.$container);
assert.strictEqual(iScrollRight, 0, "initial scroll position is set to 0");
});
// validate the denormalization against the actual scrolling results, as they are browser dependent
QUnit.test("denormalizeScrollLeftRTL", function (assert) {
var iScrollLeft = jQuery.sap.denormalizeScrollLeftRTL(0, this.container);
this.$container.scrollLeftRTL(0);
assert.strictEqual(iScrollLeft, this.container.scrollLeft, "call with 0 - " + this.container.scrollLeft);
iScrollLeft = jQuery.sap.denormalizeScrollLeftRTL(100, this.container);
this.$container.scrollLeftRTL(100);
assert.strictEqual(iScrollLeft, this.container.scrollLeft, "call with 100 - " + this.container.scrollLeft);
// the following test is browser dependent
// iScrollLeft = jQuery.sap.denormalizeScrollLeftRTL(101, this.container);
// this.$container.scrollLeftRTL(101);
// assert.strictEqual(iScrollLeft, this.container.scrollLeft, "call with 101 exceeding width - " + this.container.scrollLeft);
iScrollLeft = jQuery.sap.denormalizeScrollLeftRTL(100);
assert.strictEqual(iScrollLeft, undefined, "call without domref - undefined");
});
QUnit.test("denormalizeScrollBeginRTL", function (assert) {
// the following 3 tests are browser dependent
// var iScrollBegin = jQuery.sap.denormalizeScrollBeginRTL(0, this.container);
// assert.strictEqual(iScrollBegin, 100, "call with 0 - 100");
// iScrollBegin = jQuery.sap.denormalizeScrollBeginRTL(100, this.container);
// assert.strictEqual(iScrollBegin, 0, "call with 100 - 0");
// iScrollBegin = jQuery.sap.denormalizeScrollBeginRTL(101, this.container);
// assert.strictEqual(iScrollBegin, -1, "call with 101 (exceeding width) - -1");
var iScrollBegin = jQuery.sap.denormalizeScrollBeginRTL(100);
assert.strictEqual(iScrollBegin, undefined, "call without domref - undefined");
});
});
});
| {
"redpajama_set_name": "RedPajamaGithub"
} | 5,914 |
\subsection{What Distinguishes Our Dataset From Existing Ones?}
Researchers in \cite{aernouts2018sigfox} presented three LPWAN (Sigfox and LoRAWAN) datasets collected in an outdoor environment aimed at evaluating location fingerprinting algorithms. These datasets have been collected over three months in both rural and urban areas.
Sigfox-rural and Sigfox-urban datasets consist respectively of $25$k+ and $14$k+ Sigfox messages, whereas LoRaWAN dataset consists of $123$k+ LoRaWAN messages. These messages include a couple of protocol information, time of reception, and GPS location information but do not include device labels, and therefore, can not be used for supervised deep learning-based device classification. The closest existing work to our work is the recently released work at Northeastern University~\cite{al2020exposing}, which collected and released a massive dataset of IEEE 802.11 a/g (WiFi) standard data obtained from 20 wireless devices with identical RF circuitry over several days in (a) an anechoic chamber, (b) in-the-wild testbed, and (c) with cable connections. The focus of the Northeastern dataset is to explore the impact of the wireless channel on the performance of deep learning-based RF fingerprinting models. The dataset is limited in terms of the covered scenarios, and it is for WiFi signals only. Unfortunately, to the best of our knowledge, there are still no public datasets for LoRa device fingerprinting nor datasets that include the diverse experimental scenarios we mentioned. Our dataset presented in this paper fulfills the need for a large-scale LoRa dataset covering a wide range of diverse scenarios for a more comprehensive evaluation and validation.
\begin{table*}
\begin{adjustbox}{width=2\columnwidth, height = 0.17\columnwidth, center}
\begin{tabular}{|l|c|c|c|c|c|c|c|c|c|}
\hline
\textbf{Setups} & {\color[HTML]{333333} \textbf{\begin{tabular}[c]{@{}c@{}}Number of\\ Transmitters\end{tabular}}} & { \textbf{\begin{tabular}[c]{@{}c@{}}Number of\\ Receivers\end{tabular}}} & \textbf{Protocol} & \textbf{\begin{tabular}[c]{@{}c@{}}Number \\ of Days\end{tabular}} & \textbf{\begin{tabular}[c]{@{}c@{}}Transmissions \\ per Device\end{tabular}} & \textbf{\begin{tabular}[c]{@{}c@{}}Duration per \\ Transmission\end{tabular}} & \textbf{Distances} & \textbf{Environment} & \textbf{Representation} \\ \hline
{1) Diff Days Indoor} & {\color[HTML]{333333} 25} & {\color[HTML]{333333} 1} & LoRa & 5 & 10 & 20s & 5m & Indoor & IQ/FFT \\ \hline
{2) Diff Days Outdoor} & 25 & 1 & LoRa & 5 & 10 & 20s & 5m & Outdoor & IQ/FFT \\ \hline
{3) Diff Days Wired} & 25 & 1 & LoRa & 5 & 10 & 20s & 5m & Wired & IQ/FFT \\ \hline
{4) Diff Distances} & 25 & 1 & LoRa & 1 & 4 & 20s & 5,10,15,20m & Outdoor & IQ/FFT \\ \hline
\multicolumn{1}{|l|}{{5) Diff Configurations}} & 25 & 1 & LoRa & 1 & 4 & 20s & 5m & Indoor & IQ/FFT \\ \hline
{6) Diff Locations} & 25 & 1 & LoRa & 1 & 3 & 20s & 5m & 2 Indoor, 1 Outdoor & IQ/FFT \\ \hline
{7) Diff Receivers} & 25 & 2 & LoRa & 1 & 2 & 20s & 5m & Indoor & IQ/FFT \\ \hline
\end{tabular}
\end{adjustbox}
\caption{Summary of Experimental Setups/Scenarios.}
\label{summary}
\end{table*}
\subsection{Our LoRa Dataset in Brief}
Our RF dataset provides both time-domain I/Q samples and corresponding FFT samples collected using an IoT testbed consisting of $25$ identical Pycom IoT devices and a USRP B$210$ receiver, operating at a center frequency of $915$MHz, used for recording the received signals sampled at $1$MS/s. Recorded data in the form of both the time-domain I/Q samples and FFT samples are stored into binary files in compliance with SigMF~\cite{hilburn2018sigmf} by creating, for each binary file, a metafile written in plain-text JSON to include recording information such as sampling rate, time and day of recording, and carrier frequency, among other parameters.
This dataset covers multiple experimental setup scenarios, which are summarized in Table \ref{summary} and can be downloaded at
\href{https://research.engr.oregonstate.edu/hamdaoui/datasets/}{\color{blue}{http://research.engr.oregonstate.edu/hamdaoui/datasets}}.
The experimental scenarios are briefly described next, with more details provided in~\cite{elmaghbub2021lora}.
\begin{itemize}
\item {\bf Setup 1---Different Days Indoor Scenario:} Indoor setup with data collected over $5$ consecutive days. For each day, $10$ transmissions were captured from each of the 25 transmitters, with each transmission having a duration of $20$s. All transmissions took place $1$ minute apart from one another.
\item {\bf Setup 2---Different Days Outdoor Scenario:} Outdoor setup with data collected over $5$ consecutive days. For each day, $10$ transmissions were captured from each of the $25$ transmitters, with each transmission having a duration of $20$s. All transmissions took place $1$ minute apart from one another.
\item {\bf Setup 3---Different Days Wired Scenario:} Wired setup with data collected over $5$ consecutive days. For each day, $10$ transmissions were captured from each of the $25$ transmitters, with each transmission having a duration of $20$s. All transmissions are $1$ minute apart from one another.
