text stringlengths 14 5.77M | meta dict | __index_level_0__ int64 0 9.97k ⌀ |
|---|---|---|
Q: How to check if the user has an active subscription in Android, Google Play? I have an app with a subscription in Google Play.
When the user starts the app, I need to know if the user has an active subscription. This would seem an obvious thing to do, but from searching and trying to implement it, it seems impossible?
I am using Google's newer billing 2/3, following Google's tutorials,
class BillingManager implements PurchasesUpdatedListener
...
public void checkAsync() {
Log.e(TAG, "checkAsync");
billingClient.queryPurchaseHistoryAsync(BillingClient.SkuType.SUBS, new PurchaseHistoryResponseListener() {
@Override
public void onPurchaseHistoryResponse(BillingResult billingResult, List<PurchaseHistoryRecord> list) {
Log.e(TAG, "checkCached result: " + list);
if (list == null) {
return;
}
for (PurchaseHistoryRecord ps : list) {
//System.out.println("PAYMENT: " + ps.getSku() + " : " + ps.getPurchaseTime());
}
}
});
}
public void checkCached() {
Log.e(TAG, "checkCached");
List<Purchase> result = billingClient.queryPurchases(BillingClient.SkuType.SUBS).getPurchasesList();
Log.e(TAG, "checkCached result: " + result);
if (result == null) {
return;
}
for (Purchase purchase : result) {
handlePurchase(purchase);
}
}
This is how I think you're supposed to get a user's purchases. But it does not work at all, both calls return null always. It only returns the correct purchases when you reinstall the app or clear the data.
So how exactly is an app supposed to do this?
Purchasing works for the app once I enter internal testing, and download it through the Google Play link. (before that subscriptions do not work at all).
*** updated
So to further clarify:
I am using a valid test user, and subscriptions are working correctly. My question is on the what the API queryPurchases() or queryPurchaseHistoryAsync() are suppose to do.
What I am seeing, is that these only return purchases that have not be processed by the app. They seem to store that the purchase was processed in the apps data.
After the purchase these return null, after the app restarts these return null.
If I clear the app datam or reinstall the app then they return the purchase (once), then again null after restart.
From what I see, these are only useful to detect when a user reinstalls your app, or installs on a different phone. They cannot be used to determine the status of a subscription.
So my question is,
1 - is this something that just does not work in internal testing and will magically work differently when the app is release?
2 - is there a different API that your suppose to use to check the status of a subscription?
3 - are you suppose to manage subscriptions yourself in your app by storing a user preference/cookie when you acknowledge the subscription the first time so you know when the subscription expires?
A: You need "licenced testers". They would allow you to "sideload" your app on devices, even for debug builds. My interpretation of sideload in this case would cover installing from Android Studio build tools as well as adb install .... and other methods that don't involve the play store.
https://developer.android.com/google/play/billing/test
Ordinarily, the Google Play Billing API is blocked for apps that aren't signed and uploaded to Google Play. License testers can bypass this check, meaning you can sideload apps for testing, even for apps using debug builds with debug signatures without the need to upload to the new version of your app. Note that the package name must match that of the app that is configured for Google Play, and the Google account must be a license tester for the Google Play Console account.
I also don't see how you're using startConnection. Until that's completed successfully I wouldn't be sure you have the latest data. I wouldn't be surprised if that makes you get stale values. I would check that carefully to make sure there's no silent errors happening, by both looking at onBillingSetupFinished and onBillingServiceDisconnected. And for the time being avoid trusting queryPurchases():
https://medium.com/@NandagopalR/integrating-google-play-billing-into-an-android-application-b6eb6af176a7
The queryPurchases() method uses a cache of the Google Play Store app without initiating a network request. If you need to check the most recent purchase made by the user for each product ID, you can use queryPurchaseHistoryAsync(), passing the purchase type and a PurchaseHistoryResponseListener to handle the query result.
By the way what's the value of isReady() right before queryPurchaseHistoryAsync, and what's the value of BillingResult::getDebugMessage and BillingResult::getResponseCode?
Also, use isFeatureSupported, though it seems it's not like your problem is coming from here. But I'd advise not testing with subscriptions until you get all the moving parts working: https://developer.android.com/reference/com/android/billingclient/api/BillingClient#isFeatureSupported(java.lang.String)
A: Okay, figured it out, was my mistake.
I was calling queryPurchases() in my main activity onCreate(), but the BillingClient was not ready yet.
I moved it to onBillingSetupFinished() and it now returns the correct purchases.
Everything is now working as expected. You get the active subscriptions when you call queryPurchases() after an app restart.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 4,418 |
FY 2023 Budget Hearings
It's budget time! Speak up on behalf of NJ children.
The New Jersey Senate and Assembly Budget Committees have scheduled their hearings on the Governor's proposed FY 2023 Budget. These hearings provide stakeholders with a forum to speak up on issues impacting children and their families. Read ACNJ's summary on budget highlights for kids and families.
Children need your voice to be heard! Below are the hearing dates and times. The hearings will be conducted remotely without the possibility of in-person attendance. If you wish to register to testify, click on the Senate or Assembly hearing links below.
Senate Budget and Appropriations Committee
Wednesday, March 29, 2022 @ 10:00 AM Register
Thursday, April 21, 2022 @ 10:00 AM Register
Assembly Budget Committee
Wednesday, March 23, 2022 @ 9:30 AM Register | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 764 |
Melanichneumon pluto är en stekelart som först beskrevs av Henry Lorenz Viereck 1903. Melanichneumon pluto ingår i släktet Melanichneumon och familjen brokparasitsteklar. Inga underarter finns listade i Catalogue of Life.
Källor
Brokparasitsteklar
pluto | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 5,144 |
\section{Introduction}
Since the discovery of superconductivity in LaFeAsO$_{1-x}$F$_{x}$,\cite{kamihara2008} several types of iron-based superconductors have been reported. The so called ``11" family of FeSe superconductors attracted much attention due to their simpler crystal structure, and particular electronic and physical properties.
Since the first report of superconductivity with critical temperature $T_{c} = 8$K for PbO-type $\alpha$-FeSe$_{0.88}$ by Hsu et al.\cite{hsu2008}, a $T_{c}$ of 37 K
at a pressure of 8.9 GPa was already reached.\cite{medvedev} FeSe compounds have a similar band structure to ferropnictides.\cite{subedi2008,review2015}
FeSe$_{1-x}$ with Se deficiency was reported to exhibit anomalies related to spin density waves (SDW) and magnetic ordering at temperatures near 100 K.\cite{lee2008} On the other hand, Ref.\cite{mcqueen2009} reported that FeSe exhibited superconductivity within a narrow range of stoichiometries, Fe$_{1.01\pm0.02}$Se, without magnetic ordering.
Pure $\beta$-FeSe undergoes a structural transition from a low-temperature orthorhombic to a tetragonal phase at $T_{s}\sim 90$K, not accompanied by a SDW,
and the compound exhibits superconductivity below $T_{c} = 8.87$K.
Angle-resolved photoemission spectroscopy (ARPES) experiments in $\beta$-FeSe revealed a significant change in the electronic structure when going through the structural transition.\cite{shimojima2014} Recently it was claimed \cite{watson2015} that the observed changes in electronic structure could not be explained by the small lattice distortion, issue which we will address in our present work.
Recently, Amig\'o et al.\cite{amigo2014} reported that multiband effects are needed to describe the magnetotransport properties of $\beta$-FeSe (Fe$_{0.96}$Se) single crystals. Concretely, in the normal state below 90 K, a strongly anisotropic positive magnetoresistance, that becomes negligible above that temperature, was found.
This magnetoresistance and the upper critical field could be understood with a phenomenological uncorrelated two-band model. Also a recent ultra-high magnetic field study\cite{watson2-2015} reported that magnetotransport in FeSe results from a small multiband Fermi surface (FS) with different carrier mobilities.
In this work, to study normal state magnetotransport properties of $\beta$-FeSe superconductors, we propose to employ a minimal microscopic model, which includes two effective bands describing the low-energy electronic structure, as well as intra- and inter-orbital Coulomb interactions. Previously\cite{condmat2015} we treated the model
using perturbative techniques to determine the electron Green's functions and the temperature-dependent spectral density function.
The kinetic energy part of the Hamiltonian is represented by the effective two-orbital model proposed by Raghu et al. in Ref.\cite{raghu}, consisting of
a two-dimensional lattice for the Fe atoms, with two degenerate orbitals per site. Tight-binding parameters were fitted
to obtain an effective band structure describing the Fermi surface topology of ferropnictides.\cite{raghu,yao2009} The two-orbital model
was shown to be suitable to describe the extended s-wave pairing and other superconducting properties of these systems.\cite{Graser2009,yao2009,Ran2009,Hu2009,zhou2011,dagotto2011,jhu2012,gliu2014}
\section{Calculation of magnetotransport properties of FeSe compounds}\label{section2}
\subsection{Microscopic two-orbital minimal model for FeSe}
To describe analytically the normal state magnetotransport properties of FeSe superconductors, we will consider the following minimal model preserving the essential low-energy physics:
\begin{eqnarray}
\mathcal{H} & = & \mathcal{H}_{0} + V_{int}
\label{Hamiltonian}
\end{eqnarray}
\noindent The kinetic energy part of the Hamiltonian in Eq. \ref{Hamiltonian} is given by the uncorrelated two-orbital model by Raghu et al.\cite{raghu} mentioned in the Introduction:
\begin{eqnarray}
{\mathcal{H}}_{0} & = &\sum_{k,\sigma}{\left[{E}_{c}(k) {c}^{\dagger}_{k\sigma} {c}_{k\sigma} + {E}_{d}(k) {d}^{\dagger}_{k\sigma}{d}_{k\sigma}\right]}
\label{Hamiltonian0}
\end{eqnarray}
where ${c}^{\dagger}_{k\sigma}$ creates an electron with crystal momentum $\vec{k}$ and spin $\sigma$ in the effective band with energy $E_{ c } (\vec{k})$,
likewise for ${d}^{\dagger}_{k\sigma}$ and $E_{ d } (\vec{k})$. The effective band energies are:
\begin{equation}
E_{ \overset { d }{ c } }(\vec{k}) = \epsilon_{+}(\vec{k}) \pm \sqrt{ \epsilon_{-}^{2}(\vec{k}) + \epsilon_{xy}^{2}(\vec{k})} -\mu
\label{effectivebands}
\end{equation}
\noindent $\mu$ denotes the chemical potential at temperature $T$, and:
\begin{eqnarray}
\epsilon_{\pm}(\vec{k})&=& \frac{\epsilon_{x}(\vec{k}) \pm \epsilon_{y}(\vec{k}) }{2}; \,\,\,\, \epsilon_{xy}(\vec{k}) = -4t_{4}\sin(k_{x})\sin(k_{y}) \nonumber\\
\epsilon_{x}(\vec{k}) & = & -2t_{1}\cos(k_{x}) - 2t_{2}\cos(k_{y})- 4t_{3}\cos(k_{x})\cos(k_{y}) \nonumber \\
\epsilon_{y}(\vec{k}) & = & -2t_{2}\cos(k_{x}) - 2t_{1}\cos(k_{y}) -4t_{3}\cos(k_{x})\cos(k_{y}) \nonumber
\end{eqnarray}
\noindent The tight-binding parameters $t_{i}, i=1-4, $ denote the hopping amplitudes between sites of the two-dimensional lattice of Fe atoms,
derived in Ref. \cite{raghu} as: $ t_{1} = -1 $ eV, $t_{2} = 1.3 $ eV , $t_{3}$ = $t_{4} = -0.85 $ eV.
The electron correlations are represented by $V_{int}$ in Eq. \ref{Hamiltonian}. The effect of local intra- and inter-orbital correlations
in ferropnictides was previously studied.\cite{dagotto2011,scalapino2012,condmat2015} It was found that the inter-orbital correlation was less relevant
than the intra-orbital one. Therefore, in our minimal model for FeSe we consider only the local intra-orbital Coulomb repulsion $U$:
\begin{equation}
{V}_{int} = \sum_{i} U \left( {n}_{i\uparrow}{n}_{i\downarrow} +{N}_{i\uparrow}{N}_{i\downarrow} \right)
\label{Vint}
\end{equation}
where: $n_{i\sigma} = {c}^{\dagger}_{i\sigma}{c}_{i\sigma}$ and $N_{i\sigma} = {d}^{\dagger}_{i\sigma}{d}_{i\sigma}$, and $i$ denotes the Fe-lattice sites. Since
correlations in FeSe compounds are intermediate,\cite{aichhorn2010, craco1-2014, craco2-2014, maletz2014, condmat2015,vollhardt2015} and mainly motivated by
the fact that it had been possible to describe previous magnetotransport results in terms of a phenomenological model with two uncorrelated carrier bands,\cite{amigo2014}
here we decided to use Hartree-Fock approximation (HF) for the correlations. A recent study of the effect of correlations
in FeSe\cite{vollhardt2015}, which found no relevant qualitative differences employing density functional theory (DFT) calculations and DFT+DMFT (DFT with
dynamical mean field theory) for the FS and the low energy spectral properties, provides further justification for the level of approximation we used.
We determined the HF renormalized band structure, and self-consistently calculated $\mu(T)$ for total electron filling $n$ of the two renormalized effective bands (see Ref.\cite{condmat2015} for details).
\subsection{Calculation of the electrical conductivity tensor and Hall coefficient}
\label{calculations}
To describe magnetotransport in FeSe compounds, we evaluated the electrical conductivity tensor $\sigma_{\alpha\beta}$, defined by:
\begin{equation}
\langle j_{\alpha}(t)\rangle = \sigma_{\alpha\beta}E_{\beta}(t)
\end{equation}
\noindent where $\langle j_{\alpha}(t)\rangle$ is the average current at temperature $T$ and time $t$ flowing in the $\alpha$-direction, in response to an electric field, $E_{\beta}(t)$, applied in the $\beta$-direction.
Assuming the presence of a magnetic field $\vec{H} = H_z \hat{z} $ perpendicular to the ab-plane of FeSe, and the electric current flowing in the $x$-direction ($j_x$) as a
result of an electric field along $\hat{x}$ plus the Hall electric field along $\hat{y}$:
\begin{equation}
\langle j_{x} \rangle = \sigma_{xx}(\omega) E_{x}(t) + \sigma_{xy}(\omega) E_{y}(t)
\end{equation}
\noindent where $\sigma_{xx}(\omega)$ and $\sigma_{xy}(\omega)$, are respectively the longitudinal and transversal components of the electrical conductivity tensor.
To compare our analytical results with experiments, we determined the ab-plane dc-resistivity ($\rho_{xx}$) and the Hall resistivity ($\rho_{xy}$) as the static (zero-frequency, i.e $\omega \to 0$) limit of:
\begin{equation}
\rho_{xx} = \frac{\sigma_{xx}(\omega)}{\sigma_{xx}^{2}(\omega) + \sigma_{xy}^{2}(\omega)}; \,\,\,\,\,\,\,\, \rho_{xy} = \frac{\sigma_{xy}(\omega)}{\sigma_{xx}^{2}(\omega) + \sigma_{xy}^{2}(\omega)}
\end{equation}
In the Kubo formulation for transport,\cite{kubo,stinchcombe} $ \sigma_{\alpha\beta}$ are given by appropriate generalised susceptibilities $\chi_{AB}(\omega)$,
measuring the linear response of observable $A$ of a system to an applied external field coupling to its observable $B$. The susceptibilities, in turn, can be
calculated using retarded Green's functions, $\ll A; B \gg (\omega)$.\cite{zubarev, stinchcombe} Here:
\begin{eqnarray}
\sigma_{xx}(\omega) & = \chi_{j_{x},eX}(\omega) & = \ll j_{x}; eX \gg(\omega) \\
\sigma_{xy}(\omega) & = \chi_{j_{x},eY}(\omega) & = \ll j_{x}; eY \gg(\omega)
\end{eqnarray}
\noindent where $X$ and $Y$ are the respective components of the system's position operator.
The electron Green's functions include a sum of respective contributions from the $c$ and $d$ effective bands, which
can each be calculated from the following exact set of equations of motion (EOM)\cite{zubarev}:
\begin{align}\label{greens}
& \omega \ll j_{x}, eX \gg^{c,d} = \frac{1}{2\pi}\langle [j^{c,d}_{x},eX] \rangle + \ll [j^{c,d}_{x}, \mathcal{H} ]; eX \gg \nonumber \\
& \omega \ll j_{x}, eY \gg^{c,d} = \frac{1}{2\pi}\langle [j^{c,d}_{x},eY] \rangle + \ll [j^{c,d}_{x}, \mathcal{H} ]; eY \gg
\end{align}
\noindent where the current operator\cite{mahan} is defined as: $j^{c}_{x} = \frac{e}{m^{*}_{c}}\sum_{\vec{k},\sigma}{k_{x}c^{\dagger}_{\vec{k},\sigma}c_{\vec{k},\sigma}}$ and $j^{d}_{x} = \frac{e}{m^{*}_{d}}\sum_{\vec{k},\sigma}{k_{x}d^{\dagger}_{\vec{k}, \sigma}d_{\vec{k},\sigma}}$, being $ m^{*}_{i}$, i=c,d, the effective masses of the carriers
in each band. New higher order Green's functions appear coupled in Eqs. \ref{greens}. In order to close the system of coupled equations of motion
we used HF approximation to decouple them,
and determined $ \ll j_{x}, eX \gg $ and $ \ll j_{x}, eY \gg $ in first order of perturbations on the electron correlation $U$.
The final expressions obtained for the ab-plane electrical conductivity components, in presence of $\vec{H} = H_z \hat{z}$, read:
\begin{eqnarray}\label{sigmaxx}
\sigma_{xx}(\omega) = \frac{e^{2}}{\Omega} \sum_{\vec{k},\sigma}\left\lbrace \frac{\langle c^{\dagger}_{\vec{k}\sigma}c_{\vec{k}\sigma}\rangle}{\hbar(\omega - \omega_{c})-n_{c}E_{c}(\vec{k})-2Un_{c}^{2}} \right. \nonumber \\
\left. + \frac{\langle d^{\dagger}_{\vec{k}\sigma}d_{\vec{k}\sigma}\rangle}{\hbar(\omega - \omega_{d})-n_{d}E_{d}(\vec{k})-2Un_{d}^{2}} \right\rbrace
\end{eqnarray}
\begin{align} \label{sigmaxy}
& \sigma_{xy} (\omega) = \frac{ne}{H_{z}} + \frac{e^{2}}{\Omega}\sum_{\vec{k},\sigma} \phi(\vec{k}) \left\lbrace \frac{1}{\hbar\omega - \tilde{E_{c}}(\vec{k}) + \hbar(\omega+\omega_{c}) }\right.\nonumber \\
& \left. - \frac{1}{-\hbar\omega - \tilde{E_{c}}(\vec{k}) + \hbar(\omega-\omega_{c}) } + \frac{1}{\hbar\omega - \tilde{E_{d}}(\vec{k}) + \hbar(\omega+\omega_{d}) }\right. \nonumber \\
&\left. - \frac{1}{-\hbar\omega - \tilde{E_{d}}(\vec{k}) + \hbar(\omega-\omega_{d}) } \right\rbrace
\end{align}
\noindent where: $\Omega$ is the unit cell volume, $ \tilde{E_{i}}(\vec{k}) = E_{i}(\vec{k}) + 2 U n^{2}_{i} $ for $i= c,d$. Above:
$\phi(\vec{k}) \equiv \left(\frac{\langle c^{\dagger}_{\vec{k}\sigma}c_{\vec{k}\sigma}\rangle - \langle d^{\dagger}_{\vec{k}\sigma}d_{\vec{k}\sigma}\rangle}{E_{d}(\vec{k})-E_{c}(\vec{k})}\right)$, being $\omega_{i} \equiv \frac{eH_{z}}{c}\left(\frac{1}{m^{*}_{i}}\right)$ ($i=c,d$), i.e. the cyclotron frequency of $c$ and $d$ electrons.
$m^{*}_{i}$, $i=c,d$ represent the diagonal components of the effective mass tensor, given by:
$
\left(\frac{1}{m^{*}_{i}}\right)_{\mu\nu} = \frac{1}{\hbar^{2}} \frac{\partial^{2}E_{i}(\vec{k})}{\partial{k_{\mu}}\partial{k_{\nu}}}
$. The conductivity due to multiple band maxima or minima is proportional to the sum of the inverse of the individual masses,
multiplied by the density of carriers in each band, to take into account all contributions to the conductivity.\cite{cardona} To evaluate the conductivities, we used
the Chadi-Cohen BZ sampling method\cite{chadicohen,cunningham} for square and rectangular lattices,
to perform the required BZ summations.
The following expression for the Hall coefficient ($R_{H}$) was obtained, using Eq.\ref{sigmaxy}:
\begin{align}
&R_{H}=\frac{1}{\sigma_{xy}H_{z}}; \,\,\,\,\,\, \sigma_{xy} = \lim_{\substack{\omega \to 0 \\ \delta \to 0^{+}}}{\Re\left[ \sigma_{xy}(\omega + i\delta)\right]} \equiv \left(\frac{1}{\gamma_{c} + \gamma_{d}}\right) \nonumber \\
&\gamma_{i}\equiv \left\lbrace \left(\tfrac{en_{i}}{m^{*}_{i}}\right)\tfrac{(\omega + \omega_{i})\left[ (\omega - \omega_{i})^{2} + \delta^{2} \right] + (\omega - \omega_{i})\left[(\omega + \omega_{i})^{2} + \delta^{2} \right]}{(\omega + \omega_{i})^{2}(\omega - \omega_{i})^{2} + \delta^{2}(\omega + \omega_{i})^{2} - \delta^{2}(\omega - \omega_{i})^{2} + \delta^{4}}\right\rbrace
\end{align}
In next section, we will compare our Hall coefficient results with those obtained using the classical expression for two types of uncorrelated carriers (with charge e):\cite{smith1978}
\begin{equation}\label{eqsingleton}
R_{H} = \frac{1}{e}\frac{(\mu_{c}^{2}n_{c}+\mu_{d}^{2}n_{d})+ (\mu_{c}\mu_{d} H_{z})^{2}(n_{c}+n_{d})}{(\mu_{c}n_{c}+\mu_{d}n_{d})^{2} + (\mu_{c}\mu_{d} H_{z})^{2}(n_{c}+n_{d})^{2}}
\end{equation}
\noindent where $\mu_{i}$ , $i= c, d$, denotes the mobility in each electron band. One has: $\mu_{i} = e \tau_{i} / m^{*}_{i} = \sigma_{i} / (e n_{i})$,\cite{singleton}
being $\tau^{-1}_{i}$ and $ \sigma_{i}$ respectively the scattering rates and dc-conductivities for the electrons in each band.
\section{Results and discussion}\label{results}
We present magnetotransport results for the normal state of FeSe compounds, and compare them with those calculated as
presented in previous section. Using the optimal correlation value $ U = 3 eV$, previously found to describe best other electronic properties of these compounds,\cite{condmat2015}
we analize the dependence on temperature, doping and magnetic field $H_{z} = H$, and compare our results with new experimental data and those of Ref.\cite{amigo2014},
as well as with the results obtained assuming uncorrelated electrons.
Notice that the value $U=3$eV represents less than one third, $\sim0.29$, of the total bandwidth for uncorrelated electrons,\cite{raghu}
thus characterising FeSe compounds as systems with intermediate electron correlations as discussed in previous section.
\begin{figure}[b!]
\begin{center}
\includegraphics[width=8.5cm]{Figure1.eps}
\caption[]{$H = 0$ : ab-plane resistivity as a function temperature. $\rho_{xx}(T) / \rho_{xx}(150 K)$, calculated for different doping values (indicated in the figure):
using the temperature-dependent lattice parameters, $a(T)$ and $b(T)$, reported for FeSe.\cite{khasanov2010} Also included is the result obtained
assuming a tetragonal lattice, with constant lattice parameter: $ a = b = 3.77 \AA $ ( double dot-dashed line).
Experimental curve (dotted line): Fe$_{0.96}$Se single crystal, from Ref.\cite{amigo2014}.
Inset: Experimental $\rho_{xx}(T) / \rho_{xx}(150 K)$ (dotted line) measured for a Fe$_{0.94}$Se$_{0.98}$S$_{0.02}$ single crystal, and
calculated curve(solid line) for $n = 2.25$.
Model parameters used: $U=3$, $t_{1}=-1.0$, $t_{2}=1.3$, $t_{3}=t_{4}=-0.85$. All energies in eV. Chadi-Cohen\cite{chadicohen,cunningham} order
for BZ summations: $\nu=9$.}
\label{figure1}
\end{center}
\end{figure}
First, in Figure \ref{figure1} we study the temperature dependence of the ab-plane dc-resistivity, represented by $\rho_{xx}$(T), for Fe$_{0.96}$Se and Fe$_{0.94}$Se$_{0.98}$S$_{0.02}$ single crystals
in the absence of magnetic field, measured with a standard 4 points dc-technique. The main figure compares the experimental data (normalized at T=150 K)
with two calculations using our approach:
one for a tetragonal crystal with constant lattice parameters (the normalized resistivity plotted has negligible dependence on doping up to 150 K),
while the other, more realistic, takes into account the T-dependence of the lattice parameters $ a(T), b(T) $ of FeSe\cite{khasanov2010} and, in particular, the
structural transition,\cite{mcqueen2009-PRL,amigo2014} which occurs at $T_{s} \sim 90$ K for the Fe$_{0.96}$Se sample, and at 87 K for the Fe$_{0.94}$Se$_{0.98}$S$_{0.02}$ one.
As expected, a clear improvement of the
description of the ab-plane dc-resistivity at $H=0$ is obtained using the T-dependent lattice parameters of FeSe.\cite{khasanov2010} The best agreement to the experimental data
is obtained considering a total electron filling $ n = 2.3$ (main figure), and $2.25$ (inset), for the correlated two-orbital model,
which corresponds to an Fe-content of x=0.96, and x=0.94, respectively.
In accordance with experiment, the calculated ab-plane resistivity presents
a metallic-like behavior in the normal state with a change of slope around the structural transition temperature.
Hence, we will continue using the temperature-dependent lattice parameters in what follows.
\begin{figure}[t!]
\begin{center}
\includegraphics[width=8.5cm]{Figure2.eps}
\caption[]{Effect of a magnetic field parallel to $c$-axis: temperature dependence of the ab-plane resistivity (normalized to $\rho(150K, H=0)$) for Fe$_{0.96}$Se
( $n=2.3$ ) in the main figure. Calculated and experimental\cite{amigo2014} results for H = 8 T, 16 T, as indicated in the plot. Other parameters as in Fig.\ref{figure1}.
Inset: calculated and experimental\cite{amigo2014} ab-plane resistivity (normalized to $\rho(150K, H=0)$) of Fe$_{0.94}$Se$_{0.98}$S$_{0.02}$ for H = 16 T.
}
\label{figure2}
\end{center}
\end{figure}
\begin{figure}[h]
\begin{center}
\includegraphics[width=8.5cm]{Figure3.eps}
\caption[]{ Magnetoresistance as a function of $H$ parallel to the $c$-axis: calculated (lines) and experimental (symbols) results for temperatures $T=14$, 16, and $50$ K,
as indicated in the plot.
The experimental data at T=14K are taken from Ref.\cite{amigo2014}.
Model parameters: $U=3$ eV, $n=2.3$ and others as in Fig.\ref{figure1}. }
\label{figure3}
\end{center}
\end{figure}
In the next three figures we will present magnetotransport results obtained under applied magnetic fields
parallel to the c-axis of the Fe$_{x}$Se samples: i.e. perpendicular to the plane formed by the Fe atoms.
In Figure \ref{figure2}, the main figure exhibits the normal state ab-plane resistivity $\rho_{xx}$(T) calculated and measured at magnetic fields of 8T and 16 T,
having fixed the total band filling
at $n=2.3$ to describe Fe$_{0.96}$Se. In the inset we show $\rho_{xx}$(T) at 16 Tesla for the Fe$_{0.94}$Se$_{0.98}$S$_{0.02}$ sample, with the corresponding calculated curve using $n=2.25$.
Notice that above $T_{c} = 8.87$ K for Fe$_{0.96}$Se\cite{amigo2014}, and above $T_{c}= 10.06$ K for Fe$_{0.94}$Se$_{0.98}$S$_{0.02}$, we obtain very good agreement.
A change of slope of the resistivity at the structural transition temperature is seen,
and, in particular, our results describe the positive magnetoresistance observed below $T_{s}$\cite{amigo2014} and the negligible one above $T_{s}$.
In Figure \ref{figure3} we present calculated and experimental magnetoresistance results for Fe$_{0.96}$Se
as a function of magnetic field parallel to $c$, at three different temperatures.
Only the experimental $T = 14 K$ results included have been published before\cite{amigo2014}.
Notice the remarkable agreement at T=14 K , 16 K , and 50 K between the experimental magnetoresistance and the values calculated assuming
$ U= 3 eV$ and $ n = 2.3$.
In particular, our results describe a quadratic $ \sim H^{2}$ behavior of the magnetoresistance,
consistently with the prediction from a phenomelogical two-band model used in Ref.\cite{amigo2014}.
In the present work, we also find experimentally and describe theoretically that the magnetoresistance concavity (and therefore also its magnitude) is monotonically
reduced as temperature is increased towards $T_{s}\sim90$K, which is consistent with the results in Figure \ref{figure2}, and in agreement with recent measurements
included in an ultra-high magnetic field study of FeSe.\cite{watson2-2015}
At T = 40 K, we find effective masses: $m^{*}_{c}=2.63m_{e}$ and $m^{*}_{d}=3.46m_{e}$, in agreement with DFT+DMFT calculations by Aichhorn \etal,\cite{aichhorn2010} where a
significant orbital-dependent mass renormalization in the range of 2 - 5 was predicted, and confirmed by ARPES results at $T=40$K\cite{maletz2014}.
\begin{figure}[t!]
\begin{center}
\includegraphics[width=8.5cm]{Figure4.eps}
\caption[]{Temperature dependence of the Hall coefficient at H = 16 T. Comparison between: our experimental results for Fe$_{0.94}$Se$_{0.98}$S$_{0.02}$ (points), and two theoretical calculations: present
analytical approach (solid line) for the correlated two-orbital model( $U= 3 eV, n= 2.25$, other parameters as in Fig.1), and
phenomenological uncorrelated two-carrier model: Eq.\ref{eqsingleton} (dot-dashed line). An arrow indicates the critical temperature of the sample at $H=0$.
The inset shows the effect of temperature on the difference ($n_{c} - n_{d}$) of the partial fillings of the effective bands in our correlated two-orbital model. }
\label{Hall}
\end{center}
\end{figure}
Next, in Figure \ref{Hall}, we present experimental and theoretical results obtained for the Hall coefficient $R_{H}$ in a Fe$_{0.94}$Se$_{0.98}$S$_{0.02}$ single crystal
as a function of temperature, at $H=16$T parallel to the $c$-axis.
The Hall contribution was measured with a standard dc technique using four contacts
along two perpendicular lines, separating the small resistivity contributions by measuring in positive and negative magnetic fields along the $c$-axis.
We also included in Figure \ref{Hall} the theoretical result obtained with our analytical approach,
for the correlated two-orbital model with parameters $U=3 eV$ and filling $n = 2.25 $.
Notice the good agreement obtained with the experimental data.
We found that in our theoretical approach $R_{H}$, apart from its dependence on magnetic field, is very sensitive to total electron filling $n$,
presenting qualitative sizeable changes depending on the Fe-content. These changes are related to the
position of the Fermi level with respect to the effective model's band structure\cite{raghu,condmat2015} (which can be seen in Fig.\ref{deformation}(a)).
The theoretical curve in Figure \ref{Hall} corresponds to a multi-band situation in which the Fermi level crosses the two $c$ and $d$ correlated bands,
with unequal fillings of those bands. In particular, the inset depicts the temperature dependence of the difference ($n_{c} - n_{d}$)
between the partial fillings of these bands at total filling $n=2.25$. Notice that it is maximum at the same temperature,
$ \sim 38 K$, at which the dependence on temperature of the lattice parameters sets in. This maximum coincides with the inflection point in $R_{H} (T)$, which we checked
that also happens at $ H = 16 T$ if two uncorrelated carrier bands contributed to $R_{H} (T)$ according to Eq.\ref{eqsingleton}.
The latter case is also shown in Figure \ref{Hall}, using the carrier mobilities and densities obtained from our approach for: $U = 0$ and $ n = 2.25$.
Figure \ref{Hall} evidentiates that better agreement to the experimental data is obtained with the correlated two-orbital model, than in the absence of electron correlations.
\begin{figure}[h!]
\begin{center}
\includegraphics[width=8.5cm]{Figure5a.eps}
\includegraphics[width=8.5cm]{Figure5b.eps}
\caption[]{H=0, effect of lattice deformation $ \delta$ on the electronic structure. (a) Band structure of the correlated two orbital model in Hartree-Fock approximation
shown in the large (unfolded) BZ,\cite{raghu} i.e. one Fe/cell, at $\delta=0$ (dashed line) and $\delta=0.002$ (solid line). $T = 10 K$, $n_c = 1.87$ and $n_d = 0.43$.
Inset: amplification near $\vec{k_0}=(\pi,\pi)$, which corresponds to the zone center $\Gamma$ in the small (folded) BZ, i.e. two Fe/cell. (b) T-dependence of the band splitting at two BZ points: denoted as $\Gamma$ and $M$ in the small BZ. Concretely: T-dependence of the calculated band splitting at $\Gamma$, and for comparison we include respective ARPES data at $\Gamma$ and $M$. Inset: temperature dependence of the deformation parameter using the lattice parameters of Ref.\cite{khasanov2010}.}
\label{deformation}
\end{center}
\end{figure}
To end, we discuss the effect of the lattice deformation related to the structural transition on the electronic properties of FeSe superconductors,
in the absence of magnetic field. It has been suggested that the emergence of magnetoresistance in FeSe superconductors below $T_{s}$
might be related to changes in the electronic structure.\cite{amigo2014,shimojima2014}
On the HF renormalized band structure of our effective correlated two-orbital model for FeSe compounds,
the main effects of the deformation are found in the BZ region around $\vec{k_0}=(\pi,\pi)$ of the large BZ i.e. with one Fe/cell.\cite{raghu},
as Figure \ref{deformation}(a) shows. We include results for two values of the orthorhombicity parameter $\delta = (a-b)/(a+b)$\cite{shimojima2014},
namely, $\delta=0$ and $\delta=0.002$ .
Our results indicate that the energetically non-equivalent $xz$ and $yz$ orbitals\cite{raghu} become degenerate at and above the structural transition,
in agreement with recent ARPES experiments.\cite{shimojima2014} The symmetry breaking, manifested in the band splitting appearing at $\vec{k_0}$,
results from the lattice deformation from tetragonal to orthorhombic.
Next, Fig. \ref{deformation}(b) exhibits the temperature dependence we calculated for the band splitting at $\vec{k_0}$,
measured by: $\Phi(T)=E_{d}(\vec{k_0})-E_{c}(\vec{k_0})$. Notice that $\vec{k_0}$ of the large BZ, corresponds
to the centre of the small BZ obtained with two Fe/cell, i.e. $\Gamma$.
For comparison, in Fig. \ref{deformation}(b) we also include ARPES results for $\Phi(T)$ at $\Gamma$ and $M$ ( using the small BZ notation, as in ARPES\cite{shimojima2014,nakayama,watson2015}).
Ref.\cite{shimojima2014} mentions that the band splitting measured at $M$ is nearly comparable to that at $\Gamma$,
possibly due to the relatively large error bars for these data.
The inset of Fig. \ref{deformation}(b) depicts the T-dependence of $\delta$, resulting from the T-dependent FeSe lattice parameters of Ref.\cite{khasanov2010}.
\section{Conclusions}\label{conclusions}
We studied magnetotransport in the normal state of Fe$_{x}$Se compounds, presenting experimental data obtained in single crystals
as well as a theoretical description of the results. Using a simplified microscopic model to describe the compounds, based on two correlated effective orbitals,
we determined the normal state electrical conductivity tensor and Hall coefficient in the linear response regime, employing the Kubo formulation.
We decoupled the equations of motion for the current-current correlation functions in first-order (Hartree-Fock) approximation,
with model parameters in the range relevant for Fe-chalcogenides, previously used to describe their spectral properties.
With this simplified model we could successfully describe:
i) the effect of the structural transition from a tetragonal to an orthorhombic phase observed in the ab-plane electrical resistivity;
ii) the positive magnetoresistance in presence of a magnetic field perpendicular to the ab-plane in the orthorhombic phase, which becomes negligible above the structural transition temperature;
iii) the Hall coefficient $R_{H}$ as a function of temperature, showing that the inclusion of moderate electron correlations improves the description of the experimental results;
iv) effects of the lattice deformation related to the structural transition on the electronic properties of FeSe superconductors: we found changes in the electronic structure below the structural phase transition temperature, comparable to those reported in ARPES experiments.
Our work presents experimental and theoretical evidence confirming the key role of the structural transition on the strongly anisotropic magnetotransport
properties observed in the normal state of $\beta$-FeSe superconductors, and that moderately correlated multiband models can provide the best description
of these experimental results.
\section{Acknowledgments}
G.N. and C.I.V. are researchers of CONICET (Argentina). M.L.A. and J.D.Q.F. have CONICET fellowships.
We acknowledge support from CONICET (PIP 0448 and PIP 0702), ANPCyT (PICT Raices'2012, nro.1069) and SeCTyP-UNCuyo.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 439 |
W matematyce
137 jest trzydziestą trzecią liczbą pierwszą, następującą po 131 i poprzedzającą 139
137 jest mniejszą z liczb bliźniaczych (137, 139)
137 nie jest liczbą palindromiczną, czyli liczbą czytana w obu kierunkach, w pozycyjnych systemach liczbowych od bazy 2 do bazy 16
137 należy do dwóch trójek pitagorejskich (88, 105, 137), (137, 9384, 9385)
W nauce
liczba atomowa untriseptium (niezsyntetyzowany pierwiastek chemiczny)
galaktyka NGC 137
planetoida (137) Meliboea
kometa krótkookresowa 137P/Shoemaker-Levy
W kalendarzu
137. dniem w roku jest 17 maja (w latach przestępnych jest to 16 maja). Zobacz też co wydarzyło się w roku 137, oraz w roku 137 p.n.e.
Zobacz też
dzielnik i cechy podzielności
Przypisy
Bibliografia
0137 | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 36 |
Why P+A
Making an A in Every Way
Clients, News
Congratulations, Kansas City Public Schools, you now have:
A New Superintendent
A New School Board
A Plan to achieve Accreditation
After months of uncertainty in the wake of Dr. John Covington's departure for Detroit, there is a sense of relief and a return to stability in the Kansas City Public Schools. A respected interim superintendent with a local record of success leading the Kauffman Scholars, Dr. R. Stephen Green, Ed. D. has agreed to lead the district as superintendent with what amounts to a four-year contract. His leadership and belief in the importance of community involvement in the district's transformation plan will be supported by the new board, who this week took the next steady step by unanimously electing Airick Leonard West as president and Crispin Rea as vice president. Newly elected board members Jon Hile, Marisol Montero, and Curtis Rogers attended their first meeting and made it clear they are motivated to help Kansas City regain accreditation while maintaining local control and participation in the city's schools.
New KCPS Board member Curtis Rogers interviews with local media members.
Congratulations also to the young scholars who will be well-served by the vision and commitment of Superintendent Green, Board President West, Vice President Rea, and the passion and enthusiasm of the board.
Dr. R. Stephen GreenKanas City Public SchoolsKansas City Missouri School DistrictKCPS School BoardKCPS SuperintendentTransformation Plan
We're Social! Follow Parson + Associates
+ Service-Disabled Veteran-Owned Small Business (SDVOSB)
+ Minority Business Enterprise (MBE)
+ Disadvantaged Business Enterprise (DBE)
+ Small Local Business Enterprise (SLBE)
Visit Parson + Associates
1780 Woodland Ave.
parsonkc.com
Mission Statementt
To engage people and tailor messages to build a bridge between complex ideas and common sense solutions.
© 2020 Parson + Associates. All Rights Reserved.
Web Design and Development by Turn the Page Online Marketing
Dr. R. Stephen Green is now Superintendent of the Kansas City Public Schools
Community Connects at East Campus Networking Open House | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 381 |
Бу́же () — озеро в Браславском районе Витебской области. Относится к бассейну реки Друйка. Расположено в 7 км на северо-запад от Браслава. Входит в группу Браславских озер и находится на территории Национального парка «Браславские озера».
Описание
Площадь зеркала 3,93 км² (по другим данным - 4,18 км²), длина 3,42 км, наибольшая ширина 1,72 км, максимальная глубина 9,1 м, средняя - 3,7, м, длина береговой линии около 12,2 км. Объем воды около 14,72 млн м³, площадь водосбора около 66 км².
Озерная котловина подпрудного типа, вытянута с юго-запада на северо-восток. Озеро эвтрофное. Берега возвышенные, песчано-галечниковые и каменистые, поросшие кустарником и редколесьем. Окружено обширной поймой, которая местами заполнена валунами, заросшая кустарником. Мелководье узкое, песчаное. Почти не зарастает. Наибольшие глубины находятся в западной части озера, вблизи к берегу. Озеро соединено ручьями с озерами Савонар, Рака и Чекуть. На озере расположено 16 островов, общей площадью около 0,249 км².
В озере обитают щука обычная, окунь, лещ, плотва, краснопёрка, налим, карп, карась, линь, белый амур и другие виды рыб. Производится промысловый лов рыбы. Организовано платное любительское рыболовство.
На берегу озера расположены деревни Крюки, Коханишки, Михалишки, Бужаны.
История
Примечания
Бассейн Друйки
Озёра Браславского района | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 733 |
Q: libsoxr in Windows x64 platform with inline asm is not compiling I am trying to compile libsoxr (it is derived from libsox library by Audacity team) library. I can compile osx 32, osx 64 and win32. They use inline asm and Windows x64 platform doesn't support inline asm. Problematic code piece is this:
#if HAVE_FENV_H
#include <fenv.h>
#elif defined _MSC_VER
#define FE_INVALID 1
#define FE_DIVBYZERO 4
#define FE_OVERFLOW 8
#define FE_UNDERFLOW 16
#define FE_INEXACT 32
#define FE_ALL_EXCEPT (FE_INEXACT|FE_DIVBYZERO|FE_UNDERFLOW|FE_OVERFLOW|FE_INVALID)
static __inline int fetestexcept(int excepts)
{
short status_word;
__asm fnstsw status_word
return status_word & excepts & FE_ALL_EXCEPT;
}
static __inline int feclearexcept(int excepts)
{
int16_t status[14];
__asm fnstenv status
status[2] &= ~(excepts & FE_ALL_EXCEPT);
__asm fldenv status
return 0;
}
#endif
I don't know what fnstenv and fldenv do. May somebody guide me for making compatible with x64?
A: Assuming that you are compiling with the Microsoft compiler then you can use RTL functions instead of inline assembly.
To test for particular floating point status flags call _statusfp. To clear floating point status flags call _clearfp.
In order to use _statusfp you'll need to translate from the raw 8087 flags, to the abstract flags used by _statusfp.
Update
The code in the question is an implementation of a small part of fenv.h which is part of C99. It's needed for the MS compiler since it only implements C89. In my view you would be much better off using a real C99 compiler. That would come with an implementation of fenv.h.
A: These two functions are part of the C standard so I doubt this is the first time someone has wanted them for MSVC—have a look around to see what other projects have done. If nothing turns up and you want a quick and easy solution, you should be able to further qualify the
#elif defined _MSC_VER
directive that wraps this code for your compiler (_M_X64) to not use these inline assembly versions—default (slower) code should come into effect.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 970 |
Q: Entity Framework 7 Application Role In EF6 it was possible to use System.Data.Entity.Infrastructure.Interception to implement Application Roles for MS SQL Server. I cannot find any reference to Interception in the EF7 libraries. In the Roadmap for EF7 it lists "Simple interception mechanisms for query and updates" as a high priority. Will this provide the functionality to implement app roles? If it does that's great BUT is there some reason app roles have to be hidden behind such obscurity? Why can't we simply have DbContext properties for app role username and password?
A: Rowan Miller, Microsoft Program Manager, provided some guidance which led to this solution to the problem. In summary, add a constructor to the generated DbContext which opens the connection and executes sp_setapprole. If you don't open the connection first the connection will be closed after sp_setapprole and you will lose the app role before executing your subsequent query. At some point this capability may appear in an EF7 Interception library but this strategy works with EF7 RC1.
public partial class YourContext : DbContext
{
public YourContext()
{
try
{
this.Database.OpenConnection();
this.Database.ExecuteSqlCommand("EXEC sp_setapprole appRole, appPsw");
}
catch (Exception exception)
{
...
}
}
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 5,873 |
{"url":"https:\/\/blogs.ams.org\/visualinsight\/category\/packings\/","text":"# Pentagon-Decagon Packing\n\nTwo regular pentagons and a regular decagon meet snugly at a vertex: their interior angles sum to 360\u00b0. However, they can\u2019t tile the plane. However, they come fairly close, as shown in this picture by Greg Egan.\n\n# Packing Smoothed Octagons\n\nWhich shape is worst of all for packing the plane? That is, which has the lowest maximal packing density? Suppose we demand that our shape be convex and also centrally symmetric: that is, a subset $S \\subseteq \\mathbb{R}^2$ such that $x \\in S$ implies $-x \\in S$. Then a certain \u2018smoothed octagon\u2019 is conjectured to be the worst. Amazingly, this shape has a 1-parameter family of maximally dense packings, shown in this image created by Greg Egan.\n\n# Packing Regular Octagons\n\nThis is the densest packing of regular octagons in the plane, drawn by Graeme McRae. It is interesting because it is a counterexample to the 2-dimensional analogue of a conjecture made in 3 dimensions by Stanislaw Ulam.","date":"2022-12-01 14:10:47","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.8241313099861145, \"perplexity\": 979.5153801144886}, \"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-2022-49\/segments\/1669446710813.48\/warc\/CC-MAIN-20221201121601-20221201151601-00426.warc.gz\"}"} | null | null |
This really is free printable resume templates blank list word format resume free sample resumes lovely new blank resume format resume templats free sample resume examples legalsocialmobilitypartnership blank resume template printable – all resume should be written as if you're applying being a company owner in the organization you would like to start.
A resume is actually a one- to two-page document summarizing your career objectives, professional experiences and achievements, and academic background. The heading of your resume should contain a message, address and speak to information. The body from the resume needs to be broken into the following sections: career objective, profile/summary, professional experience, achievements, scholastics, and references. Your employment objective really should be brief, up to two sentences; it must give your potential employers a solid idea of how you want to move forward with your professional life. A concise profile or a synopsis should discuss what you do and how your skills and experience best apply to your job you have an interest in. The summary, as well as other areas of your resume, ought not contain personal data that discloses ethnicity, sexual orientation, marital status, age, living situations, or other personal data that's not directly related for your career. Personal profile/summary should only contain a number of well-written sentences that convey what you could provide the table with regards to the specific job. Use this to bring in the employer's attention, but do not overload in trying to be creative – stay professional. Your experience listing should include information using one to five jobs you've held, starting with your or last job, and listing previous positions in chronological order.
Your chance should add date range of one's employment, name of nokia's or person(s) you been employed for, and the location while stating the spot that the place of employment can be found (full address of employment will not be necessary). List your title and your primary responsibilities, with increased duties which might be applicable to the kind of work you are seeking. Your education include college, graduate and post-graduate work, and also any courses or professional certifications which might be related to your work development. Achievements, volunteer positions, publications and interests should only be listed if they apply for your professional work experience References really should be listed if requested; best practices suggest not to list generic statements about references being offered upon request because this is understood.
Within the competitive, internet-driven playing field of job searches, your resume represents someone to potential employers. It serves for your tool get noticed, get the interview and/or get a job. An awesome resume can make you differentiate yourself from other candidates by showcasing your aptitudes. Bring to mind your resume as your sales hype – it is advisable to sell yourself in perfect way. Invest some time and research into growing your resume. You will want to make sure that your resume is error free – double check your grammar and spelling, be certain that all company and school names and cities are spelled properly. A resume containing errors, regardless how minimal, can give your potential employer the idea that a sensational scene attention to detail, that you don't take the time to ensure your hard work, and that you are a poor communicator. Additionally, keep the resume is formatted well. Stick to basic fonts, like Arial and Times New Roman. Keep your font size and color standard; don't utilize large fonts or multi-colors as part of your resume. Don't overload with bold, italicized, or large-cap text. Maintain format consistent and make sure which the resume looks great when viewed online and also when printed out. Keep the resume to several pages – any extra pages give an impact that you can don't discover how to concisely summarize your education and experience, or that you are listing unnecessary information as a taking up space. If you've never written a resume before, reference books, Internet resources or seek help from a specialist resume writing service. A well-written resume can make a distinction between being stuck at your own job and obtaining a job interview to land the career of your dreams.
The most difficult and time-consuming portion of any resume may be the set of your experience, regardless of the level you reach inside your professional career. The secret is to consider your work objective and prioritize your projects with respect to the goals.
Your professional experience probably should not only showcase those activities you have carried out in your previous jobs, but should demonstrate your qualifications in the way motivates employers to need to know more. Obviously, we're also speaking about results, any tangible, measurable items which might be impacting to the underside line. Let your employers know that your chosen project came within budget, that you simply exceeded the timeline, you acquired X number of recent customers, or which you increased sales by way of double-digit percentage. Employers can wrap their minds around numbers, simply because they're focused entirely on them daily. You need to let your potential employer know that you can think such as they generally do and that you take results into thinking as your perform your livelihood on day-to-day basis.
To begin with your task history, begin each description with an influence word, such as managed, developed, communicated, etc. Perform some research and only use the energy words and phrases that work for ones industry. Make sure that the statements you list first under your work responsibilities quantify your achievements – don't hesitate to list out sales figured, customer acquisition rates, budget and timeline successes, or another figures that really help put your responsibilities in the context of the business/field you happen to be working in. Be specific. The only way your statements are truly quantified is in case you include numbers. Saying that you acquired new customers is significantly totally different from proclaiming that you increased the client database by 10%. As pointed out above, this can be a most essential part of listing your work descriptions on your own resume. Your employer would like to know besides what that you did, but wait, how well took action now it. Also, these statements really should be aligned with the career objective you included near the top of the resume. In order to get a career in project management, letting your employer know for you to managed a team of 20 people and the entire results you achieved will effectively highlight your qualifications. It is essential to quantify your livelihood description statements for your resume; however, as a thing of caution, tend not to quantify all statements, only one or two that are most critical to your task and are generally goal driven. This shows your employer for you to think regarding exceeding your goals. All subsequent descriptions of the responsibilities should keep the first a couple items on your own list.
Being a final test, put yourself in the footwear of your employer. Cross-check the task description and ensure that you address the qualifications required to complete the job with the knowledge for your resume. Let your potential employer know you might have what they are searching for, and will also be sure to have a great impression.
This entry was posted in Resume Templates and tagged Blank, Free, Resume, Template by allresume45. Bookmark the permalink. | {
"redpajama_set_name": "RedPajamaC4"
} | 6,978 |
\section{Introduction}
The aim of this paper is to highlight some relations between completions, strongly flat modules and perfect rings in the non-commutative case. We explore some connections between some notions of Homological Algebra (cotorsion modules) and topological rings (completions in some natural topologies). These connections are well known for modules over commutative rings, thanks to Matlis, who proved that the completion in the $R$-toplogy for an integral domain $R$ is closely related to the cotorsion completion functor $\operatorname{Ext}_R^1(K,-)$. Here $Q$ is the field of fractions of $R$ and $K:=Q/R$. We investigate these connections in the non-commutative case, defining a suitable $R$-topology on any module over a not-necessarily commutative ring $R$. This leads us to the study of strongly flat modules, because the completion of $R$ in its $R$-topology turns out to be a strongly flat $R$-module (Theorem~\ref{4.9}).
We consider strongly flat modules over non-commutative rings as defined in \cite[Section~3]{submitted}.
The class of strongly flat modules lies between the class of projective modules and the class of flat modules.
In particular, we study when the class of strongly flat modules is covering, because this is related to an open problem posed by Enochs, whether ``every covering class is closed under direct limit" (see for example \cite [Open problem 5.4] {approx}).
Since flat modules are direct limits of projective modules, the class of strongly flat modules is closed under direct limits if and only if flat modules are strongly flat.
Bazzoni and Salce \cite{silsal} gave a complete answer to this question for modules over commutative domains, completely determining when the class of strongly flat modules over a commutative domain is covering.
Subsequently, Bazzoni and Positselski generalized this to arbitrary commutative rings in \cite{Leonidsilvana}. They proved that, for a commutative ring $R$,
the class $ \mathcal{SF}$ of strongly flat modules is covering if and only if
flat modules are strongly flat, if and only if
$R/aR$ is a perfect ring for every regular element $a \in R$. In our Example~\ref{5.18}, we will show that there exist non-invariant chain domains $R$ for which $\operatorname{End}(R/I)$ is perfect for every non-zero principal right or left ideal
$I$ of $R$, but the class of strongly flat left $R$-modules is not covering. Very recent papers related to these topics are the articles \cite{BP,P}
by Bazzoni and Positselski.
For a commutative ring $R$, the set of regular elements is always an Ore set, and if $Q$ denotes the classical quotient ring of $R$, the class of strongly flat modules is $^\bot \{Q^\bot \} $ \cite{ FSALCE}.
The generalization of strongly flat modules to non-commutative rings given in \cite {submitted} depends on the choice of the overring $Q$ of $R$. More precisely, if
$\varphi \colon R\to Q$ is a bimorphism in the category of rings, that is, $\varphi$ is both a monomorphism and an epimorphism, we assume that ${}_RQ$ is a flat left $R$-module. We view at $R$ as a subring of $Q$ and $\varphi \colon R\to Q$ as the inclusion.
Then a left $R$-module $_RM$ is {\em Matlis-cotorsion} if $\operatorname{Ext}^1(_RQ,{}_RM)=0$ \cite{submitted}.
Let $\mathcal {MC}$ denote the class of Matlis-cotorsion left $R$-modules. For any class of left $R$-modules $ \mathcal {A}$, set $ ^ \bot \mathcal {A}:= \{ \,B\in R\mbox{\rm -Mod} \mid \operatorname{Ext}_R ^1 (B, A) = 0$ for every $A \in \mathcal {A}\,\}$ and $\mathcal {A}^ \bot := \{ \,B\in R\mbox{\rm -Mod} \mid\operatorname{Ext}_R ^1 (A, B) = 0$ for every $A \in \mathcal {A}\,\}$.
A left $R$-module is {\em strongly flat} if it is in $ ^\bot \mathcal {MC} $.
The class of strongly flat left $R$-modules will be denoted by $\mathcal {SF}$.
By
\cite [Theorem 6.11] {approx}, the cotorsion pair $(\mathcal{SF} , \mathcal{MC})$ is
complete, that is, every left $R$-module has a special
$\mathcal{MC}$-preenvelope (or, equivalently, every left module has a special $\mathcal{SF}$-precover).
Thus, by \cite [Corollary 6.13] {approx}, the class $\mathcal{SF} $ consists of
all direct summands of
modules $N$ such that $N$ fits into an exact sequence of the form
$$ 0 \to F \to N \to G \to 0,$$
where $F$ is a free $R$-module and $G$ is $\{Q\}$-filtered. For the terminology, see \cite{submitted}.
Whenever $R$ is a right Ore domain, i.e., the class of regular elements is a right Ore set,
the class of strongly flat left $R$-modules is the class $^\bot \{Q^\bot\} $, where $Q$ is the classical right quotient ring of $R$.
Several of our results about strongly flat modules are for modules over a nearly simple chain domain. Recall that a chain domain $R$, that is, a not-necessarily commutative integral domain for which the modules $R_R$ and $_RR$ are uniserial, is {\em nearly simple} if it has exactly three two-sided ideals, necessarily $R$, its Jacobson radical $J(R)$ and $0$. The reason why we concentrate on chain domains $R$ with classical quotient ring $Q$ is due to the fact that for these rings the $R$-module $_RK:=Q/R$ is uniserial, and thus, in the study of $\operatorname{End}(_RK)$, we can take advantage of our knowledge of the endomorphism rings of uniserial modules \cite{DungAlberto, DungFacchini, Facchinitransaction, Facsal, Pavel, puninsky}. In our Example~\ref{5.18}, we also take advantage of our knowledge of the endomorphism rings of cyclically presented modules over local rings \cite{Aminis}.
If $R$ is a right chain domain and the class of strongly flat $R$-modules is
covering, then
$R$ is right invariant, that is, $aR = Ra$ for every $a \in R$. In this case, flat modules are strongly flat
(equivalently, the class $ \mathcal{SF}$ of strongly flat modules is closed under direct limit).
We began this paper in September 2017, when both of us where visiting the Department of Algebra of Charles University in Prague, and continued in March 2018 when the first named author was visiting the IPM (Institute for Research in Fundamental Sciences) in Tehran. We are very grateful to both institutions for their hospitality.
\section{The $R$-topology} \label{completions}
In Sections \ref{completions}, \ref{3} and \ref{4} of this paper, we suppose that {\em we have a ring $R$ and a multiplicatively closed subset $S$ of $R$ satisfying:} (1) {\em If $a,b\in R$ and $ab\in S$, then $a\in S$.} (2) {\em $S$ is a right Ore set in $R$.} (3) {\em The elements of $S$ are regular elements of~$R$}. (4) {\em The right ring of quotients $Q:=R[S^{-1}]$ of $R$ with respect to $S$ is a directly finite ring}. That is, our setting is that of \cite[Section~4]{submitted}.
Correspondingly, we have a Gabriel topology $\Cal G$ on $R$ consisting of all the right ideals $I$ of $R$ with $I\cap S\ne\emptyset$ (cf.~\cite[\S VI.6]{20}). In particular, the Gabriel topology $\Cal G$ consists of dense right ideals of $R$, the canonical embedding $\varphi \colon R\to Q:=R[S^{-1}]$ is an epimorphism in the category of rings, we view $R$ as a subring of $Q$ and $\varphi$ as the inclusion mapping, and ${}_RQ$ turns out to be a flat left $R$-module \cite[\S XI.3]{20}.
There is a hereditary torsion theory $(\Cal T,\Cal F)$ on $\operatorname{Mod-\!} R$ in which
the torsion submodule of any right $R$-module $M_R$ consists of all the elements $x\in M_R$ for which there exists an element $s\in S$ with $xs=0$. If we indicate the torsion submodule of $M$ by
$t(M)$, then clearly $t(M) \otimes _R Q = 0$. A right $R$-module $M_R$ is in $\Cal F$, that is, is torsion-free, if and only if right multiplication $\rho_s\colon M_R\to M_R$ by $s$ is an abelian group monomorphism for every $s\in S$. Dually, we will say that a right $R$-module $M_R$ is {\em divisible} if right multiplication $\rho_s\colon M_R\to M_R$ by $s$ is an abelian group epimorphism for every $s\in S$, that is, if $Ms=M$ for every $s\in S$. Every homomorphic image of a divisible right $R$-module is divisible. If $A$ is a submodule of a right $R$-module $B_R$ and both $A_R$ and $B/A$ are divisible, then $B_R$ is divisible. Any sum of divisible submodules is a divisible submodule, so that every right $R$-module $M_R$ contains a greatest divisible submodule, denoted by $d(M_R)$. A right $R$-module $M_R$ is {\em reduced} if $d(M_R)=0$. For every module $M_R$, $M_R/d(M_R)$ is reduced.
We have that $\Cal G=\{\,I\mid I$ is a right ideal of $R$, and $\varphi(I)Q=Q\,\}$, and $\Cal G$ has a basis consisting of the principal right ideals $sR$, $s\in S$. Let $M_R$ be any right $R$-module.
By \cite [XI, Proposition 3.4] {20},
the kernel of the canonical right $R$-module morphism $M_R\to M\otimes_R Q$ is
equal to $t(M)$. Note that if we set $K : = Q/R$, then
$_RK_R$ is an $R$-$R$-bimodule and $t(M_R)\cong\operatorname{Tor}_1^R(M_R, {}_RK)$ (see (15) and (16) in \cite[Section 3]{submitted}).
\bigskip
We now define a topology on any right $R$-module in the attempt of generalizing the $R$-topology studied by
Matlis \cite{dim} for a commutative ring $R$. Our definition is as follows. Let $R$ be any ring with identity, not
necessarily commutative, and $S$ be a subset of $R$ with the properties written at the beginning of this section.
Given any right $R$-module $M_R$, the {\em $R$-topology} on $M_R$ has a
neighborhood base of $0$ consisting, for every non-empty finite set of elements $s_1,\dots,s_n\in S$, of the
submodules $$U(s_1,\dots,s_n):=\{\,x\in M_R\mid xR\subseteq Ms_1\cap\dots\cap Ms_n\,\}$$ of $M_R$. For the
regular right module $R_R$, the
$R$-topology on $R$ has a neighborhood base of $0$ consisting, for every non-empty finite set of elements $s_1,\dots,s_n\in S$, of the right ideals $$U(s_1,\dots,s_n):=\{\,x\in R\mid xR\subseteq Rs_1\cap\dots\cap Rs_n\,\}$$ of $R$.
\begin{Lemma} On the right $R$-module $R_R$, the right ideals $U(s)$ are two-sided ideals of $R$, $U(s)$ is the annihilator of the left $R$-module $R/Rs$, and the $R$-topology is a ring topology on $R$.\end{Lemma}
\begin{proof} Clearly, $U(s)=\{\,x\in R\mid xR\subseteq Rs\,\}$ is the annihilator of the cylic left $R$-module $R/Rs$, and hence $U(s)$ is a two-sided ideal. Moreover, $R$ is a right linearly topological ring \cite[p.~144]{20}, because every filter of two-sided ideals of a ring is a fundamental system of neighborhoods of $0$ for a right and left linear topology on the ring \cite[p.~144]{20}.\end{proof}
We will use $R_{R\operatorname{-top}}$ to denote the topological ring $R$ with the $R$-topology.
\begin{Lemma}\label{12.7} Every right $R$-module, with respect to its $R$-topology, is a linearly topological module over the topological ring $R_{R\operatorname{-top}}$.\end{Lemma}
\begin{proof} It suffices to check property {\em TM}\,3 in \cite[p.~144]{20}. That is, we must prove that $(U_M(s):x)\supseteq U_R(s)$ for every $s\in S$, $x\in M_R$. Equivalently, that $xU_R(s)\subseteq U_M(s)$. Now if $r\in U_R(s)$, then $rR\subseteq Rs$, so that $xrR\subseteq xRs\subseteq Ms$, i.e., $rx\in U_M(s)$.\end{proof}
\begin{Lemma}\label{Matlis} If the ring $R$ is commutative, the linear topology on any right $R$-module $M$ defined by the submodules $U(s)$, $s\in S$, coincides with the $R$-topology defined by Matlis in \cite{dim}.\end{Lemma}
\begin{proof} $U(s)=\{\,x\in M\mid xR\subseteq Ms\,\}=Ms$.\end{proof}
In the next proposition, we consider the behavior of continuity of right $R$-module morphisms when the modules involved are endowed with the $R$-topology. Recall that a submodule $M$ of a right $R$-module $N_R$ is an {\em RD-pure submodule} if $Mr=M\cap Nr$ for every $r\in R$ (equivalently, if the natural homomorphism $M\otimes R/Rr\to N\otimes R/Rr$ is injective for every $r\in R$, or if the natural homomorphism $\operatorname{Hom}(R/rR,N)\to\operatorname{Hom}(R/rR,M)$ is surjective for every $r\in R$.) See \cite[Proposition~2]{WarfPur}.
\begin{proposition}\label{easy} {\rm (a)} Every right $R$-module morphism $f\colon M_R\to N_R$ between two right $R$-modules $M_R$ and $N_R$ endowed with their $R$-topologies is continuous.
{\rm (b)} For every right $R$-module $N_R$ and every $s\in S$, the $R$-submodule $U(s)$ of $N_R$ is the largest $R$-submodule of $N_R$ contained in $Ns$.
{\rm (c)} A submodule $M_R$ of a right $R$-module $N_R$ endowed with the $R$-topology is an open submodule of $N_R$ if and only if $M_R\supseteq U(s)$ for some $s\in S$.
{\rm (d)} A right $R$-module morphism $f\colon M_R\to N_R$ between two right $R$-modules $M_R$ and $N_R$ with their $R$-topologies is an open map if and only if $f(M_R)\supseteq U(s)$ for some $s\in S$.
{\rm (e)} Every right $R$-module epimorphism $f\colon M_R\to N_R$ between two right $R$-modules $M_R$ and $N_R$ is an open continuous map.
