text
stringlengths 14
5.77M
| meta
dict | __index_level_0__
int64 0
9.97k
⌀ |
|---|---|---|
{"url":"https:\/\/stats.stackexchange.com\/questions\/146799\/is-predicted-r-squared-a-valid-method-for-rejecting-additional-explanatory-varia","text":"# Is Predicted R-squared a Valid Method for Rejecting Additional Explanatory Variables in a Model?\n\nI'm building a model to understand the important drivers from a set of possible drivers for a time series of data. In my case the possible drivers are other time series.\n\nLike most statistical models I can always add additional drivers and improve the quality of my fit (measured by variance explained). In this case I'm using forward selection to add additional drivers requiring that the variance explained improve by a least certain percentage to determine whether I should add more drivers at all. This given percentage feels arbitrary depending on its value I may overfit.\n\nI was wondering if improvement in Predicted R^2 (definition from minitab.com below) would be a more consistent and better performing method for understanding when to stop adding additional drivers?\n\nPredicted R2 is calculated by systematically removing each observation from the data set, estimating the regression equation, and determining how well the model predicts the removed observation.\n\n\u2022 Any references would help considerably. Apr 16, 2015 at 22:03\n\u2022 Look over some of the higher-voted posts on our site that mention \"AIC\". Catch a few of those that mention \"R^2\" or \"R squared,\" too. For the lowdown on forward selection you might need to resort to Google so you can pick up the numerous (but helpful and authoritative) comments on stepwise regression methods, which generally focus on why other procedures are better. Finally, search out anything here related to \"model fitting.\"\n\u2013\u00a0whuber\nApr 16, 2015 at 22:14\n\nPredicted R squared would be no different than many other forms of cross-validation estimates of error (e.g., CV-MSE).\n\nThat said, R^2 isn't a great measure since R^2 will always increase with additional variables, regardless of whether that variable is meaningful. For example:\n\n> x <- rnorm(100)\n> y <- 1 * x + rnorm(100, 0, 0.25)\n> z <- rnorm(100)\n> summary(lm(y ~ x))$r.squared [1] 0.9224326 > summary(lm(y ~ x + z))$r.squared\n[1] 0.9273826\n\n\nR^2 doesn't make a good measure of model quality because of that. Information based measures, like AIC and BIC, are better.\n\nThis is especially true in a time series application where you expect your error terms to be auto-correlated. You should probably be looking at a time series model (ARIMA would be a good place to start) with exogenous regressors to account for the auto-correlation. As is, your model is likely massively overstating the explained error and inflating your R^2.\n\nI'd strongly encourage you to look at time series modeling and AIC based measures of model fit.\n\nEDIT: I wrote a little simulation to compute PRESS and the predicted R^2 for some simulated data and compared it against AIC.\n\nsim <- function() {\nx <- rnorm(100)\ny <- 1 * x + rnorm(100, 0, .25)\nz <- rnorm(100)\n\nsummary(lm(y[-1] ~ x[-1]))$r.squared summary(lm(y[-1] ~ x[-1] + z[-1]))$r.squared\n\nd <- rep(NA, 100)\npress1 <- press2 <- rep(NA, 100)\nfor (i in 1:100) {\nyt <- y[i]\nx2 <- x[-i]\ny2 <- y[-i]\nz2 <- z[-i]\nb1 <- coef(lm(y2[-1] ~ x2[-1]))\nb2 <- coef(lm(y2[-1] ~ x2[-1] + z2[-1]))\npress1[i] <- (yt - (b1) %*% c(1, x[i]))^2\npress2[i] <- (yt - (b2) %*% c(1, x[i], z[i]))^2\n}\nsst <- sum((y - mean(y))^2)\np1 <- 1 - sum(press1)\/sst\np2 <- 1 - sum(press2)\/sst\n\na1 <- AIC(lm(y[-1] ~ x[-1]))\na2 <- AIC(lm(y[-1] ~ x[-1] + z[-1]))\nc(p1 >= p2, a1 <= a2)\n}\n\nsim()\n\nx <- replicate(100, sim())\n\n\nBoth methods preferred the better model about 85% of the time. AIC has the benefits on a stronger theoretical basis and generalizes better to other methods (e.g., GLM where R^2 is not defined).\n\nThe bigger issue here is using a linear model on something with likely autocorrelated errors (a time series).\n\nUsing a dataset (Seatbelts in R) to estimate the effect of a seatbelt law, when I use just a linear model and adjust for gas price and distance driven the law's effect is estimated as -11.89 with a standard error of 6.026.\n\nIf I account for the fact that the data is correlated with itself and estimate the law effect in the context of an ARIMA model, I estimate the law's effect as -20 and with a standard error of 7.9.\n\nBecause the linear model ignored the time series properties, the estimate was off by 2 fold and the standard error of the major variable of interest was underestimated. The same thing (but worse) happens with the gas price and distance variables.\n\n\u2022 I'm a little confused with the above. R^2 should increase with additional variables, but Predicted R^2 should in general increase than decrease with additional variables, correct? Apr 16, 2015 at 23:42\n\u2022 What are the relative advantages of AIC? From M. Stone 1977 it seems that leave one out and AIC may be asymptotically equivalent, but my understanding here is limited. Apr 17, 2015 at 0:07\n\u2022 I think iacobus is using the regular, in-sample $R^2$ while the OP refers to predictive $R^2$ and gives its definition. That may be the reason for confusion. Meanwhile, PRESS looks equivalent to predictive $R^2$ in terms of variable selection. Apr 17, 2015 at 8:31\n\u2022 Thanks. That's pretty awesome. I'll try again on the AIC, but I'm using a Kalman Filter package in Python to better deal with the noise when fitting. I know AIC should work with the package but it is tough to get the likelyhood function out of that package. I fear I may be stuck using PRESS which may well be computationally less efficient. Apr 17, 2015 at 8:33\n\u2022 Calculating a sum of squares for PRESS should be no problem computationally, even for many iterations and many models (unless you need millions of PRESSes). Apr 17, 2015 at 8:34","date":"2022-08-14 21:00:41","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 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\": 0.38357800245285034, \"perplexity\": 2124.165948480656}, \"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-2022-33\/segments\/1659882572077.62\/warc\/CC-MAIN-20220814204141-20220814234141-00208.warc.gz\"}"}
| null | null |
\section{Introduction}
The aim of this article is to study two-player zero-sum general
repeated games with signals (sometimes called ``stochastic
games with partial observation''). At each stage, each player chooses
some action in a finite set. This generates a stage reward then a new
state and new signals are randomly chosen
through a transition probability depending on the current state and
actions, and with finite support. Shapley \cite{Shapley53} studied the
special case of standard stochastic games where the players observe, at
each stage, the current state and the past actions. There are several
ways to analyze these games. We will distinguish two approaches:
Borelian evaluation and uniform value.
In this article, we will mainly use a point of view coming from the
literature on determinacy of multistage games (Gale and Stewart
\cite{Gale53}). One defines a function, called \textit{evaluation}, on the
set of plays (infinite histories) and then studies the existence of a
value in the normal form game where the payoff is given by the
expectation of the evaluation, with respect to the probability induced
by the strategies of the players. Several evaluations will be considered.
In the initial model of Gale and Stewart \cite{Gale53} of two-person
zero-sum multistage game with perfect information, there is no state
variable. The players choose, one after the other, an action from a
finite set and both observe the previous choices. Given a subset $A$ of
the set of plays (in this framework: infinite sequences of actions),
player $1$ wins if and only if the actual play belongs to the set $A$:
the payoff function is the indicator function of $A$. Gale and Stewart
proved that the game is determined: either player $1$ has a winning
strategy or player $2$ has a winning strategy, in the case where $A$ is
open or closed with respect to the product topology. This result was
then extended to more and more general classes of sets until Martin
\cite{Martin75} proved the determinacy for every Borel set. When $A$
is an arbitrary subset of plays, Gale and Stewart \cite{Gale53} showed
that the game may be not determined.
In $1969$, Blackwell \cite{Blackwell69} studied the case (still
without state variable) where the players play simultaneously and are
told their choices. Due to the lag of information, the determinacy
problem is not well defined. Instead, one investigates the
probability that the play belongs to some subset $A$. When $A$ is a
$G_{\delta}$-set, a countable intersection of open sets, Blackwell
proved that there exists a real number $v$, the \textit{value} of the
game, such that for each $\varepsilon>0$, player $1$ can ensure that
the probability of the event: ``the play is in $A$'' is greater than
$v-\varepsilon$, whereas player $2$ can ensure that it is less than
$v+\varepsilon$.
The extension of this result to Shapley's model (i.e., with a state
variable) was done by Maitra and Sudderth. They focus on the specific
evaluation where the payoff is the largest stage reward obtained
infinitely often. They prove the existence of a value, called
\textit{limsup value}, in the countable framework \cite{Maitra92}, in the
Borelian framework \cite{Maitra93a} and in a finitely additive setting
\cite{Maitra93b}. In the first two cases, they assume some finiteness
of the action set (for one of the players). Their result especially
applies to finite stochastic games where the global payoff is the
limsup of the mean expected payoff.
Blackwell's existence result was generalized by Martin \cite{Martin98}
to any Borel-measurable evaluation, whereas Maitra and Sudderth \cite
{Maitra98} extended it further to stochastic games in the finitely
additive setting. In all these results, the players observe the past
actions and the current state.
Another notion used in the study of stochastic games (where a play
generates a sequence of rewards) is the uniform value where some
uniformity condition is required. Basically, one looks at the largest
amount that can be obtained by a given strategy for a family of
evaluations (corresponding to longer and longer games). There are
examples where the uniform value does not exist: Lehrer and Sorin
\cite{Lehrer92} describe such a game with a countable set of states and
only one player, having a finite action set. On the other hand,
Rosenberg, Solan and Vieille \cite{Rosenberg2002} proved the existence
of the uniform value in partial observation Markov Decision Processes
(one player) when the set of states and the set of actions are finite.
This result was extended by Renault \cite{Renault2011} to general
action space.
The case of stochastic games with standard signaling, that is, where
the players observes the state and the actions played has been treated
by Mertens and Neyman \cite{Mertens81}. They proved the existence of a
uniform value for games with a finite set of states and finite sets of
actions. In fact, their proof also shows the existence of a value for
the limsup of the mean payoff, as studied in Maitra and Sudderth and
that both values are equal.
The aim of this paper is to provide new existence results when the
players are observing only signals on state and actions. In
Section~\ref{model}, we define the model and present several specific Borel
evaluations. We then prove the existence of a value in games where the
evaluation of a play is the largest stage reward obtained along it,
called $\mathit{sup}$ evaluation and study several examples where the limsup
value does not exist.
Section~\ref{secsymmetric} is the core of this paper. We focus on the
case of symmetric signaling structure: multistage games where both
players have the same information at each stage, and prove that a value
exists for any Borel evaluation. For the proof, we introduce an
auxiliary game where the players observe the state and the actions
played and we apply the generalization of Martin's result to standard
stochastic games. Finally, in Section~\ref{uniform}, we introduce
formally the notion of uniform value and prove its existence in
recursive games with nonnegative payoffs.
\section{Repeated game with signals and Borel evaluation}\label{model}
Given a set $X$, we denote by $\Delta_f(X)$ the set of probabilities
with finite support on $X$. For any element $x\in
X$, $\delta_x$ stands for the Dirac measure concentrated on $x$.
\subsection{Model}
A \textit{repeated game form with signals} $\Gamma=(X,I,J,C,D,\pi, q)$
is defined by a set of states $X$, two finite sets of actions
$I$ and $J$, two sets of signals $C$ and $D$, an initial distribution
$\pi\in\Delta_f(X \times C \times D)$ and a transition function $q$
from $X\times I \times J$ to $\Delta_f(X\times C \times D)$. A \textit{repeated game with signals} $(\Gamma,g)$ is a pair of a repeated game
form and a reward function $g$ from $X \times I \times J$ to
$[0,1]$.
This corresponds to the general model of repeated game introduced in
Mertens, Sorin and Zamir \cite{Mertens94}.
The game is played as follows. First, a triple $(x_1,c_1,d_1)$ is drawn
according to the probability $\pi$. The initial state is $x_1$, player
$1$ learns $c_1$ whereas player~$2$ learns~$d_1$. Then, independently,
player $1$ chooses an action $i_1$ in $I$ and player~$2$ chooses an
action $j_1$ in $J$. A new triple $(x_2,c_2,d_2)$ is drawn according to
the probability distribution $q(x_1,i_1,j_1)$, the new state is $x_2$,
player $1$ learns $c_2$, player~$2$ learns $d_2 $ and so on.
At each stage $n$ players choose actions $i_n$ and $j_n$ and a triple
$(c_{n+1},d_{n+1} ,x_{n+1})$ is drawn according to
$q(x_n,i_n,j_n)$, where $x_n$ is the current state, inducing the
signals received by the players and the state at the next stage.
For each $n\geq1$, we denote by $H_n=(X \times C \times D \times I
\times J)^{n-1} \times X \times C \times D$ the set of \textit{finite
histories} of length $n$, by $H^1_n=(C \times I )^{n-1} \times C$ the
set of histories of length $n$ for player $1$ and by $H^2_n=(D \times J
)^{n-1} \times D$ the set of histories of length $n$ for player $2$.
Let\vspace*{1pt} $H = \bigcup_{n\geq1} H_n$.
Assuming perfect recall, a \textit{behavioral strategy} for player 1 is a
sequence $\sigma=(\sigma_n)_{n \geq1}$,
where $\sigma_n$, the strategy at stage $n$, is a mapping from $H^1_n$
to $\Delta(I)$, with the interpretation that $\sigma_n(h)$ is the
lottery on actions used by player 1 after $h\in H^1_n$.
In particular, the strategy $\sigma_1$ at stage $1$
is simply a mapping from $C$ to $\Delta(I)$ giving the law of the
first action played by player 1 as a function of his initial signal.
Similarly, a \textit{behavorial strategy} for player 2 is a sequence
$\tau
=(\tau_n)_{n \geq1}$, where $\tau_n$ is a mapping from $H^2_n$ to
$\Delta(J)$. We denote by $\Sigma$ and $\mathcal{T}$ the sets of behavioral
strategies of player 1 and player 2, respectively.
If for every $n\geq1$ and $h\in H^1_n$, $\sigma_n(h)$ is a Dirac
measure then
the strategy is \textit{pure}. A \textit{mixed strategy} is a
distribution over pure strategies.
Note\vspace*{1pt} that since the initial distribution $\pi$ and the transition $q$
have finite support and the sets of actions are finite, there exists a
finite subset $H^0_n \subset H_n$ such that for all strategies $(\sigma
,\tau)$ the set of histories that are reached at stage $n$ with a
positive probability is included in $H^0_n$.
Hence, no additional measurability assumptions on the strategies are
needed. It is standard that a pair of strategies $(\sigma, \tau)$
induces a probability
$\P_{\sigma, \tau} $ on the set of \textit{plays} $H_\infty=(X\times C
\times D \times I \times J)^{\infty}$ endowed with the
$\sigma$-algebra $\mathcal{H}_\infty$ generated by the cylinders
above the
elements of $H$. We denote by $\mathbb{E}_{\sigma,\tau}$ the
corresponding expectation.
Historically, the first models of repeated games assumed that both
$c_{n+1}$ and $d_{n+1}$ determine $(i_n, j_n)$ (standard signalling on
the moves also called ``full monitoring'').
A \textit{stochastic game} corresponds to the case where in addition the
state is known: both $ c_{n+1}$ and $d_{n+1}$ contain $x_{n+1}$.
A \textit{game with incomplete information} corresponds to the case where
in addition the state is fixed: $x_1= x_n, \forall n$, but not
necessarily known by the players.
Several extensions have been proposed and studied; see, for example,
Neyman and Sorin \cite{Neyman03} in particular Chapters~3, 21, 25,
28.
It has been noticed since Kohlberg and Zamir \cite{Kohlberg74b} that
games with incomplete information, when the information is \textit{symmetric}: $c_{n+1}= d_{n+1}$ and contains $(i_n, j_n)$, could be
analyzed by introducing an auxiliary stochastic game. However, the
state variable in this auxiliary stochastic game is no longer $x_n \in
X$ but the (common) conditional probability on $X$ given the signals,
that can be computed by the players: namely the law of $x_n$ in $\Delta
(X)$. Since then, this approach has been extended; see, for example,
Sorin \cite{Sorin2003b}, Ghosh et al. \cite{Gosh} and the analysis in
the current article shows that general repeated games with symmetric
information are the natural extension of standard stochastic games.
\subsection{Borel evaluation and results}
We now describe several ways to evaluate each play and the
corresponding concepts. We
follow the multistage game determinacy literature and define an
evaluation function $f$ on infinite plays. Then we study the existence
of the value of the normal form game $(\Sigma,\mathcal{T},f)$. We
will consider
especially four evaluations: the general Borel evaluation, the sup
evaluation, the limsup evaluation and the limsup-mean evaluation.
A Borel \textit{evaluation} is a $\mathcal{H}_\infty$-measurable function
from the set of plays $H_\infty$ to $[0,1]$.
\begin{definition}
Given an evaluation $f$, the game $\Gamma$ has a \textit{value} if
\[
\sup_\sigma\inf_{\tau} \mathbb{E}_{\sigma,\tau}
( f )=\inf_\tau\sup_\sigma
\mathbb{E}_{\sigma,\tau} ( f ).
\]
This real number is called the value and denoted by $v(f)$.
\end{definition}
Given a repeated game $(\Gamma,g)$, we will study several specific
evaluations defined through the stage payoff function $g$.
\subsubsection{Borel evaluation: $\sup$ evaluation}\label{bor2}
The first evaluation is the supremum evaluation where a play is
evaluated by the largest payoff obtained along it.
\begin{definition}
$\gamma^s$ is the \textit{sup} evaluation defined by
\[
\forall h\in H_\infty, \qquad {\gamma^s}(h)=\sup
_{n\geq1} g(x_n,i_n,j_n).
\]
In $(\Sigma,\mathcal{T},\gamma^s)$, the $\max\min$, the $\min\max
$, and the
value (called the \textit{sup value} if it exists) are, respectively,
denoted by $\underline{v}^s$,
$\overline{v}^s$ and $v^s$.
\end{definition}
The specificity of this evaluation is that for every $n\geq1$, the
maximal stage payoff obtained before $n$ is a lower bound of the
evaluation on the current play. We prove that the $\sup$ value always exists.
\begin{theorem}\label{sup}
A repeated game $(\Gamma, g)$ with the $\sup$ evaluation has a value $v^s$.
\end{theorem}
For the proof, we use the following result. We call \textit{strategic
evaluation} a function $F$ from $\Sigma\times\tau$ to $[0,1]$. It is
clear that an evaluation $f$ induces naturally a strategic evaluation
by $F(\sigma,\tau)=\mathbb{E}_{\sigma,\tau} ( f )$.
\begin{proposition}\label{FF}
Let $(F_n)_{n\geq1}$ be an increasing sequence of strategic
evaluations from $\Sigma\times\tau$ to $[0,1]$ that converges to some
function $F$.
Assume that:
\begin{itemize}
\item $\Sigma$ and $\tau$ are compact convex sets,
\item for every $n\geq1$, $F_n(\sigma,\cdot)$ is lower semicontinuous and
quasiconvex on $\tau$ for every $\sigma\in\Sigma$,
\item for every $n\geq1$, $F_n(\cdot,\tau)$ is upper semicontinuous and
quasiconcave on $\Sigma$ for every $\tau\in\tau$.
\end{itemize}
Then the normal form game $(\Sigma,\tau,F)$ has a value $v$.
\end{proposition}
A more general version of this proposition can be found in Mertens,
Sorin and Zamir \cite{Mertens94} (Part A, Exercise 2, Section~1.f, page~10).
\begin{pf*}{Proof of Theorem~\protect\ref{sup}}
Let $n\geq1$ and define the strategic evaluation $F_n$ by
\[
F_n(\sigma,\tau)= \mathbb{E}_{\sigma,\tau} \Bigl(\sup
_{t\leq n} g(x_t,i_t,j_t)
\Bigr).
\]
Players remember their own previous actions so by Kuhn's theorem \cite
{Kuhn53}, there is equivalence between mixed strategies and behavioral
strategies. The sets of mixed strategies are naturally convex.
The set of histories of length $n$ having positive probability is
finite and, therefore, the set of pure strategies is finite. For every
$n\geq1$, the function $F_n(\sigma,\tau)$ is thus the linear extension
of a finite game. In particular $F_n(\sigma,\cdot)$ is lower semicontinuous
and quasiconvex on $\tau$ for every $\sigma\in\Sigma$ and upper
semicontinuous and quasiconcave on $\Sigma$ for every $\tau\in\tau$.
Finally, the sequence $(F_n)_{n\geq1}$ is increasing to
\[
F(\sigma,\tau)=\mathbb{E}_{\pi,\sigma,\tau} \Bigl(\sup_{t}
g(x_t,i_t,j_t) \Bigr).
\]
It follows from Proposition~\ref{FF} that the game $\Gamma$ with the
$\sup$ evaluation has a value.
\end{pf*}
\subsubsection{Borel evaluation: $\operatorname{limsup}$ evaluation}
Several authors have especially focused on the $\operatorname{limsup}$ evaluation and
the $\operatorname{limsup}$-mean evaluation.
\begin{definition}
$\gamma^*$ is the \textit{limsup} evaluation defined by
\[
\forall h\in H_\infty, \qquad \gamma^*(h)=\limsup_n
g(x_n,i_n,j_n).
\]
In $(\Sigma,\mathcal{T},\gamma^*)$, the $\max\min$, the $\min\max
$, and the
value (called the \textit{limsup value}, if it exists) are,
respectively,
denoted by $\underline{v}^*$, $\overline{v}^*$ and $v^*$.
\end{definition}
\begin{definition}
$\gamma_m^*$ is the \textit{limsup-mean} evaluation defined by
\[
\forall h\in H_\infty,\qquad \gamma_m^*(h)=\limsup
_n \frac{1}{n}\sum_{t=1}^n
g(x_t,i_t,j_t).
\]
In $(\Sigma,\mathcal{T},\gamma_m^*)$, the $\max\min$, the $\min
\max$, and the
value (called the \textit{limsup-mean value}, if it exists) are,
respectively, denoted by $\underline{v}_m^*$, $\overline{v}_m^*$ and $v_m^*$.
\end{definition}
The limsup-mean evaluation is closely related to the limsup evaluation.
Indeed, the analysis of the limsup-mean evaluation of a
stochastic game can be reduced to the study of the limsup evaluation of
an auxiliary stochastic game having as set of states the set of finite
histories of the original game.
These evaluations were especially studied by Maitra and Sudderth \cite{Maitra92,Maitra93a}. In \cite{Maitra92}, they proved the
existence of the limsup value in a stochastic game with a countable set
of states and finite sets of actions when the players observe the state
and the actions played. Next, they extended in \cite{Maitra93a} this
result to Borel measurable evaluation.
We aim to study potential extensions of their results to repeated game
with signals. In general, a repeated game with signals has no value
with respect to the limsup evaluation as shown in the following three
examples. In each case, we also show that the limsup-mean value does
not exist.
\begin{example}\label{guess}
We consider a recursive game where the players observe neither the
state nor the action played by the other player. We say that the
players are \textit{in the dark}.
This example, due to Shmaya, is also described in Rosenberg, Solan and
Vieille \cite{Rosenberg2009} and can be interpreted as ``pick the
largest integer.''
The set of states is finite $X=\{s_1,s_2,s_3, 0^*, 1^*,-1^*,2^*,-2^*\}
$, the action set of player $1$ is $I=\{T,B\}$, the action set of
player $2$ is $J=\{L,R\}$, and the transition is given by
\[
\begin{tabular}{cccc}
& $
\begin{array}{c@{\quad}c} L & R
\end{array}
$ & $
\begin{array}{c@{\quad}c} L \hspace{14mm} & \hspace{13mm} R
\end{array}
$ & $
\begin{array}{c@{\quad}c} L & R
\end{array}
$ \\
$
\begin{array}{cc} T \\ B
\end{array}
$
&$\lleft(
\begin{array}{c@{\quad}c} s_1 & -2^* \\ s_1 & -2^*\\
\end{array}
\rright)$ & $\lleft(
\begin{array}{c@{\quad}c} s_2 & 1/2\bigl(-1^*\bigr) +1/2 (s_3) \vspace*{2pt}\\ 1/2\bigl(1^*\bigr) +1/2 (s_1) &
0^*\\
\end{array}
\rright)$ & $\lleft(
\begin{array}{c@{\quad}c} s_3 & s_3 \\ 2^* & 2^*\\
\end{array}
\rright)$ \\
& $s_1$ & $ s_2$ & $ s_3 $
\end{tabular}\hspace*{-6pt}.
\]
The payoff is $0$ in states $s_1$,$s_2$, and $s_3$. For example, if the
state is $s_2$, player $1$ plays~$T$ and player $2$ plays $R$ then with
probability $1/2$ the payoff is $-1$ forever, and with probability
$1/2$ the next state is $s_3$. States denoted with a star are absorbing
states: if state $k^*$ is reached, then the state is $k^*$ for the
remaining of the game and the payoff is $k$.
\begin{cl*}\label{cl1}
The game which starts in $s_2$ has no limsup value:
$\underline{v}^*= -1/2< 1/2=\overline{v}^*$.
\end{cl*}
Since the game is recursive, the limsup-mean evaluation and the limsup
evaluation coincide, so there is no limsup-mean value either. It also
follows that the uniform value, defined formally in Section~\ref{uniform}, does not exist.
\begin{pf*}{Proof of \hyperref[cl1]{Claim}}
The situation is symmetric, so we
consider what player $1$ can guarantee.
After player $1$ plays $B$, the game is essentially over from player
$1$'s viewpoint: either absorption occurs or the state moves to $s_1$
where player $1$'s actions are irrelevant. Therefore, the only relevant
past history in order to define a strategy of player $1$ corresponds to
all his past actions being $T$. A strategy of player $1$ is thus
specified by the probability $\varepsilon_n$ to play $B$ for the first
time at stage $n$; let $\varepsilon^*$ be the probability that player
$1$ plays $T$ forever.
Player 2 can reply as follows: fix $\varepsilon>0$, and consider $N$
such that\break $\sum_{n=N}^{\infty} \varepsilon_n \leq\varepsilon$. Define
the strategy $\tau$ which plays $L$ until stage $N-1$ and $R$ at stage
$N$. For any $n>N$, we have
\[
\mathbb{E}_{s_2, \sigma, \tau} \bigl(g (x_n,i_n,j_n)
\bigr)\leq \varepsilon ^*(-1/2)+ \Biggl( \sum_{n=1}^{N-1}
\varepsilon_n \Biggr) (-1/2) + \varepsilon (1/2 ) \leq-1/2 +
\varepsilon.
\]
It follows that player 1 cannot guarantee more than $-1/2$ in the
limsup sense.
\end{pf*}
\end{example}
\begin{example}\label{semiguess}
We consider a recursive game where one player is more informed than
the other: player $2$ observes the state variable and the past actions
played whereas player $1$ observes neither the state nor the actions played.
This structure of information has been studied, for example, by
Rosenberg, Solan, and Vieille \cite{Rosenberg2004}, Renault \cite
{Renault2012a} and Gensbittel, Oliu-Barton and Venel \cite
{Gensbittel2013}. They proved the existence of the uniform value under
the additional assumption that the more informed player controls the
evolution of the beliefs of the other player on the state variable.
The set of states is finite $X=\{s_2,s_3, 0^*, 1/2^*,-1^*,2^*\}$, the
action set of player $1$ is $I=\{T,B\}$, the action set of player $2$
is $J=\{L,R\}$, and the transition is given by
\[
\begin{tabular}{c@{\quad}c@{\quad}c}
& $
\begin{array}{c@{\quad}c} L \hspace{13mm} & \hspace{7mm} R
\end{array}
$ & $
\begin{array}{c@{\quad}c} L & R
\end{array}
$\\
$
\begin{array}{cc} T \\ B
\end{array}
$ & $\lleft(
\begin{array}{c@{\quad}c}
s_2 & 1/2\bigl(-1^*\bigr) +1/2 (s_3) \vspace*{2pt}\\
(-1/2)^* & 0^*\\
\end{array}
\rright)$ & $\lleft(
\begin{array}{c@{\quad}c}
s_3 & s_3 \\
2^* & 2^*\\
\end{array}
\rright)$ \\
& $s_2$ & $s_3$
\end{tabular}\hspace*{-6pt}.
\]
We focus on the game which starts in $s_2$. Both players can guarantee
$0$ in the $\sup$ evaluation: player $2$ by playing $L$ forever and
player $1$ by playing $T$ at the first stage and then $B$ forever.
Since the game is recursive, the limsup-mean evaluation and the limsup
evaluation are equals.
\begin{cl*}\label{cl2}
The game which starts in $s_2$ has no
limsup value: $\underline{v}^*=-1/2<-1/6=\overline{v}^*$.
\end{cl*}
\begin{pf}
The computation of the $\max\min$ with
respect to the limsup-mean evaluation is similar to the computation of
Example~\ref{guess}. The reader can check that player $1$ cannot
guarantee more than $-1/2$.
We now prove that the $\min\max$ is equal to $-1/6$. Contrary to
Example~\ref{guess}, player $2$ observes the state and actions,
nevertheless the game is from his point of view strategically finished
as soon as $B$ or $R$ is played: if $B$ is played then absorption
occurs, if $R$ is played then either absorption occurs or the state
moves to $s_3$ where player 2's action are irrelevant. Therefore, when
defining the strategy of player $2$ at stage $n$, the only relevant
past history is $(s_2,T,L)^n$ and a strategy of player $2$ is defined
by the probability $\varepsilon_n$ that he plays $R$ for the first time
at stage $n$ and the probability $\varepsilon^*$ that he plays $L$ forever.
Fix $\varepsilon>0$, and consider $N$ such that $\sum_{n=N}^{\infty}
\varepsilon_n \leq\varepsilon$. Player 1's replies can be reduced to
the two following strategies: $\sigma_1$ which plays $T$ forever and,
$\sigma_2$ which plays $T$ until stage $N-1$ and $B$ at stage $N$. All
the other strategies are yielding a payoff smaller with an $\varepsilon
$-error. The strategy $\sigma_1$ yields $0 \varepsilon^* +
(1-\varepsilon^*)(-1/2) $ and the strategy $\sigma_2$ yields $(-1/2)
\varepsilon^* +(1-\varepsilon^*) 1/2- \varepsilon$.
The previous payoff functions are almost the payoff of the two-by-two
game where player $1$ chooses $\sigma_1$ or $\sigma_2$ and player $2$
chooses either never to play $R$ or to play $R$ at least once:
\[
\pmatrix{
0 & -1/2
\vspace*{2pt}\cr
-1/2 & 1/2}.
\]
The value of this game is $-1/6$, giving the result.
\end{pf}
\end{example}
\begin{example}\label{bigmatch}
In the previous examples, the state is not known to at least one player.
The following game is a variant of the Big Match introduced by
Blackwell and Ferguson \cite{Blackwell68}. It is an absorbing game:
every state except one are absorbing. Since there is only one state
where players can influence the transition and the payoff, the
knowledge of the state is irrelevant. Players can always consider that
the current state is the nonabsorbing state.
We assume that player $2$ observes the past actions played whereas
player $1$ does not
(in the original version, both player $1$ and player $2$ were
observing the state and past actions):
\[
\begin{array}{c@{\quad}c}
&
\hspace*{-2pt}\begin{array}{c@{\quad}c}
L & R \\
\end{array}
\\
\begin{array}{c}
T \\
B \\
\end{array}
&
\lleft(
\begin{array}{c@{\quad}c}
1^* & 0^* \\
0 & 1 \\
\end{array}
\rright).\\
\end{array}
\]
\begin{cl*}\label{cl3}
The game with the sup evaluation has a
value $v_s=1$. The game with the limsup evaluation and the game with
the limsup-mean evaluation do not have a value: $\underline{v}^*=\underline{v}_m^*=0 < 1/2=\overline{v}_m^*=\underline{v}^*$.
\end{cl*}
\begin{pf
We first prove the existence of the value
with respect to the sup evaluation. Player $1$ can guarantee the payoff
$1$. Let $\varepsilon>0$, and $\sigma$ be the strategy which plays $T$
with probability $\varepsilon$ and $B$ with probability $1-\varepsilon$. This
strategy yields a sup evaluation greater than $1-\varepsilon$. Since
$1$ is the maximum payoff, it is the value: $v^s=1$.
We now focus on the limsup evaluation and the limsup-mean evaluation.
After player $1$ plays $T$ absorption occurs. Therefore, the only
relevant past history in order to define a strategy of player $1$
corresponds to all his past actions being $B$. Let $\varepsilon_n$ be
the probability that player $1$ plays $T$ for the first time at stage
$n$ and $\varepsilon^*$ be the probability that player $1$ plays $B$ forever.
Player 2 can reply as follows: fix $\varepsilon>0$, and consider $N$
such that\break $\sum_{n=N}^{\infty} \varepsilon_n \leq\varepsilon$. Define
the strategy $\tau$ which plays $R$ until stage $N-1$ and $L$ at stage
$N$. For any $n>N$, we have
\[
\mathbb{E}_{s, \sigma, \tau} \bigl(g (x_n,i_n,j_n)
\bigr)\leq \varepsilon ^*0+ \Biggl( \sum_{n=1}^{N-1}
\varepsilon_n \Biggr)0 + \varepsilon (1 ) \leq\varepsilon.
\]
Let us compute what player $2$ can guarantee with respect to the limsup
evaluation. The computation is similar for the limsup-mean evaluation.
First, player $2$ can guarantee $1/2$ by playing the following mixed
strategy: with probability $1/2$, play $L$ at every stage and with
probability $1/2$, play $R$ at every stage.
We now prove that it is the best payoff that player $2$ can achieve.
Fix a strategy $\tau$ for player $2$ and consider the induced law $\P$
on the set $H_\infty=\{L,R\}^{\infty}$ of infinite sequences of $L$ and
$R$ induced by $\tau$ when player $1$ plays $B$ at every stage. Denote
by $\beta_n$ the probability that player $2$ plays $L$ at stage $n$. If
there exists a stage $N$ such that $\beta_{N} \geq1/2$, then playing
$B$ until $N-1$ and $T$ at stage $N$ yields a payoff greater than $1/2$
to player $1$. If for every $n$, $\beta_n\leq1/2$, then the stage
payoff is in expectation greater than $1/2$ when player $1$ plays $B$.
Therefore, the expected $\mathit{limsup}$ payoff is greater than $1/2$.
\end{pf}
\end{example}
\section{Symmetric repeated game with Borel evaluation} \label{secsymmetric}
Contrary to the $\sup$ evaluation, in general the existence of the
value for a given evaluation depends on the signaling structure. In
Section~\ref{model}, we analyzed three games without $\operatorname{limsup}$-mean
value. In this section, we prove that if the signaling structure is
symmetric as defined next, the value always exists in every Borel evaluation.
\subsection{Model and results}
\begin{definition}
A \textit{symmetric signaling repeated game form} is a repeated game
form with signals
$\Gamma=(X,I,J,C,D, \pi, q)$ such that there exists a set $S$ with
$C=D=I \times J \times S$ satisfying
\[
\forall(x,i,j)\in X\times I \times J, \qquad \sum_{s,x'}
q(x,i,j) \bigl(x',(i,j,s),(i,j,s)\bigr)=1
\]
and the initial distribution $\pi$ is also symmetric: $\pi(x,c,d)>0$
implies $c=d$.
\end{definition}
Intuitively, at each stage of a symmetric signaling repeated game form,
the players observe both actions played and a public signal $s$. It
will be convenient to write such a game form as a tuple $\Gamma
=(X,I,J,S, \pi, q)$ and since for such a game:
$q(x,i,j)(x',(i',j',s'),(i'',j'',s''))>0$ only if $i=i'=i''$ and
$j=j'=j''$ and $s'=s''$,
without loss of generality, we can and will write
$q(x,i,j)(x',s)$ as a shorthand for
$q(x,i,j)(x',(i,j,s),(i,j,s))$.
With this notation $q(x,i,j)$ and the initial distribution
$\pi$ are elements of $\Delta_f(X\times S)$. The set of observed plays
is then $V_\infty=(S\times I \times J)^\infty$.
\begin{theorem}\label{theo3}
Let $\Gamma$ be a symmetric signaling repeated game form. For every
Borel evaluation $f$, the game $\Gamma$ has a value.
\end{theorem}
\begin{corollary}\label{corotheo3}
A symmetric signaling repeated game $(\Gamma,g)$ has a limsup value and
a limsup-mean value.
\end{corollary}
\subsection{Proof of Theorem~\texorpdfstring{\protect\ref{theo3}}{8}}
Let us first give an outline of the proof. Given a symmetric signaling
repeated game form $\Gamma$ and a Borel evaluation $f$, we construct an
auxiliary standard stochastic game $\widehat{\Gamma}$ (where the
players observe the state and the actions) and a Borel evaluation
$\widehat{f}$ on the corresponding set of plays.\vspace*{1.5pt} We use the existence
of the value in the game $\widehat{\Gamma}$ with
respect to the evaluation $\widehat{f}$ to deduce the existence of the
value in the original game.
The difficult point is the definition of the evaluation $\widehat{f}$.
The key idea is to define a conditional probability with respect to the
$\sigma$-algebra of observed plays. For a given probability on plays,
the existence of such conditional probability is easy since the sets
involved are Polish. In our case, the difficulty comes from the
necessity to have the same conditional probability for any of the
probability distributions that could be generated by the strategies of
the players (Sections~\ref{secfinite}--\ref{sectrzy}). (As remarked
by a referee the observed plays generate in fact a sufficient statistic
for the plays with respect to all these distributions.) The definition
of the conditional probability is achieved in three steps: we first
define the conditional probability of a finite history with respect to
a finite observed history, then we use a martingale result to define
the conditional probability of a finite history with respect to an
observed play and finally we rely on Kolmogorov extension theorem to
construct a conditional probability on plays. Finally, we introduce the
function $\widehat{f}$ on the observed plays as the integral of $f$
with respect to this conditional probability.
After introducing few notations we prove the existence of the value by
defining the game $\widehat{\Gamma}$, assuming the existence of the
function $\widehat{f}$ (Lemma~\ref{transfer}). The next three sections
will be dedicated to the construction of the conditional probability,
then to the definition and properties of the function $\widehat{f}$ for any
Borelian payoff function~$f$.
Let $\Gamma$ be a symmetric signaling repeated game form, we do not
assume the Borel evaluation to be given.
\subsection{Notation}
Let $H_n=(X\times S \times I \times J)^{n-1}\times X \times S$, $H =
\bigcup_{n\geq1} H_n$, the set of histories and $H_\infty= (X\times S
\times I \times
J)^\infty$, the set of plays.
For all $h\in H_\infty$, define $ h |_{n} \in H_n$ as the
projection of $h$ on the $n$ first stages.
For all $h_n\in H_n$, denote by $h_n^+$ the cylinder generated by
$h_n$ in $H_\infty$: $h_n^+=\{h \in H_\infty, h
|_{n}=h_n \}$ and by ${\mathcal H}_n$ the corresponding
$\sigma$-algebra. ${\mathcal H}_\infty$ denotes the $\sigma$-algebra
generated by $\bigcup_n {\mathcal H}_n$.
Let $V_n=(S \times I \times J)^{n-1}\times S = H^1_n = H^2_n $, $V =
\bigcup_{n\geq1} V_n$ and $V_\infty=(S\times I \times J)^\infty$.
For all $v\in V_\infty$, define $ v |_{n} \in V_n$ as the
projection of $v$ on the $n$ first stages.
For all $v_n\in V_n$, denote by $v_n^+$ the cylinder generated by
$v_n$ in $V_\infty$: $v_n^+=\{v \in V_\infty, v
|_{n}=v_n \}$ and by ${\mathcal V}_n$ the corresponding
$\sigma$-algebra. ${\mathcal V}_\infty$ is the $\sigma$-algebra
generated by $\bigcup_n {\mathcal V}_n$.
We denote by $\Theta$ the application from $H_\infty$ to $V_\infty$
which forgets all the states: more precisely,
$\Theta( x_1, s_1, i_1, j_1,\ldots, x_n, s_n, i_n, j_n, \ldots) = (
s_1, i_1, j_1, \ldots,\break s_n, i_n, j_n,\ldots)$.
We use the same notation for the corresponding application
defined from $H$ to $V$.
We denote by ${\mathcal V}^*_n$ (resp., ${\mathcal V}^*_\infty$) the
image of ${\mathcal V}_n$ (resp., ${\mathcal V}_\infty$)
by $\Theta^{-1}$ which are sub $\sigma$-algebras of ${\mathcal H}_n$
(resp., ${\mathcal H}_\infty$).
Explicitly, for $v_n\in V_n$, $v_n^*$ denotes the cylinder generated by
$v_n$ in $H_\infty$: $v_n^*=\{h\in H_\infty, \Theta(h)
|_{n}=v_n \}$, ${\mathcal V}^*_n$ are the corresponding
$\sigma$-algebras and ${\mathcal V}^*_\infty$ the $\sigma$-algebra
generated by their union.
Any ${\mathcal V}_n$ (resp., ${\mathcal V}_\infty$)-measurable function
$\ell$ on $V_\infty$ induces a ${\mathcal V}^*_n$ (resp., ${\mathcal
V}^*_\infty$)-measurable function $\ell\circ\Theta$ on $H_\infty$.
Define $\alpha$ from $H$ to $[0,1]$ where for $h_n = ( x_1, s_1, i_1,
j_1,\ldots, x_n, s_n)$:
\[
\alpha( h_n) =\pi(x_1,s_1)
\prod_{t=1}^{n-1} q(x_t,i_t,j_t)
(x_{t+1},s_{t+1})
\]
and $\beta$ from $V$ to $[0,1]$ where for $v_n = (s_1,
i_1, j_1,\ldots, s_n)$:
\[
\beta(v_n)= \sum_{ h_n \in H_n ; \Theta(h_n) = v_n }
\alpha(h_n).
\]
Let
${\overline H}_n = \{ h_n \in H_n$; $ \alpha( h_{n}) >0 \}$ and
${\overline V}_n = \Theta({\overline H}_n)$ and recall that these sets
are finite.
We introduce now the set of plays and observed plays that can occur
during the game as ${\overline H}_\infty=
\bigcap_n \overline{H}_n ^+$ and ${\overline V}_\infty= \Theta
({\overline H}_\infty) = \bigcap_n \overline{V}_n$. Remark that both are
measurable subsets of $H_\infty$ and $V_\infty$, respectively.
For every pair of strategies $(\sigma,\tau)$, we denote by $\P
_{\sigma
,\tau}$ the probability distribution induced over
the set of plays $(H_\infty, {\mathcal H}_\infty)$ and by $\mathbb
{Q}_{\sigma
,\tau}$ the probability distribution over the set of
observed plays $(V_\infty, {\mathcal V}_\infty)$. Thus, $\mathbb
{Q}_{\sigma,\tau
}$ is the image of $\P_{\sigma,\tau}$ under~$\Theta$. Note that $\operatorname{supp}( \P_{\sigma,\tau} )
\subset{\overline
H}_\infty$. We denote, respectively, by $\mathbb{E}_{\P_{\sigma,\tau
}}$ and $\mathbb{E}
_{\mathbb{Q}_{\sigma,\tau}}$ the corresponding expectations.
It turns out that for technical reasons
it is much more convenient to work with the space
$\overline{V}_\infty$ rather than with $V_\infty$ (and with
$\overline
{H}_\infty$ rather than with $H_\infty$).
And then, abusing slightly the notation,
$\mathcal{V}_\infty$ and $\mathcal{V}_n$ will tacitly denote
the restrictions to $\overline{V}_\infty$
of the corresponding $\sigma$-algebras defined on $V_\infty$.
On rare occasions this can lead to a confusion and then
we will write, for example, $\overline{\mathcal{V}}_n$ to denote the
$\sigma$-algebra $\{ U\cap\overline{V}_\infty\vert U\in
\mathcal{V}_n \}$ the restriction of $\mathcal{V}_n$ to $\overline
{V}_\infty$.
\subsubsection{Definition of an equivalent game}\label{secconclusion}
Let\vspace*{1pt} us define an auxiliary stochastic game $\widehat{\Gamma}$. The sets
of actions $I$ and $J$ are the same as in $\Gamma$. The set of states is
$V=\bigcup_{n\geq1} V_n$ and the transition $\widehat{q}$ from
$V\times I
\times J$ to $\Delta(V)$ is given by
\[
\forall v_n\in V_n, \forall i\in I, \forall j\in J,
\qquad \widehat{q}(v_n,i,j)=\sum_{s \in S}
\psi(v_n, i, j, s)\delta_{v_n, i,
j, s},
\]
where\vspace*{1pt} $\psi(v_n, i, j, s)= \frac{\beta(v_n, i, j,
s)}{\beta(v_n)}$.
Note that if $v_n\in V_n$ then the support of $\widehat{q}(v_n,i,j)$ is
included in $V_{n+1}$, in particular is finite.
Moreover, if $\widehat{q}(v_n,i,j)(v_{n+1})>0$ then $v_{n+1}|_n=v_n$.
The initial distribution of $\widehat{\Gamma}$ is the marginal
distribution $\pi^S$ of $\pi$ on $S$, if $s\in S= V_1$, then $\pi
^S(s)=\sum_{x\in X}\pi(x,s)$ and $\pi^S(v)=0$ for $v\in V\setminus V_1$.
Let us note that the original game $\Gamma$ and the auxiliary game
$\widehat{\Gamma}$ have the same sets of strategies. Indeed a
behavioral strategy in $\Gamma$ is a mapping from $V$ to probability
distributions over actions. Thus, each behavioral strategy in $\Gamma$
is a stationary strategy in $\widehat{\Gamma}$. On the other hand
however, each state of $\widehat{\Gamma}$ ``contains''
all previously visited states and all played actions; thus, for all
useful purposes,
in $\widehat{\Gamma}$ behavioral strategies and stationary strategies
coincide.
Now suppose that $(v_1, i_1, j_1, v_2,i_2, j_2, \ldots)$ is a play in
$\widehat{\Gamma}$. Then $v_{n+1}|_n=v_n$ for all $n$ and there exists
$v\in V_\infty$ such that $v|_n=v_n$ for all $n$.
Thus, defining a payoff on infinite histories in $\widehat{\Gamma}$
amounts to defining a payoff on $V_\infty$.
\begin{lemma}\label{transfer}
Given a Borel function $f$ on $H_\infty$, there exists a Borel function
$\widehat{f}$ on $V_\infty$ such that
\begin{equation}
\label{eqfinal}
\mathbb{E}_{\P_{\sigma,\tau}}(f)=\mathbb{E}_{\mathbb{Q}_{\sigma
,\tau}} (
\widehat{f} ).
\end{equation}
\end{lemma}
Therefore, playing in $\Gamma$ with strategies
$(\sigma,\tau)$ and payoff $f$ is the same as playing in $\widehat
{\Gamma}$ with
the same strategies and payoff $\widehat{f}$.
By Martin~\cite{Martin98} or Maitra and Sudderth~\cite{maitra2003},
the stochastic game
$\widehat{\Gamma}$ with payoff $\widehat{f}$ has a value implying that
$\Gamma$ with payoff $f$ has the same value, which completes the proof
of Theorem~\ref{theo3}.
The three next sections are dedicated to the proof of Lemma~\ref{transfer}.
\subsubsection{Regular conditional probability of finite time
events with respect to finite observed histories}\label{secfinite}
For $m\geq n \geq1$, we define $\Phi_{n,m}$ from $ H_{\infty} \times
{\overline V}_\infty$ to $[0, 1 ] $ by
\[
\Phi_{n,m} ( h, v) =
\cases{ \displaystyle
\frac{ \sum_{h', h'|_n = h|_n, \Theta(h'|_m )= v
|_m}\alpha( h' |_{m})}{ \beta( v |_{m}) },& \quad\mbox{if }$\Theta( h |_{n})= v
|_{n}$,\vspace*{3pt}
\cr
0,& \quad\mbox{otherwise}. }
\]
This corresponds to the joint probability of the players on the
realization of the history $h$ up to stage $n$, given the observed
history $v$ up to stage $m$.
Since\vspace*{1pt} $\Phi_{n,m} ( h, v)$ depends only on $h|_n$ and $v|_m$, we can
see $\Phi_{n,m}$ as a function defined
on $H_n \times\overline{V}_m $ and note that its support is included
in $\overline{H}_n \times\overline{V}_m$.
On the other hand, since each set $U\in\mathcal{H}_n$ is a finite union
of cylinders $h_n^+$ for $h_n\in H_n$ such that $h_n^+\subset U$, $\Phi
_{n,m}$ can be seen as
a mapping from $\mathcal{H}_n\times\overline{V}_\infty$ into $[0,1]$,
where $\Phi_{n,m}(U,v)=\sum_{h_n, h_n^+\subseteq U}\Phi_{n,m}(h_n,v)$.
Bearing this last observation in mind, we have the following.
\begin{lemma}\label{lemkernel}
For every $m\geq n\geq1$, $\Phi_{n,m}$ is a probability kernel from
$(\overline{V}_\infty, {\mathcal V}_m)$ to $(H_\infty, {\mathcal H}_n)$.
\end{lemma}
\begin{pf}
Since $\sum_{h_n\in H_n}\Phi_{n,m}(h_n,v)=1$ for $v\in\overline
{V}_\infty$, $ \Phi_{n,m} ( \cdot,v)$ defines a probability on
$\mathcal{H}_n$.
Moreover, for any $U\in\mathcal{H}_n$, $\Phi_{n,m}( U, v)$ is a
function of the $m$ first components of $v$ hence is ${\mathcal V}_m$-measurable.
\end{pf}
\begin{lemma}\label{link}
Let $m\geq n \geq1$ and $(\sigma,\tau)$ be a pair of strategies.
Then, for every $v_m\in\overline{V}_m$ such that $\mathbb{Q}_{\sigma
,\tau
}(v_m^+)=\P_{\sigma,\tau}(v_m^*)>0$,
and every $h_n\in H_n$:
\[
\P_{\sigma,\tau}\bigl(h_n^+|v_m^*\bigr)=
\Phi_{n,m}(h_n,v_m).
\]
\end{lemma}
\begin{pf}
Let $v_m = ( s_1, i_1, j_1, \ldots, s_m)$ and $h_n \in H_n$,
\begin{eqnarray*}
&&\hspace*{-3pt} \P_{\sigma,\tau}\bigl(h_n^+|v_m^*\bigr)\\
&&\hspace*{-3pt}\qquad =
\frac{\P_{\sigma,\tau}(h_n^+ \cap v_m^*)}{ \P_{\sigma,\tau
}(v_m^*) }
\\
&&\hspace*{-3pt}\qquad=
\cases{ \displaystyle\frac{ \sum_{h', h'|_n = h_n, \theta( h'|_m) = v_m
{\alpha( h' |_m )
W(i_1,j_1,\ldots,j_{m-1})
{\beta(v_m) W(i_1,j_1,\ldots,j_{m-1})}, & \hspace*{-4pt}\quad$\mbox{if }
\Theta(h _{n})= v_m |_{n}$,\vspace*{3pt}
\cr
0,&
\hspace*{-4pt}$\quad\mbox{otherwise}$,}
\end{eqnarray*}
where $W(i_1,j_1,\ldots,j_{m-1})=\prod_{t\leq
m-1}\sigma(v_m|_t)(i_t)\tau(v_m|_t)(j_t)$.
After simplification, we recognize on the right the definition of
$\Phi_{n,m}(v_m,h_n)$.
\end{pf}
We deduce the following lemma.
\begin{lemma}\label{phi-m-n}
For every pair of strategies $(\sigma,\tau)$, each $W\in\overline
{\mathcal{V}}_m$ and $U\in\mathcal{H}_n$ we have
\begin{equation}
\label{eqmn} \P_{\sigma,\tau}\bigl(U \cap\Theta^{-1}(W)\bigr)=\int
_W \Phi_{n,m}(U, v)\mathbb{Q}_{\sigma,\tau}(dv) .
\end{equation}
\end{lemma}
\begin{pf}
Clearly, it suffices to prove (\ref{eqmn}) for cylinders $U=h_n^+$ and
$W=v_m^+$
with $\beta(v_m)>0$.
We have
\begin{eqnarray*}
\int_{v_m^+} \Phi_{n,m}(h_n,v)
\mathbb{Q}_{\sigma,\tau}(dv) &=& \Phi_{n,m}(h_n,v_m)
\mathbb{Q}_{\sigma,\tau}\bigl(v_m^+\bigr)
\\
&=& \P_{\sigma,\tau}\bigl(h_n^+ | v_m^*\bigr)
\mathbb{Q}_{\sigma,\tau}\bigl(v_m^+\bigr)
\\
&=& \P_{\sigma,\tau}\bigl(h_n^+ | v_m^*\bigr)
\P_{\sigma,\tau}\bigl(v_m^*\bigr)
\\
&=& \P_{\sigma,\tau}\bigl(h_n^+ \cap v_m^*\bigr).
\end{eqnarray*}
\upqed
\end{pf}
Note that (\ref{eqmn}) can be equivalently written as: for every pair
of strategies $(\sigma,\tau)$,
each $W^*\in\overline{\mathcal{V}}^*_m$ and $U\in\mathcal{H}_n$
\begin{equation}
\label{eqmn^*} \P_{\sigma,\tau}\bigl(U \cap W^*\bigr)=\int_{W^*}
\Phi_{n,m}\bigl(U, \Theta(h)\bigr)\P_{\sigma,\tau}(dh).
\end{equation}
\subsubsection{Regular conditional probability of finite time events
with respect
to infinite observed histories} \label{secdwa}
In this paragraph, we prove that instead of defining one application
$\Phi_{n,m}$ for every pair $(m,n) $ such that $m\geq n\geq1$, one can
define a unique probability kernel $\Phi_n$ from $(\Omega_n,
{\mathcal
V}_\infty)$ to $(H_\infty,{\mathcal H}_n)$, with $\mathbb{Q}_{\sigma
,\tau
}(\Omega_n)=1$, for all $(\sigma, \tau)$, such that the extension of
Lemma~\ref{phi-m-n} holds.
For $h\in H_\infty$, let
\[
\Omega_{h} = \bigl\{ v\in\overline{V}_\infty\vert\mbox{$\Phi_{n,m}(h,v)$ converges as $m\uparrow\infty$} \bigr\}.
\]
The domain $\Omega_{h}$ is measurable (see Kallenberg~\cite
{Kallenberg97}, page~6, e.g.). Recall that $\Omega_{h}$ depends only on
$h|_n$ and write also $\Omega_{h|_n}$ for $ \Omega_{h}$. Let then
\[
\Omega_n = \bigcap_{h_n\in H_n}
\Omega_{h_n}.
\]
We define $\Phi_n \dvtx H_\infty\times\overline{V}_\infty\to[0,1]$ by
$\Phi_n = \lim_{m\rightarrow\infty} \Phi_{n,m}$ on $ H_\infty
\times
\Omega_n $ and $0$ otherwise. As a limit of a sequence of measurable
mappings $\Phi_n$ is measurable (see Kallenberg~\cite{Kallenberg97},
page~6, e.g.).
\begin{lemma}\label{phi-n}
\textup{(i)} For each pair of strategies $(\sigma,\tau)$,
$\mathbb{Q}_{\sigma,\tau}(\Omega_n)=1$.
\textup{(ii)}
For each $v\in\Omega_n$, $\sum_{h_n\in H_n} \Phi_n(h_n, v)=1$.
\textup{(iii)}
For each $U\in\mathcal{H}_n$
the mapping $v \mapsto\Phi_n(U, v)$ is
a measurable mapping from $(\overline{V}_\infty,\mathcal{V}_\infty
)$ to
$\mathbb{R}$.
\textup{(iv)}
For each pair of strategies $(\sigma,\tau)$, for each $U\in\mathcal{H}_n$
and each $W\in\mathcal{V}_\infty$
\begin{equation}
\label{eqn} \P_{\sigma,\tau}\bigl(U\cap\Theta^{-1}(W)\bigr)=\int
_W \Phi_n(U,v) \mathbb{Q}_{\sigma,\tau}(dv).
\end{equation}
\end{lemma}
\begin{pf}
(i)
For $h_n\in H_n$ and
each pair of strategies $\sigma,\tau$
we define on $H_\infty$ a sequence of random variables $Z_{h_n,m}$,
$m\geq
n$,
\[
Z_{h_n,m} = \P_{\sigma,\tau} \bigl[ h_n^+ |
\mathcal{V}_m^* \bigr].
\]
As a conditional expectation of a bounded random variable with respect
to an increasing sequence of $\sigma$-algebras, $Z_{h_n,m}$ is a
martingale (with respect to
$\P_{\sigma,\tau}$), hence converges $\P_{\sigma,\tau}$-almost surely
and in $L^1$ to the random variable $Z_{h_n}=\P_{\sigma,\tau}[ h_n^+ |
\mathcal{V}_\infty^*]$.
For $m\geq n$, we define the mappings $\psi_{n,m}[h_n] \dvtx \overline
{H}_\infty\to[0,1]$,
\[
\psi_{n,m}[h_n](h)= \Phi_{n,m}
\bigl(h_n,\Theta(h)\bigr).
\]
Let us show that for each $h_n\in H_n$, $\psi_{m,n}[h_n]$ is a version
of the conditional expectation $\mathbb{E}_{\P_{\sigma,\tau
}}[\mathbh{1}_{h_n}| \mathcal
{V}_m^*] = \P_{\sigma,\tau} [ h_n^+ | \mathcal
{V}_m^* ]$.
First note that $\psi_{n,m}[h_n]$ is $(H_\infty,\mathcal{V}_m^*)$
measurable. Lemma~\ref{link} implies that, for $h\in\operatorname
{supp}( \P_{\sigma
,\tau} ) \subset\overline{H}_\infty$,
$\psi_{n,m}[h_n](h)=\Phi_{n,m}(h_n,\Theta(h))=\P_{\sigma,\tau}(h_n^+
|
v|_m^*)=\P_{\sigma,\tau}(h_n^+ | \mathcal{V}_m^* )(h)$, where
$v=\Theta(h)$. Hence, the claim.
Since $\psi_{n,m}[h_n]$ is a version of $\P_{\sigma,\tau}(h_n^+ |
\mathcal{V}_m^* )$, its limit
$\psi_n[h_n]$ exists and is a version of $\P_{\sigma,\tau}(h_n^+ |
\mathcal{V}_\infty^* )$, $\P_{\sigma,\tau}$-almost surely. In particular,
\begin{enumerate}[(C1)]
\item[(C1)]
the set $\Theta^{-1}(\Omega_{h_n}) = \{ h\in H_\infty\vert \mbox
{$\lim_m \psi_{n,m}[h_n](h)$ exists} \}$
is $\mathcal{V}_\infty^*$ measurable and has $\P_{\sigma,\tau
}$-measure $1$,
\item[(C2)]
for each $W^*\in\mathcal{V}_\infty^*$, $\int_{W^*} \psi_n[h_n](h)
\P_{\sigma,\tau}(dh)=
\int_{W^*} \mathbb{E}[\mathbh{1}_{h_n^+} | \mathcal{V}_\infty^*]
\P_{\sigma,\tau}=\break
\P_{\sigma,\tau}(W^*\cap h_n^+)$.
\end{enumerate}
Note that (C1) implies that $\mathbb{Q}_{\sigma,\tau}(\Omega
_n)=1$.
(ii)
If $v\in\Omega_n$ then, for all $h_n\in H_n$, $\Phi_{n,m}(h_n,v)$
converges to
$\Phi_n(h_n, v)$. But, by Lemma~\ref{lemkernel}, $\sum_{h_n\in H_n}
\Phi_{n,m}(h_n,v)=1$.
The\vspace*{1pt} sum being with finitely many nonzero terms one\vspace*{1pt} has $\sum_{h_n\in
H_n} \Phi_n(h_n, v)=1$.
(iii) Was proved before the lemma.
(iv)
Since
$\int_W \Phi_n(h_n,v) \mathbb{Q}_{\sigma,\tau}(dv) =
\int_{\Theta^{-1}(W)} \psi_n[h_n](h) \P_{\sigma,\tau}(dh)$ for $W
\in
\mathcal{V}_\infty$, using
(C2) we get
\[
\P_{\sigma,\tau}\bigl(h_n^+\cap\Theta^{-1}(W)\bigr)=\int
_W \Phi_n(h_n,v)
\mathbb{Q}_{\sigma,\tau}(dv)
\]
for $U\in\mathcal{V}_\infty$.
\end{pf}
\subsubsection{Regular conditional probability of infinite time events
with respect
to infinite observed histories} \label{sectrzy}
In this section, using Kolmogorov extension theorem we construct from
the sequence $\Phi_n$ of probability kernels from $(\Omega_n,
{\mathcal V}_\infty)$ to $(H_\infty,{\mathcal H}_n)$, one
probability kernel $\Phi$ from $(\Omega_\infty, {\mathcal V}_\infty
)$ to $(H_\infty,{\mathcal H}_n)$, with $\mathbb{Q}_{\sigma,\tau
}(\Omega_\infty
) =1$, for all $(\sigma, \tau)$.
\begin{lemma}\label{phi}
There exists a measurable subset $\Omega_\infty$ of $V_\infty$ such
that, for all strategies $\sigma,\tau$:
\begin{itemize}
\item
$\mathbb{Q}_{\sigma,\tau}(\Omega_\infty)=1$ and
\item
there exists a probability kernel $\Phi$ from $(\Omega_\infty,
{\mathcal V}_\infty)$ to $(H_\infty,{\mathcal H}_\infty)$ such that
for each $W\in\mathcal{V}_\infty$ and $U\in\mathcal{H}_\infty$
\begin{equation}
\label{eqfinall} \P_{\sigma,\tau}\bigl(U\cap\Theta^{-1}(W)\bigr)=\int
_W \Phi(U,v) \mathbb{Q}_{\sigma,\tau}(dv).
\end{equation}
\end{itemize}
\end{lemma}
Before proceeding to the proof, some remarks are in order.
A probability kernel having the property given above is called a
regular conditional probability.
For given strategies $\sigma$ and $\tau$, the existence of a transition
kernel $\kappa_{\alpha,\beta}$ from $(V_\infty, {\mathcal V}_\infty)$
to $(H_\infty,{\mathcal H}_\infty)$ such that for each $U\in\mathcal
{V}_\infty$ and $A\in\mathcal{H}_\infty$
\[
\P_{\sigma,\tau}\bigl(A\cap\Theta^{-1}(U)\bigr)=\int
_U \kappa_{\sigma,\tau}(A,v) \mathbb{Q}_{\sigma,\tau}(dv)
\]
is well known provided that $V_\infty$ is a Polish space and $\mathcal
{V}_\infty$ is the Borel $\sigma$-algebra. In the current framework it
is easy to introduce an appropriate metric on $V_\infty$ such that
this condition is satisfied thus the existence of $\kappa_{\sigma
,\tau
}$ is immediately assured.
The difficulty in our case comes from the fact that we look for a
regular conditional probability
which is \textit{common for all probabilities} $\P_{\sigma,\tau}$, where
$(\sigma,\tau)$ range over all strategies of both players.
\begin{pf*}{Proof of Lemma~\protect\ref{phi}}
We follow the notation of the proof of Lemma~\ref{phi-n} and define
$\Omega_\infty=\bigcap_{n\geq1} \Omega_n$. Let $(\sigma,\tau)$
be a
couple of strategies. For every \mbox{$n\geq1$}, $\mathbb{Q}_{\sigma,\tau
}(\Omega
_n)=1$, hence $\mathbb{Q}_{\sigma,\tau}(\Omega_\infty)=1$. By
Lemma~\ref{phi-n}(ii), given $v\in\Omega_\infty$, the sequence $\{\Phi
_n(\cdot,
v) \}_{n\geq1}$
of probabilities on $\{(H_\infty, {\mathcal H}_n)\}_{n\geq1}$ is well
defined. Let us show that this
sequence satisfies the condition of Kolmogorov's extension theorem.
In fact $\Phi_{n,m}(\cdot, v)$ is defined on the power set of $H_{n}$ by
\[
\forall A \subset H_{n}, \qquad \Phi_{n,m}(A, v)=\sum
_{h_n\in A} \Phi_{n,m}(h_n, v).
\]
Thus, for every $h_n \in H_n$, we have
\begin{eqnarray*}
\Phi_{n,m}(h_n, v)&=&\frac{\P_{\sigma,\tau}(v|_m^* \cap
h_n^+)}{\P_{\sigma,\tau}(v|_m^*)}
\\
&=& \frac{\P_{\sigma,\tau}(v|_m^* \cap(h_n\times I \times J \times
X\times S)^+)}{\P_{\sigma,\tau}(v|_m^*)}
\\
&=&\Phi_{n+1,m}\bigl(h_n \times(I \times J \times X\times S),
v\bigr).
\end{eqnarray*}
Taking the limit, we obtain the same equality for $\Phi_n$ and $\Phi
_{n+1}$ hence the compatibility condition.
By the Kolmogorov extension theorem for each $v\in\Omega$,
there exists a measure $\Phi(\cdot, v)$
on $(H_\infty,\mathcal{H}_\infty)$ such that
\[
\Phi\bigl(h_n^+, v\bigr)= \Phi_n\bigl(h_n^+,
v\bigr)
\]
for each $n$ and each $h_n\in H_n$.
Let us prove that, for each $U\in\mathcal{H}_\infty$, the mapping
$v\mapsto\Phi(U,v)$ is $\mathcal{V}_\infty$-measurable on $\Omega
_\infty$.
Let $\mathcal{C}$ be the class of sets $A \in\mathcal{H}_\infty$ such
that $\Phi(A,\cdot)$ has this property. By Lemma~\ref{phi-n},
$\mathcal
{C}$ contains the $\pi$-system
consisting of cylinders generating $\mathcal{H}_\infty$. To show that
$\mathcal{H}_\infty\subseteq\mathcal{C}$ it suffices to show that
$\mathcal{C}$ is a $\lambda$-system. Let $A_i$ be an increasing
sequence of sets belonging to
$\mathcal{C}$. Since, for each $v\in\overline{V}_\infty$, $\Phi
(\cdot
,v)$ is a measure, we have $\Phi(\bigcup_n A_n,v)=\sup_n \Phi(A_n,v)$.
However, $v\mapsto\sup_n \Phi(A_n,v)$ is measurable as a supremum of
measurable mappings $v\mapsto\Phi(A_n,v)$.
Let $A\supset B$ be two sets belonging to $\mathcal{C}$. Then $\Phi
(A\setminus B,v) + \Phi(B,v)=\Phi(A,v)$ by additivity of measure and
$v\mapsto\Phi(A\setminus B,v)= \Phi(A,v) - \Phi(B,v)$ is measurable as
a difference of measurable mappings.
To prove (\ref{eqfinall}), take a measurable subset $W$ of $\overline
{V}_\infty$ and consider the set function
\[
\mathcal{H}_\infty\ni U \mapsto\int_W \Phi(U,dv)
Q_{\sigma,\tau}(dv).
\]
Since $\Phi(\cdot,v)$ is nonnegative this set function is a measure on
$(H_\infty,\mathcal{H}_\infty)$. However, by Lemma~(\ref{phi-n}), this
mapping is equal to $U \mapsto\P_{\sigma,\tau}( U \cap\Theta^{-1}(W))$
for $U$ belonging to the $\pi$-system of cylinders generating
$\mathcal
{H}_\infty$. But two measures equal on a generating $\pi$-system are
equal, which terminates the proof of (\ref{eqfinall}).
\end{pf*}
A standard property of probability kernels and the fact that $\Omega
_\infty$ has measure $1$ imply:
\begin{corollary}\label{cortransfer}
Let $f : H_\infty\to[0,1]$ be $\mathcal{H}_\infty$-measurable mapping.
Then the mapping $\widehat{f} : V_\infty\to[0,1]$ defined by
\[
\widehat{f}(v) =
\cases{ \displaystyle\int_{H_\infty}
f(h)\Phi(dh, v), & \quad\mbox{if $v\in\Omega_\infty$},\vspace*{3pt}
\cr
0,
& \quad$\mbox{otherwise}$,}
\]
is $\mathcal{V}_\infty$-measurable and
\[
\mathbb{E}_{\P_{\sigma,\tau}}[ f ] = \mathbb{E}_{\mathbb
{Q}_{\sigma,\tau}}[ \widehat{f} ]\qquad
\forall\sigma, \tau.
\]
\end{corollary}
\begin{remark}
In the previous proof, we proceeded through a reduction from a
symmetric repeated game to a stochastic game in order to apply Martin's
existence result. The same procedure can be applied for $N$-player
repeated games. Let us consider a $N$-player symmetric signaling
repeated game. One defines a conditional probability and therefore
associates to all Borel payoffs $f^i$ on plays, $i \in N$ an associated
Borel evaluation $\widehat{f}^i$ on the space of observed plays, therefore,
reducing the problem to a $N$-player stochastic game with Borelian payoffs.
For example, Mertens \cite{Mertens86} showed the existence
of pure $\varepsilon$-Nash equilibrium in $N$-person stochastic games
with Borel payoff functions where at each stage at most one of the
players is playing. Using the previous reduction, one can deduce the
existence of
pure $\varepsilon$-Nash equilibrium in $N$-person symmetric repeated
games with Borel payoff functions where at each stage at most one of
the players is playing.
\end{remark}
\section{Uniform value in recursive games with nonnegative
payoffs}\label{uniform}
In Section~\ref{model} and Section~\ref{secsymmetric}, we focused on
Borel evaluations. In this last section, we focus on the family of mean
average of the $n$ first stage rewards and the corresponding uniform value.
\begin{definition}
For each $n \geq1$, the \textit{mean expected payoff} induced by
$(\sigma,\tau)$ during the first $n$ stages is
\[
\gamma_n (\sigma, \tau)=\mathbb{E}_{ \sigma, \tau} \Biggl(
\frac
{1}{n}\sum_{t=1}^n g
(x_t,i_t,j_t) \Biggr).
\]
\end{definition}
\begin{definition}\label{stocuni1} \label{stocuni}
Let $v$ be a real number.
A strategy $\sigma^*$ of player $1$ \textit{guarantees} $v$ \textit{in the
uniform sense} in $(\Gamma, g)$ if for all $\eta>0$ there exists $n_0
\geq1$ such that
\begin{equation}
\forall n\geq n_0, \forall\tau\in\mathcal{T},\qquad \gamma
_n\bigl(\sigma^*,\tau\bigr) \geq v-\eta.
\end{equation}
Player $1$ can \textit{guarantee} $v$ \textit{in the uniform sense} in
$(\Gamma, g)$ if for all $\varepsilon>0$ there exists a strategy
$\sigma^*\in\Sigma$ which guarantees $v-\varepsilon$ in the uniform
sense.
A symmetric notion holds for player $2$.
\end{definition}
\begin{definition}
The \textit{uniform $\max\min$}, denoted by $\underline{v}_\infty$, is
the supremum of all the payoff that player $1$ can guarantee in the
uniform sense. A \textit{uniform $\min\max$} denoted by $\overline
{v}_\infty$ is defined in a dual way.
If both players can guarantee $v$ in the uniform sense, then $v$ is the
\textit{uniform value} of the game $(\Gamma, g)$ and denoted by
$v_\infty$.
\end{definition}
Many existence results have been proven in the literature concerning
the uniform value and uniform $\max\min$ and $\min\max$; see, for
example, Mertens, Sorin and Zamir \cite{Mertens94} or Sorin \cite
{Sorin2002}. Mertens and Neyman \cite{Mertens81} proved that in a
stochastic game with a finite state space and finite actions spaces,
where the players observe past payoffs and the state, the uniform value
exists. Moreover, the uniform value is equal to the limsup-mean value
and for every $\varepsilon>0$ there exists a strategy which guarantees
$v_\infty-\varepsilon$ both in the limsup-mean sense and in the
uniform sense.
In general, the uniform value does not exist (either in games with
incomplete information on both sides or in stochastic games with
signals on the actions) and in particular its existence depends upon
the signaling structure.
\begin{remark}
For $n\geq1$, the \textit{$n$-stage game} $(\Gamma_n, g)$ is
the zero-sum game with normal form $(\Sigma, \mathcal{T}, \gamma_n)$
and value $v_n$.
It is interesting to note that in the special case of symmetric
signaling repeated games with a finite set of states and finite set of
signals, a uniform value may not exist, since even the sequence of
values $v_n $ may not converge (Ziliotto \cite{Ziliotto2013}), but
there exists a value for any Borel evaluation by Theorem~\ref{theo3}.
\end{remark}
We focus now on the specific case of recursive games with nonnegative
payoff defined as follows.
\begin{definition}
Recall that a state is \textit{absorbing} if the probability to stay
in this state is 1 for all actions and the payoff is also independent
of the actions played. A repeated game is \textit{recursive} if the
payoff is equal to $0$ outside the
absorbing states. If all absorbing payoffs are nonnegative, the
game is \textit{recursive} and \textit{nonnegative}.
\end{definition}
Solan and Vieille \cite{Solan2002} have shown the existence of a
uniform value in nonnegative recursive games where the players observe
the state and past actions played. We show that the result is true
without assumption on the signals to the players.
In a recursive game, the limsup-mean evaluation and the limsup
evaluation coincide.
If the recursive game has nonnegative payoffs, the sup evaluation, the
limsup evaluation and the limsup-mean evaluation both coincide. So,
Theorem~\ref{sup} implies the existence of the value with respect to
these evaluations. Using a similar proof, we obtain the stronger theorem.
\begin{theorem}\label{recursive} A recursive game with nonnegative
payoffs has a uniform value~$v_\infty$, equal to the sup value and the
limsup value. Moreover, there exists a strategy of player $2$ that
guarantees $v_\infty$.
\end{theorem}
The proof of the existence of the uniform value is similar to the proof
of Proposition~\ref{FF} while using a specific sequence of strategic
evaluations.
\begin{pf*}{Proof of Theorem~\protect\ref{recursive}}
The sequence of stage payoffs is nondecreasing on each history:
$0$ until absorption occurs and then constant, equal
to some nonnegative real number. In particular, the payoff converges
and the $\operatorname{limsup}$ can be replaced by a limit.
Let $\sigma$ be a strategy of player $1$ and $\tau$ be a strategy of
player $2$, then $\gamma_n(\sigma, \tau)$ is nondecreasing in $n$.
This implies that the corresponding sequence of values $(v_n)_{n\in
\mathbb{N}
}$ is nondecreasing in $n$. Denote $v=\sup_n v_n$ and let us show
that $v$ is the uniform value.
Fix $\varepsilon>0$, consider $N$ such that $v_N\geq v-\varepsilon$ and
$\sigma^*$ a strategy of player 1 which is optimal in $\Gamma_N$. We
have for each $\tau$ and, for every $n\geq N$,
\[
\gamma_n\bigl(\sigma^*, \tau\bigr) \geq\gamma_N\bigl(
\sigma^*, \tau\bigr)\geq v_N \geq v-\varepsilon.
\]
Hence, the strategy $\sigma^*$ guarantees $v-\varepsilon$ in the
uniform sense. This is true for every positive $\varepsilon$, thus player
$1$ guarantees $v$ in the uniform sense.
Using the monotone convergence theorem, we also have
\begin{eqnarray*}
\gamma^*\bigl(\sigma^*,\tau\bigr)&=&\mathbb{E}_{\sigma^*,\tau} \Biggl( \lim
_n \frac{1}{n}\sum_{t=1}^n
g(x_t,i_t,j_t) \Biggr) \\
&=& \lim
_n \mathbb{E}_{\sigma^*,\tau} \Biggl( \frac{1}{n}\sum
_{t=1}^n g(x_t,i_t,j_t)
\Biggr) \\
&\geq & v-\varepsilon.
\end{eqnarray*}
We now show that player $2$ can also guarantee $v$ in the uniform
sense. Consider for every $n$, the set
\[
K_n=\bigl\{\tau, \forall\sigma, \gamma_n(\sigma,
\tau)\leq v\bigr\}.
\]
$K_n$ is nonempty because it contains an optimal strategy for player 2
in $\Gamma_n$ (since $v_n\leq v$). The set of strategies of player 2 is
compact, hence by continuity of the $n$-stage payoff $\gamma_n$, $K_n$
is itself compact. $\gamma_n \leq\gamma_{n+1}$ implies
$K_{n+1}\subset
K_n$ hence $\bigcap_n K_n\neq\varnothing$: there exists $\tau^*$
such that for every strategy of player $1$, $\sigma$ and for every
positive integer $n$, $\gamma_n(\sigma,\tau) \leq v$.
It follows that both players can guarantee $v$, thus $v$ is the uniform
value.
By the monotone convergence theorem, we also have
\[
\gamma^*\bigl(\sigma,\tau^*\bigr)= \mathbb{E}_{\sigma,\tau^*} \Biggl( \lim
_n \frac{1}{n}\sum_{t=1}^n
g(x_t,i_t,j_t) \Biggr)
= \lim
_n \mathbb{E}_{\sigma,\tau^*} \Biggl( \frac{1}{n}\sum
_{t=1}^n g(x_t,i_t,j_t)
\Biggr) \leq v.
\]
Hence, $v$ is the sup and limsup value.
\end{pf*}
\begin{remark}
The fact that the sequence of $n$-stage values $(v_n)_{n\geq1}$ is
nondecreasing is not enough to ensure the existence of the uniform
value. For example, consider the Big Match \cite{Blackwell68} with no
signals: $v_n=1/2$ for each $n$, but there is no uniform value.
\end{remark}
\begin{remark}
The theorem states the existence of a $0$-optimal strategy for
player 2 but player 1 may only have $\varepsilon$-optimal
strategies. For example, in the following MDP, there are two
absorbing states, two nonabsorbing states with payoff~$0$ and two
actions $\mathit{Top}$ and $\mathit{Bottom}$:
\[
\begin{tabular}{cc}
$\lleft(
\begin{array}{c}
1/2 (s_1) + 1/2 (s_2)\vspace*{1pt} \\
0^*\\
\end{array}
\rright)$ & $\lleft(
\begin{array}{c}
s_2 \\
1^*\\
\end{array}
\rright)$. \\
$s_1$ & $s_2$
\end{tabular}
\]
The starting state is $s_1$ and player $1$ observes nothing. A good
strategy is to play $Top$ for a long time and then $Bottom$. While
playing $Bottom$, the process absorbs and with a strictly positive
probability the absorption occurs in state $s_1$ with absorbing payoff
$0$. So
player $1$ has no strategy which guarantees the uniform value of 1.
\end{remark}
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 4,472
|
Home >>Search Speakers >> Richie Contartesi
Richie Contartesi
Author of "In Spite of the Odds"
College Walk-on to Professional Football Player
Motivational Keynote Speaker
Richie Contartesi | Motivational Keynote Speaker | 5'7" Walk-on to Professional Football Player
On August 21st, 2009, Richie Contartesi achieved his first major goal, earning a full D1 football scholarship in the SEC (South Eastern Conference) at Ole Miss standing 5'7″ and 150lb. More importantly, he learned how to re-ignite a relentless resilience which has allowed him to continue to succeed in other areas of his life. He went on to achieve many of his biggest goals like becoming a starter at Ole Miss, playing Arena football professionally, writing a #1 bestseller, building a successful speaking and training business, a real estate investment company and now is the owner of a CrossFit gym. Richie doesn't just speak about football but shares how the person you become in pursuit of going after a major goal is more important than just achieving the goal itself.
Who is Richie Contartesi? He is never the smartest, the biggest, the most talented, or the most gifted and in spite of all his childhood issues like failing 5th grade, watching his parents get divorced, and almost taking his own life. You will see a man who looks in the face of adversity and says "Bring it on". You will see a kind and loving man with a big heart. You will see a man who laughs in the face of fear and you will be inspired by his relentless resilience to succeed.
It all started in May of 1998. After Richie failed 5th grade and almost took his own life. He went on the internet and looked up the academic requirements to play college football, his childhood dream. That night he had no idea what he was doing, but when he printed out those requirements, what he did was write down his goal and his life changed forever.
His dad always said to him "Have the dirtiest uniform on the field, listen to your coach, and keep your mouth shut." Richie followed that model in football and also in business. And now before he does anything, he always writes down his goal, builds relationships and finds mentors to surround himself with, he overcomes fear, and he is relentless until he wins. All lessons he learned in the face of adversity.
To book Richie Contartesi call Executive Speakers Bureau at 901-754-9404.
Reignite The Relentless Within:
Four key personal commitments to capitalize on adversity, optimize performance levels, and evolve into a goal-achieving machine.
What makes a professional successful? First, it's reigniting your relentless mindset. The mindset we all once had as kids when we didn't care what people thought and had no fear. Richie guides his audience on a self-reflective journey to set and achieve goals, capitalize on adversity, optimize performance levels, and re-ignite the relentless within. He shares his personal journey which earned him a full Division 1 football scholarship to SEC powerhouse Ole Miss. Standing at only 5'7" and weighing 150 pounds, Richie's relentless attitude, larger than life presence, and message will bring you entertainment, knowledge, laughter, joy, and even a tear. You'll have an opportunity to look deep within yourself and begin your own personal journey to success.
You'll discover how to:
Reignite Your Relentless Mindset
Build Mutually Beneficial Relationships
Easily Overcome the Fear of Rejection and Failure
Embody Your Own Relentless Resilience
Your Custom Keynote
During the assessment process you and your team will work with Richie to customize the presentation based on your needs. Whether you are looking for high content, sales oriented, motivational, and or inspirational. You will work together to deliver the right message that will help you achieve your goals the fastest. Understanding your audience, team, objectives, goals, challenges, theme and more will help to structure the perfect keynote.
Decision Strategist
Gary Bradt
John Tartaglio
First and Only Person in History to Run and complete a Marathon Without Legs
James Malinchak
Empowering Audiences to Achieve Extraordinary Results
Gary Berntsen
Best-Selling Author of Jawbreaker
Stacey Flowers
Cal Fussman
Corporate Consultant on Interviewing and Storytelling
Matt Birk
NFL Man of the Year
Brad Nieder M.D
The Healthy Humorist
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 4,900
|
\section{Multi-choice Question-Answering}
We now investigate a number of intelligent baselines for QA.
We also study inherent biases in the data and try to answer the quizzes based simply on answer characteristics such as word length or within answer diversity.
Formally, let $S$ denote the story, which can take the form of any of the available sources of information -- \eg~plots, subtitles, or video shots.
Each story $S$ has a set of questions, and we assume that the (automatic) student reads one question $q^S$ at a time.
Let $\{a_{j}^S\}_{j=1}^{M}$ be the set of multiple choice answers (only one of which is correct) corresponding to $q^S$, with $M=5$ in our dataset.
The general problem of multi-choice question answering can be formulated by a three-way scoring function $f(S,q^S,a^S)$. This function evaluates the ``quality'' of the answer given the story and the question.
Our goal is thus to pick the best answer $a^S$ for question $q^S$ that maximizes $f$:
\begin{equation}
j^* = \arg\max_{j=1\ldots M} f(S, q^S, a_{j}^S) \,
\end{equation}
Answering schemes are thus different functions $f$.
We drop the superscript $(\cdot)^S$ for simplicity of notation.
\subsection{The Hasty Student}
\label{sec:hasty_student}
We first consider $f$ which ignores the story and attempts to answer the question directly based on latent biases and similarities.
We call such a baseline as the ``Hasty Student'' since he/she is not concerned to read/watch the actual story.
The extreme case of a hasty student is to try and answer the question by only looking at the answers.
Here, $f(S, q, a_{j}) = g_{H1}(a_{j}|{\bf a})$, where $g_{H1}(\cdot)$ captures some properties of the answers.
{\bf Answer length.}
We explore using the number of words in the multiple choices to find the correct answer and explore biases in the dataset.
As shown in Table~\ref{tab:qa_stats}, correct answers are slightly longer as it is often difficult to frame long deceiving answers.
We choose an answer by:
(i) selecting the longest answer;
(ii) selecting the shortest answer; or
(iii) selecting the answer with the most different length.
{\bf Within answer similarity/difference.}
While still looking only at the answers, we compute a distance between all answers based on their representations (discussed in Sec.~\ref{sec:representation}).
We then select our answer as either the most similar or most distinct among all answers.
{\bf Q and A similarity.}
We now consider a hasty student that looks at both the question and answer, $f(S, q, a_j) = g_{H2}(q, a_{j})$.
We compute similarity between the question and each answer and pick the highest scoring answer.
\subsection{The Searching Student}
\label{sec:sliding}
While the hasty student ignores the story, we consider a student that tries to answer the question by trying to locate a subset of the story $S$ which is most similar to both the question and the answer.
The scoring function $f$ is
\begin{equation}
f(S, q, a_{j}) = g_I(S, q) + g_I(S, a_{j}) \, .
\end{equation}
a factorization of the question and answer similarity.
We propose two similarity functions:
a simple windowed cosine similarity, and another using a neural architecture.
{\bf Cosine similarity with a sliding window.} We aim to find the best window of $H$ sentences (or shots) in the story $S$ that maximize similarity between the story and question, and story and answer.
We define our similarity function:
\begin{equation}
f(S, q, a_{j}) = \max_l \sum_{k = l}^{l+H} g_{ss}(s_k, q) + g_{ss}(s_k, a_{j}) \, ,
\end{equation}
where $s_k$ denotes a sentence (or shot) from the story $S$.
We use $g_{ss}(s, q) = x(s)^T x(q)$ as a dot product between the (normalized) representations of the two sentences (shots).
We discuss these representations in detail in Sec.~\ref{sec:representation}.
{\bf Searching student with a convolutional brain (SSCB).}
Instead of factoring $f(S, q, a_{j})$ as a fixed (unweighted) sum of two similarity functions $g_{I}(S, q)$ and $g_{I}(S, a_{j})$, we build a neural network that learns such a function.
Assuming the story $S$ is of length $n$, \eg~$n$ plot sentences or $n$ video shots, $g_{I}(S, q)$ and $g_{I}(S, a_{j})$ can be seen as two vectors of length $n$ whose $k$-th entry is $g_{ss}(s_k, q)$.
We further combine all $[g_I(S, a_{j})]_j$ for the 5 answers into a $n\times 5$ matrix.
The vector $g_{I}(S, q)$ is replicated $5$-times, and we stack the question and answer matrix together to obtain a tensor of size $n \times 5 \times 2$.
Our neural similarity model is a convnet (CNN), shown in Fig.~\ref{fig:model:cnn}, that takes the above tensor, and applies couple layers of $h = 10$, $1 \times 1$ convolutions to approximate a family of functions $\phi(g_I(S, q), g_I(S, a_{j}))$.
Additionally, we incorporate a max pooling layer with kernel size $3$ to allow for scoring the similarity within a window in the story.
The last convolutional output is a tensor with shape ($\frac{n}{3}, 5$), and we apply both mean and max pooling across the storyline, add them, and make predictions using softmax.
We train our network using cross-entropy loss and the Adam optimizer~\cite{kingma2014adam}.
\begin{figure}
\vspace{-5mm}
\centering
\includegraphics[width=0.95\linewidth,trim=0 0 0 0,clip]{figs/CNN.pdf}
\vspace*{-0.5cm}
\caption{\small Our neural similarity architecture (see text for details).}
\label{fig:model:cnn}
\vspace*{-0.5cm}
\end{figure}
\subsection{Memory Network for Complex QA}
Memory Networks were originally proposed for text QA and model complex three-way relationships between the story, question and answer.
We briefly describe MemN2N proposed by~\cite{Sukhbaatar2015} and suggest simple extensions to make it suitable for our data and task.
The input of the original MemN2N is a story and question.
The answering is restricted to single words and is done by picking the most likely word from the vocabulary $\mathcal{V}$ of 20-40 words.
Note that this is not directly applicable to MovieQA, as our data set does not have perform vocabulary-based answering.
A question $q$ is encoded as a vector $u \in \mathbb{R}^d$ using a word embedding $B \in \mathbb{R}^{d \times |\mathcal{V}|}$.
Here, $d$ is the embedding dimension, and $u$ is obtained by mean-pooling the representations of words in the question.
Simultaneously, the sentences of the story $s_l$ are encoded using word embeddings $A$ and $C$ to provide two different sentence representations $m_l$ and $c_l$, respectively.
$m_l$, the representation of sentence $l$ in the story, is used in conjunction with $u$ to produce an attention-like mechanism which selects sentences in the story most similar to the question via a softmax function:
\begin{equation}
p_l = \mathrm{softmax}(u^T m_l) \, .
\end{equation}
The probability $p_l$ is used to weight the second sentence embedding $c_l$, and the output $o = \sum_l p_l c_l$ is obtained by pooling the weighted sentence representations across the story.
Finally, a linear projection $W \in \mathbb{R}^{|\mathcal{V}| \times d}$ decodes the question $u$ and the story representation $o$ to provide a soft score for each vocabulary word
\begin{equation}
a = \mathrm{softmax}(W (o + u)) \, .
\end{equation}
The top scoring word $\hat a$ is picked from $a$ as the answer.
The free parameters to train are the embeddings $B$, $A$, $C$, $W$ for different words which can be shared across different layers.
Due to its fixed set of output answers, the MemN2N in the current form is not designed for multi-choice answering with open, natural language answers.
We propose two key modifications to make the network suitable for our task.
{\bf MemN2N for natural language answers.}
To allow the MemN2N to rank multiple answers written in natural language, we add an additional embedding layer $F$ which maps each multi-choice answer $a_j$ to a vector $g_j$.
Note that $F$ is similar to embeddings $B$, $A$ and $C$, but operates on answers instead of the question or story.
To predict the correct answer, we compute the similarity between the answers $g$, the question embedding $u$ and the story representation $o$:
\begin{equation}
\label{eq:memnet_multichoice_ans}
a = \mathrm{softmax}((o + u)^T g)
\end{equation}
and pick the most probable answer as correct.
In our general QA formulation, this is equivalent to
\begin{equation}
f(S, q, a_{j}) = g_{M1}(S, q, a_{j}) + g_{M2}(q, a_{j}),
\end{equation}
where $g_{M1}$ attends to parts of the story using the question, and a second function $g_{M2}$ directly considers similarities between the question and the answer.
{\bf Weight sharing and fixed word embeddings.}
The original MemN2N learns embeddings for each word based directly on the task of question-answering.
However, to scale this to large vocabulary data sets like ours, this requires unreasonable amounts of training data.
For example, training a model with a vocabulary size 14,000 (obtained just from plot synopses) and $d = 100$ would entail learning 1.4M parameters for each embedding.
To prevent overfitting, we first share all word embeddings $B, A, C, F$ of the memory network.
Nevertheless, even one embedding is still a large number of parameters.
We make the following crucial modification that allows us to use the Memory Network for our dataset.
We drop $B$, $A$, $C$, $F$ and replace them by a fixed (pre-trained) word embedding $Z \in \mathbb{R}^{d_1 \times |\mathcal{V}|}$ obtained from the Word2Vec model and learn a shared linear projection layer $T \in \mathbb{R}^{d_2 \times d_1}$ to map all sentences (stories, questions and answers) into a common space.
Here, $d_1$ is the dimension of the Word2Vec embedding, and $d_2$ is the projection dimension.
Thus, the new encodings are
\begin{equation}
u = T \cdot Z q; \, m_l, c_l = T \cdot Z s_l; \, \mathrm{and} \, g_j = T \cdot Z a_j .
\end{equation}
Answer prediction is performed as before in Eq.~\ref{eq:memnet_multichoice_ans}.
We initialize $T$ either using an identity matrix $d_1 \times d_1$ or using PCA to lower the dimension from $d_1 = 300$ to $d_2 = 100$.
Training is performed using stochastic gradient descent with a batch size of 32.
\subsection{Representations for Text and Video}
\label{sec:representation}
{\bf TF-IDF} is a popular and successful feature in information retrieval.
In our case, we treat plots (or other forms of text) from different movies as documents and compute a weight for each word.
We set all words to lower case, use stemming, and compute the vocabulary $\mathcal{V}$ which consists of words $w$ that appear more than $\theta$ times in the documents.
We represent each sentence (or question or answer) in a bag-of-words style with an TF-IDF score for each word.
{\bf Word2Vec.}
A disadvantage of TF-IDF is that it is unable to capture the similarities between words.
We use the skip-gram model proposed by~\cite{mikolov2013efficient} and train it on roughly 1200 movie plots to obtain domain-specific, $300$ dimensional word embeddings.
A sentence is then represented by mean-pooling its word embeddings.
We normalize the resulting vector to have unit norm.
{\bf SkipThoughts.}
While the sentence representation using mean pooled Word2Vec discards word order, SkipThoughts~\cite{skipthoughts} use a Recurrent Neural Network to capture the underlying sentence semantics.
We use the pre-trained model by~\cite{skipthoughts} to compute a $4800$ dimensional sentence representation.
{\bf Video.}
To answer questions from the video, we learn an embedding between a shot and a sentence, which maps the two modalities in a common space. In this joint space, one can score the similarity between the two modalities via a simple dot product. This allows us to apply all of our proposed question-answering techniques in their original form.
To learn the joint embedding we follow~\cite{ZhuICCV15} which extends~\cite{kiros15} to video.
Specifically, we use the GoogLeNet architecture~\cite{szegedy2014going} as well as hybrid-CNN~\cite{ZhouNIPS2014} to extract frame-wise features, and mean-pool the representations over all frames in a shot.
The embedding is a linear mapping of the shot representation and an LSTM on word embeddings on the sentence side, trained using the ranking loss on the MovieDescription Dataset~\cite{Rohrbach15} as in~\cite{ZhuICCV15}.
\section{Conclusion}
\vspace{-1mm}
We introduced the MovieQA data set which aims to evaluate automatic story comprehension from both video and text.
Our dataset is unique in that it contains several sources of information -- video clips, subtitles, scripts, plots and DVS. We provided several intelligent baselines and extended existing QA techniques to analyze the difficulty of our task.
Our benchmark with an evaluation server is online at~\url{http://movieqa.cs.toronto.edu}.
\iffalse
Owing to the variety in information sources, our data set is applicable to Vision, Language and Machine Learning communities.
We evaluate a variety of answering methods which discover the biases within our data and demonstrate the limitations on this high level semantic task.
Current state-of-the-art methods do not perform well and are often only a little better than random.
Using this data set we will create an evaluation campaign that can help breach the next frontier in improved vision and language understanding.
\fi
\section{MovieQA dataset}
\label{sec:movieqa}
\vspace{-1mm}
\input{tables/qa_stats.tex}
The goal of our paper is to create a challenging benchmark that evaluates semantic understanding over long temporal data.
We collect a dataset with very diverse sources of information that can be exploited in this challenging domain.
Our data consists of quizzes about movies that the automatic systems will have to answer.
For each movie, a quiz comprises of a set of questions, each with 5 multiple-choice answers, only one of which is correct.
The system has access to various sources of textual and visual information, which we describe in detail below.
We collected 408 subtitled movies, and obtained their extended summaries in the form of plot synopses from \emph{Wikipedia}.
We crawled \emph{imsdb} for scripts, which were available for 49\% (199) of our movies.
A fraction of our movies (60) come with DVS transcriptions provided by~\cite{Rohrbach15}.
{\bf Plot synopses}
are movie summaries that fans write after watching the movie.
Synopses widely vary in detail and range from one to 20 paragraphs, but focus on describing content that is directly relevant to the story.
They rarely contain detailed visual information (\eg~character appearance), and focus more on describing the movie events and character interactions.
We exploit
plots to gather our quizzes.
{\bf Videos and subtitles.}
An average movie is about 2 hours in length and has over 198K frames and almost 2000 shots.
Note that video alone contains information about e.g., ``Who'' did ``What'' to ``Whom'', but may be lacking in information to explain why something happened.
Dialogs play an important role, and only both modalities together allow us to fully understand the story.
Note that subtitles do not contain speaker information. In our dataset, we provide video clips rather than full movies.
{\bf DVS} is a service that narrates movie scenes to the visually impaired by inserting relevant descriptions in between dialogs.
These descriptions contain sufficient ``visual'' information about the scene that they allow visually impaired audience to follow the movie.
DVS thus acts as a proxy for a perfect vision system, and is another source for answering.
{\bf Scripts.}
The scripts that we collected are written by screenwriters and serve as a guideline for movie making.
They typically contain detailed descriptions of scenes, and, unlike subtitles, contain both dialogs and speaker information.
Scripts are thus similar, if not richer in content to DVS+Subtitles, however are not always entirely faithful to the movie as the director may aspire to artistic freedom.
\vspace{-1mm}
\subsection{QA Collection method}
\vspace{-1mm}
\input{tables/dataset_comparison.tex}
\begin{figure}
\vspace{-2.5mm}
\centering
\includegraphics[width=0.93\linewidth,trim=0 0 0 20,clip]{figs/stats-qword_calength.pdf}
\vspace*{-0.4cm}
\caption{Average number of words in MovieQA dataset based on the first word in the question. Area of a bubble indicates \#QA.}
\vspace*{-0.4cm}
\label{fig:stats:qword_calength}
\end{figure}
Since videos are difficult and expensive to provide to annotators, we used plot synopses as a proxy for the movie.
While creating quizzes, our annotators only referred to the story plot and were thus automatically coerced into asking story-like questions.
We split our annotation efforts into two primary parts to ensure high quality of the collected data.
{\bf Q and correct A.}
Our annotators were first asked to select a movie from a large list, and were shown its plot synopsis one paragraph at a time.
For each paragraph, the annotator had the freedom of forming any number and type of questions.
Each annotator was asked to provide the correct answer, and was additionally required to mark a minimal set of sentences within the plot synopsis paragraph that can be used to both frame the question and answer it.
This was treated as ground-truth for localizing the QA in the plot.
In our instructions, we asked the annotators to provide context to each question, such that a human taking the quiz should be able to answer it by watching the movie alone (without having access to the synopsis).
The purpose of this was to ensure questions that are localizable in the video and story as opposed to generic questions such as ``What are they talking?".
We trained our annotators for about one to two hours and gave them the option to re-visit and correct their data.
The annotators were paid by the hour, a strategy that allowed us to collect more thoughtful and complex QAs, rather than short questions and single-word answers.
{\bf Multiple answer choices.}
In the second step of data collection, we collected multiple-choice answers for each question.
Our annotators were shown a paragraph and a question at a time, but not the correct answer.
They were then asked to answer the question correctly as well as provide 4 wrong answers.
These answers were either deceiving facts from the same paragraph or common-sense answers.
The annotator was also allowed to re-formulate or correct the question.
We used this to sanity check all the questions received in the first step.
All QAs from the ``val'' and ``test'' set underwent another round of clean up.
{\bf Time-stamp to video.}
We further asked in-house annotators to align each sentence in the plot synopsis to the video by marking the beginning and end (in seconds) of the video that the sentence describes.
Long and complicated plot sentences were often aligned to multiple, non-consecutive video clips.
Annotation took roughly 2 hours per movie.
Since we have each QA aligned to a sentence(s) in the plot synopsis, the video to plot alignment links QAs with video clips.
We provide these clips as part of our benchmark.
\subsection{Dataset Statistics}
\begin{figure}
\vspace{-4mm}
\centering
\includegraphics[width=0.9\linewidth]{figs/stats-answer_type.pdf}
\vspace*{-0.6cm}
\caption{Stats about MovieQA questions based on answer types.
Note how questions beginning with the same word may cover a variety of answer types:
\emph{Causality}: What happens ... ?; \emph{Action}: What did X do?
\emph{Person name}: What is the killer's name?; \etc
}
\vspace*{-0.5cm}
\label{fig:stats:answer_type}
\end{figure}
In the following, we present some statistics of our MovieQA dataset.
Table~\ref{tab:dataset-comparison} presents an overview of popular and recent Question-Answering datasets in the field.
Most datasets (except MCTest) use very short answers and are thus limited to covering simpler visual/textual forms of understanding.
To the best of our knowledge, our dataset not only has long sentence-like answers, but is also the first to use videos in the form of movies.
{\bf Multi-choice QA.}
We collected a total of 14,944 QAs from 408 movies.
Each question comes with one correct and four deceiving answers.
Table~\ref{tab:qa_stats} presents an overview of the dataset along with information about the train/val/test splits, which will be used to evaluate automatically trained QA models.
On average, our questions and answers are fairly long with about 9 and 5 words respectively unlike most other QA datasets.
The video-based answering split for our dataset, supports 140 movies for which we aligned plot synopses with videos.
Note that the QA methods needs to look at a long video clip ($\sim$200s) to answer the question.
Fig.~\ref{fig:stats:qword_calength} presents the number of questions (bubble area) split based on the first word of the question along with information about number of words in the question and answer.
Of particular interest are ``Why'' questions that require verbose answers, justified by having the largest average number of words in the correct answer, and in contrast, ``Who'' questions with answers being short people names.
Instead of the first word in the question, a peculiar way to categorize QAs is based on the answer type.
We present such an analysis in Fig.~\ref{fig:stats:answer_type}.
Note how reasoning based questions (Why, How, Abstract) are a large part of our data.
In the bottom left quadrant we see typical question types that can likely be answered using vision alone.
Note however, that even the reasoning questions typically require vision, as the question context provides a visual description of a scene (\eg,~``Why does John run after Mary?'').
\input{tables/textsrc_stats.tex}
{\bf Text sources for answering.}
In Table~\ref{tab:textsrc_stats}, we summarize and present some statistics about different text sources used for answering.
Note how plot synopses have a large number of words per sentence, hinting towards the richness and complexity of the source.
\section{QA Evaluation}
\label{sec:eval}
We present results for question-answering with the proposed methods on our MovieQA dataset.
We study how various sources of information influence the performance, and how different levels of complexity encoded in $f$ affects the quality of automatic QA.
{\bf Protocol.}
Note that we have two primary tasks for evaluation.
(i) {\bf Text-based}: the story takes the form of various texts -- plots, subtitles, scripts, DVS; and
(ii) {\bf Video-based}: story is the video, and with/without subtitles.
{\bf Dataset structure.}
The dataset is divided into three disjoint splits: \emph{train}, \emph{val}, and \emph{test}, based on unique movie titles in each split.
The splits are optimized to preserve the ratios between \#movies, \#QAs, and all the story sources at 10:2:3 (\eg~about 10k, 2k, and 3k QAs).
Stats for each split are presented in Table~\ref{tab:qa_stats}.
The \emph{train} set is to be used for training automatic models and tuning any hyperparameters.
The \emph{val} set should not be touched during training, and may be used to report results for several models.
The \emph{test} set is a held-out set, and is evaluated on our MovieQA server.
For this paper, all results are presented on the \emph{val} set.
{\bf Metrics.}
Multiple choice QA leads to a simple and objective evaluation.
We measure \emph{accuracy}, the number of correctly answered QAs over the total count.
\vspace{-1mm}
\subsection{The Hasty Student}
\vspace{-1mm}
\input{tables/results-cheating_baseline.tex}
The first part of Table~\ref{tab:results-cheating_baseline} shows the performance of three models when trying to answer questions based on the answer length.
Notably, always choosing the longest answer performs better (25.3\%) than random (20\%).
The second part of Table~\ref{tab:results-cheating_baseline} presents results when using within-answer feature-based similarity.
We see that the answer most similar to others is likely to be correct when the representations are generic and try to capture the semantics of the sentence (Word2Vec, SkipThoughts).
The most distinct answers performs worse than random on all features.
In the last section of Table~\ref{tab:results-cheating_baseline} we see that computing feature-based similarity between questions and answers is insufficient for answering.
Especially, TF-IDF performs worse than random since words in the question rarely appear in the answer.
{\bf Hasty Turker.}
To analyze the deceiving nature of our multi-choice QAs, we tested humans (via AMT) on a subset of 200 QAs.
The turkers were not shown the story in any form and were asked to pick the best possible answer given the question and a set of options.
We asked each question to 10 turkers, and rewarded each with a bonus if their answer agreed with the majority.
We observe that without access to the story, humans obtain an accuracy of 27.6\%.
We suspect that the bias is due to the fact that some of the QAs reveal the movie (e.g., ``Darth Vader'') and the turker may have seen this movie.
Removing such questions, and re-evaluating on a subset of 135 QAs, lowers the performance to 24.7\%.
This shows the genuine difficulty of our QAs.
\vspace{-1mm}
\subsection{Searching Student}
\vspace{-1mm}
{\bf Cosine similarity with window.}
The first section of Table~\ref{tab:results-comprehensive_baseline} presents results for the proposed cosine similarity using different representations and text stories.
Using the plots to answer questions outperforms other sources (subtitles, scripts, and DVS) as the QAs were collected using plots and annotators often reproduce words from the plot.
We show the results of using Word2Vec or SkipThought representations in the following rows of Table~\ref{tab:results-comprehensive_baseline}.
SkipThoughts perform much worse than both TF-IDF and Word2Vec which are closer.
We suspect that while SkipThoughts are good at capturing the overall semantics of a sentence, proper nouns -- names, places -- are often hard to distinguish.
Fig.~\ref{fig:results-simple_baselines_qfw} presents a accuracy breakup based on the first word of the questions.
TF-IDF and Word2Vec perform considerably well, however, we see a larger difference between the two for ``Who'' and ``Why'' questions.
``Who'' questions require distinguishing between names, and ``Why'' answers are typically long, and mean pooling destroys semantics.
In fact Word2Vec performs best on ``Where'' questions that may use synonyms to indicate places.
SkipThoughts perform best on ``Why'' questions where sentence semantics help improve answering.
\input{tables/results-comprehensive_baseline.tex}
{\bf SSCB}. The middle rows of Table~\ref{tab:results-comprehensive_baseline} show the result of our neural similarity model.
Here, we present additional results combining all text representations (\textit{SSCB fusion}) via our CNN.
We split the \emph{train} set into $90\%$ train / $10\%$ dev, such that all questions and answers of the same movie are in the same split, train our model on train and monitor performance on dev.
Both \emph{val} and \emph{test} sets are held out.
During training, we also create several model replicas and pick the ones with the best validation performance.
Table~\ref{tab:results-comprehensive_baseline} shows that the neural model outperforms the simple cosine similarity on most tasks, while the fusion method achieves the highest performance when using plot synopses as the story.
Ignoring the case of plots, the accuracy is capped at about $30\%$ for most modalities showing the difficulty of our dataset.
\vspace{-1mm}
\subsection{Memory Network}
\vspace{-0.5mm}
The original MemN2N which trains the word embeddings along with the answering modules overfits heavily on our dataset leading to near random performance on \textit{val} ($\sim$20\%).
However, our modifications help in restraining the learning process.
Table~\ref{tab:results-comprehensive_baseline} (bottom) presents results for MemN2N with Word2Vec initialization and a linear projection layer.
Using plot synopses, we see a performance closer to SSCB with Word2Vec features.
However, in the case of longer stories, the attention mechanism in the network is able to sift through thousands of story sentences and performs well on DVS, subtitles and scripts.
This shows that complex three-way scoring functions are needed to tackle such QA sources.
In terms of story sources, the MemN2N performs best with scripts which contain the most information (descriptions, dialogs and speaker information).
\input{tables/results-video_baseline.tex}
\begin{figure}
\centering
\includegraphics[width=0.8\linewidth,trim=0 0 0 12,clip]{figs/results-simple_baselines_qfw.pdf}
\vspace*{-0.6cm}
\caption{Accuracy for different feature representations of plot sentences with respect to the first word of the question.}
\label{fig:results-simple_baselines_qfw}
\vspace{-0.5cm}
\end{figure}
\vspace{-1mm}
\subsection{Video baselines}
\vspace{-1mm}
We evaluate SSCB and MemN2N in a setting where the automatic models answer questions by ``watching'' all the video clips that are provided for that movie.
Here, the story descriptors are shot embeddings.
The results are presented in Table~\ref{tab:results-video_baseline}.
We see that learning to answer questions using video is still a hard problem with performance close to random.
As visual information alone is insufficient, we also perform and experiment combining video and dialog (subtitles) through late fusion.
We train the SSCB model with the visual-text embedding for subtitles and see that it yields poor performance (22.3\%) compared to the fusion of all text features (27.7\%).
For the memory network, we answer subtitles as before using Word2Vec.
\section{Introduction}
\label{sec:intro}
\vspace{-1mm}
\begin{figure*}[t]
\vspace{-3mm}
\includegraphics[height=0.161\linewidth]{figs/et}
\includegraphics[height=0.161\linewidth]{figs/vegas1}
\includegraphics[height=0.161\linewidth]{figs/forrest}
\includegraphics[height=0.161\linewidth]{figs/10things}\\[1mm]
\begin{scriptsize}
\addtolength{\tabcolsep}{-2.1pt}
\begin{tabular}{m{0.00cm}m{3.63cm}m{0.00cm}m{4.1cm}m{0.00cm}m{3.72cm}m{0.02cm}m{4cm}}
& \hspace{-3.6mm}{\color{cornellred}{\bf Q}}: How does E.T. show his happiness that he is finally returning home? & & \hspace{-3.6mm}{\color{cornellred}{\bf Q}}: Why do Joy and Jack get married that first night they meet in Las Vegas? & & \hspace{-3.6mm}{\color{cornellred}{\bf Q}}: Why does Forrest undertake a three-year marathon? & & \vspace{-2.6mm}\hspace{-3.6mm}{\color{cornellred}{\bf Q}}: How does Patrick start winning Kat over?\\[-1mm]
& \hspace{-3.4mm}{\color{cadmiumgreen}{\bf A}}: His heart lights up & & \hspace{-3.4mm}{\color{cadmiumgreen}{\bf A}}: They are both vulnerable and totally drunk & & \hspace{-3.4mm}{\color{cadmiumgreen}{\bf A}}: Because he is upset that Jenny left him & & \hspace{-3.4mm}{\color{cadmiumgreen}{\bf A}}: By getting personal information about her likes and dislikes
\end{tabular}
\end{scriptsize}
\vspace{-3mm}
\caption{\small Examples from the MovieQA dataset.
For illustration we show a single frame, however, all these questions/answers are time-stamped to a much longer clip in the movie.
Notice that while some questions can be answered using vision or dialogs alone, most require both.
Vision can be used to locate the scene set by the question, and semantics extracted from dialogs can be used to answer it.}
\label{fig:example_questions}
\vspace{-3mm}
\end{figure*}
Fast progress in Deep Learning as well as a large amount of available labeled data has significantly pushed forward the performance in many visual tasks such as image tagging, object detection and segmentation, action recognition, and image/video captioning.
We are steps closer to applications such as assistive solutions for the visually impaired, or cognitive robotics, which require a holistic understanding of the visual world by reasoning about all these tasks in a common framework.
However, a truly intelligent machine would ideally also infer high-level semantics underlying human actions such as motivation, intent and emotion, in order to react and, possibly, communicate appropriately.
These topics have only begun to be explored in the literature~\cite{thewhy,ZhuICCV15}.
A great way of showing one's understanding about the scene is to be able to answer any question about it~\cite{malinowski14nips}.
This idea gave rise to several question-answering datasets which provide a set of questions for each image along with multi-choice answers.
These datasets are either based on RGB-D images~\cite{malinowski14nips} or a large collection of static photos such as Microsoft COCO~\cite{VQA,VisualMadlibs}.
The types of questions typically asked are ``What'' is there and ``Where'' is it, what attributes an object has, what is its relation to other objects in the scene, and ``How many'' objects of certain type are present.
While these questions verify the holistic nature of our vision algorithms, there is an inherent limitation in what can be asked about a static image.
High-level semantics about actions and their intent is mostly lost and can typically only be inferred from temporal, possibly life-long visual observations.
Movies provide us with snapshots from people's lives that link into stories, allowing an experienced human viewer to get a high-level understanding of the characters, their actions, and the motivations behind them.
Our goal is to create a question-answering dataset to evaluate machine comprehension of both, complex videos such as movies and their accompanying text.
We believe that this data will help push automatic semantic understanding to the next level, required to truly understand stories of such complexity.
This paper introduces MovieQA, a large-scale question-answering dataset about movies.
Our dataset consists of 14,944 multiple-choice questions with five deceiving options, of which only one is correct, sourced from 408 movies with high semantic diversity.
For 140 of these movies (6,462 QAs), we have timestamp annotations indicating the location of the question and answer in the video.
The questions range from simpler ``Who'' did ``What'' to ``Whom'' that can be solved by vision alone, to ``Why'' and ``How'' something happened, that can only be solved by exploiting both the visual information and dialogs (see Fig.~\ref{fig:example_questions} for a few example ``Why'' and ``How'' questions).
Our dataset is unique in that it contains multiple sources of information: video clips, subtitles, scripts, plots, and DVS~\cite{Rohrbach15} as illustrated in Fig.~\ref{fig:frontpage}.
We analyze the data through various statistics and intelligent baselines that mimic how different ``students'' would approach the quiz.
We further extend existing QA techniques to work with our data and show that question-answering with such open-ended semantics is hard.
We have created an online benchmark with a leaderboard (\url{http://movieqa.cs.toronto.edu/leaderboard}), encouraging inspiring work in this challenging domain.
\section{Introduction}
\label{sec:intro}
Fast progress in Deep Learning as well as the large amount of available labeled data has significantly pushed forward the performance in many visual tasks such as image tagging, object detection and segmentation, action recognition, and image/video captioning.
We are steps closer to applications such as visual solutions for the blind or cognitive robotics, which require a holistic understanding of the visual world by reasoning about all these tasks in a common framework.
However, a truly intelligent machine would ideally also infer high-level semantics behind people's (visual) actions such as motivation, intent and emotion, in order to react and, possibly, communicate appropriately.
These topics have only began to be explored in the literature~\cite{thewhy,ZhuICCV15}.
A great way of showing one's internal understanding about the scene is to be able to answer any question about it~\cite{malinowski14nips}.
This idea gave rise to several question-answering datasets which provide a set of questions for each image along with multi-choice answers.
These datasets are either based on RGB-D images~\cite{malinowski14nips} or a large collection of static images such as Microsoft Coco~\cite{VQA,VisualMadlibs}.
The types of questions typically asked are \emph{what} is there and \emph{where} is it, what color or shape an object has, what's its relation to other objects in the scene, and \emph{how many} objects of certain type are there in the scene.
While these questions verify the holistic nature of our vision algorithms, there is an inherent limitation to what can be asked about a static image.
High-level semantics such as motivation and intent is mostly lost and can typically only be inferred from temporal, possibly life-long visual observations.
In this paper, we argue that question-answering about movies ...
Movies provide us with snapshots from people's lives that link into stories, allowing an experienced human viewer to get a high-level understanding of the characters and their actions, the motivation behind the actions as well as the emotions they are feeling by taking them.
Our aim in this paper is to collect a large-scale question-answering dataset about movies.
Our dataset will provide a set of questions and answers with ground-truth alignment to both video and text.
This will enable the learning techniques to learn how to localize content of a particular granularity.
The challenge will then be to deal with highly varying duration of the alignment, with questions like ``What is she wearing at dinner?'' answerable with only a fraction of a second of video, while questions such as ``What is Nick and Amy's marriage like?'' requiring almost the full-length of video.
We believe that our dataset will inspire the next generation of question-answering techniques that will need to capture the true semantics of long and complex input.
We plan to create a benchmark hosted online with an active leader board.
We test a set of recently proposed QA methods on our dataset, showing...
\begin{figure*}
\centering
\includegraphics[width=1.0\linewidth,trim=0 0 0 0,clip]{figs/MovieQA_5Q.pdf}
\vspace*{-0.6cm}
\caption{intro}
\label{fig:intro}
\end{figure*}
\section{Related work}
\label{sec:relwork}
\vspace{-1mm}
Integration of language and vision is a natural step towards improved understanding and is receiving increasing attention from the research community.
This is in large part due to efforts in large-scale data collection such as Microsoft's COCO~\cite{lin2014microsoft}, Flickr30K~\cite{Flickr30k} and Abstract Scenes~\cite{Abstract15} providing tens to hundreds of thousand images with natural language captions.
Having access to such data enabled the community to shift from hand-crafted language templates typically used for image description~\cite{BabyTalk} or retrieval-based approaches~\cite{Farhadi10,Im2Txt,Yang11} to deep neural models~\cite{Zitnick14,Karpathy15,kiros15,Vinyals14} that achieve impressive captioning results.
Another way of conveying semantic understanding of both vision and text is by retrieving semantically meaningful images given a natural language query~\cite{Karpathy15}.
An interesting direction, particularly for the goals of our paper, is also the task of learning common sense knowledge from captioned images~\cite{Vedantam15}.
This has so far been demonstrated only on synthetic clip-art scenes which enable perfect visual parsing.
{\bf Video understanding via language.}
In the video domain, there are fewer works on integrating vision and language, likely due to less available labeled data.
In~\cite{lrcn2014,Venugopalan:2014wc}, the authors caption video clips using LSTMs, ~\cite{rohrbach13iccv} formulates description as a machine translation model, while older work uses templates~\cite{SanjaUAI12,Das:2013br,krishnamoorthy:aaai13}.
In~\cite{Lin:2014db}, the authors retrieve relevant video clips for natural language queries, while~\cite{ramanathan13} exploits captioned clips to learn action and role models.
For TV series in particular, the majority of work aims at recognizing and tracking characters in the videos~\cite{Baeuml2013_SemiPersonID,Bojanowski:2013bg,Ramanathan:2014fj,Sivic:2009kt}.
In~\cite{Cour08,Sankar:bv}, the authors aligned videos with movie scripts in order to improve scene prediction.
~\cite{Tapaswi_J1_PlotRetrieval} aligns movies with their plot synopses with the aim to allow semantic browsing of large video content via textual queries.
Just recently,~\cite{Tapaswi2015_Book2Movie,ZhuICCV15} aligned movies to books with the aim to ground temporal visual data with verbose and detailed descriptions available in books.
{\bf Question-answering.}
QA is a popular task in NLP with significant advances made recently with neural models such as memory networks~\cite{Sukhbaatar2015}, deep LSTMs~\cite{Hermann15}, and structured prediction~\cite{wang2015mctest}.
In computer vision,~\cite{malinowski14nips} proposed a Bayesian approach on top of a logic-based QA system~\cite{Liang13}, while~\cite{Malinowski15,mengye15} encoded both an image and the question using an LSTM and decoded an answer.
We are not aware of QA methods addressing the temporal domain.
{\bf QA Datasets.}
Most available datasets focus on image~\cite{KongCVPR14,lin2014microsoft,Flickr30k,Abstract15} or video description~\cite{Chen11,Rohrbach15,YouCook}.
Particularly relevant to our work is the MovieDescription dataset~\cite{Rohrbach15} which transcribed text from the Described Video Service (DVS), a narration service for the visually impaired, for a collection of over 100 movies.
For QA, \cite{malinowski14nips} provides questions and answers (mainly lists of objects, colors, \etc) for the NYUv2 RGB-D dataset, while~\cite{VQA,VisualMadlibs} do so for MS-COCO with a dataset of a million QAs.
While these datasets are unique in testing the vision algorithms in performing various tasks such as recognition, attribute induction and counting, they are inherently limited to static images.
In our work, we collect a large QA dataset sourced from over 400 movies with challenging questions that require semantic reasoning over a long temporal domain.
Our dataset is also related to purely text QA datasets such as MCTest~\cite{MCTest} which contains 660 short stories with 4 multi-choice QAs each, and~\cite{Hermann15} which converted 300K news summaries into Cloze-style questions.
We go beyond these datasets by having significantly longer text, as well as multiple sources of available information (plots, subtitles, scripts and DVS).
This makes our data one of a kind.
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 88
|
OTRR Curated Collection #2
By Paul Kornman · #2372 ·
Download Radio Playhouse at https://www.otrr.org/OTRRLibrary/otrrlibrary.html
That Library web structure was set up years ago, and is a minor pain to keep updated. It's one of the reasons I went to the database-driven interface that is in the new location. It's on my todo list
Error in OTRRR Certified Man Called X
I added the other file to the OTRR Library (not the Certified Set) and amended the episode record in the Library app to point to the replacement file. - Paul
It's not an index, more of a Table of Contents. From that page (https://otrr.org/mains.html?c=mags), if you select "List by Issue", you will see a long table that lists every article, its author (if i
OTRR Curated Collections
After some more back and forth with testers and Larry, a newer version of Curated Set #1 is available (in the same place): https://www.otrr.org/OTRRLibrary/curated/curated001.html
More notes: 1) fixed "Age" of Victoria. Thanks. 2) There was a technical glitch with most of the files. Now, you should be able to click any row in the table and get that particular episode's audio in
By Paul Kornman · #1897 · Edited
Typo Fixed. Thanks for the heads up. To other points - You can click on the shows in any order. Just click the show's row in the table and the audio player will load it. There was no thought (at least
Larry Maupin has researched a set of episodes called "Dramatizations of Famous Murder Cases, Trials and Unsolved Mysteries on Radio Programs". This set has been become OTRR's first Curated Collection:
Update To Yesterday's Post
Larry - I am looking forward to the first curated collection. I don't always respond quickly to this message board, but that does not mean I'm not interested. Many members read these messages, and som
Larry - I'm happy to add these file to the OTRR Library. Do I have your permission to do that? - Paul
Ottr radio station.
No, there is no app. That's beyond my skill set. - Paul
OTR has a on-demand service HOTROD (https://otrr.org/hotrod/hotrod7.html) which allows you to stream any files from our Library. You can set up playlists as well. - Paul
Request for Discussion on "Radio's Missing Masters"
I think the idea of a curated set or playlist by members would be a great addition to the Library. I tried to set up a "OTRR recommends" section, but the response from members was minimal. I'd be happ
I am capable of copying episodes into another "directory" in the library, but I'd rather not. I have tried to move all of the Singles and Doubles Files to their individual program's folders with the e
Radio Guide issues added to library
Depends on what you mean by searchable. At http://otrr.org/mains.html?c=mags, you can pick a magazine and see the contents of each issue. At https://www.otrr.org/OTRRPedia/pedia.html you can pick a pe
Data base of otr shows?
A) I'll look into the links on OTRRPedia, but I just checked the Library (http://www.otrr.org/OTRRLibrary/otrrlibrary.html) and there are 16 episodes of Future Tense with links that are working. B) Go
FYI - OTRR maintains a database of OTR shows - the OTRRPedia. Here's the URL for 'Future Tense': http://www.otrr.org/OTRRPedia/pedia.html?s=pgm&id=3935&t=0
OTRR
I made a tool to show which files have been added from the Purchasing Group: https://otrr.org/otrrPG.html - Paul
By Paul Kornman · #861 ·
Try the beta site (https://www.otrr.org/OTRRLibrary/otrrlibrary.html) and let me know what you think. You can view episodes in a 'Table' (new way - click the file to listen to it) or as a list (old wa
Fw: Archive.org to declare bankruptcy?
Q) Does OTRR Library need money? A) Not specifically. We accept PayPal donations through our three websites. Those funds are used to maintain the websites. Currently, we are running a surplus, so we'r
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 518
|
\section*{}
\section{Introduction}
To search substellar objects around white dwarf binaries are important for understanding of the interaction between companions and evolved stars \citep[]{2015ApJS..221...17Q}. Recent years have been reported several successful examples for the detection of the planets around the white dwarf binaries such as QS Vir \citep*[]{2010MNRAS.401L..34Q, 2011IAUS..276..495A}, NN Ser \citep{2014MNRAS.437..475M}, HU Aqr \citep{2011MNRAS.414L..16Q}, \citep{2015MNRAS.448.1118G} and RR Cae \citep{2012MNRAS.422L..24Q}. These substellar companions are detected by measuring the variations in the observed mid-eclipse time via the presence of the third body. As the motion of the binary near the common center of mass, the arrival eclipse time will vary periodically. This method was widely used to study other eclipsing binary systems containing a white dwarf and a red dwarf because the components are large differences in radius and luminosity \citep[]{2010MNRAS.407.2362P}.
Cataclysmic variables are semi-detached binaries containing a white dwarf accreting material
from a main-sequence star via Roche lobe overflow \citep[]{1995CVSC...562A..19V}. As one of the subclasses of CVs, dwarf novae show recurrent outbursts with the amplitude of 2-5 mag and short duration about a few days to weeks. Recently, several eclipsing dwarf novae were selected to detect the substellar companions and evolution by analyzing variations in orbital period. One of good examples is V2051 Oph, Qian et al. (2015) reported that it has a giant planet with a mass of $7.3(\pm0.7)M_{Jup}$ and an eccentricity of $e'=0.37$. Moreover, the secular decrease in orbital period of V2051 Oph suggested that magnetic braking may not entirely cease in fully convective stars. EX Dra is a long-period ($P=5.04$ h) dwarf nova with very deep eclipse with 1.5 mag in quiescence. It was discovered in the Hamburger Quasar Survey \citep[]{1989ESASP.296..883B}. Follow-up observations by \citet{1993IAU.165..89B} showed that EX Dra is a deeply eclipsing dwarf nova with an orbital period of just over 5 h. \citet{1997A&A...327..173F} presented spectroscopic and photometric observations, and estimated some basic parameters.
By using photometric observations, \citet{2000MNRAS.316..529B} found that this system has a mass ratio $q={0.72}$ and an inclination $i=85^{\circ}$, and that the $O-C$ diagram showed a periodic oscillation with a period of 4 yr and an amplitude of 1.2 min.
\citet{2003PASP..115.1105S} analyzed the eclipse profile of multi-colour light curves with a parameter-fitting model. They derived a mass ratio $q<0.81$ and an inclination of $i>83^{\circ}$. The revised ephemeris showed a cyclical variation with a period of 5 yr. Recently, the analysis by \citet{2012A&A...539A.153P} given a
period modulation with a period of 21 yr and an amplitude of 2.5 min.
In present paper, we use new eclipse timings coupled with the old data to analyse the $O-C$ diagram of EX Dra. Our results indicated that both there are two possible brown dwarfs orbiting EX Dra, and this star may be undergoing a peculiarly evolutionary stage.
\section{CCD photometric observations and new mid-eclipse times}
We started to monitor EX Dra since 09, November 2009 by using the 0.6-m reflecting telescope attached an Andor DV436 2K CCD camera at the Yunnan Observatories(YNAO). Later, this star was monitored with the 85-cm telescope mounted an Anor DW436 1K CCD camera at the XingLong station of the National Astronomical Observatories and the 2.4-m telescope at the Lijiang observational station in YNAO .
Since the May of 2014, EX Dra was continuously observed with CCD photometer on 50-cm (Apogee Alta U8300 with 528 x 512 pixels) and 60-cm (Apogee 47, field 1024 x 1024 pixels) telescopes of Sternberg Astronomical Institute Crimean Station in R-band.
Four light curves of EX Dra in quiescence are displayed in Figure 1. The phases were computed by using the linear ephemeris,
\begin{equation}
Min.I=HJD2456065.154955+0.209937316\times{E},
\end{equation}
where HJD2456065.154955 is the initial epoch from our mid-eclipse times listed in Table 1, and 0.209937316 d is the orbital period from \citet{2012A&A...539A.153P}.
The most obvious features in Figure 1 are that the light curves in quiescence exhibit double eclipse rarely and strong orbital hump. In addition, the shape and brightness are variable with time. This can be explained as the change of mass transfer rate and the unstability of accretion disc. The egress of white dwarf can be seen clearly in the light curves. For comparison, two profiles during outburst are shown in Figure 2. These curves are V-shaped and symmetric, indicating an axisymmetrical brightness distribution in the accretion disc at maximum. The orbital hump disappears during outburst.
\begin{figure}[!h]
\begin{center}
\includegraphics[width=1.0\columnwidth]{figure1.eps}
\caption{Four eclipsing light curves of EX Dra observed by using the 60cm telescope in Sternberg Astronomical Institute Crimean Station.}
\end{center}
\end{figure}
\begin{figure}[!h]
\begin{center}
\includegraphics[width=1.0\columnwidth]{figure2.eps}
\caption{Two eclipse profiles of EX Dra during outbursts obtained with 60cm telescope in Sternberg Astronomical Institute Crimean Station on 2015 March 08 and 2016 March 08, respectively.}
\end{center}
\end{figure}
By adopting the same method from \citet{2012A&A...539A.153P}, the 29 new mid-eclipse times during quiescence were obtained and listed in Table 1.
We only used the light curves during quiescence to determine the mid-eclipse times. The reason also has been discussed in \citet{2012A&A...539A.153P}. This consistency with previous published eclipse timings highly increases reliability of the following analysis.
We also computed the eclipse width of the white dwarf as $\Delta{\tau}=0.0230(1)$ days, which is very close to previous studies.
The uncertainty in determining mid-egress times depends on the time resolution and signal to noise ratio. We estimate that the error of mid-eclipse times is about quarter of the integration time. The reason is that the errors of mid-egress and mid-ingress times are about half of the time resolution, and the mid-eclipse times are average value of the mid-ingress and mid-egress times. Therefore, the errors of mid-eclipse times were determined as the combination of the errors of mid-egress and mid-ingress by using the error propagation function.
All new mid-eclipse times have been converted into the $BJD$ system and are listed in the second column of Table 1, corresponding to errors are also given in fifth column.
The exposure time for each mid-eclipse times was listed in sixth column.
The details of
the used filters could be found in seventh column where "R" and "I" refer to R-band and I-band, respectively. "N" indicates that no filters were used. "0.6m", "85cm" and "2.4m" in the eighth column of the table refer to the 0.6m, 85-cm and 2.4-m telescopes in China, while "50cm" and "60cm" refer to the 50-cm and 60-cm telescopes in Russia.
\section{The changes of the $O-C$ curve of EX Dra}
\citet{2000MNRAS.316..529B} shown a cyclical behaviour in $O-C$ diagram with a period of 4 years and a amplitude of 1.18 min. Follow-up studies by \citet{2003PASP..115.1105S} pointed out the period and amplitude are about $25\,\%$ bigger than the corresponding values from \citet{2000MNRAS.316..529B}. Recently, \citet{2012A&A...539A.153P} revised ephemeris by adding many mid-eclipse times and found a greater cyclical variation with a period of 21 years and an amplitude of 2.5 min, but a singly sinusoidal ephemeris cannot describe the complex $O-C$ change well. Therefore, it seems that there are two cyclic variations in the $O-C$ diagram.
Combining new data with the old timings from the literature \citep[]{1997A&A...327..173F, 2000MNRAS.316..529B, 2003PASP..115.1105S, 2012A&A...539A.153P}, the latest $O-C$ diagram was obtained (see Figure 3). All $O-C$ values were calculated with the linear ephemeris published by \citet{2012A&A...539A.153P},
\begin{equation}
Min.I=BJD2452474.80513+0.209937316\times{E},
\end{equation}
where BJD$2452474.80513$ is the initial epoch. New $O-C$ curve is more complex than meets the eye. Based on previous studies, we suspected the existence of two cyclic variations. To describe the overall trend of the $O-C$ curve well, a quadratic ephemeris is required. Thus, a possible model with an upward parabolic variation and double periodic terms is considered:
\begin{equation}
O-C=\Delta{T_{0}}+\Delta{P_{0}}{E}+\frac{\beta}{2}{E^{2}}+\tau_{A}+\tau_{B},
\end{equation}
where $\tau_{A}$ and $\tau_{B}$ are the two cyclic changes.
Our best fit to the $O-C$ diagram by using the Levenberg-Marquardt method shows that both of $\tau_{A}$ and $\tau_{B}$ are strictly periodic, i.e.
\begin{equation}
\tau_{A}=K_{A}\sin(\frac{2\pi}{P_{A}}{E}+\varphi_{A}),
\end{equation}
and
\begin{equation}
\tau_{B}=K_{B}\sin(\frac{2\pi}{P_{B}}{E}+\varphi_{B}).
\end{equation}
In general, the eccentricity should be taken into account in the fitting process. However, the eccentricity was close to zero ($e<0.01$) but with a larger error, which is why we set e=0 in the final fit. All fitting parameters and the corresponding values are given in Table 2.
The best-fitting results reveal a secular period increase at a rate of $\dot{P}={+7.46}\times10^{-11}{s} {s^{-1}}$.
In Figure 3, the dashed line in the upper panel refers to the linear period increase and the solid line represents the combination of two cyclic changes and the linear increase. After the long-term increase was subtracted, the superposition of a long (the dashed line) and a short (the solid line) periodic variation are displayed in the middle panel. Following both the linear increase and the two cyclic changes were removed, the residuals are plotted in the lowest panel. The two cyclic variations extracted from the middle panel of Figure 3 are displayed in Figure 4 where the periods of $\tau_{A}$ and $\tau_{B}$ are 21.40 years and 3.99 years and the corresponding amplitudes are 89.6 s and 50.1 s.