\item {\bf Setup 4---Different Distances Scenario:} Outdoor setup with data collected from $4$ different distances: $5$m, $10$m, $15$m, and $20$m away from the receiver. For each distance, one transmission was captured from each of the $25$ transmitters, with each transmission having a duration of $20$s. All transmissions are $1$ minute apart from one another.
\item {\bf Setup 5---Different Configurations Scenario:} Indoor setup with data collected from $4$ different LoRa configurations. For each configuration, one transmission was captured from each of the $25$ transmitters, with each transmission having a duration of $20$s. All transmissions are $1$ minute apart from one another.
\item {\bf Setup 6---Different Locations Scenario:} This data has been collected in $3$ different locations: room, outdoor, and office environments. At each location, one transmission was captured from each of the $25$ transmitters, with each transmission having a duration of $20$s. All transmissions are $1$ minute apart from one another.
\item {\bf Setup 7---Different Receivers Scenario:} Indoor setup with data collected using $2$ different receivers. For each receiver, one transmission was captured from each of the $25$ transmitters, with each having a duration of $20$s. All transmissions are $1$ minute apart from one another.
\end{itemize}
The rest of the paper is organized as follows. Section \ref{sec:testbed} describes the testbed components. Sections~\ref{experiment} and~\ref{dataset} describe the different experimental setups and the dataset, respectively. Section~\ref{use-case} presents a use case for the dataset. The challenges, limitations, and new opportunities are discussed in Section \ref{challenges} and the paper is concluded in Section \ref{Conclusion}.
\subsection{Hardware Description}
\begin{figure}
\centering
\includegraphics[width=1\columnwidth, height=0.57\columnwidth]{figures/dataset.pdf}
\caption{IoT Testbed: $25$ Pycom devices and USRP B$210$.}
\label{fig:dataset}
\end{figure}
\begin{figure*}
\centering
\includegraphics[width=1.7\columnwidth, height=0.5\columnwidth]{figures/testbed.pdf}
\caption{The flowgraph of our data collection.}
\label{fig:flow}
\end{figure*}
Our IoT testbed, shown in Fig.~\ref{fig:dataset}, consists of $25$ Pycom devices with Semtech SX$1276$ LoRa transceivers, $25$ Pycom sensor shields, and an Ettus USRP (Universal Software Radio Peripheral) B$210$ for data sampling, which was configured with a center frequency of $915$MHz and a sample rate of $1$MS/s. Our collection of Pycom devices is made up of $23$ Lopy4 boards and $2$ Fipy boards, which are MicroPython-enabled development boards with multiple network capabilities: LoRa, Sigfox, WiFi, Bluetooth, and NB-IoT. The sensor shield collection comprises $22$ PySense boards equipped with sensing capability, $2$ PyScan boards equipped with RFID scanning capability, and $1$ PyTrack board equipped with GPS/tracking capability. We used lipo batteries to power the devices. Each Pycom device was connected to a dedicated LoRa antenna and configured to transmit LoRa transmissions at the $915$MHz US band adapting the following configuration: Raw-LoRa mode, $125$KHz bandwidth, a spreading factor (SF) of $7$, a preamble of $8$, a TX power of $20$dBm, and a coding rate of $4/5$.
\subsection{Software Description}
\subsubsection{Transmission Subsystem}
We programmed and configured our Pycom boards using MicroPython \cite{tollervey2017programming}, which is an efficient implementation of Python3 that is composed of a subset of standard Python libraries and optimized to run on microcontrollers and constrained environments. Also, we used Pymakr plugin as a REPL console that connects to Pycom boards to run codes or upload files.
\subsubsection{Reception Subsystem}
We used the GNURadio software \cite{valerio2008open}, a real-time signal processing graphical tool, to set up and configure the USRP receiver to capture LoRa transmissions, plot their time and spectrum domains, implement some preprocessing techniques and store the samples into their files. Fig.~\ref{fig:flow} shows the general flowgraph used for data acquisition.
\subsection{LoRa Protocol Description}
We transmitted/captured LoRa modulation signals, a proprietary physical layer implementation that employs Chirp Spread Spectrum (CSS) in the sub-GHz ISM band and trades data rate for coverage range, power consumption, or link robustness. LoRa does so by providing a tunable parameter, called a spreading factor (SF), which varies from $7$ to $12$ and determines the sequence length of an encoded symbol within a fixed bandwidth. A higher spreading factor means longer ranges with lower data rates. Unlike other spread spectrum techniques, the chirp-based modulation allows LoRa to maintain the same coding gain and immunity to noise and interference while meeting the low-cost, low-power consumption requirements. A LoRa modulator generates both raw chirp signals with fixed amplitude and continuously varying frequency with constant rate and a set of modulated chirps that are cyclically time-shifted raw-chirps where the initial frequency determines the content of the chirp symbol.
\subsection{Setup 1: Different Days Indoor Scenario}
In order to enable performance evaluation while masking the impact of the outside environment, we created an indoor setup, ran experiments, and collected datasets for this setup. These indoor experiments were carried out in a typical occupied room environment over $5$ consecutive days. All devices transmitted the same message from the same location, $5$m away from the receiver so that all devices experience similar channel conditions. As shown in Fig.~\ref{fig:flow1}, for each day, each transmitter generated $10$ transmissions, each of $20$s duration, all spaced apart by $1$ minute. Hence, we collected about $200$M complex-valued samples from each device per day. We used GNURadio packages to store the sampled raw-I/Q values and their corresponding FFT-Based representation into binary files as depicted in Fig.~\ref{fig:flow1}.
\subsection{Setup 2: Different Days Outdoor Scenario}
In order to allow for performance evaluation while considering the impact of outdoor wireless channel impairments, we carried out the experiments in an outdoor environment at nighttime. Here again, all devices transmitted the same message from the exact location, situated $5$m away from the receiver, so that all devices experience similar channel conditions. Like in the indoor setup case and as shown in Fig.~\ref{fig:flow1}, for five consecutive days, each transmitter generated $10$ transmissions per day, each of $20$s duration, all spaced $1$ minute apart from one another. This resulted in about $200$M complex-valued samples per device per day. We ran this experiment over $5$ consecutive days and provided 5-day worth of data to study the robustness of deep learning models when trained on data collected on one day but tested on data captured on a different day. We used GNURadio packages to store the sampled raw-I/Q values and their corresponding FFT-based representation into binary files as depicted in Fig.~\ref{fig:flow1}.
\subsection{Setup 3: Different Days Wired Scenario} The wireless channel has a notable impact on the performance of deep learning models and presents its unique challenges \cite{al2020exposing}. Hence, to assess how well these models perform in the absence of the wireless channel's impact, we created a wired setup where the Pycom boards are directly connected to the USRP via an SMA cable and $30$dB attenuator. Similar to the Indoor and Outdoor experiments, we ran this experiment over $5$ consecutive days. For every day, each device transmitted $10$ bursts each of $20$s duration. Therefore, the total amount of collected data is $200$M complex-valued samples per device per day. We again used GNURadio packages to store the sampled raw-I/Q values and their corresponding FFT-based representation into binary files as depicted in Fig.~\ref{fig:flow1}.