{\rm (f)} Every right $R$-module isomorphism $f\colon M_R\to N_R$ is a homeomorphism when the two right $R$-modules $M_R$ and $N_R$ are endowed with their $R$-topologies.
{\rm (g)} If $M_R$ is an RD-pure submodule of a right $R$-module $N_R$ and $M_R,N_R$ are endowed with their $R$-topologies, then the embedding $M_R\hookrightarrow N_R$ is a topological embedding.\end{proposition}
The proofs are easy and we omit them.
\section{The right $R$-module $\operatorname{Hom}(K_R, M\otimes_RK)$}\label{3}
In this section, the hypotheses on $R$ and $S$ are the same as in the previous section.
For any right $R$-module $M_R$, we will be interested in the right $R$-module $$\operatorname{Hom}(K_R, M\otimes_RK).$$ Here the right $R$-module structure is given by the multiplication defined, for every $f\in \operatorname{Hom}(K_R, M\otimes_RK)$ and $r\in R$, by $(fr)(k)=f(rk)$ for all $k\in K$.
For any right $R$-module $M_R$, the right $R$-module $$\operatorname{Hom}(K_R, M\otimes_RK)$$ can be endowed with the $R$-topology, defined by the submodules $U(s_1,\dots,s_n):=U(s_1)\cap\dots\cap U(s_n)$ as a
neighborhood base of $0$. But we have that:
\begin{Lemma} For the modules $\operatorname{Hom}(K_R, M\otimes_RK)$, one has that $U(s)=V(s)$, where, for every element $s\in S$, $$V(s):=\{\,f\in \operatorname{Hom}(K_R, M\otimes_RK)\mid f((Rs^{-1})/R)=0\,\}.$$\end{Lemma}
\begin{proof} $(\subseteq)$. Let $f$ be an element of $U(s)$, so that $f\in\operatorname{Hom}(K_R, M\otimes_RK)$ and $fR\subseteq \operatorname{Hom}(K_R, M\otimes_RK)s$. In order to show that $f\in V(s)$ we have to prove that $f((Rs^{-1})/R)=0$. Fix $r\in R$. Then $fr=gs$ for some $g\in \operatorname{Hom}(K_R, M\otimes_RK)$. Hence $f(rs^{-1}+R)=(fr)(s^{-1}+R)=(gs)(s^{-1}+R)=g(ss^{-1}+R)=0$. Thus $f((Rs^{-1})/R)=0$.
$(\supseteq)$. Suppose $f\in V(s)$, so that $f((Rs^{-1})/R)=0$. In order to prove that $f\in U(s)$, we must show that, for every fixed element $r\in R$, there exists $g\in \operatorname{Hom}(K_R, M\otimes_RK)$ with $fr=gs$. Define $g\colon K_R\to M\otimes_RK_R$ by $g(q+R)=f(rs^{-1}q+R)$ for all $q\in Q$. Then $g$ is a well defined right $R$-module morphism, because if $q\in R$, then $f(rs^{-1}q+R)=f(rs^{-1}+R)q\in f((Rs^{-1})/R)R=0$, and $fr=gs$.\end{proof}
\bigskip
We will denote by $V(s_1,\dots,s_n)$ the intersection $V(s_1)\cap\dots\cap V(s_n)$, but it is necessary to remark that:
\begin{Lemma} For every $s,s'\in S$, there exists $t\in S$ such that $V(s)\cap V(s')\supseteq V(t)$.\end{Lemma}
\begin{proof}
Given $s,s'\in S$, there exist $t\in S$ and $r,r'\in R$ with $t=sr=s'r'$ \cite[Lemma~4.21]{goodwar}. Then $s^{-1}=rt^{-1}$, so that $Rs^{-1}=Rrt^{-1}\subseteq Rt^{-1}$. Therefore $V(t)\subseteq V(s)$, because if $f\in \operatorname{Hom}(K_R, M\otimes_RK)$ and $f(Rt^{-1}/R)=0$, then $f(Rs^{-1}/R)=0$, that is, $f\in V(s)$. Similarly, $V(t)\subseteq V(s')$.\end{proof}
A right (or left) $R$-module $M_R$ is {\em $h$-divisible} if every homomorphism $R_R\to M_R$ extends to an $R$-module morphism $Q_R\to M_R$ \cite[Section 2]{submitted}. Any right (or left) $R$-module $M$ contains a unique largest $h$-divisible submodule $h(M)$ that contains every $h$-divisible submodule of $M$. An $R$-module $M_R$ is {\em $h$-reduced} if $h(M_R)=0$, or, equivalently, if $\operatorname{Hom}(Q_R,{}M_R)=0$ \cite{submitted}.
Obviously, $h$-divisible right $R$-modules are divisible.
\begin{proposition} \label{equal} Divisible torsion-free right $R$-modules are
$Q$-modules.
In particular, $h(M_R)=d(M_R)$ for any torsion-free right $R$-module~$M_R$.\end{proposition}
\begin{proof} Suppose $M_R$ torsion-free and divisible. Then right multiplication by $s$ is an automorphism of the abelian group $M$ for every $s\in S$. By the universal property of $Q=R[S^{-1}]$, the canonical ring antihomomorphism $R\to\operatorname{End}_\mathbb{Z}(M)$ extends to a ring antihomomorphism $Q\to\operatorname{End}_\mathbb{Z}(M)$ in a unique way. That is, there is a unique right $Q$-module structure on $M$ that extends the right $R$-module structure of $M_R$. Thus $M$ is a right $Q$-module. In particular, it is an $h$-divisible right $R$-module.
\end{proof}
\bigskip
Let $M_R$ be a right $R$-module. For every element $x\in M_R$, there is a right $R$-module morphism $R_R\to M_R$, $1\mapsto x$. Tensoring with $_RK$, we get a right $R$-module morphism $\lambda_x\colon K_R\to M\otimes_RK$, defined by $\lambda_x(k)=x\otimes k$. The canonical mapping $\lambda\colon M_R\to\operatorname{Hom}(K_R, M\otimes_RK)$, defined by $\lambda(x)=\lambda_x$ for every $x\in M_R$, is a right $R$-module morphism, as is easily checked. In the rest of this section, all $R$-modules are endowed with their $R$-topologies.
\begin{theorem}\label{righthreduced} Let $M_R$ be an $h$-reduced torsion-free right $R$-module. Then the canonical mapping $\lambda\colon M_R\to\operatorname{Hom}(K_R, M\otimes_RK)$ is an embedding of topological modules and $\operatorname{Hom}(K_R, M\otimes_RK)$ is complete.\end{theorem}
\begin{proof} The canonical mapping $\lambda\colon M_R\to\operatorname{Hom}(K_R, M\otimes_RK)$ is injective by \cite[Theorem~4.5]{submitted}. In order to show that $\lambda\colon M_R\to\operatorname{Hom}(K_R, M\otimes_RK)$ is an embedding of topological modules, it suffices to show that $\lambda^{-1}(V(s_1,\dots,s_n))=U(s_1,\dots,s_n)$ for every $s_1,\dots,s_n\in S$. Now $x\in \lambda^{-1}(V(s_1,\dots,s_n))$ if and only if $\lambda_x\in V(s_1,\dots,s_n)$, that is, if and only if $x\otimes (Rs_1^{-1}+\dots+Rs_1^{-1}/R)=0$ in $M\otimes_RK$. Equivalently, if and only if $x\otimes (rs_i^{-1}+R)=0$ in $M\otimes_RK$ for every $r\in R$ and $i=1,2,\dots,n$. By \cite[Step 3 of the proof of Theorem~4.5]{submitted}, this is equivalent to $xr\in Ms_i$ for every $r\in R$ and $i=1,2,\dots,n$, that is, if and only if $x\in U(s_1,\dots,s_n)$.
In order to prove that $\operatorname{Hom}(K_R, M\otimes_RK)$ is complete, we must show that every Cauchy net converges. Let $A$ be a directed set with order relation $\le$ and let $\{f_\alpha\}_{\alpha\in A}$ be a Cauchy net in $\operatorname{Hom}(K_R, M\otimes_RK)$. Define a morphism $f\in \operatorname{Hom}(K_R, M\otimes_RK)$ as follows. Since we are dealing with a Cauchy net, for every $s\in S$ there exists $\alpha\in A$ such that $f_\beta-f_\gamma\in V(s)$ for every $\beta,\gamma\in A$, $\beta,\gamma\ge\alpha$. Set $f(rs^{-1}+R)=f_\alpha(rs^{-1}+R)$ for every $r\in R$. We leave to the reader the easy verification that $f$ is a well defined mapping.
Let us check that $f(kr)=f(k)r$ for every $k\in K_R$ and $r\in R$.
We have that $k=as^{-1}+R$ for some $a\in R$, $s\in S$. By the right Ore condition, there exist $r'\in R$ and $t\in S$ such that $as^{-1}r=r't^{-1}$. Since $A$ is directed, there exists $\alpha$ such that
$f ( r't^{-1} + R) = f_\alpha ( r't^{-1}+ R)$ and $f( as^{-1}+R) r = f_\alpha ( as^{-1}+R) r $.
Therefore $f (kr) = f(k) r$.
It is now easily seen that $f$ is the limit of the Cauchy net.
\end{proof}
For any right $R$-module $M_R$ endowed with its $R$-topology, the {\em (Hausdorff) completion} of $M_R$ is $\displaystyle \widetilde{M_R}:= \lim_{\longleftarrow} M/U(s_1,\dots,s_n)$. Notice that the set of all the submodules $U(s_1,\dots,s_n)$ of $M_R$ is downward directed under inclusion. Here $\{s_1,\dots,s_n\}$ ranges in the set of all finite subsets of $S$. There is a canonical mapping $\eta\colon M\to \widetilde{M_R}$, whose kernel is the closure $\overline{\{0\}}$ of $0$ in the $R$-topology of $M_R$. Clearly, $\overline{\{0\}}=\bigcap_{s_1,\dots,s_n\in S} U(s_1,\dots,s_n)=
\bigcap_{s\in S} U(s)=\{\,x\in M_R\mid xR\subseteq \bigcap_{s\in S}Ms\,\}$.
From Lemma \ref{12.7}, we get that if $M_R$ is a right $R$-module, the right $R$-module $\operatorname{Hom}(K_R, M\otimes_RK)$ with the topology defined by the submodules $V(s)$ is a topological module over the topological ring $R_{R\operatorname{-top}}$.
\begin{proposition} The right $R$-submodules $V(s)$ of the ring $\operatorname{End}(K_R)$ are two-sided ideals of $\operatorname{End}(K_R)$. The topology they define on $\operatorname{End}(K_R)$ is a ring topology. If $R$ is commutative, this topology on $\operatorname{End}(K_R)$ coincides with the topology on the completion $H$ of $R$ with respect to the $R$-topology \cite[p.~15]{dim}.\end{proposition}
\begin{proof} When we consider $M = R_R$, then, by
\cite[Step 2 of the proof of Theorem~4.5]{submitted}, the elements of $K$ annihilated by right multiplications of an element $s\in S$ are those of $Rs^{-1}/R$. It follows that $Rs^{-1}/R$ is a fully invariant submodule of $K_R$. From this we get that every $V(s)$ is a two-sided ideal of the ring $\operatorname{End}(K_R)$.
Every filter of two-sided ideals of a ring is a fundamental system of neighborhoods of $0$ for a right and left linear topology on the ring \cite[p.~144]{20}. Thus the topology defined by the two-sided ideals $V(s)$ is a ring topology on $\operatorname{End}(K_R)$. Moreover, if $R$ is commutative, the submodules $V(s)$ define the $R$-topology on the right $R$-module $\operatorname{Hom}(K_R, M\otimes_RK)$ for every module $M$ (Lemma~\ref{12.7}), which coincides with the $R$-topology defined by Matlis in \cite{dim} by Lemma~\ref{Matlis}.
Finally, Matlis' $R$-topology on $\operatorname{End}(K_R)$ coincides with the topology on the completion $H$ of $R$ with respect to the $R$-topology, because the topology on the completion $H$ coincides with the $R$-topology on $H$.
\end{proof}
\section{Torsion-free modules}\label{4}
In this section, we keep the same hypotheses and notations as in the previous two sections.
As we have seen, for any right $R$-module $M_R$, there is a right $R$-module morphism $\lambda\colon M_R\to\operatorname{Hom}(K_R, M\otimes_RK)$, defined by $\lambda(x)=\lambda_x$ for every $x\in M_R$, where $\lambda_x\colon k\to x\otimes k$, and there is a canonical mapping $\eta\colon M\to \widetilde{M_R}$ of $M_R$ with its $R$-topology into its Hausdorff completion.
\begin{proposition} \label {torsionfreeright} Let $M_R$ be a torsion-free right $R$-module. Then: {\rm (a)} $\ker\lambda$ is the closure of $0$ in the $R$-topology; {\rm (b)} $\ker\lambda$ is the kernel of the canonical mapping $\eta\colon M\to \widetilde{M_R}$; and {\rm (c)} $\ker\lambda$ is equal to $h(M_R)$.\end{proposition}
\begin{proof} We have already remarked that the kernel of $\eta$ is the closure of $\overline{\{0\}}$ of $0$. Hence (a)${}\Leftrightarrow{}$(b).
The right $R$-module $\operatorname{Hom}(K_R,N_R)$ is $h$-reduced for every right $R$-module $N_R$ \cite[Theorem~2.8]{submitted}. Let $M_R$ be a torsion-free right $R$-module. Since $$\lambda\colon M_R\to \operatorname{Hom}(K_R, M\otimes_RK)$$ is a homomorphism into an $h$-reduced $R$-module, it follows that $h(M)\subseteq \ker\lambda$.
Let us prove that $\ker\lambda\subseteq \overline{\{0\}}$. Suppose $x\in \ker\lambda$. Then $x\otimes (rs^{-1}+R)$ is equal to zero in the tensor product $M\otimes K$. By \cite[Theorem~3.1(1)]{submitted}, there exists an element $y_{r,s}\in M_R$ such that $x\otimes rs^{-1}=y_{r,s}\otimes 1$ in $M\otimes_RQ$. Thus $xr\otimes 1=y_{r,s}s\otimes 1$ in $M\otimes_RQ$. Since $M_R$ is torsion-free, it follows that $xr=y_{r,s}s$ in $M_R$ by \cite[Theorem~3.1(1)]{submitted} again. This proves that $xR\subseteq \bigcap_{s\in S}Ms$, and so $\ker\lambda\subseteq \overline{\{0\}}$.
Conversely, $\overline{\{0\}}\subseteq\ker\lambda$, because if $x\in \overline{\{0\}}$, then $xR\subseteq Ms$ for every $s\in S$, that is, for every $s\in S$ and every $r\in R$ there exists $m_{r,s}\in M$ with $xr=m_{r,s}s$. Then, for every element $rs^{-1}+R\in K$, we have that $x\otimes(rs^{-1}+R)=xr\otimes (s^{-1}+R)=m_{r,s}s\otimes (s^{-1}+R)=m_{r,s}\otimes s(s^{-1}+R)=0$ in $M\otimes_RK$. Thus $x\in\ker\lambda$. This proves that $\overline{\{0\}}=\ker\lambda$. Therefore (a) and (b) hold.
We now show that $\ker\lambda$ is divisible. For every $s\in S$, $s$ is invertible in $Q$, hence $sQ=Q$, so $sK=K$. Now if $x\in \ker\lambda$ and $t\in S$, then $x\in \overline{\{0\}}$, hence $x=yt$ for some $y\in M_R$. We must prove that $y\in \ker\lambda$, that is, that $y\otimes K=0$ in $M\otimes K$. But $y\otimes K=y\otimes sK=ys\otimes K=x\otimes K=0$ in $M\otimes K$. This proves that $\ker\lambda=\overline{\{0\}}$ is divisible.
Thus $ \ker\lambda = h(M)$ by Proposition \ref{equal}.
\end{proof}
Clearly, from Proposition~\ref{torsionfreeright}, we have that:
\begin{corollary}
If $M_R$ is a torsion-free module, then $\widetilde{M_R} \cong \widetilde{M_R / h(M) }$.
\end{corollary}
\begin{lemma} \label{fourparts} Let $M$ be torsion-free right $R$-module. Then:
{\rm (a)} Every element of $M\otimes_RK$ can be written in the form $x\otimes (s^{-1}+R)$ for suitable elements $x\in M_R$ and $s\in S$.
{\rm (b)} Let $s$ be an element of $S$. The elements $y$ of $M\otimes_RK$ such that $ys=0$ are those that can be written in the form $x\otimes (s^{-1}+R)$ for a suitable $x\in M_R$.
{\rm (c)} If $x\in M_R$, $r\in R$ and $s\in S$, then $x\otimes(rs^{-1}+R)=0$ in $M\otimes_RK$ if and only if $xr\in Ms$.
{\rm (d)} The set $\{\, U(s) \mid s \in S\,\} $ is downward directed.
\end{lemma}
\begin{proof}
In the proof of Steps 1, 2 and 3 of \cite [Theorem 4.5] {submitted}, we do not use the fact that $M$ is $h$-reduced. So the proofs of (a), (b) and (c) are like those of Steps 1, 2 and 3 in \cite [Theorem 4.5] {submitted}.
(d) Assume that $s, t \in S$. Then there exist $u \in S$ and $r_1,r_2\in R$ such that $ s ^{-1}= r_1u^{-1} $ and
$ t ^{-1}= r_2 u^{-1} $. If $m \in U(u)$ and $r \in R$, then $m \otimes (rs^{-1}+R) =
m \otimes (r r_1u^{-1} +R)=0$. Part (c) implies that $m \in U(s)$, and so $U(u) \subseteq U(s)$.
Similarly, $U(u) \subseteq U(t)$.
\end{proof}
\begin{remark}\label{remarktorsionfree}
{\rm By Lemma~\ref{fourparts}(d), for
$M$ torsion-free, we have that $$\widetilde{M}= \lim_{\longleftarrow} M/U(s).$$ Notice that the kernel of
the canonical mapping $\eta \colon M\to \widetilde{M}$ is divisible by Theorem \ref{torsionfreeright}. } \end{remark}
\bigskip
Now let $M_R$ be a torsion-free right $R$-module, so that $$\lambda\colon M_R\to\operatorname{Hom}(K_R,\linebreak M\otimes_RK)$$ is continuous with respect to the $R$-topologies (Proposition~\ref{easy}(a)) and $\operatorname{Hom}(K_R,\linebreak M\otimes_RK)$ is Hausdorff. Notice that $M\otimes_RK$ and $M/h(M)\otimes_RK$ are isomorphic, so that $\operatorname{Hom}(K_R, M\otimes_RK)$ is complete (Theorem~\ref{righthreduced}). Thus $\lambda$ extends in a unique way to a continuous morphism $\widetilde{\lambda}\colon \widetilde{M} \to \operatorname{Hom}(K_R, M\otimes_RK)$. In Theorem~\ref{completion} and Example~\ref{quasismall}, we see { that $\widetilde{\lambda}$ is a continuous monomorphism, but not necessary an
isomorphism.
\begin{theorem}\label{completion} Let $M_R$ be a torsion-free right $R$-module. Then there exists a right $R$-module monomorphism $\widetilde{\lambda}\colon \widetilde{M} \to \operatorname{Hom}(K_R, M\otimes_RK)$ such
that $\lambda=\widetilde{\lambda}\eta$.
\end{theorem}
\begin{proof}
Define $\widetilde{\lambda}$ as follows. We know that $$\displaystyle\widetilde{M}= \lim_{\longleftarrow} M/U(s)\le\prod _{s\in S} M/U(s),$$ so that every element of $\widetilde{M}$ is of the form $\widetilde{m}=(m_{s}+U(s))_{s\in S}$.
Set $\widetilde{\lambda}(\widetilde{m})(rs^{-1}+R)=m_{s}\otimes (rs^{-1}+R)$ for every $r\in R$, $s\in S$.
In order to prove that $\widetilde{\lambda}(\widetilde{m})\colon K_R\to M\otimes_RK$ is a well defined mapping and is $R$-linear, note first of all that
if $s, t \in S$ are such that $U(t) \subseteq U(s)$ and $r \in R$, then $m_s - m_{t} \in U (s)$ implies that
$m_s \otimes rs^{-1}+ R = m_{t} \otimes rs^{-1}+ R$ by Lemma~\ref{fourparts}(c). From this, it is easily shown that $\widetilde{\lambda}$ is a well defined $R$-module morphism. Also notice that $\lambda=\widetilde{\lambda}\eta$.
Now we prove that $\widetilde{\lambda}$ is a monomorphism.
Suppose that $\widetilde{m}=(m_{s}+U(s))_{s\in S}$ is in $\ker{\widetilde{\lambda}}$. Then, for any $k\in K$ and any $s\in S$ with $ks=0$, we have that $m_{s}\otimes k=0$ in $M\otimes_RK$. In particular, for every $r\in R$, $s\in S$, the identity $(rs^{-1}+R)s=0$ implies that $m_{s}\otimes (rs^{-1}+R)$ in $M\otimes_RK$. By Lemma~\ref{fourparts}(c), this means that $m_{s}r\in Ms$ for every $r$ and $s$. Hence $m_{s}\in U(s)$ for every $s\in S$. This shows that
$\widetilde{\lambda}$ is injective.
\end{proof}
}
\begin{example} \label {quasismall}{\rm
Let $R$ be the nearly simple chain domain in \cite [Example 6.5] {chainringandprimeideal}.
In that example, the $R$-module $Q/R$ can be chosen to be countably generated, because the
group $G$ is countable, and so is its positive cone $P$.
If the skew field $K$ in that example is countable, then $K[P]$
is countable. In order to construct the ring $R$, the authors consider a right and left Ore subset $S$ of
$K[P]$, which is necessarily countable because $K[P]$ is countable, and
then they set $R := K[P] S^{-1}$.
Therefore if the skew field $K$ is countable, then
$R$ is countable, and so $Q/R$ is a countably generated $R$-module.
As $R_R$ is torsion-free, its completion is $\displaystyle\lim_{\longleftarrow} R/U(s)$ by
Remark~\ref{remarktorsionfree}, and, for every non-zero element $s$ of $J(R)$,
$U(s) = 0$ because $R$ is nearly simple. So $ \displaystyle R = \lim_{\longleftarrow} R/U(s)$. Let us prove that $R \ncong \operatorname{End} (K_R) $.
The module $K_R$ is a countably generated uniserial torsion locally coherent module (that is, every finitely generated submodule is coherent). By \cite[Proposition~8.1]{puninsky}, the module $K_R$ is not quasi-small. Since uniserial modules with a local endomorphism ring are quasi-small \cite{DungFacchini}, the ring $\operatorname{End}(K_R)$ cannot be isomorphic to $R$.
The same argument applies to any nearly simple chain domain $R$ with $Q/R$ countably generated.}
\end{example}
\begin{proposition}
If $R$ is a topological ring with a basis $B$ of neighborhoods of zero consisting of two-sided ideals, and
$R/I$ is a local ring for every proper ideal $I\in B$, then the Hausdorff completion of $R$ is either $0$ or
a local ring.
\end{proposition}
\begin{remark}{\rm
The case of completion of $R$ equal to zero concernes only the trivial case of $B=\{R\}$. We will not consider this case in the proof.}
\end{remark}
\begin{proof}
Let $M_I $ be the maximal ideal of $R$ such that $M_I/I$
is the maximal ideal of $R/I$ for every proper ideal $I\in B$. If $I,J\in B$, then considering the canonical projection
$ R/I\cap J\to R/I$, one sees that
$M_{(I\cap J)}=M_I$. It follows that there exists a maximal ideal $M$ of
$ R$ such that $M_I=M$ for every proper ideal $ I\in B$. The completion of $R$ is the inverse limit of the rings
$R/I$, which is a subring of the ring $\prod_{I\in B}R/I$, which has $\prod_{I\in B}M/I$ as a two-sided ideal, whose intersection $N$ with the inverse limit is a two-sided ideal of the inverse limit. Let us prove that the inverse limit is a local ring with maximal ideal $N$. It suffices to show that every element of the inverse limit not in $N$
is invertible. Let $(x_I+I)_{I\in B}$ be an element in the inverse limit, but not in $N$. Thus
$x_I\in R$ and, for $I,J\in R$ with $I\subseteq J$, we have that $ x_I-x_J\in J$, i.e., $x_I+I $
is mapped to $ x_J+J $ via the canonical projection $R/I\to R/J.$ Also, $ x_I\notin M$ for some proper ideal $I$ of
$B$. It follows that $x_I\notin M $ for every proper ideal $ I$ of $ B$. Thus $x_I+I\notin M/I,$
hence is invertible in $R/M$. Let $y_I+I$ be the inverse of $x_I+I$ in $R/I$. Now the ring morphism
$R/I\to R/J$
maps inverses to inverses. This shows that $(y_I+I)_{I\in B}$ is an element of the inverse limit, and concludes the proof.
\end{proof}
Therefore the completion of any local ring in the $R$-topology is a local ring.
\medskip
Note that, by Theorem \ref {completion} and \cite [Proposition 2.6]{submitted}, if
$M_R$ is torsion-free, then $\widetilde{M_R}$ is torsion-free.
\begin{theorem}\label{4.9} Let $R$ be a right Ore domain and $\widetilde{R_R}$ the completion of $R_R$ in the $R$-topology. Then $\widetilde{R_R}$ is a strongly flat right $R$-module.\end{theorem}
\begin{proof} We can apply the
results of \cite[Section~3]{submitted}, which are right/left symmetric, that is, hold for both right $R$-
modules and left $R$-modules. Notice that $R_R$ is $h$-reduced. We have the short exact sequence\begin{equation}
\xymatrix{
0 \ar[r] & R_R \ar[r] & \operatorname{End}(K_R) \ar[r] & \operatorname{Ext}^1_R({}_RQ_R,R_R) \ar[r] & 0.
}\end{equation} We know
that $\widetilde{R_R}$ is a submodule of $\operatorname{End}(K_R)$ that contains $R_R$. Hence $\widetilde{R_R}/R_R$ is
isomorphic to a submodule of $\operatorname{Ext}^1_R({}_RQ_R,R_R) $. In particular,
$\widetilde{R_R}/R_R$ is torsion-free, because $\operatorname{Ext}^1_R({}_RQ_R,R_R) $ is a $Q$-module, hence torsion-free. Let us
prove that $\widetilde{R_R}/R_R$ is divisible, i.e., that $(\widetilde{R_R}/R_R)r=\widetilde{R_R}/R_R$ for every non-zero $r\in R$. Equivalently, we must prove that $\widetilde{R_R}\subseteq \widetilde{R_R}r+R_R$. Now $R_R$ is
dense in $\widetilde{R_R}$,
so that, for every $\widetilde{r}\in\widetilde{R_R}$ and every
non-zero element $s$ of $R$, we have that $(\widetilde {r} + U(s)) \cap R_R \neq \emptyset. $ In particular,
$(\widetilde{r}+U(r))\cap R_R\ne\emptyset. $
Notice that $U(r)\subseteq \widetilde{R_R}r$, because, for every $x\in U(r)$, we have that $xR\subseteq \widetilde{R_R}r$, hence $x\in \widetilde{R_R}r$. It follows that $(\widetilde{r}+\widetilde{R_R}r)\cap R_R\ne\emptyset. $ Thus there exists $\widetilde{r'}\in \widetilde{R_R}$ and $r''\in R_R$ with $
\widetilde{r}+\widetilde{r'}r=r''$. Therefore $\widetilde{r}=-\widetilde{r'}r+r''\in\widetilde{R_R}r+R_R$. This proves that $\widetilde{R_R}/R_R$ is divisible and torsion-free, hence a module over the division ring $Q$. Thus $\widetilde{R_R}/R_R\cong Q^{(X)}$ for some set $X$. The short exact sequence \begin{equation*}
\xymatrix{
0 \ar[r] & R_R \ar[r] & \widetilde{R_R}\ar[r] & Q^{(X)} \ar[r] & 0.
}\end{equation*} shows that $\widetilde{R_R}$ is strongly flat.\end{proof}
\section{Strongly flat modules}
{\em In all this section, we consider two rings $R$ and $Q$, a bimorphism $\varphi \colon R\to Q$ in the category of rings, that is, $\varphi$ is both a monomorphism and an epimorphism, and we assume that ${}_RQ$ is a flat left $R$-module. For simplicity, we will view $R$ as a subring of $Q$ and $\varphi \colon R\to Q$ as the inclusion.}
Let us recall some properties of such an inclusion $\varphi \colon R\hookrightarrow Q$. It is always possible to suppose $Q \subseteq Q_{\rm max}(R) $, the maximal ring of quotients of $R$ \cite[proof of Theorem XI.4.1]{20}. The
inclusion $\varphi \colon R\to Q$ is an epimorphism in the category of rings if and only if the canonical $R$-$R$-bimodule morphism $Q \otimes _R Q \to Q$ induced by the multiplication $\cdot\colon Q\times Q\to Q$ of the ring $Q$ is an $R$-$R$-bimodule isomorphism \cite[Proposition~XI.1.2]{20}.
The family of all the subrings $Q$ of $Q_{\rm max}(R) $ with $\varphi \colon R\hookrightarrow Q$ a bimorphism and ${}_RQ$ flat is directed under inclusion \cite[Lemma XI.4.2]{20}. Its direct limit is the ``maximal flat epimorphic right ring of quotients''
$Q_{\rm tot}(R)$ of $R$ (see the paragraph after the proof of Corollary~\ref{12}).
By \cite [Theorem 4.8] {rep},
$\operatorname{Ext}^1(_RM,{} _RN) \cong \operatorname{Ext}^1(_QM,{} _QN) $ for any pair $M,N$ of left $Q$-modules, and similarly for right $Q$-modules.
\begin{lemma}\label{Divisiblestronglyflat}
Divisible strongly flat left $R$-modules are
projective $Q$-modules.
\end{lemma}
\begin{proof}
Assume that $_RD$ is a divisible strongly flat module. Since $K \otimes Q = 0$, we have $K \otimes D = 0$. Since $_RD$ is flat, we have
$D \cong Q \otimes D$. For any exact sequence
$0 \to R^{(X)} \to D \oplus T \to Q^{(Y)} \to 0 $, the corresponding exact sequence $0 \to Q^{(X)} \to D \oplus Q\otimes T \to Q^{(Y)} \to 0 $ splits. Therefore
$D$ is a projective $Q$-module.
\end{proof}
Recall that any left perfect ring is directly finite. The following result shows that
when $_R \mathcal{SF}$ is covering, then $Q$ is left perfect. Thus the results is the same as in
the commutative case, but the proof is necessarily different.
\begin{theorem} \label{Qisperfect}
If all left $Q$-modules have a strongly flat cover as left $R$-modules,
then
$Q$ is left perfect.
\end{theorem}
\begin{proof}
Assume that $_QM$ is a left $Q$-module and
$f\colon{}_RS \to{} _RM$ is a strongly flat cover of $_RM$. Then we have
an epimorphism $1\otimes f \colon Q\otimes S \to M$, $1\otimes f\colon
q\otimes s\mapsto qf(s)$.
Since $_RS$ is strongly flat, $_QQ\otimes_R S$ is a direct
summand of a direct sum of copies of $Q$, i.e.,
it is a projective left $Q$-module. Since projective left $Q$-modules
are strongly flat left $R$-modules, the left $R$-module $_RQ\otimes S$ is
strongly flat.
But $f$ is a strongly flat precover of $M$, so that there exists $g \colon Q\otimes
S \to S$ with
$f g = 1\otimes f$.
Note that $_RS$ is flat, and so $S$ can be embedded in $Q\otimes S$, that
is, there is a left $R$-module monomorphism $h\colon _RS\to _RQ\otimes_R
S$, defined by $h\colon s \mapsto 1\otimes s$.
Then $f (gh) = f $, and thus $gh $ is an automorphism of $_RS$ because
$f\colon _RS \to _RM$ is a cover. Thus $(gh)^{-1}gh=1$, so that
$e:=h(gh)^{-1}g$ is an idempotent endomorphism of the left $R$-module
$_RQ\otimes S$. Hence $e$ is an
idempotent endomorphism of the left $Q$-module $_QQ\otimes S$. This shows that
$_QQ\otimes S$ is the direct sum of the image and the kernel of $e$,
which are $Q$-modules. But the image of $e$ is the image of $h$. Hence
the splitting monomorphism $h\colon s \mapsto 1\otimes s$ induces by
corestriction a right $R$-module isomorphism of $_RS$ onto the
$Q$-module $_Qh(S)$. By \cite[Section~2(7)]{submitted},
if a left $R$-module $_RA$ is a left $Q$-module $_QA$, then its
unique left $Q$-module structure is given by the canonical isomorphism
$\operatorname{Hom}(_RQ,_RA) \to{} _RA$. Therefore $S$ has a unique left $Q$-module
structure, which extends its left $R$-module structure, and as such
$_QS$ is a projective $Q$-module. Thus $f\colon{} _QS \to {}_QM$ is a left
$Q$-module morphism.
Note that projective $Q$-modules are strongly flat, and so $f\colon{} _QS \to{}
_QM$ is a projective cover of $_QM$. Therefore
$Q$ is left perfect.
\end{proof}
The following result has a proof similar to that of \cite[Proposition 2.4 ((1) and (2))]{silsal}.
\begin{lemma}\label{B-small}
Let $A$ be a module with a strongly flat cover and let
\begin{equation} 0 \to C \to M \to A \to 0 \label{(1)}\end{equation}
be a special strongly flat precover of $A$. Then the exact sequence~{\rm (\ref{(1)})} is a strongly flat cover
if and only if $C$ is $\mathcal{MC}$-small (i.e., $C + H = M$ and $C\cap H $ Matlis-cotorsion imply
$H = M$).
\end{lemma}
\begin{theorem} \label{twosidedidealIQ}
Let
$I$ be a two-sided ideal of $R$ such that $IQ = Q$.
If all left $R/I$-modules have a strongly flat cover as left
$R$-modules, then
$R/I$ is left perfect.
\end{theorem}
\begin{proof}
It is enough to show that every left $R/I$-module has a projective
$R/I$-cover.
Let $M$ be an $R/I$-module and $f\colon{} _RA \to {}_RM$ be a strongly flat
cover of $_RM$.
Since $IM = 0$, we have that $IA \subseteq \ker (f)$.
Since ${}_RA$ is strongly flat, there exists an exact sequence
$ 0 \to R^{(X)} \to A \oplus T \to Q^{(Y)} \to 0 $, where $X$ and $Y$ are sets.
Since $IQ = Q$, we have $R/I \otimes Q = 0$. Thus we see that $A/IA$ is a projective left $R/I$-module.
So $f$ induces a map $h\colon A/IA \to M$, $h\colon a+ IA \mapsto f(a)$, and $\ker(h) = \ker(f)/IA$.
Now
$_RA$ is strongly flat and $IQ = Q$, and so $A/IA$ is a projective
left $R/I$-module.
We now show that $h$ is a cover for $M$ or, equivalently, that
$\ker(f)/IA$ is small in $A/IA$.
Assume that $T + \ker(f) = A$, where $T$ is an $R$-module of $A$ that
$IA \subseteq T $. Since $IQ = Q$, Hom$(Q, \ker(f)/ \ker(f) \cap T) = 0$. On the other hand,
since $f\colon{} _RA \to {}_RM$ is a strongly flat
cover of $_RM$, the module $\ker(f)$ is Matlis-cotorsion by Wakamatsu Lemma (see \cite [Lemma 5.13] {approx}), and thus
$ \ker(f) \cap T$ is Matlis-cotorsion. Therefore $T = A$ by Lemma \ref{B-small}.
\end{proof}
\begin{lemma} \label{3.3}
Assume that $R$ is a local ring with Jacobson radical $J$. Let
$0 \to C \to S \to M \to 0$ be an $\mathcal{SF}$-cover for $M$. Then
$C \leq JS$.
\end{lemma}
\begin{proof} Assume that $C \nleq JS$. Then $JS \neq S$. Since $R/J$ is a division ring, there exists a proper submodule $T/JS$ of $S/JS$ such that
$T/JS + (C + JS)/JS = S/JS$. Consequently $T + C = S$.
Consider the exact sequence
$0 \to T \cap C \to C \to S/ T \to 0$.
Let us show that $\operatorname{Hom}(Q, S/T) = 0$.
Note that $R_R$ is essential in $Q_R$ (because $Q$ is a subring of $Q_{\rm max}(R)$).
Thus if $x \in Q \setminus R$,
then the right ideal of $I = \{\,r \mid xr \in R\,\}$
is proper ideal of $R$, and so $I \leq J$.
By \cite [Part (b) of Theorem 3.9]{Goodearlnonsingular},
$IQ = Q$ and so $JQ = Q$.
If Hom$(Q, S/T) \neq 0$, then there exists a proper submodule
$E$ of $Q$ such that $Q/E$ is isomorphic to a submodule of $S/T$.
Thus $ Q = JQ \leq E$, which is a contradiction. Therefore
Hom$(Q, S/T) = 0$, and so $T\cap C \in Q^\bot$.
Since $C$ is $\mathcal{MC}$-small, we have $T = S$, which is a contradiction.
\end{proof}
It is known that if $R$ is commutative, $Q$ is the field of fractions of $R$ and
$_R \mathcal{SF}$ is covering, then $\pdim(_RQ) \leq 1$. We do not know what occurs in the
non-commutative case. Therefore we now study the projective dimension of~${}_RQ$.
\begin{proposition}\label{3.4'} Suppose ${}_RQ$ is a projective left $R$-module. Then ${}_RQ$ is a finitely generated left $R$-module. \end{proposition}
\begin{proof} Since ${}_RQ$ is projective, it has a dual basis \cite[Exercise 11, pp.~202-203]{andersonfuller}, that is, there are elements $x_\alpha\in Q$ and morphisms $f_\alpha\colon {}_RQ\to {}_RR$ ($\alpha\in A)$, such that, for all $x\in Q$, $f_\alpha(x)\ne0$ for only finitely many $\alpha\in A$ and $x=\sum_{\alpha\in A}f_\alpha(x)x_\alpha$. Applying the functor ${}_QQ\otimes_R-\colon R\mbox{\rm -Mod}\to Q\mbox{\rm -Mod}$, we get left $Q$-module morphisms $1\otimes f_\alpha\colon {}_QQ\otimes_RQ\to {}_QQ\otimes_RR$. Now there are left $Q$-module isomorphisms $Q\to {}_QQ\otimes_RQ$, $q\mapsto 1\otimes q$, and ${}_QQ\otimes_RR\to {}_QQ$, $q\otimes r\mapsto qr$. Composing, we get left $Q$-module endomorphisms ${}_QQ\to {}_QQ$, which are necessarily right multiplications by elements $y_\alpha\in Q$. Now, for all $x\in Q$, $f_\alpha(x)\ne0$ for only finitely many $\alpha\in A$. For $x=1$, we get that there is a finite subset $F$ of $A$ such that $f_\alpha(1)=0$ for every $\alpha\in A\setminus F$. Thus $(1\otimes f_\alpha)(1\otimes 1)=0$ for every $\alpha\in A\setminus F$. It follows that right multiplication by $y_\alpha$ maps $1$ to $0$, that is, $y_\alpha=0$ for every $\alpha\in A\setminus F$. It follows that $1\otimes f_\alpha\colon {}_QQ\otimes_RQ\to {}_QQ\otimes_RR$ is the zero mapping for every $\alpha\in A\setminus F$. Thus $(1\otimes f_\alpha)(q\otimes q')$ is the zero element of ${}_QQ\otimes_RR$ for every $q,q'\in Q$. Hence $1\otimes f_\alpha(q')$ is the zero element of ${}_QQ\otimes_RR$. It remains to show that the mapping $_RR\to {}_QQ\otimes_RR$, $r\to 1\otimes r$, is injective, which is easily seen because $\operatorname{Tor}_1^R(K,R)=0$. This proves that $f_\alpha=0$ for every $\alpha\in A\setminus F$. As a consequence, $_RQ$ is isomorphic to a direct summand of ${}_RR^F$, so that ${}_RQ$ is a finitely generated left $R$-module. \end{proof}
\begin{corollary}\label{12} Let $R$ be a ring, $S$ a multiplicatively closed subset of regular elements of $R$, and suppose that $S$ is a right denominator set, so that the right ring of fractions $Q:=R[S^{-1}]$ exists. If ${}_RQ$ is a projective left $R$-module, then $Q=R$, that is, all the elements of $S$ are invertible in $R$. \end{corollary}
\begin{proof} By Proposition \ref{3.4'}, there are finitely many elements $r_1s_1^{-1},\dots,r_ns_n^{-1}$ that generate $Q$ as a left $R$-module. Reducing to the same denominator \cite[Lemma~4.21]{goodwar}, we find elements $r'_i\in R$ and $s\in S$ such that $s_ir'_i=s$ for every~$i$. Multiplying by $s^{-1}$ on the right and by $s_i^{-1}$ on the left, we get that $r'_is^{-1}=s_i^{-1}$. Thus $Q=\sum_{i=1}^n Rr_1s_1^{-1}\subseteq Rs_1^{-1}$. This proves that $Q=Rs_1^{-1}$. In particular, $s^{-2}\in Rs_1^{-1}$, from which $1\in Rs$. Let $t\in R$ be such that $1=ts$. Then $t=s^{-1}$ in~$Q$. Thus $Q=Rs_1^{-1}=Rt\subseteq R$, hence $Q=R$.\end{proof}
On page 235 of \cite{20}, Stenstr\"om asks for necessary and sufficient conditions
for $Q_{\rm max}(R) $ to be equal to $Q_{\rm tot}(R)$. He shows that if $Q_{\rm max}(R)$ is a
right Kasch ring (i.e., a ring that contains a copy of its simple right modules), then $Q_{\rm max}(R) = Q_{\rm tot}(R)$.
If $R$ is right hereditary right noetherian \cite[Example 3, p.~235]{20} or
commutative noetherian
\cite[Example 4, p.~237]{20} or a right Goldie ring \cite[Theorem XII 2.5] {20}, then
$Q_{\rm max}(R)$ is known to be Kasch.
\begin{example}
{\rm Here is an example of a ring $R$ for which $Q_{\rm tot}(R) = Q_{\rm max}(R)$ is a projective right and left $R$-module, but $R \neq Q_{\rm max}(R)$. Let $R$ be the ring of all lower triangular $2 \times 2 $ matrices over a field $F$.
The ring $R$ is right nonsingular and $E(R_R)=S^{0}R=Q_{\rm max}(R)$ is a projective right and left $R$-module \cite [Exercise 14 on Page 78, and Corollary 2.31] {Goodearlnonsingular} (in Goodearl's notation, $S^{0}A := E(A/ Z(A_A))$, the injective envelope of $A/ Z(A_A)$ for any ring $A$).
More precisely,
$Q_{\rm max} (R) $ is the $2 \times 2 $ matrix ring over the field $F$, which is a semisimple artinian ring, hence a right and left Kasch ring, and
so $Q_{\rm max}(R) = Q_{\rm tot}(R)$ as we have seen above.}
\end{example}
We are now ready to consider the case of $\pdim(_RQ)\le 1$. Recall that a cotorsion pair $(\Cal A,\Cal B)$ is said to be {\em hereditary} if $\operatorname{Ext}_R^i(A,B)=0$ for all $i\ge 1$, $A\in\Cal A$ and $B\in\Cal B$.
Note that if $\Cal F$ is the class of flat modules and $\Cal E \Cal C$ the class
of Enochs cotorstion modules, the the cotorsion pair
$(\Cal F,\Cal E \Cal C)$ is always a hereditary cotorsion pair.
Similarly to \cite[Lemma 7.53] {approx}, we can show that:
\begin{lemma}
The following conditions are equivalent for the pair of rings $R\subseteq Q$:
{\rm (a)} $\pdim(_RQ) \leq 1$.
{\rm (b)} The cotorsion pair $(\mathcal{SF}, \mathcal{MC})$ is hereditary.
\end{lemma}
\begin{proof}
(a)${} \Rightarrow{}$(b). Assume that $\pdim(_RQ) \leq 1$. Then strongly flat modules, which are summands of
extensions of a direct sum of copies of $Q$ by a free module, are of $\pdim$ at most $1$. Thus
the cotorsion pair $(\mathcal{SF}, \mathcal{MC})$ is hereditary.
(b)${} \Rightarrow{}$ (a). By \cite [Theorem 3.5]{AS}, it is enough to show that that $\operatorname{Ext}^1(K, M)$ is $h$-reduced Matlis-cotorsion.
Using the exact sequence $0 \to M \to E(M) \to E(M)/M \to 0$, we have the exact sequence
$0 \to A \to B \to \operatorname{Ext}^1(K, M) \to 0$, where
$A = $ Hom$(K, E(M))/$ Hom $(K, M)$ and $B = $ Hom $(K, E(M)/M)$.
Note that, for every module $N$, Hom$(K, N)$ is Matlis-cotorsion and $h$-reduced by \cite [Theorem 2.8] {submitted}.
So $\operatorname{Ext}^1(K, M)$ is $h$-reduced if and only if $A \in Q^\bot$. Now $A \in Q^\bot$
follows from the fact that $(\mathcal{SF}, \mathcal{MC})$ is hereditary and the exact sequence
$0 \to $ Hom $(K, M) \to $ Hom$(K, E(M)) \to A \to 0$. As the module
$A$ is Matlis-cotorsion, from the exact sequence $0 \to A \to B \to \operatorname{Ext}^1(K, M) \to 0$ and the fact that
$(\mathcal{SF}, \mathcal{MC})$ is hereditary, we get that $\operatorname{Ext}^1(K, M)$ is Matlis-cotorsion.
\end{proof}
As a consequence, $_R\mathcal{SF} ={} _R\mathcal{F}$ implies $\pdim(_RQ) \leq 1$.
\begin{lemma}\label{stronglyflatreduced}
Let $R$ be a right Ore domain and $Q $ the classical right quotient ring of $R$. If $S$ is a strongly flat left $R$-module, then $S/h(S)$ is also strongly flat.
\end{lemma}
\begin{proof}
Assume that $R$ is not a division ring. There exists an exact
sequence $0 \to R^{(X)} \to S \oplus C \to Q^{(Y)} \to 0$.
We claim that Hom$(Q, R) = 0$. Otherwise, i.e., if $_RQ$ can be embeded in $_RR$, there exists a monomorphism $\varepsilon\colon _RQ\to _RR$. Then
$\varepsilon$ can be viewed as a monomorphism $_RQ\to{} _RQ$. This monomorphism $\varepsilon$ is right multiplication by an element $q$ of $Q$. Now $\varepsilon$ a monomorphism implies $q\ne 0$, and $R$ right Ore domain implies $Q$ division ring.
Hence $q$ is invertible in
$Q$, so that $R=Q$, which is a contradiction. This proves our claim. Now
we have the embedding Hom$(Q, S \oplus C) \to $ Hom $(Q, Q^{(Y)})$.
So we have an exact sequence $0 \to R^{(X)} \to (S\oplus C)/h(S\oplus C) \to Q^{(Y)}/ h(S\oplus C)\to 0$.
Since $h(S\oplus C)$ is a torsion-free divisible module, it is a $Q$-module. But $Q$ is division ring, so
$h(S\oplus C)$ is a direct summand of $Q^{(Y)}$. It follows that $S /h(S)$ is strongly flat.
\end{proof}
A {\em left coherent} ring is a ring over which every finitely generated left ideal is finitely presented or, equivalently,
intersection of two finitely generated left ideals is finitely generated.
\begin{theorem}\label{ideal}
Assume that $R$ is a left coherent Ore domain with classical right quotient $Q$. A left ideal ${}_RI$ of $R$ is a strongly flat left module if and only if $_RI$ is finitely generated projective.
\end{theorem}
\begin{proof}
Assume $_RI$ a non-zero strongly flat.
We have the exact sequence of $R$-$R$-bimodules $0 \to R \to Q \to Q/R = K \to 0$.
Since $_RI$ is flat, we get the exact sequence of left $R$-modules
$0 \to R \otimes I \to Q \otimes I \to K \otimes I \to 0$.
Therefore $ K \otimes I \cong (Q \otimes I) / (R \otimes I)$.
We want to show $R/I$ embeds in $K \otimes I$ as $R$-modules.
Consider the sequence of left $R$-modules
$0 \to{}_RI\to{}_RQ\to{}_RQ/I\to 0$ and apply to it the functor $Q\otimes_R-$. Since $Q_R$ is flat, we get to
an exact sequence $0 \to Q\otimes_R I \to Q\otimes_R Q \to Q\otimes_R Q/I \to 0$.
Under the natural isomorphism $f\colon Q \otimes _R Q \to Q$, the image of $Q \otimes I $ is $QI = Q$, because $I$ is non-zero,
and the image of
$R\otimes I$ is $I$, and so $ K \otimes I \cong (Q \otimes I)/( R\otimes I) \cong Q/I$ as a left $R$-module. Now $R/I \leq Q/I$ implies that $R/I$ embeds in $K \otimes I$ as $R$-module.
There exists an exact
sequence $0 \to R^{(X)} \to I \oplus T \to Q^{(Y)} \to 0$.
Since $K \otimes _R Q = 0$,
we conclude that $K \otimes I$, and so $R/I$, embed in $K^{(X)}$ as left $R$-modules.
Consequently, there exists an element $x \in{} _R K^{(X)}$
whose annihilator is equal to $I$.
But the annihilator of an element of $ K^{(X)}$
is equal to the intersection of finitely many annihilators of
elements of $K$.
If $ab^{-1} + R \in {}_RK$, then ann$(ab^{-1} + R) = R \cap Rba^{-1}$.
Note that $ R \cap Rba^{-1} \cong Ra \cap Rb$, which is a finitely generated left ideal of $R$ because $R$ is left coherent. Thus $I$ is a finitely generated left ideal of $R$ and, since it is flat, $I$ is projective \cite [Theorem 4.30] {lam2}.
\end{proof}
\begin{lemma}\label{Sflat}
Let $R$ be a right Ore ring with classical right quotient ring $Q$. Then the strongly flat cover of any $h$-reduced flat left $R$-module is $h$-reduced.
\end{lemma}
\begin{proof}
Assume that $ M $ is a flat $h$-reduced module and $0 \to C \to S \to M \to 0$ is a strongly flat cover of
$M$. Since $M$ is $h$-reduced, we can assume that $D:= h(C) = h(S)$.
So we have an exact sequence $ 0 \to C/D \to S/D \to M \to 0$. By Lemma \ref{stronglyflatreduced}, $S/D$ is strongly flat.
We can easily see that this sequence is a strongly flat precover for $M$.
Note that $C$ is torsion-free, and so $D$ is a left $Q$-module. Thus $\operatorname{Ext}_R^1(Q, D) = 0$. Since
$C$ is $\mathcal{MC}$-small in $S$, we see that $C/D$ is
$\mathcal{MC}$-small. It follows that
$0 \to C/D \to S/D \to M \to 0$ is a strongly flat cover of $M$ by Lemma~\ref {B-small}(2), and so $S \cong S/D$. Therefore $D = 0$.
\end{proof}
\begin{proposition} \label{final}
Assume that $R$ is an Ore local domain with classical quotient ring $Q$.
Suppose that $K \otimes _R S$ is
direct sum of copies of $K$ for every strongly flat module $_RS$.
If $_R \mathcal{SF}$ is a covering class, then $_R \mathcal{SF} ={} _R \mathcal{F}$.
\end{proposition}
\begin{proof}
Firstly, notice that left $Q$-modules are
injective as $R$-modules because $R$ is both a right and a left Ore domain. So, if $M$ is flat, then
$h(M)$ is
a direct summand of $M$, and therefore $M \cong h(M) \oplus M/h(M)$.
Clearly, $Q$-modules are strongly flat, and thus it is enough to show that any flat $h$-reduced module is strongly flat.
Let $M$ be an $h$-reduced flat left module and $0 \to C \to S \to M \to 0$ be a strongly flat cover of
$M$. By Lemma \ref{Sflat}, $S$ is also $h$-reduced, and thus $C$ is an $h$-reduced flat left $R$-module.
Assume that $C \neq 0$, and let $0 \to C' \to S' \to C \to 0 $ be a strongly flat cover of $C$.
Then $S'$ is Matlis-cotorsion $h$-reduced strongly flat.
Note that by the left version of \cite [Theorem 4.6] {submitted},
we have an exact sequence
$0 \to S' \to \operatorname{Hom}(K, K \otimes S') \to \operatorname{Ext}^1 (Q, S') \to 0$.
Thus $S' \cong \operatorname{Hom}(K, K \otimes S')$.
Since $S'$ is strongly flat, it is a direct summand of a direct sum of copies of $K$,
and thus
$K \otimes S'$ is isomorphic to a direct summand of a direct sum of copies of $K$, $K \otimes S' \cong K^{(Z)}$ say.
Thus $ S' \cong \operatorname{Hom}(K, K \otimes S') \cong \operatorname{Hom}(K, K^{(Z)}) \cong \operatorname{Hom} (K, K \otimes R^{(Z)})$.
By \cite [Theorem 4.6] {submitted}, we have an exact sequence
$0 \to R^{(Z)} \to \operatorname{Hom}(K, K \otimes R^{(Z)}) \to \operatorname{Ext}^1 (Q, R^{(Z)}) \to 0$.
Since $\operatorname{Ext}^1 (Q, R^{(Z)})$ is a left $Q$-module,
$R/J \otimes \operatorname{Ext}^1 (Q, R^{(Z)}) = 0$, where $J$ denotes the Jacobson radical of $R$.
Consequently, $JS' \neq S'$.
On the other hand,
by Lemma~\ref{3.3} and considering the pure exact sequence
$0 \to C \to S \to M \to 0$, we see that $JC = C$. By
Lemma \ref{3.3} and considering the exact sequence
$0 \to C' \to S' \to C \to 0 $ again, we see that $J S' = S'$, which is a contradiction. This proves that $C = 0$, so that
$M$ is strongly flat.
\end{proof}
For any left module $_RM$, let $\Add(_RM)$ denote the class of all left $R$-modules isomorphic to direct summands of direct sums of copies of $_RM$.
We will say that $\Add(_RM)$ is {\em trivial} if every direct summand of a direct sum of copies of $_RM$ is a direct sum of copies of $_RM$.
\begin{lemma}\label{Pavel}
Let $R$ be a nearly simple chain domain and let $_RK$ be the uniserial left $R$-module $Q/R$. Suppose $\Add(_RK)$ not trivial. Then
there exists a submodule $V$ of $_RK$ that is not quasismall. Moreover, all the elements of $\Add(_RK)$ are isomorphic to $R$-modules of the form $_RK^{(X)} \oplus _RV^{(Y)}$.
\end{lemma}
\begin{proof}
See \cite[Theorem 1.1(ii)] {Pavel}.
\end{proof}
In the next proposition, we describe uniserial strongly flat modules over Ore domains.
\begin{proposition}
If $R$ is
an Ore domain with classical quotient ring $Q$, then every non-zero uniserial strongly flat left module over $R$ is isomorphic to $_RQ$ or~$_RR$.
\end{proposition}
\begin{proof}
Let $_RU$ be a non-zero uniserial strongly flat left module over an Ore domain $R$.
Since $_RU$ is flat, considering the exact sequence $0 \to R \to Q$, we have an embedding $U\to Q\otimes_RU$.
Hence the annihilator of every non-zero element of $ _RU$ is zero, and so cyclic submodules of $U$ are isomorphic to $_RR$. In particular, the ring $R$ is a left chain ring.
Moreover, $U$ is the union of cyclic submodules isomorphic to
$ _RR$, that is, a direct (linearly ordered) system of copies of $_RR$, where the connecting homomorphisms are right
multiplications by non-zero elements of $R$. Applying the functor $ _RQ\otimes_R-$, since tensor product commutes with direct limits, we get that $_RQ\otimes_RU$ is a direct limit of a direct system copies of $_RQ$, in which the
connecting isomorphisms are right multiplications by non-zero elements of $ R$, that is, the connecting isomorphisms are all left $R$-module automorphisms of $_RQ$. That is, $_RQ\otimes_RU \cong _RQ$.
Hence $_RU$ embeds into $_RQ\otimes_RU\cong{}_RQ.$ If this embedding is onto, then $_RU\cong {}_RQ$. If the embedding is not onto, then $_RU$ is isomorphic to a proper submodule of $_RQ$, hence to a left ideal of $R$.
By Theorem~\ref{ideal}, $_RU$ is cyclic, and so isomorphic to $_RR$.
\end{proof}
\begin{lemma}\label{JV}
Let $R$ be a nearly simple chain domain with Jacobson radical $J$. If $\Add(K)$ is not trivial,
$V$ is as in Lemma {\rm \ref{Pavel}} and $M := \operatorname{Hom} (K, V^{(X)}),$ where $X$ is a non-empty set,
then $JM \neq M$.
That is, $M$ has maximal submodule.
\end{lemma}
\begin{proof} The module $M=\operatorname{Hom}(_RK,{}_RV)$
is a left $R$-module because $_RK_R$ is a bimodule.
Notice that $_RM$ always has a direct summand isomorphic to
$\operatorname{Hom}(K, V)$, so that we can suppose that $X$ has exactly one element.
By \cite [(ii) of Theorem 1.1] {Pavel}, $K$
has an endomorphism whose image is contained in $V$, say $\varphi\colon {}_RK\to{}_RV $, that is injective but
not surjective. Let us show that $\varphi$ is in $M$ but not in $JM$.
For every $j \in J$ and $\psi\in\operatorname{Hom}(_RK,{}_RV)$, the left $R$-module morphism $j\psi$
is not injective. In fact,
$ j\psi$ is right multiplication by $j$ viewed as a morphism $ _RK\to{} _RK $
composed with $\psi\colon{}_RK\to{}_RV. $
Thus the first morphism annihilates the element $j^{-1}+R,$
so that the kernel of $ j\psi$ is non-zero. (This proves that $ j\psi$ is not injective for $j\ne 0.$
But also when $j=0$, $j\psi$ is not injective.)
Now every element of $JM $ a finite sum of elements of the form $j\psi,$
i.e., of non-injective homomorphisms, hence is not injective because $_RK$ is
uniserial, hence uniform.
Therefore $\varphi\colon {}_RK\to{}_RV$ is not an element of $JM$.
\end{proof}
Recall that a two sided ideal $I$ of $R$ is {\em completely prime} if $xy \in I$ implies that $x \in I $ or $y \in I$ for every $x, y \in R$.
\begin{theorem}
If $R$ is a right chain domain with classical right quotient ring $Q$ such that
$_R \mathcal{SF}$ is a covering class, then
$R$ is invariant and
$_R \mathcal{SF} = {}_R \mathcal{F}$.
\end{theorem}
\begin{proof}
If $I$ is a non-zero completely prime two-sided ideal of $R$,
$R/I$ is a left perfect domain by Theorem~\ref{twosidedidealIQ}, and so it is is a division ring. Since $J(R)/I$ is an ideal of $R/I$, we conclude that
the only proper non-zero completely prime ideal of $R$ is $J(R)$.
A chain domain $R$ is said to be of rank one if $J(R)$ is its only non-zero completely prime ideal.
By \cite{chainringandprimeideal}, such a ring
is either invariant, i.e., $aR = Ra$
for all $a \in R$, or it
is nearly simple, in which case $ 0$ and $J(R)$ are the only two-sided ideals, or
$ R$
is exceptional
and there exists a non-zero prime ideal
$P$
properly contained in $J(R)$. In this last case, $\bigcap_n {P^n} = 0$ and
there are no further ideals between $P$ and $J(R)$.
In the second and the third case, $J(R)$ is not neither right nor left finitely generated and $J^2 = J$.
Now we break the proof in three steps.
\medskip
{\em Step 1: The ring $R$ cannot be exceptional. }
The Jacobson radical of $R/P$ is $J/P$, which cannot be nilpotent because $J^2 = J$.
Thus $R/P$ cannot have a
$T$-nilpotent Jacobson radical (see for example the proof of \cite [Lemma~3.33]{quasifereb}), and so $R/P$ is neither a right nor a left perfect ring, and so the class of strongly flat left modules is not covering by Theorem~\ref{twosidedidealIQ}.
\medskip
{\em Step 2: The ring $R$ cannot be nearly simple chain domain. }
Suppose $R$ a nearly simple chain domain.
For every strongly flat module $_RS$, $K \otimes S$ is direct summand of a direct sum of copies of $K$, so that $K \otimes S$ belongs to $\Add(K)$.
We have two cases:
$\Add(K)$ is trivial or not. If $\Add(K)$ is trivial, then $_R \mathcal{SF}$ covering implies $_R \mathcal{SF} = {}_R \mathcal{F}$ by Proposition \ref{final}. But every cyclic (=\,finitely generated) ideal of $R$ is flat (= projective), so $_RJ$ must be flat, hence strongly flat (see for example \cite [Theorem 39.12(2)] {wis}). Thus
$J$ must be finitely generated by Theorem~\ref{ideal}, which is a contradiction.
Now assume that $\Add(K)$ is not trivial.
By Lemma \ref{Pavel},
there exists a uniserial module $V$ which is not quasismall and every element in $\Add(K)$ is in form of $K^{(Y)} \oplus V^{(X)}$ for suitable sets $X$ and $Y$.
Let $0 \to C \to S \to J \to 0$ be a strongly flat cover of
$J$. By Lemma \ref{Sflat}, $S$ is also $h$-reduced, and so $C$ is an $h$-reduced flat left module.
Assume $C \neq 0$, and let $0 \to C' \to S' \to C \to 0 $ be a strongly flat cover of $C$.
Then $S'$ is Matlis-cotorsion $h$-reduced strongly flat.
By the left version of \cite [Theorem~4.6] {submitted},
we have an exact sequence
$0 \to S' \to \operatorname{Hom}(K, K \otimes S') \to \operatorname{Ext}^1 (Q, S') \to 0$.
So $S' \cong \operatorname{Hom}(K, K \otimes S')$.
Since $S'$ is strongly flat, $K \otimes S'$ is a
direct summand of a direct sum of copies of $K$.
Therefore there exist sets $X$ and $Y$ such that
$K \otimes S' \cong K^{(Y)} \oplus V^{(X)}$.
So $ S' \cong \operatorname{Hom}(K, K \otimes S') \cong \operatorname{Hom}(K, K^{(Y)}) \oplus \operatorname{Hom}(K, V^{(X)})$.
As we saw in the proof of Theorem \ref{final}, if $Y$ is non-empty, we can consider
the exact sequence $ 0 \to R^{Y} \to \operatorname{Hom}(K, K \otimes R^{(Y)}) \to \operatorname{Ext}^1_R ( Q, R^{(Y)}) \to 0 $,
and conclude that
$J \operatorname{Hom}(K, K^{(Y)}) \neq \operatorname{Hom}(K, K^{(Y)})$. Similarly,
by Lemma \ref {JV}, if $X$ is non-empty, $J \operatorname{Hom}(K, V^{(X)}) \neq \operatorname{Hom}(K, V^{(X)})$.
Consequently, $JS' \neq S'$.
By Lemma \ref{3.3}, considering the pure exact sequence
$0 \to C \to S \to J \to 0$, we see that $JC = C$. By Lemma \ref{3.3} again, from the exact sequence
$0 \to C' \to S' \to C \to 0 $, we get that $J S' = S'$, which is a contradiction. This proves that $C = 0$, so that
$J$ is strongly flat, which contradicts Theorem~\ref{ideal}.
\medskip
{\em Step 3: The ring $R$ is invariant and $_R \mathcal{SF} = {}_R \mathcal{F}$.}
By Steps 1 and 2, the ring $R$ must be invariant. Therefore the endomorphism ring of every uniserial module is local
(the proof is similar to the commutative case, because, like in the proof of \cite [Corollary 3] {Facsal},
every uniserial module is unshrinkable, and so the endomorphism ring of every uniserial module is local like in the proof of
\cite [Example 2.3(e)] {Facchinitransaction}). Thus
$\operatorname{End} (K _R)$ is local and every direct summand of copies of $K$ is isomorphic to a direct sum of copies of $K$ because $K_R$ is uniserial by \cite [Proposition 2.2] {DungAlberto}. Thus $_R \mathcal{SF} ={} _R \mathcal{F}$ by Proposition~\ref{final}. \end{proof}
We conclude with an example concerning right noetherian right chain domains. In a right noetherian right chain domain $R$, all right ideals are principal and two-sided \cite[Lemma~3.2]{bessen}. In particular, $J(R)=pR$ for some $p\in R$. The right noetherian right chain domain $R$ is said to be {\em of type $\omega$} \cite[p.~26 and Lemma 3.4]{bessen} if its chain of right ideals (=\,two-sided ideals) is the chain $$R=p^0R\supset J(R)=pR\supset p^2R\supset\dots\supset 0=\bigcap_{n\ge0}p^nR.$$ Thus for every non-zero right ideal $I$ of $R$, we have that $\operatorname{End} (R_R/I)\cong R/I$ is a right artinian ring, hence a perfect ring. In the next example, we show that this is also true for every non-zero principal left ideal $I$ of a right noetherian right chain domain $R$ of type $\omega$ which is not left Ore. Notice that in our example of right noetherian right chain domain of type $\omega$ which is not left Ore, the ring is not left chain (otherwise it would be left Ore) and is not left noetherian \cite[Proposition 3.7]{bessen}. The main example of such a ring can be constructed with the skew poynomial ring with coefficients in a field $F$, where $F$ has an endomorphism that is not an automorphism of $F$.
\begin{example}\label{5.18}
{\rm Let $R$ be a right noetherian right chain domain of type $\omega$ which is not left Ore.