The derived period modulations are very close to the previous results detected by \citet{2012A&A...539A.153P} and \citet{2000MNRAS.316..529B}, respectively.
\begin{table*}[!h]
\caption{New CCD mid-eclipse times of EX Dra in quiescence.}
\begin{center}
\small
\begin{tabular}{llcllccc}\hline\hline
Min.(HJD) & Min.(BJD) & E & $O-C$ & Errors & Exp.time(s) & Filters &Telescopes \\\hline
2455144.9992 &2455144.9999 &12719 &0.0021 &0.0002 & 60 & R &0.6m\\
2455746.0491 &2455746.0498 &15582 &0.0014 &0.0001 & 40 & R &85cm\\
2456065.1550 &2456065.1557 &17102 &0.0026 &0.0002 & 60 & N &2.4m\\
2456523.0282 &2456523.0290 &19283 &0.0026 &0.0001 & 40 & I &85cm\\
2456799.3056 &2456799.3063 &20599 &0.0024 &0.0001 & 40 & R &50cm\\
2456807.2833 &2456807.2841 &20637 &0.0026 &0.0001 & 40 & R &60cm\\
2456819.4599 &2456819.4607 &20695 &0.0028 &0.0001 & 40 & R &50cm\\
2456833.3160 &2456833.3167 &20761 &0.0030 &0.0001 & 40 & R &50cm\\
2456834.3655 &2456834.3663 &20766 &0.0028 &0.0001 & 40 & R &60cm\\
2456838.3540 &2456838.3548 &20785 &0.0025 &0.0001 & 40 & R &60cm\\
2456960.3272 &2456960.3280 &21366 &0.0022 &0.0004 & 75 & R &60cm\\
2456960.5372 &2456960.5379 &21367 &0.0022 &0.0004 & 75 & R &60cm\\
2457024.1480 &2457024.1488 &21670 &0.0020 &0.0004 & 75 & R &60cm\\
2457080.2018 &2457080.2026 &21937 &0.0026 &0.0002 & 60 & R &60cm\\
2457098.2561 &2457098.2569 &22023 &0.0023 &0.0002 & 60 & R &60cm\\
2457106.2335 &2457106.2342 &22061 &0.0020 &0.0001 & 40 & R &60cm\\
2457106.4443 &2457106.4451 &22062 &0.0029 &0.0001 & 40 & R &60cm\\
2457108.3340 &2457108.3347 &22071 &0.0031 &0.0001 & 40 & R &60cm\\
2457118.4105 &2457118.4112 &22119 &0.0026 &0.0001 & 40 & R &60cm\\
2457123.2390 &2457123.2398 &22142 &0.0026 &0.0001 & 40 & R &60cm\\
2457123.4493 &2457123.4501 &22143 &0.0029 &0.0001 & 40 & R &60cm\\
2457124.4968 &2457124.4976 &22148 &0.0008 &0.0001 & 40 & R &60cm\\
2457267.2561 &2457267.2569 &22828 &0.0027 &0.0001 & 40 & R &60cm\\
2457267.4668 &2457267.4676 &22829 &0.0034 &0.0001 & 40 & R &60cm\\
2457268.3058 &2457268.3066 &22833 &0.0028 &0.0002 & 60 & R &60cm\\
2457271.2457 &2457271.2465 &22847 &0.0035 &0.0002 & 60 & R &60cm\\
2457271.4561 &2457271.4568 &22848 &0.0039 &0.0001 & 40 & R &60cm\\
2457463.3380 &2457463.3388 &23762 &0.0032 &0.0001 & 40 & R &60cm\\
2457463.5479 &2457463.5487 &23763 &0.0032 &0.0001 & 40 & R &60cm\\
\hline
\end{tabular}
\end{center}
\end{table*}
\begin{table*}
\caption{Parameters of the best fitting for $O-C$ .}
\label{tab:table2}
\begin{center}
\small
\begin{tabular}{lllllllll}
\hline
\hline
Parameters &Values \\
\hline
Correction on the initial epoch, ${\Delta{T_{0}}}$ (d) & $-6.0(\pm1.6)\times10^{-4}$ \\
Correction on the initial period, ${\Delta{P_{0}}}$ (d) & $+3.09(\pm0.28)\times10^{-8}$ \\
Rate of the linear increase, $\beta$ (d/cycle) & $+1.57(\pm0.26)\times10^{-11}$ \\
\\
Parameters & Case A &Case B \\
Semi-amplitude, $K_{A}$, $K_{B}$ (d) & $1.04(\pm0.24)\times10^{-3}$ & $5.80(\pm0.73)\times10^{-4}$ \\
Orbital period, $P_{A}$, $P_{B}$ (yr) & $21.40(\pm1.44)$ & $3.99(\pm0.11)$ \\
The orbital phase, $\varphi_{A}$, $\varphi_{B}$ (deg) & $11.17(\pm0.86)$ & $-146.57(\pm5.26)$ \\
\hline
\end{tabular}
\end{center}
\end{table*}
\begin{figure}[!h]
\begin{center}
\includegraphics[width=1.0\columnwidth]{figure3.eps}
\caption{$O-C$ diagrams of EX Dra with respect to the double-cyclic variations. The open circles and solid circles denote the data in literature and in our observation, respectively. The solid line in the upper panel refers to a combination of a upward parabolic and two cyclic changes. The dashed line represents only the upward parabolic variation that reveals a continuous increase in the orbital period. When the long-term period increase was subtracted, the superposition of a long (the dashed line) and a short (the solid line) periodic variation are displayed in the middle panel. These variations were removed, the residuals are plotted in the lowest panel.}
\end{center}
\end{figure}
\begin{figure}[!h]
\begin{center}
\includegraphics[width=1.0\columnwidth]{figure4.eps}
\caption{The two cyclic variations $\tau_{A}$ and $\tau_{B}$ extracted from the middle panel of Figure 3.}
\end{center}
\end{figure}
\section{Discussion}
The standard model predicts that the evolution of CVs is driven by angular momentum losses(AMLs). The result is that, as a CV evolves, the orbital period decreases.
However, our result show that the period of EX Dra is increasing at a rate of $\dot{P}={+7.46}\times10^{-11}{s} {s^{-1}}$. EX Dra is a long-period ($P=5.04$ h) CV containing a late-type main sequence star overfilling its Roche lobe ($M_{2}\sim0.54M_{\odot}$) and a white dwarf ($M_{1}\sim0.75M_{\odot}$) \citep[]{2000MNRAS.316..529B}, the mass transfer between two components will cause the orbit expansion. Supposing a conservation mass transfer on long time scales and adopting the parameters given by \citet{2000MNRAS.316..529B}, a calculation using the equation \citep[]{1977ARA&A..15..127T}
\begin{equation}
\frac{\dot{P}}{P}=-3\dot{M}_{2}\times(\frac{1}{M_{1}}-\frac{1}{M_{2}}),
\end{equation}
leads to a mass transfer rate of $\dot{M_{2}}=8.34\times10^{-8}M_{\odot} \\ yr^{-1}$. It is alternatively possible that the quadratic term is only a part of a longer cyclic oscillation.
Our results also reveal that there are two cyclic variations in the $O-C$ curve. To interpret cyclic period changes of EX Dra, two main mechanisms are the solar-type magnetic activity cycle in M-type secondary star \citep[]{1992ApJ...385..621A} and the light time travel effect. The Applegate mechanism built on the basis of the conclusion presented by \citet{1989SSRv...50..219H}.
They found that all cool component stars are strictly limited in the spectral types later than F5. Recently, however, a statistical investigation for the cyclic period oscillations has shown that the percentages of cyclic variations for both late-type and early type interacting binaries are very close \citep[]{2010MNRAS.405.1930L}. Thus, the conclusion proposed by \citet{1989SSRv...50..219H} may be not correct, and moreover , \citet{1992ApJ...385..621A} already noted that his model should be revised if the shell becomes a significant fraction of the star's mass ($M_{s}>0.1M_{2}$).
The secondary star in EX Dra is a late-type main sequence star with the spectral type about M1-3/5 \citep[]{2003A&A...404..301R}(update RKcat7.23 version, 2015), this star should have a very deep convective envelope. With the theoretical model calculation and statistics, the shell mass of EX Dra's donor was estimated to be $M_{s}\approx0.15M_{\odot}\approx 0.28M_2$. To explain the cyclic oscillation of the pre-CV NN Ser, moreover, \citet{2006MNRAS.365..287B} by comparing the energies required to cause the observed variation found that NN Ser's secondary star cannot provide enough energy to drive Applegate mechanism. Using the same method for EX Dra, the required energies to produce two cyclic oscillations were calculated and shown in Figure 5. The results show that the required minimum energy in Case A are larger than the total energy radiated in 21.4 yr, and in Case B the required minimum energy are also slightly larger than the total energy radiated in a whole cycle (see Figure 5). Combining the parameters presented by \citet{2000MNRAS.316..529B} with Kepler's third law
\begin{equation}
{P^{2}_{orb}}=\frac{4\pi^{2}a^{3}}{G(M_{1}+M_{2})},
\end{equation}
to yield the orbital separation as $a=1.62R_{\odot}$. Applying $T_{2}=3400K$ for the M2-3 type, the luminosity of the secondary star can be draw as $L_{2}=(\frac{R_{2}}{R_{\odot}})^{2}(\frac{T_{2}}{T_{\odot}})^{4}L_{\odot}$.
Therefore, the Applegate mechanism is difficult to explain the observed cyclic changes.
The most plausible explanation seems to be a pair of light travel time effects via the presence of two companions.
The mass function and the mass of tertiary companions were derived by using the following equation \citep[]{1985ibs..book.....P}:
\begin{equation}
f(m)_A=\frac{4\pi^{2}}{GP_{A}^{2}}(a_{A}^{'}sini^{'}_{A})^{3}=\frac{(M_{A}sini'_{A})^{3}}{(M_{1}+M_{2}+M_{A})^{2}},
\end{equation}
and
\begin{equation}
f(m)_B=\frac{4\pi^{2}}{GP_{B}^{2}}(a_{B}^{'}sini^{'}_{B})^{3}=\frac{(M_{B}sini'_{B})^{3}}{(M_{1}+M_{2}+M_{B})^{2}},
\end{equation}
where $G$ is the gravitational constant, $P_A$ and $P_B$ are the periods of $\tau_{A}$ and $\tau_{B}$, and $a'_{A}sini'_{A}$ and $a'_{B}sini'_{B}$ can be determined by
\begin{equation}
a'_{A}sini'_{A}=K_{A}\times c ,
\end{equation}
and
\begin{equation}
a'_{B}sini'_{B}=K_{B}\times c ,
\end{equation}
$K_A$ and $K_B$ are the semi-amplitude of $\tau_{A}$ and $\tau_{B}$. The results are listed in Table 3. Assuming a random distribution of orbital plane inclinations, the orbital inclination for the companion A (i.e. Case A) is larger than $22^{\circ}.96$, the mass corresponds to $M_{A}\leq0.075M_{\odot}$, it may be a brown dwarf with $74.5\,\%$ and a low-mass star with only $25.5\,\%$ probability. As for companion B (corresponding to Case B), if its orbital inclination is less than $42^{\circ}.39$, the mass is $M_B\geq0.075M_{\odot}$, it may be a brown dwarf with $52.9\,\%$ and low-mass star with $47.1\,\%$. If they are coplanar (i.e. $i'=i=85^{\circ}$) to the orbital plane of the eclipsing pair, their masses would match to two brown dwarfs.
\begin{table*}
\caption{Orbital parameters of the circumbinary substellar objects.}
\begin{center}
\small
\begin{tabular}{lllllllll}
\hline
\hline
Parameters &Companion A &Companion B \\
\hline
Eccentricity, $e_A$ and $e_{B}$ & 0 & 0 \\
Mass function,$f(m)_A$ and $f(m)_B$ $(M_{\odot})$ & $1.28(\pm0.63)\times10^{-5}$ &$6.37(\pm0.33)\times10^{-5}$ \\
The companion masses, $M_{A}\sin{i'_A}$ and $M_{B}\sin{i'_B}$ $(M_{Jup})$ & $29.3(\pm0.6)$ & $50.8(\pm0.2)$ \\
Semi-major axis of the planet, $d_{A}$ and $d_{B}$ ($au,{i'=90^{\circ}}$) & $9.83(\pm1.21)$ & $3.18(\pm0.11)$ \\
\hline
\end{tabular}
\end{center}
\end{table*}
\begin{figure}[!h]
\begin{center}
\includegraphics[width=1.0\columnwidth]{figure5.eps}
\caption{Energy required to cause two periodic changes in the $O-C$ diagram by using Applegate's mechanism. $M_s$ refers to the assumed shell mass of the secondary star. The black dashed line denotes the energy required for different shell mass in Case A, and the black solid line corresponding to Case B. The grey solid line represents the total energy radiates from the secondary in 4 years and the grey dashed line is the total radiant energy of the secondary in 21.4 years.}
\end{center}
\end{figure}
\section{Conclusions}
We have published 29 new mid-eclipse times of EX Dra in quiescence spanning from 2009 to 2016. These mid-eclipse times were used to analyze the orbital period variation.
Besides a secular increase with a rate of $\dot{P}={+7.46}\times10^{-11}{s} {s^{-1}}$, the orbital period also shows the double-cyclic changes. According to the evolutionary theory of CVs, the orbital period should decrease.
If the long-term period increase was explained as the mass transfer from the secondary to primary star, the derived mass transfer rate is $\dot{M_{2}}={8.34}\times10^{-8}M_{\odot}yr^{-1}$.
However, it is possible that the quadratic term may be just a observed part of a longer cyclic oscillation.
For double periodic oscillations, EX Dra's secondary star can not provide enough energy to satisfy the energy requirements of Applegate mechanism. The more acceptable explanation is the existence of a pair of substellar objects around EX Dra. Assuming the circumbinary objects to be in the orbital plane ($i'=i=85^{\circ}$) of the eclipsing pair, they are two brown dwarfs.
The orbital parameters of the substellar objects in Table 3 reveal some interesting features. First, both the orbits are circular. Second, the orbital periods of $21.40(\pm1.44)$ and $3.99(\pm0.11)$ years are nearly the ratio of $5:1$. This implies that the possibility exists for the mean-motion resonance between the two companions and their orbits would be stable.
From the evolutionary perspective, CVs are products of a common envelope (CE) phase \citep[]{2013A&ARv..21...59I}. The circumbinary companions may originate from a large protoplanetary disc or a fragmentation of protostellar disc.
In the former case, the formation process is similar to two hot subdwarf stars HW Vir and AA Dor \citep[]{2000A&A...356..665R, 2009ARA&A..47..211H}, and its description will not be repeated here. In the latter case, the formation process is as follows: the objects will started with the mass a few $M_{Jup}$ and then increase their mass by accreting material from the disc \citep[]{2009A&A...495..201A, 2009MNRAS.392..413S}. For the former formed objects, they would migrate inwards and gain enough mass to become stars \citep[]{2007MNRAS.382L..30S}; for the objects staying in the outer disc region, they could not gain enough mass, and become brown dwarfs \citep[]{2009MNRAS.400.1563S}. The higher-mass objects of the inner region will evolve to progenitor of the post-common envelope binaries. The circumbinary companions formed at the almost same as their hosts and survived the CE phase \citep[]{2014MNRAS.444.1698B}.
Besides, there are also other possibilities, such as the second generation substellar originated in CE event \citep[]{2014A&A...562A..19V}. However, there is only a remote possibility for EX Dra because THE substellar objects have relatively large mass($29.3$ and $50.8$$M_{Jup}$) \citep[]{2014MNRAS.444.1698B}. Moreover, the circular orbits means that they may have existed for a long time-scales before the CE phase. Certainly, in order to confirm our conclusion, further observations are needed in the future.
\acknowledgments
This work is supported by the Chinese Natural Science Foundation (Grant No. 11325315, 11133007, 11573063 and 11611530685), the Strategic Priority Research Program ``The Emergence of Cosmological Structure'' of the Chinese Academy of Sciences (Grant No. XDB09010202) and the Science Foundation of Yunnan Province (Grant No. 2012HC011). This study is also supported by the Russian Foundation for Basic Research (project No. 17-52-53200).
New CCD photometric observations of EX Dra were obtained with the 60cm and the 2.4m telescopes at the Yunnan Observatories, the 85cm telescope in Xinglong Observation base in China and 50cm and 60cm telescopes of Sternberg Astronomical Institute Crimean Station. Finally, we thank the anonymous referee for those helpful comments and suggestions.
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 1,684
|
\section{Introduction}
Supersymmetry is an elegant solution of hierarchy problem of Standard Model (SM) \cite{SUSY},
however, a simple supersymmetric extension of SM suffers from non-conservation of baryon number
and $\mu$-problem.
Therefore we must introduce new symmetry such as R-parity in MSSM,
to suppress proton decay operators and $\mu$-term.
One of the solutions of $\mu$-problem is given by introducing extra $U(1)$ gauge symmetry
\cite{extra-u1}.
In this frame work, several new superfields such as singlet $S$, exotic quarks $G,G^c$,
must be introduced to cancel gauge anomaly.
This is the elegant solution of $\mu$-problem,
however proton instability is not solved because
the baryon number violating interactions in superpotential of MSSM are replaced by single
exotic quark interactions.
In the superpotential, there is no obvious distinction between
baryon number violating trilinear terms and Yukawa interactions.
Therefore it is natural to introduce flavor symmetry in suppressing baryon number
violating operators.
Especially non-Abelian discrete symmetry is good candidate for the flavor symmetry,
because large mixing angles of Maki-Nakagawa-Sakata (MNS) mixing matrix
may be explained simultaneously, and more simply, non-Abelian symmetry can be the reason why generation exists.
At previous work,
we explained $S_4\times Z_2$ flavor symmetry not only realizes maximal mixing angle $\theta_{23}$
but also suppresses proton decay based on
$SU(3)_c\times SU(2)_W\times U(1)_Y\times U(1)_X\times U(1)_Z$ gauge symmetry \cite{s4u1}.
As the suppression mechanism of proton decay is complicated and model dependent,
in this paper, we give more detailed estimation of proton life time and
investigate several flavon sectors.
If we start from exactly flavor symmetric theory,
the spontaneous flavor symmetry breaking realizes special VEV direction of flavons,
which affects proton life time significantly.
If the Yukawa hierarchy is realized by Froggatt-Nielsen mechanism,
as $p\to e^+X$ are suppressed
due to the small coupling constants, $p\to \mu^+X$ may dominate proton decay width.
In our model, as $p\to \mu^+X$ are suppressed by cancellation,
$p\to e^+K^0$ dominates the proton decay width.
This paper is organized as follows.
In section 2, we estimate proton life time based on gauge non-singlet flavon model.
In section 3, we modify the flavon sector by adding gauge-singlet flavon and
Froggatt-Nielsen flavon.
In section 4, we eliminate gauge non-singlet flavon and construct
Dirac neutrino model.
Finally we give conclusion of our analysis in section 5.
\section{$S_4\times Z_2$ flavor symmetric extra U(1) model}
At first we explain the basic structure of our model.
We extend the gauge symmetry to $G_{32111}=SU(3)_c\times SU(2)_W\times U(1)_Y\times U(1)_X\times U(1)_Z$
which is the subgroup of $E_6$.
In order to cancel gauge anomaly, we must add new superfields, such as
SM singlet $S$, exotic quark $G,G^c$ (hereafter we call them g-quark) and right handed neutrino (RHN) $N^c$.
We can embed these superfields with MSSM superfields $Q,U^c,D^c,L,E^c,H^U,H^D$
into ${\bf 27}$ of $E_6$ \cite{e6}.
As the singlet $S$ develops VEV and breaks $U(1)_X$ gauge symmetry, $O$ (TeV) scale $\mu$-term
is induced naturally.
In order to break $U(1)_Z$ and generate a large Majorana mass of RHN,
we add SM singlet $\Phi,\Phi^c$.
The gauge representations of superfields are given in Table 1 \cite{s4u1}.
\begin{table}[htbp]
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|c||c|c|}
\hline
&$Q$ &$U^c$ &$E^c$&$D^c$ &$L$ &$N^c$&$H^D$&$G^c$
&$H^U$&$G$ &$S$ &$\Phi$&$\Phi^c$\\ \hline
$SU(3)_c$&$3$ &$3^*$ &$1$ &$3^*$ &$1$ &$1$ &$1$ &$3^*$ &$1$
&$3$ &$1$ &$1$ &$1$ \\ \hline
$SU(2)_W$&$2$ &$1$ &$1$ &$1$ &$2$ &$1$ &$2$ &$1$ &$2$
&$1$ &$1$ &$1$ &$1$ \\ \hline
$y=6Y$ &$1$ &$-4$ &$6$ &$2$ &$-3$&$0$ &$-3$ &$2$ &$3$
&$-2$&$0$ &$0$ &$0$ \\ \hline
$x$ &$1$ &$1$ &$1$ &$2$ &$2$ &$0$ &$-3$ &$-3$ &$-2$
&$-2$&$5$ &$0$ &$0$ \\ \hline
$z$ &$-1$&$-1$ &$-1$ &$2$ &$2$ &$-4$ &$-1$ &$-1$ &$2$
&$2$ &$-1$&$8$ &$-8$ \\ \hline
$R$ &$-$ &$-$ &$-$ &$-$ &$-$ &$-$ &$+$ &$+$ &$+$
&$+$ &$+$ &$+$ &$+$ \\ \hline
\end{tabular}
\end{center}
\caption{$G_{32111}$ assignment of fields.
Where the $x$, $y$ and $z$ are charges of $U(1)_X$, $U(1)_Y$ and $U(1)_Z$,
and $Y$ is hypercharge. R-parity $R=\exp\left[\frac{i\pi}{20}(3x-8y+15z)\right]$ is unbroken.}
\end{table}
Under the gauge symmetry given in Table 1, the renormalizable superpotential is given by
\eqn{
W&=& Y^UH^UQU^c +Y^DQD^cH^D +Y^EH^DLE^c +Y^NH^ULN^c +Y^M\Phi N^cN^c+\lambda SH^UH^D+kSGG^c \nonumber \\
&+&M\Phi \Phi^c +Y^{QQG}GQQ +Y^{UDG}G^cU^cD^c +Y^{EUG}GE^cU^c +Y^{QLG}G^cLQ +Y^{NDG}GN^cD^c.
}
In this superpotential, unwanted terms are included in the second line.
The first term of the second line is the mass term of singlets $\Phi, \Phi^c$ which prevent singlets
from developing VEVs. The other five terms of the second line are single g-quark interactions, which
break baryon and lepton number and induce rapid proton decay.
In the first line, we must take care of the flavor changing neutral currents (FCNCs)
induced by extra Higgs bosons \cite{e6-FCNC}.
Therefore the superpotential Eq.(1) is not consistent at the present stage.
In order to stabilize proton, we introduce $S_4\times Z_2$ flavor symmetry.
If we assign $G,G^c$ to $S_4$ triplet and quarks and leptons to doublet or singlet,
the single g-quark interaction is forbidden.
However, as the g-quark must never be stable from phenomenological reason,
we assign $\Phi^c$ to $S_4$ triplet to break the flavor symmetry slightly.
In this case, as $\Phi,\Phi^c$ play the role of flavons, we call them gauge non-singlet flavons.
In order to realize the large mixing angle of $\theta_{23}$ in the MNS matrix and
suppress the Higgs-mediated FCNCs, we assign the superfields
in our model as given in Table 2 \cite{s4pamela}.
In the non-renormalizable part of superpotential,
the single g-quark interactions which contribute to the g-quark decay are given as follows
\eqn{
W&\supset&\frac{Y^{QQG}}{M^2_P}\Phi\Phi^cQQG
+\frac{Y^{UDG}}{M^2_P}\Phi\Phi^cG^cU^cD^c
+\frac{Y^{EUG}}{M^2_P}\Phi\Phi^cGE^cU^c
+\frac{Y^{QLG}}{M^2_P}\Phi\Phi^cG^cLQ .
}
\begin{table}[htbp]
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|}
\hline
&$Q_1$ &$Q_2$ &$Q_3$ &$U^c_1$ &$U^c_2$ &$U^c_3$ &$D^c_1$ &$D^c_2$ &$D^c_3$ \\
\hline
$S_4$ &${\bf 1}$&${\bf 1}$& ${\bf 1}$&${\bf 1}$&${\bf 1}$&${\bf 1}$ &${\bf 1}$&${\bf 1}$&${\bf 1}$\\
\hline
$Z_2$ &$-$ &$-$ &$-$ &$-$ &$-$ &$-$ &$-$ &$-$ &$-$ \\
\hline
\hline
&$E^c_1$ &$E^c_2$ &$E^c_3$ &$L_i$ &$L_3$ &$N^c_i$ &$N^c_3$ &$H^D_i$ &$H^D_3$ \\
\hline
$S_4$ &${\bf 1}$&${\bf 1}$&${\bf 1'}$&${\bf 2}$&${\bf 1}$&${\bf 2}$ &${\bf 1}$&${\bf 2}$&${\bf 1}$ \\
\hline
$Z_2$ &$+$ &$-$ &$+$ &$-$ &$-$ &$+$ &$-$ &$-$ &$+$ \\
\hline
\hline
&$H^U_i$ &$H^U_3$ &$S_i$ &$S_3$ &$G_a$ &$G^c_a$ &$\Phi_i$ &$\Phi_3$ &$\Phi^c_a$\\
\hline
$S_4$ &${\bf 2}$&${\bf 1}$&${\bf 2}$ &${\bf 1}$&${\bf 3}$&${\bf 3}$ &${\bf 2}$&${\bf 1}$&${\bf 3}$\\
\hline
$Z_2$ &$-$ &$+$ &$-$ &$+$ &$+$ &$+$ &$+$ &$+$ &$+$ \\
\hline
\end{tabular}
\end{center}
\caption{$S_4\times Z_2$ assignment of superfields
(Where the index $i$ of the $S_4$ doublets runs $i=1,2$,
and the index $a$ of the $S_4$ triplets runs $a=1,2,3$.)}
\end{table}
\subsection{Higgs sector and hidden sector}
Under the flavor symmetry given in Table 2, the superpotential of Higgs sector is given by,
\eqn{
W_H&=&\lambda_1S_3(H^U_1H^D_1+H^U_2H^D_2)+\lambda_3S_3H^U_3H^D_3 \nonumber \\
&+&\lambda_4H^U_3(S_1H^D_1+S_2H^D_2)+\lambda_5(S_1H^U_1+S_2H^U_2)H^D_3.
}
where one can take, without any loss of the generalities,
$\lambda_{1,3,4,5}$ as real, by redefining four arbitrary fields of
$\{S_i, S_3, H^U_i, H^U_3, H^D_i, H^D_3\}$.
As only $S_4$ singlets $H^U_3,H^D_3$ and $S_3$ couple to quarks and g-quarks respectively,
they behave like MSSM Higgs and SM singlet respectively. $S_i$ also behaves like SM singlets.
Through the
renormalization group equations, the squared masses of $H^U_3,S_3$ become negative,
they develop VEVs and break gauge symmetry.
As the result, the A-term $A_3S_3H^U_3H^D_3$ enforces $H^D_3$ developing VEV.
However, $S_4$ doublets do not develop VEVs.
To generate the VEVs of $S_4$ doublets, we must add flavor breaking squared mass terms.
The origin of flavor breaking terms is discussed below.
Note that there is accidental $O(2)$ symmetry
induced by the common rotation of the $S_4$ doublets in $W_H$.
As it is thought that the scalar squared masses are induced
as the result of SUSY breaking in hidden sector,
we assume flavor symmetry is broken at the same time.
We assume hidden sector is described by
flavor symmetric extension of O'Raifeartaigh model \cite{OR}.
We introduce gauge singlet $A,B_+,B_i,C_+,C_i$ and assign
$Z'_2$ charges to them to separate hidden sector from observable sector.
We assume $U(1)_R$ symmetry is hold at the limit of infinite Planck scale,
$M_P\to \infty$. The representations of hidden sector superfields
under the $S_4\times Z_2\times Z'_2\times U(1)_R$ symmetry are given in table 3.
\begin{table}[htbp]
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|}
\hline
&$A$ &$B_+$ &$B_i$ &$C_+$ &$C_i$ \\
\hline
$S_4$ &$1$ &$1$ &$2$ &$1$ &$2$ \\
\hline
$Z_{2(4)}$&$+(0)$ &$+(0)$ &$-(1/2)$ &$+(0)$ &$-(1/2)$ \\
\hline
$Z'_2$ &$+$ &$-$ &$-$ &$-$ &$-$ \\
\hline
$U(1)_R$ &$2$ &$2$ &$2$ &$0$ &$0$ \\
\hline
\end{tabular}
\end{center}
\caption{$S_4\times Z_{2(4)}\times Z'_2\times U(1)_R$ assignment of superfields
(Where the index $i$ of the $S_4$ doublets runs $i=1,2$.)}
\end{table}
Under the symmetry given in Table 3, the superpotential of hidden sector is given by,
\eqn{
W_{\mbox{hidden}}&=&-M^2A+m_+B_+C_++m(B_1C_1+B_2C_2)+\frac12\lambda_+AC^2_+
+\frac12\lambda A(C^2_1+C^2_2) .
}
For the F-terms of hidden sector superfields,
\eqn{
F_A&=&-M^2+\frac12\lambda_+C^2_+ +\frac12\lambda(C^2_1+C^2_2), \\
F_{B_+}&=&m_+C_+ ,\\
F_{B_1}&=&mC_1 ,\\
F_{B_2}&=&mC_2 ,\\
F_{C_+}&=&m_+B_++\lambda_+AC_+ ,\\
F_{C_1}&=&mB_1+\lambda AC_1 ,\\
F_{C_2}&=&mB_2+\lambda AC_2,
}
as it is impossible to satisfy the equations $F_A=0$ and $F_{B_+}=F_{B_i}=0$ at the same time,
SUSY is spontaneously broken.
At the same time, flavor symmetry $S_4\times Z_2$ is also broken spontaneously.
As the superpotential has accidental $O(2)$ symmetry,
the direction of $S_4$ doublet F-term defined by,
\eqn{
F_{B_1}=Fc_B,\quad F_{B_2}=Fs_B,
}
is described by free parameter $\theta_B$. Although the spontaneous breaking of $O(2)$ results the
appearance of Nambu-Goldstone boson (NGB), as the interactions of the NGB
with observable sector particles are suppressed by Planck scale and the mixings between Higgs bosons and NGB are suppressed by large hidden sector VEV scale,
we assume this NGB does not
cause any problem. The problem of R-axion, which is the NGB of spontaneous $U(1)_R$ symmetry breaking,
may be avoided by adding explicit $U(1)_R$ symmetry breaking higher dimensional terms \cite{R-sym}.
The problem of O(2) NGB may be also solved by adding O(2) breaking
higher dimendional terms.
The SUSY breaking in hidden sector is mediated to observable sector by gravity
through the non-renormalizable terms in K\"ahler potential as given by,
\eqn{
K\supset \frac{1}{M^2_P}[a_H(H_1B_1+H_2B_2)(H_1B_1+H_2B_2)^\dagger
+(b_HB_+H_3(H_1B_1+H_2B_2)^\dagger+h.c. )],
}
where $H=H^U,H^D,S$, and flavor symmetric terms and the contributions to other superfields are omitted
\footnote{In this paper, we assume flavor symmetric SUSY breaking parameters are larger than
flavor breaking SUSY breaking parameters without any reason, because we are interested in the effects of degenerated scalar g-quark mass spectrum.}.
These terms induce scalar squared mass terms as follow,
\eqn{
V_{FB}= m^2_{BH1}|c_BH_1+s_BH_2|^2
+[m^2_{BH2}H_3(c_BH_1+s_BH_2)^\dagger+h.c.].
}
If we substitute the VEV $\left<H_1\right>=vc_H, \left<H_2\right>=vs_H,\left<H_3\right>=v'$ for $H_a$, then we get
\eqn{
V_{FB}&=&m^2_{BH1}v^2\cos^2(\theta_B-\theta_H)
+[m^2_{BH2}vv'\cos(\theta_B-\theta_H)+h.c] \nonumber \\
&=&a[\cos(\theta_B-\theta_H)-b]^2+\mbox{const} .
}
At second line, we simplified the equation, because we are interested only in direction $\theta_H$.
As we can change the sign of $b$ by the redefinition of the sign of $v$,
we can define $b>0$ without loss of generality.
The minimum of potential $V_{FB}$ is classified as follow,
\eqn{
a>0, b>1&:&\theta_H=\theta_B, \\
a>0 ,b<1&:&\theta_H=\theta_B-\arccos b, \\
a<0&:&\theta_H=\theta_B+\pi ,
}
from which one can see that the angle $\theta_H$ is controlled by
free parameter $a,b$ for the given $\theta_B$.
In this section, we assume the condition Eq.(16) is satisfied for $H^U,H^D,S$ and
select the common VEV direction $\theta_B$.
In this direction, as the flavor symmetric part of Higgs potential has accidental $O(2)$ symmetry
and depends only on $\theta_{H^U}-\theta_{H^D},\theta_{H^U}-\theta_S,\theta_{H^D}-\theta_S$ ,
$\theta_H(=\theta_{H^U}=\theta_{H^D}=\theta_S=\theta_B)$ is fixed by $V_{FB}$.
Therefore we can not assume $V_{FB}$ as perturbation
even if flavor breaking parameter is small.
\subsection{Flavon sector}
The superpotential of flavon sector is given by,
\eqn{
W_\Phi&=&\frac{Y^\Phi_1}{2M_P}\Phi^2_3\left[(\Phi^c_1)^2+(\Phi^c_2)^2+(\Phi^c_3)^2\right] \nonumber \\
&+&\frac{Y^\Phi_2}{2M_P}(\Phi^2_1+\Phi^2_2)\left[(\Phi^c_1)^2+(\Phi^c_2)^2+(\Phi^c_3)^2\right] \nonumber \\
&+&\frac{Y^\Phi_3}{2M_P}\left\{2\sqrt{3}\Phi_1\Phi_2\left[(\Phi^c_2)^2-(\Phi^c_3)^2\right]
+(\Phi^2_1-\Phi^2_2)\left[(\Phi^c_2)^2+(\Phi^c_3)^2-2(\Phi^c_1)^2\right]\right\} \nonumber \\
&+&\frac{Y^\Phi_4}{2M_P}\Phi_3\left\{\sqrt{3}\Phi_1\left[(\Phi^c_2)^2-(\Phi^c_3)^2\right]
+\Phi_2\left[(\Phi^c_2)^2+(\Phi^c_3)^2-2(\Phi^c_1)^2\right]\right\},
}
and the flavor symmetric part of potential is given by,
\eqn{
V&=&-m^2_1|\Phi_3|^2+m^2_2[|\Phi_1|^2+|\Phi_2|^2]
+m^2_3[|\Phi^c_1|^2+|\Phi^c_2|^2+|\Phi^c_3|^2] \nonumber \\
&-&\frac{A^\Phi_1}{2M_P}\Phi^2_3\left[(\Phi^c_1)^2+(\Phi^c_2)^2+(\Phi^c_3)^2\right] \nonumber \\
&-&\frac{A^\Phi_2}{2M_P}(\Phi^2_1+\Phi^2_2)\left[(\Phi^c_1)^2+(\Phi^c_2)^2+(\Phi^c_3)^2\right] \nonumber \\
&-&\frac{A^\Phi_3}{2M_P}\left\{2\sqrt{3}\Phi_1\Phi_2\left[(\Phi^c_2)^2-(\Phi^c_3)^2\right]
+(\Phi^2_1-\Phi^2_2)\left[(\Phi^c_2)^2+(\Phi^c_3)^2-2(\Phi^c_1)^2\right]\right\} \nonumber \\
&-&\frac{A^\Phi_4}{2M_P}\Phi_3\left\{\sqrt{3}\Phi_1\left[(\Phi^c_2)^2-(\Phi^c_3)^2\right]
+\Phi_2\left[(\Phi^c_2)^2+(\Phi^c_3)^2-2(\Phi^c_1)^2\right]\right\} \nonumber \\
&+&\mbox{F-term}+\mbox{D-term}.
}
As this potential does not have accidental continuum symmetry,
flavor breaking terms can be assumed as perturbation.
We assume negative squared mass of $\Phi_3$ pulls the trigger of $U(1)_Z$ breaking
and $\Phi_3$ and $\Phi^c_a$ develop VEVs along the D-flat direction.
For the VEV of $\Phi_i$, there are two possibilities in the flavor symmetry breaking.
The one of them is $S_3$ symmetric vacuum defined as,
\eqn{
\Phi_3=V,\quad \Phi_i=0,\quad \Phi^c_a=\frac{V}{\sqrt{3}}(1,1,1),
}
and the other is $S_3$ breaking vacuum defined as,
\eqn{
\Phi_3=Vc,\quad \Phi_i=Vs(0,1),\quad \Phi^c_a=\frac{V}{\sqrt{2}}(0,1,1).
}
Other degenerated vacuums are given by acting $S_4$ translations on these VEVs, respectively.
Note that spontaneous breaking of discrete symmetry causes domain wall problem.
We assume flavor symmetry is not recovered in reheating era after inflation \cite{domain-wall}
\footnote{Here we note that the reheating temerature must be lower than $10^7$ GeV
to avoid over production of gravitino in gravity mediation scenario \cite{g-lifetime}.
For such a low temperature, there is no particle such as $U(1)_Z$ gauge boson or RHN
which interacts with flavons through renormalizable operators
in thermal bath. Therefore thermal mass of flavon is induced by nonrenormalizable terms
and negligible (for example, $m_{\mbox{thermal}} \sim m_\nu T/V<10^{-6}$ eV for the effective operator $\frac{m_\nu}{V}\Phi\nu\nu$) , and flavor symmetry is not recovered.}.
As the vacuum defined in Eq.(22) is not phenomenologically allowed
because g-quark $g_1,g^c_1$ become stable,
we select the vacuum defined in Eq.(21) by tuning $m^2_2$ large enough.
From the simplified potential
\eqn{
V\sim -m^2_{SUSY}|\Phi|^2+\frac{|Y^\Phi|^2}{M^2_P}|\Phi^3|^2,
}
the size of flavon VEV is estimated as follow,
\eqn{
\Phi \sim \sqrt{\frac{m_{SUSY}M_P}{Y^\Phi}}.
}
If we put $Y^\Phi\sim 0.01$ and $m_{SUSY}\sim 10$ TeV, then we get $V\sim 10^{12}$ GeV
which is the favorable value to satisfy the constraint for g-quark and proton life time
at the same time \cite{king}.
\subsection{Quark and Lepton sector}
As the flavor symmetry reduces the number of free parameters drastically,
we can decide the value of free parameters by very few assumptions.
For the quark sector, superpotential is given by
\eqn{
W_Q=Y^U_{ab}H^U_3Q_aU^c_b+Y^D_{ab}H^D_3Q_aD^c_b.
}
As the extra Higgs $H^U_i,H^D_i$ do not couple to quarks,
Higgs mediated flavor changing neutral currents are not induced.
For the lepton sector, superpotential is given by
\eqn{
W_L&=&Y^N_2\left[H^U_1(L_1N^c_2+L_2N^c_1)+H^U_2(L_1N^c_1-L_2N^c_2)\right] \nonumber \\
&+&Y^N_3H^U_3L_3N^c_3+Y^N_4L_3(H^U_1N^c_1+H^U_2N^c_2) \nonumber \\
&+&Y^E_1E^c_1(H^D_1L_1+H^D_2L_2)+Y^E_2E^c_2H^D_3L_3+Y^E_3E^c_3(H^D_1L_2-H^D_2L_1) \nonumber \\
&+&\frac12 Y^M_1\Phi_3(N^c_1N^c_1+N^c_2N^c_2)+ \frac12 Y^M_3\Phi_3 N^c_3N^c_3 \nonumber \\
&+&\frac12Y^M_2[\Phi_1(2N^c_1N^c_2)+\Phi_2(N^c_1N^c_1-N^c_2N^c_2)].
}
Without any loss of generalities, by the field redefinition, we can define
$Y^E_{1,2,3},Y^M_{1,3},Y^N_{2,4}$ are real and non-negative and
$Y^M_2,Y^N_3$ are complex.
We tune the angle $\theta_B=\theta_{23}=\frac{\pi}{4}$ by hand and
define the VEVs of scalar fields as follows
\eqn{
&&\left<H^U_1\right>=\left<H^U_2\right>=\frac{1}{\sqrt{2}}v_u,\quad
\left<H^U_3\right>=v'_u ,\quad
\left<H^D_1\right>=\left<H^D_2\right>=\frac{1}{\sqrt{2}}v_d,\quad
\left<H^D_3\right>=v'_d ,\nonumber \\
&&\left<S_1\right>=\left<S_2\right>=\frac{1}{\sqrt{2}}v_s,\quad
\left<S_3\right>=v'_s ,\nonumber \\
&&\left<\Phi_1\right>=\left<\Phi_2\right>=0,\quad
\left<\Phi_3\right>=V , \quad
\left<\Phi^c_1\right>=\left<\Phi^c_2\right>=\left<\Phi^c_3\right>=\frac{V}{\sqrt{3}},
}
and define the mass parameters as follows \cite{kubo}
\eqn{
\begin{tabular}{llll}
$M_1=Y^M_1V$, & $M_3=Y^M_3V$, & & \\
$m^\nu_2=Y^N_2v_u$, & $m^\nu_3=|Y^N_3|v'_u$, & $m^\nu_4=Y^N_4v_u$, & \\
$m^l_1=Y^E_1v_d$, & $m^l_2=Y^E_2v'_d$, & $m^l_3=Y^E_3v_d$. &
\end{tabular}
}
With these parameters, the mass matrices are given by
\eqn{
\begin{tabular}{ll}
$M_l=\frac{1}{\sqrt{2}}\Mat3{m^l_1}{0}{-m^l_3}{m^l_1}{0}{m^l_3}{0}{\sqrt{2}m^l_2}{0}$, &
$M_D=\frac{1}{\sqrt{2}}\Mat3{m^\nu_2}{m^\nu_2}{0}{m^\nu_2}{-m^\nu_2}{0}
{m^\nu_4}{m^\nu_4}{\sqrt{2}e^{i\delta}m^\nu_3}$, \\
$M_R=\Mat3{M_1}{0}{0}{0}{M_1}{0}{0}{0}{M_3}$. &
\end{tabular}
}
Due to the seesaw mechanism, the neutrino mass matrix is given by
\eqn{
M_\nu&=&M_DM^{-1}_RM^t_D=\Mat3{\rho^2_2}{0}{\rho_2\rho_4}
{0}{\rho^2_2}{0}
{\rho_2\rho_4}{0}{\rho^2_4+e^{2i\delta}\rho^2_3},
}
where
\eqn{
\rho_2=\frac{m^\nu_2}{\sqrt{M_1}},\quad \rho_4=\frac{m^\nu_4}{\sqrt{M_1}},\quad \rho_3=\frac{m^\nu_3}{\sqrt{M_3}}.
}
The charged lepton mass matrix is diagonalized as follow
\eqn{
V^\dagger_l M^*_l M^t_l V_l&=&\mbox{diag}(m^2_e,m^2_\mu, m^2_\tau)=((m^l_2)^2,(m^l_3)^2,(m^l_1)^2), \\
V_l&=&\frac{1}{\sqrt{2}}\Mat3{0}{-1}{1}
{0}{1}{1}
{-\sqrt{2}}{0}{0},
}
and those of the light neutrinos are given by
\eqn{
V^t_\nu M_\nu V_\nu&=&\mbox{diag}(e^{i(\phi_1-\phi)}m_{\nu_1},e^{i(\phi_2+\phi)}m_{\nu_2},m_{\nu_3}), \\
V_\nu&=&
\Mat3{-\sin\theta_\nu}{e^{i\phi}\cos\theta_\nu}{0}
{0}{0}{1}
{e^{-i\phi}\cos\theta_\nu}{\sin\theta_\nu}{0}.
}
From the above equations, the MNS matrix is given by
\eqn{
V_{MNS}&=&V^\dagger_lV_\nu P_\nu
=\frac{1}{\sqrt{2}}\Mat3{-\sqrt{2}e^{-i\phi}\cos\theta_\nu}{-\sqrt{2}\sin\theta_\nu}{0}
{\sin\theta_\nu}{-e^{i\phi}\cos\theta_\nu}{1}
{-\sin\theta_\nu}{e^{i\phi}\cos\theta_\nu}{1}P_\nu,
}
where
\eqn{
P_\nu=\mbox{diag}(e^{-i(\phi_1-\phi)/2},e^{-i(\phi_2+\phi)/2},1).
}
Here it is worth mentioning that a rather large mixing angle of 1-3 component of $V_{MNS}$; $\theta_{13}$, is measured by
the recent experiments such as T2K \cite{t2k11}, Double Chooz \cite{dc11}, Daya-Bay \cite{daya12}, and RENO \cite{reno12}.
In global analysis, moreover, the values of $\sin^2\theta_{13}$ is 0.026(0.027) depending on the normal(inverted) neutrino mass ordering
\cite{global-ana}.
As the value of $\theta_B$ is correctly tuned, experimental value of
$\theta_{23}$ is realized. As there is no parameter to tune $\theta_{13}$,
if $\theta_{13}\neq 0$ as is suggested by the recent experimental results, our model is excluded.
From the experimental constraints \cite{PDG},
\eqn{
\tan\theta_\nu=\frac{1}{\sqrt{2}},\quad
m^2_{\nu_2}-m^2_{\nu_1}=7.6\times 10^{-5}(\mbox{eV}^2),\quad
m^2_{\nu_2}-m^2_{\nu_3}=2.5\times 10^{-3}(\mbox{eV}^2),
}
the phase $\phi$ is constrained by the condition
\eqn{
r\cos\phi =0.361,\quad r=\frac{\rho_2}{\rho_4}.
}
If we put the size of VEVs of Higgs fields as follow,
\eqn{
v_u=10,\quad v'_u=155.3,\quad v_d=2.0,\quad v'_d=77.8\quad (\mbox{GeV}),
\label{vev}
}
then from the charged lepton masses \cite{mass}
\eqn{
m^l_1=1.75\mbox{GeV},\quad m^l_2=487\mbox{keV}, \quad m^l_3=103\mbox{MeV},
}
we can decide Yukawa coupling constants as follow,
\eqn{
Y^E_1=0.875,\quad Y^E_3=5.15\times 10^{-2},\quad Y^E_2=6.25\times 10^{-6}.
}
For the neutrinos, if we put
\eqn{
V=10^{12}\mbox{GeV},\quad Y^M_1=Y^M_3=1,
}
and assume $\phi=0$, all Yukawa coupling constants are decided as follow (see appendix)
\eqn{
\mbox{physical quantities}&:& \phi=\phi_2=0,\quad \phi_1=\pi , \nonumber \\
&&m_{\nu_1}=5.240\times 10^{-2}\mbox{eV},\quad
m_{\nu_2}=5.312\times 10^{-2}\mbox{eV},\quad m_{\nu_3}=1.795\times 10^{-2}\mbox{eV}
\nonumber \\
\mbox{parameters}
&:&\delta=\frac{\pi}{2} ,\quad
\rho^2_2=1.795\times 10^{-2}\mbox{eV},\quad
\rho^2_3=15.51\times 10^{-2}\mbox{eV},\quad \rho^2_4=13.79\times 10^{-2}\mbox{eV} , \nonumber \\
&&m^\nu_2=4.24\mbox{GeV},\quad m^\nu_3=12.45 \mbox{GeV},\quad m^\nu_4=11.74\mbox{GeV}, \nonumber \\
&&Y^N_2=0.424,\quad Y^N_3=0.080,\quad Y^N_4=1.17 .
}
For the lepton sector, there is no flavor changing process as same as quark sector
discussed above. Considering the interactions
\eqn{
{\cal L}_l
&=&Y^E_1\tau^c\left[l_\mu\left(\frac{H^D_2-H^D_1}{\sqrt{2}}\right)
+l_\tau\left(\frac{H^D_1+H^D_2}{\sqrt{2}}\right)\right]
-Y^E_2H^D_3e^cl_e \nonumber \\
&+&Y^E_3\mu^c\left[l_\mu\left(\frac{H^D_1+H^D_2}{\sqrt{2}}\right)
+l_\tau\left(\frac{H^D_1-H^D_2}{\sqrt{2}}\right)\right],
}
as e does not couple to $\mu$ and $\tau$,
$\mu\to e\gamma$ and $\tau\to e\gamma$ processes are forbidden.
Further more, due to the unbroken $S_2$ symmetry such as
$l_\mu\to-l_\mu,\mu^c\to-\mu^c,(H^D_1,H^D_2)\to(H^D_2,H^D_1)$,
$\tau\to\mu\gamma$ process is also forbidden.
This conclusion is not modified even in the case of $\theta_B\neq \frac{\pi}{4}$.
If there is small deviation from $\frac{\pi}{4}$ in $\theta_B$, under the $S_2$ translation
$H^D_1\to H^D_1\cos2\theta_B+H^D_2\sin2\theta_B, H^D_2\to H^D_1\sin2\theta_B-H^D_2\cos2\theta_B$,
$H^D_1c_B+H^D_2s_B$ behaves like $S_2$-even field and $H^D_1s_B-H^D_2c_B$ behaves like $S_2$-odd field.
Note that the $S_4$ flavor symmetry does not help to solve SUSY flavor problem in our model
because the quarks are assigned in $S_4$-singlet.
If the gaugino mass parameters are much larger than soft scalar masses, this problem may be solved.
\subsection{g-quark sector}
As the masses of g-quarks are given by
\eqn{
W_G=kS_3(G_1G^c_1+G_2G^c_2+G_3G^c_3),
}
the g-quark mass matrix is proportional to unit matrix.
For the scalar g-quarks, as the contribution from
soft flavor breaking term should be added, the degeneracy of masses may be broken.
However, such effects can be assumed as perturbation.
It is thought that dark matter does not have strong interaction,
g-quark should not be stable if the reheating temperature is higher than g-quark mass.
Under the symmetry defined in Table 1 and Table 2,
g-quarks can decay through the non-renormalizable terms as follows,
\eqn{
W_B&=&\frac{1}{M^2_P}Y^{QQG}_{ab}\Phi_3[G_1\Phi^c_1+G_2\Phi^c_2+G_3\Phi^c_3]Q_aQ_b \nonumber \\
&+&\frac{1}{M^2_P}Y^{UDG}_{ab}\Phi_3[G^c_1\Phi^c_1+G^c_2\Phi^c_2+G^c_3\Phi^c_3]U^c_aD^c_b \nonumber \\
&+&\frac{1}{M^2_P}Y^{EUG}_a\Phi_3[G_1\Phi^c_1+G_2\Phi^c_2+G_3\Phi^c_3]E^c_2U^c_a \nonumber \\
&+&\frac{1}{M^2_P}Y^{L_sQG}_a\Phi_3[G^c_1\Phi^c_1+G^c_2\Phi^c_2+G^c_3\Phi^c_3]L_3Q_a \nonumber \\
&+&\frac{1}{M^2_P}Y^{L_dQG}_a\Phi_3[\sqrt{3}L_1(G^c_2\Phi^c_2-G^c_3\Phi^c_3)
+L_2(G^c_2\Phi^c_2+G^c_3\Phi^c_3-2G^c_1\Phi^c_1)]Q_a.
}
Here we estimate the g-quark life time by the $Y^{EUG}$ interaction,
under the assumption that only the decay to right-handed slepton is kinematically allowed,
because it is natural to expect that the right-handed sleptons
are lighter than the squarks and the left-handed sleptons in the result of
running based on renormalization group equation.
If we put
\eqn{
Y^{EUG}_1=Y^{EUG}_2=Y^{EUG}_3=1,
}
then the interaction is given by
\eqn{
{\cal L}_g=\frac{(A^{EUG}_{RF})_SV^2}{\sqrt{3}M^2_P}E^c_2g_1(u^c_1+u^c_2+u^c_3),
}
where $E^c_2$ is right handed selectron, and the renormalization factor $(A^{EUG}_{RF})_S$
is evaluated by the RGE
\eqn{
(4\pi)\frac{d\ln Y^{EUG}_a}{d\ln\mu}=-\frac{16}{3}\alpha_s ,
}
and given by
\eqn{
(A^{EUG}_{RF})_S&=&\left(\frac{M_P}{M_Z}\right)^{4\alpha_s/3\pi}
=\left(\frac{2.43\times 10^{18}}{91}\right)^{0.05008}=6.647,
}
where only QCD correction is accounted. This approximation is not bad
because the beta function of the coupling constant of strong interaction $g_s$ vanishes at 1-loop level
in our model, which makes the contribution of $\alpha_s$ dominant in the RGE of $Y^{EUG}$.
Using the interaction Eq.(49), we calculate the g-quark decay width.
For simplicity, we assume $u^c_a$ in Eq.(49) are mass eigenstates.
Requiring the life time of g-quark is shorter than 0.1 sec
(otherwise the success of BBN is spoiled \cite{g-lifetime}) as follow
\eqn{
\Gamma(g_1)=3\left(\frac{(A^{EUG}_{RF})_SV^2}{3M^2_P}\right)^2\frac{M_g}{16\pi}
>\frac{1}{0.1\ \mbox{sec}} ,
}
we get
\eqn{
\frac{M_g}{1000 \mbox{GeV}}\left(\frac{V}{M_P}\right)^4>2.25 \times 10^{-26},
}
which bounds the VEV size of flavon from below.
Finally, using the interaction
$Y^{QQG}-Y^{EUG}$, we estimate the proton decay width.
Integrating out the scalar g-quarks,
we get the effective four-Fermi interactions as follows
\eqn{
{\cal L}_{p\to e^+\pi^0}=\frac{V^4}{M^4_PM^2_G}
Y^{QQG}_{ab}Y^{EUG}_cA_{RF}\bar{q}_a\bar{q}_bu^c_ce^c,
}
where
\eqn{
A_{RF}&=&(A^{EUG}_{RF})_S(A^{QQG}_{RF})_S(A_{RF})_L \quad \cite{p-RGE},
}
and the renormalization factor $(A^{QQG}_{RF})_S$ is evaluated by the REG
\eqn{
(4\pi)\frac{d\ln Y^{QQG}_a}{d\ln\mu}=-\frac{24}{3}\alpha_s,
}
and given by
\eqn{
(A^{QQG}_{RF})_S&=&\left(\frac{M_P}{M_Z}\right)^{2\alpha_s/\pi}
=\left(\frac{2.43\times 10^{18}}{91}\right)^{0.07512}=17.139.
}
As the long distant part of renormalization factor is given by
\eqn{
(A_{RF})_L=\left(\frac{\alpha_s(1\mbox{GeV})}{\alpha_s(m_b)}\right)^{6/25}
\left(\frac{\alpha_s(m_b)}{\alpha_s(M_Z)}\right)^{6/23}=1.4 \quad \cite{p-decay},
}
we get
\eqn{
A_{RF}=159.5.
}
In the quark mass basis, Eq.(54) is rewritten as follow
\eqn{
{\cal L}_{p\to e^+\pi^0}&=&\frac{V^4}{M^4_PM^2_G}
\left[2(L^T_uY^{UDG}L_u)_{ab}(Y^{EUG}R_u)_c\right]A_{RF}\bar{u}'_a\bar{d}'_b(u^c_c)'e^c, \\
&&\bar{u}=L_u\bar{u}',\quad \bar{d}=L_d\bar{d}',\quad u^c=R_u(u^c)'.
}
For simplicity, we put
\eqn{
\left[2(L^T_uY^{UDG}L_u)_{11}(Y^{EUG}R_u)_1\right]=1,
}
then the proton decay width is given by
\eqn{
\Gamma(p\to \pi^0+e^+)&=&\frac{m_p}{64\pi f^2_\pi}
\left[\left(\frac{V}{M_P}\right)^4\frac{A_{RF}}{M^2_G}\right]^2(1+F+D)^2
\left(1-\frac{m^2_{\pi^0}}{m^2_p}\right)^2\alpha^2_p \quad \cite{PDform}.
}
If we put
\eqn{
&&F=0.47,\quad D=0.80,\quad \alpha_p=-0.012\ \mbox{GeV}^3,\quad f_\pi=130\ \mbox{MeV}, \quad
\cite{chiral}\nonumber \\
&&m_{\pi^0}=135\ \mbox{MeV},\quad m_p=940\ \mbox{MeV}, \quad \cite{PDG}
}
then we get
\eqn{
\Gamma(p\to \pi^0+e^+)&=&(5.01 \times 10^{-12}\ \mbox{GeV})
\left[\left(\frac{V}{M_P}\right)^4\frac{(1000\ \mbox{GeV})^2}{M^2_G}\right]^2 .
}
From the experimental bound $\tau(p\to \pi^0+e^+)>1600\times 10^{30}[\mbox{years}]$ \cite{PDG},
the VEV size of flavon is bounded from above as follow
\eqn{
\left[\left(\frac{V}{M_P}\right)^4\left(\frac{1000\ \mbox{GeV}}{M_G}\right)^2\right]^2
< 2.60\times 10^{-54}.
}
Hereafter we assume the approximation $M_g=M_G$ is held for simplicity.
From Eq.(53) and Eq.(66), the allowed region for $V$ is given by (see Fig.1)
\eqn{
2.25\times 10^{-26}\left(\frac{1000 \mbox{GeV}}{M_G}\right)<\left(\frac{V}{M_P}\right)^4
<1.61\times 10^{-27}\left(\frac{M_G}{1000\ \mbox{GeV}}\right)^2 .
}
This inequality holds when the mass bound,
\eqn{
M_G> 2.41 \mbox{TeV},
}
is satisfied.
\begin{figure}[ht]
\unitlength=1mm \hspace{3cm}
\begin{picture}(70,70)
\includegraphics[height=6cm,width=10cm]{v-mg22.eps}
\end{picture}
\caption{ $M_G$ versus $V$: The pink region comes from the constraint of the life time of the g-quark, which should be less than 0.1 sec. The green region comes from the constraint of the proton stability. The black region is allowed by the both constraints. The heavier of $M_G$, the wider the allowed region is.}
\label{proton}
\end{figure}
Before ending this section, we discuss the unsatisfactory point of this model.
Considering the mass spectrum of the quarks and charged leptons,
it is expected that the trilinear coupling of first generation is multi suppressed
by the suppression mechanism of Yukawa couplings and $S_4$ symmetry.
If it is true, as the proton decay width accommodates another suppression factor,
the condition Eq.(62) is never satisfied and
the experimental verification of proton decay seems to be impossible.
Therefore the bounds Eq.(66)-(68) should not be taken seriously.
In the realistic model, the suppression by the gauge non-singlet flavon may be too strong.
To improve this point, we modify the flavon sector in next section.
\section{$S_4\times Z_4\times Z_9$ flavor symmetric model}
In this section we introduce Froggatt-Nielsen mechanism to explain Yukawa hierarchy \cite{fn},
and the flavon sector is modified as follows.
To realize $O(10^2)$ hierarchy, we introduce $Z_9$ symmetry and add gauge and $S_4$ singlet $X$
as Froggatt-Nielsen (FN) flavon.
To weaken the over suppression of trilinear terms, we replace the flavon $\Phi^c_a$
by $S_4$ singlet $\Phi^c$ and gauge singlet $T$ which is assigned to $S_4$ triplet.
To forbid renormalizable terms of $T$, the $Z_2$ symmetry is replaced by $Z_4$.
The flavor representations of superfields are given in Table 4.
\begin{table}[htbp]
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|}
\hline
&$Q_1$ &$Q_2$ &$Q_3$ &$U^c_1$ &$U^c_2$ &$U^c_3$ &$D^c_1$ &$D^c_2$ &$D^c_3$
&$E^c_1$ &$E^c_2$ &$E^c_3$ &$L_i$ &$L_3$ \\
\hline
$S_4$ &${\bf 1}$&${\bf 1}$& ${\bf 1}$&${\bf 1}$&${\bf 1}$&${\bf 1}$ &${\bf 1}$&${\bf 1}$ &${\bf 1}$
&${\bf 1}$&${\bf 1}$&${\bf 1'}$&${\bf 2}$&${\bf 1}$ \\
\hline
$Z_4$ &$1/2$ &$1/2$ &$1/2$ &$1/2$ &$1/2$ &$1/2$ &$1/2$ &$1/2$ &$1/2$
&$0$ &$1/2$ &$0$ &$1/2$ &$1/2$ \\
\hline
$Z_9$ &$1/9$ &$1/9$ &$0$ &$2/9$ &$1/9$ &$0$ &$2/9$ &$1/9$ &$1/9$
&$0$ &$2/9$ &$1/9$ &$0$ &$1/9$ \\
\hline
\hline
&$N^c_i$ &$N^c_3$ &$H^U_i$ &$H^U_3$ &$H^D_i$ &$H^D_3$ &$S_i$ &$S_3$
&$G_a$ &$G^c_a$ &$T_a$ &$\Phi$ &$\Phi^c$ &$X$ \\
\hline
$S_4$ &${\bf 2}$&${\bf 1}$&${\bf 2}$ &${\bf 1}$&${\bf 2}$ &${\bf 1}$&${\bf 2}$&${\bf 1}$
&${\bf 3}$&${\bf 3}$ &${\bf 3}$&${\bf 1}$&${\bf 1'}$&${\bf 1}$\\
\hline
$Z_4$ &$0$ &$1/2$ &$1/2$ &$0 $ &$1/2$ &$0$ &$1/2$ &$0$
&$1/4$ &$3/4$ &$1/4$ &$0$ &$1/2$ &$0$\\
\hline
$Z_9$ &$0$ &$0$ &$0$ &$0$ &$0$ &$0$ &$0$ &$0$
&$0$ &$0$ &$0$ &$0$ &$0$ &$8/9$\\
\hline
\end{tabular}
\end{center}
\caption{$S_4\times Z_4\times Z_9$ assignment of superfields
(Where the index $i$ of the $S_4$ doublets runs $i=1,2$,
and the index $a$ of the $S_4$ triplets runs $a=1,2,3$.)}
\end{table}
\subsection{Flavon sector}
The leading terms of flavons are given as follows,
\eqn{
W_F&=&W_T+W_\Phi+W_X,\\
W_T&=&\frac{1}{M_P}\left[\frac14 Y^T_1(T^4_1+T^4_2+T^4_3)
+\frac12 Y^T_2(T^2_1T^2_2+T^2_1T^2_3+T^2_2T^2_3)\right] ,\\
W_X&=&\frac{1}{6M^6_P}X^9, \\
W_\Phi&=&\frac{1}{2M_P}Y^\Phi(\Phi\Phi^c)^2.
}
The VEV size of gauge non-singlet is estimated by Eq.(24).
Now we change the value of $V=\left<\Phi\right>$ to $10^{11}$ GeV which is
given by naive estimation when we put $Y^\Phi\sim 1$.
This affects neutrino Yukawa couplings given in Eq.(44) as follows
\eqn{
Y^N_2\to \frac{Y^N_2}{\sqrt{10}}=0.134,\quad
Y^N_3\to \frac{Y^N_3}{\sqrt{10}}=0.025,\quad
Y^N_4\to \frac{Y^N_4}{\sqrt{10}}=0.370.
}
The VEV size of FN flavon is estimated as follow
\eqn{
\epsilon=\left(\frac{\left<X\right>}{M_P}\right)&\sim&\left(\frac{m_{SUSY}}{M_P}\right)^{\frac{1}{7}}
\sim 10^{-2}.
}
$S_4$ symmetric part of potential of $T$ is given by
\eqn{
V_T&=&-m^2(|T_1|^2+|T_2|^2+|T_3|^2) \nonumber \\
&-&\frac{1}{M_P}\left[\frac14 B^T_1(T^4_1+T^4_2+T^4_3)
+\frac12 B^T_2(T^2_1T^2_2+T^2_1T^2_3+T^2_2T^2_3)\right] \nonumber \\
&+&\frac{1}{M^2_P}\left[|Y^T_1T^3_1+Y^T_2T_1(T^2_2+T^2_3)|^2
+|Y^T_1T^3_2+Y^T_2T_2(T^2_1+T^2_3)|^2 \right. \nonumber \\
&+&\left. |Y^T_1T^3_3+Y^T_2T_3(T^2_1+T^2_2)|^2\right],
}
which has minimum in the VEV direction given by
\eqn{
T_a=\frac{V_T}{\sqrt{3}}(1,1,1).
}
As same as the gauge non-singlet model, we can assume
flavor breaking term as perturbation.
As the size of $V_T$ is at the same order as $V$, we put
\eqn{
\frac{V_T}{M_P}=10^{-8}.
}
\subsection{Quark and Lepton sector}
Due to the $Z_9$ symmetry, the effective Yukawa coupling constants accommodate power of $\epsilon$
through the superpotential
\eqn{
W_Q=\sum_{ij}\left(\frac{X}{M_P}\right)^{9(q_i+u_j)}\left(Y^U_{ij}\right)_0H^U_3Q_iU^c_j
+\sum_{ij}\left(\frac{X}{M_P}\right)^{9(q_i+d_j)}\left(Y^D_{ij}\right)_0H^D_3Q_iD^c_j,
}
where $q_i,u_i,d_i$ are $Z_9$ charge of $Q_i,U^c_i,D^c_i$ respectively.
As the results, the mass matrices of quarks are given as follows,
\eqn{
M_u&\sim&\Mat3{\epsilon^{3}}{\epsilon^{2}}{\epsilon}
{\epsilon^{3}}{\epsilon^{2}}{\epsilon}
{\epsilon^{2}}{\epsilon}{1}
\sim \Mat3{1}{1}{\epsilon}{1}{1}{\epsilon}{\epsilon}{\epsilon}{1}
\mbox{diag}(\epsilon^3,\epsilon^2,1)
\Mat3{1}{\epsilon}{\epsilon^2}{\epsilon}{1}{\epsilon}{\epsilon^2}{\epsilon}{1}v'_u
=L_u\mbox{diag}(m_{u,c,t})R^\dagger_u, \\
M_d&\sim&\Mat3{\epsilon^{3}}{\epsilon^{2}}{\epsilon^{2}}
{\epsilon^{3}}{\epsilon^{2}}{\epsilon^{2}}
{\epsilon^{2}}{\epsilon}{\epsilon}
\sim \Mat3{1}{1}{\epsilon}{1}{1}{\epsilon}{\epsilon}{\epsilon}{1}
\mbox{diag}(\epsilon^3,\epsilon^2,\epsilon)
\Mat3{1}{\epsilon}{\epsilon}{\epsilon}{1}{1}{\epsilon}{1}{1}v'_d
=L_d\mbox{diag}(m_{d,s,b})R^\dagger_d,
}
from which we get Cabbibo-Kobayashi-Maskawa matrix
\eqn{
V_{CKM}=L^\dagger_u L_d \sim \Mat3{1}{1}{\epsilon}{1}{1}{\epsilon}{\epsilon}{\epsilon}{1} ,
}
and quark masses divided by experimental values respectively,
\eqn{
&&\frac{m_u}{(m_u)_{\mbox{exp}}}=\frac{(A^Y_{RF})_S\epsilon^3 v'_u}{1.3\times 10^{-3}}=0.79 ,\quad
\frac{m_c}{(m_c)_{\mbox{exp}}}=\frac{(A^Y_{RF})_S\epsilon^2 v'_u}{0.624}=0.17 ,\quad
\frac{m_t}{(m_t)_{\mbox{exp}}}=\frac{v'_u}{173}=0.90 , \nonumber \\
&&\frac{m_d}{(m_d)_{\mbox{exp}}}=\frac{(A^Y_{RF})_S\epsilon^3 v'_d}{2.9\times 10^{-3}}=0.18 ,\quad
\frac{m_s}{(m_s)_{\mbox{exp}}}=\frac{(A^Y_{RF})_S\epsilon^2 v'_d}{0.055}=0.94 ,\quad
\frac{m_b}{(m_b)_{\mbox{exp}}}=\frac{(A^Y_{RF})_S\epsilon v'_d}{2.89}=1.8 , \nonumber \\
}
where $\epsilon=0.01, v'_u=155.3\mbox{GeV},v'_d=77.8\mbox{GeV}, (A^Y_{RF})_S=(A^{EUG}_{RF})_S=6.647$
are used. The renormalization factor of top-Yukawa coupling is neglected because it has
infrared quasi-fixed point.
For the lepton sector, the Yukawa coupling constants divided by
required values given in Eq.(42) and Eq.(73) are given by
\eqn{
&&\frac{Y^E_1}{(Y^E_1)_{\mbox{exp}}}=\frac{1}{0.875}=1.1,\quad
\frac{Y^E_3}{(Y^E_3)_{\mbox{exp}}}=\frac{\epsilon}{5.15\times 10^{-2}}=0.19,\quad
\frac{Y^E_2}{(Y^E_2)_{\mbox{exp}}}=\frac{\epsilon^3}{6.25\times 10^{-6}}=0.16, \nonumber \\
&&\frac{Y^N_2}{(Y^N_2)_{\mbox{exp}}}=\frac{1}{0.134}=7.5,\quad
\frac{Y^N_3}{(Y^N_3)_{\mbox{exp}}}=\frac{\epsilon}{0.025}=0.40,\quad
\frac{Y^N_4}{(Y^N_4)_{\mbox{exp}}}=\frac{1}{0.370}=2.7.
}
Where we used running masses of quarks and charged leptons \cite{mass}:
\eqn{
\begin{tabular}{lll}
$m_u(m_Z)=1.28^{+0.50}_{-0.39} (\mbox{MeV})$, &
$m_c(m_Z)=624\pm 83 (\mbox{MeV})$, &
$m_t(m_Z)=172.5\pm 3.0 (\mbox{GeV})$, \\
$m_d(m_Z)=2.91^{+1.24}_{-1.20} (\mbox{MeV})$, &
$m_s(m_Z)=55^{+16}_{-15} (\mbox{MeV})$, &
$m_b(m_Z)=2.89\pm 0.09 (\mbox{GeV})$, \\
$m_e(m_Z)=0.48657 (\mbox{MeV})$, &
$m_\mu(m_Z)=102.72 (\mbox{MeV})$, &
$m_\tau(m_Z)=1746 (\mbox{MeV})$.
\end{tabular}
}
The discrepancies between the estimated values and experimental values
in Eq.(82) and Eq.(83) are easily recovered by multiplying $O(1)$ coefficients $(Y)_0$.
Here we assume $O(1)$ means $0.5<O(1)<5$, therefore $Y^N_2$ is out of this range.
However our fitting is not totally wrong.
CKM matrix
\eqn{
(V_{CKM})_{\mbox{exp}}\simeq
\Mat3{1}{0.23}{0.4\times 10^{-2}}
{0.23}{1}{4.1\times 10^{-2}}
{0.8\times10^{-2}}{3.9\times10^{-2}}{1}, \quad \cite{PDG}
}
is also recovered from Eq.(81). Therefore our procedure works well in quark and lepton sectors.
\subsection{g-quark sector}
The leading terms of single g-quark interactions are given by
\eqn{
W_B&=&\frac{1}{M_P}\left(\frac{X}{M_P}\right)^{9(u_a+d_a)}(Y^{UDG}_{ab})_0
[T_1G^c_1+T_2G^c_2+T_3G^c_3]U^c_aD^c_b \nonumber \\
&+&\frac{1}{M_P}\left(\frac{X}{M_P}\right)^{9q_a+1}(Y^{QL_sG}_a)_0
[T_1G^c_1+T_2G^c_2+T_3G^c_3]Q_aL_3 \nonumber \\
&+&\frac{1}{M_P}\left(\frac{X}{M_P}\right)^{9q_a}(Y^{QL_dG}_a)_0
[\sqrt{3}(T_2G^c_2-T_3G^c_3)L_1+(T_2G^c_2+T_3G^c_3-2T_1G^c_1)L_2]Q_a,
}
where the contribution from $E^c_1\supset \tau^c$ is omitted because $p\to\tau^+X$
is impossible. Note that $Y^{QQG},Y^{EUG}$ are suppressed by $(V_T/M_P)^3$.
In the quark mass basis, trilinear coupling matrix and vectors are given as follows,
\eqn{
(R^T_uY^{UDG}R_d)_{ab}&\sim&
\Mat3{1}{\epsilon}{\epsilon^2}{\epsilon}{1}{\epsilon}{\epsilon^2}{\epsilon}{1}
\Mat3{\epsilon^4}{\epsilon^3}{\epsilon^3}
{\epsilon^3}{\epsilon^2}{\epsilon^2}
{\epsilon^2}{\epsilon}{\epsilon}
\Mat3{1}{\epsilon}{\epsilon}{\epsilon}{1}{1}{\epsilon}{1}{1}
\sim \Mat3{\epsilon^4}{\epsilon^3}{\epsilon^3}
{\epsilon^3}{\epsilon^2}{\epsilon^2}
{\epsilon^2}{\epsilon}{\epsilon} , \\
(Y^{QL_sG}L_u)_a&\sim&\epsilon (\epsilon,\epsilon,1)
\Mat3{1}{1}{\epsilon}{1}{1}{\epsilon}{\epsilon}{\epsilon}{1}
\sim(\epsilon^2,\epsilon^2,\epsilon), \\
(Y^{QL_dG}L_u)_a&\sim&(\epsilon,\epsilon,1)
\Mat3{1}{1}{\epsilon}{1}{1}{\epsilon}{\epsilon}{\epsilon}{1}
\sim(\epsilon,\epsilon,1).
}
As the coupling constants are large enough, the problem of
long life time of g-quark is solved.
Integrating out the scalar g-quarks,
would-be the largest contribution to proton decay is given by
\eqn{
{\cal L}_{p\to \mu^+K^0}&=&\frac{\epsilon^4}{M^2_PM^2_G}
(\bar{u}^c)'(\bar{s}^c)'u'[\sqrt{3}e_1(\left<T_2\right>^2-\left<T_3\right>^2)
+e_2(\left<T_2\right>^2+\left<T_3\right>^2-2\left<T_1\right>^2)],
}
where $e_1,e_2$ are linear combinations of $\mu$ and $\tau$.
Interestingly, this interaction vanishes in the VEV direction given in Eq.(76).
This means the contributions from three scalar g-quarks are canceled.
Therefore the dominant contribution to proton decay is given by
\eqn{
{\cal L}_{p\to e^+K^0}
=\frac{\epsilon^5V^2_T}{M^2_PM^2_G}A_{RF}(\bar{u}^c)'(\bar{s}^c)'u'e,
}
from which we get
\eqn{
\Gamma(p\to e^+ +K^0)&=&\frac{m_p}{32\pi f^2_\pi}
\left[\epsilon^5\left(\frac{V_T}{M_P}\right)^2\frac{A_{RF}}{M^2_G}\right]^2
\left[-1+\frac{m_N}{m_{B'}}(F-D)\right]^2\left(1-\frac{m^2_{K^0}}{m^2_p}\right)^2\alpha^2_p.
}
Substituting
\eqn{
\epsilon=0.01,\quad
m_N=m_p=940\mbox{MeV},\quad m_{B'}=\frac{m_\Lambda+m_\Sigma}{2}=1150\mbox{MeV},\quad
m_{K^0}=498\mbox{MeV},\quad \cite{PDG}
}
and Eq.(59), Eq.(64) and Eq.(77), we get
\eqn{
\Gamma(p\to e^+ +K^0)&=&1.69\times 10^{-64}\left(\frac{1000\mbox{GeV}}{M_G}\right)^4\mbox{GeV}.
}
For the experimental bound $\tau(p\to e^+ +K^0)>150 \times 10^{30}[\mbox{years}]$ \cite{PDG},
mass bound as follows,
\eqn{
M_G&>&1.0\mbox{TeV}
}
must be satisfied. The experimental bound
$\tau(p\to \bar{\nu} +K^+)>670 \times 10^{30}[\mbox{years}]$ \cite{PDG} for the operator
\eqn{
{\cal L}_{p\to \bar{\nu}K^+}
=\frac{\epsilon^5V^2_T}{M^2_PM^2_G}A_{RF}(\bar{u}^c)'(\bar{s}^c)'d'\nu,
}
gives weaker mass bound as follows,
\eqn{
\Gamma(p\to \bar{\nu} +K^+)&=&\frac{m_p}{32\pi f^2_\pi}
\left[\epsilon^5\left(\frac{V_T}{M_P}\right)^2\frac{A_{RF}}{M^2_G}\right]^2
\left[\frac23 \frac{m_N}{m_{B'}}D\right]^2\left(1-\frac{m^2_{K^0}}{m^2_p}\right)^2\alpha^2_p \nonumber \\
&=&0.20\times 10^{-64}\left(\frac{1000\mbox{GeV}}{M_G}\right)^4\mbox{GeV} , \\
M_G&>&0.9\mbox{TeV}.
}
Note that these bounds should not be taken seriously,
because there is $O(10)$ ambiguity coming from SUSY breaking parameter
in $V_T$, this constraint also has such ambiguity.
The important point is that
we can expect the experimental observation of proton decay for TeV scale g-quark in near future
\footnote{The recent experimental mass bound for $Z'$ gauge boson
$m_{Z'}\simeq 0.515v'_s>1.52$ TeV \cite{newZp}
gives g-quark mass bound $m_g=kv'_s>3$ TeV for $g_X=g_Y,k=1$.}.
Finally we give a short comment about flavor breaking effects on cancellation.
Due to the perturbation from flavor breaking squared mass terms,
scalar g-quark squared mass and squared VEVs of flavons receive
$O(m^2_B/m^2_{SUSY})$ contaminations, where $m^2_B$ is flavor breaking squared mass.
As the results, the cancellation is spoiled.
However, if flavor breaking terms are small enough to satisfy the condition
\eqn{
r_B=\frac{m^2_B}{m^2_{SUSY}}< \epsilon,
}
the cancellation mechanism works effectively.
In the opposite case of $r_B>\epsilon$, $p\to \mu^+K^0$ dominates the proton decay width.
Therefore the size of $r_B$ affects proton decay channels significantly.
There is interesting correlation through $r_B$ between proton decay channel and
degree of degeneracy of scalar g-quark masses.
Note that too small $r_B$ causes the appearance of pseudo-Nambu-Goldstone boson (pNGB).
However this may be not a serious problem, because if there is
the VEV hierarchy such as
$v_s\gg v_u,v_d$, which makes $S_i$ dominant in pNGB, then
the interactions of this pNGB with quarks, leptons and weak bosons
are very weak.
\section{Dirac neutrino model}
As is shown in previous section, the stability of proton is realized by
strong suppression factor of $\left<T\right>/M_P\sim 10^{-8}$.
It is not unnatural to expect that the small neutrino mass is realized by same mechanism.
In this section, we construct Dirac neutrino model based on $S_4$ flavor symmetry.
We eliminate $\Phi,\Phi^c$ and change the flavor assignment as given in Table 5.
In this model, as we can not break $U(1)_Z$ gauge symmetry, only one extra U(1) gauge symmetry
is allowed to exist. As the RHNs do not have $U(1)_X$ charge, Majorana mass terms are not
forbidden by $U(1)_X$. Therefore we change the extra gauge symmetry to
one linear combination of two extra U(1) gauge
symmetries as defined by
\eqn{
X_\theta=X\cos\theta+Z\sin\theta,
}
and assume RHN and $S$ have non-zero charge of $X_\theta$. The value of $\theta$
does not affect our analysis.
\begin{table}[htbp]
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|}
\hline
&$Q_1$ &$Q_2$ &$Q_3$ &$U^c_1$ &$U^c_2$ &$U^c_3$ &$D^c_1$ &$D^c_2$ &$D^c_3$ \\
\hline
$S_4$ &${\bf 1}$&${\bf 1}$& ${\bf 1}$&${\bf 1}$&${\bf 1}$&${\bf 1}$ &${\bf 1}$&${\bf 1}$ &${\bf 1}$\\
\hline
$Z_4$ &$1/2$ &$1/2$ &$1/2$ &$1/2$ &$1/2$ &$1/2$ &$1/2$ &$1/2$ &$1/2$ \\
\hline
$Z_9$ &$1/9$ &$1/9$ &$0$ &$2/9$ &$1/9$ &$0$ &$2/9$ &$1/9$ &$1/9$ \\
\hline
\hline
&$E^c_1$ &$E^c_2$ &$E^c_3$ &$L_i$ &$L_3$ &$H^U_i$ &$H^U_3$ &$H^D_i$ &$H^D_3$ \\
\hline
$S_4$ &${\bf 1}$&${\bf 1}$&${\bf 1'}$&${\bf 2}$&${\bf 1}$&${\bf 2}$ &${\bf 1}$&${\bf 2}$ &${\bf 1}$ \\
\hline
$Z_4$ &$0$ &$1/2$ &$0$ &$1/2$ &$1/2$ &$1/2$ &$0 $ &$1/2$ &$0$ \\
\hline
$Z_9$ &$0$ &$3/9$ &$1/9$ &$0$ &$0$ &$0$ &$0$ &$0$ &$0$ \\
\hline
\end{tabular}
\begin{tabular}{|c|c|c|c|c|c|c|c|}
\hline
&$S_i$ &$S_3$ &$N^c_a$ &$G_a$ &$G^c_a$ &$T_a$ &$X$ \\
\hline
$S_4$ &${\bf 2}$&${\bf 1}$&${\bf 3}$ &${\bf 3}$&${\bf 3}$ &${\bf 3}$&${\bf 1}$\\
\hline
$Z_4$ &$1/2$ &$0$ &$3/4$ &$1/4$ &$3/4$ &$1/4$ &$0$ \\
\hline
$Z_9$ &$0$ &$0$ &$2/9$ &$0$ &$0$ &$0$ &$8/9$ \\
\hline
\end{tabular}
\end{center}
\caption{$S_4\times Z_4\times Z_9$ assignment of superfields
(Where the index $i$ of the $S_4$ doublets runs $i=1,2$,
and the index $a$ of the $S_4$ triplets runs $a=1,2,3$.)}
\end{table}
As the quark and charged lepton Yukawa interactions are not modified, we
consider only Yukawa interaction of neutrino which is given by
\eqn{
W_N&=&\frac{X^2}{M^3_P}(Y^N_1)_0(T_1N^c_1+T_2N^c_2+T_3N^c_3)(H^U_1L_1+H^U_2L_2) \nonumber \\
&-&\frac{X^2}{M^3_P}(Y^N_2)_0[\sqrt{3}(T_2N^c_2-T_3N^c_3)(H^U_1L_2+H^U_2L_1)
+(T_2N^c_2+T_3N^c_3-2T_1N^c_1)(H^U_1L_1-H^U_2L_2)] \nonumber \\
&-&\frac{X^2}{M^3_P}(Y^N_3)_0[\sqrt{3}(T_2N^c_2-T_3N^c_3)H^U_1
+(T_2N^c_2+T_3N^c_3-2T_1N^c_1)H^U_2]L_3.
}
Substituting the VEVs given in Eq.(74), Eq.(76) and Eq.(77) for $X$ and $T_a$,
we get the effective superpotential as follow
\eqn{
W_N&=&Y^N_1(H^U_1L_1+H^U_2L_2)(N^c_1+N^c_2+N^c_3) \nonumber \\
&+&Y^N_2[\sqrt{3}(N^c_3-N^c_2)(H^U_1L_2+H^U_2L_1)
+(2N^c_1-N^c_2-N^c_3)(H^U_1L_1-H^U_2L_2)] \nonumber \\
&+&Y^N_3L_3[\sqrt{3}(N^c_3-N^c_2)H^U_1+(2N^c_1-N^c_2-N^c_3)H^U_2] , \\
&&Y^N_{1,2,3}=(Y^N_{1,2,3})_0\epsilon^2\left(\frac{V_T}{\sqrt{3}M_P}\right)\sim O(10^{-12}),
}
from which Dirac neutrino mass matrix is given by
\eqn{
M_D&=&\Mat3{(m_1+2m_2)c_u}{m_1c_u-m_2(c_u+\sqrt{3}s_u)}{m_1c_u-m_2(c_u-\sqrt{3}s_u)}
{(m_1-2m_2)s_u}{m_1s_u+m_2(s_u-\sqrt{3}c_u)}{m_1s_u+m_2(s_u+\sqrt{3}c_u)}
{2m_3s_u}{-m_3(s_u+\sqrt{3}c_u)}{m_3(-s_u+\sqrt{3}c_u)} , \\
&&m_1=Y^N_1v_u,\quad m_2=Y^N_2v_u,\quad m_3=Y^N_3v_u,
}
where we can define $m_1$ and $m_3$ as real and non-negative and $m_2$ as complex
without loss of generality.
In the VEV direction $\theta_u=\theta_d=\theta_B$, two large angles of
charged lepton and neutrino mixing matrix are canceled and make it difficult
to realize two large mixing angles of MNS matrix.
Therefore we select the condition Eq.(17) and put $\theta_d=0$ and
$\theta_u=\frac{\pi}{4}$
by hand (Note that $V_{MNS}$ depends on $\theta_u,\theta_d$ only through $\theta_u-\theta_d$.
To realize maximal mixing of $\theta_{23}$, we tune $\theta_u-\theta_d=\frac{\pi}{4}$.),
then the charged lepton mass matrix is given by
\eqn{
M_l&=&
\Mat3{m^e_1}{0}{0}
{0}{0}{m^e_3}
{0}{m^e_2}{0},
}
from which we get
\eqn{
V^T_lM_lM^T_lV_l&=&\mbox{diag}((m^e_2)^2,(m^e_3)^2,(m^e_1)^2)
=\mbox{diag}(m^2_e,m^2_\mu,m^2_\tau),\quad
V_l=\Mat3{0}{0}{1}{0}{1}{0}{1}{0}{0}.
}
To realize experimental results, the conditions given as follows
\eqn{
V^\dagger_\nu M^*_DM^T_DV_\nu &=&diag(m^2_{\nu_1},m^2_{\nu_2},m^2_{\nu_3})=M^2_{\mbox{diag}}, \\
V_\nu&=&V_lV_{MNS}=\frac{1}{\sqrt{6}}V_l\Mat3{\sqrt{2}}{0}{0}{0}{1}{1}{0}{-1}{1}
\Mat3{1}{0}{-\lambda^*}{0}{1}{0}{\lambda}{0}{1}
\Mat3{\sqrt{2}}{1}{0}{-1}{\sqrt{2}}{0}{0}{0}{\sqrt{3}}, \\
M^*_DM^T_D&=&V_\nu M^2_{\mbox{diag}} V^\dagger_\nu
=\mbox{diag}(1,1,1)m^2_{\nu_2}+\Delta m^2_{32}V_\nu \mbox{diag}(-r_\nu,0,1)V^\dagger_\nu, \\
r_\nu&=&\frac{\Delta m^2_{21}}{\Delta m^2_{32}},
}
must be satisfied. Eq.(110) is rewritten as follow
\eqn{
&&\Mat3{(3/2)m^2_1+6|m_2|^2-m^2_{\nu_2}}{(3/2)m^2_1}{6m^*_2m_3}
{(3/2)m^2_1}{(3/2)m^2_1+6|m_2|^2-m^2_{\nu_2}}{0}
{6m_2m_3}{0}{6m^2_3-m^2_{\nu_2}} \nonumber \\
&=&\frac{\Delta m^2_{32}}{6}
\Mat3{3-r_\nu}{3+r_\nu}{-2r_\nu-3\sqrt{2}\lambda}
{3+r_\nu}{3-r_\nu}{2r_\nu-3\sqrt{2}\lambda}
{-2r_\nu-3\sqrt{2}\lambda^*}{2r_\nu-3\sqrt{2}\lambda^*}{-4r_\nu},
}
where $O(\lambda^2)$ terms are neglected. From this equation, we get
\eqn{
&&\sin\theta_{13}=\lambda=\lambda^*=\frac{\sqrt{2}}{3}r_\nu
=\frac{\sqrt{2}}{3}\frac{\Delta m^2_{21}}{\Delta m^2_{32}}
=0.014\quad (\theta_{13}=0.8^\circ) , \\
&&m_1=0.029\mbox{eV} ,\quad
m_2=-0.0033\mbox{eV},\quad
m_3=0.0025\mbox{eV} \nonumber \\
&&m_{\nu_1}=0.0035\mbox{eV},\quad m_{\nu_2}=0.0094\mbox{eV},\quad m_{\nu_3}=0.051\mbox{eV} , \\
&&Y^N_1=2.9\times 10^{-12},\quad Y^N_2=-0.33\times 10^{-12},\quad
Y^N_3=0.25\times 10^{-12}.
}
The small discrepancies between Eq.(103) and Eq.(115) are recovered
by multiplying O(1) coefficients $(Y^N)_0$.
By the modification of $Z_9$ charge of $L_3$,
the proton decay width is dominated by $p\to e^+K^0$ given in Eq.(91)
because suppression factor is reduced from $\epsilon^5$ to $\epsilon^4$.
From the experimental bound, we get
\eqn{
M_G>\frac{1.0\mbox{TeV}}{\sqrt{\epsilon}}=10\mbox{TeV}.
}
Note that $Y^{NDG}$ is suppressed by $(V_T/M_P)^2$.
\section{Conclusion}
We have considered proton stability based on $S_4$ symmetric extra U(1) models.
Without Froggatt-Nieslen mechanism, most stringent bound for proton decay channel
is given by $\tau(p\to e^+\pi^0)$. As the single quark interaction is doubly suppressed
by the VEV of gauge non-singlet flavon, g-quark life time become very long.
Therefore the allowed region for flavon VEV is very narrow for TeV scale g-quark.
Introducing Froggatt-Nielsen mechanism, as we can weaken the $S_4$ flavon VEV suppression,
g-quark life time becomes short enough.
From the naive power counting, we can expect
$p \to\mu^+K^0$ would dominate the proton decay width, however, corresponding operator
vanishes by cancellation and do not contribute to proton decay.
Therefore our model predicts $p\to e^+K^0$ dominates the proton decay width.
This conclusion is not modified in Dirac neutrino model.
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 5,215
|
\section{Introduction}
Let $\zeta(s)$ denote the Riemann zeta-function. In this article we are interested in obtaining lower bounds for moments of the form
\begin{equation}\label{Jk}
J_{k}(T) =\frac{1}{N(T)} \sum_{0<\gamma\leq T} \big|\zeta'(\rho)\big|^{2k}
\end{equation}
where $k\in\mathbb{N}$ and the sum runs over the non-trivial (complex) zeros $\rho=\beta+i\gamma$ of $\zeta(s)$. As usual, we let the function
\begin{equation}\label{NT}
N(T) = \sum_{0<\gamma\leq T} 1 = \frac{T}{2\pi}\log\frac{T}{2\pi}-\frac{T}{2\pi} + O(\log T)
\end{equation}
denote the number of zeros of $\zeta(s)$ up to a height $T$ counted with multiplicity.
Independently, Gonek \cite{G3} and Hejhal \cite{H} have conjectured that $J_{k}(T)\asymp (\log T)^{k(k+2)}$ for each $k\in\mathbb{R}$. By modeling the Riemann zeta-function and its derivative using characteristic polynomials of random matrices, Hughes, Keating, and O'Connell \cite{HKO} have refined this conjecture to state that $J_{k}(T)\sim C_{k}(\log T)^{k(k+2)}$ for a precise constant $C_{k}$ when $k\in\mathbb{C}$ and $\Re k >-3/2$.
However, we no longer believe this conjecture to be true for $\Re k < -3/2$. This
is since we expect there exist infinitely many zeros $\rho$ such that $|\zeta'(\rho)|^{-1} \gg
|\gamma|^{1/3-\varepsilon}$ for each $\varepsilon>0$.
Results of the sort suggested by these conjectures are only known for a few small values of $k$. See, for instance, the results of Gonek \cite{G1} for the case $k=1$ and Ng \cite{N1} for the case $k=2$. Also, Gonek \cite{G3} obtained a lower bound
in the case $k=-1$. Our main result is to obtain a lower bound for $J_{k}(T)$ for each $k\in\mathbb{N}$ of the order of magnitude that is suggested by these conjectures.
\begin{theorem}\label{th1}
Assume the Riemann Hypothesis and let $k\in \mathbb{N}$. Then for sufficiently large $T$ we have
\begin{equation*}
\frac{1}{N(T)}\sum_{0<\gamma\leq T} \big|\zeta'(\rho)\big|^{2k} \gg_{k} (\log T)^{k(k+2)}.
\end{equation*}
\end{theorem}
Under the assumption of the Riemann Hypothesis, Milinovich \cite{M} has recently shown that $J_{k}(T)\ll_{k,\varepsilon} (\log T)^{k(k+2)+\varepsilon}$ for $k\in\mathbb{N}$ and $\varepsilon>0$ arbitrary. When combined with Theorem \ref{th1}, this result lends strong support for the conjecture of Gonek and Hejhal for $k$ a positive integer.
Theorem \ref{th1} can be used to exhibit large values of $\zeta'(\rho)$. For example, as an immediate corollary we have the following result.
\begin{corollary}\label{cor}
Assume the Riemann Hypothesis and let $\rho=\tfrac{1}{2}+i\gamma$ denote a non-trivial zero of $\zeta(s)$. Then for each $A>0$ the inequality
\begin{equation} \label{large}
\big|\zeta'(\rho)\big| \geq (\log |\gamma|)^{A}
\end{equation}
is satisfied infinitely often.
\end{corollary}
\noindent This result was previously proven by Ng \cite{N3} by an application
of Soundararajan's resonance method \cite{S}. The present proof is simpler and provides many more
zeros $\rho$ such that (\ref{large}) is true. On the other hand, the resonance
method is capable of detecting much larger values of $\zeta'(\rho)$ assuming a
very weak form of the generalized Riemann hypothesis.
Our proof of Theorem \ref{th1} relies on combining a method of Rudnick and Soundararajan \cite{RS1,RS2} with a mean-value theorem of Ng (our Lemma \ref{ng}) and a well-known lemma of Gonek (our Lemma \ref{gonek}). It is likely that our proof can be adapted to prove a lower bound for $J_{k}(T)$ of the conjectured order of magnitude for all rational $k$ (with $k\geq 1$) in a manner analogous to that suggested in \cite{RS1}.
Let $k \in\mathbb{N}$ and define, for $\xi\geq 1$, the function $\mathcal{A}_{\xi}(s) = \sum_{n\leq \xi}n^{-s}$. Assuming the Riemann Hypothesis, we will estimate
\begin{equation*}\label{sigma}
\Sigma_{1} = \sum_{0<\gamma\leq T}\zeta'(\rho)\mathcal{A}_{\xi}(\rho)^{k-1}\overline{\mathcal{A}_{\xi}(\rho)}^{k} \quad \text{ and } \quad \Sigma_{2} = \sum_{0<\gamma\leq T}\big|\mathcal{A}_{\xi}(\rho)\big|^{2k}
\end{equation*}
where the sums run over the non-trivial zeros $\rho=\tfrac{1}{2}+i\gamma$ of $\zeta(s)$. H\"{o}lder's inequality implies that
\begin{equation*}
\sum_{0<\gamma\leq T}\big|\zeta'(\rho)\big|^{2k}\geq \frac{\big|\Sigma_{1}\big|^{2k}}{\big(\Sigma_{2}\big)^{2k-1}},
\end{equation*}
and so we see that Theorem \ref{th1} will follow from the estimates
\begin{equation}\label{bound}
\Sigma_{1} \gg T(\log T)^{k^{2}+2} \quad \text{ and } \quad \Sigma_{2} \ll T(\log T)^{k^{2}+1}.
\end{equation}
It is convenient to express $\Sigma_{1}$ and $\Sigma_{2}$ slightly differently. Assuming the Riemann Hypothesis, $1-\rho=\bar{\rho}$ for any non-trivial zero $\rho$ of $\zeta(s)$. Thus, $\overline{\mathcal{A}_{\xi}(\rho)} = \mathcal{A}_{\xi}(1-\rho)$. This allows us to re-write the sums in (\ref{sigma}) as
\begin{equation}\label{simp}
\Sigma_{1} =\!\sum_{0<\gamma\leq T}\zeta'(\rho)\mathcal{A}_{\xi}(\rho)^{k-1}\mathcal{A}_{\xi}(1\!-\!\rho)^{k} \quad \text{ and } \quad \Sigma_{2} =\!\sum_{0<\gamma\leq T}\mathcal{A}_{\xi}(\rho)^{k}\mathcal{A}_{\xi}(1\!-\!\rho)^{k}.
\end{equation}
It is with these representations of $\Sigma_{1}$ and $\Sigma_{2}$ that we establish the bounds in (\ref{bound}).
\section{Some preliminary estimates}
For each real number $\xi\geq 1$ and each $k\in\mathbb{N}$, we define the arithmetic sequence of real numbers $\tau_{k}(n;\xi)$ by
\begin{equation}\label{tau1}
\sum_{n\leq \xi^{k}}\frac{\tau_{k}(n;\xi)}{n^{s}} =\Big(\sum_{n\leq\xi}\frac{1}{n^{s}}\Big)^{k}= \mathcal{A}_{\xi}(s)^{k}.
\end{equation}
The function $\tau_{k}(n;\xi)$ is a truncated approximation to the arithmetic function $\tau_{k}(n)$ (the $k$-th iterated divisor function) which is defined by
\begin{equation}\label{tau2}
\zeta^{k}(s) = \Big(\sum_{n=1}^{\infty}\frac{1}{n^{s}}\Big)^{k} = \sum_{n=1}^{\infty}\frac{\tau_{k}(n)}{n^{s}}
\end{equation}
for $\Re s> 1$. We require a few estimates for sums involving the functions $\tau_{k}(n)$ and $\tau_{k}(n;\xi)$ in order to establish the bounds for $\Sigma_{1}$ and $\Sigma_{2}$ in (\ref{bound}).
We use repeatedly that, for $x\geq 3$ and $k,\ell \in \mathbb{N}$,
\begin{equation}\label{divisor1}
\sum_{n\leq x} \frac{\tau_{k}(n)\tau_{\ell}(n)}{n} \asymp_{k,\ell} (\log x)^{k\ell}
\end{equation}
where the implied constants depend on $k$ and $\ell$.
These bounds are well-known.
From (\ref{tau1}) and (\ref{tau2}) we notice that $\tau_{k}(n;\xi)$ is non-negative and $\tau_{k}(n;\xi) \leq \tau_{k}(n)$ with equality holding when $n\leq \xi$. In particular, choosing $k=\ell$ in (\ref{divisor1}) we find that, for $\xi\geq 3$,
\begin{equation}\label{divisor2}
(\log \xi)^{k^{2}} \ll_{k} \sum_{n\leq \xi}\frac{\tau_{k}(n)^{2}}{n}\leq \sum_{n\leq \xi^{k}}\frac{\tau_{k}(n;\xi)^{2}}{n}\leq \sum_{n\leq \xi^{k}}\frac{\tau_{k}(n)^{2}}{n} \ll_{k}(\log \xi)^{k^{2}}.
\end{equation}
\section{A Lower Bound for $\Sigma_{1}$}
In order to establish a lower bound for $\Sigma_{1}$, we require a mean-value estimate for sums of the form
$$ S(X,Y;T) = \sum_{0<\gamma\leq T} \zeta'(\rho)X(\rho)Y(1-\rho) $$
where
$$ X(s) = \sum_{n\leq N} \frac{x_{n}}{n^{s}} \quad\quad \text{ and } \quad \quad Y(s) = \sum_{n\leq N}\frac{y_{n}}{n^{s}} $$
are Dirichlet polynomials. For $X(s)$ and $Y(s)$ satisfying certain reasonable conditions, a general formula for $S(X,Y;T)$ has been established by the second author \cite{N2}.
Before stating the formula, we first introduce some notation. For $T$ large, we let $\mathscr{L}=\log\tfrac{T}{2\pi}$ and $N=T^{\vartheta}$ for some fixed $\vartheta\geq 0$.
The functions $\mu(\cdot)$ and $\Lambda(\cdot)$ are used to denote the usual arithmetic functions of M\"{o}bius and von Mangoldt. Also, we define the
arithmetic function $\Lambda_2(\cdot)$ by $\Lambda_2(n)
=(\mu*\log^2)(n)$ for each $n\in\mathbb{N}$.
\begin{lemma}\label{ng}
Let $x_{n}$ and $y_{n}$ satisfy $|x_{n}|,|y_{n}| \ll \tau_{\ell}(n)$ for some $\ell\in\mathbb{N}$ and assume that $0<\vartheta<1/2$. Then for any $A>0$, any $\varepsilon>0$, and sufficiently large $T$ we have
\begin{eqnarray*}
S(X,Y;T) &=& \frac{T}{2\pi} \sum_{mn\leq N} \frac{x_{m}y_{mn}}{mn}\Big(\mathcal{P}_{2}(\mathscr{L})-2\mathcal{P}_{1}(\mathscr{L})\log n + (\Lambda*\log)(n)\Big)
\\
&& \quad - \frac{T}{4\pi} \sum_{mn\leq N} \frac{y_{m}x_{mn}}{mn}\mathcal{Q}_{2}(\mathscr{L}\!-\!\log n) + \frac{T}{2\pi} \sum_{\substack{a,b\leq N \\ (a,b)=1}}\frac{r(a;b)}{ab}\sum_{g\leq \min\big(\tfrac{N}{a},\tfrac{N}{b}\big)} \frac{y_{ag} x_{bg}}{g}
\\
&& \quad + \ O_{A}\big(T(\log T)^{-A} + T^{3/4+\vartheta/2+\varepsilon}\big)
\end{eqnarray*}
where $\mathcal{P}_1,\mathcal{P}_2$, and $\mathcal{Q}_2$ are monic polynomials of degrees 1,2,
and 2, respectively, and for $a,b\in\mathbb{N}$ the function $r(a ;b)$ satisfies the bound
\begin{equation} \label{rabbd}
|r(a;b)| \ll \Lambda_2(a) + (\log T) \Lambda(a) \ .
\end{equation}
\end{lemma}
\begin{proof}
This is a special case of Theorem 1.3 of Ng \cite{N2}.
\end{proof}
Letting $\xi=T^{1/(4k)}$, we find that the choices $X(s)=\mathcal{A}_{\xi}(s)^{k-1}$ and $Y(s)=\mathcal{A}_{\xi}(s)^{k}$ satisfy the conditions of Lemma \ref{ng} with $\vartheta=1/4$, $N=\xi^{k}$, $x_{n}=\tau_{k-1}(n;\xi)$, and $y_{n}=\tau_{k}(n;\xi)$. Consequently, for this choice of $\xi$,
\begin{eqnarray*}
\Sigma_{1} &=& \frac{T}{2\pi} \sum_{\substack{mn\leq \xi^{k} \\ m\leq \xi^{k-1}}} \frac{\tau_{k-1}(m;\xi)\tau_{k}(mn;\xi)}{mn}\Big(\mathcal{P}_{2}(\mathscr{L})-2\mathcal{P}_{1}(\mathscr{L})\log n + (\Lambda*\log)(n)\Big)
\\
&& \quad \quad - \frac{T}{4\pi} \sum_{mn\leq \xi^{k-1}} \frac{\tau_{k}(m;\xi)\tau_{k-1}(mn;\xi)}{mn}\mathcal{Q}_{2}(\mathscr{L}\!-\!\log n)
\\
&& \quad \quad + \frac{T}{2\pi} \sum_{\substack{a,b\leq \xi^{k} \\ (a,b)=1}}\frac{r(a;b)}{ab}\sum_{g\leq \min\big(\tfrac{N}{a},\tfrac{N}{b}\big)} \frac{\tau_{k}(ag;\xi) \tau_{k-1}(bg;\xi)}{g}+ \ O\big(T\big)
\\
&=& \mathcal{S}_{11}+\mathcal{S}_{12}+\mathcal{S}_{13} + O(T),
\end{eqnarray*}
say. To estimate $\mathcal{S}_{11}$, notice that, for $T$ sufficiently large, $n\leq \xi^{k}=T^{1/4}$ implies that $$\Big(\mathcal{P}_{2}(\mathscr{L})-2\mathcal{P}_{1}(\mathscr{L})\log n + (\Lambda*\log)(n)\Big) \gg \mathscr{L}^{2}$$
and moreover, by (\ref{divisor2}),
$$ \sum_{\substack{mn\leq \xi^{k} \\ m\leq \xi^{k-1}}} \frac{\tau_{k-1}(m;\xi)\tau_{k}(mn;\xi)}{mn} \geq \sum_{n\leq \xi^{k}} \frac{\tau_{k}(n;\xi)^{2}}{n} \gg (\log T)^{k^{2}}. $$
Thus, $\mathcal{S}_{11}\gg T(\log T)^{k^{2}+2}.$ Since $\mathcal{Q}_{2}(\mathscr{L}\!-\!\log n) \ll \mathscr{L}^{2}$, we can bound $\mathcal{S}_{12}$ by using the inequalities $\tau_{k}(n;\xi)\leq \tau_{k}(n)$ and $\tau_{k}(mn)\leq \tau_{k}(m)\tau_{k}(n)$. In particular, by twice using (\ref{divisor1}), we find that
\begin{equation*}
\begin{split}
\mathcal{S}_{12} &\ll T\mathscr{L}^{2} \sum_{mn\leq \xi^{k}}\frac{\tau_{k}(m)\tau_{k-1}(m)\tau_{k}(n)}{mn}
\leq T\mathscr{L}^{2} \Bigg(\sum_{m\leq T}\frac{\tau_{k}(m)\tau_{k-1}(m)}{m}\Bigg)\Bigg(\sum_{n\leq T}\frac{\tau_{k-1}(n)}{n}\Bigg)
\\
&\ll T(\log T)^{2+k(k-1) + k-1}\ll T(\log T)^{k^{2}+1}.
\end{split}
\end{equation*}
It remains to consider the contribution from $\mathcal{S}_{13}$.
Again using the inequalities $\tau_{k}(n;\xi)\leq \tau_{k}(n)$ and $\tau_{k}(mn)\leq \tau_{k}(m)\tau_{k}(n)$
along with (\ref{rabbd}), it follows that $\mathcal{S}_{13}$ is bounded by
\begin{equation*}
\begin{split}
\sum_{a,b\leq \xi^{k}}&\frac{(\Lambda_2(a)+ (\log T)\Lambda(a))}{ab} \sum_{g\leq \xi^{k}} \frac{\tau_{k}(a)\tau_{k}(g)\tau_{k-1}(b)\tau_{k-1}(g)}{g}
\\
&\ll \sum_{a\leq T}\frac{(\Lambda_2(a)+(\log T)\Lambda(a))\tau_{k}(a)}{a}\sum_{b\leq T}\frac{\tau_{k-1}(b)}{b} \sum_{g\leq T} \frac{\tau_{k}(g)\tau_{k-1}(g)}{g}
\\
&\ll (\log T)^{2+(k-1)+k(k-1)} = (\log T)^{k^{2}+1}.
\end{split}
\end{equation*}
Combining this with our estimates for $\mathcal{S}_{11}$ and $\mathcal{S}_{12}$, we conclude that $\Sigma_{1}\gg T(\log T)^{k^{2}+2}$.
\section{An Upper Bound for $\Sigma_{2}$}
Assuming the Riemann Hypothesis, we interchange the sums in (\ref{simp}) and find that
\begin{equation}\label{sigma2}
\Sigma_{2} = N(T)\sum_{n\leq \xi^{k}} \frac{\tau_{k}(n;\xi)^{2}}{n} + 2\Re\sum_{m\leq \xi^{k}}\sum_{m < n\leq\xi^{k}} \frac{\tau_{k}(m;\xi)\tau_{k}(n;\xi)}{n}\sum_{0 < \gamma\leq T} \Big(\frac{n}{m}\Big)^{\rho}
\end{equation}
where $N(T)$ denotes the number of non-trivial zeros of $\zeta(s)$ up to a height $T$. Recalling that $\xi=T^{1/(4k)}$ and using (\ref{NT}) and (\ref{divisor2}), it follows that
\begin{equation}\label{mt}
N(T)\sum_{n\leq \xi^{k}} \frac{\tau_{k}(n;\xi)^{2}}{n} \ll T(\log T)^{k^{2}+1}.
\end{equation}
In order to bound the second sum on the right-hand side of (\ref{sigma2}), we require the following version of the Landau-Gonek explicit formula.
\begin{lemma}\label{gonek}
Let $x,T>1$ and let $\rho=\beta+i\gamma$ denote a non-trivial zero of $\zeta(s)$. Then
\begin{equation*}
\begin{split}
\sum_{0 <\gamma \leq T} x^{\rho} &= -\frac{T}{2\pi}\Lambda(x) + O\big(x\log(2xT)\log\log(3x)\big)
\\
&\quad + O\Big(\log x \min\Big(T,\frac{x}{\langle x \rangle}\Big)\Big) + O\Big(\log(2T)\min\Big(T,\frac{1}{\log x}\Big)\Big)
\end{split}
\end{equation*}
where $\langle x \rangle$ denotes the distance from $x$ to the closest prime power other than $x$ itself and $\Lambda(x)=\log p$ if $x$ is a positive integral power of a prime $p$ and $\Lambda(x)=0$ otherwise.
\end{lemma}
\begin{proof}
This is a result of Gonek \cite{G2,G4}.
\end{proof}
\noindent Applying the lemma, we find that
\begin{equation*}
\begin{split}
\sum_{m\leq \xi^{k}}\sum_{m<n\leq\xi^{k}} \frac{\tau_{k}(m;\xi)\tau_{k}(n;\xi)}{n}\sum_{0<\gamma\leq T}\Big(\frac{n}{m}\Big)^{\rho} &=-\frac{T}{2\pi}\sum_{m\leq \xi^{k}}\sum_{m<n\leq\xi^{k}} \frac{\tau_{k}(m;\xi)\tau_{k}(n;\xi)\Lambda(\tfrac{n}{m})}{n}
\\
&\quad +O\left(\mathscr{L}\log\mathscr{L} \sum_{m\leq \xi^{k}}\sum_{m<n\leq\xi^{k}} \frac{\tau_{k}(m;\xi)\tau_{k}(n;\xi)}{m}\right)
\\
&\quad + O\left(\sum_{m\leq \xi^{k}}\sum_{m<n\leq\xi^{k}} \frac{\tau_{k}(m;\xi)\tau_{k}(n;\xi)}{m}\frac{\log\frac{n}{m}}{\langle \frac{n}{m} \rangle}\right)
\\
&\quad + O\left(\log T\sum_{m\leq \xi^{k}}\sum_{m<n\leq\xi^{k}} \frac{\tau_{k}(m;\xi)\tau_{k}(n;\xi)}{n \log\frac{n}{m}}\right)
\\
& = \mathcal{S}_{21}+\mathcal{S}_{22}+\mathcal{S}_{23}+\mathcal{S}_{24},
\end{split}
\end{equation*}
say. Since we only require an upper bound for $\Sigma_{2}$ (which, by definition, is clearly positive), we can ignore the contribution from $\mathcal{S}_{21}$ because all the non-zero terms in the sum are negative.
In what follows, we use $\varepsilon$ to denote a small positive
constant which may be different at each occurrence.
To estimate $\mathcal{S}_{22}$, we note that $\tau_{k}(n;\xi)\leq \tau_{k}(n)\ll_{\varepsilon} n^{\varepsilon}$ which implies
$\mathcal{S}_{22} \ll T^{1/4+\varepsilon}$. Turning to $\mathcal{S}_{23}$, we write $n$ as $qm+\ell$ with $-\frac{m}{2}<\ell\leq\frac{m}{2}$ and find that
$$ \mathcal{S}_{23} \ll T^{\epsilon}
\sum_{m\leq\xi^{k}}\frac{1}{m} \sum_{q\leq\lfloor\frac{\xi^{k}}{m}\rfloor+1} \sum_{-\frac{m}{2}<\ell\leq\frac{m}{2}} \frac{1}{\langle q+\frac{\ell}{m} \rangle}$$
where $\lfloor x \rfloor$ denotes the greatest integer less than or equal to $x$. Notice that $\langle q+\frac{\ell}{m} \rangle=\frac{|\ell|}{m}$ if $q$ is a prime power and $\ell\neq0$, otherwise $\langle q+\frac{\ell}{m} \rangle$ is $\geq\frac{1}{2}$. Hence,
\begin{equation*}
\begin{split}
\mathcal{S}_{23} &\ll T^{\varepsilon} \Big( \sum_{m\leq\xi^{k}}\frac{1}{m}
\sum_{{\begin{substack}{q\leq\lfloor\frac{\xi^{k}}{m}\rfloor+1
\\ \Lambda(q) \ne 0}\end{substack}}}
\sum_{1\leq\ell\leq\frac{m}{2}} \frac{m}{\ell}
+ \sum_{m\leq\xi^{k}}\frac{1}{m} \sum_{q\leq\lfloor\frac{\xi^{k}}{m}\rfloor+1} \sum_{1\leq\ell\leq\frac{m}{2}} 1
\Big)
\\
&\ll T^{\varepsilon}
\Big( \sum_{m\leq\xi^{k}} \sum_{q\leq\lfloor\frac{\xi^{k}}{m}\rfloor+1} 1 \Big)
\ll T^{1/4+\epsilon} \ .
\end{split}
\end{equation*}
It remains to consider $\mathcal{S}_{24}$. For integers $1\leq m<n\leq \xi^{k}$, let $n=m+\ell$. Then
$$ \log\frac{n}{m} = -\log\big(1-\frac{\ell}{m}\big) > \frac{\ell}{m} . $$
Consequently,
\begin{equation}\label{s4}
\mathcal{S}_{24} \ll T^{\epsilon}\sum_{m\leq \xi^{k}}\sum_{1 \le \ell \leq \xi^{k}}
\frac{1}{(m+\ell) \frac{\ell}{m}}
\ll T^{\epsilon} \xi^k =T^{1/4+\epsilon}
.
\end{equation}
Combining (\ref{mt}) with our estimates for $\mathcal{S}_{22}$, $\mathcal{S}_{23}$, and $\mathcal{S}_{24}$ we deduce that $\Sigma_{2} \ll
T(\log T)^{k^{2}+1}$ which, when combined with our estimate for $\Sigma_{1}$, completes the proof of Theorem \ref{th1}.
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 2,165
|
Q: Convert SQL to LINQ in MVC3 with Ninject I'm using MVC3 and still learning LINQ. I'm having some trouble trying to convert a query to LINQ to Entities. I want to return an employee object.
SELECT E.EmployeeID, E.FirstName, E.LastName, MAX(EO.EmployeeOperationDate) AS "Last Operation"
FROM Employees E
INNER JOIN EmployeeStatus ES ON E.EmployeeID = ES.EmployeeID
INNER JOIN EmployeeOperations EO ON ES.EmployeeStatusID = EO.EmployeeStatusID
INNER JOIN Teams T ON T.TeamID = ES.TeamID
WHERE T.TeamName = 'MyTeam'
GROUP BY E.EmployeeID, E.FirstName, E.LastName
ORDER BY E.FirstName, E.LastName
What I have is a few tables, but I need to get only the newest status based on the EmployeeOpertionDate. This seems to work fine in SQL. I'm also using Ninject and set my query to return Ienumerable. I played around with the group by option but it then returns IGroupable. Any guidance on converting and returning the property object type would be appreciated.
Edit: I started writing this out in LINQ but I'm not sure how to properly return the correct type or cast this.
public IQueryable<Employee> GetEmployeesByTeam(int teamID)
{
var q = from E in context.Employees
join ES in context.EmployeeStatuses on E.EmployeeID equals ES.EmployeeID
join EO in context.EmployeeOperations on ES.EmployeeStatusID equals EO.EmployeeStatusID
join T in context.Teams on ES.TeamID equals T.TeamID
where T.TeamName == "MyTeam"
group E by E.EmployeeID into G
select G;
return q;
}
Edit2:
This seems to work for me
public IQueryable<Employee> GetEmployeesByTeam(int teamID)
{
var q = from E in context.Employees
join ES in context.EmployeeStatuses on E.EmployeeID equals ES.EmployeeID
join EO in context.EmployeeOperations.OrderByDescending(eo => eo.EmployeeOperationDate) on ES.EmployeeStatusID equals EO.EmployeeStatusID
join T in context.Teams on ES.TeamID equals T.TeamID
where T.TeamID == teamID
group E by E.EmployeeID into G
select G.FirstOrDefault();
return q;
}
A: Jeff, your "Edit2" helped and worked for me as well! I did it like this:
public IEnumerable<ListCases> GetListCases(int? customerId, int? languageId, int? userGroupId)
{
var query = from cfs in this.DataContext.CaseFieldSettings
join cfsl in this.DataContext.CaseFieldSettingLanguages on cfs.Id equals cfsl.CaseFieldSettings_Id
join cs in this.DataContext.CaseSettings on cfs.Name equals cs.Name
where cfsl.Language_Id == languageId && cfs.ShowOnStartPage == 1 && cfs.Customer_Id == customerId && cs.UserGroup == userGroupId && cs.Name != null
group cfs by new {
cfs.Id, cfs.Customer_Id, cfsl.CaseFieldSettings_Id, cfsl.Label, cfsl.Language_Id, cfs.ShowOnStartPage, cs.Name
} into grouped
select new ListCases
{
Id = grouped.Key.Id, CFS_Id = grouped.Key.CaseFieldSettings_Id, Label = grouped.Key.Label, Language_Id = grouped.Key.Language_Id, Customer_Id = grouped.Key.Customer_Id, Show = grouped.Key.ShowOnStartPage, CSName = grouped.Key.Name
};
return query;
}
A: I'm not exactly clear on how your SQL query is supposed to work, but I've tried to shape your LINQ query to match what it is doing and it also returns type Employee. This should set you up towards getting the exact solution you need:
var q = from E in context.Employees
join ES in context.EmployeeStatuses on E.EmployeeID equals ES.EmployeeID
join EO in context.EmployeeOperations
.OrderByDescending (x => x.EmployeeOperationDate)
on ES.EmployeeStatusID equals EO.EmployeeStatusID
join T in context.Teams on ES.TeamID equals T.TeamID
where T.TeamName == "MyTeam"
group new { E.EmployeeID, E.FirstName, E.LastName, EO.EmployeeOperationDate }
by new {E.EmployeeID, E.FirstName, E.LastName } into G
orderby G.Key.LastName,G.Key.FirstName
select new Employee
{
EmployeeID = G.Key.EmployeeID,
FirstName = G.Key.FirstName,
LastName = G.Key.LastName,
LastOperation = G.First ().EmployeeOperationDate
};
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 2,638
|
Giovani dos Santos (ur. 2 lipca 1981) – brazylijski lekkoatleta, długodystansowiec.
Osiągnięcia
Trzykrotny medalista mistrzostw kraju na 10 000 metrów: złoto (2013), srebro (2012) oraz brąz (2011).
Rekordy życiowe
Bieg na 10 000 metrów – 28:23,89 (2013)
Bibliografia
Brazylijscy długodystansowcy
Medaliści Igrzysk Panamerykańskich 2011
Urodzeni w 1981
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 8,045
|
BAAKC
ABLE 50th Anniversary Photos
the Knox College
Louis Davis, Jr.
Impacting Lives and Influencing Policy
Louis Davis, Jr., class of 1986, is the State Director of AARP District of Columbia State Office and a registered lobbyist. In these roles, Louis lobbies for 190,000 District of Columbia residents age 50+ in one of the most powerful advocacy groups in the U.S.
Louis majored in political science at Knox, and he credits the quality of the Knox education for preparing him for professional success.
"My experience at Knox did not influence my career choice, but rather, prepared me well for the career I would later choose," Louis reflected. "I think my career chose me. I believe I was born to affect positive change in the lives of people, especially those needing an advocate."
Louis is a native of Chicago and a graduate of Kenwood Academy. He was a member of ABLE and a popular DJ for WVKC, The Voice of Knox College. Louis went on to earn a Master of Public Administration from the George Washington University Trachtenberg School of Public Policy and Public Administration in 2008.
Focused service in the nation's capital
Long before Louis was a DC mover and shaker he worked in the City of Chicago mayor's offices of both Harold Washington and Eugene Sawyer. Later he served as an Appropriations Assistant and Senior Legislative Assistant for U.S. Representative José E. Serrano of New York (D-NY).
Louis quickly discovered that political outreach and grassroots advocacy fulfilled his passion to help others. In 1993 Louis was appointed to the Healthcare Reform Taskforce established by First Lady Hillary Clinton, and he was instrumental in getting legislation funded for Title IV (now called Part D) of the Ryan White CARE Act. The Ryan White CARE Act is the nation's largest federally funded program for people living with HIV-AIDS. Louis was awarded the prestigious National AIDS Alliance for Children, Youth, and Families award for his work on this project.
In the mid 90s, Louis joined the leadership at USAID to implement democracy and governance programs in Liberia, West Africa. Later, he lead an executive team in a 16-month process across five continents for World Vision International (WVI), the world's largest faith-based disaster-relief and development organization. This project to create a new vision statement and platform, engaged over 6,000 people around the world, and was approved unanimously by the WVI Board of Directors.
"At the end of each day, I feel a sense of accomplishment because I lead teams of staff and volunteers to greatness in service to others," Louis commented.
Over past decades Louis has lead human services projects and influenced public policies and improved and saved many lives. He administered a $10 million Housing and Urban Development grant that funded high school dropout-prevention programs in 21 states; and he has served on the nonprofit Boards of Directors of Whitman Walker Clinic, IONA Senior Services, and Leadership Greater Washington.
Louis served for nine years as a volunteer at the Children's National Medical Center, and he is a supporter of World Vision, LaRabida Children's Hospital in Chicago, and the High Line of New York City. His current hobbies include music, photography, and expanding his film collection.
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 9,902
|
While the move is on to become a cashless society, notes and coins are likely to be with us for some time yet. 'Touch and go' payments may be increasing, but for many small purchases most of us still rely on good old cash. And because it's easier to hand over a note for each purchase than to scramble in our pockets or purses for the correct change, by the end of the week we often end up with a hefty pile of low value coins. These coins are such a pain that, according to one survey, 93% of respondents admitted to throwing away five cent pieces, with 29% even ditching ten cent pieces.
Okay, so tossing away a dollar's worth of small change each week won't put much of a dent in your future wealth, but at least consider dropping those coins into a donation box. Combined with thousands of other donations your spare change can make a real difference to the people that charities look after.
Gas and electricity: when was the last time you shopped around for the best deal on your gas and power bills? You could save hundreds of dollars a year.
Gift cards: often end up at the back of a drawer until they expire, or you may only spend part of the total value.
Lunches: even if you skip the smashed avo, a takeaway lunch costs much more than one you make yourself.
In most of these cases the solutions are pretty obvious.
Only buy the food you will use.A few loose carrots and apples might be a better buy than the kilo bags that start to rot in the crisper. If you regularly have a surplus of some foods find recipes that use them. Soups and casseroles are a great way to use up all sorts of ingredients.
Compare what other gas and electricity retailers are offering.
Have a good look at your credit card statement. Were all your purchases necessary?
Place your gift cards in front of your credit cards to remind you to use them instead.
Make your own lunch. Many people can easily save $10 or $15 dollars per day with very little effort. Once any impulse buying habits are under control, this could be the supercharger of your savings.
Will implementing these changes make a real difference? Let's see.
Imagine that you adopt some of these suggestions and as a result save an average of $60 per week. Stashed away in a savings account earning an interest rate of 2% per annum for 20 years, those modest weekly savings will grow to over $76,700. Contributed to an investment that provides an average return of 7% pa and you could be looking at having around $136,000 in 20 years'time.
Does that give you a better idea of how much money you could really be throwing away?
What to do with your newfound savings capacity will depend on your goals and situation. Your financial adviser will be able to help you make the most of the money you don't throw away.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 3,128
|
# ember-medium-editor [](https://travis-ci.org/kolybasov/ember-medium-editor) [](https://emberobserver.com/addons/ember-medium-editor) [](https://badge.fury.io/js/ember-medium-editor)
[medium-editor](https://github.com/yabwe/medium-editor) library for Ember Apps.
#### README and docs are under reconstruction for now. Preparing it for [v1.0](https://github.com/kolybasov/ember-medium-editor/issues/9) release!
## Installation
With `ember`:
* `ember install ember-medium-editor`
With `npm`:
* `npm install --save-dev ember-medium-editor`
With `yarn`:
* `yarn add --dev ember-medium-editor`
## Configuration
```js
// ember-cli-build.js
let app = new EmberApp(defaults, {
mediumEditor: {
/**
* If true will include only JS in the build.
*
* @type Boolean
* @default false
*/
excludeStyles: false,
/**
* List of themes: https://github.com/yabwe/medium-editor/tree/master/dist/css/themes
*
* @type String
* @default 'default'
*/
theme: 'default'
}
});
```
## Usage
```handlebars
{{medium-editor
model.text
options=(hash)
onChange=(action (mut model.text))}}
```
## Issues
If you encounter any issue please report it [here](https://github.com/kolybasov/ember-medium-editor/issues).
## [API Docs](https://ember-medium-editor.mbasov.me/docs/index.html)
## Licence
[MIT](./LICENSE.md)
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 958
|
{"url":"https:\/\/love2d.org\/forums\/viewtopic.php?f=5&t=1795&sid=048bbd6365d4f70a7d6e02e4a347a532","text":"## HUMP - yet another set of helpers\n\nvrld\nParty member\nPosts: 917\nJoined: Sun Apr 04, 2010 9:14 pm\nLocation: Germany\nContact:\n\n### HUMP - yet another set of helpers\n\nThere are quite a number of helper libraries out there, ranging from vector classes to complete frameworks like P\u00f6lygamy, but I decided to release the byproducts of projects anyway.\n\nAnd thus hump - L\u00d6VE Helper Utilities for Massive Progression - was born.\nCurrently it features a matured vector class, a simple and easy class system, a gamestate system, timers and tweens, signals and slots, and a camera with locking and movement smoothing\nI may add other things like some time in the future.\n\nHump differs from other libraries in that every component is independent of the remaining ones (well, apart from camera.lua, which needs the vector class), meaning that you can pick and choose what you need and what you don't. Hump's footprint is very small and it should fit nicely into your projects.\n\nSourcecode and documentation are available on github and readthedocs:\n\nSourcecode: http:\/\/github.com\/vrld\/hump\nLast edited by vrld on Mon Oct 12, 2015 6:36 am, edited 2 times in total.\nI have come here to chew bubblegum and kick ass... and I'm all out of bubblegum.\n\nhump | HC | SUIT | moonshine\nnevon\nCommander of the Circuloids\nPosts: 938\nJoined: Thu Feb 14, 2008 8:25 pm\nLocation: Stockholm, Sweden\nContact:\n\n### Re: HUMP - yet another set of helpers\n\nNice! That's great, and I will most likely use this in upcoming projects. What license is it under?\nvrld\nParty member\nPosts: 917\nJoined: Sun Apr 04, 2010 9:14 pm\nLocation: Germany\nContact:\n\n### Re: HUMP - yet another set of helpers\n\nI knew i forgot something, so, ahem:\nHelper Utilities for a Multitude of Problems is free software under the (modified) MIT license. Do whatever you want with it\nI have come here to chew bubblegum and kick ass... and I'm all out of bubblegum.\n\nhump | HC | SUIT | moonshine\nChief\nParty member\nPosts: 101\nJoined: Fri Mar 12, 2010 7:57 am\nLocation: Norway, 67\u00b0 north\n\n### Re: HUMP - yet another set of helpers\n\nL\u00f6vely! Gotta love the open source!\nLast edited by Chief on Sat Jun 04, 2011 9:11 pm, edited 1 time in total.\nnevon\nCommander of the Circuloids\nPosts: 938\nJoined: Thu Feb 14, 2008 8:25 pm\nLocation: Stockholm, Sweden\nContact:\n\n### Re: HUMP - yet another set of helpers\n\nSo, I wanted to use the gamestate lib and the class lib. The documentation for the class one is great, but there isn't any for the other one. It seems straightforward enough for the most part; the only thing I'm wondering about is how to use the enter and leave states. Got an example to show?\nvrld\nParty member\nPosts: 917\nJoined: Sun Apr 04, 2010 9:14 pm\nLocation: Germany\nContact:\n\n### Re: HUMP - yet another set of helpers\n\nThe enter and leave functions are called when you use Gamestate.switch(to, ...).\n\nThe first argument to the enter function is the previous state, the rest are the remaining arguments passed to Gamestate.switch. The previous-state-as-argument mostly a hack to retrieve state values not passed to Gamestate.switch but also to make the switch call not too verbose.\n\nThe leave is called right before the state is switched (and before the enter of the next state). I mostly use this for stopping music or resetting fonts.\n\nAn example:\n\nCode: Select all\n\nGamestate.titlescreen = Gamestate.new()\nlocal st = Gamestate.titlescreen\n\nfunction st:enter()\nstart_music()\nlove.graphics.setBackgroundColor(10,20,30)\nlove.graphics.setColor(0,0,0)\nend\n\nfunction st:leave()\nstop_music()\nend\n\nfunction st:keyreleased(key)\nif key == 'return' then\nGamestate.switch(Gamestate.game, self.playerName)\nend\nend\n\n[...]\n\nGamestate.game = Gamestate.new()\nlocal st = Gamestate.game\n\nfunction st:enter(pre, playerName)\nif pre == Gamestate.pause then return end\nself.playerName = playerName\nend\n[...]\n\nFor a working example see here: http:\/\/github.com\/vrld\/notalone\/tree\/master\/state\/\nI have come here to chew bubblegum and kick ass... and I'm all out of bubblegum.\n\nhump | HC | SUIT | moonshine\nnevon\nCommander of the Circuloids\nPosts: 938\nJoined: Thu Feb 14, 2008 8:25 pm\nLocation: Stockholm, Sweden\nContact:\n\n### Re: HUMP - yet another set of helpers\n\nI'm using the gamestate lib now, and it is incredibly easy to use. Thanks a bunch for making and releasing this.\npekka\nParty member\nPosts: 206\nJoined: Thu Jan 07, 2010 6:48 am\nLocation: Oulu, Finland\nContact:\n\n### Re: HUMP - yet another set of helpers\n\nGood job with the documentation, vrld. You're being a model to all library makers here.\n\nThere's a cosmetic mark-up problem after function Gamestate.mousereleased(x,y,btn) that you might want to correct some time.\nSslaxx\nCitizen\nPosts: 57\nJoined: Sat Feb 14, 2009 8:54 pm\nLocation: Malvern, Worcs, UK\nContact:\n\n### Re: HUMP - yet another set of helpers\n\nI'm doing something pretty wrong with the gamestate code, I just don't know what.\n\nhttp:\/\/sslaxx.twu.net\/EggTester.love - just started to convert one section of the code so far to use gamestates, but it produces this error:\nboot [string \"Code\/gamestate.lua\"]:28: attempt to index field 'current' (a nil value) stack traceback:\n[string \"boot.lua\"]:833: in function 'error_printer'\n[string \"boot.lua\"]:768: in function <[string \"boot.lua\"]:766>\n[string \"Code\/gamestate.lua\"]:28: in function 'update'\n[string \"boot.lua\"]07: in function <[string \"boot.lua\"]:295>\n[C]: in function 'xpcall'\n[string \"boot.lua\"]:840: in main chunk\nHelp?\nvrld\nParty member\nPosts: 917\nJoined: Sun Apr 04, 2010 9:14 pm\nLocation: Germany\nContact:\n\n### Re: HUMP - yet another set of helpers\n\npekka wrote:There's a cosmetic mark-up problem after function Gamestate.mousereleased(x,y,btn) that you might want to correct some time.\nThanks, I missed that.\nI've created a git project page with a more usable documentation: http:\/\/vrld.github.com\/hump\/\nSslaxx wrote:Help?\nThat error occurs because you did not use Gamestate.switch() to enter your title gamestate.\nRather than calling Gamestate.TitleScreen:enter() you need to do Gamestate.switch(Gamestate.TitleScreen) - preferably in love.load() so you can be sure the state exists.\nI pointed that out in the new documentation and also added a dummy initial state to gamestate.lua which will throw errors at you if did not switch to a valid state.\nI have come here to chew bubblegum and kick ass... and I'm all out of bubblegum.\n\nhump | HC | SUIT | moonshine\n\n### Who is online\n\nUsers browsing this forum: No registered users and 5 guests","date":"2021-04-19 12:39:26","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 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\": 0.3790258765220642, \"perplexity\": 6643.815433920701}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2021-17\/segments\/1618038879374.66\/warc\/CC-MAIN-20210419111510-20210419141510-00434.warc.gz\"}"}
| null | null |
\chapter*{Acknowledgements}\addcontentsline{toc}{section}{Acknowledgements}}{\par}
\chapterstyle{reparticle}
\setcounter{secnumdepth}{2}
\selectlanguage{british}
\setlength{\headwidth}{\textwidth}
\copypagestyle{manaart}{plain}
\makeheadrule{manaart}{\headwidth}{0.75\normalrulethickness}
\makeoddhead{manaart}
\footnotesize\textit{Porta Mana}}{}
\footnotesize\textit{Conjectures and questions in convex geometry}
\makeoddfoot{manaart}{}{\thepage}{}
\pagestyle{manaart}
\makeatletter
\newcommand\addprintnote
\begin{picture}(0,0
\put(0,-14)
\makebox(0,0)
{\tin
This document is optimized for on-screen reading and 2-pages-on-1-sheet
printing on A4 or Letter paper}}
\end{picture
}
\makeoddfoot{plain}{}{\makebox[0pt]{\thepage}\addprintnote}
}
\makeoddhead{plain}{}{}{\footnotesize
Report no.\ pi-qf-22
}
\title{Conjectures and questions in convex geometry\\[1.5\jot]\large
of interest for quantum theory\\ and other physical statistical theorie
}
\author{\firstname{P.G.L.}\ \surname{Porta\,Mana
}
\postauthor{\\[-.5ex
\affiliation{Perimeter Institute for Theoretical Physics, Canada}
\\[-1ex]
\epost{\textless\email{lmana AT pitp.ca}\textgreate
}\end{tabular}\par\end{center}}
\predate{\begin{center}\footnotesize}
\date{16 May 201
\\ (first drafted 21 November 2009)
\postdate{\end{center}}
\usepackage{breakurl}
\usepackage{pifont}
\theoremstyle{plain}
\newtheorem{theorem}{Theorem}
\theoremstyle{remark}
\newtheorem{remark}{Remark}
\newtheorem{quest}{Question}
\theoremstyle{definition}
\newtheorem{example}{Example}
\newtheorem{cexample}[example]{Counter-example}
\newtheorem{definition}{Definition}
\newtheorem{fact}{Fact}
\newtheorem{note}{Remark}
\newtheorem{conj}{Conjecture}
\newtheorem{tconj}{Physical conjecture}
\newenvironment{innote}{\footnotesize}{}
\renewcommand{\latin}[1]{#1}
\newcommand{\tprod}{\mathop{\textstyle\prod}\nolimits}
\newcommand{\tsum}{\mathop{\textstyle\sum}\nolimits}
\newcommand{\tland}{\mathop{\textstyle\bigwedge}\nolimits}
\newcommand{\tlor}{\mathop{\textstyle\bigvee}\nolimits}
\newcommand{\T}{^\intercal
\newcommand{\yC}{\mathcal{C}}
\newcommand{\yO}[1]{\mathcal{P}_{#1}}
\newcommand{\yOC}{\yO{\yC}}
\newcommand{\yOT}{\yO{\yT}}
\newcommand{\yom}{v}
\newcommand{\yoz}{\yom_0}
\newcommand{\you}{\yom_\text{u}}
\newcommand{\yT}{\varDelta}
\newcommand{\yTT}{\bar{\yT}}
\newcommand{\yOTT}{\yO{\yTT}}
\newcommand{\yQ}{\square}
\newcommand{\yf}{F}
\newcommand{\yg}{G}
\newcommand{\yfp}{\pi}
\newcommand{\ygp}{\gamma}
\newcommand{\yff}{\bar{\yf}}
\newcommand{\ygg}{\bar{\yg}}
\newcommand{\ya}{\lambda}
\newcommand{\yga}{\alpha}
\newcommand{\ypa}{\psi_a}
\newcommand{\ypb}{\psi_b}
\newcommand{\yozz}{d_0}
\newcommand{\youu}{d_\text{u}}
\newcommand{\ydr}{\varTheta}
\newcommand{\yer}{\varEpsilon}
\newcommand{\yb}{x}
\newcommand{\yx}{\bm{q}}
\newcommand{\yll}{L}
\newcommand{\yl}{L}
\newcommand{\yc}{y_x}
\newcommand{\QEM
{\ding{167}}
\newcommand{\qem}{\leavevmode\unskip\penalty9999 \hbox{}\nobreak\hfill
\quad\hbox{\QEM}}
\DeclareMathOperator{\aff}{aff}
\DeclareMathOperator{\conv}{conv}
\setfloatadjustment{figure}{\footnotesize\centering}
\selectlanguage{british}
\begin{document}
\selectlanguage{british}
\input{hyp}
\firmlists*
\maketitle
\abslabeldelim{:\quad}
\setlength{\abstitleskip}{-\absparindent}
\abstractrunin
\begin{abstract}
Some conjectures and open problems in convex geometry are presented, and
their physical origin, meaning, and importance, for quantum theory and
generic statistical theories, are briefly discussed.