\subsection{Setup 4: Different Distances Scenario}
Some end-devices constantly change their positions, so it is critical to explore the impact of distance on the performance of classifiers and see whether or not a classifier would still recognize a device when it moves to a position that is different from the one used for training. This experiment was carried out in a typical outdoor environment in a sunny day. We considered four different distances, $5$m, $10$m, $15$m, and $20$m, and for each distance, we collected $1$ transmission of $20$s for each of the $25$ devices. We kept the receiver at the same location for all the transmissions while locating the transmitters at $4$ different distances away from the receiver base ($5$m, $10$m, $15$m, and $20$m). The transmissions were captured consecutively in time with only $60$s apart from one another. Each transmitter generated $4$ transmissions each of $20$s duration, resulting in $80$M complex-valued samples from each device. We again used GNURadio packages to store the sampled raw-I/Q values and corresponding FFT-based samples into binary files as depicted in Fig.~\ref{fig:flow2}.
\begin{figure}
\centering
\includegraphics[width=1\columnwidth]{figures/scen_b2.pdf}
\caption{Setup diagram: setups 4, 5, 6, and 7.}
\label{fig:flow2}
\end{figure}
\subsection{Setup 5: Different Configurations Scenario}
LoRaWAN uses the Adaptive Data Rate (ADR) mechanism to optimize the data rates, air-time, and energy consumption in the network to accommodate the varying RF conditions. This mechanism allows the network server to inform end devices to adjust their power consumption and data rate as needed. This is achievable by controlling the following parameters at end devices: spreading factor, bandwidth, and power consumption. Changing the spreading factor, for example, in LoRa modulation results in a change in the data rate, receiver sensitivity, time in the air, and power consumption. Fig.~\ref{fig:spec4} shows the frequency spectrum of a snapshot of the four LoRa configurations that we included in our dataset. Ideally, a classification model (e.g., a deep learning fingerprinting model) should identify a device even if it changes its configuration; i.e., models that are trained using one configuration but tested on a different configuration should still perform well. Therefore, in order to enable the assessment of how agnostic these models are to protocol configuration, we captured transmissions using $4$ different configurations, as presented in Table~\ref{con}. For this, we collected a single LoRa transmission of $20$s from each device for each configuration in an indoor setup with $5$m as the distance between the receiver and transmitters. Like other setups, for this setup, we used GNURadio packages to store the sampled raw-I/Q values and their corresponding FFT-based representation into binary files as depicted in Fig.~\ref{fig:flow2}.
\begin{figure}
\centering
\includegraphics[width=1\columnwidth,height=0.53\columnwidth]{figures/con_spec.pdf}
\caption{Spectrum of the 4 LoRa configurations from Setup 5}
\label{fig:spec4}
\end{figure}
\subsection{Setup 6: Different Locations Scenario}
Another practical scenario we consider here aims at allowing deep learning models to be trained on data captured in one location but tested on data collected in another location. For this, we captured LoRa transmissions in three different location deployments, room environment, office environment, and outdoor environment, all taken on the same day. Here, we kept the distance between the receiver and transmitters (i.e., $5$m) and the LoRa configuration the same. We captured a single transmission of $20$s from each device at each location with a $60$s period between the devices, resulting in $60$M complex samples from each device. We again used GNURadio packages to store the sampled raw-I/Q values and their corresponding FFT-based representation into binary files as depicted in Fig.~\ref{fig:flow2}.
\begin{table}
\scriptsize
\begin{tabular}{|l|c|c|l|c|c|}
\hline
\textbf{Configurations} & \multicolumn{1}{l|}{{ \textbf{\begin{tabular}[c]{@{}l@{}}Spread\\ Factor\end{tabular}}}} & \multicolumn{1}{l|}{{ \textbf{Bandwidth}}} & \textbf{Bit Rate} & \multicolumn{1}{l|}{\textbf{\begin{tabular}[c]{@{}l@{}}Tx \\ Power\end{tabular}}} & \multicolumn{1}{l|}{\textbf{\begin{tabular}[c]{@{}l@{}}Coding\\ Rate\end{tabular}}} \\ \hline
Configuration 1 & { 7} & {\color[HTML]{333333} 125KHz} & 5470 bps & 20dBm & 4/5 \\ \hline
Configuration 2 & 8 & 125KHz & 3125 bps & 20dBm & 4/5 \\ \hline
Configuration 3 & 11 & 125KHz & 537 bps & 20dBm & 4/5 \\ \hline
Configuration 4 & 12 & 125KHz & 293 bps & 20dBm & 4/5 \\ \hline
\end{tabular}
\caption{LoRa Configurations}
\label{con}
\end{table}
\subsection{Setup 7: Different Receivers Scenario}
Like transmitters, receivers also suffer from hardware impairments due to hardware imperfection. Therefore, deep learning models trained using data collected by one receiver but tested using data collected by a different receiver may not perform well due to the possible additional impairments caused by the receiver's reception.
To allow researchers to study the impact of such a change in the receiving device, we provided a dataset for the 25 Pycom transmitting devices, collected using two different USRP B$210$ receivers. In this experiment, we considered an indoor setup where the transmitters (the 25 Pycom devices) were located $5$m away from the receiver. For each receiver, we captured a single transmission of $20$s from each of the 25 Pycom transmitters, yielding a total of $40$M samples for each device. Like other setups, for this setup, we used GNURadio packages to store the sampled raw-I/Q values and their corresponding FFT-based representation into binary files as depicted in Fig.~\ref{fig:flow2}.
\subsection{Dataset Overview}
\begin{figure*}[t!]
\centering
\includegraphics[height=0.83\columnwidth, width=2\columnwidth]{figures/dataset_organization.drawio.pdf}
\caption{Online link/file organization of the datasets. Note that Diff\_Days\_Indoor\_Setup, Diff\_Days\_Outdoor\_Setup, and Diff\_Days\_Wired\_Setup directories have the same file system structure.}
\label{fig:orgnization}
\end{figure*}
Our dataset contains LoRa transmissions of $25$ devices and a total of $43$ transmissions on average for each device. An Ettus USRP B$210$ is used to record each transmission, operating at a center frequency of $915$MHz with a sampling rate of $1$MS/s. Each recording consists, on average, of $20$M I/Q samples. For each recording, we stored the time-domain I/Q samples and FFT-based samples into binary files. The binary files are encoded with Float$32$, and the complex-valued sampled are interleaved where the I-values are in the odd indices, and the Q-values are in the even ones. In order to be SigMF (Signal Metadata Format)~\cite{hilburn2018sigmf} compliant, we created a metadata file written in plain-text JSON adapting SigMF for each binary file to describe the essential information about the collected samples, the system that generated them, and the features of the signal itself. In our case, we stored in the metadata files information regarding $(i)$ the sampling rate, $(ii)$ time and day of recording, and $(iii)$ the carrier frequency, among others.
More details and use cases of these LoRa datasets can be found in~\cite{elmaghbub2021lora}. The datasets can be downloaded from NetSTAR Lab at
\href{https://research.engr.oregonstate.edu/hamdaoui/datasets/}{\color{blue}{http://research.engr.oregonstate.edu/hamdaoui/datasets}}.
Users of the datasets may refer to Fig.~\ref{fig:orgnization} for help with the file organization and notation. Specifically, these are:
\begin{itemize}
\item {\bf Diff\_Days\_Indoor\_Setup}, {\bf Diff\_Days\_Outdoor\_Setup}, and {\bf Diff\_Days\_Wired\_Setup} directories (correspond to Setups 1, 2 and 3), each having 5 subdirectories, one for each day. Each day subdirectory has $25$ subdirectories, one for each device. Each device subdirectory has $40$ files (for $10$ transmissions): $10$ I/Q data files plus their corresponding $10$ metadata files and $10$ fft data files plus their corresponding 10 metadata files.
\item {\bf Diff\_Distances\_Setup} directory (corresponds to Setup 4), having $4$ subdirectories representing the four distances. Each subdirectory includes 100 files: $25$ I/Q data files plus their corresponding $25$ metadata files and $25$ fft data files plus their corresponding $25$ metadata files.