For every non-zero principal left ideal $I$ of $R$, the endomorphism ring $\operatorname{End} (_RR/I)$ is a perfect ring. }
\end{example}
\begin{proof}
For every non-zero element $x\in R$, we have that $xR=p^nR$ for some $n\ge 0$.
Therefore $x=p^nu$ for some invertible element $u\in R$. Right multiplication by $u$
induces an isomorphism $R/Rp^n\to R/Rx$. Hence it suffices to show that
$\operatorname{End}(R/Rp^n)$ is right and left perfect for $n\ge 1$.
Notice that $ Rp^n\subseteq p^nR$. Set
$S:=\operatorname{End}(R/Rp^n)\cong E/Rp^n$, where $E: = \{\, r \in R \mid p^n r \in R p^n \,\}$ denotes the idealizer of $Rp^n$ in $R$, and set $K : = \{\, r \in R \mid p^n r \in J p^n \,\}$. By \cite [Theorem 2.1]{Aminis}, $S$ has at most two maximal ideals, the ideals $K/Rp^n$ and $(J \cap E)/ Rp^n $.
Let us show that $ K \subseteq J$.
Assume the contrary, so that $K$ contains a unit $u$ of $R$. Therefore
$p^n u = r p^n$ for some $r\in J$.
Then $ r = p^j v $ for some unit $v$ of $R$ and some $j \geq 1$.
Thus $p^n = p^j v p^n u^{-1}$.
If $j \geq n$, then $ 1 = p^{j -n } v p^n u^{-1}$, which implies $J(R)=R$, a contradiction.
If $j < n$, then $p^{n -j} = v p^n u^{-1}$. Thus $p^{n -j}$ belongs to the two-sided ideal $pR$ of $R$, which is a contradiction because $n-j<n$.
Therefore $ K \subseteq J$, so
$S$ is local with maximal ideal $(J \cap E)/ Rp^n $.
We claim that if $p^n y \in E$, then $p^ny \in Rp^n$.
To prove the claim, assume that $p^n y \in E$. Then there exists $s\in R$ such that $p^n p^n y = sp^n$. Similarly, there exists $i \geq 0$ and a unit $u$ in $R$ such that
$s = p^i u$. If $i \geq n$, then we are done, the claim is proved. Otherwise, if $i < n$, by supposing that $y = p^j v$ for some unit $v$, we get that
$ u^{-1} p^l v = p^n$, so $l > n$, which is a contradiction by \cite [Lemma 2.3] {Aminis}.
Therefore $(J\cap E)^n\subseteq J^n\cap E=p^nR\cap E\subseteq Rp^n$, to that the Jacobson radical $J(S)=(J\Cal E)/Rp^n$ of the local ring $S$ is nilpotent. It follows that $S=\operatorname{End}(R/Rp^n)$ is a right and left perfect ring.
\end{proof}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 1,084 |
HP-UX Host Intrusion Detection System (HIDS) is a host-based security software that enables administrators to proactively monitor, detect, and respond to both known and unknown attacks that originate from the network or on the host.
HP-UX HIDS includes a number of patterns called detection templates that guard host systems against exploitation of underlying system vulnerabilities. HIDS includes preconfigured detection templates that facilitate product deployment. This software identifies patterns that suggest security breaches or misuse by examining information about system activity from a variety of data sources. These activities can include a hacker attempting to break into or disrupt the system, subversive 'insider' activities, or someone trying to spread a virus.
Transport Layer Security (TLS) protocol version 1.2 for client and server authentication, integrity, and privacy.
For more information, see HP-UX Host Intrusion Detection System Version 4.9 Release Notes .
HP-UX HIDS version 4.8 is updated to provide the fix for POODLE vulnerability. For more information, see HP-UX Host Intrusion Detection System Version 4.8 Release Notes .
HP-UX HIDS version 4.7 supports Java 6.0 on HP-UX 11i v3. For more information, see HP-UX Host Intrusion Detection System Version 4.7 Release Notes.
This version supports the monitoring HP-UX Containers(HP-UX SRP) feature provided on HP-UX HIDS version 4.4. It also supports all the features and defect fixes provided on HP-UX HIDS version 4.3 and earlier.
HP-UX HIDS version 4.3 includes only defect fixes. See the HP-UX HIDS version 4.3 release notes for details.
A log file monitoring feature that enables administrators to receive alerts when log entries that match regular expression string patterns are detected in plain text log files (for example, syslog). Administrators specify the path names of the log files to monitor and the regular expression string patterns to monitor for each log file.
A critical file monitoring feature that enables administrators to receive alerts when there are failed attempts to create, delete, or modify critical files. Previous HIDS releases only monitor for successful attempts to modify files. With this feature enabled, both successful and failed attempts are detected. As with successful attempts, failed attempts to modify critical files can be indicative of an intrusion or of system misuse.
IPv6 support that allows HIDS to function in a pure IPv6 network as well as a mixed IPv6/IPv4 network.
Numeric user name support for specifying user name template property values.
An alert volume reduction feature that proactively suppresses duplicate alerts from being generated, logged, and reported to the HIDS administrator console. Using this feature, administrators can manage HIDS alerts easily by focusing their attention on fewer and more significant alerts.
A reporting feature that enables the generation of customized and consolidated alert reports that are easy to view and print. Reports can be generated in HTML, text, and raw formats.
Eliminating the time consuming and error prone process of manually generating filtering rules.
Facilitating the review of alerts from multiple agents running the same schedule by presenting an alert report that consolidates duplicate alerts and groups alerts triggered by the same program.
Performing automatic schedule updates and deployments.
Customize a predefined schedule to filter out alerts generated as part of normal system activity during the initial HIDS deployment.
Fine tune an existing schedule if new alerts that are deemed safe to ignore are generated after deployment.
The Surveillance Schedules and Surveillance Groups managed by the HIDS administrative GUI and CLUI are stored in text format only, allowing users to also edit schedules and groups using their preferred editor.
The Creation and Modification of Setuid file template now also monitors the creation and modification of privileged setgid files. | {
"redpajama_set_name": "RedPajamaC4"
} | 3,921 |
The Robert Allen Duralee Group, Inc.
Our Company is committed to taking all reasonable steps to protect and keep your personal information secure when you interact with our Web sites.
Qualified accounts of the Company are invited to access account information in order to request a memo/sample or to place an order. As part of this process, each qualified account has a unique password, which may be changed at any time. Various registration forms require users to give us contact information (including name, address and e-mail address). In addition, when purchasing products, we will collect financial information, including credit card information.
We also use your Internet Protocol ("IP") address to help diagnose any problems with our server and to monitor and administer our Web sites.
We welcome individuals who do not have accounts to visit our Web sites.
To access your account, you must enter your account number and a password. This helps to protect the privacy of your interactions with us. We will not share your password, which you may change at any time. We also request that you protect your password by not sharing it with others.
You may update or change your account information with us at any time. Please contact Customer Service at 888.233.8458.
Our company will never willfully sell, trade, rent, disclose or make available personally identifiable information about you to any third party without first receiving your permission, except when we believe in good faith that the law requires it, or to protect the rights or property of the Company. We do not disclose details of our mailing list or information about our site visitors to other companies.
Our Company utilizes "cookies," which are small pieces of information that are stored by your browser on your computer's hard drive. Cookies themselves do not contain any personally identifiable information other than noting a repeat visit from a specific computer. We use them to monitor traffic patterns and site usage, and to develop a more unique experience for you when you visit our Web sites. Our cookies do not have the capability of "stealing" information from your hard drive.
When you place an order, we collect information such as your name, e-mail address and terms of payment. This is used to process and fulfill your Product Orders and to notify you of your order status. All information submitted to us is transmitted over the Internet in an encrypted manner. We will not store your credit card information after your transaction is completed, and information from these transactions is not released to third parties.
You must be 18 years or older to use the Company's Web sites. If you are under 18 years of age, you must get permission from your parent or guardian before sending any information to any site. | {
"redpajama_set_name": "RedPajamaC4"
} | 3,038 |
Eckernförde (danska: Egernførde och Egernfjord, lågtyska: Eckernför och Eckernföör) är en stad i förbundslandet Schleswig-Holstein i Tyskland och tillhör distriktet (Kreis) Rendsburg-Eckernförde. Folkmängden uppgår till cirka invånare. Eckernförde ligger intill Eckernfördebukten (Eckernförder Bucht) vid Östersjön.
Historik
Namnet Ekerenvorde är dokumenterat sedan 1197 och år 1302 omnämns Eckernförde som stad. Bakgrunden till namnet är inte helt klarlagd. Förleden Eckern härrör möjligtvis från Bucheckern (tyska för bokollon) och efterleden Förde (Furt) betyder vadställe. En ekorre återfinns i stadens vapen.
Badort
Eckernförde är en känd Östersjöbadort med en cirka fyra kilometer lång sandstrand. Staden besöktes 2011 av cirka 40 000 övernattande gäster och antalet gästnätter var cirka 140 000.
Industrier
Företaget J.P. Sauer und Sohn GmbH har sitt säte i Eckernförde och är Tysklands äldsta existerande vapentillverkare.
Bilder
Källor
Kommuner och städer i Kreis Rendsburg-Eckernförde | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 7,798 |
\section{Introduction}\label{sec:intro}
Solar electron events are common phenomenon observed in interplanetary space.
They are often observed in the energy range of $<1$ keV to $>300$ keV with an occurrence rate near the earth of $\sim190$ events per year during solar maximum and $\sim10$ per year during solar minimum \citep{Wang2012}.
Energetic electrons are accelerated either in solar flares or in the vicinity of shocks driven by coronal mass ejections (CMEs).
The escaping accelerated electrons at energies of $2$ to $\gtrsim10$ keV \citep{Lin1985} excite type III radio bursts when propagating through the plasma of the solar corona and interplanetary space.
$98.75\%$ of the observed solar electron events in solar cycle $23$ are reported by \citet{Wang2012} to be accompanied by type III radio bursts.
\citet{Wang2006} examined three scatter-free impulsive electron events and found that $0.4$-$10$ keV electrons were injected $9.1\pm4.7$ minutes before the type III bursts, while electrons at energies of $13$-$300$ keV were injected $7.6\pm1.3$ minutes after the type III bursts.
The delayed injection of high-energy electrons after the type III radio burst has been reported by \citet{Krucker1999}, \citet{Haggerty2001}, \citet{Haggerty2002}, \citet{Haggerty2003}, \citet{Cane2003a}, \citet{Cane2003}, \citet{Simnett2002}, \citet{Maia2004}, and \citet{Klein2005}.
And the reason was suggested to be either high energy electrons are released at a later time (due to propagation of large-scale coronal transient \citep{Krucker1999}, shock waves \citep{Simnett2002}, or coronal magnetic restructuring after CMEs \citep{Maia2004, Klein2005}), or that the high energy electrons experience more interplanetary scattering \citep{Cane2003a, Cane2003}.
By calculating the extended path length traveled by high-energy electrons compared to low-energy electrons, due to the interplanetary scattering, \citet{Wang2011} suggested that such scattering is too small to explain the observed delays.
Electrons of tens of keV are great tracers of the interplanetary magnetic field topology because of their fast speed and small gyro radius, in the order of $10^{-6}$ AU near the earth.
By measuring the direction to the centroid of the type III radio emission region and using the Helios model of the interplanetary plasma density \citep{Bougeret1984}, \citet{Reames1986} were able to map out the interplanetary magnetic field path traveled by electrons.
They found that in some events, the magnetic field paths depart from Parker spiral.
Assuming the release time of energetic electrons matches that of the type III radio burst in the high frequency, \citet{Larson1997} calculated the path length in a twisted flux-rope topology of magnetic clouds and found that the path length is $\sim3$ AU near the leading edge of the magnetic cloud and $\sim1.2$ AU near the center.
Using the same method, \citet{Kahler2011, Kahler2011a} investigated the field line lengths in magnetic clouds and interplanetary coronal mass ejections. They found that there was no significant difference of the path length among magnetic clouds, ICMEs, and ambient solar wind. Moreover, the field line lengths at different parts of ICMEs do not show obvious differences.
Interplanetary magnetic field length has also been investigated using the observed magnetic turbulence power spectra as a function of resolution scale and turbulence condition.
\citet{Ragot2006} simulated the interplanetary magnetic field lines using $10^{15}$ different turbulent modes and computed the magnetic field lengths numerically.
She found that, comparing to the case of a smooth field, a turbulent field line can be $\sim 1.3$ times longer.
Besides these attempts, another widely used method for determining the interplanetary magnetic field path length is the velocity dispersion analysis (VDA). By fitting the onset time of energetic electrons or ions at different energy channels, one can obtain the particle's release time at the sun and the path length simultaneously.
The validity of the VDA is based on several assumptions including the simultaneous release of particles of different energies on the sun; the scatter-free propagation of the first arriving particles; and the accurate determination of onset time in in-situ observations.
\citet{Kahler2006} examined the $c/v$ onset plots of energetic electron events and found the fitted path length distributed between $0.15$ and $2.7$ AU. And out of $80$ events surveyed, $64$ events have path length smaller than the minimum travel distance calculated from the instantaneous in-situ solar wind speeds.
\citet{Kahler2006} attributed the reason for the unphysical short path lengths to be either the instrumental effects that produce early onset in lower energy, or the invalid assumption of impulsive and simultaneous injection, or the diffusive propagation.
The validity of VDA has also been analyzed numerically by \citet{Lintunen2004}, \citet{Saiz2005}, \citet{Laitinen2015}, and \citet{Wang2015}.
Clearly, the work of \citet{Kahler2006} indicated that not all events are suitable for VDA analysis: for example, events having very gradual rising time profiles may violate the assumption of energy-independent releasing assumption. Furthermore, both the onset times and the peak times can be very hard to determined. These add further issues to the
commonly used VDA. To address these issues, we consider a new VDA analysis approach and use this approach to study the interplanetary magnetic field path length in selected solar electron events.
As discussed above, to reduce the potential discrepancy of the release time between the low and high energy electrons, we focus on electrons at energies of $27-310$ keV. Furthermore, we choose events that have rapid rise and reasonably rapid decay phases. Such a time profile often indicates a scatter-free propagation and an impulsive, nearly symmetric injection at the sun \citep{Lin1974, Wang2006}.
\section{Event Database Description} \label{sec:data}
We use the electron measurements made by the 3-D Plasma and Energetic Particle (3DP) instrument (Silicon semiconductor telescope (SST) and electron electrostatic analyzers (EESA-L and EESA-H)) onboard WIND spacecraft \citep{Lin1995}.
The EESA-L and EESA-H detectors measure $\sim3$ eV to $30$ keV electrons with angular resolution of $22.5^\circ\times22.5^\circ$ and the SST measures $\sim 25$ to $400$ keV electrons with angular resolution of $22.5^\circ\times36^\circ$.
Between December 1994 and October 2016, there are $1944$ solar energetic electron events identified with a flux increase of 2 times the standard deviation of the background flux above the background in three or more energy channels. The presence of a velocity dispersion with faster electrons arriving earlier than slower electrons is also required in selecting these electron events.
Among those $1944$ events, we select $882$ events that are detected by at least four energy channels by the SST detector, at energies of $27-310$ keV.
Due to the instrumental effects of SST, a portion of incident high energy electrons could be scattered out of the detector and contaminate the low energy channels, which leads to the early detection of the lower energy particles \citep{Wang2006}.
\citet{Tan2013} compared the path lengths calculated with and without correcting the instrumental effect and suggested that the unphysical short path lengths calculated in \citet{Kahler2006} could be due to the contamination from the deposition energy loss.
Later, in stead of using the corrected onset times, \citet{Wang2016} used the peak times to perform the path length analysis.
Not only the contamination from higher energy channels is minimum at the peak time, the effect of the background superhalo electrons is also minimized.
However, since the peak is where the rate of flux change is zero, so deciding the correct peak time with small uncertainties can be hard.
Using peak time to perform the VDA implicitly assumes that the peak of the injection profile of different energies are at the same time and the injection duration at the sun is short.
{Due to scattering, the peak of the time intensity profile observed at 1 AU will be broader than that at the source. In the work of \citet{Wang2006}, to better obtain
the peak time, events with short duration of injection are selected. As
we will see below, in the new method, we do not need to accurately determine
the peak time. However, a short duration of injection often suggests a
rapid rising phase. And a rapid rising phase leads to a smaller uncertainties of the obtained path length. For this reason, we focus on electrons with higher energies ($27-310$ keV), which have relative shorter injection duration compared to low-energy electrons \citep{Wang2006, Wang2016}}.
Unlike the peak, which may be hard to identify, the rising phase in a majority of electron events (certainly impulsive ones and also some graduate ones) is very rapid and the shapes are similar at different energy channels. Noticing this feature, we develop a new VDA procedure to reduce the effect of a gradual decay phase on the determination of the peak times.
We first find the peak value, $j_p$, in the time intensity profile of outward-propagating electrons.
Note that while peak time is hard to determine, peak flux determination
is easy. Furthermore, our method does not require an accurate
value of the peak flux (see below).
We next find the times $t_\eta$ in the rising phase that satisfy
\begin{equation}
j(t_\eta)-j_b=\eta (j_p-j_b)
\end{equation}
with $\eta$ to be the fraction parameter. In the
above, $j_b$ is the background flux.
The background flux $j_b$ is taken to be the minimal flux in a $20\sim30$ minutes window before or after the event.
Because the rising phases are sharp, and the fact that
the rising phases for different energy channels are similar, we can use $t_{\eta}$'s for the corresponding VDA analysis.
We further assume a $\Delta j = \alpha j(t_\eta)$ to obtain the uncertainty of $t_\eta$ at corresponding $j(t_\eta)$ as illustrated in figure~\ref{fig:method}.
The red rectangular box represents the method that we determine the uncertainty of the time $t_\eta$, $\Delta t_\eta$.
The vertical boundary represents the $\pm \alpha$ of the flux uncertainty region and the corresponding horizontal boundary represents the uncertainty of $t_\eta$.
The uncertainties, $\Delta t_\eta$, are used as weights in the Chi-square fitting method that we utilize to perform the linear regression.
Therefore, different values of $\alpha$ may yield slightly different fitting results and their corresponding uncertainties.
In this work, the value of $\alpha$ is chosen to be $7.5\%$, sufficiently large to account for the time resolution of the observations ($\sim 24 s$), the uncertainty in determining the peak flux, and the contamination due to electrons from higher energy channels.
Note that the contamination from higher energy channels occur mainly at the onset so that it will lead to inaccurate identification of the onset times and a smaller path length. Indeed, as shown in \citet{Kahler2006}, in performing the traditional VDA analysis, one may obtain path lengths that are as small as $0.15$ AU due to the contamination effect. And the contamination from the high energy channel is event-based, therefore, it is hard to estimate the effect of contamination on path length in a quantitative way. As an event progresses, the contamination becomes less significant since the fraction of particles within its original energy bin increases.
In \citet{Li2013}, the time intensity profile before and after correcting the contamination from the higher energy channels is plotted and the contamination effect shows decreasing trend from the onset to the peak, in the rising phase.
In our analysis, if the profiles of the rising phase for different energy channels are similar, we can determine accurately $t_\eta$ for different energy channels. The onset time corresponds to $\eta \rightarrow 0$.
However contamination is most significant at small $\eta$. Therefore, to obtain a correct onset time, what one can do is obtain a series of $L_\eta$ for $\eta$ not so small and then extrapolate to $\eta$ = 0. Because the rising phase is rather rapid, deciding $t_\eta$ at $\eta$s which are not close to zero is easier. Consequently extrapolating to $L_\eta(\eta\rightarrow 0)$ will be more accurate than the traditional VDA analysis. Therefore our analysis is less affected by contamination than the VDA analysis.
The most notable feature of our new method is the use of multiple $\eta$'s. It, in a way, provides a self-consistent check for the result. In the scenario that the onset times are the same for all electrons at different energies, illustrated in the left panel of figure~\ref{fig:onset}, a smaller value $\eta$ will lead to a path length closer to the real situation.
And if the injection profile at different energies peak at the same time, as shown in the right panel in figure~\ref{fig:onset}, then we expect a larger $\eta$ gives more accurate path length.
Indeed, by fitting the path length as a function of $\eta$ and extrapolating it to $\eta=0$ and $\eta=1$, one can get
the corresponding path lengths for the two scenarios in figure~\ref{fig:onset}.
If the rising phase is very short, then using different $\eta$'s may yield very close results.
As discussed above, the value of $\eta$ can not be too small in order to eliminate the energy deposition effect from the instrument.
In this work, we use $\eta=$ 3/4, 1/2, and 1/3.
Hereafter, we refer to our new VDA analysis as FVDA for fractional VDA.
We note that the FVDA method is best for electrons
and may lead to overestimate of path length for protons and ions.
This is because the interplanetary waves and turbulence often affect more the transport of protons and ions than electrons. Indeed, by employing several assumptions about the interplanetary scattering and the injection profile, \citet{Saiz2005} examines numerically how the VDA analysis can lead to an over estimate of the path length for $2$ to $2000$ MeV protons. Since there is no background level in their numerical experiments, they tested three thresholds at $0.01\%$, $2\%$, and $60\%$ of the peak value and found that although the derived path length at the $0.01\%$ threshold is about $25\%$ larger than the true path length, it can be off by $100\%$ for the larger threshold of $60\%$. We point out that the analysis of \citet{Saiz2005} is very similar to our FVDA method except that
their choices of the $0.01\%$ and $60\%$ thresholds are not feasible in reality due to non-zero background. \citet{Masson2012} considered
using a single threshold of $50\%$ the peak value for proton events with rapid rise. As shown in \citet{Saiz2005}, the derived path length can be significantly larger than the actual one.
We select events that satisfy the following criteria:
(1) the time intensity profiles of the outward propagating electrons show quick rising phases (within one hour) in a broad energy range with well-defined and similar shapes.
(2) the event has good count statistics and the entire phase of the event can be clearly separated from the background.
With these two criteria, we obtain $125$ events.
\section{Analysis and Discussion} \label{sec:analysis}
We make no clear distinction between gradual events and impulsive events in our analysis. Gradual events with rapid onset phases and well-defined profiles which allow the determination of peak fluxes are also included in our selected events.
The magnetic field path length is calculated by performing the linear regression of the time $t_\eta$ and the inverse of velocity $1/v_i$:
\begin{equation}
\frac{L_\eta}{v_i}=t_{\eta,i}-t_0.
\end{equation}
$L_\eta$ is the path length calculated using time {$t_{\eta,i}$ which gives a flux equal to $\eta (j_p-j_b)+j_b$ for energy $E_i$}, and $t_0$ is the release time at the sun. In the analysis, the velocity $v_i$ is chosen such that the corresponding $E_i$ is the center energy of the energy channels. Although the earliest arriving particles in one energy channel often have the highest energy in that energy channel, electrons in one energy channel are composed mostly by the low energy electrons (due to the negative energy spectral slope).
To examine the effect of the choice of energy we use, we have also performed the linear regression assuming $v_i$ to be either the lower or higher boundary of the energy channel. The results are similar, and the overall distribution discussed below does not change much.
In the following statistical discussion, we discard events that have uncertainties larger than $0.1$ AU.
A larger uncertainty in the linear regression usually means the times, $t_{\eta,i}$, do not lie perfectly in a straight line. These could be events in which the time intensity profiles of different energy channels do not behave similarly.
With this constraint, a total of $81$ events remain.
The events and the results are summarized in table~\ref{tbl:database}. This is our main result of the paper. The first $4$ columns contain the year, month, day, and start time of the events. The $5-7$ columns contain the calculated path length and its uncertainty when $\eta$ equals $3/4$, $1/2$, and $1/3$, respectively.
We also show the peak-to-background intensity ratio ($j_p/j_b$) in the energy channel of $82-135$ keV in column $8$, as an indicator of the relative strength of the event with respect to the pre-event background.
This is the only common channel for all events we studied in this work.
The time difference dT (in seconds) between the $3/4$ and $1/2$ of the peak flux is shown in column $9$, as an indicator of the duration of the rising phase.
Panel (a) in figure~\ref{fig:dist} is a scattered plot of the calculated path lengths with uncertainties for all events. The red, blue, and green color represent the path length calculated with $\eta$ equals $3/4$, $1/2$, and $1/3$.
The grey shaded curve plot the $13$-month smoothed monthly total sunspot number obtained from \url{http://www.sidc.be/silso/datafiles}.
The distribution of the path lengths do not show clear correlation with respect to the sunspot number.
The histograms of the path lengths and their uncertainties are plotted in panels (b), (c), and (d).
The corresponding mean and standard deviation is $1.17$ AU and $0.17$ AU for $\eta=3/4$, $1.11$ AU and $0.14$ AU for $\eta=1/2$, and $1.06$ AU and $0.15$ AU for $\eta=1/3$. One may also try to fit the histogram with a Gaussian distribution. For panel (b) and (d), the
Gaussian fits are plotted by the black dashed-line (for panel (c), the Shapiro-Wilk test \citep{Shapiro1965}
rejects the Gaussian hypothesis of the histogram at $\alpha$-value of $0.05$.)
We note that as $\eta$ gets smaller, the mean path length shifts to a smaller values. For any given event, choosing a smaller $\eta$ does not necessarily lead to a more accurate result of path length. As explained before,
only when the release time at different energies are the same, choosing a smaller $\eta$ yields a better estimation of the path length. However, as mentioned above, the contamination effect from the higher energy channel is stronger when $\eta$ is smaller. This contamination will lead to earlier elevations of the low energy channels. Consequently, the calculated path length will be smaller than the actual one. Because contamination is more pronounced at smaller $\eta$, one expects that as $\eta$ becomes smaller, the path length also decreases. Indeed, this is what we found
from our analysis. This also explains to some extend
the fact that there are $11$, $17$, and $26$ (out of $81$) events which have calculated path lengths less than $1$ AU for $\eta=3/4$, $1/2$, and $1/3$, respectively.
There is another reason why we obtain path lengths less than 1 AU: if low energy electrons are released earlier than high energy electrons, our analysis will also yield a shorter path length than the actual value. One can turn this argument around and argue that the events whose path length are smaller than $1$ AU must not have the same release times for electrons at different energies. In any case, our analysis suggests that using multiple $\eta$ will allow us to obtain some constraints on the release profiles at the sun.
Although we are focusing on the rising phases of these events, electrons however undergo pitch angle scattering as they propagate out to $1$ AU due to interplanetary turbulence. \citet{Tan2011,Tan2013} suggested that the first arriving electrons may not be scatter-free due to pre-existing R-mode and L-mode waves which can scatter non-relativistic electrons.
Moreover, a transition from the scatter-free to diffusive propagation of electrons was reported by \citet{Tan2011}.
The transition energy is found to be event dependent and it is between $\sim 60$ keV and $\sim 120$ keV in one event and between $\sim 250$ keV and $\sim 500$ keV in another.
Nevertheless, there are only $2$, $1$, $1$ events
(for $\eta=3/4$, $1/2$, and $1/3$ respectively) that have calculated path length greater than $1.5$ AU.
Earlier, \citet{Kahler2006} calculated the travel distance of $80$ electrons detected by WIND SST instrument and found the path lengths ranged from $0.15$ to $2.7$ AU. However, even in their analysis, there were only two events that have path length greater than $2.0$ AU; and the majority of their events have path lengths shorter than $1.5$ AU.
Figure~\ref{fig:peakdist} shows the dependence of the calculated path length on the peak-to-background ratio ($j_p$/$j_b$) of the $82-135$ keV electrons. Panel (a) shows the histogram of $j_p$/$j_b$ and panel (b), (c), and (d) plot the calculated path length and its uncertainties with respect to $j_p$/$j_b$.
The plots do not show a clear correlation between the path length and the $j_p$/$j_b$ ratio, indicating that our method is not biased toward larger or smaller events. This is different from previous work of
\citet{Kahler2006}, who compared the path length with the corresponding $82$ keV peak-to-background ratios and found a weak correlation. In the simulations of \citet{Lintunen2004}, an association of longer path length with smaller peak intensities is obtained because of the delayed onset times of low-energy channels compared to the high-energy channels. And more intense events are associated with slightly smaller path length.
These are because the determination of the onset time is affected by the background count level as well as the spectral shape.
To determine the onset time, one often uses the criterion that the intensity exceeds a fixed fraction $k$ (from $0.001$ to $0.1$) of the peak intensity.
In the work of \citet{Lintunen2004}, the authors found that increasing the fraction, $k$, by an order of magnitude will increase the path length by $30-50\%$.
Figure~\ref{fig:dtdist} shows the dependence of the calculated path length on the duration of the rising phase of the event. We use the time difference between the $3/4$ peak time and the $1/2$ peak time as a proxy for the
duration of the rising phase, and in the following we
refer this as the onset time scale. Panel (a) plots the histogram of the onset time scale of all events and panels (b), (c), and (d) plot the correlation between the onset time scale and the calculated path length for the three choices of $\eta=$ $3/4$, $1/2$, and $1/3$. There is no clear correlation between the path length and the onset time scale. This suggests that one can extend the FVDA method to events where the rising phase does not need to be very sharp.
\section{Conclusion}\label{sec:conclusion}
Understanding the configuration of interplanetary magnetic field is important to understand the transport of energetic particles in SEP events. In particular, the configuration of interplanetary magnetic field can put strong constraints on particles' cross field diffusion. Previous studies have obtained values of $\kappa_{\perp}/\kappa_{||}$ which differ considerably for the dropout events and the wide-spreading events \citep{Mazur2000, Giacalone2000, Dresing2014}. To reconcile this discrepancy, one possible explanation is that magnetic field configurations can differ considerably in different events. For example, if there is significant field line meandering in the interplanetary medium \citep{Laitinen2016}, the same event can be seen from multiple spacecraft which have large longitudinal separation. The field line meandering effect may also vary with different interplanetary turbulence conditions.
In this work, we calculate the interplanetary magnetic path length using a newly developed method: the fraction velocity dispersion analysis (FVDA) method. This method does not require an accurate determination of the onset time of electrons as in the standard VDA method. It is therefore less affected by the background flux. It also does not require an accurate determination of the
peak time, as done in \citet{Wang2016}. Instead of considering either the peak time or the onset time, the FVDA utilizes the times in the rising phase of an event that correspond to the flux that is a fraction, $\eta$, of the peak flux. We applied the FVDA method to electron events that have a well-defined peak and relative prompt onset phase. Note that a stronger scattering usually leads to a prolonged onset phase. Therefore, our selection criteria naturally eliminate very diffusive events and events with extended injection.
Using this method, we identified $81$ electron events of which path lengths are obtained with uncertainties less than $0.1$ AU. And the calculated path lengths using different $\eta$s yield similar results.
The obtained path lengths in these events are very close to the nominal Parker field lengths, suggesting that the magnetic field themselves may also be close to the Parker configuration.
From the distribution of the path length with respect to the peak-to-background ratios and the onset time scales, our method does not show bias toward either large or small, fast or slow onset events. The mean and stand deviation of the calculated path lengths are ($1.17$ AU, $0.17$ AU) for $\eta=3/4$, ($1.11$ AU, $0.14$ AU) for $\eta=1/2$, and ($1.06$ AU, $0.15$ AU) for $\eta=1/3$. And the distribution of the calculated path lengths for $\eta=3/4$ and $1/3$ are well represented by a Gaussian distribution.
The FVDA method we developed here is an extension of VDA. Comparing to the traditional VDA analysis, it is less affected by the uncertainty in determining the onset time or the peak time of the time intensity profiles. However, because it makes use of the rapid rising phase, so if the rising phases for different energy channels are not similar, the resulting path length from FVDA can be also nonphysical. This puts some constraints in applying the FVDA method. In our current work, we find that the FVDA applies nicely to "nearly-scatter-free" electron events. We caution that one has to take extreme care when applying FVDA to other events such as ion events or events that are scatter-dominated.
For a turbulent magnetic field, \citet{Ragot2008} suggested that a correction factor should be applied to the length of the smooth magnetic field length. And the correction factor is found to be $1.16\pm0.06$ and $1.23\pm0.03$ in slow and fast solar wind at $0.3$ AU, and $1.45\pm0.25$ and $1.33\pm0.06$ in slow and fast solar wind at $1$ AU. This is not what we find in this work. Our results of the magnetic field lengths indicate that the interplanetary magnetic field paths traveled by the $27-310$ keV electrons are close to the ideal Parker length. One may expect that the path lengths are correlated with solar activity. However, the events in our study span over a full solar cycle and show no clear dependence on solar cycle. Therefore our results suggest that the interplanetary magnetic field does not differ much from the Parker's spiral field during both solar maximum or solar minimum.
This is somewhat counter intuitive in that we expect the solar wind magnetic field is affected more during solar maximum. However, we note that in our analysis we required the background to be reasonably quiet even in solar maximum: no large preceding events occurred in our selection. Although there was no large preceding events, there still could be many small (or nano) eruptions. One may expect that these small eruptions can also change the interplanetary magnetic field configurations. Our results, however, suggest that they do not.
Note that the change of the Parker spiral length from $1.14$ AU (solar wind speed of $\sim 400$ km/s) to 1.03 AU (solar wind speed of $800$ km/s) is comparable to the uncertainties of the path length calculated in this work. Our results has important implications for the study of energetic particle transport in the solar wind. Further detailed studies for selected events in table~\ref{tbl:database} will be reported in a separate paper.
Acknowledgement: L.Z. and M.Z. are supported at Florida Institute of Technology under NNX15AN72G, NNX15AB76G, 80NSSC19K0076, and 80NSSC18K0644; G.L. and A.M. are supported at University of Alabama in Huntsville under NNX17AI17G, NNX17AK25G, and 80NSSC19K0075. L.W. thanks NSFC for support under grants 41774183 and 41861134033. F.E. is supported by NNX17AK25G at the Bay Area Environmental Research Institute. {G.L. and F.E. also acknowledge supports from the International Space Science Institute (ISSI) through the team on 'Solar flare acceleration signatures and their connection to solar energetic particles'. In particular discussions with Drs. L. Klein, T. Laitinen, N. Bian, and Du Toit Strauss.}
\begin{center}
\input{table}
\end{center}
\clearpage
\begin{figure}[ht!]
\plotone{method-depict.png}
\caption{Illustration of the FVDA method. The black curve is the time intensity profile observed. The red, blue, and green asterisk symbols on the curve represent the point whose intensity is $3/4$, $1/2$, and $1/3$ of the peak intensity. The red box demonstrate the method that we determine the uncertainty associated with the time when the intensity is $3/4$ of the peak value. The vertical boundary represents the $\pm\alpha$ of the flux and the horizontal boundary represents the corresponding time uncertainty.}\label{fig:method}
\end{figure}
\clearpage
\begin{figure}[ht!]
\plotone{onset-depict.png}
\caption{Two onset models that we consider in this work. The left panel shows the case that electrons with different energies have the same onset time at the sun. The right panel shows the case that electrons with different energies have the same peak time at the sun. Red, black, and blue curves indicate the time intensity profiles for electrons with high, medium, and low energy.}\label{fig:onset}
\end{figure}
\clearpage
\begin{figure}[ht!]
\plotone{hist-filter7-apj.png}
\caption{Panel (a) plots the path lengths and their uncertainties calculated using FVDA when $\eta$ is $3/4$ (red), $1/2$ (blue), and $1/3$ (green). The shaded curve plots the monthly total sunspot number. Panel (b) plots the histogram of the path lengths and their uncertainties calculated with $\eta=3/4$. The mean and stand deviation of the path length is 1.17 AU and 0.17 AU. The black dashed curve is the fitted Gaussian distribution.
Panel (c) plots the histogram of the path lengths and their uncertainties calculated with $\eta=1/2$. The mean and stand deviation of the path length is 1.11 AU and 0.14 AU.
Panel (d) plots the path length and their uncertainty distributions calculated with $\eta=1/3$. The mean and stand deviation of the path length is 1.06 AU and 0.15 AU.
The black dashed curve is fitted Gaussian distribution.}\label{fig:dist}
\end{figure}
\clearpage
\begin{figure}[ht!]
\plotone{peak2bkgrd-path-apj.png}
\caption{Panel (a) plots the histogram of the peak-to-background intensity ratio ($j_p/j_b$) in the energy range of $82-135$ keV. Panels (b), (c), and (d) plot the calculated path length with respect to the peak-to-background ratio when $\eta$ is $3/4$, $1/2$, and $1/3$.}\label{fig:peakdist}
\end{figure}
\clearpage
\begin{figure}[ht!]
\plotone{dt-path-1-apj.png}
\caption{Panel (a) plots the histogram of the onset time scale in the energy range of $82-135$ keV. Panels (b), (c), and (d) plot the calculated path length with respect to the onset time scale when $\eta$ is $3/4$, $1/2$, and $1/3$.\label{fig:dtdist}}
\end{figure}
\clearpage
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 450 |
Incels strike again » « When will people learn that debate is just noise to distract you all?
Analyze this statement
I am accused on YouTube of being a liberal snob and parroting the radical left's blank-slate narrative that everything we do is socially constructed by someone who offers up his ironic credentials: he's a STEM student in a major university .
Dave Bloom 1 day ago 12 Subscribers
@PZ Myers By the way, your general undertone of 'you plebs lack the intelligence sufficient to grasp my brilliance' is just more liberal narcissism. I'm a STEM student in a major university. Someone apparently thought my logic was adequate.
I just found that hilarious. He is valid because he's on the bottom rung of a socially constructed hierarchy! Apparently, that hierarchy is genetic and evolutionarily deep, because Lobsters.
By the way, everything is socially constructed. Everything is genetic. You can't separate the two.
Atheism and Skepticism
Pseudonym says
What sort of hierarchy do you have in mind that would put, say, a high-school dropout on a higher rung than "a STEM student at a major [whatever that means] university"? (Other than inability to use logic, which is more of an idiosyncratic fault of this individual than endemic to major-university STEM majors in particular?)
rcs619 says
I thought it was pretty obvious that the vast majority of things we do in a society are social constructs. They may have a long history behind them, or even be a hand-me-downs from a previous civilization that the current one split off from, but that still doesn't mean they're any less artificial. Humans have instincts of course, but even those are usually filtered through the lens of whatever your particular society has decided is normal,, expected, acceptable or taboo.
I mean everything from gender roles, to customs, values and social norms vary WILDLY from society to society. That should be a pretty big clue that they're artificial constructs. There is no baseline human "normal" that is universal across all human cultures.
Also bragging about being a STEM student to someone who is literally a professor in a STEM field doesn't seem like an ideal tactic. Anyone can be a STEM student. Pretty much all you have to do is make C's or higher and apply somewhere.
I'm a STEM student in a major university.
Big deal.
So are millions of other people.
More millions of other people are…graduates in STEM fields from major universities.
Now that is a somewhat bigger deal.
I'm not sure what the point of that minor factual claim is other than to demonstrate that Dave Bloom isn't very bright.
F.O. says
Hey, everyone knows that none in STEM in a major university has ever committed any logical fallacy ever. /s
4 November 2018 at 10:06 am
And the STEM student's major university is (pause for dramatic irony) the University of Minnesota at Morris, where the Stem student probably hope to take a key biology class next fall. From whom?
Just one more logic fail from the YouTubes.
And here I was, thinking you were immune to flattery, PZ.
Knabb says
Wow. It's bad enough trying to claim rank, but it's significantly worse trying to claim rank against someone who outranks you in the exact same field (here defined as STEM in general) takes some chutzpah.
longdog says
Certainly I think it's fair to say that PZ has always been very reluctant to ascribe biological, "deterministic" reasons for human behaviour- I don't think even he would dispute that.
I understand this approach inasfar as physiology is quite variable, its mechanisms are somewhat unclear, and its expression is socially informed to some degree. But it does tend to frustrate me that I can't really find an example of PZ attributing any behaviour at all to what the layperson would call "genetics". I think the reason people claim you're espousing a blank-slate worldview is that you haven't expressed much to the contrary.
https://freethoughtblogs.com/pharyngula/2018/03/13/the-cruelest-cut-against-evolutionary-psychology/ This post in particular comes to mind- there's a big old list of EP claims (some admittedly more dubious than others) and they're not addressed all that meaningfully, in my opinion.
To give an example, I remember reading about how female apes (I think chimps?) will make vocalizations during sex, and this seems to make the male ejaculate faster. This is obviously pretty analogous to human behavior, right? There's kind of this societal expectation that women be vocal during sex. There's a societal component to this- I think it's obviously reinforced in porn, some women are more vocal than others across the board, and different cultures seem to expect different kinds of vocalizations, etc. But the fact that we independently see it in an animal closely related to us suggests to me that it's a more innate behaviour than one might otherwise think. And it's been my experience reading this blog that I kind of don't think the implications of that would really be addressed or acknowledged, here.
I'm not trying to get real rowdy in this comment section or anything, more just to express that I'd agree there's something of a "blind-spot" when it comes to PZ's interpretation of behaviour.
John Morales says
longdog:
So, this "blind-spot" (not just a blind spot), you're not very explicit about what it is.
But your reasons for inferring its existence are clear enough: you are frustrated ("I can't really find an example of PZ attributing any behaviour at all to what the layperson would call "genetics""), disappointed ("a big old list of EP claims (some admittedly more dubious than others) and they're not addressed all that meaningfully, in my opinion"), incredulous ("But the fact that we independently see it in an animal closely related to us suggests to me that it's a more innate behaviour than one might otherwise think. "), and runs counter (as you see it) to your own opinion ("But the fact that we independently see it in an animal closely related to us suggests to me that it's a more innate behaviour than one might otherwise think.").
Not the most convincing reasons, for mine, but clearly expressed.
And it's been my experience reading this blog that I kind of don't think the implications of that would really be addressed or acknowledged, here.
Be not disappointed; I've hereby acknowledged them, and about the implications thereof anyone can make their own determination.
Nerd of Redhead, Dances OM Trolls says
I'm not trying to get real rowdy in this coment section or anything, more just to express that I'd agree there's something of a "blind-spot" when it comes to PZ's interpretation of behavior.
Yet, as a scientist, I see no evidence to support your inane conclusion. Care to link? My favorite phrase to liars and bullshitters is "put up o shut the fuck up".
I can't exactly prove a negative in the scientific sense, can I? If you like I can probably find some other posts of this nature (similar to what I linked) that I find lacking.
I'm a little troubled that you've gone right away to implying I'm a liar and a bullshitter, though.
=8)-DX says
Yep. Language is a social construct, fuck anyone trying to convey information pretending they can do so without social construction. And everything that we are is a result of our genetics, which happily enough allowed for the growth of various plastic learning systems in our brains. If there is one thing you can say about a human child, it's that they're very good at adapting to whatever environment they live in.
methuseus says
@longdog #8, 11:
What the hell are you even trying to say? That PZ doesn't believe in evolution? That he doesn't believe evolution governs actions in any way? I think you know, really, that those are bald-faced lies. Sure, PZ is derisive of some ideas in EP, but that's because they're ridiculous. Even some in EP would agree with that. Evolutionary psychologists have no evidence, they only have ideas with no empirical evidence to back them up. So PZ agrees that some of it makes sense, but other things are insane to say. Therefore, EP overall makes little sense and only consists of feelings.
Seriously, what do you think this "blind-spot" is? I'm sure he does have blind-spots, as do I, and will freely admit to some of them. It's bullshit, non-existent ones that he won't admit to. And I'd be wary of anyone trying to get him to admit to that sort of thing anyway.
leerudolph says
Apparently, that hierarchy is genetic and evolutionarily deep, because Lobsters.
Marina Hyde covers several beats for the Guardian. One of her series is "Lost in Showbiz", and her most recent is devoted to Jordan Peterson—in particular, to a recent video released by Gentleman's Quarterly,
in which Jordan is interviewed by the New Statesman's Helen Lewis. It's hard to pick my favourite moment from the nearly two-hour-long encounter, but I very much enjoyed the bit where Lewis reasons: "Lobsters don't get depressed. I think you're anthropomorphising to a ridiculous degree. These are creatures that urinate out of their faces."
Then again, it must be said that Peterson spends most of the interview looking like he's about to urinate out of his face. In the entire exchange, he smiles about once, at some perceived irony in something wistfully arch that he has just said. One's primary takeout is not: here is a man who can laugh at himself. Which is such a missed opportunity.
emergence says
The main problem is with the notion that specific human behaviors must be hard-coded instincts, and that these make up the majority of human behavior. It makes more sense to me that the human brain evolved generalized processes for learning and association.
As for that YouTube comment, I've noticed that you can be accused of "narcissism" for just saying that you think your ideas are correct. I doubt the commenter can point to anything PZ does that Jordan Peterson or Ben Shapiro don't do a whole lot more.
Also, the idiot who posted that comment touted his status as a STEM student to back up his claim that he doesn't need PZ to tell him how biology works. PZ is a biology professor. It's his job to teach students about biology. Apparently the idea that professors teach students is narcissistic. | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 3,281 |
\section{Introduction}
{\bf Keywords.} Sharply 2-transitive groups, Permutation groups.\ \\
A \emph{sharply $2$-transitive group} is a pair $(G,X)$ where $G$
is a group acting on the set $X$ in such a way that for all
$x,\,y,\,z,\,t\in X$ such that $x\neq y$ and $z\neq t$ there is a
unique $g\in G$ for which $gx=z$ and $gy=t.$ From now on $(G,X)$
will stand for a sharply $2$-transitive group with $|X|\geq 3$. We
fix an element $x\in X$. We let $H:=\{g\in G: gx=x\}$ denote the
stabilizer of $x$. Finally we let $I$ denote the set of
involutions (elements of order 2) of $G$.
It follows easily from the definition that the group $G$ has an
involution, in fact any element of $G$ that sends a distinct pair
$(y,z)$ of $X$ to the pair $(z,y)$ is an involution by sharp
transitivity. It is also known that $I$ is one conjugacy class and
the nontrivial elements of $I^2$ cannot fix any point (See Lemma 1
and Lemma 4). Then one can see that $I^2$ cannot have an
involution if $H$ has an involution.
In case $H$ has no involution, one says that $\chr(G)=2$.
Let us assume that $\chr(G)\neq 2$. Then $I^2\setminus\{1\}$ is
one conjugacy class \cite[Lemma 11.45]{bn}. Since $I^2$ is closed
under power taking, either the nontrivial elements of $I^2$ all
have order $p$ for some prime $p\neq 2$ or $I^2$ has no nontrivial
torsion element. One writes $\chr(G)=p$ or $\chr(G)=0$ depending
on the case.
One says that $G$ \emph{splits} if the one point stabilizer $H$
has a normal complement in $G$. It is not known whether or not an
infinite sharply 2-transitive group splits, except for those of
characteristic 3. Results in this direction for some special cases
can be found in \cite[\S 11.4]{bn} and \cite[ch 2]{ke}. We will
prove that if $\chr(G)=3$ then $G$ splits, a result of W.\ Kerby
\cite[Theorem 8.7]{ke}. But Kerby's proof is in the language of
near domains and is not easily accessible. Here, we give a much
simpler proof of this fact, in fact an experienced reader can
directly go to the proof the Theorem, which contains only a simple
computation (all the lemmas are well-known facts).
All the results of this short and elementary paper can be found in
\cite[\S 11.4]{bn}, except for the final theorem.
\begin{lemma}
$I$ is one conjugacy class.
\end{lemma}
\begin{proof}
Let $i,j\in I$ and $x\in X$ be such that $jx\neq x$ and $ix\neq
x$. Since $G$ is $2-$transitive, there exists a $g\in G$ such that
$gx=x$ and $gjx=ix$. Then $i^gjx=x$ and $i^gj(jx)=jx$. By double
sharpness of $G$, $i^gj=1$. Hence, $i^g=j$ and we are done.
\end{proof}
\begin{lemma}
If $N$ is a nontrivial normal subgroup of
$G$ then $G=NH$.
\end{lemma}
\begin{proof}
Let $g\in G\setminus
H, a\in N, y\in X\setminus \{x\}$ be such that $ay\neq y$ and
$h\in G$ be such that $hx=y$ and $hgx=ay$. Then $(a^{-1})^hg\in
H$ and $g\in NH$. Since $1\in N$, it holds for all $g\in G$.
\end{proof}
\begin{lemma}
$H$ has at most one
involution.
\end{lemma}
\begin{proof}
Let $i,j\in H\cap I,\, y\in
X\setminus \{x\},\, g\in G $ be such that $gjy=iy$ and $gy=y$. Then
$ji^g(y)=y$ and $ji^g(jy)=jy$. Since $ji^g$ fixes two different
points and $G$ is sharply $2$-transitive, $ji^g=1$ and $j=i^g$.
One can easily see that $H\cap H^z\neq \{1\}$ if and only if $z\in H$.
Therefore $g\in H$ as $j\in H\cap H^g$. Since $g$ fixes two
points, namely $x$ and $y$, $g=1$. Hence $i=j$ and we are done.
\end{proof}
\begin{lemma}
A nontrivial element of $I^2$ cannot fix any
element of $X$.
\end{lemma}
\begin{proof}
Assume not. Then, there are distinct involutions $i,\,j$ such that
$ij$ fixes a point. Since $G$
is transitive, we may assume $ij\in H$. It follows from Lemma 3 that $j\notin H$ otherwise $i\in H$,
hence a contradiction.
On the other hand, $(ij)^{-1}=(ji)=(ij)^j$ and $(ij)^j\in
H\cap H^j$. Therefore, $j\in H$, a contradiction.
\end{proof}
\begin{lemma}
If the elements of $Ii$ commute with each other for
some $i\in I$, then $I^2$ is a normal subgroup of $G$.
\end{lemma}
\begin{proof}
It suffices
to prove that $I^2$ is closed under multiplication. Let
$i,\,j,\,k,\,w\in I$. We claim that $ijkw\in I^2$. By Lemma 1,
we may assume that the elements of $Ii$ commute with each other. Noting that $Ii = iI$, we have
$(ijk)^2=ijkijk=kiijjk=1$. So, $ijk\in I \cup \{1\}$. If $ijk\in
I$, we are done. Assume $ijk=1$. If $H$ has an involution, by
Lemma 1,
$(ij)^g=k^g\in H$ for some $g\in G$ , i.e.\ $(ij)^g$ fixes $x$, contradicting
Lemma 4. If $H$ has no involution, $ij=k\in I$ and, by Lemma 1, $I\subseteq
I^2$. Therefore, $ijkw=w\in I^2$.
\end{proof}
\begin{lemma}
If $H$ has an involution, then the action of $G$ on
$X$ is equivalent to the action of $G$ on $I$ by
conjugation.
\end{lemma}
\begin{proof}
Let $i\in H$ be an
involution. It is easy to see that the action of $G$ on $X$ is
equivalent to the action of $G$ on the left coset space $G/H$. So
we may assume that the set $X$ is the left coset space $G/H$.
Consider the map from $G/H$ to $I$ defined as $\bar{g}\mapsto i^{g^{-1}}$ for
$g\in G$. One can easily see that
this is the required equivalence.
\end{proof}
\begin{thm}
If $\chr(G)=3$ then $G$ splits.
\end{thm}
\begin{proof}
We claim that $G=I^2\rtimes H$. If $I^2$ is a normal subgroup of $G$,
then we know that $H\cap I^2=\{1\}$ by Lemma 4 and $G=I^2H$ by Lemma
2. Therefore, we just need to prove that $I^2$ is a normal subgroup of $G$.
By lemma 5, it is enough to show that the elements of $Ii$ commute with each
other for some $i\in I$. Let $i\in H\cap I$ be the (unique) involution of $H$ and let $ji,ki\in Ii$.
We may assume that $j\neq k$.
By double sharpness of $G$, it suffices to prove that $jiki$ and
$kiji$ agree on two different points. By Lemma 6, we can take
$X$ to be $I$ and the action to be the conjugation. We now claim
that $jiki$ and $kiji$ agree on $j$ and $k$ i.e.\ that
$j^{jiki}=j^{kiji}$ and $k^{jiki}=k^{kiji}$. By symmetry of the
situation, it is enough to prove one of the equalities. Since
$\chr(G)=3$, $i^j=j^i$ for all $i,j\in I$ and so we have
$$j^{jiki}=j^{(k^i)}=(k^i)^j=k^{ij}=k^{jiji}=(k^j)^{iji}=(j^k)^{iji}=j^{kiji}.$$
\end{proof}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 9,733 |
{"url":"https:\/\/math.stackexchange.com\/questions\/895298\/probability-that-fx-y-z0-given-the-variables-follow-normal-distribution","text":"# Probability that $f(x,y,z)>0$ given the variables follow normal distribution\n\nAssuming that variables $x,y$ and $z$ follow the Gaussian distribution with $\\mu_x=\\mu_y=\\mu_z=1000000$ and $\\sigma_x=\\sigma_y=\\sigma_z=200000$, what is the probability that $$f(x,y,z) = \\frac{x}{1.1}+\\frac{y}{1.1^2}+\\frac{z}{1.1^3}-2000000>0$$\n\nMy try: $P(x)=\\frac{e^{\\frac{-(x-\\mu)^2}{2\\sigma^2}}}{\\sigma \\sqrt{2\\pi}}$ The required probability is $$\\int\\int\\int P(x)P(y)P(z) dx dy dz$$ with $(x,y,z)$ such that $f(x,y,z)>0$. After this I don't know how to integrate that too over such a difficult domain.\n\nThe mean of $aX+bY+cZ-d$ is $aE(X)+bE(Y)+cE(Z)-d$. In our case we have $a=\\frac{1}{1.1}$, $b=\\frac{1}{(1.1)^2}$, and $c=\\frac{1}{(1.1)^3}$.\nThe variance of $aX+bY+cZ-d$ is $a^2\\text{Var}(X)+b^2\\text{Var}(Y)+c^2\\text{Var}(Z)$.\nNow you have everything needed for the computation of your probability. You know how to find the probability that a normal random variable with known mean and variance is $\\gt 0$.","date":"2019-08-21 07:05:15","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\": 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\": 0.9835851192474365, \"perplexity\": 51.54003450490265}, \"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-2019-35\/segments\/1566027315811.47\/warc\/CC-MAIN-20190821065413-20190821091413-00493.warc.gz\"}"} | null | null |
Q: Basic SASS - Nesting and HTML Syntax I just started using SASS however I can't seem to find a clear answer/example for this.
Say I have:
<img class="socialIcons" src="/images/facebook.png"/>
<img class="socialIcons" src="/images/google.png"/>
I now want to have some additional styles for Facebook and Google - is there some clever SASS syntax I can use for example so I get:
<img class="socialIcons-facebook" src="/images/facebook.png"/>
<img class="socialIcons-google" src="/images/google.png"/>
And in my SASS use:
.socialIcons {
max-height: 30px;
padding-right: 10px;
&.facebook {
background: blue;
}
}
So that it adopts the general socialIcons style as well as Facebook.
Can't seem to figure the syntax.
Thanks.
A: You mean this?
<img class="socialIcons facebook" src="/images/facebook.png"/>
<img class="socialIcons google" src="/images/google.png"/>
The '&' syntax will match an element that has both the outer class and the class with the & selector. You can overwrite the 'socialIcons' class with the more specific nested selector. No need to change the Sass.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 518 |
Auth0 Client Rules
========================
This small Symfony app lists all the clients and rules associated with them. If you've used Auth0 before (https://auth0.com/), you should already be familiar with the concept of rules.
By default, all Auth0 rules are applied to all client applications. Sometimes, we need to apply only some rules to some applications. This app will help you do that.
In the app, you will see that the rules which apply to some specific clients only will be displayed in green color, while the rules which apply to all clients will be displayed in a yellow color.
In order to apply rules for some clients only, you have to configure your rules like this:
```javascript
function (user, context, callback) {
if (context.clientName === 'MyAppToWhiteList' || context.clientName === 'AnotherAppToWhiteList' || context.clientID === '123456789') {
// Your rule logic
}
callback(null, user, context);
}
```
You can do it either by ```clientID``` or ```clientName``` so it's really easy to do it in any fashion you like.
Usage & requirements
--------------
Since this example app uses Docker, you will need to have it installed on your system - download and install it from here: https://www.docker.com/. After you clone this repo, take the following steps:
* Run ```make build && make up && make install``` from the project.
* Once the container is built, follow the steps from here https://auth0.com/docs/quickstart/webapp/symfony to configure the app with your data (you will need to be logged in to your Auth0 account to see pre-populated data).
* Make sure you create a valid token for calling the Auth0 Management APIv2: https://auth0.com/docs/api/management/v2/tokens - replace the following with your specific data in ```src/Controller/ClientRulesController.php```:
``` const AUDIENCE = <YOUR_AUDIENCE_URL>
const DOMAIN = <YOUR_AUTH0_DOMAIN>
const CLIENT_ID = <YOUR_CLIENT_ID>
const CLIENT_SECRET = <YOUR_CLIENT_SECRET>
```
* Once you're done, run the app at http://localhost:5500/clients/rules and login - you should then see all the rules for all clients.
* That's it! Make sure you have allowed only specific users to access the client by using the "Whitelist for specific app" rule from your app (https://manage.auth0.com/#/rules). It should look something like this:
```javascript
function (user, context, callback) {
// we just care about NameOfTheAppWithWhiteList or its id
// bypass this rule for every other app
if(context.clientName !== 'NameOfTheAppWithWhiteList' || context.clientId !== '123456789'){
return callback(null, user, context);
}
var whitelist = [ 'user1@example.com', 'user2@example.com' ]; //authorized users
var userHasAccess = whitelist.some(
function (email) {
return email === user.email;
});
if (!userHasAccess) {
return callback(new UnauthorizedError('Access denied.'));
}
callback(null, user, context);
}
```
Contributions
--------------
Feel free to fork the repo and create PR with improvements.
| {
"redpajama_set_name": "RedPajamaGithub"
} | 9,733 |
29-я церемония вручения наград премии «Оскар» за заслуги в области кинематографа за 1956 год прошла 27 марта 1957 года в RKO Pantages Theatre (Голливуд, Лос-Анджелес, Калифорния).