\\[2\jot]
\msc{52B11,14R99,81P13}\\
\pacs{02.40.Ft,03.65.Ta,05.90.+m}
\end{abstract}
\newrefsegment
\selectlanguage{british}
\chapter{Introduction}
\label{cha:intro}
In this note I present a couple of conjectures and open problems in convex
geometry, wishing that they will raise the interest of geometers and be
solved soon.
What is the origin of these conjectures and open problems? There is a
branch of Bayesian probability theory, called the theory of statistical
models, that studies the probabilistic relations among particular sets of
propositions or variables \citep[][and refs
therein]{holevo1976_t1978,holevo1980_t1982,holevo1985,portamana2011c}. Its
range of applications is therefore as vast and diverse as that of
probability theory. One of these applications, which is gaining the
interest of more and more researchers in physics, mathematics, and
statistics, is the study of the probabilistic and communication-theoretic
features of quantum theory and other physical statistical theories, and of
how these features can be mimicked by or emerge from a classical theory.
Convex geometry is one of the main mathematical structures at the core of
the theory of statistical models. So with this theory we can translate some
theorems and conjectures about quantum and non-quantum theories and their
relations with classical theories into theorems and conjectures about convex
geometry, and vice versa. It is these physical conjectures that are here
presented, translated in strictly convex-geometric terms.
The presentation follows standard notation and terminology
\citep{coxeter1948,coxeter1961_r1969,valentine1964,gruenbaum1967_r2003,klee1971,broendsted1983,webster1994,portamana2011}.
Some definitions are presented in the next section, in particular the
notion of a statistical model; then the notion of \emph{refinement} of a
statistical model is presented in \sect~\ref{sec:maps} with many
illustrative examples: this is the notion to which are connected the open
problems and conjectures presented in \sect~\ref{sec:conjectures}. Physical
motivation and meaning are discussed in an appendix.
\chapter{Definitions}
\label{sec:affintro}
In this section we consider a compact convex space $\yC$ of dimension
$n$
\begin{definition}
The \emph{convex-form space} $\yOC$ of the convex space $\yC$ is the set
of all affine forms on $\yC$ with range in $\clcl{0,1}$, called
\emph{convex forms}:
\begin{equation}
\label{eq:outcomesp}
\yOC \defd
\set{\yom \colon \yC \to \clcl{0,1} \st \text{$\yom$ is affine}}.
\end{equation}
Convex combination is easily defined on this set, which is thus a convex
space itself. A vector sum and difference can also be naturally defined
but the set is not closed under them. The forms $\yoz \colon a \mapsto 0$
and $\you \colon a \mapsto 1$ are called \emph{null-form} and
\emph{unit-form}. The action of a form $\yom$ on a point $a$ is denoted by
$\yom \inn a$. We call $\yom$ an \emph{extreme form} if it is an extreme
point of $\yOC$.
\end{definition}
Later on we shall on occasion write vector differences of convex forms when
their result is still a convex form.
It is useful to recall that a non-constant convex form $v$ on $\yC$ is
determined by an ordered pair of $(n-1)$-dimensional, parallel hyperplanes
non-intersecting the interior of $\yC$, as explained in
fig.~\ref{fig:plausform}.
\begin{figure}[!t]
\centering
\includegraphics[width=.8\columnwidth]{plausform.ps}
\caption{\small The contour hypersurfaces of an affine form are parallel
hyperplanes; such a form is completely determined by assigning the two
hyperplanes corresponding to the values $0$ and $1$. To be a convex
form, these two hyperplanes must not intersect the interior of the
convex space on which the form is defined. On the left, $v$ is a convex
form for the pentagonal convex space; two lines are indicated where $v$
has values $1/2$ and $2/3$; no points of the space yield the values $0$
or $1$. On the right, $w$ cannot be a convex form (although it is an
affine form) because it assigns strictly negative values to some points
of the convex space; this happens because the $0$-value line cuts the
convex space.}
\label{fig:plausform}
\end{figure}
\begin{fact} If the convex space $\yC$ has dimension $n$ then its
convex-form space $\yOC$ has dimension $n+1$. If $\yC$ is a polytope, so
is $\yOC$. The convex structure of $\yOC$ is determined by that of $\yC$,
and its affine span $\aff\yOC$ is the space of affine forms on $\aff\yC$.
$\yOC$ is a bi-cone whose vertices are the null-form $\yoz$ and the
unit-form $\you$, and this bi-cone is centro-symmetric with centre of
symmetry $(\yoz +\you)/2$. The number of extreme points of $\yOC$ besides
$\yoz$ and $\you$ is determined by the structure of the faces of $\yC$;
\eg, if $\yC$ is a two-dimensional polytope, the number of extreme points
of $\yOC$ equals $2m+2$, where $m$ is the number of bounding directions
of $\yC$.
\end{fact}
For example, the convex-form space of an $n$-dimensional simplex is an
$(n+1)$-dimensional parallelotope (with $2^{n+1}$ extreme points), and that
of a parallelogram is an octahedron, as shown in fig.~\ref{fig:out_sp};
that of a pentagon is a pentagonal trapezohedron, shown in
fig.~\ref{fig:pentag}.
\begin{figure}[!pt]
\centering
\includegraphics[width=9.5\columnwidth/10]{outspace_tri2.ps}
\includegraphics[width=9.5\columnwidth/10]{outspace_qua2.ps}
\caption{\small Two two-dimensional convex spaces, on the left, with their
three-dimensional convex-form spaces, on the right. They constitute two
statistical models. The convex forms $w$, $u$ are represented as pairs
of parallel lines on the convex spaces and as points on the convex-form
spaces. $v_0$ and $v_1$ are the null- and unit-forms.}
\label{fig:out_sp}
\end{figure}
The following definition introduces the most important mathematical objects
of our study:
\begin{definition}
A \emph{statistical model} is a pair $(\yC, \yOC)$ of a convex polytope
and its convex-form space; its \emph{dimension} is simply the dimension
of $\yC$. A \emph{simplicial} statistical model is one in which $\yC$ is
an $n$-dimensional simplex $\yT$, and therefore $\yOT$ is an
$(n+1)$-dimensional parallelotope.
\end{definition}
The name, especially the adjective `statistical', is admittedly sibylline,
but it rightly prophesies a connexion with probability theory.
\pagebreak[1]
\chapter{Refinement of a statistical model}
\label{sec:maps}
In this section we consider compact convex \emph{polytopes} of dimension
$n$; \ie, we are assuming that our convex spaces have a finite number of
extreme points.
In convex geometry it is a well-known fact that any polytope can be
obtained as a section or a projection of a usually higher-dimensional
simplex (Gr\"unbaum \citey{gruenbaum1967_r2003}, \sect~5.1, theorems~1 and
2). Both these kinds of `correspondence' $\yC \leftrightsquigarrow \yT$
between a polytope $\yC$ and the simplex $\yT$ from which the polytope is
obtained have the following characteristics:
\begin{enumerate}[C1.]
\item each point of $\yC$ has at least one corresponding point in $\yT$,
\ie, the correspondence $\yC \leftrightsquigarrow \yT$ is defined on all
$\yC$;
\item there may be points of $\yT$ with no corresponding points in $\yC$,
\ie, the correspondence $\yC \leftrightsquigarrow \yT$ needs not be
defined on all $\yT$;
\item several points in $\yT$ can correspond to one and the same point in
$\yC$, \ie, $\yC \leftrightsquigarrow \yT$ can be one-to-many;
\item at most one point in $\yC$ can correspond to one in $\yT$, \ie, $\yC
\leftrightsquigarrow \yT$ cannot be many-to-one;
\item a convex combination of points in $\yT$ corresponds to the same
convex combination of the corresponding points in $\yC$, when the latter
are defined, \ie, $\yC \leftrightsquigarrow \yT$ is affine.
\end{enumerate}
These characteristics mathematically pin down the $\yC \leftrightsquigarrow
\yT$ correspondence as a partial, onto, affine map from $\yT$ to $\yC$ (we
denote partial maps by hooked arrows):
\begin{align}
\yf: \yT \xhookrightarrow{\text{onto, affine}} \yC,
\label{eq:generalS}
\end{align}
\begin{itemize}
\item `onto' = it covers $\yC$, from C1;
\item `partial' = it needs not be defined on all $\yT$, from C2;
\item `affine' = if $\yf$ is defined on $a, b$ then $\yf[\ya a + (1-\ya)
b] = \ya \yf(a) + (1-\ya) \yf(b)$, from C5.
\item `map' = it is many-to-one or one-to-one, but not one-to-many, from
C3 and C4.
\end{itemize}
Intuitively, this map says that the simplex $\yT$ has a `finer' structure
than the polytope $\yC$, or that $\yC$ is a `coarser' image of $\yT$
because parts of the latter are either missing or not distinguishable in
$\yC$. We might call $\yT$ a `simplicial refinement' of $\yC$. The
projection or section of a simplex are particular cases of this map.
\bigskip
It is natural to try to generalize this kind of construction and its
associated theorems from polytopes to statistical models: given a
statistical model $(\yC, \yOC)$, one asks whether it can be obtained from a
simplicial statistical model $(\yT, \yOT)$, considered as a `refinement'.
More precisely, we want a correspondence $\yC \leftrightsquigarrow \yT$
between the convex spaces, one $\yOC \leftrightsquigarrow \yOT$ between
their convex-form spaces, and we want both correspondences to satisfy
requirements analogous to C1--C5. Moreover, we clearly want these
correspondences to preserve the action of convex forms on the respective
convex spaces in the two statistical models.
This means that the two correspondences have to be expressed by partial,
surjective, affine maps from $\yT$ to $\yC$ and from $\yOT$ to $\yOC$
\begin{align}
&\yf: \yT \xhookrightarrow{\text{onto, affine}} \yC,
\label{eq:generalF}
\\
&\yg: \yOT \xhookrightarrow{\text{onto, affine}} \yOC
\label{eq:generalG}
\end{align}
that satisfy
\begin{multline} \label{eq:compatib}
\yg(\yom)\inn \yf(a) = \yom \inn a
\\ \text{for all $a \in \yT$, $\yom \in \yOT$
on which $\yf$, $\yg$ are defined}.
\end{multline}
We are thus led to the following
\begin{definition}
A \emph{simplicial refinement} of a statistical model
$(\yC, \yOC)$ is a set $(\yT, \yOT, \yf, \yg)$ where:
\begin{enumerate}[I.]
\item $(\yT, \yOT)$ is a simplicial statistical model,
\item $\yf\colon \yT \hookrightarrow \yC$ is partial, onto, and affine,
\item $\yg\colon \yOT \hookrightarrow \yOC$ is partial, onto, and affine,
\item $\yf$ and $\yg$ are such that $\yg(\yom)\inn \yf(a) = \yom \inn
a$ on their domains of definition.
\end{enumerate}
\end{definition}
We shall often omit the adjective `simplicial' when speaking about a
simplicial refinement.
\begin{example}\label{ex:simplic}
Consider the statistical model where $\yC$ is a parallelogram and $\yOC$ an
octahedron, as at the bottom of fig.~\ref{fig:out_sp}
or~\ref{fig:stat_red}. A simplicial refinement is given by $(\yTT, \yOTT,
\yfp,\ygp)$, where: $\yTT$ is a tetrahedron; $\yOTT$ a hypercube; the map
$\yfp$ is the projection of the tetrahedron onto the parallelogram; the map
$\ygp$ maps the zero- and unit-forms of $\yOTT$ onto those of $\yOC$, while
the other four extreme forms of $\yOC$ are the images of the following
forms on $\yTT$ in the notation of fig.~\ref{fig:stat_red}:
\begin{figure}[!t]
\centering\hspace*{\fill}
\includegraphics[width=0.49\columnwidth]{squa_ref4.ps}\hfill
\includegraphics[width=0.49\columnwidth]{octa_ref3.ps}\hspace*{\fill}
\caption{\small Illustration of the refinement of the statistical model of
Example~\ref{ex:simplic}. Note how the mapping $\yfp\colon \yTT \to \yC$
is total and non-injective, and $\ygp\colon \yOTT \hookrightarrow \yOC$
is partial; both are projections. The representation of the hypercube or
four-dimensional parallelotope $\yOTT$ is explained in
fig.~\ref{fig:4cube} on the next page.}
\label{fig:stat_red}
\end{figure}
\begin{figure}[!ph]
\centering
\includegraphics[width=\columnwidth]{4dcubecol.ps}
\caption{\small A hypercube, or four-dimensional parallelotope, is a
four-dimensional polytope with eight, pairwise parallel,
three-dimensional facets, all parallelepipeds; and 24 two-dimensional
faces, all parallelograms. The figure on the top is the projection of a
4-parallelotope onto a three-dimensional space (further projected on
paper); because of the dimensional reduction some of the projected facets
intersect each other. To help you distinguish all eight of them in the
top figure, they are separately represented underneath it. As the
convex-form space of a three-dimensional simplex (tetrahedron), two
vertices of the hypercube represent the nought- and unit-forms, also
indicated in the figure.}
\label{fig:4cube}
\end{figure}
\begin{itemize}\tightlist
\item that having zero value on the vertices
$a$ and $b$ and unit value on $c$ and $d$,
\item as the previous but with zero and unit values exchanged,
\item that having zero value on the vertices
$a$ and $d$ and unit value on $b$ and $c$,
\item as the previous but with zero and unit values exchanged.
\end{itemize}
We see that $\yfp$ is a total map, defined on all $\yTT$; whereas $\ygp$ is
partial: in particular, it is not defined on the non-constant extreme forms
of $\yOTT$.\qem
\end{example}
The preceding example is based on the fact that any convex polytope with
$m$ extreme points can be obtained as the \emph{projection} of an
$(m-1)$-dimensional simplex: see again Gr\"unbaum
\citey{gruenbaum1967_r2003}, \sect~5.1, theorem~2. We have the following
\begin{fact}
The theorem just mentioned can always be used to construct a simplicial
refinement, in the guise of Example~\ref{ex:simplic}, of \emph{any}
statistical model; \cf\ Holevo \citey[\sect~I.5]{holevo1980_t1982}.
\end{fact}
We already said that another theorem of convex geometry states that any
convex polytope with $m$ facets can be obtained as the \emph{section} of an
$(m-1)$-\bd dimensional simplex \cite[\sect~5.1,
theorem~1]{gruenbaum1967_r2003}. This theorem \emph{cannot} be used to
construct a simplicial refinement, though, as shown in the following
\begin{cexample}\label{ex:counter}
The parallelogram $\yC$ of the preceding example can be obtained as the
intersection of the tetrahedron $\yTT$ and an intersecting plane parallel
to the segments $ac$ and $bd$ of fig.~\ref{fig:impossquare}. This defines
a partial, surjective, affine map $\yf\colon \yTT \hookrightarrow \yC$,
which is simply the identity, $\yf(s) =s$, in its domain of definition
$\yC \subset \yTT$. However, it is impossible to find an affine map
$\yg\colon \yOTT \hookrightarrow \yOC$ that be surjective and such that
$\yom \inn s = \yg(\yom)\inn \yf(s) \equiv\yg(\yom) \inn s$: the extreme
forms of $\yOC$, for example, cannot have any counter-image. The reason
for this is geometrically explained in fig.~\ref{fig:impossquare}. Thus
we cannot construct a simplicial refinement of $(\yC, \yOC)$ where $\yC$
is obtained by sectioning $\yTT$.\qem
\end{cexample}
\begin{figure}[!t]
\centering
\includegraphics[width=\columnwidth]{impossquare.ps}
\caption{\small An extreme convex form $v$ of the two-dimensional
parallelogram $\yC$ is represented by the two parallel dot-dashed lines
$s_1s_2$ and $s_3s_4$ lying in the same plane as $\yC$ (left figure). A
convex form $w$ of $\yTT$ is represented by two parallel planes, and if
$w$ is to correspond to $v$, $\yg(w)=v$, in such a way that $\yg(w)\inn
s_i = v\inn s_i$, these two planes must contain $s_1s_2$ and $s_3s_4$;
they cannot intersect the interior of $\yTT$, however, if $w$ is to be a
convex form. But it is impossible to satisfy both requirements for both
planes: \eg, the only plane that contains the line $s_3s_4$ and does not
cut $\yTT$ is $acd$ (right figure); then $a'c'd'$ is the parallel plane
containing $s_1s_2$, but this plane cuts $\yTT$ (meaning, \eg, that $w
\inn b<0$ or $w \inn b>1$). All other constructions one can think of have
the same problem. Thus the map $\yg$ cannot
exist.
\label{fig:impossquare}
\end{figure}
Linusson \citey{linusson2007} has shown that the previous counter-example
is generally valid:
\begin{theorem}[Linusson]
Given a \emph{non-simplicial} statistical model $(\yC, \yOC)$ it is
impossible to find a simplicial refinement $(\yT, \yOT, \yf, \yg)$ such
that $\yf$ is injective in its domain of definition (which means that
$\yf$ would represent the intersection of $\yT$ with a hyperplane,
whereby $\yC$ is obtained).
\end{theorem}
\begin{example}\label{ex:pentag1}
Let $\yC$ be a two-dimensional pentagonal convex space with extreme
points $\set{s_1, s_2, s_3, s_4, s_5}$. Its convex-form space $\yOC$ is a
pentagonal trapezohedron that has, besides the null- and unit-forms
$\yoz$ and $\you$, ten other extreme points given by the forms
\[\set{v_1,\dotsc,v_5,\you-v_1,\dotsc,\you-v_5}\] such that
\begin{equation}\label{eq:penta}
\begin{aligned}
(v_i \inn s_j) &=
\begin{pmatrix}
1&\yga&0&0&\yga \\
\yga&1&\yga&0&0 \\
0&\yga&1&\yga&0 \\
0&0&\yga&1&\yga \\
\yga&0&0&\yga&1
\end{pmatrix} \qquad\text{with $\yga\defd \frac{\sqrt{5}-1}{2}$},
\\
(\you-v_i) \inn s_j &= \you \inn s_j - v_i \inn s_j= 1- v_i \inn s_j;
\end{aligned}
\end{equation}
see fig.~\ref{fig:pentag}.
\begin{figure}[!t
\centering
\includegraphics[width=\columnwidth]{trpzhedron.ps}
\caption{\small The pentagonal convex set $\yC$ and its convex-form space
$\yOC$ from Example~\ref{ex:pentag1}. The convex form $v_3$ is shown on
$\yC$ (as a pair of parallel lines) and on $\yOC$ (as a point).}
\label{fig:pentag}
\end{figure}
A refinement of this model is given by $(\yT, \yOT, \yf, \yg)$ where $\yT$
is a nine-dimensional simplex, or decatope, with ten extreme points
$\set{e_1, \dotsc, e_{10}}$, and $ \yOT$ is a ten-dimensional parallelotope
with twelve extreme points given by the forms \[\set{\yozz, \youu, d_1,
\dotsc, d_{10}, \youu-d_1, \dotsc, \youu-d_{10}}\] such that
\begin{equation}
\yozz \inn e_j = 0,\quad \youu \inn e_j =1,\quad
d_i \inn e_j = \delt_{ij},\quad
(\youu-d_i)\inn e_j = 1-\delt_{ij}.
\label{eq:extr10}
\end{equation}
The map $\yf$ is defined on the points
\begin{equation}
\label{eq:corr_pure}
\begin{gathered}
\yf[(e_1 + e_2)/2] = s_1,\quad
\yf[(e_3 + e_4)/2] =s_2,\quad
\yf[(e_5 + e_6)/2] = s_3,\\
\yf[(e_7 + e_8)/2] = s_4,\quad
\yf[(e_9 + e_{10})/2] = s_5,
\end{gathered}
\end{equation}
and their convex combinations; \ie, it is partial and defined on a
four-dimensional simplex given by the convex span of $\set{(e_1 + e_2)/2,
(e_3 + e_4)/2,\dotsc,(e_9 + e_{10})/2}$; see fig.~\ref{fig:pentagproj}.
\begin{figure}[!t]
\centering
\includegraphics[width=\columnwidth]{pentaref1.ps}
\caption{\small Representation of the decatope $\yT$ and the map $\yf$ of
Example~\ref{ex:pentag1}. $\yT$ is represented by its graph
\citep[\sects~11.3, 8.4]{gruenbaum1967_r2003}, which can be understood
as a parallel projection of $\yT$ onto a two-dimensional plane: the
vertices represent its extreme points $\set{e_i}$, the
$\tbinom{10}{2}=45$ lines connecting any two vertices represent its
edges, and all $\tbinom{10}{3}=120$ triangles connecting any three
vertices represent its two-dimensional faces. The map $\yf$ is only
defined on a four-dimensional, simplicial convex subset (in lighter
green) of $\yT$.}
\label{fig:pentagproj}
\end{figure}
The map $\yg$ is also partial:
define the following one-dimensional convex subsets
\begin{equation}
\label{eq:corr_outc}
\begin{gathered}
\begin{aligned}
\ydr_1 &\defd \set{d_1 + d_2 + \yb (d_9 + d_3)+ \yc (d_{10} + d_4)\st \yb \in \clcl{2\yga -1,1}},\\
\ydr_2 &\defd \set{d_3 + d_4 + \yb (d_1 + d_5)+ \yc (d_2 + d_6)\st \yb \in \clcl{2\yga -1,1}},\\
\ydr_3 &\defd \set{d_5 + d_6 + \yb (d_3 + d_7)+ \yc (d_4 + d_8)\st \yb \in \clcl{2\yga -1,1}}, \\
\ydr_4 &\defd \set{d_7 + d_8 + \yb (d_5 + d_9)+ \yc (d_6 + d_{10})\st \yb \in \clcl{2\yga -1,1}}, \\
\ydr_5 &\defd \set{d_9 + d_{10} + \yb (d_7 + d_1)+ \yc (d_8 +
d_2)\st \yb \in \clcl{2\yga -1,1}},
\end{aligned}
\\
\text{with $\yc\defd 2 \yga -\yb$};
\end{gathered}
\end{equation}
then $\yg$ is defined by
\begin{equation}
\label{eq:corr_outc}
\begin{gathered}
\yg(\yozz) = \yoz,\qquad \yg(\youu) = \you,\qquad \yg(\ydr_i) =
\set{v_i},\; i=1,\dotsc,5, \\ \yg(\youu - \ydr_i) =
\set{\you-v_i},\; i=1,\dotsc,5,
\end{gathered}
\end{equation}
where $\youu-\ydr_i \defd \set{\youu-a \st a\in \ydr_i}$.
The important features of this refinement are these:
\begin{enumerate}[a.]
\item each extreme point of $\yC$ corresponds to a \emph{non-extreme} point
of $\yT$, and to that alone;
\item each of the ten non-trivial extreme points of $\yOC$ corresponds to
a non-zero-dimensional convex set of non-extreme points of $\yOT$;
\item the map $\yf$ is partial, \ie\ it is not defined on some points of
$\yT$, not even its ten pure ones $\set{e_i}$; contrast this with
Example~\ref{ex:simplic};
\item on the four-dimensional simplex on which it is defined, the map $\yf$
acts as a projection onto $\yC$, analogously to the map $\yfp$ of
Example~\ref{ex:simplic}.\qem
\end{enumerate}
\end{example}
In the example just discussed, the fact that the extreme points of $\yC$
correspond to single points of $\yT$ makes it possible for the extreme
points of $\yOC$ to correspond to one-dimensional sets in $\yOT$. But the
opposite situation is also possible:
\begin{example}\label{ex:pentag2}
The sets $\yC$, $\yOC$, $\yT$, $\yOT$ are defined as in the preceding
example, but the maps $\yf$ and $\yg$ are defined differently. Consider these
five one-dimensional faces of $\yT$:
\begin{equation}
\label{eq:corr_outc}
\begin{gathered}
\yer_1 \defd \conv\set{e_1,e_2} = \set{\yb e_1 + (1-\yb)e_2 \st x\in\clcl{0,1}}\\
\begin{aligned}
\yer_2 &\defd \conv\set{e_3,e_4}, \qquad&
\yer_3 &\defd \conv\set{e_5,e_6},\\
\yer_4 &\defd \conv\set{e_7,e_8},\qquad &\yer_5 &\defd
\conv\set{e_9,e_{10}};
\end{aligned}
\end{gathered}
\end{equation}
then define $\yf$ by
\begin{equation}
\label{eq:corr_outc}
\yf(\yer_i) = \set{s_i},\; i=1,\dotsc,5,
\end{equation}
see fig.~\ref{fig:pentag2},
\begin{figure}[!b]
\centering
\includegraphics[width=\columnwidth]{pentaref2.ps}
\caption{\small Representation of the decatope $\yT$ and the map $\yf$ of
Example~\ref{ex:pentag2}; \cf\ fig.~\ref{fig:pentagproj}. The map $\yf$
now maps five edges of $\yT$ to the $\set{s_i}$ and by convex
combination is defined on all of $\yT$.}
\label{fig:pentag2}
\end{figure}
and $\yg$ by
\begin{equation}
\label{eq:corr_outc_fix}
\begin{gathered}
\yg(\yozz) = \yoz,\quad
\yg(\youu) = \you,\\
\begin{aligned}
&\yg[d_1 + d_2 + \yga (d_3 + d_9)+ \yga (d_4 + d_{10})] = r_1,\\
&\yg[ d_3 + d_4 + \yga (d_1 + d_5)+ \yga (d_2 + d_6)]= r_2,\\
&\yg[ d_5 + d_6 + \yga (d_3 + d_7)+ \yga (d_4 + d_8)]=r_3,\\
&\yg[ d_7 + d_8 + \yga (d_5 + d_9)+ \yga (d_6 + d_{10})]=r_4,\\
&\yg[ d_9 + d_{10} + \yga (d_1 + d_7)+ \yga (d_2 + d_8)]=r_5.
\end{aligned}
\end{gathered}
\end{equation}
This refinement differs from the one in the previous Example in that
\begin{enumerate}[a.]
\item the five extreme points of $\yC$ correspond to five
one-dimensional faces of $\yT$;
\item the ten non-trivial extreme points of $\yOC$ correspond to single,
non-extreme points of $\yOT$;
\item the map $\yf$ is total, it is indeed a parallel projection of $\yT$
onto $\yC$.
\end{enumerate}
This refinement is in fact more similar to that of
Example~\ref{ex:simplic}, with the difference that the simplex from which
the polytope is obtained by projection is not the one of the least possible
dimension. \qem
\end{example}
\chapter{Conjectures and questions}
\label{sec:conjectures}
I do not know of any study of the general properties of simplicial
refinements of a statistical model. Apart from Linusson's theorem, general
theorems are lacking.
It seems that if we try to construct a refinement of a statistical model
$(\yC, \yOC)$ with a simplex $\yT$ having fewer extreme points than $\yC$,
the construction runs into problems similar to those of Counter-\bd
example~\ref{ex:counter} for the map $\yg$. Moreover, even considering
higher-\bd dimensional simplices, it seems that if we try to construct
$\yf\colon \yT\hookrightarrow \yC$ in such a way that only some
\emph{non-extreme} points of $\yT$ map onto some extreme points of $\yC$,
as in Example~\ref{ex:pentag1}, then those non-extreme points have to be
enough `far apart' face-wise, otherwise we cannot construct $\yg$, again
for problems like those in Counter-example~\ref{ex:counter}.
These remarks naturally lead to the following conjectures. Unfortunately I
have not been able to prove or disprove them, and wish that convex
geometers will take note of them:
\bigskip
Consider a non-simplicial statistical model $(\yC, \yOC)$,
where $\yC$ has $m$ extreme points:
\begin{conj}\label{con:convconj}
All simplicial refinements of $(\yC, \yOC)$ have the form $(\yT, \yOT,
\yff\circ \yfp, \ygg\circ\ygp)$, where:
\begin{enumerate}[a.]
\item $(\yTT, \yOTT, \yfp, \ygp)$ is the refinement of $(\yC, \yOC)$ where
$\yTT$ is the $(m-1)$-dimensional simplex from which $\yC$ is obtained by
parallel projection $\yfp$, as in Example~\ref{ex:simplic},
\item $(\yT, \yOT, \yff, \ygg)$ is a refinement of $(\yT, \yOT)$, where
$\yT$ is a simplex of dimension larger than $(m-1)$ and $\yff\colon \yT
\hookrightarrow \yTT$ is an affine, onto, partial map (a partial
projection, an intersection, or a combination of the two).
\end{enumerate}
In other words, the refinement obtained by projection of an
$(m-1)$-dimensional simplex is the lowest-dimensional one. See
fig.~\ref{fig:convconj}.
\begin{figure}[!b]
\centering
\includegraphics[width=\columnwidth]{convconj.ps}
\caption{\small Conjecture~\ref{con:convconj} says that any refinement
$(\yT, \yOT, \yf, \yg)$ of $(\yC, \yOC)$ has the schema of the above
figure, with $\yf=\yff\circ\yfp$ and $\yg=\ygg\circ \ygp$. In other words,
the refinement obtained by projection is the one of lowest dimension.}
\label{fig:convconj}
\end{figure}
\end{conj}
\begin{conj}\label{conj:ep}
No simplicial refinement $(\yT, \yOT, \yf, \yg)$ exists with $\yT$
having fewer extreme points than $\yC$. This is also a corollary of the
previous conjecture.
\end{conj}
\begin{conj}\label{conj:faces}
Let $(\yT, \yOT, \yf, \yg)$ be any refinement of $(\yC, \yOC)$, and
$a,b$ be any two extreme points of $\yC$. Let $\yf^{-1}({a}),
\yf^{-1}({b})$ be their counter-images in $\yT$, and $A,B$ the minimal
faces of $\yT$ containing these counter-images. Then $A\cap B
=\emptyset$. In other words, no refinement exists such that two extreme
points of $\yC$ correspond to points in $\yT$ lying on adjacent faces.
\end{conj}
\bigskip The following are very important questions for the application of
the theory of statistical models to quantum theory and general physical
statistical theories. Given a non-simplicial statistical model $(\yC,
\yOC)$:
\begin{quest}
Does a simplicial refinement exist with $\yT$ having fewer extreme points
than $\yC$? (\Cf\ Conjecture~\ref{conj:ep}.)
\end{quest}
\begin{quest}
What is the least dimension of $\yT$ among all simplicial refinements?
\end{quest}
\begin{quest}
For each simplicial refinement, can the affine partial map $\yf$ be
extended to a map $\aff\yT \to \aff\yC$? If so, is the extension unique?
What about an analogous extension of $\yg$?
\end{quest}
\begin{quest}
How to extend the theory of simplicial refinements to statistical models
$(\yC,\yOC)$ where $\yC$ has a continuum of extreme points or even
infinite dimension?
\end{quest}
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 9,617
|
Home News The countdown begins for Macfrut Digital
The countdown begins for Macfrut Digital
on: August 30, 2020 In: News
The numbers of the first and only digital trade fair for the fresh produce industry, to be held from 8 to 10 September, are outstanding: 530 exhibition spaces, 40% of which will be hosted by foreign companies, 600 registered buyers from all over the world and 4,000 registered visitors to date, 70% of which are from abroad.
The first edition of Macfrut Digital, the virtual trade fair for the fresh produce industry to be held from 8 to 10 September 2020, born from the intuition of Cesena Fiera, is drawing near. And the numbers recorded so far, just a few days before its official opening, prove that it was a good intuition: 530 exhibition units have been sold to 400 exhibitors (when the event was launched, the target was 200 units for a total of 150 exhibitors), registrations have sold out and companies have been put on a waiting list, with many foreign participants equal to 40% of the exhibiting companies and 600 buyers that have already registered on the platform, of which 550 from abroad. The ICE – Italian Trade Agency offices around the world and Cesena Fiera's network of foreign agents played a key role in attracting buyers.
The most striking numbers are those relating to foreign participants. Nearly one in two exhibitors comes from abroad, with China leading the way with as many as 87 exhibitors, thanks to the collaboration with ATPC (Agricultural Trade Promotion Centre), the promotional body of the Chinese Ministry of Agriculture and Rural Affairs, which was one of the supporters of the event. In addition to China, there will be several collective stands in the "Country Pavilion", the international pavilion that has been added to the nine pavilions initially planned. Participants who have registered come from Central and South America (Colombia, Chile and the Dominican Republic) and there will be a large number of participants from Africa also in this digital edition of the event, thanks to partnerships established with UNIDO (United Nations Organisation for Industrial Development), Lab Innova and AICS (Italian Association for Culture and Sport). Not to mention Europe, with participants from Belgium, France, Germany, Greece, the Netherlands, Spain and Switzerland. There will also be many participants from Eastern Europe (Albania, Bulgaria and Ukraine).
Exhibitors will have a privileged channel where they will be able to "communicate" with the 600 buyers that have already registered on the platform (550 from abroad and 50 from Italy) and organise B2B meetings. In fact, Macfrut Digital will offer two levels of interaction: one for visitors, who will be able to visit the virtual stands, communicate with each other, ask for information and make contact to schedule business meetings, and the other for exhibitors, who will be able to schedule B2B meetings with buyers through a programmed agenda, with the support of Cesena Fiera's staff in order to align the agendas and meet the specific interests of the buyers and companies involved. Macfrut Digital can be accessed from any Internet device (PC or smartphone), while B2B meetings will be accessible online but only from PC, via a dedicated link provided by the staff.
The Macfrut Digital platform can be accessed subject to registration, which is free of charge. As of today, more than 4,000 visitors have already registered, 70% of whom are from abroad. In the last few days, participation requests have increased exponentially. Therefore, those who are interested in participating should register as soon as possible to secure their participation, bearing in mind that a maximum number of 30,000 visitors is allowed.
Renzo Piraccini, President of Macfrut, explains: 'We have embarked on a new adventure and we can safely say that the fruit and vegetable sector has responded extremely well. We believe that we are pioneering a project that has huge potential, which was unimaginable until a few months ago. The fact that the health emergency is ongoing, resulting in a level of uncertainty at international level, demonstrates that we made the right decision, which has changed the way we approach the sector.'
For more information, visit macfrutdigital.com
Lambert Peat Moss Inc. offers special screening and blending processes to achieve a unique consistency for growers
IGS announces referral partnership with IREP in Middle East
BRANDT NUTRITION PRODUCTS HELP DAVID HULA SET NEW CORN YIELD WORLD RECORD
EXPLORING THE NAÏO TEST PLOTS
Agritela Lux© by Arrigoni protects apple and pomegranate trees
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 9,752
|
I don't know where April went...but wow it flew by! If you missed anything on Silhouette School - and we were jam packed with Silhouette tutorials, tips, tricks and free designs - we've got ya covered!
Like we do every month I'm sharing a wrap up of everything we covered. So here are the 21 Silhouette CAMEO tips, tricks, helpful hints, free Silhouette designs and more that were shared in April 2017!
Game Changing Way to Organize Commercial Fonts!
I am curious about these beautiful Bundle packages that are available for purchase. How do people use all these fonts, as the Silhouette simply cuts and does not have a great way to fill in. I am new to the print and cut. But do people mostly use them more to print out wording for cards and such? I am just wondering, thanks.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 7,831
|
Q: Navigation buttons with a CSS slider I have a CSS slider that works for the most part except for two things I cannot figure out. The first being, how to get my slider navigation buttons to work the way I am after. Right now, if you click on one, my addClass adds .active to all of the buttons instead of only the one clicked on. Along with that I cannot figure out how to associate the slider position with the navigation menu. What I mean is if the user does not click any of the buttons I still want the buttons to show which slide is active.
MY second issue is whenever the buttons are clicked it stops the slide show.
What am I doing wrong in my attempts?
Off-topic, but would it be difficult to transition what I have now to fade in the slides rather than sliding them?
$('.control-button').click(function() {
button = $(this).attr('id');
id = button.replace('slide', '');
pos = (id - 1) * 100;
$('div#slider figure').css('animation-play-state', 'paused');
$('div#slider figure').removeClass('figure2');
$('.control-button').addClass('active');
posStr = '-' + pos + '%';
$('.figure').css('left', posStr);
});
$('img').click(function() {
$('div#inner').css('left', '0px');
$('div#slider figure').addClass('figure2');
$('div#slider figure').css('animation-play-state', 'running');
})
div#slider {
width: 100%;
overflow: hidden;
}
div#slider .figure {
position: relative;
width: 400%;
margin: 0;
padding: 0;
font-size: 0;
text-align: left;
/*animation: 20s company-slider infinite;*/
}
.figure2 {
animation: 20s company-slider infinite;
}
@keyframes company-slider {
0% {
left: 0%;
}
30% {
left: 0%;
}
35% {
left: -100%;
}
55% {
left: -100%;
}
60% {
left: -200%;
}
90% {
left: -200%;
}
95% {
left: -300%;
}
100% {
left: -300%;
}
}
div#slider figure img {
width: 25%;
min-height: 100%;
float: left;
}
/*div#slider figure:hover { animation-play-state:paused; }*/
div#slider li {
list-style: none;
}
div#slider label {
background-color: #111;
bottom: .5em;
cursor: pointer;
display: block;
height: .5em;
position: absolute;
width: .5em;
z-index: 10;
}
#controls {
width: 100%;
height: auto;
}
#control-container {
padding: 25px 12%;
}
.control-button {
display: inline;
margin: 0 2%;
width: 25%;
background: gray;
height: 10px;
border: none;
}
.control-button.active {
display: inline;
margin: 0 2%;
width: 25%;
background: black;
height: 10px;
border: none;
}
<script src="https://ajax.googleapis.com/ajax/libs/jquery/1.11.1/jquery.min.js">
</script>
<div id="slider">
<figure class="figure figure2">
<img src="https://iso.500px.com/wp-content/uploads/2016/02/stock-photo-139669245.jpg" alt>
<img src="http://i.cbc.ca/1.3376224.1450794847!/fileImage/httpImage/image.jpg_gen/derivatives/4x3_620/tundra-tea-toss.jpg" alt>
<img src="https://static.pexels.com/photos/22804/pexels-photo.jpg" alt>
<img src="https://iso.500px.com/wp-content/uploads/2016/02/stock-photo-139669245.jpg" alt>
</figure>
<div id="controls">
<div id="control-container">
<button id="slide1" class="control-button"></button>
<button id="slide2" class="control-button"></button>
<button id="slide3" class="control-button"></button>
</div>
</div>
</div>
A: This makes all of the buttons active:
$('.control-button').addClass('active');
Replace it with:
$('.control-button').removeClass('active');
$(event.target).addClass('active');
Add the event parameter to the function so you can use event.target:
$('.control-button').click(function(event) {
EDIT:
Making the control buttons activate "naturally" as the images slide is a bit harder.
Make each image have an attribute that says which slide it is:
<figure class="figure figure2">
<img data-number="slide1" src="https://iso.500px.com/wp-content/uploads/2016/02/stock-photo-139669245.jpg" alt>
<img data-number="slide2" src="http://i.cbc.ca/1.3376224.1450794847!/fileImage/httpImage/image.jpg_gen/derivatives/4x3_620/tundra-tea-toss.jpg" alt>
<img data-number="slide3" src="https://static.pexels.com/photos/22804/pexels-photo.jpg" alt>
<img data-number="slide4" src="https://iso.500px.com/wp-content/uploads/2016/02/stock-photo-139669245.jpg" alt>
</figure>
Create a system to detect when the image comes into view:
window.setInterval(function() {
Detect which image is being shown:
var activeImage = document.elementFromPoint($(window).width()/2, 10); // The image at the horizontal midpoint of the screen
Set the class of the corresponding control button to active:
$('.control-button').removeClass('active');
$("#"+$(activeImage).attr("data-number")).addClass('active'); // Sets whichever control button that corresponds to the image under the horizontal midpoint of the screen as active
Set how often you want to check at the closing of the setInterval:
}, /*time interval in miliseconds*/);
A: As far as adding 'active' to all buttons you should replace this:
$('.control-button').addClass('active');
with this:
$('.control-button').removeClass('active'); //this removes all '.active' first
$(this).addClass('active');
What's happening here is that within the .control-button click function, $(this) represents the current .control-button element.
A: https://jsfiddle.net/q3j01xrs/
What about this? I improved your script. I took out your CSS animation and replaced it by another JavaScript animation. Have a look!
$('.control-button').click(function() {
clearInterval(setInt);
$('.control-button').removeClass('active');
$(this).addClass('active');
var getIndex = $(this).index('.control-button');
$('.figure img').hide();
$('.figure img').eq(getIndex).show();
setInt = setInterval(slide, 5000);
});
function slide() {
var getIndex = $('.figure img:visible').index('.figure img');
$('.control-button').removeClass('active');
$('.figure img:visible').animate({
width: 'toggle'
}, 350);
getIndex++;
if ((getIndex) == $('.control-button').length) {
$('.control-button').eq(0).addClass('active');
$('.figure img').eq(0).animate({
width: 'toggle'
}, 350);
} else {
$('.control-button').eq(getIndex).addClass('active');
$('.figure img').eq(getIndex).animate({
width: 'toggle'
}, 350);
}
};
var setInt = setInterval(slide, 5000);
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 500
|
import {ActivatedRoute, Router} from '@angular/router';
import {UserService} from './../shared/user.service';
import {GoogleAuthService} from './../../auth/shared/google-auth.service';
import {Component, OnInit} from '@angular/core';
@Component({
selector: 'app-user-create-player',
templateUrl: './user-create-player.component.html',
styleUrls: ['./user-create-player.component.css']
})
export class UserCreatePlayerComponent implements OnInit {
private leagueId: string;
constructor(
private googleAuthService: GoogleAuthService,
private userService: UserService,
private route: ActivatedRoute,
private router: Router
) { }
ngOnInit() {
this.getLeagueId();
}
getLeagueId() {
this.route.params.map(param => param['league_id'])
.forEach(league_id => this.leagueId = league_id);
}
show(): boolean {
return this.googleAuthService.isAuthorized() && this.hasNotPlayer();
}
private hasNotPlayer(): boolean {
return this.googleAuthService.getCurrentPlayerId() == null;
}
create() {
let currentUser = this.googleAuthService.getCurrentUser();
this.userService.createPlayer(this.leagueId, currentUser.id)
.then(user => {
sessionStorage.setItem(this.googleAuthService.USER, JSON.stringify(user));
this.goToPlayer();
});
}
private goToPlayer() {
let playerId = this.googleAuthService.getCurrentPlayerId();
this.router.navigate(['/leagues', this.leagueId, 'players', playerId]);
}
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 2,030
|
\section{Introduction}\label{intro}
Modeling chemostats is a really interesting and important problem with special interest in mathematical biology, since they can be used to study recombinant problems in genetically altered microorganisms (see e.g. \cite{freter,rfreter}), waste water treatment (see e.g. \cite{dans,lariviere}) and play an important role in theoretical ecology (see e.g. \cite{bungay,cunnin,fredrickson,jannash,taylor,veldcamp,w83,w80}). Derivation and analysis of chemostat models are well documented in \cite{sm,swalt,w90} and references therein.\newline
Two standard assumptions for simple chemostat models are 1) the availability of the nutrient and its supply rate are fixed and 2) the tendency of the microorganisms to adhere to surfaces is not taken into account. However, these are very strong restrictions as the real world is non-autonomous and stochastic, and this justifies the analysis of stochastic chemostat models.\newline
Let us first consider the simplest chemostat model
\begin{eqnarray}
\frac{dS}{dt} &=& (S^0-S)D-\frac{mSx}{a+S}, \label{I1}
\\[1.3ex]
\frac{dx}{dt} &=& x\left(\frac{mS}{a+S}-D\right), \label{I2}
\end{eqnarray}
\noindent where $S(t)$ and $x(t)$ denote concentrations of the nutrient and the microbial biomass, respectively; $S^0$ denotes the volumetric dilution rate, $a$ is the half-saturation constant, $D$ is the dilution rate and $m$ is the maximal consumption rate of the nutrient and also the maximal specific growth rate of microorganisms. We notice that all parameters are positive and we use a function Holling type-II, $\mu(S)=mS/(a+S)$, as functional response of the microorganism describing how the nutrient is consumed by the species (see \cite{sree2} for more details and biological explanations about this model).\newline
However, we can consider a more realistic model by introducing a white noise in one of the parameters, therefore we replace the dilution rate $D$ by $D+\alpha \dot{W}(t)$, where $W(t)$ is a white noise, i.e., is a Brownian motion, and $\alpha\geq 0$ represents the intensity of noise. Then, system \eqref{I1}-\eqref{I2} is replaced by the following system of stochastic differential equations understood in the It\^o sense
\begin{eqnarray}
dS &=& \left[(S^0-S)D-\frac{mSx}{a+S}\right]dt+\alpha(S^0-S)dW(t), \label{I3}
\\[1.3ex]
dx &=& x\left(\frac{mS}{a+S}-D\right)dt-\alpha xdW(t). \label{I4}
\end{eqnarray}
System \eqref{I3}-\eqref{I4} has been analyzed in \cite{xu} by using the classic techniques from stochastic analysis and some stability results are provided there. However, as in our opinion there are some unclear points in the analysis carried out there, our aim in this paper is to use an alternative approach to this problem, specifically the theory of random dynamical systems, which will allow us to partially improve the results in \cite{xu}. In addition, we will provide some results which hold almost surely while those from \cite{xu} are said to hold in probability.\newline
Firstly, thanks to the well-known conversion between It\^o and Stratonovich sense, we obtain from \eqref{I3}-\eqref{I4} its equivalent Stratonovich formulation which is given by
\begin{eqnarray}
dS&=&\left[(S^0-S)\bar{D}-\frac{mSx}{a+S}\right]dt+\alpha(S^0-S)\circ dW(t),\label{5}
\\[1.3ex]
dx&=&\left[-\bar{D}x+\frac{mSx}{a+S}\right]dt-\alpha x\circ dW(t),\label{6}
\end{eqnarray}
\noindent where $\bar{D}:=D+\frac{\alpha^2}{2}$.\newline
In Section \ref{preliminaries} we recall some basic results on random dynamical systems. In Section \ref{randomchemostat} we start with the study of equilibria and we prove a result related to the existence and uniqueness of global solution of system \eqref{5}-\eqref{6}, by using the so-called Ornstein-Uhlenbeck (O-U) process. Then, we define a random dynamical system and prove the existence of a random attractor giving an explicit expression for it. Finally, in Section \ref{nsfc} we show some numerical simulations with different values of the parameters involved in the model and we can see what happens when the amount of noise $\alpha$ increases.
\section{Random dynamical systems}\label{preliminaries}
In this section we present some basic results related to random dynamical systems (RDSs) and random attractors which will be necessary for our analysis. For more detailed information about RDSs and their importance, see \cite{arnold}.\newline
Let $(\mathcal{X},\Vert \cdot \Vert _\mathcal{X})$ be a separable Banach space and let $(\Omega, \mathcal{F}, \mathbb{P})$ be a probability space where
$\mathcal{F}$ is the $\sigma-$algebra of measurable subsets of $\Omega$ (called
``events") and $\mathbb{P}$ is the probability measure. To connect the state $\omega$
in the probability space $\Omega$ at time 0 with its state after a time of $t$
elapses, we define a flow $\theta = \{\theta_t \}_{t \in \mathbb{R}}$ on
$\Omega$ with each $\theta_t$ being a mapping $\theta_t: \Omega \to \Omega$ that
satisfies
\begin{itemize}
\item[(1)] $\,\,\,\,$ $\theta_0 = \mbox{Id}_\Omega$,
\item[(2)] $\,\,\,\,$ $\theta_s \circ \theta_t = \theta_{s+t}$ for all $s, t \in \mathbb{R}$,
\item[(3)] $\,\,\,\,$ the mapping $(t, \omega) \mapsto \theta_t \omega$ is measurable,
\item[(4)] $\,\,\,\,$ the probability measure $\mathbb{P}$ is preserved by $\theta_t$, i.e., $\theta_t \mathbb{P} = \mathbb{P}$.
\end{itemize}
This set-up establishes a time-dependent family $\theta$ that tracks the noise,
and $(\Omega, \mathcal{F}, \mathbb{P}, \theta)$ is called a \textit{metric dynamical system} (see \cite{arnold}).
\begin{definition}\label{RDS}
A stochastic process $\{\varphi(t,\omega )\}_{t\geq 0,\omega \in
\Omega }$ is said to be a \textrm{continuous RDS over $(\Omega ,\mathcal{F},%
\ensuremath{\mathbb{P}},\{\theta _t\}_{t\in \ensuremath{\mathbb{R}}})$ with
state space $\mathcal{X}$} if $\varphi:[0,+\infty )\times \Omega \times \mathcal{X}\rightarrow \mathcal{X}$ is $(%
\mathcal{B}[0,+\infty )\times \ \mathcal{F}\times \mathcal{B}(\mathcal{X}),\ \mathcal{B%
}(\mathcal{X}))$- measurable, and for each $\omega \in \Omega $,
\begin{itemize}
\item[(i)] $\,\,\,\,$ the mapping $\varphi(t,\omega ): \mathcal{X}\rightarrow \mathcal{X}$, $x\mapsto \varphi(t,\omega
)x$ is continuous for every $t\geq 0$,
\item[(ii)] $\,\,\,\,$ $\varphi(0,\omega )$ is the identity operator on $\mathcal{X}$,
\item[(iii)] $\,\,\,\,$ (cocycle property) $\varphi(t+s,\omega )=\varphi(t,\theta _s\omega)\varphi(s,\omega )$ for all $s,t\geq 0$.
\end{itemize}
\end{definition}
\begin{definition}
Let $(\Omega ,\mathcal{F},\ensuremath{\mathbb{P}})$ be a probability space. A random set $K$ is a measurable subset of $\mathcal{X}\times\Omega$ with respect to the product $\sigma-$algebra $\mathcal{B}(\mathcal{X})\times\mathcal{F}$.\newline
The $\omega-$section of a random set $K$ is defined by
$$K(\omega)=\{x\,:\,(x,\omega)\in K\},\quad \omega\in\Omega.$$
In the case that a set $K\subset \mathcal{X}\times \Omega$ has closed or compact $\omega-$sections it is a random set as soon as the mapping $\omega\mapsto d(x,K(\omega))$ is measurable (from $\Omega$ to $[0,\infty)$) for every $x\in \mathcal{X}$, see \cite{crauel2}. Then $K$ will be said to be a closed or a compact, respectively, random set. It will be assumed that closed random sets satisfy $K(\omega)\neq\emptyset$ for all or at least for $\ensuremath{\mathbb{P}}-$almost all $\omega\in\Omega$.
\end{definition}
\begin{remark}
It should be noted that in the literature very often random sets are defined provided that $\omega\mapsto d(x,K(\omega))$ is measurable for every $x\in \mathcal{X}$. Obviously this is satisfied, for instance, when $K(\omega)=N$ for all $\omega$, where $N$ is some non-measurable subset of $\mathcal{X}$, and also when $K=(U\times F)\cup(\overline{U}\times F^c)$ for some open set $U\subset \mathcal{X}$ and $F\notin\mathcal{F}$. In both cases $\omega\mapsto d(x,K(\omega))$ is constant, hence measurable, for every $x\in \mathcal{X}$. However, both cases give $K\subset \mathcal{X}\times\Omega$ which is not an element of the product $\sigma-$algebra $\mathcal{B}(\mathcal{X})\times\mathcal{F}$.
\end{remark}
\begin{definition}
A bounded random set $K(\omega) \subset \mathcal{X}$\ is said to be \textrm{%
tempered with respect to }$\{\theta _t\}_{t\in \ensuremath{\mathbb{R}}}$ if
for a.e. $\omega \in \Omega $, $$\lim_{t\rightarrow \infty } e^{-\beta
t}\sup\limits_{x\in K(\theta _{-t}\omega )}\Vert x\Vert _\mathcal{X}=0, \quad \mbox{for all} \ \beta > 0;$$ a random variable $\omega \mapsto r(\omega )\in \ensuremath{\mathbb{R}}$%
\textrm{\ }is said to be \textrm{tempered with respect to }$\{\theta
_t\}_{t\in \ensuremath{\mathbb{R}}}$ if for a.e. $\omega \in \Omega $, $$%
\lim_{t\rightarrow \infty} e^{-\beta t}\sup\limits_{t\in %
\ensuremath{\mathbb{R}}}|r(\theta _{-t}\omega )|=0, \quad \mbox{for all} \ \beta >0.$$
\end{definition}
In what follows we use ${\mathcal{E}}(\mathcal{X})$ to denote the set of all tempered
random sets of $\mathcal{X}$.
\begin{definition}
A random set $B(\omega )\subset \mathcal{X}$ is called a
\textrm{random absorbing set} in ${\mathcal{E}}(\mathcal{X})$ if for any $E\in
{\mathcal{E}}(\mathcal{X})$ and a.e. $\omega \in \Omega $, there exists $T_E(\omega )>0
$ such that $$\varphi(t,\theta _{-t}\omega)E(\theta _{-t}\omega )\subset
B(\omega), \quad \forall t\geq T_E(\omega ).$$
\end{definition}
\begin{definition}
Let $\{\varphi(t,\omega )\}_{t\geq 0,\omega \in \Omega }$ be an RDS
over $(\Omega ,\mathcal{F},\ensuremath{\mathbb{P}},\{\theta _t\}_{t\in %
\ensuremath{\mathbb{R}}})$ with state space $\mathcal{X}$ and let $A(\omega
)(\subset \mathcal{X})$ be a random set. Then $\mathcal{A}=\{A(\omega)\}_{\omega\in\Omega}$ is called a \textrm{global random ${\mathcal{E}}-$attractor (or pullback $\mathcal{E}-$attractor)
} for $\{\varphi(t,\omega )\}_{t\geq 0,\omega \in \Omega }$ if
\begin{itemize}
\item[(i)] $\,\,\,\,$ (compactness) $A(\omega )$ is a compact set of $\mathcal{X}$ for
any $\omega \in \Omega $;
\item[(ii)] $\,\,\,\,$ (invariance) for any $\omega \in \Omega $ and all $t\geq 0$, it holds $$%
\varphi(t,\omega)A(\omega)=A(\theta _t\omega );$$
\item[(iii)] $\,\,\,\,$ (attracting property) for any $E\in {\ \mathcal{E}}(\mathcal{X})$ and
a.e. $\omega \in \Omega $,
$$
\lim\limits_{t\rightarrow \infty }\mathrm{dist}%
_\mathcal{X}(\varphi(t,\theta _{-t}\omega )E(\theta _{-t}\omega ),A(\omega ))=0,
$$
where $$\mathrm{dist}_\mathcal{X}(G, H) = \sup_{g \in G} \inf_{h \in H} \|g - h\|_\mathcal{X}$$ is the Hausdorff semi-metric for $G, H \subseteq \mathcal{X}.$
\end{itemize}
\end{definition}
\begin{proposition} \label{attractor} \cite{CLR2006,FS1996} Let $B \in \mathcal{E}(\mathcal{X})$ be a closed absorbing set for the continuous random dynamical
system $\{\varphi(t,\omega)\}_{t \geq 0,\omega\in\Omega}$ that satisfies the asymptotic compactness condition for $a. e. \ \omega \in \Omega$, i.e., each sequence
$x_n \in \varphi(t_n, \theta_{-t_n}\omega) B (\theta_{-t_n} \omega)$ has a convergent subsequence in $\mathcal{X}$ when $t_n \to \infty$. Then $\varphi$ has a unique
global random attractor $\mathcal{A}=\{A(\omega)\}_{\omega\in\Omega}$ with component subsets
$$
A(\omega) = \bigcap_{\tau \geq T_B(\omega)} \overline{\bigcup_{t \geq \tau} \varphi(t, \theta_{-t} \omega) B(\theta_{-t} \omega)}.
$$
If the pullback absorbing set is positively invariant, i.e., $\varphi(t,\omega) B(\omega)$ $\subset$ $B(\theta_{t} \omega)$ for all $t$ $\geq$ $0$, then
$$
A(\omega) = \bigcap_{t \geq 0} \overline{\varphi(t, \theta_{-t} \omega) B(\theta_{-t} \omega)}.
$$
\end{proposition}
\begin{remark}\label{remark1}When the state space $\mathcal{X}$ $=$ $\mathbb{R}^d$ as in this paper, the asymptotic compactness follows trivially. Note that the random attractor is path-wise attracting
in the pullback sense, but does not need to be path-wise attracting in the forward sense, although it is forward attracting in probability, due to some possible large deviations, see e.g. \cite{arnold}.
\end{remark}
The next result ensures when two random dynamical systems are conjugated (see also \cite{caraballoconjugated2,caraballo-book,caraballoconjugated1}).
\begin{lemma}\label{lemmaconjugation}
Let $\varphi_u$ be a random dynamical system on $\mathcal{X}$. Suppose that the mapping $\mathcal{T}:\Omega\times \mathcal{X}\rightarrow \mathcal{X}$ possesses the following properties: for fixed $\omega\in\Omega$, $\mathcal{T}(\omega,\cdot)$ is a homeomorphism on $\mathcal{X}$, and for $x\in \mathcal{X}$, the mappings $\mathcal{T}(\cdot,x)$, $\mathcal{T}^{-1}(\cdot,x)$ are measurable. Then the mapping
$$(t,\omega,x)\rightarrow\varphi_v(t,\omega)x:=\mathcal{T}^{-1}(\theta_t\omega,\varphi_u(t,\omega)\mathcal{T}(\omega,x))$$
is a (conjugated) random dynamical system.
\end{lemma}
\section{Random chemostat}\label{randomchemostat}
In this section we will investigate the stochastic system \eqref{5}-\eqref{6}. To this end, we first transform it into differential equations with random coefficients and without white noise.\newline
Let $W$ be a two sided Wiener process. Kolmogorov's theorem ensures that $W$ has a continuous version, that we will denote by $\omega$, whose canonical interpretation is as follows: let $\Omega$ be defined by $$\Omega =\{\omega \in \mathcal{C}(\mathbb{R}, \mathbb{R}): \omega(0) = 0\} = \mathcal{C}_0(\mathbb{R}, \mathbb{R}),$$ $\mathcal{F}$ be the Borel $\sigma-$algebra on $\Omega$ generated by the compact open topology (see \cite{arnold} for details) and $\mathbb{P}$ the corresponding Wiener measure on $\mathcal{F}$. We consider the Wiener shift flow given by $$\theta_t \omega(\cdot) = \omega(\cdot + t) - \omega(t),\quad t\in \ensuremath{\mathbb{R}},$$ then $(\Omega, \mathcal{F}, \mathbb{P}, \{\theta_t\}_{t \in \mathbb{R}})$ is a metric dynamical system. Now let us introduce the following Ornstein-Uhlenbeck process on $%
(\Omega,\mathcal{F},\ensuremath{\mathbb{P}},\{\theta _t\}_{t\in %
\ensuremath{\mathbb{R}}})$
\begin{equation*}
z^*(\theta _t\omega )=-\int\limits_{-\infty }^0e^s\theta _t\omega (s)%
ds,\quad t\in \ensuremath{\mathbb{R}},\quad \omega \in \Omega, \label{delta}
\end{equation*}
which solves the following Langevin equation (see e.g. \cite{arnold,CL})
\begin{eqnarray*}
dz+zdt =d\omega(t),\quad t\in\mathbb{R}. \label{OU}
\end{eqnarray*}
\begin{proposition}
\label{property-delta-lm1} (\cite{arnold,CL}) There exists a $
\theta _t$-invariant set $\widetilde{\Omega }\in \mathcal{F}$
of $\Omega$ of full $\ensuremath{\mathbb{P}}$ measure such that for $\omega \in
\widetilde{\Omega },$ we have
\begin{itemize}
\item [(i)] $\,\,$ the random variable $|z^*(\omega )|$ is tempered.
\item [(ii)] $\,\,$ the mapping
\[
(t,\omega )\rightarrow z^*(\theta _t\omega )=-\int\limits_{-\infty
}^0e^s\omega (t+s)\mathrm{d}s+\omega(t)
\]
is a stationary solution of \eqref{OU}
with continuous trajectories;\newline
\item [(iii)] $\,\,$ in addition, for any $\omega \in \tilde \Omega$:
\begin{eqnarray*}
\lim_{t\rightarrow \pm \infty }\frac{|z^*(\theta _t\omega )|}%
t&=& 0;\\
\lim_{t\rightarrow \pm \infty }\frac 1t\int_0^tz^*(\theta _s\omega
)ds&=&0;\\
\lim_{t\rightarrow \pm \infty }\frac 1t\int_0^t |z^*(\theta _s\omega
)| ds&=& \mathbb{E}[z^*] < \infty.
\end{eqnarray*}
\end{itemize}
\end{proposition}
In what follows we will consider the restriction of the Wiener shift $\theta$ to the set $\tilde \Omega$, and we restrict accordingly the metric dynamical system to this set, that is also a metric dynamical system, see \cite{caraballoconjugated2}. For simplicity, we will still denote the restricted metric dynamical system by the old symbols $(\Omega,\mathcal{F},\ensuremath{\mathbb{P}},\{\theta _t\}_{t\in %
\ensuremath{\mathbb{R}}})$.
\subsection{Stochastic chemostat becomes a random chemostat}
In what follows we use the Ornstein-Uhlenbeck process to transform \eqref{5}-\eqref{6} into a random system. Let us note that analyzing the equilibria we obtain that the only one is the axial equilibrium $(S^0,0)$ and then we define two new variables $\sigma$ and $\kappa$ by
\begin{eqnarray}
\sigma(t) &=& (S(t)-S^0)e^{\alpha z^*(\theta_t\omega)}, \label{vcsigma}
\\[1.3ex]
\kappa(t) &=& x(t)e^{\alpha z^*(\theta_t\omega)}. \label{vckappa}
\end{eqnarray}
For the sake of simplicity we will write $z^*$ instead of $z^*(\theta_t\omega)$, and $\sigma$ and $\kappa$ instead of $\sigma(t)$ and $\kappa(t)$.\newline
On the one hand, by differentiation, we have
\begin{eqnarray}
\nonumber
d\sigma&=&e^{\alpha z^*(\theta_t\omega)}\cdot dS+\alpha(S-S^0)e^{\alpha z^*(\theta_t\omega)}[-z^*dt+dW]
\\[1.3ex]
\nonumber
&=&-\bar{D}\sigma dt-\frac{m(S^0+\sigma e^{-\alpha z^*(\theta_t\omega)})}{a+S^0+\sigma e^{-\alpha z^*(\theta_t\omega)}}\kappa dt-\alpha z^*\sigma dt.
\end{eqnarray}
On the other hand, we obtain
\begin{eqnarray}
\nonumber
d\kappa&=&e^{\alpha z^*(\theta_t\omega)}\cdot dx+\alpha xe^{\alpha z^*(\theta_t\omega)}[-z^*dt+dW]
\\[1.3ex]
\nonumber
&=&\frac{m(S^0+\sigma e^{-\alpha z^*(\theta_t\omega)})}{a+S^0+e^{-\alpha z^*(\theta_t\omega)}}\kappa dt-\bar{D}\kappa dt-\alpha z^*\kappa dt.
\end{eqnarray}
Thus, we deduce the following random system
\begin{eqnarray}
\frac{d\sigma}{dt}&=&-(\bar{D}+\alpha z^*)\sigma -\frac{m(S^0+\sigma e^{-\alpha z^*(\theta_t\omega)})}{a+S^0+\sigma e^{-\alpha z^*(\theta_t\omega)}}\kappa,\label{7}
\\[1.3ex]
\frac{d\kappa}{dt}&=&-(\bar{D}+\alpha z^*)\kappa+\frac{m(S^0+\sigma e^{-\alpha z^*(\theta_t\omega)})}{a+S^0+\sigma e^{-\alpha z^*(\theta_t\omega)}}\kappa.\label{8}
\end{eqnarray}
\subsection{Random chemostat generates an RDS}
Next we prove that the random chemostat given by \eqref{7}-\eqref{8} generates an RDS. From now on, we will denote $\mathcal X:=\{(x,y)\in\mathbb{R}^2\,:\, x\in\mathbb{R},\, y\geq 0\}$, the upper-half plane.
\begin{theorem}\label{theorem1}
For any $\omega\in\Omega$ and any initial value $u_0:=(\sigma_0,\kappa_0)\in \mathcal X$, where $\sigma_0:=\sigma(0)$ and $\kappa_0:=\kappa(0)$, system \eqref{7}-\eqref{8} possesses a unique global solution $u(\cdot;0,\omega,u_0):=(\sigma(\cdot;0,\omega,u_0) , \kappa (\cdot;0,\omega,u_0))\in\mathcal{C}^1([0,+\infty),\mathcal X)$ with $u(0;0,\omega,u_0)=u_0$. Moreover, the solution mapping generates a RDS $\varphi_u:\mathbb R^+\times \Omega\times \mathcal X \rightarrow \mathcal X$ defined as
$$\varphi_u(t,\omega)u_0:=u(t;0,\omega,u_0),\quad \text{for all}\,\, t\in \mathbb R^+, \, u_0\in\mathcal X,\, \omega\in\Omega,$$
\noindent the value at time $t$ of the solution of system \eqref{7}-\eqref{8} with initial value $u_0$ at time zero.
\end{theorem}
\begin{proof}Observe that we can rewrite one of the terms in the previous equations as
\begin{eqnarray*}
\frac{m(S^0+\sigmae^{\alpha z^*(\theta_t\omega)})}{a+S^0+\sigmae^{\alpha z^*(\theta_t\omega)}}\kappa &=& \frac{m(S^0+\sigmae^{\alpha z^*(\theta_t\omega)}+a-a)}{a+S^0+\sigmae^{\alpha z^*(\theta_t\omega)}}\kappa=m\kappa-\frac{ma\kappa}{a+S^0+\sigmae^{\alpha z^*(\theta_t\omega)}}
\end{eqnarray*}
and therefore system \eqref{7}-\eqref{8} turns into
\begin{eqnarray}
\frac{d\sigma}{dt} &=& -(\bar{D}+\alpha z^*)\sigma-m\kappa+\frac{ma}{a+S^0+\sigmae^{\alpha z^*(\theta_t\omega)}}\kappa, \label{a1}
\\[1.3ex]
\frac{d\kappa}{dt} &=& -(\bar{D}+\alpha z^*)\kappa+m\kappa-\frac{ma}{a+S^0+\sigmae^{\alpha z^*(\theta_t\omega)}}\kappa. \label{a2}
\end{eqnarray}
Denoting $u(\cdot;0,\omega,u_0):=(\sigma(\cdot;0,\omega,u_0) , \kappa (\cdot;0,\omega,u_0))$, system \eqref{a1}-\eqref{a2} can be rewritten as
\begin{eqnarray*}
\frac{du}{dt} &=& L(\theta_t\omega)\cdot u+F(u,\theta_t\omega), \label{9}
\end{eqnarray*}
\noindent where
\begin{eqnarray*}
L(\theta_t\omega) &=& \left(\begin{array}{cc}
-(\bar{D}+\alpha z^*) & -m \\
0 & -(\bar{D}+\alpha z^*)+m
\end{array}\right)
\end{eqnarray*}
\noindent and $F:\mathcal X\times[0,+\infty)\longrightarrow\mathbb{R}^2$ is given by
\begin{eqnarray*}
F(\xi,\theta_t\omega) &=& \left(\begin{array}{c}
\displaystyle{\frac{ma}{a+S^0+\xi_1e^{-\alpha z^*(\theta_t\omega)}}\xi_2}\\
\displaystyle{\frac{-ma}{a+S^0+\xi_1e^{-\alpha z^*(\theta_t\omega)}}\xi_2}
\end{array}\right),
\end{eqnarray*}
\noindent where $\xi=(\xi_1,\xi_2)\in\mathcal X$.\newline
Since $z^*(\theta_t\omega)$ is continuous, $L$ generates an evolution system on $\mathbb{R}^2$. Moreover, we notice that
\begin{eqnarray*}
\frac{\partial}{\partial \xi_2}\left[\pm\frac{am}{a+S^0+\xi_1e^{\alpha z^*(\theta_t\omega)}}\xi_2\right] &=& \pm\frac{am}{a+S^0+\xi_1e^{\alpha z^*(\theta_t\omega)}}
\end{eqnarray*}
and
\begin{eqnarray*}
\frac{\partial}{\partial \xi_1}\left[\pm\frac{am}{a+S^0+\xi_1e^{\alpha z^*(\theta_t\omega)}}\xi_2\right] &=& \mp\frac{ame^{\alpha z^*(\theta_t\omega)}}{(a+S^0+\xi_1e^{\alpha z^*(\theta_t\omega)})^2}\xi_2
\end{eqnarray*}
\noindent thus $F(\cdot,\theta_t\omega)\in\mathcal{C}^1(\mathcal X \times[0,+\infty);\mathbb{R}^2)$ which implies that it is locally Lipschitz with respect to $(\xi_1,\xi_2)\in\mathcal X$. Therefore, thanks to classical results from the theory of ordinary differential equations, system \eqref{7}-\eqref{8} possesses a unique local solution. Now, we are going to prove that the unique local solution of system \eqref{7}-\eqref{8} is in fact a unique global one.\newline
By defining $Q(t):=\sigma(t)+\kappa(t)$ it is easy to check that $Q$ satisfies the differential equation
\begin{eqnarray}
\nonumber
\frac{dQ}{dt}&=&-(\bar{D}+\alpha z^*)Q,
\end{eqnarray}
\noindent whose solution is given by the following expression
\begin{eqnarray}
Q(t;0,\omega,Q(0))&=&Q(0)e^{-\bar{D}t-\alpha\int_0^tz^*(\theta_s\omega)ds}.\label{Q}
\end{eqnarray}
The right side of \eqref{Q} always tends to zero when $t$ goes to infinity since $\bar{D}$ is positive, thus $Q$ is clearly bounded. Moreover, since
$$\left.\frac{d\sigma}{dt}\right|_{\sigma=0}=-\frac{mS^0}{a+S^0}\kappa<0$$
\noindent we deduce that, if there exists some $t^*>0$ such that $\sigma(t^*)=0$, we will have $\sigma(t)<0$ for all $t>t^*$. Because of the previous reasoning, we will split our analysis into two different cases.\newline
\begin{itemize}
\item {\bf Case 1. $\sigma(t)>0$ for all $t\geq 0$:} in this case, from \eqref{7} we obtain
\begin{eqnarray}
\nonumber
\frac{d\sigma}{dt}&\leq& -(\bar{D}+\alpha z^*)\sigma
\end{eqnarray}
\noindent whose solutions should satisfy
\begin{eqnarray}
\sigma(t;0,\omega,\sigma(0))&\leq&\sigma(0)e^{-\bar{D}t-\alpha\int_0^tz^*(\theta_s\omega)ds} \label{sigma}.
\end{eqnarray}
Since $\bar{D}$ is positive, we deduce that $\sigma$ tends to zero when $t$ goes to infinity, hence $\sigma$ is bounded.
\item {\bf Case 2. there exists $t^*>0$ such that $\sigma(t^*)=0$:} in this case, we already know that $\sigma(t)<0$ for all $t>t^*$ and we claim that the following bound for $\sigma$ holds true
\begin{equation}
\sigma(t;0,\omega,\sigma(0))>-(a+S^0)e^{\alpha z^*(\theta_t\omega)}.\label{sigmabound}
\end{equation}
To prove \eqref{sigmabound}, we suppose that there exists $\bar{t}>t^*>0$ such that $$a+S^0+\sigma(\bar{t})e^{-\alpha z^*(\theta_{\bar{t}}\omega)}=0,$$
\noindent then we can find some $\varepsilon(\omega)>0$ small enough such that $\sigma(t)$ is strictly decreasing and
\begin{equation}
-(\bar{D}+\alpha z^*(\theta_t\omega))-\frac{m(S^0+\sigma(t)e^{-\alpha z^*(\theta_t\omega)})}{a+S^0+\sigma(t)e^{-\alpha z^*(\theta_t\omega)}}\kappa(t)>0\label{c}
\end{equation}
\noindent holds for all $t\in[\bar{t}-\varepsilon(\omega),\bar{t})$. Hence, from \eqref{c} we have
$$\frac{d\sigma}{dt}(\bar{t}-\varepsilon(\omega))>0,$$
\noindent thus there exists some $\delta(\omega)>0$ small enough such that $\sigma(t)$ is strictly increasing for all $t\in[\bar{t}-\varepsilon(\omega),\bar{t}-\varepsilon(\omega)+\delta(\omega))$, which clearly contradicts the uniqueness of solution. Hence, \eqref{sigmabound} holds true for all $t\in\mathbb{R}$ and we can also ensure that $\sigma$ is bounded.
\end{itemize}
Since $\sigma+\kappa$ and $\sigma$ are bounded in both cases, $\kappa$ is also bounded. Hence, the unique local solution of system \eqref{7}-\eqref{8} is a unique global one. Moreover, the unique global solution of system \eqref{7}-\eqref{8} remains in $\mathcal{X}$ for every initial value in $\mathcal{X}$ since $\kappa\equiv 0$ solves the same system.\newline
Finally, the mapping $\varphi_u: \mathbb R^+\times \Omega \times \mathcal X \rightarrow \mathcal X$ given by
\begin{eqnarray*}
\varphi_u(t,\omega)u_0 &:=& u(t;0,\omega,u_0),\quad \text{for all}\,\, t\geq 0, \,\, u_0\in\mathcal X,\,\, \omega\in\Omega, \label{ds}
\end{eqnarray*}
\noindent defines a RDS generated by the solution of \eqref{7}-\eqref{8}. The proof of this statement follows trivially hence we omit it.\newline
\qed
\end{proof}
\subsection{Existence of the pullback random attractor}
Now, we study the existence of the pullback random attractor, describing its internal structure explicitly.
\begin{theorem}\label{t2}
There exists, for any $\varepsilon>0$, a tempered compact random absorbing set $B_\varepsilon(\omega)\in\mathcal{E}(\mathcal X)$ for the RDS $\{\varphi_u(t,\omega)\}_{t\geq 0,\,\omega\in\Omega}$, that is, for any $E(\theta_{-t}\omega)\in\mathcal{E}(\mathcal{X})$ and each $\omega\in\Omega$, there exists $T_E(\omega,\varepsilon)>0$ such that $$\varphi_u(t,\theta_{-t}\omega)E(\theta_{-t}\omega)\subseteq B_\varepsilon(\omega),\quad\quad\text{for all}\,\, t\geq T_E(\omega,\varepsilon).$$
\end{theorem}
\begin{proof}Thanks to \eqref{Q}, we have
\begin{eqnarray}
\nonumber
Q(t;0,\theta_{-t}\omega,Q(0))&=&Q(0)e^{-\bar{D}t-\alpha\int_{-t}^0z^*(\theta_s\omega)ds}\stackrel{\quad t\rightarrow+\infty\quad}{\longrightarrow}0.
\end{eqnarray}
Then, for any $\varepsilon>0$ and $u_0\in E(\theta_{-t}\omega)$ there exists $T_E(\omega,\varepsilon)>0$ such that, for all $t\geq T_E(\omega,\varepsilon)$, we obtain
$$-\varepsilon\leq Q(t;0,\theta_{-t}\omega,u_0)\leq \varepsilon.$$
If we assume that $\sigma(t)\geq 0$ for all $t\geq 0$, which corresponds to {\bf Case 1} in the proof of Theorem \ref{theorem1}, since $\kappa(t)\geq 0$ for all $t\geq 0$, we have that
$$B^1_\varepsilon(\omega):=\left\{(\sigma,\kappa)\in\mathcal{X}\,\,:\,\, \sigma\geq 0,\, \sigma+\kappa\leq\varepsilon\right\}$$
\noindent is a tempered compact random absorbing set in $\mathcal{X}$.\newline
In the other case, i.e., if there exists some $t^*>0$ such that $\sigma(t^*)=0$, which corresponds to {\bf Case 2} in the proof of Theorem \ref{theorem1}, we proved that
\begin{equation*}
\sigma(t;0,\theta_{-t}\omega,u_0)>-(a+S^0)e^{\alpha z^*(\omega)}.
\end{equation*}
Hence, we obtain that
$$B^2_\varepsilon(\omega):=\left\{(\sigma,\kappa)\in\mathcal{X}\,\,:\,\, -\varepsilon-(a+S^0)e^{\alpha z^*(\omega)}\leq\sigma\leq 0,\, -\varepsilon\leq\sigma+\kappa\leq\varepsilon\right\}$$
\noindent is a tempered compact random absorbing set in $\mathcal{X}$.\newline
In conclusion, defining
\begin{equation*}
B_\varepsilon(\omega)=B^1_\varepsilon(\omega)\cup B^2_\varepsilon(\omega)=\left\{(\sigma,\kappa)\in\mathcal{X}\,\,:\,\,-\varepsilon\leq\sigma+\kappa\leq\varepsilon,\, \sigma\geq-(a+S^0)e^{\alpha z^*(\omega)}-\varepsilon\right\},
\end{equation*}
\noindent we obtain (see Figure \ref{figure1}) that $B_\varepsilon(\omega)$ is a tempered compact random absorbing set in $\mathcal{X}$ for every $\varepsilon>0$.
\begin{multicols}{2}
\begin{figure}[H]
\begin{center}
\psscalebox{0.75 0.75}
{
\begin{pspicture}(0,-4.430296)(10.480869,4.430296)
\definecolor{colour0}{rgb}{0.0,0.6,1.0}
\definecolor{colour1}{rgb}{1.0,0.2,0.2}
\definecolor{colour2}{RGB}{0,102,0}
\pspolygon[linecolor=white, linewidth=0.04, fillstyle=solid,fillcolor=blue](2.3836365,-1.6571428)(5.637391,-1.6650479)(1.1915416,2.8242803)(1.2018182,-0.16623364)(1.2018182,-0.45714274)
\rput[l](10.2808695,-1.5828345){\large $\sigma$}
\rput[b](3.9796839,4.2242804){\large $\kappa$}
\psline[linecolor=colour1, linewidth=0.04, linestyle=dashed, dash=0.17638889cm 0.10583334cm](1.2018182,3.4883118)(1.2018182,-3.238961)
\rput[t](1.18917,-3.40655){\large $-(a+S^0)e^{\alpha z^*(\omega)}-\varepsilon$}
\pspolygon[linecolor=white, linewidth=0.04, fillstyle=solid,fillcolor=colour2](3.9931226,0.004517342)(3.992332,-1.6753246)(5.637391,-1.666629)
\psline[linecolor=black, linewidth=0.04, dotsize=0.07055555555555555cm 2.0,arrowsize=0.05291666666666667cm 2.0,arrowlength=1.4,arrowinset=0.0]{*->}(0.0,-1.6571428)(10.14,-1.6571428)
\psline[linecolor=black, linewidth=0.04, dotsize=0.07055555555555555cm 2.0,arrowsize=0.05291666666666667cm 2.0,arrowlength=1.4,arrowinset=0.0]{*->}(3.9836364,-4.4302654)(4.001818,4.1246753)
\rput[t](5.532253,-1.85){\large $\varepsilon$}
\rput[t](2.4160473,-1.76){\large $-\varepsilon$}
\rput[t](2.5,0.1){\large \bf \textcolor{white}{$B^2_\varepsilon(\omega)$}}
\rput[t](4.5,-1){\bf \textcolor{white}{$B^1_\varepsilon(\omega)$}}
\end{pspicture}
}
\end{center}
\caption{Absorbing set \bf $B_\varepsilon(\omega):=\textcolor[RGB]{0,102,0}{B^2_\varepsilon(\omega)}\cup\textcolor{blue}{B^1_\varepsilon(\omega)}$}
\label{figure1}
\end{figure}
\columnbreak
\begin{figure}[H]
\begin{center}
\psscalebox{0.75 0.75}
{
\begin{pspicture}(0,-4.430296)(10.480869,4.430296)
\definecolor{colour1}{rgb}{1.0,0.2,0.2}
\rput[l](10.2808695,-1.5828345){\large $\sigma$}
\rput[b](3.9796839,4.2242804){\large $\kappa$}
\psline[linecolor=colour1, linewidth=0.04, linestyle=dashed, dash=0.17638889cm 0.10583334cm](1.2018182,3.4883118)(1.2018182,-3.238961)
\rput[t](1.18917,-3.40655){\large $-(a+S^0)e^{\alpha z^*(\omega)}$}
\psline[linecolor=black, linewidth=0.04, dotsize=0.07055555555555555cm 2.0,arrowsize=0.05291666666666667cm 2.0,arrowlength=1.4,arrowinset=0.0]{*->}(0.0,-1.6571428)(10.14,-1.6571428)
\psline[linecolor=black, linewidth=0.04, dotsize=0.07055555555555555cm 2.0,arrowsize=0.05291666666666667cm 2.0,arrowlength=1.4,arrowinset=0.0]{*->}(3.9836364,-4.4302654)(4.001818,4.1246753)
\psline[linecolor=blue, linewidth=0.04, dotsize=0.07055555555555555cm 2.0]{*-*}(3.984706,-1.6924368)(1.1964706,1.2016808)
\rput[bl](2.5,0.1){\large \bf \textcolor{blue}{$B_0(\omega)$}}
\end{pspicture}
}
\end{center}
\caption{Absorbing set \textcolor{blue}{\bf $B_0(\omega)$}}
\label{figure2}
\end{figure}
\end{multicols}
\qed
\end{proof}
Then, thanks to Proposition \ref{attractor}, it follows directly that system \eqref{7}-\eqref{8} possesses a unique pullback random attractor given by
$$\mathcal{A}(\omega)\subseteq B_\varepsilon(\omega),\quad\quad \text{for all}\,\, \varepsilon>0,$$
\noindent thus
$$\mathcal{A}(\omega)\subseteq B_0(\omega),$$
\noindent where
$$B_0(\omega):=\left\{(\sigma,\kappa)\in\mathcal{X}\,\,:\,\, \sigma+\kappa=0,\, \sigma\geq-(a+S^0)e^{\alpha z^*(\omega)}\right\}$$
\noindent is a tempered compact random absorbing set (see Figure \ref{figure2}) in $\mathcal{X}$.\newline
The following result provides information about the internal structure of the unique pullback random attractor.
\begin{proposition}\label{pe}
The unique pullback random attractor of system \eqref{7}-\eqref{8} consists of a singleton component given by $\mathcal{A}(\omega)=\{(0,0)\}$ as long as
\begin{equation}
\bar{D}>\mu(S^0)\label{ce}
\end{equation}
\noindent holds true.
\end{proposition}
\begin{proof}We would like to note that the result in this proposition follows trivially if $\sigma$ remains always positive ({\bf Case 1} in the proof of Theorem \ref{theorem1}) since in that case both $\sigma$ and $\kappa$ are positive and $\sigma+\kappa$ tends to zero when $t$ goes to infinity, thus the pullback random attractor is directly given by $\mathcal{A}(\omega)=\{(0,0)\}$.\newline
Due to the previous reason, we will only present the proof in case of there exists some $t^*>0$ such that $\sigma(t^*)=0$ which implies that $\sigma(t)<0$ for all $t>t^*$ whence $S(t)<S^0$ for all $t>t^*$ then $\mu(S)\leq\mu(S^0)$ for all $t>t^*$ since $\mu(s)=ms/(a+s)$ is an increasing function. Hence, from \eqref{8} we have
\begin{eqnarray}
\nonumber
\frac{d\kappa}{dt}&\leq&-(\bar{D}+\alpha z^*)\kappa+\frac{mS^0}{a+S^0}\kappa,
\end{eqnarray}
\noindent which allows us to state the following inequality
\begin{eqnarray}
\nonumber
\kappa(t;t^*,\theta_{-t}\omega,\kappa(t^*))&\leq&\kappa(t^*)e^{-\left(\bar{D}-\frac{mS^0}{a+S^0}\right)(t-t^*)-\alpha\int_{-t}^{t^*}z^*(\theta_s\omega)ds},
\end{eqnarray}
\noindent where the right side tends to zero when $t$ goes to infinity as long as \eqref{ce} is fulfilled, therefore the unique pullback random attractor is given by $\mathcal{A}(\omega)=\{(0,0)\}.$\newline
\qed
\end{proof}
\subsection{Existence of the pullback random attractor for the stochastic chemostat}
We have proved that the system \eqref{7}-\eqref{8} has a unique global solution $u(t;0,\omega,u_0)$ which remains in $\mathcal X$ for all $u_0\in\mathcal X$ and generates the RDS $\{\varphi_u(t,\omega)\}_{t\geq 0,\omega\in\Omega}$.\newline
Now, we define a mapping
$$\mathcal{T}: \Omega\times \mathcal X \longrightarrow \mathcal X$$
as follows
$$\mathcal{T}(\omega,\zeta)=\mathcal{T}(\omega,(\zeta_1,\zeta_2))= \left(\begin{array}{c}
T_1(\omega,\zeta_1) \\
T_2(\omega,\zeta_2)\\
\end{array}\right)=\left(\begin{array}{c}
(\zeta_1-S^0)e^{\alpha z^*(\omega)} \\
\zeta_2 e^{\alpha z^*(\omega)}\\
\end{array}\right)$$
whose inverse is given by
$$\mathcal{T}^{-1}(\omega,\zeta)= \left(\begin{array}{c}
S^0+\zeta_1 e^{-\alpha z^*(\omega)} \\
\zeta_2 e^{-\alpha z^*(\omega)}\\
\end{array}\right).$$
We know that $v(t)=(S(t),x(t))$ and $u(t)=(\sigma(t),\kappa(t))$ are related by (\ref{vcsigma})-(\ref{vckappa}). Since $T$ is a homeomorphism, thanks to Lemma \ref{lemmaconjugation} we obtain a conjugated RDS given by
\begin{eqnarray}
\nonumber \varphi_v(t,\omega)v_0 &:=& \mathcal{T}^{-1}(\theta_t \omega,\varphi_u(t,\omega)\mathcal{T}(\omega,v_0))
\\[1.3ex]
\nonumber &=& \mathcal{T}^{-1}\left(\theta_t\omega,\varphi_u (t,\omega)\left(\begin{array}{c}
(S(0)-S^0)e^{\alpha z^*(\omega)} \\
x(0) e^{\alpha z^*(\omega)}\\
\end{array}\right)\right)
\\[1.3ex]
\nonumber &=& \mathcal{T}^{-1}(\theta_t\omega,\varphi_u(t,\omega)u_0)
\\[1.3ex]
\nonumber &=& \mathcal{T}^{-1}(\theta_t\omega,u(t;0,\omega,u_0))
\\[1.3ex]
\nonumber &=& \left(\begin{array}{c}
S^0+\sigma(t)e^{-\alpha z^*(\theta_t \omega)} \\
\kappa(t) e^{-\alpha z^*(\theta_t \omega)}\\
\end{array}\right)
\\[1.3ex]
\nonumber &=& v(t;0,\omega,v_0)
\end{eqnarray}
which means that $\{\varphi_v(t,\omega)\}_{t\geq 0,\omega\in\Omega}$ is an RDS for our original stochastic system \eqref{5}-\eqref{6} whose unique pullback random attractor satisfies that $\widehat{\mathcal{A}}(\omega) \subseteq \widehat{B}_0(\omega)$, where
\begin{equation}
\widehat{B}_0(\omega):=\left\{(S,x)\in\mathcal{X}\,\,:\,\, S+x=S^0,\,\, S\geq-a\right\}.\label{absb}
\end{equation}
In addition, under \eqref{ce}, the unique pullback random attractor for \eqref{5}-\eqref{6} reduces to a singleton subset $\widehat{\mathcal{A}}(\omega)=\{(S^0,0)\}$, which means that the microorganisms become extinct.\newline
We remark that it is not possible to provide conditions which ensure the persistence of the microbial biomass even though our numerical simulations will show that we can get it for many different values of the parameters involved in the system, as we will present in Section \ref{nsfc}.
\section{Numerical simulations and final comments}\label{nsfc}
To confirm the results provided through this paper, in this section we will show some numerical simulations concerning the original stochastic chemostat model given by system \eqref{5}-\eqref{6}. To this end, we will make use of the Euler-Maruyama method (see e.g. \cite{maru} for more details) which consists of considering the following numerical scheme:
\begin{eqnarray*}
S_j &=& S_{j-1}+f(x_{j-1},S_{j-1})\Delta t+g(x_{j-1},S_{j-1})\cdot(W(\tau_j)-W(\tau_{j-1})),
\\[1.3ex]
x_j &=& x_{j-1}+\widetilde{f}(x_{j-1},S_{j-1})\Delta t+\widetilde{g}(x_{j-1},S_{j-1})\cdot(W(\tau_j)-W(\tau_{j-1})),
\end{eqnarray*}
\noindent where $f$, $g$, $\widetilde{f}$ and $\widetilde{g}$ are functions defined as follows
\begin{eqnarray*}
f(x_{j-1},S_{j-1}) &=& \left[(S^0-S_{j-1})D-\frac{mS_{j-1}x_{j-1}}{a+S_{j-1}}\right],
\\[1.3ex]
g(x_{j-1},S_{j-1}) &=& \alpha(S^0-S_{j-1}),
\\[1.3ex]
\widetilde{f}(x_{j-1},S_{j-1}) &=& x_{j-1}\left(\frac{mS_{j-1}}{a+S_{j-1}}-D\right),
\\[1.3ex]
\widetilde{g}(x_{j-1},S_{j-1}) &=& \alpha x_{j-1},
\\[1.3ex]
\end{eqnarray*}
\noindent and we remark that
$$W(\tau_j)-W(\tau_{j-1})=\sum_{k=jR-R+1}^{jR}dW_k,$$
where $R$ is a nonnegative integer number and $dW_k$ are $\mathcal{N}(0,1)-$distributed independent random variables which can be generated numerically by pseudorandom number generators.\newline
From now on, we will display the phase plane $(S,x)$ of the dynamics of our chemostat model, where the blue dashed lines represent the solutions of the deterministic (i.e., with $\alpha=0$) system \eqref{I1}-\eqref{I2} and the other ones are different realizations of the stochastic chemostat model \eqref{5}-\eqref{6}. In addition, we will set $S^0=1$, $a=0.6$, $m=3$ and we will consider $(S(0),x(0))=(2.5,5)$ as initial pair. We will also present different cases where the value of the dilution rate and the amount of noise change in order to obtain different situations in which the condition \eqref{ce} is (or is not) fulfilled.\newline
On the one hand, in Figure \ref{sim1} we take $D=3$ and we choose $\alpha=0.1$ (left) and $\alpha=0.5$ (right). In both cases, it is easy to check that $\bar{D}=1.5050$ (left), $\bar{D}=1.6250$ (right) and $\mu(S^0)=1.8750$ thus, thanks to Proposition \ref{pe}, we know that the microorganisms become extinct, as we show in the simulations.
\begin{figure}[H]
\begin{center}
\noindent\includegraphics[scale=0.28]{corr1.eps}\hspace{-0.5cm}\includegraphics[scale=0.28]{corr2.eps}
\caption{Extintion. $\alpha=0.1$ (left) and $\alpha=0.5$ (right)}
\label{sim1}
\end{center}
\end{figure}
On the other hand, in Figure \ref{sim2} we take $D=3$ but, in this case, $\alpha=1$ (left) and $\alpha=1.5$ (right). Then, it follows that $\bar{D}=2$ (left) and $\bar{D}=2.6250$ (right) then, since $\mu(S^0)=1.8750$ and thanks to Proposition \ref{pe}, we also obtain the extinction of the species.
\begin{figure}[H]
\begin{center}
\noindent\includegraphics[scale=0.28]{corr3.eps}\hspace{-0.5cm}\includegraphics[scale=0.28]{corr4.eps}
\caption{Extinction. $\alpha=1$ (left) and $\alpha=1.5$ (right)}
\label{sim2}
\end{center}
\end{figure}
Now, in Figure \ref{sim3} we will take $D=1.5$ and we choose $\alpha=0.1$ (left) and $\alpha=0.5$ (right). Then, we can check that $\bar{D}=1.5050$ (left), $\bar{D}=1.6250$ (right) and $\mu(S^0)=1.8750$ thus, although it is not possible to ensure mathematically the persistence of the microbial biomass, we can get it for the previous values of the parameters, as we can see in the simulations.
\begin{figure}[H]
\begin{center}
\noindent\includegraphics[scale=0.28]{corr5.eps}\hspace{-0.5cm}\includegraphics[scale=0.28]{corr6.eps}
\caption{Persistence. $\alpha=0.1$ (left) and $\alpha=0.5$ (right)}
\label{sim3}
\end{center}
\end{figure}
However, in Figure \ref{sim4} we take $D=1.5$, $\alpha=1$ (left) and $\alpha=1.5$ (right). Since condition \eqref{ce} holds true, it is not surprising to obtain the extinction of the microorganisms.
\begin{figure}[H]
\begin{center}
\noindent\includegraphics[scale=0.28]{corr7.eps}\hspace{-0.5cm}\includegraphics[scale=0.28]{corr14.eps}
\caption{Extinction. $\alpha=1$ (left) and $\alpha=1.5$ (right)}
\label{sim4}
\end{center}
\end{figure}
Finally, in Figure \ref{sim5} we will take $D=0.8$ and we will choose $\alpha=0.1$ (left) and $\alpha=0.5$ (right). It is easy to check that $\bar{D}=0.8050$ (left), $\bar{D}=0.9250$ (right) and $\mu(S^0)=1.8750$ thus, although it is not possible to guarantee mathematically the persistence of the species, since \eqref{ce} is not fulfilled, we can obtain it in this case.
\begin{figure}[H]
\begin{center}
\noindent\includegraphics[scale=0.28]{corr8.eps}\hspace{-0.5cm}\includegraphics[scale=0.28]{corr9.eps}
\caption{Persistence. $\alpha=0.1$ (left) and $\alpha=0.5$ (right)}
\label{sim5}
\end{center}
\end{figure}
\begin{remark}
We would like to mention that the fact that the substrate $S$ (or its corresponding $\sigma$) may take negative values does not produce any mathematical inconsistence in our analysis, in other words, our mathematical analysis is accurate to handle the mathematical problem. However, from a biological point of view, this may reflect some troubles and suggests that either the fact of perturbing the dilution rate with an additive noise may not be a realistic situation, or that we should try to use a some kind of switching system to model our real chemostat in such a way that when the dilution may be negative we use a different equation to model the system. This will lead us to a different analysis in some subsequent papers by considering a different kind of randomness or stochasticity in this parameter or designing a different model for our problem.\newline
On the other hand, it could also be considered a noisy term in each equation of the deterministic model in the same fashion as in the paper by Imhof and Walcher \cite{imhof}, which ensures the positivity of both the nutrient and biomass, although does not preserve the wash out equilibrium from the deterministic to the stochastic model (see e.g. \cite{CGLii} for more details about this situation). \end{remark}
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 4,533
|
{"url":"https:\/\/ncatlab.org\/nlab\/show\/temporal+algebra","text":"Temporal algebras\n\nIdea\n\nA temporal algebra is a modal algebra with two operations that reflect the future and past modal operators of a temporal logic.\n\nDefinitions\n\nDefinition\n\nA temporal algebra is a Boolean algebra, $(\\mathbb{B}, m_0,m_1)$, with operators, of type $2$, with the condition that the operators are conjugate:\n\n$m_0 x \\cdot y = 0$ if, and only if, $m_1 y \\cdot x = 0$.\n\nEquivalently (and equationally) this can be written as\n\n$x \\leq l_0 m_1 x \\cdot l_1 m_0 x$\n\nwhere $l_i$ is the dual of $m_1$.\n\nRevised on December 24, 2010 07:26:36 by Toby Bartels (75.88.75.53)","date":"2016-07-25 04:19:06","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 7, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.8926247954368591, \"perplexity\": 629.3006880880824}, \"config\": {\"markdown_headings\": false, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 5, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2016-30\/segments\/1469257824204.27\/warc\/CC-MAIN-20160723071024-00095-ip-10-185-27-174.ec2.internal.warc.gz\"}"}
| null | null |
We currently have a full time opening for a Grounds Specialist at Victory Memorial Park in Surrey, British Columbia. This is the opportunity to be part of the Dignity Memorial® provider network and grow your career in the funeral, cremation and cemetery services business. For us, there is no greater responsibility than celebrating each life like no other and making a difference in the lives of people we serve.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 4,516
|
{"url":"http:\/\/tex.stackexchange.com\/questions\/51002\/how-can-a-title-be-placed-for-a-group-of-pgfplots\/51008","text":"# How can a title be placed for a group of pgfplots?\n\nUsing the groupplots library in pgfplots, how can a title be set for a group of plots? The following code (basically from the manual) is an example layout. I don't want each plot to have a title, but rather a title be placed centered on top of the group.\n\n\\documentclass[crop,tikz]{standalone}\n\n\\usepackage{pgfplots}\n\\usepgfplotslibrary{groupplots}\n\n\\begin{document}\n\n\\begin{tikzpicture}\n\\begin{groupplot}[\ngroup style={\n{group size=2 by 2}},\nheight=3cm,width=3cm,\ntitle={Title}]\n\n\\nextgroupplot\n\\addplot coordinates {(0,0) (1,1) (2,2)};\n\\nextgroupplot\n\\addplot coordinates {(0,2) (1,1) (2,0)};\n\\nextgroupplot\n\\addplot coordinates {(0,2) (1,1) (2,1)};\n\\nextgroupplot\n\\addplot coordinates {(0,2) (1,1) (1,0)};\n\\end{groupplot}\n\\end{tikzpicture}\n\n\\end{document}\n\n\n-\n\nYou can use a TikZ node as a title for a quick solution.\n\n\\documentclass{article}\n\\usepackage{pgfplots}\n\\usetikzlibrary{calc}\n\\usepgfplotslibrary{groupplots}\n\\begin{document}\n\\begin{tikzpicture}\n\\begin{groupplot}[group style={group size=2 by 2},height=3cm,width=3cm]\n\\nextgroupplot[title=One]\n\\addplot coordinates {(0,0) (1,1) (2,2)};\n\\nextgroupplot[title=Two]\n\\addplot coordinates {(0,2) (1,1) (2,0)};\n\\nextgroupplot[title=Three]\n\\addplot coordinates {(0,2) (1,1) (2,1)};\n\\nextgroupplot[title=Four]\n\\addplot coordinates {(0,2) (1,1) (1,0)};\n\\end{groupplot}\n\\node (title) at ($(group c1r1.center)!0.5!(group c2r1.center)+(0,2cm)$) {THE Title};\n\\end{tikzpicture}\n\\end{document}\n\n\n-\nSince I don't intend on having individual plot titles, I used \\node (title) at ($(group c1r1.north)!0.5!(group c2r1.north)$) [above, yshift=\\pgfkeysvalueof{\/pgfplots\/every axis title shift}] to simulate how pgfplots places the title using axis description cs. \u2013\u00a0sappjw Apr 6 '12 at 18:30\n@sappjw Yep, that's a neat solution. \u2013\u00a0percusse Apr 6 '12 at 18:35\n\nYou can also use the \\matrix command to group plots together inside of a single tickzpicture, and add the title to the created tickzpicture:\n\n% Preamble: \\pgfplotsset{width=7cm,compat=1.9}\n\\begin{tikzpicture}\n\\pgfplotsset{small}\n\\matrix {\n\\begin{axis}\n\\end{axis}\n&\n% differently large labels are aligned automatically:\n\\begin{axis}[ylabel={$f(x)=x^2$},ylabel style={font=\\Huge}]\n\\end{axis}\n\\\\\n%\n\\begin{axis}[xlabel=$x$,xlabel style={font=\\Huge}]\n\n(the title isn't added here but you just need to add it on top of the matrix command) More information about the PGFPLOTS package in the PGFPlot Manual.","date":"2016-02-14 23:30:29","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 1, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.9719589948654175, \"perplexity\": 5336.205453298481}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.3, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2016-07\/segments\/1454702039825.90\/warc\/CC-MAIN-20160205195359-00033-ip-10-236-182-209.ec2.internal.warc.gz\"}"}
| null | null |
{"url":"https:\/\/gmatclub.com\/forum\/if-m-and-n-are-integers-is-mn-an-odd-integer-277649.html","text":"GMAT Question of the Day - Daily to your Mailbox; hard ones only\n\n It is currently 16 Feb 2019, 16:35\n\n### GMAT Club Daily Prep\n\n#### Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.\n\nCustomized\nfor You\n\nwe will pick new questions that match your level based on your Timer History\n\nTrack\n\nevery week, we\u2019ll send you an estimated GMAT score based on your performance\n\nPractice\nPays\n\nwe will pick new questions that match your level based on your Timer History\n\n## Events & Promotions\n\n###### Events & Promotions in February\nPrevNext\nSuMoTuWeThFrSa\n272829303112\n3456789\n10111213141516\n17181920212223\n242526272812\nOpen Detailed Calendar\n\u2022 ### Free GMAT Algebra Webinar\n\nFebruary 17, 2019\n\nFebruary 17, 2019\n\n07:00 AM PST\n\n09:00 AM PST\n\nAttend this Free Algebra Webinar and learn how to master Inequalities and Absolute Value problems on GMAT.\n\u2022 ### Free GMAT Strategy Webinar\n\nFebruary 16, 2019\n\nFebruary 16, 2019\n\n07:00 AM PST\n\n09:00 AM PST\n\nAiming to score 760+? Attend this FREE session to learn how to Define your GMAT Strategy, Create your Study Plan and Master the Core Skills to excel on the GMAT.\n\n# If m and n are integers, is mn an odd integer?\n\nAuthor Message\nTAGS:\n\n### Hide Tags\n\nMath Revolution GMAT Instructor\nJoined: 16 Aug 2015\nPosts: 6949\nGMAT 1: 760 Q51 V42\nGPA: 3.82\nIf m and n are integers, is mn an odd integer?\u00a0 [#permalink]\n\n### Show Tags\n\n30 Sep 2018, 23:39\n00:00\n\nDifficulty:\n\n45% (medium)\n\nQuestion Stats:\n\n66% (01:51) correct 34% (01:38) wrong based on 65 sessions\n\n### HideShow timer Statistics\n\n[Math Revolution GMAT math practice question]\n\nIf $$m$$ and $$n$$ are integers, is $$mn$$ an odd integer?\n\n1) $$m(n+1)$$ is even\n2) $$(m+1)n$$ is even\n\n_________________\n\nMathRevolution: Finish GMAT Quant Section with 10 minutes to spare\nThe one-and-only World\u2019s First Variable Approach for DS and IVY Approach for PS with ease, speed and accuracy.\n\"Only $149 for 3 month Online Course\" \"Free Resources-30 day online access & Diagnostic Test\" \"Unlimited Access to over 120 free video lessons - try it yourself\" Director Joined: 18 Jul 2018 Posts: 670 Location: India Concentration: Finance, Marketing WE: Engineering (Energy and Utilities) Re: If m and n are integers, is mn an odd integer? [#permalink] ### Show Tags 30 Sep 2018, 23:50 From statement 1: m(n+1) is even. Let's consider numbers for easy simplification. m=1 n=3 then m(n+1) is even and mn = 3. Odd If m=2 n=4 then m(n+1) is even but mn = 8. Even. Insufficient. From statement 2: Same case as above. Insufficient. Combining both gives 2mn+m+n as even. 2mn will always be even. So m+n has to be even. If m=1 n=3. m+n is even Or If m=2 n=4. m+n is even. But mn will be even and odd. Hence Combining also doesn't give a unique answer. E is the answer. _________________ Press +1 Kudo If my post helps! CEO Status: GMATINSIGHT Tutor Joined: 08 Jul 2010 Posts: 2775 Location: India GMAT: INSIGHT Schools: Darden '21 WE: Education (Education) Re: If m and n are integers, is mn an odd integer? [#permalink] ### Show Tags 01 Oct 2018, 01:59 MathRevolution wrote: [Math Revolution GMAT math practice question] If $$m$$ and $$n$$ are integers, is $$mn$$ an odd integer? 1) $$m(n+1)$$ is even 2) $$(m+1)n$$ is even Question: Is mn odd? Question: Are each if m and n odd? Statement 1: $$m(n+1)$$ is even i.e. either m is even and n+1 is even OR m is odd and n+1 is even or m and n+1 both are even i.e. either m is even and n is odd OR m is odd and n is odd i.e. mn may be even or mn may be Odd NOT SUFFICIENT Statement 2: $$(m+1)n$$ is even i.e. either n is even and m+1 is even OR n is odd and m+1 is even or m and n+1 both are even i.e. either n is even and m is odd OR n is odd and m is odd i.e. mn may be even or mn may be Odd NOT SUFFICIENT Combining the two statements i.e. either n is even and m is odd OR n is odd and m is odd i.e. mn may be even or mn may be Odd NOT SUFFICIENT Answer: Option E _________________ Prosper!!! GMATinsight Bhoopendra Singh and Dr.Sushma Jha e-mail: info@GMATinsight.com I Call us : +91-9999687183 \/ 9891333772 Online One-on-One Skype based classes and Classroom Coaching in South and West Delhi http:\/\/www.GMATinsight.com\/testimonials.html ACCESS FREE GMAT TESTS HERE:22 ONLINE FREE (FULL LENGTH) GMAT CAT (PRACTICE TESTS) LINK COLLECTION CEO Joined: 11 Sep 2015 Posts: 3431 Location: Canada Re: If m and n are integers, is mn an odd integer? [#permalink] ### Show Tags 01 Oct 2018, 06:03 Top Contributor MathRevolution wrote: [Math Revolution GMAT math practice question] If $$m$$ and $$n$$ are integers, is $$mn$$ an odd integer? 1) $$m(n+1)$$ is even 2) $$(m+1)n$$ is even Some important rules: #1. ODD +\/- ODD = EVEN #2. ODD +\/- EVEN = ODD #3. EVEN +\/- EVEN = EVEN #4. (ODD)(ODD) = ODD #5. (ODD)(EVEN) = EVEN #6. (EVEN)(EVEN) = EVEN Target question: Is mn an odd integer? Given: m and n are integers Statement 1: m(n+1) is even Let's test some values. There are several values of m and n that satisfy statement 1. Here are two: Case a: m = 1 and n = 1. Notice that m(n + 1) = 1(1+1) = 2, which is even. In this case, mn = (1)(1) = 1. So, the answer to the target question is YES, mn IS odd Case b: m = 2 and n = 2. Notice that m(n + 1) = 2(2+1) = 6, which is even. In this case, mn = (2)(2) = 4. So, the answer to the target question is NO, mn is NOT odd Since we cannot answer the target question with certainty, statement 1 is NOT SUFFICIENT Statement 2: (m+1)n is even Let's test some values (again). IMPORTANT: When testing values a second time, check to see if you ran reuse either of the cases you used in statement 1. If we do that here, we'll see that we can reuse both cases: Case a: m = 1 and n = 1. Notice that (m+1)n = (1+1)1 = 2, which is even. In this case, mn = (1)(1) = 1. So, the answer to the target question is YES, mn IS odd Case b: m = 2 and n = 2. Notice that (m+1)n = (2+1)2 = 6, which is even. In this case, mn = (2)(2) = 4. So, the answer to the target question is NO, mn is NOT odd Since we cannot answer the target question with certainty, statement 1 is NOT SUFFICIENT Statements 1 and 2 combined Notice that we were able to use the same counter-examples to show that each statement ALONE is not sufficient. So, the same counter-examples will satisfy the two statements COMBINED. In other words, Case a: m = 1 and n = 1. So, the answer to the target question is YES, mn IS odd Case b: m = 2 and n = 2. So, the answer to the target question is NO, mn is NOT odd Since we cannot answer the target question with certainty, the combined statements are NOT SUFFICIENT Answer: E Cheers, Brent _________________ Test confidently with gmatprepnow.com Math Revolution GMAT Instructor Joined: 16 Aug 2015 Posts: 6949 GMAT 1: 760 Q51 V42 GPA: 3.82 Re: If m and n are integers, is mn an odd integer? [#permalink] ### Show Tags 03 Oct 2018, 01:23 => Forget conventional ways of solving math questions. For DS problems, the VA (Variable Approach) method is the quickest and easiest way to find the answer without actually solving the problem. Remember that equal numbers of variables and independent equations ensure a solution. The first step of the VA (Variable Approach) method is to modify the original condition and the question. We then recheck the question. Modifying the question: mn is odd only when both m and n are odd. So, the question asks if both m and n are odd. Conditions 1) and 2), when applied together, tell us that either both m and n are odd numbers or both m and n are even numbers. Since we don\u2019t have a unique solution, both conditions, taken together, are not sufficient. Therefore, E is the answer. Answer: E _________________ MathRevolution: Finish GMAT Quant Section with 10 minutes to spare The one-and-only World\u2019s First Variable Approach for DS and IVY Approach for PS with ease, speed and accuracy. \"Only$149 for 3 month Online Course\"\n\"Free Resources-30 day online access & Diagnostic Test\"\n\"Unlimited Access to over 120 free video lessons - try it yourself\"\n\nRe: If m and n are integers, is mn an odd integer? \u00a0 [#permalink] 03 Oct 2018, 01:23\nDisplay posts from previous: Sort by","date":"2019-02-17 00:35:18","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 1, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.38135620951652527, \"perplexity\": 13506.741449829404}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2019-09\/segments\/1550247481249.5\/warc\/CC-MAIN-20190216230700-20190217012700-00177.warc.gz\"}"}
| null | null |
Vitex benuensis est une espèce de plantes de la famille des Lamiacées et du genre Vitex, endémique du Cameroun.
Étymologie
L'épithète spécifique benuensis fait référence à la rivière Bénoué sur les rives de laquelle les premiers spécimens furent découverts à Lagdo par Carl Ludwig Ledermann lors de son expédition au Cameroun en 1909.
Description
C'est un arbuste ou petit arbre.
Notes et références
Bibliographie
Liens externes
Espèce d'Angiospermes (nom scientifique)
Lamiaceae
Flore endémique du Cameroun
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 9,452
|
Domestic offsets in the American Power Act: preserving the integrity?
This post was co-written with Sasha Lyutse. NRDC staff have posted assessments of a number of key elements of the Kerry-Lieberman American Power Act discussion draft. Here, we'll take a closer look at the domestic offsets program. I'll give a brief overview of the program, as outlined in Part D of the bill (Sec. 731-741), […]
Jake Schmidt
This post was co-written with Sasha Lyutse.
NRDC staff have posted assessments of a number of key elements of the Kerry-Lieberman American Power Act discussion draft. Here, we'll take a closer look at the domestic offsets program. I'll give a brief overview of the program, as outlined in Part D of the bill (Sec. 731-741), highlight key strengths, and discuss several areas where we believe further improvements are needed.
The draft bill provides a solid foundation for an environmentally sound offset system, but there some important modifications are needed to ensure that the system meets key standards of environmental integrity.
Strong Safeguards for Offsets are Critical. Before digging into the details of the draft bill's offset provisions, we wanted to first step back and remind ourselves why it is so critical that the rules for offsets are correctly designed. Every offset used for compliance represents one ton of reduction that isn't achieved by regulated companies/facilities within their own operations (or within the cap). So if we don't get offsets right then the atmosphere doesn't see the intended reductions. And since we don't have any spare space in the atmosphere, it is critical that we ensure that every offset ton represents the same environmental benefit as the "other tons"—that "a ton is a ton", not a half-ton or a three-quarter ton. After all, the last thing we need is "subprime offsets" as it will lose confidence of the public and policymakers, increase global warming pollution, and lead to the loss of a low-cost compliance option if confidence in their integrity collapses.
Ensuring that the offset system is environmentally credible requires good answers both in the legislation and in the ensuing rules on: who develops the rules; whether those rules are guided by credible science; how the rules are applied to ensure the environmental integrity of each offset; what kinds of mechanisms are included to ensure third-party oversight, public scrutiny, and regular adjustment of the program.
How does the draft bill handle offsets? As Dan Lashof noted in his post, the American Power Act allows covered entities to use up to 2 billion tons per year of "offsets" (emissions reduced or carbon sequestered by sources not covered by the bill's pollution limits)—split evenly between domestic and international offsets (see my colleagues post for how international offsets are handled)—for compliance with the emissions cap.
The bill establishes criteria, administered by EPA (or, for domestic farm and forestry offsets, by the Department of Agriculture in consultation with EPA), for crediting offsets, so as to assure that offset credits are earned only for real, verifiable and permanent actions that would not happen anyway. Strong implementation of these principles is essential to ensuring the environmental integrity of the offset system and the cap (as Dan Lashof discussed).
In addition, the bill calls on EPA and USDA to jointly create a Greenhouse Gas Emission Reduction and Sequestration Advisory Committee to advise both agencies on the establishment and implementation of the domestic offsets program (a separate Committee is established to do the same for international offsets). The Advisory Committee is to make recommendations on which projects should be eligible as offsets, provide relevant scientific data, recommend methodologies for each project type, and issue regular reports to EPA, USDA and the general public (more on this below) on how the environmental integrity of offset projects can be ensured. In addition, every 5 years, the Advisory Committee must provide a public scientific review of the offset program, including a review of methodologies being used, offset project verification reports and audits, the net emissions impact of the program (i.e. a "true up" to see if offsets are indeed delivering their promised reductions) and recommend changes to improve the program's environmental performance.
The bill creates a very long list of "eligible project types" that the Advisory Committee must consider for offset eligibility. In developing program regulations, EPA and USDA are directed to give priority to those projects with well-established methodologies and must take into consideration the recommendations of the Advisory Committee. In establishing a list of eligible project types, the agencies must provide an explanation if their list differs from the list recommended by the Advisory Committee. In addition, the bill lays out a series of quality standards, including:
requirements to set baselines using conservative estimates of "business-as-usual";
guidance on determining if a project meets the additionality criteria (i.e., whether a reduction would have happened anyway); and
obligation to account for the risk of leakage (i.e. does the project cause emissions simply to shift to another location); and
rules to address reversals (for example, if a forest burns down and the carbon reduced "goes up in smoke" in a later period), with minimum mechanisms that ensure the environment is compensated if a reversal takes place, as well as penalties for intentional reversals.
It also establishes crediting periods for sequestration offsets on agricultural and forestry land, requires third-party verification and regular auditing of offset projects, and lays out a process for creating an early offsets supply.
So that is what it does, but what are its strengths? As we've discussed above, much of the overall structure of the offsets program in the draft bill is designed to guide the implementation of strong rules to ensure the environmental credibility of the system.
1. Having science (not politics) drive key offset designs. The independent Advisory Committee is tasked with making scientifically driven recommendations. Ensuring that science is at the heart of the offset system requires that this body is effectively staffed, operates on the basis of hard science (not political science), and that the Agencies appropriately handle its recommendations. Providing outside scrutiny (as we'll discuss below) is critical to providing an extra scientific check.
2. Placing environmental integrity at the heart of the system. The draft bill establishes a commitment to assuring environmental performance. It requires that the rules lead to an offset credit that is equivalent to a ton of emission reduction in the sectors with firm limits on their pollution. It also emphasizes the need for standardized methodologies (as opposed to case-by-case review) to address concerns about additionality, measurement, leakage accounting and discounting for uncertainty. In addition, the Agencies are directed to use conservative methodologies that provide a "science-based margin of safety to ensure the emissions integrity".
3. Reviews, Audits, Revisions. The
draft includes annual randomized performance audits of offset projects, offset credits and the auditors themselves, creating essential accountability for environmental integrity in the offset market. And it allows for the removal of offset project types that are found to fall short in delivering environmentally sound reductions. These regular checks on the system will be critical to provide ongoing oversight of whether the program is developing offsets which help us address global warming pollution (not make it worse).
3. Public Transparency. The bill requires that EPA and USDA establish a process to accept and respond to public comments on the program rules, as well as procedures for public appeal and review of individual project approvals. In addition, EPA and USDA decisions and the information relevant to making their decisions is to be made publicly available. Transparency in rulemaking, public availability of information, and strong public accountability are key to ensuring that not only are regulations written well, but that they are also implemented well.
The draft also includes an important modification over previous proposals. It provides strong guidance for cooperation between the EPA and USDA with respect to offsets in the domestic agriculture and forestry sectors. USDA is named the lead agency on all farm and forestry offsets, but it must carry out its work "in consultation and coordination" with EPA. Because EPA is charged with guaranteeing the overall environmental performance of the bill, it is critical that EPA have a clear and established role in ensuring consistency of the overall offsets program.
But there are also key areas that need to be strengthened, including the following:
1. Eliminating/refining the list of eligible offsets. The bill includes a long list of project types which are supposed to be allowed to generate offsets. Many of these project types are untested and inherently difficult or impossible to implement in an environmentally sound manner (in fact some of these have never been applied anywhere in the world under a credible system). The list of presumptively eligible project types should therefore be eliminated or shortened to include only those most likely to produce high quality reductions.
2. Tightening the requirements for carbon reversals. Offset credits issued for sequestering carbon in soils or forests only remain valid as long as the carbon stays out of the atmosphere. Offset purchasers (not the project developer) should be ultimately responsible to make up tons lost through reversal of the carbon storage in agriculture and forestry projects (e.g. due to renewed plowing or tree cutting) if they are not replaced through a buffer. Responsibility to monitor and compensate for reversals should also extend well beyond the crediting period.
3. Ensuring that only good early offsets count. While it is important not to penalize farmers and other landowners that have taken action to reduce emissions prior to the implementation of the offset program, compensating early actors must be done in a way that ensures the environmental integrity of the emissions cap. In the case of early actors who received offset credit through existing state or voluntary program, only credits obtained through programs that use project standards, methodologies and protocols established through a process that involves both public transparency and peer review should qualify for the federal program. Early actors who have implemented qualifying projects or practices on their land in the years prior to the offsets program, but have not registered those practices or projects with any existing program, should be compensated via a set aside fund dedicated to the domestic agriculture and forestry sectors (as discussed below).
4. Providing a dedicated source of funding, separate from offsets, for "good" but not "creditable" activities. Many farm and forestry practices present difficulties in terms of meeting rigorous additionality standards and are more appropriately compensated outside the offsets market (so as not to "bust the cap"). In addition, there are examples of practices believed to deliver important environmental benefits that scientists simply cannot yet measure or verify with enough certainty to meet offset criteria, like projects in coastal marine areas and certain agricultural practices. A domestic agriculture, forestry, and coastal marine program would allow EPA and USDA to nonetheless incentivize these practices, all the while building important scientific and technical knowledge that would allow them to possibly be included in the offsets program in the future. Such a program could also be used for early actions which reduce emissions, but aren't eligible under the early offset system. The addition of a robust, well-funded and well-regulated program would greatly enhance our ability to reduce emissions in certaub sectors and help improve the environmental integrity of the offsets program.
Offsets will almost certainly be one of the key elements of a climate and energy bill under debate in the coming weeks and months. Of course passage of the strong offset provisions in a climate and energy bill isn't the final say, as ensuring the program's integrity will require the responsible agencies to properly implement these rules and provide appropriate due diligence.
While far from perfect, the domestic program established in the bill has many strong features which must be retained to ensure that the offset system preserves the integrity of the emissions limits. Key improvements can and should be make to enhance the environmental integrity of the offsets market and help ensure that offsets are helping (not hurting) our efforts to address global warming.
Sustainable seafood takes center stage at Seattle dinner theater
Darby Minow Smith Food
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 5,727
|
The global spread of the coronavirus is disrupting travel. Stay up to date on the science behind the outbreak>>
City life: Tallinn
A medieval treasure trove, with Soviet heritage and flashes of sleek Scandinavian design, the Estonian capital is one of Europe's most striking cities
By Daniel Allen
Published 9 Apr 2019, 00:18 BST, Updated 12 Jul 2021, 10:14 BST
An aerial view of the Old Town, including St Nicholas' Church tower and Toompea
Photograph by Getty
All it takes is a visit to the sauna in Tallinn to discover the city's Communist-era heritage — caviar-laden blinis, pickled mushrooms and endless vodka toasts are still the order of the day in these steamy institutions. Yet those who think 'former Soviet' means monolithic architecture and an absence of joie de vivre should think again — the Estonian capital is vibrant, upbeat and unabashedly romantic.
Tallinn, first and foremost, is a medieval masterpiece. With its well-preserved melange of Hanseatic buildings, sublime Russian churches, conical red roofs and crenellated battlements, the Old Town bewitches and bewilders in equal measure. Wander the streets here and marvel at the strange and wonderful architecture, the magnificently archaic clocks and the array of beautiful stained glass windows. Tallinn has one of Europe's most complete historic centres, and there's very little ugly modernity here to break the spell, while after dark the flickering candlelight that still illuminates merchant houses only heightens the magic.
But for all its dramatic past — its has been ruled by the Danes, Swedes and Germans — this is also a place with an eye on the future. The city's creative Kalamaja district is knee-deep in fashionable hangouts, fancy hotels and restaurants boast minimalist, Scandinavian decor, and its tech-savvy citizens were among the first to use their phones as debit cards. Home to the headquarters of Skype, the Estonian capital is a surfer's paradise, and we aren't talking about waves in the nearby Gulf of Finland.
Of course, this interaction between the antiquated and the modern is being played out in cities the world over. But it's hard to think of many places where old and new sit together so gracefully. Throw in several leafy parks and the Pirita Promenade, where the Baltic Sea breeze slaps waves at your feet, and you've got a very strong first impression.
And those who venture into Karjavarava Square can rub the buttons of the bronze chimney sweep statue here for good fortune. Whether it works or not, you can still count yourself lucky to be in one of Europe's finest small cities.
Kalma Sauna: Just like their Finnish and Russian neighbours, Estonians are sauna addicts. A grand neoclassical complex with art deco flourishes, the Kalma opened in 1928, making it Tallinn's oldest public sauna. Self-flagellation with birch branches (said to be good for the health) is optional.
Seaplane Harbour: A great venue for kids, this interactive maritime museum is housed in a converted seaplane hangar. Inside, view the British-built Lembit submarine, a Short Type 184 seaplane, the wreck of the Maasilinn (the oldest sunken ship discovered in Estonian waters), and a cornucopia of other naval artefacts and equipment.
Raeapteek: The fascinating Raeapteek (Town Hall Pharmacy) was founded in the 15th century and is one of the oldest continuously running pharmacies in Europe. While it sells modern drugs, time-honoured panaceas on display (not for sale) include dried frogs' legs, pikes' eyes, burned bees and rabbit hearts, plus a marzipan remedy for heartbreak.
Kalev Chocolate Shop & Maiasmokk Cafe: The perfect venue for a mid-morning coffee and second breakfast, the Maiasmokk ('Sweet Tooth') cafe is housed inside the Kalev Chocolate Shop. The oldest operational cafe in Estonia, it's been an Old Town institution since 1864. Be sure to check out the Kalev Marzipan Museum Room, where visitors can learn about the history of marzipan (Estonia claims to have invented it) and watch it being made.
Kalamaja District: Situated close to the Balti Jaam train station, Kalamaja ('fish house') was once home to fishermen, fishmongers and shipwrights. The district's wooden houses have a faded charm, while the area also has something of a bohemian vibe to it, thanks to the numerous cafes and restaurants that have recently sprung up.
Alexander Nevsky Cathedral: This ornate Orthodox cathedral was built in 1900, when Estonia was part of the tsarist Russian empire. Its onion domes dominate Toompea, a hill in the city centre, while its mosaic- and icon-filled interior is spectacular.
Old Town: With its narrow, cobblestoned passageways, soaring spires and gorgeous Hanseatic architecture, Old Town (Vanalin, in Estonian) is the capital's top attraction. A UNESCO World Heritage Site, it's also home to many bars and cafes. Climb St Olaf's Church tower for a great view of the surrounding medieval roofs cape.
Nomme Market: When the weather is good, the outdoor stalls groan under the weight of pickled vegetables, forest berries, mushrooms, sausages, pastries and sides of smoked meat. Dating back to 1908, it's a vivid insight into Tallinn's culinary culture.
Pikk Tanav: The shops on Pikk Tanav are stocked with traditional Estonian clothing, which tends to be woollen, knitted and covered in interesting patterns. This is the place for beautifully handcrafted sweaters, mittens and socks. The best outlets have a sign outside saying Eesti Kasitoo ('Estonian handicraft').
Vana Tallinn: This sweet, dark brown, rum-based liqueur, created in the 1960s by the Estonian company Liviko, is flavoured with citrus oil and various spices, including cinnamon and vanilla.
Balti Jaam Market: Visitors looking for a souvenir with a difference should head to the train station flea market. Stallholders at the Balti Jaam Market peddle a wide range of items — everything from clothes and shoes to old Soviet memorabilia. The market opens at around 10am. Get there early for the choicest picks.
Kolmas Draakon: For lunch with a historical twist, check out Kolmas Draakon, a medieval-themed tavern, housed in the Town Hall. There's no table service, it's wholly candlelit and diners get free pickles. The menu includes elk broth, six types of pastry, game sausages and flagons of mead. You also have to clean the table yourself.
Moon: Run by renowned Estonian chef Roman Zastserinski, the award-winning Moon ('Poppy' in Estonian) is one of Tallinn's best and most reasonably priced restaurants. It offers a fusion of Russian flavours and Estonian produce, with classics like blinis and Siberian pelmeni (dumplings).
Mon Repos: The villa housing Mon Repos was an elegant restaurant and casino back in the 1920s. It recently reopened as an intimate, upscale restaurant. Head chef Vladislav Djatšuk offers a menu of historic recipes recreated with a modern twist. The pate with green-apple jelly, bouillabaisse and sturgeon comes highly recommended.
City Hotel Tallinn: Located at the foot of Toompea Hill, just a short walk from Toompea Castle and Alexander Nevsky Cathedral, this is a great choice for the budget traveller. There's free wi-fi and a couple of ground-floor lounges, while a buffet breakfast is available for an extra charge.
Savoy Boutique Hotel: A short walk from the Niguliste Museum, Estonian Theatre and Music Museum and Old Town, the five-star Savoy Boutique Hotel is a good option for those looking to explore local culture. With only 44 rooms and exquisite art deco interiors, an intimate vibe is guaranteed.
Hotel Telegraaf: Just a few metres from Tallinn's main square, the opulent Hotel Telegraaf is the best base for those wishing to enjoy the romance and historical charm of the city's medieval heart. Older rooms have wooden floors and king-sized beds, while all guest rooms offer stunning views over the Old Town.
Get your bearings: If the thought of getting lost among the passages and alleyways of the Old Town fills you with panic, fear not — free guided tours of Tallinn leave at noon, and also at 10am and 3pm from May to September, from in front of the city tourist information centre at Niguliste 2.
Central Market: If you're in the mood for a hearty snack, visit the Keskturg (Central Market), where local bakers offer traditional East European lavash bread — a cheap and tasty lunch with local cheese.
Soviet insights: For an off-the-wall and eye-opening look at life in Estonia under the Soviet occupation, visit the quirky KGB Museum in Sokos Hotel Viru.
Philly Joe's Jazz Club: Tallinn's only daily-opening jazz venue, Philly Joe's is what all great underground jazz clubs should be — dark, intimate and a temple of smooth grooves. There's live music every day, a great selection of cocktails and micro brews, and the staff are some of the friendliest around.
Valli Baar: If there's one Tallinn watering hole that everyone should check out, it's Valli Baar. This legendary throwback boasts a 1970s-style drinking environment, with an interior that's actually under cultural heritage protection. The (in)famous house shot is the potent, spicy millimallikas ('jellyfish'), a lethal concoction of sambuca, tequila and tabasco sauce. A great way to start the evening.
Club Hollywood: Housed in a former Russian cinema, this trendy Tallinn institution on the edge of the Old Town can accommodate up to 1,500 people over two floors. Some of Estonia's best DJs play here regularly, although things don't really get going till after midnight.
Getting there & around
Ryanair (four weekly flights from Stansted), EasyJet (14 weekly flights from Gatwick) and British Airways (two weekly flights from Heathrow) all fly direct to Tallinn from London.
Average flight time: 2h 50m.
Tallinn city centre, including the Old Town, is compact and best negotiated on foot, while destinations beyond are served by a good bus, trolley and tram network. Tallinn Airport is under three miles from the centre — bus routes 2 and 65 stop here. Taxis are reasonable.
The best time to visit Tallinn and Estonia is late spring or summer, when average temperatures of around 18C allow for pleasant walking (Tallinn can get busy in July and August). Winters are long, dark and can be bitterly cold, with minus temperatures common, although Christmas markets and a blanket of snow make the Old Town undeniably romantic. Always pack for changeable weather.
visittallinn.ee/eng
Berlitz: Tallinn Pocket Guide. RRP: £5.99.
EasyJet has seven days in Tallinn from £540 per person, including two-star hotel accommodation and return flights from Gatwick. Excludes baggage and transfers.
Published in the September 2017 issue of National Geographic Traveller (UK)
Twitter | Facebook | Instagram | Flipboard
Six unusual tours of Rome that reveal a different side to the city
City life: Kyoto
Like a local: Rotterdam
Eat: Berlin
Rooms under £100: Verona
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 5,284
|
Press Speaker Resources
NBA All-Time Leading Scorer (38,387)
Presidential Medal of Freedom Recipient
6x NBA MVP
19x NBA All-Star
New York Times Bestselling Author of 15 books
Award-Winning Film Producer/Writer
2x NAACP Image Award Winner (What Color Is My World & On the Shoulders of Giants)
Medal of Freedom Recipient
3x Columnist of the Year from the National Arts and Journalism Awards
Political Activist & Educational Advocate
Features in HBO's most watched documentary of all time – Karreem: Minority of One
Kareem Abdul-Jabbar is a global icon that changed the game of professional basketball and went on to become a celebrated author, filmmaker, and ambassador of education.
For All Press Inquiries:
info@iconomy.com
Request Fees and Availability
Download Press Kit (ZIP)
Tweets by kaj33
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 590
|
Warning: Not to be used during pregnancy. Not to be used while nursing.