\item {\bf Diff\_Configurations\_Setup} directory (corresponds to Setup 5), having $4$ subdirectories representing the four configurations.
Each of these 4 subdirectories includes 100 files: $25$ I/Q data files plus their corresponding $25$ metadata files and $25$ fft data files plus their corresponding $25$ metadata files.
\item {\bf Diff\_Locations\_Setup} directory (corresponds to Setup 6), having $3$ subdirectories representing the three locations.
Each subdirectory includes 100 files: $25$ I/Q data files plus their corresponding $25$ metadata files and $25$ fft data files plus their corresponding $25$ metadata files.
\item {\bf Diff\_Receivers\_Setup} directory (corresponds to Setup 7), having $2$ subdirectories representing the two receivers.
Each subdirectory includes 100 files: $25$ I/Q data files plus their corresponding $25$ metadata files and $25$ fft data files plus their corresponding $25$ metadata files.
\end{itemize}
\begin{figure}[h!]
\centering
\includegraphics[width=1\columnwidth, height=0.75\columnwidth]{figures/Data_Representation.pdf}
\caption{Top: I/Q time-domain representation, Middle: Frequency spectrum representation, Bottom: Spectrogram representation from Device 1, Indoor Setup.}
\label{fig:vis}
\end{figure}
To visualize the captured LoRa signals, we \hl{plotted}, in Fig.~\ref{fig:vis}, the time-domain, frequency spectrum, and spectrogram of a LoRa transmission captured in an Indoor environment. The top figure in Fig.~\ref{fig:vis} represents the in-phase (I) and quadrature (Q) components of the time-domain signal from device 1 in an indoor setup, while the figure in the middle shows its frequency spectrum and the figure in the bottom shows the upward chirps in the spectrogram of the same LoRa signal.
\subsection{Challenges}
\begin{itemize}
\item {\bf Same channel condition.} The time between a given device's first and last transmissions may not be short enough for all transmissions to be assumed to be taken under the same condition. This makes it challenging to study the performance of a learning model with training and testing being done under the same channel condition. This is because all 25 devices must first be sampled for their first transmissions before moving to the subsequent transmission. Even if one tries to collect all 10 transmissions sequentially for each device to minimize this timing effect between transmissions, we run into the problem of increasing the experiment time between devices.
\item {\bf Same power levels.} Maintaining the same power level for all devices is important so that to mask the power impact. In our experiments, to mitigate this issue, we start with a full-charged battery every time, though this still cannot guarantee that all have the same power level.
\item {\bf Concurrent transmissions.} One interesting scenario but difficult to realize is the ability to collect data from multiple devices while all transmitting concurrently, as it often occurs in random access procedures for wireless base stations. Excelling on this scenario would open the door for incorporating deep learning-based fingerprinting into next-generation cellular networks.
\item {\bf Devices at scale.} Increasing the number of transmitters in our testbed is another to-do-item that would add more credibility to the evaluation and allow us to assess the scalability performance of the proposed models.
\item {\bf Beyond RF fingerprinting.} Leveraging the multi-bearer capability of our testbed, one can use the testbed to collect datasets for modulation recognition using other technologies like LoRaWAN, SigFox, Bluetooth, WiFi, and NB-IoT technologies. Another use is for creating datasets for studying indoor device positioning problems.
\end{itemize}
\section{Introduction}
\input{2-Introduction}
\label{intro}
\section{Testbed}
\label{sec:testbed}
\input{3-testbed}
\section{Experimental Setups}
\label{experiment}
\input{4-experiment}
\section{Dataset Description}
\label{dataset}
\input{5-dataset}
\section{Use Case: LoRa Device Fingerprinting}
\label{use-case}
\input{use-case}
\section{Challenges, Limitations and Opportunities}
\label{challenges}
\input{7-challenges}
\section{Conclusion}
\label{Conclusion}
\input{9-Conclusion}
\section{Acknowledgment}
\label{ack}
\input{8-acknow}
\vspace{12pt}
\color{black}
\bibliographystyle{IEEEtran}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 8,216 |
{"url":"https:\/\/www.chemeurope.com\/en\/encyclopedia\/Airy_disc.html","text":"My watch list\nmy.chemeurope.com\n\n# Airy disc\n\nThe Airy disc (or Airy disk) is a phenomenon in optics. Owing to the wave nature of light, light passing through an aperture is diffracted and forms a pattern of light and dark regions on a screen some distance away from the aperture (see interference).\n\nThe diffraction pattern resulting from a uniformly illuminated circular aperture has a bright region in the center, known as the Airy disc which together with a series of concentric rings is called the Airy pattern (both named after George Airy). The diameter of this disc is related to the wavelength of the illuminating light and the size of the circular aperture.\n\nThe most important application of this concept is in cameras and telescopes. Due to diffraction, the smallest point to which one can focus a beam of light using a lens is the size of the Airy disk. Even if one were able to make a perfect lens, there is still a limit to the resolution of an image created by this lens. An optical system in which the resolution is no longer limited by imperfections in the lenses but only by diffraction is said to be diffraction limited.\n\nThe Airy disc is of importance in physics, optics, and astronomy.\n\n## Size of the Airy disc\n\nFar away from the aperture, the angle at which the first minimum occurs, measured from the direction of incoming light, is given by\n\n$\\sin \\theta = 1.22 \\frac{\\lambda}{d}$\n\nwhere \u03bb is the wavelength of the light and d is the diameter of the aperture. The Rayleigh criterion for barely resolving two objects is that the center of the Airy disc for the first object occurs at the first minimum of the Airy disc of the second. This means that the angular resolution of a diffraction limited system is given by the same formula.\n\n## Examples\n\n### Cameras\n\nIf two objects imaged by a camera are separated by an angle small enough that their Airy disks on the camera detector start overlapping, the objects can not be clearly separated any more in the image, and they start blurring together. Two objects are said to be just resolved when the maximum of the first Airy pattern falls on top of the first minimum of the second Airy pattern (the Rayleigh criterion).\n\nTherefore the smallest angular separation two objects can have before they significantly blur together is given as stated above by\n\n$\\sin \\theta = 1.22\\ \\frac{\\lambda}{d}$\n\nSince \u03b8 is small we can approximate this by\n\n$\\frac{x}{f} = 1.22\\ \\frac{\\lambda}{d}$\n\nwhere x is the separation of the images of the two objects on the film and f is the distance from the lens to the film. If we take the distance from the lens to the film to be approximately equal to the focal length of the lens, we find\n\n$x = 1.22\\ \\frac{\\lambda f}{d}$\n\nbut $\\frac{f}{d}$ is the f-number of a lens. A typical setting for use on an overcast day would be f\/8.[1] For blue visible light, the wavelength \u03bb is about 420 nanometers.[2] This gives a value for x of about 0.004 mm. In a digital camera, making the pixels of the image sensor smaller than this would not actually increase image resolution.\n\n### The human eye\n\nThe smallest f-number for the human eye is about 2.1.[3] The resulting resolution is about 1\u00a0\u03bcm. This happens to be about the distance between optically sensitive cells, photoreceptors, in the human eye.[citation needed]\n\n### Focused laser beam\n\nA circular laser beam with uniform intensity across the circle (a flattop beam) focused by a lens will form an Airy disk pattern at the focus. The size of the Airy disk determines the laser intensity at the focus.\n\n## Conditions for observation of the Airy disk\n\nLight from a uniformly illuminated circular aperture (or from a uniform, flattop beam) will exhibit an Airy diffraction pattern far away from the aperture due to Fraunhofer diffraction (far-field diffraction).\n\nThe conditions for being in the far field and exhibiting an Airy pattern are: the incoming light illuminating the aperture is a plane wave (no phase variation across the aperture), the intensity is constant over the area of the aperture, and the distance R from the aperture where the diffracted light is observed (the screen distance) is large compared the aperture size, and the radius a of the aperture is not too much larger than the wavelength \u03bb of the light. The last two conditions can be formally written as R > a2 \/ \u03bb .