Фильмы, получившие несколько номинаций
Список лауреатов и номинантов
Основные категории
{| class="wikitable" width="100%" border="1" cellpadding="5" cellspacing="0" align="centre"
|-
! width="280px"|Категории
!colspan=2| Лауреаты и номинанты
|-
| rowspan=5|<center>Лучший фильм
|colspan=2 style="background:#EEDD82"|★ Вокруг света за 80 дней / Around the World in Eighty Days (продюсер: Майкл Тодд)
|-
|colspan=2|• Дружеское увещевание / Friendly Persuasion (продюсер: Уильям Уайлер)
|-
|colspan=2|• Гигант / Giant (продюсеры: Джордж Стивенс и Генри Гинсберг)
|-
|colspan=2|• Король и я / The King and I (продюсер: Чарльз Брэкетт)
|-
|colspan=2|• Десять заповедей / The Ten Commandments (продюсер: Сесиль Б. де Милль)
|-
| rowspan=5|<center>Лучший режиссёр
|colspan=2 style="background:#EEDD82"|★ Джордж Стивенс за фильм «Гигант»
|-
|colspan=2|• Майкл Андерсон — «Вокруг света за 80 дней»
|-
|colspan=2|• Уильям Уайлер — «Дружеское увещевание»
|-
|colspan=2|• Уолтер Лэнг — «Король и я»
|-
|colspan=2|• Кинг Видор — «Война и мир»
|-
| rowspan=5|<center>Лучшая мужская роль
| rowspan=5 align="center" width="95px"|
|style="background:#EEDD82"|★ Юл Бриннер — «Король и я» (за роль короля Сиама Монгкута)
|-
|• Джеймс Дин (посмертно) — «Гигант» (за роль Джетта Ринка)
|-
|• Кирк Дуглас — «Жажда жизни» (за роль Винсента ван Гога)
|-
|• Рок Хадсон — «Гигант» (за роль Джордана «Бика» Бенедикта мл.)
|-
|• Лоренс Оливье — «Ричард III» (за роль Ричарда III)
|-
| rowspan=5|<center>Лучшая женская роль
| rowspan=5 align="center"|
|style="background:#EEDD82"|★ Ингрид Бергман — «Анастасия»''' (за роль Анны Коревой / Анастасии)
|-
|• Кэрролл Бейкер — «Куколка» (за роль «Куколки» Мейган)
|-
|• Кэтрин Хепбёрн — «Продавец дождя» (за роль Лиззи Карри)
|-
|• Нэнси Келли — «Дурная кровь» (за роль Кристин Пенмарк)
|-
|• Дебора Керр — «Король и я» (за роль Анны Леонуэнс)
|-
| rowspan=5|<center>Лучшая мужская роль второго плана
| rowspan=5 align="center"|
|style="background:#EEDD82"|★ Энтони Куинн — «Жажда жизни» (за роль Поля Гогена)
|-
|• Дон Мюррей — «Автобусная остановка» (за роль Борегарда «Бо» Декера)
|-
|• Энтони Перкинс — «Дружеское увещевание» (за роль Джоша Бёрдуэлла)
|-
|• Микки Руни — «Дерзкий и смелый» (за роль Дули)
|-
|• Роберт Стэк — «Слова, написанные на ветру» (за роль Кайла Хэдли)
|-
| rowspan=5|<center>Лучшая женская роль второго плана
| rowspan=5 align="center"|
|style="background:#EEDD82"|★ Дороти Мэлоун — «Слова, написанные на ветру»''' (за роль Мэрили Хэдли)
|-
|• Милдред Даннок — «Куколка» (за роль тётушки Роуз Комфорт)
|-
|• Айлин Хекарт — «Дурная кровь» (за роль Гортензии Дэйгл)
|-
|• Мерседес Маккэмбридж — «Гигант» (за роль Луз Бенедикт)
|-
|• Патти Маккормак — «Дурная кровь» (за роль Роды Пенмарк)
|-
| rowspan=5|<center>Лучший оригинальный сценарий
|colspan=2 style="background:#EEDD82"|★ Альбер Ламорис — «Красный шар»
|-
|colspan=2|• Роберт Левин — «Дерзкий и смелый»
|-
|colspan=2|• Эндрю Л. Стоун — «Джулия»
|-
|colspan=2|• Федерико Феллини и Туллио Пинелли — «Дорога»
|-
|colspan=2|• Уильям Роуз — «Замочить старушку»
|-
| rowspan=5|<center>Лучший адаптированный сценарий
|colspan=2 style="background:#EEDD82"|★ Джеймс По, Джон Фэрроу и C. Дж. Перельман — «Вокруг света за 80 дней» {{nobr|(по одноимённому роману Жюля Верна)}}
|-
|colspan=2|• Теннесси Уильямс — «Куколка» (по пьесам автора «Двадцать семь вагонов с хлопком» и «Несъедобный ужин»)
|-
|colspan=2|• Майкл Уилсон — «Дружеское увещевание» (по роману Джессамин Уэст «Friendly Persuasion»)
|-
|colspan=2|• Фред Гиол и Иван Моффат — «Гигант» (по одноимённому роману Эдны Фербер)
|-
|colspan=2|• Норман Корвин — «Жажда жизни» (по одноимённому роману Ирвинга Стоуна)
|-
| rowspan=5|<center>Лучший литературный первоисточник (Best Writing, Motion Picture Story)
| rowspan=5 align="center"|
|style="background:#EEDD82"|★ Далтон Трамбо — «Отважный»
|-
|• Лео Кэтчер — «История Эдди Дучина»
|-
|• Жан-Поль Сартр — «Гордецы»
|-
|• Чезаре Дзаваттини — «Умберто Д.»
|-
|style="background:#FFE4E1"|• Эдвард Берндс и Элвуд Уллман — «»(неофициальная номинация)
|-
| rowspan=5|<center>Лучший фильм на иностранном языке
|colspan=2 style="background:#EEDD82"| ★ Дорога / La strada (Италия), продюсеры: Дино Де Лаурентис и Карло Понти
|-
|colspan=2| • / Der Hauptmann von Köpenick (ФРГ), продюсеры: Гьюла Требич и Вальтер Коппель
|-
|colspan=2| • Жервеза / Gervaise (Франция), продюсер: Анни Дорфманн
|-
|colspan=2| • Бирманская арфа / ビルマの竪琴 (Япония), продюсер: Масаюки Такаки
|-
|colspan=2| • / Qivitoq (Дания), продюсер: О. Дальсгаард Олсен
|-
|}
Другие категории
{| class="wikitable" width="100%" border="1" cellpadding="5" cellspacing="0" align="centre"
|-
! width="280px"|Категории
! Лауреаты и номинанты
|-
| rowspan=5|<center>Лучшая музыка: Саундтрек к драматическому или комедийному фильму
|style="background:#EEDD82"|★ Виктор Янг (посмертно) — «Вокруг света за 80 дней»
|-
|• Альфред Ньюман — «Анастасия»
|-
|• Хьюго Фридхофер — «»
|-
|• Дмитрий Тёмкин — «Гигант»
|-
|• Алекс Норт — «Продавец дождя»
|-
| rowspan=5|<center>Лучшая музыка: Саундтрек к музыкальному фильму
|style="background:#EEDD82"|★ Альфред Ньюман и Кен Дэрби — «Король и я»
|-
|• Лайонел Ньюман — «»
|-
|• Моррис Столофф и Джордж Данинг — «История Эдди Дучина»
|-
|• Джонни Грин и Сол Чаплин — «Высшее общество»
|-
|• Джордж Столл и Джонни Грин — «»
|-
| rowspan=5|<center>Лучшая песня к фильму
|style="background:#EEDD82"|★ Que Sera, Sera (Whatever Will Be, Will Be) — «Человек, который слишком много знал» — {{nobr|музыка и слова: Джей Ливингстон и Рэй Эванс}}
|-
|• Friendly Persuasion (Thee I Love) — «Дружеское увещевание» —
|-
|• Julie — «Джулия» — музыка: Лейт Стивенс, слова: Том Адэр
|-
|• True Love — «Высшее общество» — музыка и слова: Коул Портер
|-
|• Written on the Wind — «Слова, написанные на ветру» —
|-
| rowspan=5|<center>Лучший монтаж
|style="background:#EEDD82"|★ Джин Руджеро, Пол Везервакс — «Вокруг света за 80 дней»
|-
|• Мэррилл Дж. Уайт — «Отважный»
|-
|• Уильям Хорнбек, Филип В. Андерсон, Fred Bohanan — «Гигант»
|-
|• Альберт Экст — «Кто-то там наверху любит меня»
|-
|• Энн Боченс — «Десять заповедей»
|-
| rowspan=5|<center>Лучшая операторская работа (Чёрно-белый фильм)
|style="background:#EEDD82"|★ Джозеф Руттенберг — «Кто-то там наверху любит меня»
|-
|• Борис Кауфман — «Куколка»
|-
|• Гарольд Россон — «Дурная кровь»
|-
|• Бёрнетт Гаффи — «Тем тяжелее падение»
|-
|• Уолтер Стрендж — «»
|-
| rowspan=5|<center>Лучшая операторская работа (Цветной фильм)
|style="background:#EEDD82"|★ Лайонел Линдон — «Вокруг света за 80 дней»
|-
|• Гарри Стрэдлинг ст. — «История Эдди Дучина»
|-
|• Леон Шамрой — «Король и я»
|-
|• Лойал Григгс — «Десять заповедей»
|-
|• Джек Кардифф — «Война и мир»
|-
| rowspan=5|<center>Лучшая работа художника (Чёрно-белый фильм)
|style="background:#EEDD82"|★ Седрик Гиббонс, Малкольм Браун (постановщики),
|-
|• Хэл Перейра, А. Эрл Хедрик (постановщики),
|-
|• Такаси Мацуяма — «Семь самураев»
|-
|• Росс Белла (постановщик), Уильям Кирнан, Луи Диэдж (декораторы) — «"Кадиллак" из чистого золота»
|-
|• Лайл Р. Вилер, Джек Мартин Смит (постановщики),
|-
| rowspan=5|<center>Лучшая работа художника (Цветной фильм)
|style="background:#EEDD82"|★ Лайл Р. Вилер, Джон ДеКуир (постановщики), Уолтер М. Скотт, Пол С. Фокс (декораторы) — «Король и я»
|-
|• Джеймс В. Салливан, Кен Адам (постановщики), Росс Дауд (декоратор) — «Вокруг света за 80 дней»
|-
|• Борис Левен (постановщик), Ральф С. Херст (декоратор) — «Гигант»
|-
|• Седрик Гиббонс, Ганс Питерс, Э. Престон Амес (постановщики),
|-
|• Хэл Перейра, Уолтер Х. Тайлер, Альберт Нозаки (постановщики),
|-
| rowspan=5|<center>Лучший дизайн костюмов (Чёрно-белый фильм)
|style="background:#EEDD82"|★ Жан Луи — «"Кадиллак" из чистого золота»
|-
|• Кохэй Изаки — «Семь самураев»
|-
|• Хелен Роуз — «»
|-
|• Эдит Хэд — «Гордый и светский»
|-
|• Чарльз Ле Мэр, Мэри Уиллс — «Мятежный подросток»
|-
| rowspan=5|<center>Лучший дизайн костюмов (Цветной фильм)
|style="background:#EEDD82"|★ Ирен Шарафф — «Король и я»
|-
|• Майлз Уайт — «Вокруг света за 80 дней»
|-
|• Мосс Мэбри, Марджори Бест — «Гигант»
|-
|• Эдит Хэд, Ральф Джестер, Джон Дженсен, Дороти Джикинс и Арнольд Фриберг — «Десять заповедей»
|-
|• Мария Де Маттеи — «Война и мир»
|-
| rowspan=5|<center>Лучший звук
|style="background:#EEDD82"|★ Карлтон У. Фолкнер (20th Century-Fox SSD) — «Король и я»
|-
|• Бадди Майерс (King Bros. Productions, Inc. SD) — «Отважный»
|-
|• Джон П. Ливадари (Columbia SSD) — «История Эдди Дучина»
|-
|• Гордон Р. Гленнан (Westrex Sound Services), Гордон Сойер (Samuel Goldwyn SSD) — «Дружеское увещевание»
|-
|• Лорен Л. Райдер (Paramount SSD) — «Десять заповедей»
|-
| rowspan=2|<center>Лучшие спецэффекты
|style="background:#EEDD82"|★ Джон П. Фултон — «Десять заповедей»
|-
|• А. Арнольд Гиллеспи, Ирвинг Дж. Риес, Уэсли Си Миллер — «Запретная планета»
|-
| rowspan=3|<center>Лучший документальный полнометражный фильм
|style="background:#EEDD82"|★ В мире безмолвия / Le Monde du silence (продюсер: Жак Ив Кусто)
|-
|• (продюсер: Луис Клайд Стоумен)
|-
|• Где плавают горы / Hvor bjergene sejler (The Government Film Committee of Denmark)
|-
| rowspan=5|<center>Лучший документальный короткометражный фильм
|style="background:#EEDD82"|★ / The True Story of the Civil War (продюсер: Луис Клайд Стоумен)
|-
|• (Charles Guggenheim & Associates, Inc.)
|-
|• (продюсер: Джон Хили)
|-
|• / The House Without a Name (продюсер: Валентайн Дейвис)
|-
|• / Man in Space (продюсер: Уорд Кимбалл)
|-
| rowspan=3|<center>Лучший короткометражный фильм, снятый на 1 бобину
|style="background:#EEDD82"|★ / Crashing the Water Barrier (продюсер: Константин Калзер)
|-
|• / I Never Forget a Face (продюсер: Роберт Янгсон)
|-
|• (продюсер: Седрик Френсис)
|-
| rowspan=4|<center>Лучший короткометражный фильм, снятый на 2 бобины
|style="background:#EEDD82"|★ / The Bespoke Overcoat (Romulus Films)
|-
|• Корова и собака / Cow Dog (продюсер: Ларри Лансбург)
|-
|• (продюсер: Джон Хили)
|-
|• Самоа / Samoa (продюсер: Уолт Дисней)
|-
| rowspan=3|<center>Лучший короткометражный фильм (мультипликация)
|style="background:#EEDD82"|★ / Magoo's Puddle Jumper (продюсер: Стивен Босустоу')
|-
|• Джеральд Макбоинг! Боинг! на планете Му / Gerald McBoing-Boing on Planet Moo (продюсер: Стивен Босустоу)
|-
||• The Jaywalker (продюсер: Стивен Босустоу)
|-
|}
Специальные награды
Научно-технические награды
См. также
«Золотой глобус» 1957 (премия Голливудской ассоциации иностранной прессы)
BAFTA 1957 ''(премия Британской академии кино и телевизионных искусств)
Примечания
Ссылки
Лауреаты и номинанты 29-й церемонии на официальном сайте американской киноакадемии
Лауреаты и номинанты премии «Оскар» в 1957 году на сайте IMDb
Организаторы и участники 29-й церемонии на сайте IMDb
База данных по всем номинантам и победителям
1957
События 27 марта
Март 1957 года
Кинопремии 1957 года
1957 год в Калифорнии | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 9,617 |
Шалажі () — село у Урус-Мартановському районі Чечні Російської Федерації.
Населення становить 5313 осіб (2019). Входить до складу муніципального утворення Шалажінське сільське поселення.
Історія
Згідно із законом від 14 липня 2008 року органом місцевого самоврядування є Шалажінське сільське поселення.
Населення
Примітки
Населені пункти Урус-Мартановського району
Села Чечні | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 1,488 |
Q: $3ab + a^3 - 2b^3 - 4a + 5b - 7 = 0$ I came across this problem:
Prove there arent't any $a$, $b$ integers that satisfy
equation $3ab + a^3 - 2b^3 - 4a + 5b - 7 = 0$
Firstly, I've thought something like this:
$$(a^3 + b^3)-3b^3 + 3ab - 4a + 4b - 4 + b = 3$$
$$(a+b)(a^2-ab+b^2) - 3(b^3 - ab) -4(a-b+1)+b = 3$$
I think it's just a tricky exercise, but I can't find a way to solve it.
Some help would be apreciated.
Thanks!
A: $3ab + a^3 - 2b^3 - 4a + 5b - 7 = 0$
Consider the equation mod $3$. Since $x^3\equiv x\pmod 3$ for all $x$, we have
$3ab+a-2b-4a+2b-7\equiv 0$ which is the same as $2\equiv 0\pmod 3$.
Contradiction.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 4,271 |
\section{Introduction}
The tree property for a regular cardinal $\kappa$ is the statement that there is no $\kappa$ - Aronszajn tree or equivalently every $\kappa$ - tree has a cofinal branch. In general constructing a model for tree property on a regular cardinal $\kappa$ is not trivial and needs large cardinal assumptions. The problem becomes even harder and needs stronger large cardinal assumptions when one tries to get tree property on several successive regular cardinals. In this direction we have:
\begin{proposition}\label{results} The following results are known about tree property:
\item[(1)] (Konig) The tree property holds on $\aleph_0$.
\item[(2)] (Aronszajn) The tree property does not hold on $\aleph_1$.
\item[(3)] (Specker) For every infinite cardinal $\kappa$ if $\kappa^{<\kappa}=\kappa$ then the tree property does not hold on $\kappa^+$. Specially if $CH$ holds then $\aleph_2$ does not have the tree property.
\item[(4)] (Silver - Mitchell) The tree property on $\aleph_2$ is equiconsistent with the existence of a weakly compact cardinal.
\item[(5)] (Abraham) Assuming the consistency of a supercompact cardinal and a weakly compact above it, it is consistent to have tree property on both $\aleph_2$ and $\aleph_3$.
\item[(6)] (Magidor) The consistency of tree property on both $\aleph_2$ and $\aleph_3$ implies the consistency of "$0^\sharp$ \textit{exists}".
\item[(7)] (Cummings - Foremann) Assuming the existence of an $\omega$-sequence of supercompact cardinals, it is consistent that the tree property holds for all $\aleph_{n}$'s, $1<n<\omega$.
\end{proposition}
\begin{proof}
For (1), (2), (3) see \cite{jech}. (4) is proved in \cite{mitchell}. For (5) and (6) see \cite{abraham}. The result (7) is proved in \cite{foremann}.
\end{proof}
An importnat point about the Aronszajn's result in proposition \ref{results} is the essential use of AC in his construction. Thus the existing $\aleph_1$ - Aronszajn tree is \textit{not} definable. Amir Leshem \cite{leshem} proved that assuming existence of a $\Pi_{1}^{1}$ - reflecting cardinal, it is consistent that a definable version of tree property (definition \ref{tree property}) holds on $\aleph_1$.
\begin{definition}\label{reflecting cardinals}
An inaccessible cardinal $\kappa$ is $\Pi_{n}^{m}$ - reflecting, if for every $A\subseteq V_{\kappa}$ definable over $V_{\kappa}$ with parameters from $V_{\kappa}$ and for every $\Pi_{n}^{m}$ - sentence $\Phi$, if $(V_{\kappa}, \in, A)\models \Phi$ then there is an $\alpha<\kappa$ such that $(V_{\alpha}, \in, A\cap V_{\alpha})\models \Phi$.
\end{definition}
\begin{definition}\label{tree property}
Let $\kappa$ be a regular cardinal. A $\kappa$ - tree $(T, <_T)$ is definable if its underlying set is $\kappa$, and the relation $<_T$ is $\Sigma_n$ - definable in the structure $(H_{\kappa}, \in)$ for some natural number $n$. We say the definable tree property holds on $\kappa$ if every definable $\kappa$ - tree has a cofinal branch.
\end{definition}
\begin{remark}\label{different forms of definable tree property}
In his paper \cite{leshem}, Leshem considers several variants of definable tree property, including what he calls definable tree property in the strict, wide and very wide sense. His results are about definable tree property in the strict sense which is exactly what we stated in the definition \ref{tree property}. According to Leshem's definitions, every definable $\kappa$ - tree in the strict sense is definable in the wide sense and every definable $\kappa$ - tree in the wide sense is definable in the very wide sense. Also every definable $\kappa$ - tree $(T, <_T)$ in the wide sense is isomorphic to a $\kappa$ - tree $(\kappa, <^*)$ that is definable in the strict sense. So it follows that without losing generality one can assume that the definable tree property in the strict and wide sense are identical while the definable tree property in the very wide sense is different from them.
\end{remark}
\begin{theorem}\label{leshem's main result} (Leshem)
The following statements are equiconsistent:
\begin{enumerate}
\item[(1)] The definable tree property holds on $\aleph_1$.
\item[(2)] There is a $\Pi_{1}^{1}$ - reflecting cardinal.
\end{enumerate}
\end{theorem}
\begin{proof}
\cite{leshem}.
\end{proof}
In section 2 we generalize Leshem's result to the consistency of definable tree property for proper class of all successors of regular cardinals using the existence of proper class many $\Pi_{1}^{1}$ - reflecting cardinals, a large cardinal assumption weaker than the existence of a Mahlo cardinal and much weaker than what is theoretically expected for achieving tree property in the usual sense for this class of regular cardinals.
\begin{main theorem}\label{main theorem}
The following statements are equiconsistent:
\begin{enumerate}
\item[(1)] The definable tree property on successor of every regular cardinal.
\item[(2)] There are proper class many $\Pi_{1}^{1}$ - reflecting cardinals.
\end{enumerate}
\end{main theorem}
The situation for the consistency of holding tree property at successor of a singular cardinal is generally more complicated than the case of regulars. By a result of Magidor and Shelah \cite{magidor} it is known that if $\lambda$ is the singular limit of $\lambda^+$ - supercompact cardinals then $\lambda^+$ has the tree property. This fact is used by them to prove the consistency of tree property on $\aleph_{\omega+1}$ from a very strong large cardinal assumption. Later Sinapova \cite{sinapova} decreased the necessary large cardinal assumption for proving the consistency of tree prperty on $\aleph_{\omega+1}$ to the existence of $\omega$ - many supercompact cardinals.
On the other hand, answering an old question of Woodin, Neeman \cite{neeman} produced, assuming the existence of $\omega$-many supercompact cardinals, a model in which $SCH$ fails at a singular strong limit cardinal $\kappa$ of cofinality $\omega$ and $\kappa^+$ has the tree property. But in Neeman's model, $GCH$ fails cofinally often below $\kappa$, and it is still an open problem if we can have a singular cardinal $\kappa$ such that $GCH$ holds below $\kappa$, $2^\kappa > \kappa^+$, and $\kappa^+$ has the tree property.
In section 3 we prove the main theorem \ref{definable tree property at a singular} which gives an affirmative answer to this question if the tree property is replaced with the definable tree property. Our proof also reduces the large cardinal strength from the existence of infinitely many supercompact cardinals to the existence of a supercompact cardinal and a measurable above it.
\begin{main theorem}\label{definable tree property at a singular}
Assume $GCH$ holds, $\kappa$ is supercompact and $\lambda >\kappa$ is measurable. Then there is a generic extension of the universe in which:
\begin{enumerate}
\item[(1)] $\kappa$ is a strongly limit singular cardinal of cofinality $\omega$,
\item[(2)] No bounded subsets of $\kappa$ are added, in particular $GCH$ holds below $\kappa$,
\item[(3)] $\lambda=\kappa^+$ and the definable tree property holds at $\lambda$,
\item[(4)] $2^\kappa = |j(\lambda)|$, in particular if (in $V$) $|j(\lambda)|> \lambda^+$, then $SCH$ fails at $\kappa$.
\end{enumerate}
\end{main theorem}
The generic extension in which the above theorem holds is essentially the extension obtained by supercompact extender based Prikry forcing introduced by Merimovich in \cite{mer4}.
Our results show that the \textit{definable} version of tree property is so different in nature from its original form and needs much weaker large cardinal assumptions for proving its consistency.
\section{Definable tree property at successor of all regular cardinals}
The entire argument in this section is for proving the main theorem \ref{main theorem}.
\subsection{From definable tree property to reflecting cardinals} \label{section for from definable tree property to reflecting cardinals}
In this subsection we prove the (1) to (2) part of the main theorem \ref{main theorem} by showing that assuming definable tree property for successors of regular cardinals in $V$, $\Pi_{1}^{1}$ - reflecting cardinals form an unbounded subclass of cardinals in $L$ (theorem \ref{from tree property to reflecting}). First let's review some facts and definitions from \cite{leshem}.
\begin{definition}\label{extension property}
A cardinal $\kappa$ has the extension property if and only if for every natural number $n$ and for every set $A\subseteq V_{\kappa}$ definable over $V_{\kappa}$ with parameters from $V_{\kappa}$, there is a transitive set $X$, and a subset $A^X$ of $X$ such that $\kappa\in X$ and $(V_{\kappa}, \in, A)\prec_{n} (X, \in, A^X)$.
\end{definition}
\begin{proposition}\label{extension and end extension}
For a cardinal $\kappa$ the following statements are equivalent:
\begin{enumerate}
\item[(1)] $\kappa$ has the extension property.
\item[(2)] For every natural number $n$, there is a transitive set $X$ which $\kappa\in X$ and the structure $(X,\in)$ is a $\Sigma_n$ - elementary end extension of $(V_{\kappa}, \in)$.
\end{enumerate}
\end{proposition}
\begin{proof}
\cite{leshem}.
\end{proof}
\begin{proposition}\label{reflecting and extension}
For a cardinal $\kappa$ the following statements are equivalent:
\begin{enumerate}
\item[(1)] $\kappa$ is $\Pi_{1}^{1}$ - reflecting.
\item[(2)] $\kappa$ is inaccessible and has the extension property.
\end{enumerate}
\end{proposition}
\begin{proof}
\cite{leshem} theorem 3.2.
\end{proof}
\begin{proposition}\label{reflecting and tree property}
The definable tree property holds on every $\Pi_{1}^{1}$ - reflecting cardinal.
\end{proposition}
\begin{proof}
\cite{leshem} lemma 3.3.
\end{proof}
\begin{lemma}\label{tree property in V and reflecting in L}
Let $\kappa$ be a successor of a regular cardinal, if $\kappa$ has the definable tree property in $V$ then $\kappa$ is $\Pi_{1}^{1}$ - reflecting in $L$.
\end{lemma}
\begin{proof}
Similar to the proof of theorem 5.1. in \cite{leshem}.
\end{proof}
\begin{theorem}\label{from tree property to reflecting}
If the definable tree property holds for proper class many regular cardinals in $V$ then there are proper class many $\Pi_{1}^{1}$ - reflecting cardinals in $L$.
\end{theorem}
\begin{proof}
Assume that $\Pi_{1}^{1}$ - reflecting cardinals in $L$ are bounded below a cardinal $\lambda$. There is a regular cardinal $\kappa > \lambda$ such that definable tree property holds for $\kappa$ in $V$. By lemma \ref{tree property in V and reflecting in L}, $\kappa$ is a $\Pi_{1}^{1}$ - reflecting cardinal in $L$ greater than $\lambda$, a contradiction.
\end{proof}
\subsection{From reflecting cardinals to definable tree property}\label{section for from reflecting cardinals to definable tree property}
In this subsection we are going to prove the (2) to (1) part of the theorem \ref{main theorem} using an Easton reverse iteration of Levy collapses of reflecting cardinals (theorem \ref{from reflecting to tree property}). At the first setp we need to prove that small forcings preserve the $\Pi_{1}^{1}$ - reflecting cardinals.
\begin{lemma}\label{preservation of reflecting cardinals under small forcings}
If $\kappa$ is a $\Pi_{1}^{1}$ - reflecting cardinal and $\mathbb{P}$ is a notion of forcing which $|\mathbb{P}|<\kappa$ then $\kappa$ remains $\Pi_{1}^{1}$ - reflecting in $V^{\mathbb{P}}$.
\end{lemma}
\begin{proof}
Assume that $\kappa$ is a $\Pi_{1}^{1}$ - reflecting cardinal and $|\mathbb{P}|<\kappa$. As small forcings preserve inaccessibility of $\kappa$, by proposition \ref{reflecting and extension} it suffices to show that $\kappa$ has the extension property in $V[G]$. Using the equivalence in proposition \ref{extension and end extension} it suffices to show that in $V[G]$ for every natural number $n$, there is a transitive set $Y$ such that $\kappa\in Y$ and the structure $(Y,\in)$ is a $\Sigma_n$ - elementary end extension of $(V_{\kappa}, \in)$. Note that by smallness of forcing notion we have $V_{\kappa}^{V[G]}=V_{\kappa}[G]$. Thus it is sufficient to show that for every natural number $n$, there is a transitive set $Y\in V[G]$ which $\kappa\in Y$ and the structure $(Y,\in)$ is a $\Sigma_n$ - elementary end extension of $(V_{\kappa}[G], \in)$.
Fix the natural number $n$, without losing generality we may assume that the forcing notion $\mathbb{P}$ in $V$ is defined by a formula of complexity $\Sigma_{m}$. Choose the sufficiently large natural number $t\geq m, n$. By extension property of $\kappa$ in $V$ as a $\Pi_{1}^{1}$ - reflecting cardinal, we get a transitive set $X\in V$ and a set $\mathbb{P}^X\subseteq X$ such that $\kappa\in X$ and the structure $(X, \in, \mathbb{P}^X)$ is a $\Sigma_{t}$ - elementary extension of $(V_{\kappa}, \in, \mathbb{P})$. In fact $\mathbb{P}^X=\mathbb{P}$ because by elementary extension the structure $(X, \in, \mathbb{P}^X)$ is agree with $(V_{\kappa}, \in, \mathbb{P})$ on the notion of $\in$.
Now we show that $(V_{\kappa}[G], \in)\prec_{n}(X[G], \in)$ which completes the proof because $X[G]$ is a transitive set in $V[G]$ with our required property for $Y$. In order to do this fix a first order $\Sigma_{n}$-formula $\varphi (x_1,\cdots, x_n)$. We have $V[G]\models \varphi (a_1,\cdots, a_n)$ iff $\exists p\in G~~~p\Vdash_{\mathbb{P}}^{V}\varphi (\dot{a}_1,\cdots, \dot{a}_n)$. Note that by smallness of forcing we may assume that $\mathbb{P}\in V_{\kappa}$ and so we can consider the forcing relation $\Vdash^{V}$ as $\Vdash^{V_{\kappa}}$, thus the last statement is equivalent to $\exists p\in G~~~(V_{\kappa}, \in, \mathbb{P})\models p\Vdash_{\mathbb{P}}\varphi (\dot{a}_1,\cdots, \dot{a}_n)$. As $t$ was chosen sufficiently large we may assume that it exceeds the complexity of the formula $p\Vdash_{\mathbb{P}}\varphi (\dot{a}_1,\cdots, \dot{a}_n)$ which is a $\Sigma_{s}$ - formula like $\psi_{\varphi} (p, \mathbb{P}, \dot{a}_1,\cdots, \dot{a}_n)$. Thus by $\Sigma_{t}$ - elementary extension, $\exists p\in G~~~(V_{\kappa}, \in, \mathbb{P})\models p\Vdash_{\mathbb{P}}\varphi (\dot{a}_1,\cdots, \dot{a}_n)$ holds iff $\exists p\in G~~~(X, \in, \mathbb{P})\models p\Vdash_{\mathbb{P}}\varphi (\dot{a}_1,\cdots, \dot{a}_n)$. Equivalently $X[G]\models \varphi (a_1,\cdots, a_n)$ which means $(V_{\kappa}[G], \in)\prec_{n}(X[G], \in)$ and so $\kappa$ is a $\Pi_{1}^{1}$ - reflecting cardinal in $V[G]$.
\end{proof}
We need to work with the notion of a weakly homogenous forcing that is defined as follows:
\begin{definition}\label{homogeneous forcing}
A notion of forcing $\mathbb{P}$ is called weakly homogeneous if and only if for every two conditions $p, q$ in $\mathbb{P}$ there is an automorphism $\pi$ of $\mathbb{P}$ such that $\pi (p)$ and $q$ are compatible.
\end{definition}
An important property of weakly homogeneous forcings is that they don't add any new definable set with parameters from the ground model.
\begin{lemma}\label{homogeneous forcing adds no definable}
Let $V[G]$ be a forcing extension of $V$ by a weakly homogeneous forcing notion and $S\in V[G]$ is a subset of $V$ definable in $V[G]$ using parameters from $V$. Then $S\in V$.
\end{lemma}
\begin{proof}
\cite{jech forcing} proposition 2.2.
\end{proof}
The next observation is that $\kappa^+$ - closed weakly homogeneous forcings preserve definable tree property on $\kappa^+$.
\begin{lemma}\label{preservation of definable tree property under closed homogeneous forcings}
If definable tree property holds on $\kappa^+$ and $\mathbb{P}$ is a $\kappa^+$ - closed weakly homogeneous notion of forcing then in $V^\mathbb{P}$, $\kappa^+$ has the definable tree property.
\end{lemma}
\begin{proof}
Assume the definable tree property holds on $\kappa^+$ in $V$ and $T$ is a $\kappa^+$ - tree in $V[G]$ which is definable in the structure $(H_{\kappa^+}^{V[G]}, \in)$. Thus there is a first order formula with parameters from $H_{\kappa^+}^{V[G]}$ which defines $T$. By $\kappa^+$ - closure of forcing we have $H_{\kappa^+}^{V[G]}=H_{\kappa^+}^{V}$ and so $T$ is definable in $V[G]$ with parameters from $V$. Thus by homogeneity of forcing $\mathbb{P}$ and lemma \ref{homogeneous forcing adds no definable}, $T\in V$.
$T$, $dom(<_T)$, $ran(<_T)$ are sets of ordinals. All these sets are definable in $V[G]$ and so lie in $V$. Since for homogeneous forcings every set of ordinals definable in $V[G]$ with parameters from $V$, then both $T$ and $<_T$ are definable in $V$ as well.
Now by $\kappa^+$ - closure property of forcing we know that cardinals $\leq \kappa^+$ are preserved and so $T$ is a $\kappa^+$ - tree in the ground model. Consequently by definable tree property for $\kappa^+$ in $V$, $T$ has a cofinal branch $b$ in $V$. Again by $\kappa^+$ - closure of forcing, $b$ is a cofinal branch for $T$ in the generic extension too. So in $V[G]$ the definable tree property holds on $\kappa^+$.
\end{proof}
\begin{lemma}\label{generalization of leshem's result}
Let $\kappa$ be a regular cardinal and $\lambda>\kappa$ is a $\Pi_{1}^{1}$ - reflecting cardinal, then in $V^{Col(\kappa, <\lambda)}$ we have $\kappa^+ = \lambda$ and the definable tree property holds on $\kappa^+$.
\end{lemma}
\begin{proof}
A straightforward modification of the proof of theorem \ref{leshem's main result}.
\end{proof}
\begin{theorem}\label{from reflecting to tree property}
If there are proper class many $\Pi_{1}^{1}$ - reflecting cardinals in $V$, then there is a generic extension of $V$ by a weakly homogeneous forcing such that $GCH$ holds and successor of every regular cardinal has the definable tree property.
\end{theorem}
\begin{proof}
Let $\langle \kappa_\alpha: \alpha \in Ord \rangle$ be an increasing continuous sequence of cardinals such that $\kappa_0=\aleph_0$, and for each successor ordinal $\alpha, \kappa_\alpha$ is a $\Pi_{1}^{1}$ - reflecting cardinal and no $\kappa_\alpha,$ for limit ordinal $\alpha,$ is inaccessible (otherwise cut the universe).
Let $\mathbb{P}=\langle\langle\mathbb{P}_{\alpha}~|~\alpha\leq Ord\rangle, \langle\dot{\mathbb{Q}}_{\alpha}~|~\alpha\in Ord\rangle\rangle$ be the reverse Easton iteration such that
\begin{enumerate}
\item[(1)] $\mathbb{P}_0$ is the trivial forcing,
\item[(2)] For $\alpha=0,$ or $\alpha$ a successor ordinal, $ \Vdash_{\alpha}$ " $\dot{\mathbb{Q}}_\alpha=\dot{C}ol(\kappa_\alpha, < \kappa_{\alpha+1})$ '',
\item[(3)] For limit ordinal $\alpha, \Vdash_{\alpha}$ " $\dot{\mathbb{Q}}_\alpha=\dot{C}ol(\kappa^+_\alpha, < \kappa_{\alpha+1})$ ".
\end{enumerate}
Our defined forcing notion has the following properties:
\begin{lemma}\label{properties of forcing}
Let $G$ be $\mathbb{P}$-generic over $V$. Then
\begin{enumerate}
\item[(1)] $CARD^{V[G]}= \{\kappa_\alpha: \alpha \in Ord \} \cup \{ \kappa^+_{\alpha}: \alpha \in Ord, \alpha $ is a limit ordinal $\}$,
\item[(2)] If $\lambda$ is successor of a regular cardinal in $V[G],$ then $\lambda=\kappa_{\alpha+1},$ for some $\alpha$,
\item[(3)] If $\alpha=0$ or $\alpha$ is a successor ordinal, then $\mathbb{P}\simeq \mathbb{P}_\alpha * \dot{\mathbb{P}}_{[\alpha, \infty)},$ where
$\Vdash_\alpha$ " $\dot{\mathbb{P}}_{[\alpha, \infty)}$ is $\kappa_\alpha$ - closed and weakly homogeneous ''.
\item[(4)] If $\alpha$ is a limit ordinal, then $\mathbb{P}\simeq \mathbb{P}_\alpha * \dot{\mathbb{P}}_{[\alpha, \infty)},$ where
$\Vdash_\alpha$ " $\dot{\mathbb{P}}_{[\alpha, \infty)}$ is $\kappa^+_{\alpha}$-closed and weakly homogeneous ''.
\item[(5)] $GCH$ holds in $V[G]$.
\end{enumerate}
\end{lemma}
\begin{proof}
The proof is standard. The homogeneity part follows from the work of Friedman-Dobrinen \cite{friedman}.
\end{proof}
Now note that in $V[G]$ the definable tree property holds for successor of every regular cardinal. To see this let $\lambda$ be the successor of a regular cardinal in $V[G]$. By part (2) of lemma \ref{properties of forcing}, there is an ordinal $\alpha$ such that $\lambda=\kappa_{\alpha+1}$. Then we have the following cases:
\textit{Case 1:} $\alpha=0$ or $\alpha$ is a successor ordinal.
\noindent As $\kappa_{\alpha+1}$ is a $\Pi_{1}^{1}$ - reflecting cardinal in $V$ and all steps of our forcing up to $\mathbb{P}_{\alpha}$ are small with respect to cardinal $\kappa_{\alpha+1}$, it follows from lemma \ref{preservation of reflecting cardinals under small forcings} that $\kappa_{\alpha+1}$ remains $\Pi_{1}^{1}$ - reflecting in $V^{\mathbb{P}_{\alpha}}$. By definition of our iteration, we force with $\dot{C}ol(\kappa_\alpha, < \kappa_{\alpha+1})$ in $V^{\mathbb{P}_{\alpha}}$. By lemma \ref{generalization of leshem's result}, $\lambda$ will have definable tree property in $V^{\mathbb{P}_{\alpha+1}}$. Also if we split our iteration at $\alpha$ as $\mathbb{P}\simeq \mathbb{P}_\alpha * \dot{\mathbb{P}}_{[\alpha, \infty)}$, then by part (3) of lemma \ref{properties of forcing} the tail forcing at step $\alpha$ is $\kappa_{\alpha}$ - closed and weakly homogeneous. If $\alpha$ is a successor ordinal like $\beta+1$ then by lemma \ref{preservation of definable tree property under closed homogeneous forcings} it follows that the already forced definable tree property on other successors of regular cardinals less than $\lambda$ which are in the form $\theta=\kappa_{\gamma+1}$ for some $\gamma<\beta$, won't be destroyed by tail forcing because it is weakly homogeneous and has enough closure. Also in the case $\alpha = 0$ there is no successor of a regular cardinal below $\lambda$ and so we have nothing to prove.
\textit{Case 2:} $\alpha$ is a limit ordinal.
\noindent Note that by continuity of the sequence $\langle \kappa_\alpha: \alpha \in Ord \rangle$, we have $\kappa_{\alpha}=sup\{\kappa_{\beta}~|~\beta<\alpha\}$. By smallness of forcing up to stage $\alpha$ with respect to $\kappa_{\alpha+1}$, $\kappa_{\alpha+1}$ remains $\Pi_{1}^{1}$ - reflecting in $V^{\mathbb{P}_{\alpha}}$. Thus the inequality $\kappa_{\alpha}<\kappa_{\alpha}^{+}<\kappa_{\alpha+1}$ holds in $V^{\mathbb{P}_{\alpha}}$. By definition of our iteration we force with $\dot{C}ol(\kappa^+_\alpha, < \kappa_{\alpha+1})$ in this stage which by lemma \ref{generalization of leshem's result} makes the definable tree property on $\kappa_{\alpha+1}$ true in $V^{\mathbb{P}_{\alpha+1}}$. By part (4) of lemma \ref{properties of forcing} if we split our iteration as $\mathbb{P}\simeq \mathbb{P}_\alpha * \dot{\mathbb{P}}_{[\alpha, \infty)}$, the tail forcing is weakly homogeneous and $\kappa_{\alpha}^{+}$ - closed which by lemma \ref{preservation of definable tree property under closed homogeneous forcings} is sufficient to preserve the already forced definable tree property on all successors of regular cardinals less than $\lambda=\kappa_{\alpha+1}$.
\end{proof}
\section{Definable tree property at successor of a singular cardinal}
In this section we give the proof of the main theorem \ref{definable tree property at a singular}.
\subsection{Supercompact extender based Prikry forcing}
In this subsection, we present Merimovich's supercompact extender based Prikry forcing which appeared in \cite{mer4}. We present it in some details as we need it for later use.
For each $\alpha<j(\lambda)$ let
$\lambda_\alpha$ be minimal $\eta<\lambda$ such that $\alpha < j(\eta),$
and let $E(\alpha) \subseteq P(\lambda)$ be defined by
\begin{center}
$A\in E(\alpha) \Leftrightarrow \alpha \in j(A).$
\end{center}
Note that each $E(\alpha)$ is a $\kappa$-complete ultrafilter on $\lambda$ and it has concentrated on $\lambda_\alpha$. Also let
\begin{center}
$i_\alpha: \text{V} \rightarrow \text{N}_\alpha \simeq \text{Ult(V}, E(\alpha)).$
\end{center}
Finally put
\begin{center}
$E= \langle \langle E(\alpha): \alpha<j(\lambda) \rangle, \langle \pi_{\beta, \alpha}: \beta, \alpha< j(\lambda), \alpha\in range(i_\beta) \rangle \rangle$
\end{center}
to be the extender derived from $j$, where $\pi_{\beta, \alpha}: \lambda \rightarrow \lambda$ is such that $j(\pi_{\beta, \alpha})(\beta)=\alpha$ (such a $\pi_{\beta, \alpha}$ exists as $\alpha \in range(i_\beta)$). Let $i: \text{V} \rightarrow \text{N }\simeq \text{Ult(V, E)}$ be the resulting extender embedding. We may assume that $j=i.$
\begin{definition}
Let $d\in [j(\lambda)]^{<\lambda}$ be such that $\kappa, |d| \in d.$ Then $\nu \in \text{OB(d)}$ if the following conditions hold:
\begin{enumerate}
\item $\nu: \dom(\nu) \rightarrow \lambda,$ where $\dom(\nu) \subseteq d,$
\item $\kappa, |d| \in \dom(\nu),$
\item $|\nu| \leq \nu(|d|),$
\item $\forall \alpha<\lambda$ $(j(\alpha)\in \dom(\nu) \Rightarrow \nu(j(\alpha))=\alpha ),$
\item $\alpha\in \dom(\nu) \Rightarrow \nu(\alpha) < \lambda_\alpha,$
\item $\alpha < \beta$ in $\dom(\nu) \Rightarrow \nu(\alpha) < \nu(\beta).$
\end{enumerate}
Also for $\nu_0, \nu_1 \in \text{OB(d)},$ set $\nu_0 < \nu_1$ if and only if
\begin{enumerate}
\item [(6)]$\dom(\nu_0) \subseteq \dom(\nu_1),$
\item [(7)] For all $\alpha\in \dom(\nu_0)\setminus j[\lambda], \nu_0(\alpha) < \nu_1(\alpha).$
\end{enumerate}
\end{definition}
We now define the forcing notion ${\mathbb{P}}^*(E, \kappa, \lambda)$ as follows:
\begin{definition}
${\mathbb{P}}^*(E, \kappa, \lambda)$ consists of all functions $f: d \rightarrow \lambda^{<\omega}$, where $d\in [j(\lambda)]^{<\lambda}$, $\kappa, |d| \in d,$ and such that
$(1)$ For any $j(\alpha) \in d, f(j(\alpha))=\langle \alpha \rangle,$
$(2)$ For any $\alpha\in d\setminus j[\lambda], $ there is some $k<\omega$ such that
\begin{center}
$f(\alpha)= \langle f_0(\alpha), \dots, f_{k-1}(\alpha) \rangle \subseteq \lambda_\alpha $
\end{center}
$\hspace{0.7cm}$ is a finite increasing subsequence of $\lambda_\alpha.$ For $f, g\in {\mathbb{P}}^*(E, \kappa, \lambda),$
\begin{center}
$f \leq^*_{{\mathbb{P}}^*(E, \kappa, \lambda)} g \Leftrightarrow f \supseteq g.$
\end{center}
\end{definition}
\begin{remark}
$\langle {\mathbb{P}}^*(E, \kappa, \lambda), \leq^*_{{\mathbb{P}}^*(E, \kappa, \lambda)} \rangle \approx Add(\lambda, |j(\lambda)|).$
\end{remark}
\begin{definition}
Assume $d\in [j(\lambda)]^{<\lambda}$ and $\kappa, |d| \in d.$
Let $T \subseteq OB(d)^{<\xi} (1<\xi \leq \omega)$ and $n<\omega.$ Then
$(1)$ $Lev_n(T)=T \cap \text{OB(d)}^{n+1},$
$(2)$ $\Suc_T(\langle \rangle) = \Lev_0(T),$
$(3)$ $\Suc_T(\langle \nu_o, \dots, \nu_{n-1} \rangle)=\{\mu\in OB(d): \langle \nu_o, \dots, \nu_{n-1}, \mu \rangle \in T \}.$
\end{definition}
\begin{definition}
Assume $d\in [j(\lambda)]^{<\lambda}$ and $\kappa, |d| \in d.$
Let $T \subseteq OB(d)^{<\xi} (1<\xi \leq \omega)$. For $\langle \nu \rangle \in T,$ let
\begin{center}
$T_{\langle \nu \rangle}=\{ \langle \nu_o, \dots, \nu_{k-1} \rangle: k<\omega, \langle \nu, \nu_o, \dots, \nu_{k-1} \rangle \in T \}$
\end{center}
and define by recursion for $\langle \nu_o, \dots, \nu_{n-1} \rangle \in T,$
\begin{center}
$T_{\langle \nu_o, \dots, \nu_{n-1} \rangle}= (T_{\langle \nu_o, \dots, \nu_{n-2} \rangle})_{\langle \nu_{n-1} \rangle}.$
\end{center}
\end{definition}
\begin{definition}
Assume $d\in [j(\lambda)]^{<\lambda}$ and $\kappa, |d| \in d.$
We define the measure $E(d)$ on $\text{OB(d)}$ by
\begin{center}
$E(d)=\{ X \subseteq \text{OB(d)}: \text{mc(d)}\in j(X) \},$
\end{center}
where $\text{mc(d)}=\{\langle j(\alpha), \alpha \rangle : \alpha\in d \}.$
\end{definition}
\begin{definition}
Assume $d\in [j(\lambda)]^{<\lambda}$ and $\kappa, |d| \in d.$ Let $T \subseteq \text{OB(d)}^{<\omega}$ be a tree.
$T$ is called an $E(d)$-tree, if
\begin{enumerate}
\item $\forall \langle \nu_0, \dots, \nu_{n-1} \rangle \in T$ $(\nu_0 < \dots < \nu_{n-1}),$
\item $\forall \langle \nu_0, \dots, \nu_{n-1} \rangle \in T$ $(\Suc_T(\langle \nu_0, \dots, \nu_{n-1} \rangle)\in E(d)).$
\end{enumerate}
\end{definition}
\begin{definition}
Assume $c\in [j(\lambda)]^{<\lambda}$ and $A \subseteq \text{OB(d)}^{<\omega}.$ Then
\begin{center}
$A \upharpoonright c =\{\langle \nu_0 \upharpoonright c, \dots, \nu_{n-1} \upharpoonright c \rangle: n< \omega, \langle \nu_0, \dots, \nu_{n-1} \rangle \in A \}.$
\end{center}
\end{definition}
\begin{remark}
For $f\in {\mathbb{P}}^*(E, \kappa, \lambda),$ we use $\text{OB(f)}, E(f)$ and $\text{mc(f})$ to denote $\text{OB}(\dom(f)), E(\dom(f))$ and $\text{mc}(\dom(f))$ respectively.
\end{remark}
We are now ready to define our main forcing notion, ${\mathbb{P}}(E, \kappa, \lambda).$
\begin{definition}
$p\in {\mathbb{P}}(E, \kappa, \lambda)$ iff $p= \langle f^p, A^p \rangle$ where
$(1)$ $f^p \in {\mathbb{P}}^*(E, \kappa, \lambda),$
$(2)$ $A^p$ is an $E(f^p)$-tree.
\end{definition}
\begin{definition}
Let $p, q\in {\mathbb{P}}(E, \kappa, \lambda).$ Then $p \leq^* q$ ($p$ is a Prikry extension of $q$) iff:
$(1)$ $f^p \leq^*_{{\mathbb{P}}^*(E, \kappa, \lambda)} f^q,$
$(2)$ $A^p \upharpoonright \dom(f^q) \subseteq A^q.$
\end{definition}
\begin{definition}
Let $f\in {\mathbb{P}}^*(E, \kappa, \lambda), \nu \in OB(f)$ and suppose $\nu(\kappa) > \max(f(\kappa)).$ Then $f_{ \langle \nu \rangle}\in {\mathbb{P}}^*(E, \kappa, \lambda)$ has the same domain as $f$ and
\begin{center}
$f_{ \langle \nu \rangle}(\alpha) = \left\{ \begin{array}{l}
f(\alpha)^{\frown} \langle \nu(\alpha) \rangle \hspace{1.1cm} \text{ if } \alpha\in \dom(\nu), \nu(\alpha) > \max(f(\alpha)),\\
f(\alpha) \hspace{2.5cm} \text{Otherwise}.
\end{array} \right.$
\end{center}
Given $\langle \nu_0, \dots, \nu_{n-1} \rangle \in OB(f)^n$ such that $\nu_0(\kappa) > \max(f(\kappa))$ and $v_0 < \dots < \nu_{n-1},$ define $f_{\langle \nu_0, \dots, \nu_{n-1} \rangle}$ by recursion as
\begin{center}
$f_{\langle \nu_0, \dots, \nu_{n-1} \rangle}=(f_{\langle \nu_0, \dots, \nu_{n-2} \rangle})_{\langle \nu_{n-1} \rangle}.$
\end{center}
Let $p\in {\mathbb{P}}(E, \kappa, \lambda),$ and suppose $\langle \nu_0, \dots, \nu_{n-1} \rangle \in A^p$ is such that $\nu_0(\kappa) > \max(f^p(\kappa))$ and $v_0 < \dots < \nu_{n-1}.$ Then
\begin{center}
$p_{\langle \nu_0, \dots, \nu_{n-1} \rangle}=\langle f^p_{\langle \nu_0, \dots, \nu_{n-1} \rangle}, A^p_{\langle \nu_0, \dots, \nu_{n-1} \rangle} \rangle.$
\end{center}
\end{definition}
\begin{remark}
Whenever the notation $\langle \nu_0, \dots, \nu_{n-1} \rangle$ is used, where $\nu_0, \dots, \nu_{n-1} \in OB(f),$ it is implicitly assumed $\nu_0(\kappa) > \max(f(\kappa))$ and $v_0 < \dots < \nu_{n-1}.$
\end{remark}
\begin{definition}
Let $p, q\in {\mathbb{P}}(E, \kappa, \lambda).$ Then
\begin{center}
$p \leq q \Leftrightarrow \exists \langle \nu_0, \dots, \nu_{n-1} \rangle \in A^q$ $(p \leq^* q_{\langle \nu_0, \dots, \nu_{n-1} \rangle}).$
\end{center}
\end{definition}
Let us state the main properties of the forcing notion ${\mathbb{P}}(E, \kappa, \lambda).$ The proof can be found in \cite{mer4}.
\begin{theorem}\label{properties of prikry forcing}
Let $\text{G}$ be ${\mathbb{P}}(E, \kappa, \lambda)$-generic over $\text{V}$. Then
\begin{enumerate}
\item $\langle {\mathbb{P}}(E, \kappa, \lambda), \leq \rangle$ satisfies the $\lambda^+-c.c.,$
\item $\langle {\mathbb{P}}(E, \kappa, \lambda), \leq, \leq^* \rangle$ satisfies the Prikry property,
\item $\langle {\mathbb{P}}(E, \kappa, \lambda), \leq^* \rangle$ is $\kappa$-closed,
\item $cf^{\text{V[G]}}(\kappa)=\omega,$
\item All $\text{V}$-cardinals in the interval $(\kappa, \lambda)$ are collapsed,
\item $\lambda$ is preserved in $\text{V[G]},$
\item In $V[G], 2^\kappa=|j(\lambda)|.$
\end{enumerate}
\end{theorem}
It follows that $\text{V}$ and $\text{V[G]}$ have the same bounded subsets of $\kappa$ and $(\kappa^+)^{\text{V[G]}}=\lambda.$
\subsection{Projection of forcing notions}
Recall that we assumed $\lambda> \kappa$ is a measurable cardinal. Let $i: V \rightarrow N$ witnesses this; so $crit(i)=\lambda$ and
$^{\lambda}N \subseteq N.$ Consider the forcing notions ${\mathbb{P}}(E, \kappa, \lambda)$
and $i({\mathbb{P}}(E, \kappa, \lambda)).$ Also note that
by closure of $N$ under $\lambda$-sequences, we have
\[
{\mathbb{P}}(E, \kappa, \lambda)={\mathbb{P}}(E, \kappa, \lambda)_N,
\]
also it is clear that
\[
i({\mathbb{P}}(E, \kappa, \lambda)) = {\mathbb{P}}(i(E), \kappa, i(\lambda))_N.
\]
Now by working in $N$, define $\pi: i({\mathbb{P}}(E, \kappa, \lambda)) \rightarrow {\mathbb{P}}(E, \kappa, \lambda)$ as follows: let $p=\langle f^p, A^p \rangle \in i({\mathbb{P}}(E, \kappa, \lambda)).$ Set
\[
\pi(p)= \langle f^p \upharpoonright (\dom(f^p) \cap j(\lambda)), A^p \upharpoonright (\dom(f^p) \cap j(\lambda)) \rangle.
\]
The next lemma can be proved easily.
\begin{lemma}\label{projection lemma}
(In N) $\pi$ is a projection of forcing notions, in fact
$(1)$ $\pi(1_{i({\mathbb{P}}(E, \kappa, \lambda)) })=1_{{\mathbb{P}}(E, \kappa, \lambda)},$
$(2)$ $\pi$ is order preserving with respect to both $\leq$ and $\leq^*$ relations,
$(3)$ If $p \in {\mathbb{P}}(E, \kappa, \lambda), q \in i({\mathbb{P}}(E, \kappa, \lambda))$ and $p \leq \pi(q),$ then there exists $q^* \leq q$
such that $\pi(q^*) \leq^* p.$
\end{lemma}
\begin{proof}
Parts $(1)$ and $(2)$ can be proved easily, so we prove the part $(3)$. Thus let $p \in {\mathbb{P}}(E, \kappa, \lambda), q \in i({\mathbb{P}}(E, \kappa, \lambda))$ and
suppose that $p \leq \pi(q).$ Let $q^*=\langle f^*, A^* \rangle \in i({\mathbb{P}}(E, \kappa, \lambda))$ be such that:
\begin{enumerate}
\item $\dom(f^*)=\dom(f^p) \cup \dom(f^q),$
\item For $\alpha \in \dom(f^p), f^*(\alpha)=f^p(\alpha),$
\item For $\alpha \in \dom(f^q)\setminus \dom(f^p), f^*(\alpha)=f^q(\alpha),$
\item $A^*$ is an $E(f^*)$-tree,
\item $A^* \upharpoonright \dom(f^p) \subseteq A^p,$
\item $A^* \upharpoonright \dom(f^q) \subseteq A^q.$
\end{enumerate}
Then it is clear that $q^* \leq q$
and that $\pi(q^*) \leq^* p.$
The lemma follows.
\end{proof}
\subsection{Homogeneity of the quotient forcing}
Assume $H$ is $i({\mathbb{P}}(E, \kappa, \lambda))$-generic over $V$ and let $G$ be the filter generated by $\pi[H].$ By Lemma \ref{projection lemma} $G$ is ${\mathbb{P}}(E, \kappa, \lambda)$-generic over $V$, and in $V[G]$, we can consider the quotient forcing:
\[
i({\mathbb{P}}(E, \kappa, \lambda))/ G = \{p \in i({\mathbb{P}}(E, \kappa, \lambda)): \pi(p) \in G \}.
\]
In the next lemma we show that the above forcing has enough homogeneity properties. We will use this to show that some objects which are in $V[H]$
were already in $V[G].$
For a forcing notion ${\mathbb{P}}$ and a condition $p\in {\mathbb{P}},$ set ${\mathbb{P}} \downarrow p =\{q\in {\mathbb{P}}: q \leq p \}$ consists of all extensions of $p$ in ${\mathbb{P}}.$ The homogeneity of our quotient forcing follows from the next theorem.
\begin{lemma}\label{homogeneity lemma} (Homogeneity lemma)
Suppose $p, q \in i({\mathbb{P}}(E, \kappa, \lambda))$ so that $\pi(p)=\pi(q).$ Then there are $p^* \leq p, ~ q^* \leq q$ and an isomorphism
\[
\Phi: i({\mathbb{P}}(E, \kappa, \lambda))\downarrow p^* \cong i({\mathbb{P}}(E, \kappa, \lambda)) \downarrow q^*.
\]
\end{lemma}
\begin{proof}
Let $p_1 \leq p$ and $q_1 \leq q$ be such that
\begin{enumerate}
\item $\dom(f^{p_1})=\dom(g^{q_1}),$ call it $d$,
\item $A^{p_1} = A^{q_1},$ call it $A$.
\end{enumerate}
For each $n<\omega$ and every $\langle \nu_0, \dots, \nu_{n-1} \rangle \in A$ let $T ( \nu_0, \dots, \nu_{n-1}) \subseteq A_{\langle \nu_0, \dots, \nu_{n-1} \rangle}$
be such that
for all $\langle \nu \rangle \in T ( \nu_0, \dots, \nu_{n-1})$ and all $\alpha \in \dom(\nu),$
\[
\nu(\alpha) > \max(f^{p_1}(\alpha)) \Leftrightarrow \nu(\alpha) > \max(f^{q_1}(\alpha)).
\]
By Lemma 3.12 \cite{mer4}, there are $p^* \leq^* p_1$ and $q^* \leq^* q_1$ such that
\begin{enumerate}
\item [(3)] $f^{p^*}=f^{p_1}$ and $f^{q^*}=f^{q_1}$,
\item [(4)] For each $n<\omega$ and $\langle \nu_0, \dots, \nu_{n-1} \rangle \in A^{p^*},$
\begin{center}
$p^*_{\langle \nu_0, \dots, \nu_{n-1} \rangle} \leq^* \langle f^{p_1}_{\langle \nu_0, \dots, \nu_{n-1} \rangle}, T( \nu_0, \dots, \nu_{n-1}) \rangle$,
\end{center}
\item [(5)] For each $n<\omega$ and $\langle \nu_0, \dots, \nu_{n-1} \rangle \in A^{q^*},$
\begin{center}
$q^*_{\langle \nu_0, \dots, \nu_{n-1} \rangle} \leq^* \langle f^{q_1}_{\langle \nu_0, \dots, \nu_{n-1} \rangle}, T( \nu_0, \dots, \nu_{n-1}) \rangle$.
\end{center}
\end{enumerate}
We now define an isomorphism $\Phi$ from $i({\mathbb{P}}(E, \kappa, \lambda))\downarrow p^*$ onto $i({\mathbb{P}}(E, \kappa, \lambda))\downarrow q^*$
as follows: Assume $r \in i({\mathbb{P}}(E, \kappa, \lambda))$ and $r \leq p^*.$ Let $\Phi(r) \in i({\mathbb{P}}(E, \kappa, \lambda))$ be such that
\begin{enumerate}
\item [(6)] $\dom(f^{\Phi(r)})=\dom(f^r),$
\item [(7)] $\forall \alpha \in \dom(f^r)\setminus \dom(f^{p^*}), f^{\Phi(r)}(\alpha)=f^r(\alpha),$
\item [(8)] $\forall \alpha \in \dom(f^{p^*}), f^{\Phi(r)}(\alpha)=f^{q^*}(\alpha) \cup (f^r(\alpha) \setminus f^{p^*}(\alpha)),$
\item [(9)] $A^{\Phi(r)}=A^r.$
\end{enumerate}
By our choice of $T( \nu_0, \dots, \nu_{n-1})$'s, $\Phi(r)$ is well-defined and it extends $q^*,$ so $\Phi(r) \in i({\mathbb{P}}(E, \kappa, \lambda))\downarrow q^*$
and
\[
\Phi: i({\mathbb{P}}(E, \kappa, \lambda))\downarrow p^* \rightarrow i({\mathbb{P}}(E, \kappa, \lambda)) \downarrow q^*
\]
is well-defined. It is also easily seen that $\Phi$
is in fact an isomorphism. The lemma follows.
\end{proof}
\subsection{Completing the proof of main theorem \ref{definable tree property at a singular}}
Finally we are ready to complete the proof of theorem \ref{definable tree property at a singular}. Let $V[G]$ be the generic extension obtained by ${\mathbb{P}}(E, \kappa, \lambda).$
By theorem \ref{properties of prikry forcing}, in $V[G],$ $\kappa$ is strong limit singular of cofinality $\omega$
and $\kappa^+=\lambda.$ Further if $|j(\lambda)|> \lambda,$ then $2^\kappa > \kappa^+$ in $V[G].$ So it suffices to show that the definable tree
property holds in $V[G]$ at $\kappa^+=\lambda.$
Note that $H^{V[G]}(\lambda)=H^{N[G]}(\lambda)$. Now let $T \in V[G]$ be a $\lambda$-tree which is definable in $H^{V[G]}(\lambda)$ using parameters from $H^{V[G]}(\lambda)$.
Also consider the forcing $i({\mathbb{P}}(E, \kappa, \lambda))$, and let $H$ be $i({\mathbb{P}}(E, \kappa, \lambda))$-generic over $V$ so that $G$
is the filter generated by $\pi[H];$ this is possible as $\pi$ is a projection map. We have $i[G]=G \subseteq H,$ so we can lift $i$ to an elementary embedding
\[
i^*: V[G] \rightarrow N[H]
\]
which is definable in $V[H].$
Then $i^*(T) \in N[H]$ is an $i^*(\lambda)$-tree, and since $i^*(\lambda)=i(\lambda)> \lambda,$ we can take some $x \in i^*(T)_\lambda$, the $\lambda$-th level of $i^*(T)$.
Now consider
\[
b=\{y \in i^*(T): y <_{i^*(T)} x \}.
\]
Then $b$ is a branch of $T$ which lies in $N[H] \subseteq V[H].$ But $b$ is definable in $V[H]$ using parameters from $V[G],$
and hence using the homogeneity lemma \ref{homogeneity lemma}, $b \in V[G].$ Thus $T$ has a cofinal branch in $V[G]$, and the result follows.
\section{Open questions}
We proved that the consistency strength of having definable tree property for successor of every regular cardinal is exactly the consistency strength of having proper class many $\Pi_{1}^{1}$ - reflecting cardinals. As it is stated in the part (7) of proposition \ref{results}, the existing proof for the consistency of usual tree property for a much smaller subclass of successors of regular cardinals, namely $\{\aleph_n~|~1<n< \omega\}$, uses a very strong large cardinal assumption in order of $\omega$ - many supercompacts. We also decreased the large cardinal assumption necessary for proving the consistency of definable tree property at successor of a singular cardinal.
The question regarding the consistency and consistency strength of usual tree property for successors of all regular cardinals is still open. The questions related to the consistency of tree property for successors of all singular cardinals and also for all regular cardinals in general are also open. Inspired by these open problems regarding the usual tree property, the following similar questions about definable tree property arise:
\begin{question}
Is it consistent to have definable tree property for successor of every singular cardinal? What is the consistency strength of this statement?
\end{question}
\begin{question}
Is it consistent to have definable tree property for all regular cardinals? What is the precise consistency strength of it?
\end{question}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 2,470 |
F-1 World Grand Prix II és un videojoc de curses de Fórmula 1 per la Nintendo 64 que va ser llançat només a Europa l'any 2000. Aquest videojoc és la continuació del F-1 World Grand Prix. Hi ha la possibilitat de córrer amb cotxes de la Fórmula 1 en diversos circuits amb els conductors amb la llicència de l'any 2000.
Videojocs del 2000
Videojocs de curses per a Nintendo 64 | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 1,402 |
Q: Oracle:Why is select from user_ind_columns so slow? Oracle version: Oracle Database 11g Enterprise Edition Release 11.2.0.3.0 - 64bit Production
I'm trying to get metadata from my Oracle database (I'm a bit of a noob at Oracle, more of a Sybase man). When I do
select * from USER_IND_COLUMNS
it takes 4 minutes 41 seconds to return just 7211 rows! Can anyone tell me why this should be? In fact, getting metadata generally (in Toad or otherwise) seems incredibly slow. All suggestions appreciated, thanks.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 1,224 |
Antigua is the main gateway to St Martin Grand-Case for passengers from London and United Kingdom. Flights from London are operated by British Airways and Virgin Atlantic.
ST BARTH COMMUTER offers a Premium charter flights service between the airports of Antigua (ANU) and St Martin Grand-Case (SFG).
With our executive charter flights, enjoy the experience of chartering one of our aircrafts for yourself, your family and also your friends !
Enjoy the possibility to be FLEXIBLE : depart when you want from/to Antigua ! | {
"redpajama_set_name": "RedPajamaC4"
} | 2,232 |
Copyright © 2011 by Margaret Hoover
All rights reserved.
Published in the United States by Crown Forum, an imprint of the
Crown Publishing Group, a division of Random House, Inc., New York.
www.crownpublishing.com
CROWN FORUM with colophon is a registered trademark of Random House, Inc.
Library of Congress Cataloging-in-Publication Data
Hoover, Margaret.
American individualism : how a new generation of conservatives can save the Republican Party/
Margaret Hoover.
p. cm.
1. Republican Party (U.S.: 1854– ) 2. Conservatism—United States.
3. Generation Y—United States. I. Title.
JK2356.H66 2011
324.2734—dc22 2011008362
eISBN: 978-0-307-71817-4
Jacket design by Jean Traina
Jacket photograph by Deborah Feingold
v3.1
_For my husband, John Avlon,
the love of my life_
# **CONTENTS**
_Cover_
_Title Page_
_Copyright_
_Dedication_
**Introduction**
**1. Growing Up Hoover**
**2. Conservative Tribalism**
**3. Meet the Millennials**
**4. Generational Theft**
**5. Freedom Means Freedom for Everyone**
**6. Education Reform: A Civil Rights Win for the Millennial Generation**
**7. A New Republican Feminism**
**8. The Choice Dilemma**
**9. Conservative Environmentalism**
**10. A Nation of Immigrants, a Nation of Borders**
**11. Islamist Supremacy: A Millennial's Worst Nightmare**
**12. America the Exceptional**
**Acknowledgments**
**Notes**
_About the Author_
# **INTRODUCTION**
This book has its origins in a lightning-strike moment I experienced during the presidential campaign of 2004. At the time I was just another bubbly young junior staffer, still savoring my good fortune at having secured a position with the Bush-Cheney 2004 reelection campaign. I bounded through the halls of the redbrick office building in Arlington, Virginia, that housed President George W. Bush's campaign headquarters. After a morning staff meeting, I decided to swing by my office-mate's desk, tucked in a corner facing south toward the Potomac River, with a clear view of Georgetown and the rest of Washington, D.C. She was a coordinator in the political department, and that morning she seemed troubled.
"Look," she said, pointing to supporting documents that spelled out what became known as the anti-same-sex-marriage strategy.
The regional political directors of President Bush's campaign had been tasked with ensuring that battleground states sponsored ballot initiatives defining marriage exclusively as a union of one man and one woman, thus prohibiting same-sex marriage. As an additional measure to boost political enthusiasm, President Bush would ask Congress to pass an amendment to the Constitution that would federally define all marriages as being between a man and a woman. President Bush's plan was to campaign in these battleground states in support of the Federal Marriage Amendment, in a joint effort with statewide candidates to energize social conservatives, who, it was feared, might otherwise not come out to vote. While they were in the voting booth casting a ballot against same-sex marriage, the thinking went, they would also pull the lever for candidate George W. Bush, the man for whom I worked.
That moment remains vivid in my memory. After an instant of confusion, I felt a wave of disappointment crash over me. A series of questions raced through my mind: Why on earth did the campaign care about defining marriage as being between a man and a woman? Did President Bush really believe it was important to make laws that discriminate against gays and lesbians? Was this strategy necessary to ensure the president's reelection? Did President Bush think that mobilizing people against gay rights was a good thing? I looked up from the papers on my office-mate's desk and stared out through the window over the treetops toward the nation's capital, feeling sick to my stomach.
That was my Ms.-Hoover-Goes-to-Washington moment. I suddenly realized as never before that the Republican Party— _my_ party—was falling seriously out of step with a rising generation of Americans. These up-and-coming young voters value the ideal of individual freedom when it comes to gay rights, as they value some degree of reproductive freedom. And they do not support conservative activists' hard-line positions on immigration and environmentalism. It was on these questions, I felt, that the Republican Party was turning young voters away. In the years since 2004, the problem has only worsened. Unless the party can connect with a younger generation and, at the same time, offer solutions to meet the challenges of modern America, it is destined to remain at best a minority party—or worse, to fade into irrelevance. That would be tragic, because a modern brand of American conservatism is more urgently needed now than ever before.