Black Cohosh Root - 180 capsules Black Cohosh Root Organic - Cut/Sifted Black Cohosh 1 oz. - Herb Pharm Black Cohosh Root - Glycerine Extract 2 oz.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 1,527
|
{"url":"https:\/\/www.physicsforums.com\/threads\/a-ballistic-pendulum.215822\/","text":"# A Ballistic Pendulum\n\n[SOLVED] A Ballistic Pendulum\n\n## Homework Statement\n\nA bullet of mass 0.01 kg moving horizontally strikes a block of wood of mass 1.5 kg which is suspended as a pendulum. The bullet lodges in the wood, and together they swing upward a distance of 0.40 m. What was the velocity of the bullet just before it struck the wooden block? The length of the string is 2 meters.\n\n## Homework Equations\n\nMomentum Conservation:\n\n$$m_{a}v_{a1}+m_{b}v_{b1}=(m_{a}+m_{b})*v_{2}$$\n\nEnergy Conservation:\n\n$$1\/2mv^{2}=mgy$$\n\n## The Attempt at a Solution\n\nSince the block is at rest before the bullet hit, if we use the momentum conservation formula, we only have to deal with the initial speed of the bullet. The resultant formula is\n\n$$m_{b}v_{b}=(m_{w}+m_{b})*v_{2}$$\n\nOnce the bullet is embedded in the block, it will have a potential energy of zero and a kinetic energy of\n\n$$K=1\/2(m_{b}+m_{w})v_{2}^{2}$$\n\nand the block with the bullet travels up a height .40m, and comes to a rest. At this point, the block\/bullet unit has a kinetic energy of zero, and a potential energy of\n\n$$U=(m_{b}+m_{w})gy$$\n\nUsing energy conservation we get:\n\n$$1\/2(m_{b}+m_{w})v_{2}^{2}=(m_{b}+m_{w})gy$$\n\nwe can solve for velocity here and get the speed after the bullet hit the block. The masses should cancel out, leaving:\n\n$$v_{2}=\\sqrt{2gy}$$\n\nWe sub in this expression for v back into the first momentum formula, getting:\n\n$$m_{b}v_{b}=(m_{b}+m_{w})*\\sqrt{2gy}$$\n\nsolving for the initial velocity $$v_{b}$$, we get $$v_{b}=(m_{b}+m_{w})\/m_{b}*\\sqrt{2gy}$$\n\nAt this point I just plugged and chugged, using the given values in the problem and came up with 423 m\/s, but it turned out to be the wrong answer. Can anyone help me figure out what I did wrong? Much thanks in advance!\n\n#### Attachments\n\n\u2022 7.1 KB Views: 654\n\n## Answers and Replies\n\nIf the bullet lodged into the wood, you cant use conservation of Kinetic Energy. Rather than that...try using what you know about centripetal forces and acceleration.\n\nHow could you use centripetal force if the pendulum isn't moving in a circle?\n\nDoc Al\nMentor\nAt this point I just plugged and chugged, using the given values in the problem and came up with 423 m\/s, but it turned out to be the wrong answer. Can anyone help me figure out what I did wrong? Much thanks in advance!\nYour solution looks good to me. Who says it's wrong?\n\nOur assignments are given online and it grades it the instant you input an answer. In this case, the answers are given in multiple choice, so I'm usually cautious to choose an answer as the chances to get it right are obviously limited to the amount of choices available. But I did choose the 423 m\/s and it was shown as incorrect.\n\nHere are my answer choices:\n\n250 m\/s\n423 m\/s\n66.7 m\/s\n646 m\/s\n366 m\/s\n\nAnd yes, I felt the same way too, doc. I'm pretty sure I didn't fumble on any of the calculations and that I used the correct formula to derive the speed, unless \"the initial speed of the bullet\" and \"the speed of the bullet just before it hits the block\" mean totally different things.\n\nWait...I see what you did there...nevermind...everything looks to be alright...dont know whats wrong with your answer.\n\nLast edited:\nDoc Al\nMentor\nFor inelastic collisions, energy is lost in the collision due to deformation. Because of that, Ki = Pf + Ediss...To avoid dealing with the dissipated energy, the best thing to do would be to use the conservation of momentum. What you do is set Pf = Ka, in which Ka is the energy right after collision (because gravitational energy is interconvertable with kinetic energy). So it would be (1\/2)*(Ma + Mb)*(v2)^2 = (Ma+Mb)gh. Solve for v2, plug it into your momentum equation, and then solve for v1.\nThat's exactly what Arejang did!\n\nDoc Al\nMentor\nI'm pretty sure I didn't fumble on any of the calculations and that I used the correct formula to derive the speed, unless \"the initial speed of the bullet\" and \"the speed of the bullet just before it hits the block\" mean totally different things.\nI doubled checked your arithmetic; I would have chosen the same answer.\n\nI just guessed all the answers and the right one was 66.7 m\/s......What the heck?\n\nSorry, this problem has been irking me to no end. Would the length of the string, 2m, play any part of this problem? That would be the only factor I could see that might alter the answer somewhat. But if that were the case, then I don't know how else to approach this problem.\n\nDoc Al\nMentor\nWould the length of the string, 2m, play any part of this problem?\nNot that I can see. Your solution is perfectly correct.\n\nthanks, I'm going to go ahead and marked this solved for the time being. If I find anything else, I'll post it back up again. Thanks for all your help!\n\nDoc Al\nMentor\nIf this homework is graded (or even if it's not), make sure your instructor sees your solution.\n\nI plan on doing so when I see him tomorrow.","date":"2021-01-22 22:30:25","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\": 0.7254629731178284, \"perplexity\": 561.9228210059034}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.3, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2021-04\/segments\/1610703531429.49\/warc\/CC-MAIN-20210122210653-20210123000653-00669.warc.gz\"}"}
| null | null |
Beniamino Bufano
by Amy from Rancho Santa Margarita
Beniamino Bufano in 1935.
Beniamino Benvenuto Bufano (Benny) is my great-great uncle. He is also a famous artist who was born in 1898 and died in 1970. He migrated to America from Italy when he was three years old. He moved to San Francisco, California in 1915, which is where most of his sculptures are.
Bufano The Penguins, Sidney Walton Square.
Benny is my hero because when I traveled to San Francisco and looked at his sculptures, I was in awe at the fact that I was related to someone so famous. His art inspired me to become more artistic.
Beniamino Bufano's Madonna protects the children of the world, Great Meadow, Upper Fort Mason.
I think my hero influenced his community when he was alive because he taught at UC-Berkeley and the California College of Arts and Crafts in Oakland. He didn't agree with the traditional way of teaching. Instead, Benny taught the new age of art.
This St. Francis by Beniamino Bufano is made from melted guns. Located at City College, Ocean & Phelan Avenues.
If I could speak to Benny, I would tell him how his art inspired me and how honored I am to be related to him. If I were to be a hero, I would want to be a bass guitar player in a famous band and give money to charities. If that doesn't happen--which it probably won't--I want to be an art teacher and inspire students.
Beniamino Bufano's Sun Yat-Sen, St. Mary's Square.
Benny is famous for his sculptures, but he also created poems, drawings, prints, paintings, and mosaics. He traveled all around the world; to New York, France, Italy, China, Cambodia, and Malaysia. While he was in China, Benny learned the Chinese porcelain glazing technique, which he applied to his terracotta sculpture. Benny married Virginia Howard in 1925. This was also the year Benny opened his first gallery.
Benny paved the road for new age art. He was a sculptor, a world traveler, and a professor of art. Benny has been called a dreamer and a man of action because of his free spirit. He was a great artist; a statement told through his still appreciated and displayed art.
Last edited 8/26/2018 7:00:01 PM
CalEXPLORnia - Bufano Peace Statue Monument: The Expanding Universe at Timber Cove
Beniamino Bufano - Wikipedia
One of Benny''s Faces: A Study of Benjamino Bufano (1886-1970), the Man Behind the Artist
Virginia B. Lewin
#peace art #sculptor
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 8,536
|
I mentioned this briefly in my previous post, but I just have to declare it again: I cannot stress enough the effectiveness of cornering your significant other with important discussions while he/she is in the shower.
Constructive criticism to give? Got a loaded question to ask? Need to address that rude comment he/she made that you've been ruminating on for two weeks? Discuss away!
He/she can literally not escape you unless he/she jumps out of the shower, getting water everywhere and experiencing the dramatic shock of a temperature change. And no one wants that. I'm telling you, I get more answers out of Bryan when I trap him in the bathroom than I do any other time of the day.
That's just healthy marriage advice right there, pure and simple. You're welcome. And, uh, let's not tell Bryan we had this conversation, ok?
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 9,097
|
El monzón es un viento estacional que se produce por el desplazamiento de la Zona de Convergencia Intertropical. En verano los vientos soplan de sur a norte, cargados de lluvias. En invierno, son vientos del interior que vienen secos y fríos, especialmente en el océano Índico y en el sur de Asia. El monzón del suroeste que arranca de la costa de Kerala, en la India, comienza generalmente en la primera quincena de junio.
Pese a esta definición original del término, posteriormente se ha usado como sinónimo de lluvia en verano, independientemente de su causa o lugar donde sucede.
Etimología
Monzón deriva del portugués monção, tomado del árabe mawsim (موسم "temporada / estación").
Procesos
Los monzones son provocados porque la tierra se enfría y se calienta más rápido que el agua, de acuerdo con el proceso de la alternancia del aire. Por lo tanto, en verano, la tierra alcanza una temperatura mayor que el océano. Como el viento sopla desde áreas de alta presión (anticiclones) hacia áreas de baja presión (ciclones) con el fin de igualar ambas presiones, la lluvia es producida por el aire húmedo elevándose y enfriándose por ese ascenso en las montañas. Cuando el sol calienta las tierras, las brisas soplan en sentido inverso, del mar a la tierra.
Está a una temperatura mayor, así, el aire se eleva, causando un área de baja presión en el océano. El viento ahora sopla desde la tierra hacia el océano. Pero como la diferencia de temperaturas es menor que durante el verano, el viento que sopla desde anticiclón a la borrasca no es tan constante.
Los monzones se producen típicamente en las costas meridionales asiáticas en el Océano Índico y, sobre todo, en las laderas meridionales de la cordilleras más elevadas del mundo (Himalaya y Karakorum) donde se producen las lluvias más intensas de nuestro planeta, con más de 10 m de agua al año (Cherrapunji, Assam), solo comparables a las que se registran en el noroeste de Colombia, en la depresión del Quibdó y en la selva de Darién, ya en la frontera con Panamá.
Sistemas monzónicos
A medida que se ha podido comprender mejor los monzones, su definición se ha ampliado para incluir casi todos los fenómenos asociados con el ciclo meteorológico anual en los continentes tropicales y subtropicales de Asia, Australia, América del Sur y África junto con sus mares y océanos adyacentes. En estas regiones es donde ocurren los cambios climáticos más dramáticos en la Tierra.
En un sentido más amplio, en el pasado geológico, los sistemas monzónicos han acompañado siempre la formación de supercontinentes como Pangea, con sus climas continentales extremos.
Monzón en la India
Monzón de verano
En el desierto de Thar en el norte de subcontinente indio, la temperatura diurna en verano es muy alta y el aire de la superficie se eleva en altitud causando una depresión local. Este es el origen (lugar) de la circulación que se establece con las costas del Océano Índico. El aire cálido y húmedo procedente del mar llega tanto del este como del oeste y converge en el Himalaya. Esta cadena de montañas fuerza al aire a elevarse y se enfría por la ley de los gases ideales y la humedad se condensa en forma de nubes y de lluvia. El flujo constante de aire húmedo produce abundantes lluvias y se pueden llegar a recoger hasta 10000 mm de lluvia al año en algunos lugares [cita requerida].
Este monzón, que llega desde el suroeste, se divide en dos ramas debido a la orografía de la India. Estos son: el monzón del suroeste del Mar Arábigo y el del Golfo de Bengala. El viento llega primero a la región de las montañas Ghats en la costa del estado de Kerala en el suroeste de la India. La circulación del viento se divide en dos: la primera rama se mueve al norte a lo largo de la vertiente occidental de las montañas, mientras que la segunda pasa por el lado este de la meseta del Decán y sufre un efecto foehn, que lo deseca y produce solo lluvias ligeras y de distribución variable en la península del Decán.
El viento monzón de esta segunda rama pasa por la bahía de Bengala, donde se humedece por la evaporación de la superficie del mar, y luego corre hacia la desembocadura del Ganges y remonta la pendiente del Himalaya al este de las montañas de Birmania. Esta rama del monzón lleva la lluvia al noreste de la India, el Estado de Bengala Occidental, Bangladés y Birmania.
La elevación del viento monzón se acentúa en esta región por la forma de embudo del delta del Ganges y las escarpadas montañas. El viento monzón, bloqueado por las montañas, debe girar hacia el oeste en la Llanura Indo-Gangética y la riega en abundancia. Cherrapunji, en el estado de Meghalaya, situada en la ladera sur del Himalaya, es uno de los lugares más húmedos de la Tierra. La humedad contenida en el monzón va desaguando gradualmente a lo largo de su trayecto y el noroeste de la India no recibe casi nada de lluvia, siendo una región muy árida.
Este proceso de desarrollo de lluvias del monzón de verano se establece de forma gradual en el subcontinente indio, por lo que la fecha de su comienzo puede variar entre marzo y junio dependiendo de la región, y la de su término, de septiembre a noviembre. A veces sucede que se debilita durante algunos años, o que se interrumpe por diferentes períodos.
Monzón de invierno
A partir de septiembre, las temperaturas diurnas disminuyen en el norte del subcontinente, con días más cortos y la temperatura desciende por la noche en estas zonas del desierto. Un gran anticiclón térmico llamado anticiclón siberiano, se forma en la región del lago Baikal. Las áreas de subsidencia que lo recubren son alimentadas en altitud por vientos de ascendencia que mantienen entonces la ZCIT en las regiones húmedas del Hemisferio Sur, principalmente por encima de Indonesia, al noreste de Australia y las costas orientales de África.
En estas condiciones, los vientos alisios nacen al sur del anticiclón de Siberia y van en dirección sureste para dirigirse hacia la ZCIT, que está al sur del ecuador. Debido a que el océano Índico se enfría más lentamente que el continente que lo rodea, estos vientos alisios se mezclan con la advección de aire polar que rodea al anticiclón de Siberia y forma con ellos corrientes del noreste que soplan de la tierra al mar. Antes de llegar a la India, el aire debe franquear los Himalayas y sufre, por lo tanto, un fuerte efecto foehn que lo deseca aún más y lo recalienta considerablemente. La circulación de los vientos se establece así por los mismos corredores que el monzón de verano utilizó durante el verano en los valles del Ganges y el Indo, dando lugar al monzón del noreste o «monzón seco».
Este viento despeja el cielo en el norte del continente, pero una vez que pasa por el océano Índico acumula humedad por la evaporación de la superficie del golfo de Bengala. Este monzón de invierno a continuación, pasará por las islas y el sureste de la India y forma nubes al subir las laderas de estas regiones. Estas lluvias son menos abundantes que durante el monzón de verano, pero ciudades como Madrás y los estados, tales como Tamil Nadu se benefician de él. Estos lugares recibe el 50% a 60% de la precipitación anual durante este monzón.
Impacto en la economía y cultura indias
El monzón de verano produce un ochenta por ciento de la precipitación total en las zonas afectadas. El regreso del monzón tiene un ritmo desigual ya que, de un año para otro, las lluvias tienen una duración y una intensidad diferentes. El monzón es beneficioso, ya que riega la tierra, y a la vez perjudicial, cuando inunda las aldeas. Es irregular e impredecible.
El eterno retorno de los monzones es una sorpresa permanente: ¿Será temprano o tardío, abundante o débil, regular o brutal? Así, la agricultura en la India, que representa el veinticinco por ciento del producto nacional bruto y el setenta por ciento del empleo, depende del monzón. Cultivos como el algodón o el arroz, tienen una alta demanda de agua. Un monzón débil, el retraso del mismo o interrupciones prolongadas se convierten en un giro dramático para cientos de millones de indios y bangalíes, cuya vida económica depende completamente de la contribución de estas lluvias monzónicas. Durante los años 1990, la sequía causada por un cambio en el patrón clásico de la temporada de los monzones causó daños humanitarios y financieros importantes.
En sus oraciones, una nación de agricultores pide un buen monzón, sin el cual el país se sumirá en la hambruna. La peregrinación se utiliza para obtener un mundo doblemente mejor, más rico y más justo. Pues, todo el año sujetos a la jerarquía de castas, los peregrinos quieren, durante el tiempo de su devoción, vivir en un mundo igualitario en el que todos los creyentes sean iguales ante los ojos de los dioses.
El monzón es también popular entre los habitantes de la ciudad, ya que enfría la atmósfera. En efecto, el cielo nublado deja pasar menos radiación solar y mantiene la temperatura ligeramente más baja que durante el periodo precedente al monzón. Sin embargo, la humedad aumenta considerablemente y la lluvia inunda las calles. Las lluvias, de este modo, dañan muchos edificios, sobre todo en las calles en cuesta. Cada año se producen muertes por ahogamiento y por las enfermedades transmitidas por insectos que se reproducen bien en estas condiciones. Algunos años, como el 2005, hubo miles de muertos debido a las inundaciones. Recientemente, algunas zonas áridas como el desierto de Thar han sufrido inundaciones cuando se prolongó la estación del monzón.
Ampliación del concepto
Desde que el sistema del monzón es mejor comprendido, su definición se ha ampliado para incluir a casi todos los fenómenos relacionados con el ciclo anual del Clima en las regiones tropicales y las regiones subtropicales de Asia, Australia, América del Sur, África y en los mares y océanos regionales. Todas estas regiones tienen los ciclos climáticos más potentes y más espectaculares de nuestro planeta, la Tierra, y es especialmente el monzón de verano que predomina en estas áreas. El monzón en el sur de China y Asia del Sur se inscribe en el mismo ciclo que el de la India. Se produce con alguna diferencia en otras regiones y no se puede hablar en general de monzones de invierno en ninguna de estas regiones excepto en la India. Por último, los fenómenos de los monzones siguen siendo marginales en las zonas tropicales y subtropicales de América, pero el término es utilizado con bastante frecuencia por el Servicio Meteorológico Nacional para designar a la temporada de lluvias en los desiertos del Oeste americano.
Monzón africano
El caso más llamativo en este sentido es el del África subsahariana. En el suroeste de esta región de África, hay un monzón que está relacionado con el desplazamiento semi-anual de la Zona de Convergencia Intertropical (ZCIT) y con la diferencia de recalentamiento del Sáhara y la costa del Atlántico ecuatorial en el Golfo de Guinea. Los vientos alisios secos del nordeste, y en especial su forma más intensa de harmattan se cortan por el movimiento hacia el norte en verano de la ZCIT donde los vientos son ligeros. El cinturón húmedo de la costa africana se amplía, sin introducirse en el interior del continente, a diferencia de lo que sucede en la India o China.
El monzón de África occidental difiere en muchos aspectos del monzón asiático. El fenómeno es muy simétrico de oeste a este a gran escala, mientras que al comienzo en la India el flujo es más complejo. Otra diferencia importante, entre muchas otras, se encuentra en el hecho de que el monzón de la India parece más constante en términos de la precipitaciones que el monzón africano. La India nunca ha conocido más de dos años consecutivos de sequía durante el siglo XX, mientras que la región del Sahel ha sufrido la sequía desde principios de 1990.
El monzón de África sigue siendo un tema de estudio. De hecho, varía hasta un 40% de un año a otro, mientras que el monzón de la India oscila solo en un 10 %. Las regiones semiáridas del Sahel y Sudán tienen un período de lluvia muy aleatorio dentro de un flujo del sur, del que depende la supervivencia de la población.
Monzón del sudeste asiático y Oceanía
En Asia del Sur, los monzones se producen desde junio hasta septiembre, con vientos del noreste. La temperatura en el centro de Asia es menor de 25 °C porque es invierno, lo que crea un anticiclón sobre la región. La corriente en chorro en esta región se divide en una rama subtropical y otra polar. La primera corriente sopla sobre todo desde el noreste, con lo que aporta aire seco a la India y Asia meridional. Al mismo tiempo, una baja presión se está desarrollando en Asia sudoriental y Australia, cuyos vientos se dirigen hacia Australia, formando una convergencia húmeda.
Monzón de América del Sur
El Litoral Argentino se ve afectado por el monzón de verano, en especial la provincia de Corrientes. La mayor parte de Brasil se ve influenciada por un monzón de verano. Río de Janeiro es famosa por sus inundaciones durante el mismo.
Monzón de América del Norte
En América del Norte, la diferencia de temperatura entre los grandes desiertos del oeste de Estados Unidos y México y el Golfo de California sirve de motor a un monzón que se extiende desde finales de junio hasta finales de septiembre. Comienza a lo largo de la costa y se extiende hasta el desierto durante este período. Afecta en México, a la Sierra Madre Occidental, y en Estados Unidos a los estados de Arizona, Nuevo México, Nevada, Utah, Colorado, Texas e incluso la parte sureste de California. Rara vez llega a la costa del Pacífico.
Este se asocia con episodios de tormentas breves pero torrenciales y no con lluvias continuas. De hecho, hace que grandes cantidades de humedad del golfo de México den lugar a un aire cálido e inestable. Esta humedad no se distribuye por territorios amplios, y las tormentas se producen cuando se presentan desencadenantes suplementarios. En general, las tormentas se producen y causan crecidas súbitas de los arroyos secos de estas áreas si el nivel del agua "precipitable" es superior a 34 mm. Hasta el setenta por ciento de la precipitación total anual en estas regiones cae durante el monzón. Las plantas se han adaptado a esta precipitación y los desiertos de Sonora y Mojave, se consideran desiertos "húmedos". Estas lluvias también juegan un papel en el control de incendios forestales.
Véase también
Clima monzónico
Clima extremo
Inundación
Crecida
Referencias
Bibliografía adicional
International Committee of the Third Workshop on Monsoons. The Global Monsoon System: Research and Forecast.
Chang, C.P., Wang, Z., Hendon, H., 2006, The Asian Winter Monsoon. The Asian Monsoon, Wang, B. (ed.), Praxis, Berlin, pp. 89–127.
Enlaces externos
Artículo: Viajar con monzones, cómo prepararse para las lluvias
East Asian Monsoon Experiment
página con conceptos básicos del Monzón de Arizona
Vientos del mundo
Patrones climáticos
Estaciones del año
Clima de India
Clima de Bangladés
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 6,072
|
Sports Briefs
Korea readies itself for matchup against Canada
Korea will meet the Canadian national team for an international friendly next month.
According to the Korea Football Association, the Korean national team will host Canada on Nov. 11 at Cheonan Stadium in Cheonan, South Chungcheong, before meeting Uzbekistan four days after in the fifth match of the final round of the Asian qualifiers for the 2018 World Cup.
Against Canada, ranked 103th in FIFA rankings, No. 47 Korea has a head-to-head record of one win, one draw and two losses.
The last time Korea met Canada was in 2002 when Korea lost 2-1 in the third-place match of Concacaf Gold Cup.
With many football teams around the world competing in qualifying round for the World Cup, a friendly against Canadians would provide an opportunity for a tune-up before the match against Uzbekistan.
By Choi Hyung-jo
James Harden's preseason stats point to bright future
Preseason statistics often can be misleading.
That might not be the case with James Harden right now.
After three preseason games, Harden has three points-assists double-doubles - the most recent one Sunday in Shanghai, when he had 26 points and 15 assists to help the Houston Rockets beat the New Orleans Pelicans 123-117 in an NBA Global Games matchup.
"We're treating every game of preseason like the regular season in order to build good habits," Harden said. "We don't take the preseason lightly."
Clearly not.
The Rockets are 3-0 in exhibitions so far this fall, scoring 131, 130 and now 123 points in those games. Harden is averaging 23.3 points and 12 assists in those three games.
And fans in China, where the Rockets are enormously popular anyway - thanks in large part to Yao Ming - have taken to Harden as well, with many sporting fake beards on Sunday to mimic his look.
"Great atmosphere," Harden said.
They'll be seeing even more NBA in China this season. NBA Commissioner Adam Silver announced Sunday an expansion of the NBA's partnership with BesTV, one that will see up to 1,300 games per season broadcast in China.
"It's a multimedia partnership that brings the very best in television and technology to the NBA in China," Silver said.
Toronto Blue Jays defeat Texas Rangers 7-6
TORONTO - The Toronto Blue Jays beat the Texas Rangers 7-6 on Sunday night to sweep their AL Division Series while the Washington Nationals defeated the Los Angeles Dodgers 5-2 to even their NL series at 1-all.
Josh Donaldson, the reigning AL MVP, led off the 10th for Toronto with a double into the right-center field gap then raced home from second base after Rougned Odor bounced a double-play relay.
The wild-card Blue Jays are headed back to the AL Championship Series after beating Texas in an ALDS for a second straight year and will face the winner of the Cleveland-Boston series.
The Indians lead 2-0, with Game 3 scheduled for Monday following a postponement Sunday.
Donaldson had two doubles among his three hits and is batting .538 through four postseason games, all wins for the Blue Jays, who had to beat Baltimore in a wild-card game to get to the ALDS.
Closer Roberto Osuna threw two perfect innings to get the win. His appearance capped 4 and 1/3 scoreless innings of one-hit ball by Toronto's bullpen after starter Aaron Sanchez allowed six runs.
Edwin Encarnacion had a two-run homer and Russell Martin a solo shot in the first inning for Toronto, which swept a postseason series for the first time.
Shortstop Elvis Andrus hit a solo homer for Texas in the third and Odor added a two-run shot in the fourth. They were the only two homers of the series for the Rangers.
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 4,448
|
Q: NMI watchdog messages, ie. 'Shutting down hard lockup detector on all cpus' When NMI watchdog has been "disabled" it is still chatty.
Does anyone know where the docs for these messages live? I'd like to see what is actually happening.
For example, verified that it is disabled:
$ cat /proc/sys/kernel/nmi_watchdog
0
YET, we still see messages like the following on shutdown or boot:
$ journalctl -xn 100000 | grep "NMI watchdog"
Oct 23 14:29:31 hostname-us kernel: NMI watchdog: disabled (cpu0): hardware events not enabled
Oct 23 14:29:31 hostname-us kernel: NMI watchdog: Shutting down hard lockup detector on all cpus
Now I know that this isn't a RESET, it's something else and I'd like to have the documented answer, not a best guess.
Tried looking through kernel.org and debian.org, man pages with no luck, only archived bugzilla pages.
We'd like to know what these messages actually mean, not make assumptions. Does anyone know where the decoder ring lives ?
A: Should have known it wasn't an exact match but managed to finally find it on kernel.org
https://www.kernel.org/doc/html/latest/admin-guide/sysctl/kernel.html
A: From http://slacksite.com/slackware/nmi.html
The NMI watchdog is a kind of timer event handler, it checks Local APIC or IO-APIC interrupt counter register when it is called on every local timer event of each CPU. Generally speaking there could be hundreds of device and timer interrupts are received per second. If there are no interrupts received in a 5 second interval, the NMI watchdog assumes that the system has hung and initiates a kernel panic. This is very helpful when you need some data for investigating the issue , but occasionally it may have such undesirable effects.
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 1,562
|
Accueil › Janet Jackson "When I Think Of You" Single (1986)
Janet Jackson "When I Think Of You" Single (1986)
Details: Used 7" for Janet Jackson's hit, "When I Think Of You," on A&M Records in 1986. I can't think of that song now without singing the extra lyrics (whispered), "When I think of you..." Solid track from Janet from her third album, "Control."
Description: "When I Think of You" is a song by American singer Janet Jackson from her third studio album, Control (1986). It was released on July 28, 1986, as the album's third single. Written by Jackson, Jimmy Jam and Terry Lewis, and produced by Jam and Lewis, the song is about a person who finds relief and fun in a lover. It was Jackson's first number-one single on the US Billboard Hot 100 and also peaked at number 10 in the United Kingdom. Pitchfork included the song on their "The 200 Best Songs of the 1980s" list at number 48. The song was resurrected in 1995 when released on two limited-edition CD single formats in the United Kingdom, one containing remixes by Deep Dish and Heller & Farley, and the other containing remixes by David Morales. That same year these remixes were included on certain releases of "Runaway". "When I Think of You" has been included in each of Jackson's greatest hits albums: Design of a Decade: 1986–1996 (1995), Number Ones (2009), and Icon: Number Ones (2010).
Grade: VG (Cover) / VG+ (Record)
1. When I Think Of You
1. Pretty Boy
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 4,048
|
Located in the Deutz district, this hotel is in the heart of the Cathedral City and near the Cologne Arena and Kolnmesse. The hotel is just a short walk from the city centre and the InterCity Express Koln-Deutz Train Station.
Spacious and fully equipped with all standard amenities, the rooms at the hotel are designed to be elegant as well as comfortable.
Radisson Blu Hotel in Cologne has an onsite Paparazzi Restaurant that offers Italian and Milanese cuisines. The Paparazzi Lounge Bar invites you to enjoy cocktails from around the world and various freshly prepared Italian coffees.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 6,516
|
{"url":"https:\/\/brilliant.org\/discussions\/thread\/prove-it-8\/","text":"\u00d7\n\n# Prove it\n\nProve that $$2^{4m}-1$$ is divisible by 15,where $$m$$ is any integer.\n\nNote by Ravi Ranjan\n1\u00a0year, 1\u00a0month ago\n\nSort by:\n\n2^4 = 1 (mod 15) or, 2^4m = 1 (mod 15) or, 2^4m -1 = 0 (mod 15)\n\nAnd it's proved. \u00b7 1\u00a0year ago\n\nPut m=1 2^4=16 16-1=15 15 is divisible on 15... Answer is yes......(2^4m)-1 is divisible on 15..... \u00b7 1\u00a0year, 1\u00a0month ago","date":"2017-01-22 03:58:15","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\": 0.4130999743938446, \"perplexity\": 10108.321559750071}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2017-04\/segments\/1484560281332.92\/warc\/CC-MAIN-20170116095121-00335-ip-10-171-10-70.ec2.internal.warc.gz\"}"}
| null | null |
Q: How to highlight neighbouring nodes in Cytoscape.js I am using cytoscape.js and would like to add the feature on mouseover or tap of a node to apply a style to:
*
*change the style of the neighbouring nodes - 1st degree
*fade out the nodes that are not connected
I seem to be able to get the neighbours, any ideas how I apply a style to non neighbours?
cy.on('tap', 'node', function(e) {
var node = e.cyTarget;
var directlyConnected = node.neighborhood();
directlyConnected.nodes().addClass('connectednodes');
});
A: Addition to Polosson's answer since I am not aloud to comment:
The api seems to have changed, it's target instead of cyTarget now (Verison 3.2.17).
Also, you might have to add the incomers to highlight all neighbours:
cy.on('mouseover', 'node', function(e) {
var sel = e.target;
cy.elements()
.difference(sel.outgoers()
.union(sel.incomers()))
.not(sel)
.addClass('semitransp');
sel.addClass('highlight')
.outgoers()
.union(sel.incomers())
.addClass('highlight');
});
cy.on('mouseout', 'node', function(e) {
var sel = e.target;
cy.elements()
.removeClass('semitransp');
sel.removeClass('highlight')
.outgoers()
.union(sel.incomers())
.removeClass('highlight');
});
A: Use the set difference: http://js.cytoscape.org/#collection/building--filtering/eles.difference
cy.elements().difference( dontIncludeTheseEles )
A: If ever you haven't found the solution to highlight children of a node when mouse hover one, here is my attempt and it works nice:
Event handler:
cy.on('mouseover', 'node', function(e){
var sel = e.target;
cy.elements().difference(sel.outgoers()).not(sel).addClass('semitransp');
sel.addClass('highlight').outgoers().addClass('highlight');
});
cy.on('mouseout', 'node', function(e){
var sel = e.target;
cy.elements().removeClass('semitransp');
sel.removeClass('highlight').outgoers().removeClass('highlight');
});
Basically, all elements are filtered if they're not the hovered node or its direct children ("outgoers") and the class 'semitransp' is added to them.
Then, the class 'highlight' is added to the hovered node and all its children.
Example of style for 'highlight' and 'semitransp' class:
var cy = cytoscape({
elements: [ {...} ]
style: [
{...},
{
selector: 'node.highlight',
style: {
'border-color': '#FFF',
'border-width': '2px'
}
},
{
selector: 'node.semitransp',
style:{ 'opacity': '0.5' }
},
{
selector: 'edge.highlight',
style: { 'mid-target-arrow-color': '#FFF' }
},
{
selector: 'edge.semitransp',
style:{ 'opacity': '0.2' }
}
]
});
A: This code implements the click functionality instead of hover to highlight the node. It´s an extension of Polosson answer.
var previous_node;
var previous_sel;
cy.on("click","node",(e)=>
{
var sel = e.target;
var id = e.target.id();
if ((id != previous_node) && (previous_node != undefined ) && (previous_sel != undefined))
{
cy.elements().removeClass("semitransp");
previous_sel.removeClass("highlight").outgoers().union(previous_sel.incomers()).removeClass("highlight");
cy.elements().difference(sel.outgoers().union(sel.incomers())).not(sel).addClass("semitransp");
sel.addClass("highlight").outgoers().union(sel.incomers()).addClass("highlight");
previous_sel = sel;
previous_node = id;
}
else
{
cy.elements().difference(sel.outgoers().union(sel.incomers())).not(sel).addClass("semitransp");
sel.addClass("highlight").outgoers().union(sel.incomers()).addClass("highlight");
previous_sel = sel;
previous_node = id;
}
})
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 673
|
\section{INTRODUCTION}
In the context of gravity the General Theory of Relativity is a successful accomplishment. This theory parallel with quantum field theory are the basis of modern physics. General Relativity speaks with the language of differential geometry which is a different approach rather than quantum field theory and it is no any exaggeration to say that this theory actually has been revolutionized our understanding of the Universe.
The Einstein field equations which are completely unchanged since November 1915 when Einstein proposed them are still the best description of space-time behaviour in large scales. These equations are:
\begin{equation}
G_{\mu\nu}=\frac{8\pi G}{c^4}T_{\mu\nu}
\end{equation}
Where $G_{\mu\nu}$ is the Einstein tensor, $T_{\mu\nu}$ is the energy-momentum tensor, c is the speed of light and G is the Newton's gravitation constant (we set here G=c=1). It is believed that these equations govern the expansion of the universe, formation of all structures in the universe from planets, stars and solar-systems up to cluster and super-cluster galaxies, black holes behaviour, gravitational waves and so more.
The limitations of General Relativity come to existence from such scenarios like the 'dark universe'. For almost thirty years the impressive amounts of astronomical data indicate that our universe is spatially flat and experiences a phase of accelerated expansion \cite{ex0}. In the context of standard cosmology, based on General Relativity, this accelerated expansion can be explained by an unknown form of energy which is called dark energy \cite{da0}, \cite{da00}. Another way of explaining this acceleration is based on some sort of modifications for the Einstein theory of gravity. There are various kind of modifications of gravity which have been proposed in the literature since past few decades. Some of them are as Lovelock gravity \cite{lo0}, scalar-tensor theories \cite{st0} \cite{st00} \cite{ts0} \cite{ts00} \cite{h10} \cite{h11}, $f(R)$ gravity \cite{fr0}, etc.
Among the scalar fields, dilaton field is a kind of scalar field that is coupled to the Einstein gravity in the low energy limit of string theory. In past few years this scalar field has been appeared in various theories like dimensional reduction, different models of supergravity, etc. Also, it is important to see the effect of dilaton field on the structures of the solutions. It was considered that the presence of the dilaton field can change the causal structure of space-time geometry. Although, this scalar field can change the asymptotic behaviour of the space-time as well as the thermodynamical properties of charged black hole(BH) solutions.
Nonlinear electrodynamics has been introduced for the sake of solving some problems. This theory can eliminate the divergences in self-energy of point-like charges(The singularity of the electrical field of point-like charges) in the linear Maxwell theory. The Born - Infeld nonlinear electrodynamic actually was proposed to resolve these divergences from the linear case\cite{n1}, \cite{n2}, \cite{n3}, \cite{n4}, \cite{n5}. Also, this nonlinear model can help to remove the curvature singularities in the black hole investigations \cite{bh0}.
Another new nonlinear electrodynamics model is known as the power-law Maxwell electrodynamics. In this model, the electromagnetic part of the Lagrangian density is $\mathcal{F}=(F_{\mu\nu}F^{\mu\nu})^p$, where p is the nonlinearity parameter. The power-law Maxwell Lagrangian is conformally invariant in higher dimensions. This kind of Lagrangian is invariant under the conformal transformation $g_{\mu\nu}\rightarrow \Omega^2 g_{\mu\nu}$ and $A_{\mu\nu}\rightarrow A_{\mu\nu}$. In the last few years, the power of Maxwell invariant(PMI) has been considered as an interesting study area in the context of geometrical physics for different dimensionality ranging \cite{d0} \cite{d1} \cite{d2} \cite{d3} \cite{d4} and more clearly in the framework of Einstein- Maxwell dilaton gravity \cite{em0} \cite{em1} \cite{em2}.
The paper in organized as the following order. In section \textcolor{red}{2}, we first introduce the Einstein-Generalized Maxwell gravity which is coupled to a dilatonic field. After that, by varying this action with respect to the metric $g_{\mu\nu}$, dilaton field $\Phi$ and the gauge field $A_{\mu}$, we have written the equations of motion. Also, We have solved these equations in order to obtain the slowly rotating dilatonic black holes inspired by power-law electrodynamics. In the end of this section we have also showed that the dialton potential can be written as the generalized three Liouville-type potentials. In section \textcolor{red}{3}, the conserved quantities as total mass, angular momentum, electric charge and electric potential have been obtained. In section \textcolor{red}{4}, we have showed that the asymptotic behavior of the obtained solutions are neither flat nor (A)dS. Also, we have plotted the electric field diagram versus r. In this case, the electric field goes to zero when $r\rightarrow 0$ and diverges at $r\rightarrow\infty$. Section \textcolor{red}{5} is devoted to investigate the thermodynamic properties of this new slowly rotating black hole solution. The black hole temperature and entropy(S) are obtained in this section and their corresponding diagrams have been investigated as well. Through the redefinition of mass as a function of the extensive quantities Q and S and their conjugates, we have showed the validity of the first law of thermodynamics. Section \textcolor{red}{6} is dedicated to determine the local and global stability of our black hole solution. In local stability analysis due to the heat capacity calculation, we determine the type-1 and type-2 phase transition points and also the impact of the dilaton field on the black hole stability through diagrams. Although, in global stability analysis via the Gibbs free energy, the Hawking-Page phase transition is determined as well as the global stability of our solution in the presence of dilaton field. In section \textcolor{red}{7}, we study the Smarr formula. We have showed that the presence of dilaton field brings a new term in the Smarr formula. Final section as is clear is devoted to our summary and closing remarks.
\section{FIELD EQUATIONS AND THE BH SOLUTIONS}
Here we introduce the action of charged black holes in the Einstein-Generalized Maxwell gravity which is coupled to a dilaton field. The general form of this action is as \cite{a1} \cite{a2}:
\begin{equation}
S=\frac{1}{16\pi}\int{d^4x\sqrt{-g}}\left(\mathcal{R}-2g^{\mu\nu}\nabla_{\mu}\Phi \nabla_{\nu}\Phi-V(\Phi)+\mathcal{L}(\mathcal{F},\Phi)\right)
\end{equation}
Where $\mathcal{R}$ is the Ricci scalar, $V(\Phi)$ is the potential of dilaton field. The last term in this relation is actually the representation of coupling between electrodynamics and scalar field. This term is as $\mathcal{L}(\mathcal{F},\Phi)=\big(-\mathcal{F}e^{-2\alpha\Phi}\big)^p$ where $\mathcal{F}=F_{\mu\nu}F^{\mu\nu}$(In electromagnetic context, $F_{\mu\nu}$ is defined as $F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}$), $\alpha$ is the coupling constant and power p is the nonlinearity parameter. In special case p=1, this action reduces to the Einstein-Maxwell-dilaton gravity action. Varying this action with respect to the metric $g_{\mu\nu}$, dilaton field $\Phi$ and the gauge field $A_{\mu}$ after some simplifications, yields:
\begin{equation}
\mathcal{R}_{\mu\nu}=2\partial_{\mu}\Phi\partial_{\nu}\Phi+\frac{1}{2}g_{\mu\nu}V(\Phi)+\bigg[(p-\frac{1}{2})g_{\mu\nu}+ \frac{p}{\mathcal{F}}F_{\mu\gamma}F_{\nu}^{\gamma}\bigg]\mathcal{L}(\mathcal{F},\Phi)
\end{equation}
\begin{equation}
\nabla_{\mu}(\sqrt{-g}\mathcal{L}_{\mathcal{F}}(\mathcal{F},\Phi) F^{\mu\nu})=0
\end{equation}
\begin{equation}
\partial_{\mu}\partial^{\mu}\Phi=\frac{1}{4}\frac{\partial{V(\Phi)}}{\partial{\Phi}}+\frac{\alpha p}{2}\mathcal{L}(\mathcal{F},\Phi)
\end{equation}
In these set of equations we have $F_{\mu\gamma}F_{\nu}^{\gamma}$$\equiv$$g^{\beta\gamma}F_{\nu\beta}F_{\mu\gamma}$ and $\mathcal{L}_{\mathcal{F}}(\mathcal{F},\Phi)\equiv\frac{\partial}{\partial\mathcal{F}}\mathcal{L}(\mathcal{F},\Phi)$.
Now, we consider the following ansatz as a 4-dimensional spherical rotating metric
\begin{equation}
ds^2=-X(r)dt^2+\frac{dr^2}{X(r)}+2a H(r)sin^2\theta dt d\phi+f(r)^2d\Omega_{2}^2
\end{equation}
Where $d\Omega_{2}^2$ is the line element of a 2-dimensional hypersurface. In this assuming metric, X(r) H(r) and f(r) are three unknown functions of r that is our goal to find. For small rotation case, we are able to solve Eqs.(\textcolor{blue}{3})-(\textcolor{blue}{5}) to first order in angular momentum parameter a. From solving Einstein field equations, the only term which is added to non-rotating case is $g_{t\phi}$ that contains the first order of a.
Here, we consider this gauge potential:
\begin{equation}
A_{\mu}=h(r)\big(\delta_{\mu}^{t}-asin^2\theta\delta_{\mu}^{\phi}\big)
\end{equation}
Also, one can show that for the infinitesimal angular momentum
\begin{equation}
\mathcal{F}=F_{\mu\nu}F^{\mu\nu}=-2\big(h^\prime(r)\big)^2
\end{equation}
Where h(r) is an unknown function that we try to find it(For proving this relation one can see more details in the appendix). Integration of Maxwell equation(Eq.(\textcolor{blue}{4})) gives:
\begin{equation}
F_{tr}=\frac{q e^{\frac{2\alpha p\Phi(r)}{2p-1}}}{f(r)^{\frac{2}{2p-1}}}
\end{equation}
Where q is an integration constant in this relation which is related to the electric charge.
By using the above ansatz(Eq.(\textcolor{blue}{6})) in Eq.(\textcolor{blue}{3}), we find the following field equations. These field equations are for tt, rr, $\theta\theta$ and $t\phi$ components, respectively.
\begin{equation}
2f^{\prime}(r)X^{\prime}(r)+f(r)X^{\prime\prime}(r)=f(r)V(\Phi)+(3p-1)f(r)\mathcal{L}(\mathcal{F},\Phi)
\end{equation}
\begin{align*}
2f^{\prime}(r)X^{\prime}(r)+f(r)X^{\prime\prime}(r)+4f^{\prime\prime}(r)X(r)=4f(r)X(r) {\Phi^{\prime}}^2+ f(r)V(\Phi)+\\(3p-1)f(r)\mathcal{L}(\mathcal{F},\Phi) \tag{11}
\end{align*}
\begin{equation}
\bigg[X(r)(f^2(r))^\prime\bigg]^\prime-2=f^2(r)V(\Phi)+(2p-1)f^2(r)\mathcal{L}(\mathcal{F},\Phi) \tag{12}
\end{equation}
\begin{align*}
f^2(r)X(r)H^{\prime\prime}(r)+2H(r)\bigg(f(r)f^\prime(r)X^\prime(r)-1\bigg)=f^2(r)H(r)V(\Phi)+\\ \bigg((2p-1)H(r)+pX(r)\bigg)f^2(r)\mathcal{L}(\mathcal{F},\Phi) \tag{13}
\end{align*}
From Eqs. (\textcolor{blue}{10}) and (\textcolor{blue}{11}) we have
\begin{equation}
\frac{f^{\prime\prime}(r)}{f(r)}=\Phi^\prime(r)^2 \tag{14}
\end{equation}
We further assume this ansatz as well:
\begin{equation}
f(r)=\beta r^N \tag{15}
\end{equation}
Where $\beta$ and N are just two constants. By putting this ansatz in Eq.(\textcolor{blue}{14}), $\Phi(r)$ will be:
\begin{equation}
\Phi(r)=\pm \sqrt{N(N-1)}ln(r)+\Phi_{0} \tag{16}
\end{equation}
Here $\Phi_{0}$ is also an integration constant. Without the loss of generality we set $\Phi_{0}=0$. As is clear in this relation, $N$ parameter is related to the dilaton field and if we set $N=1$, dilaton field will be vanished($N\geq1$).
From Eqs. (\textcolor{blue}{7}, \textcolor{blue}{9}, \textcolor{blue}{15}, \textcolor{blue}{16}), the non-vanishing components of the electromagnetic field tensor are:
\begin{equation}
F_{tr}=-F_{rt} = \begin{cases}
\frac{q}{\beta^{\frac{2}{2p-1}}r^{\frac{2}{2p-1}}} & \text{for $N=1$} \\
\frac{q}{\beta^{\frac{2}{2p-1}}}r^{\frac{2N[2p(N-1)-1]}{2p-1}}& \text{otherwise} \tag{17}
\end{cases}
\end{equation}
\begin{equation}
F_{\phi r}=-{a sin\theta}F_{tr}\quad,\quad F_{\theta \phi}=-a h(r) sin2\theta \tag{18}
\end{equation}
For h(r) in the gauge potential Eq.(\textcolor{blue}{7}), we have:
\begin{equation}
h(r) = \begin{cases}
\frac{(2p-1)q}{\beta^{\frac{2}{2p-1}}(2p-3)}r^{\frac{2p-3}{2p-1}} & \text{for $N=1$} \\
\frac{(2p-1)q}{\beta^{\frac{2}{2p-1}}[2p(2N^2-2N+1)-3]}r^{\frac{2p(2N^2-2N+1)-3}{2p-1}} & \text{otherwise} \tag{19}
\end{cases}
\end{equation}
We have assumed in this work that $\alpha$=$2\sqrt{N(N-1)}$.
By using Eqs.(\textcolor{blue}{8}, \textcolor{blue}{15}, \textcolor{blue}{16}, \textcolor{blue}{19}) in Eqs.(\textcolor{blue}{10}-\textcolor{blue}{13}), one can obtain
\begin{equation}
X(r)=\frac{r^{2-2N}}{\beta^2(2N-1)}-{m} r^{1-2N}-\frac{\Lambda}{N(4N-1)} r^{2N}+\Upsilon(r) \tag{20}
\end{equation}
\begin{equation}
H(r)=-{m} r^{1-2N}-\frac{\Lambda}{N(4N-1)} r^{2N}+\Upsilon(r) \tag{21}
\end{equation}
Where
\begin{equation}
\Upsilon(r) = \begin{cases}
-\frac{2^{p}(2p-1)^2 q^{2p} r^{-\frac{2}{2p-1}}}{4\beta^{\frac{4p}{2p-1}}(2p-3)} & \text{for $N=1$} \\
\frac{2^{p}(2p-1)^2 p q^{2p} r^{2\frac{(2 p(N-1)^2-1)}{2p-1}}}{2\beta^{\frac{4p}{2p-1}}(2 p N^2-6pN+N+2p-1)(4pN^2-4 pN-2N+2p-1)} & \text{otherwise} \tag{22}
\end{cases}
\end{equation}
Without the loss of generality, we can set $\beta=1$ in the above equations.
One property of these solutions is that if we set $N=1$, the slowly rotating Kerr metric(Lense-Thirring metric) recovers. Also, by setting $a=0$ and $p=1$ the Reissner-Nordstrom metric recovers, too \cite{gr0}.
The curvature singularity for this solution is located at r=0(We have showed this fact by using the Ricci scalar diagrams in appendix).
If dilaton potential considers as a three Liouville-type potentials as:
\begin{equation}
V(\Phi)=2\Lambda_{1} e^{2\zeta_{1}\Phi}+2\Lambda_{2} e^{2\zeta_{2}\Phi}+2\Lambda e^{2\zeta_{3}\Phi} \tag{23}
\end{equation}
Then, we have
\begin{equation}
\Lambda_{1}=-\frac{N-1}{\beta^2(2N-1)}\quad,\quad \Lambda_{2}=-\frac{(2p-1)2^{p-1}q^{2p}(2pN^2-7pN+N+2p-1)}{\beta^{\frac{4p}{2p-1}}[(2N^2-6N+2)p+N-1]} \tag{24}
\end{equation}
\begin{equation}
\zeta_{1}=-\sqrt{\frac{N}{N-1}}\quad,\quad \zeta_{2}=\frac{2p(N-2)}{2p-1}\sqrt{\frac{N}{N-1}}\quad,\quad \zeta_{3}=\sqrt{\frac{N-1}{N}} \tag{25}
\end{equation}
In Eq.(\textcolor{blue}{23}), $\Lambda$ is a free parameter where plays the role of the cosmological constant.
\section{CONSERVED QUANTITIES}
The first conserved quantity that we introduce here in terms of the mass parameter m is the mass of black hole. As is clear, the asymptotic behaviour of the metric functions(Eqs.(\textcolor{blue}{20}) and (\textcolor{blue}{21})) is unusual. In this case, the quasilocal formalism of Brown and York must be applied for obtaining the quasilocal mass of dilaton black holes \cite{qas} \cite{adm}. Using this formalism and after some calculation we find the quasilocal mass as:
\begin{equation}
M=\frac{N \beta^2}{2}m \tag{26}
\end{equation}
Also, through the use of Komar conserved quantities \cite{j1} \cite{j2} \cite{j3}, we obtain
\begin{equation}
J=\frac{(4N-1) \beta^2}{3}m a \tag{27}
\end{equation}
Using Eq.(\textcolor{blue}{4}), the electric charge can be obtained through the modified Gauss law. This modified law can be written as:
\begin{equation}
Q=\frac{1}{4\pi}\int e^{-2\alpha p \Phi}\beta^2 r^{2N}(-\mathcal{F})^{p-1}F_{\mu\nu}n^{\mu}u^{\nu}d\Omega \tag{28}
\end{equation}
Where $n^{\mu}$ and $u^{\nu}$ are the space-like and time-like unit normals to a hypersurface of radius r, respectively. With the help of Eqs.(\textcolor{blue}{9}), (\textcolor{blue}{15}) and (\textcolor{blue}{16}) and after some calculations, the electric charge is given by:
\begin{equation}
Q=\frac{2^{p-1}q^{2p-1}}{4\pi} \tag{29}
\end{equation}
The electric potential U, measured by an observer at infinity with respect to the event horizon $r_{+}$, is defined by:
\begin{equation}
U=A_{\mu}\chi^{\mu}|_{r\rightarrow\infty}-A_{\mu}\chi^{\mu}|_{r=r_{+}} \tag{30}
\end{equation}
Here, $\chi=C \partial_{t}$ is the null generator of the event horizon and C is just a constant which may be fixed. Using Eq.(\textcolor{blue}{17}), the electric potential on the horizon will be as:
\begin{equation}
U= \begin{cases}
\frac{C q}{\beta^{\frac{2}{2p-1}}}\frac{2p-1}{2p-3}r_{+}^{\frac{2p-3}{2p-1}} & \text{for $N=1$} \\
\frac{C q}{\beta^{\frac{2}{2p-1}}}\frac{2p-1}{4pN^2-4pN-2N+2p-1}r_{+}^{\frac{4pN^2-4pN-2N+2p-1}{2p-1}}& \text{otherwise} \tag{31}
\end{cases}
\end{equation}
\section{PHYSICAL PROPERTIES OF THE SOLUTIONS}
In order to investigate the asymptotic behaviour of the obtaining solutions, we work on the $r\rightarrow\infty$ limit of X(r) function.
\begin{align*}
\lim_{r\rightarrow\infty}X(r)=-\frac{\Lambda}{N(4N-1)} r^{2N}+\qquad\qquad\qquad\qquad\qquad\qquad\qquad \\ \frac{2^{p-1}(2p-1)^2 p q^{2p} r^{2\frac{(2 p(N-1)^2-1)}{2p-1}}}{(2 p N^2-6pN+N+2p-1)(4pN^2-4 pN-2N+2p-1)} \tag{32}
\end{align*}
In this limit as is clear, the asymptotic behaviour of the obtained solutions are neither flat nor (A)dS, but when we set $N=1$, the dilaton field will disappear and someone can find:
\begin{equation}
\lim_{r\rightarrow\infty}X(r)=1-\frac{\Lambda}{3} r^{2} \tag{33}
\end{equation}
This relation tells the asymptotic behaviour is flat($\Lambda=0$), AdS($\Lambda<0$) or dS($\Lambda>0$). As we can see, the unusual asymptotic behaviour of these solutions is a direct consequence of the dilaton field not even the non-linearity of the electrodynamics source.
Now, we want to study the behaviour of the electric field in the presence of the dilaton field. One bizarre property of the electric field is that the electric field goes to zero when $r\rightarrow 0$ and diverges at $r\rightarrow\infty$. In this case, we can say the presence of the dilaton filed reverses the behaviour of the electric field, because in the normal cases($N=1$) the electric field diverges at $r\rightarrow0$ and goes to zero when $r\rightarrow\infty$. To have a more precise understanding of this behaviour, we plot $E(r)$ versus r for different parameters.
\begin{center}
\includegraphics[width=14cm]{E0.jpg}
\captionof{figure}{E versus $r$ where $N$ and p parameters change. In the left diagrams we set q=1, p=2. For the right diagrams we set q=1, N=3.}
\end{center}
\section{THERMODYNAMICS}
From the proposing relation for the black holes temperature via Hawking and Bekenstein, we can calculate this quantity for the previous black hole solution. The Hawking-Bekenstein temperature relation for a given static spherically symmetric black hole is as follow through the definition of surface gravity($\kappa$):
\begin{equation}
T=\frac{\kappa}{2\pi}=\frac{1}{2\pi}\sqrt{-\frac{1}{2}(\nabla_{\mu}\chi_{\nu})(\nabla^{\mu}\chi^{\nu})}=\frac{X^{\prime}(r_{+})}{4\pi} \tag{34}
\end{equation}Where $\chi=\partial/\partial{t}$ is the killing vector of event horizon. This formula means we first take the derivative of X(r) with respect to r and then put $r_{+}$(the largest real root of the metric function $X(r_{+})=0$) instead of r. The Hawking-Bekenstein temperature for this black hole solution can be written as:
\begin{equation}
T=-\frac{(N-1)}{2\pi \beta^2(2N-1)}r_{+}^{1-2N}+\frac{(2N-1)m}{4\pi}r_{+}^{-2N}-\frac{\Lambda}{2\pi (4N-1)}r_{+}^{2N-1}+\frac{\Upsilon^{\prime}(r_{+})}{4\pi} \tag{35}
\end{equation}
Where we have
\begin{equation}
\Upsilon^{\prime}(r_+) = \begin{cases}
\frac{2^{p}(2p-1) q^{2p} r_{+}^{-\frac{2p+1}{2p-1}}}{2\beta^{\frac{4p}{2p-1}}(2p-3)} & \text{for $N=1$} \\
\frac{2^{p}(2p-1) p q^{2p}(2pN^2-4pN+2p-1) r_{+}^{\frac{4pN^2-8pN+2p-1}{2p-1}}}{\beta^{\frac{4p}{2p-1}}(2 p N^2-6pN+N+2p-1)(4pN^2-4 pN-2N+2p-1)} & \text{otherwise} \tag{36}
\end{cases}
\end{equation}
In order to see the effect of some parameters on temperature, we present the following diagrams to show the effect of $\Lambda$ and N parameters on this thermodynamical quantity.
\begin{center}
\includegraphics[width=14cm]{T0.jpg}
\captionof{figure}{T versus $r_{+}$ where $\Lambda$ and k parameters change. In the left diagrams we set m=1, q=1, N=2, p=2, $\beta=1$. For the right diagrams we set m=1, q=1, $ \Lambda=-1$, p=2, $\beta=1$.}
\end{center}
The semiclassical black hole entropy(S) in n+1 dimensions from the Bekenstein-Hawking formula brings:
\begin{equation}
S=\frac{A_{+}}{4}=\frac{\omega_{n-1}}{4}(\beta r_{+}^N)^{n-1} \tag{37}
\end{equation}
Where $\omega_{n-1}$\footnote{The surface area of a unit hypersphere in n-1 dimensions} is given by:
\begin{equation}
\omega_{n-1}=\frac{2\pi^{n/2}}{\Gamma(n/2)} \tag{38}
\end{equation}
In 3+1 dimensions we have:
\begin{equation}
S=\frac{A_{+}}{4}=\pi \beta^2 r_{+}^{2N} \tag{39}
\end{equation}
Now we introduce the mass parameter m in terms of horizon radius $r_{+}$ which is not a difficult task
\begin{equation}
m(r_{+})=\frac{1}{\beta^2(2N-1)}r_{+}-\frac{\Lambda}{N (4N-1)}r_{+}^{4N-1}+m_{k}(r_{+}) \tag{40}
\end{equation}
Here, $m_{k}(r_{+})$ is as follow:
\begin{equation}
m_{k}(r_{+}) = \begin{cases}
-\frac{2^{p-1} q^{2p}(2p-1)^2}{2\beta^{\frac{4p}{2p-1}}(2p-3)}r_{+}^{\frac{2p-3}{2p-1}} & \text{for $N=1$} \\
\frac{2^{p-1}(2p-1)^2 p q^{2p} r_{+}^{\frac{4pN^2-4pN+2p-2N-1}{2p-1}}}{\beta^{\frac{4p}{2p-1}}(2pN^2-6pN+N+2p-1)(4pN^2-4p N-2N+2p-1)} & \text{otherwise} \tag{41}
\end{cases}
\end{equation}
Also, one can define the mass as a function of the extensive quantities Q and S. This function can be written as $M(S,Q)$ where the intensive quantities conjugate to S and Q are:
\begin{equation}
T=\bigg(\frac{\partial M(S,Q)}{\partial S}\bigg)_{Q} \quad,\quad U=\bigg(\frac{\partial M(S,Q)}{\partial Q}\bigg)_{S} \tag{42}
\end{equation}
Here, it is worthwhile to check the validity of the first law of thermodynamics. In differential form, we have
\begin{equation}
d M(S,Q)=\bigg(\frac{\partial M(S,Q) }{\partial S}\bigg)_{Q}dS+\bigg(\frac{\partial M(S,Q) }{\partial Q}\bigg)_{S}dQ \tag{43}
\end{equation}
After some algebraic manipulation and simplification, the above equations confirm that the first law of thermodynamics is valid, which is:
\begin{equation}
dM=TdS+UdQ \tag{44}
\end{equation}
As a matter of calculation and using Eqs.(\textcolor{blue}{31}, \textcolor{blue}{39}, \textcolor{blue}{40}, \textcolor{blue}{42}), we have
\begin{equation}
C= \begin{cases}
{-2\pi p} & \text{for $N=1$} \\
\frac{4\pi N p^2}{2pN^2-6pN+N+2p-1} & \text{otherwise} \tag{45}
\end{cases}
\end{equation}
\section{LOCAL STABILITY, PHASE TRANSITION, GLOBAL STABILITY}
In this section, we study the stability(local and global) and phase transition of the slowly rotating charged black holes in dilaton gravity. First, we investigate the local stability by calculating the heat capacity and its corresponding diagrams. In this case, one can use the heat capacity relation which is
\begin{equation}
C_{Q}=T\bigg(\frac{\partial S}{\partial r_{+}}\bigg)_{Q}\bigg(\frac{\partial T}{\partial r_{+}}\bigg)_{Q}^{-1}=T\bigg(\frac{\partial S}{\partial T}\bigg)_{Q}=T\bigg(\frac{\partial^2 M}{\partial S^2}\bigg)_{Q}^{-1} \tag{46}
\end{equation}
In order to ensure that the black hole is thermally stable, the heat capacity required to be positive. In other words, this positivity confirms that the black hole is locally stable(When $C_{Q}>0$, the black hole is in a stable phase, and when $C_{Q}<0$, the black hole experiences an unstable phase). From equations (\textcolor{blue}{35}), (\textcolor{blue}{39}) and the heat capacity relation we have
\begin{align*}
C_{Q}=\bigg(2N\pi \beta^2 r_{+}^{2N-1}\bigg)\times\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\\ \bigg(-\frac{(N-1)}{2\pi \beta^2(2N-1)}r_{+}^{1-2N}+\frac{(2N-1)m}{4\pi}r_{+}^{-2N}+\frac{\Lambda}{2\pi (4N-1)}r_{+}^{2N-1}+\frac{\Upsilon^{\prime}(r_{+})}{4\pi}\bigg)\\ \times \bigg(\frac{(N-1)}{2\pi \beta^2}r_{+}^{-2N}-\frac{2N(2N-1)m}{4\pi}r_{+}^{-2N-1}+\frac{(2N-1)\Lambda}{2\pi (4N-1)}r_{+}^{2N-2}+\frac{\Upsilon^{\prime\prime}(r_{+})}{4\pi}\bigg)^{-1} \tag{47}
\end{align*}
Where $\Upsilon(r_{+})$ is the Eq.(\textcolor{blue}{22}).
It is worthwhile to introduce two important thermodynamical points. These points are bounded point and phase transition point where:
\begin{equation}
\begin{cases}
T=0& \text{bounded point or type one phase transition} \\
(\frac{\partial T}{\partial r_{+}})_{Q}=0 & \text{phase transition point or type two phase transition} \tag{48}
\end{cases}
\end{equation}
In other words, the bounded point is where the heat capacity or temperature vanishes. The phase transition point is where the heat capacity diverges. Here, we show the heat capacity diagrams for different parameters. In these diagrams the phase transition points are clear to see.
\begin{center}
\includegraphics[width=14cm]{C0.jpg}
\captionof{figure}{C versus $r_{+}$ where N changes. we set $\Lambda=-1$, m=0.1, p=2, q=0.1 for left diagrams. $\Lambda=-0.1$, m=2, p=2, q=1 for right diagrams.}
\end{center}
\begin{center}
\includegraphics[width=14cm]{C.jpg}
\captionof{figure}{C versus $r_{+}$ where N changes. we set $\Lambda=-1$, m=2, p=2, q=1 for left diagrams. $\Lambda=-1$, m=2, p=2, q=1 for right diagrams.}
\end{center}
In Fig.3, we study the effect of dilaton field on the local stability of our solutions. In the left case, we compare two cases. First, when we do not have dilaton field(N=1) , second when we have this field($N\geq2$). For the first case, it goes under a unstable phase until it enjoys a type-2 phase transition(around $r=0.7$). In second case which contains the blue and green colors, we can see the stability before $r=0.7$. In the right case, we compare two cases as well. First, when we do not have dilaton field($N=1$) and second we have dilaton field with $N=1,2$. As is clear, the red case is behaves as the left diagram case. It is unstable until it enjoys a type-2 phase transition(around $r=7$). When we have dilaton field($N=1, 2$), the blue and green color diagrams are unstable until they experience a type-1 phase transition($r=1$). After this point, they will be stable completely. From the Fig.3, we can say that the presence of the dilaton field makes the solutions to be stable near the origin in compare with the ordinary case(when we do not have dilaton field $N=1$). This investigations are true for Fig.4 as well.
Now, we investigate the global stability by considering the Gibbs free energy.
\begin{equation}
G=M-TS-QU \tag{49}
\end{equation}
In order to say that the black hole is globally stable depends on the positivity of the Gibbs free energy with positive temperature. Now, it is important to emphasis that the Hawking-Page phase transition occurs in places where the Gibbs free energy vanishes. The idea of global stability and phase transition was first proposed by Hawking and Page \cite{HP}.
Here, we plot the Gibbs free energy diagrams to see the effect of some parameters on this quantity, schematically.
\begin{center}
\includegraphics[width=14cm]{G.jpg}
\captionof{figure}{G versus $r_{+}$. In the left diagram $\Lambda$ changes and we set q=1, N=2, m=1, p=2, $\beta=1$. In the right diagram where N changes, we set $\Lambda=-1 $, q=1, m=1, p=2, $\beta=1$.}
\end{center}
In Fig.5, we want to see the effect of dilaton field on the global stability of our solutions compare to the ordinary case($N=1$). In the left diagrams, the ordinary case(N=1) is unstable until it goes under the Hawking-Page phase transition(around r=1). After that it is stable completely. When we have the dilaton field($N\geq2$), the $N=2$ case behaves like the ordinary case, but for the case of $N=3$, we encounter the complete global unstability. In the right diagrams for the greater values of N, there is no any global stability. In sum, we can say the presence of diaton field makes the solutions to be globally unstable in compare to the ordinary case($N=1$).
\section{SMARR FORMULA}
In this section, we introduce the smarr formula for our solution. As is clear, this solution leads to a new kind of relation for the smarr formula. Smarr's formula has been studied a lot in the case of non-linear electrodynamics \cite{sn0} \cite{sn1}. From Eqs.(\textcolor{blue}{26}, \textcolor{blue}{31}, \textcolor{blue}{35}, \textcolor{blue}{39}, \textcolor{blue}{45}) and after some tedious calculations, we obtain:
\begin{align*}
M=\frac{2N}{2N-1}TS+\frac{N(N-1)}{(2N-1)^2}\bigg(\frac{S}{\pi \beta^2}\bigg)^{\frac{1}{2N}}-\frac{\beta^2N\Lambda}{(2N-1)(4N-1)}\bigg(\frac{S}{\pi \beta^2}\bigg)^{\frac{4N-1}{2N}}\\ -\frac{\big(2pN^2-4pN+2p-1\big)}{ p(2N-1)}Q U \tag{50}
\end{align*}
We name this equation as the smarr formula for the dilatonic black holes inspired by power-law electrodynamics. In this relation, the second term is added to the Smarr formula due to the presence of dilaton field, because when we set $N=1$ the dilaton field will disappear(one can check the Eq.(\textcolor{blue}{16})). Now, we want to see the changes of M(mass parameter) versus S(entropy). As is clear, we find that the dilaton field makes the M(mass parameter) to decrease for every fix values of S. This fact is obvious in the following diagrams. We have showed the fixed value(just one case for example) with a vertical yellow line. As someone can see the red line is belongs to the ordinary case(no dilaton field $N=1$) and the other colors represent the presence of the dilaton field($N=1, N=2$). In the presence of this field, the mass would be small and smaller.
We note that this statement is not always true for the temperature T, electric potential U and the electric charge.
\begin{center}
\includegraphics[width=14cm]{M.jpg}
\captionof{figure}{M versus S. we set T=1, U=1, Q=1, p=2, $\Lambda=-10$(left diagrams) when N changes. we set T=1, U=3, Q=1, p=3, $\Lambda=-20$(right diagrams) when N changes.}
\end{center}
\section{CONCLUDING REMARKS}
In this paper, we have presented a new class of slowly rotating black hole solutions in dilation gravity where dilaton field is coupled with nonlinear Maxwell invariant. From the equations of motion, we obtained the different components of field equations. We used these field equations to find an exact solution for this special case. For this solution, we showed that the dilaton field can be written as generalized three Liouville-type potentials. From different formalisms, we obtained some conserved quantities like the total mass, angular momentum, electric charge and the electric potential. In thermodynamic analysis, we have obtained the temperature and entropy(S) relations from geometrical methods. The validity of the first law of thermodynamics is also proved in these analysis. Also, we introduced the heat capacity and the Gibbs free energy to study the local and global stability of our solution, respectively. After some tedious calculation, we have obtained the Smarr formula as well. Due to the presence of dilaton field, a new term has been added to the smarr formula. We also showed that the dilaton field makes the black hole(AdS) mass to decrease for every fix values of S(entropy).
\section{APPENDIX}
\subsection{1}
The Ricci scalar relation for our solution is as follow:
\begin{equation}
\mathcal{R}=\frac{f^2(r)X^{\prime\prime}(r)+4f(r)X(r)f^{\prime\prime}(r)+4f(r)f^{\prime}(r)X^{\prime}(r)+2X(r)(f^{\prime}(r))^2-2}{f^2(r)} \tag{51}
\end{equation}
We have plotted the Ricci scalar for different parameters. In these diagrams this geometrical parameter diverges at r=0
\begin{center}
\includegraphics[width=14cm]{R.jpg}
\captionof{figure}{$\mathcal{R}$ versus r. we set m=1, q=1, N=2, p=2, $\beta=1$ when $\Lambda$ changes and m=1, q=1, $ \Lambda=-1$, p=2, $\beta=1$ when N changes. }
\end{center}
\subsection{2}
Here, we want to prove the following relation for our proposing gauge potential Eq.(\textcolor{blue}{7}) and for the slowly rotating case
\begin{equation}
\mathcal{F}=F_{\mu\nu}F^{\mu\nu}=-2\big(h^\prime(r)\big)^2 \tag{52}
\end{equation}
The non-vanishing components of this relation are:
\begin{equation}
\mathcal{F}=F_{\mu\nu}F^{\mu\nu}=F_{tr}F^{tr}+F_{rt}F^{rt}+F_{r\phi}F^{r\phi}+F_{\phi r}F^{\phi r}+F_{\theta \phi}F^{\theta \phi}+F_{\phi \theta}F^{\phi \theta} \tag{53}
\end{equation}
From the anti-symmetric property $F_{\mu\nu}=-F_{\nu\mu}$, $F^{\mu\nu}=-F^{\nu\mu}$ we have
\begin{equation}
\mathcal{F}=F_{\mu\nu}F^{\mu\nu}=2\bigg(F_{tr}F^{tr}+F_{r\phi}F^{r\phi}+F_{\theta \phi}F^{\theta \phi}\bigg) \tag{54}
\end{equation}
Also by using $F^{\mu\nu}=g^{\mu \alpha} g^{\nu \beta} F_{\alpha \beta}$, one can write
\begin{align*}
\mathcal{F}=F_{\mu\nu}F^{\mu\nu}=\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\\2\bigg(g^{tt} g^{rr}( F_{tr})^2+g^{t\phi} g^{rr} F_{tr}F_{\phi r}+g^{rr} g^{\phi t} F_{r\phi}F_{rt}+g^{rr} g^{\phi \phi}( F_{r\phi})^2+g^{\theta\theta} g^{\phi \phi}( F_{\theta \phi})^2\bigg) \tag{55}
\end{align*}
Using the metric ansatz Eq.(\textcolor{blue}{6}) we obtain
\begin{align*}
\mathcal{F}=F_{\mu\nu}F^{\mu\nu}=\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\\2\bigg(-(h^{\prime}(r))^2+\frac{2a^2H(r)}{f^2(r)}(h^{\prime}(r))^2sin^2\theta-\frac{a^2}{X(r)f^2(r)}(h^{\prime}(r))^2sin^2\theta+\frac{4a^2}{f^4(r)}h^2(r)cos^2\theta\bigg) \tag{56}
\end{align*}
For the slowly rotating case we assume $a^2=0$(we drop the second order or higher orders of a), then we arrive:
\begin{equation}
\mathcal{F}=F_{\mu\nu}F^{\mu\nu}=-2\big(h^\prime(r)\big)^2 \tag{57}
\end{equation}
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 6,978
|
Our growth depends within the innovative equipment, fantastic talents and repeatedly strengthened technology forces for Slurry Pump Part D3036hs1 , Slurry Pump Part D3036HS1 , Slurry Pump Parts , should you be intrigued in any of our products and solutions.
We have by far the most highly developed manufacturing equipment, experienced and qualified engineers and workers, regarded high quality handle systems along with a friendly qualified revenue team pre/after-sales support for Slurry Pump Part D3036hs1 , Slurry Pump Part D3036HS1 , Slurry Pump Parts , With good quality reasonable price and sincere service we enjoy a good reputation. Products are exported to South America Australia Southeast Asia and so on. Warmly welcome customers at home and abroad to cooperate with us for the brilliant future.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 2,029
|
Reinhold Schlothauer (* 12. Februar 1948 in Eschwege) ist ein deutscher Strafverteidiger und Rechtswissenschaftler.
Leben
Schlothauer studierte von 1967 bis 1972 Jura an den Universitäten Marburg und Bonn. 1976 promovierte Schlothauer an der Universität Bremen zum Thema Zur Krise der Verfassungsgerichtsbarkeit und ist seit demselben Jahr am Landgericht Bremen als Rechtsanwalt zugelassen. 1982 erhielt er die Zulassung als Anwalt beim Hanseatischen Oberlandesgericht Bremen. Er lehrt seit 1988 an der Universität Bremen und ist dort seit 1994 Honorarprofessor. Schlothauer ist ferner Mitglied des Strafrechtsausschusses der Bundesrechtsanwaltskammer und Beiratsmitglied beim Fachinstitut Strafrecht des Deutschen Anwaltsinstituts.
Schlothauer wurde als Fachanwalt für Strafrecht zugelassen und gehört der Bremer Rechtsanwalts- und Notars-Sozietät Joester & Partner an. Er war unter anderem als Strafverteidiger in dem Strafprozess in der Folge des Konkurses der Bremer Großwerft Bremer Vulkan AG und im Prozess um den Unfall auf der Transrapid-Versuchsanlage Emsland vom 22. September 2006 tätig.
Als Wissenschaftler befasst er sich mit Straf- und Strafprozessrecht sowie mit der Kriminalpolitik in Europa. Schlothauer verfasste, teils als Co-Autor, mehrere Fachbücher vor allem zur Strafverteidigung, die zum Teil mehrere Auflagen erfuhren. Zudem verfasste er mehrere Beiträge zum Münchener Anwaltshandbuch Strafverteidigung. Er war 1980 Mitgründer der juristischen Fachzeitschrift Strafverteidiger, deren Redaktion er seitdem angehört.
Ehrungen
2001: Max-Alsberg-Preis der Vereinigung Deutsche Strafverteidiger e. V.; zusammen mit Klaus Lüderssen und Hans-Joachim Weider für das gemeinsame langjährige Engagement in der Redaktion der Fachzeitschrift Strafverteidiger.
Publikationen
Zur Krise der Verfassungsgerichtsbarkeit. Neuere Ansätze zur Methodik der Verfassungsinterpretation. Untersucht am Beispiel Horst Ehmke, Peter Häberle, Konrad Hesse, Martin Kriele und Friedrich Müller. Europäische Verlagsanstalt, Köln u. a. 1979, ISBN 3-434-25108-1 (zugleich Dissertation, Universität Bremen, 1976).
Vorbereitung der Hauptverhandlung durch den Verteidiger. Mit notwendiger Verteidigung und Pflichtverteidigung (= Praxis der Strafverteidigung, Bd. 10). 2., neu bearbeitete und erweiterte Auflage. C.F. Müller Verlag, Heidelberg u. a. 2001, ISBN 3-8114-1798-3.
zusammen mit Martin Niemöller und Hans-Joachim Weider: Gesetz zur Verständigung im Strafverfahren. Kommentar. Verlag C. H. Beck, München 2010, ISBN 978-3-406-59777-0.
zusammen mit Hans-Joachim Weider: Untersuchungshaft. Mit Erläuterungen zu den UVollzG der Länder (= Praxis der Strafverteidigung, Bd. 14). 4., neu bearbeitete und erweiterte Auflage. C.F. Müller Verlag, Heidelberg u. a. 2010, ISBN 978-3-8114-3494-3. (Rezension von Michael Hettinger im Fachbuchjournal).
zusammen mit Hans-Joachim Weider: Verteidigung im Revisionsverfahren – Praxis der Strafverteidigung, Bd. 23. C.F. Müller Verlag, Heidelberg u. a. 2013, ISBN 978-3-8114-4105-7.
Weblinks
auf der Website der Universität Bremen
Einzelnachweise
Rechtsanwalt (Deutschland)
Rechtswissenschaftler (21. Jahrhundert)
Rechtswissenschaftler (20. Jahrhundert)
Hochschullehrer (Universität Bremen)
Deutscher
Geboren 1948
Mann
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 263
|
Q: ntoskrnl.exe+7211eb 0x00000124 BSOD Windows 10 64bit. I've been having consistent crashes when running intensive applications. I have tried running memtest with no results and updating drivers.
Minidump
https://www.dropbox.com/s/xz1m0v0l3ljapcu/090517-48859-01.dmp?dl=0
CPU-Z report
https://www.dropbox.com/s/b6u1wqfdo4yfvyo/DESKTOP-K1EQJ94.txt?dl=0
A: Bugcheck 0x124 means Fatal Hardware issue
The WHEA_UNCORRECTABLE_ERROR bug check has a value of 0x00000124. This
bug check indicates that a fatal hardware error has occurred.
Analyzing the dump with Windbg.exe shows that you get BUSLG_GENERIC_ERR_*_TIMEOUT_ERR error.
*******************************************************************************
* *
* Bugcheck Analysis *
* *
*******************************************************************************
WHEA_UNCORRECTABLE_ERROR (124)
A fatal hardware error has occurred. Parameter 1 identifies the type of error
source that reported the error. Parameter 2 holds the address of the
WHEA_ERROR_RECORD structure that describes the error conditon.
Arguments:
Arg1: 0000000000000000, Machine Check Exception
Arg2: ffff890e96f6c038, Address of the WHEA_ERROR_RECORD structure.
Arg3: 0000000000000000, High order 32-bits of the MCi_STATUS value.
Arg4: 0000000000000000, Low order 32-bits of the MCi_STATUS value.
Debugging Details:
------------------
DUMP_CLASS: 1
DUMP_QUALIFIER: 400
BUILD_VERSION_STRING: 10.0.15063.540 (WinBuild.160101.0800)
DUMP_TYPE: 2
BUGCHECK_P1: 0
BUGCHECK_P2: ffff890e96f6c038
BUGCHECK_P3: 0
BUGCHECK_P4: 0
BUGCHECK_STR: 0x124_AuthenticAMD
CPU_COUNT: 6
CPU_MHZ: dbc
CPU_VENDOR: AuthenticAMD
CPU_FAMILY: 15
CPU_MODEL: 2
CPU_STEPPING: 0
CUSTOMER_CRASH_COUNT: 1
DEFAULT_BUCKET_ID: WIN8_DRIVER_FAULT
PROCESS_NAME: System
CURRENT_IRQL: 0
STACK_TEXT:
00 nt!WheapCreateLiveTriageDump
01 nt!WheapCreateTriageDumpFromPreviousSession
02 nt!WheapProcessWorkQueueItem
03 nt!WheapWorkQueueWorkerRoutine
04 nt!ExpWorkerThread
05 nt!PspSystemThreadStartup
06 nt!KiStartSystemThread
MODULE_NAME: AuthenticAMD
FAILURE_BUCKET_ID: 0x124_AuthenticAMD_PROCESSOR_BUS_PRV
You use the F2 BIOS for your GA-78LMT-USB3:
DMI BIOS
vendor Award Software International Inc.
version F2
date 11/25/2014
ROM size 4096 KB
DMI System Information
manufacturer Gigabyte Technology Co. Ltd.
product GA-78LMT-USB3
so update the BIOS and look if this improves stability.
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 3,045
|
//
// RankingListViewController.m
// Bage
//
// Created by Lori on 14-1-10.
// Copyright (c) 2014年 Duger. All rights reserved.
//
#import "RankingListViewController.h"
#import "BageCell.h"
#import "DetailViewController.h"
@interface RankingListViewController ()
@end
@implementation RankingListViewController
- (id)initWithStyle:(UITableViewStyle)style
{
self = [super initWithStyle:style];
if (self) {
// Custom initialization
}
return self;
}
- (void)viewDidLoad
{
[super viewDidLoad];
// Uncomment the following line to preserve selection between presentations.
// self.clearsSelectionOnViewWillAppear = NO;
// Uncomment the following line to display an Edit button in the navigation bar for this view controller.
// self.navigationItem.rightBarButtonItem = self.editButtonItem;
}
- (void)didReceiveMemoryWarning
{
[super didReceiveMemoryWarning];
// Dispose of any resources that can be recreated.
}
#pragma mark - Table view data source
//- (NSInteger)numberOfSectionsInTableView:(UITableView *)tableView
//{
//#warning Potentially incomplete method implementation.
// // Return the number of sections.
// return 10;
//}
- (NSInteger)tableView:(UITableView *)tableView numberOfRowsInSection:(NSInteger)section
{
// Return the number of rows in the section.
return 10;
}
- (UITableViewCell *)tableView:(UITableView *)tableView cellForRowAtIndexPath:(NSIndexPath *)indexPath
{
static NSString *CellIdentifier = @"Cell";
BageCell *cell = [tableView dequeueReusableCellWithIdentifier:CellIdentifier];
if (cell == nil) {
cell = [[[BageCell alloc] initWithStyle:UITableViewCellStyleDefault reuseIdentifier:CellIdentifier cellType:BageCellTypeWithRankingList] autorelease];
}
// Configure the cell...
return cell;
}
- (CGFloat)tableView:(UITableView *)tableView heightForRowAtIndexPath:(NSIndexPath *)indexPath
{
return 50;
}
- (void)tableView:(UITableView *)tableView didSelectRowAtIndexPath:(NSIndexPath *)indexPath
{
DetailViewController *detailVC = [[DetailViewController alloc] init];
[self.navigationController pushViewController:detailVC animated:YES];
}
/*
// Override to support conditional editing of the table view.
- (BOOL)tableView:(UITableView *)tableView canEditRowAtIndexPath:(NSIndexPath *)indexPath
{
// Return NO if you do not want the specified item to be editable.
return YES;
}
*/
/*
// Override to support editing the table view.
- (void)tableView:(UITableView *)tableView commitEditingStyle:(UITableViewCellEditingStyle)editingStyle forRowAtIndexPath:(NSIndexPath *)indexPath
{
if (editingStyle == UITableViewCellEditingStyleDelete) {
// Delete the row from the data source
[tableView deleteRowsAtIndexPaths:@[indexPath] withRowAnimation:UITableViewRowAnimationFade];
}
else if (editingStyle == UITableViewCellEditingStyleInsert) {
// Create a new instance of the appropriate class, insert it into the array, and add a new row to the table view
}
}
*/
/*
// Override to support rearranging the table view.
- (void)tableView:(UITableView *)tableView moveRowAtIndexPath:(NSIndexPath *)fromIndexPath toIndexPath:(NSIndexPath *)toIndexPath
{
}
*/
/*
// Override to support conditional rearranging of the table view.
- (BOOL)tableView:(UITableView *)tableView canMoveRowAtIndexPath:(NSIndexPath *)indexPath
{
// Return NO if you do not want the item to be re-orderable.
return YES;
}
*/
/*
#pragma mark - Table view delegate
// In a xib-based application, navigation from a table can be handled in -tableView:didSelectRowAtIndexPath:
- (void)tableView:(UITableView *)tableView didSelectRowAtIndexPath:(NSIndexPath *)indexPath
{
// Navigation logic may go here, for example:
// Create the next view controller.
<#DetailViewController#> *detailViewController = [[<#DetailViewController#> alloc] initWithNibName:@"<#Nib name#>" bundle:nil];
// Pass the selected object to the new view controller.
// Push the view controller.
[self.navigationController pushViewController:detailViewController animated:YES];
}
*/
@end
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 4,084
|
Draconic is a Serbian heavy metal band.
Draconic may also refer to:
Of or pertaining to a dragon
Of or pertaining to the constellation Draco
A harsh punishment, in reference to the Greek lawgiver Draco
The fictional language used in the video game The Elder Scrolls V: Skyrim
The fictional language used in the table top role-playing game franchise Dungeons and Dragons
Draconic period, an orbital period
See also
Draco (disambiguation)
Draconian (disambiguation)
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 2,516
|
Burton Albion against Brentford on Tuesday 6th March 2018. Burton Albion have scored 0 times so far this season while Brentford have scored 0 goals. In defence Burton Albion have conceded 0 while Brentford have let in 0.
Burton Albion have 3 wins out of their previous 8 fixtures. Brentford have won 4 from the previous eight fixtures.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 7,535
|
\section{Introduction}
Conformal transformations play an interesting role in physics and
mathematics\cite{FMS}. Conventionally, there are two meanings of
the term ``conformal transformation'' in the literature. One is
what we call the conformal coordinate transformation, which is a special
kind of coordinate transformations and whose infinitesimal generators
are the conformal Killing vectors (CKVs). The other is the local rescaling
(more commonly called the Weyl transformation), which acts directly
on the metric and can be regarded as a kind of gauge transformation.
These two meanings certainly have some relationship, but it is more
appropriate to clearly distinguish them.Unless otherwise
specified, we mean in this paper by the term ``conformal transformation''
as the conformal coordinate transformation\emph{.}
It has been shown that Maxwell's equations are invariant under the
larger conformal group \cite{E. Cunningham,Bateman-1,Rosen,FN}.
Codirla and Osborn pointed out that one can easily obtain the
electromagnetic field associated with uniformly accelerated charged
particles by the conformal invariance of the Maxwell equations with
point-like charges \cite{C. Codirla and H. Osborn}. And conformal
transformations in 2-dimension also play an important role in string
theory (see e.g. \cite{J. Polchinski}). However, as what we will
introduce in the next section, conformal transformations can not be
globally defined on the Minkowski spacetime, which means that the
conformal invariance of electrodynamics can only be regarded as
local. In fact, conformal transformations can always globally act
\textcolor{black}{on a}\textcolor{blue}{{}
}\textcolor{black}{$d$-dimensional} compactification spacetime
($d>2$),\textcolor{black}{{} which is called the conformally
compactified spacetime.} This idea inspires us to compactify the Minkowski, even dS/AdS,
spacetime.\textcolor{black}{{} In fact}, electrodynamics even cannot
be well defined globally on the ordinary conformal compactification
of the Minkowski spacetime. Instead, electrodynamics globally
defined on the double covering of that conformal compactification is
possible, which has long been noticed as mentioned in Ref. \cite{C.
Codirla and H. Osborn}. We call this double covering the doubled
conformal compactification.Interestingly, Penrose had shown the doubled Penrose diagrams in
\cite{penrose}, which is closely related to the doubled conformal
compactification here. In this paper, we try to construct the
doubled conformal conpactification to make the electrodynamics
globally defined.
Considering that the zero-radius pseudo-sphere in (4+2)-dimensional
Minkowski space can always simplify the problem and give an obvious
map about conformal compactification (see for example \cite{C. Codirla and H. Osborn,arXiv:,weinberg,P. A. M. Dirac,M. S. Costa}),
we briefly give a review of conformal compactification in the viewpoint
of the projective cone in Section 2. We show the breakdown of the
Maxwell equations on the compactification of Minkowski dS/AdS spacetime
in Section 3 and give Penrose diagrams to describe the problem in
details, in which one can find out the double covering solution, in
Section 4.
\section{General Theory and the Projective Cone}
The conformal transformations form a group, named the conformal group,
under certain conditions. It is well known that the conformal group
on the $d$-dimensional Euclidean space $\mathbb{R}^{d}$ is of $(d+1)(d+2)/2$
dimensions for $d>2$ and of infinite dimensions for $d=2$. But general
conformal transformations cannot be globally defined on such a flat
space, so this well-known result has only considered the local aspects
of conformal transformations. In fact, nontrivial conformal transformations
can be globally defined on the $d$-sphere $S^{d}$ instead, where
the conformal group is of $(d+1)(d+2)/2$ dimensions whether $d>2$
or $d=2$. There exists a conformal mapping between $\mathbb{R}^{d}$
and $S^{d}$, which smoothly extends the conventional conformal transformations
on the former to the globally defined ones on the latter. For $d=2$
only a $(d+1)(d+2)/2=6$ dimensional subgroup of the infinite-dimensional
``conformal group'' on $\mathbb{R}^{2}$ can be so extended, which
can be regarded as the ``global'' conformal group on $\mathbb{R}^{2}$.
\footnote{These two distinct conformal groups for the $d=2$ case can also be identified as the ``angle-preserving'' one and the ``circle-preserving''
one, respectively. See Ref. \cite{HS}. For another elucidation of
this problem, see Ref. \cite{FMS}, Chapter 5.}
There is exactly one point on $S^{d}$ that has no image on $\mathbb{R}^{d}$
under the conformal mapping. That point, though actually not on $\mathbb{R}^{d}$,
is called the infinity point (also known as the conformal boundary
for the case that the space has a non-positive-definite signature)
of $\mathbb{R}^{d}$. The procedure that adds the infinity point to
$\mathbb{R}^{d}$, so that the conformal transformations can be globally
defined, is known as the conformal compactification of $\mathbb{R}^{d}$.
The resulting compactified space ($S^{d}$ here) is also called the
conformal compactification (of $\mathbb{R}^{d}$). $\mathbb{R}^{d}$
and $S^{d}$ are both constant curvature (or maximal symmetry) spaces.
In differential geometry it is known that only
constant curvature spaces have $(d+1)(d+2)/2$ independent CKVs ($d>2$).
In fact, all the constant curvature spaces, with any metric signatures,
have $(d+1)(d+2)/2$-dimensional ``global'' conformal groups and
the corresponding conformal compactifications for $d\ge2$.
\footnote{$S^{d}$ is its own conformal compactification.}
To see how the doubly conformal compactification arises, let us first
have a simple review of the ordinary conformal compactification of
the Minkowski spacetime. As constant curvature spaces with the same
signature, de Sitter (dS) and anti-de Sitter (AdS) space times also
have that conformal compactification, which can be all treated from
the viewpoint of the projective cone
\footnote{Also called the null cone or the Lie sphere.
}. From now on, we concentrate on the 4-dimensional case.
The projective cone $[\mathcal{N}]$ is defined as a zero-radius pseudo-sphere
$\mathcal{N}$ in a ($4+2$)-dimensional Minkowski space:
\begin{equation}
\eta_{AB}\zeta^{A}\zeta^{B}=0,\quad(\eta_{AB})=\diag(-1,1,1,1,1,-1),\label{zero-sphere}
\end{equation}
modulo the projective equivalence relatio
\footnote{In order for $[\mathcal{N}]$ to be a (4-dimensional) manifold, the
origin $(\zeta^{A})=0$ must be excluded
}
\begin{equation}
(\zeta^{A})\sim\lambda(\zeta^{A}),\quad\lambda\neq0.\label{projective}
\end{equation}
The equivalence class corresponding to the point $(\zeta^{A})$ is
denoted by $[\zeta^{A}]$. The whole ($4+2$)-dimensional Minkowski
space $\mathbb{R}^{6}$ modulo the projective equivalence relation
(\ref{projective}) is the 5-dimensional projective space $\mathbb{R}P^{5}$,
so $[\mathcal{N}]$ is a submanifold of $\mathbb{R}P^{5}$. It is
obvious that the pseudo-sphere (\ref{zero-sphere}) and the equivalence
relation (\ref{projective}) are both invariant under general $O(2,4)$
transformations, among which a $\mathbb{Z}_{2}$ antipodal reflection
\begin{equation}
(\zeta^{A})\rightarrow-(\zeta^{A})\label{antipodal}
\end{equation}
acts trivially on $[\mathcal{N}]$.
Then it can be shown that $[\mathcal{N}]$ is the conformal compactification
of all the Minkowski, dS and AdS space times, where the $O(2,4)/\mathbb{Z}_{2}$
transformations act as conformal transformations on these space times.
These different space times can be regarded as different choices of
representatives in the equivalence classes on $\mathcal{N}$. In fact,
these constant curvature space times correspond to choose representative
points by intersecting $\mathcal{N}$ with hyperplanes in $\mathbb{R}^{6}$,
where the metrics on them are naturally induced from $\eta_{AB}$.
For the hyperplane $\mathcal{P}_{a}$:
\begin{equation}
a_{A}\zeta^{A}=1\label{}
\end{equation}
with $(a_{A})\neq0$ the normal vector, it can be shown that the intersection
manifold is characterized by
\begin{equation}
S=\eta^{AB}a_{A}a_{B}\label{}
\end{equation}
as follows:
\begin{itemize}
\item $S<0$: dS spacetime,
\item $S=0$: Minkowski spacetime,
\item $S>0$: AdS spacetime,
\end{itemize}
where in any case $S$ is just the scalar curvature of the intersection
manifold .
More concretely, to get the Minkowski spacetime we just choose a light-like
normal vector
\begin{equation}
(a_{A})=(0,0,0,0,1,1).\label{}
\end{equation}
Thus the intersection $M$ of $\mathcal{N}$ and the corresponding
$\mathcal{P}_{a}$ is flat with respect to the metric induced from
$\eta_{AB}$, and can be parametrized by
\begin{equation}
x^{\mu}=L\frac{\zeta^{\mu}}{\zeta^{+}},\quad\mu=0,\cdots,3,\label{}
\end{equation}
with $L$ an arbitrary length scale parameter and $\zeta^{\pm}=\zeta^{5}\pm\zeta^{4}$
the lightcone coordinates. The induced metric is proportional to vect
\begin{equation}
\rmd s^{2}=\eta_{\mu\nu}\rmd x^{\mu}\rmd x^{\nu},\label{}
\end{equation}
so $x^{\mu}$ is just the Cartesian coordinates on $M$. Then it is
straightforward to show that the $O(2,4)/\mathbb{Z}_{2}$ transformations
act as conformal transformations on $M$ (\textcolor{black}{for more
details, see \cite{penrose}}). Note that some equivalence classes
on $\mathcal{N}$, which correspond to the infinity points of $M$,
have no representatives on $\mathcal{P}_{a}$. They constitute precisely
the intersection of $\mathcal{N}$ and the hyperplane
\begin{equation}
a_{A}\zeta^{A}=0,\label{}
\end{equation}
which is parallel to $\mathcal{P}_{a}$. Adding those infinity points
to $M$ produces its conformal compactification $[\mathcal{N}]$,
whose $O(2,4)/\mathbb{Z}_{2}$ action is fully well-defined.
The dS and AdS space times can be obtained by choosing typically
\begin{equation}
(a_{A})=(0,0,0,0,0,1)\label{}
\end{equation}
and
\begin{equation}
(a_{A})=(0,0,0,0,1,0),\label{}
\end{equation}
respectively. However, they also have infinity points, or conformal
boundary, to be included for a fully well-defined $O(2,4)/\mathbb{Z}_{2}$
conformal action, since no single hyperplane can contain representatives
of all the equivalence classes on $\mathcal{N}$. To remedy this problem,
one may use general hypersurfaces (of antipodal symmetry) to intersect
$\mathcal{N}$. It can be shown that these intersection manifolds
are all conformally flat
\footnote{So they are all related by Weyl transformations (also called conformal
mappings), at least locally
} and that the $O(2,4)/\mathbb{Z}_{2}$ transformations induce conformal
transformations on them. The simplest choice of this hypersurface
is a 5-sphere
\begin{equation}
\delta_{AB}\zeta^{A}\zeta^{B}=2L^{2},\label{5-sphere}
\end{equation}
which intersects all the equivalence classes on $\mathcal{N}$ precisely
twice. The intersection of $\mathcal{N}$ and the 5-sphere (\ref{5-sphere})
is an $S^{1}\times S^{3}$:
\begin{equation}
\left\{ \begin{array}{rcl}
(\zeta^{0})^{2}+(\zeta^{5})^{2} & = & L^{2},\\
(\zeta^{1})^{2}+(\zeta^{2})^{2}+(\zeta^{3})^{2}+(\zeta^{4})^{2} & = & L^{2}.
\end{array}\right.\label{intersection}
\end{equation}
Upon the antipodal identification (\ref{antipodal}), one sees that
$[\mathcal{N}]$ is of topology
\begin{equation}
S^{1}\times S^{3}/\mathbb{Z}_{2},\label{}
\end{equation}
which is actually homeomorphic to $S^{1}\times S^{3}$.
Although there is no natural metric defined on $[\mathcal{N}]$, there
is an induced metric on the intersection manifold (\ref{intersection})
(modulo the antipodal identification), which is called $N$ hereafter.
$N$ is conformally flat and has a globally defined $O(2,4)/\mathbb{Z}_{2}$
conformal group, so it can be regarded as a metrical realization of
$[\mathcal{N}]$, being the conformal compactification of all the
Minkowski, dS and AdS space times.
The above construction of conformal compactification can be extended
to any dimension and any metric signature. However, for even $d$
it can be shown that $S^{1}\times S^{d}/\mathbb{Z}_{2}$ is not homeomorphic
to $S^{1}\times S^{d}$ but is a non-orientable manifold, so the conformal
compactification of ($1+d$)-dimensional spacetime is not simple and
in some sense not suitable to be a spacetime. One immediately sees
that this shortcoming can be simply overcome by discarding the $\mathbb{Z}_{2}$
antipodal identification. That is the doubly conformal compactification,
which can be realized by replacing the projective equivalence relation
(\ref{projective}) with the pseudo-projective one:
\begin{equation}
(\zeta^{A})\sim\lambda(\zeta^{A}),\quad\lambda>0.\label{}
\end{equation}
For the $d=3$ case, we use $[\mathcal{N}]_{+}$ to denote ${\cal N}$
modulo the above equivalence relation, i.e., the doubly conformal
compactification of all the Minkowski, dS and AdS space times, whose
metrical realization is exactly the intersection manifold (\ref{intersection}),
denoted by $2N$. Correspondingly, the conformal group on (\ref{intersection})
is $O(2,4)$ instead of $O(2,4)/\mathbb{Z}_{2}$.
\section{The breakdown of the Maxwell Equation on the Compactification of
Minkowski Spacetime}
We have shown the geometrical process of compactification.
It is the most important that physical considerations support the
introduction of the doubly conformal compactification. At this section
we will introduce the discontinuity of electrodynamics caused by compactifing
the Minkowski spacetime. The action functional of electrodynamics
in general space times is
\begin{equation}
S_{\mathrm{EM}}=\int d^{4}x\sqrt{-g}\left(-\frac{1}{4}g^{\mu\alpha}g^{\nu\beta}F_{\mu\nu}F_{\alpha\beta}+g^{\mu\nu}J_{\mu}A_{\nu}\right),\quad F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}.\label{action}
\end{equation}
In this paper the metrics always have a $(-,+,+,+)$ signature. For
the Minkowski spacetime with Cartesian coordinates ($g_{\mu\nu}=\eta_{\mu\nu}$),
the above action functional is invariant (up to a boundary term) under
the conformal transformations provided that $A_{\mu}$ and $J_{\mu}$
transform as
\begin{equation}
A_{\mu}(x)=\frac{\partial\tilde{x}^{\nu}}{\partial x^{\mu}}\tilde{A}_{\nu}(\tilde{x}),\quad J_{\mu}(x)=\Omega^{2}\frac{\partial\tilde{x}^{\nu}}{\partial x^{\mu}}\tilde{J}_{\nu}(\tilde{x}),\label{}
\end{equation}
respectively, where the transformation of $A_{\mu}$ is just trivial
and that of $J_{\mu}$ contains a conformal factor $\Omega^{2}$ defined
by
\begin{equation}
\frac{\partial\tilde{x}^{\mu}}{\partial x^{\rho}}\frac{\partial\tilde{x}^{\nu}}{\partial x^{\sigma}}\eta_{\mu\nu}=\Omega^{2}\eta_{\rho\sigma}.\label{}
\end{equation}
In fact, the action functional (\ref{action}) in general space times
is invariant under, in addition to the diffeomorphism, the Weyl transformation
\begin{equation}
g_{\mu\nu}(x)=k^{-2}(x)g'_{\mu\nu}(x),\label{}
\end{equation}
provided that $A_{\mu}$ (and thus $F_{\mu\nu}$) is invariant and
$J_{\mu}$ transforms as
\begin{equation}
J_{\mu}(x)=k^{2}(x)J'_{\mu}(x).\label{J_Weyl}
\end{equation}
Since the dS, AdS and $N$ space times can all be obtained by Weyl
transformations from the Minkowski spacetime, where the Cartesian
coordinates $x^{\mu}$ become the conformally flat coordinates on
these space times, we can easily map the electrodynamics from the
Minkowski spacetime to them.
The conformally flat coordinates on the dS spacetime can be obtained
by the stereographic projection \cite{Tian}, with the Weyl factor
\begin{equation}
k_{{\rm dS}}^{-2}(x)=\left(1+\frac{x^{2}}{4R^{2}}\right)^{2},\quad x^{2}=\eta_{\mu\nu}x^{\mu}x^{\nu},\label{dS_factor}
\end{equation}
with $R$ the dS radius. The AdS case is simply achieved by replacing
$R^{2}$ with $-R^{2}$ for most of the dS expressions, so we only
mention the dS case. The Weyl factor for certain conformally flat
coordinates on $N$ can be shown to be
\begin{equation}
k_{N}^{-2}(x)=1+\frac{x^{0}x^{0}+\sum_{i}x^{i}x^{i}}{2L^{2}}+\left(\frac{x^{2}}{4L^{2}}\right)^{2},\label{N_factor}
\end{equation}
which is positive definite reflecting the fact that $N$ has no conformal
boundary.
The action functional (\ref{action}) with $g_{\mu\nu}=\eta_{\mu\nu}$
\begin{equation}
S_{\mathrm{EM}}=\int d^{4}x\left(-\frac{1}{4}\eta^{\mu\alpha}\eta^{\nu\beta}F_{\mu\nu}F_{\alpha\beta}+\eta^{\mu\nu}J_{\mu}A_{\nu}\right)\label{flat action}
\end{equation}
leads to the familiar equation of motion
\begin{equation}
\eta^{\alpha\mu}\partial_{\alpha}F_{\mu\nu}=J_{\nu}.\label{EOM}
\end{equation}
The simplest solution to the above equation is
the Coulomb field for a static point charge:
\begin{equation}
E_{i}=F_{i0}=\frac{e}{4\pi}\frac{x^{i}}{|\vect x|^{3}},\quad B_{i}=\half\epsilon^{ijk}F_{jk}=0,\quad J_{0}=e\delta^{3}(\vect x),\quad J_{i}=0.\label{fundamental}
\end{equation}
General solutions to equation (\ref{EOM})
are just linear combinations of this fundamental
solution. Of special interest is the electromagnetic field
associated with uniformly accelerated point charge, which can be
obtained from the solution (\ref{fundamental}) by conformal
transformations \cite{C. Codirla and H. Osborn}. There one sees
that, however, an additional sign factor $\epsilon(\Omega)$ has to
be inserted (or equivalently, discarded) to attain a globally
defined solution. That, in fact, already indicates the doubly
conformal compactification. In the following, we will use Weyl
transformations to map the solution (\ref{fundamental}) onto the dS
and $N$ space times, where the doubly conformal compactification is
shown to be necessary. For simplicity, we take $R=1/2$ in equation
(\ref{dS_factor}) and massless $L=1/2$ in equation (\ref{N_factor}).
First we consider the dS case. In this case we have from equations
(\ref{J_Weyl}) and (\ref{dS_factor})
\begin{equation}
J'_{0}=e(1+x^{2})^{2}\delta^{3}(\vect x),\quad J'_{i}=0,\label{dS_J}
\end{equation}
while $E'_{i}$ and $B'_{i}$ are the same as $E_{i}$ and $B_{i}$ in
equation (\ref{fundamental}), respectively. Note that there are
actually two antipodal point charges here, due to the conformal
boundary $1+x^{2}=0$ separating the world line $\vect x=0$ into two
parts. Properly speaking, the line $\vect x=0$ is
separated into three segments, but the outer two of them are joined
through the Minkowski conformal infinity, as shown in figure
\ref{Penrose-CFdS}, where the dash line KN can be
regarded as the world line of the charges in dS spacetime. For
convenience, we omit the prime in these notations in the following
equations. Since one patch of conformally flat coordinates
cannot cover the whole dS spacetime
\footnote{The uncovered part corresponds to (part of) the conformal boundary
of the Minkowski spacetime
} other coordinate patches must also be used to check whether the above
solution is globally defined on the dS spacetime. It is easy to see
that one more (conformally-flat-coordinate) patch is sufficient, which
can be viewed as the inversion
\begin{equation}
x^{\mu}=\frac{\tilde{x}^{\mu}}{\tilde{x}^{2}}\label{inversion}
\end{equation}
of the original conformally flat coordinates. ( One can define
special conformal transformations by combining an inversion with a
translation and then another inversion \cite{C. Codirla and H.
Osborn}.) Under the coordinate transformation (\ref{inversion}), we
have
\begin{equation}
\tilde{J}_{\mu}(\tilde{x})=\frac{\partial x^{\nu}}{\partial\tilde{x}^{\mu}}J_{\nu}(x)=\left[\frac{\delta_{\mu}^{\nu}}{\tilde{x}^{2}}-2\frac{\tilde{x}^{\nu}\tilde{x}_{\mu}}{(\tilde{x}^{2})^{2}}\right]J_{\nu}(x),\quad\tilde{x}_{\mu}=\eta_{\mu\nu}\tilde{x}^{\nu},
\end{equation}
which means
\begin{eqnarray}
\tilde{J}_{0}(\tilde{x}) & = & \frac{J_{0}(x)}{\tilde{x}^{2}}-2\tilde{x}_{0}\frac{\tilde{x}^{0}J_{0}(x)+\tilde{x}^{i}J_{i}(x)}{(\tilde{x}^{2})^{2}}\nonumber \\
& = & e[1+(\tilde{x}^{2})\inv]^{2}\left[\frac{1}{\tilde{x}^{2}}+\frac{2\tilde{t}^{2}}{(\tilde{x}^{2})^{2}}\right]\delta^{3}\left(\frac{\tilde{\vect x}}{\tilde{x}^{2}}\right)\nonumber \\
& = & e(1-\tilde{t}^{2})^{2}\delta^{3}(\tilde{\vect x})\label{dS_inverse_J}
\end{eqnarray}
with $\tilde{t}=\tilde{x}^{0}$, and
\begin{equation}
\tilde{J}_{i}(\tilde{x})=\frac{J_{i}(x)}{\tilde{x}^{2}}-2\tilde{x}_{i}\frac{\tilde{x}^{0}J_{0}(x)+\tilde{x}^{i}J_{i}(x)}{(\tilde{x}^{2})^{2}}=0.\label{}
\end{equation}
At the same time, the electromagnetic field transforms as
\begin{equation}
\tilde{F}_{\mu\nu}=\left[\frac{\delta_{\mu}^{\rho}}{\tilde{x}^{2}}-2\frac{\tilde{x}^{\rho}\tilde{x}_{\mu}}{(\tilde{x}^{2})^{2}}\right]\left[\frac{\delta_{\nu}^{\sigma}}{\tilde{x}^{2}}-2\frac{\tilde{x}^{\sigma}\tilde{x}_{\nu}}{(\tilde{x}^{2})^{2}}\right]F_{\rho\sigma}=\frac{F_{\mu\nu}}{(\tilde{x}^{2})^{2}}-2\frac{\tilde{x}^{\rho}\tilde{x}_{\mu}}{(\tilde{x}^{2})^{3}}F_{\rho\nu}-2\frac{\tilde{x}^{\sigma}\tilde{x}_{\nu}}{(\tilde{x}^{2})^{3}}F_{\mu\sigma},\label{}
\end{equation}
which means
\begin{eqnarray}
\tilde{F}_{i0} & = & (\tilde{x}^{2})^{-3}[\tilde{x}^{2}F_{i0}-2\tilde{x}_{i}\tilde{x}^{j}F_{j0}+2\tilde{t}(\tilde{t}F_{i0}+\tilde{x}^{j}F_{ij})]\nonumber \\
& = & \frac{e}{4\pi}(\tilde{x}^{2})^{-3}\left[(\tilde{t}^{2}+\tilde{\vect x}^{2})\frac{\tilde{x}^{i}/\tilde{x}^{2}}{|\tilde{\vect x}/\tilde{x}^{2}|^{3}}-2\tilde{x}^{i}\frac{\tilde{\vect x}^{2}/\tilde{x}^{2}}{|\tilde{\vect x}/\tilde{x}^{2}|^{3}}\right]\nonumber \\
& = & -\epsilon(\tilde{x}^{2})\frac{e}{4\pi}\frac{\tilde{x}^{i}}{\tilde{\vect x}^{2}}\label{tilde_E}
\end{eqnarray}
with $\epsilon(\tilde{x}^{2})$ the sign of $\tilde{x}^{2}$ and
\begin{equation}
\tilde{F}_{ij}=-2(\tilde{x}^{2})^{-3}\tilde{t}(\tilde{x}_{i}F_{0j}+\tilde{x}_{j}F_{i0})=0.\label{tilde_B}
\end{equation}
The discontinuity of $\tilde{F}_{i0}$ from the $\epsilon(\tilde{x}^{2})$
factor in equation (\ref{tilde_E}) indicates the breakdown of the
Maxwell equation at $\tilde{x}^{2}=0$. In fact, one may roughly understand
this breakdown by the usual argument that there can be no net charge
on compact spaces, since the dS spacetime can be viewed as an expanding
$S^{3}$ and it can be shown from equations (\ref{dS_J}) and (\ref{dS_inverse_J})
that there are two charges of the same signature on any one of the
$S^{3}$ simultaneity hypersurfaces.
Also, it's interesting to consider about Maxwell equation with the
magnetic and electric charges. The solution (\ref{fundamental}) is
changed to be:
\[
E_{i}=F_{i0}=\frac{e}{4\pi}\frac{x^{i}}{\left|\vect x\right|^{3}},\quad B_{i}=\frac{1}{2}\epsilon^{ijk}F_{jk}=\frac{e_{m}}{4\pi}\frac{x^{i}}{\left|\vect x\right|^{3}},\quad J=e\delta^{3}\left(\vect x\right),\quad J_{0}^{m}=e_{m}\delta^{3}\left(\vect x\right)
\]
where the $e_{m}$ and $J^{m}$ are the magnetic charge and current.
With the second equation, one can easily see that the field
\[
F_{ij}=\epsilon_{kij}B_{k}=\frac{e_{m}}{4\pi}\frac{\epsilon_{kij}x^{k}}{\left|\vect x\right|^{3}}\neq0
\]
So the transformation of the field seems different
with(\ref{tilde_E}) and (\ref{tilde_B}). In fact, after adding the
magnetic charge, there will be a none zero term in $\tilde{F_{i0}}$,
which means that the magnetic charges will contribute to the electric
field in the conformal transformation. But the discontinuity of the
$\tilde{F_{i0}}$ at $\vect x=0$ still exists since the discontinuity
is \textcolor{black}{actually caused by the $\left|\vect x\right|^{3}$
term. Evidently, the magnetic field is not zero, but a function containing
the $\left|\vect x\right|^{3}$ term also has a $\varepsilon\left(\tilde{x}^{2}\right)$
factor.} This indicates the discontinuity of the magnetic field at
$\tilde{x}^{2}=0$. One can roughly understand this by thinking about
the symmetry of electric and the magnetic charges.
\section{Penrose Diagrams and the Doubled Covering}
The above discussion can be illustrated with Penrose diagrams, as
in figure \ref{Penrose}. The lightcone $\tilde{x}^{2}=0$ corresponds
to the dashed line segments $JG$ and $IH$ in figure \ref{Penrose-InvdS}.
Then it is easy to see that the discontinuity of $\tilde{F}_{i0}$
at $\tilde{x}^{2}=0$ can be removed by reversing the sign of the
electromagnetic field and electric current:
\begin{equation}
A_{\mu}\rightarrow-A_{\mu},\quad J_{\mu}\rightarrow-J_{\mu},\label{reversion}
\end{equation}
in the regions ``I'' to ``IV'', so that the Maxwell equation
\begin{equation}
\eta^{\alpha\mu}\partial_{\alpha}F_{\mu\nu}=k_{{\rm dS}}^{2}(x)J_{\nu}\label{}
\end{equation}
can be satisfied on the whole dS spacetime. We explicitly write this
global solution as
\begin{equation}
E_{i}=\epsilon(1+x^{2})\frac{e}{4\pi}\frac{x^{i}}{|\vect x|^{3}},\quad B_{i}=0,\quad J_{0}=\epsilon(1+x^{2})e(1+x^{2})^{2}\delta^{3}(\vect x),\quad J_{i}=0.\label{global}
\end{equation}
For this solution the two antipodal point charges on the dS spacetime
have opposite signs
\footnote{It is interesting if this fact has something to do with the arguments
given in \cite{PSV}, which is from a completely different point of
view
}
\begin{figure}[htbp]
\centering\subfigure[Penrose diagram of the dS spacetime in
conformally flat coordinates, with identification $KL=MN$ and
$KM=LN$, which can be compared with figure 1 in \cite{Tian}. The
solid line segments $GH$ and $IJ$ are its conformal boundary. The
dashed line segment stands for the world line(s) of the point
charge(s).]{\label{Penrose-CFdS
\begin{minipage}[c]{0.8\textwidth
\centering\includegraphics[scale=0.9]{Penrose-CFdS.eps}
\end{minipage}}\vspace{0.4cm}
\subfigure[Inversion (\ref{inversion}) of figure
\ref{Penrose-CFdS}. The points $K$, $L$, $M$ and $N$ are all
transformed to the origin, and the triangles (numbered ``I" to
``IV") in figure \ref{Penrose-CFdS} to the corresponding positions
in this figure, respectively.]{\label{Penrose-InvdS
\begin{minipage}[c]{0.8\textwidth
\centering\includegraphics[scale=0.9]{Penrose-InvdS.eps}
\end{minipage}}\caption{An illustration of the point charge(s) in the dS spacetime with Penrose
diagrams.}
\label{Penrose}
\end{figure}
The Penrose diagram of dS spacetime as figure \ref{Penrose-CFdS}
is not the familiar one, but has the shape of that of the Minkowski
spacetime. It turns out to be interesting that we superpose the familiar
Penrose diagrams of dS and Minkowski space times, as in figure \ref{Penrose-MdS}.
There the cylinder $AEFB$ (with identification $AB$=$EF$) is the
Penrose diagram of dS spacetime, while the diamond $KLNM$ is that
of Minkowski spacetime. If we identify the regions ``I'' to ``IV''
with ``I$'$'' to ``IV$'$'', respectively, we obtain the ordinary
conformal compactification of dS and Minkowski space times (see also
\cite{Tian}). However, we have seen that for globally defined solutions
on dS and Minkowski space times (\ref{fundamental}) and (\ref{global})
the electromagnetic field and electric current in the regions ``I''
to ``IV'' differ from that in ``I$'$'' to ``IV$'$'' by a sign,
up to positive Weyl factors. In order to find a conformal compactification
of the Minkowski and dS/AdS space times where the electrodynamics
can be globally defined, then, instead of identifying the regions
``I'' to ``IV'' with ``I$'$'' to ``IV$'$'' immediately,
one should take them as, actually, antipodal region
\footnote{This ``antipodal'' refers to the 6-dimensional one (\ref{antipodal}),
as can be seen more clearly in the following discussion
} on the doubly conformal compactification of these space times. In
other words, we use Minkowski spacetime and dS spacetime to cover
different parts of this doubly conformal compactification.
\begin{figure}[htbp]
\centering \includegraphics[scale=0.9]{Penrose-MdS.eps}
\caption{Superposition of the familiar Penrose diagrams of dS and
Minkowski space times, where extension of the ordinary conformal
compactification arises.}
\label{Penrose-MdS}
\end{figure}
Although we have not seen in figure\ref{Penrose-MdS}
the whole of the doubly conformal compactification of Minkowski and
dS/AdS space times, it is rather straightforward to construct it based
on the above analysis. From figure \ref{Penrose-MdS} it is clear
that antipodal points have relative coordinates $(\pm1,\pm1)$ on
the Penrose diagram, where we have taken $AB$ as the length unit.
The antipode of antipode, with relative coordinate $(\pm2,0)$ or
$(0,\pm2)$, should be itself. In fact, the electromagnetic field
and electric current are of the same value, up to positive Weyl factors,
at these points, so it is safe to identify these points in the conformal
sense. A thus extended version of figure \ref{Penrose-MdS} is figure
\ref{Penrose-Ext}. The doubly conformal compactification of dS and
Minkowski space times is shown, more clearly, in figure \ref{Penrose-Double}.
Note also that any pair of points related by an inversion-like transformation
\begin{equation}
x^{\mu}\rightarrow-\eta_{\mu\nu}\frac{x^{\nu}}{x^{2}}=-\frac{x_{\mu}}{x^{2}}\label{inversion-like}
\end{equation}
can be viewed as to have relative coordinates $(\pm1,0)$ or $(0,\pm1)$
on these diagrams.
\begin{figure}[htbp]
\centering \includegraphics[scale=0.9]{Penrose-Ext.eps}
\caption{Double extension of figure \ref{Penrose-MdS}.}
\label{Penrose-Ext}
\end{figure}
\begin{figure}[htbp]
\centering\subfigure[Double extension of the Penrose diagram of dS
spacetime, with the usual cylindrical identification $B'B=F'F$,
which is conformally compactified by identifying $B'F'=BF$.]{\label{Penrose-DdS
\begin{minipage}[c]{0.8\textwidth
\centering\includegraphics[scale=0.9]{Penrose-DdS.eps}
\end{minipage}}\vspace{0.4cm}
\subfigure[Double extension of the Penrose diagram of Minkowski
spacetime, which is conformally compactified by identifying
$OP=QR$ and $OQ=PR$.]{\label{Penrose-DMink
\begin{minipage}[c]{0.8\textwidth
\centering\includegraphics[scale=0.9]{Penrose-DMink.eps}
\end{minipage}}\caption{An illustration of the doubly conformal compactification of dS and
Minkowski space times with Penrose diagrams.}
\label{Penrose-Double}
\end{figure}
Then we consider the $N$ case. Similarly, we have from equations
(\ref{J_Weyl}) and (\ref{N_factor})
\begin{equation}
J_{0}=e[1+2(t^{2}+\vect x^{2})+(x^{2})^{2}]\delta^{3}(\vect x)=e(1+t^{2})^{2}\delta^{3}(\vect x),\quad J_{i}=0,\label{N_J}
\end{equation}
with $E_{i}$ and $B_{i}$ still given by equation (\ref{fundamental}),
where we have omitted the prime in these notations. The region of
$N$ uncovered by the conformally flat coordinates corresponds to
the ordinary conformal boundary of the Minkowski spacetime. By the
inversion (\ref{inversion}), we can examine the uncovered region
(actually a ``compactified'' light cone). For the electric current,
we have
\begin{equation}
\tilde{J}_{0}(\tilde{x})=e(1+\tilde{t}^{2})^{2}\delta^{3}(\tilde{\boldmath x}),\quad\tilde{J}_{i}(\tilde{x})=0.\label{N_J_inverse}
\end{equation}
For the electromagnetic field, we have also equations (\ref{tilde_E})
and (\ref{tilde_B}). So we can see the breakdown of the Maxwell equation
at $\tilde{x}^{2}=0$, similar to the dS case. Unlike the dS case,
however, since $N$ has no (conformal) boundary, this breakdown cannot
be remedied by the sign reversion (\ref{reversion}) of $A_{\mu}$
and $J_{\mu}$ in certain regions of $N$. Although one can see equations
(\ref{N_J}) and (\ref{N_J_inverse}) as the only correct form of
a point-like source that satisfies the continuity equation, there
is no corresponding electromagnetic field that globally satisfies
the Maxwell equation. In other words, one cannot find fundamental
solutions to the Maxwell equation on $N$. This problem can be resolved
by cutting open $N$ along $\tilde{x}^{2}=0$ and sewing another $N$
(also cut open) onto it, which yields the double covering $2N$, as
expected.
\section{Concluding Remarks}
In Section 2, we review the ordinary conformal compactification of
the Minkowski, dS and Ads space times, where one can see that the
pseudo-sphere really can help him to understand the conformal compactification.
First, we get the intersection $\mathcal{M}$, which in fact is a
Minkowski spacetime, of the hyperplane$\mathcal{P_{\mathit{a}}}$
and the zero radius pseudo-sphere $\mathcal{N}$ in a (4+2)-dimensional
Minkowski spacetime. It's not difficult to see that the infinity points
of $\mathcal{M}$ lie on the hyperplane parallel to $\mathcal{P_{\mathit{a}}}$.
The compactification {[}$\mathcal{N}${]}, in which the $\mathit{O}$(2,4)/$\mathbb{Z}_{2}$
action can be well defined, is generated by adding those infinity
points to $\mathcal{M}$. That is the compactification of Minkowski
spacetime. For the dS and Ads case, we use a general hypersurface
(of antipodal symmetry) to intersect $\mathcal{N}$ to get some conformally
flat manifolds.
In Section 3, we map the solution to Maxwell equation
on Minkowski spacetime (\ref{fundamental}) to the dS spacetime to
get (\ref{dS_J}), from which one can find that there are really two
antipodal point charges. This is caused by the fact that the conformal
boundary $1+x^{2}$=0 separates the world line $\vect x=0$
into two parts. With the inversion (\ref{inversion}), one can find
that the field (\ref{tilde_E}) is discontinuous at $\tilde{x}^{2}=0$,
which shows the breakdown of the Maxwell equation on the compactification
of Minkowski spacetime. It is not hard to see that the breakdown is
caused by the modulus term in equation (\ref{fundamental}).
One should take care that in (\ref{dS_inverse_J}) there is not a
simple $\tilde{x}^{2}$ factor but a Jacobian factor when one writes
$\delta^{3}\left(\tilde{\vect x}/\tilde{x}^{2}\right)$ into $\delta^{3}$$\left(\tilde{\vect x}\right)$.
And the condition for magnetic charge is not different .
In Section 4, the discussion in Section 3 is illustrated with Penrose
diagrams. Reversing the sign of the electromagnetic field and current
(see (\ref{reversion})) in ``I'' to ``IV'' in figure (\ref{Penrose-InvdS}),
one can see the Maxwell equations can be well defined on the whole
dS spacetime. Equation (\ref{global}) is the global solution,
and one can see that there are two antipodal point charges. We also
show that the ordinary compactification, identifying ``I'' to ``IV''
with ``I$'$'' to ``IV$'$'', can not give a global defined electrodynamics.
This leads us to consider the doubled conformal
compactification, and we reveal that superposing
the familiar Penrose diagrams of dS and Minkowski space times (see
figure 2), and identifying the regions ``I'' to ``IV'' with ``I$'$''
to ``IV$'$'' as antipodal regions, respectively will give the doubled
compactification. Figures 3 and 4 show this process
clearly. And the doubled conformal compactification for the electrodynamics
with a magnetic charge is very similar to the electron, which we don't
analysis here.
Since our attention is solely paid to the classical
case here, one should further consider the conformal invariant quantum
field theory containing electrodynamics correspondingly \cite{M.BAKER AND K.JONSON,H.Osborn}.
And zero-mass systems \cite{McLennan1,MN,JV} can also be considered,
including the Lienard-Wiechert field of massless charges \cite{Francesco Azzurli and Kurt Lechner16}.
These problems should be left for future works. The CPT invariance
, and causality are also some interesting aspects
for our further studying.
\subsection*{Acknowledgments}
We would like to thank C.-G. Huang, X.-N. Wu, Z. Xu and B. Zhou for
helpful discussions. This work is partly supported by the National
Natural Science Foundation of China under Grant Nos. 11075206 and
11175245.
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 450
|
package org.elasticsearch.xpack.core.ml.inference.trainedmodel.ensemble;
import org.apache.lucene.util.RamUsageEstimator;
import org.elasticsearch.common.ParseField;
import org.elasticsearch.common.io.stream.StreamInput;
import org.elasticsearch.common.io.stream.StreamOutput;
import org.elasticsearch.common.xcontent.ConstructingObjectParser;
import org.elasticsearch.common.xcontent.XContentBuilder;
import org.elasticsearch.common.xcontent.XContentParser;
import org.elasticsearch.xpack.core.ml.inference.trainedmodel.TargetType;
import java.io.IOException;
import java.util.Arrays;
import java.util.List;
import java.util.Objects;
public class Exponent implements StrictlyParsedOutputAggregator, LenientlyParsedOutputAggregator {
public static final long SHALLOW_SIZE = RamUsageEstimator.shallowSizeOfInstance(Exponent.class);
public static final ParseField NAME = new ParseField("exponent");
public static final ParseField WEIGHTS = new ParseField("weights");
private static final ConstructingObjectParser<Exponent, Void> LENIENT_PARSER = createParser(true);
private static final ConstructingObjectParser<Exponent, Void> STRICT_PARSER = createParser(false);
@SuppressWarnings("unchecked")
private static ConstructingObjectParser<Exponent, Void> createParser(boolean lenient) {
ConstructingObjectParser<Exponent, Void> parser = new ConstructingObjectParser<>(
NAME.getPreferredName(),
lenient,
a -> new Exponent((List<Double>)a[0]));
parser.declareDoubleArray(ConstructingObjectParser.optionalConstructorArg(), WEIGHTS);
return parser;
}
public static Exponent fromXContentStrict(XContentParser parser) {
return STRICT_PARSER.apply(parser, null);
}
public static Exponent fromXContentLenient(XContentParser parser) {
return LENIENT_PARSER.apply(parser, null);
}
private final double[] weights;
Exponent() {
this((List<Double>) null);
}
private Exponent(List<Double> weights) {
this(weights == null ? null : weights.stream().mapToDouble(Double::valueOf).toArray());
}
public Exponent(double[] weights) {
this.weights = weights;
}
public Exponent(StreamInput in) throws IOException {
if (in.readBoolean()) {
this.weights = in.readDoubleArray();
} else {
this.weights = null;
}
}
@Override
public Integer expectedValueSize() {
return this.weights == null ? null : this.weights.length;
}
@Override
public double[] processValues(double[][] values) {
Objects.requireNonNull(values, "values must not be null");
if (weights != null && values.length != weights.length) {
throw new IllegalArgumentException("values must be the same length as weights.");
}
assert values[0].length == 1;
double[] processed = new double[values.length];
for (int i = 0; i < values.length; ++i) {
if (weights != null) {
processed[i] = weights[i] * values[i][0];
} else {
processed[i] = values[i][0];
}
}
return processed;
}
@Override
public double aggregate(double[] values) {
Objects.requireNonNull(values, "values must not be null");
double sum = 0.0;
for (double val : values) {
if (Double.isFinite(val)) {
sum += val;
}
}
return Math.exp(sum);
}
@Override
public String getName() {
return NAME.getPreferredName();
}
@Override
public boolean compatibleWith(TargetType targetType) {
return TargetType.REGRESSION.equals(targetType);
}
@Override
public String getWriteableName() {
return NAME.getPreferredName();
}
@Override
public void writeTo(StreamOutput out) throws IOException {
out.writeBoolean(weights != null);
if (weights != null) {
out.writeDoubleArray(weights);
}
}
@Override
public XContentBuilder toXContent(XContentBuilder builder, Params params) throws IOException {
builder.startObject();
if (weights != null) {
builder.field(WEIGHTS.getPreferredName(), weights);
}
builder.endObject();
return builder;
}
@Override
public boolean equals(Object o) {
if (this == o) return true;
if (o == null || getClass() != o.getClass()) return false;
Exponent that = (Exponent) o;
return Arrays.equals(weights, that.weights);
}
@Override
public int hashCode() {
return Arrays.hashCode(weights);
}
@Override
public long ramBytesUsed() {
long weightSize = weights == null ? 0L : RamUsageEstimator.sizeOf(weights);
return SHALLOW_SIZE + weightSize;
}
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 2,039
|
Hammond to Woodman to Old England.
The approximate monthly Principal & Interest payment for this property would be $21,620.49. This payment is based on a 30-year loan at a fixed rate of 5.0% with a down payment of $447,500. Actual lender interest rates and loan programs may vary.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 4,004
|
Q: Mongoose Aggregate and connect foreign Keys I have two tables.
The first one is the master collection,
The second collection is the stat collection
Im summing up the recodes in stat table with aggregate, when I'm doing that I want to use the foreign key and get the title from the master collection. Following is the code I have used.
const totalClicks = await StatModel.aggregate([
{ $match: { campaignId: id } },
{
$group: {
_id: { $dateToString: { format: '%Y-%m-%d', date: '$createAt' } },
count: { $sum: 1 },
},
},
{
$lookup: {
from: 'campaign.channels',
localField: 'channel',
foreignField: '_id',
as: 'channel',
},
},
{ $sort: { _id: 1 } },
]);
Output comes as follows
[{"_id":"2022-06-04","count":8,"channel":[]}]
Desired out put is
[{"_id":"2022-06-04","count":8,"channel":"General"}]
A: You can do like this to get the title from populated channel,
const totalClicks = await StatModel.aggregate([
{ $match: { campaignId: id } },
{
$group: {
_id: { $dateToString: { format: '%Y-%m-%d', date: '$createAt' } },
count: { $sum: 1 },
},
},
{
$lookup: {
from: 'campaign.channels',
localField: 'channel',
foreignField: '_id',
as: 'channel',
},
},
{ $addFields: { channel: { $arrayElemAt: ['$channel', 0] } } },
{ $project: { _id, 1, count: 1, channel: '$channel.title' } },
{ $sort: { _id: 1 } },
]);
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 7,306
|
In early 2017 I created a series of individual art designs on 6 side streets off Goodwood Road, Goodwood, South Australia. The pieces were fabricated and installed by Creative Pavements and MPS Paving Systems Australia .
Each design embodies a characteristic of the area and celebrates the local community. Read more about each design here.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 1,655
|
Home/Curation/Women Make Strides in Business Ownership [Wall Street Journal]
Women Make Strides in Business Ownership [Wall Street Journal]
The overall figures provide the latest evidence of a worrisome decline in business formation. The number of firms with paid employees fell by 5% to 5.4 million between 2007 and 2012 and is 2% below 2002 levels, according to the new census data.
"It's an indicator of just how weak this particular recovery has been for small businesses, generally, and for new businesses, young businesses," said E.J. Reedy, director of research and policy for the Ewing Marion Kauffman Foundation, a nonprofit that focuses on entrepreneurship.
U.S. gross domestic product, adjusted for inflation, declined at an average annual rate of 1.5% from 2007 to 2009 as the economy contracted during the recession, then increased by an average rate of 2.1% during the next three years, according to the Bureau of Economic Analysis.
Women showed big gains in business ownership in the census data, with the share of female-owned firms rising to 36% of all firms in 2012 from 29% in 2007. The number of self-employed women and the number of women running businesses with employees both increased, pushing the total number of women-owned firms to 9.9 million in 2012.
The total number of U.S. firms edged up 2% to 27.6 million between 2007 and 2012, according to preliminary U.S. Census Bureau data released this week. But the number of women-owned firms grew much faster, rising 27% during that time.
Curated from Women Make Strides in Business Ownership [Wall Street Journal]
Carla Harris medium businesses Wall Street Journal women executives
CBNation helps entrepreneurs and business owners succeed with visibility, resources and connections. CEO Blog Nation is a community of blogs for entrepreneurs and business owners. Started in much the same way as most small businesses, CEO Blog Nation captures the essence of entrepreneurship by allowing entrepreneurs and business owners to have a voice. CEO Blog Nation provides news, information, events and even startup business tips for entrepreneurs, startups and business owners to succeed.
Minority-owned firms are powering business growth in the US [Quartz]
Baby Boomers Ready to Sell Businesses to the Next Generation [The New York Times]
6 Online Communities Entrepreneurs & Small Business Owners Should Join & More
How to Crush It in Your Startup's First Year [Cosmopolitan]
The Freelancer Generation: Why Startups And Enterprises Need To Pay Attention [Tech Crunch]
11 Practical Things That Will Make You a Better Entrepreneur Almost Instantly [Forbes]
Can Anyone Become An Entrepreneur? [FOX News Talk Show]
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 3,094
|
HSC exams to be delayed for five days, with results before Christmas
By Jordan Baker
May 14, 2020 — 10.01pm
The Higher School Certificate exams will begin five days later than originally scheduled this year to give students extra time with their teachers, while the International Baccalaureate will go ahead as planned in November.
The written HSC exams had been due to commence on October 15 but will now begin on October 20, while oral language exams have been delayed by a week to August 15, the NSW Education Standards Authority said.
HSC examinations will be delayed by five days in 2020. Credit:Marina Neil
The examination period will be held over the same timeframe as previous years and HSC results will be sent to students at 6am on December 18. The Australian Tertiary Admission Rank will be released at 9am on the same day.
NESA chief executive Paul Martin said the release of the timetable on Friday would give HSC students certainty.
"Delaying the start of the exams balances giving students some extra time at school after the October holiday period with keeping to the original timeframe for releasing results so students can move on to the next stage in their lives," he said.
"Students can use this time with their teachers to revise, ask questions and seek support directly before the exams start.
"I want to reassure parents and students that the exams will be conducted in line with the expert health advice at the time of the exams, which are still five months away."
Some schools have given year 12 more face-to-face teaching days than other schools since students began to return to classrooms this term, leaving students worried the disparity could put them at a disadvantage in the HSC.
The International Baccalaureate's May examination session - the test sat by students in the northern hemisphere - was cancelled but the November session involving Australian students is going ahead.
However some subjects are only tested in May, so the cancellation will affect Australian students sitting those subjects. Their results will now be based on their class assessment marks, Antony Mayrhofer, from IB Schools Australasia, said.
At the start of the COVID-19 crisis in Australia, there were fears the HSC and IB written exams would have to be called off, as they have been in many countries such as Britain, where final results will be based on assessments.
Some HSC performance examinations have been cancelled, such as the group performance in Drama and the ensemble performance in music.
Meanwhile, the University Admissions Centre has announced that students whose parents receive the JobKeeper or JobSeeker allowance during their HSC year will be considered disadvantaged and will get extra help with university entry.
They will get a higher Australian Tertiary Admission Rank or be given a place reserved for Educational Access Scheme students.
Sign up to our Coronavirus Update newsletter
Get our Coronavirus Update newsletter for the day's crucial developments at a glance, the numbers you need to know and what our readers are saying. Sign up to The Sydney Morning Herald's newsletter here and The Age's here.
Jordan Baker is Education Editor of The Sydney Morning HeraldConnect via Twitter or email.
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 8,101
|
Calvilla San Salvador es el nombre de una variedad cultivar de manzano (Malus domestica). Esta manzana está cultivada en la colección de germoplasma de manzanas del CSIC. Así mismo está cultivada en diversos viveros entre ellos algunos dedicados a conservación de árboles frutales en peligro de desaparición. Esta manzana es originaria de Cantabria, donde tuvo su mejor época de cultivo comercial antes de la década de 1960, y actualmente en menor medida aún se encuentra.