\n\nIn practice, the conditions for uniform illumination can be met by placing the source of the illumination far from the aperture. If the conditions for far field are not met (for example if the aperture is large), the far-field Airy diffraction pattern can also be obtained on a screen much closer to the aperture by using a lens right after the aperture (or the lens itself can form the aperture). The Airy pattern will then be formed at the focus of the lens rather than at infinity.\n\nHence, the focal spot of a uniform circular laser beam (a flattop beam) focused by a lens will also be an Airy pattern.\n\nIn a camera or imaging system an object far away gets imaged onto the film or detector plane by the objective lens, and the far field diffraction pattern is observed at the detector. The resulting image is a convolution of the ideal image with the Airy diffraction pattern due to diffraction from the iris aperture or due to the finite size of the lens. This leads to the finite resolution of a lens system described above.\n\n## Mathematical details\n\nThe intensity of the Fraunhofer diffraction pattern of a circular aperture (the Airy pattern) is given by:\n\n$I(\\theta) = I_0 \\left ( \\frac{2 J_1(ka \\sin \\theta)}{ka \\sin \\theta} \\right )^2 = I_0 \\left ( \\frac{2 J_1(x)}{x} \\right )^2$\n\nwhere I0 is the intensity in the center of the diffraction pattern, J1 is the Bessel function of the first kind of order one, k = 2\u03c0 \/ \u03bb is the wavenumber, a is the radius of the aperture, and \u03b8 is the angle of observation, i.e. the angle between the axis of the circular aperture and the line between aperture center and observation point. $x = ka \\sin \\theta = \\frac{2 \\pi a}{\\lambda} \\frac{q}{R} = \\frac{\\pi q}{\\lambda N}$, where q is the radial distance from the optics axis in the observation (or focal) plane and N = R \/ d (d=2a is the aperture diameter, R is the observation distance) is the f-number of the system. If a lens after the aperture is used, the Airy pattern forms at the focal plane of the lens, where R = f (f is the focal length of the lens). Note that the limit for $\\theta \\rightarrow 0$ (or for $x \\rightarrow 0$) is I(0) = I0.\n\nThe zeros of J1(x) are at $x = ka \\sin \\theta \\approx 0, 3.8317, 7.0156, 10.1735, 13.3268, 16.4706...$. From this follows that the first dark ring in the diffraction pattern occurs where\n\n$\\sin \\theta = \\frac{3.83}{ka} = \\frac{3.83 \\lambda}{2 \\pi a} = 1.22 \\frac{\\lambda}{2a} = 1.22 \\frac{\\lambda}{d}$.\n\nThe radius q1 of the first dark ring on a screen is related to \u03b8 by\n\nq1 = Rsin\u03b8\n\nwhere R is the distance from the aperture. The half maximum of the central Airy disk (where J1(x) = 1 \/ 2) occurs at x = 1.61633...; the 1\/e2 point (where J1(x) = 1 \/ e2) occurs at x = 2.58383..., and the maximum of the first ring occurs at x = 5.13562....\n\nThe intensity I0 at the center of the diffraction pattern is related to the total power P0 incident on the aperture by[4]\n\n$I_0 = \\frac{\\Epsilon_A^2 A^2}{2 R^2} = \\frac{P_0 A}{\\lambda^2 R^2}$\n\nwhere \u0395 is the source strength per unit area at the aperture, A is the area of the aperture (A = \u03c0a2) and R is the distance from the aperture. At the focal plane of a lens, I0 = (P0A) \/ (\u03bb2f2). The intensity at the maximum of the first ring is about 1.75% of the intensity at the center of the Airy disk.\n\nThe expression for I(\u03b8) above can be integrated to give the total power contained in the diffraction pattern within a circle of given size:\n\n$P(\\theta) = P_0 [ 1 - J_0^2(ka \\sin \\theta) - J_1^2(ka \\sin \\theta) ]$\n\nwhere J0 and J1 are Bessel functions. Hence the fractions of the total power contained within the first, second, and third dark rings (where J1(kasin\u03b8) = 0) are 83.8%, 91.0%, and 93.8% respectively.[5]\n\n## Obscured Airy pattern\n\nSimilar equations can also be derived for the obscured Airy diffraction pattern[6][7], which is the diffraction pattern from an annular aperture or beam, i.e. a uniform circular aperture (beam) obscured by a circular block at the center.\n\n$I(\\theta) = \\frac{I_0}{ (1 - \\epsilon ^2)^2} \\left ( \\frac{2 J_1(x)}{x} - \\frac{2 \\epsilon J_1(\\epsilon x)}{x}\\right )^2$\n\nwhere \u03b5 is the annular aperture obscuration ratio, or the ratio of the diameter of the obscuring disk and the diameter of the aperture (beam). $\\left( 0 \\le \\epsilon < 1 \\right)$, and x is defined as above: $x=ka\\sin(\\theta) \\approx \\frac {\\pi R}{\\lambda N}$ where R is the radial distance in the focal plane from the optical axis, \u03bb is the wavelength and N is the f-number of the system. The encircled energy (the fraction of the total energy contained within a circle of radius R centered at the optical axis in the focal plane) is then given by:\n\n$E(R) = \\frac{I_0}{ (1 - \\epsilon ^2)^2 } \\left( 1 - J_0^2(x) - J_1^2(x) + \\epsilon ^2 \\left[ 1 - J_0^2 (\\epsilon x) - J_1^2(\\epsilon x) \\right] - 4 \\epsilon \\int_0^x \\frac {J_1(t) J_1(\\epsilon t)}{t}\\,dt \\right)$\n\nFor $\\epsilon \\rightarrow 0$ the formulas reduce to the unobscured versions above.\n\n## Comparison of Airy disk and Gaussian beam focus\n\nA circular laser beam with uniform intensity profile, focused by a lens, will form an Airy pattern at the focal plane of the disk. The intensity at the center of the focus will be I0,Airy = (P0A) \/ (\u03bb2f2) where P0 is the total power of the beam, A = \u03c0D2 \/ 4 is the area of the beam (D is the beam diameter), \u03bb is the wavelength, and f is the focal length of the lens.\n\nA Gaussian beam with 1 \/ e2 diameter of D focused through an aperture of diameter D will have a focal profile that is nearly Gaussian, and the intensity at the center of the focus will be 0.924 times I0,Airy.[7]","date":"2022-05-20 16:24:11","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\": 18, \"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\": 0.705640435218811, \"perplexity\": 414.8189341719723}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2022-21\/segments\/1652662533972.17\/warc\/CC-MAIN-20220520160139-20220520190139-00099.warc.gz\"}"} | null | null |
<div>
<clr-dropdown>
<button class="btn btn-link btn-font" clrDropdownToggle (click)="onclick()">
{{ 'PUSH_IMAGE.TITLE' | translate | uppercase}}
<clr-icon shape="caret down"></clr-icon>
</button>
<clr-dropdown-menu *clrIfOpen [clrPosition]="'bottom-right'" class="dropdown-width">
<div class="commands-container">
<section>
<span><h5 class="h5-override">{{ 'PUSH_IMAGE.TITLE' | translate }}</h5></span>
<span>
<clr-tooltip>
<clr-icon clrTooltipTrigger shape="info-circle" class="info-tips-icon" size="24"></clr-icon>
<clr-tooltip-content [clrPosition]="'top-right'" [clrSize]="'md'" *clrIfOpen>
{{ 'PUSH_IMAGE.TOOLTIP' | translate }}
</clr-tooltip-content>
</clr-tooltip>
</span>
</section>
<section>
<hbr-inline-alert #copyAlert></hbr-inline-alert>
</section>
<section>
<article class="commands-section">
<hbr-copy-input #tagCopy (onCopyError)="onCpError($event)" inputSize="50" headerTitle="{{ 'PUSH_IMAGE.TAG_COMMAND' | translate }}" defaultValue="{{tagCommand}}"></hbr-copy-input>
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"redpajama_set_name": "RedPajamaGithub"
} | 2,859 |
require 'rails/generators'
module Redde
module Generators
class LayoutGenerator < ::Rails::Generators::Base
source_root File.expand_path('../templates', __FILE__)
desc 'Standart redde admin generator'
attr_reader :app_name
def make_views
template '../../../../../app/views/admin/redde/_sidebar.html.haml', 'app/views/admin/redde/_sidebar.html.haml'
template '../../../../../app/views/admin/redde/_main_menu.html.haml', 'app/views/admin/redde/_main_menu.html.haml'
end
def make_js
template 'assets/javascripts/admin.js', 'app/assets/javascripts/admin.js'
end
def make_css
template 'assets/stylesheets/admin.css', 'app/assets/stylesheets/admin.css'
end
def fix_routes
route("devise_for :managers, controllers: { registrations: 'managers/registrations' }")
end
private
def ext
'.html.haml'
end
def app_name
Rails.application.class.to_s.split('::').first
end
end
end
end
| {
"redpajama_set_name": "RedPajamaGithub"
} | 35 |
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"redpajama_set_name": "RedPajamaC4"
} | 6,100 |
Q: jQuery-ui Autocomplete with subitems I'm trying to implement autocomplete with extra info showing next to the suggestion.