Today, the United States faces a daunting array of challenges that threaten to imperil the American dream. Skyrocketing deficits and debt that amount to generational theft are staking a claim to the future prosperity of the youngest Americans. Our economy has lost its vibrancy, a quality that is increasingly associated with our less democratic trading partners in foreign markets. America's status as a world leader has diminished at a time when the world's most volatile region, the Middle East, is in a state of upheaval and the threat of Islamic supremacy looms. A failure to reconcile our twin needs for secure borders and new immigrants has led us into a protracted and divisive immigration crisis. Our schools, rather than facilitating equality of opportunity, increasingly constrict the upward mobility of young people. America's challenges all have one thing in common: They will likely be with us for a long time, and the next generation of Americans will have to solve them, or face American decline.
I believe that the next generation of Americans—the first to come of age in the new millennium—understands our situation well. The "millennials," born roughly between the years 1980 and 1999, perceive our political system at an impasse. They fear that as a nation we are incapable of addressing our problems. From every corner, they hear exhausted ideological rhetoric and see political gamesmanship at the expense of practical solutions.
Millennials thought they had found a candidate to break through the rhetorical divisions and excuses for inaction when they voted overwhelmingly for Barack Obama in 2008. They believed they were electing a man who, as he had promised, would bring change to Washington. His rhetoric spoke to their desire to move beyond the partisan divide of red states and blue states, to unify the country in order to solve problems. They have been disappointed.
As a result, millennials have yet to solidly commit to a political party. As a group, they are confident, open to change, globally oriented, techno-savvy, hyperconnected, and 50 million strong. By political orientation, the largest bloc are Independents, followed by Democrats, with Republicans a distant third. Though likely to call themselves liberal, millennials are not proponents of the big-government orthodoxy of modern liberalism. And yet, they are _socially liberal_ , adhering to the least traditional views of family, homosexuality, and gender roles. In this sense, they are passionate about expanding individual freedom. They also have idealistic expectations about what government can and should do, and are optimistic about the competence of their elected leaders.
Yet the millennials also demonstrate decidedly conservative tendencies, even though relatively few call themselves Republicans. They show signs of fiscal conservatism and cherish individual freedom, self-expression, and the ability to choose their own way in life. They have favorable attitudes toward business and individual entrepreneurship and are less likely than their parents to say that the government should take on more debt in order to help those in need.
Some might call them "fiscally conservative but socially liberal." They are ripe for a political party to come along and make the case for maximum freedom, fiscal responsibility, equality of opportunity, social mobility, individual responsibility, and service to community and country. They are likely to frustrate the ambitions of old-line political purists, because they do not fit neatly into the traditional partisan or ideological boxes.
Neither party, in my view, has secured a connection with millennial sensibilities. While Barack Obama succeeded in appealing to them in the 2008 election campaign, his party failed to do so during the 2010 midterms. Republicans, meanwhile, have never managed to connect. Nor have they seriously tried. That's the purpose of this book: to make the case to millennials that they should give the Republican Party a fair hearing and to make the case to my fellow Republicans that millennials are not a lost cause.
Republicans have generated some of the best ideas for tackling the most pressing problems facing millennials—the debt and the deficit, education reform, immigration reform, market-based health-care reform, and practical approaches to environmental conservation. And when it comes to protecting individual freedom, the Republican Party has always prided itself on taking the lead.
But let's face facts: The Republican Party's brand is damaged. The perception that the Religious Right and _social_ conservatives dominate the party apparatus is part of what has caused millennials to tune us out. Increasingly disconcerted by this widening gap between the perception of the Republican Party and the expectations of millennials, I undertook a journey in search of a fresh way for my party to appeal to the millennial generation.
I arrived at my destination with the help of an unexpected source: the writings of my great-grandfather, President Herbert Hoover, the thirty-first president of the United States. Growing up a Hoover, I had plenty of insults thrown my way simply on account of my family name. When you're related to one of the great mythical villains of American political history, you grow up constantly on the defensive. My friends' parents, my teachers in grade school and high school, my professors in college—they all pilloried my great-grandfather's presidency, indeed his entire career, as a failure of the highest order.
I won't say that this didn't cost a few tears or leave me without emotional scars, but being a direct descendant of Herbert Hoover has given me a special connection to an extraordinary individual. I never met my great-grandfather, but through family stories and my own exploration of the historical record, I learned about his orphan childhood, his success as a mining engineer, his globe-trotting years in business as a self-made millionaire before the age of forty, and his unprecedented achievement in building up nongovernmental organizations. Herbert Hoover was responsible for saving more lives from hunger and disease than anyone who has ever lived. His career as the most effective secretary of commerce in our nation's history and his efforts to build a public-private coalition to provide power to the nation's western states led to one of the most successful infrastructure projects in history: the Hoover Dam.
Herbert Hoover was millennial in spirit long before the term came into existence. He was like the Bill Gates or Mark Zuckerberg of his day—an innovator who believed in practical solutions, who dedicated his material and intellectual wealth to the service of the world, who believed in a philosophy encapsulated in the title of a little book he published, "American Individualism."*
I discovered this modest work, published in 1922, and was struck by its immediacy. It seemed it could have been written just yesterday, so contemporary were its themes. Largely overlooked by historians who hold forth on my great-grandfather's presidency, it is a powerful document: a broad and forceful statement of political philosophy and an extended essay on the relationship between the individual and the state. What I found most extraordinary was the relevance of its message and how it made the case for modern conservative thinking in an original way. Its author never refers to himself as a "conservative," yet he offers a compelling explanation for the vital importance of limited but _energetic_ government in America's democratic system. His book celebrates America's diversity of religious traditions and heritages. It emphasizes the individual's responsibility to serve his or her community. It presents what modern-day conservatives would appreciate as a fundamentally individual-centered view of society, and it defends that view in a refreshing and convincing manner. The influential American historian Frederick Jackson Turner said it "contains the New and Old Testament of the American gospel."
That earlier booklet inspired me to write this book. This is a moment when modern conservatism needs to be fresh and convincing. It has lately become deeply nostalgic for the Reagan years, which makes sense, as Reagan was the last conservative leader to enjoy broad political support. He was also the last leader to unite the various tribes of the conservative nation—the neocons, paleocons, social conservatives, fiscal conservatives, and libertarians. Conservatism has always been tribal, but in the absence of a charismatic leader, it has become a contentious and self-cannibalizing movement. And within the conservative nation are plenty of tribal warlords who devote their energies to eliminating rivals who they think aren't conservative enough. It's hardly the description of a political movement ready to lead America forward.
American individualism is not only a philosophy that can appeal to the millennial generation, but a prescription for how the conservative movement and the Republican Party can rise above their internecine feuding. By invoking the principles of American individualism, we have a template for addressing the challenges of the twenty-first century in a way that can make modern conservatism relevant for the rising generation.
I recognize that in writing this book, I am taking some risks. Anytime someone writes anything about an entire generation, they risk overgeneralization—and usually commit it. At the outset, let me say that millennials are like every other generation in certain respects. They are for one thing a diverse group. Just as baby boomers were not all wild-eyed, drugged-out attendees at Woodstock, neither are all millennials the hyper-texting Obamaites they are often made out to be. That said, those who study generational shifts in America can describe demographic groups by their shared experiences, by the historical context of their formative childhood years, and perhaps more important, by their outlook, as measured by opinion polls. In these ways, millennials are indeed different from those who came before them, and they do have certain features in common, by and large.
Experts will tell you that generational dividing lines are never carved in stone, that there is an arbitrariness to any generational divide. As someone born on the cusp of Gen X and the millennial generation, I share sensibilities of both but don't entirely identify with either. This makes my position advantageous, as an observer who can translate from one group to another and help bridge the divide between an older generation of Republicans and what I hope will become a new one.
This book is the culmination of that quest I undertook in search of a Republican-rooted philosophy that can appeal to a broad section of Americans, especially millennials. That a major source of my inspiration proved to be a work written nearly a century ago might be surprising. But millennials might also be surprised to discover fresh thinking and new ideas within the Republican Party, and in these sources find the hope and change they are looking for.
*For those of you interested in reading Herbert Hoover's "American Individualism" in its entirety, I've posted it on my website, margarethoover.com.
# CHAPTER 1
# GROWING UP HOOVER
_"You have to do your own growing, no matter how tall your grandfather was."_
—IRISH PROVERB
THIS PAST CHRISTMAS I read a letter from my great-grandmother to my grandfather written on White House stationery. It was September 1931 and the envelope was addressed simply: "Allan Hoover, Stanford University." The nation's capital had awoken to a morning thunderstorm, she reported. In Europe, the banks and financial markets were in disarray. Here in America, investors were panicking, borrowers were in default or imminent danger of default, increasing numbers of loans once considered solid were becoming worthless, and American companies could not find the capital to stay in business. America's economy—the global economy, in fact—was grinding to a halt due to factors and forces that no economist had ever before witnessed. And my great-grandfather, the president of the United States, was trying to manage it all. This letter, from mother to son, cast the historic moment in personal terms: "He certainly has had his hand on the tiller in a hard storm and one still wonders what is going to happen next."
In most American families, the personal and the political rarely overlap so completely. Notes from mothers to sons are not typically written on White House stationery. After all, there have been only forty-four presidents. And as a direct descendant of one of them, I have been extraordinarily privileged to hear stories about how history was made from those who were in the room. I have read, and held in my hands, many letters like the one above detailing the inner thoughts of the people who were confronted with the greatest challenge in our nation's economic history.
And yet this privilege has come with a great cost. To be a direct descendant of Herbert Hoover is to inherit the full weight of history's disapproval. My eighth-grade textbook blamed my great-grandfather for everything from Black Tuesday's stock market crash in October 1929 to the 25 percent unemployment rates and breadlines that followed in the Great Depression. Countless history books have detailed "Hoovervilles," the cardboard shelters of the homeless; "Hoover hogs," edible armadillos; "Hoover flags," penniless pockets turned inside out; and "Hoover blankets," newspapers repurposed for outdoor sleeping. Hoover's name was so synonymous with hard times that the midwestern drought that led to the dust storms and failed harvests of the Dust Bowl somehow seemed to have been his fault.
We Hoovers have been in a defensive crouch for eight decades. Today, people say that our political culture has become coarse, uncivil, and even violent. But after Franklin Roosevelt defeated my great-grandfather for the presidency in 1932, my family bore the psychological scars for decades. I remember talking to my grandmother about what it was like in the 1930s when Hoover was in the political wilderness and Roosevelt had captured the nation's heart. She reminisced next to a photo of her father-in-law, "the Chief," smiling and smoking a favorite pipe at the Bohemian Grove. She spoke of his kindness and intelligence—and of the million dollars that the Democratic National Committee spent to destroy his reputation. The worst part about the attacks was that they worked. Roosevelt did not just win an election: he won the approval of history. Now my great American ancestor gets invoked every time a politician wants to score easy points by calling an opponent the "worst president since Herbert Hoover."
When my father was growing up in the 1950s, the Depression was still fresh in Americans' minds. My dad's experience was searing, as he was forced to defend himself in fistfights on school playgrounds. "Your granddad caused the Depression!"— _whack!_ Compared with that, I had it easy. By the early 1990s, when I was in school, I found that just the mention of my family name could still evoke negative feelings among some of my friends' parents, especially the committed Democrats. They didn't even try to hide their disapproval when it came to my political heritage.
I suppose some children might have tried to distance themselves from the family name. Some presidential descendants do exactly that in an attempt to wriggle out from under the shadow of history. But I knew about another Herbert Hoover. I knew from my grandparents about the private man, about his indomitable will, his surprising wit, and his capacity to put ideas into action. I knew that he was from modest means and had been orphaned at a young age, and that despite this background, he grew up to become one of the wealthiest self-made men in the world. I knew that before he was president, he was regarded as one of the leading lights of his generation. I knew he had risked his fortune in order to save Belgium from starvation during the First World War. I knew he had helped organize a massive effort to feed the people of Central and Eastern Europe after the war. I knew he was called "the master of emergencies," was considered a pioneer of international nongovernmental organizations, and was a man known to his contemporaries as the Great Engineer and the Great Humanitarian. And he achieved all that before the age of forty-five!
As a teenager, I could see that the world perceived Herbert Hoover as a cartoon villain. And like any teenager aroused to defend the defenseless, I thought it profoundly unjust. Sure, I had a distinct interest in seeing his name redeemed, but I also felt something deeper: the determination to help set history's record straight.
But to defend him, I had to learn far more about Herbert Hoover's life and times. I would have to peel back the layers of conventional wisdom that often obscure complex events. I would have to question those who offered a simplified explanation for a decade of economic upheaval. I also would have to rein in my instinct for defending the family name at all costs. I knew that I would have to forget at times that my last name was Hoover, and just see where the facts led me.
As it turns out, this quest ended up changing me profoundly. I learned not to blindly trust what others proclaimed to be historically true, because even the best scholars make errors and all "intellectuals" have their own biases. I learned not only that history is often written to serve the interests of the winners, but also to explain everything that follows in a coherent narrative—even when the facts are as elusive, ambiguous, and difficult to interpret as they were for the people living through them and looking for clarity at the time.
The advantage I had is that I started my work from a position of absolute skepticism about the conventional wisdom. I began to think independently. I discovered early on that anyone could be wrong in her or his assumptions. I learned to distrust groupthink and ideological orthodoxy. I saw how even self-described critics of our society—feminists at the women's college I attended, avowed Marxists in the Latin American Studies programs where I was enrolled—were often incurable go-along-to-get-along types who would never dare challenge their own ideological conventions. These were conformists parading around as nonconformists; they all dressed in the same clothes, read the same books, knew the same people, and voted for the same politicians.
And I learned something else: The more I pressed and the more I questioned, the more I liked uncovering the fuller, forgotten truth. It became infectious—a habit of mind that I began to turn to other parts of my life. I didn't just go along to get along. I was my own person. And that turned out to be the greatest gift my great-grandfather bestowed upon me across the generations: the courage to think independently.
I began my quest by reading the works of others: historians such as George H. Nash, Eugene Lyons, Richard Norton Smith, and great writers like William F. Buckley Jr. They had looked more seriously at Hoover's life and presidency, and they had discovered a picture far different from the one often presented by the traditional historical narrative. The picture of Hoover that emerged was of an organizational wunderkind, an extraordinary American whose life story had contemporary resonance. There is an abundance of objective information that can be used to vindicate his legacy. Enough time has passed that it is now possible to view Herbert Hoover through a nonpartisan lens as an American original.
Hoover's path was hardly easy. Unlike his contemporaries among the Eastern elite, he was born with no material advantages. He was the second son of an Iowa frontier blacksmith who died of a heart attack when young "Bertie" was six years old. Three years later typhoid fever took his mother, an outspoken Quaker minister, and the orphaned boy was separated from his siblings and sent to live with an eccentric uncle in Newberg, Oregon. In 1891, at age seventeen, he demonstrated sufficient character and promise to be admitted to the first class of Stanford University, which became his spiritual home. The rest of his life was shaped by his years at Stanford. He graduated with a degree in geology, a field valuable to some of the fastest-growing global industries of that day. It was there that he met his future wife, Lou Henry, who became Stanford's first woman to graduate with a degree in geology.
After Stanford, Hoover got his start on the lowest rung of the mining industry, pushing an ore cart in California mines, earning two dollars a week working ten-hour night shifts. He graduated to an office job as an assistant to a prominent mining engineer who then recommended him to an English firm that hired him to explore undeveloped mines in western Australia for possible investment. Good fortune shone upon him: Hoover recommended a site for a mine that turned into one of the largest gold veins in western Australia, and it remained active for more than six decades.
At age twenty-four, at the close of the nineteenth century, Hoover was a true global citizen. After Bert cabled a marriage proposal to Lou Henry from Australia, the couple wed in Monterey, California, and the next day set sail for China, where Hoover established new coal mines for his firm. There the newlyweds had front-row seats at one of the watershed events that ushered in the twentieth century: the Boxer Rebellion. My great-grandmother swept bullets from her porch each morning and passed the days avoiding artillery fire and studying Mandarin Chinese, adopting the character "Hu" as her name, which became my Chinese name when I studied Mandarin in Beijing one hundred years later. Trapped in the final encampment of foreigners in China, protected from thousands of Boxers by just two thousand Russian and British soldiers, Bert organized food supplies and Lou tended to the wounded. Tianjin was relieved by an international force in July of 1900, and the Hoovers escaped to England in a German mail boat. My great-grandmother kept amazing notes detailing her time in China in a journal her father had given her as a wedding present. She had a pioneering spirit to match her husband's.
Herbert Hoover referred fondly to his life prior to America's entrance into the First World War as his "years of adventure." This was no exaggeration. By the time he was in his late twenties, he had circumnavigated the globe five times by steamship, overseen mining operations on virtually every continent, and barely escaped the political upheaval that marked the end of colonial empires in China.
Hoover was put in charge of building and managing mines on every continent except Antarctica. He was like one of the tech tycoons of our age—by the time he was twenty-eight, he was the highest paid person in the world under the age of thirty, according to the _San Francisco Chronicle_ , which reported his annual salary as $33,000 (equivalent to more than $850,000 in 2010). Later, at the height of his career as an international businessman, he employed more than 100,000 people on four continents.
But Hoover's run of business success was halted by the outbreak of world war, after Germany invaded Belgium and France in 1914. With as many as nine million French and Belgian citizens in imminent danger of starvation, Hoover was asked by the American ambassador in London to organize what would become the first-ever international republic of relief, the Commission for Relief in Belgium (CRB). Hoover oversaw an effort to deliver food relief to these millions, and thanks to his success he became an international hero. Operating under its own flag, and with a monthly budget of $12 million supplied by voluntary donations and government grants, the CRB enjoyed a support system that included navy ships, factories, mills, and railroads. The CRB was managed so efficiently that after the war, Hoover was able to draw on its surplus operating funds and transform them into a scholarship fund to enable Belgian and American exchange students and scholars to pursue advanced degrees in the partner country. Since that time, the Belgian American Educational Foundation (BAEF) has provided more than three thousand Belgians and nine hundred Americans with the opportunity to spend a period of advanced study in the United States and Belgium. The ninety-year-old BAEF both commemorates and perpetuates the special Belgian-American friendship launched by Herbert Hoover.
After the United States entered the war, President Woodrow Wilson appointed him food administrator, in charge of managing the country's, and the army's, food supply. After the war he became, in essence, the food administrator for the world, as he oversaw the distribution of food relief to more than twenty countries throughout Europe and the Near East. In 1921 he led a successful campaign to combat a catastrophic famine in Soviet Russia. Hoover despised the system of Soviet Communism but insisted, "Twenty million are starving. Whatever their politics, they shall be fed!" and managed to secure a $20 million grant from the U.S. Congress for Russian food relief. PBS recently documented this heroic undertaking in the American Experience film _The Great Famine_ , which tells the story of how Hoover-led relief saved the lives of many millions, and how the Soviet regime later thoroughly erased the episode from Russia's history books. It was the first case of massive humanitarian aid being delivered to the population of an ideologically hostile government.
Seventy years later, I had the privilege of meeting one of those men whose lives he helped save. He was an eighty-year-old Russian man, who had seven decades earlier walked ten miles daily to a food distribution point for condensed milk and "Hoover rolls." He wept as he grasped my young hand. My brother and I stammered, and tried to tell him how grateful we were for his thanks. But the truth is, we felt so unworthy of his thanks and were humbled to be related to a man whose compassion and resourcefulness had saved this man's life, and the lives of millions of others like him. Through this old man's still-thick Russian accent, he told us how his entire outlook on life had been transformed by the example of Herbert Hoover's generosity. He decided to come to America, believing that a country that fed its enemies must be great. He went on to become an inventor and developed a substance that was used to remove static from the Space Shuttle's surfaces. With a wink, he informed me that if we rubbed a golf ball with it before teeing off, it would fly fifty yards farther than normal. I was thirteen at the time, and since then I have often contemplated how radically different this man's life would have turned out without the intervention of Herbert Hoover.
This man was hardly alone. According to Hoover's biographer, George Nash, Hoover was directly responsible for saving the lives of as many as one-third of Europe's population during and immediately after World War I.
Here was someone who could have devoted himself completely to making money. Instead, he spent years trying to rescue millions from hunger and starvation, making an emphatic decision to "let the fortune go to hell." That impulse—to use one's skills for as elevated a purpose as possible—reminds me of the values cherished by many of my friends in their twenties and thirties today, millennials with a commitment to public service. Like Bill Gates, who stepped down as the head of Microsoft in order to pursue worldwide philanthropy, Hoover decided to become an international activist in order to advance the greater well-being of mankind. He was an enormously successful businessman who believed that the best way to build on his success was to serve society as a whole.
As a Quaker with pacifist instincts, Hoover found himself profoundly impacted by his experience as a witness to the human suffering and "rivers of blood" caused by Europe's wartime carnage and revolutionary aftermath. He was also influenced, during the First World War, by the autobiography of Andrew White. White, the first president of Cornell University, was also a historian and diplomat whose personal collection of artifacts and documents relating to the French Revolution greatly contributed to posterity's understanding of that watershed event. Reading White's autobiography, Hoover realized that he himself was in a position to collect artifacts and documents relating to the tumultuous military, political, and economic events unfolding all around him. While overseeing postwar food relief efforts throughout Europe, he began to organize the collection of such materials, and in 1919 he made a gift of $50,000 to Stanford University to house what began as the Hoover War History Collection and would become the world's largest private repository of documents relating to twentieth-century political history. The Hoover Institution on War, Revolution and Peace, as it is known today, is my great-grandfather's proudest legacy. It remains dedicated to its original purpose. The Hoover Institution's mission, as Hoover stated it in 1959, is, "from its records, to recall the voice of experience against the making of war, and by the study of these records and their publication, to recall man's endeavors to make and preserve peace, and to sustain for America the safeguards of the American way of life."
Hoover returned home a hero after the war, one of the best-known men in America. After serving in Woodrow Wilson's Democratic administration, he was encouraged to run for president by people in both parties, including one young assistant secretary of the Navy named Franklin Roosevelt, who wanted him to run as a Democrat. Instead, after the 1920 election, Republican president Warren Harding made him commerce secretary. Harding's successor, Calvin Coolidge, called him "Wonder Boy." A newspaper cartoonist joked that his title was "Secretary of Commerce and Under-Secretary of Everything Else."
Hoover was not a man brought up in politics and patronage. As secretary of commerce, he took no interest or pleasure in settling scores or carving out special deals for political allies. Instead, he set out to solve some of the basic problems of a modern industrial economy. We take for granted that businesses will compete on a level playing field. We assume that every electrical appliance will use the same AC/DC current and that most products will be sized using the same standard. But in Hoover's day, that wasn't the case. So he aimed to bring order out of chaos.
In his role as commerce secretary, he sought to turn the federal government into a kind of referee for the free market. Ever wonder why you purchase eggs by the dozen? Or why bricks are all 3⅝ X 2¼ X 8 inches? Or why milk is pasteurized and sold in quarts and gallons, why tires for automobiles are of standard sizes, or why traffic lights and highway safety standards don't vary from state to state? All these standardizations occurred because Hoover believed the government could make industry more efficient. He didn't believe in forcing industries into compliance through legislative action, but instead initiated hundreds of conferences and meetings to build consensus about the best path forward. He focused on how voluntary cooperation between industry and government, driven by common goals, could improve industry's performance.
Hoover recognized that government has an important role to play. He championed efforts to significantly reduce the incidence of child labor. He pushed to eliminate the seven-day workweek in the steel industry. Both efforts succeeded not because Hoover jammed the legislation down the throats of businesses, but because he used conferences, studies, and public exposure to show industries how they could benefit from improving efficiency through standardization. For example, he encouraged the home construction industry to standardize their key building materials. The result was a significant decrease in the cost of building a home, making home ownership more affordable for Americans of modest means, and thus increasing the overall size of the home construction market.
In his work, Hoover drew on a diversity of thinking and ideas. He wanted the government to be a _catalyst_ for solutions, not a designer of them. He delighted in the search for fresh approaches that emerge from genuine collaboration. Eight decades later, this approach to problem solving is the same one preferred by members of the millennial generation.
Hoover was also a believer in the power of technology. He recognized the capacity of radio to transform communications, and was the first person ever to appear via television transmission (in 1927!). He anticipated the future of air travel, and the first airfield in the Washington, D.C., area was called Hoover Field in recognition of his role in establishing aviation standards.
When his governing philosophy emerged, it was different from what had prevailed in the Harding and Coolidge administrations: laissez-faire policies that he scornfully regarded as "every man for himself and the devil take the hindmost." He saw a vigorous—but limited—role for government to play, establishing the rules of the road: helping to ensure a fair-framework of equal opportunity within which people would be free to live their lives to the best of their abilities.
In 1922, he articulated his philosophy in a commencement address that he never delivered but which was published as a small book called "American Individualism." It was a statement of his beliefs and a defense of the American system at a time when ideologies like communism and fascism had begun to challenge old-world assumptions. At the time of its publication, the _New York Times Book Review_ wrote that it was "among the few great formulations of American political theory" and Fredrick Jackson Turner wrote that "it contains the New and the Old Testament of the American gospel." "American Individualism" not only offered a sketch of Hoover's political philosophy. It suggested a path forward for America, one that remains relevant enough today to have animated my own thinking on the subject—and to have inspired the title of this book.
In the spring of 1927, Hoover was again called on to oversee humanitarian relief, this time in the American heartland, when the Mississippi River flooded well beyond its banks. The Great Mississippi River Flood of 1927 was the Hurricane Katrina of the 1920s: it displaced more than one and a half million Americans, destroyed two million acres of crops, and killed thousands of cattle and other livestock. Hoover left Washington and went to the Midwest to coordinate with local governments to build tent cities complete with beds, electricity, running water, hospitals, and kitchens. He managed to have these facilities in place by the time the floodwaters had crested.
Hoover did all this without federal dollars, choosing instead to raise private funds for the relief efforts. The fund-raising drive succeeded, bringing in $25 million in donations and low-interest loans, all of which were repaid. Later, in his memoirs, he remarked about the rescue efforts that "those were the days when citizens expected to take care of one another in time of disaster and it had not occurred to them that the Federal Government should do it."
Hoover's heroic coordination of the Mississippi River flood relief catapulted him to new heights in American politics. The chorus at the 1928 Republican National Convention was "Who but Hoover?"
Toward the end of his triumphant 1928 campaign for president, Hoover gave a speech that summed up his Republican Party's philosophy. He called it "Rugged Individualism." It is a speech that is still cited today—and it was in many ways a campaign-style distillation of the ideas expressed in "American Individualism" six years earlier.
"The American system," Hoover said, "is founded upon the conception that only through ordered liberty, freedom, and equal opportunity to the individual will his initiative and enterprise spur on the march of progress. And in our insistence upon equality of opportunity has our system advanced beyond all the world."
Hoover saw the 1928 presidential election, and the differences between the Republican and Democratic parties, as a "choice between the American system of rugged individualism and a European philosophy of diametrically opposed doctrines—doctrines of paternalism and state socialism.... Every step of bureaucratizing of the business of our country poisons the very roots of liberalism—[namely] political equality, free speech, free assembly, free press, and equality of opportunity. It is not the road to more liberty, but to less liberty."
The "Rugged Individualism" speech endures, as do the underlying differences between the Republican and Democratic parties. On the one hand, a focus on the individual and the free market, and on the other hand an emphasis on government intervention into private industry by means of regulation and bureaucratization—a contrast highlighted in our current political debates and at Tea Party rallies. The choice remains between an American model and a European model of governance, and the long-term stakes remain the same: the road to liberty or the road to serfdom.
Hoover won the presidency by the largest landslide in American history, capturing almost 60 percent of the popular vote, the only civilian to have ascended to the presidency without previously holding elective office.
But his run of good fortune and success soon ended. Seven months after he assumed office in 1929, the stock market crashed, triggering a series of events that led to the Great Depression. Hoover hardly sat on his hands in the wake of the sudden decline in the economy—instead he set to work launching or accelerating public works efforts. He proposed to Congress a $160 million tax cut along with a doubling of outlays for public buildings and dams, highways and harbors. By the spring of 1930, Hoover's response to the crisis had received widespread acclaim, as the _New York Times_ editorialized: "No one in his place could have done more.... Very few of his predecessors could have done as much."
But it was not enough. The collapse of foreign banks and international trade along with persistent drought conditions in the Midwest drove unemployment up from five million to more than eleven million by 1931. The economy would not recover for more than another decade.
Circumstances beyond Hoover's control greatly complicated his efforts to revive the economy, and there was one area where my great-grandfather was truly at a disadvantage. He was trying to cope with a global depression in a modern industrial economy without the benefit of some of the core theories of modern economics. Most major theories of macroeconomics were developed, in fact, by studying what happened during the Great Depression. John Maynard Keynes, Friedrich von Hayek, Milton Friedman, and other prominent economists emerged from that period with critical theories about monetary and fiscal policy, trade, and the interrelationship of taxes and the economy as well as the value of countercyclical policies. But none of this expertise was available to Hoover in 1930.
Not surprisingly, he made some mistakes. He signed the Smoot-Hawley legislation that increased U.S. tariffs on imported goods, contributing to a global decline in trade (although this affected only 4.2 percent of the U.S. economy, so its impact, it has been argued, has been overstated by later economists). He pressed employers to maintain wages, which made it harder for employers to hire workers. And he cooperated with Congress to balance the budget with the expiration of Coolidge-era tax cuts in 1932, which resulted in the largest tax increase in history. The fact is, a modern-day conservative would find none of these mistakes easy to explain, let alone defend. But in each of these cases, he was responding to political pressures without the ability to gauge how his actions might affect the economy, and his actions were certainly less radical than those he was urged to adopt by his political rivals or those later pursued by his successor. In our current debates, the _context_ for Hoover's decision making has been entirely lost.
I believe that the most effective defense of Hoover is not just that he did better that anyone could have, but that his successor did no better. Roosevelt's administration enacted dozens of laws, in the process creating an alphabet soup of agencies to attend to almost every aspect of American life and transforming the relationship between government and the individual. But Roosevelt's economic record was no better than Hoover's. The little-known truth is that the American economy actually _worsened_ during Roosevelt's second term, and did not begin to recover until the Second World War jump-started industry. Yet, somehow, Hoover still gets all the blame.
Roosevelt did succeed in one important respect: he _appeared_ to care more than his predecessor. Politics is perception, and this image of FDR was a direct outgrowth of his well-oiled political machine. During the 1932 campaign, Charles Michelson and the Democratic National Committee were armed with a million-dollar budget to organize a smear operation to destroy Hoover's reputation. Michelson was the hyperpartisan hack who coined the term "Hoovervilles" and continually attacked Hoover as an uncaring, do-nothing, apathetic leader who ignored the hardships suffered by "the little man." Historian Thomas Fleming characterizes Michelson's attacks as a series of "atrocious assaults on President Hoover, portraying him as a vicious egotist who had self-promoted his greatest living American title out of raw ambition for power." Nothing could have been further from the truth, and the smear campaign drove my great-grandmother to write long letters to her children for posterity detailing how deeply her husband did indeed care for "the little man."
During the Bonus Marches of 1932, when World War I veterans marched on Washington demanding their war bonuses early (which Congress categorically denied), twenty thousand men were camping out at the Anacostia Flats in Washington, D.C. Hoover secretly arranged for tents, food, and water; yet unlike today's politicians, he took no credit for this effort. Brought up in the Quaker tradition, he did not believe that glory should follow those who did good works. In this case, he didn't have to worry.
After his defeat in the 1932 election, Hoover retreated to his Palo Alto home (now occupied by the president of Stanford University). Unlike most ex-presidents up to that point, he remained active in public policy. Yet the bitter partisanship of the 1932 campaign did not let up. Immediately following his swearing-in ceremony, FDR personally rescinded Herbert Hoover's Secret Service detail. As William F. Buckley Jr. wrote, "Mr. Hoover went away unguarded, discredited, unloved." Harold Ickes, secretary of the interior, changed the name of one of Hoover's greatest achievements, the Colorado River Project, from Hoover Dam to Boulder Dam when it was dedicated in 1936, an injustice so petty and grotesque that Harry Truman corrected it upon assuming the presidency after Roosevelt's death.
Within one month of entering the Oval Office, Harry Truman summoned Herbert Hoover from the shadows and into the White House, rekindling his career in public service. Truman dispatched Hoover to Europe on a thirty-eight-nation tour to oversee food relief in the wake of the Second World War, thereby empowering him to reprise the role he had played so effectively three decades earlier. In 1947, Truman established the first of two "Hoover Commissions," each helmed by my great-grandfather and dedicated to streamlining the postwar executive branch and improving government efficiency. The two presidents' bipartisan friendship inaugurated an extended period of energetic activity during which Hoover wrote a dozen books and became a much sought-after counsel for political leaders of both parties, a performance that shaped the modern postpresidency.
And so Hoover turned the page to a new chapter in his life. No longer the mining engineer, the relief organizer, the cabinet officer, or even the president, Hoover became, in the words of historian Richard Norton Smith, "a philosopher of modern conservative thought." He had begun to play this role with the publication of "American Individualism" in 1922.
Hoover had seen firsthand Europe's experiments with socialism and Bolshevism and the rise of fascism, and he felt he ought to affirm why America must resist the temptation to follow those paths. He believed fundamentally that America's greatness lay in the individual, using God-given talents and working with others, as the essential building block of society. He believed in America's inherent dynamism—that the absence of old-world social castes in the United States, which allowed an individual to rise from extreme poverty to fabulous wealth within a lifetime, was a unique gift to Americans. He believed that the nation's diversity of faiths was a strength, endowing it with a richness of spiritual traditions, heritages, and beliefs from which it could continue to draw inspiration and new energy. He believed that America's record of welcoming new cultures and initiating new traditions set it apart. He believed that our civic tradition of volunteerism was the backbone of every community and could not be matched by centralized government action. At the core of his philosophy was the profound belief that the individual was the engine driving it all. And ten years later, he had a a stark philosophical counterpoint in Roosevelt's policies.
In 1934, spurred by his alarm at where Roosevelt was taking the country, Hoover wrote a direct rebuke of the New Deal in a book called _The Challenge to Liberty_. He saw in the New Deal a dramatic expansion of the federal government's role in the life of the individual. He saw this as a dangerous encroachment that would "cripple or abandon the heritage of liberty for some new philosophy which must mark the passing of freedom."
Hoover understood that promises extended by the government come with a cost. With every promise to ease the pain of loss comes the price to be paid by ordinary citizens. Once the government takes upon itself by force the role normally performed by the individual, he wrote, it becomes "the master of the man." And indeed, Hoover cited several areas where government had become the master: it devalued the currency and devalued existing debt; it forced collective bargaining on employers; it concentrated corporate power in oligopolies and trusts; it fixed prices; it levied taxes on food and clothing and other essentials; it began to engage in business activity that had always been reserved for the private sector, such as power generation; it told farmers what to grow and how much to grow; and it restricted expansion of business in specific industries. And it did all this with the power to prosecute and jail individuals found in violation of the new rules.
This concentration of power alarmed Hoover. Unfortunately, his words did not persuade Americans to turn away from the New Deal. Far from it: during the eight years before the start of World War II, the federal government grew enormously. And once in place, the new bureaucracy began to generate its own reason for being. Inertia would set in; no argument, even one rooted deeply in constitutional principles of limited government, could defeat it. The legacy of Roosevelt is not the individual programs introduced under the New Deal, but the idea that the federal government should remain a permanent fixture in the life of the individual, from birth until death. It is the legacy that every Democratic president and even some Republican presidents have sought to enlarge upon.
In contrast to the top-down vision of the New Deal bureaucrats, my great-grandfather understood that the proper role of the government was to support the individual's pursuit of opportunity, not to guarantee a particular outcome. He saw the danger in allowing a powerful government to replace many of the essential institutions of civic life—churches, community organizations, families—thus depriving America of the diversity of solutions and ideas that had made it great. He helped lay the foundation for what has become one of the central tenets of modern conservative thinking: "The government that is powerful enough to give you everything you need is powerful enough to take it all away."
Vindicating Herbert Hoover's legacy is an uphill battle, because today Democrats are _still_ running against my great-grandfather. Senator Joe Biden offered this remark during the 2008 presidential campaign: "I'm proud to say that we Democrats aren't experts at Herbert Hoover depression economics like John McCain and his pals. From Franklin Roosevelt to Bill Clinton, we just get elected to clean up the economic mess these Republicans leave behind." Senator Harry Reid had this to say about my great-grandfather: "For Herbert Hoover, I guess ignorance was bliss. It wasn't until the American people replaced this out-of-touch Republican president with a Democrat, Franklin Delano Roosevelt, that our nation's economic recovery began."
More disturbing, conservatives and Republicans have joined this chorus. In 2008, John McCain made history by becoming the first _Republican_ nominee to run against Hoover, when he said, "My friends, the last president to raise taxes during tough economic times was Herbert Hoover, and he practiced protectionism as well..." Mitt Romney piled on as recently as the 2011 Conservative Political Action Conference with "Obama's Hoovervilles," and even Rush Limbaugh has shamefully called our current president "Barack 'Hoover' Obama."
I happen to have an ongoing argument with pundit Glenn Beck, whom I have gotten to know a bit from our shared perch at Fox News over the past few years.
If you've watched Glenn Beck's television show during the past year, or listened to his radio program, you've been exposed to his crusade against progressivism. Beck has launched a movement to identify and expel progressives from government, and has framed it in a historical narrative that begins with Teddy Roosevelt's Bull Moose candidacy for president in 1912 and extends straight through to Barack Obama's White House. Beck has plopped Herbert Hoover into the middle of this narrative, mischaracterizing him as just another progressive.
I don't fault Beck for making this mistake once, or even twice. After all, Hoover did call himself an "independent progressive in the Republican tradition." He believed, for example, that children shouldn't work in factories, and that government had a responsibility to prevent child labor and unsafe working conditions. But does that make him a socialist? Not at all. Hoover was no progressive in the continuum from Woodrow Wilson to Franklin Delano Roosevelt. He was instead FDR's most prominent and consistent philosophical opponent. He detailed his opposition to socialism, big government, and, later, the New Deal in successive essays and books. Glenn Beck completely overlooks this evidence, and although I have brought it to his attention, he continues to repeat his mistake. I suppose it's easier to hammer away at Herbert Hoover. But on this, Glenn Beck is worse than Joe Biden: he gets it wrong even when he knows better. Certainly the liberal image of Hoover as an uncaring and out-of-touch, do-nothing president was always wrong. But conservatives who dismiss Hoover out of embarrassment, ignorance, or a misplaced sense of principle are just as misguided.
There are signs, however, that the tide is finally beginning to turn. The financial crisis of 2008 and the unprecedented experiments in federal takeovers of banks and auto companies, as well as the creation of penalties and taxes regulating the private health insurance market and now the federal effort to regulate carbon—all these measures have given conservatives, as well as independents, a reason to reconsider their vilification of Hoover. They are taking a fresh look at the history and the economics of the Great Depression and the New Deal.
The columnist and political thinker Thomas Sowell writes that "what was widely believed then and later was that the stock market crash of 1929 was a failure of the free market and the cause of the massive unemployment that persisted for years during the 1930s. Given the two most striking features of that era—the stock market crash and a widespread government intervention in the economy—it is not immediately obvious which was more responsible for the dire economic conditions. But remarkably little effort has been made by most of the intelligentsia to try to sort out the cause or causes. It has been largely a foregone conclusion that the market was the cause and the government intervention was the saving grace."
Amity Shlaes's 2006 bestseller, _The Forgotten Man: A New History of the Great Depression_ , inspired a wave of scholarship that has begun chipping away at the perception of FDR as the country's economic savior during the Great Depression. Other books, such as historian Burton Folsom Jr.'s 2008 _New Deal or Raw Deal: How FDR's Economic Legacy Has Damaged America_ and the 2009 work by Robert Murphy, _The Politically Incorrect Guide to the Great Depression and the New Deal_ , have challenged the predominant narrative that FDR's New Deal saved America from Herbert Hoover's Great Depression. In a _Wall Street Journal_ article titled "Did FDR End the Depression?" Folsom answered in the negative: "It's a myth. FDR did not get us out of the Great Depression—not during the 1930s, and only in a limited sense during World War II."
In the reevaluation of Herbert Hoover, Americans are becoming acquainted with his life and career prior to and after leaving the White House, when he made some of his most lasting achievements. Hoover's legacies are as diverse as the electrification of the neon skyline of the Las Vegas Strip, the vast agricultural economy of California, the Hoover Institution's contributions to public policy, and the descendants of the millions of Europeans he saved from starvation. Those who fixate only on making money or winning elections will find it an unhappy existence much of the time. My great-grandfather understood this, and that's why he chose to dedicate his life to serving others.
He was always oriented toward the future. He was, after all, the first president born and raised west of the Mississippi River, which was still considered America's great frontier. It is there where he was laid to rest, on the sunrise side of a hill in the humble hamlet of West Branch, Iowa, overlooking the cottage in which he was born and his presidential library.
Hoover was a globalist and a technologist, and he understood America's rising position in the world. He believed that America could extend its power not just with arms, but also with assistance. Surely no nation in the history of the world had ever done so much to help civilians in other nations as America did under Hoover's guidance. And that is a tradition that continues to this day.
These are all values that I see as familiar, because they are the values of my generation. In some ways, Herbert Hoover can be considered a millennial in spirit: young at the turn of the century, aware of America's past but deeply committed to building its future. His greatest passion and highest calling was service to others, and he measured his life's successes not in dollars and votes but in results achieved. He lived a life that millennials today would embrace, and I believe he gave voice to their interests, and those of every generation committed to the ideals of American individualism.
# CHAPTER 2
# CONSERVATIVE TRIBALISM
_"The term conservatism has come to cover so wide a range of views, and views so incompatible with one another, that we shall no doubt see the growth of hyphenated designations, such as libertarian-conservative and aristocratic-conservative."_
—MILTON FRIEDMAN, 1962
## **Which Tribe Do You Belong To?**
Paleocons, neocons, lib-cons, enviro-cons, Crunchy Cons, so-cons, Religious-Right conservatives, traditionalist conservatives, southern conservatives, western conservatives, Goldwater conservatives, Tea Party conservatives—if you are somewhere right of center, which tribe do you belong to?
Growing up in Colorado, I was reared in the spirit of what my father described at my wedding as "western conservatism": individualism tempered by responsibility for the community; a predilection for limited federal government, lower taxes, the entrepreneurial spirit, and individual initiative; and an appreciation for the idea of American exceptionalism. Or put more crassly, western conservatives are the ranchers who pump their shotguns before yelling, "Keep your government off my land and out of my bedroom." That's the tribe in which I was raised, and yes, my dad gave me my first shotgun when I was twelve.
But in my journey across the conservative universe—through America's West Coast and East Coast conservative think tanks and its activist groups inside the Beltway and beyond, and as a result of my employment as a staffer on Capitol Hill, in political campaigns, and in the White House—I have discovered that Milton Friedman's description of conservatism is especially apt: Conservatism is a nation of tribes, governed by warlords, spouting often incompatible and irreconcilable philosophies and principles.
Thirty years ago, Ronald Reagan was able to bring harmony to the cacophony of conservative interests, and for a brief time they all sang from the same song sheet. But since then, for most of the time, tribalism has ruled, with one tribe ascending in national influence as another descends. Without a unifying leadership or a pragmatic campaign to rally around, conservatives have spent a lot of time over the past two decades in a proverbial circular firing squad, engaged in a deadly squabble over who is and who isn't conservative enough.
But millennials don't see all that. The truth is, most millennials think conservatism means "social conservatism," and to put it mildly, they are not impressed.
Millennials are the most ethnically diverse, nonwhite generation in American history. They are the most socially liberal. A sound majority of them believes that homosexuality should be accepted by society. They are highly urban and suburban, not small-town or rural. They are worldly, and not necessarily besotted with the idea of American exceptionalism. As a generation, they have the lowest level of affiliation with organized religion, despite being very "spiritual." So what they see, however mistakenly—a Republican Party run by a bunch of old white guys and a few gals from nonurban, mostly southern and midwestern America bent on restricting gay rights and abortion and populated by flag-waving America-firsters—doesn't leave them with a warm and fuzzy feeling.
Of course, the millennials aren't getting the full picture. There is far more to conservatism than the social conservative activists—or so-cons, as I like to call them. Conservatism is actually a pretty vibrant nation, with all our different tribes. So for the benefit of my millennial readers, and anyone else who thinks conservatism is just a one-note movement, in the pages that follow I offer a panoramic view of today's conservative movement. Call it a field guide to modern American conservatism.
A disclaimer first: Serious studies of the American conservative movement, its history and its many facets, have been written by giants such as George H. Nash, William F. Buckley Jr., and Lee Edwards, among others. I encourage anyone interested in the movement to read their works. This chapter isn't meant as a substitute for those works, merely an overview.
The biggest thing to remember is that _conservatism_ is a deceptive word. First, conservatism isn't necessarily trying to conserve anything. Second, conservatism isn't an _ism_. It isn't a single political ideology but a movement made up of several mini-movements, each with its own governing philosophy. The movement's most esteemed historian, George Nash, says this: "Perhaps the most important thing to understand about modern American conservatism is that it is not, and has never been, univocal. It is a _coalition_ , with many points of origin and diverse tendencies that are not always easy to reconcile with one another. Historically, it has been a river of thought and activism fed by many tributaries: a wide and sometimes muddy river, but one with great power, so long as the tributaries flowed into the common stream."
So where did modern American conservatism begin? The fact is, many elements of America's modern conservative movement have been around forever. You will find strains of modern conservative thinking in the ancient classics of Greece and Rome, the Hebrew scriptures, the Christian Gospel, classical liberalism spun from the French Revolution, the writings of Edmund Burke, and of course among the writings of America's Founding Fathers.
But the event that marked the emergence of an organized modern conservative coalition in America was the publication of the first issue of William F. Buckley Jr.'s _National Review_ on November 19, 1955. In his mission statement for _National Review_ , Buckley spelled out the priorities for which "conservatism" should stand: resisting government expansion to protect individual freedom; championing "the competitive price system," or classical economic liberalism; confronting concentrations of power in corporations and syndicates; defeating communism; supporting a two-party political system that practices transparent and honest debate; understanding the reality of the human condition; challenging intellectual conformity in the arts, culture, and education; opposing internationalism when it undermines the United States and its autonomy.
What is striking is that Buckley was not trying to promote some kind of preservation of an old-world order, as the word _conservative_ misleadingly implies. In the mid-1950s, the dominant political philosophy was the liberalism that had established itself during the twelve years of Franklin Delano Roosevelt's presidency and that had retained prominence afterward, despite the election of Republican president Dwight D. Eisenhower. When Buckley famously declared that conservatives were standing athwart history, yelling "Stop!" he was referring to the seeming inevitability of the growth of government, and what he felt was the transformation of the American character by the fundamental altering of the relationship between the individual and the state. From the very start, the modern conservative movement was defined by its unwillingness to go along with the status quo. It was, at its heart, a rebellious movement identified by opposition to the accepted political establishment of the time. Not exactly conservative—in fact, somewhat revolutionary.
Buckley's mission statement touched on all facets of public and private life: international policy, domestic policy, philosophy, religion, and economics. Buckley's ambition was to bring diverse groups resistant to liberal dogma together under a single banner. The three primary components of this coalition—economic libertarians, traditionalists, and anticommunists—were later joined by new groups, including neoconservatives and the Protestant evangelical Religious Right. And that diversity persists today.
Here are the principal groups that have joined this loose confederation of modern conservatism from its founding down to the present day:
## **Economic Libertarians and Fiscal Conservatives**
The New Deal had its opponents, even during the depths of the Great Depression. People besides my great-grandfather saw Roosevelt's experiments with the economy as unlikely to end the Great Depression. Instead, they saw in them the roots of a radical realignment of the relationship between the individual and the federal government. Economic libertarians, or "classical liberals," were one of the first three coalition partners in the budding conservative movement. They believed that the ever-expanding federal government would be the root of economic stagnation and a permanent impediment to individual initiative and individual liberty. They longed for an unfettered free market, where those with ambition could benefit fully from their talents and wits. To economic libertarians, modern liberalism was merely a softer version of socialism and communism. Indeed, they believed that many aspects of the New Deal were leading to central planning and warmed-over socialism and that the state, through taxation and regulation, would gradually sap society of its productive and creative energies. At the core of this philosophy was a concern for the protection of private property—things owned by individuals should not be appropriated by the state for any reason.
The leading lights of this branch of the movement were thinkers, not populists. They were not even self-described conservatives. Friedrich von Hayek's _The Road to Serfdom_ , Milton Friedman's _Capitalism and Freedom_ , and Ayn Rand's fiction and nonfiction (which deeply influenced economists like Alan Greenspan) became and remain to this day among the most important works of this tribe, and are by far the most influential economic contributions of the conservative movement as a whole.
At the same time, a cadre of economists in the last fifty years has reevaluated the state-driven, Keynesian theories that underpinned the New Deal and subsequent liberal experiments in economics. Most notably, Milton Friedman, Ludwig von Mises, and, later, Arthur Laffer have shown that historically the greater the government's involvement in the economy, the smaller the resulting economy turns out to be. They have also shown that if you raise taxes, the economy shrinks somewhat as people react to the disincentives of higher tax rates, while tax revenue doesn't rise as expected. On the other hand, these conservative economists have shown if you _cut_ tax rates, people see more reason to work and to invest, and the economy grows faster as a result. In the long run, this course of action produces greater tax revenues. The underlying assumption is that the economy is not some static machine. Rather, it is a somewhat temperamental beast and must be approached cautiously. This simple idea—that the economy can't be managed very well, and that the more you try to "fine-tune" it, the less it is apt to grow—is at the core of economic conservatism and the approach most Republicans take when it comes to taxes, regulations, and economic theory.
Today, Libertarians cluster at the CATO Institute and _Reason_ magazine, which proclaims support for "free minds and free markets... by making a principled case for liberty and individual choice in all areas of human activity." Populated by economic libertarians, this tribe today is also known for its less stringent approach to social issues like gay marriage, abortion, and drug legalization, advocating limited government intervention.
## **So-Cons, Traditionalists, and the Religious Right**
We all know that the Republican Party has a strong contingent of social and religious conservatives, but not everyone knows how this came to be, or that this element is actually far weaker today than it has been in the past.
At the founding of America's modern conservative movement, the second major partner of Buckley's three-part coalition were the traditionalist conservatives. Historian George Nash calls them the most authentically "conservative" partners because they joined the coalition out of alarm at liberalism's moral relativism. They were mostly Roman Catholic, and some were converts to Catholicism. They were religious and academic elites, such as Russell Kirk and Richard Weaver, and opinion elites, such as Robert Novak (though he came later). They believed that the state was in perpetual competition with God's authority, and that liberalism would naturally try to help the state win that battle. Thus traditionalists focused in particular on battling secular culture. They advocated "a revival of Christian orthodoxy, classical natural law, pre-modern political philosophy, and mediating institutions between the citizen and the state." Traditionalists believed liberalism was "eating away not only at our liberties but also at the ethical and institutional foundations of traditional society, thereby creating a vast spiritual vacuum into which totalitarianism could enter."
By the late 1970s, traditionalists had made common cause with the rising tide of Protestant evangelicals who later came to be known as the Religious Right. These voters had been either apolitical or Democratic-leaning since the New Deal, as were most rural voters in the South. They voted for a Democrat, Jimmy Carter, in 1976. Carter himself was a born-again Christian and drew upon the language of the Southern Baptist Church. But during the Cold War, Democrats were divided between those who remained determined to defeat communism and those who sought to negotiate with it—an issue that did not divide evangelicals, who viewed godless communism as a special threat to Christianity.
Until the Cold War years, the religiosity of these Americans had not informed their politics. But with the schism over communism, and then after the Supreme Court decision _Roe v. Wade_ legalized abortion in the United States, religious Christians felt that it was no longer acceptable for them to remain silent on political matters. They took on not only the abortion issue, but also other social issues, including, in Nash's words, "school prayer, pornography, drug use, sexual deviancy, [and] the vulgarization of mass entertainment." By 1980, leaders such as Jerry Falwell, Pat Robertson, and James Dobson were urging their congregations to participate in the political process and exercise their voting rights against what these leaders saw as America's moral decline. These evangelicals came out in droves to support Ronald Reagan, a moment that marked what scholars now call the "great awakening" of the Religious Right, when they joined the conservative coalition.
Social conservatives also took a strong stance against the feminism of the 1960s and 1970s, which encouraged women to throw off the traditional restrictions of the domestic sphere and to pursue professional careers and sexual independence. Social conservative female leaders such as Phyllis Schlafly affirmed that women were different from men, and that women should not compete with men economically and should cherish traditional gender role divides. To this day, they believe that they are the true feminists, fighting for women to be respected and honored precisely because they have a special gift for domesticity and family building.
In recent years, however, the Religious Right has diversified its political portfolio. For example, the evangelical ministry of Rick Warren's Saddleback Church has focused less on the traditional social issues of the Religious Right and more on a broad ministry of social action that includes fighting HIV/AIDS, caring for orphans, and helping addicts recover.
Today's social conservatives comprise a mix of multiple faiths, not just the evangelicals and traditional Catholics but also Orthodox Jews and religious immigrant groups from South Asia. This variegated group, which could never come to an agreement on theological issues but manages to find common cause in the political arena, has brought moral populism into the broader conservative movement and has helped deliver election victories to Republicans. They turned out in large numbers for the 1994 Gingrich revolution, and were heavily courted in 2004 by George W. Bush after it became clear that its members hadn't supported him with enthusiasm in 2000.
Having worked on the Bush reelection campaign, I can testify to the intensity of the campaign's effort to woo the Religious Right. While these efforts—specifically President Bush's support for the Federal Marriage Amendment and various state ballot initiatives banning same-sex marriage—were intended to drive social conservatives to the polls, further analysis has shown that this election strategy failed. The central myth about the "conservagenzia," a term introduced by Bush campaign chief strategist Matthew Dowd to describe the self-anointed political leaders of the Religious Right, is that social conservative voters care solely about social issues. Dowd's numbers tell a different story: today's so-cons, just like other tribes in the movement, make economic and national security issues, not social issues, their top priority. More on this in chapter five.
## **Anticommunists and Paleocons**
The third tribe of William F. Buckley's new conservative movement in the 1950s were the anticommunists. Largely irrelevant today, thanks to the collapse of communism and socialism as serious economic and political alternatives to capitalism, this was actually in some ways the most important tribe of the early conservative movement because it supplied the glue that bound the various ideological groupings together. Led by a group of former Trotskyites and confessed former communists—people such as Whittaker Chambers, John Chamberlain, James Burnham, and Frank Meyer—this tribe denounced communism as a corrupt moral system, and recognized in free-market economics the core of freedom. Both of the other conservative factions—economic libertarians and traditionalists—were alarmed by the ascendance of communism around the globe because it threatened what was most important to them, be it the fire of individual initiative and productivity or the primacy of man's relationship with God.
Anticommunism proved to be the common thread of this loose coalition: It made dealing with the existential communist threat to America and the West the overriding priority. And that gave the sometimes competing, sometimes incoherent conservative factions a common purpose. This strategy, outlined by Frank Meyer, was called "fusionism," and it had the effect of curbing the incessant infighting that characterized the early conservative movement. For roughly four decades, the fusionism strategy proved to be remarkably effective. But since the fall of the Berlin Wall in 1989, a moment when conservative anticommunists reached the apex of their influence within the movement, that glue has dissolved and unity has eluded the movement. Without a commonly perceived existential threat to America's values, conservatism has been as factionalized as it was at its founding.
Within the anticommunist wing, and largely left behind in the postcommunist era, are paleoconservatives such as Pat Buchanan, who are crossbreeds of social conservatives and foreign policy isolationists. They oppose concepts like world government and internationalism, resist the authority of the United Nations, and are "defiantly nationalist, [and] skeptical of global democracy." In a way, this group predates the modern conservative movement because they had earlier dominated the Republican Party. In the era after World War I, Republicans were deeply distrustful of further international entanglements, and until the Japanese attack on Pearl Harbor they opposed entry into World War II. Today paleoconservatives are ardent nationalists who have increasingly found common cause with the neoisolationist left.
## **Neocons and National Security Conservatives**
At the opposite end of the foreign policy spectrum is the neoconservative tribe. As an intellectual group, they began as mainly Jewish liberals who migrated rightward in the 1960s and 1970s, disenchanted by the excesses of the Great Society and unhappy about the American Left's defense of European socialism and softness on Soviet Communism. Their definition was most famously summarized by Irving Kristol's quip: a neoconservative is "a liberal who has been mugged by reality."
Irving Kristol, Norman Podhoretz, Midge Decter, and Jeane Kirkpatrick were all part of the intellectual migration from Left to Right, and through their perches at _Commentary_ magazine and _The Public Interest_ , they added intellectual, urban, and even counterrevolutionary heft to the conservative movement. While they certainly recognized the evils of communist totalitarianism, they were often at odds with the social conservative and paleoconservative factions in the movement when it came to specific issues, such as the value of church-state separation and the vital role of American power and alliances around the world. But thanks to their staunch anticommunism and anti-liberalism, they became de facto conservatives.
Because neocons came to the movement after embracing liberalism, they have less patience for some of the more unsavory aspects of conservatism's early history, particularly on the issue of civil rights. The leaders of the conservative movement—Barry Goldwater, in particular—voted against the 1964 Civil Rights Act and related laws, even though he was clearly not racist (his had been among the first businesses in Arizona to desegregate). But the neocons were impatient with such resistance and helped traditionalist conservatives to rethink their opposition to the civil rights movement. While conservatives remained opposed to affirmative action, mainstream conservatives have come to recognize that the civil rights movement deserved their support. Conservatives such as Jack Kemp and George W. Bush made some inroads within the African-American community, helping to diminish this obvious blot on the conservative movement's reputation.
Today, the label "neoconservative" has come to designate someone who supports a strong national security policy. Although pilloried for advocating the 2003 Iraq War, they have been steadfast in combating Islamist terrorism and promoting democracy abroad. And at a time when popular revolutions are erupting in the Middle East, Bush's "Freedom Agenda," which was derided by Democrats and the Left, appears to be bearing fruit as of the Arab Spring uprisings of 2011.
## **Conservative Populists: Tea Partiers, Dittoheads, and Mama Grizzlies**
Rush Limbaugh's fans and Sarah Palin's followers are not one and the same, but they have a lot in common. They have a similar outlook on the world; they are more rural than urban and are more likely to be blue-collar than white-collar. They resent the elitism of liberalism to the point where they associate its key elements with moral degeneration. They deeply distrust government, and resent paying taxes when they see so much of it wasted on what they think are useless government programs. They tend to be pro-defense, and when America is at war they are among the most vocal in supporting our troops.
While Limbaugh's Dittoheads came into political consciousness in the Clinton era, and Palin's Mama Grizzlies in the 2010 campaign season, these groups have been united by two political twists: the first was the bailouts initiated by George W. Bush and accelerated by Barack Obama; the second was the vast spending programs launched by President Obama and the Democratic House and Senate immediately after President Obama took office. With these two events, Dittoheads and Mama Grizzlies joined forces with the Tea Party. This grassroots movement was a conservative populist revolt against the irresponsible fiscal policies of the late Bush and early Obama administrations. It grew out of Ron Paul's Tea Party protests during the 2008 presidential campaign.
While a return to fiscal conservatism is the animating idea behind these Tea Party protests, they have focused on other major policy areas, and some activists are running for local county councils and school boards. These groups can easily steer Republican nominations to favored candidates, and in certain cases, can help them win elections. But not always: the one clear lesson of the 2010 campaign is that a Tea Party–backed candidate like Sharon Angle or Christine O'Donnell, no matter how loyal her supporters, will lose unless she can appeal beyond the base of the Republican Party. The power of the Tea Party movement will be tested again in the 2012 presidential election campaign.
## **Crunchy Cons and Enviro-cons**
The rise of the green movement has largely been a phenomenon of the Left. And no wonder—there is almost nothing that resonates more strongly with the liberal worldview than the need for government regulation to meet the threat of private industrial pollution. Much of what the Left has produced follows that pattern: punish private enterprise to accomplish goals, without regard to what individual taxpayers may want and be willing to pay for. So, in classic conservative fashion, there have sprung up, in response, environmentally conscious conservatives who look for free-enterprise solutions to environmental problems—like cap-and-trade credits for fighting the pollutants that cause acid rain. And then there is a group known as Crunchy Cons, "political right-wingers with countercultural sensibilities." They tend to be pro-green, anti–urban sprawl, and anti–strip mall. They base their opposition on what they see as the moral corrosion and the environmental ugliness imposed on America by the consumer culture. The countercultural elements of this movement strike me as fundamentally conservative—for example, they don't trust large institutions such as schools, so they homeschool. They are deeply distrustful of authority, so they resent nanny-state ideas, such as anti-smoking laws. In fundamental ways, Crunchy Cons are a wonderful part of the movement.
## **Reagan, Rush, RINOs, and Me**
All this tribalism leads to the question, is there any common theme among conservatives today? How can we reconcile the social conservative impulses to highly regulate sexuality with the libertarian instincts to keep government out of the bedroom? How can the neocons and paleocons agree on a path forward in Afghanistan? Can free-market pro-business types coexist with off-the-grid Crunchy Cons?
Sure, some say, but we need a Ronald Reagan to be our uniter. Always Reagan—he looms large whenever conservatives wax nostalgic for that period of American conservatism's halcyon days. It's as if Reagan were not just a great president, but a conservative saint, someone who performed the miracle of achieving ideological and political unity.
I don't blame people for thinking this way. When Reagan left office, the movement and the Republican Party were in great shape. And since then, conservatives have either lost politically or lost philosophically—often both.
But this nostalgia has left conservatives in a state of inertia, bogged down in a fruitless effort to discover something called "Reagan conservatism," as though there is a pure version of conservatism. But there isn't. The movement is too rife with factions and tribes for there to be one true version of conservatism.
And the truth is that there never was. What the Reagan yearners and Reagan revisionists forget is that Reagan wasn't a purist. Reagan, for example, followed conservative economic orthodoxy by cutting taxes, but he later raised them. Reagan didn't like terrorists and launched air strikes against them, but he also did side deals with the mullahs in Iran. He could be accused of cutting and running from Beirut after our Marines were attacked there in 1983. He signed a massive amnesty bill giving citizenship to illegal immigrants. I point out these facts not to denigrate Reagan. He was the most successful conservative president the United States has ever had. But he wasn't a purist. His politics were pragmatic, in a way that would appeal to today's millennials.
There actually is no such thing as "pure" conservatism. Successful conservative leadership is about balancing the competing factions of an inherently diverse and at times factional and self-cannibalizing movement. The modern American conservative movement does not hold together the way many political movements often do. It is not based on one coherent set of principles, but several groups, each with its own set of principles, are united in opposition to various aspects of modern liberalism. As a sometimes knee-jerk response, it can trip over itself: Fiscal conservatives favor smaller government above all and are especially hostile to foreign aid and international organizations. Neocons, meanwhile, support U.S. government aid to key allies as a means of advancing America's interests overseas. These two elements within the movement struggle to coexist, and they often collide—as they do today when some Tea Partiers would like to cut foreign aid as a whole, while neocons would preserve military and economic aid to America's stalwart friends, such as Israel and our NATO allies.
Another example is immigration policy. Most economic conservatives have no interest in restricting the flow of labor into the United States, believing, usually, that immigration adds significantly to the vitality of the American economy. But Tea Party paleoconservatives and even some national security conservatives are alarmed at our porous borders, which they believe imperil our national character and our safety, security, and economy.
These issues divide America, but they also create deep schisms within the conservative movement. They can also turn contentious. Just this past year, some social conservatives opposed even the presence of a gay conservative organization called GOProud at the annual Conservative Political Action Conference (CPAC) convention, one of the most important movement events of the year. For some social conservatives, the mere presence of gays within the conservative movement is problematic. They want to see the gay conservatives banned from participating in the movement, believing that it's impossible to be both gay and conservative. In the interest of full disclosure, I am honored to serve on GOProud's advisory council and to support their work advocating limited government, individual liberty, free markets, and confident foreign policy.
Such battles condemn the movement to weakness. We are best served by being lively and argumentative but always keeping our eye on the fact that we are a coalition. I think that these debates, rather than becoming destructive, can help make room for new ideas and opinions within the Republican Party.
The genius of Ronald Reagan's leadership was best demonstrated by his view that if someone is conservative on eight of ten major conservative principles, that's sufficient. Reagan believed in building a "big tent," capable of accommodating people who broadly agreed on most major issues, although they might occasionally differ on a few. And he drew on the formula of Frank Meyer's fusionism to link the various tribes together—something that few conservative leaders have been able to do since. For example, for the first time he was able to link the neoconservatives, anticommunists, and national security conservatives with social conservatives. He knew that social conservatives opposed communism because of its moral evils; he knew as well that national security conservatives opposed communism because of its security threat to America and the West. So he gave his famous Evil Empire speech to the National Association of Evangelicals. This helped several tribes of conservatism see that their efforts were complementary, and that they needed one another in order to achieve a practical governing coalition. This was Reagan's practical realization of Meyer's fusionism. People on the Right remember only that they agreed with Reagan. They fail to recall how remarkable it was that they _all_ agreed with Reagan. Successful conservative leaders have drawn on fusionism, and have found a way to unify conservatism's tribes—not by attempting to define "the true conservatism" but by dexterously managing the people, personalities, and competing interests of conservatism's various strands within a governing coalition tied together by a common purpose.