Sinónimos
"Manzana Calvilla San Salvador",
Historia
Cantabria presenta unas condiciones de clima y de suelos excelentes para el cultivo del manzano. De hecho, hasta mediados del siglo XX Cantabria tenía una gran variedad de manzanas tradicionales que surtían la demanda de manzanas de mesa en la zona. A partir de la década de 1960 estas fueron decayendo paulatinamente en su comercialización, en detrimento de variedades selectas extranjeras que dominan el mercado actual. Hay varias manzanas tradicionales que se están intentando recuperar por el CIFA, en Muriedas (Centro de Investigación y Formación Agraria de Cantabria).
'Calvilla San Salvador' está considerada incluida en las variedades locales autóctonas muy antiguas, cuyo cultivo se centraba en comarcas muy definidas. Se caracterizaban por su buena adaptación a sus ecosistemas y podrían tener interés genético en virtud de su adaptación. Se encontraban diseminadas por todas las regiones fruteras españolas, aunque eran especialmente frecuentes en la España húmeda. Estas se podían clasificar en dos subgrupos: de mesa y de sidra (aunque algunas tenían aptitud mixta).
'Calvilla San Salvador' es una variedad mixta, clasificada como de mesa, también se utiliza en la elaboración de sidra; difundido su cultivo en el pasado por los viveros comerciales y cuyo cultivo en la actualidad se ha reducido a huertos familiares y jardines privados.
Características
El manzano de la variedad 'Calvilla San Salvador' tiene un vigor medio; florece a inicios de mayo; tubo del cáliz cónico, más o menos pronunciado, y con los estambres situados en su mitad.
La variedad de manzana 'Calvilla San Salvador' tiene un fruto de tamaño grande o medio; forma ovoide o tronco-cónica, angulosa y rebajada de un lado desde su ápice, contorno algo irregular; piel feblemente fina; con color de fondo amarillo limón, sobre color rosado, intensidad de sobre color lavado, reparto del sobre color en Chapa, presenta chapa en zona de insolación, rosada más o menos vivo, acusa punteado abundante, ruginoso o gris verdoso y vistoso, y una sensibilidad al "russeting" (pardeamiento áspero superficial que presentan algunas variedades) débil; pedúnculo de variada longitud, a veces marcadamente engrosado en la parte superior, anchura de la cavidad peduncular medianamente estrecha, profundidad de la cavidad pedúncular de profundidad leve o marcada, sin chapa o situada en el fondo de ruginosidad marrón grisáceo, bordes irregularmente ondulados, y con importancia del "russeting" en cavidad peduncular fuerte; anchura de la cavidad calicina estrecha o media, profundidad de la cav. calicina profunda, bordes mamelonados con el fondo notablemente arrugado, y con importancia del "russeting" en cavidad calicina débil; ojo medio, cerrado o entreabierto; sépalos carnosos en su base, casi siempre separados levemente en su nacimiento, de puntas agudas y vueltas hacia fuera.
Carne de color blanca; textura tierna, jugosa; sabor dulce, algo acidulado y aromático; corazón pequeño, centrado, desplazado hacia el ojo, de forma alargada y estrecha, a veces bulbiforme; eje cóncavo; celdas grandes y alargadas, cartilaginosas; semillas alargadas y oscuras.
La manzana 'Calvilla San Salvador' tiene una época de maduración y recolección muy tardía en el invierno, se recolecta en diciembre-enero. Tiene uso mixto pues se usa como manzana de mesa fresca, y también como manzana para elaboración de sidra.
Véase también
Referencias
Bibliografía
Enlaces externos
Malus domestica
Calvilla San Salvador
Calvilla San Salvador
Calvilla San Salvador
Calvilla San Salvador
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 1,616
|
Q: ¿Por qué mi informacion de contacto PHP, siempre llega a "Correo no Deseado"? Hola amigos el problema es que los datos llegan a "Correo no Deseado"; según yo he escrito bien los parámetros para que no suceda esto, qué puedo hacer aquí les dejo parte mi código php, saludos y gracias por sus comentarios.
if ($_POST) {
$nombre = $_POST['nombre'];
$email = $POST['email'];
$telefono = $POST['telefono'];
$mensaje = $POST['mensaje'];
//
$email_to = "contacto@destino.com";
$email_subject = "Contacto desde Pagina Web";
//
$headers = "From: $nombre <$email> \r\n";
$headers .= 'Content-type: text/html; charset=iso-8859-1'."\r\n";
$headers .= 'MIME-Version: 1.0'."\r\n";
$headers .= 'X-Mailer: PHP/'.phpversion();
//
$email_message = "Detalles del formulario de contacto"."<br>";
$email_message .= "Nombre: " . $_POST['nombre'] . "<br>";
$email_message .= "e-mail: " . $_POST['email'] . "<br>";
$email_message .= "Teléfono: " . $_POST['telefono'] . "<br>";
$email_message .= "Mensaje: " . $_POST['mensaje'] . "<br>";
@mail($email_to, $email_subject, $email_message, $headers);
}
A: En el caso que comentas, una de las razones por la cual el cliente determina que el email que recibe lo envia a correo no deseado es generalmente porque el email que se recibe no tienen datos en el header o estos no son validos.
Como ejemplo este script que agrega los valores básicos en el header, usando mail()
<?php
...
...
$to = 'contacto@destino.com';
$subject = 'Contacto desde Pagina Web';
$email_message = 'Detalles del formulario de contacto'.'<br>';
$email_message .= 'Nombre: ' . $_POST['nombre'] . '<br>';
$email_message .= 'e-mail: ' . $_POST['email'] . "<br>";
$email_message .= 'Teléfono: ' . $_POST['telefono'] . "<br>";
$email_message .= 'Mensaje: ' . $_POST['mensaje'] . "<br>";
//HEADERS!
$headers = 'From: wtfchamp@wtfchamp.com' . "\r\n" .
'Reply-To: wtfchamp@wtfchamp.com' . "\r\n" .
'X-Mailer: PHP/' . phpversion();
mail($to, $subject, $email_message, $headers);
...
?>
Asegura enviar valores correctos en el header.
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 4,528
|
Q: Is there any way to rotate a magic trackpad? I want to use my magic trackpad while it's rotated, as if in 'portrait' mode.
I can't find any option to make it work like this, unlike a display Rotation option.
Is there a way for me to use it this way? If so, how?
A: You have three options that I'm aware of, but depending on the version of macOS you're running, probably only one of them is a real chance of doing exactly what you want.
In summary:
*
*Use Terminal commands to rotate the trackpad orientation. However, the commands I'm aware of only work to rotate it 180° instead of 90°. It also doesn't work for macOS Sierra or macOS High Sierra. If you're running an earlier version of macOS and rotating 180° is an option for you, then see details below for Terminal commands.
*Try using MagicPrefs, however this is now on its last legs as an option. It's worked brilliantly for many users for a long time, but for various reasons is no longer maintained
*Try using BetterTouchTool. This is probably your best bet. You can install it for free for 45 days to test how well it meets your needs.
Terminal commands
If you decide that rotating your trackpad 180° will meet your needs, and assuming you're running an older version of macOS, then follow these steps:
*
*Launch Terminal (usually found in Applications > Utilities)
*Enter the following command (or copy and paste it):
sudo sudo defaults write com.apple.MultitouchSupport ForceAutoOrientation YES
*Press Enter
*Enter the following command (or copy and paste it):
sudo defaults write com.apple.trackpad.orientation TrackpadOrientationMode 1
*Press Enter
Once you've entered both commands you should be able to just turn Bluetooth off and then on again to activate the change in orientation.
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 3,279
|
\section{Calculation of the neutralino nucleon cross section}
\label{Technical}
\subsection{Composition of direct detection rate}
\label{DDFormulas}
In this subsection, we briefly review the standard formulas for the calculation of neutralino direct detection rates. The desired quantity is the rate of events $\mathrm{d}R$ per energy interval $\mathrm{d}E$. This differential event rate is typically expressed in terms of counts per kg and day and keV. It can be written as
\begin{equation}
\label{DDRate2}
\frac{\mathrm{d}R}{\mathrm{d}E} = \sum_i c_i \frac{\sigma_i}{2m_{\tilde{\chi}^0_1}\mu_i^2}\rho_0 \eta_i.
\end{equation}
The sum runs over all detector nuclides $i$, and the factor $c_i$ denotes the mass fraction of the nuclear species $i$ in the detector. Let $m_i$ be the mass of the nucleus of species $i$. Then $\mu_i$ is the reduced mass
\begin{equation}
\mu_i = \frac{m_{\tilde{\chi}^0_1}m_i}{m_{\tilde{\chi}^0_1} + m_i}.
\end{equation}
The local dark matter density is described by $\rho_0$. Before using the canonical value of 0.3 GeV/cm$^3$, one should calculate the neutralino relic density to ensure that its value is in agreement with the experimental constraints and that the neutralinos can solely account for dark matter. $\eta_i$ contains the integration over the dark matter velocity relative to the detector $\vec{v}$,
\begin{equation}
\eta_i = \int_{v_{\mathrm{min},i}}^{v_{\mathrm{esc}}}\mathrm{d}^3v\frac{f(\vec{v})}{v}\quad\mathrm{with}\quad v_{\mathrm{min},i} = \sqrt{\frac{m_iE}{2\mu_i^2}}.
\end{equation}
The lower integration limit $v_{\mathrm{min},i}$ is given by the minimal neutralino velocity, which can cause a recoil energy $E$. The upper integration limit is fixed by the galactical escape speed $v_{\mathrm{esc}}$, which is usually set to 544 km/s. Faster particles are not gravitationally bound in the Milky Way. More details on the integration limits can be found in Refs.\ \cite{SmithRAVE, vesc, CirelliTools}. $f(\vec{v})$ is the local velocity distribution, which is typically assumed to be Maxwellian. However, several studies have unveiled that this simplification might not describe the situation properly, see e.g. \cite{vDistrib, GreenDistribution, DD_ProtonNeutron}. All the particle physics is contained in the cross sections for elastic nucleus-neutralino scattering $\sigma_i$, where we distinguish between spin-independent and spin-dependent contributions.
The spin-independent cross section can be written as
\begin{equation}
\label{sigmaSI}
\sigma_i^{\mathrm{SI}} = \frac{\mu_i^2}{\pi}\left|Z_i g_p^{\mathrm{SI}} +(A_i-Z_i)g_n^{\mathrm{SI}}\right|^2|F_i^{\mathrm{SI}}(Q_i)|^2,
\end{equation}
where $F_i^{\mathrm{SI}}(Q_i)$ is the spin-independent structure function for the nucleus $i$. It depends on the momentum transfer $Q_i = \sqrt{2m_iE}$, can be understood as the Fourier transform of the nucleon density, and is normalised to $F_i^{\mathrm{SI}}(0) = 1$. The nucleus $i$ consists of $Z_i$ protons and $A_i -Z_i$ neutrons, where $Z_i$ is its atomic number and $A_i$ is its mass number. To enable a comparison of direct detection results, that is independent of the detector material and technology, the experimental collaborations typically publish constraints on the cross section of the dark matter particle and a single nucleon\footnote{At this point, the typical assumption is that the interaction strength of neutralinos is the same for protons and neutrons. This is not necessarily fulfilled in a non-minimal model like the MSSM. Therefore we keep our calculations general and distinguish between protons and neutrons.} $N$, which simply reads
\begin{equation}
\label{sigmaSINucleon}
\sigma_N^{\mathrm{SI}} = \frac{\mu_N^2}{\pi}\left|g_N^{\mathrm{SI}}\right|^2.
\end{equation}
Here, the neutralino-nucleus reduced mass $\mu_i$ is replaced by the neutralino-nucleon reduced mass $\mu_N$ in complete analogy. The nucleon masses $m_N$ are given by
\begin{equation}
\label{Nucleonmasses}
m_p = 0.9383\ \mathrm{GeV}\quad\mathrm{and}\quad m_n = 0.9396\ \mathrm{GeV}.
\end{equation}
The effective spin-independent four-fermion couplings among neutralinos and protons $p$ or neutrons $n$ are denoted by $g_p^{\mathrm{SI}}$ and $g_n^{\mathrm{SI}}$. They can be determined via
\begin{equation}
g_N^{\mathrm{SI}} = \sum_q \langle N |\bar{q}q| N\rangle \alpha_{q}^{\mathrm{SI}},
\end{equation}
where the nucleon index $N$ stands either for a proton or a neutron and where the sum runs over all quark types $q$.\footnote{We are summing over all quark types, as we do not include gluon operators yet. Alternatively, one could replace the heavy-quark contributions by loop-induced gluon processes including heavy quarks as virtual particles.} The spin-independent interaction between quarks and neutralinos is denoted by $\alpha_{q}^{\mathrm{SI}}$. The quark matrix element $\langle N |\bar{q}q| N\rangle$ can be qualitatively understood as the probability to find a quark $q$ in the nucleon $N$. We write it as
\begin{equation}
\label{fTDef}
\langle N |m_q \bar{q}q| N\rangle = f_{Tq}^N m_N,
\end{equation}
where $m_N$ denotes the nucleon mass and $m_q$ the quark mass. The scalar coefficients $f_{Tq}^N$ are determined experimentally or via lattice QCD. We point out that especially $f_{Ts}^N$ is affected by experimental uncertainties, which mainly stem from the determination of the pion-nucleon sigma term \cite{NuclearUncertainties, EllisHadron, BottinoNucleon}. We use the values given in Refs.\ \cite{CrivellinNucleon, Hoferichter, Junnarkar} which differ from the ones implemented in {\tt DarkSUSY}\ \cite{DarkSUSY} or {\tt micrOMEGAs}\ \cite{micrOMEGAs}. We list all values for comparison in Tab.\ \ref{fTTable}.\footnote{We are working with {\tt micrOMEGAs}\ \texttt{2.4.1} to benefit from our established relic density interface. However, we have updated the nuclear input values to the most recent version manually. Hence the values given in Tab.\ \ref{fTTable} correspond to {\tt micrOMEGAs}\ \texttt{4.2.5}.} The factors $f_{Tq}^N$ of the heavy quarks are linked to those of the light quarks via \cite{Shifman}
\begin{equation}
f_{Tc}^N = f_{Tb}^N = f_{Tt}^N= \frac{2}{27}\left(1-\sum_{q=u,d,s} f_{Tq}^N\right).
\end{equation}
\begin{table}
\caption{Scalar coefficients $f_{Tq}^N$ used in different codes.}
\begin{center}
\begin{tabular}{|c|ccc|}
\hline
Scalar coefficient & {\tt DM@NLO} & {\tt DarkSUSY} & {\tt micrOMEGAs}\\
\hline
$f_{Tu}^p$ & 0.0208 & 0.023 & 0.0153 \\
$f_{Tu}^n$ & 0.0189 & 0.019 & 0.0110 \\
$f_{Td}^p$ & 0.0411 & 0.034 & 0.0191\\
$f_{Td}^n$ & 0.0451 & 0.041 & 0.0273\\
$f_{Ts}^p = f_{Ts}^n$ & 0.043 & 0.14 & 0.0447\\
$f_{Tc}^p = f_{Tb}^p = f_{Tt}^p$ & 0.0663 & 0.0595 & 0.0682\\
$f_{Tc}^n = f_{Tb}^n = f_{Tt}^n$ & 0.0661 & 0.0592 & 0.0679\\
\hline
\end{tabular}
\end{center}
\label{fTTable}
\end{table}
The spin-dependent cross section can be cast into the form
\begin{eqnarray}
\label{sigmaSD}
\sigma_i^{\mathrm{SD}} & = & \frac{4\mu_i^2}{2J +1}\big(|g_p^{\mathrm{SD}}|^2S_{\mathrm{pp},i}(Q_i) + |g_n^{\mathrm{SD}}|^2S_{\mathrm{nn},i}(Q_i)\nonumber\\
&& + |g_p^{\mathrm{SD}}g_n^{\mathrm{SD}}|S_{\mathrm{pn},i}(Q_i)\big),
\end{eqnarray}
where $J$ denotes the nuclear spin. Details on the spin structure functions $S_{\mathrm{pp},i}(Q_i)$, $S_{\mathrm{nn},i}(Q_i)$ and $S_{\mathrm{pn},i}(Q_i)$ can be found in Ref.\ \cite{VogelStructure}. The spin-dependent cross section for a neutralino and a single nucleon $N$ reads
\begin{equation}
\label{sigmaSDNucleon}
\sigma_N^{\mathrm{SD}} = \frac{3\mu_N^2}{\pi}|g_N^{\mathrm{SD}}|^2.
\end{equation}
The effective spin-dependent four-fermion couplings among neutralinos and protons $p$ ($g_p^{\mathrm{SD}}$) or neutrons $n$ ($g_n^{\mathrm{SD}}$) are given by
\begin{equation}
g_N^{\mathrm{SD}} = \sum_{q= u,d,s} (\Delta q)_N \alpha_{q}^{\mathrm{SD}}.
\end{equation}
In contrast to the spin-independent case, we sum only over the light quarks $u$, $d$ and $s$, as mainly these flavors contribute to the spin of the nucleon.\footnote{Note, however, that it was recently claimed that bottom quarks may also contribute to the spin-dependent interaction \cite{LiBottom}.} $(\Delta q)_N$ can be seen as the fraction of the nucleon spin carried by the quark $q$. More precisely, it describes the second moment of the polarized quark density and is related to the nucleon spin vector $s_\mu$ via
\begin{equation}
\langle N | \bar{q}\gamma_\mu\gamma_5 q| N\rangle = 2s_\mu(\Delta q)_N.
\end{equation}
We choose the default values of {\tt micrOMEGAs}\ for the polarized quark densitites
\begin{eqnarray}
(\Delta u)_p = (\Delta d)_n & = & 0.842,\label{Delta1}\\
(\Delta d)_p = (\Delta u)_n & = & -0.427,\label{Delta2}\\
(\Delta s)_p = (\Delta s)_n & = & -0.085,
\end{eqnarray}
constrained by isospin symmetry, i.e.\ $(\Delta u)_p = (\Delta d)_n$ and $(\Delta d)_p = (\Delta u)_n$.
\subsection{Renormalization scheme}
\label{Renormalization}
Our QCD calculations at next-to-leading order (NLO) and beyond are performed within a hybrid on-shell/$\overline{\rm DR}$ renormalization scheme, described in detail in Refs.\ \cite{ChiChi2qq3, NeuQ2qx1, NeuQ2qx2}. In the quark sector, the top and bottom quark masses are defined on-shell and in the $\overline{\rm DR}$ scheme, respectively. Note that through the Yukawa coupling to (in particular the neutral pseudoscalar) Higgs boson resonances, the bottom quark mass can have a sizeable influence on the dark matter annihilation cross section and must therefore be treated with particular care. We obtain it from the SM $\overline{\rm MS}$ mass $m_b(m_b)$, determined in an analysis of $\Upsilon$ sum rules, through evolution to the scale $\mu_R$, transformation to the SM $\overline{\rm DR}$ and then MSSM $\overline{\rm DR}$ scheme \cite{ChiChi2qq3, NeuQ2qx1}. In the squark sector, we have five independent parameters
\begin{equation}
m_{\tilde{t}_1}, \quad m_{\tilde{b}_1}, \quad m_{\tilde{b}_2}, \quad A_t \quad\mathrm{and}\quad A_b=0.
\label{eq:RenInput}
\end{equation}
The lighter stop mass and the two sbottom masses are taken to be on-shell, while the stop and sbottom trilinear coupling parameters are taken in the $\overline{\rm DR}$ scheme. From these parameters, we compute as dependent quantities the stop and sbottom mixing angles $\theta_{\tilde{t}}$ and $\theta_{\tilde{b}}$ and $m_{\tilde{t}_2}$ for the heavier stop \cite{NeuQ2qx1}. The masses of the first- and second-generation squarks are taken on-shell. The strong coupling constant $\alpha_s(\mu_R)$ is renormalized in the MSSM $\overline{\rm DR}$ scheme with six active flavors and obtained after evolution of the world-average, five-flavor SM $\overline{\rm MS}$ value at the $Z^0$-boson mass to the renormalization scale $\mu_R$ and an intermediate transformation to the SM $\overline{\rm DR}$ scheme \cite{NeuQ2qx2}.
Although EFT calculations are usually performed in a minimal scheme such as {$\overline{\mathrm{MS}}$}\ or its SUSY equivalent {$\overline{\mathrm{DR}}$}, we continue to use the hybrid scheme presented above for three main reasons: First, we want to combine our direct detection calculations with our relic density analysis, where this scheme has proven very reliable. In particular, the on-shell description of the top quark leads to improved perturbative stability and better fits our supersymmetric processes and top quark final states in comparison to a definition in the {$\overline{\mathrm{DR}}$}\ scheme \cite{Scalepaper}. A second reason is that the hybrid scheme also leads to improved perturbative stability for direct detection as described below in Sec.\ \ref{ScenarioC}. The last reason is that using this hybrid scheme allows for simpler comparison of the leading order result with {\tt micrOMEGAs}. This is due to fact that both, our calculation and {\tt micrOMEGAs}, use the same on-shell squark masses calculated by {\tt SPheno}\ as described in section \ref{Numerics}.
\subsection{Matching of the full and effective theory}
\begin{figure}
\includegraphics[width=0.49\textwidth]{DDTree.pdf}
\caption{Tree-level processes in the full theory.}
\label{fig:DDTree}
\end{figure}
\begin{figure}
\includegraphics[width=0.18\textwidth]{DDEFTTree2.pdf}
\includegraphics[width=0.18\textwidth]{DDEFTVertex.pdf}
\caption{Tree-level process (left) and virtual correction (right) in the effective theory.}
\label{fig:DDEFT}
\end{figure}
This subsection is devoted to the matching of the full theory, namely the MSSM, valid at high energies ($\mu_\mathrm{high}\sim 1$ TeV) onto the effective energy valid at low energies ($\mu_\mathrm{low}\sim 5$ GeV). The tree-level diagrams of the scattering process $\tilde{\chi}^0_1q\rightarrow\tilde{\chi}^0_1q$ within the MSSM are shown in Fig.\ \ref{fig:DDTree}. The corresponding amplitudes have to be evaluated at vanishing relative velocity and mapped onto the yet unknown Wilson coefficients $c_1$ and $c_2$ of the effective Lagrangian
\begin{equation}
\label{Leff}
\mathcal{L}_\mathrm{eff} = c_1Q_1 + c_2Q_2 = c_1\bar{\chi}\chi\bar{q}q + c_2\bar{\chi}\gamma_\mu\gamma_5\chi\bar{q}\gamma^\mu\gamma_5q.
\end{equation}
We stress that in this convention a factor $m_q$ has to be factored out of $c_1$ when replacing the nuclear matrix elements via Eq.\ (\ref{fTDef}). Both of the operators $Q_1$ and $Q_2$ given above lead to an effective four-fermion interaction as shown in the left diagram of Fig.\ \ref{fig:DDEFT}. The Higgs processes contribute solely to the scalar operator and the $Z^0$ processes solely to the axial-vector operator. We include only the scalar Higgs bosons $h^0$ and $H^0$ and not the pseudoscalar Higgs boson $A^0$, since the latter leads to the kinematically suppressed operator $\bar{\chi}\gamma_5\chi\bar{q}\gamma_5q$. The squark processes contribute to both operators. To bring the spinor fields into the desired order, a Fierz transformation has to be performed in this case.
The aforementioned mapping onto the Wilson coefficients is governed by the matching condition. This condition demands that the amplitude of the full theory $\mathcal{M}_\mathrm{full}$ is reproduced by the effective theory at the high scale $\mu_\mathrm{high}$. At tree level we have
\begin{equation}
\mathcal{M}_\mathrm{full}^\mathrm{tree} \overset{!}{=} \mathcal{M}_\mathrm{eff}^\mathrm{tree} = c_1^\mathrm{tree}Q_1^\mathrm{tree} + c_2^\mathrm{tree}Q_2^\mathrm{tree}\label{TreeMatching}
\end{equation}
which leads to
\begin{eqnarray}
c_1^\mathrm{tree} = \alpha_q^{\mathrm{SI}} & = & \sum_{\phi = h^0,H^0}\frac{g_{\tilde{\chi}\tilde{\chi}\phi}^Rg^L_{qq\phi}}{m_\phi^2} - \frac{1}{4}\sum_{i=1}^2\frac{g_{\tilde{\chi}\tilde{q}_iq}^Lg_{\tilde{\chi}\tilde{q}_iq}^{R*}}{m_{\tilde{q}_i}^2 - s}\nonumber\\
&& - \frac{1}{4}\sum_{i=1}^2\frac{g_{\tilde{\chi}\tilde{q}_iq}^Lg_{\tilde{\chi}\tilde{q}_iq}^{R*}}{m_{\tilde{q}_i}^2 - u},\\
c_2^\mathrm{tree} = \alpha_q^{\mathrm{SD}} & = & \frac{1}{2}\frac{g_{\tilde{\chi}\tilde{\chi}Z^0}^R(g^L_{qqZ^0} - g^R_{qqZ^0})}{m_{Z^0}^2}\nonumber\label{CSITree}\\
&& + \frac{1}{8}\sum_{i=1}^2\frac{|g_{\tilde{\chi}\tilde{q}_iq}^L|^2 + |g_{\tilde{\chi}\tilde{q}_iq}^{R}|^2}{m_{\tilde{q}_i}^2 - s}\nonumber\\
&& + \frac{1}{8}\sum_{i=1}^2\frac{|g_{\tilde{\chi}\tilde{q}_iq}^L|^2 + |g_{\tilde{\chi}\tilde{q}_iq}^{R}|^2}{m_{\tilde{q}_i}^2 - u}.\label{CSDTree}
\end{eqnarray}
In the limit of vanishing relative velocity, the Mandelstam variables $s$ and $u$ simplify to $(m_{\tilde{\chi}^0_1} \pm m_q)^2$, respectively. The elementary couplings between three particles $a$, $b$ and $c$ are denoted by $g_{abc}$. Using the chirality projectors $P_{L/R} = ({\mathchoice {\rm 1\mskip-4mu l} {\rm 1\mskip-4mu l}{\rm 1\mskip-4.5mu l} {\rm 1\mskip-5mu l}} \mp \gamma_5)/2$, they can be decomposed into left- and right-handed parts via
\begin{equation}
g_{abc} = g_{abc}^LP_L + g_{abc}^RP_R.
\end{equation}
Explicit expressions for the couplings can be found, e.g., in Ref.\ \cite{DreesBook}. The tree-level results have been analytically compared with those implemented in {\tt DarkSUSY}. Taking into account that {\tt DarkSUSY}\ does not distinguish between $s$- and $u$-channels, we find perfect agreement.
\begin{figure}
\includegraphics[width=0.49\textwidth]{DDProp.pdf}
\includegraphics[width=0.49\textwidth]{DDVertex.pdf}
\includegraphics[width=0.49\textwidth]{DDBox.pdf}
\caption{Virtual corrections in the full theory.}
\label{fig:DDCorrections}
\end{figure}
So far we have basically reproduced already available results. The next step is to improve on the tree-level calculation by including all $\mathcal{O}(\alpha_s)$ corrections to the leading operators. The corresponding diagrams within the full theory are shown in Fig.\ \ref{fig:DDCorrections}. We distinguish between propagator corrections (the first row), vertex corrections (the second and the third row) and box contributions (the last row).
We have calculated all of the loop amplitudes in full generality using dimensional reduction. The gluon propagator correction shown as the third diagram in the first row then vanishes, as it is proportional to the scaleless scalar integral $A_0(0) = 0$. In the case of the other propagator and the vertex corrections, we were able to benefit from previous loop calculations performed in the context of Ref.\ \cite{ChiChi2qq3}. The box amplitudes were calculated from scratch. These amplitudes lead to a plethora of effective operators. We keep only the most relevant, namely those of Eq.\ (\ref{Leff}). In case of the gluon boxes, Fierz transformations are necessary again. Explicit expressions for all involved loop amplitudes will be given in Ref.\ \cite{myPhD}.
In contrast to our relic density calculations, the loops are evaluated at zero relative velocity in the context of direct detection. This leads to additional problems, namely vanishing Gram determinants. We illustrate this technical issue separately in App.\ \ref{GramDet}.
The propagator and vertex corrections give rise to ultraviolet divergences. These divergences are removed via renormalization (cf.\ Sec.\ \ref{Renormalization}), i.e.\ by adding the corresponding counterterms. A detailed description of the counterterms involved here is given in Ref.\ \cite{NeuQ2qx1}. As we always distinguish ultraviolet and infrared poles ($\epsilon_{UV}$ and $\epsilon_{IR}$) when evaluating loop integrals, we were able to explicitly check the ultraviolet safety of our calculation.
Having the renormalized amplitudes of the full theory at hand, we can start with the matching procedure at NLO. The matching condition remains basically unchanged and reads
\begin{eqnarray}\hspace*{-1cm}
\mathcal{M}_\mathrm{full}^\mathrm{NLO} & \overset{!}{=} & \mathcal{M}_\mathrm{eff}^\mathrm{NLO}.\\
\hspace*{-1cm}\Leftrightarrow\mathcal{M}_\mathrm{full}^\mathrm{tree} + \mathcal{M}_\mathrm{full}^\mathrm{1loop} & \overset{!}{=} & c_1^\mathrm{NLO}Q_1^\mathrm{NLO} + c_2^\mathrm{NLO}Q_2^\mathrm{NLO}.\label{NLOMatching}
\end{eqnarray}
In this convention, the full NLO result consists of the tree-level result and its $\mathcal{O}(\alpha_s)$ one-loop correction. The latter includes all the virtual corrections depicted in Fig.\ \ref{fig:DDCorrections}. The crucial point in Eq. (\ref{NLOMatching}) is that there is a one-loop correction to the Wilson coefficients \textit{and} the effective operators. We neglect terms of $\mathcal{O}(\alpha_s^2)$ and write
\begin{widetext}
\begin{eqnarray}
\mathcal{M}_\mathrm{full}^\mathrm{tree} + \mathcal{M}_\mathrm{full}^\mathrm{1loop} & \overset{!}{=} & (c_1^\mathrm{tree} + c_1^\mathrm{1loop})(Q_1^\mathrm{tree}+ Q_1^\mathrm{1loop}) + (c_2^\mathrm{tree} + c_2^\mathrm{1loop})(Q_2^\mathrm{tree} + Q_2^\mathrm{1loop})\nonumber\\
& = & c_1^\mathrm{tree}Q_1^\mathrm{tree} + c_2^\mathrm{tree}Q_2^\mathrm{tree} + c_1^\mathrm{1loop}Q_1^\mathrm{tree} + c_2^\mathrm{1loop}Q_2^\mathrm{tree} + c_1^\mathrm{tree}Q_1^\mathrm{1loop} + c_2^\mathrm{tree}Q_2^\mathrm{1loop}.
\end{eqnarray}
\end{widetext}
At $\mathcal{O}(\alpha_s^0)$, we reproduce the tree-level matching condition Eq. (\ref{TreeMatching}). At $\mathcal{O}(\alpha_s)$, we obtain
\begin{eqnarray}
&&\mathcal{M}_\mathrm{full}^\mathrm{1loop} - c_1^\mathrm{tree}Q_1^\mathrm{1loop} - c_2^\mathrm{tree}Q_2^\mathrm{1loop}\nonumber\\
&=& c_1^\mathrm{1loop}Q_1^\mathrm{tree} + c_2^\mathrm{1loop}Q_2^\mathrm{tree}.
\end{eqnarray}
Before we can calculate the $\mathcal{O}(\alpha_s)$ corrections to the Wilson coefficients, i.e.\ determine $c_1^\mathrm{1loop}$ and $c_2^\mathrm{1loop}$, we have to identify the one-loop corrections to the effective operators $Q_1^\mathrm{1loop}$ and $Q_2^\mathrm{1loop}$. These can be written as
\begin{eqnarray}
Q_1^\mathrm{1loop} & = & (\mathcal{K}_\mathrm{EFTV1} + \mathcal{K}_\mathrm{EFTVC1})Q_1^\mathrm{tree}\quad\mathrm{and}\quad\\
Q_2^\mathrm{1loop} & = & (\mathcal{K}_\mathrm{EFTV2} + \mathcal{K}_\mathrm{EFTVC2})Q_2^\mathrm{tree},
\end{eqnarray}
i.e.\ they can be expressed as the tree-level operators multiplied with correction factors describing vertex corrections and vertex counterterms in the effective field theory. The vertex correction in the effective field theory is depicted on the right of Fig.\ \ref{fig:DDEFT}. This allows us to explicitly write down the one-loop Wilson coefficients as
\begin{eqnarray}
c_1^\mathrm{1loop} & = & \alpha^\mathrm{SI}_{q,\mathrm{P}} + \alpha^\mathrm{SI}_{q,\mathrm{PC}} + \alpha^\mathrm{SI}_{q,\mathrm{V}} + \alpha^\mathrm{SI}_{q,\mathrm{VC}}\nonumber\\
& + & \alpha^\mathrm{SI}_{q,\mathrm{B}} - c_1^\mathrm{tree}(\mathcal{K}_\mathrm{EFTV1} + \mathcal{K}_\mathrm{EFTVC1}).\\
c_2^\mathrm{1loop} & = & \alpha^\mathrm{SD}_{q,\mathrm{P}} + \alpha^\mathrm{SD}_{q,\mathrm{PC}} + \alpha^\mathrm{SD}_{q,\mathrm{V}} + \alpha^\mathrm{SD}_{q,\mathrm{VC}}\nonumber\\
& + & \alpha^\mathrm{SD}_{q,\mathrm{B}} - c_2^\mathrm{tree}(\mathcal{K}_\mathrm{EFTV2} + \mathcal{K}_\mathrm{EFTVC2}).
\end{eqnarray}
Here $\alpha^\mathrm{SI}_{q,\mathrm{P}}$, $\alpha^\mathrm{SI}_{q,\mathrm{PC}}$, $\alpha^\mathrm{SI}_{q,\mathrm{V}}$, $\alpha^\mathrm{SI}_{q,\mathrm{VC}}$ and $\alpha^\mathrm{SI}_{q,\mathrm{B}}$ denote the contributions to the spin-independent four-fermion coupling stemming from the propagator corrections, propagator counterterm, vertex correction, vertex counterterms and box diagrams, respectively. The spin-dependent contributions are labeled analogous. All of these terms will be given explicitly in Ref. \cite{myPhD}.
We stress that $\alpha^\mathrm{SI}_{q,\mathrm{P}} + \alpha^\mathrm{SI}_{q,\mathrm{PC}}$, $\alpha^\mathrm{SI}_{q,\mathrm{V}} + \alpha^\mathrm{SI}_{q,\mathrm{VC}}$, $\alpha^\mathrm{SI}_{q,\mathrm{B}}$ and $\mathcal{K}_\mathrm{EFTV1} + \mathcal{K}_\mathrm{EFTVC1}$ are separately ultraviolet finite, and the same holds for the spin-dependent case and the associated correction factors $\mathcal{K}_\mathrm{EFTV2} + \mathcal{K}_\mathrm{EFTVC2}$. However, there are also infrared divergences involved, which have not been discussed yet. Although most of the individual terms given above are infrared divergent, $c_1^\mathrm{1loop}$ and $c_2^\mathrm{1loop}$ as a whole are infrared finite, which is an essential feature of the matching procedure. The appearance of infrared divergences is connected with massless particles like gluons. These particles are likewise degrees of freedom in the full and the effective theory. In other words: The infrared regime of both theories is the same. Whenever there occurs an infrared divergence in the full theory, the very same infrared divergence occurs in the effective theory as well, and both cancel during the matching procedure. In our calculation, this cancellation is due to the correction factors $\mathcal{K}_\mathrm{EFTV1}$, $\mathcal{K}_\mathrm{EFTVC1}$, $\mathcal{K}_\mathrm{EFTV2}$ and $\mathcal{K}_\mathrm{EFTVC2}$ which we list now.
The vertex correction factor $\mathcal{K}_\mathrm{EFTV1}$ is obtained by calculating the diagram shown on the right of Fig.\ \ref{fig:DDEFT} involving the effective operator $Q_1^\mathrm{tree}$. We get
\begin{eqnarray}
\mathcal{K}_\mathrm{EFTV1} & = & \frac{\alpha_sC_F}{4\pi}\Big(4B_0 - 2 + 4p_bp_2(C_0 + C_1 + C_2)\Big),\nonumber\\
\end{eqnarray}
where the two- and three-point functions possess the arguments $B = B(p_b - p_2, m_q^2, m_q^2)$ and $C = C(p_2, p_b, 0, m_q^2, m_q^2)$. Here the four-momentum of the ingoing quark is denoted by $p_b$ and that of the outgoing quark by $p_2$. In the limit of vanishing relative velocity, we simply have $p = p_b = p_2$. Moreover, $C_F = 4/3$ denotes the usual color factor. This vertex correction is algebraically identical to the Higgs-gluon vertex shown on the very left in the second row of Fig.\ \ref{fig:DDEFT}, which has two important consequences. On the one hand, the Higgs-gluon vertex completely cancels in the matching procedure. The gluon is likewise a degree of freedom in the full and the effective theory and therefore the corresponding vertex correction occurs in both theories. It is included in the effective operator, not the Wilson coefficient. Moreover the correction factor $\mathcal{K}_\mathrm{EFTV1}$ is ultraviolet divergent, as it includes the two-point function $B_0$. To allow for a consistent matching procedure, we have to renormalize the effective theory in the same way as the full theory. This means that we have to add a counterterm $\delta c_1$ to the four-fermion coupling. This counterterm has to be of the same form as $\delta g_{\phi qq}$ (with $\phi = h^0,H^0$) and reads
\begin{equation}
\delta c_1^L = c_1^{\mathrm{tree},L}\left(\frac{\delta Z_m}{m_q} + \frac{1}{2}\delta Z_q^L + \frac{1}{2}\delta Z_q^{R*}\right),
\end{equation}
where $\delta Z_m$ denotes the mass and $\delta Z_q$ the wave function counterterm. For more details on these counterterms we refer the reader again to Ref.\ \cite{NeuQ2qx1}. The associated right-handed part of $\delta c_1$ is obtained by the substitution $L\leftrightarrow R$. The correction factor $\mathcal{K}_\mathrm{EFTVC1}$ is then simply given by
\begin{equation}
\mathcal{K}_\mathrm{EFTVC1} = \frac{\delta c_1^L/c_1^{\mathrm{tree},L} + \delta c_1^R/c_1^{\mathrm{tree},R}}{2}.
\end{equation}
Remember that $c_1^\mathrm{tree}$ does not only incorporate Higgs contributions, but that squark processes contribute as well (cf.\ Eq.\ (\ref{CSITree})). Whereas the Higgs-gluon vertex correction and its associated counterterm completely vanish in the matching procedure, this is not true for the vertex corrections to the squark processes shown in the third row of Fig.\ \ref{fig:DDCorrections} and their counterterms. However, the infrared divergences of these corrections and the ones stemming from the boxes shown in the last row of Fig.\ \ref{fig:DDCorrections} are precisely cancelled by the correction factors. This is an important consistency check of the whole calculation. Thanks to our generic implementation of loop integrals and the discrimination between ultraviolet and infrared poles, we could verify this cancellation explicitly.
We continue with the determination of $\mathcal{K}_\mathrm{EFTV2}$, i.e.\ the vertex correction factor for the spin-dependent operator $Q_2$. The associated diagram is shown on the right of Fig. \ref{fig:DDEFT} again, the only difference to the previous case is the included four-fermion coupling. Keeping only the relevant effective operators, we obtain
\begin{eqnarray}
\mathcal{K}_\mathrm{EFTV2} & = & \frac{\alpha_sC_F}{4\pi}\Big(2B_0 + 4p_bp_2(C_0 + C_1 + C_2)\nonumber\\
&& - 4C_{00} -1\Big)
\end{eqnarray}
where the two- and three-point functions possess the same arguments as before. The missing piece is the counterterm $\delta c_2$, which renders the vertex correction given above ultraviolet finite. This counterterm is constructed in analogy to $\delta g_{Z^0qq}$ and reads
\begin{equation}
\delta c_2^L = c_2^{\mathrm{tree},L}\left(\frac{1}{2}\delta Z_q^{\mathrm{SM},L} + \frac{1}{2}\delta Z_q^{\mathrm{SM},L*} + \frac{\alpha_sC_F}{\pi}\right).
\label{EFTVertexCSD}
\end{equation}
As before, the correction factor $\mathcal{K}_\mathrm{EFTVC2}$ is obtained via
\begin{equation}
\mathcal{K}_\mathrm{EFTVC2} = \frac{\delta c_2^L/c_2^{\mathrm{tree},L} + \delta c_2^R/c_2^{\mathrm{tree},R}}{2}.
\end{equation}
Note that we have included the additional finite part $\frac{\alpha_sC_F}{\pi}$ to retain a conventional axial current divergence which is in agreement with Refs. \cite{Hill2} and \cite{Larin}.\footnote{The results given in Ref. \cite{Larin} were obtained using the {$\overline{\mathrm{MS}}$}\ scheme and dimensional regularization. Transferring results from this scheme to the {$\overline{\mathrm{DR}}$}\ scheme and dimensional reduction -- which we are using -- is nontrivial in general. Discrepancies may arise due to the treatment of $\gamma_5$ in $D$ dimensions. However, these problems should occur at the three-loop order for the first time and do neither affect the finitite contribution included in Eq. (\ref{EFTVertexCSD}) nor the running of the axial-vector operator presented in the next section \cite{LarinMail}.} Moreover we incorporate just Standard Model contributions to $\delta Z_q$ in this case. The reason is as follows: In case of the Higgs vertex corrections including the gluon and the gluino, only the former is ultraviolet divergent. The whole counterterm $\delta g_{\phi qq}$ is responsible for the cancellation of this divergence. As the gluon vertex correction occurs likewise in the effective theory, we have constructed its associated counterterm $\delta c_1$ in complete analogy to $\delta g_{\phi qq}$. In case of the $Z^0$ vertex corrections including the gluon and the gluino, both are ultraviolet divergent. The divergences of the first diagram are removed by the Standard Model part of $\delta g_{Z^0 qq}$ and the latter by the SUSY part of $\delta g_{Z^0 qq}$. During the matching procedure, the gluon vertex correction and its corresponding counterterm has to cancel, whereas the vertex correction including the gluino and its counterterm contributes to the Wilson coefficient. Hence we only include Standard Model contributions to the spinor field counterterms in $\delta c_2$. This completes our matching calculation at NLO.
\subsection{Running of effective operators and associated Wilson coefficients}
\label{RunningSection}
The matching calculation presented in the last subsection is performed at the high scale $\mu_\mathrm{high}\sim 1$ TeV. In contrast, the nuclear matrix elements are defined at a low scale $\mu_\mathrm{low}\sim 5$ GeV. This is the energy regime we finally aim to describe with our effective field theory. To connect the two energy regimes, we have to evolve the effective operators and associated Wilson coefficients from the high scale down to the low scale by solving the corresponding renormalization group equations (RGEs). This part of the calculation is briefly referred to as ``running'' and is presented in this subsection.
The scale dependence of the Wilson coefficients is inverse to that of the corresponding operators. Therefore it cancels in the product, which is an essential feature of any operator product expansion. In the effective Lagrangian introduced in Eq.\ (\ref{Leff}), we have neglected higher-dimensional operators in our operator product expansion, i.e.\
\begin{equation}
\mathcal{L}_\mathrm{eff} = \sum_{i=1}^\infty c_iQ_i \approx c_1\bar{\chi}\chi\bar{q}q + c_2\bar{\chi}\gamma_\mu\gamma_5\chi\bar{q}\gamma^\mu\gamma_5q +\ldots
\end{equation}
As we are interested only in QCD effects, the running of the two operators given above is solely determined by their respective quark parts.
The scalar operator $m_q\bar{q}q$ is scale independent. As a consequence, the running calculation in the spin-indepependent case is rather simple. We have to factor out the quark mass $m_q(\mu_\mathrm{high})$ from the coefficient $c_1$. This quark mass has to be evolved down to the low scale $\mu_\mathrm{low}$ in the usual way, i.e.\ by solving its RGE. We then replace the combination $m_q(\mu_\mathrm{low})\bar{q}q$ via Eq.\ (\ref{fTDef}).
In contrast to that, the renormalization and the resulting running of the axial-vector operator is not trivial. This calculation has first been performed in Ref.\ \cite{Larin}. The relevant renormalization constant reads
\begin{eqnarray}
Z_A^\mathrm{Singlet} & = & 1 + \frac{\alpha_s}{\pi}C_F - \frac{1}{\epsilon_{UV}}\left(\frac{\alpha_s}{4\pi}\right)^2\left(\frac{20}{9}n_f + \frac{88}{3}\right)\nonumber\\
&& + \mathcal{O}(\alpha_s^3),
\end{eqnarray}
where $n_f$ denotes the number of active flavors and an additional finite term has been included to cure the axial anomaly. It is precisely this term which has been included in Eq.\ (\ref{EFTVertexCSD}) as well. Finite terms of order $\mathcal{O}(\alpha_s^2)$ have been neglected, as they are irrelevant for the running up to the desired order. Given this constant, we can calculate the corresponding anomalous dimension via
\begin{equation}
\gamma_A^\mathrm{Singlet} = (Z_A^\mathrm{Singlet})^{-1}\frac{\mathrm{d}}{\mathrm{d}\log\mu}Z_A^\mathrm{Singlet}
\end{equation}
and obtain
\begin{equation}
\gamma_A^\mathrm{Singlet} = \left(\frac{\alpha_s}{4\pi}\right)^2 16n_f + \mathcal{O}(\alpha_s^3).
\end{equation}
To arrive at this result one has to insert the RGE of the strong coupling constant including its divergent part, namely
\begin{equation}
\frac{\mathrm{d}g}{\mathrm{d}\log\mu} = -\epsilon_{UV}g + \beta(g),
\end{equation}
where $\beta(g)$ is the usual QCD beta function
\begin{eqnarray}
\frac{\beta(g)}{g} & = & -\beta_0\frac{\alpha_s}{4\pi} + \mathcal{O}(\alpha_s^2) = -(11-\frac{2}{3}n_f)\frac{\alpha_s}{4\pi} + \mathcal{O}(\alpha_s^2).\nonumber\\
\end{eqnarray}
The remaining step is to determine the running of the Wilson coefficient $c_2$ via
\begin{equation}
\frac{\mathrm{d}}{\mathrm{d}\log\mu}c_2(\mu) = \gamma_A^\mathrm{Singlet}c_2(\mu).
\end{equation}
We finally obtain
\begin{equation}
\frac{c_2(\mu_\mathrm{low})}{c_2(\mu_\mathrm{high})} = \exp\left(\frac{2n_f(\alpha_s(\mu_\mathrm{high}) - \alpha_s(\mu_\mathrm{low}))}{\beta_0\pi}\right),
\end{equation}
which agrees with the result given in Ref.\ \cite{Hill2}. Note that in general different operators may mix under renormalization. This is fortunately not the case here, but it will happen when one includes e.g.\ the gluon operator $G_{\mu\nu}G^{\mu\nu}$ \cite{Hill2}.
\section{Tensor reduction for vanishing Gram determinant}
\label{GramDet}
In the course of the {\tt DM@NLO}\ project, we have computed a large collection of loop integrals and associated special cases in generic form. In addition, we always distinguish between infrared and ultraviolet divergences. As these divergences have to vanish when determining physical observables, this discrimination allows for powerful checks of our calculations. Hence it is desirable to use the same thoroughly tested routines for the new direct detection calculation. However, in the context of direct detection, all the amplitudes are evaluated at zero momentum transfer.\footnote{To determine the relic density, only cross sections including a finite relative velocity are needed, as those with zero relative velocity are weighted by zero in the thermal averaging procedure.} This causes problems for the tensor reduction of loop amplitudes which we employ \cite{PVIntegrals}.
In this appendix we present our alternative approach, which is partially based on Ref.\ \cite{Ganesh}. To keep the discussion transparent and to stress the general idea, we restrict ourselves to the simple case of determining the tensor coefficients $C_1$ and $C_2$. All other necessary tensor coefficients can be worked out analogously.
We start by setting up our notation. The scalar and tensor integrals relevant for our discussion are defined via
\begin{eqnarray}
B_0(p_1, m_0^2, m_1^2) & = & \frac{(2\pi\mu_R)^{4-D}}{i\pi^2}\int\mathrm{d}^Dq\frac{1}{\mathcal{D}_0\mathcal{D}_1},\label{B0}\nonumber\\
&&\\
B_\mu(p_1, m_0^2, m_1^2) & = & \frac{(2\pi\mu_R)^{4-D}}{i\pi^2}\int\mathrm{d}^Dq\frac{q_\mu}{\mathcal{D}_0\mathcal{D}_1},\label{Bmu}\nonumber\\
&&\\
C_0(p_1, p_2, m_0^2, m_1^2, m_2^2) & = & \frac{(2\pi\mu_R)^{4-D}}{i\pi^2}\int\mathrm{d}^Dq\frac{1}{\mathcal{D}_0\mathcal{D}_1\mathcal{D}_2},\label{C0}\nonumber\\
&&\\
C_\mu(p_1, p_2, m_0^2, m_1^2, m_2^2) & = & \frac{(2\pi\mu_R)^{4-D}}{i\pi^2}\int\mathrm{d}^Dq\frac{q_\mu}{\mathcal{D}_0\mathcal{D}_1\mathcal{D}_2},\label{Cmu}\nonumber\\
&&\\
C_{\mu\nu}(p_1, p_2, m_0^2, m_1^2, m_2^2) & = & \frac{(2\pi\mu_R)^{4-D}}{i\pi^2}\int\mathrm{d}^Dq\frac{q_\mu q_\nu}{\mathcal{D}_0\mathcal{D}_1\mathcal{D}_2}.\label{Cmunu}\nonumber\\
\end{eqnarray}
Here $\mu_R$ denotes the renormalisation scale which has been introduced to fix the mass dimension of the integrals. The denominators are given by $\mathcal{D}_i = (q+p_i)^2 - m_i^2 + i\epsilon$ with $p_0 = 0$. The idea of the tensor reduction method is to decompose the tensor integrals into a linear combination of all possible Lorentz structures accompanied by yet unknown tensor coefficients. Omitting the arguments, we have
\begin{eqnarray}
B_\mu & = & p_{1,\mu}B_1,\\
C_\mu & = & p_{1,\mu}C_1 + p_{2,\mu}C_2,\\
C_{\mu\nu} &= & g_{\mu\nu}C_{00} + p_{1,\mu}p_{1,\nu}C_{11} + p_{2,\mu}p_{2,\nu}C_{22} \nonumber\\
&& + (p_{1,\mu}p_{2,\nu} + p_{2,\mu}p_{1,\nu})C_{12}.\label{StandardDecomposition}
\end{eqnarray}
The tensor coefficents are obtained by multiplying both sides with the available Lorentz invariants. In this way, the tensor integrals are reduced to a combination of scalar integrals. When determining $C_1$ and $C_2$ we have to solve a set of linear equations which results in
\begin{eqnarray}
\begin{pmatrix} C_1 \\ C_2 \end{pmatrix} & = & A^{-1}\begin{pmatrix} R_1 \\ R_2 \end{pmatrix} = \frac{1}{\det{A}}\begin{pmatrix} p_2^2 & -p_1p_2\\ -p_1p_2 & p_1^2\end{pmatrix}\begin{pmatrix} R_1 \\ R_2 \end{pmatrix}\label{C1-C2},\nonumber\\
&&
\end{eqnarray}
where we have introduced
\begin{eqnarray}
\det(A) & = & p_1^2p_2^2 -(p_1p_2)^2,\\
R_1 & = & \frac{1}{2}\left(B_0(0,2) - B_0(1,2) - f_1C_0\right),\label{R1}\\
R_2 & = &\frac{1}{2}\left(B_0(0,1) - B_0(1,2) - f_2C_0\right),\label{R2}\\
f_i & = & p_i^2 -m_i^2 + m_0^2\quad\mathrm{with}\quad i=1,2.\label{fi}
\end{eqnarray}
Furthermore we define the shorthand notation $B_0(i,j) = B_0(p_j-p_i, m_i^2, m_j^2)$, which we use analogously for $B_1$.
The method illustrated above breaks down when the matrix $A$ is not invertible, i.e.\ when $\det{(A)}$ vanishes. However, instead of using Eq. (\ref{C1-C2}) for determining the tensor coefficients $C_1$ and $C_2$, we can assume that these coefficients still exist and express $C_0$ in terms of two-point functions by writing $p_2^2R_1 -p_1p_2R_2 = 0$ and $-p_1p_2R_1 + p_1^2R_2 = 0$ and solving these (equivalent) equations for $C_0$.
The main idea of Ref.\ \cite{Ganesh} is to repeat this procedure for every tensor rank successively. We can write down the expressions determining the tensor coefficients of second rank, i.e.\ $C_{00}$, $C_{11}$, $C_{12}$ and $C_{22}$. The ultraviolet divergent coefficient $C_{00}$ is not directly\footnote{The coefficient $C_{00}$ is indirectly plagued by problems in the limit of a vanishing Gram determinant, as it is composed of the problematic coefficients $C_1$ and $C_2$.} affected by problems of vanishing Gram determinants and found to be
\begin{equation}
C_{00} = \frac{m_0^2C_0}{D-2} + \frac{B_0(1,2) + f_1C_1 + f_2C_2}{2(D-2)}.
\end{equation}
\\
\noindent
In contrast, the remaining tensor coefficients can not be obtained via standard tensor reduction for vanishing Gram determinant. Instead of that, the corresponding equations can be used to determine the tensor equations of rank one, i.e.\ $C_1$ and $C_2$. The result can be written in compact form as
\begin{equation}
\begin{pmatrix} C_1 \\ C_2 \end{pmatrix} = Z_i^{-1}\begin{pmatrix} R_{3,i} \\ R_{4,i}\end{pmatrix}\quad\mathrm{with}\quad i=1,2.\label{C1C2Alternative}
\end{equation}
\\
\noindent
The abbreviations used here are
\begin{widetext}
\begin{eqnarray}
R_{3,i} & = & x_{i1}\left(B_1(1,2) + B_0(1,2) -\frac{2m_0^2}{D-2}C_0 - \frac{1}{D-2}B_0(1,2)\right) + x_{i2}\left(B_1(0,1) + B_1(1,2) + B_0(1,2)\right),\\
R_{4,i} & = & x_{i1}\left(B_1(0,2) - B_1(1,2)\right) + x_{i2}\left(-B_1(1,2) -\frac{2m_0^2}{D-2}C_0 - \frac{1}{D-2}B_0(1,2)\right),\\
Z_i & = &\begin{pmatrix} Y_i + \frac{x_{i1}}{D-2}f_1 & \frac{x_{i1}}{D-2}f_2 \\ \frac{x_{i2}}{D-2}f_1 & Y_i + \frac{x_{i2}}{D-2}f_2 \end{pmatrix}\quad\mathrm{with}\quad Y_i = x_{i1}f_1 + x_{i2}f_2\quad\mathrm{and}\quad \begin{pmatrix} p_2^2 & -p_1p_2\\ -p_1p_2 & p_1^2\end{pmatrix} = \begin{pmatrix} x_{11} & x_{12}\\ x_{21} & x_{22}\end{pmatrix}.
\end{eqnarray}
\end{widetext}
To summarize, the presented method allows to determine the tensor coefficients of rank $n$ by investigating the equations for tensor coefficients of rank $n+1$ in the limit of vanishing Gram determinant. This works in an algorithmic manner. In comparison to the standard tensor reduction method, the expressions are more lengthy. However, note that the algebraic form of Eqs.\ (\ref{C1-C2}) and (\ref{C1C2Alternative}) is the same. One might ask what happens when $\det(Z_i)$ vanishes. This is of interest, as we precisely run into this situation in the course of our direct detection calculations when evaluating e.g.\ the three-point function $C_0(p,p,m_0^2,m_1^2,m_2^2)$.
\begin{figure*}
\centering
\includegraphics[width=0.49\textwidth]{C1-C2.pdf}
\includegraphics[width=0.49\textwidth]{Ca.pdf}
\includegraphics[width=0.49\textwidth]{C00.pdf}
\includegraphics[width=0.49\textwidth]{Caa.pdf}
\caption{Numerical stability of the three-point tensor coefficients in the limit $v\rightarrow 0$ or equivalently $t\rightarrow 0$.}
\label{fig:AltReduction}
\end{figure*}
There are basically three ways to proceed. First, note that there are actually two sets of equations hiding behind Eq.\ (\ref{C1C2Alternative}). In some lucky cases it might happen that only one variant fails while the other is still working. The second possibility is to apply l'H\^opital's rule without encountering $\det(Z_i)$ again. This is an improvement in comparison to the standard tensor reduction method, where $\det(A)$ usually reappears when taking the limit.
We took a closer look at the problematic cases involved in our calculation and found a third way out of this dilemma. We illustrate this by referring to the three-point function $C_0(p,p,m_0^2,m_1^2,m_2^2)$. Remember that $p_1 = p_2 = p$ is a stronger condition than just $p_1^2 = p_2^2 = p^2$ which happens frequently for identical external particles. The crucial observation is that the tensor coefficients $C_1$ and $C_2$ are no longer uniquely defined in this situation; only their sum $C_a = C_1 + C_2$ is. Instead of Eq.\ (\ref{StandardDecomposition}), we get
\begin{equation}
C_\mu = p_{1,\mu}C_1 + p_{2,\mu}C_2 \rightarrow p_\mu(C_1 + C_2) = p_\mu C_a.
\end{equation}
It is precisely this combination which remains in all the amplitudes in the limit $p_1\rightarrow p_2$. Hence we replace this sum by $C_a$. This coefficient can be easily obtained in the usual way and reads
\begin{equation}
C_a = \frac{1}{2p^2}\left(B_0(0,2) - B_0(1,2) - f_1C_0\right) \label{Ca}.
\end{equation}
By taking into account that tensor coefficients may coalesce under certain kinematical circumstances and applying the method of Ref.\ \cite{Ganesh} as illustrated above, we were able to stabilize the tensor reduction method for vanishing Gram determinant for all loop amplitudes occuring in our direct detection analysis. This is particularly true for the four-point functions needed for the box contributions. Although the basic idea remains unchanged, the corresponding expressions become very large and were therefore calculated with the help of \texttt{Mathematica}.
All tensor coefficients obtained in this way have been tested extensively. We have numerically compared them with the corresponding coefficients resulting from the standard tensor reduction method for small, but non-vanishing Gram determinant. Some examples are shown in Fig.\ \ref{fig:AltReduction}.
The upper left plot of Fig.\ \ref{fig:AltReduction} shows the numerical stability of the tensor coefficients $C_1$ and $C_2$ in the limit of equal momenta or equivalently $(p_b - p_2)^2 = t \rightarrow 0$. More precisely, we show the real parts of $C_1$ (in red) and $C_2$ (in blue) obtained by the regular tensor reduction method divided by $C_a/2$ obtained via Eq.\ (\ref{Ca}) and subtracted by one. In this representation, the black null line corresponds directly to $C_a/2$. We observe, as expected, that both $C_1$ and $C_2$ are relatively stable for $t \leq -0.5$ GeV$^2$ and marginally differ from $C_a/2$. The regular tensor reduction is still working here and the small, but finite velocity leads to a small shift relative to the black reference line. However, when we approach the limit $t\rightarrow0$ the regular tensor method fails and both of the coefficients become numerically unstable.
As explained before, the individual tensor coefficients $C_1$ and $C_2$ are no longer uniquely defined in this limit, only their sum is. The real part of this sum divided by $C_a$ and subtracted by one is shown in the upper right plot of Fig.\ \ref{fig:AltReduction}. It is more stable than the individual coefficients, but still becomes noisy at very small relative velocities. For larger (but still small) relative velocities, the agreement between $C_1 + C_2$ and $C_a$ is excellent, which justifies our approach.
We show analogous plots for the tensor coefficient $C_{00}$ and the combination $C_{11} + 2C_{12} + C_{22}$ in the lower part of Fig.\ \ref{fig:AltReduction}. The main features are similar to the ones discussed before. Using the original tensor reduction method, the tensor coefficients become numerically unstable at very small relative velocities. When using the alternative approach described in this section, we obtain a stable result for $v = 0$ which is in perfect agreement with the standard method for small, but non-zero relative velocities. The black reference line in the lower right plot of Fig.~\ref{fig:AltReduction} is defined by
\begin{eqnarray}
C_b & = &\frac{1}{3p^2}\big(B_0(1,2) - m_0^2C_0 + 2B_1(0,1)\nonumber\\
&& - 2f_2C_a - \frac{1}{2}\big)
\end{eqnarray}
Although of minor importance for the tensor reduction itself, we list all the masses used in the plots above for completeness. They are $m_b = 2.3$ GeV, $m_t = 148.0$ GeV, $m_{\tilde{g}} = 1170.7$ GeV, $m_{\tilde{b}_1} = 1007.3$ GeV, $m_{\tilde{b}_2} = 1071.9$ GeV, $m_{\tilde{t}_1} = 827.9$ GeV and finally $m_{\tilde{t}_2} = 1042.6$ GeV. Note that we have $p_2^2 = p_b^2 = m_b^2$ in the first three plots, whereas $p_2^2 = p_b^2 = m_t^2$ in the lower right plot. More details on the presented alternative tensor reduction method can be found in Ref.\ \cite{myPhD}.
\section{Conclusion}
\label{Conclusion}
In this paper, we presented a NLO SUSY-QCD calculation for the scattering of neutralino dark matter off of the partonic constituents of nucleons, which required a novel tensor reduction method of loop integrals with vanishing relative velocities and Gram determinants. We consistently matched these one-loop corrections to the scalar and axial-vector operators, which govern the spin-independent and spin-dependent scattering proceses in the effective field theory approach. As a result, the operators and Wilson coefficients aquired a scale dependence, which was taken into account by applying renormalization group running to the Wilson coefficients. Our formalism is valid for general compositions of bino, wino, or higgsino dark matter.
We investigated three benchmark scenarios, which satisfy current Higgs mass, relic density, flavor-changing neutral current and direct SUSY particle search constraints from the LHC, but which were not tuned to be particularly sensitive to the new NLO corrections for direct detection. Despite the fact that the first- and second generation squark masses were at the TeV scale, we observed corrections that were of similar size or in some cases larger than the currently estimated nuclear uncertainties. This could be explained by small neutralino-squark mass differences governing the propagator denominators at low velocity. In general, large corrections can be expected in the spin-independent case for Higgs bosons coupling to winos and heavy quarks, in the spin-dependent case for $Z$-bosons coupling to higgsinos and light (potentially also heavy) quark flavors, and in both cases from squarks with small masses or mass differences or scenarios with destructive interference at tree level. In the first case, our calculation is complementary to the explicit generation of heavy quarks from gluon operators at one loop, similarly to the complementarity of variable and fixed flavor schemes that are both employed in deep-inelastic scattering. The calculation for gluon operators has been performed previously elsewhere; its implementation in DM@NLO and a comparison of the two approaches is left for future work, as is a numerical study for light or nearly neutralino mass-degenerate squarks.
Through the implementation of direct detection at NLO as a second dark matter observable in DM@NLO, consistent investigations of correlations between direct detection and the relic density at NLO are now possible. First examples have been given in this paper in the three mentioned reference scenarios. Systematically, shifts in the extracted dark matter mass from NLO corrections to the relic density implied different NLO corrections to be expected in direct detection experiments.
\section{Introduction}
\label{Intro}
Nowadays, the existence of dark matter is well established by experimental observations on many different length scales. In particular, on cosmological length scales, measurements of the temperature anisotropies of the Cosmic Microwave Background (CMB) allow a very precise determination of the relic density of dark matter. The most recent value obtained by the Planck collaboration \cite{Planck}, including polarization data from the Wilkinson Microwave Anisotropy Probe \cite{WMAP9}, is
\begin{equation}
\Omega_{\mathrm{CDM}}h^2 = 0.1199 \pm 0.0022,
\label{Planck}
\end{equation}
where $h$ denotes the present Hubble expansion rate in units of 100 km s$^{-1}$ Mpc$^{-1}$. Even though the quantity of dark matter in the Universe is known very accurately, its nature remains concealed. The reason for this unfortunate situation is that so far all experimental evidence for dark matter stems exclusively from its gravitational interaction.
Among the numerous attempts to explain dark matter, postulating the existence of a yet unknown Weakly Interacting Massive Particle (WIMP) is a widely adopted paradigm. This approach is attractive because a WIMP with typical weak scale interactions and a mass of $\sim$ 100 GeV naturally leads to the observed relic density via thermal freeze-out \cite{Klasen:2015uma}. The canonical example for a WIMP is the lightest neutralino $\tilde{\chi}^0_1$, which is the lightest supersymmetric particle in many scenarios of the Minimal Supersymmetric Standard Model (MSSM). In the following, we refer to it simply by ``the neutralino''. Remember that giving rise to a suitable dark matter candidate is only a positive byproduct of introducing Supersymmetry (SUSY) as the most general space-time symmetry, which is furthermore motivated by its elegant solution to the hierarchy problem and the possible unification of gauge and Yukawa couplings. Alternatively, more minimal extensions to the SM with additional Higgs doublets \cite{Barbieri:2006dq}, neutrinos \cite{Ma:2006km} or other scalars and fermions \cite{Esch:2013rta} may be considered.
Assuming that dark matter actually consists of WIMPs, additional non-gravitational detection possibilities open up. First, one can try to directly produce WIMPs at a collider. As the WIMPs themselves are not detectable with current collider detectors, the typical observable of such a process consists of a single jet or gauge boson and missing transverse energy. The second possibility is to look for Standard Model annihilation products of WIMPs in very dense astronomical objects such as the Sun or the center of the Galaxy, where the WIMPs might have accumulated. The observational challenge of this indirect detection approach is to distinguish between the astrophysical background and a possible WIMP signal. Finally, one can try to observe the rare interactions of a WIMP with a nucleus by detecting its recoil in the so-called direct detection experiments. The technical difficulty here is to detect a very weak signal, while simultaneously excluding all non-dark matter sources \cite{Klasen:2015uma}.
The direct detection rate, i.e.\ the number of events per time and per detector mass, depends on the dark matter-nucleus interaction. On the microscopic level, this corresponds to the interaction of the WIMP with the quarks and gluons inside the nucleons of the nucleus. However, as the typical process energies are much smaller than the mediator masses\footnote{As this condition is not necessarily fulfilled at a collider, EFT methods are under debate in this context, and so-called simplified models should be used \cite{Buchmueller, Busoni, DeSimone, Matsumoto}.} of the microscopic theory, it is customary to calculate the corresponding cross sections in the framework of effective field theories (EFT) \cite{Hill1, Hill2, HisanoEFT, JijiEFT}. In the EFT approach, the heavy particles which mediate the interaction between dark matter and the constituents of the nucleus are integrated out.
Integrating-out heavy particles translates different Lorentz structures of the microscopic theory to different effective contact interactions expressed in terms of effective operators. Not all of the effective interactions contribute in the non-relativistic limit which is relevant for direct detection. In the MSSM, the dominant effective operators for neutralino dark matter are the scalar operator $m_q\bar{\chi}\chi\bar{q}q$ and the axial-vector operator $\bar{\chi}\gamma_\mu\gamma_5\chi\bar{q}\gamma^\mu\gamma_5q$, as the vector and tensor operators vanish in the case of a Majorana fermion. These operators lead to coherent spin-independent (SI) and spin-dependent (SD) contributions, respectively.
The tree-level contributions of neutralino dark matter to these operators have first been calculated in Ref.\ \cite{Griest}. Since then, several improvements have been made by either including additional operators like e.g.\ gluon operators \cite{Drees, HisanoGluon} or by calculating electroweak radiative corrections for pure wino, higgsino or bino dark matter \cite{Hisanoelw, Berlin1, Berlin2}.
In this paper, we perform a full $\mathcal{O}(\alpha_s)$ calculation for the two dominant operators listed above. In contrast to previous approaches, we allow for a general neutralino admixture and calculate the radiative corrections using fully general loop integrals. By doing so, we implement a second, loop-improved dark matter observable in our numerical package {\tt DM@NLO}, the first one being the relic density \cite{DMNLOPage, AFunnel, ChiChi2qq1, ChiChi2qq2, ChiChi2qq3, NeuQ2qx1, NeuQ2qx2, QQ2xx, Scalepaper}. Combining these calculations allows to effectively constrain the MSSM parameter space and precisely predict the direct detection rate.
The remainder of this paper is organized as follows: In Sec.\ \ref{Technical}, we briefly remind the reader how the direct detection rate is computed in general, and we describe our renormalization scheme. We present the underlying EFT calculation, specify the matching of full and effective theory, and describe the running of the operators and their associated Wilson coefficients. In order to use the same tensor reduction method for our direct detection and relic density calculation, the tensor reduction method had to be modified to account for vanishing Gram determinants. As this technical aspect might be interesting on its own, we illustrate it separately in App.\ \ref{GramDet}. Our numerical results are then given in Sec.\ \ref{Numerics}. We analyze the impact of the radiative corrections and contrast them with the nuclear uncertainties. We also study the influence of the neutralino composition on the resulting neutralino-nucleus cross sections. Furthermore, we combine our direct detection and relic density routines to obtain precise predictions for the neutralino-nucleon cross section in a given scenario. Finally, we conclude in Sec.\ \ref{Conclusion}. We do not present any technical details of our relic density calculations here, but instead refer the reader to our previous papers and in particular Ref.\ \cite{ChiChi2qq3}.
\section{Numerical results}
\label{Numerics}
\begin{table*}
\caption{pMSSM input parameters for three selected reference scenarios. All parameters except $\tan\beta$ are given in GeV. }
\begin{tabular}{|c|ccccccccccc|}
\hline
$\quad$ & $\quad\tan\beta\quad$ & $\quad\mu\quad$ & $\quad m_A\quad$ & $\quad M_1\quad$ & $\quad M_2\quad$ & $\quad M_3\quad$ & $\quad M_{\tilde{q}_{1,2}}\quad$ & $\quad M_{\tilde{q}_3}\quad$ & $\quad M_{\tilde{u}_3}\quad$ & $\quad M_{\tilde{\ell}}\quad$& $\quad A_t\quad$ \\
\hline
A & 13.4 & 1286.3 & 1592.9 & 731.0 & 766.0 & 1906.3 & 3252.6 & 1634.3 & 1054.4 & 3589.6 & -2792.3\\
B & 13.7 & 493.0 & 500.8 & 270.0 & 1123.4 & 1020.3 & 479.9 & 1535.5 & 836.7 & 3469.4 & -2070.9\\
C & 7.0 & 815.0 & 1452.8 & 675.3 & 1423.4 & 1020.3 & 809.9 & 1835.5 & 1436.7 & 3469.4 & -2670.9\\
\hline
\end{tabular}
\label{ScenarioList}
\end{table*}
\begin{table*}
\caption{Gaugino and squark masses and other selected observables corresponding to the reference scenarios of Tab.\ \ref{ScenarioList}. All masses are given in GeV.}
\begin{tabular}{|c|cc|cc|ccccc|ccc|}
\hline
$\quad$ & ~~ $m_{\tilde{\chi}^0_1}$~~ & ~~$m_{\tilde{\chi}^0_2}$~~ & ~~$m_{\tilde{\chi}^{\pm}_1}$~~ & ~~$m_{\tilde{\chi}^{\pm}_2}$~~ & ~~$m_{\tilde{u}_1}$~~ & ~~$m_{\tilde{d}_1}$~~ & ~~$m_{\tilde{t}_1}$~~ & ~~$m_{\tilde{b}_1}$~~ & ~~$m_{\tilde{g}}$~~ & ~~$m_{h^0}$~~ & ~~$\Omega_{\tilde{\chi}^0_1} h^2$~~ & $\mathrm{BR}(b\rightarrow s\gamma)$ \\
\hline
A & 738.1 & 802.4 & 802.3 & 1295.1 & 3270.9 & 3271.6 & 993.9 & 1622.9 & 2049.9 & 126.3 & 0.1244 & $3.0\cdot 10^{-4}$ \\
B & 265.7 & 498.4 & 495.7 & 1135.3 & 549.5 & 555.7 & 802.9 & 1531.0 & 1061.2 & 124.8 & 0.1199 & $3.6\cdot 10^{-4}$\\
C & 669.2 & 826.6 & 819.6 & 1438.9 & 865.0 & 868.4 & 1389.1 & 1832.3 & 1090.7 & 125.2 & 0.1179 & $3.3\cdot 10^{-4}$\\
\hline
\end{tabular}
\label{ScenarioProps}
\end{table*}
\begin{table}
\caption{Most relevant (co)annihilation channels in the reference scenarios of Tab.\ \ref{ScenarioList}. Channels which contribute less than 1\% to the thermally averaged cross section and/or are not implemented in our code are not shown.}
\begin{tabular}{|rl|cccc|}
\hline
& & ~~~~ A ~~~~ & ~~~~ B ~~~~ & ~~~~ C ~~~~ & \\
\hline
$\tilde{\chi}^0_1 \tilde{\chi}^0_1 \to$ & $t\bar{t}$ & 1\% & 10\% & 52\% & \\
& $b\bar{b}$ & 9\% & 78\% & 40\% & \\
$\tilde{\chi}^0_1 \tilde{\chi}^0_2 \to$ & $t\bar{t}$ & 3\% & & & \\
& $b\bar{b}$ & 23\% & & & \\
$\tilde{\chi}^0_1 \tilde{\chi}^{\pm}_1 \to$ & $t\bar{b}$ & 43\% & & & \\
\hline
\multicolumn{2}{|c|}{Total} & 79\% & 88\% & 92\% & \\
\hline
\end{tabular}
\label{ScenarioChannels}
\end{table}
In this section we describe our numerical setup and present numerical results for three selected reference scenarios. These scenarios are defined in a phenomenological MSSM (pMSSM) with eleven free parameters, which we have already used in our previous analyses. This setup was designed for relic density calculations including light stops \cite{NeuQ2qx1, NeuQ2qx2, QQ2xx}. As it has proven sufficient for finding interesting direct detection scenarios, we stick to it for consistency and keep in mind, that a more specific pMSSM setup may lead to considerably larger loop contributions.
The aforementioned eleven free parameters are as follows: The Higgs sector is fixed by the higgsino mass parameter $\mu$, the ratio of the vacuum expectation values of the two Higgs doublets $\tan\beta$, and the pole mass of the pseudoscalar Higgs boson $m_A$. The gaugino sector is defined by the bino ($M_1$), wino ($M_2$) and gluino ($M_3$) mass parameters, which in our setup are not related through any assumptions stemming from Grand Unified Theories. Moreover we define a common soft SUSY-breaking mass parameter $M_{\tilde{q}_{1,2}}$ for the first- and second-generation squarks. The third-generation squark masses are controlled by the parameter $M_{\tilde{q}_3}$ associated with sbottoms and left-handed stops and by the parameter $M_{\tilde{u}_3}$ for right-handed stops. The trilinear coupling in the stop sector is given by $A_t$, while the trilinear couplings of the other sectors, including $A_b$, are set to zero. Since the slepton sector is not at the center of our attention, it is parametrized by a single soft parameter $M_{\tilde{\ell}}$. The most interesting parameters for the following discussion are those determining the neutralino decomposition ($\mu$, $M_1$ and $M_2$) and $M_{\tilde{q}_{1,2}}$.
These eleven pMSSM input parameters are defined in the {$\overline{\mathrm{DR}}$}\ scheme at the scale $\tilde{M} = 1$ TeV according to the SPA convention \cite{Spa}. We identify this scale with our renormalization scale $\mu_{R}$, which simultaneously corresponds to the high scale $\mu_{\mathrm{high}}$ of our EFT calculation. The input parameters are handed over to the numerical package {\tt SPheno}\ \cite{SPheno} to calculate the associated physical spectrum.
We neglect the masses of the quarks of the first two generations in the kinematics to improve numerical stability. On the other hand, we keep those masses in the Yukawa couplings to allow for Higgs exchange processes. Remember that the Yukawa masses are basically factored out of the amplitudes and replaced by the nuclear matrix elements via Eq.\ (\ref{fTDef}). It has been checked explicitly that the effect of this simplification on the final results is negligible.
Our three reference scenarios are listed in Tab.\ \ref{ScenarioList}. Table \ref{ScenarioProps} contains the corresponding relevant gaugino and squark masses\footnote{We are not showing the squark masses $m_{\tilde{u}_2}$, $m_{\tilde{d}_2}$, $m_{\tilde{c}_1}$, $m_{\tilde{c}_2}$, $m_{\tilde{s}_1}$ and $m_{\tilde{s}_2}$. However, as we are working with a common soft mass parameter $M_{\tilde{q}_{1,2}}$, all squark masses of the first two generations are roughly the same.} as well as the obtained mass of the lightest neutral (and thus SM-like) Higgs boson, the neutralino relic density computed at tree level with {\tt micrOMEGAs}\ and the important branching ratio of the rare $B$-meson decay $b\to s\gamma$ computed with {\tt SPheno}. Moreover, Tab.\ \ref{ScenarioChannels} lists the most relevant (co)annihilation channels for determining the relic density. Other important parameters are the neutralino mixing angles, i.e.\ its bino, wino and higgsino admixture. As the phenomenology of the three reference scenarios is to a large extent driven by these parameters, we explore them in more detail in the following. We devote an individual subsection to each scenario.
\subsection*{Scenario A -- Bino-wino dark matter}
We start by investigating scenario A. This scenario has been introduced in Ref.\ \cite{ChiChi2qq3} and studied again in Ref.\ \cite{Scalepaper}. Its main feature are sizeable gaugino coannihilation contributions to the relic density calculation as listed in Tab.\ \ref{ScenarioChannels}. The direct detection in this scenario is in no way special and we include this scenario as an arbitrary conservative case.
The decomposition of the neutralino in dependence of the pMSSM input parameter $M_1$ is shown in Fig.\ \ref{fig:MixingScenA}. As long as $M_1<M_2$, the neutralino is mostly bino. It turns into mostly wino when $M_1 > M_2 = 766$ GeV while the higgsino content always stays small due to $M_1, M_2 < \mu$. Note that scenario A itself, i.e the cosmologically preferred region, sits near the turnover ($M_1 = $ 731 GeV). This situation is encountered in many pMSSM scenarios and clearly calls for a general treatment of the neutralino admixture. We also show the associated neutralino mass on the top of each plot as a derived parameter. This connects our theoretical predictions to experimental exclusion limits, which are usually given in dependence of the WIMP mass. Note that the correspondance between $M_1$ and $m_{\tilde{\chi}^0_1}$ is basically 1:1 for $M_1$ up to 800 GeV, but for larger values of $M_1$ the neutralino becomes mostly wino, so that its mass is almost independent of $M_1$.
\begin{figure}
\includegraphics[width=0.49\textwidth]{Mixing1.pdf}
\caption{Neutralino decomposition in scenario A.}
\label{fig:MixingScenA}
\end{figure}
\begin{figure*}
\includegraphics[width=0.49\textwidth]{SI-p1.pdf}
\includegraphics[width=0.49\textwidth]{SI-n1.pdf}
\includegraphics[width=0.49\textwidth]{SD-p1.pdf}
\includegraphics[width=0.49\textwidth]{SD-n1.pdf}
\caption{Spin-independent (top) and spin-dependent (bottom) neutralino-nucleon cross sections in scenario A for protons (left) and neutrons (right).}
\label{fig:CrossSectionsScenA}
\end{figure*}
We continue with the discussion of the neutralino-nucleon cross sections, which are displayed in Fig.\ \ref{fig:CrossSectionsScenA}. The upper left plot of Fig.\ \ref{fig:CrossSectionsScenA} illustrates the spin-independent neutralino-proton cross section. This quantity has been calculated by {\tt micrOMEGAs}\ (orange solid line), our code at tree level (black solid line) and our code including full $\mathcal{O}(\alpha_s)$ corrections to the dominant effective operators (blue solid line). The shift between our tree-level calculation and {\tt micrOMEGAs}\ is mainly due to different nuclear input values (cf.\ Tab.\ \ref{fTTable}). After adjusting the nuclear input, our tree-level calculation agrees quite well with {\tt micrOMEGAs}, which is shown by the dotted black line. In absolute numbers, as expected, the neutralino-proton cross section is rather small (10$^{-47}$ - 10$^{-46}$ cm$^{2}$), as long as the neutralino is mostly bino. The tree-level couplings to Higgs bosons are supressed in this case, and so are the squark processes because of the heavy squark masses (cf.\ Tab.\ \ref{ScenarioProps}). The shift between our tree-level and our full NLO calculation is of similar size as the shift between our tree level and {\tt micrOMEGAs}. In the present case, the first shift is mainly caused by SUSY-QCD corrections to the Higgs exchange process including third generation squarks as the other squarks are much heavier.
Furthermore we show the improved\footnote{More precisely, the green dotted line corresponds to the choice \texttt{MSSMDDTest(loop=1, ...)}, whereas the orange solid line corresponds to \texttt{MSSMDDTest(loop=0, ...)}.} tree-level calculation of {\tt micrOMEGAs}\ as the green dotted line. Among other improvements, this choice is supposed to replace the heavy quark contributions by the gluon one-loop processes as given in Ref.\ \cite{Drees}. However, we could not find a significant difference in comparison to the pure tree-level calculation in any scenario. Therefore, the green dotted and orange full lines are indistinguishable also in this plot.
We also show the resulting relic density obtained with {\tt micrOMEGAs}\ as the dashed orange line (right ordinate). Note that this curve is roughly inverse to the cross section curves. This correlation is not completely unexpected. Larger gaugino (co)annihilation cross sections into final quark states leading to a smaller relic density are linked to larger neutralino-nucleon cross sections. The crucial condition for this correlation is that the neutralinos annihilate dominantly into quark final states. In the present case this is given for $M_1 > 200$ GeV. For smaller $M_1$, neutralinos prefer to annihilate into electroweak final states, and the resulting bump in the relic density has no counterpart in the neutralino-nucleon cross section. The orange vertical band marks the region of $M_1$ leading to a relic density compatible with the Planck limits as given in Eq.\ (\ref{Planck}). We investigate this region in greater detail later.
The upper right plot of Fig.\ \ref{fig:CrossSectionsScenA} shows the spin-independent neutralino-neutron cross section. No major difference in comparison to the proton case is found in this scenario, since the isospin-dependent contributions from first-generation quarks are suppressed by large squark masses.
We continue with the lower left plot of Fig.\ \ref{fig:CrossSectionsScenA} where the spin-dependent neutralino-proton cross section is given. Here the blue and black solid lines completely overlap, signalizing that the NLO corrections are negligible. This is indeed the case in this scenario. Remember that only light quarks ($u,d,s$) and corresponding squarks contribute to the spin-dependent cross section (cf.\ subsection \ref{DDFormulas}). These squarks are very heavy in this scenario (cf.\ Tab.\ \ref{ScenarioProps}) and loops including them are strongly suppressed. The small shift ($\sim + 7\%$) between our results and {\tt micrOMEGAs}\ is not due to the nuclear input values this time -- by default we are using the same input in the spin-dependent case. It is rather due to the running of the operator and associated Wilson coefficient described in subsection \ref{RunningSection}, which is not implemented in {\tt micrOMEGAs}. If we deactivate the running in our code, we find perfect agreement with {\tt micrOMEGAs}. The spin-dependent neutralino-neutron cross section is shown in the lower right plot of Fig.\ \ref{fig:CrossSectionsScenA}. As before, no major difference in comparison to the proton case is found in this scenario.
\begin{figure}
\includegraphics[width=0.49\textwidth]{SI-p_Zoom1.pdf}
\caption{Combined relic density and direct detection calculation in scnenario A.}
\label{fig:ZoomScenA}
\end{figure}
We take a closer look at the cosmologically preferred region now, i.e.\ we zoom into the region 700 GeV $< M_1 <$ 800 GeV of the upper left plot of Fig.\ \ref{fig:CrossSectionsScenA}. The result is shown in Fig.\ \ref{fig:ZoomScenA}. Apart from the previously introduced three solid lines, we depict the relic density obtained with {\tt micrOMEGAs}\ (orange dashed line), our code at tree level (black dashed line) and our code at NLO (blue dashed line). These three calculations lead to different cosmologically preferred regions as indicated by the orange, black and blue vertical band, respectively. Assuming that the neutralinos solely account for dark matter, we can combine these calculations to constrain the pMSSM parameter space and to precisely predict the resulting neutralino-nucleon cross section. This corresponds to identifying the intersections of the vertical bands and solid lines of the same color. The results are given in Tab.\ \ref{PredictionsScenA} where we also list the relative shifts of the {\tt micrOMEGAs}\ and our full NLO result with respect to our tree-level calculation. The shifts are in opposite directions and of similar size in this case.
\begin{table}
\caption{Resulting $M_1$ and spin-independent neutralino-proton cross section when combining direct detection and relic density routines in scenario A.}
\begin{tabular}{|c|ccc|}
\hline
$\quad$ & $M_1$ [GeV] & $\sigma^{\mathrm{SI}}_p$ [$10^{-46}$cm$^2$]& Shift of $\sigma^{\mathrm{SI}}_p$\\
\hline
{\tt micrOMEGAs}\ & 731 & $1.68$ & $-15\%$ \\
Tree level & 734 & $1.98$ & \\
Full NLO & 733 & $2.26$ & $+14\%$ \\
\hline
\end{tabular}
\label{PredictionsScenA}
\end{table}
\subsection*{Scenario B -- Bino-higgsino dark matter}
\begin{figure}
\includegraphics[width=0.49\textwidth]{Mixing2.pdf}
\caption{Neutralino decomposition in scenario B.}
\label{fig:MixingScenB}
\end{figure}
When varying $M_1$ in scenario B, the neutralino decomposition changes again, this time from mostly bino into mostly higgsino as shown in Fig.\ \ref{fig:MixingScenB}. The turning point is at $M_1 \sim \mu \sim 500$ GeV. The neutralino mass depends only weakly on $M_1$ for larger values of $M_1$. In comparison to the previous scenario, the remaining dependence is larger which is in agreement with the softer admixture transition (compare Figs.\ \ref{fig:MixingScenA} and \ref{fig:MixingScenB}). This decomposition and the relatively light squarks (cf.\ Tab.\ \ref{ScenarioProps}) are the essential phenomenological properties of this scenario.
\begin{figure*}
\includegraphics[width=0.49\textwidth]{SI-p2.pdf}
\includegraphics[width=0.49\textwidth]{SI-n2.pdf}
\includegraphics[width=0.49\textwidth]{SD-p2.pdf}
\includegraphics[width=0.49\textwidth]{SD-n2.pdf}
\caption{Spin-independent (top) and spin-dependent (bottom) neutralino-nucleon cross sections in scenario B for protons (left) and neutrons (right).}
\label{fig:CrossSectionsScenB}
\end{figure*}
The neutralino-nucleon cross sections for scenario B are shown in Fig.\ \ref{fig:CrossSectionsScenB}. The first thing to note is that there are three vertical orange bands now, corresponding to three regions which lead to a relic density compatible with Eq.\ (\ref{Planck}). Apart from scenario B itself ($M_1 = 270$ GeV), there is a second line on the other side of the peak of the dashed orange line and a third one at $M_1 \sim 475$ GeV. The peak is due to a Higgs resonance caused by $2m_{\tilde{\chi}^0_1} \sim m_{H^0}, m_{A^0} \sim 500$ GeV, which heavily increases the neutralino cross section into bottom quarks and in turn heavily reduces the resulting relic density. Bottom quarks are favored over top quarks, as $\tan\beta = 13.7$ is rather large here. The peak does not show up in the neutralino-nucleon cross sections. This is as expected, as the Higgs process has turned from a resonant $s$-channel to a non-resonant $t$-channel. The third vertical band lies precisely in the region where the neutralino admixture changes from bino to higgsino, stressing again the necessity to treat the general neutralino admixture.
The spin-independent nucleon cross sections are shown in the upper plots of Fig.\ \ref{fig:CrossSectionsScenB}. Once again, no major difference is found between the proton and neutron case. The relative shifts between our tree-level calculation (black solid line) and {\tt micrOMEGAs}\ (orange solid line) or our NLO calculation (blue solid line) are roughly as before. No significant change is found when activating the improved tree-level calculation of {\tt micrOMEGAs}\ (green dotted line). The agreement between our tree-level calculation using the nuclear input values of {\tt micrOMEGAs}\ (black dotted line) and the {\tt micrOMEGAs}\ result is slightly worse. The remaining discrepancy is mainly due to the use of effective couplings in {\tt micrOMEGAs}\ and a different treatment of the top quark mass and of the associated stop sector (cf.\ subsection \ref{Renormalization}). Moreover {\tt micrOMEGAs}\ does not kinematically distinguish between the $s$- and the $u$-channels shown in Fig.\ \ref{fig:DDTree}. Although these differences are present in general, the resulting discrepancy depends on the concrete scenario. In this scenario they lead to a small, but visible shift, whereas they do not in the other two scenarios.
New features show up in the spin-dependent case, i.e.\ in the lower plots of Fig.\ \ref{fig:CrossSectionsScenB}. Here, the proton and neutron cross sections differ by almost one order of magnitude in the small $M_1$ regime. Moreover the tree-level and NLO results clearly separate in the proton case for small $M_1$. This large splitting is absent in the neutron case. The reason is as follows: In the small $M_1$ regime, the neutralino is mostly bino (cf.\ Fig.\ \ref{fig:MixingScenA}). Moreover, the squarks of the first two generations are rather light in this scenario (cf.\ Tab.\ \ref{ScenarioProps}). The former leads to a suppression of the usually dominant $Z^0$ processes, while the latter kinematically favors the squark processes. As a result, the squark processes contribute sizeably in the small $M_1$ regime. In contrast to the $Z^0$ processes, these processes strongly depend on the involved quark flavor and are sensitive to different choices of $(\Delta_q)_N$ as given in Eqs.\ (\ref{Delta1}) and (\ref{Delta2}). In the case of the proton, this leads to a partial cancellation of the individually large squark contributions, which is much less pronounced in the neutron case. This explains the difference between the proton and neutron cross sections. The rather large impact of the NLO corrections on the proton cross section has a related origin. As the leading squark contributions cancel in the proton case, the cross section becomes more sensitive to the subleading virtual corrections. Due to the rather light squark masses in this scenario, these virtual corrections are not negligible. For large $M_1$, the $Z^0$ processes dominate and the virtual corrections are less important.
\begin{figure}
\includegraphics[width=0.49\textwidth]{SD-p_Zoom2.pdf}
\caption{Combined relic density and direct detection calculation in scnenario B.}
\label{fig:ZoomScenB}
\end{figure}
We take a closer look at the cosmologically preferred region around the Higgs resonance in the case of the spin-dependent neutralino-proton cross section in Fig.\ \ref{fig:ZoomScenB}. As before, we are showing the resulting relic density obtained with {\tt micrOMEGAs}\ (orange dashed line), our tree-level calculation (black dashed line) and our NLO calculation (blue dashed line). The vertical bands of the respective colors correspond to the $M_1$ regions leading to a relic density compatible with Eq.\ (\ref{Planck}). These bands are very thin here, as the relic density is changing rapidly near the resonance, which allows to effectively constrain the pMSSM parameter space. Subsequently we can read off the predicted cross section. The results are shown in Tab.\ \ref{PredictionsScenB}.
\begin{table}
\caption{Resulting $M_1$ and spin-dependent neutralino-proton cross section when combining direct detection and relic density routines in scenario B.}
\begin{tabular}{|c|ccc|}
\hline
$\quad$ & $M_1$ [GeV] & $\sigma^{\mathrm{SD}}_p$ [$10^{-43}$cm$^2$]& Shift of $\sigma^{\mathrm{SD}}_p$\\
\hline
{\tt micrOMEGAs}\ & 226 & $2.78$ & $+3\%$ \\
Tree level & 228 & $2.70$ & \\
Full NLO & 227 & $1.65$ & $-39\%$ \\
\hline
{\tt micrOMEGAs}\ & 270 & $4.14$ & $+8\%$\\
Tree level & 267 & $3.84$ & \\
Full NLO & 269 & $2.47$ & $-36\%$ \\
\hline
\end{tabular}
\label{PredictionsScenB}
\end{table}
As we are using the same nuclear input as {\tt micrOMEGAs}\ in the spin-dependent case, the shift between our tree-level prediction and {\tt micrOMEGAs}\ is smaller than in scenario A, where we investigated the spin-independent neutralino-proton cross section. Note that the relative position of the vertical bands, i.e.\ the relic density constraint, can influence this shift in both directions. The effect of reading off the cross section at different $M_1$ reduces the shift in the first case ($M_1= 226$ GeV and $M_1 = 228$ GeV) and increases the shift in the second case ($M_1 = 270$ GeV and $M_1 = 267$ GeV), as the order of the bands has changed. The exact opposite occurs when comparing our tree-level and our NLO results. Here both relative shifts are large, reaching almost $-40\%$.
\subsection*{Scenario C -- Higgsino-bino dark matter}
\label{ScenarioC}
\begin{figure}
\includegraphics[width=0.49\textwidth]{Mixing3.pdf}
\caption{Neutralino decomposition in scenario C.}
\label{fig:MixingScenC}
\end{figure}
In scenario C, we vary the higgsino mass parameter $\mu$, which changes the neutralino decomposition from higgs\-ino to bino as shown in Fig.\ \ref{fig:MixingScenC}. The turning point is at $\mu \sim M_1 \sim 675$ GeV. Concerning the neutralino admixture, scenario C can be understood as a mirrored version of scenario B (cf.\ Fig.\ \ref{fig:MixingScenB}).
\begin{figure*}
\includegraphics[width=0.49\textwidth]{SI-p3.pdf}
\includegraphics[width=0.49\textwidth]{SI-n3.pdf}
\includegraphics[width=0.49\textwidth]{SD-p3.pdf}
\includegraphics[width=0.49\textwidth]{SD-n3.pdf}
\caption{Spin-independent (top) and spin-dependent (bottom) neutralino-nucleon cross sections in scenario C for protons (left) and neutrons (right).}
\label{fig:CrossSectionsScenC}
\end{figure*}
The neutralino-nucleon cross sections in scenario C are shown in Fig.\ \ref{fig:CrossSectionsScenC}. As the slope of the relic density (orange dashed line) is smaller than in the previous scenarios, the region compatible with the Planck limits is larger, which leads to a thicker vertical orange band. No essential new features are found in the spin-independent cross sections shown in the upper plots of Fig.\ \ref{fig:CrossSectionsScenC}. In particular, the relative shift between our tree-level calculation (black solid line) and {\tt micrOMEGAs}\ (orange solid line) for a given $M_1$ is roughly as big as the shift between our tree-level and our NLO calculation (blue solid line) amounting to $\sim-16\%$ and $\sim+13\%$, respectively. No significant difference is found between the proton and the neutron case.
In contrast to that, the spin-dependent cross sections shown in the lower plots of Fig.\ \ref{fig:CrossSectionsScenC} obviously depend on the nucleon type. This difference is caused by a similar phenomenon as the one described in the previous subsection. In the large $\mu$ region, the neutralino becomes mostly bino (cf.\ Fig.\ \ref{fig:MixingScenC}), which suppresses the $Z^0$ processes. Although the squarks are not as light as in scenario B here (cf.\ Tab.\ \ref{ScenarioProps}), the squark processes are kinematically favored again. Remember that the squark processes occur in the $s$- and $u$-channel. The denominators of the tree-level processes read $s/u - m^2_{\tilde{q}_i}$ which simplifies to $(m_{\tilde{\chi}^0_1} \pm m_q)^2 - m^2_{\tilde{q}_i}$ in the limit of vanishing relative velocity. Hence it is not the total squark mass, but the neutralino-squark mass difference that matters. This difference decreases with increasing $\mu$. As a result, the squark processes contribute sizeably to the spin-dependent cross sections for large $\mu$. These processes depend on the involved flavor and in turn the chosen nuclear input values as given in Eqs.\ (\ref{Delta1}) and (\ref{Delta2}). In the case of the proton, we encounter a destructive interference of the individual terms, which leads to the drop observed at $\mu\sim 850$ GeV. Here the associated four-fermion coupling changes its sign and the resulting cross section vanishes. A similar situation would be encountered in the neutron case for larger values of $\mu$. However, as this region leads to a too large relic density, we are not investigating this in more detail.
\begin{figure}
\includegraphics[width=0.49\textwidth]{SD-p-Zoom3.pdf}
\caption{Combined relic density and direct detection calculation in scnenario C.}
\label{fig:ZoomScenC}
\end{figure}
Instead, we zoom into the region $700 < \mu < 900$ GeV and analyze the spin-dependent neutralino-proton cross section in Fig.\ \ref{fig:ZoomScenC}. As before, we are showing the resulting relic density obtained with {\tt micrOMEGAs}\ (orange dashed line), our tree-level (black dashed line) and our NLO routines (blue dashed line). These three calculations lead to different regions compatible with the Planck limits, as indicated by the vertical bands of the corresponding colors. The bands are broader than before, as the relic density is increasing less rapidly when changing $\mu$. Note that the blue and orange bands overlap to a large extent, which signals that the effective couplings used in the relic routines of {\tt micrOMEGAs}\ are able to approximate the dominant NLO contributions quite well in this scenario. This may happen, but is not necessarily the case, as studied e.g. in Ref. \cite{ChiChi2qq3}. On the other hand, {\tt micrOMEGAs}\ does not include radiative corrections to the spin-dependent cross section. Hence the orange solid line is closely following the black solid line. As mentioned before, the remaining difference is due to the running of the operator and associated Wilson coefficient.
It is again interesting to combine the relic density and direct detection calculations. Our tree-level and NLO routines lead to different preferred regions along the $\mu$ axis. Simultaneosly the shift between the cross section obtained at tree level and at NLO is very large for a given $\mu$ (more than $-50\%$ near the drop). However, when combining both calculations these effects cancel each other. This is not the case for the comparison of {\tt micrOMEGAs}\ with our tree-level result, where {\tt micrOMEGAs}\ predicts a larger cross section.
The aforementioned regions of $\mu$ and the corresponding cross sections are listed in Tab.\ \ref{PredictionsScenC}. The broader vertical bands result in a range of allowed $\mu$ values and an associated range of cross sections. Note that these ranges exist in principle in every scenario. However, as they are very small in the previous scenarios we omitted them for simplicity. The shifts given in Tab.\ \ref{PredictionsScenC} are exemplary and have been obtained by combining the mean values of the cross sections.
\begin{table}
\caption{Resulting $\mu$ and spin-dependent neutralino-proton cross section when combining direct detection and relic density routines in scenario C.}
\begin{tabular}{|c|ccc|}
\hline
$\quad$ & $\mu$ [GeV] & $\sigma^{\mathrm{SD}}_p$ [10$^{-43}$cm$^2$]& Shift of $\sigma^{\mathrm{SD}}_p$\\
\hline
{\tt micrOMEGAs}\ & 815 - 821 & 1.80 - 2.43 & $+63\%$ \\
Tree level & 823 - 829 & 1.06 - 1.53 & \\
Full NLO & 813 - 819 & 1.08 - 1.62 & $+4\%$ \\
\hline
\end{tabular}
\label{PredictionsScenC}
\end{table}
\begin{figure}
\includegraphics[width=0.49\textwidth]{SD-nDRBar.pdf}
\caption{Spin-dependent neutralino-neutron cross section in scenario C using a pure {$\overline{\mathrm{DR}}$}\ scheme}
\label{fig:DRbar}
\end{figure}
Before concluding, we take a small detour and briefly comment on the renormalization scheme dependence. As we have described in subsection \ref{Renormalization}, we are working with a hybrid on-shell/{$\overline{\mathrm{DR}}$}\ scheme. In particular the squark masses of the first two generations are treated on-shell just like in {\tt micrOMEGAs}. Our code optionally also supports a pure {$\overline{\mathrm{DR}}$}\ scheme. When studying the differences between the two schemes, the spin-dependent neutralino-neutron cross section shown in the lower right plot of Fig.\ \ref{fig:CrossSectionsScenC} has proven very useful. This plot is shown again in Fig.\ \ref{fig:DRbar}, this time using a pure {$\overline{\mathrm{DR}}$}\ scheme. No visible differences between the two plots occur for small $\mu$. The virtual corrections to the spin-dependent cross section are negligible in this regime which is not affected by the choice of the scheme. For larger values of $\mu$, our tree-level result (black solid line) -- now using {$\overline{\mathrm{DR}}$}\ squark masses -- clearly separates from {\tt micrOMEGAs}\ (orange solid line) -- still using on-shell squark masses -- which has previously not been the case (cf.\ Fig.\ \ref{fig:CrossSectionsScenC}).
Remember that the cross section in this region is heavily influenced by the squark processes as explained in the beginning of this subsection. These processes benefit from the decreasing neutralino-squark mass difference appearing in the denominators of the corresponding propagators. This mass difference is sensitive to the choice of the scheme. To investigate this in greater detail, we write the scale-independent on-shell squark mass $m_{\tilde{q}}^{\mathrm{OS}}$ as a sum of two individually scale-dependent terms, the scale-dependent {$\overline{\mathrm{DR}}$}\ mass $m_{\tilde{q}}^{\mathrm{\overline{DR}}}(\mu_R)$ and an additional finite term resumming virtual corrections $\Delta m_{\tilde{q}}(\mu_R)$,
\begin{equation}
m_{\tilde{q}}^{\mathrm{OS}} = m_{\tilde{q}}^{\mathrm{\overline{DR}}}(\mu_R) + \Delta m_{\tilde{q}}(\mu_R).
\end{equation}
If we replace the on-shell squark masses by their smaller {$\overline{\mathrm{DR}}$}\ masses, i.e.\ if we discard the finite $\Delta m_{\tilde{q}}(\mu_R)$ terms, the neutralino-squark mass difference decreases even further. This leads to the observed steep drop of our tree-level result. However, at NLO this effect diminishes again and the blue lines shown in the lower right plot of Fig.\ \ref{fig:CrossSectionsScenC} and in Fig.\ \ref{fig:DRbar} roughly agree. The reason behind that is that the leading corrections incorporated in $\Delta m_{\tilde{q}}(\mu_R)$ reappear again, this time as virtual corrections to the squark propagators. In other words, the tree-level result heavily depends on the definition of the squark mass in this special situation, but the NLO result is much more stable. The main difference between the two schemes is that the virtual corrections are partially included at tree level in the on-shell mass in the first case, whereas they show up as large propagator corrections in the second case. We prefer the first scheme, which leads to smaller virtual corrections and an improved perturbative stability. Let us finally mention that the resulting differences between the two schemes in other cases, i.e.\ for other cross sections, are less pronounced. A similar study in the context of the relic density can be found in Ref.\ \cite{Scalepaper}.
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 8,482
|
Morawa kan syfta på följande platser:
Australien
Morawa (ort), Western Australia,
Morawa (region), Western Australia,
Polen
Morawa (vattendrag), Nedre Karpaternas vojvodskap,
Robotskapade Australienförgreningar
Robotskapade Polenförgreningar
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 3,777
|
<?xml version="1.0" encoding="utf-8"?>
<RelativeLayout xmlns:android="http://schemas.android.com/apk/res/android"
android:layout_width="match_parent"
android:layout_height="match_parent" >
<LinearLayout
android:id="@+id/layout_bottom_control"
android:layout_width="match_parent"
android:layout_height="wrap_content"
android:orientation="horizontal"
android:layout_alignParentBottom="true">
<Button
android:id="@+id/button_detail"
android:layout_width="0dp"
android:layout_height="wrap_content"
android:layout_weight="1"
android:gravity="center"
android:paddingLeft="15dp"
android:paddingRight="15dp"
android:paddingTop="5dp"
android:paddingBottom="5dp"
android:text="@string/button_detail"
android:textSize="18sp"
android:drawableLeft="@android:drawable/ic_menu_edit"
style="?android:attr/buttonBarButtonStyle"
/>
<Button
android:id="@+id/button_comment"
android:layout_width="0dp"
android:layout_height="wrap_content"
android:layout_weight="1"
android:gravity="center"
android:paddingLeft="15dp"
android:paddingRight="15dp"
android:paddingTop="5dp"
android:paddingBottom="5dp"
android:text="@string/button_comment"
android:textSize="18sp"
android:drawableLeft="@android:drawable/sym_action_chat"
style="?android:attr/buttonBarButtonStyle"
/>
<Button
android:id="@+id/button_share"
android:layout_width="0dp"
android:layout_height="wrap_content"
android:layout_weight="1"
android:gravity="center"
android:paddingLeft="15dp"
android:paddingRight="15dp"
android:paddingTop="5dp"
android:paddingBottom="5dp"
android:text="@string/button_share"
android:textSize="18sp"
android:drawableLeft="@android:drawable/ic_menu_share"
style="?android:attr/buttonBarButtonStyle"
/>
</LinearLayout>
<ListView
android:id="@+id/list_weibo_content"
android:layout_width="match_parent"
android:layout_height="match_parent"
android:descendantFocusability="afterDescendants"
android:layout_above="@id/layout_bottom_control"
></ListView>
</RelativeLayout>
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 50
|
function dragon() {
}
module dragon {
}
export = dragon;
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 9,902
|
{"url":"http:\/\/mathematica.stackexchange.com\/questions\/33899\/total-weight-of-a-graph","text":"# Total weight of a graph\n\nI need to find the total weight of an WeightedAdjacencyGraph. I have a graph in that form\n\na = WeightedAdjacencyGraph[{...},{...},{...}....etc]\n\n\nHow can I find its total weight?\n\n-\n\nYou could just do this: if a graph is undirected:\n\nTotal[UpperTriangularize[WeightedAdjacencyMatrix[g]], 2]\n\n\nif a graph is directed:\n\nTotal[WeightedAdjacencyMatrix[g], 2]\n\n\nor by PropertyValue\n\nTotal[PropertyValue[g, EdgeWeight]]\n\n-\nDon't forget the self loops! \u2013\u00a0 Szabolcs Oct 12 '13 at 19:23\n@Szabolcs thanks, I missed that. \u2013\u00a0 halmir Oct 13 '13 at 4:39\ntotalweight=Total@(PropertyValue[{g, #}, EdgeWeight] & \/@ EdgeList[g])","date":"2014-07-24 17:44:26","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 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\": 0.5699431896209717, \"perplexity\": 6893.948227171211}, \"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-2014-23\/segments\/1405997890181.84\/warc\/CC-MAIN-20140722025810-00151-ip-10-33-131-23.ec2.internal.warc.gz\"}"}
| null | null |
Lars Linus Persson, född 29 december 1985 i Hagfors, Värmland, är en svensk före detta professionell ishockeyspelare som spelade sammanlagt 342 SHL-matcher för Frölunda HC, Luleå HF, Färjestads BK och Leksands IF. Under slutspelet säsongen 2012-2013 vann Persson skytteligan och delade den totala poängligasegern.
Perssons moderklubb är IK Viking där han började spela ishockey i sin hemort Hagfors. Han vann TV-pucken med Värmland år 2000 och stod för 11 poäng (varav 6 mål) på 8 spelade matcher. Han flyttade senare till Göteborg för gymnasiestudier och spel i Frölunda HC juniorlag. Han fick sitt genombrott i BIK Karlskoga i Hockeyallsvenskan, för vilka han säsongen 2011/2012 noterades för 51 poäng på lika många matcher. Påföljande säsong värvades han till Luleå HF i SHL där han spelade två säsonger. Inför säsongen 2014/2015 skrev han på ett tvåårskontrakt med Färjestad BK. Efter fyra säsonger i klubben fick han inte ett förnyat kontrakt, varför han istället värvades till Leksands IF inför säsongen 2018/2019. Samma år var han med att spela upp Leksand till SHL.
Klubbar
Frölunda HC 2001–2004
Hammarby IF 2004–2006
Leksands IF 2006–2007
Rögle BK 2007–2008
BIK Karlskoga 2008–2012
Luleå HF 2012-2014
Färjestad BK 2014-2018
Leksands IF 2018-2020
Statistik
Klubbkarriär
Referenser
Externa länkar
Födda 1985
Spelare i Frölunda HC
Spelare i Hammarby Hockey
Spelare i AIK Ishockey
Spelare i Leksands IF
Spelare i IFK Arboga
Spelare i Rögle BK
Spelare i BIK Karlskoga
Spelare i Luleå HF
Spelare i Färjestads BK
Svenska ishockeyspelare
Män
Levande personer
Personer från Hagfors
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 5,371
|
Q: Classification report de Sklearn, mi precisión es siempre 0 Estoy teniendo problemas con los resultados de classification_report() de sklearn en una clasificación binaria en un data set desbalanceado, este es mi código:
*
*dividir la data en 60% train, 20% validation, 20% teset
from sklearn.model_selection import train_test_split
X_train, X_test, y_train, y_test = train_test_split(x1, y, test_size=0.2, random_state=1)
X_train, X_val, y_train, y_val = train_test_split(X_train, y_train, test_size=0.25, random_state=1)
*Sobremuestrear o submuestrear la clase minoritaria (en este caso submuestrear):
from imblearn.under_sampling import RandomUnderSampler
rus = RandomUnderSampler(random_state=1)
X_resampled, y_resampled = rus.fit_resample(X_train, y_train)
*Entrenar el modelo y predecir con el test (después de haberlo probado con validación cruzada)
model=LogisticRegression()
model.fit(X_sm, y_sm)
y_pred=model.predict(X_test)
*Métricas de sklearn (matriz de confusión y classification report)
matrix = confusion_matrix(y_test, y_pred)
print(matrix)
sns.heatmap(matrix, annot=True, cbar=False, linecolor='black',linewidths=0.7, cmap=plt.cm.Blues,fmt='g' )
plt.xlabel('Predicted')
plt.ylabel('Expected')
plt.show()
report = classification_report(y_test, preds)
print(report)
Dando como resultado lo que se observa en la siguiente imagen:
Entonces mi duda es porque teniendo esa matriz de confusión la precisión de la clase positiva es 0, ¿Estoy haciendo algo mal?
Gracias de antemano por la ayuda
A: ¡Tu programación y la representación que haces está bien! Lo único que te falta es saber exactamente como se calculan, y para que sirven las métricas de clasificación que estás usando, además de entender la salida de classification_report().
Voy a dar una respuesta amplia de todo lo que está sucediendo, y explicando la función de precisión por si le sirve a más usuarios de la comunidad.
Output de classification_report()
classification_report() te da un output con dos decimales como máximo, si quieres ampliarlo a más puedes hacer esto.
classification_report(y_test, preds, digits=4) #ampliamos a 4 decimales
Cálculo de "precisión"
Esta formula sirve para saber como de bien, está clasificando tu modelo los resultados positivos. Simplemente selecciona los que han sido clasificados como positivos de forma correcta y los divide entre los positivos totales 197 / (197 + 56.211).
*
*Tus True negative (verdaderos negativos) son 120.979. Es decir el modelo ha clasificado 120.979 negativos, y efectivamente ha acertado porque eran negativos.
*Tus false negatives (negativos falsos) son 56.211. Es decir el modelo ha clasificado 56.211 negativos, y ha fallado porque en realidad son positivos.
*Tus False Negative (falsos negativos) son 70. Es decir el modelo ha clasificado 70 positivos, y ha fallado porque en realidad son negativos
*Tus True positive (verdaderos positivos) son 197. Es decir el modelo ha clasificado 197 positivos, y efectivamente ha acertado porque eran positivos.
Teniendo toda la información anterior en cuenta, aplicamos la fomula de la precisión
True positive (197) / True positive (197) + False Positive (56.211).
Esto nos arroja un resultado de 0.0035. Como classification report por defecto solo nos devuelve dos dígitos, tu resultado es 0.00
Espero haberte ayudado :)
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 8,480
|
{"url":"http:\/\/tztermpaperjjds.blinkingti.me\/thesiscls-latex.html","text":"# Thesis.cls latex\n\nAs i already mentioned in my previous question, i intend to give a latex introduction to phd students most of them will already be using latex somehow but their. An online latex editor that's easy to use no installation, real-time collaboration, version control, hundreds of latex templates, and more. La t e x thesis class for university of colorado the le thesiscls contains the de nitions needed to \\input macrostex % my file of latex macros \\input. Add \u201cappendix\u201d before \u201ca\u201d in thesis toc but the latex thesis cls file i use generates only the letter a followed by the appendix title: a (title for. Preparing a thesis with latex you rst need to download the le thesiscls % run latex or pdflatex on this file to produce your thesis. Thesiscls latex since 1989 our certified professional essay writers have assisted tens of thousands of clients to land great jobs and advance their careers through.\n\nWhen i am trying to use the thesis class file for my thesis i am getting this error: file `thesiscls' not found \\usepackage my codes \\documentclass[12pt]{thesis. This latex template is used by many universities as the basis for thesis and dissertation submissions, and is a great way to get started if you haven't been. Direc\u00adtory macros\/latex\/contrib\/gatech make sure that the gatech-thesiscls file is on your texinputs search path and use the following command at the. Typical problems that arise while writing a thesis with latex and suggests improved solutions by handling easy packages many suggestions can be.\n\nHarvard gsas phd thesis latex template also maybe relevant is a harvard thesiscls page, which i did not find useful at all (it seems to have vanished anyway. Purchase a thesis paper phd thesis class latex essayforumo phd writer what causes a friendship to break apart the file thesiscls. Location: ctan tex-archive macros latex contrib thesis direc\u00adtory macros\/latex\/contrib\/thesis down\u00adload the con\u00adtents of this pack\u00adage in one zip archive.\n\n\u2022 The university of maryland electronic thesis and dissertation (etd) style guide latex template thesiscls should not be the thesis and dissertation templates.\n\u2022 A latex document class that conforms to the computer laboratory's phd thesis formatting guidelines.\n\u2022 I am trying to work on my undergraduate thesis as a latex document so i downloaded a thesiscls file installing cls file in miktex miktex help please.\n\nWhat's really happening with thesis cls file latex, college wellness articles, anticipation is often greater than realization essay definition, why a mba essay. Thesis cls latex download ranked #1 by 10,000 plus clients for 25 years our certified resume writers have been developing compelling resumes, cover letters.\n\nThesis.cls latex\nRated 5\/5 based on 21 review","date":"2018-08-18 00:19:38","metadata":"{\"extraction_info\": {\"found_math\": false, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.8426349759101868, \"perplexity\": 6526.42960147009}, \"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-2018-34\/segments\/1534221213247.0\/warc\/CC-MAIN-20180818001437-20180818021437-00608.warc.gz\"}"}
| null | null |
Католицькі королі () — збірний титул королеви Кастилії Ізабелли I (1451—1504) та короля Арагону Фернандо II (1452—1516), який був також Фернандо V Кастильським.
Їхнє весілля, що відбулося 1469 року, стало підґрунтям для об'єднання королівств Арагону та Кастилії і заклало фундамент для утворення сучасної Іспанії. Титул Католицьких королів надав подружжю у 1496 іспанський уродженець папа Олександр VI як засіб відзначити успішне закінчення Реконкісти та заморські завоювання Іспанії на славу Христа. Крім цього, подружжя відоме за вигнання євреїв з Іспанії здійснене у 1492 році підписанням Альгамбрського едикту. Саме під час їхнього правління в Іспанії було організовано іспанську інквізицію.
Література
О. А. Борділовська. Іспанська унія 1479 // Українська дипломатична енциклопедія: У 2-х т. /Редкол.:Л. В. Губерський (голова) та ін. — К: Знання України, 2004 — Т.1 — 760с. ISBN 966-316-039-X
Середньовічна Іспанія
Королі Іспанії
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 2,516
|
14K Solid Yellow Gold Heart Necklace w/ Dangling Natural Pearl
Save $607.51 Save $607.51 Save $607.51 Save $607.51 Save $607.51 Save $607.51 Save $607.51 Save $607.51 Save $607.51 Save $607.51 Save $607.51 Save $607.51 Save $607.51 Save $607.51 Save $607.51 Save $607.51 Save $607.51 Save $607.51 Save $607.51 Save $607.51 Save $607.51 Save $607.51 Save $607.51 Save $607.51 Save $607.51 Save $607.51 Save $607.51 Save $607.51 Save $607.51 Save $607.51 Save $607.51 Save $607.51 Save $607.51 Save $607.51 Save $607.51 Save $607.51 Save $607.51 Save $607.51 Save $607.51 Save $607.51 Save $607.51 Save $607.51 Save $607.51 Save $607.51 Save $607.51 Save $607.51 Save $607.51 Save $607.51 Save $607.51 Save $607.51 Save $607.51 Save $607.51 Save $607.51 Save $607.51 Save $607.51 Save $607.51 Save $607.51 Save $607.51 Save $607.51 Save $607.51 Save $607.51 Save $607.51 Save $607.51 Save $607.51 Save $607.51 Save $607.51 Save $607.51 Save $607.51 Save $607.51 Save $607.51 Save $607.51 Save $607.51
Rope Chain Option 0.68 mm thickness 18 inch 0.68 mm thickness 18 inch0.68 mm thickness 20 inch0.68 mm thickness 22 inch0.68 mm thickness 24 inch1.15 mm thickness 18 inch1.15 mm thickness 20 inch1.15 mm thickness 22 inch1.15 mm thickness 24 inch0.68 mm thickness 14 inch0.68 mm thickness 16 inch1.15 mm thickness 14 inch1.15 mm thickness 16 inch
0.68 mm thickness 18 inch / White Gold / Don't send - $326.81 USD 0.68 mm thickness 18 inch / White Gold / Send appraisal - $330.14 USD 0.68 mm thickness 20 inch / White Gold / Don't send - $330.22 USD 0.68 mm thickness 20 inch / White Gold / Send appraisal - $333.55 USD 0.68 mm thickness 22 inch / White Gold / Don't send - $333.62 USD 0.68 mm thickness 22 inch / White Gold / Send appraisal - $336.95 USD 0.68 mm thickness 24 inch / White Gold / Don't send - $337.03 USD 0.68 mm thickness 24 inch / White Gold / Send appraisal - $340.36 USD 1.15 mm thickness 18 inch / White Gold / Don't send - $331.40 USD 1.15 mm thickness 18 inch / White Gold / Send appraisal - $334.73 USD 1.15 mm thickness 20 inch / White Gold / Don't send - $334.81 USD 1.15 mm thickness 20 inch / White Gold / Send appraisal - $338.14 USD 1.15 mm thickness 22 inch / White Gold / Don't send - $342.81 USD 1.15 mm thickness 22 inch / White Gold / Send appraisal - $346.14 USD 1.15 mm thickness 24 inch / White Gold / Don't send - $350.81 USD 1.15 mm thickness 24 inch / White Gold / Send appraisal - $354.14 USD 0.68 mm thickness 14 inch / White Gold / Don't send - $326.81 USD 0.68 mm thickness 14 inch / White Gold / Send appraisal - $330.14 USD 0.68 mm thickness 16 inch / White Gold / Don't send - $326.81 USD 0.68 mm thickness 16 inch / White Gold / Send appraisal - $330.14 USD 1.15 mm thickness 14 inch / White Gold / Don't send - $326.81 USD 1.15 mm thickness 14 inch / White Gold / Send appraisal - $330.14 USD 1.15 mm thickness 16 inch / White Gold / Don't send - $326.81 USD 1.15 mm thickness 16 inch / White Gold / Send appraisal - $330.14 USD 0.68 mm thickness 18 inch / Yellow Gold / Don't send - $311.81 USD 0.68 mm thickness 18 inch / Yellow Gold / Send appraisal - $315.14 USD 0.68 mm thickness 20 inch / Yellow Gold / Don't send - $315.22 USD 0.68 mm thickness 20 inch / Yellow Gold / Send appraisal - $318.55 USD 0.68 mm thickness 22 inch / Yellow Gold / Don't send - $318.62 USD 0.68 mm thickness 22 inch / Yellow Gold / Send appraisal - $321.95 USD 0.68 mm thickness 24 inch / Yellow Gold / Don't send - $322.03 USD 0.68 mm thickness 24 inch / Yellow Gold / Send appraisal - $325.36 USD 1.15 mm thickness 18 inch / Yellow Gold / Don't send - $316.40 USD 1.15 mm thickness 18 inch / Yellow Gold / Send appraisal - $319.73 USD 1.15 mm thickness 20 inch / Yellow Gold / Don't send - $319.81 USD 1.15 mm thickness 20 inch / Yellow Gold / Send appraisal - $323.14 USD 1.15 mm thickness 22 inch / Yellow Gold / Don't send - $327.81 USD 1.15 mm thickness 22 inch / Yellow Gold / Send appraisal - $331.14 USD 1.15 mm thickness 24 inch / Yellow Gold / Don't send - $335.81 USD 1.15 mm thickness 24 inch / Yellow Gold / Send appraisal - $339.14 USD 0.68 mm thickness 14 inch / Yellow Gold / Don't send - $311.81 USD 0.68 mm thickness 14 inch / Yellow Gold / Send appraisal - $315.14 USD 0.68 mm thickness 16 inch / Yellow Gold / Don't send - $311.81 USD 0.68 mm thickness 16 inch / Yellow Gold / Send appraisal - $315.14 USD 1.15 mm thickness 14 inch / Yellow Gold / Don't send - $311.81 USD 1.15 mm thickness 14 inch / Yellow Gold / Send appraisal - $315.14 USD 1.15 mm thickness 16 inch / Yellow Gold / Don't send - $311.81 USD 1.15 mm thickness 16 inch / Yellow Gold / Send appraisal - $315.14 USD 0.68 mm thickness 18 inch / Rose Gold / Don't send - $316.81 USD 0.68 mm thickness 18 inch / Rose Gold / Send appraisal - $320.14 USD 0.68 mm thickness 20 inch / Rose Gold / Don't send - $320.22 USD 0.68 mm thickness 20 inch / Rose Gold / Send appraisal - $323.55 USD 0.68 mm thickness 22 inch / Rose Gold / Don't send - $323.62 USD 0.68 mm thickness 22 inch / Rose Gold / Send appraisal - $326.95 USD 0.68 mm thickness 24 inch / Rose Gold / Don't send - $327.03 USD 0.68 mm thickness 24 inch / Rose Gold / Send appraisal - $330.36 USD 1.15 mm thickness 18 inch / Rose Gold / Don't send - $321.40 USD 1.15 mm thickness 18 inch / Rose Gold / Send appraisal - $324.73 USD 1.15 mm thickness 20 inch / Rose Gold / Don't send - $324.81 USD 1.15 mm thickness 20 inch / Rose Gold / Send appraisal - $328.14 USD 1.15 mm thickness 22 inch / Rose Gold / Don't send - $332.81 USD 1.15 mm thickness 22 inch / Rose Gold / Send appraisal - $336.14 USD 1.15 mm thickness 24 inch / Rose Gold / Don't send - $340.81 USD 1.15 mm thickness 24 inch / Rose Gold / Send appraisal - $344.14 USD 0.68 mm thickness 14 inch / Rose Gold / Don't send - $316.81 USD 0.68 mm thickness 14 inch / Rose Gold / Send appraisal - $320.14 USD 0.68 mm thickness 16 inch / Rose Gold / Don't send - $316.81 USD 0.68 mm thickness 16 inch / Rose Gold / Send appraisal - $320.14 USD 1.15 mm thickness 14 inch / Rose Gold / Don't send - $316.81 USD 1.15 mm thickness 14 inch / Rose Gold / Send appraisal - $320.14 USD 1.15 mm thickness 16 inch / Rose Gold / Don't send - $316.81 USD 1.15 mm thickness 16 inch / Rose Gold / Send appraisal - $320.14 USD
This gorgeous, affordable pearl necklace is perfect for you or a loved one. Forged by hand with passion and precision, this piece is a pure example of how beautiful it is when gemstones and gold come together to form exquisite jewelry that will dazzle the eye and last for generations to come. Backed by our customer satisfaction guarantee. Available in 14K yellow, white or rose gold.
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 4,661
|
\section{Introduction}
Maxwell's theory of electromagnetism is arguably one of the best established classical field theories of Nature.
Classically the coupling of electromagnetic fields to gravity are described by the Einstein-Maxwell field equations.
This implies in the weak field approximation a quantum field theory of massless spin-1 photons coupled to massless spin-2 gravitons.
At the tree level the photon propagation takes place along the null geodesics of the Minkowski spacetime.
In perturbative QED in curved spacetimes though many unexpected new phenomena may occur.
At the 1-loop level there exists for instance vacuum polarization effects for which a photon produces a virtual electron-positron pair
that survives for a very short time interval before it pair annihilates back into a photon.
Such quantum vacuum processes confer a length scale to the photon of the order of the Compton wavelength of an electron.
Thus a photon in motion in a curved spacetime will be affected by the gravitational fields, even at the 1-loop order. Perturbative effects such as these
in curved spacetimes can be conveniently described by an effective field theory at the classical level through certain non-minimal couplings of the electromagnetic fields to gravity\cite{B1,B2,B3,B4}.
In general, in an effective field theory defined at a mass scale $\mu$ , only those fields with masses lighter than or of the of order $\mu$ are taken into account.
But the derivation of such an effective theory involves more than integrating out heavy particles of masses $M >\mu$ \cite{georgi,neubert}.
Suppose one considers a renormalizable field theory at scales $\mu >> M$ with fields included at all the mass scales.
Then the theory is scaled down to $\mu \approx M$ by the renormalization group flow.
On the other hand, an effective field theory at scales $\mu <M$ with only those fields with masses less than or equal to $\mu$ is introduced
which may include higher derivative couplings and possibly many other types of non-renormalizable interactions.
Then the short-distance physics that is incorporated into the coefficients of the effective Lagrangian
is disentangled from the long-distance physics that remains explicit in the actual low energy theory. Such a disentanglement can be achieved order by order
in perturbation theory by matching the physical parameters of the high energy theory at $\mu >>M$ to the corresponding parameters of the low energy theory at
mass scales $\mu \approx M$.
In QED, it is found that when the electron field is integrated out, the resulting effective Lagrangian
at the 1-loop level involves certain $F^4$-type self couplings of the electromagnetic fields. The corresponding long-distance theory complements Maxwell electrodynamics by the higher derivative Heisenberg-Euler terms \cite{B5}:
$$
{\mathcal{L}}= \frac{1}{2} F \wedge *F + \frac{2\alpha^2}{45 m^4} \left (X^2 + \frac{7}{4}Y^2 \right ) *1.
$$
In a similar way, the effective action density 4-form at the 1-loop level in a curved spacetime for QED had been calculated \cite{B7}:
\begin{eqnarray}
{\mathcal{L}}_{eff} &=& \frac{12 \alpha}{45\pi m^2} (d*F) \wedge *(d*F) + \frac{\alpha}{45\pi m^2} R_{ab} F^{ab} \wedge *F \nonumber \\ & &-\frac{13\alpha}{45\pi m^2} Ric_{a} \wedge \iota^aF \wedge *F +\frac{5\alpha}{90\pi m^2} R F \wedge *F. \nonumber
\end{eqnarray}
where $\alpha$ is the fine structure constant and $m$ is the free electron mass. The 2-photon 2-graviton vertices are settled in perturbative QED by the conventional $RF^2$ interactions in the effective action. These interaction vertices were found to mean photon graviton oscillations at low energies relative to the Planck scale and in weak gravitational fields \cite{B8}. This process may be significant in the presence of strong magnetic fields near heavy magnetars and pulsars \cite{B9,D1,D2}.
In fact a curved spacetime acts as an optically active medium for the photon with an index of refraction determined by the higher-loop QED corrections \cite{shore3,dereli,hollowood}.
Then dispersion effects such as gravitational rainbows or gravitational bi-refringence may occur.
The distinctive feature of a non-Abelian gauge field theory is the self - interactions among its gauge bosons. In QCD, the exchange potentials between color charges give rise to asymptotically free forces. Then in the UV limit constituent quarks behave as if they are free particles inside the hadrons. Perturbative quantization is viable in this case and the high-energy QCD tests agree with observations up to logarithmic corrections. On the other hand the perturbation theory fails altogether in the IR limit where the interaction potential becomes very complicated. The corresponding low-energy limit of QCD is not easy to define. A possible approach to deal with such long distance effects of QCD is to work with an effective field theory that would be valid at lower energy scales. A perturbative 1-loop effective field theory for QCD includes the generic types of $F^3$ and $(\nabla_{A}F)^2$ Yang-Mills self-interaction terms that show up as exotic strong-interaction contributions in colour condensate models \cite{baskal-dereli}. Then it should be expected in curved spacetimes, the QCD vacuum polarization effects shall also contribute, as it is in the case of perturbative QED, non-minimal coupling terms of the generic type $RF^2$ at the 1-loop level to an effective Einstein-Yang-Mills theory \cite{dereli-ucoluk}.
\medskip
We consider in Section:2 a simple non-minimally coupled Einstein-$SU(2)$ Yang-Mills effective field theory model.
Variational field equations will be derived in the coordinate-independent language of exterior differential forms.
We refer to our previous papers for the notation and conventions \cite{B10b,B7-e,B7-f}.
We provide in Section:3 a brief discussion of the unique conformally flat, static, spherically symmetric Bertotti-Robinson solution of the source-free Einstein-Maxwell equations.
The celebrated Wu-Yang magnetic pole potentials are introduced in Section:4.
In particular we give a non-generic class of regular static, spherically symmetric solutions of the coupled Einstein-Yang-Mills field equations.
\section{Non-minimally Coupled Einstein-Yang-Mills \\Fields}
\noindent We first consider the Yang-Mills potential 1-form\footnote{We set the gauge coupling constant $e=1$}
\begin{equation}
A=A^{j}T_{j} = A^{j}_{\;\;\mu} dx^{\mu} \otimes T_{j}
\end{equation}
where $\{T_j : j=1,2,3\}$ are the anti-hermitian generators of the Lie algebra $su(2)$ that satisfy the structure equations
\begin{equation}
\left [ T_j , T_k \right ] = \epsilon_{jkl} T_{l}.
\end{equation}
One may also introduce a local coordinate chart $\{x^\mu : \mu=0,1,2,3 \}$, even though we will be using extensively the coordinate independent
formalism of exterior differential forms over the spacetime.
Then the Yang-Mills field 2-form is determined by
\begin{equation}
F=dA + A\wedge A = F^{j} T_j = \frac{1}{2} F^{j}_{\;\;\mu \nu} dx^\mu \wedge dx^\nu \otimes T_{j}.
\end{equation}
We work out
$$
F^{j} = dA^j + \epsilon_{jkl} A^k \wedge A^l \;\; , \;\;
F^{j}_{\;\;\mu \nu} = \partial_{\mu} A^{j}_{\;\;\nu} - \partial_{\nu} A^{j}_{\;\;\mu} + \epsilon_{jkl} A^{k}_{\;\;\mu}A^{l}_{\;\;\nu}.
$$
Under a local gauge transformation $U = e^{\theta^{j}T_{j}} \in SU(2)$, the Yang-Mills fields transform as
$$
A \rightarrow U A U^{-1} + U dU^{=1} \quad , \quad F \rightarrow U F U^{-1}
$$
The integrability of $F$ implies the Bianchi identity
\begin{equation}
\nabla_{A}F \equiv dF + A \wedge F - F \wedge A =0.
\end{equation}
We label the quadratic Yang-Mills invariants by
\begin{equation}
X = *\hspace{0.5mm}Tr(F\wedge *F), \quad Y = *\hspace{0.5mm}Tr(F\wedge F).
\end{equation}
\noindent The Lagrangian density 4-form of Einstein-Yang-Mills theory with a cosmological constant is given by
\begin{equation}
{\mathcal{L}}_0 = \left ( \frac{1}{2\kappa^2} R + \frac{1}{2} X + \Lambda \right ) *1
\end{equation}
where
\begin{equation}
R =-*(R_{ab} \wedge *e^{ab}).
\end{equation}
denotes the scalar curvature of space-time. $\kappa^2=4\pi G$ with $G$ being the Newton's universal constant and $\Lambda$ is a cosmological constant.
\noindent We consider, for the sake of simplicity, only the following non-minimal coupling terms:
\begin{equation}
{\mathcal{L}}_1 = \frac{\gamma}{2} X R *1 + \frac{\gamma^\prime}{2} Y R *1.
\end{equation}
$\gamma$ and $\gamma^{\prime}$ are two arbitrary coupling constants.
Furthermore we assume that the connection is the unique, torsion-free Levi-Civita connection.
This constraint will be imposed by the method of Lagrange multipliers
\begin{equation}
{\mathcal{L}}_C = \left (de^a + \omega^{a}_{\;\;b} \wedge e^b \right ) \wedge \lambda_a.
\end{equation}
Therefore the total action that is to be varied becomes
\begin{equation}
I[ e^a, \omega^{a}_{\;\;b}, A, \lambda_a] = \int_{M} \left ( {\mathcal{L}}_0 + {\mathcal{L}}_1 + {\mathcal{L}}_C \right )
\end{equation}
After a long computation, the final form of the gravitational field equations turn out to be
\begin{eqnarray}
( \frac{1}{\kappa^2} + \gamma X + \gamma^{\prime} Y ) G_a &-& (\Lambda -\frac{\gamma}{2} X R -\frac{\gamma^\prime}{2} Y R ) *e_a \\
&=& \left ( 1+\gamma R \right ) \tau_{a}[F] + \gamma D( \iota_a *dX) + \gamma^{\prime} D( \iota_a *dY), \nonumber \label{einstein}
\end{eqnarray}
where the Einstein 3-forms
\begin{equation}
G_a = -\frac{1}{2} R^{bc} \wedge *e_{abc} = G_{ab} *e^b \nonumber
\end{equation}
and the Yang-Mills stress-energy-momentum 3-forms
\begin{equation}
\tau_{a}[F] = \frac{1}{2}Tr \left ( \iota_a F \wedge *F - F \wedge \iota_a *F \right ) = T_{ab}[F] *e^b .