Whenever I hover over the item, it should show some extra details about items.
I tried adding extra div with autocomplete's dynamic positions, but no luck.
If anyone has any idea, I'm all ears!
Thanks
A: You can take a look here -> https://jqueryui.com/autocomplete/#custom-data on how to create a jQuery UI Autocomplete with extra info. (make sure to use their defined names "label, desc" etc).
After you create the jquery ui autocomplete with the info you need you can use jquery .hover() to display information however you need (in case you dont like the default example on UI page) . https://api.jquery.com/hover/
Edit: Just to make it easier for you to understand, instead of appending the item.desc while UI autocompletes you can user hover(item.desc)
I'm talking about this part of the code ->
.autocomplete( "instance" )._renderItem = function( ul, item ) {
return $( "<li>" )
.append( "<div>" + item.label + "<br>" + item.desc + "</div>" )
.appendTo( ul );
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 2,029 |
<p>Calibration of plots other than 2D XY charts was affected in the recent Version 3.7 update. The build has now been fixed. Sorry about the inconvenience!</p>
<br/>
<p>I have also updated the downloadable version to the latest version 3.7 now.</p>
| {
"redpajama_set_name": "RedPajamaGithub"
} | 8,482 |
O distintivo Panzer (em alemão, Panzerkampfabzeichen) foi uma honraria do Terceiro Reich instituída pelo coronel-general Walther von Brauchitsch em 20 de dezembro de 1939 e destinada aos militares dos destacamentos blindados.
Bibliografia
Heinrich Doehle: Die Auszeichnungen des Grossdeutschen Reiches, ISBN 3-931533-43-3
Hans-Ulrich Krantz: Orden und Ehrenzeichen der Bundesrepublik Deutschland, Maximilian-Verlag, Köln 1958
Kurt-Gerhard Klietmann: Auszeichnungen des Deutschen Reiches 1936–1945, ISBN 3-8794-3689-4
Ordens e condecorações nazistas | {
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} | 6,779 |
Place in la' criminal underworld's belly, Arc could be the narrative of Paris Pritchert, a former police officer turned addict and drug dealer, who embarks on a quest to discover a missing child in the hope of redeeming his personality. The only catch is, like all addicts, Paris' optimism completely relies upon the medication in his system and -- in cases like this -- his own firm belief he is able to succeed in his assignment when they could just stay high 24/7 and living long enough to see it through. Paris enlists the Assistance of Gibbs, an American prostitute cautioned in the words of Maya Angelou and Nadine Gordimer, but in the terminology of this street to aid in the endeavor. Along with the course of the duo spans with people of the kid's parents, a physician with a penchant for soliciting"Street Boys", a self-ascribed King Of Porn, a drug supplier having a talent in earning impeccable hors d'oeuvres, and a hardened cop having scams compared to the most adept street hustler.
Genres: Thriller, Mystery, Drama
Casts: Ann Cusack, Ken Howard, Peter Facinelli, Billy Lush, Jonah Blechman
Production: N/A
Blood and Wine
1996 101m Movie
2014 92m Movie
Taken Too Far
Stalked by My Doctor: The Return
Operation Condor
Watch Arc Full Movie Online Free | MovieOrca | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 735 |
Q: How do I determine $A^{50}$, given the eigenvalues of $A$? I understand that the eigenvalues of $A^{50}$ will just be the eigenvalues of $A$ raised to the $50$th power. I also tried working backwards from this conclusion, but I am unable to obtain the characteristic polynomial for $A^{50}$.
My eigenvectors here are (1, 1) and (2, 3).
What is the key insight to progressing with this question?
A: The idea is that (high) powers of matrices are, in general, annoying to compute. Powers of diagonal matrices however, are very easy: each diagonal element is simply raised to this power.
Now if you can diagonalize your matrix $A$ as $PDP^{-1}$, with $D$ diagonal and the invertible matrix $P$ containing the corresponding eigenvectors in the columns, then:
$$\begin{align}A^n
& = \left( PDP^{-1} \right)^n \\
& = PD\underbrace{P^{-1}P}_{I}D\underbrace{P^{-1}P}_{I}DP^{-1} \ldots PDP^{-1} \\
& = PD^nP^{-1}
\end{align}$$
For $A^{50}$, you simply compute $D^{50}$ and perform the two extra matrix multiplications with $P$ and $P^{-1}$.
In your case with
$$A = \begin{pmatrix} 4 & -2 \\ 3 & -1 \end{pmatrix}$$
you found eigenvectors $v_1 = (1,1)$ and $v_2 = (2,3)$ with corresponding eigenvalues $\lambda_1=2$ and $\lambda_2=1$; so you get $P$ and $D$ as follows:
$$P = \begin{pmatrix} 1 & 2 \\ 1 & 3 \end{pmatrix} \quad , \quad
D = \begin{pmatrix} 2 & 0 \\ 0 & 1 \end{pmatrix}$$
Find $P^{-1}$ and use the 'trick' from above.
A: Let $TAT^{-1} = B$ be diagonal. Then $B^{50} = (TAT^{-1})^{50} = TA^{50}T^{-1}$, so
$A^{50} = T^{-1}B^{50}T$, and powers of a diagonal matrix are easy to calculate.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 7,203 |
Danuta Straszyńska (née le à Ostrowiec Świętokrzyski) est une athlète polonaise, spécialiste du 100 mètres haies.
Titrée sur 100 mètres haies lors des Universiade d'été de 1965, elle remporte la médaille d'or du relais 4 × 100 mètres aux championnats d'Europe 1966, en compagnie de Elżbieta Żebrowska, Irena Kirszenstein et Ewa Kłobukowska.
Elle se classe sixième du 80 mètres haies lors des Jeux olympiques de 1968, et sixième du 100 m haies lors des Jeux olympiques de 1972.