Which brings me to millennials. If the Republican Party, backed by the conservative movement, is ever going to connect with this generation, it will need a leader who can inspire a new kind of fusionism.
Such a leader has yet to emerge. Many conservatives say that the leader already exists in Rush Limbaugh, but I'm afraid I can't agree. And yet, it's true that among millennials, the people most associated with the conservative movement are entertainers like Rush and Glenn Beck. This is not a good thing for the viability of conservatism with the next generation.
It pains me to say this because I grew up with Rush. I came to political awareness mostly during the Clinton administration, in a Republican household during an age when AM radio appeared to be in its dying days. Then an AM radio host in Sacramento appeared on the scene, and instead of playing golden oldies, he talked. And talked and talked and talked. Rush Limbaugh devoted his entire show to talking about politics mostly, and occasionally American football. He had plenty of material. President Bill Clinton was then stumbling through the first two years of his presidency, and so between noon and three p.m. every day, Rush teed off on Clinton.
Every radio and cable news talking head—conservative, liberal, and in between—owes a debt of gratitude to Rush Limbaugh. He made it possible for politics to be fun, irreverent, and interesting. Before Rush, politics was the province of PBS and mainstream networks, who were deferential to politicians. And cozy with them, too.
Rush broke all those rules. Nobody in Washington knew this guy, and yet he had the attention of millions of people who took his political commentary and analysis quite seriously. He didn't kowtow to politicians of any stripe—he hit Democrats hard, but he would go after liberal Republicans, too. He also was Clinton's biggest and most listened to critic. To the extent that the conservative movement seemed to have died when Reagan left office and the Berlin Wall crumbled, its tribes revived in opposition to Clinton in 1992, thanks largely to the rallying of Rush.
Rush's listeners, for the most part, came to the show with some center-right leanings, but he turned them into a pretty well-informed army of conservatives. They became known as Dittoheads, people who simply said "ditto" to indicate that they agreed with everything he said. Rush would not tolerate dissent or deviation from his orthodoxy, often verbally attacking callers who said they agreed with 99 percent of what he said. "What," he demanded to know, "was the one percent they did not agree with?" And then he would badger them into total agreement. This is what it meant to be a Dittohead.
In the process, Rush transformed conservatism. No longer were conservative ideas batted about only by highbrow intellectual readers of _National Review_ and _Commentary_. Rush was a populist who appealed broadly to regular folks and the conservative elites alike. And for the first time, conservatism had a daily megaphone to millions. Rush's success proved that there was an enormous audience hungry for conservative ideas.
I grew up a Dittohead. I appreciated the stark contrast Rush presented to the steady drumbeat of conventional liberalism that dominated the major networks, newspapers, and magazines and my schooling. I actually used the word _Dittohead_ proudly on my first political résumé. Listening to Rush was a big part of my political education. While most kids in the liberal midsize American city I grew up in loved Bill Clinton, Rush offered a clever and entertaining philosophical antidote. He didn't make politics taste like medicine but made it fun and appealing. Aside from my parents' rightward political leanings and Herbert Hoover's "American Individualism," Rush's show was the most important influence in my political coming-of-age.
Rush's personality was large, his voice sonorous, his optimism contagious, and his outrage punctuated with humor, creativity, and the notion of the persecuted white male under attack. I found his parodies of Bill Clinton's reckless and decidedly unpresidential behavior to be brilliant.
More important, Rush stood for the important themes of the conservative movement: limited government, lower taxes, the importance of the entrepreneurial spirit, American exceptionalism, liberalism's degradation of individual initiative and dignity through the creation and maintenance of the welfare state, and liberalism's hypocrisy. He argued for a strong national defense prudently deployed. Most of all, he delighted in goosing the mainstream media—he would later coin the phrase "drive-by media"—for their perpetual inability to understand conservative ideas because they were so blind to conservative thinking (because there were few if any conservatives in their midst).
Rush was hardly an evangelical, but he stood with social conservatives. He could be antagonistic toward groups he found distasteful. I was a new student at Bryn Mawr, a women's liberal arts college with more than a few lesbian students, when I first heard his "dykes on bikes" parody, and it seemed cruel, to say the least. I could find no humor in this parody, which seemed to reinforce the notion that gay people should just stay quiet about their sexuality. (He was many years away from the day when, for his third wedding, he invited Sir Elton John, a famously gay activist and artist, to perform.)
But despite his vast influence, to most in the millennial generation Rush Limbaugh is a loud, bombastic white guy with annoying opinions. He is no longer a fresh face. He has been around, as a leading figure in political entertainment and commentary, for more than two decades. Everyone I knew growing up watched David Letterman, not Johnny Carson. Today's young adults watch Jon Stewart; they don't listen to Rush.
Furthermore, while Rush is a force in the conservative movement, he ultimately must build and nurture his audience. He has to be entertaining—that's his primary goal. That means he has to be provocative. If Jon Stewart weren't funny, nobody would watch him. If Rush didn't stir the pot, nobody would tune in. And so just as liberals would be foolish to look to Jon Stewart as some kind of political leader, conservatives need to find someone to build the movement as a way to shape a better America, not just to maintain a loyal radio audience.
The conservative movement owes Rush Limbaugh an enormous debt of gratitude for reinvigorating conservatism in the 1990s, and for pioneering the market for center-right news analysis in the heartland. While I admire what Rush Limbaugh has achieved as an entertainer, the Maha Rushie (as he calls himself somewhat mockingly) is wrong when he says he represents "true conservatism." In this, he is no more than another conservative warlord who wants his version of conservatism to become "the accepted version" in its purest sense. He has used his radio platform to define for others who is conservative and who isn't. His influence has served to consolidate and perpetuate the schisms within the movement. As a result, on issues such as immigration or gay marriage or even the proper role of government, everyone needs to be holier than the pope—or be drummed out of the movement. But if we are going to call out those among us who we think are insufficiently loyal to the movement, we might as well hang it up right now, because ultimately we'll become an ideology of one.
Which leads me to RINOs and other so-called squishes. Nobody identifies himself as a RINO (Republican in Name Only) on purpose. It's an insult hurled at various members of the broader conservative coalition or Republican Party as a bullying tactic when they aren't adhering to whatever conservative orthodoxy the more vocal members of the coalition demand. Astute members of the conservative movement have been called RINOs in recent years, including some of the smartest and most thoughtful members of the various tribes. The modern political etymology of the word _squish_ dates to Nixon's vice president Spiro Agnew, who used it to refer to radical liberals in the 1960s and '70s. Later, in the Reagan administration, members of the broader conservative movement co-opted the word _squish_ to refer to James Baker, Reagan's chief of staff, and to George H. W. Bush, the vice president.
This instinct is exactly the opposite of Reagan's 80 percent/20 percent rule—my 80 percent ally isn't my 20 percent enemy—or his eleventh commandment (Thou shall not speak ill of another Republican). "RINO hunters" think that if you agree with the conservative movement only on selected issues, or only to a certain degree, you might _think_ you're a conservative, but you are not. Instead, they say you are a "squish."
A contemporary example of someone RINO hunters deride as a squish is Senator Scott Brown, the sole Republican elected statewide in liberal Massachusetts—because he voted for the financial regulatory reform bill and is pro-choice. Conservatives' sudden disapproval of Brown marks an amazing reversal. It was not so long ago that conservative activists who now call him a RINO flooded Massachusetts to pound the pavement in support of his election to what used to be Ted Kennedy's Senate seat. Brown ran on the simple message of promising to vote against "Obamacare" (a vow he kept, although Democrats used special rules in the Senate to bypass the Republican threat of filibuster) and to fight terrorism without compunction. After winning, Senator Brown reached such heights of stardom that the conservative website _Drudge Report_ even heralded him as a possible presidential candidate. But even though the senator did what he was elected to do—he opposed what he, his constituents, Senate Republicans, and the majority of the American electorate viewed as a bad health-care bill—he has been labeled a RINO because he is "not conservative enough."
But the RINO hunters have it backward. William F. Buckley Jr. always said that the rule was to support the most conservative candidate in the race who could get elected. And while Alabama or Oklahoma Republicans can put up a social and fiscal conservative who can easily win, in Delaware or Massachusetts they can't. Some conservatives argue that they would rather have thirty senators like Jim DeMint than sixty-five like Scott Brown. But that's a formula for perpetual minority status, even extinction. In the 1990s there were Republican mayors and governors in overwhelmingly Democratic states, including New York, Massachusetts, New Jersey, Vermont, Connecticut, and Rhode Island. Many of them governed with conservative principles in mind, especially as fiscal conservatives. These Republicans included Rudy Giuliani of New York City, Dick Reardon of Los Angeles, and Stephen Goldsmith of Indianapolis. Others, like governors Bill Weld, Christine Todd Whitman, and George Pataki, represented a breed of northeastern Republican that is practically extinct today.
In my view, it's much better to have a diverse but bigger Republican Party than a smaller monolithic one. This is especially true in Congress when there are some issues on which Republicans need every vote in order to prevail—think about the effort to stop Obamacare, or the votes for the Bush tax cuts in 2001 and 2003. Conservatives who rush into the arms of "purists" such as Christine O'Donnell and Sharron Angle, two failed Senate candidates, are making a fool's bet. They get a candidate they like the most, but a candidate who loses nonetheless. Had Republicans nominated electable candidates instead of these two in 2010, they would be within reach of winning significant votes in the Senate. Instead, Harry Reid remains majority leader in 2011.
To be sure, conservatives are not alone in dealing with these issues. Liberals also have loose confederations and coalitions. That's how American politics works. But no party can hold on to power unless it works with coalitions and welcomes diverse thinking. RINO hunting is right-wing political correctness—if you're not with us 100 percent, you might as well be a traitor and are diminished to irrelevance.
The history of the conservative movement tells a story different from what the RINO hunters would have you believe. Hayek wasn't a social conservative; Irving Kristol and Milton Friedman weren't either. Reagan signed an amnesty bill, approved a tax increase, and at one time supported pro-choice legislation. Were these stalwart lions of the movement also traitors to the cause?
Some say yes. But Buckley thought otherwise. At one point early into _National Review_ 's publication, at a time when the various factions were feuding with one another, Buckley famously exploded: "Conservatives must get their philosophical house in order!" This was the animating idea behind fusionism, in which the various factions learned to coexist by underscoring shared goals and muting their disagreements in order to maximize their political influence as a movement. This was the guiding spirit behind Reagan's notion of the "Big Tent."
In the coming chapters, I will lay out which conservative principles Republicans should emphasize in order to appeal to the millennial generation. I realize that others within the movement may disagree with me on some of these issues. But I know we can agree on certain fundamental points: the government that governs least governs best; lower taxes and simpler tax codes will strengthen the economy and create more jobs; deficits are a spending problem, not a revenue problem; entitlement reforms are essential if America is to avoid fiscal disaster; America's rich diversity of spiritual traditions strengthens our cultural fabric; the United States must remain the world's leading military power in order to secure our freedom at home and that of our allies abroad.
Of course, there will be areas where my conservative principles do not lead me to endorse the same policy positions as some other conservatives—especially on matters of personal privacy and individual freedoms. Our ability to appeal to the hearts and minds of millennials does not depend on everyone agreeing with me. But it does depend on people like me being able to make their case and others like me being able to say, "I'm a conservative, but I don't agree with some conservatives on certain issues." If we can't get at least that far, we won't go anywhere.
Clearly, we will need to establish priorities as a movement, much as fusionism unified the movement before the fall of Soviet Communism by making anticommunism the unifying theme of social conservatives, economic conservatives, and national security conservatives. At a time when fiscal disaster looms, thanks to an impending debt crisis and decades of irresponsible fiscal behavior in Washington, fiscal responsibility as a prescription for restoring American prosperity is a theme that, like anticommunism, will help unify conservatism's particular tribes around a central goal.
Social conservatives champion fiscal discipline because there is a _moral_ imperative to restraining the federal government's intrusions into the private lives of citizens. Social conservatives no more want an expanded federal government telling them how to live their lives or educate their children than economic conservatives want the federal government's largesse to diminish economic opportunity and dynamism. Just as the urgent threat of communism brought social conservatives, economic conservatives, and national security conservatives together under one tent, so too can the existential threat of fiscal ruin serve to rally these factions as one. Introducing the reforms required to rein in the nation's runaway debt will lead to a stronger, more secure American economy, and that means a stronger national defense—an outcome that national security hawks and neoconservatives, who advocate American leadership in the world, will strongly support.
I believe that this brand of fusionism, with its emphasis on fiscal conservatism, should be embedded in the broader concept of American individualism—a framework that will appeal especially to the millennial generation. With American individualism as our integrating philosophy, and with fiscal responsibility as our action-oriented unifying political theme, I am convinced that, in reaching out to the next generation, we stand a good chance of making an enduring connection.
This modern restatement of my great-grandfather's philosophy identifies American individualism as a potent mix of rugged individualism and community spirit. It rests on three pillars.
First, it is the individual who stands at the center of American society. The freedom of the individual to pursue happiness, as immortalized in the Declaration of Independence and guaranteed by the protections of the Constitution, is sacrosanct. Yet encroachments on individual freedom constantly threaten, from government or even in the name of a majority of citizens. In every area of life, the protection of individual freedoms should be cherished, and we must never forget that we abridge the freedoms of one at great peril to freedom for all.
There are several components to individual freedom. There is the moral component: the ability of the individual to make moral choices on his or her own behalf. There is the economic component: the ability of the individual to make the best use of his or her creative spark, and to enjoy the benefits of hard work and special talents. And there is the responsibility component: the responsibility of the individual for the choices he or she makes. Freedom does not mean doing only what one wants without regard to the consequences. American individualism places a heavy responsibility on the individual not only to benefit from freedom but to act as a responsible free person should—to be completely subject to the results of one's decisions and actions.
There is another role for the state here, and it is this: protection of the individual. A strong defense is vital to the rights of the individual. Freedom, as they say, isn't free, and that is particularly true today. After all, the greatest enemies of America are particularly appalled by our individual freedoms, which we sometimes take for granted—the right of women to marry whomever they want or not at all, the right of religious minorities to worship fully and publicly, the right of free speech, and so on. Each of these rights is exercised, fundamentally, by individuals because of the protection of the state. And so American individualism makes clear that the state must exist and must serve the cause of freedom.
The second dimension of American individualism is participation in the community. American individualism does not mean that people are free from obligation to serve the community. In fact, American individualism rests squarely on the notion that the individual is the first line of defense for the community, that the individual, rather than government, must take the lead in addressing the problems of society and community. The existence of volunteer-based social and civic organizations is something that politicians often extol but do very little to support. That is because these organizations do not derive their power from political support—they exist, and get stronger, because individuals want to be part of them, not because they are required to by government.
America has always had a vibrant culture of community-based organizations and institutions. Liberals tend to believe that the best social improvements emanate from government, but in fact most of the great reforms have sprung from the individual minds of men and women acting alone or in concert with one another, oftentimes cooperating under the auspices of civic or economic organizations. America is not a nation "bowling alone," as one critic has claimed. It is connected in all kinds of ways, and a revamped American individualism will recognize that there is no difference between a Rotary Club and a Facebook group devoted to solidarity with the Iranian people—both serve a valuable social purpose, both bring together diverse peoples devoted to a worthy cause, and both make possible things that individuals could not accomplish on their own. One might meet physically, the other virtually, but they are both civic organizations, broadly defined. I believe strongly that millennials will find great appeal in this dimension of community participation based on the free choices of individuals rather than the dictates of government.
The third and final dimension of American individualism is the diversity of America's religious traditions, the spiritualism that is part of American life. A spiritual component is a necessary resource for individuals to draw upon for strength and stability, and spiritual cultivation both in private and in communities helps strengthen the fabric of society. American individualism embraces people's natural spiritual yearning and gives it room to grow, and it inspires the individual's search for faith and truth. Spiritual traditions help lead individuals toward life's intangible riches—and help balance the human experience. This is a force for good in our culture.
Cast in this way, American individualism can be a framework for communicating a new conservative sensibility to the next generation, while spotlighting the crucial fact that the most pressing threat facing America today is our fiscal disorder. An unwavering emphasis on fiscal responsibility offers a way to unify conservative factions while building a bridge to the next generation of civic-minded, politically engaged citizens—the millennial generation.
# CHAPTER 3
# MEET THE MILLENNIALS
_"Every generation discovers the world all new again and knows it can improve it."_
—HERBERT HOOVER
MILLENNIALS ARE THE generation of tattoos, iTunes, texting, and Twitter. Born between the beginning of the Reagan presidency and the end of the Clinton presidency, they are the rising generation in America, fifty million strong. They are independent-minded and distrust hyperpartisan politics. And they made all the difference in Barack Obama's election to the White House.
In the 2008 election, Senator Obama and Senator McCain polled close to even among almost every major age group of the electorate. But millennials, who composed 18 percent of voters, went for Obama by a two-to-one margin. Sixty-six percent of youth voted for Obama, to 31 percent for McCain.
Now, it is conventional wisdom that the young tend to vote liberal, then come around to conservative thinking as they age. But that's not always so—young adults are not reliably liberal. In the presidential elections of 1980 and 1984, youth voted consistently for Ronald Reagan. In 2000, the year the first millennials voted in a presidential election, they split their votes evenly between George W. Bush and Al Gore. It was only in 2004 that eighteen- to twenty-nine-year-olds broke decidedly toward the Democratic Party, gains that increased in the 2006 off-cycle elections.
Democrats had hoped that Barack Obama would solidify gains and deliver a new generation to the Democratic Party's voter rolls, just as Ronald Reagan's "Republican Revolution" did and as Franklin Delano Roosevelt and John F. Kennedy had decades earlier. Democrats now count on the votes of millennials in elections on the state and local level. They have courted these voters in new and effective ways, and have found millennials to be especially responsive to their attention.
But millennials are in no way irretrievably lost to the Republican Party. The Pew Research Center reports that only 37 percent call themselves Democrats, while 22 percent say they're Republicans, and 38 percent identify themselves as Independents. And as we have seen in the 2010 elections, when people call themselves Independents, they often have distinctly _conservative_ leanings on certain issues. This presents both an opportunity and a challenge for Republicans. In order to bring Independent millennials into the party, Republicans will have to appeal to them on those issues where they are already conservative leaning. At the same time, the party will have to build a bigger tent on social issues to allow for differences of opinion where Independent millennials are more liberal leaning.
Millennials have conservative instincts when it comes to fiscal responsibility and economic policy. And against the backdrop of the astounding fiscal mismanagement of the first years of the Obama administration, there is an opening for Republicans to make headway with the millennials. Between the elections of 2008 and 2010, President Obama and the Democrats saw a decided drop in support from millennials.
To win over millennials, Republicans need to take some time to understand them, and to figure out just what attracted them to candidate Obama in the first place. If they do, Republicans will find that they stand a real chance of capturing the millennials' imaginations—and their votes.
## **A Snapshot of Millennials**
First, let's focus on the characteristic features of the millennial generation as a whole.
* They are racially and ethnically the most diverse generation in America. Forty percent of them are nonwhite, and a full 10 percent are less white than the generation before them, while 20 percent have at least one immigrant parent. And 93 percent are comfortable with interracial dating, the highest rating of any generation.
* They are on track to become the most educated generation in America; more have attended four-year colleges or at least completed some college education than any previous generation. On average they scored higher on standardized tests than did Generation Xers. During their youth, the nation saw its rates of juvenile crime, teen pregnancy, and abortion decline.
* They are less conventionally religious but more spiritual: They identify less with organized religions than any generation before them (one in four is unaffiliated with any particular faith), yet they pray as often as their GenX predecessors and their baby boom parents did at the same point in their life cycles. According to a _Reader's Digest_ poll, 67 percent say that religion is important to them, and 34 percent say they've become more spiritual in recent years.
* Millennials use digital technology the way their parents used telephones to stay socially connected to their peers. Eighty percent of them have created a profile on a social networking site such as Facebook, Twitter, MySpace, or LinkedIn. Twenty percent have posted a video of themselves online.
* Ninety-four percent of millennials rely on cell phones, but they don't use them to talk to one another as much as to text-message their friends. I remember the shock delivered to a friend's family when the cell phone bill arrived and his younger tween brother had managed to send seven hundred texts in one month—at a rate of 25 cents per message. The result: a $175 cell phone bill without a word having ever been spoken!
* Their parents spent a lot of time thinking about parenting. During their childhood, more than nine thousand books about children and parenting appeared in print, and their parents often told them they were "special" or they were "great," no matter what. Critics complain that the self-esteem movement went haywire with this generation, with kids receiving stickers and even trophies just for showing up. The result is a generation accustomed to praise and uncomfortable with criticism. Employers report that their young millennial workers require so much coddling that they have had to retrain their managers to perform essential oversight functions in ways that won't undermine their young employees' loyalty and energy.
* They volunteer in droves and are highly civic-minded. In 2005, 83 percent of new freshmen in college had volunteered regularly in high school. Sixty percent have volunteered in the last twelve months, the most of any generation. In the months and years after 9/11, this generation idolized firefighters, policemen, and soldiers. Many of them subsequently joined the ranks of those professions. They have not only a highly elevated sense of civic duty but also high expectations for the competence of their elected leaders. They are not as deeply cynical of authority as the baby boomers. Millennials have significant faith in the power and ability of government to do good. Seasoned pollsters say this is unlikely to change.
* Millennials vote. On account of their sheer numbers, in 2004 "as many raw votes were cast by those thirty and under as by those over sixty-five." Over half of all eligible millennials voted in 2008, which added up to two million more votes than in 2004. While it is true that younger voters tend to cast their ballots _less predictably_ than their elders, this generation votes _in greater numbers_ than previous generations at this point in their lives.
* Millennials cannot abide hyperpartisanship in their politics, which they view as divisive and destructive. If we are going to try to sell them on our politics, it will have to be a politics overstuffed with proactive solutions to solving problems. Criticism in our politics will have to be constructive if we are going to have any hope of appealing to the millennial generation.
* Millennials are slow to start their own families—just one in five millennials (21 percent) is currently married, which amounts to half the share of their parents' generation at the same stage in their lives. And even those who would benefit from marriage and family have been reluctant to take the plunge. About a third (34 percent) are parents. In 2006, more than a third of women between the ages of eighteen and twenty-nine who had given birth were _unmarried_. This is a far higher share of unwed mothers than that experienced by any previous generation.
* And yet millennials are devoted to the families they were born into. The oldest of them, who often live on their own, have friendly relationships with their parents and communicate with them often (45 percent talk on the phone with one or both parents daily). What's more, 52 percent of millennials say being a good parent is one of the most important things in life, even though only 60 percent were raised with both parents.
* They don't have major hang-ups when it comes to sexual orientation. More than two-thirds of them think homosexuality should be accepted by society. Majorities of them that affiliate with an organized religion tend to have no moral objections to homosexuality. They are the only generation where a majority favors the legalization of same-sex marriage.
* Nearly 40 percent of them have tattoos, and half of those have more than one. Yet more than two-thirds of those tattoos are hidden beneath their clothing. For all their enthusiasm for self-expression, they still value privacy and discretion.
* Millennials don't settle. Because this group often used their high school and college summers to burnish their résumés and transcripts rather than earn money at menial jobs, they do not take easily to base-level employment. Books have been written, articles published, and consultancies spawned to deal with this new generation's entry into the workforce, and the challenges they present to managers. As one author writes, "They've been down to Machu Picchu to help excavate it. But they've never punched a time clock." Worldly and well-read young employees are always in demand, yet a lack of basic office skills turns out to be a significant handicap, especially when those basic office tasks are the first things that need to get done.
* Millennials have been hit harder by the Great Recession than any other group—37 percent of people aged eighteen to twenty-nine are unemployed or have dropped out of the workforce, the highest share among this age group in more than three decades. In 2009, only 20 percent of college students who had hoped to have a job upon graduation were successful, a drop from 51 percent two years earlier.
So here's the group portrait: Millennials are expressive, but they understand the importance of boundaries in the face of "too much information." They are innovative, tech-savvy, and entrepreneurial, but they apply those assets to more than just their careers. They tend to be strongly civic-minded. They resist ideology, and they focus more on fixing problems and making things work. They build large networks of friends, remain close to their families, and are nonjudgmental about people from different backgrounds and lifestyles. They are quietly spiritual and have demonstrated a pronounced moral sensibility, and they also place great value on doing good works. They think they can _change_ the world, not just talk about changing it—and, if anything, they are determined to do the hard work themselves. And, I believe, millennials are open to certain core conservative messages.
## **Politics and Millennials**
The oldest millennials were alive when Reagan was president, although they barely remember him in office. Vietnam, Nixon, Watergate, the Iran hostage crisis, Iran-Contra, and even the Cold War—these are all chapters in their history books. They know and like Bill Clinton, not so much on account of his presidency, which was tainted by scandal, but because of his postpresidential efforts on behalf of world health, development projects, and human rights around the world. Many of them liked George W. Bush at first, but after the mismanaged war in Iraq and the bungled federal response to Hurricane Katrina (not to mention Republican scandals in Congress through most of Bush's eight years in office), this generation soured on him, and on Republicans, dramatically.
Barack Obama's candidacy offered millennials not so much a competing political philosophy but, more important to them, an alternative to hyperpartisan politics as usual. He exuded great optimism about his ability to reduce political gridlock in Washington, to unify the nation, and to restore confidence in our governing institutions. That he has largely failed on these fronts, while chalking up mountains of debt in the process, may prove difficult for him to explain in 2012 to those millennials who supported him in 2008. Republicans have a chance to make up ground with millennials merely by focusing on those unmet promises instead of on the parties' ideological differences. The percentages of youth who during the 2010 cycle self-identified as Democrats and Republicans were back down to 2004 levels: 54 percent Democratic to 40 percent Republican. While the Democrats still maintain a large lead, Republicans have an opening.
And while millennials are liberal on social issues, such as gay rights, gender roles, or the traditional family, and while they tend to trust government more than any other generation, this doesn't mean they are reflexively liberal across the board. On a range of issues, such as the relationship between the individual and government, or the appropriate rates of taxation and of government spending, or how much government regulation is necessary, millennials are decidedly not liberal.
In fact, millennial attitudes on business regulation and government welfare are indistinct from the attitudes of baby boomers and Generation X. Millennials don't believe that government should provide an even broader safety net if it means sinking the country further into debt. They believe, more than other demographic groups, that businesses take fair profits and are not too powerful. And on certain issues, millennials are starting to catch up to other demographic groups. While they tend to favor, more than any previous generation, affirmative action programs to help minorities with preferential treatment, their support for such programs has dropped significantly—to below 50 percent—in the last few years.
So Democrats would be mistaken to interpret votes cast for John Kerry or Barack Obama as signs of a strong affinity for ideological liberalism. In significant ways, these votes represent a rejection of Republican _leadership_ , not of all conservative principles. In the 2006 and 2008 election cycles, Republicans, who had dominated the executive and legislative branches, had to pay a price for their poor performance in office. And to millennials, that meant voting Democratic.
But that was before millennials felt the effects of the worst economic crisis in seventy years, and before a liberal Congress got to run the show. Evidently, millennials didn't like what they saw—although not enough to bring them to the polls in significant numbers in 2010. While they still favored Democrats in the 2010 congressional races, they were much more likely not to vote at all, with their turnout rate dropping to 20 percent, from 26 percent in 2006.
I would expect that as millennials age, fiscal conservatism will resonate even more deeply. Millennials are not just young biologically. They are young financially. To the extent they pay taxes, they are likely to be in the lowest tax brackets. They haven't yet become business owners making payrolls or investors preparing for retirement. Even if they have conservative leanings when it comes to such matters, they haven't yet reached the point where they view the price of going along with Democrats as prohibitive.
Even so, Republicans must remember that millennials retain a "liberal" view of government. They believe that government can be effective. While half of millennials agree that regulation of business does more harm than good, only 42 percent of millennials think that when something is run by the government it's usually managed inefficiently and wastefully. Or to look at it another way: 58 percent of millennials think government is good at running things.
You would have thought that after the failures of Katrina and the Iraq War, millennials would have become deeply cynical about the ability of government to do anything right. But the lesson they drew wasn't that government is incapable of handling big issues, or that government is ineffective and wasteful. They simply concluded that the _Bush administration_ was incompetent. By comparison,
GenXers—at a similar stage in their lives—were much more skeptical of _government's_ ability to operate efficiently. So while Republicans can appeal to millennials on certain pocketbook issues, they have to be mindful that the negative rhetoric of the past—even Reagan's immortal line that "government is the problem"—will not resonate with this new generation. The idea of limited but _energetic_ government might.
## **Hope and Change... and Choice**
For about the past four decades, there has been a gender gap in American politics. Men have favored Republicans, while women have favored Democrats. Today, in addition to a gender gap, there is a generation gap. Yet, while the two major political parties tend to split the support of every demographic group between them, the Democrats have held a two-to-one advantage among millennials.
_For Republicans, the key to closing that yawning generational gap is to emphasize economic values, and de-emphasize social issues_. The Republican Party has to continue to be the party of economic growth and opportunity. It has to represent the ideas that promise a better tomorrow—more wealth and prosperity for all Americans. Millennials grew up amid greater wealth than any previous generation, yet thanks to the Great Recession they have learned not to take that wealth for granted. If Republicans are to be successful with millennials, they have to make a credible case that they are the best guardians of American prosperity.
The ground is fertile for such an appeal. Far fewer eighteen- to twenty-nine-year-olds identify themselves as Democrats today compared with 2008. Considering the economic struggles millennials have experienced these past few years—living at home with their parents, sleeping on the couches of friends, putting off some of the purchases (like homes and cars) that usually come with adulthood—millennials appreciate all too well that without a strong economy, very few things are possible.
It is on social issues that Republicans will find themselves on shaky ground when it comes to millennials. On issues like gay rights, reproductive freedom, and religion in the public square, the most vocal representatives of the Religious Right have made it difficult for Republicans to be perceived as anything other than the fire-and-brimstone party. But it's not as if the Republicans can't alter this image. For one thing, they can talk about moral values as civic values. There is no reason for Republicans to shy away from saying that our civic life needs to be morally anchored.
After all, it is a terrible tragedy when a child goes without a father, or when a teenage girl seeks an abortion because she doesn't have the means or the support to raise a baby. What Republicans can do is focus less on the explicit commandments of the scriptures and more on the spirit of the scriptures. They can say that America needs responsible citizens to care for one another, for their families, and for their communities. Republicans need to argue that while government has a role to play in meeting certain social needs, it cannot always be a substitute for the actions of ordinary citizens and civic organizations. Republicans can and should make the case that while government is part of the solution, it can't be the only solution, because it can never be as smart and as forward-thinking as individuals acting together or alone.
Republicans would do well to adopt the "Big Society" concept introduced by a successful center-right coalition leader on the other side of the pond. As formulated by British prime minister David Cameron, the Big Society emphasizes the virtues of sacrifice and service. It recognizes that, while government should offer compassion to its neediest citizens, bloated government cannot serve the people. It holds that individuals are ultimately responsible for their lives, and must be trusted and encouraged to look out for themselves and to help others in need. Cameron describes the Big Society as "the spirit of activism, dynamism, people taking the initiative, working together to get things done." This is, essentially, a restatement of the core premise of American individualism.
Liberals, by comparison, offer a very different view of the roles and responsibilities of government and individuals. Liberalism believes that government is something you pay for with taxes and from which you can then extract the benefits, like some kind of bank. It's as if people are _owed_ something by the government. And while in one sense government does owe much to its citizens, in modern liberalism's worldview the notions of personal sacrifice, initiative, and individual responsibility are overshadowed by the expectations of government largesse.
What's more, liberals have pursued the "government as bank" analogy to the point where the cost of government has become far greater than people are willing to pay. Some of the core elements of the financial contract between the citizens and the government—especially Social Security—are already in deep fiscal distress. Few millennials believe they will ever be able to collect the retirement benefits to which they are just beginning to contribute. And because Democrats, through massive spending and borrowing programs, have solidified their traditional reputation as the Party of Debt, they have exposed themselves to the criticism of millennials as a result of their having bankrupted the government "bank."
What should the Republican message be? That government is a _necessary_ player in our civic life but not the _essential_ one. There are certain things the government must do that individuals cannot do: defend our borders, protect our citizens, enforce our laws, keep the playing field level by ensuring the principle of equal opportunity. There are certain social benefits that government oversight makes possible: a way to save for retirement and a way for the poor to pay for their health care, principles that the New Deal and Great Society programs introduced and which most Americans now accept as part of the American system. The question that surrounds these programs today has to do with their size and how much individual initiative they allow. Republicans can and should argue that the dynamism of the individual needs to be allowed to flourish. In this context that means programs to maximize individual choice, such as private accounts within Social Security and health savings accounts for health-care spending, both of which maximize the individual's ability to make his or her own choices.
This would offer a sharp contrast to liberalism's assumption that the state must guide people's lives from cradle to grave. Millennials can do the math, and they know that our current entitlement spending and debt obligations will ultimately deprive them of the benefits they've been promised. They are open to a message of limited government responsibilities, limited government ambitions, and more freedom for individuals to handle their own affairs and build their own futures.
Finally, Republicans must make the case that government has to be willing to try new things, and in multiple ways. Instead of offering one-size-fits-all solutions to the problems of society, as Democrats often do, Republicans can and should be the party of government _customization_ —giving citizens more individual choices. This is particularly true in the areas of health-care reform and education reform, where conservatives generally press for market-driven choices over government rules and restrictions.
Because of the importance of the Internet in their lives, millennials have been imbued with a sense that anything can be customized to their individual tastes and needs. Liberalism generally holds that the state can make better choices for people than people can make for themselves—on things like retirement savings, health insurance plans, and even what kinds of cars they drive. Republicans have always had a winning position on these issues: more competition means more choices, which means lower prices and better quality.
Where the party has traditionally stood for individual choices and responsibilities—education, economy and taxes, health care, retirement security and savings, personal conduct—it will have to get better at communicating what it believes, and pursue policy goals that will make those beliefs the law of the land.
But where the party has traditionally stood _against_ permitting certain individual choices—be it reproductive freedom or whether two same-sex adults should be allowed to legally marry—we Republicans ought to be more philosophically consistent, recognizing that the rising generation prefers more choices and less governmental involvement in these formerly hot-button social issues. I say "formerly" because to this next generation these aren't the polarizing culture-war questions that they were to earlier generations. In fact, the majority of millennials are in agreement about these issues.
I realize that for some Republicans and social conservatives this will be a painful, if not impossible, adjustment. But if Republicans don't pledge themselves to the virtue of individual choice applied _broadly_ , they may well have it thrust upon them. After all, millennials are known to manufacture their own choices. I would not be surprised if millennials embraced a dynamic third-party candidate in a national election if they remain dissatisfied with the offerings of the two major parties. A third-party candidate could easily present himself or herself as a better choice than the candidates of the major parties, drawing on the best of their respective political philosophies and casting aside the rest.
That's why neither party can take millennials for granted. The sensibilities of this cohort are diverse and sometimes, to outside eyes, appear contradictory. But what millennials will always want is more choice, because they have come to expect it. They have grown up in an age when they have had more consumer choices available to them than anyone before them, more ways to express themselves, and more sources of information, entertainment, and social friendships. Why should they expect less from their politics?
Freedom of choice is the underlying spirit of American individualism—the belief that individuals are not the same, do not want the same things, and do not necessarily want the same outcomes. That's what choice is all about—you get to choose the life you want to live, the career or interests you want to pursue, the friends you keep, and the places you live. American individualism recognizes the need to balance rights and responsibilities. To the extent that the Republican Party can become the party of individual choice and the party of civic involvement, it has a good chance to connect with millennials. If Republicans explore these values fully, they will see the need to adopt them in matters of social policy. The political payoff of consistency in the matter of individual freedom of choice will be enormous.
The next year is critical for Republicans to make this case. After all, under President Obama, trust in the government has gone down. Millennials have seen the economy stagnate. They have waited and watched as Washington has grown even more polarized, not less. And they have seen a significant increase in government power, a significant rise in government spending, and a significant increase in government debt—all without any comparable increase in their options and choices at home, at work, and elsewhere. Republicans today have their best opportunity to appeal to the sensibilities of millennials and bring them into the big tent.
# CHAPTER 4
# GENERATIONAL THEFT
_"Blessed are the young, for they shall inherit the national debt."_
—HERBERT HOOVER, JANUARY 1936
IMAGINE YOU ARRIVE late to an elegant restaurant to meet your entire extended family—parents, siblings, grandparents, aunts and uncles, and cousins—as they are wrapping up an epic celebratory dinner. They have ordered cocktails, bottles of Dom Pérignon, and a sumptuous seven-course meal, complete with appetizers, savory soups, delicious entrées, and a rich soufflé for dessert. As tuxedo-clad waiters clear away the used china, you join the celebration, steal a nibble of dessert, and delight in tasting the remnants of the expensive wines your elders ordered before you arrived. When the check arrives, you notice that most of your family has disappeared. Your grandparents left shortly after you arrived, your parents left during dessert, and as you and your siblings stay to polish off the wine, you notice that you are the only ones left at the table. Having no choice, you pay the enormously expensive bill. Although your parents and relatives haven't literally stolen money out of your pocket, they have left you with a huge tab to pay.
Right now, in America today, seniors and near retirees are essentially stealing from millennials. They have thrown themselves an extravagant party, but they won't be sticking around to pay the bill. The millennials will have to do that.
Since the 1960s, baby boomers have been enjoying what amounts to a huge party at the expense of their children and grandchildren. The parents (and grandparents) of millennials took the New Deal programs and the Great Society programs and expanded them. Social Security went on steroids. Medicare and Medicaid metastasized. The cost of government as a share of the economy ballooned.
No one should be more concerned about out-of-control government spending and deficits than members of the millennial generation. After all, they are the ones who will ultimately get stuck with the bill. Every time you hear someone argue that government isn't doing enough and has to spend more than it already does, what that person is not saying is that millennials and their children will need to be taxed more, live in a smaller economy, and be protected by a smaller military as a direct result. If there is one issue that pits old against young, generation against generation, it is cascading deficits and rising national debt. One generation spends more than it is able to afford, and demands more than it is able to pay; the next generation has to cover the costs.
Every dollar of debt today will have to be paid back, with interest, at some point in the future. Every dollar of debt accumulated today will have to be recovered by taxing someone at some point in the future. Millennials must know—and if they don't know, they will soon have to learn—that the future bill for today's spending will be paid by them, either in the form of higher taxes or diminished economic opportunity, or some combination of the two.
Now, I don't have a problem with people, or even nations, incurring responsible degrees of debt from time to time. America has carried debt for virtually its entire history. Debt financed America during the Revolution, the Civil War, both world wars, and the Cold War. Like a mortgage or a student loan, debt allows us to do something today that accrues value over time—as long as we manage the debt over time.
America, in fact, has been a great investment for people willing to lend to it. Historically, we have always paid our bills on time. And we have been the driving force for freedom and prosperity in the world for more than two hundred years, thanks in part to things we did that were financed by debt. And let's not forget about our assets. If America were truly a household or a business, we could show an incredible pile of assets that we have stored up and nurtured over time: the world's strongest military, massive natural resources and mineral rights, and some of the world's most beautiful natural wonders. Our nation has a wealth of assets.
The problem is that we are moving well past the point where all of these assets, which provided us with security in decades past, can sustain us in the future. In the past, our deficits—what we owe each year—have often remained relatively modest, on average about 2 percent of GDP. In the last few years, however, the deficit has risen to an alarming 10 percent of GDP. If this were just one bad year, America could manage. Even if it were two years, America's economy would produce enough to eventually work that debt off. But thanks to the policies of the first two years of the Obama administration, which included massive spending increases in the form of "stimulus" and the added long-term costs of "Obamacare," the government is destined to run deficits of this size for many years to come. On top of this, underfunded entitlement programs created eighty years ago are ballooning to the point where in order to pay for them, we are adding exponentially to the debt. The combination of all those years when we spent more than we had, combined with growing deficits and mounting underfunded entitlement programs, creates the kind of national debt that we see tallied in that scary debt clock near Times Square—$14 trillion and counting.
While debt and deficits represent a serious threat to our future, the fact is that people just aren't as agitated by the debt as they are by the more palpable threat of a terrorist attack, a disease, a famine, or a war. It is easy to become committed to solving any of those problems, because emotionally, when we see pain, suffering, or injustice, we want to stop it. Debt doesn't create the same sense of panic. It's a number on a piece of paper. It remains hidden. It just doesn't seem that menacing.
This isn't willful denial; it's a completely understandable lack of reaction to an invisible danger. And this is hardly a generational phenomenon. Based on the relative disinterest exhibited by politicians across the political spectrum, and the fact that politicians are rarely punished by voters for overspending (2010 is the recent exception), the rational thing for politicians to do is to ignore deficits and debt. Solutions seem more painful than the problems in many cases, so the solutions go ignored. The reality is that elected representatives will take action to fix the nation's problems only when their reelection depends on it.
I learned this firsthand working in the Bush White House in 2005, when President Bush admirably attempted to rally the Republican Senate and House to reform the ticking time bomb that is Social Security. But even with a unified legislature, mobilizing the will to fix Social Security proved impossible. Politicians ignore problems until they become crises, and even then the political will might not emerge to fix them. Unfortunately, in the absence of obvious evidence that our deficits and our debt have reached the crisis stage, there is no political will to tackle the problem. But make no mistake: a crisis is looming.
Ignoring debt and deficits is incredibly stupid and, for millennials in particular, shortsighted. Here is what deficits will do to America—especially to its youngest workers in the millennial generation. Imagine that every American household has to pay a "deficit tax" on its earnings, on top of all the other taxes it pays right now (including for Social Security, Medicare, state income, and state unemployment). If the government wanted to pay off the total national debt in five years, every American household would have to pay roughly $475 a week extra in taxes, or about $24,800 a year. Let's say the median household income is $50,000 per year, or $960 per week. That household already loses about $250 in various withholding taxes, leaving it with $710 a week. In order to pay off the deficit over five years, every American household would have to forfeit another 67 percent of its take-home pay, which would drop to $235 a week.
And here is the thing: even if everyone signed on to this deficit tax, we wouldn't be free of future debt. Because the government has piled up so many promises and continues to spend beyond what it takes in by more than $1 trillion a year, American households would have to keep paying the deficit tax, to the tune of an additional $8,850 a year, forever. Or at least until changes are made to eliminate our astronomical deficit spending.
The only good news here, from the perspective of millennials, is that they are beginning to understand this harsh reality. Some of them care deeply about the skyrocketing national debt and our inability to manage federal budgets responsibly.
Perhaps the best evidence for the rising concern about debt is the growing number of debt-related websites run by millennials. One of these websites is called WeCantPayThatTab.org. Its cofounders, Ryan Schoenike and Brandon Aitchison, track the national debt and its implications for the next generation. In typical millennial fashion, they don't discuss the issue in partisan terms. Remember, millennials are a civic generation that want to solve problems, not point fingers. They don't care what Republicans or Democrats say; they want to see the problem fixed.
They point out that unless we are willing to make hard choices to change our fiscal behavior, we'll be facing interest payments on our national debt in 2020 of more than $916 billion, a number that represents "more than we spent on education, energy, homeland security and the wars in Iraq and Afghanistan in 2009." The site tabulates the total debt of the United States at "over $73,000,000,000,000 including unfunded liabilities." That's $73 trillion in case you lost count of the zeroes.
Seventy-three trillion dollars—it's a big number. But if $1 trillion is big, what's $73 trillion? Why not go to $173 trillion? If debt hasn't hurt us yet, why should it hurt us in the future? Unfortunately, the thing with debt is that it takes two to tango. First, you need someone to spend their way into debt—in this case, the United States. That's the easy part. The harder part is finding someone who has money to lend. In this case, it's individuals and governments that are running surpluses, and they would rather invest what they have in a country that has never failed to pay on time. Countries like Germany, Japan, and China, along with their wealthier citizens, have invested in America for decades. They have had no problem writing checks for America to cash because we were very good at paying them back with interest.
But in recent years our debts have swollen so uncontrollably that those lenders are starting to have second thoughts about giving us any more of their money. Anyone who has ever applied for a loan knows that the lender is going to look at a person's credit report and income level to make a decision about whether a loan is worth the risk. For years our country's credit report was pretty solid. But now it's looking sketchy. We owe a lot of money. If our economy were bigger, this might not be as big an issue. But our economy just isn't big enough or growing fast enough—compared with the emerging markets in Asia or Latin America—to pay off our debt. To pay off what we owe will take a lot more effort and a lot more time than it did when our debts were smaller and our economy was growing.
Lenders will think, "Why give America more? What if there is a better investment out there?" And gradually, the money spigot will start to tighten. The flow of credit will begin to slow. Other borrowers less creditworthy than Uncle Sam, such as state governments, will have a tougher time getting loans. The economy as a whole will labor under the weight of a heavier load of debt and will have fewer incoming resources for investment and future growth. When countries carry debt loads of up to 30 percent of GDP, their GDP tends to grow in a healthy upward trajectory. But if their debt load balloons to 90 percent of GDP, their economy invariably slows to a trickle. That's a hard truth that Ireland, Italy, Greece, and Spain have come to know. How close are we to that breaking point? America's debt level is now at about 85 percent of its GDP. If we are not yet at that breaking point, we aren't far off.
There is a reason why Democrats are known as the party of big spending. The Democrats created the New Deal. The Democrats created Medicare and Medicaid. The Democrats just passed a massive new health care entitlement in "Obamacare" that will drive the federal budget toward even more spending in the next ten years and beyond. When Republicans suggest cutting government spending, Democrats typically protest. The central premise of modern liberalism is the virtue of government spending, the core philosophy of the left wing of the Democratic Party. The Blue Dogs, the coalition of centrist Democrats who championed fiscal responsibility through spending restraint, were largely voted out of office in 2010. There are fewer and fewer fiscally responsible Democrats in office. This could leave an opening for Republicans to bridge the generational gap. Millennials have thus far voted for Democrats, but it's not farfetched to believe that their allegiance might now shift over the issue of deficits and debt.
Yet until the Obama administration blew up the deficit and debt, Republicans' reputation had been lacking when it came to fiscal responsibility. By any measure of spending, deficits, or debt, over the last three decades Republicans have been only slightly better managers of taxpayer dollars. I worked in Congress when the Republicans, acting on the request of President George W. Bush, passed the Medicare prescription drug bill. I remember thinking, "What are we doing? How does it make any sense, now that we are in power, for us to pass the largest federal entitlement program since LBJ?" Later, I worked in the Bush Office of Management and Budget as deficits and spending spiked to more than 4 percent of GDP. No fiscal conservative can be proud of the recent Republican fiscal record.
President Bush got a lot of criticism from fiscal conservatives for never vetoing any of the Republican Congress's colossal spending bills. But in President Bush's defense, if he had vetoed any of the spending bills he would have risked congressional support for the Iraq War. People forget that times were tough in 2005 and the war wasn't going well. Al Qaeda had established itself in the northern Anbar province and was executing regular deadly attacks on Americans and Iraqis. We were losing American lives, and congressional Democrats weren't the only ones clamoring for a withdrawal of our troops from Iraq. Against this backdrop, President Bush couldn't afford a fight with his own party over spending. Congressional Republicans granted the president more time and money to turn the war around, and in exchange he went along with their irresponsible appropriations and out-of-control earmarks. They incorrectly thought it would lead to their perpetual reelection, so they decided to just go with the flow of the spending culture of Washington.
By the end of the Bush presidency, those of us who go on TV to defend principles of fiscal responsibility—those of us who believe that the Republican Party doesn't share the tax-and-spend impulses of the Left and who believe it's our job to rein in spending and be responsible fiscal stewards of the national economy—were silenced by facts. President Bush gave the go-ahead to the Troubled Asset Relief Program (TARP), a $700 billion bank bailout program, as it morphed from a program to buy up toxic bank assets to one aimed at reliquidifying bank balance sheets. While TARP may end up costing taxpayers a fraction of its original pricetag, the view from Main Street was one of puzzlement: Why would a Republican president approve such a vast expansion of federal largesse?
In December 2008, after Obama had won the election but before he took office, Congress agreed to lend American automakers $15 billion in an effort to prevent their collapse. It was clear at the time that this was a Band-Aid applied by the Bush administration in order to get automakers through the end of the year and into the beginning of the Obama presidency. These Bush administration initiatives also proved to be too much for Republicans to defend. Some conservatives remained silent, although many others spoke out, making it clear that these moves violated conservative fiscal principles. But by then, opposition didn't matter anymore. The Democrats were in charge of the House, the Senate, and the White House, and they enjoyed the freedom to do as they pleased.
Within his first hundred days in office, President Obama signed spending bills whose total cost equaled that of the debt accumulated by every president from George Washington to George W. Bush combined. His spending spree totaled $1.2 trillion—that's roughly $24 billion a day, or $1 billion an hour. In his first two years in office, President Obama presided over the largest debt increase in history, created a vast new health insurance benefit, and considered plans for a massive energy tax. His biggest initiative to _reduce_ government spending was a call for his cabinet agencies to identify $100 million in spending cuts. Now, $100 million is a lot to you and me, but it is not even one-half of 1 percent of the federal budget. Let's be honest, in the White House budget office, $100 million in spending cuts is geekspeak for a rounding error: it's "budget dust." But for Barack Obama and congressional Democrats, this served as a fig leaf of fiscal responsibility.
So now the Republicans have a chance to prove to voters of all stripes that they are, in fact, more responsible with taxpayer dollars. And because the federal deficit and the economy regularly rank among the most important issues to voters, Republicans can know that as they fight to restore their reputation for fiscal responsibility, they will also be fighting for the issues that voters care about most. It is a clarifying moment, because for many years Republican leaders thought that cutting government spending was risky politics. Now they know that cutting government spending can be popular. That is a significant awakening.
Because spending and the deficit is an issue of prime importance to millennials, Republicans can join this battle knowing that if they do what's right by fighting to bring down the deficit and opposing new forms of spending and new taxes, they will have a chance to win over millennials on fiscal issues. Because these issues are a source of concern for millennials, Republicans have an opening to earn their votes by practicing fiscal responsibility. Millennials will vote Republican if they believe the economy will be stronger and will create more jobs, and that the country's fiscal future will be in better hands.
But while most voters now say they want the government to spend less, it will be difficult for ordinary Americans of all ages to get used to smaller government. After all, when you try to cut services and programs that people have become accustomed to, they get angry. Look at Europe.
We've seen students rioting in the United Kingdom because of increases in school tuition in the face of government cutbacks in education subsidies.
In France, riots occurred in response to changes in the retirement age for those eligible for government pensions and because of shortages of gasoline for automobiles.
In Spain, protesters against austerity measures picketed government offices and staged a twenty-four-hour general strike.
Students and unions in Greece threw Molotov cocktails and beat back riot police as state benefits were decreased.
People wonder if such riots could ever happen here. Absolutely. Just look at the tens of thousands of public-sector union protesters who gathered in Madison, Wisconsin, and, defying official orders, occupied the state capitol to protest Governor Scott Walker's plan to limit their collective bargaining power. Proposals to address mounting state budget deficits in Ohio, Indiana, and elsewhere across the nation generated threats of further, escalating protests.
The longer it takes to address our unfunded mandates, the harder it will become to fix them. But we must. Here are the stakes: in 2030—if we continue on the current trajectory—we'd have to implement draconian cuts, up to half of all discretionary spending, or else impose suffocating tax increases in order to close the fiscal gap.
Even today, close to 50 percent of Americans don't pay any federal income tax because we have continually lowered the rate at which people pay federal taxes. Yes, you read that correctly. Almost half of the American population pays no federal taxes. None. Zero. Zilch. The big 0. By the year 2030, there will be more than seventy million Americans eligible to receive Social Security, but the system will have such a huge deficit that we won't have enough new workers paying into it to cover those benefits. Medicare and Medicaid, both of which are administered through state budgets and are already operating with significant deficits, will be bloated with patients, thanks to a combination of more retirees and additional enrollees through President Obama's health-care overhaul.
Thanks to the Tea Party, the protests against government fiscal policies have been caused by the fact that government spends not too little, but rather too much. There is a broad and influential constituency in America in favor of restoring fiscal sanity. And the protests we will face if we do nothing are not the protests of those who want more government largesse but of those who want less of it.
But as Republicans reach out to millennials on this issue, a few things must be remembered. First, it will be necessary to keep the tone civil, and focus on getting the work done. Pointing fingers at Democrats as the sole culprits of our financial and economic mess will ring hollow to this generation. The party that will appeal to millennials is the party that rolls up its sleeves and shows it's willing to get the job done. Millennials are hardwired to tune out partisan bickering.
Republicans need to start fresh, by credibly owning up to their past mistakes and committing to working to build coalitions across party lines to seriously address America's looming fiscal doom. Millennials want leaders who are willing to make tough choices now, so that they won't have to pay a debilitating price in the future. Republicans need to demonstrate to millennials that we are serious.
If millennials and conservatives can agree on anything, it should be restoring limits on the government's responsibilities, embracing individual freedoms, and constantly restraining federal spending and the size and power of the federal government so that it lives within its means. Government should not do things that individuals can and should do for themselves. Our objective should be to foster a robust economy that provides for an opportunity society with a safety net for those in need.
The relevant questions for the next generation are: How big a safety net do we need and who should qualify? Will it be a shared federal and state responsibility? What role should charitable organizations have? At what point is it immoral for the federal government to punish tomorrow's workers in order to pay for today's overpromised retirees? These are the questions millennials need to ask themselves. Somebody has to. The current generation of leaders is largely unwilling to do so, or to make sacrifices, even when those sacrifices appear unavoidable. Millennials, by their very nature, are frustrated when nobody is willing to do what plainly has to be done.
I believe that the modern liberal movement is not prepared to supply millennials with practical solutions to these questions. Modern liberalism, especially in its post–Great Society incarnation, as we saw in the last Democratic Congress governed by Nancy Pelosi, is based on the premise that when there is a problem, the solution is to be found in a government spending program. The bigger the problem, the more government needs to spend taxpayer money. If students are illiterate, liberalism says we need to spend more on school buildings and existing teachers, not give parents a choice of schools. If people are out of work, liberalism says we need to write them checks so they can replace some of their take-home pay, not cut taxes on corporations so they can create new jobs. And if the economy is in the tank, liberalism—which embraces Keynesian economics—says government must create vast public works projects to fill the order books of companies, not cut regulations and taxes, which prevent the economy from growing.
Remember: Every dollar we spend comes from collecting taxes from Americans, so each time liberalism calls on government to provide a service, the taxpayer is on the hook. And with each new commitment, taxes go higher, and people keep less of what they earn.
The problem is that government, by writing checks to support those who need help, is taking money away from people who are earning their own living, oftentimes struggling to do so. Now, it's one thing when government seeks to help those who are destitute or physically unable to help themselves. But it's quite another thing when roughly half the population is subsidized, in one way or another, by the other half. That's what we have in this country. And here's the thing. It's not just a questionable moral arrangement for a society. It also happens to be far more expensive than we can afford. That's why we have deficits. The government has agreed to support many people, for many reasons, but is unable to collect enough taxes from the rest of us.
The challenge for the millennial generation is to say: "Enough." Millennials already believe debts and deficits must be controlled, that government is trying to do too much, and that it's time for the government to focus on meeting the promises it is committed to meet, instead of loading on even more commitments.
So what has to change? Here is an agenda for millennials to embrace, an agenda of fiscal sanity. Some of it, admittedly, is a little wonky. One thing I learned in the Bush administration's Office of Management and Budget is that the devil is in the details. You can tell Washington to stop spending, but Washington has a million little ways to tell you it can't. Those million little ways are what preoccupy the budget wonks. And if you want to get anything done in Washington, you have to prove your budget policy chops. Otherwise, it's like trying to operate in a strange land without speaking the language.
Here are three big areas—spending reform, tax reform, and entitlement reform—where the Republican Party can take the most effective action in order to restore fiscal sanity and make our case to the millennial generation.
## **Spending Reform**
The first thing we can do to rein in spending is implement real PAYGO. Despite the sound of it, this isn't some kind of highway toll. It's a rule that forces legislators to actually pay for new spending by cutting spending somewhere else. While President Obama praised the new PAYGO legislation in February 2010 as a way to force Congress to "pay for what it spends, just like everybody else," in fact it hasn't quite worked out that way, because Congress can ignore its own rule just by getting enough votes. In the 110th and 111th Congresses, federal lawmakers have either waived or ignored PAYGO twenty-five times. Even if a real PAYGO covering all spending is implemented, we still need lawmakers who have the integrity to play by their own rules.
We can also eliminate _earmarks_ , mini-spending programs that would not pass muster if they had to compete with other government priorities but get inserted into legislation because they have a powerful sponsor in Congress. Each time an earmark is inserted into a spending bill, its cost could have gone to support something more deserving or gone unspent. More important, earmarks are a way to buy votes for legislation that might otherwise not pass or to buy votes toward future elections. In the Omnibus Spending Bill that was proposed by the lame-duck Congress in December 2010, Tea Partiers publicized $8 billion in Republican and Democratic earmarks before debate on the bill could even begin. The resulting uproar forced Senate majority leader Harry Reid to pull the bill. For years, politicians thought earmarks were good politics; not anymore, it turns out. Republicans, who are some of the worst offenders—Senator Cochrain, I'm talking to you—need to understand that if they want to connect with millennials, they need to end earmarks.
## **Tax Reform**
The first time millennials filled out their taxes, they probably thought that the task seemed awfully complex. It's true. And, in fact, tax forms are so forbidding and impenetrable that even the most patient and the most intelligent among us are forced to hire someone to do them for us. This is wrong. It is one thing for government to ask citizens to pay their fair share (whatever that is); it's quite another to impose an additional burden on citizens as part of the process. Conservatives have always had the upper hand on this issue, and they can press their advantage with some combination of two of my favorite ideas: House budget chairman Paul Ryan's plan to simplify the tax code on personal and corporate taxes and the flat tax, first proposed by Alvin Rabushka and Robert Hall of the Hoover Institution in 1992.
Ryan's tax plan dramatically simplifies the federal tax code. It proposes only two levels of taxation: 10 percent on annual income up to $50,000 for single filers and $100,000 for couples, and 25 percent for filers with incomes above those levels. It also provides a generous standard deduction to replace itemized deductions. It further eliminates the now out-of-control alternative minimum tax, and it eliminates taxes on savings accounts, CDs, money market accounts, capital gains, dividends, and the estates of people who have already paid their taxes and are dead. In addition, it replaces the corporate income tax (which is among the highest in the world) with a business consumption tax of 8.5 percent.
The flat tax, on the other hand, is just that: a single tax rate that is paid by everyone who files an income tax return. What started out as a "crazy" proposal and "out of touch with reality" (to quote the idea's detractors) began to gain support when Steve Forbes endorsed it in the presidential election of 1992. Today, if you walk into Rabushka's office at the Hoover Institution at Stanford University, you'll see the map he keeps with flag pins denoting those nations that have adopted the flat tax. To date, more than twenty have done so—including Russia, Ukraine, Latvia, and Iceland. Millennials surely can see the appeal of either idea because they both promise the same thing: a simple way for Americans to pay their fair share, knowing they have complied with the law, and without gaming the system. It also ensures that everyone pays into the system and that everyone actually has a stake in the game.
## **Entitlement Reform**
About 60 percent of the federal budget is spoken for and very difficult to cut because it pays for things that are already promised: Social Security, Medicare, Medicaid, and interest payments on the debt. The reality is that all the spending and tax reforms in the world won't solve our fiscal problems if we don't take on these behemoths. We have to address the problem of out-of-control entitlement programs. The bill for the promises made in the past through Social Security, Medicare, and Medicaid is coming due, and we can't afford to pay it. Our projected unfunded liability for these entitlement programs alone reached $107 trillion in 2009, according to the National Center for Policy Analysis.
But it's not enough for Republicans to merely complain about the problem. Millennials will grow weary of such complaints unless viable solutions are forthcoming. Otherwise, they will simply conclude, correctly, that Republicans are merely whining and not offering realistic proposals to fix the problem. Fortunately, fresh faces in the Republican Party have proposed detailed solutions for the pending crisis. Paul Ryan's "Roadmap for America's Future" outlines three easy changes that could make Social Security solvent for decades. The first component would gradually increase the retirement age over time to reflect more accurately when workers actually need to stop working. When Social Security was created, the average lifespan for male workers was sixty years and the retirement age was fifty-five. Now life expectancy for men is approaching seventy-five years, but the retirement age is sixty-five. The program was designed to assist retirees for an average of five years after their retirement. Now that people are living ten or more years beyond their retirement, the program has to adjust. Ryan's plan wouldn't affect anyone who is fifty-five or older today, but it would start to raise the retirement age for those younger and give them enough time to plan to supplement their Social Security income. Part of that plan, if they choose, would include a government-managed personal account, similar to the program available to elected members of Congress and federal employees.
The final reform to Social Security would simply adjust the way we calculate future benefits. Right now, the government increases future benefits according to what people earn and multiplies that by the increase in average wages. This makes sense, but as it turns out, wages rise faster than the rate of inflation, which is what really counts to retirees. By using the inflation rate rather than the projected increase in wages to determine future benefits, we could chop off a huge chunk of what the Social Security system currently owes future retirees. People will still get what they paid into the system, and that amount will cover the way inflation eats away at savings. But those benefits will be less, as they must be, in order to make sure the program continues to exist for everyone.
Likewise, Chairman Ryan's 2011 budget that was passed by the U.S. House of Representatives courageously tackled the problem of saving Medicare. Ryan's bold plans prove that Republicans are serious about reforming entitlements and can do more than demogogue debt and deficits. This is _exactly_ what millennials need to see.
But ultimately, addressing the structural problems in Medicare and Medicaid won't resolve the increasingly urgent underlying problem—the rising cost of health care. That will require reforms to the health-care marketplace, in order to make the purchase of health insurance and of health care itself more responsive to market forces through increased competition and a more empowered consumer. For example, people should be allowed to purchase health insurance across state lines, which would increase competition and lower the costs of health insurance plans, while health savings accounts should be made more portable and tax-deductible. Incentives should be in place to encourage more catastrophic health insurance plans to cover younger adults at lower costs, while medical malpractice reform would liberate doctors from the excessive costs that come with unnecessary tests and out-of-control insurance premiums. In contrast to Obamacare, free-market reforms that increase individual choice and lower costs can become a hallmark of Republican health-care reform, one that will appeal to the millennial generation.
A government can run a deficit for many reasons, and deficits are not necessarily evil. We run deficits to fight wars when winning those wars is essential to guaranteeing our freedom and the security of free nations around the world. But when we run _structural_ deficits—deficits that exist regardless of whether the economy is performing well, regardless of whether tax revenues are up or down—we have committed a moral wrong. We have allowed our promises to exceed our ability to pay, and we have allowed government to occupy a place in our lives much bigger than it should be. Government that is bigger than our capacity to pay for it is, by definition, a violation of the compact between the government and the governed and between the citizens of today and the citizens of tomorrow. Millennials understand this. They may favor big government in certain cases, but they do not favor government bigger than we can afford. They don't want to spend their lives paying for their parents' entitlement-spending party. And they don't want America to go bankrupt on their watch.
President Obama has used soaring rhetoric in speaking of the importance of reducing deficits, but he has made those deficits significantly larger—in just two years—and he has offered no plan to curb future deficits. Here is what Andrew Sullivan, who usually supports Obama, had to say to millennials about the president's recent budgets: "To all those under 30 who worked so hard to get this man elected, know this: he just screwed you over. He thinks you're fools. Either the U.S. will go into default because of Obama's cowardice, or you will be paying far far more for far far less because this president has no courage when it counts. He let you down. On the critical issue of America's fiscal crisis, he represents no hope and no change. Just the same old Washington politics he once promised to end."
Our character as a nation is at stake: Will we be a debtor nation, forever requiring others to fund our deficits? Will we be forever unable to face the dangerous imbalances in our budget? Will we be unable to afford the things we want, or even the things we need, in future years, because we will be paying for the things previous generations enjoyed? Other nations have been in this position. Other nations have chosen to ignore the problems. Those nations—once great and powerful—are now mere shadows of their former selves. America must not allow this to happen. We must face this issue squarely and face it now. And Republicans must take the lead. Millennials will see these ideas, and they will see who is proposing them. They will have a choice between a party that is simply adding to the debt, and a party that is putting forward a plan to bring it down. The choice will be clear, and no matter what other issues help shape millennials' political loyalties, no choice they face will be as stark, and as consequential for their economic fortunes and way of life, as this one.