\end{equation}
The Yang-Mills field equations are also modified according to
\begin{equation}
\nabla_{A}F=0, \quad \nabla_{A}*\left ( F +\gamma R F + \gamma^\prime R *F \right ) =0. \label{maxwell}
\end{equation}
In what follows we will be looking for the solutions of the coupled field equations (11) and (14).
\section{Bertotti-Robinson Spacetimes}
\noindent The Bertotti-Robinson spacetimes \cite{bertotti,robinson,stephani,dolan,tariq-tupper,dadhich,garfinkle-glass} admit a product topology
$ M^{1,3} = AdS_2 \times S^2.$
They were first discussed as the unique conformally flat, static spherically symmetric solution
of the source-free Einstein-Maxwell equations with a non-null electromagnetic potential.
In particular, we consider in spherical polar coordinates $(t,r,\theta, \varphi)$ a static, spherically symmetric metric
\begin{equation}
g = -f_{1}^2(r) dt^2 + f_{2}^2(r) dr^2 + r_{0}^2 \left ( d\theta^2 + \sin^2 \theta d\varphi^2 \right ),
\end{equation}
together with a static, spherically symmetric electric potential
\begin{equation}
A = q(r) dt.
\end{equation}
Then the source-free Einstein-Maxwell field equations
\begin{equation}
-\frac{1}{2} R^{bc} \wedge *e_{abc} = \frac{\kappa^2}{2} \left ( \iota_aF \wedge *F - F \wedge \iota_a*F \right ),
\end{equation}
\begin{equation}
dF=0 \quad, \quad d*F=0
\end{equation}
are satisfied for the choice
$$
f_{1}(r) = f_{2}(r) =\frac{r_0}{r} \quad, \quad q(r) = -\frac{Q}{r} .
$$
Hence the metric
\begin{equation}
g = \frac{r_0^2}{r^2} \left [ -dt^2 + dr^2 + r^2 (d\theta^2 +\sin^2\theta d\varphi^2) \right ]
\end{equation}
is conformally flat and the Weyl curvature 2-forms identically vanish:
\begin{equation}
C_{ab} = R_{ab} -\frac{1}{2} ( e_a \wedge Ric_b - e_b \wedge Ric_a ) +\frac{R}{6} e_a \wedge e_b =0.
\end{equation}
The curvature scalar $R=0$ vanishes as well. $A$ turns out to be the Coulomb electric potential so that the net electric charge
\begin{equation}
\frac{1}{4\pi} \oint_{S^2} *F = Q.
\end{equation}
\noindent We also make note of an alternative way of writing down the Bertotti-Robinson solution with a re-definition of the time coordinate as
\begin{equation}
dt =(\frac{r}{r_0})^2 d\tilde{t}.
\end{equation}
Then the conformally flat metric reads
\begin{equation}
g = -\left (\frac{r^2}{r_0^2} \right )^2 d{\tilde{t}}^2 + \left (\frac{r_0^2}{r^2} \right )^2 \left [ dr^2 + r^2 (d\theta^2 + \sin^2\theta d\varphi^2 ) \right ],
\end{equation}
and the electric potential 1-form becomes
\begin{equation}
A= -\frac{Q}{r_0^2} r d{\tilde{t}} .
\end{equation}
\section{Wu-Yang Magnetic Pole Solutions}
We now look for static, spherically symmetric solutions of our Einstein-Yang-Mills system of equations for which the space-time metric in terms of isotropic coordinates $\{t,x,y,z\}$
is given by
\begin{equation}
g = - \left ( \frac{r}{r_0}\right )^{2k} dt^2 + \left ( \frac{r_0}{r}\right )^{2} \left ( dx^2 + dy^2 + dz^2 \right ),
\end{equation}
where $r=\sqrt{x^2+y^2+z^2}$.
We introduced a real parameter $k$ so that our Bertotti-Robinson metric is no longer conformally flat unless $k^2=1$.
Yet the spacetime still admits a product topology $AdS_2 \times S^2$.
We calculate the curvature scalar which turns out to be
\begin{equation}
R =-\frac{2}{r_0^2}(1-k^2).
\end{equation}
\noindent Let us take the Wu-Yang potential 1-forms \cite{wu-yang,bartnik-mckinnon,balakin1}
\begin{eqnarray}
A^1 = \left ( \frac{K(r) - 1}{r^2} \right ) (ydz-zdy), \nonumber \\
A^2 = \left ( \frac{K(r) - 1}{r^2} \right )(zdx-xdz), \nonumber \\
A^3 = \left ( \frac{K(r) - 1}{r^2} \right ) (xdy-ydx).
\end{eqnarray}
Then we calculate
\begin{eqnarray}
F^1&=&\big[(y^2+z^2) \frac{1}{r} \left ( \frac{K(r) - 1}{r^2} \right )^{\prime} +2 \left ( \frac{K(r) - 1}{r^2} \right ) + 2e^2 x^2 \left ( \frac{K(r) - 1}{r^2} \right )^2\big]dy \wedge dz \nonumber \\
&&+\left [ \frac{1}{r}\left ( \frac{K(r) - 1}{r^2} \right )^{\prime} -2 \left ( \frac{K(r) - 1}{r^2} \right )^2 \right ] xdx \wedge (ydz - zdy),
\end{eqnarray}
and by the cyclic permutations of $(xyz)$, we also write down
\begin{eqnarray}
F^2&=&\big[(z^2+x^2) \frac{1}{r} \left ( \frac{K(r) - 1}{r^2} \right )^{\prime} +2 \left ( \frac{K(r) - 1}{r^2} \right ) + 2e^2 y^2 \left ( \frac{K(r) - 1}{r^2} \right )^2\big]dz \wedge dx \nonumber \\
&&+\left [ \frac{1}{r}\left ( \frac{K(r) - 1}{r^2} \right )^{\prime} -2 \left ( \frac{K(r) - 1}{r^2} \right )^2 \right ] ydy \wedge (zdx - xdz),
\end{eqnarray}
and
\begin{eqnarray}
F^3&=&\big[(x^2+y^2) \frac{1}{r} \left ( \frac{K(r) - 1}{r^2} \right )^{\prime} +2 \left ( \frac{K(r) - 1}{r^2} \right ) + 2e^2 z^2 \left ( \frac{K(r) - 1}{r^2} \right )^2\big]dx \wedge dy \nonumber \\
&&+\left [ \frac{1}{r}\left ( \frac{K(r) - 1}{r^2} \right )^{\prime} -2 \left ( \frac{K(r) - 1}{r^2} \right )^2 \right ] zdz \wedge (xdy - ydx).
\end{eqnarray}
Suppose special solutions of the Yang-Mills equations for which $K(r)=K$ is a constant are sought. In this case we calculate
\begin{equation}
d*F^1 + A^2 \wedge *F^3 - A^3 \wedge *F^2 = \left ( \frac{r}{r_0} \right )^{k+1} \frac{ K(K^2 - 1)}{ r^3} dt \wedge dr \wedge dx =0
\end{equation}
and the other two field equations are similarly determined by cyclic permutations of $(123)$ and $(xyz)$.
There exists three constant solutions: $K=\pm1$ and $K=0$. In the first two cases the Yang-Mills fields vanish, $F=0$, so that both of these cases correspond to the Yang-Mills vacuum. The non-trivial Wu-Yang magnetic pole solution corresponds to the choice $K=0$.
On the other hand, substituting in the Yang-Mills field 2-forms above and tracing over the Lie algebra, we get
\begin{equation}
X = -\left ( \frac{r}{r_0} \right )^4 \Big( \frac{{2K'}^2}{r^2}+\frac{ (K+1)^2(K-1)^2}{r^4} \Big), \quad Y =0.
\end{equation}
Therefore it can easily be verified for the Wu-Yang solution that
\begin{equation}
X=-\frac{1}{r_0^4}, \quad R=-\frac{2}{r_0^2}(1-k^2).
\end{equation}
Finally the complete set of non-minimally coupled Einstein-Yang-Mills field equations with a cosmological constant are solved provided
\begin{equation}
\Lambda = \frac{1}{2r_0^4} \quad , \quad \gamma = \frac{r_0^4}{\kappa^2} .
\end{equation}
Then the metric parameter should be bound according to
\begin{equation}
0 < k^2 = 1 - \frac{\kappa^2}{2 r_0^2} < 1.
\end{equation}
It is interesting to note that under the assumption that the coupling constants are fixed by
\begin{equation}
\kappa^2 \Lambda =\frac{1}{2 \gamma } =\frac{1}{2 \gamma^{\prime} },
\end{equation}
the total Lagrangian density 4-form factorizes as follows:
\begin{equation}
{\mathcal{L}}_0 + {\mathcal{L}}_1 = \left ( \frac{1}{2\kappa^2} R + \Lambda \right ) \left ( 1+ \gamma \kappa^2 X + \gamma^{\prime}\kappa^2 Y \right ) *1.
\end{equation}
For the solution above, both of these factors vanish. Hence we get a very special type of an exact solution of the Einstein-Yang-Mills field equations
derived from a Lagrangian density that admits a product structure. Then
under independent field variations with respect to the metric and the gauge potential, the variational field equations will always contain either the first factor or the second factor.
For our solution, it is evident that the coupled field equations are satisfied since both the gravity and gauge factors vanish on their own.
\section{Concluding Remark}
Here a simple model of non-minimally coupled Einstein-Yang-Mills field equations are considered. A very special class of exact static, spherically symmetric
solutions which are determined by the Wu-Yang magnetic pole potentials in a non-conformally flat Bertotti-Robinson spacetime are found.
Such solutions may be related physically to regular black holes
with colour \cite{balakin2,liu et al, zhang-mann}. However, they are in no way generic. We regard the existence of these non-generic solutions in a simple
case of non-minimal couplings induced by 1-loop quantum effects in curved spacetimes as an indication that
Yang-Mills vacuum polarization effects might in general imply unexpected new results.
\section{Acknowledgement}
Y.\c{S}. is grateful to Ko\c{c} University for its hospitality and partial support.
\newpage
{\small
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 8,955
|
The South African Maritime Safety Authority (SAMSA) is a South African government agency established on 1 April 1998 as a result of the 1998 South African Maritime Safety Authority Act 5. as such it is responsible for the implementation of current International & National Regulations regarding the Maritime Industry as well as upon all recreational marine vessels within its jurisdiction.
SAMSA via the administration and/or management of all things marine related is in effect the governing authority and as such is required to investigate maritime accidents/incidents & to provide various marine related services both on behalf of Government as well as to Government.
Overview of services
To & on behalf of Government:
Advise Government on maritime issues relating to or affecting South Africa
Administer current legislation & policies, submit additional proposals thereon as & when required so as to flag State Implementation
Represent South Africa at international forums, liaise with foreign governments & other International institutions on behalf of South African Government
On behalf of the Minister of Transport liaise with other South African institutions & various State Departments
Administration of government maritime contracts
Provide a maritime Search and Rescue (SAR) capability within the South African area of responsibility - via the management (on behalf of DOT) of the Maritime Rescue Coordination Centre (MRCC)
Conduct Accident investigations and provide Emergency Casualty Response
Control of State Ports, including management of the DOT contracted pollution prevention and response capability
To Maritime Industry (local & International):
Conduct Statutory surveys and issue Safety certification of vessels
Certification of Seafarers
Provide Assistance and advice on maritime legislation
Provide advice & grant approval in construction and refitting of vessels, including the evaluation & approval of fittings & equipment used.
Consultancy to industry on technical matters, safety and qualifications
To Stakeholders:
Safety equipment approval
Port State Control Inspections
Inspections of ships and cargoes of timber, grain and hazardous goods
Accreditation of maritime training institutions and maritime training programmes
Monitoring of South African seafarers' welfare and conditions of service
Provision of maritime safety information to shipping & ensuring a reliable radio service
Ensuring that navigational aids are in place around the South African coastline
Assimilation and maintenance of shipping information and statistics
It is subordinate to the Minister of Transport, who heads the Department of Transport. Despite it being a marine authority its head office is over 500 km away from the nearest ocean in Pretoria.
SAMSA administers the South African ship register.
In July 2012 the authority acquired the former Antarctic supply vessel S. A. Agulhas as a training ship.
References
External links
South African Maritime Safety Authority
Government agencies of South Africa
Maritime safety organizations
1998 establishments in South Africa
Government agencies established in 1998
Ship classification societies
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 360
|
Network Magazine Names Red Hat Linux Product of the Year
Continues Widespread Industry Praise for Red Hat Products and Services for the Enterprise and Internet Infrastructure
RALEIGH, N.C — 26 de abril de 2000 —
Red Hat, Inc. (NASDAQ: RHAT), the leader in open source internet infrastructure solutions, today announced that Network Magazine has named the Red Hat Linux 6.1 operating system (OS) Product of the Year in its May issue.
"Wherever it resides, the value Linux adds is changing networking," write Network Magazine editors. "Red Hat Linux has become the distribution of choice for enterprise customers. This phenomenon isn't based on Linux's easy, graphical installer, but rather for its comprehensive support, which includes a free installation hotline, and its many enterprise-oriented enhancements. Red Hat's Linux is the best, safest choice."
The Network Magazine award continues industry-wide praise for Red Hat Linux, including InfoWorld's Product of the Year for three years in a row. Other recent awards include: InformationWeek's "Product of the Year," Federal Computer Week's Government Best Buy, one of Computer Shopper's "Top 100 Products" and Software Development's Jolt Award. InternetWeek also named Red Hat "King of the Linux Server Hill," praising its unmatched technical support.
"Once again leading, independent publications have validated Red Hat's powerful open source solutions and services for today's enterprise Internet infrastructure," said Michael Tiemann, CTO, Red Hat, Inc. "Red Hat continues to receive widespread industry adoption and praise for its technical excellence, leadership and knowledgeable, innovative solutions for Internet computing."
About Network Magazine
Network Magazine delivers information about network technology, product offerings, the competitive environment, and tangible user experiences to the IT managers and executives charged with the extremely complex tasks of designing, constructing, maintaining, upgrading, and managing the network infrastructure. On April 14, 2000 Network Magazine received a Maggie award as Best Computer/Trade Publication from the Western Publishing Association in a ceremony night in Los Angeles. This comes one year after winning the PC Expo: "Most Improved Trade Publication" with a circulation over 50,000.
There is no default locale or translation available for this content.
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 3,522
|
Puppy Yoga Edinburgh: Adorable puppy yoga classes launch in Edinburgh with cuddles and play
Edinburgh Jenners fire: Firefighter Barry Martin dies from injuries sustained in fire at Jenners building in Princes Street
Edinburgh's newest park hit by fly-tipping as row continues over pledge on free bulky uplifts
A huge pile of half-burned wooden boards in a city park is the latest evidence of the growing problem of fly-tipping across the Capital.
By Ian Swanson
17th Nov 2021, 4:55am - 3 min read
Let us know what you think and join the conversation at the bottom of this article.
The unsightly mess was dumped in Edinburgh's newest park at Little France and spotted by Evening News reader Anna Michna.
She said: "I was walking in Little France Park and I discovered this massive pile of half-burned kitchen furniture in the middle of cycle path.
Fly-tipping at Little France Park
Edinburgh Traffic: Brunswick Street closed by police to pedestrians and vehicles over 'unsafe building'
"It looks like whoever dumped this must have driven in from Craigmillar Castle Road because one of the black poles had been removed and left lying next to the bin.
"I haven't seen anything like that before in Little France Park."
It comes after Lib Dem councillor Kevin Lang claimed the council's SNP-Labour administration was preparing to abandon a pledge to bring back free bulky uplifts to tackle fly-tipping.
And today he repeated his call for the scrapping of the current £5 per item charge for special uplifts.
The removed pole suggests how the fly-tipper gained access
Official figures show that in the first five months of this year the council received 272 reports of fly-tipping compared with 486 for the whole of 2020/21.
At last week's environment committee, Councillor Lang challenged convener Lesley Macinnes on whether the promise made by the SNP at the last council elections and included among the commitments in the coalition deal with Labour was still going to be delivered.
Councillor Macinnes said the matter would be covered in a report due to come to the committee next year, but she added it had been found elsewhere that making the service free did not make much difference. And she claimed free uplifts would involve an "enormous" cost.
But today Councillor Lang said: "The problem of fly-tipping across the city is bad and it's getting worse every year. It seems as though the SNP-led administration think this is a problem that will solve itself, but it won't. It needs changes to policies if we're going to get changes in behaviour.
"That's why there was a promise to get rid of the special uplifts charges and that's a promise the administration are reneging on.
"It is a simple, straightforward and very effective change that would would make it easier and cheaper for people to have bulky waste collected and recycled from their homes."
And he questioned Councillor Macinnes's claims about the expense involved in the policy.
"The Lib Dem group looked into the costs of it and in our budget proposals earlier this year we included the money to cover the costs of it – and this was on the advice of officers who said it was affordable and reasonable for us to get rid of these charges.
"So I do not understand what has changed, first from the time the administration made the promise because I presume they made the promise with the facts and figures before them, but also what has changed from earlier in the year when we made it a priority in our costed budget."
A council spokesperson said: "It's always really disappointing to see fly-tipping spoiling our beautiful city as the selfish actions of a few can really affect those living in a local community. There is no excuse for it. We have a simple booking system for residents to access our recycling centres and a special uplift service that costs only £5 per item to legally dispose of unwanted bulky household waste.
"Enforcement action will be taken whenever possible and there is sufficient evidence to identify who is responsible for fly-tipping and we step up patrols where we can in areas where it is reported."
Edinburgh council chiefs accused of abandoning pledge to reintroduce free bulky ...
Thank you for reading this article. We're more reliant on your support than ever as the shift in consumer habits brought about by coronavirus impacts our advertisers.
If you haven't already, please consider supporting our trusted, fact-checked journalism by taking out a digital subscription.
Kevin LangEdinburghSNP
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 9,910
|
Many entrepreneur clients complain about being pulled in too many directions at once. Especially in the early stages of a new venture, you are the CEO, COO, CMO, CHR, CTO and more, all rolled into one. You know you need to focus on the big picture, on achieving your business goal, but you just get dragged down into the detail on a daily basis.
Sound familiar…? Well, here are my top 5 tips to take back control of your day, get your head above water, and get the important stuff done on time.
Prioritize and focus on what's important.
It's vital to have clear goals and identify what absolutely must happen if you are to deliver your goals. Prioritize and focus on these essentials above all else. It's easy to get caught up in minutiae, responding to other people's requests, losing focus on the work only you can do. If, instead, you focus on the important tasks that relate to your vision and values and deal with them before they become urgent, it's easier to do them thoughtfully and to a high standard. If you have more to do than you have time available, you need to feel comfortable that you've tackled the priority tasks first.
Get in the zone and get it done.
How often do you get "in flow" or "in the zone"? This is the mental state when you're fully immersed in the task and completely absorbed by what you do, resulting in the loss of your sense of space and time. To get in the flow you need to focus on one challenging task and give it your full attention without interruption for at least 30 minutes. If you allow interruptions or try to multi-task you'll break your train of thought. Multitasking does not increase efficiency, attempting several tasks at once can result in lost time and productivity.
If you want to get things done, block out some time, find a quiet space, turn off your phone and email notifications, and make a start. Focus on one priority task at a time and do some quality work.
Work in your strengths zone and mitigate your weaknesses.
Rather than stressing or procrastinating about tasks you find difficult or routine, delegate to someone who has the skill to do it. If you delegate or outsource ensure you assign the task to the right person or company, one you trust to have the skills to do the job and get it done on time and on budget.
Use your own time working in your strengths zone, tapping into your natural talents and enthusiasms, and you'll be more productive, more engaged and happier. Spending extended periods of time doing work in areas outside your strengths zone will be harder, energy draining, demotivating and make you feel disengaged.
Often when under pressure, or when things don't go to plan, it's easy to rush things, to make make poor decisions and settle for quick fixes. As we all know, quick fixes don't solve the root cause of problems. Slow down, take time to think things through, and find a permanent solution. Rework is inefficient. If you don't have time to do it right the first time, you probably don't have time to do it over again.
As an entrepreneur, running your own business, it may feel awkward to say no to requests for help or to decline invitations to meetings and networking. Most meetings are a drain of time, yet we all continue to schedule them, attend them and complain about them. Before attending a meeting, ask yourself if it's necessary and if you can achieve the same result via email, phone, or a web-based meeting. If your schedule is full, and you can't squeeze more in, then you must set some boundaries, prioritize and say no to low priority activities. If you don't, you'll feel drained, overextended, anxious and in time your productivity will decline and your stress levels increase.
In summary, to be productive, connect to and get energy and motivation from the vision you have for your business. Prioritize the important work you need to do to achieve your vision, and commit to protect your time to work on it. I'm sure you chose to embark on your journey as an entrepreneur for a reason and this is your time to make it happen. Good luck!
We all have grandious ideas about what we want to achieve in our business. Not the day to day stuff but the projects you've had on your agenda for quite a while that get pushed to the side because the day to day just takes over. There is a way to get those projects done each and every year and throw a big celebration at year end for your accomplishments.
You may have seen how to be more productive tomorrow and how to get the important stuff done each week but lets see how you can be more productive over the entire year.
The little things that make you more productive today and next week go along way to adding up to a year. But how do you tie it all together to make the power of many into the sum of the whole.
Clarify your Vision. What are you and what are you becoming?
Clarify your mission. What do you do? What business are you in?
Set some bold goals. What goals can you set that aren't comfy but big, hairy, audacious goals? They make you want to squirm and the prospect of achieving it makes you want to jump up and down with glee, but the thought of the journey involved to get there makes you want to vomit.
Break It Down. Take each bold goal and write down the steps involved in getting there. Depending on the goal you may have from 2 to 6 steps. Otherwise known as a milestone.
Break It down again. With each step you've arrived at ask yourself what are the tasks involved to achieve that step. Write them as if you had to explain it to a 5 year old.
Estimate time. How much time will each task take? (and be realistic) Add up the estimated time to achieve each milestone.
Set a BHAG Date. Lock It in. When exactly do you want to achieve your BHAG? Perhaps it's a completed book, launching a new program, new website launched? Pick a date. If you'd read this post you will already have your business development days and have plotted them onto your wall calendar. Lock in your BHAG date. Really lock it in. Circle it, highlight it, star it. This is achievement day!
Plot Your Milestones. Now work backwards, plot in your milestones by assigning dates based on the estimated time to achieve each milestone.
Run out of time to Schedule? No business development days left when plotting out the milestones? Good to know isn't it. You have a few choices, push out your BHAG date, schedule in more business development days or get more resources. Pick one and adjust accordingly.
Record Your Tasks. Take the tasks and put into your task management software, notebook or whatever you use. Your due dates for these tasks are dependent on their estimated time allocation and when the milestone is due to achieve, write it down, schedule it in, make it happen!
Now get to it. See you in a year!!!
Like to see how this works with an actual business? I've created a free download of a case study for this formula above. Simply click on this link to access your free download.
Forget about your mental picture of an accountant sitting behind a desk crunching numbers all day, stopping only to peer at you over the top of his or her wire-rimmed glasses.
A trained accountant, yes, but also much more. Alycia Edgar does not fit the picture painted above; instead she is a latte-sipping, beach-loving business owner who happens to be great at numbers and systems.
Alycia is a leading expert in creating systems for profitable business growth for her clients. Nothing makes her happier than looking at your business numbers and helping you to understand what the numbers are saying, implementing systems to make your business more productive, or processes to make everything happen quicker and smoother.
Alycia is passionate about the idea that the right systems are the key to driving business growth.
She can share clear strategies that can help business owners become more productive and profitable immediately.
Alycia works with business owners to achieve this result through one-on-one mentoring and consulting on how to roll out innovative systems, procedures and processes. And will be launching her revamped Bizfficiency course later in the year.
A lover of social media, you can follow Alycia on twitter @alyciaedgar, Facebook – Alycia Edgar, instagram, Alycia Edgar.
To find out more visit www.businessperformancehq.com.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 2,484
|
Quicksie Service Club
Bowl For Kids' Sake
Impossible Question
WQXE News
Team Quicksie
On-Air Team
Local Sports Update
Big 3 at 3
Showtime with Scott
Praise & Shine
Brian and Trisha
Adam in the Afternoon
The weekend is upon us, and you're looking for something to do. Scott has you covered with Showtime, new release box office, DVD and Blu-ray hits to keep you entertained. Showtime with Scott every Friday in the 10a and 5p hours only on Quicksie 98.3!
Showtime With Scott Ep. 63
Quicksie 98.3 11/08/2019 Showtime
Opening today, from author Stephen King, is "Doctor Sleep". Up next is "Last Christmas". Also at theaters now is "Midway". This week's Blu-ray and DVD releases include "Fast & Furious: Hobbs & Shaw" and "scary stories to tell in the dark". See you at the movies!
With the arrival of the first weekend of November comes the 6th installment in the Terminator series…"Terminator: Dark Fate". Also at theaters now is "Harriet". This week's Blu-ray and DVD releases include "Ten Minutes Gone" and "A Cinderella Story: Christmas Wish". We have exited October, and now we dive in to November! See you at the movies!
More chills await on this weekend before Halloween! At theaters now is "Countdown". Also showing now is "Black and Blue". New releases on DVD and Blu-ray this week include the new 2019 version of "The Lion King", "Angel of Mine", and "Bloodline". And The State Theater has lined up two great movies for this pre-Halloween weekend! Tonight at 7, it's the Don Knotts classic "The Ghost and Mr. Chicken", and tomorrow at 4pm is "Goosebumps 2: Haunted Halloween". Tickets are just $3 each. Visit thestate270.org Take your pick, and catch a great flick!
At theaters now, from Disney Studios, "Maleficent: Mistress of Evil". Also opening nationwide today is "Zombieland: Double Tap. DVD and Blu-ray releases this week include "Crawl", "Stuber", and "The Art of Self Defense". The weekend and movies await!
Now showing at theaters, and just in time for Halloween, it's "The Addam's Family", an animated version of Charles Addams' series of cartoons about a peculiar, ghoulish family. Up next is "Gemini Man". New DVD's and Blu-ray out this week include "Annabelle Comes Home", and "Toy Story 4". Have fun this weekend!
WQXE Quicksie 98.3
Studio Phone 1-800-905-0983
Email: quicksie@wqxe.com
233 West Dixie Avenue
Advertise on Quicksie
Copyright © 2020 Skytower Communications. All Rights Reserved.
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 2,997
|
Q: Starting Help First Android app with c# I'm trying to write an android app using VS Community 2015 in c#
The App should send a message to a websocket an display it received answer.
This is my MainActivity.cs
public class MainActivity : Activity
{
protected override void OnCreate(Bundle bundle)
{
base.OnCreate(bundle);
// Set our view from the "main" layout resource
SetContentView(Resource.Layout.Main);
// Get our button from the layout resource,
// and attach an event to it
Button button = FindViewById<Button>(Resource.Id.MyButton);
TextView t1 = FindViewById<TextView>(Resource.Id.textView1);
button.Click += delegate
{
string msg = SetStatusText();
t1.SetText(msg,TextView.BufferType.Normal);
};
}
private string SetStatusText()
{
Connector c = new Connector();
c.Connect();
return c.msg;
}
}
The Connector goes like this
class Connector
{
public string msg { get; set; }
WebSocket websocket = new WebSocket("ws://192.168.1.103:2012/");
public void Connect()
{
websocket.Opened += new EventHandler(websocket_Opened);
websocket.MessageReceived += new EventHandler<MessageReceivedEventArgs>(websocket_MessageReceived);
websocket.Open();
}
private void websocket_MessageReceived(object sender, MessageReceivedEventArgs e)
{
msg = e.Message;
}
private void websocket_Opened(object sender, EventArgs e)
{
websocket.Send("Status");
}
}
In debug mode I see the message the websocket returned in msg variable of the Connector, but the way I'm returning it to the MainActivity doesn't work. It stays NULL.
A: You are using the "msg" variable just after connecting to the WebSocket, that will fail as you still did not received a message, you need to wait in some way to websocket_MessageReceived be executed.
If you want to wait until the message is received you can use an event:
class Connector
{
public string msg { get; set; }
public event EventHandler MessageReceived;
WebSocket websocket = new WebSocket("ws://192.168.1.103:2012/");
public void Connect()
{
websocket.Opened += new EventHandler(websocket_Opened);
websocket.MessageReceived += new EventHandler<MessageReceivedEventArgs>(websocket_MessageReceived);
websocket.Open();
}
private void websocket_MessageReceived(object sender, MessageReceivedEventArgs e)
{
msg = e.Message;
if(MessageReceived != null)
MessageReceived(this, EventArgs.Empty);
}
private void websocket_Opened(object sender, EventArgs e)
{
websocket.Send("Status");
}
}
Then, when you create your connection you can do:
c.MessageReceived += (o,e) => t1.SetText(c.msg,TextView.BufferType.Normal);
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 3,522
|
Tag: ##Unmarried
Biography Writer
Ruth Pfau Biography
Jeeya September 13, 2019 Leave a comment
Ruth Pfau was a German-Pakistani Catholic nun of the Society of the Daughters of the Heart of Mary, and a physician. She moved from Germany to Pakistan in 1961 and devoted more than 55 years of her life to fighting leprosy in Pakistan. Pakistan honored her with a range of awards including the Hilal-i-Pakistan, Hilal-i-Imtiaz,...
Winnie Harlow Biography
The gorgeous Winnie Harlow has become of the trending names in the American modeling industry; all thanks to her resilience, courage, and determination. The model has become an inspiration and a source of hope to vitiligo patients who have lost their self-confidence and sense of belonging through the deplorable skin condition. Despite her condition, Winnie...
Caroline Calloway Biography
Social media serves to be a great platform to make one's career shine. Some celebrities came to the spotlight, accepting the craze that the social media offers. Caroline Calloway is one of the stars, who shined through social media.The famous face on social media, Caroline Calloway, is the biggest celebrity now. She came to fame...
Kieran Gibbs Biography
Kieran Gibbs is an English professional footballer who plays as a left-back for Championship club West Bromwich Albion and the English national team. He began his senior career with Arsenal in 2007, after joining the club from the Wimbledon academy in 2004. Gibbs began as a winger, moving to left-back after a loan spell with...
Camryn Grimes Biography
The 2018 Emmy Awards remains a memorable event for the likes of Camryn Grimes who was able to win one more award for her outstanding role in the drama series, The Young and the Restless. The charming and elegant American is among the many performers who found their place in the entertainment business from a...
Mark Halperin Biography
Mark Halperin is an American journalist, most recently known for his position as a senior political analyst for MSNBC and as a contributor, and former co-managing editor with John Heilemann of Bloomberg Politics. He previously worked as the political director at ABC News, where he worked as the editor of the Washington, D.C., newsletter The...
Luca Schaefer-Charlton Biography
Luca Schaefer-Charlton is a Canadian internet celebrity, best known for posting lip-sync videos on TikTok (musical.ly). Soon after creating his TikTok account, Luca started accumulating fans because of his funny and unique content. On his TikTok account, Luca has posted a variety of videos including imitation videos of popular celebrities. One such video, in which...
Jamie Laing Biography
Jamie Laing is a British actor, entrepreneur, and social media influencer. He rose to fame for his role in the award-winning television series 'Made in Chelsea.' The series aired on the UK's channel 'E4,'. He joined the cast in the second season of the series. Jamie is also one of the two presenters of the...
Stephanie Gilmore Biography
Stephanie Gilmore is a renowned Australian professional surfer. She ruled as a world champion for seven-time world champion on the Women's ASP World Tour (2007, 2008, 2009, 2010, 2012, 2014, 2018). Similarly, she also has claimed more than 30 elite World Tour victories. Keeping this on the note, she is the first woman in the...
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 9,447
|
Q: How to Read a Row into Pandas--that has been broken by a return character I'm trying to read a census building permits text file with several rows that look like the below.
At times, the description field is too long which causes a line break in the row--which screws up pandas.
533 45220 Tallahassee, FL 1613 810
999 13980 Blacksburg-Christiansburg-Radford,
VA 543 455
108 11100 Amarillo, TX 740 718
The the below code will read the file into pandas––but many rows are shifted. How do you parse a file text file like this? Many thanks in advance.
testdf = pd.read_table('./csv/bldg_permits/metro/tb3u2016.txt', header='infer',
encoding="ISO-8859-1",skiprows=9,
delimiter = '\s+', skipinitialspace=True,
error_bad_lines=False)
A: Pandas won't be able to patch lines together like that as part of read_csv().
I'd recommend doing a first pass to clean the data (separators are also an issue), and then a second pass to load into Pandas.
First, get data from the URL (I'm using requests but any URL parser will do):
import pandas as pd
import re
import requests
url = "https://www.census.gov/construction/bps/txt/tb3v2016.txt"
r = requests.get(url)
Now iterate over lines, writing each line to lines.
lines = []
begin_data = 10
backup_by = 1
for i, l in enumerate(r.text.split("\n")[begin_data:]):
line = (pd.Series(l).str.replace("(,|,\\*) ", "\\1_")
.str.replace("([A-z\\.]) ([A-z])", "\\1_\\2", n=-1))
if line.str.match("\d")[0]: # normal line
lines.append(line[0])
elif len(lines) > 0: # not a normal line, add to previous line
lines[i-backup_by] = lines[i-backup_by].strip() + line[0].strip()
backup_by += 1
fname = "census_data.txt"
f = open(fname, "w")
_ = [print(line, file=f) for line in lines]
Notes on above block:
*
*Since we're going to read this table into Pandas with a \s+ delimiter, replace spaces with _ when they're not part of column delimiters. We're looking for two of these edge cases in particular:
*
*Ex. Alexandria, LA --> Alexandria,_LA
*Ex. Minneapolis-St. Paul-Bloomington --> Minneapolis-St._Paul-Bloomington
*If one line looks funny (meaning it doesn't start with a numeric CSA code), assume that's actually meant to be part of the line before it and add it to that previous line.
*We need to keep track of the index of lines that represents the previous line we want to add to. Each time we iterate over a line of the original data and don't add a new line to lines, the difference between our loop counter (i) and the index of the last element in lines increments by 1. So we use a counter (backup_by) that figures out the correct index of lines to append to.
Now read the cleaned text file into Pandas:
colnames = ["CSA", "CBSA", "Name", "Total", "1 Unit", "2 Units",
"3 and 4 Units", "5 Units or more"]
df = pd.read_table(fname, header=None, names=colnames, encoding="ISO-8859-1",
engine='python', delim_whitespace=True, skipfooter=3)
df.head()
CSA CBSA Name Total 1 Unit 2 Units \
0 999 10180 Abilene,_TX 55593 55193 400
1 184 10420 Akron,_OH 226669 226169 0
2 999 10500 Albany,_GA 28679 23686 0
3 440 10540 Albany,_OR 98763 97926 0
4 104 10580 Albany-Schenectady-Troy,*_NY 512058 361454 10605
3 and 4 Units 5 Units or more
0 0 0
1 500 0
2 360 4633
3 0 837
4 26585 113414
At this point, you can go back and remove the _ placeholders for spaces that were inserted into the Name field, if so desired.
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 3,381
|
Q: Ember DS.Store error while submitting form I'm a newbie to ember and I'm trying to create a basic sign-up form.
Relevant model:
App.NewUser = DS.Model.extend({
user_name: DS.attr('string'),
password: DS.attr('string'),
confirm_password: DS.attr('string'),
email: DS.attr('string'),
first_name: DS.attr('string'),
last_name: DS.attr('string'),
});
Relevant controller:
App.SignupController = Ember.ArrayController.extend({
actions: {
signup: function() {
var data = this.getProperties('first_name', 'last_name', 'email', 'user_name', 'password', 'confirm_password');
var newUser = this.store.createRecord('newUser', data);
newUser.save();
},
},
});
When the "signup" action executes, I get the following error:
Error: Attempted to handle event `didSetProperty` on <App.NewUser:ember332:null> while in state root.deleted.saved. Called with {name: last_name, oldValue: undefined, originalValue: undefined, value: undefined}.
What am I doing wrong?
A: This is a bug, Ember Data is setting the record state incorrectly if you're setting a value to what it's currently set to (undefined on createRecord)
You'll want to either coerce your values into empty strings or not set undefined values while creating the record.
for(var key in data){
if(!data[key]) delete data[key];
}
http://emberjs.jsbin.com/OxIDiVU/124/edit
https://github.com/emberjs/data/issues/1648
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 513
|
Avondale Shopping Village, located at the end of College Drive, offers a range of services including general medical, pathology services and a pharmacy. Other nearby health services include general and specialist practitioners, dental services, optometrists, chiropractors and physiotherapists.
Wyong Hospital is the closest public hospital to the Lake Macquarie campus.
Students on the Sydney campus have access to a range of health services at the Sydney Adventist Hospital and the Fox Valley Medical and Dental Centre. Other nearby health services include general medical and specialist practitioners, dental services, optometrists, chiropractors and physiotherapists.
Hornsby Ku-ring-gai Hospital is the closest public hospital to the Sydney campus. The Sydney Adventist Hospital also offers urgent emergency treatment to private patients.
To receive medical treatment as a domestic student, you must be registered with Medicare, Australia's government healthcare system. Membership with Medicare provides access to free treatment as a patient in a public hospital as well as subsidised treatment by medical practitioners including general practitioners and specialists nominated by the hospital. Ambulance services are not covered by Medicare.
You can apply for a Medicare card independent of your parents, and will need to do so before arriving at Avondale. You may need to produce proof of your student status when applying for a Medicare card, so remember to keep your acceptance letter from Avondale.
Private medical insurance offers more comprehensive health cover than standard medicare cover, and includes the benefit of private hospital usage and extras. ACA Health Benefit Fund is available to students who are connected to the Seventh-day Adventist Church and/or associated entities.
New Zealand citizens who come to Australia for the specific purpose of study are ineligible for Medicare benefits. However, New Zealand students who do not have permanent residence are eligible to be treated at all public hospitals under Reciprocal Health Care arrangements. Presentation of a New Zealand passport will be required. A refund for doctor services can be obtained by presenting a paid tax invoice, along with a passport, to a Medicare office.
To receive a visa, the Australian Government requires international students to have Overseas Student Health Cover (OSHC). Cover is required for the full length of the course and must be paid up front. OSHC is organised through the Admission Enquiry Centre.
Ambulance cover is not covered by Medicare. Students are responsible for ensuring they are covered through membership in a private health fund for ambulance transport in New South Wales.
Overseas Student Health Cover (OSHC), which is compulsory for international students, covers the full charge for emergency ambulance transport. However, routine journeys in an ambulance, even if offered by medical personnel, are not covered.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 8,013
|
\section{Introduction} \label{sec:intro}
Integrated optical microresonators continue to develop as a promising platform for generating, controlling, and measuring quantum states of light \cite{Xu2008,Xia2007,Ferrera2008,Ferrera2009,Azzini2012,Azzini2012a, Preble2015, Gentry2015,Kumar2015,Jung2013}. Advances in fabricating such chip-based structures is enabling the construction of micron-scale optical ring resonators with reported quality factors Q of over one million \cite{Ferrera2009,Razzari2010}. By exploiting the nonlinear optical response of the ring medium, combined with the massive enhancement of intraring pump intensity made possible by the large Q values of the resonator, a wide variety of nonlinear optical phenomena can be realized using relatively modest input powers. Entangled photon pair generation in silicon microrings has been demonstrated using mere $\mathrm{\mu W}$ of pump power \cite{Azzini2012,Gentry2015}, and optical parametric oscillation in a silicon nitride microring has been observed using a pump power of only 50 mW \cite{Levy2010}. Arrays of coupled silicon microrings have also been investigated as a potential source of heralded single photons \cite{Davanco2012}.
Such high-Q microrings are ideal for investigations of strongly driven nonlinear optical effects. Depending on the application, these effects can be undesirable or highly sought after: multi-pair production from an entangled photon pair source contaminates the sought after energy correlation, whereas optical parametric oscillation (OPO) \emph{only} arises in the strongly driven regime. Theoretical studies of quantum nonlinear optics in integrated microresonators have typically treated the nonlinearity perturbatively \cite{Vernon2015,Chen2011,Camacho2012,Helt2010, Yang2007,Yang2007a,Helt2012,Liscidini2012}, which limits calculations to quantities relating to a single generated photon pair.
Recently we presented a general theoretical treatment of photon pair generation arising from spontaneous four-wave mixing (SFWM) in microring resonators, fully accounting for the quantum effects of scattering losses within the resonator \cite{Vernon2015}. As our focus was on the effects of such losses, we limited ourselves to the low power regime in which a perturbative solution of the relevant equations of motion provides an adequate description of the pair generation process. In this work we extend our analysis to a more strongly driven regime, where perturbative strategies are inadequate and competing nonlinear effects, including self-phase modulation (SPM) and cross-phase modulation (XPM), become important. We restrict ourselves to one of the most common pump states used in experiment, that of a coherent, narrowband continuous wave pump beam, for which a nonperturbative, analytic solution to the semiclassical equations of motion is achievable within the undepleted pump approximation. This approximation limits us to pump intensities below the onset of OPO, the threshold for which is clearly indicated by our equations; OPO in such structures will be the subject of a later communication. Even below the OPO threshold, the subtle interplay between the various nonlinear terms that couple the ring modes, as well as the effects of multiple photon pair generation, give rise to a rich variety of nonlinear optical phenomena that are accessible by varying only two input parameters, namely the pump intensity and detuning.
In Sec. \ref{sec:hamiltonian_and_fields} we begin by assembling the relevant Hamiltonian and field operators for the ring-channel system. In Sec. \ref{sec:eqns_of_motion} a brief review and summary of our earlier \cite{Vernon2015} theoretical framework is presented, wherein the system's dynamics are reduced to a set of coupled ordinary differential equations for the ring operators alone. Steady state solutions for the pump mode, incorporating the effects of SPM and scattering losses, are developed in Sec. \ref{sec:pump_dynamics}, and the stability of those solutions is studied. The equations of motion for the signal and idler modes are then solved in Sec. \ref{sec:signal_idler_dynamics}, enabling the calculation of physical quantities including the photon generation rate, single photon power spectrum, and joint spectral intensity distribution. For each of these measurable quantities the corresponding predictions at low and high pump powers are compared, and we identify a set of experimental features, or ``smoking guns," that distinguish the qualitative behaviour at high pump powers from that at low pump powers.
\section{Hamiltonian and fields}\label{sec:hamiltonian_and_fields}
We consider an integrated microring resonator side-coupled to a channel waveguide, as illustrated in Fig. \ref{fig:ring_diagram}. We assume the ring size and quality factor Q have been chosen such that the ring accommodates individual resonant modes which are well separated in frequency; that is, we are in the high finesse limit. While a simple generalization of our framework can be used to treat arbitrarily many ring resonances, we restrict our model for the time being to contain only three ring modes of interest. The full system Hamiltonian can then be written \cite{Vernon2015} as
\begin{eqnarray}\label{main_hamiltonian}
H = H_{\mathrm{channel}} + H_{\mathrm{ring}} + H_{\mathrm{coupling}} + H_{\mathrm{bath}},
\end{eqnarray}
wherein $H_{\mathrm{channel}}$ refers to the channel fields, $H_{\mathrm{ring}}$ to the ring modes, $H_{\mathrm{coupling}}$ to the coupling between the channel and ring, and $H_{\mathrm{bath}}$ to any modes into which ring photons may be lost, as well the coupling of those modes to the ring modes. Introducing channel fields $\psi_J(z)$, the channel Hamiltonian is
\begin{eqnarray}\label{H_channel_defn}
H_{\mathrm{channel}} &=& \sum_J \Bigg[ \hbar\omega_J \int dz\; \psi_J^\dagger(z)\psi_J(z) \nonumber \\
&+& \frac{i\hbar v_J}{2}\int dz\; \left( \frac{d\psi_J^\dagger(z)}{dz}\psi_J(z) - \mathrm{H.c.}\right)\Bigg],
\end{eqnarray}
where the fields satisfy the usual commutation relations
\begin{eqnarray}\label{commutators}
\left[\psi_J(z),\psi_{J'}(z')\right] &=& 0,\nonumber \\
\left[\psi_J(z),\psi_{J'}^\dagger(z')\right] &=& \delta(z-z')\delta_{JJ'}.
\end{eqnarray}
The index $J\in\{P,S,I\}$ runs over three fields of interest, respectively labelled $P$, $S$ and $I$ for pump, signal and idler, with corresponding reference frequencies $\omega_J$ and propagation speeds $v_J$. Each field $\psi_J$ contains frequency components centred at $\omega_J$, taken to be the resonant frequency of the corresponding ring mode, and ranges over a bandwidth that does not overlap with those of other fields $\psi_{J'}$, but involving excitation over sufficiently long distances that the Dirac $\delta$ function in (\ref{commutators}) is a good approximation \cite{Vernon2015}. By allowing the fields to have different propagation speeds we include the possibility of group velocity dispersion between the different channel fields. The Hamiltonian (\ref{H_channel_defn}) does assume group velocity dispersion within each channel field is negligible, but it is straightforward to include arbitrary dispersion. The spatial co-ordinate $z$ ranges from $z=-\infty$ to $z=+\infty$ with the coupling to the ring assumed to take place at a single point $z=0$. Within this point coupling approximation the coupling Hamiltonian becomes
\begin{eqnarray}
H_{\mathrm{coupling}} = \sum_J\left(\hbar\gamma_J b_J^\dagger \psi_J(0) + \mathrm{H.c.}\right),
\end{eqnarray}
in which we have introduced ring-channel coupling coefficients $\gamma_J$, as well as discrete ring mode annihilation operators $b_J$. In addition to the physical channel, to simulate scattering losses in the ring we include an extra ``phantom channel" into which ring photons can be lost. The phantom channel similarly accommodates three fields $\phi_J(z)$ with respective propagation speeds $u_J$ and coupling coefficients $\mu_J$, and is represented as $H_\mathrm{bath}$ by a channel and coupling Hamiltonian identical to those for the physical channel \cite{Vernon2015}.
The Hamiltonian for the ring modes can be written as
\begin{eqnarray}
H_{\mathrm{ring}} = \sum_J \hbar\omega_J b_J^\dagger b_J + H_{\mathrm{NL}},
\end{eqnarray}
where $H_{\mathrm{NL}}$ includes all the nonlinearity in the system. Since the fields will be most intense within the ring resonator, we neglect channel nonlinearities and take $H_{\mathrm{NL}}$ to contain only ring mode operators. In this work we consider effects arising from the third-order nonlinear susceptibility in the ring, taking
\begin{eqnarray}
H_{\mathrm{NL}} &=& \left(\hbar\Lambda b_P b_P b_S^\dagger b_I^\dagger + \mathrm{H.c.}\right) + \hbar\eta b_P^\dagger b_P^\dagger b_P b_P \nonumber \\
&+& \hbar\zeta\left(b_S^\dagger b_P^\dagger b_S b_P + b_I^\dagger b_P^\dagger b_I b_P\right).
\end{eqnarray}
The first term is responsible for SFWM, in which two pump photons are converted to a signal and idler photon pair. The second leads to SPM of the pump, while the latter two are responsible for XPM between the pump and signal and idler modes. It is safe to neglect SPM and XPM terms that involve only the signal and idler modes, since the power in those modes will be small compared to that in the pump mode. While we focus in this work on SFWM involving a single pump mode, it is straightforward to incorporate multiple pump modes into our model. The nonlinear coupling coefficients $\Lambda$, $\eta$ and $\zeta$ are not independent, as they arise from the same nonlinear susceptibility, but we formally leave them arbitrary for the time being so that the effects of each term in $H_{\mathrm{NL}}$ can more easily be identified. Obtaining expressions for these constants depends on the approximations used to derive the nonlinear sector of the ring Hamiltonian. We present our derivation of this Hamiltonian and the associated constants $\Lambda$, $\eta$ and $\zeta$ in Appendix \ref{appendix:lambda}, arriving at an estimate of
\begin{eqnarray}
\Lambda \approx \frac{\hbar\omega_P^2cn_2}{n^2 V_{\mathrm{ring}}},
\end{eqnarray}
with $\eta=\Lambda/2$ and $\zeta=2\Lambda$. In this expression $n_2$ refers to the nonlinear refractive index of the ring material, $n$ to the linear refractive index, and $V_{\mathrm{ring}}$ to the volume of the ring mode. For the silicon nitride rings used in typical experiments \cite{Levy2010}, with $n_2\approx 2.4\times 10^{-19}\mathrm{m^2/W}$ \cite{Ikeda2008} this yields $\Lambda\sim 10\;\mathrm{Hz}$. For typical silicon rings \cite{Azzini2012}, with $n_2 \approx 2.7\times 10^{-18}\mathrm{m^2/W}$ \cite{Boyd2008} this calculation predicts $\Lambda \sim 10^3\;\mathrm{Hz}$.
\begin{figure}
\includegraphics[width=1.0\columnwidth]{ringchannel.png}
\caption{Integrated ring-channel system geometry with labelled ring modes and incoming and outgoing outgoing channel fields. Photons generated in the ring may exit to the physical channel or be lost to the upper effective ``phantom channel".} \label{fig:ring_diagram}
\end{figure}
\section{Equations of motion}\label{sec:eqns_of_motion}
The Heisenberg equations of motion for the field operators $\psi_J(z,t)$ and $\phi_J(z,t)$ and the ring operators $b_J(t)$ follow from the Hamiltonian (\ref{main_hamiltonian}), and can be simplified by the introduction of auxiliary quantities\cite{Vernon2015}; here we summarize the results.
The equations of motion for the channel fields are
\begin{eqnarray}\label{basic_field_eqn}
\left(\frac{\partial}{\partial t} + v_J\frac{\partial}{\partial z} + i\omega_J\right)\psi_J(z,t) = -i\gamma_J b_J(t) \delta(z),
\end{eqnarray}
with similar expressions obeyed by the phantom channel fields $\phi_J(z,t)$. Note that the solutions to these equations contain a discontinuity at $z=0$, which is a consequence of our point-coupling assumption. To avoid explicitly dealing with this discontinuity, it is helpful to introduce formal channel fields which we identify as those fields which are incoming and outgoing with respect to the coupling point. We define the incoming field $\psi_{J<}(z,t)$ by
\begin{eqnarray}
\psi_{J<}(z,t) = \psi_J(z,t)\;\;\mathrm{for}\; z<0,
\end{eqnarray}
and extend it to $z\geq 0$ by requiring everywhere that it satisfy the homogeneous version of (\ref{basic_field_eqn}),
\begin{eqnarray}
\left(\frac{\partial}{\partial t} + v_J\frac{\partial}{\partial z} + i\omega_J\right)\psi_{J<}(z,t) = 0.
\end{eqnarray}
This confers a false future on $\psi_{J<}(z,t)$, corresponding to the free evolution of the incoming field without any coupling to the ring. We similarly define the outgoing field $\psi_{J>}(z,t)$ by taking
\begin{eqnarray}
\psi_{J>}(0,t) = \psi_J(z,t)\;\;\mathrm{for}\; z>0,
\end{eqnarray}
and demanding that for all $z$
\begin{eqnarray}
\left(\frac{\partial}{\partial t} + v_J\frac{\partial}{\partial z} + i\omega_J\right)\psi_{J>}(z,t) = 0,
\end{eqnarray}
giving $\psi_{J>}(z,t)$ a false past to the left of the coupling point. By an identical procedure we may define the incoming and outgoing phantom channel fields $\phi_{J<}(z,t)$ and $\phi_{J>}(z,t)$. Since we will primarily be concerned with the properties of the photons generated in the ring, which exit to one of the channels and propagate to positive $z$, all calculations involving the ring's output will be carried out on the outgoing fields $\psi_{J>}(z,t)$. Our goal is therefore to construct an explicit solution for these fields in terms of the incoming fields $\psi_{J<}(z,t)$. Indeed, since these fields freely propagate, the field at large positive $z$ (where any measurements on the generated photons would occur) is entirely determined by the outgoing field at $z=0$,
\begin{eqnarray}
\psi_J(z,t) = e^{-i\omega_J z/v_J}\psi_{J>}(0,t-\nicefrac{z}{v_J})\;\;\mathrm{for}\;z>0.
\end{eqnarray}
It therefore suffices to construct a solution for $\psi_{J>}(0,t)$, which can be very simply related to the incoming field $\psi_{J<}(0,t)$ and the corresponding ring operator $b_J(t)$ \cite{Vernon2015} via
\begin{eqnarray}\label{channel_transformation}
\psi_{J>}(0,t) = \psi_{J<}(0,t) - \frac{i\gamma_J}{v_J}b_J(t).
\end{eqnarray}
For each operator $\mathcal{O}_J(t)$ it will be convenient to define the corresponding slowly-varying barred operator $\overline{\mathcal{O}}_J(t)$,
\begin{eqnarray}
\overline{\mathcal{O}}_J(t) = e^{i\omega_J t}\mathcal{O}_J(t).
\end{eqnarray}
In terms of these quantities and the incoming and outgoing fields, the equations for the ring mode annihilation operators $\overline{b}_J(t)$ are found to satisfy
\begin{widetext}
\begin{subequations}\label{ring_eqns_master}
\begin{eqnarray}
\left(\frac{d}{dt} + \overline{\Gamma}_P + 2i\eta\overline{b}_P^\dagger(t)\overline{b}_P(t)\right)\overline{b}_P(t) &=& -i\gamma_P^*\overline{\psi}_{P<}(0,t) - i\mu_{P}^*\overline{\phi}_{P<}(0,t) - 2i\Lambda^*\overline{b}_P^\dagger(t)\overline{b}_S(t)\overline{b}_I(t)e^{-i\Delta_\mathrm{ring} t}, \label{ring_pump_master} \\
\left(\frac{d}{dt} + \overline{\Gamma}_S + i\zeta\overline{b}_P^\dagger(t)\overline{b}_P(t)\right)\overline{b}_S(t) &=& -i\gamma_S^*\overline{\psi}_{S<}(0,t) - i\mu_S^*\overline{\phi}_{S<}(0,t) -i\Lambda\overline{b}_P(t)\overline{b}_P(t)\overline{b}_I^\dagger(t)e^{i\Delta_\mathrm{ring} t}, \label{ring_signal_master}\\
\left(\frac{d}{dt} + \overline{\Gamma}_I + i\zeta\overline{b}_P^\dagger(t)\overline{b}_P(t)\right)\overline{b}_I(t) &=& -i\gamma_I^*\overline{\psi}_{I<}(0,t) - i\mu_I^*\overline{\phi}_{I<}(0,t) -i\Lambda\overline{b}_P(t)\overline{b}_P(t)\overline{b}_S^\dagger(t)e^{i\Delta_\mathrm{ring} t}, \label{ring_idler_master}
\end{eqnarray}
\end{subequations}
\end{widetext}
where we have introduced the ring mode detuning
\begin{eqnarray}
\Delta_\mathrm{ring} = \omega_S + \omega_I - 2\omega_P,
\end{eqnarray}
as well as the total effective linewidths $\overline{\Gamma}_J$,
\begin{eqnarray}
\overline{\Gamma}_J=\Gamma_J + M_J,
\end{eqnarray}
where $\Gamma_J$ and $M_J$ denote the damping rates associated with the physical channel and phantom channel couplings, respectively:
\begin{eqnarray}
\Gamma_J &=& \frac{|\gamma_J|^2}{2v_J}, \nonumber \\
M_J &=& \frac{|\mu_J|^2}{2u_J}.
\end{eqnarray}
These total damping rates can be simply related to the quality factors $Q_J$ of the resonator modes; for example, for the pump resonance
\begin{eqnarray}
Q_P=\frac{\omega_P}{\overline{\Gamma}_P},
\end{eqnarray}
which yields $Q_P\sim 10^6$ for a ring with $\overline{\Gamma}_P=1$ GHz given a pump with wavelength $\lambda=1550$ nm. The coupled set of driven, damped ordinary differential equations (\ref{ring_eqns_master}) fully describes the nonlinear dynamics of the ring-channel system. Combined with the channel transformation (\ref{channel_transformation}), a solution to this system of equations permits the calculation of any measurable quantities on the outgoing photons in the channel.
It is important to note at this stage that our treatment neglects the effect of ring heating due to the large circulating pump power present in the ring. Such thermal effects are routinely observed in experimental investigations of microring systems, and typically manifest as an effective power-dependent drift in the resonant frequencies of the ring as it undergoes thermal expansion \cite{Levy2010}. For slowly varying and cw pumps a simple way to account for this is through the addition of a pump photon number-dependent correction to each resonance. Our model already incorporates a similar effect: SPM and XPM of each mode are represented by precisely such terms. The inclusion of thermal resonance drift can therefore be modelled by altering the coefficients $\eta$ and $\zeta$ in Eqs. (\ref{ring_eqns_master}), which would be replaced by effective constants $\eta_{\mathrm{eff}}$ and $\zeta_\mathrm{eff}$,
\begin{eqnarray}\label{thermal_substitution}
\eta_\mathrm{eff} = \eta + \eta_\mathrm{thermal}, \nonumber \\
\zeta_\mathrm{eff} = \zeta + \zeta_\mathrm{thermal}.
\end{eqnarray}
While $\eta$ and $\zeta$ are both positive, $\eta_\mathrm{thermal}$ and $\zeta_\mathrm{thermal}$ would be negative, since as the ring expands the resonant frequencies are typically lowered \cite{Almeida2004}. Depending on the relative magnitude of the thermal drift coefficients compared to the SPM and XPM strengths, in some circumstances $\eta_\mathrm{eff}$ and $\zeta_\mathrm{eff}$ may become negative. While for the remainder of this work we neglect thermal drift of the ring resonances, so that $\eta_\mathrm{thermal}=\zeta_\mathrm{thermal}=0$, we emphasize that our conclusions do not depend sensitively on this assumption unless otherwise stated.
\section{Steady state pump solution}\label{sec:pump_dynamics}
The set of coupled equations (\ref{ring_eqns_master}) treats both the pump and signal and idler modes quantum mechanically, retaining the operator nature of $b_J(t)$ for each $J$. While this is necessary if one wishes to fully account for the nonclassical properties of the pump mode, in typical experiments \cite{Azzini2012a, Azzini2012} the system is pumped by a coherent laser beam or pulse. In such situations the initial pump state is described by setting each incoming pump mode to a coherent state. The pump field can then be well approximated by its expectation value, which is a classical function of time. To implement this semiclassical approximation we take
\begin{eqnarray}
\overline{b}_P(t)\rightarrow \overline{\beta}_P(t) = \langle \overline{b}_P(t) \rangle.
\end{eqnarray}
In addition to treating the pump classically, we also implement the undepleted pump approximation. In the equation for the ring pump mode (\ref{ring_pump_master}) the term involving $\overline{b}_P^\dagger \overline{b}_S \overline{b}_I$ accounts for the effect on the pump mode when a signal-idler photon pair is produced. Neglecting such effects, we drop this term and instead take the semiclassical pump amplitude $\overline{\beta}_P(t)$ to satisfy
\begin{eqnarray}\label{intermediate_pump_eqn}
\bigg( \frac{d}{dt} + \overline{\Gamma}_P &+& 2i\eta|\overline{\beta}_P(t)|^2 \bigg)\overline{\beta}_P(t)\nonumber \\ &=& -i\gamma_P^*\langle\overline{\psi}_{P<}(0,t)\rangle,
\end{eqnarray}
in which we have assumed $\langle \overline{\phi}_{P<}(0,t) \rangle = 0$, so that there is no incoming pump energy in the phantom channel. Note that while this approximation amounts to neglecting pump depletion due to photon pair generation, linear pump losses are still accounted for in our model, as evidenced by the presence of the damping term $\overline{\Gamma}_P$ in Eq. (\ref{intermediate_pump_eqn}).
In this work we consider the case of a continuous wave (cw) pump beam injected in to the channel, so that
\begin{eqnarray}
\langle \overline{\psi}_{P<}(0,t) \rangle = \frac{p}{\gamma_P^*}e^{-i\Delta_P t},
\end{eqnarray}
where $\Delta_P$ is the detuning of the injected pump from the ring pump resonance, and $p$ is a constant related to the input pump power $P_{\mathrm{in}}$ in the channel at the coupling point via
\begin{eqnarray}
p = \sqrt{\frac{2\Gamma_P P_\mathrm{in}}{\hbar\omega_P}}.
\end{eqnarray}
In steady state, after the ring pump mode has come to equilibrium with the channels, we expect there to be a constant average number of pump photons $N_P$ in the ring, where
\begin{eqnarray}
N_P=\lim_{t\to\infty}|\overline{\beta}_P(t)|^2.
\end{eqnarray}
Defining $\widetilde{\beta}_P(t) = e^{i\Delta_P t}\overline{\beta}_P(t)$, from Eq. (\ref{intermediate_pump_eqn}) we have
\begin{eqnarray}\label{pump_tilde_eqn}
\left( \frac{d}{dt} + \overline{\Gamma}_P + i(2\eta|\overline{\beta}_P(t)|^2 - \Delta_P) \right)\widetilde{\beta}_P(t) = -ip.
\end{eqnarray}
It is not difficult to show that $N_P$ will be constant only when $\widetilde{\beta}_P(t)$ has both constant amplitude and constant phase, so that $d\widetilde{\beta}_P(t)/dt=0$. Setting this time derivative to zero in the above equation and taking the modulus squared of the result, we find that in steady state $N_P$ must be a root of the cubic equation
\begin{eqnarray}\label{pump_cubic_root_eqn}
C_P(N_P)=0,
\end{eqnarray}
where
\begin{eqnarray}\label{pump_cubic}
& &C_P(N_P) \equiv \nonumber \\ &4&\eta^2N_P^3 - 4\eta\Delta_P N_P^2 + (\overline{\Gamma}_P^2 + \Delta_P^2)N_P - |p|^2.
\end{eqnarray}
In the absence of SPM (when $\eta \to 0$, or when the input power is very small), $N_P$ is related to the incoming power by a simple linear function,
\begin{eqnarray}
N_P = \frac{|p|^2}{\overline{\Gamma}_P^2 + \Delta_P^2}.
\end{eqnarray}
The presence of SPM, however, complicates the task of determining $N_P$ as a function of $|p|^2$ for a given detuning $\Delta_P$ and nonlinearity $\eta$. The cubic equation (\ref{pump_cubic_root_eqn}) has in general as many as three real, positive roots. Furthermore, only some of these may correspond to \emph{stable} solutions of (\ref{intermediate_pump_eqn}). Before solving for the roots of $C_P(N_P)$, we first derive a set of criteria to assess the stability of any such solution.
To determine whether or not a given root of (\ref{pump_cubic}) is stable, we conduct an analysis similar to that of Hoff, Nielsen and Andersen \cite{Andersen2015}. For a given constant solution $\widetilde{\beta}_P^{(0)}$ to (\ref{pump_tilde_eqn}), we define the fluctuation amplitude $\delta\beta_P(t)$ via
\begin{eqnarray}
\widetilde{\beta}_P(t) = \widetilde{\beta}_P^{(0)} + \delta\beta_P(t).
\end{eqnarray}
Keeping terms up to first order in $\delta\beta_P$, the equations of motion satisfied by $\delta\beta_P(t)$ and $\delta\beta_P^*(t)$ can be written as
\begin{eqnarray}
\frac{d}{dt}\begin{pmatrix}
\delta\beta_P(t) \\ \delta\beta_P^*(t)
\end{pmatrix} = F\begin{pmatrix}
\delta\beta_P(t) \\ \delta\beta_P^*(t)
\end{pmatrix},
\end{eqnarray}
where $F$ is the $2\times 2$ matrix given by
\begin{eqnarray}
\lefteqn{F =}\nonumber \\ &&\begin{pmatrix}
-\overline{\Gamma}_P - i(4\eta N_P - \Delta_P) & -2i\eta\left[\widetilde{\beta}_P^{(0)}\right]^2 \\
2i\eta\left[\widetilde{\beta}_P^{(0)*}\right]^2 & -\overline{\Gamma}_P + i(4\eta N_P - \Delta_P)
\end{pmatrix}.
\end{eqnarray}
For a given solution to be stable, we require the real part of both eigenvalues of $F$ to be negative, so that the fluctuation term $\delta\beta_P(t)$ will decay with time. These eigenvalues are
\begin{eqnarray}
f_\pm = -\overline{\Gamma}_P \pm \sqrt{4\eta^2N_P^2 - (4\eta N_P - \Delta_P)^2}.
\end{eqnarray}
Now, $\mathrm{Re}(f_-)<0$ automatically; demanding that $\mathrm{Re}(f_+)<0$ yields the condition
\begin{eqnarray}
4\eta^2 N_P^2 - (4\eta N_P - \Delta_P)^2 < \overline{\Gamma}_P^2.
\end{eqnarray}
Solving this inequality, we find that any solution $N_P$ for Eq. (\ref{pump_cubic_root_eqn}) corresponds to a stable solution of Eq. (\ref{pump_tilde_eqn}) if $|\Delta_P|$ is below a ``critical detuning", $|\Delta_P| < \Delta_{\mathrm{critical}}$, where
\begin{eqnarray}
\Delta_{\mathrm{critical}} = \sqrt{3}\;\overline{\Gamma}_P.
\end{eqnarray}
When $|\Delta_P| > \Delta_{\mathrm{critical}}$, a solution $N_P$ of (\ref{pump_cubic_root_eqn}) corresponds to a stable solution of (\ref{pump_tilde_eqn}) if and only if $N_P$ lies outside a certain interval, $N_P \notin (N_-,N_+)$, where
\begin{eqnarray}\label{stability_eqn}
N_{\pm} = \frac{1}{3\eta}\left(\Delta_P \pm \frac{1}{2}\sqrt{\Delta_P^2 - \Delta_\mathrm{critical}^2}\right).
\end{eqnarray}
\begin{figure}
\includegraphics[width=1.0\columnwidth]{DZero.png}
\includegraphics[width=1.0\columnwidth]{DSub.png}
\includegraphics[width=1.0\columnwidth]{DSup.png}
\caption{(Colour online) Steady state average photon number in the ring pump mode as a function of channel input power with $\overline{\Gamma}_P=1\;\mathrm{GHz}$, $\eta=1\;\mathrm{Hz}$ for (a) zero detuning, (b) subcritical detuning, (c) supercritical detuning. Red and blue curves indicate stable solutions, green unstable. The dashed line respresents choice of optimal detuning to maximize $N_P$ at each input power ($\Delta_P=\Delta_P^\mathrm{opt}(N_P)=2\eta N_P$).}
\label{fig:photon_number}
\end{figure}
Having established the stability criteria for a given $N_P$, we return to the task of finding real, positive roots of (\ref{pump_cubic_root_eqn}). While analytic expressions for the roots exist, it is more instructive to use indirect arguments to study their nature. Taking the derivative of $C_P$ with respect to $N_P$, we find that $dC_P/dN_P=0$ at $N_P=N_{\pm}$. Thus, for subcritical detunings ($|\Delta_P| < \Delta_\mathrm{critical})$, where the $N_{\pm}$ are not purely real, there are no local extrema -- it is easy to show that a graph of $C_P(N_P)$ is monotonically increasing and intersects the $N_P$ axis only once, leading to a single real, positive root $N_P$ which corresponds to a stable solution. On the other hand, for supercritical detunings ($|\Delta_P| > \Delta_\mathrm{critical})$, the function $C_P(N_P)$ goes through a local maximum at $N_-$ and minimum at $N_+$. The number of times $C_P(N_P)$ intersects the $N_P$ axis is then determined by the power parameter $|p|^2$; varying it translates the graph of $C_P(N_P)$ vertically. If $C_P(N_-)>0$ and $C_P(N_+)<0$, the graph of the function intersects the $N_P$ axis three times, indicating the existence of three real, positive values of $N_P$. The outer two correspond to stable solutions, while the inner root is unstable. These multiple cases are illustrated in Fig. \ref{fig:photon_number}, in which $N_P$ is plotted as a function of input power for various detunings. For values of $|\Delta_P|$ above the critical detuning of $\sqrt{3}\;\overline{\Gamma}_P$ there exists a region of optical bistability, in which two stable equilibrium average pump photon numbers for a given input power are permitted, a phenomenon that has been observed experimentally in microring systems \cite{Almeida2004}. In this region the two stable solutions are separated by an unstable (and therefore physically inaccessible) range of $N_P$. Also plotted in this figure is the case of ``optimal detuning", $\Delta_P=\Delta_P^\mathrm{opt}(N_P)$, in which $\Delta_P$ is not taken to be fixed, but chosen to exactly cancel the effect of SPM as $P_\mathrm{in}$ is increased,
\begin{eqnarray}\label{optimal_detuning}
\Delta_P=\Delta_P^{\mathrm{opt}}(N_P) = 2\eta N_P,
\end{eqnarray}
which restores the simple linear relationship between $N_P$ and $|p|^2$,
\begin{eqnarray}
N_P = \frac{|p|^2}{\overline{\Gamma}_P^2}.
\end{eqnarray}
This behaviour is indicated by the dashed line in Fig. \ref{fig:photon_number}, which corresponds to a stable pump solution for all input powers, always lies on or above the fixed-detuning curves, and at each input power corresponds to the choice of detuning that maximizes $N_P$.
\section{Signal and idler dynamics}\label{sec:signal_idler_dynamics}
Having developed the steady state pump solution, we return to the signal and idler equations of motion. We first develop an exact solution to these equations, valid for a cw pump of arbitrary intensity, and then use this solution to calculate the photon pair generation rate, as well as the one- and two-photon spectra of the generated photons.
\subsection{Exact solution}
We begin by writing the equations (\ref{ring_signal_master}--\ref{ring_idler_master}) for the signal and idler ring operators in the presence of a classically described cw pump that leads to a ring pump amplitude of the form $\overline{\beta}_P(t) = \overline{\beta}_P e^{-i\Delta_P t}$, where $\overline{\beta}_P$ is a constant. Letting $\widetilde{b}_x(t) = e^{i\Delta_P t}\overline{b}_x(t)$ for $x=S,I$ we obtain
\begin{eqnarray}
\frac{d}{dt}\begin{pmatrix}
\widetilde{b}_S(t) \\ \widetilde{b}_I^\dagger(t)
\end{pmatrix} = M \begin{pmatrix}
\widetilde{b}_S(t) \\ \widetilde{b}_I^\dagger(t)
\end{pmatrix} + D(t),
\end{eqnarray}
where $M$ is the $2\times 2$ coupling matrix defined by
\begin{eqnarray}\label{M_defn}
\lefteqn{M =} \nonumber \\
& &\begin{pmatrix}
-\overline{\Gamma}_S - i(\zeta|\overline{\beta}_P|^2 - \Delta_P) & -i\Lambda \overline{\beta}_P^2 \\
i\Lambda\overline{\beta}_P^{*2} & -\overline{\Gamma}_I + i(\zeta|\overline{\beta}_P|^2 - \Delta_P)
\end{pmatrix},\;\;\;
\end{eqnarray}
and $D(t)$ the driving term responsible for quantum fluctuations from the physical and phantom channels,
\begin{eqnarray}
D(t) = \begin{pmatrix}
-ie^{i\Delta_P t}(\gamma_S^*\overline{\psi}_{S<}(0,t) + \mu_S^*\overline{\phi}_{S<}(0,t)) \\
ie^{-i\Delta_P t}(\gamma_I\overline{\psi}_{I<}^\dagger(0,t) + \mu_I\overline{\phi}_{I<}^\dagger(0,t)) \\
\end{pmatrix}.
\end{eqnarray}
In obtaining $M$ we have assumed the ring resonances are equally spaced, so that $\Delta_\mathrm{ring}=\omega_S + \omega_I - 2\omega_P=0$; the pump detuning, however, is left arbitrary. Previously \cite{Vernon2015} we employed a perturbative approach in the frequency domain to solve these equations, while ignoring the effects of SPM and XPM. While this provides an adequate description of the pair generation process for low pump powers, a nonperturbative strategy is needed to treat the strongly driven case. In the cw regime, where $M$ is time-independent, this coupled set of linear ordinary differential equations can be solved exactly in the time domain for arbitrary pump intensities by taking
\begin{eqnarray}
\begin{pmatrix}
\widetilde{b}_S(t) \\ \widetilde{b}_I^\dagger(t)
\end{pmatrix} = \int\limits_{-\infty}^{\;\;t} dt' G(t,t') D(t'),
\end{eqnarray}
where the $2\times 2$ matrix Green function $G(t,t')$ is given by
\begin{eqnarray}
G(t,t') &=& e^{\int_{t'}^t M dt''} = e^{M\cdot(t-t')} \nonumber \\
&=&\begin{pmatrix}
g_D(t,t') & g_A(t,t') \\ g_A^*(t,t') & g_D^*(t,t')
\end{pmatrix}.
\end{eqnarray}
For simplicity we henceforth assume the ring-channel coupling constants and propagation speeds for each mode are the same, $\gamma_J=\gamma$, $v_J=u_J=v$ and $\mu_J=\mu$ so $\overline{\Gamma}_J=\overline{\Gamma}$ for each $J$. The matrix elements $g_D$ and $g_A$ are then given by
\begin{eqnarray}
&g_D&(t,t') = e^{-\overline{\Gamma}(t-t')} \\
\;\;\;\;&\times& \bigg(\cosh[\overline{\rho}(t-t')] - i\frac{\zeta N_P - \Delta_P}{\overline{\rho}}\sinh[\overline{\rho}(t-t')]\bigg) \nonumber
\end{eqnarray}
and
\begin{eqnarray}
g_A(t,t') = \frac{-i\Lambda\overline{\beta}_P^2}{\overline{\rho}}\sinh[\overline{\rho}(t-t')],
\end{eqnarray}
in which we have introduced the dynamical parameter $\overline{\rho}$,
\begin{eqnarray}\label{rhobar_defn}
\overline{\rho} = \sqrt{\Lambda^2 N_P^2 - (\zeta N_P - \Delta_P)^2}.
\end{eqnarray}
Depending on the pump photon number $N_P$ and detuning $\Delta_P$, $\overline{\rho}$ may be either purely real, purely imaginary, or exactly zero. Indeed, as will become clear in the following sections, $\overline{\rho}$ serves as an important parameter in characterizing the system's behaviour.
With explicit solutions written down for the ring operators $\widetilde{b}_J(t)$, we can make use of the incoming-outgoing channel field relation (\ref{channel_transformation}) to determine $\overline{\psi}_{J>}(0,t)$. We find for the signal
\begin{eqnarray}
\lefteqn{\overline{\psi}_{S>}(0,t) =} \\
\int dt' &\bigg[&q_{SS}(t,t')\overline{\psi}_{S<}(0,t') + p_{SS}(t,t')\overline{\phi}_{S<}(0,t') \nonumber \\
&+& q_{SI}(t,t')\overline{\psi}_{I<}^\dagger(0,t') + p_{SI}(t,t')\overline{\phi}_{I<}^\dagger(0,t')\bigg], \nonumber
\end{eqnarray}
where we have introduced the temporal response functions $q_{xx'}(t,t')$ for the physical channel and $p_{xx'}(t,t')$ for the phantom channel:
\begin{eqnarray}
& &q_{SS}(t,t') = \delta(t-t') \nonumber \\
&-& \frac{|\gamma|^2}{v}\theta(t-t')e^{-(\overline{\Gamma} + i\Delta_P)(t-t')} \\
&\times& [\cosh[\overline{\rho}(t-t')] - i\frac{\zeta|\overline{\beta}_P|^2 - \Delta_P}{\overline{\rho}}\sinh[\overline{\rho}(t-t')], \nonumber
\end{eqnarray}
and
\begin{eqnarray}
\lefteqn{q_{SI}(t,t') =} \\
& &\frac{-\gamma^2\Lambda\overline{\beta}_P^2}{v\overline{\rho}}\theta(t-t')e^{-i\Delta_P(t+t')}e^{-\overline{\Gamma}(t-t')}\sinh[\overline{\rho}(t-t')]. \nonumber
\end{eqnarray}
The phantom channel response functions are related to these via
\begin{eqnarray}
p_{SS}(t,t') = \frac{\mu^*}{\gamma^*}(q_{SS}(t,t') - \delta(t-t'))
\end{eqnarray}
and
\begin{eqnarray}
p_{SI}(t,t') = \frac{\mu}{\gamma}q_{SI}(t,t').
\end{eqnarray}
Similar response functions $p_{Ix}(t,t')$ and $q_{Ix}(t,t')$ can be introduced for the idler fields, which, due to our assumption of equal coupling coefficients and propagation speeds for the signal and idler fields, are identical to those for the signal: $p_{IS} = p_{SI}$, $p_{II}=p_{SS}$, $q_{IS}=q_{SI}$ and $q_{II}=q_{SS}$.
\subsection{Photon generation rate}\label{sec::pair_gen_rates}
Armed with explicit expressions for the outgoing fields $\overline{\psi}_{S>}$ and $\overline{\psi}_{I>}$, we can calculate any measurable quantity related to the generated signal and idler photon pairs. Of particular interest is the photon pair generation rate, one of the primary figures of merit used in assessing the practical utility of the ring-channel system. The steady state outgoing flux of signal photons $J_S$ into the physical channel can be calculated via
\begin{eqnarray}
J_S &=& \lim_{t\to\infty} v \langle \overline{\psi}_{S>}^\dagger(0,t)\overline{\psi}_{S>}(0,t)\rangle \nonumber \\
&=& \lim_{t\to\infty} \frac{2\overline{\Gamma}}{\Gamma}\int dt' |q_{SI}(t,t')|^2.
\end{eqnarray}
Computing the integral, we find
\begin{eqnarray}
J_S = \frac{2\Gamma\Lambda^2 N_P^2}{\overline{\Gamma}^2 - \overline{\rho}^2}.
\end{eqnarray}
The nature of the scaling of $J_S$ with pump photon number $N_P$ depends intimately on the character of $\overline{\rho}$, the behaviour of which as a function of $N_P$ for various detunings is illustrated in Fig. \ref{fig::rhobar_plot}. Recalling (\ref{rhobar_defn}), we find that $\overline{\rho}$ is real when $N_P \in [\Delta_P/3\Lambda,\Delta_P/\Lambda]$, with $\overline{\rho}=0$ at the endpoints of this interval, and imaginary otherwise.
\begin{figure}
\includegraphics[width=1.0\columnwidth]{Rhobar_plot.png}
\caption{(Colour online) Real and imaginary parts of $\overline{\rho}$ as a function of $N_P$ for zero, subcritical and supercritical detunings, indicating that $\overline{\rho}$ is always either purely real or purely imaginary. The red line indicates $\overline{\rho}=\overline{\Gamma}$. The transition between $\overline{\rho}$ being purely imaginary and purely real occurs at the points $N_P=\Delta_P/3\Lambda$ and $N_P=\Delta_P/\Lambda$. For supercritical detunings, there exist points where $\overline{\rho}=\overline{\Gamma}$ (represented on the plot by red diamonds), indicating the onset of OPO behaviour. The nonlinear parameter $\Lambda$ taken as $\Lambda=10$ Hz.}\label{fig::rhobar_plot}
\end{figure}
\begin{figure}
\includegraphics[width=1.0\columnwidth]{JSplot.png}
\caption{(Colour online) Photon pair generation rate as a function of channel input power for various detunings. The dashed curve indicates the optimal detuning case. System parameters for this plot are $\eta=1\;\mathrm{Hz}$, $\overline{\Gamma}=1\;\mathrm{GHz}$.}\label{fig:pair_rate_plot}
\end{figure}
For low enough $N_P$, when $\overline{\rho} \approx i|\Delta_P|$, $J_S$ scales quadratically with the number of pump photons $N_P$. Since in this regime $N_P$ is directly proportional to the channel input power $P_{\mathrm{in}}$, the overall scaling of $J_S$ with $P_{\mathrm{in}}$ remains quadratic, in agreement with experiment \cite{Azzini2012}. As the pump power increases, however, the scaling of $J_S$ is affected by several separate power-dependent processes.
First, for a fixed detuning $\Delta_P$, the SPM-induced drift of the pump resonance slows the scaling of $N_P$ with channel input power $P_\mathrm{in}$, as demonstrated in Fig. \ref{fig:photon_number}. Second, XPM between the pump, signal and idler modes effectively shifts the resonance lines of the signal and idler modes, compromising the resonance enhancement of the pair generation process. Finally, for supercritical detunings $|\Delta_P|>\Delta_\mathrm{critical}$, $\overline{\rho}\to\overline{\Gamma}$ when $N_P\to N_{\pm}$, where $N_{\pm}$ are the same two photon numbers that define the stability of the pump solution (\ref{stability_eqn}). In that limit the photon flux $J_S$ formally diverges. This unphysical prediction corresponds to the onset of optical parametric oscillation\cite{Levy2010,Razzari2010}. As this threshold is approached, stimulated emission leads to photon pairs being generated faster than the rate at which they are removed from the ring, preventing the system from reaching a steady state within our model. We are prevented from treating this case by our assumption of an undepleted pump. Our results are expected to be valid when the intraring conversion efficiency $E_\mathrm{ring}$ is much less than unity; this efficiency, defined as the ratio between the steady state signal (or idler) and pump photon numbers, can be expressed as
\begin{eqnarray}
E_\mathrm{ring} = \frac{J_S}{2\Gamma N_P} = \frac{\Lambda^2N_P}{\overline{\Gamma}^2-\overline{\rho}^2}.
\end{eqnarray}
While in future work we intend to investigate the OPO regime and the associated effects of pump depletion, for the time being we restrict ourselves to regimes where $E_\mathrm{ring} \ll 1 $; in all examples presented below this inequality is satisfied.
Perhaps most remarkable is the special regime of optimal detuning, wherein $\Delta_P$ is chosen to maximize $N_P$ at each channel input power, $\Delta_P=\Delta_P^\mathrm{opt}(N_P)$ as defined in Eq. (\ref{optimal_detuning}). For this choice $\overline{\rho}=0$ identically for all $N_P$,
\begin{eqnarray}\label{rhobar_special}
\overline{\rho} &=& \sqrt{\Lambda^2 N_P^2 - (\zeta N_P - \Delta_P)^2} \nonumber \\
&=& \sqrt{\Lambda^2 N_P^2 - \left(2\Lambda N_P - 2\frac{\Lambda}{2}N_P\right)^2} \nonumber \\
&=& 0.
\end{eqnarray}
The photon pair flux then maintains its quadratic scaling with both $N_P$ and channel input power $P_{\mathrm{in}}$ over its entire domain:
\begin{eqnarray}
J_S = \frac{8\Gamma^3\Lambda^2}{(\hbar\omega_P)^2\overline{\Gamma}^6}P_{\mathrm{in}}^2.
\end{eqnarray}
This cancellation between the effects arising from photon pair generation, XPM, and the SPM-dependent detuning strategy $\Delta_P^\mathrm{opt}(N_P)$ arises from the simple fixed relationship between the associated nonlinear coupling strengths $\Lambda$, $\eta$ and $\zeta$. Crucial for this phenomenon is that the strength of the photon pair generation process scale quadratically with the pump photon number $N_P$. This cancellation effect would therefore not be possible using, for example, spontaneous parametric downconversion, the strength of which would scale linearly with $N_P$. The presence of thermal resonance drift would not compromise the existence of an $N_P$-dependent detuning strategy that yields $\overline{\rho}=0$ over all $N_P$, though such a strategy would no longer correspond to that which also linearizes and maximizes the relationship between $N_P$ and channel input power.
The photon pair generation rate as a function of channel input power is plotted in Fig. \ref{fig:pair_rate_plot} for various values of $\Delta_P$ alongside this optimal detuning case. For lower powers, when SPM and XPM are negligible, a pump beam with $\Delta_P=0$ gives the best scaling of $J_S$. For intermediate powers the detuning may be tweaked to combat SPM and XPM in order to maximize $J_S$, while for high powers the optimal detuning strategy of $\Delta_P=\Delta_P^\mathrm{opt}(N_P)$ beats any fixed subcritical detuning. The behaviour of these curves suggests a simple experiment to identify the presence of nonperturbative, strongly driven effects: one could simply measure the outgoing signal or idler power as a function of pump input power for a set of fixed, subcritical pump detunings. For fixed nonzero detunings $\Delta_P<\Delta_\mathrm{critical}$, strongly driven effects are indicated by the presence of a global maximum of generated signal power at intermediate pump input power, followed by decreasing signal power approaching an asymptotic value of
\begin{eqnarray}
\hbar\omega_S\lim_{N_P\to\infty} J_S(N_P) = \hbar\omega_S\frac{2\Gamma}{3}.
\end{eqnarray}
For critically coupled ring systems $\Gamma\approx\overline{\Gamma}/2$ \cite{Vernon2015}, so the asymptotic signal power can be related to the total effective ring linewidth $\overline{\Gamma}$ as simply $\hbar\omega_S\overline{\Gamma}/3$. If thermal detuning of the ring resonances is included this asymptotic power will be different; however, the qualitative behaviour of the signal power as a function of pump power will be unchanged.
\subsection{Single photon spectrum}
Another physical quantity of interest is the spectral lineshape of signal and idler photons that are emitted from the ring. For low power cw pumps these single photon spectra typically exhibit a Lorentzian lineshape \cite{Azzini2012,Azzini2012a,Vernon2015} with a characteristic width determined by the total effective linewidths of the microring cavity resonances. As we now demonstrate, these spectral characteristics are significantly different in the strongly pumped regime. We develop results for the signal field spectrum; the idler field will have identical properties.
\begin{figure}
\includegraphics[width=1.0\columnwidth]{Splitting.png}
\caption{(Colour online) Origin of splitting in the signal and idler lineshapes. Green, blue, and red curves indicate pump, signal and idler resonances, respectively, and could respresent the enhancement factor \cite{Heebner2008} that would characterize the ratio of the intensity in the ring to the incident channel intensity in a linear experiment. The dashed green line indicates a pump detuned by $\Delta_P$, which leads to a generated pair having either its signal or idler photon detuned by $\sim 2\Delta_P$ from the corresponding resonance, as indicated by the signal and idler pairs connected by dashed black lines. The presence of pairs from both cases leads to a doublet structure for both the signal and idler lineshapes.}\label{fig::splitting}
\end{figure}
We define the power spectrum \cite{Mandel1995} for the signal channel field as
\begin{eqnarray}\label{nu_S_defn1}
\lefteqn{\nu_S(\omega_s; t) =} \\
& & \lim_{T\to\infty}\frac{1}{T}\int\displaylimits_{t-T/2}^{t+T/2} dt \int \frac{d\tau}{\sqrt{2\pi}} g^{(1)}(t,t+\tau) e^{i\omega_s \tau} \nonumber
\end{eqnarray}
where the first-order temporal coherence function $g^{(1)}(t_1,t_2)$ is defined by
\begin{eqnarray} \label{g1_defn}
g^{(1)}(t_1,t_2) &=& v \langle \overline{\psi}_{S>}^\dagger(0,t_1)\overline{\psi}_{S>}(0,t_2)\rangle \nonumber \\
&=& \frac{\overline{\Gamma}}{\Gamma}\int dt' q_{SI}^*(t_1,t')q_{SI}(t_2,t').
\end{eqnarray}
In writing (\ref{nu_S_defn1}) we have introduced the relative frequency co-ordinate $\omega_s$, which corresponds to a frequency offset from the ring reference $\omega_S$. The physical frequency $\overline{\omega}_s$ associated with $\omega_s$ is therefore
\begin{eqnarray}
\overline{\omega}_s = \omega_S + \omega_s.
\end{eqnarray}
In the remaining sections we adopt this notation of lowercase subscripts for frequency offsets: $\omega_s$ for the signal, $\omega_p$ for the pump and $\omega_i$ for the idler.
\begin{figure*}
\includegraphics[width=1.0\columnwidth]{LineshapeZeroDetuning.png}
\includegraphics[width=1.0\columnwidth]{LineshapeSomeDetuning.png}
\includegraphics[width=1.0\columnwidth]{LineshapeSupercriticalDetuning.png}
\includegraphics[width=1.0\columnwidth]{LineshapeOptimalDetuning.png}
\caption{(Colour online) Spectral lineshape $\nu_S(\omega_s)$ scaled to unit maximum vs. ring pump photon number $N_P$. For each plot we take $\overline{\Gamma}=1$ GHz and $\Lambda=10$ Hz. (a) $\Delta_P=0$, (b) $\Delta_P=0.8\Delta_\mathrm{critical}\approx 1.4$ GHz, (c) $\Delta_P=1.5\Delta_\mathrm{critical}\approx2.6$ GHz, and (d) $\Delta_P=\Delta_P^\mathrm{opt}(N_P)=2\eta N_P$. The origin of the frequency axis corresponds to the ring resonance at $\omega_S$. }\label{fig::lineshapes}
\end{figure*}
Evaluating (\ref{g1_defn}) and setting $t_1=t$, $t_2=t+\tau$ we obtain
\begin{eqnarray}
\lefteqn{g^{(1)}(t,t+\tau) =}\\
& & \frac{\Gamma\Lambda^2 N_P^2}{\overline{\rho}}e^{i\Delta\tau}e^{-\overline{\Gamma} |\tau|}\frac{\overline{\rho}\cosh[\overline{\rho}|\tau|] + \overline{\Gamma}\sinh[\overline{\rho}|\tau|]}{\overline{\Gamma}^2 - \overline{\rho}^2}\nonumber,
\end{eqnarray}
which is independent of $t$, depending only on the relative time difference $\tau$, as would be expected for a cw pump. Taking the Fourier transform, we arrive at an expression for the lineshape,
\begin{eqnarray}\label{nu_S_defn}
\lefteqn{\nu_S(\omega_s)=} \\
& & \frac{4\Gamma\overline{\Gamma} \Lambda^2 N_P^2}{\sqrt{2\pi}|\overline{\Gamma} - \overline{\rho} +i(\omega_s - \Delta_P)|^2|\overline{\Gamma}+\overline{\rho}+i(\omega_s-\Delta_P)|^2} \nonumber,
\end{eqnarray}
with an identical equation for the idler lineshape $\nu_I(\omega_i)$. This expression takes the form of a product of two Lorentzians. We consider first subcritical detunings. When $\overline{\rho}$ is imaginary, these Lorentzians have identical characteristic widths $\delta\omega=\overline{\Gamma}$ and are centred on $\omega_s=\Delta_P\pm|\overline{\rho}|$. For low powers, when $\overline{\rho} \approx i|\Delta_P|$ the spectrum is therefore peaked at $\omega_s=0$ and $\omega_s=2\Delta_P$, in agreement with the perturbative calculation \cite{Vernon2015}. This splitting is easily understood as a consequence of the tradeoff between energy conservation and resonance enhancement of the pair generation process. As illustrated in Fig. \ref{fig::splitting}, when a photon pair is produced with a detuned pump, either the signal photon \emph{or} idler photon in a pair, but not both, can be generated within a ring resonance; energy conservation then requires the other to be generated with a frequency that lies away from its corresponding resonance. This is seen in the $N_P\to 0$ limit of Fig. \ref{fig::lineshapes}(b). At sufficient pump photon number $N_P$ a similar splitting can arise from the effective XPM-induced detuning of the signal and idler ring resonances even for a pump with $\Delta_P=0$, as seen in Fig. \ref{fig::lineshapes}(a) for large $N_P$. When $\Delta_P=0$ the lineshape begins as a singly-peaked Lorentzian, eventually splitting to a doublet structure when $\overline{\rho}$ becomes imaginary as a consequence of XPM.
For nonzero $\Delta_P$, as $N_P$ increases, XPM effectively counters the pump detuning and the extent of this splitting is reduced as $|\overline{\rho}|$ decreases, eventually vanishing when $N_P=\Delta_P/3\Lambda$. If $N_P$ is increased further, $\overline{\rho}$ becomes real and ceases to contribute to spectral splitting, resulting instead in an effective correction to the linewidth. The lineshape then takes the form of a product of two Lorentzians, both centred on $\omega_s=\Delta_P$, with respective widths $\delta\omega_\pm = \overline{\Gamma} \pm \overline{\rho}$. As $\overline{\rho}$ becomes comparable to $\overline{\Gamma}$ the smaller of these two widths becomes dominant, leading to a lineshape with overall effective width $\delta\omega\approx\overline{\Gamma}-\overline{\rho}$. For subcritical detunings, as demonstrated at large $N_P$ in Fig. \ref{fig::lineshapes}(b), the spectral splitting is then resumed as $\overline{\rho}$ once again becomes imaginary. \
For supercritical detunings, as the threshold for optical parametric oscillation is approached $\overline{\rho}\to\overline{\Gamma}$ and the bandwidth of the emitted signal and idler photons becomes arbitrarily narrow, as seen in Fig. \ref{fig::lineshapes}(c). This follows from our idealization of the pump as an indefinitely coherent cw beam; in actual experiments the bandwidth of the generated photons will become comparable to that of the pump, a phenomenon that has been observed in strongly pumped experiments on silicon nitride microrings \cite{Levy2010}.
Finally, as shown in Fig. \ref{fig::lineshapes}(d), in the special case of optimal detuning when $\Delta_P=\Delta_P^\mathrm{opt}(N_P)$, so that $\overline{\rho}=0$, the lineshape remains peaked at a single $N_P$-dependent frequency for each $N_P$, with unchanging characteristic width $\delta\omega=\overline{\Gamma}$, precisely mimicking the low-power result at zero detuning.
Experimentally, measuring the signal or idler lineshape as a function of input power for a nonzero, subcritical detuning as in Fig. \ref{fig::lineshapes}(b) would reveal the richness of the strongly driven regime, and illustrate the behaviour of the $\overline{\rho}$ paramater, which incorporates the effects of both XPM and pair generation.
\subsection{Joint spectral intensity}
To assess the degree of spectral correlation between the signal and idler modes, it is instructive to study the joint spectral intensity distribution of the generated photon pairs. While it is straightforward to define this quantity for a system driven by a train of weak pump pulses, in which multi-pair generation can be neglected, it is a more subtle task to craft a sensible measure of spectral correlation in the strongly driven cw regime. In particular, there is no single function that characterizes a joint probability amplitude of signal and idler photons, since in general there will be far more than two photons in the quantum state of the signal and idler modes. Furthermore, even for weak cw pumps, if one introduces outgoing channel annihilation operators $c_J(\omega_j)$ via
\begin{eqnarray}\label{amplitude_defn}
\overline{\psi}_{J>}(0,t) = \int \frac{d\omega_j}{\sqrt{2\pi}}c_J(\omega_j)e^{-i\omega_j t}
\end{eqnarray}
and naively calculates expectation values of the form
$\langle c_S^\dagger(\omega_s)c_I^\dagger(\omega_i)c_S(\omega_s)c_I(\omega_i)\rangle$, the idealization of a zero-bandwidth cw pump leads to ill-defined expressions involving the square of Dirac $\delta$ distributions.
To resolve these difficulties, in Appendix \ref{appendix:JSI} we develop a model of a typical experiment used to characterize the JSI for weakly driven systems, in which the coincidence count rate of signal and idler photons at respective frequencies $\omega_s$ and $\omega_i$ is measured. We the extend the definition of the JSI to strongly driven systems by defining the JSI to equal the calculated outcome of such an experiment for arbitrary input power. This definition reduces to the usual result for single-pair output states, and serves as a sensible measure of spectral correlation between the signal and idler fields. This coincidence rate can be written as
\begin{eqnarray}\label{JSI_definition}
I_\mathrm{corr}(\omega_s,\omega_i) &=& \frac{v^2\delta t}{(2\pi)^2}\int dt_1...\int dt_4 \\
&\bigg[& e^{i\omega_s(t_3-t_1)}e^{i\omega_i(t_4-t_2)}T(t_1)T(t_2)T(t_3)T(t_4)\nonumber \\
&\times&\langle \overline{\psi}_{S>}^\dagger(t_1)\overline{\psi}_{I>}^\dagger(t_2)\overline{\psi}_{S>}(t_3)\overline{\psi}_{I>}(t_4)\rangle\bigg],\nonumber
\end{eqnarray}
where $T(t)$ is the Fourier transform of a transmission function $\hat{T}(\omega)$ that resolves the frequencies of the signal and idler photons prior to detection,
\begin{eqnarray}\label{T_defn}
T(t) = \int \frac{d\omega}{\sqrt{2\pi}}\hat{T}(\omega)e^{-i\omega t},
\end{eqnarray}
and $\delta t$ is the temporal resolution of the coincidence counter. In this expression the spatial dependence of the field operators $\overline{\psi}_{J>}(z,t)$ has been suppressed; the signal and idler arms of the JSI measurement are assumed to occur at balanced distances from the ring-channel coupling point.
The four-time expectation value $\langle \overline{\psi}_{S>}^\dagger(t_1)\overline{\psi}_{I>}^\dagger(t_2)\overline{\psi}_{S>}(t_3)\overline{\psi}_{I>}(t_4)\rangle$ is found to naturally split into two parts,
\begin{eqnarray}
&v&^2\langle \overline{\psi}_{S>}^\dagger(t_1)\overline{\psi}_{I>}^\dagger(t_2)\overline{\psi}_{S>}(t_3)\overline{\psi}_{I>}(t_4)\rangle = \\
& &A^*(t_1,t_2)A(t_3,t_4) + g^{(1)}(t_1,t_3)g^{(1)}(t_2,t_4),\nonumber
\end{eqnarray}
where
\begin{eqnarray}
A(t_1,t_2) = \int dt'& \bigg[&q_{SI}(t_1,t')q_{II}(t_2,t') \\
&+& p_{SI}(t_1,t')p_{II}(t_2,t')\bigg].\nonumber
\end{eqnarray}
The function $g^{(1)}$ is precisely the first-order coherence function defined in Eq. (\ref{g1_defn}) used to calculate the single photon spectrum,
\begin{eqnarray}
g^{(1)}(t_1,t_3) &=& \frac{\Gamma\Lambda^2 N_P^2}{\overline{\rho}}e^{i\Delta_P(t_3-t_1)}e^{-\overline{\Gamma}|t_3-t_1|} \\
&\times&\frac{\overline{\rho}\cosh[\overline{\rho}|t_3-t_1|] + \overline{\Gamma}\sinh[\overline{\rho}|t_3-t_1|]}{\overline{\Gamma}^2-\overline{\rho}^2}.\nonumber
\end{eqnarray}
The $A(t_1,t_2)$ term, after computing the integrals, is given by
\begin{eqnarray}
A(t_1,t_2)&=&\frac{\gamma^2\Lambda\overline{\beta}_P^2}{2v}e^{-\overline{\Gamma}|t_2-t_1|}e^{-i\Delta_P(t_1+t_2)} \\
&\times&\frac{[a_1\sinh[\overline{\rho}|t_2-t_1|] + a_2\cosh[\overline{\rho}|t_2-t_1|}{\overline{\Gamma}^2-\overline{\rho}^2},\nonumber
\end{eqnarray}
where the constants $a_1$ and $a_2$ are defined by
\begin{eqnarray}
a_1 &=& \overline{\rho} - i\frac{\zeta N_P -\Delta_P}{\overline{\rho}}\overline{\Gamma}, \nonumber \\
a_2 &=& \overline{\Gamma} -i(\zeta N_P - \Delta_P).
\end{eqnarray}
The JSI can therefore be expressed as the sum of correlated and uncorrelated terms,
\begin{eqnarray}
I(\omega_s,\omega_i) = I_{\mathrm{corr}}(\omega_s,\omega_i) + I_\mathrm{uncorr}(\omega_s,\omega_i),
\end{eqnarray}
where
\begin{eqnarray}\label{phi_corr_defn}
\lefteqn{I_{\mathrm{corr}} (\omega_s,\omega_i) =} \\
& & \frac{\delta t}{(2\pi)^2}\bigg\vert\int d\nu_1 \int d\nu_2\hat{A}(\nu_1,\nu_2) \hat{T}(\omega_s-\nu_1)\hat{T}(\omega_i-\nu_2)\bigg \vert^2 \nonumber
\end{eqnarray}
and
\begin{eqnarray}\label{phi_uncorr_defn}
\lefteqn{I_\mathrm{uncorr}(\omega_s,\omega_i) =\frac{\delta t}{(2\pi)^2}} \\
&\times&\int d\nu_1 \int d\nu_2\left[\hat{g}^{(1)}(\nu_1,-\nu_2)\hat{T}(\omega_s-\nu_1)\hat{T}(\omega_s - \nu_2)\right] \nonumber \\
&\times& \int d\nu_1' \int d\nu_2'\left[\hat{g}^{(1)}(\nu_1',-\nu_2')\hat{T}(\omega_i-\nu_1')\hat{T}(\omega_i - \nu_2')\right]. \nonumber
\end{eqnarray}
As indicated by their labels, $I_\mathrm{uncorr}$ can be expressed as a separable product of functions of $\omega_s$ and $\omega_i$, while $I_\mathrm{corr}$ cannot. Each is expressed as a convolution of the Fourier transforms $\hat{A}(\nu_1,\nu_2)$ and $\hat{g}^{(1)}(\nu_1,\nu_2)$ of the $A(t_1,t_2)$ and $g^{(1)}(t_1,t_2)$ functions,
\begin{eqnarray}
\hat{A}(t_1,t_2)=\int \frac{dt_1}{\sqrt{2\pi}} \int \frac{dt_2}{\sqrt{2\pi}}A(t_1,t_2)e^{i\nu_1t_1}e^{i\nu_2t_2},
\end{eqnarray}
and similarly for $g^{(1)}(\nu_1,\nu_2)$, with the transmission filter function $\hat{T}(\nu_1)\hat{T}(\nu_2)$. The $A$ and $g^{(1)}$ functions are determined by the dynamics of the signal and idler modes in the ring, while their convolution with the $T$ functions reflects the frequency averaging that arises from the finite resolution of a realistic JSI measurement.
Computing the Fourier transform, we find for $\hat{A}$
\begin{widetext}
\begin{eqnarray}
\hat{A}(\nu_1,\nu_2) = \Gamma^2\Lambda^2 N_P^2 \delta(\nu_1+\nu_2-2\Delta_P)\left(\frac{1-i\frac{\zeta N_P-\Delta_P}{\overline{\rho}}}{(i\Delta\nu - \overline{\Gamma} + \overline{\rho})(-i\Delta\nu-\overline{\Gamma}+\overline{\rho})} + \frac{1+i\frac{\zeta N_P-\Delta_P}{\overline{\rho}}}{(i\Delta\nu - \overline{\Gamma} - \overline{\rho})(-i\Delta\nu-\overline{\Gamma}-\overline{\rho})}\right),
\end{eqnarray}
with $\Delta\nu = (\nu_1 - \nu_2)/2$. The term in parentheses multiplying the $\delta$ function varies on the scale of $\overline{\Gamma}$. Assuming that the measurement frequency resolution $\delta\omega_\mathrm{trans}$ is much narrower than this, the slowly varying term can be pulled out of the integrals in (\ref{phi_corr_defn}), leaving
\begin{eqnarray}\label{phi_corr_expression}
I_\mathrm{corr}(\omega_s,\omega_i) &\approx& \delta t\Gamma^2\Lambda^2N_P^2 [D(\omega_s-\Delta_P,\omega_i-\Delta_P)]^2 \\
&\times& \bigg\vert\frac{1-i\frac{\zeta N_P-\Delta_P}{\overline{\rho}}}{(i(\omega_i-\Delta_P) - \overline{\Gamma} + \overline{\rho})(-i(\omega_i-\Delta_P)-\overline{\Gamma}+\overline{\rho})} + \frac{1+i\frac{\zeta N_P-\Delta_P}{\overline{\rho}}}{(i(\omega_i-\Delta_P) - \overline{\Gamma} - \overline{\rho})(-i(\omega_i-\Delta_P)-\overline{\Gamma}-\overline{\rho})}\bigg\vert^2 \nonumber
\end{eqnarray}
\end{widetext}
where
\begin{eqnarray}
\lefteqn{D(\omega_s,\omega_i) =} \\
& &\frac{1}{2\pi}\int d\nu_1 \int d\nu_2 \delta(\nu_1+\nu_2)\hat{T}(\omega_s-\nu_1)\hat{T}(\omega_i-\nu_2).\nonumber
\end{eqnarray}
The function $D(\omega_s,\omega_i)$ can be interpreted as the ``smoothed" version of the Dirac $\delta(\omega_s + \omega_i)$ distribution, and arises from the finite bandwidth of the JSI measurement scheme; $D(\omega_s-\Delta_P,\omega_i-\Delta_P)$ is sharply peaked and uniform along the energy-conserving antidiagonal line $\omega_s+\omega_i-2\Delta_P=0$ with characteristic width $\delta\omega_\mathrm{trans}$ (the measurement resolution) in the direction orthogonal to that line.
Finally, taking the Fourier transform of $g^{(1)}(t_1,t_3)$, we find
\begin{eqnarray}
\lefteqn{\hat{g}^{(1)}(\nu_1,-\nu_2) = \delta(\nu_1-\nu_2) }\\
&\times& \frac{4\overline{\Gamma}\Gamma\Lambda^2N_P^2}{|\overline{\Gamma}-\overline{\rho}+i(\nu_1-\Delta_P)|^2|\overline{\Gamma}+\overline{\rho}+i(\nu_1-\Delta_P)|^2}.\nonumber
\end{eqnarray}
As with $\hat{A}$, apart from the $\delta$ function this is slowly varying compared to the measurement resolution; the term multiplying the $\delta$ function can be pulled out of the integral in Eq. (\ref{phi_uncorr_defn}). The uncorrelated contribution to the JSI $I_\mathrm{uncorr}$ is therefore well approximated by
\begin{eqnarray}\label{phi_uncorr_expression}
I_\mathrm{uncorr}(\omega_s,\omega_i)\approx \frac{\delta t}{2\pi}\bigg\vert \int d\omega|\hat{T}(\omega)|^2\bigg\vert^2\nu_S(\omega_s)\nu_I(\omega_i),\;\;\;
\end{eqnarray}
where $\nu_S(\omega_s)$ and $\nu_I(\omega_i)$ are precisely the single-photon lineshape functions given by Eq. (\ref{nu_S_defn}) as derived in the previous section. The uncorrelated part of the JSI is therefore proportional to the simple product the signal and idler lineshapes.
For low power cw pumps, wherein multi-pair generation is insignificant, the uncorrelated part of the JSI $I_\mathrm{uncorr}$ is negligible and $I_\mathrm{corr}$ dominates. The JSI then takes the form of a narrow antidiagonal line corresponding to the energy-conserving condition $\omega_s+\omega_i-2\Delta_P=0$. For $\Delta_P=0$, the line is singly peaked, as illustrated in Fig. \ref{fig::JSI_splitting_plots}(a). For nonzero $\Delta_P$ the line is distributed among two peaks separated by $2\Delta_P$, as evident in Fig. \ref{fig::JSI_splitting_plots}(b), consistent with the single photon spectrum derived in the previous section. At higher powers, such as in Fig. \ref{fig::JSI_splitting_plots}(c), this splitting can also arise from XPM-induced signal and idler detuning even for a pump with $\Delta_P=0$. When the splitting is due to XPM-induced signal and idler detuning, the JSI remains centred on the unperturbed ring resonances. On the other hand, when pump detuning is responsible for the splitting, the JSI is translated by $\Delta_P$ along both frequency axes.
In Fig. \ref{fig::JSI_uncorrelated_piece} the uncorrelated contribution $I_\mathrm{uncorr}$ to the JSI is plotted for the same pump parameters as in Fig. \ref{fig::JSI_splitting_plots}. The weight of the uncorrelated contribution is extremely small compared to the correlated contribution at low powers, as indicated by the scales in Figs. \ref{fig::JSI_splitting_plots} and \ref{fig::JSI_uncorrelated_piece}, but grows to an appreciable level at high powers. For $\Delta_P=0$, as in Fig. \ref{fig::JSI_uncorrelated_piece}(a), at low $N_P$ the uncorrelated part of the JSI displays a single peak centred at the origin. In the regimes that give rise to split lineshapes, as illustrated in Fig. \ref{fig::JSI_uncorrelated_piece}(b) and \ref{fig::JSI_uncorrelated_piece}(c), the uncorrelated contribution takes the form of four distinct peaks, symmetrically placed about the centre of the overall distribution. Two of these peaks lie on the antidiagonal, overlapping with the correlated contribution. The remaining two lie on the diagonal, and would therefore appear to violate energy conservation if assumed to correspond to signal and idler photons that originated from the same pair. It is therefore natural to interpret these peaks as corresponding to signal and idler photons that are detected from \emph{separate} pairs. As these uncorrelated, ``non-energy conserving" peaks are well separated from the correlated part of the JSI, they are uncontaminated by the correlated contribution to the JSI. The properties of photon pairs detected in these peaks would therefore be expected to differ from those detected in the antidiagonal peaks. We intend to investigate such properties in future work.
The form of the JSI depends qualitatively on whether $\overline{\rho}$ is imaginary or real, a behaviour we saw earlier in the single photon spectrum. When $\overline{\rho}$ is imaginary, and thus contributes to the frequency terms in the denominators of Eqs. () and (), a splitting in the JSI appears. When $\overline{\rho}$ is real, and acts as an effective correction to the linewidth $\overline{\Gamma}$, the JSI is localized to a single line arising from the overlap of $I_\mathrm{corr}$ with a single peak in $I_\mathrm{uncorr}$. For sufficiently detuned pumps, in the regime of real $\overline{\rho}$ the uncorrelated contribution can be large enough to be visible on the JSI plot without exaggeration or scaling. Indeed, for supercritical detunings $|\Delta_P|>\Delta_\mathrm{critical}$, as the OPO threshold is approached and $\overline{\rho}\to\overline{\Gamma}$ the uncorrelated contribution vastly dominates over the correlated contribution, as seen in Fig. \ref{fig::JSI_real_rhobar}. This is an expected consequence of the rapid growth in the photon pair generation rate in this regime -- multiple photon pairs are generated in sufficiently large quantities that joint detection of a signal and idler photon originating from the same energy-conserving pair is unlikely relative to the probability of detecting a signal and idler photon which originated from separate pairs and thus obey no relationship in energy. Another effect seen as $\overline{\rho}\to\overline{\Gamma}$ is the narrowing of the entire JSI distribution to a small point-like peak centred on $(\omega_s,\omega_i)=(\Delta_P,\Delta_P)$. Within our idealization of a zero-bandwidth cw pump the area of this point would be limited only by the frequency resolution of the JSI measurement scheme, though in actual experiments the finite pump bandwidth would serve as a fundamental lower bound for the overall extent of the JSI.
Perhaps the most definitive experimental indication of strongly driven effects lies in the top-right and bottom-left uncorrelated peaks of the JSI distribution for a detuned pump as in Fig. \ref{fig::JSI_uncorrelated_piece}(b). For sufficient detunings these peaks are well separated from the antidiagonal and thus easily distinguished from the correlated part of the JSI. For low powers, wherein only one photon pair is generated in the ring at any given time, they would be entirely absent from the measured JSI. As the power increases, \emph{any} non-spurious coincidence detection of photons in these regions indicates multi-pair generation, as photons generated in those peaks do not conserve energy and therefore must be associated with separate, independently produced pairs.
\begin{figure}
\includegraphics[width=1.0\columnwidth]{JSI_corr_0_10.png}
\includegraphics[width=1.0\columnwidth]{JSI_corr_3Gamma_10.png}
\includegraphics[width=1.0\columnwidth]{JSI_corr_0_2e8.png}
\caption{(Colour online) Correlated part $I_\mathrm{corr}$ of joint spectral intensity distribution, scaled to unit maximum, of signal and idler photon pairs for a pump with (a) $\Delta_P=0, N_P=10$, (b) $\Delta_P=3\overline{\Gamma}, N_P=10$, and (c) $\Delta_P=0, N_P=2\times 10^8$. Ring parameters are taken as $\overline{\Gamma}=1$ GHz and $\Lambda=10$ Hz. The splitting evident in (b) arises from the pump detuning, whereas in (c) the XPM-induced detuning of the signal and idler ring modes is responsible.}\label{fig::JSI_splitting_plots}
\end{figure}
\begin{figure}
\includegraphics[width=1.0\columnwidth]{JSI_uncorr_0_10.png}
\includegraphics[width=1.0\columnwidth]{JSI_uncorr_3Gamma_10.png}
\includegraphics[width=1.0\columnwidth]{JSI_uncorr_0_2e8.png}
\caption{(Colour online) Uncorrelated part $I_\mathrm{uncorr}$ of joint spectral intensity distribution, scaled to unit maximum, of signal and idler photon pairs. Pump parameters are (a) $\Delta_P=0, N_P=10$, (b) $\Delta_P=3\overline{\Gamma}, N_P=10$, and (c) $\Delta_P=0, N_P=2\times 10^8$. Ring parameters are taken as $\overline{\Gamma}=1$ GHz and $\Lambda=10$ Hz.}\label{fig::JSI_uncorrelated_piece}
\end{figure}
\begin{figure}
\includegraphics[width=1.0\columnwidth]{JSI_supercrit_90.png}
\includegraphics[width=1.0\columnwidth]{JSI_supercrit_95.png}
\caption{(Colour online) Joint spectral intensity distribution, scaled to unit maximum, of signal and idler photon pairs for (a) $\Delta_P=1.5\Delta_\mathrm{critical}$ with $N_P=9.8\times 10^7$ (90\% of OPO threshold) and (b) $\Delta_P=1.5\Delta_\mathrm{critical}$ with $N_P=1\times 10^8$ (95\% of OPO threshold). Ring parameters are taken as $\overline{\Gamma}=1$ GHz and $\Lambda=10$ Hz.}\label{fig::JSI_real_rhobar}
\end{figure}
\section{Conclusion}
We have investigated the strongly driven regime of spontaneous four-wave mixing in microring resonators for a cw pump input. A nonperturbative, exact analytic solution to the semiclassical equations of motion within the undepleted pump approximation was developed, which permits the calculation of any physical quantity related to the outgoing signal and idler fields while fully taking into account intraring scattering losses. The effects of self- and cross- phase modulation, as well as multi-pair generation, were found to drastically alter the nature of the photon pair generation process at high powers. A critical pump detuning of $\Delta_\mathrm{critical}=\sqrt{3}\;\overline{\Gamma}$, where $\overline{\Gamma}$ is the total effective linewidth of the ring resonances, was found to divide the behaviour of the system into two regimes. For supercritically detuned pumps, a region of optical bistability of the pump mode is predicted, and a threshold emerges for optical parametric oscillation of the signal and idler modes. Pump power-dependent splitting of the generated signal and idler photon spectra was uncovered, arising from both pump detuning and cross-phase modulation. In certain intermediate-power regimes, dramatic narrowing of the spectral linewidth of generated signal and idler photons associated with the approach to optical parametric oscillation was found. The joint spectral intensity distribution (JSI) was analysed, and found to consist of separate uncorrelated and correlated contributions. The correlated contribution is negligible at low powers, but becomes significant as multi-pair generation becomes appreciable at higher powers. In the regime of spectral splitting, the uncorrelated part of the JSI displays an intriguing quadruplet of peaks, two of which are well separated from the correlated part. An optimal detuning strategy was derived in which the pump detuning is chosen to exactly cancel the effect of self-phase modulation at each input power, maximizing the intraring pump intensity. By detuning the pump in this manner the effects of both spectral splitting and bandwidth reduction are eliminated, and the photon pair generation rate continues to scale quadratically with the pump input even for arbitrarily high powers.
Three simple experimental tests of our predictions in the strongly driven regime were proposed:
\begin{enumerate}
\item For fixed subcritical nonzero detunings the photon pair generation rate as a function of input pump power is predicted to have a local maximum at intermediate powers, followed by a decreasing approach to an asymptotic level at high powers.
\item The single photon spectra of the outgoing signal and idler fields are predicted to show spectral splitting proportional to the pump detuning at low powers, followed by a regime of a singly peaked spectrum with pump power-dependent narrowing of bandwidth at intermediate powers, and finally resuming a doublet structure at high powers.
\item The presence of two non-energy conserving peaks lying on the diagonal of the joint spectral intensity distribution, which are a consequence of multi-pair generation, is predicted to occur for sufficiently large pump powers.
\end{enumerate}
Our analysis was restricted to cw pump inputs; studying how these strongly driven phenomena are altered for short pulses requires a numerical approach. Additionally, a slightly more sophisticated solution is required to fully study the regime of optical parametric oscillation, in which the undepleted pump approximation breaks down. We intend to extend our techniques to treat these regimes future publications.
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 567
|
\section{Introduction}
\label{sec:xsecns}
\vspace*{-0.5pt}
\noindent
HERA was an electron(positron)-proton collider located at DESY, Hamburg.
It ran in two phases HERA-I from 1992-2001 and HERA-II 2003-2007.
Two similar experiments, H1 and ZEUS, took data. In HERA-1 running each
experiment collected $\sim 100$pb$^{-1}$ of $e^+p$ data
and $\sim15$pb$^{-1}$ of $e^-p$ data with electron beam energy $27.5$GeV and
proton beam energies $820,920~$GeV. In HERA-II running each experiment took
$\sim 140$pb$^{-1}$ of $e^+p$ data
and $\sim180$pb$^{-1}$ of $e^-p$ data with the same electron beam energy and
proton beam energies $920~$GeV. In addition to this, before the shut-down in
2007, each experiment took $\sim 30$pb$^{-1}$of data with reduced proton
beam energies $460,575~$GeV.
Deep inelastic lepton-hadron
scattering data has been used both to investigate the theory of
the strong interaction and to determine the momentum distributions of the
partons within the nucleon. The data from the HERA collider now dominate
the world data on deep inelastic scattering since they cover an unprecedented
kinematic range: in $Q^2$, the (negative of the)
invariant mass squared of the virtual exchanged boson,
$0.045 < Q^2 < 3\times 10^{-5}$; in Bjorken $x$, $6\times10^{-7} < x < 0.65$.
Futhermore, because the HERA experiments investigated
$e^+p$ and $e^-p$, charge current(CC) and neutral current (NC) scattering,
information can be gained on flavour separated up- and down-type quarks and
antiquarks and on the gluon- from its role in the scaling violations of
perturbative quantum-chromo-dynamics.
From 2008, the H1 and ZEUS experiments
began to combine their data in order to provide the most complete and
accurate set of deep-inelastic data as the legacy of HERA. Data on
inclusive cross-sections have been combined for the HERA-I phase of running
and a preliminary combination has been made also using the HERA-II data.
This latter exersize also includes the data run at lower proton beam
energies in 2007. Combination of $F_2^{c\bar{c}}$ data is also underway,
and combination of $F_2^{b\bar{b}}$ data and of jet data is foreseen.
The HERA collaborations have used these combined data to
determine parton distribution functions (PDFs). These analyses had resulted
in the HERAPDF sets. The present review
concentrates on the information on proton structure which has been
gained from these HERA data.
\section{Formalism}
\label{formalism}
In the quark parton model deep inelastic lepton-hadron scattering is pictured
as in Fig.~\ref{fig:qpm}.
\begin{figure}[ht]
\centerline{\psfig{figure=qpm-fig.eps,height=0.2\textheight}}
\caption{Schematic diagram of lepton-hadron scattering in the
quark-parton model}
\label{fig:qpm}
\end{figure}
\noindent
where $l,l'$ represent leptons (lepton is taken to include
antileptons, unless it is necessary to distinguish them), and $N$ represents
the nucleon. The associated four vectors are $k,k'$ for the
incoming and outgoing leptons respectively, and $p$ for the target
(or incoming) nucleon. The process is mediated by the exchange of a
virtual vector boson, $V^*$($\gamma, W$ or $Z$), with four
momentum given by
\[
q = k - k'.
\]
Various Lorentz invariants are useful in the description of the kinematics of
the process:
\[
s = ( p + k )^2,
\]
the centre of mass energy squared for the $lp$ interaction,
\[
Q^2 = -q^2,
\]
the (negative of) the invariant mass squared of the virtual exchanged boson,
\[
x = Q^2/2p.q,
\]
the Bjorken $x$ variable,
which is interpreted in the quark-parton model as the fraction
of the momentum of the incoming nucleon taken by the struck quark,
and
\[
y = p.q/p.k,
\]
which gives
a measure of the amount of energy transferred between the lepton and the hadron
systems.
Note that (ignoring masses),
\[
Q^2 = s x y,
\]
so that only two of these quantities are independent.
Finally the centre of mass of the $V^*p$ system (or equivalently the invariant
mass of the final state hadronic system) is often denote by $W$
\[
W^2=(q+p)^2.
\]
Neutral current (NC) deep inelastic scattering is mediated by $\gamma$ and $Z$
exchange and the NC deep inelastic $e^{\pm}p$ scattering cross sections can
be expressed as
\begin{eqnarray} \label{eq:ncsi}
\tilde{\sigma}^{\pm}_{NC} = \frac{Q^4 x}{2\pi \alpha^2 Y_+}
\sigma^{\pm}_{NC}
= F_2 \mp \frac{Y_-}{Y_+} xF_3 -\frac{y^2}{Y_+} F_L~,
\end{eqnarray}
where the electromagnetic coupling constant $\alpha$, the photon
propagator and a helicity factor are absorbed
in the definition of a reduced cross section $\tilde{\sigma}$, and
$Y_{\pm}=1 \pm (1-y)^2$.
The structure functions $F_2$, $F_L$ and $xF_3$
are given by
\begin{eqnarray} \label{strf}
F_2 &=& F_2^{\gamma} - \kappa_Z v_e \cdot F_2^{\gamma Z} +
\kappa_Z^2 (v_e^2 + a_e^2 ) \cdot F_2^Z~, \nonumber \\
F_L &=& F_L^{\gamma} - \kappa_Z v_e \cdot F_L^{\gamma Z} +
\kappa_Z^2 (v_e^2 + a_e^2 ) \cdot F_L^Z~, \nonumber \\
xF_3 &=& \kappa_Z a_e \cdot xF_3^{\gamma Z} -
\kappa_Z^2 \cdot 2 v_e a_e \cdot xF_3^Z~.
\end{eqnarray}
where $v_e$ and $a_e$ are the vector and axial-vector weak couplings of
the electron
and $\kappa_Z(Q^2) = Q^2 /[(Q^2+M_Z^2)(4\sin^2 \theta_W \cos^2
\theta_W)]$.
At low $Q^2$, the contribution of $Z$
exchange is negligible and $xF_3 = 0, F_2 = F_2^{\gamma}, F_L = F_L^{\gamma}$ and $\tilde{\sigma} = F_2 - y^2 F_L/Y_+$.
The contribution of the term containing the structure function $F_L$ is
only significant for large values of $y$.
In the Quark Parton Model (QPM),
$F_L=0$, and the
other strcuture functions are given by
\begin{eqnarray} \label{ncfu}
(F_2^{\gamma}, F_2^{\gamma Z}, F_2^Z) &=& [(e_u^2, 2e_uv_u,
v_u^2+a_u^2)(xU+ x\bar{U})
+ (e_d^2, 2e_dv_d, v_d^2+a_d^2)(xD+ x\bar{D})]~,
\nonumber \\
(xF_3^{\gamma Z}, xF_3^Z) &=& 2 [(e_ua_u, v_ua_u) (xU-x\bar{U})
+ (e_da_d, v_da_d) (xD-x\bar{D})]~,
\end{eqnarray}
such that at low $Q^2$
\begin{equation}
F_2^{\gamma} = [e_u^2(xU+ x\bar{U}) + e_d^2(xD+ x\bar{D})]~,
\end{equation}
where $e_u,e_d$ denote the electric charge of up- or
down-type quarks while $v_{u,d}$ and $a_{u,d}$ are
the vector and axial-vector weak couplings of the up- or
down-type quarks.
Here $xU$, $xD$, $x\bU$ and $x\bD$ denote
the sums of up-type, of down-type and of their
anti-quark momentum distributions, respectively.
In the QPM these ditributions are
functions of Bjorken $x$ only, and not also of $Q^2$ as they would be in full
generality- this is what is meant by Bjorken scaling.
Below the $b$ quark mass threshold,
these sums are related to the quark distributions as follows
\begin{equation} \label{ud}
xU = xu + xc\,, ~~~~~~~~
x\bU = x\bu + x\bc\,, ~~~~~~~~
xD = xd + xs\,, ~~~~~~~~
x\bD = x\bd + x\bs\,,
\end{equation}
where $xs$ and $xc$ are the strange and charm quark distributions.
Assuming symmetry between sea quarks and anti-quarks,
the valence quark distributions result from
\begin{equation} \label{valq}
xu_v = xU -x\bU\,, ~~~~~~~~~~~~~ xd_v = xD -x\bD\,.
\end{equation}
Charge current (CC) deep inelastic scattering is mediated by $W^+$ and $W^-$
exchange and the CC deep inelastic $e^{\pm}p$ scattering cross sections can
be expressed as
\begin{equation}
\label{Rnc}
\tilde{\sigma}^{\pm}_{CC} =
\frac{2 \pi x}{G_F^2}
\left[ \frac {M_W^2+Q^2} {M_W^2} \right]^2 \sigma^{\pm}_{CC}
\end{equation}
where analogously to Eq~\ref{eq:ncsi},
\begin{equation}
\label{ccsi}
\tilde{\sigma}^{\pm}_{CC}=
\frac{Y_+}{2}W_2^\pm \mp \frac{Y_-}{2} xW_3^\pm - \frac{y^2}{2} W_L^\pm.
\end{equation}
In the QPM, $W_L^\pm = 0$,
and the CC structure functions represent sums and differences
of quark and anti-quark-type distributions depending on the
charge of the lepton beam:
\begin{eqnarray}
\label{ccstf}
W_2^{+} = x\bU+xD\,,\hspace{0.05cm} ~~~~~~~
xW_3^{+} = xD-x\bU\,,\hspace{0.05cm} ~~~~~~~
W_2^{-} = xU+x\bD\,,\hspace{0.05cm} ~~~~~~~
xW_3^{-} = xU-x\bD\,.
\end{eqnarray}
From these equations it follows that
\begin{equation}
\label{ccupdo}
\tilde{\sigma}^+ = x\bU+ (1-y)^2xD\,, ~~~~~~~
\tilde{\sigma}^- = xU +(1-y)^2 x\bD\,.
\end{equation}
Therefore the NC and CC measurements may be used
to determine the combined sea quark distribution functions, $x\bU$ and
$x\bD$,
and the valence quark distributions, $xu_v$ and $xd_v$.
Perturbative QCD extends the formalism of the QPM such that the parton
momentum distributions (PDFs) become functions of $Q^2$ as well as $x$. However
this scaling violation induces only a logarithmic dependence on $Q^2$,
as described by the DGLAP equations
~\cite{Gribov:1972ri,Lipatov:1974qm,Dokshitzer:1977sg,Altarelli:1977zs}.
The DGLAP equations are coupled equations for the change of the quark,
antiquark and gluon densities as $\ln Q^2$ changes
\begin{eqnarray}
{\partial\over \partial\ln Q^2}\left(\matrix{q_i(x,Q^2)\cr g(x,Q^2)}\right) &=&
{\asq\over 2\pi}\sum_j\int_x^1{d\xi\over \xi} \nonumber \\
&&\left(
\matrix{P_{q_iq_j}({x\over \xi},\asq)&P_{q_ig}({x\over \xi},\asq)\cr
P_{gq_j}({x\over \xi},\asq)&P_{gg}({x\over \xi},\asq)\cr}
\right)
\left(\matrix{q_j(\xi,Q^2)\cr g(\xi,Q^2)}\right),\nonumber \\
\label{eqn:ap_gen}
\end{eqnarray}
where the $q_i,q_j$ are taken to include both quarks and antiquark
distributions. The splitting functions are expanded as power series
in the strong coupling $\as$,
\begin{eqnarray}
P_{q_iq_j}(z,\as)&=&\delta_{ij}P^{(0)}_{qq}(z)+
{\as\over 2\pi}P^{(1)}_{qq}(z)+\dots \nonumber \\
P_{qg}(z,\as)&=&P^{(0)}_{qg}(z)+
{\as\over 2\pi}P^{(1)}_{qg}(z)+\dots \nonumber \\
P_{gq}(z,\as)&=&P^{(0)}_{gq}(z)+
{\as\over 2\pi}P^{(1)}_{gq}(z)+\dots \nonumber \\
P_{gg}(z,\as)&=&P^{(0)}_{gg}(z)+
{\as\over 2\pi}P^{(1)}_{gg}(z)+\dots \nonumber
\end{eqnarray}
and are calculable within pQCD. Thus the gluon momemtum distribution
influences the quark distributions through its contribution to their scaling
violations, and the gluon PDF is determined by analysing the $Q^2$ dependence
of the data.
To leading order in pQCD the equations for the structure functions in terms
of the PDFs are still given by the QPM expressions. However beyond leading
order a convolution of parton distributions and QCD-calculable coefficient
functions is necessary.
\begin{eqnarray}
{F_2(x,Q^2)\over x}&=&\int{d\xi\over \xi}\left[\sum_ie^2_iq_i(\xi,Q^2)
C_q\left({x\over \xi},\as\right)+
\bar{e}^2g(\xi,Q^2)C_g\left({x\over \xi},\as\right)\right],\nonumber \\
&&
\label{eqn:dglap_f2}
\end{eqnarray}
where, $\bar{e}^2=\sum_ie^2_i$, and the sums run over all active quark
and antiquark flavours. $C_q$ an $C_g$ are the coefficient functions,
which may also be expanded as power series in $\as$,
\begin{eqnarray}
C_q(z,\as)&=&\delta(1-z)+{\as\over 2\pi}C^1_q(z)+\dots \nonumber \\
C_g(z,\as)&=&{\as\over 2\pi}C^1_g(z)+\dots \nonumber.
\end{eqnarray}
In the QPM the transverse momentum of the partons is assumed to be zero and
one of the consequences of this for spin ${1\over 2}$ quarks is that the
longitudinal structure function ($F_L=F_2-2xF_1$) is zero. However this is
no longer true beyond leading order, and $F_L$ is given by.
\begin{equation}
{F_L(x,Q^2)\over x}={\as\over 2\pi}\int_x^1{d\xi\over \xi}\left[
\sum_ie^2_i{8\over 3}\left({x\over \xi}\right)q_i(\xi,Q^2)+
\bar{e}^2 4\left({x\over \xi}\right)^2\left(1-{x\over \xi}\right)g(\xi,Q^2)
\right]
\label{eqn:fl_qg}
\end{equation}
Thus the gluon distribution also influences the longitudinal structure function
particularly at low $x$.
\section{Data sets}
The deep inelastic $ep$ scattering cross sections
depend on the centre-of-mass energy $s$ and on two other independent
kinematic variables, usually taken to be $Q^2$ and $x$. The
salient feature of the HERA collider experiments is the possibility
to determine the $x$ and $Q^2$ from the scattered
electron, or from the hadronic final state, or using a combination
of the
two. The choice of the most appropriate kinematic reconstruction method
for a given phase space region is
based on resolution, measurement accuracy and radiative correction effects
and has
been optimised differently for the two HERA experiments H1 and ZEUS, as
described in the original publications.
The use of different reconstruction techniques by the two experiments
contributes to improved accuracy when the data sets are combined, since
although
the detectors were built following similar physics considerations they
opted for different technical solutions, both for the calorimetric
and the tracking measurements. Thus the experiments can calibrate each
other.
\subsection{The combined inclusive HERA-I data set}
\label{sec:datacomb}
The inclusive cross-sections data collected by each experiment in the
HERA-I running period have been combined~\cite{h1zeuscomb}.
A summary of the data used in this analysis is given in
Table\,\ref{tab:data}.
The NC data cover a wide range in $x$ and $Q^2$.
The lowest $Q^2 \ge 0.045$~GeV$^2$ data come from the measurements of ZEUS
using
the BPC and BPT~\cite{Breitweg:1997hz,Breitweg:2000yn}. The $Q^2$ range
from $0.2$~GeV$^2$ to $1.5$~GeV$^2$
is covered using special HERA runs, in which the interaction vertex
position was shifted forward allowing for larger angles
of the backward scattered electron
to be accepted~\cite{Breitweg:1998dz,Collaboration:2009bp}.
The lowest $Q^2$ for the shifted vertex data
was reached using events in which the effective electron
beam energy was reduced by initial state
radiation~\cite{Collaboration:2009bp}.
Values of $Q^2\ge 1.5$~GeV$^2$ are measured using the nominal vertex
settings.
For $Q^2 \le 10$~GeV$^2$, the cross section is very high
and the data were collected using dedicated
trigger setups~\cite{Chekanov:2001qu,Collaboration:2009bp}.
The highest accuracy of the cross-section measurement is achieved
for $10 \le Q^2 \le 100$~GeV$^2$
\cite{Chekanov:2001qu,Collaboration:2009kv}.
For $Q^2\ge 100$~GeV$^2$, the statistical uncertainty of the data becomes
relatively large.
The high $Q^2$ data included here were collected with
positron~\cite{Adloff:1999ah,Chekanov:2001qu,Adloff:2003uh,Chekanov:2003yv}
and with
electron~\cite{Adloff:2000qj,Chekanov:2002ej} beams.