Elle termine au pied du podium du 100 m haies lors des championnats d'Europe d'athlétisme 1971.
Notes et références
Liens externes
Hurdleuse polonaise
Sprinteuse polonaise
Coureuse de 100 mètres haies
Championne d'Europe d'athlétisme
Athlète (femme) aux Jeux olympiques d'été de 1968
Athlète (femme) aux Jeux olympiques d'été de 1972
Naissance en février 1942 | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 9,334 |
Gmina Mawrowo i Rostusza (mac. Општина Маврово и Ростуша) – gmina w zachodniej Macedonii Północnej (połoski region statystyczny).
Skład etniczny
Według spisu powszechnego z 2002 r. w gmina liczyła wówczas 8618 mieszkańców, w tym następujące grupy etniczne:
50,46% – Macedończycy
31,09% – Turcy
17,2% – Albańczycy
1,25% – pozostali
W skład gminy wchodzą:
42 wsie: Adżiewci, Bełiczica, Bibaje, Bitusze, Bogdewo, Bołetin, Cerowo, Duf, Galicznik, Grekaje, Jancze, Kiczinica, Krakornica, Łazaropole, Leunowo, Mawrowi Anowi, Mawrowo, Niczpur, Nikiforowo, Nistrowo, Niwiszte, Nowo Seło, Orćusze, Prisojnica, Ribnica, Rosoki, Rostusza, Sełce, Sence, Skudrinje, Sretkowo, Suszica, Tanusze, Trebiszte, Tresoncze, Wełebrdo, Widusze, Wołkowija, Wrben, Wrbjani, Żirownica, Żużnje.
Zobacz też
kościół św. Mikołaja w Mawrowie
Przypisy
Mawrowo-Rostusza | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 1,688 |
Q: Joining multiple thread results using callback method How we can use callback method in java to join the result of multiple threads which are processed in parallel?
A: You can use a CompletionService. It receive Future results in a queue as they become available from the executor threads. It is not technically a callback but acts like one. For example:
public class Main {
public static void main(String... args) {
CompletionService<SomeResult> completionService =
new ExecutorCompletionService<SomeResult>(executor);
ResultProcessor processor = new ResultProcessor(completionService);
processor.start();
completionService.submit(new Callable<SomeResult>() {
public SomeResult call() {
...
return result;
}
});
}
}
class SomeResult {}
class ResultProcessor extends Thread {
private CompletionService<SomeResult> completionService;
ResultProcessor(CompletionService completionService) {
this.completionService = completionService;
}
public void run() {
while(...) {
Future<SomeResult> resultFuture = completionService.take(); //blocks if none available
SomeResult result = resultFuture.get();
... // result processing
}
}
}
A: It's not using callbacks, but Java's fork/join framework would be a good way to start, it's exactly meant for synchronizing on parallel executing tasks.
But if you want to build something 'callback based' you would pass an interface to the threads that is executed once they are finished that will do the synchronization and make sure the code is thread-safe.
A: you can call a synchronized method to 'join' result in each individual thread, and you can use a barrier or a countdownlatch to synchronize those threads.
A: With fork/join, the main thread waits until all threads/forks to finish, then process all the results; with callback, each thread calls a method in the main thread to publish its result once the result is ready. So callback may be slightly faster when threads' execution time aren't equal.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 8,113 |
Q: Get permissions of a file or folder Iam trying to get permissions on file/folder for current user. I found nice article about it here. I have tried to run this program but I got few erros and I dont know where I can find solutions to them. I tried to make my own version. Unfortunately when i try:
LPWSTR lpszPrimaryDC = NULL;
NetGetDCName(NULL, L"A", (LPBYTE *)&lpszPrimaryDC);
I got error: NERR_DCNotFound. How I can solve this problem?
A: The documentation says that error is returned when it "Could not find the domain controller for the domain specified in the domainname parameter." Do you have a domain called "A"? If not, the function is right to fail (and you need to rethink why/how you are calling it).
A: The only way the code didn't crash and gave me correct answer to question: is file or folder readable?
`
FILE *myFile = fopen(dirPath, "r");
if (myFile == 0) {
// "File or Dir is not readable
}
`
Hope this helps. You can use the same for writing test with "w".
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 2,039 |
Showing posts with label Vaduz. Show all posts
Diocese of Chur: Conflict Regarding the Successor of the Faithful Bishop Huonder Ignites
Progressive Insurrection in the Diocese of Chur
(Bern) Progressive Swiss Catholics are again attempting an uprising. Several movements have launched a petition "It's enough!" [Es Reicht!]. In this way they demand the appointment of an Apostolic Administrator for the Bishopric of Chur by Pope Francis. The leader of the insurrection is the regional Vicar General, Martin Kopp, who has been responsible for central Switzerland since 2002.
The Diocese of Chur in Switzerland is one of the oldest and most important diocese in the German-speaking world. It also includes Romansh Switzerland and the Italian valleys of Graubünden. Once, parts of Tyrol and Vorarlberg and Liechtenstein belonged to the diocese.
Progressive fight against Chur's Bishop
Not only is Graubünden subject to the Bishop of Chur, but also other Swiss cantons, namely Schwyz, Nidwalden, Obwalden, Uri, Glarus, and the canton of Zurich, which has been reformed for 500 years and has been liberal for 200 years. This has not a little to do with the attacks by progressive Church circles against the Chur bishops, which have lasted for decades.
Bishop Wolfgang Haas, who was enthroned in 1990, had so much resistance against him that the Vatican withdrew him. For this, the Principality of Liechtenstein was separated from Chur and designated as an independent diocese. In order to express the disapproval with the Swiss progressives, the Vatican made Vaduz an Archdiocese and Haas its first archbishop.
As Bishop of Chur he was followed for ten years by the hermit Benedictine Amédée Grab. The Liberals, whom he tried to involve, calmed down somewhat, but then, with all the greater intolerance, have rebeled against Bishop Vitus Huonder, who has been in office since 2007.
Yesterday: Own Diocese of Zurich - Today: no Bishop
Occasions are always found when the basic consensus is denied, because a "different" church has long been desired. Since it was not possible to force Bishop Huonder to his knees, the canton of Zurich wants to secede and establish itself as an independent diocese. 30 per cent of Zurichers profess themselves as Reformed, 27 per cent as Catholics.
Regional Vicar General Martin Kopp, the author of the initiative "Es reicht!"
On April 21, 2017, Bishop Huonder will complete his 75th year of life. In Chur no one really expects that Pope Francis would extend his office.
It's about the succession. The progressive church circles dream of a "reformed" church, which is remote from Rome. In this picture the Pope is especially disturbing to these same circles, even if he is called Francis. For this reason, there is a demand for a separate diocese of Zurich has now been abandoned. Chur is an ancient diocese, Zurich would be a new diocese and would be more directly under the supervision and control of Rome. They do not want that at all!
This is why the new suggestion by Bishop Sikar Kopp is not to appoint a successor but to appoint an apostolic administrator. The Progressives want to prevent a second bishop Haas or a second Bishop Huonder and would like to have a bishop separate from Rome.
"Without a new beginning the Diocese of Chur is dead," exclaimed episcopal vicar Martin Kopp on 24 October in the Daily News, which repeatedly turned to attacks against Bishop Huonder and the Catholic Church. "It can not go on like this so far," says Kopp, who had already conflicted in the past with his bishop and closest collaborators. He has defended "homo-blessings" and liturgical slapstick.