# CHAPTER 5
# FREEDOM MEANS FREEDOM FOR EVERYONE
_"It's time America realized that there is no gay exemption in the right to life, liberty, and the pursuit of happiness in the Declaration of Independence."_
—BARRY GOLDWATER
EVERY IDEALISTIC YOUNG adult who goes to Washington, D.C., to work in public service sooner or later confronts the unavoidable reality that she's working for someone she doesn't agree with 100 percent of the time. For me, that moment arrived when I learned about the Bush-Cheney reelection campaign's anti-same-sex-marriage strategy.
I had come to the nation's capital in the spirit of patriotism after the attacks of September 11, 2001, was driven by a strong desire to serve my country, and was honored to be working for President George W. Bush. But when I learned the campaign would be supporting a divisive strategy to mobilize socially conservative voters, I suddenly realized that serving this president meant supporting a man who was willing to take measures that were detrimental to my gay friends and acquaintances, people I love and respect.
I was deeply troubled by the president's proposal to amend the Constitution to define marriage as being exclusively between a man and a woman and by the campaign's strategy. But as a twenty-five-year-old junior staffer, I wasn't in a position to argue the point. I believed then, as I believe today, that we needed to support our nation's leadership in wartime. American troops were in harm's way in Iraq and Afghanistan; radicalized Islamist terrorists, although weakened, were still plotting 9/11-style attacks against America; and the world was less stable than it had ever been before in my lifetime. For me, it came down to the fact that I trusted President Bush to lead the country through the war more than I trusted his opponent, Senator John Kerry. I also supported President Bush's domestic policy priorities, including his tax cuts in 2001 and 2003, his dedication to improving America's failing education system, and his outreach to Hispanic Americans. But I still wondered whether this strategy—winning key states by advocating anti-gay policies—was the only way, or the best way, for a wartime president to get reelected.
I could have stood on principle and left the campaign for another job, but my departure wouldn't have made so much as a ripple. People who quit campaigns usually burn enough bridges to prevent their participation in politics in the future. I believed that remaining on the campaign staff would be better in the long run, presuming that I would find an opportunity to one day advocate within the party for a different approach to marriage freedoms. And so, even though I supported gay rights and found the campaign's tactics on this issue to be inconsistent with a conservatism that champions individual liberty, I worked hard to reelect George W. Bush.
We learned after the election that the anti-same-sex-marriage initiatives played no decisive role in securing President Bush's victory. While the marriage amendments passed in all eleven states, not all of those voters pulled the lever for Bush. For example, Michigan's rust-belt Catholics voted against same-sex marriage but still voted for John Kerry. Overall, the initiatives mobilized people _across_ party lines rather than ensuring that only social conservatives showed up to vote as a bloc for President Bush.
As it turned out, the 2004 presidential election hinged on the state of Ohio, and analysis of the results in Ohio's eighty-eight counties finds no conclusive evidence that the 136,000 votes that won the state for Bush were directly tied to the ballot initiative.
This strategy was bogus from the beginning, in the opinion of the Bush-Cheney campaign's chief strategist, Matthew Dowd, who conducted extensive polling before and after the election. As Dowd later told me, "No social issues or themes were in the top five messages that motivated [social conservative] voters." Instead, they were most concerned with national security issues, such as terrorism, and economic issues, such as tax cuts. Comparing battleground states that featured anti-same-sex-marriage initiatives with those battleground states that did not, Dowd's postelection analysis found zero difference among Religious Right or social conservative voter turnout. Social conservatives voted for George W. Bush in equal numbers, regardless of whether there was an anti-gay measure on the ballot.
What the anti-same-sex-marriage initiative accomplished instead was to brand the Republican Party as the anti-gay party. The campaign decision to follow this strategy had the unfortunate effect of cementing a false narrative within the political establishment that social issues are the primary mobilizer of the Republican Party's base.
Empowered by this narrative, an elite group of prominent and vocal social conservatives, the "conservagenzia," as Dowd has dubbed them, have gained outsize influence within the Republican Party infrastructure and misrepresent the most urgent concerns of the party's base. This mythology now holds the Republican Party hostage and prevents us from reaching out to millennials and Independents—who, absent the emphasis on social issues, are quite likely to agree with the Republican Party's message on fiscal and national security issues.
The widespread acceptance of this mythology has been terrible, not only for the party's ability to attract younger voters, but because it represents a departure from one of the most important principles of the Republican Party and the conservative movement.
Yes, you heard that correctly. The Republican Party has violated one of its core premises—that it is the party of individual freedom—and by doing so, has jeopardized its own future.
When I tell people who are my age or younger that I'm in favor of gay rights, including the freedom to marry the person they love, or same-sex marriage, they seem stunned. Not because supporting equal rights for gays and lesbians strikes them as unusual. Rather, they assume it's impossible for someone to be a Republican and to support gay rights. They wonder why I have not been kicked out of the Republican Party.
From the point of view of the majority of millennials, when it comes to gay rights, the Republican Party is on the wrong side of history. They don't know about the many prominent Republicans who have publicly _supported_ the freedom to marry, including former first lady Laura Bush, former vice president Dick Cheney, former solicitor general Ted Olson, and former chairman of the Republican National Committee Ken Mehlman.
Ken's experience is especially compelling. He recently came out and in the time since, he has dedicated himself to achieving equal rights for gay and lesbian Americans. Some of his efforts are behind the scenes, making the case to his fellow conservatives. But much of his energy is devoted to public efforts to raise money and awareness, just the kind of thing he did as a political operative for President Bush.
The Republican Party as a whole has yet to get credit for the individual efforts of an increasing number of Republicans, people like me, who advocate for gay rights within the GOP. Much better known to the broad public are the efforts of special interest groups such as Concerned Women for America, the Family Research Council, and the National Organization for Marriage, which aggressively lobby against equal rights for gays and lesbians. Until very recently, their message and their tone defined the Republican Party's approach toward gay rights and its image on this issue. But anyone who values the long-term interests of the Republican Party should understand that this approach is harmful.
Recall what we know about millennials. They are by no means uninterested in issues of morality; in fact they typically have strong and consistent ideas about what constitutes moral behavior. And the majority of them just don't see a correlation between sexual orientation and morality. They have grown up with gay friends, gay family members, gay coaches, and gay teachers. Millennials understand intrinsically that sexual orientation is not an active choice people make. As they see it, sexual orientation is a part of someone's personal makeup and should not in any way influence his or her rights as a citizen. And indeed, the gay rights movement has achieved its greatest success by insisting on the ordinary rights of citizens: matters such as the legal status of domestic partnerships and civil unions, hospital visitation rights, and freedom to serve in the nation's armed forces openly. These are elemental rights for most Americans, yet they are the kinds of rights for which gays and lesbians have had to struggle. No wonder millennials support their struggle: There is no reason to deprive any citizen of these basic rights.
Millennials are almost twice as likely as Americans who are sixty-five years and older to think that homosexuality should be accepted. People under thirty who identify with specific religious traditions are more likely than their older co-religionists to view homosexuality as acceptable. One might assume that acceptance of homosexuality would be much more common among the non-religious, yet even among religiously affiliated millennials the numbers prove that this generation is less likely to perceive homosexuality through the lens of religion. As many as 39 percent of young evangelicals, 69 percent of young mainline Protestants, 72 percent of young Catholics, and 51 percent of the youth in historic black Protestant churches believe homosexuality should be accepted by society. For the majority of youth, homosexuality is neither a religious nor a secular "sin" that justifies discrimination.
One of American society's most inspiring features is its ability to improve itself over time. On the issue of gay rights and acceptance of gays, as a society we have made remarkable progress in my own lifetime. When I was young, there was still strong prejudice against open homosexuality. I recall distinctly what happened when a dear friend's father came out in the late 1980s. Not only was being gay still considered taboo, but prejudices were newly reinforced by the outbreak of the AIDS virus, which was then considered to be a disease associated solely with the "gay lifestyle." At that time, the movie _Philadelphia_ , which would so movingly dramatize the struggle of a gay man dying of AIDS, hadn't even been made. Magic Johnson still appeared invincible (and he is still with us today, but who knew that antiretroviral medicines were in our future?). I was in sixth grade, and another classmate hurled an insult at my friend, making fun of her gay dad. The pain she suffered from such insults affected me deeply. Her father loved her just as my father loved me, and I decided then and there that there was no reason our fathers should be viewed differently by society. Why should their love and their positions in society as fathers be held to different standards of respect and honor? Why should one man be mocked and another man be left unscathed? Why should my friend suffer because of who her father was? And why would anyone choose a "lifestyle" that would invite a lifetime of discrimination? Clearly sexual orientation wasn't a choice. These emotions raged within me, and as I worked them out in my heart and head, my sense of injustice mounted.
Over the past twenty years millions of Americans have had similar personal experiences having to do with their gay friends, family members, and acquaintances, and I believe that their individual experiences are largely responsible for the sweeping change in attitudes about gays and lesbians we have seen in this country. Politics is personal and the more that people have personal interactions with gays and lesbians, the more they support equal rights for them. Polling confirms that the number of people who say that they have a friend, relative, or acquaintance who is gay mirrors the number of people who are in favor of expanding civil rights to gay individuals.
This sea change is reflected in our popular culture. Every day Ellen DeGeneres is invited into living rooms all across America to talk about ordinary issues. She is one of America's most loved and lovable talk show hosts—and her sexuality happens to be irrelevant to her success as an entertainer. Shows such as _Queer Eye for the Straight Guy_ and _Will & Grace_ have made even more stereotypically flamboyant elements of gay culture more accepted throughout straight America. The situation comedy _Modern Family_ debuted in 2009 and features a male couple learning the ropes of parenting after adopting a Vietnamese girl. Even the fact that Elton John, a lesbian, gay, bisexual, transgender (LGBT) movement icon, performed at Rush Limbaugh's wedding (images are available on Rush's Facebook page) proves that discrimination against gays and lesbians from all sides of the political spectrum has markedly diminished within American culture.
I started out as a Republican in part because of a family legacy. But my own personal journey, in politics and in life, has convinced me that the Republican Party, with the conservative movement's emphasis on individual freedom and its spirit of American individualism, is the natural home for equal rights. And why not? After all, this is the party that gave political force to the abolitionist movement and the woman suffrage movement. The Republican Party has individual freedom built into its DNA.
Conservatives believe that people, not government, make the best decisions for themselves, and Republicans act on this belief by advancing policy prescriptions on a variety of issues, from welfare reform to entitlement reform, health-care reform, education reform, and tax policy. The freedom of people to live their lives as they choose, and to enjoy the full benefits of their rights as enshrined in the Constitution, is entirely consistent with the central tenet of conservatism: individual freedom.
Still, there is a stark difference in how distinct generations within the Republican Party, and within all of American society, view homosexuality. I didn't fully appreciate what a generational issue this was until the time I had the chance to talk with Bill O'Reilly about California's Proposition 8, the discriminatory law in California that defines marriage as being exclusively between a man and a woman.
Let me say at the outset that even though Bill O'Reilly is the top-rated personality in cable news, most of the people who talk about him negatively have never watched his show for more than a few minutes at a time. They would rather believe everything that his detractors such as Rachel Maddow and Keith Olbermann have said about him. The truth is, Bill is a registered Independent, and sometimes his views fall more toward the center than the right. Yes, he is old-school, from a working-class background in Levittown, New York. He is culturally conservative and Catholic. He is hostile to modern liberalism precisely because he thinks liberals tend to undermine the values that have made America great, such as freedom of religion and the rights and responsibilities of families. Bill is also remarkably compassionate, and he raises and donates millions of dollars to charities annually. He genuinely cares about his viewers and sees his job as looking out for people whom the government or special interests are ignoring or mistreating. He looks out for the people in Levittown and the countless other places like it across America.
Once when I appeared on his show, Bill casually mentioned that I, as a Republican, must be opposed to same-sex marriage and in favor of Proposition 8 in California. Because TV segments fly by quickly, I had no idea how I was going to explain, in such a short amount of time, that I support the freedom of gays and lesbians to marry. I believe preventing them from marrying amounts to legal discrimination that reduces them to second-class citizens. Most people my age don't view homosexuality as a "lifestyle choice" or something that the liberal elite are pressuring us to accept as a form of political correctness. How could I fit all of this into a sound-bite-size response? So instead I took the easy way out and told him that there was no way that he'd get it, that he was just too old. At that instant I heard a loud noise and realized it was a producer from the control room who was whistling into Bill's earpiece. Bill's a pro, so he didn't flinch; but I had zinged him and he knew it. I had made my point. Unfortunately, I appeared to have tagged him as an old fuddy-duddy in front of his beloved fans. I was certain I'd never be invited back, but when I was—happily—I wondered whether this exchange had any influence on Bill, who now seems to understand that my views on gay rights reflect the sensibility of the next generation.
Ensuring that gays and lesbians attain the same rights as all other Americans has always seemed to me like one of the most obvious and most morally clear causes that Republicans can make their own, just as Republicans historically stood on the side of freedom against slavery and on the side of women's suffrage.
I am in good company defending this position. My support for gay rights is entirely consistent with American individualism and the conservative movement's most important principle: maximizing individual freedom and protecting equal opportunity for all Americans. Ronald Reagan was an early opponent of discrimination against gays and lesbians. In 1978, Reagan spoke out and campaigned against California's Proposition 6, which sought to ban homosexual teachers from public schools, as depicted in the Academy Award–winning film _Milk_. In the words of biographer Craig Shirley, "Ronald Reagan opposed any form of discrimination based on homosexuality.... The true American conservatism as articulated and embraced by the Gipper celebrated the individual, privacy and 'maximum freedom consistent with law and order.' "
As we all know, politics is rarely driven by a single social issue. In the 2010 campaign, the hot-button issues were fiscal responsibility, curbing federal spending, and President Obama's health-care overhaul. In 2008, it was the economy. In 2006, it was the Iraq War and Bush fatigue. It is doubtful that the issue of gay rights will ever rise to that level. But it's safe to say that as millennials assume positions of authority and leadership throughout society, the Republican Party cannot afford to be perceived as the party of bias and discrimination.
The GOP needs to recognize that supporting gay rights is historically consistent with the party's fight to expand freedoms to previously disenfranchised segments of society. Republican emphasis on individual freedom and individual choice should not be limited to our economic policy. Why should the conservative movement, which says it supports individual freedom, support it only for certain people? In the words of former vice president Dick Cheney when discussing his views on same-sex marriage and his lesbian daughter, "Freedom means freedom for everyone."
There are some on the right, however, who suggest that a person can't be a true conservative if he or she supports equal rights for gays and lesbians. While this may ring true among the traditionalists and religious conservatives, there are others within the conservative movement—economic conservatives, libertarian conservatives, national security conservatives—who support the full integration of gays and lesbians as equal citizens in society. There is considerably more diversity among Republicans and conservatives than you might imagine from watching MSNBC.
But to some conservatives, people like me who support gay rights are Republicans in Name Only—RINOs, the targets of RINO hunters, who claim to have a corner on what conservatism means. But RINO hunting on the issue of gay rights is bound to lead to a smaller conservative movement. Drumming out dissent instead of supporting what Ronald Reagan called a "big tent" will only turn people away from the GOP. And it sends a signal to all Americans—and especially millennials—that the Republican Party is close-minded and complacent, a party going nowhere.
Yet history shows us that it can be otherwise. During the past century, the Republican Party has repeatedly evolved in response to a changing world. It was at one time isolationist; it is now largely internationalist. It used to represent powerful moneyed monopolies, but now is the home of Main Street populists. The Republican Party is defined by a set of ideals and principles, including the protection of individual freedom from concentrations of power, which was essential to America's founding, has been vital to its prosperity, and remains essential to its future. If Republicans fall out of touch with those principles today, they have a diminished future as a party or a movement.
It is true that the modern conservative movement—and thus the Republican Party—has always been a bit schizophrenic on social policies. Historically the conservative movement has been most unified when it was in opposition. Mostly, though, conservatism is a cacophonous aggregation, like one of those big, happy, and somewhat offbeat families gathered for Thanksgiving dinner. Someone will always be arguing with someone about something. In the 1950s its coalition of traditionalists, economic libertarians, and anticommunists fought constantly with one another about the proper balance between progress and tradition. It was such bickering that led the godfather of the movement, William F. Buckley Jr., to say that "the conservative movement in America has got to put its theoretical house in order."
Let me suggest how today's Republicans can put their house in order and come to an understanding with gay rights advocates. For most Americans, the rights of gays and lesbians to be employed and to have access to public and private accommodations are not in dispute. What is in dispute is the question of whether gays and lesbians should be allowed to form the same bond and enter the same sacred relationship—marriage—that heterosexuals do. Those in favor of and those opposed to marriage equality engage this issue with equal intensity. Opponents of marriage freedom insist that they are concerned for the institution of marriage. They believe that marriage is an institution unique to one man and one woman, ideally for the purposes of procreation. This, they say, is the best arrangement for a healthy society.
The proponents of freedom to marry, like myself, make the same argument. I agree that marriage is humanity's most vital social institution. Marriage creates a legal and moral bond for two individuals to support each other throughout life. Indeed, many of our social institutions rely on this fundamental commitment. Next to the sanctity of the individual, the sanctity of the marriage bond is a foundational element in public life, an expression of our simultaneously self-reliant and mutually reliant society. The fabric of society is woven together and held securely by the bonds individuals form with one another and with their communities. The stronger the connective tissue between all these components, the stronger we are as a nation.
If conservatives truly cherish these values, there is no better way to support them than through policies that encourage cohesive marriages and families. Making "the big tent" a bit bigger by welcoming same-sex couples into the tradition of marriage will serve to strengthen the institution of marriage, not weaken it.
This argument should have special appeal to libertarian conservatives and others whose primary concern is to prevent the intrusion of the state into the private realm. For them, broken families are costly: delinquency, poor health, illiteracy, recidivism, and all manner of social ills stem directly from broken families. If conservatives favor a society in which individuals are less reliant on the government, then they should encourage individuals—gay or straight—to forge lifelong commitments to each other, and grant these commitments every advantage under the law. Expanding the benefits of marriage to same-sex couples strengthens the fabric of our society by cultivating an increasingly self-reliant and interdependent society, with the result that individuals become less reliant on the government.
Some conservatives believe that same-sex marriages will somehow taint heterosexual marriages, but they have failed to explain how exactly that would happen. And the truth is, as most conservatives will admit, the preponderance of no-fault divorce has done more to harm the institution of marriage than same-sex marriages.
Others argue that civil unions, which grant gay and lesbian couples all the same rights and protections under the law as marriage, should be a sufficient compromise. But by fighting for the exclusive use of the word _marriage_ , they implicitly acknowledge that there is something special about the word and thus about the act itself. Indeed, the reason people oppose the use of the word _marriage_ to describe same-sex commitments is because they would prefer to create a lesser category—proof that they themselves recognize that marriage is magical and distinct from every other relationship. In my view, that specialness, with all its societal implications and significance, is something that should be available to all Americans. If you agree that gays and lesbians should be able to form permanent bonds, then why not allow them to experience all the magic encapsulated in that word: marriage?
I do not believe that those who support civil unions but not legal marriage are bigots; I just don't think they have considered why nothing less than the right to marriage is what is appropriate for gays and lesbians and why something society deems special should be denied to certain citizens just because of who they are.
We've also learned in America to beware of something called separate but equal under the law. Separate is not equal. And insisting on a separate category of union other than marriage gives some of our citizens an excuse to treat other citizens differently, and that invariably leads to discrimination.
Two landmark cases, _Loving v. Virginia_ in 1967, which legalized interracial marriage, and _Turner v. Safley_ in 1987, which allowed prisoners the right to marry, affirmed that marriage is a right enshrined in the United States Constitution. But as conservative constitutional expert Theodore "Ted" Olson has pointed out, the Supreme Court has ruled fourteen times since 1888 that marriage is a fundamental right, the same as those enshrined in the Bill of Rights—freedom of speech, freedom of assembly, freedom to bear arms, freedom of the press. If the Supreme Court has ruled that marriage is a fundamental right, then laws forbidding marriage between homosexuals are unconstitutional. In singling out a group of citizens and denying them the freedom to marry, our laws have created a second class of citizens and institutionalized discrimination against them. When you consider that imprisoned _criminals_ are allowed to marry, it seems outrageously unjust that law-abiding gays and lesbians are denied the same freedom.
The day when a majority of Americans recognize that the freedom to marry is a fundamental constitutional right is not far off. It's coming not just because millennials are rising to positions of authority, but also because the law is taking us in that direction.
I remember a gray January morning in San Francisco inside courtroom number five of the U.S. District Court, Northern District of California, where I watched from a wooden gallery bench the legal dream team of David Boies and Ted Olson argue the constitutionality of same-sex marriage in the landmark civil rights case _Perry v. Schwarzenegger_. I was asked to join the Advisory Council of the American Foundation for Equal Rights, a bipartisan group of advocates for the nonprofit entity that brought the plaintiffs' case to court. I had come to San Francisco to witness the final days of a three-week trial.
This was not a cut-and-dried, Left-versus-Right, secular-versus-religious kind of contest. Olson is a respected constitutional conservative, a founder of the Federalist Society who successfully argued _Bush v. Gore_ before the Supreme Court (among fifty-five other Supreme Court cases). He served as President George W. Bush's solicitor general. Boies was the opposing counsel in _Bush v. Gore_ and is a well-known Democratic Party activist. But now these two men stood side by side in a legal quest to prove that marriage equality is a constitutional right, not a partisan issue.
The plaintiffs' legal team assembled a thorough record of evidence that Proposition 8, the California law that defined marriage as being between a man and a woman, unreasonably discriminates against gays and lesbians, relegating them to second-class citizenship.
Among the seventeen witnesses called to the stand were experts in the fields of psychology, political science, economics, sociomedical sciences, and history. Economists testified about the financial harm done to same-sex couples and their children; political scientists about their political vulnerability; sociologists and psychologists about the societal stigma attached to homosexuality; historians about the history of marriage having become available to more and more groups over time.
As the judge was about to enter the chamber, one of the lawyers in a black suit whispered to me: "You're lucky; you're right on time to see David do a cross." Indeed, Boies is known as one of the great litigators of our age. His ability to cross-examine a witness, dismantling testimony brick by brick until the entire edifice crumbles, is something his wife has described as an aphrodisiac. Boies's target that day was one of two witnesses for the defense who had earlier testified against allowing same-sex couples to marry.
The witness was David Blankenhorn, who runs the Institute for American Values. He is the author of two books on the subject, _Fatherless in America_ and _The Future of Marriage_. Boies made short work of Blankenhorn. The witness found himself agreeing that marriage is better for children, regardless of the gender of the parents, and that growing up in same-sex households is better than in a single-parent home. He admitted that "adopting same-sex marriage would be likely to improve the well-being of gay and lesbian households and their children." I wondered to myself, _This man represents the people who disagree with same-sex marriage?_ Finally, Boies read from Blankenhorn's book _The Future of Marriage:_
This still-revolutionary principle—"all men [persons] are created equal"—deeply informs the American experience and character and is increasingly viewed globally as the essential universal moral law. On the issue of same-sex marriage, is this profound principle of equality and dignity the heart of the matter? After all, part of the reason why the principle is so revolutionary is that it can grow and deepen over time. Groups that had long been considered effectively outside of its moral reach—African Americans, women, people of certain colors or languages or religions—can over time, and often as a result of great struggle, enter into its protective sphere. I believe that today the principle of equal human dignity must apply to gay and lesbian persons. In that sense, insofar as we are a nation founded on this principle we would be _more_ American on the day we permitted same-sex marriage than we were the day before.
These were indeed the words of this star witness, and he could offer no argument or reason at that trial, as he spoke under oath, for why our society or its laws ought to prohibit gays and lesbians from enjoying the benefits of marriage.
Just two months earlier, I had been married in California—and it struck me on that day in the San Francisco courtroom that I had taken for granted my freedom to marry the person I loved. At the Proposition 8 trial, the arguments for marriage made by people who are not legally able to enjoy its benefits presented a clearer and more compelling justification for marriage than any that my new husband and I had considered in our prenuptial preparation. I was forcefully reminded exactly why marriage is so special and why every individual in America, gay or straight, deserves equal access to this sacred institution.
I believe the Republican Party will come around. On September 22, 2010, prominent Republicans including two former governors, one RNC finance chairman, business leaders, and several Bush-Cheney 2004 campaign staff alumni hosted a high-profile fund-raiser in support of the freedom to marry. The fund-raiser was chaired by our old boss Ken Mehlman, the campaign's manager and the former Republican National Committee chairman who had only recently come out. We raised $1.3 million for marriage freedom that day, the first public manifestation of a fundamental shift happening inside the Republican Party. More and more Republicans are realizing that gay rights are a simple issue of freedom.
Though Republicans have been justifiably maligned for their sponsorship of anti-same-sex-marriage initiatives, the Democratic Party has fallen short as well. Once again, you heard that right. Despite liberal support for same-sex marriage, no major Democrat running for national office has openly supported freedom to marry. New York State Democrats famously failed to pass marriage equality legislation in December 2009. And don't forget that a Democratic president, Bill Clinton, signed the Defense of Marriage Act (DOMA), which prohibits states from legally recognizing same-sex marriages or civil unions conducted in other states. Even David Boies, the Democratic lawyer representing the plaintiffs in the Proposition 8 trial, chided President Obama for not supporting the court case. After all, Boies pointed out, the president's parents would not have been able to legally wed in fifteen states at the time of their marriage, and only thanks to the Supreme Court's decision in _Loving v. Virginia_ was the prohibition on interracial marriage lifted in 1967.
It will take many courageous Republicans to make the case that a conservatism that champions individual freedom must also support gay rights. Democrats are very good at playing identity politics. But I believe that in the end, identity politics is a losing proposition. It may work in certain instances, but what gays and lesbians want—and what millennials expect gays and lesbians to receive—is respect and honor as citizens and individuals. That means not more privileges or pledges but the expectation of full participation in public life.
The Republican Party is changing not only thanks to leaders like Ken Mehlman but also to people like the former police chief and mayor of San Diego, Jerry Sanders. Mayor Sanders, the father of a lesbian daughter, didn't support marriage equality before his election and believed civil unions to be an acceptable compromise. The issue didn't figure in his campaign. He ran for mayor on a platform of cleaning up San Diego's legal and financial troubles after having served on the San Diego police force for twenty-six years.
In 2007, the San Diego City Council forced Mayor Sanders to take a stand after it passed a resolution supporting a court challenge to California's ban on same-sex marriage. By city law, Mayor Sanders had ten days to sign or veto the resolution supporting marriage equality. As the father of a lesbian but the mayor of a conservative city, Sanders agonized over his decision. Politically, it was a dangerous one: The mayor faced a serious reelection battle the following year and the odds against him would grow longer if he chose not to veto the resolution. He waited until the night before the deadline before making his decision. He decided to veto. While his daughter disagreed with him, she supported his decision because she believed it was more important for San Diego that her father continue to serve as the city's mayor.
The night before the ten-day deadline expired, he hosted his closest gay friends and supporters at his home, so he could explain why he intended to veto the resolution. He then listened as they expressed their disappointment. In his words, "About fifteen people spoke that night. But before the first one was finished, I shared their disappointment. It was then that I realized that all opposition to same-sex marriage, including my own opposition, was grounded in prejudice."
The next day, Mayor Sanders signed the resolution intead of vetoing it. The video of the press conference, posted online, instantly went viral and has since received more than a million views on YouTube. Mayor Sanders went on to testify for the plaintiffs in _Perry v. Schwarzenegger_ , arguing that California's Proposition 8 is unconstitutional. About that testimony, and his own evolution on the issue of gay rights, he later wrote:
I hope that everyone will find someone they love deeply, someone with whom they can share life's experiences and grow old together. I cannot look anyone in the face and tell them that their relationships, their very lives, are any less meaningful than the marriage I share with my wife.
Sometimes I find it hard to believe that I came so close to making the wrong decision, and to endorsing government-sanctioned discrimination.... I was reelected to a second term the next year. My position on marriage equality definitely made it more difficult.... As someone who has spent most of his lifetime in public service, I understand that when government tolerates discrimination against any class of people, it makes it easier for citizens to do the same thing.... History tells us that the first step toward true equality has always been equality under the law. Denying gays and lesbians the right to marry is no different than denying black people the right to sit in a "whites only" section of the restaurant. The law and our own experience tell us that "separate but equal" is an oxymoron. Separate is never equal.
This is the testimony of someone who not only has compassion for friends and loved ones, but who has gotten closer to the heart of the American credo of personal liberty. It is also the testimony of someone who recognizes that it is a distinctly American trait to improve our system when it fails to live up to its promise, when the words _freedom, justice_ , and _equality_ do not apply everywhere and to everyone. This is the strength of our system, and we are stronger for identifying our own failings. I am proud to say that my great-grandfather, Herbert Hoover, wrote precisely about this process of renewal in his "American Individualism": "Many people confuse the exposure of wrongs which were below the surface with degeneration; [but in fact] their very exposure is progress.... A considerable experience leads me to the conviction that while we do wash our dirty linen in public, most others never wash it."
Supporting equal rights for gays and lesbians is at the core of the struggle to reinvigorate conservatism as a movement of personal freedom and responsibility. I know that the Republican Party has not led on this issue. I know, in fact, that it has been on the wrong side of this issue. But the time for such political gamesmanship is over. It didn't work before; it will never work. And it will never be right.
The Republican Party that I want to be part of proudly supports gay rights. And in doing so, it opens its ranks to a new generation that shares its enduring commitment to individual liberty.
# CHAPTER 6
# EDUCATION REFORM
_A Civil Rights Win for the Millennial Generation_
_"Education spending will be most effective if it relies on parental choice and private initiative—the building blocks of success throughout our society."_
—MILTON FRIEDMAN
THROUGHOUT THIS BOOK, when I refer to millennials, I have in mind that supercharged, highly educated, techno-savvy cohort whose parents and teachers have hovered over and doted on them. These millennials have shelves groaning with trophies, framed newspaper articles, and countless stickers and ribbons recognizing their achievements. They have flooded colleges and universities with résumés that have impressed admissions officers. Their community service projects have been inventive and productive. Their extracurricular activities have been awe-inspiring in their breadth. These millennials are a by-product of some of the most intensive parenting and schooling America has ever seen.
And yet there are other millennials who haven't been so fortunate. They have been raised in single-parent homes, in urban ghettos, and in rural backwaters. They struggle as all poor and forgotten people struggle—often silently, without allies, and without institutions to support them.
Ideally, America's public school system would have given these "other" millennials a way out. This is, after all, the role America's public school system was designed to play: to create an educated citizenry and to give every child a chance to learn skills and develop talents. The American education system was to be the great equalizer of opportunity—not a leveler but a common springboard from which all American youth could rise to the best of their ability, regardless of race, religion, or economic background.
The underlying ideal of equal opportunity is the basis of America's meritocracy, and it was the foundation of my great-grandfather's "American Individualism." A public school system premised on the principle that every student will have the same opportunity to receive an excellent education helps us avoid the stagnancy of societies based on class or caste or tribe. It promises to let the most talented individuals rise to the top, regardless of where, and to whom, they are born.
Tragically, this is a promise America makes but is failing to keep. We are falling dreadfully short of this ideal, and nowhere is this clearer than in the case of the millennial generation: a generation of educational haves and have-nots, a generation subject to a public school system that too often ends up cementing instead of rectifying the inequalities of birth. It would be one thing if our public school system simply had deep flaws. But the problem is far bigger: our public school system is separate and unequal. There is one educational system for children who are born into good zip codes, and another educational system for those who live in the bad zip codes.
While teens in our best public schools learn advanced math, attend language immersion classes, and rack up advanced placement credits, the rest of the education system is drifting further behind the rest of the world. Among industrialized countries, American students now rank fifteenth in reading, fourteenth in science, and nineteenth in math. What's more, seven thousand millennials drop out of high school _each day_. That's one student every nine seconds, or 1.2 million teens per year.
Since the early 1990s, approximately 70 percent of American students have graduated from high school. That means that roughly 30 percent of millennials are unlikely to have anything close to a decent career because most high-paying jobs require at least a high school diploma. We know that those leaving school before attaining a high school diploma are disproportionately African-American, Hispanic-American, or Native-American, all of whose graduation rates hover around 50 percent. This racial achievement gap flies in the face of the ideal of equal opportunity that was at the heart of the civil rights movement in America—and central to the philosophy of American individualism.
These "other" millennials who do not attain even the most basic level of education are not only disproportionately African-American, Hispanic-American, and Native-American, but they are also concentrated in a handful of schools.
Half of our nation's dropouts come from only 15 percent of America's high schools—roughly two thousand schools. These schools have earned the dubious distinction of being assigned a special name—"dropout factories"—because their rate of success is so bad. Students attending these schools have only a fifty-fifty chance of graduating. Almost 50 percent of African-Americans and nearly 40 percent of Latinos—but just 11 percent of white students—attend dropout factories.
Income is now a predictor of school achievement—how much money parents earn can determine whether a student succeeds in school, or even graduates. High school students from the lowest-income families drop out of school at six times the rate of their peers from higher-income families. Some observers have been all too willing to blame this failure on the students and their parents, and to assume that because poor neighborhoods can't support good schools, there is no sense in trying harder with poor kids. But this complacency has bred a cycle of poverty and failure. Failing neighborhoods are blamed for creating failing schools, which are blamed for creating failing kids, who go on to live in failing neighborhoods. Too many people are willing to accept this cycle, and while they may suggest we "do something" about it, they are unwilling to challenge their own assumption that a poor neighborhood means a poor school.
As a result, we have one educational system that produces whiz kids who are mostly white and from affluent or semi-affluent homes, and another educational system that cycles through poor minorities without teaching them the skills they need to survive in a global economy. Is this a system that gives its youngest citizens equal opportunity? No way.
For all the focus among the lucky millennials on the importance of diversity, and for all their ease around people of different backgrounds, they are part of a generation of Americans that has seen a resegregation of schools and a resegregation of economic opportunity—due, in part, to our school system's unequal and unfair treatment of students. If millennials truly want to promote diversity in our nation, they will resolve to reform our public school system. They should call the system what it is: separate, unequal, and unfair. They should agitate for a school system where one's zip code does not determine one's future earning power.
Many of the lucky millennials are ashamed of the educational establishment and its inequalities, not only because those inequalities are so obviously unfair, but also because those inequalities violate the core principles millennials subscribe to. Thousands stand up against those inequalities by volunteering for Teach For America, the nation's largest supplier of excellent teachers to poor urban schools. Millennials believe that government should be competent, but public schools are not competent, at least not across the board. Millennials believe that public service is supposed to be a noble calling, but they see that many of our public servants in the classroom, our teachers, do their jobs unevenly and sometimes poorly. Millennials have made Teach For America one of the most impressive and successful nongovernmental organizations in the country, and yet the reason Teach For America exists is that there is a fundamental failure of our public school establishment.
To look at it another way, imagine that someone started a new, largely volunteer nonprofit security service to supplement the work of Homeland Security. Our reaction would be, "What is so wrong with Homeland Security that someone had to come up with an alternative?" In the most basic sense, our government has failed the millennial generation in educational opportunities, and they are rising to challenge its failure.
I want to be clear. Of course there are many thousands of excellent teachers in America—and they deserve to be celebrated as American heroes. But just as with any large group, not all teachers are great, or even good. And bad teachers—and the way they are protected by union rules—are having a demonstrably negative effect on the entire educational system.
If Republicans get serious about education reform, which I believe we are in a unique position as a party to do, we will find that we have a wonderful opportunity to connect with millennials. Of all the public policy issues treated in this book—social issues, national security, the environment—the one issue that millennials understand better than anyone else is education. They are the closest to the K–12 experience and they know there are some schools that are fantastic, some teachers who are heroes, and some students who are talented beyond words. But they also know there are other schools that are beyond hopeless, other teachers who are incompetent, and other students who are trapped. If there is a single issue discussed in this book that offers an opportunity for millennials to lead, and on which they have already demonstrated leadership, it is in the crisis in American education.
Republicans can point out to millennials that our unfair educational system is a threat to their long-term economic security. Too often we think of bad schools as a social issue, as if there were no financial cost to a failing school, save the cost of teacher salaries and building maintenance. But the macroeconomic cost of dropouts is devastating. According to a recent McKinsey & Company report, if America had closed the minority achievement gap in 1998—which simply means that if we had brought all student achievement, regardless of race or ethnicity, up to the same level as that of white students—then GDP in 2009 would have been 2 to 4 percent higher. That's equal to about $310 billion to $525 billion of additional economic activity. The same report says that the achievement gap in the United States is equivalent to having "a permanent national recession." The report concludes that "cutting the dropout rate in half would yield $45 billion annually in new federal tax revenues."
The Organization for Economic Cooperation and Development (OECD) has stated that if the United States were to boost its reading, math, and science scores to the levels of those in Finland, the result would be GDP gains "on the order of $103 trillion." These are staggering figures, reflecting a troubling reality.
For millennials struggling to find their footing in a beaten-down U.S. economy, these numbers should be a wake-up call. The past failure of American policy makers to reform our school system has deprived all millennials of a more prosperous economy with more high-paying jobs. So not only have the students in failing schools suffered directly, but everyone else has as well, even those who come from the good zip codes.
The question is: What do we do about it? The first instinct is to spend more money on our schools. This is a completely understandable reaction because we believe, as Americans often believe, that we get what we pay for. But the truth is that in education, we not only don't always get what we pay for, we often get less.
Since World War II, America's spending on students has increased 40 percent per decade, nearly doubling every twenty years. But we're still not even close to leading in math or reading scores at any level. We spend between 41 and 50 percent more money on education than the average OECD country, yet we are near the bottom in the ranking of OECD student test results.
It turns out—yet again—that throwing money at a problem doesn't solve it. Liberals and Democrats have resisted this argument for years, but it has become harder to defend the status quo—let alone to invest more money in it. People are starting to wonder where all those education dollars go. And they are finding out.
If you haven't seen the movie _Waiting for "Superman,"_ you are missing out on a powerful depiction of what is wrong with our public schools.
What's remarkable is that this film was made not only by a Democrat but by the same man who made Al Gore's _An Inconvenient Truth_. Davis Guggenheim was determined to present an honest, nonideological picture of what happens in bad schools. He didn't go into the project intending to point fingers but to show how things truly are. He describes how incompetent teachers are protected from dismissal by tenure. He shows how a culture of inertia has taken hold in many schools, scaring away ambitious teachers or forcing them to sell out to mediocrity.
One scene focuses on six hundred New York City teachers who have been suspended, with full pay, for violations ranging from incompetence to sexual abuse of students. These teachers report each day to a central location known as the "rubber room" and spend seven hours sitting around reading newspapers, playing cards, talking—all on the taxpayers' dime. They're impossible to fire because the procedures for terminating their employment are labyrinthine. So each year, these teachers are paid a total of $100 million of New York City and State taxpayer money to do nothing.
At the other extreme are schools that receive less money per pupil than their public counterparts—Catholic schools, charter schools, private academies for poor youth funded by wealthy donors—but have managed to buck the trend. They teach thousands of students from poor neighborhoods each year, and these kids, against incredible odds, manage to learn and manage to succeed. Even in the poorest neighborhoods, education reformers have proved that it is possible to educate disadvantaged students, and by doing so, to break the cycle of poverty.
What these examples demonstrate is not that money is irrelevant—of course schools need money. But far more important is accountability—making sure that teachers are held accountable for their students' learning, that administrators are held accountable for creating a safe and productive learning environment, that schools are held accountable to the parents and the community, and that the students are held accountable to their teachers.
What we have learned from these examples, above all, is that the spirit that drives the lucky and privileged millennials is precisely the spirit we need to see every day throughout our entire school system—a focus on creative solutions, a fundamental belief in the importance of each individual, and an unyielding sense of civic duty, because our schools are a reflection of our national strength and character. Let me share with you some examples of the millennial spirit at work in the area of education reform:
## **Geoffrey Canada**
If the American equivalent of being knighted is being kissed on national TV by Oprah Winfrey, then Geoffrey Canada is a knight of the education reform roundtable.
Canada, who was born in Harlem, has dedicated his life to closing the education achievement gap in the neighborhoods where he grew up. His warm personality is balanced by a quiet determination. After his graduation from Harvard, Canada wanted to give something back to his community by working to turn around New York's schools. He quickly discovered that the teachers' unions would not budge, so he decided to focus on a one-hundred-square-block section of East Harlem that included the poorest neighborhoods in New York City.
In 1990 Canada launched the Harlem Children's Zone. Canada guarantees each of the eleven thousand children in the zone that if they stay with his program, he and his dedicated staff will support them and that they will get into college. The students attend public schools or charter schools, which are publicly chartered schools run without the oversight of the public school system. While it is true that charter schools spend slightly more per pupil—$16,000 versus $14,500 in a traditional public school—they also come with three assets: a 30 percent longer school year, more teacher involvement, and a guarantee against failure. The students are predominantly African-American and Latino. In Harlem, as a result of the good works of the Harlem Children's Zone, they have closed the achievement gap.
## **Michelle Rhee**
When you meet Michelle Rhee, you know exactly what she is focused on: fixing public education in America.
In her first classroom in Baltimore, twenty years ago, she learned that despite the violence, poverty, and broken homes typical of the backgrounds of the majority of her students, excellent teachers could make a profound difference in student performance. She reached a basic conclusion: if the difference between a good school and a bad one is the quality of teachers, then the way to fix failing schools is to replace bad teachers with good ones. At first she dedicated herself to this goal through Teach For America, and then later through a nonprofit she founded called the New Teacher Project. When she met Adrian Fenty, the newly elected Democratic mayor of Washington, D.C., he offered her an opportunity to administer an entire school system. Here was her opportunity to put excellent teachers into an entire system and demonstrate improved results. When she received the power to shut down failing schools and replace bad teachers with good ones, what had previously been the worst school system in the nation was within two years leading the country in gains in math and reading at the fourth- and eighth-grade levels.
Sadly, Rhee's efforts were cut short when Mayor Fenty's rival beat him in a closed Democratic primary, thanks largely to more than $1 million in donations from teachers' unions. Sure enough, the new mayor has promised to roll back some of Rhee's reforms.
Nonetheless, Rhee's lessons endure. She demonstrated that putting excellent teachers in classrooms and incentivizing them to stay can actually improve results in the nation's worst school systems. She often poses this fact to audiences: if you took the bottom 5 percent of teachers in America's schools and replaced them with average teachers, America would go from the bottom half of educational results to being number one in education in the world. She then asks, "What CEO would not choose to fire the bottom 5 percent of underperforming employees to get that better top result in his or her bottom line?"
Rhee was willing to innovate in order to improve educational results. She questioned the teacher pay system in Washington, D.C., which rewarded teachers equally, regardless of the results they achieved. She proposed a merit pay system, especially for younger teachers who were earning $40,000 to $50,000 annually, in which they could more than double their salaries on the basis of excellent performance, with compensation reaching as high as $130,000. In return for this new pay system, teachers would have to give up tenure and thus lifelong job security. Many senior teachers favored the idea, but the teachers' union would not even let its members vote on the proposal.
Rhee's latest project is even more audacious. She has started a political organization called Students First. Inspired by the political arms of teachers' unions, it will raise money to donate to local, state, and national races for mayor and for positions on school boards. In addition, it will organize activists to demonstrate their support on behalf of students. What Rhee has learned is that the next frontier for education reform is not the classroom or the school system, but the ballot box.
## **Chris Christie**
New Jersey's Republican governor has been sharply focused on education reform since he entered office, forming a partnership across partisan lines with Newark mayor Cory Booker to empower local education reformers and do an end run around teachers' union regulations that stifle innovation. Together, they support reforms like merit pay for teachers, charter schools, and school vouchers.
Christie has been willing to close down failing schools and reward a more entrepreneurial approach to education. Facebook founder Mark Zuckerberg donated a $100 million challenge grant in support of local reform efforts. Christie has not been shy about taking on the unions, arguing that we need to put schoolchildren first and focus on classroom results, rather than assume that simply throwing money at the problem is the key to reform, especially during a budget crisis.
Although the state spends nearly $14,000 per pupil, on average—and $24,000 per pupil in Newark—the money invested has failed to produce adequate results. In return for his political courage in taking on the teachers' unions, Christie has suffered vehement political attacks from New Jersey's formidable political Left. In a 2010 e-mail from Bergen County teachers' union president Joe Coppola to seventeen thousand members, he wrote, "Dear Lord, this year you have taken away my favorite actor, Patrick Swayze, my favorite actress, Farrah Fawcett, my favorite singer, Michael Jackson, and my favorite salesman, Billy Mays.... I just wanted to let you know that Chris Christie is my favorite governor." This public declaration of what amounts to a death wish directed at a sitting governor for bucking the failing status quo in education illustrates how deeply entrenched is the opposition of the teachers' unions to education reform. On Governor Christie's watch, education reform has been given the priority it deserves, but his efforts, like Michelle Rhee's, prove that politics is the final frontier of true education reform. Christie's leadership demonstrates that in politics Republicans have a unique opportunity to save the American educational system because Democrats, who rely on teachers' union funding to support their reelection bids, are simply unable to challenge them. When courageous Democrats do embrace education reform, they find themselves made an example of by teachers' unions (see Fenty, Adrian).
But we can't all be pioneers like Geoffrey Canada, Michelle Rhee, or Chris Christie. What can ordinary Republicans do to support the cause of education reform? I suggest we focus on two things.
## **Educational Democracy**
We know that when people have a choice, whether it's between presidential candidates or brands of soap, there is healthy competition. Choices are good, no matter what kind. One of the most fascinating trends in the lives of millennials is the rise of educational democracy: people voting with their feet to increase their educational options. Most strikingly, millennials have been the first generation in more than a century to see many of its members schooled at home. Parents of millennials had good reason to believe that schools were deficient in critical ways. A motley amalgam of parents—including highly conservative Christian families, some "crunchy" liberals, and even some Crunchy Conservatives—independently came to the conclusion that our school system was no place for their children. These parents took on the job of homeschooling, providing their children with the quality of education they felt the government was incapable of delivering. From 1994 to 1999, the number of home-schooled children doubled to nearly 900,000. By 2007, the number had nearly doubled again, to 1.5 million.
That's one form of educational democracy in action, but it shouldn't be only for those parents who have the time and energy to devote to their children's educations. All parents should be able to vote with their feet. That means we need to support every effort to open up the school system to competition.
We need measures like the KIPP (Knowledge Is Power Program) academies and the charter school movement. KIPP is a national network of ninety-nine schools in twenty states and Washington, D.C., that serve more than twenty-six thousand students, from fifth grade through high school. After four years at KIPP schools, 100 percent of eighth-grade classes outperformed their district averages in both mathematics and reading, based on state tests. After four years at KIPP, these same students are performing at the eightieth percentile in math and the fifty-eighth percentile in reading. More than 85 percent of KIPP graduates are attending college. But KIPP schools are focused on lower-income students. We need competition at all levels of education.
School systems should welcome charter schools and promote magnet programs that focus on special curricula and even voucher systems that allow parents to move their tax dollars to the school of their choice.
I'll concede that vouchers are controversial. Some people oppose them because if parents use them at parochial schools, this might breach constitutional barriers against public tax money going to religious institutions. But most resistance comes from teachers' unions that don't want to see tax dollars shifted from unionized public schools to nonunionized private ones. I also suspect that some opposition is driven by people living in wealthy areas, who benefit from the geographic desirability of good school districts. When you live in a high-quality school district, it amounts to an artificial cushion inflating the value of your home. Families will always want to live in the neighborhoods that feed into the best schools.
What we have is a system in which those with high incomes and pricey homes get the best schools, which are both publicly subsidized and unlikely to admit students from low-income families. This amounts to a state-sanctioned limitation on social mobility.
Democrats have sought for years to block vouchers. President Bush and the Republican Congress created the Opportunity Scholarship Program, a small voucher program to allow Washington's underprivileged children to attend top-quality private schools, including the school that President Obama's daughters attend. But as soon as Democrats regained control of Congress, they phased out the program. One has to wonder what the Democrats were so afraid of: a few thousand poor kids attending private schools on taxpayer dollars? Indeed, what could be more in keeping with the American ideal of giving every child an opportunity to receive a high-quality education?
Such is the orthodoxy that is perpetuated by teachers' unions, which pledge themselves to one purpose: protecting public schoolteachers' jobs. And so another top priority must be to end the monopoly of the unions in order to empower education reformers.
## **Checking the Teachers' Unions**
After Hurricane Katrina in 2005, roughly one-third of the city of New Orleans moved away. But this tragedy turned out to be an opportunity for the city's failing schools.
Before Katrina, 64 percent of New Orleans schools had been classified as "academically unacceptable" by the state of Louisiana. The school board president at the time, Ellenese Brooks-Simms, had been convicted for taking bribes; the FBI was investigating fiscal wrongdoing by the New Orleans Parish School Board, and the system had been officially declared financially bankrupt. As the city's public employees temporarily vacated New Orleans, so too did the teachers' unions, which had been the primary impediment to education reform. In that small window of time, education reformers seized their opportunity to implement meaningful reforms.
Three months after Katrina, a plan for the Louisiana Recovery School District was implemented by the state. The school district would be funded publicly, but many schools would be run privately, and they would compete with one another. The schools were given the authority to hire and fire their own teachers, without interference from the teachers' unions. The district found itself deluged with applications from teachers eager to work in a system that embraced an education reform agenda and rewarded outstanding performance.
Almost immediately, educational performance improved. The number of academically unacceptable schools is now down by more than a third, and student test scores have improved dramatically. A report by the Cowen Institute, a Tulane University think tank that monitors the school district's progress, tells a positive story: "Over half of all voters and over two-thirds of parents in the Cowen Institute's poll agreed" that "teachers are improving education in New Orleans." And the unions are not part of this story.
The opportunity New Orleans experienced in the absence of its teachers' unions in the months immediately following Hurricane Katrina illustrates an emerging consensus in America about education reform.
Public education is a highly regulated, union-dominated public monopoly that discourages the innovation necessary to solve its problems. Extraordinary individuals from all sides of the political spectrum are arriving at the same conclusions: the teachers' unions are failing our students, and the best way to fix education is to support competitive networks of private, charter, and magnet schools as an alternative. Many of today's education reformers, such as Geoffrey Canada and organizations like KIPP schools, are committed to developing programs that are insulated from the outsize influence of teachers' unions so they can focus on teaching students and avoid the political battles that have come to characterize the education agenda in America.
But others, such as Michelle Rhee, think this is not enough. They believe that the political influence of the unions has grown so strong that networks that circumvent the teachers' unions will only help a small percentage of America's disadvantaged youth. They think the time has come for the political power of the unions to be confronted head-on and for alternative political organizations to be built to compete and challenge their monopoly on the education agenda.
In this debate, Republicans have a unique opportunity to be the champions of meaningful education reform precisely because the Democrats' hands are tied. Teachers' unions are among the largest and most influential donors to the Democratic Party. In 2008, 10 percent of the delegates to the Democratic convention were representatives of two unions: the American Federation of Teachers (AFT) and the National Education Association (NEA). These two national teachers' unions donate more money to Democratic candidates and causes than any other special interest group in politics—more than the National Rifle Association, more than the pharmaceutical lobby, more even than the bigger Service Employees International Union (SEIU). Just follow the money and you'll find that in 2010, the NEA, the largest teachers' union in the country, contributed nearly $40 million to Democratic candidates in races throughout the country.
Education reform is the civil rights issue of our time, and the Republican Party can engage the millennial generation by declaring that the status quo is fundamentally unfair and dangerous for our long-term economic prosperity.
In becoming the party that champions education reform, Republicans will connect with a new generation of voters by demonstrating that we can deliver them a fairer, more effective system. Because Democrats think of themselves as the party of civil rights, the tables will turn, forcing Democrats to choose between protecting the interests of teachers' unions and the interests of African-Americans, Latinos, Native-Americans, and every other community unfairly treated by the public school system. It will give Republicans a chance to remind the entire nation that it is the Party of Lincoln, the party that abolished slavery in the nineteeth century, that is best suited to end the gross injustice of our school system in the twenty-first century.
Education reform is also an issue that can unite the Republican coalition. Christian conservatives, libertarians, and mainstream Republicans can all come together behind the education reform movement, a movement that champions individual freedom, choice, competition, and innovation. This united coalition should not be afraid to stick up for Democrats like Michelle Rhee and Adrian Fenty and thousands of other reformers in communities across America who are willing to stand up to the teachers' unions.
The issue of education has traditionally been a Democratic issue. But when it comes to improving our schools, Democrats have only two ideas: pump more money into public school budgets and defend teachers' unions under all circumstances. After decades of declining performance and rising costs, this is a stale argument. Americans of all ages, races, and economic backgrounds are desperate for new ideas and new thinking. They are ready for a fresh start. Now is the moment for Republicans to embrace a bold education reform agenda, take on the teachers' unions, and harness the enthusiasm of the millennial generation.
# CHAPTER 7
# A NEW REPUBLICAN FEMINISM
_"The Independent Girl prefers to fight her own battles in this life, and sallies forth to each encounter with a martial spirit which is quite startling."_
—LOU HENRY HOOVER, 1890
FOR AS LONG as I've been alive, the word _feminism_ has been used as an expletive among conservatives and Republicans.
It probably didn't help that the background noise of my formative political years was punctuated by Rush Limbaugh's tirades about "feminazis," his crude code word for radical feminists. And my parents certainly thought the feminists of the late 1960s and 1970s went too far, by pushing an agenda that seemed to denigrate a woman's traditional role in the family as mother, wife, and household leader—all positions my mother and grandmother were proud to hold. But I was hardly alone in my aversion to the "feminist" label. The fact is, most women do not think of themselves as feminists. The last time Gallup asked the question, in 2001, only 25 percent of American women called themselves feminists.
And yet, most women sympathize broadly with some of the core successes of the modern feminist movement, specifically that women are empowered to have careers and the same social standing and legal rights enjoyed by men, in the classroom, the workplace, and all other areas of public and private life.
At the heart of contemporary feminism is something that most modern women hold dear: reproductive freedom, meaning control over when they become pregnant and whether to carry a pregnancy to term.
But "feminism" itself continues to get bad press. Lingering associations with Far Left feminists of the 1960s and 1970s are the main reason. These have stigmatized the movement, identifying it with a radical agenda that has often had little to do with the daily lives of ordinary women, and much to do with the political ideology and pet causes of its leadership.
Feminist organizations alienated more people than they attracted with bra-burning protests and negative obsessions with the "male patriarchy." They made abortion rights a core issue, perhaps with sound motives, but too often they displayed a knack for offending pro-life women. They spoke about America itself being "oppressive." And too seldom did they have positive things to say about men.
Recently, I was in a used-book store and came across a classic example of why this strain of feminism degenerated into an absurd stereotype. It was a 1972 issue of _Ms. Magazine_ , the feminist bible of the Gloria Steinem era. In bold across the top of the cover was a headline article titled "Body Hair—The Final Frontier for Female Liberation." The article took the position that women should follow the example of men and stop shaving their legs, armpits, et cetera. The piece made me laugh and cringe at the same time. If we women try to assert our equality by not shaving our legs, I thought, we won't get equality, we'll just get more hair!
The next generation of feminists—the millennial generation—is more interested in individual liberty than in the hazy concept of "women's liberation." We want to _embrace_ our femininity, not try to become more like men. We recognize that men and women are different—although we expect equality of opportunity in American society. We understand that we have benefited from the extraordinary effort of the pioneering women who pushed for equal rights for more than a century before us. What is interesting, and important, is that the millennial approach to feminism echoes the approach of the first wave of feminist suffragists, who framed their case for a woman's right to vote in the individual freedom enshrined in the American republic's founding documents.
I am humbled when I read about the accomplishments of women such as Elizabeth Cady Stanton and Susan B. Anthony, who agitated for the right to vote in the nineteenth and early twentieth centuries. In order to popularize their movement, they formed alliances with contemporary evangelical figures and prohibitionists such as Frances Willard. It has largely been forgotten that many of these pioneering feminists were _Republicans_ , proud members of the Party of Lincoln. Christina Hoff Sommers, of the American Enterprise Institute, has written extensively about this first wave of egalitarian and conservative feminists. They were strong, independent-minded women with the courage of their convictions.
My great-grandmother Lou Henry Hoover belonged to this first wave of American feminists—although she would have preferred the label "American individualist." She was the first woman to graduate with a degree in geology from Stanford University. She was an early supporter of and lifelong leader in the Girl Scout movement, which helped to expose girls to much more than traditional household duties. She promoted the idea that there was a life for girls and women in the outdoors, in small business, and in charitable service to others. Challenging the conventional wisdom of the era, she endorsed the idea that girls should have access to the same experiences and opportunities as boys—and be treated as individuals and equals.
I certainly recognize the victories won by the second wave of feminists, in the 1960s and 1970s. While this wave had its political radicals, and generated the inevitable popular backlash, Americans have by and large internalized their core arguments as our culture has witnessed the progressive integration of women into society.
The change has been particularly evident in the workplace. In the early 1960s, when my mom joined the workforce, to the extent that women worked at all, they became secretaries, nurses, and teachers. If they happened to be especially pretty, they might become an airline stewardess, as my mother did. As glamorous as it was at the time to fly the friendly skies, it was by no means a work environment that millennial women would consider attractive.
Airline stewardesses had regular weigh-ins to ensure they stayed slim, and they donned miniskirts for work on all-male flights. My mother tells of female supervisors patting her down to make sure she wore a corset (she didn't; at 105 pounds, it was pointless). When these young women decided to marry, company policy forced them to quit their jobs. Many simply didn't tell their employers that they had gotten married—an act of dishonesty, to be sure, but one forced on them by an unfair and deeply sexist policy.
It's hard to believe these were standard practices at any American company just forty years ago. Today shows like _Mad Men_ turn the memories of that era's misogyny and sexism into period entertainment, costume dramas about the distant past. Not only have women come a long way; the whole nation has.
When I decided to attend Bryn Mawr, an all-women's college, my father feared that bra-burning, male-hating, radicalized "feminazis" would have a pernicious influence on his daughter. And when my parents heard that Gloria Steinem had visited our campus, they were convinced I'd end up an acolyte of these 1970s radicals.
Much to their relief, I never embraced the radical feminism championed by faculty at my college; nor did the overwhelming majority of my friends. I recognize fully that there are challenges unique to me as a woman—not the least the challenge of someday balancing children and a career—but I never blamed an oppressive male patriarchy, as some professors no doubt hoped I would. I saw the challenges of being a woman as _personal_ challenges, not social injustices, and I refused to see myself as a victim because of my gender.
What's more, I have found that my friends and most millennial women have made the same choice. When a guy treats a woman poorly, she doesn't detect a patriarchal plot to repress women but a particular jerk who needs to be put in his place.
Challenges remain: for example, even as women's freedom in the United States and the West is at an all-time high, women still haven't achieved full parity with men on the salary scale. While reveling in women's hard-won freedoms, we should not indulge in complacency. Even so, it's worth appreciating that American women today are fortunate to have been born in this country at this time in history.
So the aims of second-wave 1970s feminism have largely been internalized, while its excesses—and the label itself—have for the most part been abandoned. Nonetheless, many feminists who came of age in the 1960s and 1970s are still wedded to the identity politics of their youth, when any random issue might serve to feed the fires of group grievance.
But the millennial generation doesn't warm to identity politics. In fact, they are a "postgrievance" generation concerned with finding solutions and solving problems rather than angry finger-pointing. They are individualists first and foremost. They are uncomfortable blaming an entire group of people for any particular social ill. In addition, they're disenchanted with the hyperpartisanship they hear in the media on both the Left and the Right, especially from commentators who demonize their opponents instead of giving them the benefit of the doubt and trying to solve problems together in a constructive spirit of compromise. A feminism that appeals to the millennials isn't going to look anything like the earlier version of feminism, which they soundly reject.
Perhaps the most compelling recent example of this generational split was evident in the 2008 presidential nomination battle within the Democratic Party. Hillary Clinton's campaign was a milestone for second-wave feminists. She was, after all, a product of that wave, and firmly anchored in female identity politics. Women made up her most loyal base of support. The best-remembered line of her concession speech celebrated her supporters as "18 million cracks in the glass ceiling." And to this day, many of Clinton's longtime supporters believe that 2008 was the greatest opportunity of their lifetime to see a woman elected to the White House.
Contrast this with Barack Obama's campaign. He, not Hillary, won the hearts, minds, and votes of millennials. He accomplished this not because he was the African-American candidate or because he was the younger candidate. Neither his race nor his youth defined him. He did not campaign as the "black candidate" for president, as had other African-Americans before him. Candidate Obama was able to transcend identity politics, to the extent that some older African-American leaders initially distrusted him: they weren't sure he was "black enough," and Jesse Jackson even accused him of "acting white." Candidate Obama's postracial, postgrievance, and postpartisan rhetoric appealed to the sensibilities of millennials in a way that Hillary Clinton's female identity politics could not.
One of these days, when a woman is finally elected president (and I believe we're closer to crossing that threshold than many people think), she will have won not by pandering to a sense of grievance or by identifying herself politically as a woman but simply by demonstrating outstanding leadership. While millennial women are perfectly happy to revel in "girl power," they do not think of women as a special interest group. They understand that, in a fluid society, special interest groups end up disempowering people by confining them to group definitions, and thereby undermining their individuality. Just as the sexism of the 1960s and earlier robbed women of their dreams and their voices, radical feminism's groupthink deprived individual women of the ability to think for themselves, act for themselves, and be themselves.
Millennial women have discovered that life is not an all-or-nothing, zero-sum game where women gain by men losing. For example, if women assert their sexual freedom by making sexual spectacles of themselves, we don't become freer—we just give men a cheap thrill. The increase in sexual freedom for women since the sexual revolution has done nothing to reduce the sexual objectification of women in our culture, on college campuses in particular. A sexual double standard is still in place. The negative consequences of all this are often deeply psychological, but they are also physical—the rise of sexually transmitted diseases among college-educated millennial women is one of the most underreported demographic stories today. People will say we need to practice safe sex, but the deeper lesson here is that we must use our sexual freedom more responsibly.
Millennials are the first generation raised with the concept of co-parenting, where both parents share parenting duties. The millennial experience has ushered in a reevaluation of traditional gender roles in the home, a rejection of the traditional division-of-duties family model of male breadwinner and female caretaker.
And, finally, in a comical epilogue to the _Ms. Magazine_ issue that called for the free growth of women's body hair as a political final frontier, it's hard to escape the fact that women's grooming has actually gone in the opposite direction. Millennial generation women have been far more aggressive in _removing_ unwanted hair than their mothers and grandmothers. But, ironically, so have men, many of whom engage in a bit of "man-scaping." So, while we didn't follow the prescriptions of the 1970s feminists when it came to body hair, we did end up achieving more egalitarian grooming practices! Who knew?
What's the bottom line? The zero-sum formulas—when women gain, men give in—that may have been effective in helping women win basic legal rights, employment rights, and social rights become counterproductive when the issues get more complicated and come down to individual tastes and preferences.
The direction of feminism in America has huge implications for the major political parties and their platforms. That the Democratic Party has had a lock on the hearts of women is a truism. But beneath the surface a change is under way. Millennial women have made it clear that they aren't sold on the Democratic Party's adherence to female identity politics. If Republicans recognize this fact, and then act on it, we have an opportunity to capture the votes of millennial women.
I believe that millennial women are looking for a new approach to issues, one that recognizes the differences between men and women, and between women and other women—a philosophy that is deeply individualized. We need, in other words, a modern feminism rooted in the principles of American individualism.
And here the Republican Party has an opening. The GOP has an opportunity to build a new _Republican_ feminism that can speak to the next generation of women. This new Republican feminism should reject the limits and arguments of identity politics and recognize that all issues are "women's issues." New Republican feminism should support the spectrum of life choices available to women, such as whether to stay at home as a primary parent or pursue a high-powered career. If men and women have equal opportunities to make successful careers, then either parent can opt to stay home as the primary caregiver. Each woman, and each family, will have the freedom to figure it out individually.
The new Republican feminism would see women as individuals first and would value each woman's unique and God-given combination of intelligence, character, skill, and creativity. Republican feminism would call upon each woman to stand up, apply her talents, contribute to society, and earn the full reward for her efforts in the free and open market. This is what millennial women want and expect—no special favors, no preferences, no barriers.
Millennial women also need new role models as diverse as the many paths they can follow—women who have both careers and families, or only families, or only careers. These include women who are scientists, nurses and physicians, astronauts, artists, journalists, and stay-at-home moms. All these role models—and many, many others I haven't listed—are essential to helping younger women determine which career and life paths might be most viable and rewarding for them. Republicans should encourage women to explore this entire spectrum of choices.