The CC data for $e^+p$ and $e^-p$ scattering cover the range $300\le Q^2\le
30000$~GeV$^2$~\cite{Adloff:1999ah,zeuscc97,Chekanov:2002zs,Adloff:2003uh,
hekanov:2
003vw}.
\begin{table}
\begin{center}
\begin{scriptsize}
\begin{tabular}{|lr|lr|lr|c|c|c|c|c|}
\hline
\multicolumn{2}{|c|}{Data Set} &
\multicolumn{2}{|c|}{$x$ Range} &
\multicolumn{2}{|c|}{$Q^2$ Range} &
${\cal L}$ & $e^+/e^-$ & $\sqrt{s}$ & Reference \\
\multicolumn{2}{|c|}{ } &
\multicolumn{2}{|c|}{ } &
\multicolumn{2}{|c|}{GeV$^2$} &
pb$^{-1}$ & \\
\hline
H1~svx-mb & $95$-$00$ & $5\times 10^{-6}$ & $0.02$ & $0.2$ & $12$ &
$2.1$ &$e^+p$ & $301$-$319$ & \cite{Collaboration:2009bp} \\
H1~low~$Q^2$ & $96$-$00$ & $2\times 10^{-4}$ & $0.1$ & $12$ &
$150$ & $22$ &$e^+p$ & $301$-$319$ &\cite{Collaboration:2009kv}\\
H1~NC & $94$-$97$ & $0.0032$ &$0.65$ &$150$ &$30000$ &
$35.6$ & $e^+p$ & $301$ & \cite{Adloff:1999ah}\\
H1~CC & $94$-$97$ & $0.013$ &$0.40$ &$300$ &$15000$ &
$35.6$ & $e^+p$ & $301$ & \cite{Adloff:1999ah}\\
H1~NC & $98$-$99$ & $0.0032$ &$0.65$ &$150$ &$30000$ &
$16.4$ & $e^-p$ & $319$ & \cite{Adloff:2000qj}\\
H1~CC & $98$-$99$ & $0.013$ &$0.40$ &$300$ &$15000$ &
$16.4$ & $e^-p$ & $319$ & \cite{Adloff:2000qj}\\
H1~NC HY & $98$-$99$ & $0.0013$ &$0.01$ &$100$ &$800$ &
$16.4$ & $e^-p$ & $319$ & \cite{Adloff:2003uh}\\
H1~NC & $99$-$00$ & $0.0013$ &$0.65$ &$100$ &$30000$ &
$65.2$ & $e^+p$ & $319$ & \cite{Adloff:2003uh}\\
H1~CC & $99$-$00$ & $0.013$ &$0.40$ &$300$ &$15000$ &
$65.2$ & $e^+p$ & $319$ & \cite{Adloff:2003uh} \\
\hline
ZEUS~BPC & $95$ & $2\times 10^{-6}$ & $6\times 10^{-5}$
&$0.11$ & $0.65$ & $1.65$ & $e^+p$ & $301$ &
\cite{Breitweg:1997hz} \\
ZEUS~BPT & $97$ & $6\times 10^{-7}$ & $0.001$ & $0.045$ &
$0.65$ & $3.9$ & $e^+p$ & $301$ & \cite{Breitweg:2000yn}\\
ZEUS~SVX & $95$ & $1.2\times 10^{-5}$ & $0.0019$ & $0.6$ &
$17$ & $0.2$ & $e^+p$ & $301$ & \cite{Breitweg:1998dz}\\
ZEUS~NC & $96$-$97$ & $6\times10^{-5}$ &$0.65$& $2.7$ &
$30000$ & $30.0$ & $e^+p$ & $301$ \cite{Chekanov:2001qu}\\
ZEUS~CC & $94$-$97$ & $0.015$ & $0.42$ & $280$ & $17000$
&$47.7$ & $e^+p$ & $301$ & \cite{zeuscc97}\\
ZEUS~NC & $98$-$99$ & $0.005$ & $0.65$ & $200$ & $30000$
&$15.9$ & $e^-p$ & $319$ & \cite{Chekanov:2002ej} \\
ZEUS~CC & $98$-$99$ & $0.015$ & $0.42$ & $280$ & $30000$
&$16.4$ & $e^-p$ & $319$ & \cite{Chekanov:2002zs} \\
ZEUS~NC & $99$-$00$ & $0.005$ & $0.65$ & $200$ & $30000$
&$63.2$ & $e^+p$ & $319$ & \cite{Chekanov:2003yv} \\
ZEUS~CC & $99$-$00$ & $0.008$ & $0.42$ & $280$ & $17000$
&$60.9$ & $e^+p$ &$319$ & \cite{Chekanov:2003vw}\\
\hline
\end{tabular}
\end{scriptsize}
\end{center}
\caption{H1 and ZEUS data sets used for the combination. }
\label{tab:data}
\end{table}
The full details of the combination procedure are given in
ref~\cite{h1zeuscomb}.
The combination of the data sets uses the
$\chi^2$ minimisation method described in~\cite{Collaboration:2009bp}.
The $\chi^2$ function takes into account the correlated systematic
uncertainties
for the H1 and ZEUS cross-section measurements.
Global normalisations of the data sets
are split into an overall normalisation uncertainty of $0.5\%$, common to
all data sets, due to uncertainties of higher order corrections
to the Bethe-Heitler process used for the luminosity calculation,
and experimental uncertainties which are treated as correlated systematic
sources.
Some sources of point-to-point correlated uncertainties
are common for CC and NC data and for several data sets of the same
experiment. The systematic uncertainties were treated as independent
between H1 and ZEUS apart from the $0.5\%$ overall normalisation
uncertainty.
All the NC and CC cross-section data from H1 and ZEUS are combined in one
simultaneous minimisation.
Therefore resulting shifts of
the correlated systematic uncertainties propagate coherently to both CC and
NC data.
There are in total $110$ sources of correlated systematic uncertainty,
including global normalisations, characterising
the separate data sets. None of these systematic sources
shifts by more than $2\, \sigma$ of the nominal value in the averaging
procedure.
The absolute normalisation of the combined data set
is to a large extent defined by the most precise measurements of
NC $e^+p$ cross-section in the $10\le Q^2\le 100$~GeV$^2$ kinematic range.
Here
the H1~\cite{Collaboration:2009kv} and ZEUS~\cite{Chekanov:2001qu} results
move towards each other and
the other data sets follow this adjustment.
The influence of several correlated systematic uncertainties is reduced
significantly for the averaged result.
For example, the uncertainty due to the H1 LAr calorimeter energy scale
is halved while the uncertaintydue to the
ZEUS photoproduction background is reduced by a factor of 3.
There are two main reasons for thess significant reductions.
Since H1 and ZEUS use different reconstruction methods
similar systematic sources influence
the measured cross section differently as a function of $x$ and $Q^2$.
Therefore, requiring the cross sections to agree at all $x$ and $Q^2$
constrains
the systematics efficiently. In addition, for some regions of the phase
space, one of the two experiments has superior precision compared to the
other.
For these regions, the less precise measurement is fitted to the more
precise one, with a simultaneous reduction of the correlated systematic
uncertainty.
This reduction propagates to the other average points, including those
which are based solely on the measurement from the less precise experiment.
In addition to the 110 sources of systematic uncertainty which result from
the separate data sets there are three sources of procedural uncertainty deriving
from the choices made in the combination. Firstly all systematic
uncertainties were treated as multiplicative, this has been varied by treating
all sources bar the normalisations as additive, and the difference is used
to estimate a procedural systematic error. Secondly, the correlated
systematics from H1 and ZEUS were treated as uncorrelated between the
experiments, but this may not be
completely true due to some similarity of methods. An alternative combination
procedure treats 12 sources of similar systematics as correlated.
This results in some differences in the result for the photo-producton background and the
hadronic energy scale and these differences are use to estimate two further
procedural systematic errors.
The data averaging procedure results in a set of measurements for each
process:
the average cross section value at a point $i$, its relative
correlated systematic, relative statistical and relative uncorrelated
systematic uncertainties, respectively.
The number of degrees of
freedom, $ndf$, is calculated as the difference between the total
number of measurements and
the number of combined data points. The value of $\chi^2_{\rm min}/ndf$ is
a measure of the consistency of the data sets.
Tabulated results for the average NC and CC cross sections and the
structure function $F_2$
together with statistical, uncorrelated systematic and procedural
uncertainties are given in ref~\cite{h1zeuscomb}.
The total integrated luminosity of the combined data set corresponds to
about $200$~pb$^{-1}$ for $e^+p$ and
$30$~pb$^{-1}$ for $e^-p$.
In total $\totalave$ data points are combined to $\uniqueave$ cross-section
measurements.
The data show good consistency, with $\chi^2/ndf = \chiave/\dofave$, and
there are no tensions between the input data sets.
For $Q^2\ge 100$~GeV$^2$ the precision of the H1 and
ZEUS measurements is about equal and thus the systematic uncertainties are
reduced uniformly.
For $2.5\le Q^2 < 100$~GeV$^2$ and $Q^2<1$~GeV$^2$ the precision is
dominated by the H1~\cite{Collaboration:2009bp,Collaboration:2009kv}
and ZEUS~\cite{Breitweg:2000yn} measurements, respectively.
Therefore the overall reduction of the uncertainties is smaller,
and it is essentially obtained from the reduction of the correlated
systematic uncertainty.
The total uncertainty of the combined measurement is typically smaller than
$2\%$ for $3 < Q^2 < 500$~GeV$^2$ and reaches $1\%$ for $20 < Q^2 <
100$~GeV$^2$.
The uncertainties are larger for high inelasticity $y>0.6$
due to the photoproduction background.
In Fig~\ref{fig:quality} averaged data are compared to the
input H1 and ZEUS data, illustrating the improvement in precision. Because
of the reduction in size of the systematic error this improvement is far
better than would be expected simply from the rough doubling of statistics
which combining the two experiments represents.
In Fig~\ref{fig:vsQ2l}, the combined NC $e^+p$ data at very low $Q^2$ are
shown.
In Fig~\ref{fig:scal} the NC reduced cross section, for $Q^2 > 1$\,GeV$^2$,
is shown
as a function of $Q^2$ for the
HERA combined $e^+p$ data and for fixed-target data~\cite{bcdms,nmc} across
the whole of the measured kinematic plane.
The combined NC $e^{\pm}p$ reduced cross sections are compared in the
high-$Q^2$
region in Fig~\ref{fig:ncepem}.
In Figs~\ref{fig:dataCCp} and \ref{fig:dataCCm}
the combined data set is shown for CC scattering at high $Q^2$.
The HERAPDF1.0 fit, described in Sec.~\ref{sec:pdfchap}, used these data as input.
It is superimposed on the data
in the kinematic region suitable for the application of perturbative QCD.
\begin{figure}[tbp]
\vspace{-0.5cm}
\centerline{
\epsfig{figure=xbins-v2a.eps,height=0.4\textheight}}
\caption {HERA combined NC $e^+p$ reduced
cross section as a function of
$Q^2$ for six $x$-bins compared to the separate
H1 and ZEUS data input to the averaging procedure.
The individual measurements are displaced horizontally for a better
visibility.
}
\label{fig:quality}
\end{figure}
\begin{figure}[tbp]
\vspace{-0.5cm}
\centerline{
\epsfig{figure=ncepa.eps,height=0.4\textheight}
}
\caption {HERA combined NC $e^+p$ reduced cross section at very low $Q^2$.
}
\label{fig:vsQ2l}
\end{figure}
\begin{figure}[tbp]
\vspace{-0.5cm}
\centerline{
\epsfig{figure=scal.eps,height=0.4\textheight}}
\caption {HERA combined NC $e^+p$ reduced cross section
and fixed-target data as a function of $Q^2$.
The HERAPDF1.0 fit is superimposed.
The bands represent the total uncertainty of the fit.
The dashed lines are shown for $Q^2$ values not included in the QCD
analysis.
}
\label{fig:scal}
\end{figure}
\begin{figure}[tbp]
\vspace{-0.5cm}
\centerline{
\epsfig{figure=nc-epem.eps,height=0.4\textheight}}
\caption {HERA combined NC $e^{\pm}p$ reduced cross sections
at high $Q^2$.
The HERAPDF1.0 fit is superimposed.
The bands represent the total uncertainty of the fit.
}
\label{fig:ncepem}
\end{figure}
\begin{figure}[tbp]
\centerline{
\epsfig{figure=ccep.eps,height=0.4\textheight}
}
\caption {HERA combined CC $e^+p$ reduced cross section.
The HERAPDF1.0 fit is superimposed.
The bands represent the total uncertainty of the fit.
}
\label{fig:dataCCp}
\end{figure}
\begin{figure}[tbp]
\centerline{
\epsfig{figure=ccem.eps,height=0.4\textheight}}
\caption {HERA combined CC $e^-p$ reduced cross section.
The HERAPDF1.0 fit is superimposed.
The bands represent the total uncertainty of the fit.
}
\label{fig:dataCCm}
\end{figure}
\subsection{ HERA-II inclusive data sets}
The published inclusive data combination does not included the data
from the HERA-II running period 2003-2007. These data were collected
for both electron and positron beams, polarised both positively
and negatively, at $\surd{s}$ = 318 GeV.
The polarised data can be used to measure electroweak
parameters~\cite{ewpapers}. This is beyond the scope of the present review.
For investigation of the parton distribution functions
these data have been combined into unpolarised cross-sections.
Details of the luminosities collected are given in
Table~\ref{tab:lumi}
\begin{table}[tbp]
\centerline{
\begin{tabular}{|l|r|r|}
\hline
New Data Set& Luminosity in pb$^{-1}$ & Reference \\
\hline
CC $e^-p$ 04/06 ZEUS & $175$ & \cite{zeusccem}\\
CC $e^-p$ 04/06 H1 & $180$ & \cite{h1cc}\\
NC $e^-p$ 05/06 ZEUS & $170$ & \cite{zeusncem} \\
NC $e^-p$ 05/06 H1 & $180$ & \cite{h1nc}\\
CC $e^+p$ 06/07 ZEUS & $132$ & \cite{zeusccep} \\
CC $e^+p$ 06/07 H1 & $149$ & \cite{h1cc}\\
NC $e^+p$ 06/07 ZEUS & $ 135$ & \cite{zeusncep} \\
NC $e^+p$ 06/07 H1 & $149$ & \cite{h1nc}\\
\hline
\end{tabular}}
\caption{Luminosities of the HERA-II data sets.
}
\label{tab:lumi}
\end{table}
A preliminary combination of these data with the HERA-I data has been made~\cite{hera2comb}
using the same $\chi^2$ minimisation method, such that a new a set of
measurements for each process, NC and CC $e^+p$ and $e^-p$, results.
There are in total $131$ sources of correlated systematic uncertainty,
characterising the separate data sets, and three sources of procedural
uncertainty, plus the overall normalisation uncertainty of $0.5\%$, as before.
These preliminary combined data are shown in
Figs.~\ref{fig:ncepem2}-~\ref{fig:dataCCm2}. Comparison of these figures
with Figs.~\ref{fig:ncepem}-~\ref{fig:dataCCm} shows how much the addition
of the HERA-II data improves precision at high $x$ and $Q^2$.
The HERAPDF1.5 fit~\cite{hera2fit}, described in Sec.~\ref{sec:pdfchap}, used these data as input.
It is superimposed on the data in the figures.
\begin{figure}[tbp]
\vspace{-0.5cm}
\centerline{
\epsfig{figure=nc-epem2.eps,height=0.4\textheight}}
\caption {HERA I+II preliminarycombined NC $e^{\pm}p$ reduced cross sections
at high $Q^2$.
The HERAPDF1.5 fit is superimposed.
The bands represent the total uncertainty of the fit.
}
\label{fig:ncepem2}
\end{figure}
\begin{figure}[tbp]
\centerline{
\epsfig{figure=ccep2.eps,height=0.4\textheight}
}
\caption {HERA I+II preliminary combined CC $e^+p$ reduced cross section.
The HERAPDF1.5 fit is superimposed.
The bands represent the total uncertainty of the fit.
}
\label{fig:dataCCp2}
\end{figure}
\begin{figure}[tbp]
\centerline{
\epsfig{figure=ccem2.eps,height=0.4\textheight}}
\caption {HERA I+II preliminary combined CC $e^-p$ reduced cross section.
The HERAPDF1.5 fit is superimposed.
The bands represent the total uncertainty of the fit.
}
\label{fig:dataCCm2}
\end{figure}
\subsection{$F_L$ data}
\label{sec:lowE}
During the final running period the proton beam ran at three different
proton beam energies ($920, 575, 460~$GeV) and NC $e^+p$ data were collected.
These data access high-$y$ and have been used to measure
the longitudinal structure function $F_L$~\cite{h1fl,zeusfl}.
The luminosities for the input data for the combination are specfied in
Table~\ref{tab:fl}
\begin{table}[tbp]
\centerline{
\begin{tabular}{|l|r|r|}
\hline
New Data Set& Luminosity in pb$^{-1}$ & Reference \\
\hline
ZEUS $E_p=460~$GeV& $13.9$ & \cite{zeusfl}\\
ZEUS $E_p=575~$GeV& $7.1$ & \cite{zeusfl}\\
ZEUS $E_p=920~$GeV& $45$ & \cite{zeusfl}\\
H1 $E_p=460~$GeV& $12.4$ & \cite{h1fl}\\
H1 $E_p=575~$GeV& $6.2$ & \cite{h1fl}\\
H1 $E_p=920~$GeV& $21.6$ & \cite{h1fl}\\
\hline
H1 $E_p=460~$GeV& $12.2$ & \cite{h1newfl}\\
H1 $E_p=575~$GeV& $5.9$ & \cite{h1newfl}\\
H1 $E_p=920~$GeV& $97.6,5.9$ & \cite{h1newfl}\\
\hline
\end{tabular}}
\caption{Luminosities of the data sets for the low energy running.
The first 6 data sets have already been combined. The final 3 will be combined.
}
\label{tab:fl}
\end{table}
The reduced cross-section data from these runs have been combined~\cite{lowEcomb}
and the combination is shown in Fig~\ref{fig:lowE}.
\begin{figure}[tbp]
\vspace{-0.5cm}
\centerline{
\epsfig{figure=LowExsecn.eps,height=0.4\textheight}}
\caption {HERA combined NC $e^{+}p$ reduced cross sections from running at
three different proton beam energies. the predictions of HERAPDF1.0 are
superimposed.
}
\label{fig:lowE}
\end{figure}
Using these data a combined measurement of $F_L$ can be made~\cite{lowEcomb}.
Recall that the NC $e^+p$ reduced cross section is given by,
$\tilde{\sigma} = F_2 - y^2 F_L/Y_+$, for $Q^2 \ll M_Z^2$.
Since $Q^2=s x y$ one needs measurements at different $s$ values in order
to access different
$y$ values for the same $x,Q^2$ point.
The structure function $F_L$ is measured as a slope of a linear fit of
$\tilde{\sigma}$ versus $f(y)= y^2/Y_+$, in $x,Q^2$ bins.
Fig~\ref{fig:lowEslope} shows an
example of such a fit, for various $x$ values, at $Q^2 = 32~$GeV$^2$.
The measured $F_L$ is shown, averaged in $x$ as a function of $Q^2$, in
Fig~\ref{fig:fl}, with various theoretical predictions superimposed.
At low-$x$, NLO QCD
in the DGLAP formalism predicts that
this structure function is strongly related to the gluon PDF,
see Eqn.~\ref{eqn:fl_qg}.
\begin{figure}[tbp]
\vspace{-0.5cm}
\centerline{
\epsfig{figure=LowExsecnvsy.eps,height=0.4\textheight}}
\caption {The slope of $\tilde{\sigma}$ vs $f(y)= y^2/Y_+$ for various $x$
bins at $Q^2=32$GeV$^2$.
}
\label{fig:lowEslope}
\end{figure}
\begin{figure}[tbp]
\vspace{-0.5cm}
\centerline{
\epsfig{figure=flcombined.eps,height=0.4\textheight}}
\caption {The HERA combined measurement of $F_L$ avberaged in $x$ at a
given value of $Q^2$.
}
\label{fig:fl}
\end{figure}
This combination will be updated to include the recently
published
H1 data, which extend to lower $Q^2$~\cite{h1newfl}.
The luminosities used for this extension are given in Table~\ref{tab:fl}
and the $F_L$
measurement from these H1 data is shown in Fig~\ref{fig:h1newfl}.
\begin{figure}[tbp]
\vspace{-0.5cm}
\centerline{
\epsfig{figure=H1newFL.eps,height=0.4\textheight}}
\caption {The H1 2011 measurement of $F_L$ averaged in $x$ at a
given value of $Q^2$.
}
\label{fig:h1newfl}
\end{figure}
\subsection{$F_2^{c\bar c}$ and $F_2^{b\bar b}$ data sets}
\label{sec:charmdata}
A preliminary combination has been made of data on $F_2^{c\bar c}$~\cite{f2ccomb} from
various different methods of tagging charm: using the $D^*$, using
the vertex detectors to see the displaced decay vertex, using direct
$D_0, D^+$ production identified using the vertex detectors, and indentifying
semi-leptonic charm decays via muons, also using the vertex detectors.
The details of the data sets used in the combination are given in Table~\ref{tab:f2c}.
\begin{table}[tbp]
\centerline{
\begin{tabular}{|l|r|r|}
\hline
Data Set& Luminosity in pb$^{-1}$ & Reference \\
\hline
$D^*$ 99/00 H1 & $47$ & \cite{h1d99}\\
$D^*$ 04/07 H1 & $340$ & \cite{h1d07}\\
Vtx. 99/00 H1 & $57.4$ & \cite{h1vtx99} \\
Vtx. 06/07 H1 & $189$ & \cite{h1vtx07}\\
$D^*$ 98/00 ZEUS & $82$ & \cite{zeusd00} \\
$D^*$ 96/97 ZEUS & $37$ & \cite{zeusd97}\\
$D^0,D^{\pm}$ 04/05 ZEUS & $ 134$ & \cite{zeusd0dp} \\
muons 05 ZEUS & $126$ & \cite{zeusmuon}\\
\hline
\end{tabular}}
\caption{Luminosities of the $F_2^{c\bar c}$ data sets.
}
\label{tab:f2c}
\end{table}
The results of the $F_2^{c\bar c}$ combination compared to the
separate measurements which
went into it are shown in Fig~\ref{fig:f2ccomb}.
\begin{figure}[tbp]
\vspace{-0.5cm}
\centerline{
\epsfig{figure=f2cdata.eps,height=0.4\textheight}}
\caption {The HERA combined measurement of $F_2^{c\bar c}$ compared to the
data sets of H1 and ZEUS used for the combination. these data sets are
slightly displaced in $x$ for visibility.
}
\label{fig:f2ccomb}
\end{figure}
The $F_2^{c\bar c}$ combination is shown compared to the predictions of
HERAPDF1.0 in Fig.~\ref{fig:f2cfit}
\begin{figure}[tbp]
\vspace{-0.5cm}
\centerline{
\epsfig{figure=f2cfit.eps,height=0.4\textheight}}
\caption {The HERA combined measurement of $F_2^{c\bar c}$ compared to the
predictions of HERAPDF1.0
}
\label{fig:f2cfit}
\end{figure}
Data on $F_2^{b\bar b}$ have not yet been combined. A recent comparison of
H1~\cite{h1f2b}
and ZEUS~\cite{zeusf2b} separate results is shown in Fig.~\ref{fig:f2b}.
\begin{figure}[tbp]
\vspace{-0.5cm}
\centerline{
\epsfig{figure=f2b2010.eps,height=0.4\textheight}}
\caption {The H1 and ZEUS measurements of $F_2^{b\bar b}$.
}
\label{fig:f2b}
\end{figure}
\subsection{ Jet data sets}
\label{sec:jetdata}
Jet data may also be used to constrain the PDFs.
So far H1 and ZEUS jet data have not
been combined but some separate H1 and ZEUS jet data sets have been input to
the HERAPDF fits in order to exploit their ability to constrain the gluon PDF
and to make a determination of the value of $\alpha_s(M_Z)$ simultaneously
with the PDF determination~\cite{herapdf16}. The jet data which have been used are summarised in
Table~\ref{tab:jetdata}
\begin{table}[tbp]
\centerline{
\begin{tabular}{|l|r|r|}
\hline
Data Set& Luminosity in pb$^{-1}$ & Reference \\
\hline
High $Q^2$ norm. inc. jets 96/07 H1 & $395$ & \cite{h1hq2}\\
Low $Q^2$ inc. jets 96/00 H1 & $43.5$ & \cite{h1lq2}\\
High $Q^2$ inc. jets 96/97 ZEUS & $38.6$ & \cite{zeus97}\\
High $Q^2$ inc. jets 98/00 ZEUS & $82$ & \cite{zeus00}\\
\hline
\end{tabular}}
\caption{Luminosities of the jet data sets.
}
\label{tab:jetdata}
\end{table}
These data are illustrated in Figs~\ref{fig:jetzeus1},~\ref{fig:jetzeus2},~\ref{fig:jeth11},~\ref{fig:jeth12}.
\begin{figure}[tbp]
\vspace{-0.5cm}
\centerline{
\epsfig{figure=DESY-02-112_4.eps,height=0.4\textheight}}
\caption {ZEUS 96/97 measurements of tne inclusive jet cross section, as a function of $E_T$ jet in the Breit frame for various $Q^2$ bins.
}
\label{fig:jetzeus1}
\end{figure}
\begin{figure}[tbp]
\vspace{-0.5cm}
\centerline{
\epsfig{figure=DESY-06-128_11.eps,height=0.4\textheight}}
\caption {ZEUS 98/00 measurements of the inclusive jet cross section, as a function of $E_T$ jet in the Breit frame for various $Q^2$ bins. .
}
\label{fig:jetzeus2}
\end{figure}
\begin{figure}[tbp]
\vspace{-0.5cm}
\centerline{
\epsfig{figure=H1_highQ2_4.eps,height=0.4\textheight}}
\caption {H1 HERA-I+II measurements of tne normalised inclusive jet cross section, as a function of $p_T$ for various $Q^2$ bins.
}
\label{fig:jeth11}
\end{figure}
\begin{figure}[tbp]
\vspace{-0.5cm}
\centerline{
\epsfig{figure=H1lowQ2-3.eps,height=0.4\textheight}}
\caption {H1 HERA-I measurements of tne low-$Q^2$ inclusive jet cross section, as a function of $p_T$ jet for various $Q^2$ bins.
}
\label{fig:jeth12}
\end{figure}
\section{Extraction of parton densities}
\label{sec:pdfchap}
The section discusses how parton momentum densities are extracted from the
HERA data\footnote{Open access code for the HERA PDF fits,
and many other useful utilities, are available from the HERAFitter website
http://herafitter.hepforge.org}. There are several PDF fits to different
HERA data sets.
The HERAPDF1.0~NLO set used only the HERA-I combined data~\cite{h1zeuscomb}.
However there are
studies, using the same fit formalism, including the preliminary low-energy
combined data~\cite{lowEpdf} and using the preliminary combined $
F_2^{c\bar{c}}$ data~\cite{charmpdf}. The
HERAPDF1.5~NLO set~\cite{hera2fit} used the HERA-I+II preliminary combined
data~\cite{hera2comb} and the same
fit formalism. Studies were also made using the same data but
extending the parametrisation HERAPDF1.5f~NLO~\cite{herapdf16}.
This extended parametrisation
was then used for the HERAPDF1.5~NNLO fit~\cite{herapdf15nnlo}.
Subsequent fits also used the
extended parametrisation: HERAPDF1.6~NLO fit~\cite{herapdf16}
which included the HERA-I+II
inclusive data and separate H1 and ZEUS jet data;
HERAPDF1.7~NLO fit~\cite{herapdf17} which used all the data
described in the previous sections (HERA-I+II
inclusive data and low energy inclusive data, $F_2^{c\bar{c}}$ data and H1 and
ZEUS jet data sets).
The relationship of the measured cross-sections to the parton
distributions, presented in Sec.~\ref{sec:xsecns}, is not so
straightforward beyond LO since the evolved parton
distributions must be convoluted with coefficient functions and all types
of
parton may contribute to a particular structure function through the
evolution.
However, the simple LO formulae still give a good guide to the major
contributions.
The cross-sections for NC and CC, $e^+$ and $e^-$ scattering on protons
provide enough information to extract the $u$ and $d$ valence PDFs and the
$\bar{U}$ and $\bar{D}$ PDFs, as well as the gluon PDF from scaling
violation. Briefly:
\begin{itemize}
\item
HERA NC $e^+$ reduced cross-section data at low $Q^2$ give information on
the shape of the sea
distribution at low $x$ region, whereas the high $Q^2$ NC $e^+$ and $e^-$
cross-sections not only extend the coverage to high $x$ ($x < 0.65$) but
also provide information on the valence combination
$xF_3 = x(B^0_u u_v + B^0_d d_v)$, which is extracted from the
difference between the high-$Q^2$ NC $e^+$ and $e^-$ cross sections.
The $x$ range of the valence measurement is, $0.01 < x < 0.65$.
\item
HERA CC data gives further information on flavour separation. The
$e^-$ cross-section at high-$x$ is $u$-valence dominated and the $e^+$
cross-section is $d$-valence dominated, giving unique information on the
$d$ quark. Fixed proton target data are
$u$-quark dominated, and so historically information on the $d$-quark has
been extracted from deuterium target
data or from neutrino scattering. However, each of these methods has
difficulties. The neutrino data uses heavy isoscalar targets such that
uncertain nuclear corrections~\cite{cjstuff} are necessary.
The deuterium target data also
needs some nuclear binding corrections~\cite{jlabstuff} and extraction of
the $d$-quark is
dependent on the assumption of strong isospin invariance.
\item
NC data on $F_2$ have also been used to constrain the gluon distribution.
Since the gluon does not couple to the
photon it does not enter the expressions for the structure functions at all
in the QPM. However, it is constrained by the momentum sum rule, and by the
way that gluon to quark-antiquark splitting feeds into the sea distributions
(from the $P_{qg}$ term in the DGLAP equations).
The shape of the gluon distribution
extracted from a DGLAP QCD fit will be correlated with
the value of $\alpha_s$, since
an increase in $\alpha_s$ increases the negative contribution from the
$P_{qq}$ term but this may be compensated by a positive contribution from
the $P_{qg}$ term if the gluon is made harder.
Hence, a fixed value of $\alpha_s$,
as determined from independent data, has often been assumed in PDF fits.
HERA data are invaluable in constraining the low-$x$gluon distribution,
since at small $x$ QCD evolution becomes gluon
dominated and the uncertainties referred to above are reduced.
This is because $F_2$ is essentially given
by the singlet sea quark distribution for $x \leqsim 0.01$, and this in
turn is driven by the gluon through the $P_{qg}$ term in
Eqn.~\ref{eqn:ap_gen}. The approximate LO relationship
\begin{equation}
xg(x)\simeq \frac{3\pi}{\bar{e}^2\alpha_s}\frac{\partial F_2(x/2,Q^2)}{\partial lnQ^2}
\end{equation}
illustrates how the gluon distribution depends on the scaling violation of
$F_2$ at low $x$.
Hence in this kinematic region the gluon distribution may be obtained
almost directly from the $F_2$ scaling violation data.
\item
Jet production data from HERA can give more direct information on the
gluon since vector-boson gluon fusion (BGF) to quark-anti-quark pairs makes a
significant
contribution to final state jet production. Such data has also been input to
the PDF fits to
constrain the gluon distribution in the $x$ range, $0.01 < x < 0.1$,
and to simultaneously determine
$\alpha_s(M_Z)$, see Sec.~\ref{sec:jets}.
\item
The longitudinal structure function $F_L$
can also give information on the gluon as can be seen from
Equation~\ref{eqn:fl_qg}. At low $x$ the dominant contribution comes from the
gluon and the integral over the gluon distribution approximates to a $\delta$
function such that a measurement of $F_L(x,Q^2)$ is almost a direct
measurement of the gluon distribution $yg(y,Q^2)$ at $y=2.5x$~\cite{amcs1987}.
The heavy quark structure functions $F_2^{c\bar{c}}$ and $F_2^{b\bar{b}}$
may also yield information on
the gluon since heavy quarks are generated by the BGF process. However,
currently such data are most useful for distinguishing between different
schemes for heavy quark production and fixing the value of the heavy quark mass
parameters that enter into these schemes, see Sec.~\ref{sec:charm}
\end{itemize}
Perturbative QCD predicts the $Q^2$ evolution of the parton distributions,
but
not the $x$ dependence.
The parton distributions are extracted by
performing a direct numerical integration of the DGLAP equations at NLO and
NNLO~\cite{nnloevol}.
For most PDF extractions (the notable exception is the NNPDF analysis)
a parametrised analytic shape for the parton distributions (valence, sea
and gluon) is assumed to be valid at some starting value of $Q^2 = Q^2_0$.
This starting
value is arbitrary, but should be large enough to ensure that
$\alpha_s(Q^2_0)$ is small enough for perturbative calculations to be
applicable. For the HERAPDF the value $Q^2_0 = 1.9$GeV$^2$ is chosen such
that the starting scale is below the charm mass threshold, $Q_0^2 < m_c^2$.
Then the DGLAP equations
are used to evolve the parton distributions up to higher $Q^2$ values,
where they are convoluted with coefficient functions to make predictions
for the structure functions and cross sections.
These predictions are then fitted to data to
determine the PDF parameters, and thus the shapes of the parton distributions
at the starting scale and, through evoution, at any other value of $Q^2$.
The QCD evolution is performed using the programme
QCDNUM~\cite{qcdnum}. The HERADF uses the \msbar renormalisaton
scheme, with the renormalisation and factorisation scales chosen to be
$Q^2$. The light quark coefficient functions~\cite{nlo,nnlo}
are calculated using the programme QCDNUM. The heavy quark
coefficient functions are calculated in the general-mass
variable-flavour-number scheme of \cite{Thorne:1997ga}, with recent
modifications and extension to NNLO~\cite{Thorne:2006qt,ThornePrivComm}.
(This scheme will be called the RT-VFN scheme).
The heavy quark masses for the central fit were chosen to be $m_c=1.4~$GeV
and $m_b=4.75~$GeV and the strong coupling constant was fixed to
$\asmz = 0.1176$. These choices are varied to evaluate model
uncertainties. The predictions are then fitted to the
combined HERA data sets on differential cross sections
for NC and CC $e^+p$ and $e^-p$ scattering.
A minimum $Q^2$ cut, $Q^2_{min} = 3.5$~GeV$^2$, was imposed
to remain in the kinematic region where
perturbative QCD should be applicable. This choice is also varied when
evaluating model uncertainties. It is also conventional to apply a
minimum cut
on $W$, invariant mass of the hadronic system, to avoid sensitivity to
target mass and large-$x$ higher-twist contributions. However the HERA data
have $W>15$\,GeV and $x< 0.65$, so that no further cuts are necessary.
PDFs were parametrised at the input scale by the generic form
\begin{equation}
xf(x) = A x^{B} (1-x)^{C} (1 + \epsilon\surd x + D x + E x^2).
\label{eqn:pdf}
\end{equation}
The parametrised PDFs are the gluon distribution $xg$, the valence quark
distributions $xu_v$, $xd_v$, and the $u$-type and $d$-type anti-quark
distributions
$x\bar{U}$, $x\bar{D}$. Here $x\bar{U} = x\bar{u}$,
$x\bar{D} = x\bar{d} +x\bar{s}$, at the chosen starting scale.
The normalisation parameters, $A_g, A_{u_v}, A_{d_v}$, are constrained
by the quark number sum-rules and momentum sum-rule.
The $B$ parameters $B_{\bar{U}}$ and $B_{\bar{D}}$ are set equal,
$B_{\bar{U}}=B_{\bar{D}}$, such that
there is a single $B$ parameter for the sea distributions.
The strange quark distribution is expressed
as $x$-independent fraction, $f_s$, of the $d$-type sea,
$x\bar{s}= f_s x\bar{D}$ at $Q^2_0$. For $f_s=0.5$ the
$s$ and $d$ quark densities would be the same, but the value $f_s=0.31$
is chosen to be consistent with determinations
of this fraction using neutrino-induced di-muon
production~\cite{Martin:2009iq,Nadolsky:2008zw}.
This choice is varied when evaluating model uncertainties.
The further constraint
$A_{\bar{U}}=A_{\bar{D}} (1-f_s)$, together with the requirement
$B_{\bar{U}}=B_{\bar{D}}$, ensures that
$x\bar{u} \rightarrow x\bar{d}$ as $x \rightarrow 0$.
For the HERAPDF1.0 and 1.5 NLO central fits, the valence $B$ parameters,
$B_{u_v}$ and $B_{d_v}$ are also set equal,
but this assumption is dropped for fits using the extended paramterisation.
The form of the gluon parametrisation is also extended for these latter
fits such that a term of the form $A_g'x^{B_g'}(1-x)^{C_g'}$ is subtracted
from the standard parametrisation, where $C_g' = 25$ is
fixed and $A_g'$ and $B_g'$ are fitted. This allows for the gluon
distribution to become negative at low $x,Q^2$,
although it does not do so within the kinematic range of the fitted data.
The central
fit is found by first setting the $\epsilon$, $D$ and $E$ parameters to
zero and then varying them, one at a time, the best fit is achieved for
$E_{u_v}\not= 0$. This is then adopted as standard and the other $\epsilon$, $D$ and $E$ parameters are then varied, one at a
time, However these fits do not
represent a significant improvement in fit quality for the HERAPDF1.0, 1.5 and
1.7 NLO fits, and thus
a central fit with just $E_{u_v}\not= 0$ is chosen.
For the HERAPDF1.5f, 1.6 and HERAPDF1.5NNLO fit an extra parameter, $D_{u_v}\not= 0$ is used.
The HERAPDF1.0 and 1.5 NLO fits have $10$ parameters,
and the 1.5f, 1.6 NLO and 1.5NNLO fits have $14$ parameters and the
HERAPDF1.7NLO fit has $13$ parameters.
The assumptions made in setting the parameters for this central fit are now
discussed:
\begin{itemize}
\item In common with most PDF fits it is assumed that $q_{sea} = \bar{q}$.
\item
The HERAPDF parametrizes $\bar U$ and $\bar D$ separately to allow for the
fact that $\bar u \ne \bar d$ at high $x$, but the restriction
$x\bar u \to x\bar d$ as $x \to 0$ is imposed.
\item
The strange sea is suppressed. However determinations of the degree of
suppression are not very accurate and hence model uncertainty on this
fraction is evaluated by allowing the variation, $0.23 < f_s < 0.38$.
\item
The $u$-valence and $d$-valence shapes are parametrized separately, but the
form of the parametrization imposes $d_v/u_v = (1-x)^p$ as $x \to 1$.
\item The heavy quarks are treated using a General-Mass-Variable-Flavour
Number-Scheme. There is some model uncertainty in the choice of the heavy
quark masses. The ranges $1.35 < m_c < 1.65~$GeV and $4.3 < m_b < 5.0~$GeV
are considered as model variations. There are also different heavy
quark schemes. The ACOT
scheme~\cite{acot} has been used as a cross-check to the Thorne-Roberts scheme.
\item All PDF extractions make choices concerning the fitted kinematic
region, i.e the minimum values of $Q^2$, $W^2$, $x$.
These choices can have small systematic effects on the PDF shapes
extracted.
The choice of $Q^2_{min}$ is varied in the range $2.5 < Q^2_{min} < 5.0$.
\item The PDFs
extracted for $Q^2 \gg Q^2_0$ lose sensitivity to the exact form of the
parametrisation at $Q^2_0$. However the choices of $Q^2_0$ and of the form
of parametrisation represent a parametrisation uncertainty. The HERAPDF
uses the technique of saturation of the $\chi^2$, increasing the number of
parameters systematically until the $\chi^2/ndf$ no longer decreases
significantly. However, a number of variations on the central fit
parametrisation, which have similar fit quality, are considered in order
to give an estimate of parametrization uncertainty. The value
of $Q^2_0$ is also varied in the range $1.5 < Q^2_0 < 2.5~$GeV$^2$
for the same
purpose.
\end{itemize}
Table~\ref{tab:model} summarizes the variations in numerical values
considered when evaluating model uncertainties on the HERAPDF.
\begin{table}[tbp]
\centerline{
\begin{tabular}{|l|l|l|r|}
\hline
Variation& Standard Value & Lower Limit & Upper Limit \\
\hline
$f_s$ & $0.31$ & $0.23$ & $0.38$ \\
$m_c$ [GeV] & $1.4$ & $1.35\, (Q^2_0=1.8)$ & $1.65$ \\
$m_b$ [GeV] & $4.75$ & $4.3$ & $5.0$ \\
$Q^2_{min}$ [GeV$^2$] & $3.5$ & $2.5$ & $5.0$ \\
$Q^2_0$ [GeV$^2$] & $1.9$ & $1.5\,(f_s=0.29)$ & $2.5\,(m_c=1.6,f_s=0.34)$
\\
\hline
\end{tabular}}
\caption{Standard values of input parameters and the variations
considered.
}
\label{tab:model}
\end{table}
Note that the variations of $Q^2_0$ and $f_s$ are not independent, since
QCD evolution
will ensure that the strangeness fraction increases as $Q^2_0$ increases.
The value $f_s=0.29$ is used for $Q^2_0=1.5~$GeV$^2$ and the value
$f_s=0.34$
is used for $Q^2_0=2.5~$GeV$^2$ in order
to be consistent with the choice $f_s=0.31$ at $Q^2_0=1.9~$GeV$^2$.
The variations of $Q^2_0$ and $m_c$ are also not independent,
since $Q_0 < m_c$ is required in the fit programme. Thus when $m_c =
1.35~$\,GeV,
the starting scale used is
$Q^2_0=1.8~$\,GeV$^2$. Similarly, when $Q^2_0 = 2.5~$GeV$^2$ the
charm mass used is $m_c=1.6~$GeV. In practice, the variations of $f_s$,
$m_c$,
$m_b$, mostly affect the model uncertainty of the $x\bar{s}$, $x\bar{c}$,
$x\bar{b}$, quark distributions, respectively, and have little effect on
other parton flavours.
The difference between the central fit and the fits corresponding to model
variations of $m_c$, $m_b$, $f_s$, $Q^2_{min}$ are
added in quadrature, separately for positive and negative deviations, to
represent the model uncertainty of the HERAPDF sets.
The variation in $Q^2_0$ is regarded as a parametrisation uncertainty,
rather than a model uncertainty. The variations of $Q^2_0$ mostly increase
the PDF uncertainties of the sea and gluon at small $x$.
At the starting scale the gluon shape is valence-like, so
for the downward variation of the starting scale, $Q^2_0 = 1.5~$GeV$^2$,
a gluon parametrisation
which explicitly allows for a negative gluon contribution at low $x$
is considered for the 1.0 and 1.5 NLO fits- in all other HERAPDF fits it
is already a standard part of the parametrisation.
Similarly a parametrisation variation, $B_{u_v}\not= B_{d_v}$,
which is standard for the 1.5f, 1.6 and 1.7 NLO and the 1.5NNLO fits,
is also allowed for the 1.0 and 1.5 NLO fits. This increases the
uncertainties on the valence quarks at low $x$. Finally, variation of
the number of terms in
the polynomial $(1 + \epsilon\surd x + D x + E x^2)$ is considered for each
fitted parton distribution. In practice only a small number of these
variations have significantly different PDF shapes from the central fit,
notably: $D_{u_v}\not= 0$ (standard for 1.5f, 1.6 NLO and 1.5NNLO), $D_{\bar{U}}\not= 0$
and $D_{\bar{D}}\not=0$.
These variations mostly increase the PDF uncertainty at high $x$,
but the valence PDFs at low $x$ are also affected
because of the constraints of the quark number sum rules.
The difference between all these parametrisation
variations and the central fit is stored
and an envelope representing the maximal deviation at each $x$ value
is constructed
to represent the parametrisation uncertainty.
The HERAPDF uses a form of the $\chi^2$ specified in ref~\cite{h1zeuscomb}
to perform the fit of the predictions to the HERA data.
The consistency of the input data
justifies the use of the conventional
$\chi^2$ tolerance, $\Delta\chi^2=1$, when determining the $68\%$C.L.
experimental uncertainties on the HERAPDF1.0 fit.
Modern deep inelastic scattering experiments
have very small statistical uncertainties, so that the contribution of
correlated
systematic uncertainties has become dominant for individual data sets and
consideration of the treatment of such errors is essential.
However, the HERA data combination has changed this situation. The
combination of the H1 and ZEUS data sets has resulted in a data set for NC
and CC $e^+p$ and $e^-p$ scattering with correlated systematic
uncertainties which are smaller or comparable to the statistical and
uncorrelated uncertainties.
Thus the central values and experimental uncertainties on the
PDFs which are extracted from the combined data are not much dependent
on the method of treatment of correlated systematic uncertainties in the
fitting procedure. For the HERAPDF1.0(1.5) NLO central fit,
the $110(131)$ systematic uncertainties
which result from the ZEUS and H1 data sets are combined
in quadrature, and the three sources of uncertainty which result from
the combination procedure are treated as correlated
by the Offset method~\cite{offhesse}.
The resulting experimental uncertainties on the PDFs are small.
For the HERAPDF1.5f, 1.6, 1.7 NLO fits and the HERAPDF1.5 NNLO fit it
was decided
to treat the three procedural errors as correlated by the Hessian
method~\cite{offhesse}.
This has a negligible effect on the size of the experimental uncertainties
and a small effect on the resulting $\chi^2$ value,
see Sec.~\ref{sec:herapdf1.5}.
The total PDF uncertainty is obtained by adding in quadrature experimental,
model and parameterisation uncertainties.
\subsection{Results from the HERAPDF fit}
\label{sec:results}
\subsubsection{HERAPDF1.0}
We first discuss results from the published HERAPDF1.0 fit. This fit has a
$\chi^2$ per degree of freedom of $574/582$.
Fig~\ref{fig:summary} shows summary plots of the HERAPDF1.0 PDFs at
$Q^2=10~$GeV$^2$.
\begin{figure}[tbp]
\vspace{-0.5cm}
\centerline{
\epsfig{figure=herapdf1.0_summary_10.eps ,width=0.5\textwidth}}
\vspace*{-0.6cm}
\caption {
The parton distribution functions from
HERAPDF1.0, $xu_v,xd_v,xS=2x(\bar{U}+\bar{D}),xg$, at
$Q^2 = 10$~GeV$^2$.
The experimental, model and parametrisation
uncertainties are shown separately. The gluon and sea
distributions are scaled down by a factor $20$.
}
\label{fig:summary}
\end{figure}
Figs~\ref{fig:pdfs19}-\ref{fig:pdfs10000} show the HERAPDF1.0
distributions,
$xu_v,xd_v,xS,xg$, as a function of $x$ at
$Q^2=10, 10000$~GeV$^2$,
where $xS =2x (\bar{U} + \bar{D})$ is the sea PDF. Note that for $Q^2 >
m_c^2$,
$x\bar{U} = x\bar{u} + x\bar{c}$, and for $Q^2 > m_b^2$,
$x\bar{D} = x\bar{d} +x\bar{s} +x\bar{b}$, so that the heavy quarks are
included in the sea distributions. The break-up of $xS$ into the flavours
$xu_{sea}=2x\bar{u}$, $xd_{sea}=2x\bar{d}$, $xs_{sea}=2x\bar{s}$,
$xc_{sea}=2x\bar{c}$, $xb_{sea}=2x\bar{b}$ is illustrated so that the
relative
importance of each flavour at different $Q^2$ may be assessed.
Fractional uncertainty bands are shown below each PDF. The experimental,
model and parametrisation uncertainties are shown separately.
The model and parametrisation uncertainties are asymmetric. For the sea
and gluon distributions, the variations in parametrisation which have
non-zero $\epsilon$, $D$ and $E$ affect the large-$x$
region, and the uncertainties arising from the
variation of $Q^2_0$ and $Q^2_{min}$ affect the small-$x$
region. For the valence distributions the non-zero $\epsilon$, $D$ and $E$
parametrisation uncertainty is important
for all $x$, and is their dominant uncertainty.
The total uncertainties at low $x$ decrease with increasing $Q^2$ due to
QCD evolution resulting, for instance,
in $2\%$ uncertainties for $xg$ at $Q^2=10000$~GeV$^2$ for $x<0.01$.
\begin{figure}[tbp]
\vspace{-0.3cm}
\centerline{
\epsfig{figure=herapdf1.0_uvdvSg_1.9.eps ,width=0.4\textheight}}
\caption {
The parton distribution functions from
HERAPDF1.0, $xu_v,xd_v,xS=2x(\bar{U}+\bar{D}),xg$, at
$Q^2=1.9~$GeV$^2$. The break-up of the Sea PDF, $xS$, into the flavours,
$xu_{sea}=2x\bar{u}$, $xd_{sea}=2x\bar{d}$, $xs_{sea}=2x\bar{s}$,
$xc_{sea}=2x\bar{c}$ is illustrated. Fractional uncertainty bands are shown
below each PDF.
The experimental, model and parametrisation
uncertainties are shown
separately.
\label{fig:pdfs19}}
\end{figure}
\begin{figure}[tbp]
\vspace{-0.3cm}
\centerline{
\epsfig{figure=herapdf1.0_uvdvSg_10_sea.eps ,width=0.4\textheight}}
\caption {
The parton distribution functions from
HERAPDF1.0, $xu_v,xd_v,xS=2x(\bar{U}+\bar{D}),xg$, at
$Q^2=10~$GeV$^2$. The break-up of the Sea PDF, $xS$, into the flavours,
$xu_{sea}=2x\bar{u}$, $xd_{sea}=2x\bar{d}$, $xs_{sea}=2x\bar{s}$,
$xc_{sea}=2x\bar{c}$ is illustrated. Fractional uncertainty bands are shown
below each PDF.
The experimental, model and parametrisation
uncertainties are shown
separately.
\label{fig:pdfs10}}
\end{figure}
\begin{figure}[tbp]
\vspace{-0.3cm}
\centerline{
\epsfig{figure=herapdf1.0_uvdvSg_10000_sea.eps ,width=0.4\textheight}}
\caption {
The parton distribution functions from
HERAPDF1.0, $xu_v,xd_v,xS=2x(\bar{U}+\bar{D}),xg$, at
$Q^2=10,000~$GeV$^2$. The break-up of the Sea PDF, $xS$, into the flavours,
$xu_{sea}=2x\bar{u}$, $xd_{sea}=2x\bar{d}$, $xs_{sea}=2x\bar{s}$,
$xc_{sea}=2x\bar{c}$, $xb_{sea}=2x\bar{b}$ is illustrated.
Fractional uncertainty bands are shown
below each PDF. The experimental, model and parametrisation
uncertainties are shown
separately.}
\label{fig:pdfs10000}
\end{figure}
The break-up of the PDFs into different flavours is further illustrated in
Fig~\ref{fig:pdfsudcs}, where the quark distributions
$x\bar{u}, x\bar{d}, x\bar{c}, x\bar{s}$ are shown at $Q^2=10$~GeV$^2$.
The $u$ flavour is better constrained than the $d$ flavour because of
the dominance of this flavour in all interactions except $e^+p$ CC
scattering.
The quark distribution $x\bar{s}$ is derived from $x\bar{D}$
through the assumption on the value of $f_s$, and the uncertainty on
$x\bar{s}$
directly reflects the uncertainty on this fraction.
The charm PDF, $x\bar{c}$, is strongly related to the
gluon density such that it is affected by the same variations which
affect the gluon PDF (variation of $Q^2_0$ and
$Q^2_{min}$) as well as by the variation of $m_c$.
The uncertainty on the
bottom PDF, $x\bar{b}$ (not shown), is dominated by the variation of $m_b$.
\begin{figure}[tbp]
\vspace{-0.3cm}
\centerline{
\epsfig{figure=herapdf1.0_udcs_10.eps ,width=0.4\textheight}}
\vspace{0.5cm}
\caption {
The parton distribution functions from
HERAPDF1.0, $x\bar{u}, x\bar{d}, x\bar{c}, x\bar{s}$ at
$Q^2=10~$GeV$^2$.
Fractional uncertainty bands are shown
below each PDF. The experimental, model and parametrisation
uncertainties are shown
separately.}
\label{fig:pdfsudcs}
\end{figure}
The shapes of the gluon and the sea distributions can be compared by
considering Figs~\ref{fig:pdfs19}-\ref{fig:pdfs10000}.
For $Q^2 \geqsim 10\,$GeV$^2$, the gluon density rises dramatically
towards low $x$
and this rise increases with increasing $Q^2$. This rise is one of the most
striking discoveries of HERA.
However, at low $Q^2$ the gluon shape flattens at low $x$. At
$Q^2 = 1.9\,$GeV$^2$, the gluon shape
becomes valence like and the parametrisation variation which includes a
negative
gluon term increases the uncertainty on the gluon at low $x$.
However the gluon distribution itself is not negative in the fitted
kinematic region.
The uncertainty in the sea distribution
is considerably less that that of the gluon distribution.
For $Q^2 > 5$~GeV$^2$, the gluon density becomes much
larger than the sea density, but for lower $Q^2$ the sea density continues
to
rise at low $x$, whereas the gluon density is suppressed. This may be a
signal
that the application of the DGLAP NLO formalism for $Q^2 \leqsim 5$~GeV$^2$
is questionable. Kinematically low $Q^2$ HERA
data is also at low $x$ and the DGLAP formalism may be indequate at
low $x$ since it is missing
$ln(1/x)$ resummation terms and possible non-linear effects
- see Ref.~\cite{lowx}.
Discussion of this topic is beyond the scope of the
present review. PDF fits within the DGLAP formalism are successful down to
$Q^2\sim 2$~GeV$^2$ and $x\sim 10^{-4}$ and this is the kinematic region
considered in the present review.
\subsubsection{Including Heavy Quark data in PDF fits}
\label{sec:charm}
The HERA combined charm data have been presented in Sec.~\ref{sec:charmdata}.
Fig.\ref{fig:f2cfit} shows the comparison of the HERA combined measurements
of $F^{c\overline{c}}_2$ with the predictions of the HERAPDF1.0 fit.
These data can of course be included in the fit. There are $41$ jet data points
and, for the preliminary combination, these are provided with
uncorrelated systematic errors and a single combined source of correlated
error which was treated by the Offset method. The $\chi^2$ for the inclusive
data is hardly changed by the addition of the charm data but for the $\chi^2$
for the charm data is very sensitive to the charm mass and the scheme used
for heavy flavour treatment~\cite{hqscan}.
It is found that in order to obtain a good fit using the standard
Thorne-Roberts variable Flavour Number Scheme (RT-VFN)~\cite{} it
is necessary to increase the standard value of the charm mass.
Fig~\ref{fig:mcchiscan} shows a scan of the $\chi^2$ of the HERAPDF1.0 fit to
the inclusive
HERA-I data vs the charm quark mass parameter entering into the standard RT-VFN
scheme. In the same figure a scan for a similar fit to the inclusive
HERA-I data plus
the combined $F^{c\overline{c}}_2$ data is shown.
The sensitivity of the charm data to
the charm quark mass parameter is clear.
\begin{figure}[tbp]
\vspace{-0.5cm}
\centerline{
\epsfig{figure=mscan_nocharm.eps ,width=0.4\textwidth}
\epsfig{figure=mcscan_charm.eps ,width=0.4\textwidth}}
\caption { $\chi^2$ scan vs the charm quark mass $m_c$ for the HERAPDF1.0 fit
to just HERA-I inclusive data (left) and a fit which also inlcudes combined
HERA $F^{c\overline{c}}_2$ data (right). Both of these fits use the RT-VFN heavy quark scheme
}
\label{fig:mcchiscan}
\end{figure}
However the Standard RT-VFN scheme is not the only possible heavy quark
scheme.
The fit to HERA-I inclusive plus $F^{c\overline{c}}_2$ data has been repeated
for the Optimized RT-VFN scheme~\cite{rtopt}, the full ACOT scheme,
the S-ACOT-$\chi$ scheme~\cite{sacot}
and the Zero-Mass Variable Flavour Number Scheme (ZM-VFN) in which
light-quark coefficient functions are used for the heavy quarks,
which are simply turned on at threshold $Q^2\sim m_c^2$.
Fig.~\ref{fig:allcharm_wp}
shows the $\chi^2$ scan for these different heavy quark schemes. It can be seen
that all schemes, bar the ZM-VFN, give acceptable fits, and that each scheme
has its own preferred value of the charm quark mass. These values and the
correspondin $\chi^2$ values are given in
Table~\ref{tab:mc}. Fig~\ref{fig:f2cfitall} shows these fits compared to the
charm data.
\begin{figure}[tbp]
\vspace{-0.5cm}
\centerline{
\epsfig{figure=mcscan_allcharm.eps ,width=0.4\textwidth}
\epsfig{figure=wp_allcharm.eps ,width=0.4\textwidth}}
\caption {Left: $\chi^2$ scan vs the charm quark mass $m_c$ for a fit to
HERA-I inclusive and $F^{c\overline{c}}_2$ data for various heavy quark
schemes. Right: Predictions for the $W^+$ cross-section at the LHC (7TeV)
for these schemes vs $m_c$. The value of the charm mass parameter which gives
the minimum $\chi^2$ is marked by a star for each scheme.
}
\label{fig:allcharm_wp}
\end{figure}
\begin{table}[tbp]
\centerline{
\begin{tabular}{|l|r|r|}
\hline
Scheme& $m_c$(minimum)& $\chi^2$/ndp $F^{c\overline{c}}_2$ \\
\hline
RT Standard & $1.58^{+0.02}_{-0.03}$ & $42.0/41$\\
RT Optimized & $1.46^{+0.02}_{-0.04}$ & $46.5/41$\\
ACOT-full & $1.58^{+0.03}_{-0.04}$ & $59.9/41$ \\
S-ACOT $\chi$ & $1.26^{+0.02}_{0.04}$ & $68.5/41$\\
ZM-VFN & $1.68^{+0.06}_{-0.07}$ & $88.1/41$ \\
\hline
\end{tabular}
}
\caption{Charm mass parameters and $\chi^2$ values per number of data points
($ndp$) for fits to
$F^{c\overline{c}}_2$ data using various heavy quark schemes.
}
\label{tab:mc}
\end{table}
\begin{figure}[tbp]
\vspace{-0.2cm}
\centerline{
\epsfig{figure=f2cfit_allcharm.eps ,width=0.5\textwidth}}
\caption {
$F^{c\overline{c}}_2$ data compared to the predictions of various
heavy quark schemes, within the HERAPDF fit formalism.
}
\label{fig:f2cfitall}
\end{figure}
Predictions for $W^+, W^-, Z$ production at the LHC are sensitive to the value
of the charm mass and to the heavy quark scheme used, as illustrated
in Fig.~\ref{fig:allcharm_wp}. For any chosen value of the charm mass the
spread of predictions for different schemes is $\sim 7\%$. However if each
prediction is used at its own favoured value of the charm mass then this
spread is reduced to $\sim2\%$ and, if the disfavoured ZM-VFN is excluded
to$ \leqsim 1\%$.
\subsubsection{Including Low Energy run data in PDF fits}
\label{sec:lowEfit}
The preliminary combined data from the low energy running described in
Sec~\ref{sec:lowE} have been input to the HERAPDF fit together with the
HERA-I combined high energy data. Thess data have $25$ sources of correlated
systematic uncertainty from the individula experiments, and $3$ procedural
sources of systematic uncertainty similarly to the high energy combination.
These correlated errors are added
in quadrature except for the $3$ procedural which are treated as fully
correlated by the Hessian method.
There are $224$ combined data points on the NC $e^+ p$ cross section from the
low energy proton beam running and
when they are fit together with the $592$ combined data points from the HERA-I
running the
$\chi^2/ndf$ is $845.7/806$ for $10$ parameters. The partial $\chi^2/ndp$ are
$588/592$ for the high energy inclusive data and $257.6/224$ for the
low energy inclusive
data. These data are sensitive to the minimum $Q^2$
cut imposed, as illustrated in Fig.~\ref{fig:460575}. A better fit is obtained
with a larger, $Q^2>5$~GeV$^2$, cut.
The partial $\chi^2/ndp$ after this cut are
$527.1/566$ and $200/215$ for the low energy data.
The data at low $x,Q^2$ access high $y$ and thus
sensitive to the longitudinal structure function $F_L$. Because of the close
relationship of $F_L$ and the gluon PDF these data should affect the
gluon PDF. This is illustrated in Fig.~\ref{fig:lowEpdf} which
shows that variation of the $Q^2$-cut
affects the gluon PDF more for the fit including low energy data, since the
result is outside the error bands which include this cut-variation for the
HERAPDF1.0 fit.
Kinematically cutting out low $Q^2$ data also implies cutting out data at the
lowest $x$ and the data are similarly sensitive to an $x>0.0005$ cut. Data at
low $x$ may not be well fit by the DGLAP formalism since this is missing
$ln(1/x)$ resummation terms and possible non-linear effects.
Fig~\ref{fig:lowEpdf} also illustrates sensitivity to
a 'saturation' inspired cut of $Q^2 > 1.0x^{-0.3}~$GeV$^2$.
\begin{figure}[tbp]
\vspace{-0.5cm}
\centerline{
\epsfig{figure=460.eps ,width=0.4\textwidth}
\epsfig{figure=575.eps ,width=0.4\textwidth}}
\caption {The combined HERA data from running with proton beam energies
$E_p=460$~GeV and $E_p=575$~GeV is shown for a few low-$Q^2$ bins, compared to
predictions from PDF fits to these data and the combined HERA-I high energy
data. Predictions are shown for data subject to two different minimum
$Q^2$ cuts: $3.5~$GeV$^2$ and $5.0~$GeV$^2$.
}
\label{fig:460575}
\end{figure}
\begin{figure}[tbp]
\vspace{-0.5cm}
\centerline{
\epsfig{figure=lowenergy_summary.eps ,width=0.4\textwidth}
\epsfig{figure=acut10.eps,width=0.4\textwidth}}
\caption { The parton distribution functions from
HERAPDF1.0, $xu_v,xd_v,xS=2x(\bar{U}+\bar{D}),xg$, at
$Q^2 = 10$~GeV$^2$.
The total uncertainties are shown. The gluon and sea
distributions are scaled down by a factor $20$. (Left) The lines overlayed
show the results of fits to the HERA-1 data plus
the low energy running data with the standard minimum $Q^2$ cut of
$3.5~$GeV$^2$ and with a harder cut of $5.0~$GeV$^2$. (Right)
The lines overlayed
show the results of fits to the HERA-1 data and to the HERA-1 plus
the low energy running data, with the 'saturation inspired'
cut of $Q^2 > 1.0 x^{-0.3}$GeV$^2$.
}
\label{fig:lowEpdf}
\end{figure}
However, one cannot claim that any break-down of the DGLAP formalism has yet
been observed, since if the HERAPDF1.0 formalism is generalised to
the extended parametrisation with $14$ parameters,
then the increased uncertainty in the low-$x$
gluon, illustrated in Fig.~\ref{fig:herapdf1.5f}, covers the sensitivity of
the low energy data to the low $x,Q^2$ cuts.
\subsubsection{HERAPDF1.5}
\label{sec:herapdf1.5}
The HERAPDF1.5~NLO fit uses the same formalism as HERAPDF1.0 but includes
preliminary HERA-I+II data.
The $\chi^2$ per degree of freedom for the HERAPDF1.5~NLO central fit is
$760/664$, where the increased $\chi^2$ reflects the greater accuracy of the
HERA-I+II combination. This fit has already been compared to the data in
Figs.~\ref{fig:ncepem2}-~\ref{fig:dataCCm2}.
The improvement to the PDFs is illustrated in Fig.~\ref{fig:herapdf15},
which shows the HERAPDF1.5 in a format
such that it may be directly compared with
HERAPDF1.0 in Fig.~\ref{fig:summary}. Fig.~\ref{fig:herapdf1510} shows the
HERAPDF1.5 overlayed on HERAPDF1.0 on a linear $x$ scale, such that the
improvement at high $x$ may be clearly seen.
\begin{figure}[tbp]
\vspace{-0.2cm}
\centerline{
\epsfig{figure=herapdf1.5_summary_10.eps ,width=0.5\textwidth}}
\caption {The parton distribution functions from
HERAPDF1.5, $xu_v,xd_v,xS=2x(\bar{U}+\bar{D}),xg$, at
$Q^2 = 10$~GeV$^2$. The experimental, model and parametrisation
uncertainties are shown separately. The gluon and sea
distributions are scaled down by a factor $20$.
}
\label{fig:herapdf15}
\end{figure}
\begin{figure}[tbp]
\vspace{-0.2cm}
\centerline{
\epsfig{figure=herapdf1.5_1.0_10.eps,width=0.5\textwidth}}
\caption {The parton distribution functions from
HERAPDF1.5, $xu_v,xd_v,xS=2x(\bar{U}+\bar{D}),xg$, at
$Q^2 = 10$~GeV$^2$, overlayed on the parton distributions from HERAPDF1.0.
The total uncertainties for each PDF are shown. A linear scale in $x$ is used
to emphasize the reduction in uncertainties for HERAPDF1.5 at high $x$.
}
\label{fig:herapdf1510}
\end{figure}
The fit formalism was also extended
to included more PDF parameters,
as already described in Sec.~\ref{sec:pdfchap}.
This fit, called HERAPDF1.5f NLO fit, has $\chi^2$ per degree of freedom
$730/664$, where the improvement to the $\chi^2$ is mostly due to the
treatment of the three procedural systematic errors by the
Hessian rather than Offset method. There is a small decrease in $\chi^2$,
$\Delta\chi^2=-5$ due to the increase of the number of parameters from $10$
to $14$. The PDFs of the HERAPDF1.5f and 1.5 NLO fits are
compared in Fig.~\ref{fig:herapdf1.5f},
where one can see that the extra freedom in the parametrisation does not
change the central values of the PDFs significantly. The total size of the PDF
uncertainties are also not changed significantly, although some of the
parametrisation
uncertainty in HERAPDF1.5 is now included in the
experimental uncertainty in HERAPDF1.5f.
The most significant change to the uncertainties is a modest
increase in the uncertainty of the low-$x$ gluon. This covers the sensitivity
to low-$x, Q^2$ cuts found in the low energy data combination,
see Sec.~\ref{sec:lowEfit}.
\begin{figure}[tbp]
\vspace{-0.5cm}
\centerline{
\epsfig{figure=herapdf1.5.eps ,width=0.4\textheight}}
\vspace*{-0.2cm}
\centerline{
\epsfig{figure=herapdf1.5f.eps ,width=0.4\textheight}}
\caption {
The parton distribution functions from
HERAPDF1.5 and HERAPDF1.5f, $xu_v,xd_v,xS=2x(\bar{U}+\bar{D}),xg$, at
$Q^2 = 10$~GeV$^2$. Fractional uncertainty bands are shown
below each PDF.
The experimental, model and parametrisation
uncertainties are shown separately.
}
\label{fig:herapdf1.5f}
\end{figure}
The HERAPDF1.5 NNLO fit was performed on the same preliminary combined
HERA I+II data. The $\chi^2$ per degree of freedom for
for the HERAPDF1.5~NNLO central fit it is $740/664$. For this NNLO
fit the addition
of extra parameters made a significant difference to the $\chi^2$,
The change from a 10 to 14 parameter fit, results in a change of
$\Delta\chi^2=-32$ with the
largest difference coming from the addition of the term which allows
freedom in the low $x$ gluon. Fig.~\ref{fig:herapdf15nnlovs10} compares the
HERAPDF1.5~NNLO fit to HERAPDF1.0~NNLO which was an NNLO version of the
HERAPDF1.0 using just $10$ parameters and fitting just HERA-I data. One can
see that the extra parameters give somewhat different shapes to the valence
quarks and a much harder high-$x$ gluon PDF.
\begin{figure}[tbp]
\vspace{-0.2cm}
\centerline{
\epsfig{figure=herapdf1.5nnlo_vs1.0nnlo.eps,width=0.5\textwidth}}
\caption {The parton distribution functions from
HERAPDF1.5 NNLO, $xu_v,xd_v,xS=2x(\bar{U}+\bar{D}),xg$, at
$Q^2 = 10$~GeV$^2$, compared to HERAPDF1.0 NNLO.
A linear scale in $x$ is used
to emphasize the differences at high $x$.
}
\label{fig:herapdf15nnlovs10}
\end{figure}
Fig.~\ref{fig:herapdf15nnlovsnlo}
compares the HERAPDF1.5NNLO fit to the corresponding NLO fit HERAPDF1.5f.
These fits have the same number of parameters.
The change from NLO to NNLO gives a somehat steeper sea and softer gluon at
low $x$ consistent with the different rates of evolution at NNLO. The
most striking difference is the greater level of uncertainty at low $x$ for
the NNLO fit. This is mostly due to sensitivity to the low $Q^2$ cut on the
data. One might have expected that an NNLO fit would fit low $x,Q^2$ data
better than an NLO fit, however this would seem not to be the case,
see also ref.~\cite{caola}.
\begin{figure}[tbp]
\vspace{-0.2cm}
\centerline{
\epsfig{figure=herapdf1.5_nnlovsnlo.eps,width=0.5\textwidth}}
\caption {The parton distribution functions from
HERAPDF1.5NNLO, $xu_v,xd_v,xS=2x(\bar{U}+\bar{D}),xg$, at
$Q^2 = 10$~GeV$^2$, compared to the HERAPDF1.5f NLO fit.
The gluon and sea
distributions are scaled down by a factor $20$.
}
\label{fig:herapdf15nnlovsnlo}
\end{figure}
\subsubsection{Including jet data in PDF fits: HERAPDF1.6}
\label{sec:jets}
The gluon PDF contributes only indirectly to the
inclusive DIS cross sections. However,
the QCD processes that give rise to scaling violations in the
inclusive cross sections, namely the QCD-Compton (QCDC) and
boson-gluon-fusion
(BGF) processes, can also be
observed as events with distinct jets in the final
state provided that the energy and momentum transfer are large enough. The
cross section for QCDC scattering depends on $\alpha_s(M_Z)$ and the quark
PDFs. The cross section for the
BGF process depends on $\alpha_s(M_Z)$ and the gluon PDF. These two processes
are dominant in different kinematic regions. Thus jet cross
sections give new information about the PDFs. For the inclusive data,
the correlation between
$\alpha_s(M_Z)$ and the gluon PDF limits the accuracy with which either can be
determined. The jet data bring new information which helps to reduce the
overall correlation.
In the HERAPDF1.6 NLO PDF fit the jet data sets presented in
Sec.~\ref{sec:jetdata} are fitted together with the preliminary HERA I+II
combined inclusive data. These data sets have $12$ correlated systematic
errors, which are treated as fully correlated by the Hessian method.
The predictions for the jet cross sections have been calculated to NLO in
QCD using the NLOjet++ program~\cite{nlojet} and have been input to the fit
by the FASTNLO interface~\cite{fastnlo}. The calculation of the NLO jet cross
sections is too slow to be used iteratively in a fit.
Thus NLOjet++ is used to compute LO and NLO weights
which are independent of $\alpha_s$ and the
PDFs. The FASTNLO program then calculates the NLO QCD cross sections,
by convoluting these weights with the PDFs and $\alpha_s$.
The predictions must be multiplied by
hadronisation corrections before they can be used to fit the data.
These were determined by using Monte Carlo (MC) programmes,
which model parton hadronisation to estimate the ratio of
the hadron- to parton-level cross sections for each bin.
The hadronisation corrections are generally within a few
percent of unity. The predictions for jet production were also corrected for
$Z^0$ contributions.
The fit is done with the same settings as for the HERAPDf1.5f fit.
The $\chi^2/ndf$ for the fit is $812/766$, for a fit to $674$ inclusive
data points and $106$ jet data points with $14$ parameters.
The partial $\chi^2$ of the data sets is $730/674$ for the inclusive data and
$82/106$ for the jet data.
Fig.~\ref{fig:herapdf1.6} shows the parton distributions and their
uncertainties for the HERAPDF1.6 fit. HERAPDf1.5f is also shown on this plot
as a blue line. The fit with jets has rather similar central PDFs values to
the fit without jets, apart from having a somewhat less hard high-$x$ sea.
The uncertainties
are also similar to those of HERAPDf1.5f, with a slightly reduced uncertainty
on the high-$x$ gluon. The quality of the fit to the jet data
establishes that NLO QCD is able simultaneously to describe both
inclusive cross sections and jet cross sections, thereby
providing a compelling demonstration of QCD factorisation.
\begin{figure}[tbp]
\vspace{-0.5cm}
\centerline{
\epsfig{figure=herapdf16_frac.eps ,width=0.4\textheight}}
\caption {
The parton distribution functions from
HERAPDF1.6, $xu_v,xd_v,xS=2x(\bar{U}+\bar{D}),xg$, at
$Q^2 = 10$~GeV$^2$. Fractional uncertainty bands are shown
below each PDF.
The experimental, model and parametrisation
uncertainties are shown separately. the central fit of
HERAPDF1.5f is shown as a blue line
}
\label{fig:herapdf1.6}
\end{figure}
The standard value of $\alpha_s(M_Z)$ used in the fits has been
$\alpha_s(M_Z)=0.1176$.
The correlation between $\alpha_s(M_Z)$ and the gluon
PDF is too strong to make an accurate detrmination of $\alpha_s(M_Z)$
using purely inclusive data, but the jet data are sensitive to $\alpha_s(M_Z)$
such that one may let it be a free parameter of the fit. The value of
$\alpha_s(M_Z)$ which results is
$
\alpha_S(M_Z) = 0.1202 \pm 0.0013(exp) \pm 0.0007(model/param) \pm 0.0012 (had) +0.0045/-0.0036(scale).
$
We estimate the model and parametrisation uncertainties for $\alpha_S(M_Z)$
in the same way
as for the PDFs and we also add the uncertainties in the hadronisation
corrections applied to the jets. The scale uncertainties are estimated by
varying the renormalisation and factorisation scales chosen in the jet
publications by a factor of two up and down. The dominant contribution to the
uncertainty comes
from the jet renormalisation scale variation.
Fig.~\ref{fig:chiscan} shows a $\chi^2$ scan vs $\alpha_S(M_Z)$ for the fits
with and without jets, illustrating how much better $\alpha_S(M_Z)$ is
determined when jet data are included. The model and parametrisation errors
are also much better controlled.
\begin{figure}[htb]
\begin{center}
\includegraphics[width=0.4\textwidth]{chiscan.eps}
\caption {The difference between $\chi^2$ and its minimum value for the
HERAPDF1.5f and HERAPDf1.6 fits as a function of $\alpha_s(M_Z)$
}
\end{center}
\label{fig:chiscan}
\end{figure}
The $\chi^2$ for the HERAPDF1.6 fit
with free $\alpha_S(M_Z)$ is 807.6 for 765 degrees of freedom. The
partial-$\chi^2$
for the inclusive data has barely changed but the partial-$\chi^2$
for the jet data decreases to 77.6 for 106 data points.
Fig.~\ref{fig:jetnojetalph} shows the
summary plots of the PDFs for HERAPDF1.5f and HERAPDF1.6,
each with $\alpha_S(M_Z)$ left
free in the fit. It can be seen that without jet data the uncertainty on
the gluon PDF at low $x$ is large due to the strong
correlation between the low-$x$ shape of
the gluon PDF and $\alpha_S(M_Z)$. However once jet data are
included the extra information on gluon induced processes reduces this
correlation and the resulting
uncertainty on the gluon PDF is not much larger than it
is for fits with $\alpha_S(M_Z)$ fixed.
\begin{figure}[htb]
\begin{tabular}{cc}
\includegraphics[width=0.45\textwidth]{herapdf15f_alpha.eps} &
\includegraphics[width=0.45\textwidth]{herapdf16_alpha.eps}
\end{tabular}
\caption {The parton distribution functions
$xu_v,xd_v,xS=2x(\bar{U}+\bar{D}),xg$, at $Q^2 = 10$~GeV$^2$, from
HERAPDF1.5f and HERAPDf1.6, both with $\alpha_S(M_Z)$
treated as a free parameter of the fit.
The experimental, model and parametrisation
uncertainties are shown separately. The gluon and sea
distributions are scaled down by a factor $20$.
}
\label{fig:jetnojetalph}
\end{figure}
Direct
photoproduction dijet cross sections have also been used in PDF fits to
constrain the gluon as for example
in the ZEUS-jets analysis of ZEUS inclusive
cross-section data and jet data~\cite{zeusjets}
However, such data have net yet been used in the HERAPDF fits because
the cross-section predictions for photoproduced jets are sensitive to the
choice of the input photon PDFs. In order to minimise sensitivity to this
choice, the analysis can be
restricted to use only the `direct' photoproduction
cross sections. These are defined by the cut $x^{\rm obs}_\gamma > 0.75$,
where $x^{\rm obs}_\gamma$ is a measure of the fraction of the photon's
momentum that enters into the hard scatter. This is a direction for further
study
\subsubsection{Bringing it all together: HERAPDF1.7}
Finally an NLO fit has been made bringing together all the data sets: HERA I+II
combined high energy data, combined low energy running data,
$F^{c\overline{c}}_2$ data and jet data. This fit is called
HERAPDF1.7~\cite{herapdf17}. The
charm data are fit in the optimized version of the RT heavy quark scheme,
with its preferred
value of $m_c=1.5$. The value of $\alpha_s(M_Z)=0.119$ is fixed. This value
gives the best fit to all the data in this fit, with the jet data
dominating the sensitivity. Other settings are as for the HERAPDF1.6 fit
except that the parameter $D_{uv}$ was found to be consistent with zero and
hence only $13$ parameters have been used for the HERAPDF1.7 fit.
The correlated systematic uncertainties
of the data sets are treated as for the individual
fits: for the inclusive combined data sets at both low and high energy only the
procedural errors are treated as correlated by the Hessian method.
For the $F^{c\overline{c}}_2$ data one source is treated as correlated by the
offset method and for the jet data all $12$ sources are treated as correlated
by the Hessian method.
The overall $\chi^2/ndf$ is $1097.6/1032$ with partial $\chi^2/ndp$ of:
$44.1/41$ for $F^{c\overline{c}}_2$ data; $226.6/224$ for low energy data;
$80.6/106$ for jet data and $746/674$ for HERA-I+II high energy data.
The data are all very compatible. The results of this combined fit are
illustrated in Fig.~\ref{fig:herapdf1.7}
\begin{figure}[tbp]
\vspace{-0.5cm}
\centerline{
\epsfig{figure=herapdf17.eps,width=0.35\textheight}}
\caption {
The parton distribution functions from
HERAPDF1.7, $xu_v,xd_v,xS=2x(\bar{U}+\bar{D}),xg$, at
$Q^2 = 10$~GeV$^2$.
The experimental, model and parametrisation
uncertainties are shown separately. The sea and gluon
distributions are scaled down by a factor $20$.
}
\label{fig:herapdf1.7}
\end{figure}
\subsection{Comparison of HERAPDF to other PDFs}
\label{compare}
\begin{figure}[tbp]
\vspace{-1.0cm}
\centerline{\psfig{figure=hera15_mstw08_cteq66.eps,width=0.45\textwidth}~~
\psfig{figure=hera15_ct10_nnpdf.eps,width=0.45\textwidth}}
\centerline{\psfig{figure=hera15_nnlo_mstw08_nnpdf.eps,width=0.45\textwidth}~~
\psfig{figure=hera15_nnlo_abkm09_jr09.eps,width=0.45\textwidth}}
\caption {HERAPDF1.5 compared to other PDFs. Top: NLO PDFs from MSTW08, CTEQ66,
NNPDF2.1, CT10. Bottom: NNLO PDfs from MSTW08, NNPDF2.1, ABKM09, JR09
}
\label{fig:pdfs}
\end{figure}
Fig.~\ref{fig:pdfs} compares HERAPDF1.5 to MSTW08~\cite{Martin:2009iq},
CTEQ6.6~\cite{cteq66}, CT10~\cite{ct10}, NNPDF2.1~\cite{nnpdf21},
ABKM09~\cite{abkm09}, JR09~\cite{jr09} at $Q^2=10~$GeV$^2$.
All PDFs are shown with $68\%$ CL uncertainties. The top row compares
NLO PDFs and the bottom row compares NNLO PDFs. These PDF sets have been
chosen for comparison because they have been selected for benchmarking by
the PDF4LHC group~\cite{Alekhin:2011sk} (though CT10 and NNPDF2.1 are updates
of the benchmarked PDFs).
All PDFs are broadly compatible but
there are differences of detail which can have
important consequences for predictions of LHC cross sections.
A concise way to compare predictions for various LHC
cross sections is to compare parton-parton
luminosities for quark-antiquark and gluon-gluon interactions.
\begin{equation}
\frac{\partial L_{gg}}{\partial\hat{s}} = \frac{1}{s} \int_\tau^1 \frac{dx_1}{x_1} f_g(x_1,\hat{s}) f_g(x_2,\hat{s})
\end{equation}
\begin{equation}
\frac{\partial L_{\Sigma(\bar{q}q)}}{\partial\hat{s}} = \frac{1}{s} \int_\tau^1 \frac{dx_1}{x_1}~ \Sigma_{q=d,u,s,c,b}\left[ f_q(x_1,\hat{s}) f_{\bar{q}}(x_2,\hat{s}) + f_{\bar{q}}(x_1,\hat{s}) f_{q}(x_2,\hat{s})\right]
\end{equation}
where $s$ is the centre of mass energy squared of the proton-proton collision
and
$x_1$ and $x_2$ are the fractional momenta of the partons in each proton, such
that the centre of mass energy squared of the parton-parton collision is, $\hat{s}= \tau s$, where $\tau=x_1 x_2$.
Fig.~\ref{fig:lumi} shows $q-\bar{q}$ and $g-g$
luminosities for $p-p$ interactions at the LHC \footnote{Plots on top and middle rows from G.Watt http://projects.hepforge.org/mstwpdf/pdf4lhc} with
$\surd{s} = 7~$TeV,
in ratio to those of the MSTW2008 PDF,
for PDFs issued by CTEQ, NNPDF and HERAPDF.
This figure also shows the corresponding luminosity plots for the
HERAPDF1.5, 1.6, 1.7 NLO updates described in this review.
\begin{figure}[tbp]
\vspace{-1.0cm}
\centerline{\psfig{figure=ratioqqbarlumi1_68cl.eps,width=0.30\textwidth}~~
\psfig{figure=ratiogglumi1_68cl.eps,width=0.30\textwidth}}
\centerline{\psfig{figure=ratioqqbarlumi1_68cl_2011.eps,width=0.30\textwidth}~~
\psfig{figure=ratiogglumi1_68cl_2011.eps,width=0.30\textwidth}}
\centerline{\psfig{figure=lumi_herapdf17.eps,width=0.30\textwidth}~~
\psfig{figure=lumigg_herapdf17.eps,width=0.30\textwidth}}
\caption{Left hand side: the $q-\bar{q}$ luminosity in ratio to that of
MSTW2008 for various PDFs. Right hand side: the same for the $g-g$
luminosities.
Upper row, PDFs as benchmarked by the PDF4LHC group:
middle row, updates for
CT10 and NNPDF2.1 issued in 2011: bottom row, the HERAPDF NLO updates
described in this review.
}
\label{fig:lumi}
\end{figure}
Fig.~\ref{fig:luminnlo} shows similar luminosity comparison plots for
NNLO PDFs from MSTW, ABKM, JR and HERAPDF.
\begin{figure}[tbp]
\vspace{-1.0cm}
\centerline{\psfig{figure=lumi_nnlo.eps,width=0.30\textwidth}~~
\psfig{figure=lumigg_nnlo.eps,width=0.30\textwidth}}
\centerline{\psfig{figure=lumi_hera_nnlo.eps,width=0.30\textwidth}~~
\psfig{figure=lumigg_hera_nnlo.eps,width=0.30\textwidth}}
\caption {Left hand side: the $q-\bar{q}$ luminosity in ratio to that of
MSTW2008 for various PDFs. Right hand side: the same for the $g-g$
luminosities.
Upper row, NNLO PDFs as benchmarked by the PDF4LHC group ({\it Plots from G.Watt http://projects.hepforge.org/mstwpdf/pdf4lhc/}): bottom row,
the HERAPDF1.5 NNLO PDFs described
in this review compared to HERAPDF1.0 NNLO
}
\label{fig:luminnlo}
\end{figure}
There are several reasons why the PDF predictions differ. It
is beyond the scope of the
current review to describe all the other PDFs in detail. However a few
remarks can be made
on the main differences between HERAPDF and other PDFs.
Firstly they are
based on different data sets and different choices of cuts on these data sets
and this is closely related to the differing way in
which the PDF uncertainties are estimated since the use of many different data
sets has led to the use of increased $\chi^2$ tolerances for some of the PDF
sets. Secondly, different choices of PDF parametrisation are made and this
impacts on the size of the uncertainties. Thirdly, PDFs use different central
values of $\alpha_s(M_Z)$ and this affects the shape of the PDFs,
particularly the gluon PDF.
Fourthly, the PDF analyses differ in the schemes
used to account for heavy quark production and different heavy quark masses.
\subsubsection{Correlated systematic uncertainties and $\chi^2$ tolerance.}
Most modern data used in PDF fits are statistically very precise such that
systematic errors dominate. Thus the correct treatment of correlated
systematic errors becomes very important.
In PDF fits done prior to the year 2000 point-to-point correlated
systematic errors were not specifically treated. They were
added in quadrature to the uncorrelated errors. This can lead to biassed
results. The correct treatment of correlated systematic errors is discussed in
Ref~\cite{offhesse}. The consensus amongst PDF fitters is that
the uncertainty due to correlated systematic errors
should be included in the theoretical prediction such that
\[
F_i(p,s) = F_i^{\rm NLOQCD}(p) +
\sum_{\lambda} s_{\lambda} \Delta^{\rm sys}_{i\lambda}
\]
where $p$ are the PDF parameters, $s_\lambda$
represent independent (nuisance) variables for each source of
systematic uncertainty and $\Delta^{\rm sys}_{i\lambda}$ represents the
one standard deviation correlated systematic error on data point $i$ due to correlated error source $\lambda$.
A representative form of the $\chi^2$ is then given by
\begin{equation}
\chi^2 = \sum_i \frac{\left[ F_i(p,s)-F_i(\rm meas) \right]^2}{\sigma_i^2}
+ \sum_\lambda s^2_\lambda
\label{eq:chi2}
\end{equation}
where $\sigma_i$ is the uncorrelated error on each data point.
Thus the nuisance parameters are fitted together with the PDF parameters.
This method of treatment of correlated systematic has been termed the Hessian
method. An alternative is the Offset method in which $s_\lambda=0$ for the
central fit but the nuisance parameters are varied when determining the
error on the PDF parameters~\cite{offhesse}.
In the PDF fits of CTEQ/CT, MSTW and GJR/JR the Hessian method
is used with
increased $\chi^2$ tolerances such that a $68\%(90\%)$CL is not set by a
variation of
$\Delta\chi^2 = 1(2.73)$ but by a larger variation.
The reason for the use of such increased $\chi^2$
tolerances arises when using many different input data sets which are not
all completely consistent.
The tolerances are set so as to ensure that all the separate data
sets are fit to within their $68\%(90\%)$CL.
The tolerance
can differ according to the parameter being fitted (or more exactly
according to an eigenvector combination of parameters,
see Sec.~\ref{sec:eigen}) but as a rough
guide the CTEQ $90\%$CL tolerance is $\Delta\chi^2 \sim 100$
and the MSTW $90\%$CL tolerance is $\Delta\chi^2 \sim30$ for the MSTW2008
analysis\footnote{Note that MSTW provide both $68\%$ CL
and $90\%$CL uncertainties, whereas for CTEQ/CT a factor of $1./\surd{2.73}$
must be applied to the $90\%$CL uncertainties to
obtain $68\%$CL uncertainties.}.
The GJR/JR analyses use $\Delta\chi^2 \sim 50$.
The use of these increased $\chi^2$ tolerances has caused great controversy.
For example, Pumplin~\cite{pumplin} argues that a
$\chi^2$ tolerance of at most $\sim 10$ can be justified on the grounds of
data incompatibility and that the
more inflated values implicitly account for parametrisation
variations.
The ABKM group does not use increased tolerances and that is why their
PDF uncertainties are generally smaller than those of other groups. HERAPDF
also does not use an increased tolerance but considers additional
model and parametrisation uncertainties, see Sec.~\ref{sec:pdfchap}.
The NNPDF group use a completely different method of estimating
PDF uncertainties, see Sec~\ref{sec:paramunc}.
For the HERAPDF the Hessian procedure has already been applied to the data
combination to set the best values for the systematic shifts and the
combination procedure itself results in greatly reduced systematic errors,
such that there is no longer a significant difference in the PDF
uncertainties obtained using the Offset and Hessian methods of treating
systematic uncertainties. The good $\chi^2$ for the combination fit also
establishes that the resulting data set is very consistent, see
Sec.~\ref{sec:datacomb}, such that in the HERAPDF the conventional tolerance
$\Delta\chi^2 = 1$
is appropriate for setting $68\%$CL uncertainties.
\subsubsection{Diagonalisation and Eigenvector PDF sets}
\label{sec:eigen}
In either the Hessian or the Offset method, the Hessian matrices and
covariance matrices are not, in general,
diagonal. The variation of $\chi^2$ w.r.t. some parameters is much more rapid
than that of others, but because the parameters are correlated to each other
the effect of each parameter is not clear. When evaluating uncertainties
on physical observables it
can be an advantage to use an eigenvector basis of PDFs, which provide an
optimized representation of parameter space in the neighbourhood of the
minimum.
The eigenvalues of the covariance matrix represent the squares of the errors
on the combination of parameters which gives the corresponding eigenvector.
An eigenvector basis of PDFs is the usual way of summarizing the results
of a PDF analysis including its
error estimates. Two sets of PDF parameters
must be supplied for each eigenvalue, representing
displacement up and down along its eigenvector direction by the $\chi^2$
tolerance. The symmetric error on a quantity $F$ which is a function of the PDF
parameters is then simply given by
\[
<\sigma_F^2>= \sum_j \left[\frac{F(p_j^+) - F(p_j^-)}{2}\right]^2
\]
where $F(p_j^+)$, $F(p_j^-)$ are the values of $F$ evaluated up and down
along eigenvector $j$. Asymmetric errors may be evaluated by the prescription:
\[
<\sigma_F^2(+)>= \sum_j \left[max(F(p_j^+) - F(p_j^0),F(p_j^-) - F(p_j^0),0)\right]^2
\]
\[
<\sigma_F^2(-)>= \sum_j \left[max(F(p_j^0) - F(p_j^-),F(p_j^0) - F(p_j^-),0)\right]^2
\]
where $F(p_j^0)$, is the central value of $F$.
The PDFs from the HERAPDF are made public in this format via the LHAPDF
(http://lhapdf.hepforge.org) interface. As well as the eigenvector sets,
which give the experimental uncertainty of the HERAPDF,
further sets are provided to
cover the model and parametrisation variations. These should be combined with
the experimental errors as specified in Sec.~\ref{sec:results}.
Further PDF sets are also provided
for a range of fixed $\alpha_s(M_Z)$ values,
so that uncertainty due to $\alpha_s(M_Z)$ variation may also be evaluated.
The LHAPDF library is also the repository for the PDF sets from other PDF fitting groups.
\subsubsection{Choice of data sets and kinematic cuts}
The CTEQ6.6, MSTW2008, ABKM09 and JR09 PDF analyses do
not use the recently combined inclusive cross section data from
HERA-I~\cite{h1zeuscomb} which are up to three times more accurate than the
separate H1 and ZEUS data sets used by previous PDF analsyses.
These combined HERA data are shifted in normalisation by $\sim 3\%$ with
respect to the previous HERA data, and this explains the higher
luminosity of the HERAPDF at low $\tau$.
Conversely the HERAPDF analysis uses only HERA data, whereas the CTEQ, MSTW
and NNPDF analyses are 'global' PDFs which also use: older fixed target data,
both from DIS and from Drell-Yan production; Tevatron $W,Z$ cross section data
and jet production data. The ABKM and JR PDFs each use some but not all of
these non-HERA data sets.
The use of a single consistent data set with a clear statistical interpretation
of uncertainty limits was one of the primary motivations behind the HERAPDF.
However there are other reasons why the use of some of the
other data sets may lead to
further uncertainties. Firstly, the neutrino-$Fe$ fixed-target scattering data
from CCFR and NuTeV, which is often used to help to determine the valence
densities, needs
corrections for nuclear effects (the 'EMC effect').
Although such nuclear corrections are made in the global PDF analyses, they are not perfectly determined
and the uncertainty due to these corrections is not fully
accounted~\cite{cjstuff}.
More recently similar critisms have been made of the use
deuterium target data (either in DIS or Drell-Yan).
Accardi et al~\cite{jlabstuff} have reconsidered deuterium
corrections for the fixed target data. They find large uncertainties in these
corrections and this results in greater (unaccounted for) uncertainty in
the high-$x$ $d-$quark for fits where the deuterium data is the principal
source of information on the $d-$quark. (For the HERAPDF the information on
$d-$quark comes from CC $e^+ p$ scattering).
Fixed proton target data do not suffer from these problems, but the kinematic
reach of such data does extend into the high-$x$, low-$Q^2$ region, where
the theoretical interpretation of the data requires consideration of
target mass
corrections and higher twist terms. Most PDF analyses make a
$W^2\geqsim 15$GeV$^2$ cut to avoid this region
(for the HERAPDF this is unnecessary since all HERA data is at
large $W$). The ABKM analysis choses to include the low $W$ data
and model the higher twist terms. The high-$x$ region is also receiving
attention from the CTEQ-JLAb group~\cite{keppel}.
A further problem in the use of older fixed target
data is that results were often presented
and used in terms of $F_2$ rather than reduced cross sections.
ABM~\cite{abmf2} have examined the use
of NMC $F_2$ data in the global fits. The NMC
extraction of $F_2$ relied on assumptions
on the value of $F_L$ which are not consistent with modern QCD calculations.
ABM find that using NMC published values of $F_2$, rather than the NMC
cross section data, raises their extracted
values of $\alpha_S$ erroneously.
The HERAPDF avoids bias from erroneous assumptions about heavy target
corrections,
deuterium corrections, higher twist corrections and $F_L$ corrections,
by using only HERA pure proton target
cross section data, but a price is paid in terms of the uncertainties of the
high-$x$ parton distributions, which are generally larger than those of the
other groups.
It is also notable that the HERAPDFs have a harder high-$x$ sea
and a softer high-$x$ gluon PDF at NLO. It has been suggested that this may be
because Tevatron jet data are not included in the HERAPDF fit. However the
story is not quite so simple.
Global fits use Tevatron high-$E_T$ jet production data to help to
pin down the
high-$x$ gluon. The HERAPDF analysis uses HERA-jet data for the same
purpose, although the HERA jet data do not extend to as high $x$ values as the
Tevatron jet data. These Tevatron jet data have very large
correlated systematic uncertainties compared to HERA jet data
such that much trust must be put in the evaluation of systematic
uncertainties. Tevatron Run-I jet
data suggested a hard high-$x$ gluon, but Run-II data soften this. The MSTW
analysis uses only
Run-II data whereas the CT/CTEQ analyses use both Run-I and Run-II data.
These choices can explain the harder gluon luminosities of the CT PDFs
at high-$x$.
Watt and Thorne~\cite{wattthorne} obtain poor $\chi^2$ when comparing the
Tevatron jet data to the
HERAPDF1.0, 1.5 predictions. However their fits only
compare to the central predictions of the HERAPDF. A more valid
comparison would account for the HERAPDF error bands. If the
Tevatron jet data are input to the HERAPDF1.5 fit a much better $\chi^2$
($\chi^2/ndp = 1.48$ for CDF and $1.35$ for $D0$ jets) is obtained.
Significantly, the resulting PDFs do not lie outside the HERADF1.5 error bands
(although they do imply a harder high-$x$ gluon- on the upper edge of the error
band). The reason that the HERAPDF can give a reasonable description of
Tevatron jet data, while still having a relatively soft high-$x$ gluon PDF, is
that high-$E_T$ jets result not only
from the high-$x$ gluon but also from high-$x$ quarks and HERAPDF has a rather
hard high-$x$ quark PDF.
The ABKM analysis also choses not to use Tevatron jet data, partly because
new physics effects may be hidden in the data, biassing the PDFs.
Consequently, ABKM has a soft high-$x$ gluon luminosity. Nevertheless,
ABM gives a good description of Tevatron jet data~\cite{abmjets}.
A further issue regarding the use of Tevatron jet
data concerns their use together
with deuterium fixed-target data. The greater uncertainty in
the high-$x$ $d-$quark, due to uncertain deuterium corrections,
will feed into the high-$x$ gluon PDF, since the $d-g$ process provides a
substantial part of the Tevatron jet
cross section. However this larger uncertainty is usually not
accounted for~\cite{jlabstuff}.
\subsubsection{Parametrisation and model uncertainty}
\label{sec:paramunc}
HERAPDF central fits have a relatively small number of parameters $\sim 14$.
However, parametrisation uncertainty is estimated by making fits with
additional parameters freed, or with a change of the choice of the starting
scale, $Q^2_0$, which is equivalent to a re-parametrisation. The comparison of
HERAPDF1.5, which uses $10$ free parameters and HERAPDF1.5f which uses $14$
free parameters in Fig.~\ref{fig:herapdf1.5f}, shows that this procedure for
accounting for parametrisation uncertainty largely accounts for the
uncertainty introduced when the the extra parameters are freed in the central
fit.
The HERAPDF results in a similar central value and uncertainty estimates to
those of the global PDFs in many kinematic regions. In the case of the central
values this is because the HERA data dominate the global input data.
In the case of the uncertainty
estimates it is partly due to the fact that the HERAPDF experimental
uncertainties are augmented by estimates of the model and parametrisation
variations, which are not accounted in the CT and MSTW analyses.
This lends support to the idea that the
increased $\chi^2$ tolerances of MSTW and CT partly cover some of these
additional model and parametrisation uncertainties.
The NNPDF global analysis uses a completely different approach both to PDF
parametrisation and to the determination of PDF uncertainties.
All errors (statistical, systematic and normalisation)
as given by experimental collaborations are represented by
Monte Carlo replica sets of artificial data.
A neural net is used to learn the shape of these replicas rather than
using a fixed parametrisation at the starting point. This can be regarded as
equivalent to using a very large number of parameters.
The PDFs are not
determined by a $\chi^2$ fit but by stopping the learning algorithm before
overlearning occurs. The results are not presented in terms of eigenvectors
of the fit but in terms of a set of replicas such that their mean
gives the best estimate of the central PDF and the standard deviation from
this mean gives the
$68\%$CL uncertainty estimate. It is remarkable that this entirely different
procedure gives broadly similar central values and uncertainty estimates as
those of the MStW and CTEQ global fits. To some extent this
vindicates the standard
procedure, in particular with regard to the use of increased $\chi^2$
tolerances
to set the $68\%$CL uncertainties.
\subsubsection{The value of $\alpha_s(M_Z)$}
Some groups (HERAPDF, CTEQ, NNPDF) adopt a fixed value of $\alpha_s(M_Z)$,
inspired by the PDG value, and others (ABKM, GJR, MSTW) fit $\alpha_s(M_Z)$
simultaneously with the PDF parameters and use their best fit value.
All groups bar GJR use values
$\sim 0.118-0.120$ at NLO but there is a definite low($0.113$)- high($0.117$)
split at NNLO. HERAPDF, CT(EQ), NNPDF and MSTW provide PDFs at different
$\alpha_s(M_Z)$ values so that the effect of variation of $\alpha_s(M_Z)$
on cross section predictions can be evaluated.
MSTW obtain the highest value of $\alpha_S(M_Z)$, at both NLO and NNLO,
and these high values have been
atributed to the use of Tevatron jet data in their fits. However, ABM have
tried inputting these jet data to their fit and have found that
this has only a small effect on their extraction of a low value of
$\alpha_s(M_Z)$~\cite{abmjets}. There is also a 'folk-lore'
that DIS data prefer
lower values of $\alpha_s(M_Z)$. However both MSTW~\cite{Martin:2009bu} and NNPDF~\cite{Lionetti:2011pw} have performed DIS only
fits in which they find that only the BCDMS data prefer low $\alpha_s(M_Z)$
values. The HERA data actually prefer quite high values as shown in
Sec.~\ref{sec:jets}. The effect of this on the gluon-gluon luminosity
may be seen in Fig.~\ref{fig:lumi}
by comparing the HERAPDF1.6 curve, with fixed $\alpha_s(M_Z)=0.1176$, to that
of the HERAPDF1.6 free $\alpha_s(M_Z)$ curve,
which has $\alpha_s(M_Z)=0.1202$.
The larger $\alpha_s(M_Z)$ value leads to a smaller low-$x$ gluon and a
somewhat harder high-$x$ gluon such that the
gluon-gluon luminosity is then in better agreement with that of MSTW2008,
which also use a large $\alpha_s(M_Z)$ value.
\subsubsection{Heavy Quark Schemes}
The ABKM and GJR groups use
Fixed-Flavour-Number (FFN) treatments, HERAPDF, CTEQ and MSTW use various
General-Mass-Variable-Flavour-Number (GMVFN) treatments and
NNPDF2.0~\cite{nnpdf20} used
a Zero-Mass-Variable-Flavour-Number treatment(ZMVFN). These heavy quark schemes
are discussed in Ref.~\cite{bookch}.
The use of the zero-mass treatment explains why the NNPDF2.0
luminosities lie lower than those of
CTEQ, MSTW and HERAPDF at low $\tau$.
This may be seen by comparing the top row of Fig.~\ref{fig:lumi}
to the middle row where the NNPDF2.1
luminosity, which used a GMVFN, is seen to be in much better agreement
with the other PDFs. This is because, when charm mass is accounted for, charm
is suppressed at threshold and the light quark densities must be somewhat
larger in order to describe the deep inelastic cross-section.
However not all GMVFNs are the same.
Predictions for $F_2^c$ differ
between schemes~\cite{Rojo:2010gv} and the choice of scale within a scheme can
also affect predictions.
The value of the charm and beauty masses also differ between the PDF analyses.
HERAPDF, NNPDF and MSTW now provide PDFs at different charm and beauty mass
values so that the effect of this can be evaluated. In future the
combined data on
$F_2^{c\bar{c}}$, discussed in Sec.~\ref{sec:charm}, should help to reduce the
uncertainty on PDFs coming from the choice of scheme and the value of the
charm mass.
The heavy quark mass schemes described in Sec.~\ref{sec:charm}
all use a charm quark mass parameter which should be the pole-mass.
However the pole-mass has a strong dependence on the order of the
perturbative calculation and
may best be regarded as a parameter.
It may be better to consider the \msbar running-mass. HERA data on
$F_2^{c\bar{c}}$ has also been used for a determination of this
mass~\cite{moch}
\subsection{Comparisons of HERAPDF predictions to Tevatron and LHC data}
Finally we present some representative
comparisons of HERAPDF predictions to PDF sensitive
data from the Tevatron and LHC colliders. Fig.~\ref{fig:tevWZ} presents
comparisons to CDF data on the direct $W$-asymmetry~\cite{cdfwasym} and $Z0$
rapidity spectrum~\cite{cdfz0}.
\begin{figure}[tbp]
\vspace{-1.0cm}
\centerline{\psfig{figure=tev_wasym.eps,width=0.30\textwidth}~~
\psfig{figure=tev_z0.eps,width=0.30\textwidth}}
\caption {Left hand side: data on the direct $W$-asymmetry from CDF; right
hand side: data on the $Z0$ rapidity spectrun from CDF; compared
to NLO predictions from CTEQ6.6, MSTW08 and HERAPDF1.5. The blue band indicates
the uncertainties on the HERAPDF prediction.
}
\label{fig:tevWZ}
\end{figure}
These data are well described by the HERAPDF1.5 prediction\footnote{The predictions of the HERAPDF1.6 and 1.7 PDFs are very similar to that of HERAPDF1.5}.
A fit of the data to the central value of the prediction yields a $\chi^2$ of
36 for $28$ data points for the $Z0$ data and of $41$ for $13$ data points for
the asymmetry data. These descriptions are improved if the data is input to the
HERAPDF fit, to $\chi^2/ndp = 26/28$ for the $Z0$ data and $21/13$ for the
asymmetry data\footnote{Note that the $\chi^2/ndp$ for these asymmetry data are as well described by the HERAPDF as they are by other PDFs which have used them, e.g. NNPDF.}. The resulting PDFs lie well within the HERAPDF1.5 error bands.
The HERAPDF uncertainty bands could be reduced by input of these data.
This is a future project beyond the scope of the current review.
Fig.~\ref{fig:tevjets} presents
comparisons of HERAPDF1.0 predictions to D0 data on the inclusive jet production ~\cite{d0jets}
\begin{figure}[tbp]
\vspace{-1.0cm}
\psfig{figure=d0jets.eps,width=0.60\textwidth}
\caption {D0 data on inclsuive jet production compared to NLO
predictions from HERAPDF1.0}
\label{fig:tevjets}
\end{figure}
Because of the large correlated systematics of these data it is not possible
to assess the quality of the description by eye. If these data are input to
the HERAPDF1.5 fit a $\chi^2/ndp=145/110$ can be obtained.
Similary if CDF inclusive jet production data~\cite{cdfjets}
are input to the HERAPDF1.5
NLO fit a $\chi^2/ndp=113/76$ is obtained. In both cases the resulting PDFs
move to the edge of the HERAPDF1.5 error band- tending to favour a harder
high-$x$ gluon. For this reason, the HERAPDF1.6 $\alpha_s(M_Z)=0.1202$ fit,
which already has a harder gluon than the 1.5 fit,
gives the best description of these data out of all the NLO HERAPDF sets.
The HERAPDF1.5NNLO PDF fit gives a better description of these
data than the NLO PDFs- the central PDF of HERAPDF1.5NNLO yields a $\chi^2$
per data point of
$\chi^2/ndp=72/76$. However this can only be approximate since the
theoretical description of the jet data itself contains only an approximate
calculation for the NNLO jet cross-section.
Fig.~\ref{fig:atlasWZ} presents
comparisons of various PDFS, including HERAPDF1.5,
to ATLAS data on the $W$-lepton decay pseudorapidity distributions
and the $Z0$ rapidity distribution, as well as on the $W$-lepton
asymmetry~\cite{atlasWZ} .
\begin{figure}[tbp]
\vspace{-1.0cm}
\centerline{\psfig{figure=atlaswm.eps,width=0.30\textwidth}~~
\psfig{figure=atlaswp.eps,width=0.30\textwidth}}
\centerline{\psfig{figure=atlasz.eps,width=0.30\textwidth}~~
\psfig{figure=atlas_asym.eps,width=0.30\textwidth}}
\caption {Comparisons of ATLAS data on $W^+$ and $W^-$ decay lepton
pseudorapidity spectra, $Z0$ rapidity spectra and $W$ decay lepton asymmetry
data to NNLO predictions from MSTW08, HERAPDF1.5, ABKM09, JR09.
}
\label{fig:atlasWZ}
\end{figure}
Fig.~\ref{fig:cmsW} presents
comparisons of HERAPDF1.5 predictions to $234$pb$^{-1}$ of preliminary
CMS 2011 data on the $W$ decay lepton asymmetry ~\cite{cmswasym}.
\begin{figure}[tbp]
\vspace{-1.0cm}
\psfig{figure=cms_asym.eps,width=0.60\textwidth}
\caption {CMS data on $W$ decay lepton asymmetry compared to NLO
predictions from HERAPDF1.5, MSTW08 and CT10W}
\label{fig:cmsW}
\end{figure}
These LHC $W$ and $Z$ cross section data
are well described by the HERAPDF. However, a detailed study by the ATLAS
Collaboration~\cite{atlastrange} using the ATLAS $W$ and $Z$ data and the
HERA-I combined data has indicated a preference of the ATLAS data
for unsuppressed strangeness at $x\sim 0.01$.
Further discussion of this is beyond the scope of the present review.
Fig.~\ref{fig:atlasjets} presents
comparisons of various PDF predictions, including HERAPDF1.5, to ATLAS data on the inclusive jet production ~\cite{atlasjets}.
\begin{figure}[tbp]
\vspace{-1.0cm}
\centerline{\psfig{figure=atlasjets_central.eps,width=0.45\textwidth}~~
\psfig{figure=atlasjets_forward.eps,width=0.45\textwidth}}
\caption {ATLAS data on inclusive jet production in central and forward
rapidity regions in ratio to the NLO predictions of CT10 and compared to NLO
predictions from HERAPDF1.5, NNPDF2.1 and MSTW08}
\label{fig:atlasjets}
\end{figure}
Fig.~\ref{fig:cmsjets} presents
comparisons of various PDF predictions, including HERAPDF1.5, to CMS data on the inclusive jet production ~\cite{cmsjets}.
\begin{figure}[tbp]
\centerline{\psfig{figure=fnl2342b_pdfcomp3_bin1.eps,width=0.45\textwidth}~~
\psfig{figure=fnl2342b_pdfcomp3_bin6.eps,width=0.45\textwidth}}
\caption {CMS data on inclusive jet production in central and forward
rapidity regions in ratio to the NLO predictions of CT10 and compared to NLO
predictions from HERAPDF1.0, ABKM09, MSTW08 and NNPDF2.1}
\label{fig:cmsjets}
\end{figure}
Because of the large correlated systematics of these data
it is not possible
to assess the quality of the description by eye. The ATLAS jet data are
published with information on these correlations and a $\chi^2$ per data point
of $\sim 60/90$ can be obtained for each of the HERAPDFs, and the $\chi^2$
for the MSTW, CT and NNPDFs are similar.
Thus the data are not yet very discriminating,
however they indicate a preference for a somewhat less hard high-$x$ gluon
than the Tevatron jet data.
\section{Summary}
Deep inelastic lepton-hadron
scattering data from the HERA collider now dominate
the world data on deep inelastic scattering since they cover an unprecedented
kinematic range.
The H1 and ZEUS experiments
are combining their data in order to provide the most complete and
accurate set of deep-inelastic data as the legacy of HERA.
Data on
inclusive cross-sections have been combined for the HERA-I phase of running
and a preliminary combination has been made also using the HERA-II data.
This latter exersize also includes the data run at lower proton beam
energies in 2007. Combination of $F_2^{c\bar{c}}$ data is underway,
and combination of $F_2^{b\bar{b}}$ dat and of jet data is foreseen.
The HERA collaborations have used these combined data to
determine parton distribution functions (PDFs) in the proton.
Because the HERA experiments investigated
$e^+p$ and $e^-p$, charge current(CC) and neutral current (NC) scattering,
the inclusive HERA data provide infromation on
flavour separated up- and down-type quarks and
antiquarks and on the gluon- from its role in the scaling violations of
perturbative quantum-chromo-dynamics. The lower proton beam energy data
provide further information on the gluon at small $x \leqsim 0.01$
since they allow a determination of
the longitudinal structure function. The charm data provide additional
information on heavy quark schemes and heavy quark mass values. The jet data
(separate data from H1 and ZEUS at the time of writing) provide additional
information on the gluon PDF in the $x$ range, $0.01 \leqsim x \leqsim 0.1$ and on $\alpha_s(M_Z)$.
The analysis of these data sets has resulted in the
the HERAPDF parton distribution functions. In this review we have described
and compared these sets with each other and with PDF sets from other groups.
We have also demonstrated that the HERAPDF sets give successful descriptions
of data on $W$ and $Z$ production and on jet production from the Tevatron
and the LHC.
The currently recommended version
of these PDFs, which are available on LHAPDF, are the HERAPDF1.5 NLO and
NNLO sets.
----
----
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 4,271
|
Q: Does Any keyword in sql gives distinct records when used in a subquery I looked at a query of "ANY" from a tutorial site which was like:
SELECT ProductName
FROM Products
WHERE ProductID = ANY (SELECT ProductID
FROM OrderDetails
WHERE Quantity = 10);
This query is returning 31 rows and no duplicates.
After this I tried to apply same query using Joins but I was unable to get result coming from above query.
Join query I used:
SELECT Products.ProductName
FROM Products
LEFT JOIN OrderDetails ON Products.ProductID = OrderDetails.ProductID
WHERE OrderDetails.Quantity = 10
ORDER BY Products.ProductName;
This is returning 44 rows, and has duplicates included.
After I used DISTINCT in this join query with ProductName, I got the desired result.
Hence I want to know - does "ANY" clause produce distinct records?
PS: Same record came in both Join queries (with and without distinct) with Inner Join as Well.
A: A join is a completely different operation to that of any (or similar all).
any is a logical operator and in your example is used to determine whether each row in Products should be returned.
The most rows that could be returned is equal to the number of rows in Products if the boolean result of the any operator is true for each ProductId.
By joining the tables, the two inputs to the join operator are compared and matching rows are output, which means if a single productId is input from Products and the input from Orderdetails has two rows with the same ProductId values ie with Quantity=10 the result is 2 rows are output, 1 for each matching row.
A:
Hence I want to know - does "ANY" clause produce distinct records?
No. It is actually the opposite. The records being chosen are those in the FROM clause. So, in the first query, there are no duplicates in Products. The WHERE clause is never going to generate duplicate records. That is not a property of ANY in particular; it is also true of IN and EXISTS and any other comparison operation.
What is opposite is that the JOIN does produce duplicate records. That is what you are seeing in the second query. The table OrderDetails has multiple rows for a given product.
Note that ANY (and IN) do actually implement a type of JOIN called a semi-join. So, there is a relationship between what these operators do and JOINs in the FROM clause. However, semi-joins and anti-joins are different from inner and outer joins that are defined in the FROM clause.
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 1,068
|
Q: ScyllaDB Not getting Installed on Ubuntu 16.04 I have tried the steps on the website of ScyllaDB - http://www.scylladb.com/doc/getting-started-ubuntu16.04/
Also, I have went through this step of deb installation - http://www.scylladb.com/download/#fndtn-deb
But I didn't got anything. I want to install scylla, but not able to.
Whenever I try to install using - sudo apt-get install scylla I get errors where it says:
E: Unable to locate package scylla
Please if anyone has any information why it is happening then do let me know. I have even gone through the other related questions on Stackoverflow.
A: Yes, I got my mistake. I have added the repository and was instantly trying to install scylla. I forgot to update the repository.
sudo apt-get update
And finally it worked...:)
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 7,450
|
\section{Introduction and specific requirements}
FAZIA (\textit{Forward-angle A and Z Identification Array}) is a modern and innovative three-layer telescope [Si + Si + CsI(Tl)] array.
This review paper follows the work of R. Bougault \textit{et al.} \cite{Bougault14} where the R\&D phase
of the FAZIA project was carefully described.
In this paper we will give a detailed overview on the technological aspects of the final detector array;
in particular, we will focus on the electronics, the data flow and the mechanical aspects of the apparatus.
The multi-detector FAZIA aims at detecting and identifying particles and fragments produced
in heavy-ion reactions around Fermi energy. The main requirement of FAZIA is the modularity
and portability: in fact, FAZIA was designed to measure in various laboratories,
in different setups and coupled to several detectors. Another important objective is to
maximise unit identification for charges and masses of detected nuclei.
In the present situation, we clearly discriminate charges up to $Z\sim 55$ and masses up to $Z\sim 25$. This goal
was achieved using custom detectors produced following a well-studied recipe \cite{VonAmmon92,Bardelli09}
and using original electronics with novel \textit{pulse-shape discrimination} (PSD) techniques \cite{Ammerlaan63,Barlini09,Bardelli11} based
on high speed analog-to-digital converters with rates up to \SI{250}{MS/s}
and 14-bit resolution. The whole electronics is embedded in the proximity of the telescopes
inside the vacuum chamber.
The commissioning runs of FAZIA proved the capability to integrate inside the scattering
chamber all the electronics required for silicon detectors and scintillators read-out by photodiodes.
This very innovative electronics includes pre-amplifiers, analogue chains, high speed
converters, read-out logic, high voltage devices and pulse generator for
analogue chains. Indeed, the first experimental campaigns proved that it is possible to integrate, on the same
multi-layer card, some potentially noisy power-supplies (such as switching regulators) with sensitive low-noise
pre-amplifiers whose power-supply rejection ratio is not high. This integration has been possible by
applying strict electromagnetic compatibility design guidelines.
The great complexity and the fine granularity of the apparatus implies a difficult scalability and thus a relatively poor angular coverage:
in fact a 16-telescope block, which is the smallest independent FAZIA unit, covers only around \SI{0.05}{\%} of the full solid angle at \SI{1}{m} distance from the target.
Indeed, the next FAZIA schedule foresees the mounting of a 12-block array in 2019, coupled with INDRA detector array \cite{Pouthas95} at GANIL to increase the angular coverage.
The electronic cards are described in Sec.~\ref{sec:electronics}.
Both the ``block'' electronics, that is mounted directly next to detectors
inside the scattering chamber (in vacuum) and the devices outside the chamber are detailed.
Afterwards, a functional description of the apparatus follows (Sec.~\ref{sec:functional}): in particular
the clock distribution (Sec.~\ref{ssc:clock}), the data packet structure (Sec.~\ref{ssc:packet}), the trigger logic (Sec.~\ref{ssc:trigger}), the data flow (Sec.~\ref{ssc:data}),
the possibility of coupling (Sec.~\ref{ssc:coupling}), the acquisition (Sec.~\ref{ssc:acq}), and the slow control (Sec.~\ref{ssc:slow})
are presented. In Sec.~\ref{sec:mechanics} the adopted mechanical solutions necessary
to hold and to cool the block electronics and to align the detectors are described. Finally, in Sec.~\ref{sec:conclusions},
we give an overview on all the innovative aspects of the FAZIA modular array and the next improvements we are going to realise.
\section{Description of the electronic boards}\label{sec:electronics}
The first feature which could be noticed when looking at the FAZIA apparatus
is the absolute scarcity of electronic racks outside the scattering chamber (Fig.~\ref{fig:schema}).
In fact, the so-called regional board (RB) is the only electronic card placed outside (see Sec.~\ref{ssc:rb})
and it performs the functions of a standard ``event building'' card.
Only two connections per block are necessary: a \SI{48}{V} (\SI{6}{A}) power supply line and a \SI{3}{Gbit/s} full-duplex
optical link used to transfer data, to synchronise the clocks, to send triggers, and to manage any block parameter via slow control.
\begin{figure}[htbp]
\centering
\includegraphics[width=0.8\textwidth]{s_full.pdf}
\caption{Schematic representation of the FAZIA electronic cards inside and outside the scattering chamber. For a detailed description see the text.}
\label{fig:schema}
\end{figure}
Inside the vacuum chamber, the basic element of the FAZIA array is the block, which consists of 16 detector telescopes.
The telescopes are connected to 8 front-end electronic (FEE) cards (see Sec.~\ref{ssc:fee}),
which feature, among other components, charge sensitive pre-amplifiers, ADCs, Si bias voltage regulators, and FPGAs for data handling.
Up to two telescopes can be connected to each front-end card.
All the FEE cards are connected to a ``Block Card'' (see Sec.~\ref{ssc:bc}) via a common backplane, which hosts
also two power supply boards (see Sec.~\ref{ssc:ps}). The Block Card (BC) is mainly devoted to handle I/O
operations and to merge data coming from the FEE cards. Power Supply (PS) and Half Bridge (HB) cards produce and monitor the voltages
needed to the other boards on the block.
The FAZIA blocks communicate with the event building electronics via the Block Card
through the \SI{3}{Gbit/s} optical link.
A metallic housing covers all the block; there is also a metal screen around the BC, PS and HB cards to avoid that electromagnetic emissions
from DC/DC converters reach the FEE cards. All the electronic cards of the block are supported by (and firmly screwed to) a copper plate in which water can flow for cooling.
The FEE cards and the electromagnetic shield structure have been designed at IPN Orsay; BC, PS and HB boards have been developed at INFN -- Naples;
the copper plates are built at INFN -- Bologna and Firenze; the RB has been designed at Jagiellonian University in Krak\'{o}w (Poland) in collaboration with INFN -- Naples.
\subsection{Power supply}\label{ssc:ps}
The DC levels needed by the block electronics (BC and FEEs) are produced by two cards that
are both connected to the backplane: Half Bridge and Power Supply.
As previously mentioned, the FAZIA block needs only an external \SI{48}{V} DC line to work. This voltage
is provided to the Half Bridge card, which performs a high power conversion to \SI{22}{V} DC (\SI{14}{A} max) and \SI{5.5}{V} DC (\SI{70}{A} max)
by means of two commercial switching converters (CUI VHB350 series). These devices have an insulation,
with low parasitic capacity, between primary and secondary stage to avoid ground loops.
To reduce the electromagnetic interferences produced by the converters, a particular two stage filter was
added at the \SI{48}{V} input line: the first stage is a BALUN (\textit{balanced-to-unbalanced}) transformer
used to balance the interferences and the second stage contains a common-mode transformer to filter them.
The output \SI{5.5}{V} voltage is split into two lines: one (\SI{4}{A} max) is dedicated to the Block Card and it is equipped
with an over-current protection circuit; the other (\SI{40}{A} max) is used to power the FEE cards and it is controlled
by a power MOSFET. The \SI{22}{V} output line, together with the \SI{5.5}{V} voltage monitor line,
is connected to the Power Supply card via the backplane.
The PS card main task is to produce and monitor the voltages required by the Block Card and the FEE cards.
In particular, it uses DC/DC switching converters to transform the \SI{22}{V} input to \SI{+13}{V} DC (\SI{2}{A}), \SI{-9}{V} DC (\SI{1}{A}),
\SI{+6}{V} DC (\SI{5}{A}), and \SI{-6}{V} DC (\SI{2}{A}). The \SI{+13}{V} and \SI{-9}{V} lines, which are used to bias the pre-amplifiers,
have an additional linear regulator to reduce the high frequency ripple. The PS card produces also a negative HV line, currently
used to bias the CsI(Tl) crystal photodiodes, which is user settable in the range \SIrange{0}{600}{V} DC (\SI{4}{mA} max).
Due to the huge range of the output voltage, it was not possible to use only a flyback converter. In particular,
a step down power supply stage (controlled by the same driver) is placed before the standard flyback circuit implemented with the UC3842 driver.
In this way, we obtained that the voltage at the input of the flyback stage is reduced when the output voltage decreases.
Moreover, to achieve a better commutation precision of the driver \textit{Pulse-Width Modulation} (PWM) stage, a circuit that modulates the switching frequency
in function of the set voltage was implemented.
The PS card is also equipped with a PIC microcontroller which continuously measures the voltages and the current flowing through the power supply lines;
it also monitors the temperature of the board and, in the case of overheating, it shuts down the FEE card lines.
\subsection{FEE cards}\label{ssc:fee}
The core of the FAZIA block is the front-end electronic card \cite{Salomon16}. Up to 8 FEEs can be mounted on each block.
The area of these boards (\SI{299x88}{mm}) is subdivided into three parts (Fig.~\ref{fig:fee}).
The first stage embeds all the low-noise analogue electronics (pre-amplifiers, amplifiers and anti-aliasing filters).
At the opposite side, the FEE hosts the switching power-supplies which are sources of electromagnetic disturbances and
thus are placed, by design, far from the analogue stages. In the middle part, among many other components,
two Xilinx Virtex-5 (model XC5VLX50) FPGA chips (one for each telescope and called ``A'' and ``B'' from now on),
a PIC microcontroller, and 12 ADCs are mounted.
The printed circuit board has 16 layers on which 1700 components are located on the top surface only.
The power consumption of a card is about \SI{30}{W}. The connection toward the heat sink is done by
two aluminium plates: one pressed and sticked on the upper side of both FPGAs and one entirely covering the back side of the card.
This second sheet has a shelf that is screwed on the main cooled copper plate of the block (see Sec.~\ref{ssc:cooling}).
\begin{figure}[htbp]
\centering
\includegraphics[width=\textwidth]{fee_stages.jpg}
\caption{Picture of a front-end electronic card without the protection and dissipation plates.
The three stages in which the card is subdivided are highlighted.}
\label{fig:fee}
\end{figure}
\paragraph{Analogue stage}
Six charge pre-amplifiers (three per telescope) are placed on each front-end card just next to the detector connectors.
Their architecture is based on a folded cascode amplifier and the output dynamic range is \SI{8}{V} for
a total energy of \SI{4}{GeV} ($\sim\SI{300}{MeV}$ Si-equivalent for the CsI(Tl) channels),
providing a sensitivity of \SI{2}{mV/MeV} for both Si1 and Si2.
The detector signals are AC coupled with a \SI{10}{nF} capacitor to block the bias voltage.
A circuitry (containing a DAC) is added to the pre-amplifier output stage to tune the output offset voltage.
This function allows one to set the analogue chain baseline close to the bottom level of the ADC input range,
in order to better exploit the available dynamic range of the ADC.
The baseline level could be remotely controlled in any moment via a slow control command, since the
DAC input buses are directly connected to the FPGAs.
The analogue lines then split to have multiple channels per detector. In particular, for the
first Si stage, we have three paths: high range charge signal (QH1), low range charge signal (QL1) and current signal (I1).
For the second Si stage we have the high range charge signal (Q2) and the current signal (I2). For the CsI stage
we have only one charge signal (Q3).
The high range (low gain) signals (QH1, Q2 and Q3) are attenuated by a factor 4, to adapt the \SI{8}{V} dynamic
range of the pre-amplifiers to the \SI{2}{V} input range of the ADCs.
On the contrary, the low range (high gain) signal (QL1) is amplified by a factor 3. The current signals (I1 and I2)
are obtained by analogue differentiation of the Si1 and Si2 pre-amplifier outputs.
All the six signals described above pass through an anti-aliasing filter before being sampled by the ADCs.
A square pulse generator has been built on the front-end card in order to test the analogue chain response.
It is also useful to verify possible amplification changes during data taking (which, incidentally, have never been observed up to now).
The generator is based on a simple MOSFET clocked switch connected to the inputs of all the
pre-amplifiers through a capacitor.
The user can change the amplitude, the frequency, and the duty cycle of the pulse generator
via slow control instructions. The pulser amplitude is set by a DAC, whose input bus is connected to
the front-end PIC microcontroller. Frequency and duty cycle depend on the switch clock, that is generated
by the FPGA ``B''. Alternatively, an external clock could be used by setting the appropriate slow control register.
\paragraph{Digital stage}
In the previous paragraph, the twelve signals (six per telescope) that are generated in the analogue stage of the card were described.
QH1, Q2 and Q3 signals are connected to 14-bit, \SI{100}{MS/s} analog-to-digital converters with a \SI{2}{V} input range.
Since we have a \SI{2}{mV/MeV} sensitivity from the pre-amplifier and then we reduce the signal by a factor 4,
at the end we get an energy conversion factor of about 4.1 ADC units per MeV for QH1 and Q2 signals.
The relatively slow sampling rate is more than sufficient for energy measurements.
Moreover, the very high effective number of bits ($ENOB=11.4$) of the ADCs guarantees an accurate reconstruction of the released
energy inside the detector.
QL1, I1 and I2 signals are sampled by 14-bit, \SI{250}{MS/s} ADCs with a \SI{1.5}{V} input range.
Considering the QL1 signal, we have again a \SI{2}{mV/MeV} sensitivity from the pre-amplifier but in this case we amplify
the signal by a factor 3, so at the end we get an energy conversion factor of about 66 ADC units per MeV
(16 times larger with respect to the corresponding high range signal).
The low range line is thus very useful to identify and measure energy of particles that produce small energy losses
within the first Si layer, such as light fragments up to $Z\sim10$.
Another important feature of the QL1 signal is the possibility to use it for timing, because
it is acquired by a fast sampling ADC.
Current signals are also acquired by \SI{250}{MS/s} 14-bit ADCs.
In fact, because of their fast time evolution, their shape cannot be faithfully reconstructed by interpolation if sampled at \SI{100}{MS/s}.
Indeed, the current signal represents an accurate image
of the charge collection process within the detector and allows the best identification
performances via pulse-shape discrimination for ions stopping in the first stage of the telescope \cite{Carboni12,LeNeindre13,Pastore17}.
The twelve analog-to-digital converters are read by the two FPGAs: the six signals from the first telescope
are handled by FPGA ``A'' and the six signals from the second telescope are handled by FPGA ``B''.
The two programmable arrays compute in real time the energy through digital filtering.
They also generate local triggers, data packing and transmission to the acquisition (see Sec.~\ref{ssc:data}).
Also the 8-bit PIC microcontroller lies in the digital part of the FEE card.
It is capable to accept and execute commands received through a serial slow control link and
it can subsequently write and read the slow control FPGA registers by Serial Peripheral Interface (SPI) link.
As previously said, the PIC controls the pulser amplitude.
It controls also the high voltage devices (described in the next paragraph) and reads, by means of a dedicated 16-bit ADC,
the current flowing through the Si detectors.
When this reverse current increases, the PIC automatically increases the HV device output to compensate
the voltage drop on the \SI{10}{M\ohm} bias resistor and to maintain the biasing voltage of the silicon detectors at constant value.
This function is quite important to preserve good PSD during an experiment.
Finally, the PIC also gets temperature values coming from embedded sensors located in several critical positions
on the cards.
\paragraph{Converters stage}
The voltage conversion stage is placed just close to the backplane connector.
The low voltage power supply part includes four switching converters and more than 20 linear
regulators. Four high voltage devices are also embedded on the board. The high voltages
are generated from a common \SI{5.5}{V} input and their purpose is to bias the silicon detectors
of the two telescopes. The architecture of the HV modules is based on a switching regulator and a transformer.
The high voltage devices for Si1 can ramp up to \SI{300}{V} with a precision of less than \SI{0.1}{V}.
The HV device for Si2 can ramp up to \SI{500}{V} with the same absolute precision.
The HV functions and values are remotely controlled via the PIC microcontroller.
The integrated high voltage device represents a new and original solution within the nuclear physics
community, as it is embedded into one single electronic card operating under vacuum.
\subsection{Block card}\label{ssc:bc}
The main task of the Block Card (shown in Fig.~\ref{fig:blkcard}) is to retrieve all the data coming from the FEE cards and to build from them a
partial event. The main communication path between the BC and FEE cards is implemented with 24 serial
buses in a ``star'' configuration: each front-end card is reached by three \SI{400}{Mbit/s} buses.
Two buses connect each FEE card to the BC and one goes from the BC to each FEE.
The implementation of these fast serial links was done using Xilinx ``ISERDES'' and ``OSERDES'' cores.
To align and synchronize data, a programmable delay was also added before the ISERDES unit.
Moreover, other common signals reach every FEE card from the Block Card: i.e. the validation signal,
the external pulser clock and the slow control line. Every kind of communication between
BC and FEEs passes through the backplane. On the contrary, the communication with the event building electronics
takes place through a \SI{3}{Gbit/s} optical link. In particular, the Block Card hosts a \textit{small form-factor pluggable} (SFP)
optical transceiver that supports two (RX and TX) SX fibers (\SI{850}{nm} wavelength) with a LC connector.
\begin{figure}[htbp]
\centering
\includegraphics[width=0.8\textwidth]{blkcard.jpg}
\caption{
Picture of a Block Card. In the center, the FPGA is covered by the heat sink. In the right side, the SFP transceiver
used to communicate with the Regional Board can be seen.
}
\label{fig:blkcard}
\end{figure}
Communications and data sorting are handled by a Xilinx Virtex-5 FPGA (model XC5VFX70T), together with
the fundamental task of capturing a special packet from the optical link to produce a synchronized clock (see Sec.~\ref{ssc:clock}).
The Block Card contains also a photodiode which reads another optical fiber and a \SI{50}{MS/s} ADC to sample
the signal coming from it. To exploit the full ADC range, a variable gain amplification circuit was also implemented.
This system is used as an extra synchronization method, as it will be described in Sec.~\ref{ssc:clock}.
Finally, the BC features a microSD slot. If a memory card with a special file is present in the slot when the board is switched on,
the block identification number (ID) is read from that file. In this way it's very easy to change the block ID during the experiment.
The block ID is the only way to distinguish the blocks when a slow control command is sent. The microSD card could also be
used to reload the FPGA firmware with a ``.xsvf'' file.
\subsection{Event building electronics}\label{ssc:rb}
The regional board (Fig.~\ref{fig:rb}) is a 6U size VME card (operating outside the reaction chamber, in air) whose main tasks are: to read data from the FAZIA blocks
and form a complete event, to analyze the triggers from the blocks and possibly send validations to them,
to handle slow control requests from many PCs and eventually to communicate with acquisition servers.
The RB is an evolution of the ``test card'' developed by INFN -- Naples group. That card featured an USB protocol to transmit data to the acquisition system and it was capable
to handle up to 8 blocks. The test card was widely used to develop many features implemented in the regional board, such as the internet protocol on the on-board FPGA.
The RB can manage up to 36 blocks and, in case of configurations that need a higher number of blocks,
multiple regional boards may be interconnected using optical links and/or the VME bus. However, this feature
has not been yet implemented.
\begin{figure}[htbp]
\centering
\includegraphics[width=0.6\textwidth]{rb_tag.jpg}
\caption{Picture of the regional board with highlights of the most important features: (a) VME connectors and CPLD,
(b) FPGA and crystal oscillators, (c) special connectors for block communication; (d) SFP slots; (e) LEMO connectors and CENTRUM port.}
\label{fig:rb}
\end{figure}
In addition to the VME bus, regional board connections include six special connectors for block communications,
three SFP slots for RB interconnection and communication with acquisition (through Ethernet protocol),
four LEMO connectors (``veto in'', ``veto out'', ``trigger in'' and ``trigger out'') in LVTTL logic
for trigger coupling with other devices and a port for event synchronization among many compatible detectors.
The connectors for block communications allow the installation of an optical translator over each of them.
At the other end of the optical translator, a compact 12-fiber connector is placed. Each translator
handles up to twelve mono-directional fiber connections (RX or TX), thus two translators are needed to operate up to twelve blocks
(four translators for up to 24 blocks and all the six translators for up to 36).
The port for event synchronization uses the CENTRUM technology developed at the GANIL laboratory (Caen, France).
This technology was chosen in view of the forecast coupling of FAZIA with the INDRA detector array,
which is installed at GANIL and already uses CENTRUM protocol.