"Tensions" provoked by Progressives
The faithful Chur Society of Priests replied on the 26th of October that although we share Kopp's view that "tensions in the diocese of Chur" must be overcome. "However, regardless of whether the diocese is governed by a newly elected bishop or an apostolic administrator, this person must fulfill the mission of the Church, which every bishop assumes upon his consecration." And further: "To reduce tension in the diocese of Chur, Chur's priesthood pleads for more prudence and self-reflection, with the help of the Gospel. The latter moves us to contradict the regional Vicar-General: Hope dies last, and for the diocese of Chur, the proverb is that those declared dead live longer!"
The "tensions" in the Bishopric of Chur are attributed above all to the public-law Landeskirchen. Their existence is a peculiarity of Switzerland, which is connected with the Reformation and with the victory of the Liberals in 1847. In contrast to the state, the diocese is not a corresponding partner (church tax, parish property, finances, annual budget, employees), but to the canton which a corporation in public law, called a "Landeskirche", which must be democratically constituted by law. What applies to the cantonal level continues in every parish. In addition to the church-legal structure there is a state-church structure parallel to all levels (federation, cantons, municipalities).
In the historically Catholic cantons, the church has more room for maneuver. In the historically reformed cantons, including Zurich, the Protestant constitution is imposed upon the Catholic Church. The tone is not given by the diocese, but by the public-law body "Roman-Catholic Landeskirche". As is known in the Catholic Union in the Federal Republic of Germany and in Austria, the bodies of these committees are very progressive. And 500 years of the formation of the Reformation seem to encourage new elements of Protestantism imposed upon parts of the Catholic Church.
The progressive initiative wants petition to continue by the end of the year, then the petition, including signatures will be handed over to the Apostolic Nuncio and the Swiss Bishops' Conference.
Image: Diocese of Chur / SMM (screenshots)
Posted by Tancred at 6:35 AM 6 comments:
Labels: Bergoglio Effect, Bishop Huonder, Chur, Lichtenstein, Switzerland, Vaduz
Spagna Despedida a Eloy Tato, el obispo del Concilio Vaticano II | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 5,130 |
Sparattosperma is a genus of flowering plants belonging to the family Bignoniaceae.
Its native range is Southern Tropical America.
Species:
Sparattosperma catingae A.H.Gentry
Sparattosperma leucanthum (Vell.) K.Schum.
References
Bignoniaceae
Bignoniaceae genera | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 33 |
Ulrike Flach geb. Franzen (* 1. Januar 1951 in Oberhausen) ist eine deutsche Politikerin (FDP). Sie war von 1998 bis 2013 Mitglied des Bundestages und 2011 bis 2013 Parlamentarische Staatssekretärin im Bundesministerium für Gesundheit. Zuvor war sie seit 2009 stellvertretende Vorsitzende der FDP-Bundestagsfraktion.
Von 2014 bis 2017 war sie Mitglied des Aufsichtsrates der CompuGroup Medical AG.
Ausbildung und Beruf
Nach dem Abitur 1969 in Oberhausen absolvierte Ulrike Flach ein Studium der Angewandten Sprachwissenschaften in Mainz, London und Genua, welches sie 1973 als Diplom-Übersetzerin beendete. Seit 1974 war sie als Übersetzerin bei der Kraftwerk Union in Mülheim an der Ruhr tätig. Ihr Vertrag ruhte seit Januar 2005 und wurde 2013 beendet.
Familie
Ulrike Flach ist verheiratet und hat zwei Kinder und fünf Enkelkinder.
Partei
Seit 1975 ist sie Mitglied der FDP. Von 1976 bis 1978 war sie Vorsitzende des FDP-Kreisverbandes Oberhausen. Von 1992 bis 2010 war sie Vorsitzende des FDP-Kreisverbandes Mülheim/Ruhr, von 1995 bis 2005 stellvertretende Landesvorsitzende der FDP Nordrhein-Westfalen. Sie gehörte 1997 bis 2005 dem FDP-Bundesvorstand an. 2006 wurde sie erneut in den Landesvorstand der FDP NRW gewählt. Sie war außerdem von 2006 bis 2010 stellvertretende Landesvorsitzende der Vereinigung Liberaler Kommunalpolitiker (VLK) NRW.
Abgeordnete
Von 1998 bis 2013 war Ulrike Flach Mitglied des Deutschen Bundestages. Hier war sie seit Juni 2000 Vorsitzende des Ausschusses für Bildung, Forschung und Technikfolgenabschätzung. Das Amt als Ausschussvorsitzende legte sie nach Kritik wegen Zahlungen der Siemens AG in Höhe von 60.000 € im Jahr, für die sie nebenher weiter als Übersetzerin gearbeitet hatte, am 24. Januar 2005 nieder. Ihre Nachfolgerin wurde Cornelia Pieper.
Von 2005 bis 2009 war Ulrike Flach Obfrau der FDP-Bundestagsfraktion im Haushaltsausschuss und Technologiepolitische Sprecherin der Fraktion. Nach der Bundestagswahl 2009 war sie bis zum Wechsel in die Regierung im Jahre 2011 Mitglied des Haushalts- und Gesundheitsausschusses, gesundheitspolitische Sprecherin und stellvertretende Vorsitzende der FDP-Fraktion.
Vom 12. Mai 2011 bis 17. Dezember 2013 war Flach Parlamentarische Staatssekretärin im Bundesministerium für Gesundheit. Sie folgte auf Daniel Bahr, der zum Bundesgesundheitsminister ernannt wurde.
Ulrike Flach zog über die FDP-Landesliste von Nordrhein-Westfalen in den Deutschen Bundestag ein.
Zur Bundestagswahl 2013 trat Flach nicht wieder an.
Ehrungen
2010: Verdienstkreuz am Bande der Bundesrepublik Deutschland
2010: Ehrenring der Stadt Mülheim an der Ruhr
2013: Ehrendoktorwürde der Nicolae-Testemițanu-Universität für Medizin und Pharmazie, Chisinau (Moldawien)
Weblinks
Homepage
Einzelnachweise
Parlamentarischer Staatssekretär (Bundesrepublik Deutschland)
Bundestagsabgeordneter (Nordrhein-Westfalen)
FDP-Mitglied
Politiker (Essen)
Politiker (Mülheim an der Ruhr)
Politiker (20. Jahrhundert)
Politiker (21. Jahrhundert)
Träger des Bundesverdienstkreuzes am Bande
Ehrenringträger der Stadt Mülheim an der Ruhr
Deutscher
Geboren 1951
Frau
Person (Fernuniversität in Hagen) | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 8,500 |
/**
* Created by alykoshin on 08.12.16.
*/
'use strict';
var miniFullScreen = require('mini-fullscreen');
function printStatus(result) {
console.log('result:', result);
console.log('miniFullScreen.getEnabled():', miniFullScreen.getEnabled());
console.log('miniFullScreen.getActive():', miniFullScreen.getActive());
console.log('miniFullScreen.getElement():', miniFullScreen.getElement());
}
function init() {
var htmlElement = document.getElementById('full-element');
miniFullScreen.on('change', function(event) {
console.log('miniFullScreen.on(change):', event);
});
miniFullScreen.on('error', function(event) {
console.log('miniFullScreen.on(error):', event);
});
document.getElementById('action-start').addEventListener('click', function() {
var result = miniFullScreen.start(htmlElement);
printStatus(result);
});
document.getElementById('action-stop').addEventListener('click', function() {
var result = miniFullScreen.stop(htmlElement);
printStatus(result);
});
document.getElementById('action-toggle').addEventListener('click', function() {
var result = miniFullScreen.toggle(htmlElement);
printStatus(result);
});
}
window.addEventListener('load', function load(event){
// remove listener, as it is not needed any more
window.removeEventListener('load', load, false);
// call main method
init();
}, false);
| {
"redpajama_set_name": "RedPajamaGithub"
} | 7,626 |
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