All the features and communications (except the VME bus management) are handled by a Xilinx Virtex-5 FPGA (model XC5VTX150T).
The VME bus is operated by a Xilinx CPLD (model XC95144XL) that is directly connected to the FPGA.
The \SI{5}{V} operating voltage is supplied by the VME bus.
\section{Functional description of the electronics}\label{sec:functional}
\subsection{Clock distribution}\label{ssc:clock}
FAZIA is designed to measure over a broad range of beam energies. On some application, mainly with low energy beams,
the time of flight (ToF) technique is very useful and the clock synchronization among all the ADCs is necessary.
In fact, if the ADCs in different blocks had independent clocks,
the accuracy of the time measurement could not be better than one clock cycle (\SI{4}{ns} or \SI{10}{ns} depending on the kind of ADC).
So, to synchronize all the sampling ADCs, they must be provided with exactly the same clock on all the cards.
The clock distribution tree is schematically represented in Fig.~\ref{fig:clock} and it is detailed here.
The primary clock is generated on the regional board: there are two crystal oscillators
set at \SI{125}{MHz} and \SI{150}{MHz} and both are connected to the FPGA chip. The former is used only to
clock the Ethernet part of the FPGA project. The latter is used to generate (inside the FPGA) a \SI{25}{MHz} clock,
and to clock the built-in GTX transceivers, which are used to send and receive data from the blocks through the
optical link. The transceivers are devices embedded inside the FPGA. In our case
they convert 16-bit data at \SI{150}{MHz} (\SI{2.4}{Gbit/s}) to a serial line at \SI{3}{Gbit/s} and vice versa. The missing 1.25 factor
in the data rate comes from the 8b/10b coding (see Sec.~\ref{ssc:packet}) of the serial line.
Xilinx GTP and GTX transceivers are very suitable to implement a connection with fixed latency,
as they can provide an extremely reduced clock skew \cite{Giordano11}.
\begin{figure}[htbp]
\centering
\includegraphics[width=0.6\textwidth]{clock.pdf}
\caption{Schematic representation of the FAZIA clock distribution, which was designed in order to sample all the signals with the same phase. Further details in the text.}
\label{fig:clock}
\end{figure}
The optical fibers leaving the regional board enter the scattering chamber and reach the various blocks.
They are connected to the SFP optical translators on the block cards. The signals eventually arrive inside the
block card FPGAs, where they are de-serialized by GTX transceivers. As briefly introduced in the previous section,
the BC has a peculiar system to recover the \SI{25}{MHz} clock from the optical link:
on the card there is a voltage controlled crystal oscillator (VCXO), connected to a phase-locked loop (PLL) device, which is used to clock the FPGA.
The PLL reference is the signal recovered from the fiber by the FPGA itself. So, when the BC is switched on,
the PLL is not locked and the FPGA is clocked by a \SI{25}{MHz} signal that is uncorrelated with the
\SI{25}{MHz} signal generated on the regional board. Then the block card starts to catch the special K28.5 sequences (see Sec.~\ref{ssc:packet})
from the optical link and generates the synchronized \SI{25}{MHz} signal that enters into the PLL device.
At this point the PLL is locked and so is the FPGA clock. The PLL output is also split into eight outputs
that reach the FEE cards through the backplane. The front-end cards use the phase-locked
\SI{25}{MHz} signal to produce (using VCXO and PLL devices) \SI{100}{MHz} and \SI{250}{MHz} frequencies.
Finally, these signals are used to clock the ADCs and the FPGAs. Due to the physiological delays between clock edges and ADC sampling,
practically the various sampled signals are not exactly synchronous, since the delays are not identical among the ADCs. The methods to reduce
the residual asynchronism, of the order of \SI{100}{ps}, are currently under study and they will be the subject of another work.
The phase-locked \SI{25}{MHz} clock on the FEEs is used also to increment a universal 15-bit timestamp counter.
To ensure that all the cards are assigning the same timestamp at the same moment, a ``time tag zero'' signal (TTSync)
is sent on the optical fibers by the RB every time that the timestamp counter goes overflow (\SI{1.31}{ms}).
To improve the block cross-synchronization, an external sinusoidal signal (the same for all blocks)
could be sampled by the block cards at \SI{50}{MS/s}. Time marks extracted from sinusoidal waveforms
are very precise, even when the sampling rate is not very high \cite{Bardelli07}. This method
allows to check the clock synchronization among blocks with a precision down to \SI{10}{ps}.
The \SI{50}{MHz} clock is generated by means of VCXO and PLL from the synchronized \SI{25}{MHz} clock,
as also the \SI{150}{MHz} reference used to clock the GTX devices on the Block Card.
\subsection{Packet structure}\label{ssc:packet}
Since each block is connected with the regional board using only an optical link, the packet structure
should be optimized to reduce the data overhead while transmitting all the needed information.
The optical link is composed of two fibers: one is used to transmit data to the block (TX) and the other
to receive (RX). In both directions data are structured in packets of six 16-bit words and the 8b/10b encoding is used.
That means that each byte, when is sent on the fiber, is converted into a 10-bit frame: this conversion helps
to provide enough state changes to allow clock recovery. Converting 8-bit sequences to 10 bits means also
that some 10-bit sequences have no corresponding 8-bit data. One of these sequences (called ``K28.5'') is
used to distribute and synchronize the master clock among all the blocks (see Sec.~\ref{ssc:clock}).
\begin{table}[htbp]
\centering
\begin{center}
\begin{tabular}{ccc}
\toprule
\textbf{Word} & \textbf{TX} & \textbf{RX}\\
\midrule
0 & sync and control & sync and control\\
1 & event number & data\\
2 & - & data\\
3 & block acq. busy & data\\
4 & block acq. busy & data\\
5 & block acq. busy & data\\
\bottomrule
\end{tabular}
\end{center}
\caption{Summary of the packet structure used in the data transmission between the RB and the blocks. More details in the text.}
\label{tab:packet}
\end{table}
The first 16-bit word of the packet is always a special synchronization and control frame, both in TX and RX transmission.
In particular, the first 8 bits are not defined and they are substituted by the K28.5 sequence in the 8b/10b conversion.
The whole packet structure is summarized in Tab.~\ref{tab:packet} and detailed in the next two paragraphs.
\subsubsection{TX packet structure}\label{sss:tx}
As said before, the first frame of the transmitted packet is used by the block card to synchronize the master clock via the K28.5 sequence.
The remaining 8 bits have the following use:
\begin{description}
\item[bit 7] validation signal generated by the RB (see Sec.~\ref{ssc:trigger});
\item[bit 6] slow control TX line (see Sec.~\ref{ssc:slow});
\item[bit 5] TTSync signal (see Sec.~\ref{ssc:clock});
\item[bit 4] global reset of all blocks;
\item[bit 3:0] not assigned.
\end{description}
When a validation signal is sent, the second frame of the TX packet contains the 12-bit event number used to check the consistency
in the event building phases (see Sec.~\ref{ssc:data}). The third frame is not used. The last three frames are used to selectively block
the data transmission from the blocks. In fact, when a regional board FIFO dedicated to a specific block is about to be completely full,
the RB must block the data acquiring process only from that block. That is achieved by writing on a TX frame
a block ID followed by its acquisition status (enabled/disabled). The involved block always reads the last three TX frames
searching for its ID: if it is found, the block reads the acquisition enable bit and updates its own status.
\subsubsection{RX packet structure}
The first frame of the received packet is, also in this case, a synchronization and control word and the first part is again the K28.5 sequence.
The remaining 8 bits of the first RX frame are the following:
\begin{description}
\item[bit 7] GTT flag (see Sec.~\ref{ssc:data});
\item[bit 6] slow control RX line (see Sec.~\ref{ssc:slow});
\item[bit 5:1] block trigger multiplicity (see Sec.~\ref{ssc:trigger});
\item[bit 0] not assigned.
\end{description}
The other five frames of the packet contain the data flow coming from the block.
\subsection{Trigger logic}\label{ssc:trigger}
On both FPGAs of the front-end cards, fast trapezoidal shaping filters are implemented
on the QH1, Q2 and Q3 signals in order to generate local triggers.
There are also slow control registers to adjust, for each channel, the filter parameters (rising edge and flat top lengths),
the low threshold and the high one. Usually, the filter is set at a \SI{200}{ns} rising edge and a \SI{200}{ns} flat top:
these values do not affect the sustainable event rate, which is limited by the data acquisition (see Sec.~\ref{ssc:data}).
For each front-end, the user can also choose the trigger timeout,
the trigger source (logic OR among any combination of Si1, Si2 and CsI) and the kind of threshold: i.e.
one may use the low threshold only (trigger is produced when the maximum amplitude of shaped signal is larger than it) or both low and high
thresholds (trigger is produced when the maximum amplitude of shaped signal is between them).
The local triggers generated by the FEEs reach the block card through 16 dedicated lines (one per telescope) on the backplane.
FAZIA trigger system is multiplicity based: on each block, the BC counts the local triggers and sends the total
to the regional board every \SI{40}{ns} through the optical link.
The RB collects all the multiplicity values coming from each
block and applies up to eight programmable rules. For each one the user can choose (via slow control) the blocks checked
by the rule, the multiplicity threshold and the downscale factor $K$. The regional board will then integrate inside a time window
the multiplicities coming only from the blocks specified by the rule, and it will produce a ``rule trigger''
only if the integrated value overcomes the multiplicity thresholds. The trigger is then accepted once every $K$ occurrences.
The logic OR among all the rule triggers is eventually the global trigger signal.
The RB then checks if there are any alerts: FPGA data buffers are almost full, the GTT flag (see Sec.~\ref{ssc:data}) from any block is true
or there is an external veto from the ``veto in'' LEMO connector; if there is at least one alert, then a veto flag is issued.
In these cases, except when there is only an external veto, the flag is also sent to the ``veto out'' LEMO connector.
The ``trigger out'' LEMO connector is true when there is a global trigger without the veto flag.
The main output of the trigger component on the regional board is the validation signal: it is
produced when the veto flag is false and the global trigger or the external trigger from the ``trigger in''
LEMO connector are true. The user may also choose to work in ``slave'' mode (when coupling FAZIA with other devices):
in this case the validation is only produced when the veto flag is false and the external trigger is true.
In any case the validation signal is sent at a \SI{25}{MHz} rate to every block via the optical links,
captured by the block cards and distributed to every FEE through the backplanes.
Together with the validation signal, also an event number, generated by a counter on the regional board FPGA,
is always sent to the FEEs using the same path.
\subsection{Data flow}\label{ssc:data}
On both FPGAs of the front-end cards, four trapezoidal shaping filters are implemented,
in addition to the trigger shapers described in the previous section, to calculate in real time the energy
released in each stage of the telescope. There is one shaper on the QH1 signal, one on Q2 and two on Q3.
In fact, the CsI(Tl) signal has a ``fast'' and a ``slow'' filter in order to exploit the well known $fast-slow$ technique
\cite{Gal95} to identify the light ions that stop
in the last stage of the telescope. As in the case of the trigger shapers, all the rising edge and
flat top lengths of the filters are user adjustable.
Usually, the filters on QH1 and Q2 were set at a rise time of \SI{2}{\micro s} and a flat top of \SI{1}{\micro s}.
Rise time and flat top for Q3 shapers were set respectively at \SI{2}{\micro s} and \SI{10}{\micro s} for the slow filter
and at \SI{2}{\micro s} and \SI{500}{ns} for the fast one. These values do not affect the sustainable rate because
it is limited by the maximum acquisition rate ($\sim$\SI{2000}{ev/s}). Since a faster rate is not affordable, the beam intensity can be accordingly chosen to avoid pile-up and dead time.
In addition to being shaped, all the signals (including also QL1, I1 and I2 that have no shaper) are continuously stored in circular buffers with dimension $N\leq 1024$, settable via slow control.
When a front-end card FPGA receives a validation, all the raw signals are transferred from their circular
buffers to FIFO memories, whose lengths are adjustable via slow control up to a maximum of 4096
samples ($\sim\SI{41}{\micro s}$) for QH1, Q2 and Q3 signals and 8192 samples ($\sim\SI{33}{\micro s}$) for QL1, I1 and I2.
At the same time, the acquisition thresholds are checked.
These thresholds are again user programmable via slow control and act on the energy shaped signals.
If any of the QH1, Q2, Q3 ``fast'' and Q3 ``slow'' shaped signals exceeds its respective acquisition threshold,
then all the telescope signals are marked for acquisition.
A single large FIFO memory is finally used to store the whole local event from the telescope handled by the FPGA.
On this memory, if the telescope is marked for acquisition,
the six waveforms are transferred together with the four maximum values of the shaped signals and
the event number sent by the RB. Between each front-end card and the block card,
the data travel on two \SI{400}{MHz} serial buses (introduced in Sec.~\ref{ssc:bc}),
thus capable to offer a throughput of \SI{800}{Mbit/s}. The buses are connected to the FPGA ``A'' on the FEE side,
so data sampled by the telescope ``B'' must pass through FPGA ``A''. If any of the FIFO memories on the front-end card
is about to fill up, the \textit{Global Trigger Throttle} (GTT) flag is raised and the RB is vetoed.
When a block card FPGA receives the validation signal, it enters a state where it starts to read the FIFO memories
on the front-end cards.
The event number written inside the data coming from each FEE is checked and the event is discarded if the number is
less than expected. If the actual number is greater than expected, instead, the FEE is skipped
but the data are kept for the next event. In this way the block card builds a coherent partial event and stores it in
a FIFO buffer waiting to transfer it to the regional board. Inside the partial event, also the external sinusoidal
signal (see Sec.~\ref{ssc:clock}) is stored.
After the generation of the validation signal, the regional board enters a state where it starts to read the FIFO memories
on the block cards.
In complete analogy to the block card behavior, the RB checks the event number written inside the data coming
from each block and the event is discarded if the number is
less than expected. If the actual number is greater than expected, instead, the block is skipped
but the data are kept for the next event. In this way the regional board builds a full coherent event and stores it in
a large FIFO buffer. The RB adds also some trigger information (i.e. number of accepted and vetoed triggers in the last ten seconds)
to allow the calculation of the dead time.
The FIFO memory is continuously read by a component of the FPGA code that produces Ethernet frames
using the \textit{User Datagram Protocol} (UDP) and send them to the acquisition system through a optical fiber connected to a SFP translator.
The user may specify up to 16 machines (sending their IP and MAC addresses via slow control) to which data will be sent.
The RB will send an event to the first computer, then another to second and so on. Then it will start again from the first.
In this way the dead time due to computing time is minimized. The maximum throughput obtained with the Ethernet
communication is about \SI{800}{Mbit/s}.
\subsection{Coupling with other detectors}\label{ssc:coupling}
The regional board features some programmable auxiliary connections which can be used to couple FAZIA with
other detectors. In particular, they will be used soon to measure together with INDRA at GANIL.
The coupling is done on different levels. The lowest is the trigger level: to ensure
a common dead time between FAZIA and INDRA we need to properly interconnect the trigger in/out and veto in/out
LEMO connections (see Sec.~\ref{ssc:trigger}) between the two detectors.
The second level consists in the generation of a timestamp: this is performed thanks to the CENTRUM module,
which is an absolute timestamp generator connected to both apparatuses.
When FAZIA generates a validation signal, it also sends a timestamp request to the CENTRUM system, which dispatches to the RB a frame containing a 48-bit timestamp and a 32-bit event number,
plus a 16-bit checksum. A similar packet is sent to INDRA when it produces a request.
The FAZIA regional board, after checking the checksum, inserts the CENTRUM frame inside the data flow that is sent to the acquisition.
The third and highest level is done by NARVAL, an acquisition system developed by IPN Orsay (France). NARVAL receives
data from both INDRA and FAZIA acquisitions and merges those events with timestamp differences smaller than
a pre-defined (and reaction-dependent) coincidence window, producing global events composed of data from both the apparatuses.
Of course this system is designed to be as general as possible and it may be used in the future also to couple
FAZIA with other detectors than INDRA. Moreover, both CENTRUM and NARVAL technologies support the connections
of many apparatuses, so one may also think to easily couple three or four different devices with FAZIA.
\subsection{Acquisition}\label{ssc:acq}
The FAZIA acquisition (DAQ) has been developed and it is currently maintained by INFN -- Naples.
FAZIA DAQ is a multi-threaded and multi-machine system, written in C++ language and consisting of different classes (DAQ modules) that exchange
messages and events through ZeroMQ sockets \cite{zmq}. The main DAQ modules are the following:
\begin{description}
\item[FzReader] It acquires raw data coming from the Regional Board by listening to the dedicated UDP socket. Then it forwards the data to FzParser thread pool.
\item[FzParser] Each FzParser includes a \textit{Finite State Machine} (FSM) able to analyze and validate each acquired event
in order to put all the information inside a structured format based on the Google Protocol Buffers \cite{protobuf}.
Multiple FzParser threads can run on a multi-core machine in order to benefit from parallel execution of tasks.
Each thread eventually forwards data to the FzWriter module through a \SI{10}{Gbit/s} dedicated network.
\item[FzWriter] This module stores data in files and directories with Google Protobuf data format.
It also runs a data spy in order to allow on-line data processing and analysis by external data visualization tools.
\item[FzNodeManager] It is a local supervisor for FzReader/FzParser or FzWriter that run on each FAZIA DAQ deployed machine.
It sends a report on module status to FzController and it receives run control and setup commands for module management.
\item[FzController] It is a global supervisor for all FzNodeManager modules.
It offers a global view on whole cluster status and it accepts commands for FAZIA DAQ setup and run control.
\end{description}
\begin{figure}[htbp]
\centering
\includegraphics[width=0.5\textwidth]{daq.pdf}
\caption{Schematic representation of the FAZIA DAQ system}
\label{fig:daq}
\end{figure}
Fig.~\ref{fig:daq} shows the overall architecture of FAZIA DAQ. Some plugins have also been developed to interact
with different Run Control systems. These plugins profit of the ZeroMQ network layer of FzController in order to
control the data acquisition from different clients (e.g. EPICS or SOAP). This feature of FAZIA DAQ makes it
very suitable and flexible for integration and coupling with other experiments and detectors.
For example, a GANIL plugin was developed to couple FAZIA with INDRA: it is a C++ class of the DAQ which allows the remote control of FAZIA
acquisition from NARVAL system using the SOAP protocol. At the same time, the plugin sends data to NARVAL using a TCP/IP connection.
\subsection{Slow control}\label{ssc:slow}
As illustrated in the previous sections, slow control instructions permit to control almost every aspect of the electronics.
The commands, in the form of UDP packets, can be sent by any PC in the same subnet where the regional board is located. The RB analyzes the frame
to check if the instruction is for the regional board itself. If it is, then the board executes the command and immediately sends a reply.
If not, the RB forwards to all the blocks the message via the optical links. On every block, the instruction is dispatched via the backplane
to every card containing a PIC (FEEs, PS, and BC). Since the slow control frame uniquely identifies the card
to which it is intended, only that card answers and the reply is returned to regional board. In every case, the RB transmits back the reply message
to the PC that has sent the request.
When the slow control request is not for the RB, it must be converted into a standard \SI{115.2}{kbit/s} serial signal which can be correctly read by the PIC devices
on the PS, BC or the FEEs. In particular, in the regional board FPGA code, a \textit{universal asynchronous receiver-transmitter} (UART) device is implemented.
The UART converts the slow control frames received via the Ethernet device into a slow serial data flow which is sent on the optical fibers
to all the blocks. Of course, since the slow control bit is sent on the fiber at a \SI{25}{MHz} rate (see Sec.~\ref{sss:tx}), the \SI{115.2}{kbit/s} serial data flow is oversampled.
Vice versa, when the RB receives a slow control reply from a block as a serial data flow, this flow is deserialized by the same UART component described above.
\section{Mechanical solutions}\label{sec:mechanics}
\subsection{Detector holding mechanics}\label{ssc:block}
A rendering of the FAZIA detectors is shown in Fig.~\ref{fig:det}, where silicon pads and CsI(Tl) crystals can be spotted.
The 16 telescopes, which form a block, are arranged in a $2\times 2$ matrix of \textit{quartettos}.
A ``quartetto'' is a self-consistent sub-structure of four telescopes in a $2\times 2$ configuration.
In the figure, the pad precise alignment is also shown. Indeed, each quartetto axis (perpendicular to the surface of the four Si pads)
points to the target. This configuration has been achieved through a fixed geometry and thus without degrees of freedom.
Our choice has been to build the various supports for a distance of \SI{100}{cm} between the target and the first Si layer.
This value has been chosen as a compromise between detector granularity and solid angle coverage. Moreover,
\SI{100}{cm} is the minimum distance from the target which guarantees a negligible channeling contribution (see below) and
a sufficient flight base for time of flight identification.
\begin{figure}[htbp]
\centering
\includegraphics[width=0.6\textwidth]{det.jpg}
\caption{
Rendering of the detectors and their support mechanics. Orientation axes of quartettos are also shown.
}
\label{fig:det}
\end{figure}
The detector supporting mechanics has been designed focusing on two main goals. The first is the reduction of all the dead
regions between quartettos and on external edges. As a matter of fact, considering a single block,
the active area is \SI{84}{\%} of the total front side. The second important goal is the precision of the telescope
orientation, in order to reduce the channeling effects. In fact, our silicon detectors are cut in such a way that the particles which perpendicularly impinge
on the sensors do not travel along crystal axes \cite{Bardelli09,Bardelli11}.
By means of a laser mirroring method (see Sec.~\ref{ssc:laser}) applied to the reflective Si pad surfaces, we obtained that the orientation misalignment is typically less
than \SI{1.5}{\degree}; larger values have been rarely observed in pads with misplacements or bad gluing. Those two objectives imposed the
choice of a proper material to build silicon pad holders. We chose the 7075 aluminium alloy, also used in avionics,
because it is precisely machinable and light but, at the same time, it is robust even in thin layers.
The supports for the $2\times 2$ Si pad matrices (Fig.~\ref{fig:telaio}) were built using \textit{wire electrical discharge machining}.
This technique, which softly acts on the bulk, reduces the strains so that the final piece maintains its planarity and details, needed for a good and
precise gluing of the silicon pads and the matching between different parts.
We remind that a telescope is composed of two silicon detectors followed by a CsI(Tl) crystal so, to form a quartetto,
two similar $2\times 2$ silicon frames should be locked one on top of the other by means of very thin grooves and ridges.
The four CsI(Tl) scintillators are fixed to a central cross-shaped support, which is again made of 7075 aluminium alloy. The strength of this alloy is
particularly important for this piece, since it must hold the four crystals (weighing \SI{0.72}{kg}) and all the silicon pads as a cantilever.
\begin{figure}[htbp]
\centering
\includegraphics[width=0.5\textwidth]{telaio.jpg}
\caption{
A silicon pad holder with mounted detectors is shown: the grooves and the ridges needed to lock two holders in position could be seen. The kapton
strips, used to connect the sensors to the front-end electronics, are soldered to the metal and $\mu$-bonded to Si pads.
}
\label{fig:telaio}
\end{figure}
The fastening between the detectors and the rest of the block is relatively simple, in order to
allow an easy replacement of the detector ``nose''. In this way, when some detectors are
damaged, one can replace the whole nose in a reasonable time. Usually, the insertion of the kapton strips
into the FEE female connectors is the longest operation.
The above described design and its practical implementation are particularly delicate and many
technicians and engineers, from Bologna, Florence and Naples INFN departments, contributed with their work and expertise.
\subsection{Frame for the INDRA-FAZIA campaign}
\begin{figure}[htbp]
\centering
\includegraphics[width=0.5\textwidth]{indra-fazia.jpg}
\caption{
INDRA-FAZIA mechanical drawing. It consists of the usual $4\pi$ INDRA multi detector setup, where the first five rings have been removed
to ensure the backward angular coverage from \SI{14}{\degree} to \SI{176}{\degree} and twelve FAZIA blocks, from \SI{1.5}{\degree} to \SI{14}{\degree},
divided in four triplets. The whole set-up is hold in the already existing INDRA vacuum chamber at GANIL.
}
\label{fig:indrafazia}
\end{figure}
The INDRA-FAZIA experimental phase at GANIL is foreseen during the period \numrange{2019}{2023}. It consists of several nuclear physics experiments
with different beams (e.g. Ca, Ni, Kr, Sn and Xe) at various energies (from 25 to \SI{80}{MeV/u}) on a large variety of targets.
It will use the coupling between the multi detectors INDRA and FAZIA.
This set up will be fitted in the already existing INDRA vacuum chamber at GANIL.
The forward part of the $4\pi$ INDRA multi detector is going to be replaced by the new FAZIA array in order to benefit from its better isotopic,
energy and angular resolution. The first five INDRA rings (from \SI{1.5}{\degree} to \SI{14}{\degree})
will be replaced by twelve FAZIA blocks (Fig.~\ref{fig:indrafazia}) as described previously. Those twelve blocks will be divided in four triplets.
A triplet consists of three FAZIA blocks in a ``L'' configuration. The blocks are mounted inside a metallic structure and fixed by brackets.
This triplet frame holds many handles, various towing rings and also two removable arms in order to help the mounting
and the handling inside the vacuum chamber. The main chassis, holding the four triplets, is constituted by a vertical square Elcom based structure,
where the four angles have been reinforced with thick stainless steel plates. The triplets are then fixed to each four angles of the vertical main square
frame via screws on the thick plates. An intermediate angled brace between the triplets and each angle of the main square frame fixes
the geometry of the blocks, i.e. the distance from the target and the polar angles respect to the beam axis where the detectors pointed out.
For the first experiment at GANIL this intermediate brace has been designed to ensure a one meter distance to the target and a minimum (maximum) polar angles of
\SI{1.5}{\degree} (\SI{14}{\degree}). The vertical main square frame is eventually screwed to a movable platform which uses the already existing INDRA vacuum chamber rails,
enabling the right positioning distance to the target. Two remotely controlled thick disk shields, with different diameters
(a small one protecting the angles from \SI{0.85}{\degree} to \SI{3.1}{\degree} and a big one from \SI{0.85}{\degree} to \SI{5}{\degree}),
have been designed to protect the silicon detectors during beam focalization or data taking to avoid radiation damage \cite{Barlini13} with stopped heavy ions.
The whole final setup with twelve FAZIA blocks weights around \SI{230}{kg} (a complete single FAZIA block weights slightly less than \SI{15}{kg})
and it is foreseen to be towed by a crane if necessary. The CAD design and building of the whole support frame for the twelve blocks have been designed at
the Laboratoire de Physique Corpusculaire de Caen, LPC Caen (France).
\subsection{Cooling}\label{ssc:cooling}
Since a single FAZIA block absorbs almost \SI{300}{W}, a very efficient cooling solution must be adopted to operate under vacuum.
The final setup consists in a thick copper plate, on which all the cards are screwed on. The conduction is ensured by thermal grease
between each card and the copper surface.
The copper plate has been designed in order to efficiently distribute the liquid flow along the entire surface which holds
the 8 FEE cards. Solving the conflict between the internal pipes for the liquid flow and the many screw holes
needed to ensure mechanical and thermal coupling of the electronic boards has been a difficult issue.
In total, we have about 100 screws. The adopted solution has been to start from a copper slab \SI{8}{mm}
thick having two main internal in-out pipes running longitudinally on
the two sides of the plate. The liquid distribution is then ensured in the copper volume through eight transversal
holes, drilled in the plate and joining the input-output main lateral pipes.
These transversal holes have variable diameters along the plate length in
order to compensate for the pressure drop at the various distances from the entrance/exit tube fittings.
Outside the scattering chamber a powerful chiller (ACW LP60) is mounted to refrigerate the water (with \SI{30}{\%} alcohol or glycol) flowing through all the blocks.
The temperature of the cooling liquid is kept at about \SI{10}{\celsius}.
To ensure an independent cooling for each block a so called ``clarinet'' device has been built to dispatch the fluid to all the cooling circuits.
It was designed at the \textit{Grand Accélérateur National d'Ions Lourds} (GANIL).
\subsection{Laser angle measurement}\label{ssc:laser}
To conclude this review on the FAZIA technological solutions, a notable mention goes to a method that the collaboration implemented to
precisely measure the polar angles $\vartheta$ and $\varphi$ of the detectors with respect to the beam direction.
First of all, a piece of beam line beyond the scattering chamber is dismounted to allow the mounting of a laser.
This laser, aligned with the centre of beam line and the target, is fired in the opposite direction with respect to the beam.
In place of the target a mirror is then mounted in such a way that its centre is exactly on the path of the laser.
The mirror equipment is mounted in a gimbal configuration allowing the rotation around two orthogonal axes:
the inclination and declination can be controlled via two precise stepping motors. The fine regulation of the ``zero'' position
(when the laser bounces back in the same direction from where it impinged on the mirror) is set once via micrometric screws.
Then, by regulating the two mirror angles (using the stepping motors) up to when the reflected
laser beam impinges on the centre of a telescope, it is easy to obtain its $\vartheta$ and $\varphi$ angles
via a reading of the encoded current position of the motors.
The centre of the telescope is determined by visual inspection with an estimated accuracy of the order of \SI{1}{mm},
corresponding to $\sim$\SI{3}{'} accuracy on angle measurement.
The described technique is in fact a simplified version of a method used by the collaboration for
the FIASCO experiment (see Sec~2.1 of \cite{Bini03}).
\section{Conclusions and future improvements}\label{sec:conclusions}
In this review paper we examined the most peculiar and innovative features of the FAZIA telescope array from the technological point of view.
To summarise, some of the most important characteristics of FAZIA are the compactness and modularity: since FAZIA is structured
in independent blocks, various geometry configurations are possible; moreover, all the analogue chains operate under vacuum, very close
to the detectors, so that electronic noise pick-up and signal distortion along the transmission lines are greatly reduced. Another factor that should not be neglected is the optimisation
of the analogue stages done in the last years: in fact, the pre-amplifiers were designed after many tests performed in the past with
prototype telescopes \cite{Hamrita04}; a similar consideration is valid also for the choice of the ADCs. The combination of pre-amplifiers which have
a very high dynamic range and low noise with converters that have a well balanced design in term of ENOB and sampling rate permits
to obtain a good energy resolution in a very wide spectrum of particle energies and charges (from Am $\alpha$ sources to heavy
nuclei with energies of the order of \SI{4}{GeV}).
The clock distribution is another fundamental part of the FAZIA apparatus. Thanks to a very complex design that uses fixed latency
optical connections and many PLL devices on each electronic card, it is possible to achieve a synchronisation among all the
acquired channels within \SI{100}{ps}.
Finally, other important aspects of FAZIA are its flexibility and upgradability. In fact, all the electronic boards
contain programmable devices which are steadily maintained and updated to include new features.
Indeed, one can implement new firmware solutions on the on-board FPGA after that a given algorithm has been tested
off-line using the entire sampled waveforms previously acquired.
For example, in this respect we are going to implement the pulse-shape discrimination of silicon signals directly
on-board, by adding the search of the current signal maximum on the front-end card FPGA code.
Another feature which may be added to the same code is an energy shaper on the QL1 signal, to have
the high gain energy measurement without the need to send the whole signal to the acquisition.
To improve the data bandwidth, especially in view of experiments that will use many FAZIA blocks, we are also going to implement
two Ethernet connections (instead of one as we have now) between the regional board and the acquisition system.
This work was partly supported by the Polish Ministry of Science and Higher Education under Contract No. 778N - FAZIA/2010/0,
the Polish National Science Centre under Contracts No. 2013/08/M/ST2/00257 (COPIGAL) and No. 2014/14/M/ST2/00738 (COPIN-INFN Collaboration).
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 4,031
|
Home » Learning Center » Jewelry Design » Behind the Design » The Jewelry of Glenda Arentzen
The Jewelry of Glenda Arentzen
by Patricia Malarcher
At mid-career, goldsmith Glenda Arentzen still produces jewelry with the freshness that has been her hallmark for a quarter of a century. It permeates whatever she makes, appearing not only in lively, asymmetrical profiles, but also in details: the crisp edges of a 24k gold earring; the apparently spontaneous shape of a brooch; the loose but precise fit of a textured scrap of gold to its structural support. Arentzen's light but authoritative touch never emphasizes metal as a precious commodity, but rather as a means to illuminate ideas.
In addition to the characteristics above, the work now is tempered with a new ingredient, the artist's seasoned reflections on human life and nature. During the past decade, this enriching content has been mined primarily from Arentzen's experience of living and working in rural New Hampshire.
On many counts, Arentzen typifies the generation of artists in craft mediums, who came of age in the early 1960s. In 1962, the year she received a bachelor's degree in art from Skidmore College, representation in "Young Americans" at the Museum of Contemporary Crafts (now the American Craft Museum) launched her exhibition career; a two-year stint as an assistant in Adda Husted-Andersen's New York studio was followed by a Fulbright year in Denmark; after earning a master's degree from Teacher's College of Columbia University, she taught part-time at the Fashion Institute of Technology and other schools to supplement her income from commissions and other sales of work.
Landscape Ring, 14k yellow gold, sterling silver, 1991
Each step forward was a pioneering venture along paths that, at the time, were only sketchily defined. As the field's horizons have widened, her development of knowledge and skills has kept pace with opportunities, among which is a gift for Queen Elizabeth to present at Ascot, and awards that include three from Diamonds International. Her extensive research into patination and the work that resulted from it introduced a new vocabulary of color in silver and gold.
Although some time ago Arentzen reached the professional level where jewelry sales earn her a living, she has managed to retain her pioneering spirit. And, although her career might be read as a blueprint for success as a metalsmith, the zest for life that overflows into her work is a singular, nontransferable trait. She once described object-making as the "visual record of my adventures."
One morning last spring at the Aaron Faber Gallery, Arentzen was interviewed amidst an installation of 75 new works amassed for a solo show. Just emerging from "the frenzy of the final pieces," she seemed still energized by the process of creation.
As she shifted her attention from one object to another, her conversation sparkled with images that have influenced her forms: the patterning of rocks on a hillside; the cracks in a macadam road with something oozing out from beneath; a wet leaf hitting her face in the fall; the downward thrust of a waterfall, echoed in the metal "rain" falling from an earring. She conveyed the impression of an alchemist who had transformed the New England landscape into gold. In particular, a large percentage of the pieces had been inspired by the flow of water. "I seem to have become attuned to my surroundings in motion," she observed. Ironically, it was not beside a brook, but, rather, on the platform of a New York City subway station that she became poignantly aware of this attunement.
"I remember the feeling of revelation when, a couple of years ago, a woman, an underground street performer with a microphone, was singing a rap song with a cadence like a flow. It went on for many verses; each had something to do with a civilization, starting with Egypt and going all through history. The chorus was about a river, and had a feminist viewpoint: woman is the river. There was a kind of feeling that connected with what I see in rivers – for instance, when I'm in a plane flying low enough to watch how they flow."
Box Bracelet, sterling silver, 14k gold, 1988
River is the title of the necklace that served as the show's centerpiece. Its fat, textured curves of gold narrow and widen as they separate and join, compelling the eye to trace a circular course. Although it has a raw simplicity, its gentle rise in the center as well as its construction in segments suggest a fluidity intended to accommodate the body. "Some people have the misconception that something can be either art or wearable, that it's impossible to make art jewelry that is also comfortable. With River, l wanted to make an art statement, but also wanted it to rest on the shoulders properly, to meet its obligations in terms of concept."
This new work is mostly organic images based on natural forms that hold emotion for Arenzen. Inspired by both micro and macro views of landscape, the pieces are not objective translations, but rather, abstractions reflecting the attempt to get inside ideas, to capture essences that would offer viewers, and particularly those who wear the jewelry, access to fantasy. "I wanted to make comforting, provoking, stimulating, 'places to be' – in many ways they are retreats," she said. "When pieces really work, you feel a passion in them that seems to be getting at some elemental areas of life." Arentzen's notion of her work as "places to be" was expressed in an article in Craft Horizons in 1976. Since then, however, her personal frame of reference for "place" has radically changed, and this has affected her ideas.
Although her business is centered in New York, and she still has a New York apartment, 10 years ago she headed For New Hampshire to enter a new phase of life. The object: to merge her life with that of Rick Harkness, a glassblower she had met at the Haystack School when she was learning to blow glass herself. Harkness's country house was equipped with a "wood stove, dirt cellar, cows and step-children." The jolt that accompanied the move was more than that from the simplicity of living alone to the complexity of a household. It also meant disconnecting, for long periods, from the stimulation of urban life and interaction with peers and customers who appreciated the expressive character of her work.
Although New Hampshire is a craft-conscious state, dotted with guild-sponsored shops, Arentzen realized that her artistic orientation veered from the regional norm. "There, I noticed that the idea of body ornament is little pieces of metal on people's ears and hands. Even with people who have money and travel a lot, jewelry isn't ostentatious. It seems that in urban areas the need is stronger for statements of personal expression. In New Hampshire, it's impolite to stand out."
The lack of response to her work there has been disconcerting. "When I wore some of these new pieces, they provoked no comments. There is always an occasional comment in New York. I'm not looking for a positive, or even a constructive reaction, but when people comment by looking, they complete a piece with their eyes.
"Feedback from colleagues and galleries helps shape your work. I had to reorganize the way I touch base. Now, I regularly spend a 'library night' looking at art magazines. I've also been making the effort to read books on philosophy of design."
Early influences that have continued to sustain her are a graduate course with Rudolf Arnheim at Columbia and painting as well as jewelry courses with Earl Pardon at Skidmore. "Earl Pardon had certain principles for guiding you through a critique of what you were doing – not calling things 'good' or 'bad,' but evaluating your work in terms of connecting forms to concepts. Arnheim went from one element of design to another and explained why things happen in terms of his psychological and visual research. That gave me a solid grounding in Gestalt psychology, in the way things work."
Indeed, this seems to be reflected in Arentzen's ability to distill visual ideas in configurations that seem uncontrived but irreducible. For example, in a necklace called, Stepping Stones, in which angled units just meet at irregular intervals, one immediately feels a tenuous balance, a precarious dance from one section to another.
Box Bracelet, 14k, 24k yellow gold, sterling silver, gemstones, c. 1988
Although two years ago Arentzen's 25-year retrospective at Faber was critically acclaimed, the spring show challenged her to reach beyond past successes. "This was different from other shows," she said. "When we were setting the date, the people at the gallery encouraged me. They said, 'make the stuff that speaks to your heart, and we'll sell it.'"
While welcoming the freedom to create a collection without the commercial constraint of satisfying budgets or tastes of potential customers, Arentzen also recognized that some determinants in the normal process of design had been eliminated. "Ordinarily, when I'm organizing a show for a gallery or a big fair, I take a concept and expand it by trying to make accessories for certain prices and occasions, for shorter or taller people. Although I was still concerned with wearability, this work was not made for the market but in the service of an idea."
Permission to follow her intuition led her to concentrate on a few major works but to forego "drop-dead exhibition pieces" in favor of "things that were smaller and quieter, that made a statement, pleasant or not. Stones were used sparingly and usually seemed to emerge from the colors of the metal around them, rather than to assert an independent presence. The gallery's directive also allowed the simplicity that best exploits the use of 24k gold, which in combination with 14k gold and sometimes silver is contained in a fifth of the pieces. Most often, it occurs in unpretentious, almost casual forms that subtly glow in relation to the metals around it. "It has a warmth that leads the eye in, with a lot of impact in a wonderful way," Arentzen observed.
Although she seemed a bit concerned that, technically, the jewelry world would find nothing new in the show, it does reflect her efforts to deal with some problematic aspects of this softest, purest gold. "There's no place to hide the rough spots; fittings have to be perfect because you can't file away the excess." To offset its softness, she Found she could back it with a thin layer of a different gold.
As a spinoff from the show, she intends to continue working with the contrast between 24k gold and other metals. In particular, she wants to pursue a series of pins she refers to as "secrets." Suggestive of samples from the layers of a wooded terrain, these are made with fragmentlike shapes of 24k gold, loosely set in frames and repousséd in a way that both hides and reveals stones and other elements just beneath the surface. "Although we live in an earring culture, pins are the things that lend themselves to themes," Arentzen said. "They are more easily perceived as fine art, and most readily related to the framed form."
She also expressed a special satisfaction with several earrings and pins in which wire gridlike structures held what appeared to be casually formed "shards" of 24k gold. Lightly held in place, they were reminiscent of leaves blown into bramble bushes. "They're like impressions of the flotsam and jetsam of life," she observed. "The geometric wires could be seen as a manmade way of trying to pull something organic together."
In striving for elemental qualities, Arentzen has become aware of certain forms with universal appeal. "All kinds of people have a positive emotional reaction to a wide ring that encloses the finger," she noted. Even when her rings are as dramatic as Landscape, a disc that rests on three fingers like a lily pad, the fit of the unseen band is a critical component. However, she also made a choker designed to feel constricting – "like you're jumping out of your skin" – that seems to contradict her usual regard for comfort. This piece emerged from her experience of feeling the press of life's demands from many different directions.
Stepping Stones Necklace, 14k yellow gold, triangular diamond, 1991
Between the cases of jewelry on the gallery walls hung several finger-painted works on paper, executed with sweeps of black paint and gold-leaf, veiled with tracing paper. "A lot of those are sketches that later may turn out in wax," she said, alluding to the paintings as "background noise." The brisk strokes and swirls suggested warmup exercises for the linear patterns, the ridges and incisions carved in the wax Arentzen prepares for casting. "Drawing is where the idea becomes tangible for the first time. It may be just a reference, but it tells a lot about the way you work."
For Arentzen, casting is a critical tool for making textured parts, which she later assembles, interspersing them with smooth components. She values the handmade quality of the wax units that sometimes carry fingerprints or drops of melted wax into the metal. Her reliance on casters in New York is a major drawback in living far away from the city. The easy procedure of delivering wax locally and picking up the metal on the same day has been replaced with a mail-order system that takes two weeks to complete. The time gap, however, has not vet discouraged Arentzen from using this approach to working out her ideas.
Noticeably absent from the recent show were the elegant, sleekly sophisticated, geometric-patterned forms that Arentzen had perfected, using colors achieved through patination in the married-metals technique. Two years ago, an article in Ornament discussed this work, along with Arentzen's conviction that their complex, pieced-fabrication served as a refreshing complement to the textured work. Arentzen's present distance from casting facilities suggests that this might have been the logical direction to follow, but her ideas have moved away from it. "I've nearly stopped doing it, not because I feel negative about it, but because I'm finished with it. I resisted the temptation to do it; the fat, graphic quality didn't seem as important as the landscape ideas."
Perhaps a key to the vitality of Arentzen's work is that, in addition to continually scrutinizing and reflecting on the visual world, she also keeps her inner sense of texture and pattern alive by a regular routine of both tap and ice dancing. With the exhibition's opening behind her, she was looking forward to her 50th birthday, an occasion for a carwheel. Predictably, the embodied memory of that arc in space will someday find expression in gold.
Smith, Dido, "Glenda Arentzen's Magic Metal," Craft Horizons, February 1976.
Blauer, Ettagale, "Glenda Arentzen: Contemporary Goldsmith," Ornament, 12 (3), 1989.
Patricia Malarcher is a fiber artist and craft critic.
Metalsmith Magazine – 1991 Fall
In association with SNAG's
Metalsmith magazine, founded in 1980, is an award winning publication and the only magazine in America devoted to the metal arts.
Behind the Design,
Business and Marketing,
Features,
Jewelry Design,
Jewelry Displays,
Patricia Malarcher
Previous Thomas Gentille: Performance of the Ephemeral
Next Rebekah Laskin: Material Voice
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 7,923
|
\section{Introduction}
Let $\g$ be a simple finite dimensional complex Lie algebra. For the loop algebras, $\g\otimes\C[t,t^{-1}]$, the global Weyl modules were introduced by Chari and Pressley, \cite{CP}. Feigin and Loktev extended these global Weyl modules to the case where the Laurent polynomials above were replaced by the coordinate ring of a complex affine variety, \cite{FL}. Chari, Fourier and Khandai then generalized this definition to the map algebras, $\g\otimes A$, where $A$ is a commutative, associative complex unital algebra, \cite{CFK}. Feigin and Loktev also gave an isomorphism, which explicitly determines the structure of the global Weyl modules for the map algebras of $\sl_n$ of highest weight $m\omega_1$, \cite{FL}.
The goal of this work is to use the structure isomorphism given by Feigin and Loktev to give nice bases for the global Weyl modules for the map algebras of $\sl_n$, $\sl_n\otimes A$, of highest weight $m\omega_1$. These bases will be given in terms of specific elements of $\bu(\sl_n\otimes A)$ acting on the highest weight vector. This was done in \cite{Cham} in the case $n=2$, but the case $n>2$ has not previously appeared in the literature.
\section{Preliminaries}
\subsection{The Structure of $\sl_n$}
Recall that $\sl_n$ is the Lie algebra of all complex traceless matrices
The Lie bracket is the commutator bracket given by $[A,B]=AB-BA$.
Given any matrix $\left[b_{i,j}\right]$ define $\varepsilon_k\left(\left[b_{i,j}\right]\right):=b_{k,k}$. For $i\in\{1,\ldots,n-1\}$ define $\alpha_i:=\varepsilon_i-\varepsilon_{i+1}$. Define $$R^\pm:=\left\{\displaystyle\pm\left(\alpha_i+\dots+\alpha_j\right)\bigg| 1\leq i<j\leq n-1\right\}$$ to be the positive and negative roots respectively, and define $R=R^+\cup R^-$ to be the set of roots. Let $e_{i,j}$ be the $n\times n$ matrix with a one in the $i$th row and $j$th column and zeros in every other position. Define $h_i:=h_{\alpha_i}=e_{i,i}-e_{i+1,i+1}$, for $i\in\{1,\ldots,n-1\}$. Then $\h:=\span\{h_i|1\leq i\leq n\}$ is a Cartan sub-algebra of $\sl_n$. Given $\alpha=\alpha_i+\cdots+\alpha_j\in R^+$ define $x_\alpha:=e_{i,j}$ and $x_{-\alpha}:=e_{j,i}$. Then $\{h_i,\ x_{\pm\alpha}\ |\ 1\leq i\leq n-1,\ \alpha\in R\}$ is a Chevalley basis for $\sl_n$. Given $i\in\{1,\ldots,n-1\}$, define $x_i:=x_{\alpha_i}=e_{i,i+1}$, $x_{-i}:=x_{-\alpha_i}=e_{i+1,i}$. Note that, for all $1\leq i\leq n-1$, $\span\{x_{-i},h_i,x_i\}\cong\sl_2$.
Define nilpotent sub-superalgebras $\n^{\pm}:=\span\{x_{\alpha}|\alpha\in R^{\pm}\}$ and note that $\sl_n=\n^{-}\oplus\h\oplus\n^{+}.$
Define the set of fundamental weights $\{\omega_1,\ldots,\omega_{n-1}\}\subset\h^\ast$ by $\omega_i(h_j)=\delta_{i,j}$ for all $i,j\in\{1,\ldots,n-1\}$. Define $P^+:=\span_{\Z_{\geq0}}\{\omega_1,\ldots,\omega_{n-1}\}$ to be the set of dominant integral weights.
\subsection{Map Algebras and Weyl Modules}
For the remainder of this work fix a commutative, associative complex unital algebra $A$. Define the map algebra of $\sl_n$ to be $\sl_n\otimes A$ with Lie bracket given by linearly extending the bracket
$$[z\otimes a,w\otimes b]=[z,w]\otimes ab$$
for all $z,w\in\sl_n$ and $a,b\in A$.
Define $\bu(\sl_n\otimes A)$ to be the universal enveloping algebra of $\sl_n\otimes A$.
As in \cite{CFK} we define the global Weyl model for $\sl_n\otimes A$ of highest weight $\lambda\in P^+$ to be the module generated by a vector $w_\lambda$, called the highest weight vector, with relations:
$$(x\otimes a)w_\lambda=0,\hskip.5in (h\otimes 1)w_\lambda=\lambda(h)w_\lambda,\hskip.5in (x_{-i}\otimes 1)^{\lambda(h_i)+1}.w_\lambda=0$$
for all $a\in A$, $x\in\n^+$, $h\in\h$, and $1\leq i\leq n-1$.
\subsection{Multisets}
Given any set $S$ define a multiset of elements of $S$ to be a multiplicity function $\chi:S\to\Z_{\geq0}$. Define $\f(S):=\{\chi:S\to\Z_{\geq0}:|\supp\chi|<\infty\}$. For $\chi\in\f(S)$ define $|\chi|:=\sum_{s\in S}\chi(s)$. Notice that $\f(S)$ is an abelian monoid under function addition. For $\psi,\chi\in\f(S)$, $\psi\subseteq\chi$ if $\psi(s)\leq\chi(s)$ for all $s\in S$. Define $\f(\chi)(S):=\{\psi\in\f(S)\ |\ \psi\subseteq\chi\}$. In the case $S=A$ the $S$ will be omitted from the notation. So that $\f:=\f(A)$ and $\f(\chi)=\f(\chi)(A)$.
If $\psi,\chi\in \f$ with $\psi\in\f(\chi)$ we define $\chi-\psi$ by standard function subtraction. Also define $\pi:\f-\{0\}\to A$ by
$$\pi(\psi):=\prod_{a\in A}a^{\psi(a)}$$
and extend $\pi$ to $\f$ be setting $\pi(0)=1$. Define $\m:\f\to\Z$ by
$$\m(\psi):=\frac{|\psi|!}{\prod_{a\in A}\psi(a)!}$$
For all $\psi\in\f$, $\m(\psi)\in\Z$ because if $\supp\psi=\{a_1,\ldots,a_k\}$ then $\m(\psi)$ is the multinomial coefficient
$$\binom{|\psi|}{\psi(a_1),\ldots,\psi(a_k)}$$
For $s\in S$ define $\chi_s$ to be the characteristic function of the set $\{s\}$. Then for all $\chi\in\f(S)$
$$\chi=\sum_{s\in S}\chi(s)\chi_s$$
\subsection{The Symmetric Tensor Space}
Given any vector space $W$, there is an action of the symmetric group $S_k$ on $\underbrace{W \otimes W \otimes \cdots \otimes W}_\text{$k$-times}$ given by
$$\sigma(w_1\otimes w_2\otimes\cdots\otimes w_k)=v_{\sigma^{-1}(1)}\otimes v_{\sigma^{-1}(2)}\otimes\cdots\otimes v_{\sigma^{-1}(k)}\text{ where }v_1,\dots,v_k\in W.$$
For any vector space $W$, define its $k$th symmetric tensor space
$$S^k(W)=\span\left\{\sum_{\sigma \in S_k}\sigma(w_1\otimes\cdots\otimes w_k)\bigg|w_1,\dots,w_k\in W\right\}$$
Define $V\cong\C^n$ to be an $\sl_n$-module via left matrix multiplication, and write the basis as $v_1:=(1,0,\ldots,0)$, and for $i\in\{1,\ldots,n+m-1\}$, $v_{i+1}:=x_{-i}v_i$. Then $V\otimes A$ is an $\sl_n\otimes A$ module under the action $(z\otimes a)(w\otimes b)=zw\otimes ab$.
Given $\varphi_1,\dots,\varphi_n\in\f$ with $k:=\sum_{i=1}^n|\varphi_i|$ define
$$w(\varphi_1, \dots, \varphi_n):=\bigotimes_{a_1\in\supp\varphi_1}(v_1\otimes a_1)^{\otimes\varphi_1(a_1)}\otimes\cdots\otimes\bigotimes_{a_n\in\supp\varphi_n}(v_n \otimes a_n)^{\otimes\varphi_n(a_n)}\in (V\otimes A)^{\otimes k}$$ and
$$v(\varphi_1, \dots, \varphi_n) := \sum_{\sigma\in S_k}\sigma(w(\varphi_1, \dots, \varphi_n))\in S^k(V\otimes A).$$
We will need the following theorem of Feigin and Loktev, which is Theorem 6 in \cite{FL}.
\begin{thm}[Feigin--Loktev, 2004]\label{FL}
For all $m\in\N$
$W_A(m\omega_1) \cong S^m(V \otimes A)$ via the map given by
$$w_{m\omega_1} \mapsto (v_1 \otimes 1)^{ \otimes m}.$$
\end{thm}
We will also need the following lemma.
\begin{lem}\label{smbasis}
Let $\B$ be a basis for $A$. Then the set
$$\fB:=\left\{v(\varphi_1,\dots,\varphi_n)\ \bigg|\ \varphi_1,\ldots,\varphi_n\in\f(\B),\ \sum_{i=1}^n|\varphi_i|=m\right\}$$
is a basis for $S^m(V\otimes A)$.
\end{lem}
\begin{proof}
$\fB$ spans $S^m(V\otimes A)$ because $\B$ spans $A$ and $v_1,\ldots,v_n$ spans $V$. $\fB$ is linearly independent because the set
$$\{(v_{j_1}\otimes b_1)\otimes\dots\otimes(v_{j_m}\otimes b_m)\ |\ j_1,\ldots,j_m\in\{1,\ldots,n\},\ b_1,\ldots,b_m\in\B\}$$
is a basis for $(V\otimes A)^{\otimes m}$ and hence is linearly independent.
\end{proof}
Given $k\in\N$ define $\Delta^{k-1}: \bu(\sl_n \otimes A) \to \bu(\sl_n \otimes A)^{\otimes k}$ by extending the map $\sl_n\otimes A\to\bu(\sl_n\otimes A)^{\otimes k}$ given by
\begin{eqnarray*}
\Delta^{k-1}(z \otimes a)&=&\sum_{j = 0}^{k-1} 1^{\otimes j} \otimes (z \otimes a) \otimes 1^{\otimes k-1-j}
\end{eqnarray*}
Note that $\Delta^{k-1}(1)=1^{\otimes k}$ not $k1^{\otimes k}$.
Since $V\otimes A$ is a $\bu(\sl_n\otimes A)$ module, $(V\otimes A)^{\otimes m}$ is a left $\bu(\sl_n\otimes A)$-module with $u$ acting as $\Delta^{m-1}(u)$ followed by coordinatewise module actions. Moreover $S^m(V\otimes A)$ is a submodule under this action. Thus $S^m(V\otimes A)$ is a left $\bu(\sl_n\otimes A)$-module under this $\Delta^{m-1}$ action.
\subsection{}
For all $i=1,\dots,n-1$ and $\chi,\varphi\in\f$ recursively define $q_i(\varphi,\chi)\in\bu(\sl_n\otimes A)$ as follows
\begin{eqnarray*}
q_i(0, 0) &:=& 1\\
q_i(0, \chi) &:=& -\frac{1}{|\chi |}\sum_{0\neq\psi\in\f(\chi)} \m(\psi)(h_i\otimes\pi(\psi))q_i(0,\chi-\psi)\\
q_i(\varphi,\chi)&:=&-\frac{1}{|\varphi |}\sum_{\psi\in\f(\chi)}\sum_{d\in\supp\varphi} \m(\psi)(x_{-i}\otimes d\pi(\psi))q_i(\varphi-\chi_d,\chi-\psi)
\end{eqnarray*}
Given $\varphi_n, \dots, \varphi_n\in\f$, define
$$\hskip-.5in q(\varphi_1, \dots, \varphi_n)
:=q_{n-1}(\varphi_n,\varphi_{n-1})q_{n-2}((|\varphi_n|+|\varphi_{n-1}|)\chi_1, \varphi_{n-2})\dots q_{2}\left(\left(\sum_{j=3}^n|\varphi_j|\right)\chi_1, \varphi_{2}\right)q_{1}\left(\left(\sum_{k=2}^n|\varphi_j|\right)\chi_1, \varphi_{1}\right)$$
\begin{rem}
Note that the $q_i(0,\chi)$ coincide with the $p_i(\chi)$ defined in \cite{BC}.
\end{rem}
\section{Main Theorem}
The main result of this work is the theorem stated below.
\begin{thm}\label{thm}
Given a basis $\B$ for $A$ and $m\in\Z_{>0}$, the set
$$\left\{q(\varphi_1,\ldots,\varphi_n)w_{m\omega_1}\ \bigg|\ \varphi_1,\ldots,\varphi_n\in\f(\B),\ \sum_{i=1}^n|\varphi_i|=m\right\}$$
is a basis for $W_A(m\omega_1).$
\end{thm}
The proof of this theorem will be given after several necessary lemmas and propositions.
\subsection{Necessary Lemmas and Propositions}
\begin{prop}\label{deltak}
For all $k\in\N$ $\Delta^k=(1^{\otimes k-1}\otimes\Delta^1)\circ\Delta^{k-1}$.
\end{prop}
\begin{proof}
The case $k=1$ is trivial. For $k\geq2$ and $u\in\bu(\sl_n\otimes A)$ we have
\begin{eqnarray*}
\left(1^{\otimes k-1}\otimes\Delta^1\right)\left(\Delta^{k-1}(u)\right)&=&\left(1^{\otimes k-1}\otimes\Delta^1\right)\left(\sum_{j=0}^{k-1}1^{\otimes j}\otimes u\otimes1^{\otimes k-1-j}\right)\\
&=&\left(1^{\otimes k-1}\otimes\Delta^1\right)\left(\sum_{j=0}^{k-2}1^{\otimes j}\otimes u\otimes1^{\otimes k-1-j}+1^{\otimes k-1}\otimes u\right)\\
&=&\sum_{j=0}^{k-2}1^{\otimes j}\otimes u\otimes1^{\otimes k-2-j}\otimes\Delta^1(1)+1^{\otimes k-1}\otimes\Delta^1(u)\\
&=&\sum_{j=0}^{k-2}1^{\otimes j}\otimes u\otimes1^{\otimes k-2-j}\otimes1\otimes1+1^{\otimes k-1}\otimes(u\otimes1+1\otimes u)\\
&=&\sum_{j=0}^{k-2}1^{\otimes j}\otimes u\otimes1^{\otimes k-j}+1^{\otimes k-1}\otimes u\otimes1+1^{\otimes k-1}\otimes1\otimes u\\
&=&\sum_{j=0}^k1^{\otimes j}\otimes u\otimes1^{\otimes k-j}\\
&=&\Delta^k(u)
\end{eqnarray*}
\end{proof}
Given $\chi\in\f$ and $k\in\N$ define
$$\comp_k(\chi)=\left\{\psi:\{1,\ldots,k\}\to\f(\chi)\ \Bigg|\ \sum_{j=1}^k\psi(j)=\chi\right\}$$
\begin{lem}\label{deltaqi}
For all $i\in\{1,\ldots,n-1\}$
$$\Delta^{k-1}\left(q_i(\varphi, \chi)\right)=\sum_{\substack{\psi\in\comp_k(\chi)\\\phi\in\comp_k(\varphi)}}
q_i(\phi(1),\psi(1))\otimes\cdots\otimes q_i(\phi(k),\psi(k))$$
\end{lem}
\begin{proof}
This can be proven by induction on $k$. The case $k=1$ is trivial. In the case $k=2$ the lemma becomes
$$\Delta^1(q_i(\varphi, \chi)) = \sum_{\substack{\psi \in \f(\chi) \\ \phi \in \f(\varphi)}} q_i(\phi, \psi) \otimes q_i(\varphi - \phi, \chi - \psi)$$
This can be proven by induction on $|\varphi|$. For $k>2$ use Proposition \ref{deltak}. The details in the $\sl_2$ case can be found in \cite{Cham}. This can be extended to the $\sl_n$ case via the injection $\Omega_i:\sl_2\otimes A\to\sl_n\otimes A$ given by
$$\Omega_i(x ^-\otimes a)=x_{-i}\otimes a,\hskip.5in\Omega_i(h\otimes a)=h_i\otimes a,\hskip.5in\Omega_i(x^+\otimes a)=x_i\otimes a$$
For all $i\in\{1,\ldots,n-1\}$ and $a\in A$.
\end{proof}
\begin{lem}\label{action produces 0}
For all $\varphi,\chi \in \f$ with $|\varphi | + |\chi| > 1$ and all $i\in\{1,\ldots,n-1\}$ $ \ q_i(\varphi, \chi)(v_i \otimes 1) = 0$.
\end{lem}
\begin{proof}
Assume that $\varphi = 0$. This case will proceed by induction on $|\chi |>1$. If $ |\chi | = 2$ (so that $\chi=\{a,b\}$ for some $a,b\in A$) we have
\begin{eqnarray*}
q_i(0,\{a, b\})(v_i \otimes 1) &=&\left[ (h_i \otimes a)\otimes(h_i \otimes b) - (h_i \otimes ab)\right](v_i \otimes 1) \\
&=& (h_i \otimes a)\otimes (v_i \otimes b) - (v_i \otimes ab)\\
&=& (v_i \otimes ab) - (v_i \otimes ab) \\
&=& 0
\end{eqnarray*}
For the next case assume that $|\chi|>2$ then
$$q_i(0, \chi)(v_i \otimes 1) = -\frac{1}{|\chi |} \sum_{\emptyset \neq \psi \in \mathcal{F}(\chi)} {\mathcal{M}(\psi)(h_i \otimes \pi(\psi)) q_i(\chi - \psi)(v_i \otimes 1}) = 0$$
by induction.
Now assume that $|\varphi | = 1$ (or $\varphi = \chi_b$ for some $b\in A$). Then
\begin{eqnarray*}
q_i(\chi_b, \chi)(v_i \otimes 1) &=& -\sum_{\psi \in \f(\chi)} \m(\psi)(x_{-i} \otimes b \pi(\psi)) q_i(0, \chi-\psi)(v_i \otimes 1)\\
&=& - \m(\chi)(x_{-i} \otimes b\pi(\chi))(v_i \otimes 1)- \sum_{a\in \supp \chi} \m(\chi-\chi_a)(x_{-i} \otimes b \pi(\chi-\chi_a)) q_i(0, \chi_a)(v_i \otimes 1)\\
&=& - \m(\chi)(v_{i+1} \otimes b\pi(\chi)) - \sum_{a \in \supp \chi} \m(\chi - \chi_a)(x_{-i} \otimes b \pi(\chi - \chi_a))(-h_i \otimes a)(v_i \otimes 1)\\
&=& - \m(\chi)(v_{i+1} \otimes b\pi(\chi)) + \sum_{a \in \supp \chi} \m(\chi - \chi_a)(x_{-i} \otimes b \pi(\chi - \chi_a))(v_i \otimes a)\\
&=& - \m(\chi)(v_{i+1} \otimes b\pi(\chi)) + \sum_{a \in \supp \chi} \m(\chi - \chi_a)(v_{i+1} \otimes b\pi(\chi))\\
&=& - \m(\chi)(v_{i+1} \otimes b\pi(\chi)) + \sum_{a \in \supp \chi} \frac{(|\chi|-1)!}{\prod_{c \in \supp (\chi - \chi_a)} (\chi-\chi_a)(c)!} (v_{i+1} \otimes b\pi(\chi))\\
&=& - \m(\chi)(v_{i+1} \otimes b\pi(\chi)) + \sum_{a \in \supp \chi} \frac{(|\chi|-1)!}{\prod_{\substack{c \in \supp \chi \\ c\neq a}} \chi(c)!(\chi(a)-1)!} (v_{i+1} \otimes b\pi(\chi))\\
&=& - \m(\chi)(v_{i+1} \otimes b\pi(\chi)) + \sum_{a \in \supp \chi}\frac{\chi(a)}{|\chi|} \m(\chi)(v_{i+1} \otimes b\pi(\chi))\\
&=& - \m(\chi)(v_{i+1} \otimes b\pi(\chi)) + \m(\chi)(v_{i+1} \otimes b\pi(\chi))\\
&=& 0
\end{eqnarray*}
Finally assume that $|\varphi | > 1$. Then
\begin{eqnarray*}
q_i(\varphi, \chi)(v_i \otimes 1) &=& - \frac{1}{|\varphi|} \sum_{\psi \in \f(\chi)} \sum_{d\in \supp \varphi} \m(\psi)(x_{-i} \otimes d\pi(\psi))q_i(\varphi - \chi_d, \chi-\psi)(v_i \otimes 1)\\
&=& - \frac{1}{|\varphi |} \sum_{\psi \in \f(\chi)} \sum_{ d \in \supp \varphi} \m(\psi)\\
&& \Bigg( -\frac{1}{|\varphi | - 1} \sum_{\psi_1 \in \f(\chi-\psi)} \sum_{d_1 \in \supp(\varphi-\chi_d)} \m(\psi_1) \\
&& (x_{-i} \otimes d \pi(\psi))(x_{-i} \otimes d_1 \pi(\psi_1)) q_i(\varphi-\chi_d-\chi_{d_1}, \chi - \psi - \psi_1)\Bigg)(v_i \otimes 1)\\
&=& 0
\end{eqnarray*}
because at least two $x_{-i}$ terms act on a single $v_i$ as 0.
\end{proof}
\begin{lem}\label{qivi}
For all $i\in\{1,\ldots,n-1\}$ and $\varphi,\chi\in\f$ with $|\varphi|+|\chi|=k$
$$q_i(\varphi,\chi)\left(v_i\otimes 1\right)^{\otimes k}=(-1)^kv\left(0,\ldots,0,\underbrace{\chi}_i,\underbrace{\varphi}_{i+1},0,\ldots,0\right)$$
\end{lem}
\begin{proof}
\begin{eqnarray*}
q_i(\varphi,\chi)\left(v_i\otimes 1\right)^{\otimes k}&=&\Delta^{k-1}\left(q_i(\varphi,\chi)\right)\left(v_i\otimes 1\right)^{\otimes k}\\
&=&\left(\sum_{\substack{\psi\in\comp_k(\chi)\\\phi\in\comp_k(\varphi)}}
q_i(\phi(1),\psi(1))\otimes\cdots\otimes q_i(\phi(k),\psi(k))\right)\left(v_i\otimes 1\right)^{\otimes k}\\
&&\text{by Lemma \ref{deltaqi}}\\
&=&\sum_{\substack{\psi\in\comp_k(\chi)\\\phi\in\comp_k(\varphi)}}
\left(q_i(\phi(1),\psi(1))\left(v_i\otimes 1\right)\right)\otimes\cdots\otimes \left(q_i(\phi(k),\psi(k))\left(v_i\otimes 1\right)\right)
\end{eqnarray*}
By Lemma \ref{action produces 0} we see that the only potentially nonzero terms in the sum are those for which $|\phi(j)|+|\psi(j)|\leq1$ for all $j\in\{1,\ldots,k\}$. Since $|\varphi|+|\chi|=k$ if we have $|\psi(j)|+|\phi(j)|=0$ for some $j\in\{1,\ldots,n-1\}$ then there is a $r\in\{1,\ldots,n-1\}$ such that $|\psi(r)|+|\phi(r)|>1$. So the only potentially nonzero terms in the sum are those for which $|\phi(j)|+|\psi(j)|=1$ for all $j\in\{1,\ldots,k\}$.
Suppose that $\phi(j)=\chi_a$ and $\psi(j)=0$ for some $j\in\{1,\ldots k\}$ and some $a\in A$. Then
$$q_i\left(\chi_a,0\right)\left(v_i\otimes 1\right)=-\left(x_{-i}\otimes a\right)\left(v_i\otimes 1\right)=-\left(v_{i+1}\otimes a\right)$$
Suppose that $\phi(j)=0$ and $\psi(j)=\chi_a$ for some $j\in\{1,\ldots k\}$ and some $a\in A$. Then
$$q_i\left(0,\chi_a\right)\left(v_i\otimes 1\right)=-\left(h_i\otimes a\right)\left(v_i\otimes 1\right)=-\left(v_i\otimes a\right)$$
So $-\left(v_{i+1}\otimes a\right)$ and $-\left(v_i\otimes a\right)$ are the only possibilities for factors in the tensor product above. Since we are summing over all possible submultisets of $\varphi$ and $\chi$ we have the result.
\end{proof}
\begin{lem}\label{qonv}
For all $m\in\mathbb{N}$ and all $\varphi_1,\ldots,\varphi_n\in\f$ with $\sum_{i=1}^n|\varphi_i|=m$
$$q(\varphi_1,\ldots,\varphi_n)(v_1\otimes 1)^{\otimes m} = (-1)^{\sum_{j=1}^nj|\varphi_j|}v(\varphi_1,\ldots,\varphi_n)$$
\end{lem}
\begin{proof}
Since for all $j\in\{1,\ldots,n-1\}$ and $k\in\{1,\ldots,n\}$
$$x_{-j}v_k=\delta_{j,k}v_{j+1},\hskip.5in h_jv_k=\delta_{j,k}v_j-\delta_{j+1,k}v_{j+1}$$
by Lemma \ref{qivi} we have\\
$q(\varphi_1,\ldots,\varphi_n)(v_1\otimes 1)^{\otimes m}$
\begin{eqnarray*}
&=&q_{n-1}(\varphi_n,\varphi_{n-1})q_{n-2}((|\varphi_n|+|\varphi_{n-1}|)\chi_1, \varphi_{n-2})\dots q_1\left(\left(\sum_{j=2}^n|\varphi_j|\right)\chi_1, \varphi_1\right)(v_1\otimes 1)^{\otimes m}\\
&=&(-1)^m q_{n-1}(\varphi_n,\varphi_{n-1})\dots q_2\left(\left(\sum_{j=3}^n|\varphi_j|\right)\chi_1, \varphi_2\right)v\left(\varphi_1,\left(\sum_{j=2}^n|\varphi_j|\right)\chi_1,0,\ldots,0\right)\\
&=&(-1)^{|\varphi_1|+2\sum_{j=2}^n|\varphi_j|}q_{n-1}(\varphi_n,\varphi_{n-1})\dots q_3\left(\left(\sum_{j=4}^n|\varphi_j|\right)\chi_1, \varphi_3\right)v\left(\varphi_1,\varphi_2,\left(\sum_{j=3}^n|\varphi_j|\right)\chi_1,0,\ldots,0\right)\\
&=&(-1)^{\sum_{j=1}^{n-2}j|\varphi_j|}q_{n-1}(\varphi_n,\varphi_{n-1})v\left(\varphi_1,\dots,\varphi_{n-2},\left(|\varphi_{n-1}|+|\varphi_n|\right)\chi_1,0\right)\\
&=&(-1)^{\sum_{j=1}^nj|\varphi_j|}v\left(\varphi_1,\dots,\varphi_n\right)
\end{eqnarray*}
\end{proof}
\subsection{The Proof of Theorem \ref{thm}}
\begin{proof}
By Lemmas \ref{qonv} and \ref{smbasis}
$$\left\{q(\varphi_1,\dots,\varphi_n)(v_1\otimes 1)^{\otimes m}\ \bigg|\ \varphi_1,\ldots,\varphi_n\in\f(\B),\ \sum_{i=1}^n|\varphi_i|=m\right\}$$
is a basis for $S^m(V\otimes A)$. Therefore by Theorem \ref{FL}
$$\left\{q(\varphi_1,\dots,\varphi_n)w_{m\omega_1}\ \bigg|\ \varphi_1,\ldots,\varphi_n\in\f(\B),\ \sum_{i=1}^n|\varphi_i|=m\right\}$$
is a basis for $W_A(m\omega_1)$.
\end{proof}
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 8,003
|
Q: Creating Delphi Objects at runtime based on class type Is it possible to create objects at runtime based on it's type by calling a method.
What I am trying to achieve is
var
lForm1 : TForm;
lForm2 : TForm;
begin
CreateObjects([lForm1, lForm2]);
// After this call I have the variables initialized and I can use them.
end;
A: There is insufficient information in the question.
Where does the "type" of the form objects (in the question) come from? Is it simply a type name? How does CreateObjects() discover the type that is required for each object?
It cannot come from the "type" of the object reference passed in, as this may be (and almost certainly will be, as in your example) merely a base type from which the required concrete type will ultimately derive.
Without more detailed information about your specific implementation goals and constraints, a complete, concrete answer is not possible.
However, in general terms what you seek may be achieved by a combination of virtual constructors and the RegisterClass / FindClass infrastructure provided by the VCL.
In simple terms, you would have a base class that introduces the common constructor used to instantiate your classes [for TComponent derived classes this already exists in the form of the Create(Owner: TComponent) constructor].
At runtime you can then obtain a reference to any (registered) class using FindClass('TClassName'). This will return a class reference with which you can then invoke the appropriate virtual constructor:
type
TFoo = class ....
TFooClass = class of TFoo;
// etc
var
someClass: TFooClass;
someObj: TFoo;
begin
someClass := TFooClass(FindClass('TFooDerivedClass'));
someObj := someClass.Create(nil);
:
Note in the above that TFooDerivedClass is a class that ultimately derives from TFooClass (and is assumed for simplicity to derive in turn from TComponent and is instantiated with a NIL owner in this case). Classes that are already registered with the type system can be found using FindClass(). This includes any control or component class that is referenced by some DFM in your application. Any additional classes that need to be registered may be explicitly registered using RegisterClass().
How your specific application identifies the types of objects involved and any mapping of type names onto other arbitrary system of identification is an implementation detail that you must take care of.
A: Untested concept code:
function instantiate(var instancevars : array of tobject;
const classtypes : array of TBaseClassType):boolean;
begin
if (length(instancevars)=0) or (length(instancevars)<>length(classtypes)) then
exit(false);
for i:=0 to length(instancevars)-1 do
instancevars[i]:=classtypes[i].create;
result:=true;
end;
Then use
instantiate([lform1,lform2],[tform1,tform2]);
to make it work.
Note for this to work "TBaseClassType" must be some baseclass for all classes used for this function, and have a virtual constructor (e.g. TPersistent?). Possibly you also need to correct the .create line (e.g. add (NIL) )
You can't get a type from a variable, the information is only available compiletime.
A: Quoting your comment on Henk's answer :
That's what I don't want to do. I have a lot of server side methods where I create a lot of controls at runtime and I was wondering is creating objects as above would reduce the code.
What do you mean by "a lot"?
If you mean a lot of components of exactly the same type (e.g : "but1, but2, but3, .. but55 : TButton;" ) then change your code and use an array to represent your variables - you can then make a simple loop to create them.
If you mean a lot of components of different types (e.g : but1 : TAnimatedButton; but2 : TFlatButton; but3 : T3DButton;), I don't see any simple method to do this, and I think you would create a small debugging hell more than anything else.
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 8,645
|
Reception will only be open to answer queries from Monday 8th to Thursday 18th April during the Easter break.
Reception will only be open from 9:00 – 12:00 to answer queries, from Monday 8th to Thursday 18th April during the Easter break.
Reception will be closed on bank holiday Friday19th and Monday 22nd April.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 6,434
|
/**
* @interface
*/
WebInspector.ProjectSearchConfig = function() {}
WebInspector.ProjectSearchConfig.prototype = {
/**
* @return {string}
*/
query: function() { },
/**
* @return {boolean}
*/
ignoreCase: function() { },
/**
* @return {boolean}
*/
isRegex: function() { },
/**
* @return {!Array.<string>}
*/
queries: function() { },
/**
* @param {string} filePath
* @return {boolean}
*/
filePathMatchesFileQuery: function(filePath) { }
}
/**
* @constructor
* @param {string} parentPath
* @param {string} name
* @param {string} originURL
* @param {string} url
* @param {!WebInspector.ResourceType} contentType
*/
WebInspector.FileDescriptor = function(parentPath, name, originURL, url, contentType)
{
this.parentPath = parentPath;
this.name = name;
this.originURL = originURL;
this.url = url;
this.contentType = contentType;
}
/**
* @interface
* @extends {WebInspector.EventTarget}
*/
WebInspector.ProjectDelegate = function() { }
WebInspector.ProjectDelegate.Events = {
FileAdded: "FileAdded",
FileRemoved: "FileRemoved",
}
WebInspector.ProjectDelegate.prototype = {
/**
* @return {string}
*/
type: function() { },
/**
* @return {string}
*/
displayName: function() { },
/**
* @return {string}
*/
url: function() { },
/**
* @param {string} path
* @param {function(?Date, ?number)} callback
*/
requestMetadata: function(path, callback) { },
/**
* @param {string} path
* @param {function(?string)} callback
*/
requestFileContent: function(path, callback) { },
/**
* @return {boolean}
*/
canSetFileContent: function() { },
/**
* @param {string} path
* @param {string} newContent
* @param {function(?string)} callback
*/
setFileContent: function(path, newContent, callback) { },
/**
* @return {boolean}
*/
canRename: function() { },
/**
* @param {string} path
* @param {string} newName
* @param {function(boolean, string=, string=, string=, !WebInspector.ResourceType=)} callback
*/
rename: function(path, newName, callback) { },
/**
* @param {string} path
* @param {function()=} callback
*/
refresh: function(path, callback) { },
/**
* @param {string} path
*/
excludeFolder: function(path) { },
/**
* @param {string} path
* @param {?string} name
* @param {string} content
* @param {function(?string)} callback
*/
createFile: function(path, name, content, callback) { },
/**
* @param {string} path
*/
deleteFile: function(path) { },
remove: function() { },
/**
* @param {string} path
* @param {string} query
* @param {boolean} caseSensitive
* @param {boolean} isRegex
* @param {function(!Array.<!WebInspector.ContentProvider.SearchMatch>)} callback
*/
searchInFileContent: function(path, query, caseSensitive, isRegex, callback) { },
/**
* @param {!WebInspector.ProjectSearchConfig} searchConfig
* @param {!Array.<string>} filesMathingFileQuery
* @param {!WebInspector.Progress} progress
* @param {function(!Array.<string>)} callback
*/
findFilesMatchingSearchRequest: function(searchConfig, filesMathingFileQuery, progress, callback) { },
/**
* @param {!WebInspector.Progress} progress
*/
indexContent: function(progress) { }
}
/**
* @constructor
* @extends {WebInspector.Object}
* @param {!WebInspector.Workspace} workspace
* @param {string} projectId
* @param {!WebInspector.ProjectDelegate} projectDelegate
*/
WebInspector.Project = function(workspace, projectId, projectDelegate)
{
/** @type {!Map.<string, !{uiSourceCode: !WebInspector.UISourceCode, index: number}>} */
this._uiSourceCodesMap = new Map();
/** @type {!Array.<!WebInspector.UISourceCode>} */
this._uiSourceCodesList = [];
this._workspace = workspace;
this._projectId = projectId;
this._projectDelegate = projectDelegate;
this._url = this._projectDelegate.url();
this._displayName = this._projectDelegate.displayName();
projectDelegate.addEventListener(WebInspector.ProjectDelegate.Events.FileAdded, this._fileAdded, this);
projectDelegate.addEventListener(WebInspector.ProjectDelegate.Events.FileRemoved, this._fileRemoved, this);
}
/**
* @enum {string}
*/
WebInspector.Project.Events = {
DisplayNameUpdated: "DisplayNameUpdated"
};
WebInspector.Project.prototype = {
/**
* @return {string}
*/
id: function()
{
return this._projectId;
},
/**
* @return {string}
*/
type: function()
{
return this._projectDelegate.type();
},
/**
* @return {string}
*/
displayName: function()
{
return this._displayName;
},
/**
* @param {string} displayName
*/
setDisplayName: function(displayName)
{
if (this._displayName === displayName)
return;
this._displayName = displayName;
this.dispatchEventToListeners(WebInspector.Project.Events.DisplayNameUpdated);
},
/**
* @return {string}
*/
url: function()
{
return this._url;
},
/**
* @return {boolean}
*/
isServiceProject: function()
{
return this._projectDelegate.type() === WebInspector.projectTypes.Debugger || this._projectDelegate.type() === WebInspector.projectTypes.Formatter || this._projectDelegate.type() === WebInspector.projectTypes.Service;
},
/**
* @param {!WebInspector.Event} event
*/
_fileAdded: function(event)
{
var fileDescriptor = /** @type {!WebInspector.FileDescriptor} */ (event.data);
var path = fileDescriptor.parentPath ? fileDescriptor.parentPath + "/" + fileDescriptor.name : fileDescriptor.name;
var uiSourceCode = this.uiSourceCode(path);
if (uiSourceCode)
return;
uiSourceCode = new WebInspector.UISourceCode(this, fileDescriptor.parentPath, fileDescriptor.name, fileDescriptor.originURL, fileDescriptor.url, fileDescriptor.contentType);
this._uiSourceCodesMap.set(path, {uiSourceCode: uiSourceCode, index: this._uiSourceCodesList.length});
this._uiSourceCodesList.push(uiSourceCode);
this._workspace.dispatchEventToListeners(WebInspector.Workspace.Events.UISourceCodeAdded, uiSourceCode);
},
/**
* @param {!WebInspector.Event} event
*/
_fileRemoved: function(event)
{
var path = /** @type {string} */ (event.data);
this._removeFile(path);
},
/**
* @param {string} path
*/
_removeFile: function(path)
{
var uiSourceCode = this.uiSourceCode(path);
if (!uiSourceCode)
return;
var entry = this._uiSourceCodesMap.get(path);
var movedUISourceCode = this._uiSourceCodesList[this._uiSourceCodesList.length - 1];
this._uiSourceCodesList[entry.index] = movedUISourceCode;
var movedEntry = this._uiSourceCodesMap.get(movedUISourceCode.path());
movedEntry.index = entry.index;
this._uiSourceCodesList.splice(this._uiSourceCodesList.length - 1, 1);
this._uiSourceCodesMap.delete(path);
this._workspace.dispatchEventToListeners(WebInspector.Workspace.Events.UISourceCodeRemoved, entry.uiSourceCode);
},
_remove: function()
{
this._projectDelegate.removeEventListener(WebInspector.ProjectDelegate.Events.FileAdded, this._fileAdded, this);
this._projectDelegate.removeEventListener(WebInspector.ProjectDelegate.Events.FileRemoved, this._fileRemoved, this);
this._workspace.dispatchEventToListeners(WebInspector.Workspace.Events.ProjectRemoved, this);
this._uiSourceCodesMap = new Map();
this._uiSourceCodesList = [];
},
/**
* @return {!WebInspector.Workspace}
*/
workspace: function()
{
return this._workspace;
},
/**
* @param {string} path
* @return {?WebInspector.UISourceCode}
*/
uiSourceCode: function(path)
{
var entry = this._uiSourceCodesMap.get(path);
return entry ? entry.uiSourceCode : null;
},
/**
* @param {string} originURL
* @return {?WebInspector.UISourceCode}
*/
uiSourceCodeForOriginURL: function(originURL)
{
for (var i = 0; i < this._uiSourceCodesList.length; ++i) {
var uiSourceCode = this._uiSourceCodesList[i];
if (uiSourceCode.originURL() === originURL)
return uiSourceCode;
}
return null;
},
/**
* @return {!Array.<!WebInspector.UISourceCode>}
*/
uiSourceCodes: function()
{
return this._uiSourceCodesList;
},
/**
* @param {!WebInspector.UISourceCode} uiSourceCode
* @param {function(?Date, ?number)} callback
*/
requestMetadata: function(uiSourceCode, callback)
{
this._projectDelegate.requestMetadata(uiSourceCode.path(), callback);
},
/**
* @param {!WebInspector.UISourceCode} uiSourceCode
* @param {function(?string)} callback
*/
requestFileContent: function(uiSourceCode, callback)
{
this._projectDelegate.requestFileContent(uiSourceCode.path(), callback);
},
/**
* @return {boolean}
*/
canSetFileContent: function()
{
return this._projectDelegate.canSetFileContent();
},
/**
* @param {!WebInspector.UISourceCode} uiSourceCode
* @param {string} newContent
* @param {function(?string)} callback
*/
setFileContent: function(uiSourceCode, newContent, callback)
{
this._projectDelegate.setFileContent(uiSourceCode.path(), newContent, onSetContent.bind(this));
/**
* @param {?string} content
* @this {WebInspector.Project}
*/
function onSetContent(content)
{
this._workspace.dispatchEventToListeners(WebInspector.Workspace.Events.UISourceCodeContentCommitted, { uiSourceCode: uiSourceCode, content: newContent });
callback(content);
}
},
/**
* @return {boolean}
*/
canRename: function()
{
return this._projectDelegate.canRename();
},
/**
* @param {!WebInspector.UISourceCode} uiSourceCode
* @param {string} newName
* @param {function(boolean, string=, string=, string=, !WebInspector.ResourceType=)} callback
*/
rename: function(uiSourceCode, newName, callback)
{
if (newName === uiSourceCode.name()) {
callback(true, uiSourceCode.name(), uiSourceCode.networkURL(), uiSourceCode.originURL(), uiSourceCode.contentType());
return;
}
this._projectDelegate.rename(uiSourceCode.path(), newName, innerCallback.bind(this));
/**
* @param {boolean} success
* @param {string=} newName
* @param {string=} newURL
* @param {string=} newOriginURL
* @param {!WebInspector.ResourceType=} newContentType
* @this {WebInspector.Project}
*/
function innerCallback(success, newName, newURL, newOriginURL, newContentType)
{
if (!success || !newName) {
callback(false);
return;
}
var oldPath = uiSourceCode.path();
var newPath = uiSourceCode.parentPath() ? uiSourceCode.parentPath() + "/" + newName : newName;
var value = /** @type {!{uiSourceCode: !WebInspector.UISourceCode, index: number}} */ (this._uiSourceCodesMap.get(oldPath));
this._uiSourceCodesMap.set(newPath, value);
this._uiSourceCodesMap.delete(oldPath);
callback(true, newName, newURL, newOriginURL, newContentType);
}
},
/**
* @param {string} path
* @param {function()=} callback
*/
refresh: function(path, callback)
{
this._projectDelegate.refresh(path, callback);
},
/**
* @param {string} path
*/
excludeFolder: function(path)
{
this._projectDelegate.excludeFolder(path);
var uiSourceCodes = this._uiSourceCodesList.slice();
for (var i = 0; i < uiSourceCodes.length; ++i) {
var uiSourceCode = uiSourceCodes[i];
if (uiSourceCode.path().startsWith(path.substr(1)))
this._removeFile(uiSourceCode.path());
}
},
/**
* @param {string} path
* @param {?string} name
* @param {string} content
* @param {function(?string)} callback
*/
createFile: function(path, name, content, callback)
{
this._projectDelegate.createFile(path, name, content, innerCallback);
function innerCallback(filePath)
{
callback(filePath);
}
},
/**
* @param {string} path
*/
deleteFile: function(path)
{
this._projectDelegate.deleteFile(path);
},
remove: function()
{
this._projectDelegate.remove();
},
/**
* @param {!WebInspector.UISourceCode} uiSourceCode
* @param {string} query
* @param {boolean} caseSensitive
* @param {boolean} isRegex
* @param {function(!Array.<!WebInspector.ContentProvider.SearchMatch>)} callback
*/
searchInFileContent: function(uiSourceCode, query, caseSensitive, isRegex, callback)
{
this._projectDelegate.searchInFileContent(uiSourceCode.path(), query, caseSensitive, isRegex, callback);
},
/**
* @param {!WebInspector.ProjectSearchConfig} searchConfig
* @param {!Array.<string>} filesMathingFileQuery
* @param {!WebInspector.Progress} progress
* @param {function(!Array.<string>)} callback
*/
findFilesMatchingSearchRequest: function(searchConfig, filesMathingFileQuery, progress, callback)
{
this._projectDelegate.findFilesMatchingSearchRequest(searchConfig, filesMathingFileQuery, progress, callback);
},
/**
* @param {!WebInspector.Progress} progress
*/
indexContent: function(progress)
{
this._projectDelegate.indexContent(progress);
},
__proto__: WebInspector.Object.prototype
}
/**
* @enum {string}
*/
WebInspector.projectTypes = {
Debugger: "debugger",
Formatter: "formatter",
Network: "network",
Snippets: "snippets",
FileSystem: "filesystem",
ContentScripts: "contentscripts",
Service: "service"
}
/**
* @constructor
* @extends {WebInspector.Object}
* @param {!WebInspector.FileSystemMapping} fileSystemMapping
*/
WebInspector.Workspace = function(fileSystemMapping)
{
this._fileSystemMapping = fileSystemMapping;
/** @type {!Object.<string, !WebInspector.Project>} */
this._projects = {};
this._hasResourceContentTrackingExtensions = false;
}
WebInspector.Workspace.Events = {
UISourceCodeAdded: "UISourceCodeAdded",
UISourceCodeRemoved: "UISourceCodeRemoved",
UISourceCodeContentCommitted: "UISourceCodeContentCommitted",
ProjectAdded: "ProjectAdded",
ProjectRemoved: "ProjectRemoved"
}
WebInspector.Workspace.prototype = {
/**
* @return {!Array.<!WebInspector.UISourceCode>}
*/
unsavedSourceCodes: function()
{
/**
* @param {!WebInspector.UISourceCode} sourceCode
* @return {boolean}
*/
function filterUnsaved(sourceCode)
{
return sourceCode.isDirty();
}
var unsavedSourceCodes = [];
var projects = this.projectsForType(WebInspector.projectTypes.FileSystem);
for (var i = 0; i < projects.length; ++i)
unsavedSourceCodes = unsavedSourceCodes.concat(projects[i].uiSourceCodes().filter(filterUnsaved));
return unsavedSourceCodes;
},
/**
* @param {string} projectId
* @param {string} path
* @return {?WebInspector.UISourceCode}
*/
uiSourceCode: function(projectId, path)
{
var project = this._projects[projectId];
return project ? project.uiSourceCode(path) : null;
},
/**
* @param {string} originURL
* @return {?WebInspector.UISourceCode}
*/
uiSourceCodeForOriginURL: function(originURL)
{
var projects = this.projectsForType(WebInspector.projectTypes.Network);
projects = projects.concat(this.projectsForType(WebInspector.projectTypes.ContentScripts));
for (var i = 0; i < projects.length; ++i) {
var project = projects[i];
var uiSourceCode = project.uiSourceCodeForOriginURL(originURL);
if (uiSourceCode)
return uiSourceCode;
}
return null;
},
/**
* @param {string} type
* @return {!Array.<!WebInspector.UISourceCode>}
*/
uiSourceCodesForProjectType: function(type)
{
var result = [];
for (var projectName in this._projects) {
var project = this._projects[projectName];
if (project.type() === type)
result = result.concat(project.uiSourceCodes());
}
return result;
},
/**
* @param {string} projectId
* @param {!WebInspector.ProjectDelegate} projectDelegate
* @return {!WebInspector.Project}
*/
addProject: function(projectId, projectDelegate)
{
var project = new WebInspector.Project(this, projectId, projectDelegate);
this._projects[projectId] = project;
this.dispatchEventToListeners(WebInspector.Workspace.Events.ProjectAdded, project);
return project;
},
/**
* @param {string} projectId
*/
removeProject: function(projectId)
{
var project = this._projects[projectId];
if (!project)
return;
delete this._projects[projectId];
project._remove();
},
/**
* @param {string} projectId
* @return {!WebInspector.Project}
*/
project: function(projectId)
{
return this._projects[projectId];
},
/**
* @return {!Array.<!WebInspector.Project>}
*/
projects: function()
{
return Object.values(this._projects);
},
/**
* @param {string} type
* @return {!Array.<!WebInspector.Project>}
*/
projectsForType: function(type)
{
function filterByType(project)
{
return project.type() === type;
}
return this.projects().filter(filterByType);
},
/**
* @return {!Array.<!WebInspector.UISourceCode>}
*/
uiSourceCodes: function()
{
var result = [];
for (var projectId in this._projects) {
var project = this._projects[projectId];
result = result.concat(project.uiSourceCodes());
}
return result;
},
/**
* @param {boolean} hasExtensions
*/
setHasResourceContentTrackingExtensions: function(hasExtensions)
{
this._hasResourceContentTrackingExtensions = hasExtensions;
},
/**
* @return {boolean}
*/
hasResourceContentTrackingExtensions: function()
{
return this._hasResourceContentTrackingExtensions;
},
__proto__: WebInspector.Object.prototype
}
/**
* @type {!WebInspector.Workspace}
*/
WebInspector.workspace;
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 1,123
|
goog.provide("jchemhub.view.Renderer");
goog.require("goog.structs.Map");
/**
* Abstract Class to render a model object to a graphics object
*
* @constructor
* @param graphics
* {goog.graphics.AbstractGraphics} graphics to draw on.
* @extends {jchemhub.view.Renderer}
*/
jchemhub.view.Renderer = function(controller, graphics, opt_config, defaultConfig) {
this.controller = controller;
this.graphics = graphics;
this.config = new goog.structs.Map(defaultConfig);
if (opt_config) {
this.config.addAll(opt_config); // merge optional config into
// defaults
}
}
jchemhub.view.Renderer.prototype.render = goog.abstractMethod;
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 5,837
|
namespace TomasHorvath.BlogEngine.Domain
{
public enum EntityType : int
{
BlogPost = 1 ,
Page = 2
}
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 8,619
|
/*global angular*/
(function () {
'use strict';
var dependencies = [];
dependencies.push("ngMaterial");
dependencies.push("ui.router");
angular.module("app", dependencies);
})();
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 2,273
|
The Non-GMO Project Standard
The NOn-GMO Project
Our Mission and HistoryCareersContact
OrganizationCommittees
Privacy PolicyTerms of UseAccessibility
How to Earn the Butterfly Label
Steps to Get Non-GMO Project VerifiedApply for VerificationTechnical AdministratorsTesting Labs
Verification FAQsConsultantsRetailer SupportEquitable Transfer Program
RenewalProduct Verification ResourcesComplaints & Appeals
What is the Non-GMO Project Standard and how does it work?
What is the Non-GMO Project Standard?
Read The Standard
Download the Standard in EnglishDownload the Standard in SpanishPublic Comment
What You Can DoShopBlogNon-GMO Month
Learn more about GMOs
LearnGMO FactsFAQs
Non-GMO Verified PRoducts
Find Non-GMO Verified ProductsVerification Request
OrganzationCommittees
Verification FAQ'sConsultantsRetailer SupportEquitable Transfer Program
What is the Non-GMO Project and how does it work?
What You Can DoBlogNon-GMO Month
LearnGMO FactsFAQ's
Verified ProductsVerification Request
GMOs and Heritage Corn: Protecting the Source of Life
Melissa Waddell
This article is part of a 3-part series on familiar foods with surprising backstories. Part Three: Mexico is the birthplace […]
This article is part of a 3-part series on familiar foods with surprising backstories. Part Three: Mexico is the birthplace of corn, and corn is the "source of life." But the unique genetic resources of native maize — and the social structure and cultural identity that evolved along with the crops — are under threat from powerful agribusiness, global trade agreements and GMOs.
Read Part One: Is Synbio Vanilla "Natural"? Heck, No! and Part Two: What Does Bill Gates Have To Do With Ethiopian Chickens?
Did you know a full third of the human population depends on corn as a staple food? It's one of the most commonly grown grains in the world, second only to rice. Corn is also considered a high-risk for being GMO.
Genetically modified corn became available in 1996, engineered to tolerate chemical weed killers or produce their own insecticide. Today, at least 92% of U.S.-grown corn and more than 80% of Canadian-grown corn are genetically modified to do one or both of these things. But the ubiquity of GMO corn stops at the southern border. Mexico is both the birthplace of corn and the repository of thousands of invaluable, locally-adapted varieties.
Cultivating GMO corn for commercial use is prohibited on Mexican soil, and President Lopez Obrador has pledged to end imports of GMO corn — most of which come from the U.S. It takes some serious grit to banish the pet technology of powerful agri-chemical corporations. The reasons behind this bold move reach deep into the soil, into the rural landscapes of Mexico's small-holder farmers, and into the past.
Maize in Mexico
Corn is part of a family of cereal grain domesticated in Mexico close to 9,000 years ago. Indigenous Taino people called it mahiz, meaning "source of life" in the local dialect, from which we get the modern term, "maize."
Today, maize production is critical to food security and political stability in Mexico. It is at the heart of Mexican cultural, agronomic and gastronomic life.
This spiritual and social importance contrasts deeply with genetically modified corn's commodification, degradation and devaluation — a difference that was already palpable when the North American Free Trade Agreement ("NAFTA") nearly destroyed traditional Mexican agriculture.
NAFTA and native maize
In the early 1990s, NAFTA — the first iteration of a trade agreement between Mexico, Canada and the United States — was a mere glint in the eye of North American leaders and lobbyists. Mexican farmers grew enough maize for most domestic consumption, saving and sharing seed as part of the stewardship of small-holder farming. The government protected the market by only allowing foreign corn imports if the domestic supply faced a shortfall.
North of the border, American farmers also grew corn. U.S corn, however, was a far cry from the 21,000+ native varieties grown in Mexico. It was a commodity crop grown from high-yield hybrid seeds, destined for biofuels, animal feed and highly processed packaged goods, or sold to overseas markets. Robust federal insurance programs and subsidies made U.S. corn cheap.
NAFTA opened Mexican markets to highly subsidized U.S. corn. American agribusiness giants flooded the Mexican market at less than the cost of production. The Counter describes NAFTA's impact during its first decade, when "U.S. corn exports to Mexico quadrupled, while the price of domestically-grown corn in Mexico crashed by nearly 70 percent." With their livelihood all but wiped out, many agricultural workers abandoned farming altogether, migrating to urban centers — and eventually across the border — searching for jobs. And when farmers leave the land, native maize loses its key caretakers.
Native maize varieties have much to recommend them: They often perform better under difficult conditions, in poor soil and mountainous areas. On the other hand, modern hybrids prefer flat plains and mechanized harvesting. Native maize varieties are optimized for a range of local conditions, fostering unique traits that are crucial as we adapt to climate change.
Diversity is the bedrock of native maize varieties, but in global markets that favor consistent output, it sometimes works against producers. Foreign buyers look for massive quantities of identical ears and kernels that can be processed and packaged at scale. The things that make native maize genetically valuable can also make it a niche product.
And once GMOs arrived on the scene, that cornucopia faced a new threat.
Contamination nation
Foreign seed has long posed a threat to native maize. In 2005, Mexico passed a biosecurity law to limit genetically modified corn cultivation in Mexico. Sadly, the law was not iron-clad, and Big Ag already had its eye on the Mexican market. Before long, biotech corporations planted experimental plots of GMO corn in Mexico's northern states, and pollen from the genetically modified corn ultimately contaminated native maize.
Wind pollination isn't the sole source of contamination. According to agribusiness writer and researcher Tim Wise, "The most pervasive form of [contamination] isn't pollen on the wind, it's kernels of maize in people's pockets." When people carry corn seed over longer distances, it becomes that much harder to maintain the integrity of native maize varieties. After all, no one can tell if a kernel nestled in the palm of their hand contains patented DNA. And once that kernel grows, it can contaminate nearby stalks. GMOs that are released into the environment cannot be recalled.
While GMOs spread north of the border, the kernels traveled south. Contamination by modified and patented DNA is well documented, threatening the genetic resources of one of the world's most important crops.
Glyphosate, be gone!
The preservation of Mexico's cultural and genetic heritage is only gaining steam. On the last day of 2020, Mexico's president announced the phasing out of GMO corn imports as well as glyphosate, the weedkiller most commonly used with GMO crops. Mexican courts rejected the move by corporate giants to lift Mexico's biosecurity restrictions, ruling instead to protect biodiversity and the right to a healthy environment.
These decisions reflect the knowledge that the "source of life" is inextricable from Mexico's cultural heritage and social fabric. The team at A Growing Culture argues convincingly that culture and agriculture are inseparable:
"The gateway to environmental erosion is cultural erosion. When the fabric of communities is weakened through industrialization, the careful stewardship of the land, the embeddedness, and the knowledge of these communities are weakened as well."
A truly nourishing and equitable food system emerges from the essential interconnectedness of people and land, tradition and innovation. Industrial-style agriculture driven by transnational corporate interests disrupts this interconnectedness, with devastating consequences for both people and the planet. The triumph of culture and biodiversity over capitalism and Big Ag signals a brighter future.
Join our newsletter. Catch the butterfly.
About the ProjectTake ActionVerificationThe Standard
Contact UsCareers
Support the Non-GMO Project
Your tax deductible donation helps to build and protect our non-GMO future. Thank you!
© 2022 The Non-GMO Project
closechevron-downbarsmagnifiercrossarrow-right linkedin facebook pinterest youtube rss twitter instagram facebook-blank rss-blank linkedin-blank pinterest youtube twitter instagram
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 2,388
|
Central Coast is not the only one location we can offer you. There is a lot more locations to find and to explore. Just give it a try!
Choose from one of these locations: Geraldton, Lompoc, Derby, Pretoria, Hof, a random location or browse all locations.
If you want to improve your result on finding Central Coast on a blind map, then just play the Central Coast quiz again.
We are sorry that we cannot give you any hint on where Central Coast is located on the map. Just play the game and give it a try! If you want to know, how to play the Central Coast Map Game, then we can help you.
Did someone else find Central Coast better than you? Did you set the record on Central Coast? Did you improve? Check your results from this game session and compare them to other players' scores.
Let's say some words about this website, the intention of this quiz and take a look behind the scenes. Why did we create this Central Coast quiz?
Where is Central Coast located on the map? Find Central Coast now!
Have you ever been to Australia or even Central Coast before? Central Coast in Australia is an attractive destination for holidays and weekend trips. You will find nice hotels and restaurants there. So when do you start? Australia has so much interesting cities, but Central Coast is surely one of the top locations in Australia. Central Coast is in Australia - but where exactly? Rather in the centre or in the west?
NEW: Play our new funny game The Distance between Central Coast and ... It's free, easy and fun to play.
Where is Central Coast located on this blind map? Just click on the map right at the position where you think Central Coast is located. The map can be zoomed and shifted - as you are used to do with Google Maps - to ensure you can find Central Coast. You can drag and drop the red marker or just click anywhere else to change the location of Central Coast.
When you are done, confirm your choice by clicking just on the red marker. You then will see, how far you missed Central Coast on the map: the yellow marker will show you the correct position of Central Coast.
Did you here about Central Coast in the news? Or read tweets about Central Coast? Maybe you just want to improve your geography knowledge about Central Coast - then it is a good idea to log in using Twitter and keep track of your results. Whenever someone else finds Central Coast better than you, we send you a direct message on Twitter.
This is not only a geography quiz about Central Coast - we have lots of other locations from all around the world. Just try a random location or choose from the menu above any location you search for or you might know. What about finding Derby on a blind map?
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 9,568
|
const React = require('React');
const react = require('react');
// React is treated as object of named exports
("" : React.Node);
("" : react.Node);
("" : React.NotAType); // error NotAType missing in React
("" : react.NotAType); // error NotAType missing in react
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 1,691
|
Do you have old Six Flags Magic Mountain photos that you'd like to share with the world but don't know how? One of my missions with this site is to share as much history of Six Flags Magic Mountain as possible, even going back to the beginning in 1971 when it was known as just Magic Mountain. Let's partner up and help each other!
The park has changed a lot in the last 40+ years, yet not very much of it has been documented anywhere. I'm looking for old Magic Mountain photos of just about everything, from the buildings, to the rides, to the uniforms, and even the trolls. If you have old photos that you're willing to share, I'd love to publish them on this site, with full credit to you of course. You will retain full copyright on any photos that you took and were willing to share.
If you have old photos that are hardcopy only, I can help with that as well. If you mail them to me, regardless of the format (photos, slides, negatives, etc), I will convert them into high-resolution, full-color digital images and mail both the originals and digital copies back to you.
I'm also interested in old memorabilia besides just photos. If you have old ticket stubs, park maps, promotional flyers, etc., that's all great stuff as well. Six Flags Magic Mountain has a very rich and diverse history, and anything that helps tell their story would be appreciated. Please help me tell their story. If you have some Magic Mountain photos that you would like to share, or just have questions, please contact me directly at Kurt@TheCoasterGuy.com.
I have a box in my basement somewhere with old park maps, souvenier year books and other stuff. Whats the deadline Kurt?
Awesome, I'll get right to work!
I think I have a 1992 or 1993 wall size souvenir park map somewhere in the attack at my parents place. I'll take a look next time I'm there.
I haven't found a good way to post images of the wall maps due to their enormous size. However, I'd love to get some detailed photos of it for my research on the park's history. I have a huge void in my map collection from 91-97. Better yet, would you want to sell it?
You still have the photos I sent you? I hope you stored them in a good place. You should create a classic SFMM photo page.
I am enjoying your updates on Full Throttle. I may be going to the park next month, hopefully the trains will be testing by then.
Without knowing your last name, which photos are you referring to, Tim?
What a great idea! Will you be making a new page for all of it or will you only keep it on your facebook?
They won't be going on Facebook. What I'd like to do is finish out my Ride Profile series for all of the rides that are no longer there, like Mountain Express, Psyclone, Freefall, Galaxy, etc.
That's cool! I wasn't born yet when those rides were at the mountain, so those ride profiles would be a great read!
Tell you what Kurt, if I DO find it back at my folks place when I go looking for it next time I'm over there ill give it to you gratis! My worry is that I hope it's still there! I stashed it there over a dozen years ago when I originally left the nest for college. My mother is clueless about it when I called her the other day! Fingers crossed. But yea if I find it, it's yours no prob.
Kurt, I posted in your 1973 map thread. Go to my friend's website http://www.themeparkmemories.com for some old pics. Permission will need to be granted to use however.
When I was a kid I had a kidsongs vhs (kidsongs.com, can't believe they still exist), that was filmed at Magic Mountain. Possibly there's something of value in there, if you can bear the cheesy songs. Of course you'll have to get permission if there is.
I just sent four photos from the early '70s. Please let me know what you think.
Very cool…thank you, Eric! I think I can clean those up a bit. I'll definitely use the Mountain Express photo when I write that profile.
I'm really interested of knowing the past attractions.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 2,510
|
Who did the research that was used to compare Enlafax with other brands?
I have read the information provided by Medsafe quoting all the research facts and figures regarding Enlafax, however it does not clearly state who did the research. Could you please tell me who and where the research was done? Was it research that the company selling the drug did or was it confirmed by independent researchers. This question should be answerable without breaking any confidence as it is not requesting how the product was made but how the research was conducted.
Thank you for your email regarding the research that was used to compare Enlafax with other brands.
This email is to acknowledge that your request for information will be treated as such under the Official Information Act (1982). Your request is currently being worked on and we will have a reply to you as soon as possible and no later than the due date of 7 June 2018.
Please find attached PHARMAC's respond to your recent OIA request.
but how the research was conducted."
June 2018 as required under the Official Information Act 1982.
if such an extension is necessary.
Enlafax or the data relating to adverse event reporting.
Medsafe you are refering to.
which is their information sheet about Enlafax, on their website.
Reply to Maima Koro re Request H201803736.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 1,785
|
\section{Introduction}
In this paper, just finite and simple graphs are taken into account.
The sets of edges and vertices of a graph $G$ are denoted, respectively, by $E(G)$ and $V(G)$.
The degree of a vertex $v\in V(G)$ is indicated by $d_v(G)$ or just $d_v$ if the graph being discussed is unambiguous.
We utilize the conventional notation and nomenclature of (chemical) graph theory, and we refer readers to the relevant books, for example, \cite{ga,gc}.
Graphs are being used to model chemical structures by replacing atoms and bonds of the structures with vertices and edges, respectively. In this way, it is possible to study the chemical structures using the concepts of graph theory. Such a field of study is usually referred to as the chemical graph theory.
Graph invariants that adopt quantitative values are widely termed as topological indices in chemical graph theory.
The connectivity index (also known as the Randi\'c index) \cite{Gutman-13}, a well-known topological index, was devised in the 1970s by the chemist Milan Randi\'c under the name ``branching index'' \cite{g1}. Soon after its discovery, the connectivity index quickly found a variety of uses \cite{g2,g3,g4} in chemistry and consequently it become one of the most applied and well-researched topological indies.
For a graph $G$, the connectivity index is defined as
\[
R(G) = \sum_{vw\in E(G)} \frac{1}{\sqrt{d_vd_w}}.
\]
The Randi\'c index has be
modified in several ways. Here, we mention two topological indices which were introduced by taking into consideration the definition of the Randi\'c index, namely the ``sum-connectivity (SC) index'' \cite{g5} and the ``atom-bond connectivity (ABC) index'' \cite{g7}. These indices have the following definitions for a graph $G$:
\[
SC(G) = \sum_{vw\in E(G)} \frac{1}{\sqrt{d_v+d_w}}
\]
and
\[
ABC(G) = \sum_{ab\in E(G)} \sqrt{\frac{d_v+d_w-2}{d_vd_w}}.
\]
Detail regarding the mathematical aspects of the SC and ABC indices may be found in the review papers
\cite{Ali-19} and \cite{g9}, respectively.
By utilizing the definitions of the ABC and SC indices, a novel topological index -- the atom-bond sum-connectivity (ABS) index -- has recently been proposed in \cite{g10}.
For a graph $G$, this index is defined as
$$
ABS(G) = \sum_{uv\in E(G)} \left(\frac{d_u+d_v-2}{d_u+d_v}\right)^{\frac{1}{2}}\,.
$$
In the paper \cite{g10}, graphs possessing the maximum and minimum values of the ABS index were characterized over the classes of graphs and (chemical) trees of a given order; such kind of extremal results regarding unicyclic graphs were found in \cite{ABS-EJM}, where also chemical applications of the ABS index were reported. The preprint \cite{Alraqad-arXiv} is concerned with the problems of determining graphs possessing the minimum ABS index among all trees of a fixed order and/or a given number of pendant vertices; see also \cite{Maitreyi-arXiv} where one of these two problems is attacked independently.
A pendant vertex in a graph is a vertex of degree $1$. The least number of colors required to color the vertices of a graph, so that no two adjacent vertices have the same color, is termed as the chromatic number. A subset $S$ of the vertex set of $G$ is said to be independent if the vertices of $S$ are pairwise non-adjacent in $G$. The maximum number among cardinalities of all independent sets of $G$ is known as the independence number of $G$ and it is denoted by $\alpha(G)$. This paper characterizes the graphs attaining the greatest values of the ABS index over the classes of graphs of a given order and with a fixed (i) chromatic number (ii) independence number (iii) number of pendant vertices.
\section{Results}\label{Sec2}
In order to avoid trivialities, throughout this section, we consider only connected graphs.
To prove our results, we need few technical lemmas.
\begin{lemma}[see \cite{g10}]\label{lem-AZI+e}
Let $u$ and $v$ be non-adjacent vertices in a connected graph $G$. If $G+uv$ is the graph obtained from $G$ by adding the edge $uv$ in $G$ then
\[
ABS(G + uv) > ABS(G).
\]
\end{lemma}
\begin{lemma}\label{l1}
Let $$f(x,y)=\left(\dfrac{x+y-2}{x+y}\right)^{\frac{1}{2}},$$
where $\min\{x,y\}\geq 1$. For every positive real number $s$, define the function $g_s(x,y)=f(x+s,y)-f(x,y)$. Then $f$ is strictly increasing in $x$ and in $y$. The function $g_s$ is strictly decreasing and convex in $x$ and in $y$.
\end{lemma}
\begin{proof}
The first and second partial derivatives function $\frac{\partial f}{\partial x}$ of $f$ with respect to $x$ and $y$ are calculated as
$$\frac{\partial f}{\partial x}(x,y)=\frac{\partial f}{\partial y}(x,y)=(x+y-2)^{-\frac{1}{2}}(x+y)^{-\frac{3}{2}},$$
$$\frac{\partial^2 f}{\partial x^2}(x,y)=\frac{\partial^2 f}{\partial y^2}(x,y)=-\frac{1}{2}(x+y-2)^{-\frac{3}{2}}(x+y)^{-\frac{3}{2}}-\frac{3}{2}(x+y-2)^{-\frac{1}{2}}(x+y)^{-\frac{5}{2}}$$
Clearly $\frac{\partial f}{\partial x}(x,y)>0$, whenever $x>1$, and thus $f$ is strictly increasing in $x$ and in $y$.
Since $\frac{\partial^2 f}{\partial x^2}(x,y)<0$, whenever $x>1$, we get $\frac{\partial f}{\partial x}(x,y)$ is strictly decreasing in $x$ when $x\geq1$. This implies that $\frac{\partial g_s}{\partial x}(x,y)=\frac{\partial f}{\partial x}(x+s,y)-\frac{\partial f}{\partial x}(x,y)<0$ when $x\geq 1$, and thus $g_s(x,y)=f(x+s,y)-f(x,y)$, is strictly decreasing in $x$ when $x\geq 1$. Additional, $\frac{\partial^2 f}{\partial x^2}(x,y)$ is strictly increasing $x\geq 1$. So $\frac{\partial^2 g_s}{\partial x^2}(x,y)=\frac{\partial^2 f}{\partial x^2}(x+s,y)-\frac{\partial^2 f}{\partial x^2}(x,y)>0$, and hence $g_s(x,y)$, is convex in $x$ when $x\geq 1$.
\end{proof}
\begin{lemma}\label{l2}
Let $M$ and $N$ be real numbers satisfying $1\leq M\leq N$. Then for every positive real number $s$, the function $h_s(x)=g_s(x,M)-g_s(x,N)$ is increasing in $x$ when $x\geq 1$.
\end{lemma}
\begin{proof}
When $x\geq 1$, we have $\frac{\partial^2 f}{\partial x \partial y}$ is strictly increasing in $x$. So
$\frac{\partial^2 g_s}{\partial x \partial y}(x,y)=\frac{\partial^2 f}{\partial x \partial y}(x+s,y)-\frac{\partial^2 f}{\partial x \partial y}(x,y)>0$.
Thus $\frac{\partial g_s}{\partial x}$ is increasing in $y$, and hence $h_s'(x)=\frac{\partial g_s}{\partial x}(x,N)-\frac{\partial g_s}{\partial x}(x,M)>0$. Therefore $h_s(x)$ is increasing in $x$ when $x\geq 1$.
\end{proof}
The next theorem gives a sharp upper bound on the $ABS$ value of all connected graphs of a fixed order and fixed chromatic number. A graph whose vertex set can be partitioned into $r$ sets $V_1,V_2,\ldots, V_r$ in such a way that all the vertices in every $V_i$ (with $1\le i\le r$) are pairwise non-adjacent is known as an $r$-partite graph, where $r\ge2$ and the sets $V_1,V_2,\ldots, V_r$ are called the partite sets. If, in addition,
every vertex of partite set $V_i$ is adjacent to all the vertices of the other partite sets for $i=1,2,\ldots,r$, then the graph is called the complete $r$-partite graph. We denote, by $T_{n,\chi}$, the complete $\chi$-partite graph of order $n$ such that $|n_i - n_j| \le 1$, where $n_i$, with $i = 1, 2, \cdots , \chi,$ is the number of vertices in the $i$-th partite set of $T_{n,\chi}$.
\begin{theorem}\label{t1}
If $G$ be a connected graph of order $n\geq5$ and having chromatic number $\chi\geq3$, Then
\begin{align}\label{eq1} ABS(G)\leq& \frac{r(r-1)q^2}{2}\sqrt{\frac{n-q-1}{n-q}}+r(\chi-r)q(q+1)\sqrt{\frac{2n-2q-3}{2n-2q-1}}\nonumber\\
&+\frac{(\chi-r)(\chi-r-1)(q+1)^2}{2}\sqrt{\frac{n-q-2}{n-q}},\end{align}
where $q$ and $r$ are non negative integers such that $n=q\chi+r$ and $r<\chi$. Moreover, the equality holds in (\ref{eq1}) if and onlt if $G\cong T_{n,\chi}$.
\end{theorem}
\begin{proof}
Let $G$ be a graph having the maximum $ABS$ in the class of all connected graphs of a fixed order $n$ and with a fixed chromatic number $\chi$, where $3\le \chi \le n-1$ and $n\ge5$. Note that the vertex set $V(G)$ of $G$ can be partitioned into $\chi$ independent subsets, say $V_1,V_2,\cdots,V_\chi$ such that $|V_i|=n_i$ for $i=1,2,\cdots,\chi$, provided that $n_1\leq n_2 \leq \cdots \leq n_{\chi}$. Consequently, $G$ is isomorphic to a $\chi$-partite graph and hence, by Lemma \ref{lem-AZI+e}, it must be isomorphic to the complete $\chi$-partite graph $K_{n_1,n_2,\cdots,n_{\chi}}$. To complete the proof, we have to show that $n_{\chi}-n_1\leq 1$. Contrarily, assume that $n_{\chi}-n_1\geq 2$. Let $G'\cong K_{n'_1,n'_2,\cdots,n'_{\chi}}$, where $n'_1=n_1+1$, $n'_{\chi}=n_{\chi}-1$, and $n'_i=n_i$ for every $i\in \{2,\cdots,\chi-1\}$
\begin{align}\label{eq-01A}
ABS(G')-ABS(G)&= (n_1+1)(n_{\chi}-1)f(n-n_1-1,n-n_{\chi}+1) - n_1n_{\chi}f(n-n_1,n-n_{\chi}) \nonumber\\
&\ \ \ \ +\sum_{i=2}^{\chi-1}\left[n_i(n_1+1)f(n-n_1-1,n-n_i) - n_1n_if(n-n_1,n-n_i)\right]\nonumber\\
&\ \ \ \ +\sum_{i=2}^{\chi-1}\left[n_i(n_{\chi}-1)f(n-n_{\chi}+1,n-n_i) - n_{\chi}n_if(n-n_{\chi},n-n_i)\right]\nonumber\\
&=(n_{\chi}-n_1-1)f(n-n_1,n-n_{\chi})\nonumber\\
&\ \ \ \ +\sum_{i=2}^{\chi-1}n_i\left[f(n-n_1-1,n-n_i) - f(n-n_{\chi}+1,n-n_i)\right]\nonumber\\
&\ \ \ \ +\sum_{i=2}^{\chi-1}n_i\left[n_{\chi}g_1(n-n_{\chi},n-n_i)-n_1g_1(n-n_1-1,n-n_i)\right]\nonumber
\end{align}
Since $n-n_1-1\geq n-n_{\chi}+1$, from Lemma \ref{l1} we get that for each $i=2,\cdots,\chi-1$,
$f(n-n_1-1,n-n_i) - f(n-n_{\chi}+1,n-n_i)\geq0$ and
$n_{\chi}g_1(n-n_{\chi},n-n_i)-n_1g_1(n-n_1-1,n-n_i)>n_1\left[g_1(n-n_{\chi},n-n_i)-g_1(n-n_1-1,n-n_i)\right]\geq0.$
So $ABS(G')-ABS(G)$, a contradiction. Thus $n_{\chi}-n_1\leq 1$.
\end{proof}
The next theorem gives a sharp upper bound on the $ABS$ value of all connected graphs of a fixed order and fixed independent number.
\begin{theorem}\label{t3}
If $G$ is a connected graph of order $n$ and independent number $\alpha$ then
$$ABS(G)\leq \alpha\sqrt{(n-\alpha)(n-\alpha-1)}+\frac{1}{2}(n-\alpha)\sqrt{(n-\alpha-1)(n-\alpha-2)},$$ with equality holds if and only if $G\cong N_{\alpha}+K_{n-\alpha}$
\end{theorem}
\begin{proof}
Let $G$ be a connected graph which has the maximum $ABS$ value among all connected graphs of order $n$ and independent number $\alpha$. Let $S$ be an independent set in $G$ with $|S|=\alpha$. Assume that there is a vertex $u\in S$ that is not adjacent to a vertex $v\in V(G)-S$. Then $G+uv$ has order $n$ and and indepenedent number $\alpha$ and $ABS(G+uv)>ABS(G)$, a contradiction. Thus, each vertex in $S$ is adjacent to every vertex in $G-S$. Furthermore, every pair of vertices in $G-S$ are adjacent, yielding $G[V(G)-S]\cong K_{n-\alpha}$. Thus $G\cong N_{\alpha}+K_{n-\alpha}$.
Therefore, $ABS(G)= \alpha\sqrt{(n-\alpha)(n-\alpha-1)}+\frac{1}{2}(n-\alpha)\sqrt{(n-\alpha-1)(n-\alpha-2)}$
\end{proof}
The next theorem gives a sharp upper bound on the $ABS$ value of all connected graphs of a fixed order and fixed number of pendant vertices. We denote by $S_{n-1}$ the star graph of order $n$ and by $S_{m,n-m}$ the double star graph of order $n$ where the internal vertices have degrees $m$ and $n-m$. We also denote by $K_{m}^p$ the graph of order $m+p$ and $p$ pendant vertices such that the induced subgraph on the internal vertices is complete graph and all pendant vertices are adjacent to the same internal vertex.
\begin{theorem}\label{t5}
Let $G$ be a graph of order $n$ having $p$ pendant vertices.
\begin{enumerate}
\item\label{t5p1} if $p=n-1$ then $G\cong ABS(S_{n-1})$, and thus $ABS(G)=\frac{(n-1)\sqrt{n-2}}{n}$.
\item\label{t5p2} if $p=n-2$ then $ABS(G)\leq \frac{1}{\sqrt{3}}+\frac{\sqrt{n-2}}{n}+\frac{(n-3)\sqrt{n-3}}{n-1}$ with equality holds if and only if $G\cong S_{2,n-2}$
\item\label{t5p3} if $p \leq n-3$ then $$ABS(G)\leq p\sqrt{\frac{n-2}{n}}+(n-p-1)\sqrt{\frac{2n-2p-3}{2n-2p-1}}+\frac{1}{2}\sqrt{n-p-1}(n-p-2)^{\frac{3}{2}}$$ with equality holds if and only if $G\cong K_{n-p}^p$.
\end{enumerate}
\end{theorem}
\begin{proof}
(\ref{t5p1}) Straightforward.\\
(\ref{t5p2}) Let $u,v$ be the internal vertices of $G$. We may assume that there are $t$ pendant vertices adjacent to $u$ and $p-t$ pendant vertices adjacent to $v$. Thus
\begin{align}
ABS(G)&= tf(1,d_u)+(p-t)f(1,d_v)+f(d_u,d_v)\nonumber\\
&=tf(1,t+1)+(p-t)f(1,p-t+1)\nonumber
\end{align}
Consider the function $h(t)=tf(1,t+1)+(p-t)f(1,p-t+1)$.
$$h'(t)=\frac{M-N}{(t+2)(p-t+2)\sqrt{(t+2)(p-t+2)}},$$ where $M=((p-1)t-t^2+3p+6)\sqrt{pt-t^2+2t}$ and $N=(p-t+3)(t+2)\sqrt{(p-t)(t+2)}$. Clearly, both $M>0$ and $N>0$ when $1\leq t\leq p-1$. Thus the sign of $h'(t)$ is determined by the sign of $(M-N)(M+N)=M^2-N^2$. Now
$$M^2-N^2=(2t-p)(3tp(p-t)+10t(p-t)+8p^2+48p+72).$$ Hence $h'(t)<0$ when $1\leq t<p/2$ and $h'(t)>0$ when $p/2 < t \leq p-1$. Thus $h(t)$ has maximum value at $t=1$ or $t=p-1$. Thus $$ABS(G)\leq h(1)=h(p-1)=\frac{1}{\sqrt{3}}+\frac{\sqrt{p}}{p+2}+\frac{(p-1)\sqrt{p-1}}{p+1}$$ with equality holds if and only if $G\cong S_{2,p}=S_{2,n-2}$.
(\ref{t5p3}) Let $P$ be the set of pendant vertices in $G$. If there are two non adjacent vertices $u,v\in V(G)\setminus P$ then $G+\{uv\}\in \Gamma {n,p}$ and by Lemma \ref{lem-AZI+e}, $ABS(G+\{uv\})>ABS(G)$, a contradiction. Thus the induced subgraph $G[V(G)\setminus P]$ is $K_{n-p}$. Label the vertices of $G[V(G)\setminus P]$ by $v_1,...,v_{n-p}$, and for each $i=1,...,n-p$, let $a_i=|N(v_i)\cap P|$ so that $a_1\geq a_2\geq ... \geq a_{n-p}$. To obtain the desired result we want need to show that $a_1=p$ and $a_2=...=a_{n-p}=0$. So seeking a contradiction assume that $a_i\geq 1$ for some $i\geq 2$. Then $a_1\geq a_2 \geq 1$. Let $x\in P\cap N(v_2)$ and take $G'=G-\{xv_2\}+\{xv_1\}$. Note that for each $i=1,...,n-p$, $deg_G(v_i)=a_i+n-p-1$. Then
\begin{align}
ABS(G')-ABS(G)&=f(1,a_1+n-p)-f(1,a_2+n-p-1)\nonumber\\
&+a_1(f(1,a_1+n-p)-f(1,a_1+n-p-1))\nonumber\\
&-(a_2-1)(f(1,a_2+n-p-1)-f(1,a_2+n-p-2))\nonumber\\
&\sum_{i=3}^{n-p}(f(a_i+n-p-1,a_1+n-p)-f(a_i+n-p-1,a_1+n-p-1))\nonumber\\
&-\sum_{i=3}^{n-p}(f(a_i+n-p-1,a_2+n-p-1)-f(a_i+n-p-1,a_2+n-p-2))\nonumber\\
&=f(1,a_1+n-p)-f(1,a_2+n-p-1)\nonumber\\
&+a_1g_1(1,a_1+n-p-1)-(a_2-1)g_1(1,a_2+n-p-2)\nonumber\\
&\sum_{i=3}^{n-p}(g_1(a_1+n-p-1,a_i+n-p-1)-g_1(a_2+n-p-2,a_i+n-p-1))\nonumber
\end{align}
Now for each $i=3,...,n-p$, $a_i\geq0$ and so by Lemma \ref{l2},
\begin{align}
g_1(a_1+n-p-1,a_i+n-p-1)-&g_1(a_2+n-p-2,a_i+n-p-1)\geq \nonumber\\
&g_1(a_1+n-p-1,n-p-1)-g_1(a_2+n-p-2,n-p-1).\nonumber
\end{align}
Thus
\begin{align}
ABS(G')-ABS(G)\geq &f(1,a_1+n-p)-f(1,a_2+n-p-1)\nonumber\\
&+a_1g_1(a_1+n-p-1,1)-(a_2-1)g_1(a_2+n-p-2,1)\nonumber\\
&+(n-p-2)(g_1(a_1+n-p-1,n-p-1)-g_1(a_2+n-p-2,n-p-1))\nonumber\\
&=w(a_1)-w(a_2-1)\nonumber
\end{align}
Where $$w(t)=f(t+n-p,1)+tg_1(t+n-p-1,1)+(n-p-2)g_1(t+n-p-1,n-p-1)$$
Our next aim is to show that $w(t)$ is increasing in $t$. Note that
\begin{align}
w'(t)=&\frac{\partial f}{\partial t}(t+n-p,1)+g_1(t+n-p-1,1)+t\frac{\partial g_1}{\partial t}(t+n-p-1,1)\nonumber\\
&+(n-p-2)\frac{\partial g_1}{\partial t}(t+n-p-1,n-p-1)\nonumber
\end{align}
Since $\frac{\partial g_1}{\partial x}(x,y)$ is increasing in $y$ when $y\geq1$, we get $$\frac{\partial g_1}{\partial t}(t+n-p-1,n-p-1)\geq \frac{\partial g_1}{\partial t}(t+n-p-1,1),$$ So
\begin{align}
w'(t)\geq& \frac{\partial f}{\partial t}(t+n-p,1)+g_1(t+n-p-1,1)+(t+n-p-1)\frac{\partial g_1}{\partial t}(t+n-p-1,1)\nonumber\\
&=L-K,\nonumber
\end{align}
where $$L=\frac{(t+n-p)^2+(t+n-p)-1}{(t+n-p-1)^{\frac{1}{2}}(t+n-p+1)^{\frac{3}{2}}}\text{ and }K=\frac{(t+n-p)^2-(t+n-p)-1}{(t+n-p-2)^{\frac{1}{2}}(t+n-p)^{\frac{3}{2}}}.$$
Since $$L^2-K^2=\frac{2(t+n-p)^5-3(t+n-p)^4-8(t+n-p)^3+3(t+n-p)^2+4(t+n-p)+1}{(t+n-p-1)(t+n-p+1)^3(t+n-p-2)(t+n-p)^3}>0,$$ we get $w'(t)\geq L-K>0$ and thus $w(t)$ is increasing in $t$ as desired. This implies that $ABS(G')-ABS(G)>w(a_1)-w(a_2-1)>0$, a contradiction. So $a_1=p$ and $a_i=0$ for all $i=2,...,n-p-1$, and hence $G\cong K_{n-p}^p$.
\end{proof}
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 2,348
|
Dean Stephen Edwards (born 25 February 1962) is an English former professional footballer. After starting his career at Shrewsbury Town, he played for Kuopion Palloseura and Vaasan Palloseura in Finland, before returning to England to play for Telford United, Wolverhampton Wanderers, Exeter City, Torquay United and Northampton Town, scoring for Torquay in two different Wembley appearances. After ending his career at Sliema Wanderers and Hong Kong Rangers, he managed several clubs in non-league football.
Playing career
Edwards was born in Wolverhampton and began his career as an apprentice with Shrewsbury Town, turning professional in February 1980. He played only 13 times, in the old 2nd division (now Championship) scoring twice for the Shrews before joining Finnish side Kuopion Palloseura. From there he moved to Vaasan Palloseura. He subsequently moved to Telford United from where he joined Wolverhampton Wanderers in October 1985, playing for Wolves and forming a partnership with Andy Mutch.
He left Molineux in March 1987, having scored 10 times in 30, joining Exeter City. He became a regular both on the teamsheet and the scoresheet at Exeter, hitting the net 17 times in 54 league games before moving to Torquay United in August 1988.
He became an integral part of Cyril Knowles' side, scoring Torquay's goal at Wembley in the Football League Trophy final defeat against Bolton Wanderers, 1989 and scored again at Wembley two years later v Blackpool as Torquay won promotion via the play-offs to league 1 1991
In December 1991 he moved back to Exeter City on a free transfer, having played 116 league games and scored 26 league goals for the Gulls. He stayed only a few months at Exeter before another short spell at Northampton Town to finish his league career. He played in Hong Kong, and Malta with Sliema Wanderers,
Managerial career
In Autumn 1996 he became player-manager of Bideford in the Western League, but resigned on 30 July 1998 out of despair at the lack of commitment to pre-season training amongst his squad.
In January 2004, Edwards was appointed as player-manager of non-league side Pelsall Villa
In March 2006 he became assistant manager, under former Wolves teammate Mel Eves, at Willenhall Town and in September 2006, at the age of 44, was still registered as a player for Willenhall. After Eves departed in October 2007, Edwards took over as manager.
Edwards resigned as Willenhall manager in April 2008 and was appointed manager of Northern Premier League side Hednesford Town the following month. After a strong start took them to the top of the table, a dramatic dip in form saw the Pitmen eventually finish outside the play-offs, despite spending the majority of the season in the top five. He won the Birmingham senior cup for the first time in 73 years, signing players Tyrone Barnett, (Peterborough), Ross Draper (Inverness Caledonian), and Elliott Durrell (Wrexham),
The 2009–10 season saw Hednesford return to the Southern League, after the league restructured, with Edwards bringing in a number of experienced names in a bid to push the club back into the Conference North. Shortly after being knocked out of the FA Cup by Hellenic League minnows Pegasus Juniors at the first qualifying round stage, Edwards departed the club on 15 September 2009.
In June 2015 he was appointed as Director of Football at Torquay United. However, he left the club in September 2015. In October 2019 Edwards was appointed the manager of Barnstaple Town. He resigned from his post at Barnstaple in September 2021.
References
Sources
1962 births
Living people
Footballers from Wolverhampton
English footballers
Shrewsbury Town F.C. players
Kuopion Palloseura players
Vaasan Palloseura players
Telford United F.C. players
Wolverhampton Wanderers F.C. players
Exeter City F.C. players
Torquay United F.C. players
Northampton Town F.C. players
Sliema Wanderers F.C. players
Hong Kong Rangers FC players
Bideford A.F.C. players
Bideford A.F.C. managers
Pelsall Villa F.C. players
Pelsall Villa F.C. managers
Willenhall Town F.C. players
Willenhall Town F.C. managers
Hednesford Town F.C. managers
Torquay United F.C. non-playing staff
Barnstaple Town F.C. managers
English Football League players
Association football forwards
English football managers
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 1,501
|
Sira ist ein wenig verbreiteter weiblicher Vorname.
Herkunft und Bedeutung
Der Vorname Sira ist im Orient, aber auch in Westafrika verbreitet und war der Name einer persischen Prinzessin. In Deutschland wird er sehr selten mit einer Quote von 1 auf 100.000 etwa seit 1993 zunehmend verwendet. Das Wort kommt in vielen Sprachen vor und hat somit verschiedene Bedeutungen.
in Algerien, kabylisch,: Syra 'Stern'
italienisch, auch spanisch: Sirena (Vorname) (griech./italien.), Kurzform 'Sira'Entsprechend wird das griechische Fabelwesen Sirene als Ursprung für Sira genannt.
in Finnland ist Siira relativ verbreitet.
Bekannte Namensträgerinnen
Sira Rabe (Pseudonym der Louise Laurent), Erotik-Schriftstellerin
Sira Abed Rego (* 1973), spanische Politikerin der Izquierda Unida (IU), der Vereinigten Linken
Sonstiges
Zira ist eine Figur im Film Der König der Löwen 2 – Simbas Königreich sowie im Roman und der gleichnamigen Filmreihe Der Planet der Affen.
Einzelnachweise
Weblinks
Weiblicher Vorname
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 5,321
|
Storage companies in the city of Guelph that offer storage for indoors are listed on this page.
The city of Guelph is in the Hamilton and Niagara Area. View additional Hamilton and Niagara Area Indoor Storage businesses that are nearby.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 9,866
|
The main objective of GRECA Trustmark and Ecommerce Europe Trustmark is to increase consumers' trust in electronic market. After a successful evaluation, each e-shop receives a budge which is recognizable by the Greek, as well as the European market. The evaluation is based on 120 criteria and is conducted by ELTRUN researchers. The criteria are divided into 12 categories relevant with legislation, Code of Ecommerce Ethics, site usability and best practices.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 8,132
|
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.