A new Republican feminism will be built upon a foundation of equality of opportunity, as well as a recognition that great gains have been made for women over the last quarter-century. But Republican feminism must also recognize that equal opportunity doesn't guarantee equal outcomes, as evidenced by the fact that there aren't yet nearly as many women as men running major organizations, occupying corner offices, or walking the halls of Congress. Despite this persistent inequality, Republican feminism should respect men as _partners_ in achieving equality of opportunity and should avoid blaming men and the "oppressive male patriarchy" as we continue to work together in order to achieve genuinely equal opportunity for all women and men.
A new Republican feminism should also recognize that the most urgent battles in the twenty-first century for feminism are not those fought in the arena of American politics, but the ongoing struggles for fundamental freedom in the developing world.
Radical campus feminists sometimes forget that there are corners of the world where questions such as "Is Barbie encouraging bulimia?" are absurdly irrelevant to the more severe oppression women still endure; places where women's genitalia are mutilated, where women are sold as property, trafficked as sex slaves, or murdered by their own family members in order to satisfy antiquated notions of family honor.
That's why a new Republican feminism should be focused on exporting the rights that American women have already achieved to ever larger numbers of women around the world. It is time to put feminism in a broader, more global context by championing not just equality of opportunity at home, but the cause of human rights in countries where women are too often still second-class citizens.
The next wave of feminism is global feminism, and it's being fought on the front lines by the most inspiring of women who are surviving and triumphing against the most barbaric societal and cultural inequalities.
Consider Somaly Mam, a survivor of Cambodia's sex trade, who was sold into slavery at the age of ten and raped daily for several years. Today she is a fully rehabilitated survivor who has devoted herself to eliminating sexual slavery worldwide and whose individual efforts have rescued more than six thousand girls from the horrors of sexual trafficking and empowered them to pursue productive lives within their societies.
A new Republican feminism that is global in scope will also support women like Ayaan Hirsi Ali, a Somali who was the victim of the most misogynistic of cultural practices when she was forced to undergo genital mutilation at a young age. She managed to flee an arranged marriage as a young women by seeking political asylum in the Netherlands, where she achieved citizenship and became a member of parliament. After radicalized Islamists murdered Theo Van Gogh and threatened Ali's life for a film they made criticizing the lack of women's rights in certain Muslim communities, she has become an international icon. Now she is an outspoken advocate for liberating young women from the repressive practices of radicalized Islamist men who perpetrate "honor crimes" on Muslim women living in the West, and who refuse to assimilate into Western society and treat Muslim women with the dignity and respect that Western laws demand.
Women such as Nobel Peace Prize winner Aung San Suu Kyi, who until recently was a political prisoner in her home country of Myanmar, is another icon, a vocal opponent of her country's tyrannical government who has paid a high price for her leadership role in Burma's National League for Democracy (NLD) Party, standing in firm opposition to the ruling military junta. Her recent release from a seven-year detainment has reenergized the majority of Burmese citizens who hope that her leadership can help bring democracy to their country.
All of these women are survivors who have transcended the most difficult of human circumstances and who serve as role models for a new generation of feminists. A new Republican feminism will focus its energies on achieving freedom and equal opportunity for women everywhere—which means not just achieving full parity with men in America but also standing with women who struggle for basic human rights beyond our shores.
A new Republican feminism, in order to connect with millennial women, should understand that women can have diverse views about the traditional centerpiece of the feminist agenda: reproductive freedom. Millennial women feel that abortion, while an essential choice for women, is often not the right choice for individual women, and they value independence on this most personal of decisions. For example, a prominent millennial Republican, Meghan McCain, is pro-life, but she would never insist that her view, or the law, should prevent other millennials from being pro-choice. Republicans will have a much better chance of winning the support of millennial women if the party does not impose a pro-life (or pro-choice) litmus test.
In some ways, this new Republican feminism has already gained a foothold in Republican politics. Look at the new class of women elected in 2010: governors such as Susana Martinez, Nikki Haley, and Mary Falin—and members of Congress such as Nan Hayworth, Ann Marie Buerkle, and Senator Kelly Ayotte. These women represent the diversity of choices enjoyed by many modern women: they are mothers and grandmothers, they are wives and career women, and they don't represent a monolithic opinion on the issue of reproductive freedom.
This new Republican feminism is already making the case that _all_ issues are women's issues. When Republican congresswoman Cathy McMorris Rodgers ran for reelection in 2010 while pregnant with her second child, she campaigned to overturn President Obama's health-care legislation by addressing women who were small-business owners. Since women are responsible for two-thirds of all small-business start-ups in the United States, women were disproportionately hurt by the exorbitant costs associated with the health insurance premium spikes that resulted from implementation of the unpopular health-care law.
Of course, it's impossible to talk about a new Republican feminism without bringing in Sarah Palin, who has herself refused to reject the feminist label. Forget that you might not agree with everything Palin says. She is the first Republican woman to appear on a national ticket, yet as a candidate she never once played the woman card. She embraced her femininity and her motherhood, but never made these the most important features of her candidacy.
The Feminist Left's hypocrisy rose to new heights when Palin hit the national scene. She was criticized for her participation in beauty contests and ridiculed as "Caribou Barbie." (Apparently, objectifying women is fine with left-wing feminists if those women don't share your politics.) Sandra Bernhard, the activist and feminist actress, even warned Sarah Palin that she'd be "gang-raped by my big black brothers if she enters Manhattan." Such ugly utterances about another woman by self-proclaimed feminists demonstrate exactly why so many women reject the label "feminist."
In the end, Sarah Palin's image may have been too maligned during the campaign for her to have any chance to become a hero to millennial women. To be sure, some of that damage was self-inflicted, but much of it resulted from a smear campaign orchestrated, ironically, by women. When it came down to it, Palin represented the greatest threat to old-line feminism—a woman who had benefited from its earlier activism but who did not share its political views, especially on the question of reproductive freedom.
I hope the variety of the women whom Republicans have elected in 2010 will catalyze the formation of a new Republican feminism, one that respects the instincts of individualism intrinsic to the first-wave American feminism and helps Republicans build a bridge to the millennial generation. In this way the Republican Party can return feminism to its roots, to its genesis in the Party of Lincoln, by engaging the political, professional, and personal challenges and choices faced by American women in the twenty-first century. Republicans have an opportunity to reach out to the millennial generation with a new Republican feminism, grounded in American individualism. _Feminism_ doesn't _have_ to be a dirty word, least of all to Republicans.
# CHAPTER 8
# THE CHOICE DILEMMA
_"The federal government has no business deciding the wrongness or rightness of a woman having an abortion."_
—BARRY GOLDWATER
I WAS NINETEEN WHEN I learned that a friend of mine had had an abortion. She was older than I was and confided that when she was sixteen, she had become pregnant, and her family had taken her to a clinic to undergo the procedure. Since then, she told me, every time she saw a little girl, she felt overcome by sadness. Something deep within her sensed that she had lost her own little girl.
Her sense of loss struck me. It had been six years since she had had the abortion, but to her it still felt like yesterday. For all the pro-choice arguments that focus on women's rights and reproductive freedom, I had never heard any "cons" that spoke of the emotional hardships of a would-be mother after undergoing an abortion. Surely a fetus was more than just tissue, and an abortion more than a simple medical procedure, if a young woman's emotions pulsed so strongly six years later. This personal connection with a friend's residual emotional pain from having an abortion was, for me, dramatic evidence that the choice to have an abortion wasn't as straightforward as the pro-choice activists would have us believe.
Five years later, I learned that a colleague of mine had become pregnant as a teen. Her religious beliefs had prevented her from considering an abortion, but her parents' disapproval forced her to hide her pregnancy until the school year was over and she was able to go away for the summer to have her baby. She arranged for help through a Catholic charity that supported her with room and board in a neighboring state, and also arranged for an adoption after her son was born. Tears streamed down her face as she recounted this story to me. She told me that she was tortured by the sight of stretch marks every morning in the shower. Where was her son now? she wondered. How had his life turned out? Giving up a baby to whom she had given birth, forfeiting the joy and fulfillment of holding the child she felt kick inside her for weeks, proved to be the most heart-wrenching experience of her life. This friend went on to finish high school, graduate from a top-tier university, and be accepted by one of America's premier law schools. But she never stopped wondering, _Did I make the right choice?_
These two women represent the two outcomes of the pro-choice/pro-life "best-case scenario" for teen pregnancy. One terminated an untimely pregnancy; the other carried the baby to term but gave it up. But both choices left painful emotional scars. From these two friends I learned that the practical reality of abortion and teenage motherhood is far more personal and complex than the polarized abortion debates in America suggest. It's easy to form hypothetical opinions about teenage parenthood, teenage sex, and abortion, but these real-life personal experiences exposed layers of the issue that I hadn't previously considered. And real-life experience _must_ inform our discussion about abortion. My awareness of each friend's tragic situation caused me to reevaluate the pro-life/pro-choice absolutists who dominate this political debate in America while observing the issue's deeply personal and individual dimensions.
From my friends' practical experiences I learned that neither option, abortion or adoption, is a total win-win situation for the would-be teen mother. I could see that the black-and-white terms of the pro-choice versus pro-life debate fail to capture the simple but essential fact that there are no easy answers to teen pregnancy, or to the question of abortion in general. And so, like many women, I have views on abortion that are shaped by my personal interactions and experiences.
As I mentioned earlier, Irving Kristol once joked that a neoconservative was a liberal who'd been mugged by reality. I think that on the abortion issue, people on one or another side of the issue find themselves mugged by reality—pro-lifers with a pregnant teenage daughter, pro-choicers staring at an ultrasound three months after conception. The only option for me and, as it turns out, for most Americans, is to take a position that fully considers every side of this highly charged debate.
I'm with those people who adhere to a position that can be defined as "personally pro-life but politically pro-choice"—which is to say that I believe abortion is wrong, but nevertheless I think it should be legal, limited, and safe. This point of view pleases neither the pro-choice crowd nor the pro-life crowd, but it puts me in agreement with 78 percent of Republican voters, according to a 2008 study, who believe that "a woman, not the government, should make the decision to have an abortion."
A conservative movement that hopes to appeal to the millennial generation cannot make abortion a simple rallying cry, a litmus test, or a wedge issue, as have previous generations. Millennials take a mature view of this moral dilemma. While teen pregnancy first spiked and then recently hit new lows in their lifetimes, they've been inundated by a hypersexualized pop culture, and confronted by celebrity teen parents from Jamie Lynn Spears to Bristol Palin, pro-life movies like _Juno_ , and reality TV series like _16 and Pregnant_. Millennials don't pay attention to the tired, predictable hyperpartisanship of the second-wave feminists and their detractors, who have been going at it since long before they were born. Remember that millennials abhor perfunctory partisanship; as soon as they hear someone spouting one-sided opinions, they become suspicious.
Which means the call-and-response of "Baby killer!" and "Take your laws out of my uterus!" screamed back and forth by activist extremists on each side of this debate are bound to alienate millennials, who, like most Americans, understand that the simplistic labels "pro-life" and "pro-choice" don't begin to describe the moral complexity of abortion.
The truth is that millennials' sensibilities on abortion mirror those of a majority of Americans. Recently there has been an uptick in the popularity of the term _pro-life_ —though it hasn't been paired with a comparable rise in support for the effort to make abortion illegal. So while more than half of Americans of all ages view abortion as morally unacceptable, they are unwilling to see it outlawed in all circumstances.
Meghan McCain best describes the sensibilities of her generation on this issue: she is "pro-sex" and "pro-life" (and pro–gay marriage, incidentally). But she also specifies that this is her _personal_ decision, and she wouldn't want to impose her views on anyone else. This position does not fit neatly into the old polarized, all-or-nothing activist position, but it reflects a generational sensibility that allows for personal stands of conscience without demonizing those who hold different viewpoints. A full range of opinion is respected, and there is an appreciation for the diversity of real-life decisions and compassion for those who must make them.
## **The Two Extremes**
Pro-life absolutists leave no room for doubt. They believe that abortion should be illegal in every circumstance, including cases of incest and rape. Only when the mother's life is in danger do these ideologues pause to reflect on the complexity of this question. Suddenly "pro-life" is forced to confront a choice: which life is more valuable, the mother's or the unborn child's?
Remember, the _life_ in the "pro-life" political position doesn't refer to the already living and independently functioning woman. It refers to the fetus developing in utero, which science has demonstrated to be more than what the radical pro-choicers dismiss as mere cells and tissue, but is still unable to survive outside the womb. From the most extreme pro-life perspective, the mother's well-being is considered secondary.
But what about a woman's ability to judge her own emotional, economic, and physical capacity to carry a child to term, and then to raise it or give it up for adoption instead? Does it make sense for the federal government to decide unilaterally, on behalf of all pregnant women, that under no circumstances should they ever have the right to choose to have an abortion? I say no.
Fundamentally, the pro-life position places the state's moral judgment ahead of the individual's. While it is clear that the state has the right—even the obligation—to protect life, the matter of abortion is unique. The question of when life begins and what protections should be accorded to the unborn are open to significant debate. And given that individuals will draw their own conclusions about the starting point of human life, an absolutist position is necessarily arbitrary. Should the federal government have the authority to make that decision? The conservatism that I've always subscribed to, that is consistent with maximum individual freedom and choice, would hold that it does not. While there is significant disagreement among the conservative movement's factions on this point, a conservatism that will appeal to the millennial generation will argue that within limits this is an individual decision among a woman, her family, her doctor, and her God, not the government.
How, then, did it come to pass that the Republican Party, that staunch champion of freedom, autonomy, and individual responsibility, supports the idea that the federal government should make these moral and biological decisions for all its citizens? In the case of abortion, the Republican Party assumes that the government can make a uniform choice for all women, in all circumstances, always. But conservatives generally say they believe that individuals make the best choices for themselves.
Is it any wonder that there is a gender gap in American politics? According to Gallup, 41 percent of women identify themselves as Democrats, 29 percent as Independents, and only 27 percent as Republicans. Perhaps this gap exists because the Republican Party, which pledges itself to smaller government, makes an exception in the case of women's most private and personal decisions.
But if moral certainty is the illness of the Right, moral surrender is the illness of the Left. Pro-choice absolutists believe that a woman should always be able to have an abortion, at any point in her pregnancy, even in the third trimester when a fetus is sufficiently developed to survive outside the womb. Consider the practice of "partial-birth" abortion. The procedure, which is now banned in most cases, strikes a majority of Americans as barbaric. Yet pro-choice absolutists insist that it is essential to reproductive rights, and they reject any effort to restrict it. Pro-choice absolutists also argue that taxpayers should help pay for abortions, both domestically and internationally. While most Americans oppose such subsidies, and the Hyde Amendment bans federal taxpayer funding of abortion, pro-choice absolutists say that denying such support is tantamount to restricting a woman's freedom.
The truth is that both extremes are wrong. An unwanted pregnancy is a dilemma, not a straightforward choice. It is a dilemma because all the alternatives are terrible. Rarely does a pro-choicer candidly address the _tragedy_ of abortion—not just the tragedy of the act but the often long-lasting impact on the mother, manifested in profound feelings of remorse and regret. And similarly, rarely does a pro-lifer seriously address the gritty realities of teenage sex and unintended pregnancies, other than to preach the virtues of abstinence education. In a pro-lifer's ideal society there are no abortions and no birth control—and no sex before marriage.
Each side argues that Americans can't afford to equivocate on this issue. On the one hand, if life begins at conception, abortion is state-sanctioned infanticide. On the other hand, if a woman has a right to make decisions about her body and her future, what difference does it make whether she is two months pregnant or seven or even nine months pregnant? When do the rights of the life _inside_ the mother outweigh the rights of the life of the mother? To pro-choice absolutists, the answer is never. To pro-life absolutists, the answer is always. For the majority of Americans, the answer is somewhere in between—in the first or second trimester, but not in the third.
For the hyperpolarized set, these all-or-nothing pro-life/pro-choice debates are good for business. They push people in the middle away from the kind of consensus that, opinion polls indicate, actually exists on the issue and toward implacably hostile poles of opinion. But what's good for the absolutists on either side is bad for those with unplanned pregnancies, and it's bad for our ability to talk openly about the financial, emotional, and practical challenges related to one of life's most painful personal dilemmas.
Most Americans, including millennials, reject the extremes on this issue, and instead have arrived organically at a consensus. Slim majorities of Americans _call_ themselves pro-life, but most Americans—by a slightly larger majority—agree that abortion should remain legal. Most Americans take a pragmatic rather than an ideological approach, one that falls between the extremes: abortion should be rare, but a woman's decision should be left up to her.
While most Americans view abortion as morally wrong (Gallup 2010: 50 percent–38 percent), they nonetheless have a positive view of _Roe v. Wade_ , the Supreme Court ruling that says a woman has a constitutional right to get an abortion within the first trimester. When Gallup last asked the question "Would you like to see the Supreme Court overturn its 1973 _Roe versus Wade_ decision concerning abortion, or not?," only 33 percent of respondents replied in favor, while 52 percent said they were against overturning, and 15 percent had no opinion.
Polling also shows that Americans favor reasonable restrictions on the right to have an abortion. The majority are opposed to abortions in the final trimester: months seven, eight, and nine. They favor parental notification laws when minors seek abortions, and they don't think people's tax dollars should have to pay for abortions.
Even most Republicans agree with these views, according to Republican Majority for Choice, a pro-choice Republican group. Here are some surprising results from their 2008 poll of Republicans on the issue of abortion:
* "54% of self-described pro-life Republicans believe that women should have access to the full range of reproductive options including education, contraception, motherhood, adoption and abortion."
* "74% of Republican voters do not support an addition to the GOP platform that calls for a Constitutional Amendment that would ban all abortion, even without exceptions for rape and the life and health of the mother."
* "81% of Republicans support a GOP platform that states 'members of the GOP have differing views on the issue of abortion, and we should respectfully agree to disagree.' " [In fact, 78% of _pro-life_ Republicans support a GOP platform that states that members of the GOP have differing views on the issue of abortion, and we should respectfully agree to disagree.]
* "50% of Republicans believe that platform language that takes a specific position on the issue of abortion or other personal or moral choices is polarizing and contributes to the wedge within the Republican Party."
Despite this considerable common ground, there is a small but dedicated wing of social conservatives whose religious beliefs forbid them from conceding an inch. These anti-abortion activists are sincere in their beliefs and guided by the certainty that comes with believing they are doing God's will. This does not leave much room for disagreement or even civil conversation. The sole purpose of one of these groups, RNC for Life, is to ensure that the Republican Party remains the pro-life party, and moreover that it maintains its support for a human life amendment to the Constitution. Forget the fact that throughout its history the conservative movement has supported economic and domestic policies that put a premium on individual freedom and has sought to limit the intrusion of government into people's lives. When it comes to abortion politics, the Republican Party is beholden to its special interests.
## **Bad Politics**
The irony, of course, is that every moment spent on the disagreement within the party on the subject of abortion is a moment lost for pursuing larger goals as a unified group. Compared with issues like the economy, fighting Islamist terrorism, reforming entitlements, and reining in debt and deficits, abortion is a second-tier issue. For the vast majority of American voters, it is one of many issues they consider when voting for a candidate. When the question was last asked by Gallup, only 15 percent of pro-lifers say a candidate must share their view. Eleven percent of pro-choicers say the same thing. That leaves 74 percent of the electorate willing to vote for someone who doesn't share their view on abortion.
Nevertheless, abortion remains a first-tier issue for an influential fraction of the Republican Party, a group that has considerable sway over the rest of the party apparatus. It conditions its support for the party on the presence in the party platform of a pledge to introduce a constitutional amendment on human life. Since 1976 the Republican Party platform has contained a commitment to the pro-life position.
Since 1980 no Republican candidate for president or vice president has been pro-choice. In the 2010 election, only a few northeastern congressional districts elected pro-choice Republicans. The fact remains that for most of the last decade pro-choice Republican politicians seemed to be extinct, and if they did exist, they knew to keep their pro-choice positions under wraps, lest they arouse the ire of the pro-life base of the party. This situation tends to minimize the opportunities within the Republican Party for pro-choice women to run for office. Even those elected Republican women who are personally pro-life but do not favor making abortion illegal, are hounded by the pro-life wing, labeled RINOs by the talk radio crowd, and accused of being "not conservative enough."
A Republican Party that would appeal to the millennial generation would tolerate diversity and rid itself of its abortion litmus test. Recent polls show that 60 percent of millennials favor keeping abortion legal, even though 53 percent feel abortion is morally wrong. What millennials don't like is the ideological rigidity that characterizes the Republican Party's stance on this issue.
Millennials also don't like a political party that seems out of touch with reality. Part of that reality is that 95 percent of Americans have sex before they get married. Forty-seven percent of teenagers have sex while they are still in high school, and the number jumps to 65 percent of college students. Most young people have sex for the first time at about age seventeen, but they do not marry until they're in their mid- to late twenties.
Republicans need to get comfortable with the fact that the sexual revolution came and went, and in its wake millennials are sexually active far earlier than any previous generation. If Republicans are going to connect with the next generation, we need to come to terms with the fact that Meghan McCain isn't the only young Republican who is "pro-sex." We also need to be honest about the fact that being pro-sex doesn't necessarily put a young woman at enormous risk of pregnancy. Ms. McCain, like the majority of women in the millennial generation, is also pro–birth control.
I believe that conservatism, as viewed through the lens of American individualism, provides a solid foundation for responding to this challenge. Because American individualism places the individual's wisdom at the center of all issues of political life, we must start there. Do we trust individuals to make the best possible decisions for themselves, especially in moments of crisis? If so, then no matter what our moral positions might be, we must give priority to the individual's right to make these decisions for herself and himself.
Moreover, as a conservative who would like to see less power and authority centralized in the federal government, I think that a proper role for mediating institutions in our culture—religious organizations, community centers, and other nonprofits—is to help women prevent unintended pregnancies, to advise them about alternatives open to them, and to pay for their abortions if that is their decision.
To be a constructive force in Republican politics, in a way that attracts the millennial generation, social conservatives and libertarian conservatives should form coalitions and work together through the country's vast network of nongovernmental institutions, including churches, synagogues, mosques, and secular and nonsecular nonprofits, in order to transform our culture and diminish the number of abortions. Pro-choice and pro-life conservatives can agree on the urgent need to reduce abortions, and by advocating policies that increase adoptions and prevent unintended pregnancies we'll achieve concrete results that will demonstrate to millennials that we are committed to solving problems. Pro-life activists would do better to advocate policies such as these rather than pursuing a human life amendment to the Constitution.
The conservative impulse to elevate our moral environment should be aligned with the conservative impulse to empower individual freedom. Those who feel strongly that abortion is immoral should use persuasion, not legislation, to win the argument.
The best outcome would be preventing _all_ unplanned pregnancies—helping women avoid the dilemma of either having an abortion or carrying an unwanted child to term. To this end, talking with women, even girls, about the risks of sex is essential, as is talking to them about birth control methods. Teaching them exclusively about abstinence is simply not going to be effective. It's not a realistic solution. Anyone who assumes that teens can be restrained by self-control has forgotten what it's like to be a teen. And if our message stays stuck on abstinence-only, we'll remind millennials that we're out of touch, and we'll become increasingly irrelevant.
I think we also ought to recognize that teen pregnancy carries a cost to society. It tends to lead to significant economic hardships for the mothers involved and for their children. Those children are much more likely to impact society negatively. So while the state's role must be limited, we must take the responsibility upon ourselves and our civil institutions to play an active role in helping teens understand the consequences of their actions in a rigorous, fact-based, and morally centered sex education effort. We expect our schools to help our kids prepare for a twenty-first-century economy, but teenage motherhood leads to high school dropouts and government assistance, and often to delinquency among children of teen parents. We should focus our efforts as conservatives on supporting sex education through mediating institutions, in order to help our kids understand the risks of unprotected sex, sexually transmitted diseases, and how to make choices that will keep them on track to benefit from the opportunities of the twenty-first-century American economy.
## **Palin, Millennials, and the Way Forward**
For most Republicans, the Palin family looms large in the abortion debate. Sarah Palin is, in many respects, a pro-life icon. Beyond her political positions opposing abortion rights, she clearly practices what she preaches. She chose to carry to term a Down syndrome baby, Trig, whom America met during the 2008 presidential campaign and who has become an endearing symbol of the special needs community. Today, expectant parents can, and often do, have their fetuses genetically tested for Down syndrome; those with the genetic mutation are aborted roughly 90 percent of the time. Palin made it abundantly clear by the decision she made, in a situation where most women would have decided the other way, that she is strongly pro-life. Some people found her decision admirable.
Enter teenage daughter Bristol Palin. During the campaign, Bristol announced that she had become pregnant, and planned to marry the father. The announcement raised hackles in the pro-choice community. They judged Bristol to be the victim of her mother's politics, unable to abort and therefore the classic example of why teenagers should not be forced to consult their parents in order to go through with the procedure. The pro-life community applauded Bristol Palin's decision, calling her a hero to the movement.
But Bristol Palin is neither the hero nor the victim the politicized pro-lifers and pro-choicers, respectively, would have us believe she is. She did not face the choices confronted by ordinary teen mothers. Her problem was not whether she could afford to raise a child on her own or whether her parents would disown her. She's a single teenage mom who has the support of her family and has been able to boost her own fortune by trading on her family's fame. She owns a condo, she dances with the stars, she's a celebrity who gets paid for public appearances, especially on the abstinence speakers' circuit, which enables her to pay her mortgage. She glamorously hosts traditional tea parties and dons $7,000 Carolina Herrera dresses to pose for _Harper's Bazaar_ centerfolds with her baby perched on her hip.
In short, Bristol Palin is the exception to the rule. Most teenage mothers are from the poorest communities. They are statistically much more likely to be minorities and from the lowest socioeconomic backgrounds, and they're much more likely than not to come from families that can't afford to help raise their unplanned-for children. Sons of teen mothers are 13 percent more likely to serve time in prison, and daughters of teen mothers are 22 percent more likely to become unwed teen mothers themselves.
Fortunately, in the lifetime of millennials, conservatives have had much to cheer about. The teen birth rate hit a new low in 2009. And the culture is changing. The show _16 and Pregnant_ on MTV has done a lot of good to reach girls and explore the unglamorized reality of being a teenage parent. According to the _Christian Science Monitor_ , "82 percent of the teens who watch it say the show helps them better understand the challenges of teen pregnancy and parenthood—and why they should avoid it." What MTV has discovered is that _stories_ about teenage parents—not adults lecturing them—are the most compelling way to explore the full impact of pregnancy.
This is the model we should follow when it comes to talking with teenagers about sex, pregnancy, and the abortion dilemma—not by trying to break down issues into oversimplified and out-of-touch categories but by focusing on the realities millennials face, and examining the full consequences of their options and their actions.
As conservatives, we hold fast to the idea that some individuals can handle adversity better than others; so why do we think differently about teens? Perhaps some can raise a child on their own. But to expect all teens to do so, or to bring a child to term and give it up for adoption as gracefully as did the title character in the movie _Juno_ , is utterly unrealistic. And that expectation ends up becoming destructive when we condemn those who make mistakes, but we don't help them avoid those mistakes, as when we deny them the basic knowledge of the risks involved with being sexually active in the first place.
At the heart of this issue is the belief that conservatism trusts in the individual. We may not be ready to sell alcohol to a teenager, but we trust them with an extraordinary amount of authority over their own lives and their own bodies. We do them no favors by pretending they won't exercise that authority; we also do ourselves no favors by failing to teach them how to use that authority wisely. This is at the heart of American individualism: an expectation that the individual, acting with the support of the community, will make the right choices for himself or herself.
# CHAPTER 9
# CONSERVATIVE ENVIRONMENTALISM
_"The spiritual uplift, the goodwill, cheerfulness and optimism that accompanies every expedition to the outdoors is the peculiar spirit that our people need in times of suspicion and doubt.... No other organized joy has values comparable to the outdoor experience."_
—HERBERT HOOVER, 1926
## **The Making of a Republican Environmentalist**
I was raised in Colorado. Growing up in big-sky country in a family that valued outdoor life, I was blessed with a childhood that included an abundance of fishing, camping, hiking, and, yes, shooting.
My father was a devoted outdoorsman and hunter. He gave me my first 20-gauge shotgun when I turned twelve years old. I might have preferred a pony, but he taught me to shoot and to handle my gun responsibly—and, as it happens, I'm still a pretty decent shot. But my dad also taught me, through his words and his deeds, that responsible hunters are the most conscientious environmentalists you'll ever meet. By spending hours outdoors, they learn to appreciate, more than most people, the natural beauty of this country. Good hunters want to be sure that the animals they hunt are plentiful. And best of all, good hunters do not waste a thing.
I remember my father—who eventually graduated to hunting with only a bow and arrow—coming home with everything from turkey to deer tied down on his truck. I recall family dinners that featured the meat he brought home from his hunting trips. Everything was used up somehow, including my mother's patience. For days on end our kitchen became a meat-processing factory. My dad insisted on stuffing his own deer sausages, and would later serve them on Christmas Day.
I also knew that my family had a rich heritage of respect for nature. My great-grandmother Lou Henry Hoover learned to hunt, fish, and camp in the hills of Monterey, California, in the 1880s with her own father. She subsequently went on to devote her energies to the Girl Scouts of America, which has introduced millions of young girls to the outdoors. In a late-Victorian era that expected little more from young ladies than sewing, childbearing, and serving as ornaments for their husbands, she wanted women to experience, and to feel confident in, nature's marvels.
Herbert Hoover was a lifelong devotee of fly-fishing. There are photos of him, dressed in a suit and tie and waders, standing knee-deep in water at the edge of a stream fishing for trout. I've come to believe that the river was his church, the place where he communed with God. Although he was a devout Quaker, he stopped attending meetings while residing in Washington, D.C., because too many parishioners were "moved by the spirit" to voice their objections to his policies. Nature became his refuge, and so it remained for the rest of his life.
As a result of my family's traditions, I was raised to be an environmentalist, at least the kind who hunts, eats meat, and cherishes fresh air, fresh water, and the outdoors.
And yet, at a young age, I began to resist and resent some aspects of the environmentalist movement. I learned that there was a difference between reverence for nature and reverence for environmentalism. Reverence for nature involved simple things like not littering, leaving nature as we found it, preserving sources of clean water, changing individual habits to conserve energy, and balancing the protection of natural habitats with a responsible use of natural resources. Reverence for environmentalism seemed to be rule-bound and legalistic, and it completely discounted the fact that for millennia nature has served the needs of humanity, not vice versa. My family taught me to revere nature, yet the modern environmentalist movement aroused skepticism within me.
In Denver, where I grew up, an ominous brown cloud settled over the city in between storms, the result of car exhaust, fires, chimney smoke, and industrial pollution. When fronts blew in from the Rocky Mountains, the cloud would dissipate, but it always returned. To try to control pollution, the city introduced a "no wood-burning" policy. On specific days, as a way to reduce the smoke exacerbating the pollution problem, we weren't allowed to burn wood in the fireplace. These regulations were enforced by city employees scanning the sky with infrared binoculars to enable them to detect the heat rising from chimneys in order to determine if anyone was breaking the law, a violation that could earn you a $500 fine. Now think about that. Here were government employees using high-tech equipment to spy on ordinary citizens burning wood, and the wood burners—enjoying the crackle of a fire, the warm hearth, and the coziness of it all—were the lawbreakers, while the binocular-toting bureaucrats were playing the role of environmentalists. Go figure!
One year, the city had the nerve to declare a "no-wood-burning" day on Christmas Eve, the day when our family tradition was to gather by the fireplace, exchange gifts, and write letters to Santa that my father would "send up the chimney" for Santa to read (so we were told). My dad is a responsible, law-abiding citizen, but that year he decided enough was enough. The family tradition would go on, come what may. Would the city have suffered unacceptable pollution damage by waiving this prohibition on just this one special day, out of respect for holiday traditions? Probably not. While I'm all for encouraging individuals to change personal habits to help the environment, in the end it's better if those changes are voluntary, not enforced by the government. Anyway, I wondered what kind of Scrooge would decide to enforce a "no-wood-burning" day on Christmas Eve.
That episode left an impression on me. Normally, parents try to be a good example for their children, and I don't think my dad was trying to tell his outdoorsy kids that the environment doesn't matter. Rather, he was telling us that sometimes environmentalists focus on the wrong things. Instead of protecting nature, they worry about controlling the actions of others. They see a problem in nature, and they simply assume that humanity is the cause. But sometimes, even if people might be the cause of the problem, they can also be the solution. If you keep blaming people, and reining them in, telling them what they can or can't do, you breed resentment against your cause—and in this case perhaps, even worse, resentment against protecting the environment.
There really are different ways to be an environmentalist: One is the way of my dad, someone who uses nature but who protects it from overuse, who is a steward of the environment, who recognizes that humanity has a responsibility to treat it with respect. A second way is the way of those city bureaucrats with the infrared binoculars, those who view the American way of life—with our centrally cooled and heated homes, our two-car garages, our meat-based diets—as inherently problematic and requiring significant controls and reforms enforced by the government. In the first view, the responsibility to be a good environmental steward rests with the individual. In the second view, the responsibility rests with the state. It doesn't take much imagination to see where most conservatives, and where I as an American individualist, come down on this issue.
Historically, however, the Republican Party has been on the other side of the divide on this issue. Two of its most famous presidents—Theodore Roosevelt and Richard Nixon—were very active on environmental issues, and both used the tools of the state to pursue their policy goals.
An avid outdoorsman, Roosevelt believed that America's West—an untamed region of wild beauty—represented the essence of America's national character. He believed in the frontier, and understood how it shaped American values, behavior, and culture. As president, he sought to protect large swaths of the country from development and settlement. He was guided in this process by the nation's earliest leading environmentalists. Sierra Club founder John Muir, Boone and Crockett Club cofounder George Bird Grinnell, forestry advocate Gifford Pinchot, and buffalo breeder William Hornaday were part of Roosevelt's circle, and they advised him on what areas to protect and why—places such as the Grand Canyon, Muir Woods, Devils Tower, and Crater Lake. These lands were unique, they argued, and if not preserved against development, their pristine beauty would be lost forever. During his second term, Roosevelt dramatically expanded the federal government's control over land in the United States, creating 150 national forests and more than quadrupling acreage protected from development to nearly two hundred million acres. Included in this acreage were five national parks, eighteen national monuments, and fifty-one wildlife refuges.
But Roosevelt's actions did not forestall the rising levels of pollution caused by America's rapid industrialization. Our air and water were increasingly dirty and dangerous, and there was little to prevent companies and individuals from simply dumping whatever they needed to dispose of into public rivers, streams, and the atmosphere. And then the nation reached a breaking point: in the early summer of 1969, the Cuyahoga River caught fire near Cleveland, Ohio. It is believed that the fire was started by sparks from the wheels of a passing train, which ignited an oil slick on the river, causing it to burst into flames.
What was becoming all too apparent was that while the United States had become the most powerful and wealthiest industrialized nation in the world, it had allowed pollution to rise to dangerous levels, and we were beginning to pay the price in the alarming deterioration of our environment.
In response to the Cuyahoga River's catching fire, and just three months after the first Earth Day in 1970, another Republican president, Richard Nixon, created the Environmental Protection Agency.
Today the EPA is regarded by some conservatives as among the most dangerous and destructive of all agencies in the federal government. In fact, if Nixon were with us today, conservatives would probably call him a RINO for having created a massive government agency to regulate the environment. But as Nixon saw it, the problem of pollution was not going to be solved by the mechanics of the free market. Economists often talk about the "tragedy of the commons": when a public space like a park is owned collectively and used freely, and no individual has an incentive to take responsibility for its upkeep. There is no profit or gain in picking up litter today, after all, when the litter will return tomorrow. The result is not just an abundance of litter, but a public space that nobody wants to occupy. The solution to the tragedy of the commons is either private ownership or collective action—either a volunteer effort or a state-sponsored one—to assert control over the public space.
The nation as a whole was faced with such a problem, on a far bigger scale, and Nixon decided to take collective action. Our environment was without protection precisely because nobody owned it. And so the federal government, acting on behalf of all of us, stepped in. Perhaps economic conservatives still choke on this, but the situation was dire, as demonstrated by the blazing Cuyahoga River.
But in the process of turning environmental protection into a responsibility of the state, modern environmentalists have devalued the importance of actually experiencing nature. If you can be an environmentalist by writing laws but without ever getting outside, you almost inevitably lose touch with what nature really is about. I think many western conservatives in particular enjoy seeing liberal city-dwelling "environmentalists" go camping and struggle to cope with nature as it is: raw, uncomfortable, and challenging. And that's the thing. Sometimes what makes sense to an office-dwelling professional environmentalist makes no sense to people who actually work and live outdoors.
Think about the rules that restrict logging in our national forests. While we can all appreciate that clear-cutting a mountainside is a catastrophic way to manage our natural resources, some loggers and many forest experts say that selective logging—removing trees but not entire groves—is a smart thing to do. When you remove some trees, you reduce the underbrush and heavy growth that can feed forest fires. Indeed, those mountains that have been actively managed and selectively reduced have fared far better than those that have been managed by bureaucrats in Washington, D.C. Sometimes you have to live in nature in order to understand how best to protect it.
But over time the modern environmental movement has placed more emphasis on issues like driving a hybrid or recycling cans as opposed to the importance of getting out (and getting active!) in nature and appreciating the great outdoors. As a result, more and more Americans don't understand the vital significance of our environment in their daily lives. Teddy Roosevelt used to ride his horse from the White House up Rock Creek, where he would dismount and go swimming for relaxation before returning to the office. If he did that today, well, needless to say, he would be cited for multiple violations. Modern environmentalists have gone so far as to suggest that we should forbid car and bus access to our national parks. But if we do that, fewer people will get to see what we are trying so hard to protect.
In general, environmentalists have placed so much emphasis on what people do to harm the environment, that they have made the movement seem like those Malthusians who used to argue that there was no way the planet's population could continue to grow without causing catastrophic famines (of course, they were spectacularly wrong). There is actually a school of modern environmentalism that argues that in order to protect the planet, people must produce fewer children.
Now there is even a television series called _Life After People_ , which depicts what would happen to cities, roads, landmarks, and other inhabited spaces if all of humanity were to disappear. Buildings would collapse. Animals would forage in supermarkets. Nearly extinct species would begin to flourish. Domesticated animals would struggle and die off. While this sounds like something out of a horror film, to a radical environmentalist it is closer to a vision of utopia.
Perhaps the restoration of the earth to its natural state before the rise of human civilization is not seriously the utopia of modern environmentalism, but that is the clear implication of so much of what modern left-wing environmentalism tells us to do. Don't eat meat. Don't travel on airplanes. Don't build big houses. Don't drive your car if you can avoid it. And no, don't have wood fires on Christmas Eve. If you do, we'll catch you. And if we don't, doom will descend on Planet Earth.
Looking back, the environmental philosophy of Roosevelt and Nixon was practical. Open and beautiful spaces should be protected so they can be enjoyed for generations. Clean air and clean water are valuable and need to be available to everyone. The environment is our public space, and every public space needs a caretaker, and everyone who uses it needs to be responsible.
Most Americans really do want to help protect nature and make sure the earth's water and air keep getting cleaner. They are concerned about the risks of climate change and are willing to change their ways to address it. But they have also seen how environmental challenges of great magnitude presented to them with urgency in the past—declining resources, rising pollution, questions over how to balance the needs of civilization and nature—have been answered by people developing solutions. They understand that innovation is an instinct of civilization—especially American civilization. It's not the American way to be confronted with problems and simply surrender. We figure them out. We adapt.
That's the opportunity for environmental conservatives. Most Americans really don't want the government to make decisions for them about how to treat the environment. Most Americans are willing to make sacrifices for the environment, but they want to pick which ones they will make, and when. Americans would rather be shown how innovation and creativity might steer us clear of climate change. They don't want to rely on the government to develop the solution. If conservatives position themselves as interested in environmental conservation and interested in solutions to challenges like climate change, pollution, and energy efficiency, they have a chance to prove themselves worthy to lead on those issues. And as it happens, the people who will be most responsive to these kinds of effort are millennials, the most environmentally minded generation in American history.
## **The Millennial Challenge**
Millennials have been exposed to more information about environmental issues than any previous generation. They have been trained from a young age, often in the classroom, to be environmental warriors. School projects on endangered species, school trips focused on ecology, perhaps even vacations devoted not to the great capitals of Europe but to the great rain forests and animal preserves of the world—all these have given this generation an ingrained sense of the imminent danger facing our environment.
One of the formative events in the experience of many older millennials was the _Exxon Valdez_ oil spill of 1989. I remember the heart-wrenching images, on television screens and in newspapers across the country, of birds, otters, and other wildlife smothered in oil. More recently, the BP Gulf of Mexico oil spill reinforced among younger millennials a sense of outrage about the loss of wildlife as a result of our seemingly unquenchable thirst for oil.
Growing up, millennials read Dr. Seuss's _The Lorax_ and followed the exploits of the Once-ler, an industrialist who acquires great wealth by ravaging the environment, depleting the land of Truffula trees. My class even acted this story out in our fourth-grade musical. The story was a cartoon version of the standard view offered by the environmentalist movement: that industrial progress invariably comes at the expense of nature. Over time, many millennials have reached the conclusion that most human activity distorts and damages nature. But because few of them are willing to abandon the comforts of modern life, they have made it their mission to "green" human activity whenever and wherever possible. In fact, three-quarters of them demand that the products they buy be manufactured in an environmentally responsible way—using recycled or compostable materials, organic ingredients, and minimal packaging. Whether something is "green" or not has become a symbol of quality, even if the "green" feature actually serves to make it an inferior product (toilet paper made from recycled paper stock is a perfect example).
But while millennials are generally active in environmental causes, they do not always go the distance. They drive cars no less than Generation Xers. They are enthusiastic users of energy-hogging portable devices such as iPhones and laptops, all of which end up adding to the problem of dangerous chemicals leaking from the nation's trash heaps. Their zeal for going paperless creates massive demand for heat-emitting, power-sucking data centers to store all their documents, photos, and other digital information.
I am not pointing all this out in order to paint the millennials as hypocrites. Far from it. I only wish to show that millennials have decided, like the rest of us, that in some ways they will be environmentally conscious, and in some ways they won't. They pick and choose. They happen to be pickier than earlier generations of Americans. But the critical fact is that their environmentalism is driven by their own choices. They refuse, so far, to embrace a "one size fits all" approach to the problem.
They are, in other words, the ideal audience for a _conservative_ approach to environmentalism, one that emphasizes the principles of American individualism: voluntary action inspired by a sense of responsibility and service to the community and the environment.
Let's start with the issue of climate change and what we need to do about it. First of all, let's agree that climate change is real. The earth's climate is getting warmer. In fact, it's been getting warmer for the past century and a half, since the end of the Little Ice Age. During that period, which lasted from about the middle of the thirteenth century to the middle of the nineteenth, temperatures were two to three degrees cooler than they are now. The Little Ice Age caused widespread crop failure and loss of life throughout the Northern Hemisphere.
But let's also stipulate that climate change is an inexact science. _Newsweek_ once published a cover story titled "The Cooling World," which warned that global temperatures had dropped significantly over the course of the previous thirty-five years. The decrease was so troubling that the article wondered if we might not be entering a "new little ice age." That article was published in 1975. The lesson here is that while we can conclude that climate change is real, we should also build in some margin for human error.
Yet members of the modern environmental movement, especially those taking their cues from Al Gore, act as if they are 100 percent certain that climate change is not only real, but apocalyptic, and that the evidence is all around us. If we have a heat wave, that's climate change. If we get lots of snow, that's climate change. Deadly hurricanes? Climate change. Lack of hurricanes? Climate change.
This kind of heads-I-win, tails-you-lose logic shouldn't fool anyone. But it does. It misleads people who care deeply about the environment into thinking that we have no options left but to adopt wholesale the policy agenda of the modern left-wing environmental movement, whose top agenda item is restricting carbon emissions through the government implementation of a complex regulatory scheme that would result in a massive backdoor tax on America's middle class.
Defeating the modern left-wing environmentalist plan for implementing such a program is a fight that conservative environmentalists should absolutely take on. According to a Heritage Foundation analysis of the bill that the Democrats in the House approved in 2009, known to some as Cap-and-Trade but to conservatives as Cap-and-Tax, electricity prices would rise by 90 percent and gas prices by 74 percent. Even if those estimates overshoot the mark, we should beware of creating a new government regulatory bureaucracy that cannot even prove its effectiveness in addressing climate change, since climate scientists won't be able to produce evidence to show climate change has been halted until many, many years into the future. And, in any case, the law would only have the effect of reducing global temperatures by a mere one-tenth of one degree over the next one hundred years!
In addition, these new laws would only regulate the United States. But we are not the world's biggest polluter, and we have even decreased our carbon emissions voluntarily over the last two decades. The American economy is far less pollution- and energy-intensive than China's, India's, and other nations' economies. These rising nations are far less interested than America is in regulating carbon. They are playing economic catch-up, and that requires a lot of energy. Roughly 80 percent of the world's people want to live with the same level of comfort as the other 20 percent experience. And who can blame them?
The modern left-wing environmental movement has put forward ideas that are expensive, will lead to massive government spending and regulation, and are ineffective in solving climate change globally. How can environmental conservatives lose this argument?
Sadly, we are losing it every day. Everyone assumes that conservatives have no serious plan to address climate change. During their two-plus decades on earth, millennials have heard environmentalists sounding alarms about pollution, habitat loss, and global warming. Meanwhile, they have heard conservatives merely guffaw at the very notion of climate "science." If you were an environmentally oriented millennial, why would you listen to conservatives?
We can do better. And here are a few ways how. First, let's turn down the heat—both literally and metaphorically. There is nothing wrong with conservation. Using less energy is a smart thing for this nation, especially if it means importing less oil from the Middle East. Conservatives should recognize that the cheapest form of energy is the energy you don't use. Meanwhile, let's also turn the heat down when we talk about the issue. Instead of focusing on insisting that climate change is a myth foisted upon us by left-wing radicals attempting to reengineer the American economy, let's confront those vocal skeptics on the Right and restore our credibility with a fact-driven platform that demonstrates a healthy respect for science. We can accept that climate change exists without embracing the Left's solutions for how to combat it, and even offer alternative solutions.
Second, let's pluck the low-hanging fruit. Climate scientist Bjorn Lomborg points out that humanity has always adapted to climate change, and suggests that this time should be no different. He proposes that we take relatively simple measures to respond to the threat _right now_. For example, because cities tend to be much warmer than less populated areas, we should focus on cooling cities by planting more trees in them and by painting roofs, roads, and other large heat-absorbing dark surfaces white and other light colors. At a cost of $1 billion we could negate any potential warming in the Los Angeles basin for the next ninety years, a reasonable price compared with the massively expensive legislation proposed by congressional Democrats.
Third, we should create and sustain as many viable renewable energy technologies as possible. This includes not only advanced solar photovoltaics but also biofuels such as algae. We should encourage clean coal plants and fourth-generation nuclear reactors. Even if all the technology bets don't pay off, we need to find out what might replace fossil fuels, and at what cost. The federal government should have a hand in this, but it should be a light one. The government's history of picking technology winners and losers is checkered at best. The free market can serve us better. But there needs to be some kind of self-funding mechanism to support energy technologies—a "green bank" to provide early-stage funding for promising ideas. If the technology reduces emissions, it gets support. As its price comes down, the support is withdrawn.
Fourth, we have to champion oil, coal, and natural gas as "bridge" energy sources. That means we need to prepare our own economy for the end of affordable fossil fuels over the next fifty to one hundred years. It's important to remember that it took roughly a century for oil to become as essential as it is today. It may take another century for its significance to diminish. But like all commodities that were once highly valued and later fell into disuse (like salt's essential role as a preservative before the arrival of refrigeration), oil will one day play a lesser role. It won't happen tomorrow, but we need to start preparing for that day.
Meanwhile, we can't ignore our own energy assets. We are the Saudi Arabia of coal. With hydraulic fracturing technology, we may also be the Saudi Arabia of natural gas. And we shouldn't ignore our own considerable oil reserves in places like Alaska. We can extract significant energy stores from our own soil, a valuable way to keep energy prices as affordable as possible for as long as possible while we make the transition to new energy technologies. The central premise here is simple—no energy should be squandered. We could power our country every day for millions of years if we could harness the power of the sun efficiently. Every night, wind blows, uncaptured as a power source. The same is true for just about any energy source you can think of. There is energy to be tapped almost everywhere if we can develop the technology we need to do the job, whether it involves driving down the cost of solar panels or wind turbines, using hydraulic fracturing technology to get at tough-to-reach fossil fuels, or introducing ultracapacitors to curb line losses in electricity transmission. More than 90 percent of the electricity produced is lost along the way before it reaches the lightbulb in our homes—what if we cut that figure to 50 percent?
I'm all in favor of a conversion to clean energy sources, including wind and solar power. But I also recognize that none of these technologies is even remotely close to supplying our country's and the world's energy needs. And they're not likely to be close for at least a couple of decades. The idea that we can somehow magically turn a switch and convert the world's infrastructure to running on these technologies is ridiculous. We need to give these technologies the time and financial resources they require to be developed into viable alternatives to our current energy sources, both technologically and economically. We do need to incentivize developers of these technologies to accelerate research and development, but we also need to stress that any lasting change in our energy economy has to be market driven. Whatever replaces oil and coal will have to be as efficient and as affordable. It will have to stand on its own in the marketplace.
Will this approach resonate with millennials? I believe it will. But only if we can demonstrate that we are serious about it. So often when it comes to the environment, Republicans discuss their ideas grudgingly, even unenthusiastically. We act as if we don't really believe what we are saying when it comes to protecting the environment. Even when we do believe it, we sometimes back away for fear of being perceived as no different from the left-wing environmental radicals. But this is easily solved if Republicans will be bold enough to stake out a conservative environmental agenda that dares to talk openly about all of America's and the world's environmental challenges, from climate change to kicking our addiction to foreign oil.
More than anything else, we ought to adopt the attitude of happy warriors and have confidence that America has the capacity to figure out this problem. The government has an important role to play when it comes to protecting the environment, but it cannot—and should not—be trusted to deliver a low-carbon, energy-independent future. That has to come from individuals, working together and driven by the realities of the market. We have to insist that effective environmentalism _—conservative_ environmentalism—is about results. It's about giving our children and grandchildren an environment as healthy as the one we inherited. It's about giving our economy a future independent of foreign oil. It's about creating a new energy economy that does not rely on subsidies and tax dollars. By emphasizing innovation, free-market principles, and individual initiative, conservative environmentalism holds great promise.
# CHAPTER 10
# A NATION OF IMMIGRANTS, A NATION OF BORDERS
_"The Republican Party, unfortunately, has been cast as the anti–illegal immigration party. It is not the anti–illegal immigration party. It is the pro–legal immigration party... and having a legal immigration system that works begins with border security."_
—SENATOR MARCO RUBIO
DURING MY JUNIOR year of college I lived in Cochabamba, Bolivia, with an incredibly loving Bolivian family while I studied at a local university. In many ways, my host family resembled my own in the United States. Delia, my "sister," was my age and studying architecture at the local university. Her brother, Alejandro, was the same age and had the same name as my younger brother in America. My Bolivian "parents" shared the same anniversary as my mom and dad. So much about my Bolivian family's structure and closeness resembled that of my own family. I lived with my Bolivian family while I studied, and secured a summer internship after the semester ended so I could extend my time in Bolivia. Despite Bolivia's status as South America's poorest country, I fell in love with all things Bolivian: my friends, the food, and the culture and the geographical richness.
While in Cochabamba, I became acquainted with several of Delia's friends, including Emilia, whose mother operated a modest restaurant on the side of a highway staffed by Emilia's siblings and featuring her mother's recipes and cooking. In this way Emilia's mother managed to eke out a living, despite an abusive husband who drank and spent his family into oppressive debt.
Later, after returning home, I learned that Emilia's mother had arrived in the United States to try to earn enough money to pay off her husband's debts. I was eager to see her again, and to greet her with the same hospitality she had extended to me in Bolivia. By the time we reconnected, she was employed cleaning houses, was learning English, and had begun to earn far more income than her delicious _silpancho_ servings in Cochabamba could ever have brought in. At an age when most women would prefer to downshift to a more comfortable pace and enjoy the company of grandchildren, this woman was a stranger in a foreign land, working a hard job, learning a new language, and tidying up a financial mess back home by cleaning houses in America.
I had no illusions about how Emilia's mother had entered the country—like nearly 40 percent of unauthorized workers, she had arrived legally and overstayed her visa. But I found it difficult to begrudge this sixty-year-old grandmother her desire to do the best she could to support her family with her limited resources. I couldn't guarantee that I wouldn't do the same thing if I were in her position. I also knew that Emilia's mother had no intention of staying in America or becoming a longtime drain on the economy through her use of public services. She would earn what she could, pay down her debts, and return home to Bolivia.
The story of Emilia's mother, one of millions of such stories, illustrates America's epidemic of illegal immigration and undocumented labor. It is a story that is part economic—the search for better-paying work. It is a story that is part national security—the inability of our nation to protect its borders against those who would enter, or remain here, without legal permission. It is a story that is part cultural—the culture of America, which has traditionally welcomed those who wanted to do better but also expects those who come temporarily to come legally and work legally, and expects those who stay to learn English and assimilate into American culture.
The case of Emilia's mother touches on all these issues. She came here to work, not to live. She arrived effortlessly and stayed as long as she needed. She provided for her family in a way she could never have in her native Bolivia. Yet what one might find admirable about one part of her story, one might resent about another. There is a reason why illegal immigration is a highly charged emotional issue for many Americans.
I tell this story as a way to convey my understanding of the compelling human element at the heart of the debates about unauthorized workers in America and the problem of illegal immigration.
I also know, however, that we cannot continue on the trajectory we are currently on as a nation—failing to secure our borders while ignoring the fact that millions of people come here to work illegally, people like Emilia's mother. The facts are troubling and cannot be ignored. Here are some prime examples from U.S. government statistics:
* There were 10.8 million undocumented immigrants living in the United States on January 1, 2009. An estimated 6.7 million were from Mexico.
* Arizona had 460,000 undocumented immigrants in 2009, out of a total population of 6.6 million.
* The U.S. Border Patrol made 650 arrests a day in its Tucson, Arizona, sector alone in 2010.
* The Phoenix Police Department reported 357 kidnappings in 2007, targeting individuals with ties to Mexican drug-smuggling gangs.
The reality is that illegal immigration exacts steep human and economic costs. When people come to the country illegally, they require schools for their children, more buses for public transportation, and other public facilities for which they rarely pay. Arizona state treasurer Dean Martin estimates that his state's government loses between $1.3 billion and $2.5 billion each year providing services for undocumented immigrants, plus other associated costs. Nationally, the costs reach $113 billion a year, according to a 2010 study by the Federation for American Immigration Reform.
Here is the dilemma that confronts us: How do we welcome immigrants who are eager to embrace the American way of life, while at the same time implementing tough and effective border security and visa enforcement? How can we encourage immigrants to contribute their unique cultural and personal attributes to the American melting pot, while also ensuring that these new arrivals learn English and assimilate into American culture?
In facing this challenge, Republicans need to frame the debate in a way that doesn't alienate the millennial generation. Millennials are the most ethnically diverse generation in American history. More than 40 percent are nonwhite, and of that group, the largest ethnic group is Hispanic. A comprehensive 2009 Pew study on millennials found that "younger people [are] more tolerant of immigrants than are older people," and that younger Americans are less apt to say immigrants have a negative impact on American customs and values. But while the study found that millennials are "much less supportive of further restrictions on immigration than other cohorts," 59 percent of millennials still believe that the federal government should secure and police our borders and manage immigration fairly and effectively.
Republicans are capable of reaching out to millennials even on the difficult issue of immigration reform. Millennials generally may have faith in the government, but they can't help but notice its utter failure to secure our borders. Also, the status quo isn't only a failure, it's compassionless: people on the border are completely at the mercy of the smugglers—just as undocumented workers in this country are victimized by criminals and unethical employers alike.
Like millennials who value competence over ideology, Republicans can express sincere outrage about the federal government's lackluster efforts at border security. Such criticism must be followed by equally sincere and realistic proposals for actually securing the border. Remember, millennials want pragmatic solutions, not rhetorical posturing. Border security must be the starting point. If the federal government could control the border, real progress could be accomplished in other key areas of comprehensive immigration reform.
A few years ago, legislation was being considered on Capitol Hill to put our country's unauthorized immigrants on a pathway to citizenship. It was a thoughtful proposal, and it had meaningful support from key members of the Senate.
I had the honor of working for the president who worked with Congress to introduce the bill. As a young Republican staffer, I had many reasons to be proud to serve under George W. Bush, but chief among them was his determined effort to meet the needs of American Latinos. As governor of Texas, Bush captured 70 percent of the Hispanic vote in his campaign for reelection in 1998. He sought to replicate that success nationally in his 2004 presidential reelection campaign. His chief strategist, Karl Rove, developed a series of outreach initiatives to Hispanics with the intention of capturing at least 40 percent of their vote—a significant target for a national Republican campaign.
Working on the Bush-Cheney 2004 reelection campaign, I took Rove's 40 percent goal seriously. As a member of the campaign's finance staff, I developed a program to organize grassroots fund-raisers in the Hispanic communities throughout the country. "Viva Bush" was our motto, and our small effort was but one of many important campaign initiatives that reflected President Bush's seriousness on behalf of Republicans to build support within the Hispanic community.
After his reelection, President Bush tackled immigration reform through the Secure Borders, Economic Opportunity and Immigration Reform Bill of 2007. Championed by President Bush, Arizona senator John McCain, and Massachusetts senator Edward Kennedy, this bill was an effort to bring the twelve million unauthorized immigrants out of the shadows. These millions were to be put on the pathway to citizenship, assuming that they had begun to learn English and that they had not committed any crimes.
The bill also included funding for a security barrier along the U.S.-Mexican border, for the installation of increased surveillance technology, and for an additional twenty thousand border patrol agents to help ramp up security.
In short, had it become law, this bill would have provided a way for unauthorized immigrants to participate in our economy legally and to become assimilated into our culture while we made the necessary investments to secure our border. The bill failed even to be brought to a vote on the floor of the Senate chamber. As it turned out, America's border states and their Republican representatives were not prepared to support a bill giving unauthorized workers a shot at citizenship until the federal government could first demonstrate that it was capable of securing the border. The bill also failed because Democrats let partisanship get in the way of serious reform and were unprepared to support a bill that would give President Bush the credit for introducing historic immigration reform.
Americans living in border states are justifiably angry that our federal government has failed for so long to live up to its constitutional obligation to secure the border. What has become obvious since 2007 is that no legislation to reform immigration will ever be politically acceptable until the federal government has first secured the border.
Even a bill as innocuous as the DREAM Act (the Development, Relief and Education for Alien Minors Act) failed to pass in December 2010. This bill would have created a pathway to citizenship for the children of unauthorized immigrants (many in the millennial generation) who are in America illegally through no fault of their own. Despite massive Democratic majorities in both houses of Congress during a lame-duck session, even this seemingly harmless bill failed to become law. The fate of the DREAM Act is the ultimate proof that the politics of immigration reform is paralyzed beyond hope until measurable success can be achieved on border security.
Our focus, therefore, has to be on securing the border, and also on ensuring that those who come here on short-term visas leave on schedule. If we do not take steps to differentiate between authorized and unauthorized immigrants, we will end up tarnishing the proud immigrant tradition of this country by placing a cloud of suspicion over every law-abiding newcomer. Enforcement of the law and protection of our border will redound not just to our benefit but to the benefit of all immigrants in America.
What can be done? As it turns out, plenty.
There have, in fact, been isolated successes in securing the border.
Yuma County, Arizona, provides a textbook example of how border security can be accomplished. It has gone from having some 138,500 unauthorized aliens enter the country across its section of border before serious border security was implemented to only 7,000 a year after implementation, a 95 percent decrease over a five-year period.
The success in Yuma County demonstrates that border security isn't rocket science. Yuma County's solution to securing the border consisted of three straightforward components: first, building miles of fencing; second, hiring a sufficient number of border patrol agents; and third, implementing Operation Streamline, a set of guidelines for how authorities should prosecute _on the spot_ anyone caught trying to cross the border. The first time individuals are found crossing the border illegally they spend fourteen days in jail; the second time, thirty days; the third time, sixty days—until eventually they realize they cannot successfully cross. Imagine the results if this system could be implemented across the entire southwestern border!
This approach isn't cheap—it takes federal and state funds to pay for portable detention spaces, judges, lawyers, and court clerks. But the investment is inexpensive compared with the cost of the elaborate cat-and-mouse games involved in catching and prosecuting unauthorized immigrants elsewhere, or the cost they exact on the system as a whole.
Some of our liberal and Democratic friends have a hard time with the notion of Americans building a fence along the southern border. They think it will send a message to the world that we are isolationist and xenophobic, that we are unwelcoming of other cultures, and even that we are racists. Some have likened such a fence to the Berlin Wall, while others have dubbed it _elmuro de odio_, "the wall of hate."
Any such suggestion is part of a systematic effort to paint the Republican Party as anti-immigrant, conflating the concepts of legal immigration and illegal immigration in an attempt to distort the debate. It is intended to perpetuate negative stereotypes about the Republican Party that continue to hamper its efforts to win support among not only Hispanics but also among millennials. In the eyes of many millennials, a person's (or party's) stand on immigration is more than a reflection of a particular policy. It is seen as a broader statement about attitudes toward diversity, globalism, and even civil rights. Republicans must push back on this attack by marshaling the facts.
Let's take the example of Arizona's law to enforce federal immigration rules and procedures. Despite the fact that the law has the support of 77 percent of Arizonans and 73 percent of all Americans, immediately after the bill's passage the Obama Justice Department declared that it would challenge the legislation. Attorney General Eric Holder admitted that he hadn't even read the Arizona bill when he decided to take this action. The irony of the challenge was that the administration claimed that the law infringed on the responsibilities of the federal government, despite the fact that it was precisely _because_ the feds were failing to execute their constitutional responsibilities that the law had to be passed in the first place.
But what's in this law? Is it, as President Obama says, a violation of civil rights and a form of racial profiling?
Not at all. In fact, the law specifically forbids racial profiling. It merely empowers local and state law enforcement to question someone's immigration status _if that person has already broken a law_. Only at that point, and only if there is _reasonable suspicion_ that the person detained might be in the country without authorization, is a law enforcement official allowed to question that person about his or her citizenship. And there is an extended list of protections defining exactly what constitutes "reasonable suspicion."
The Arizona law actually has more civil rights protections in it than federal laws. It specifies that race _cannot_ be one of the criteria for "reasonable suspicion." The federal immigration law, meanwhile, does not even stipulate that a person first has to commit a crime in order to be detained or questioned about his or her immigration status. It would seem that the only thing Arizona did that might offend the feds was to say, we will enforce a law that you won't—and by the way, we will do it more fairly and equitably than you might if you ever get around to it.
Nevertheless, even with those clarifications, there is no doubt that the Arizona law is not an attempt at a comprehensive approach to immigration reform.
It does nothing about the employment of unauthorized immigrants by unscrupulous employers. It leaves out any discussion of allowing unauthorized immigrants an opportunity to resolve their status—something only the federal government has the authority to do. Its focus on "attrition through enforcement" is only one dimension of what has to be a far broader approach to the immigration problem. It is a desperate measure, by a desperate state, in the face of the federal government's failure to secure the borders.
Neither party has the confidence of the American people when it comes to border security. Republicans tried and failed to pass immigration reforms when they had control of both the White House and Capitol Hill. Democrats in the same position were unable to pass the DREAM Act.
By now, the answer should be obvious. Those who would like to see comprehensive immigration reform must first support serious improvements in border security before they attempt to do anything else. It's that simple—we must secure the borders first. Only after we have successfully secured our borders will there be a reasonable chance for comprehensive immigration reform.
Here are some ideas that can help make comprehensive immigration reform innovative and successful:
## **Implement Operation Streamline Broadly**
As Yuma County has proved, by immediately detaining and jailing those people who try to cross the border illegally, we make it much less likely that they will keep trying. Yes, this will cost money. But it is far easier and less costly to hire judges, court clerks, prison guards, and lawyers to deal with the illegals near the border than it is to identify them once they are already in the country. Given the success of this program, it should be implemented wherever possible across the entire southwestern border.
## **Prioritize Skills-Based Immigration**
Nobody begrudges immigrants the right to come to this country to do work Americans are unwilling to do. Yet we do not do enough to match immigrant workers with jobs in industries where there are constant labor shortages. For example, Bill Gates and other tech executives have spoken out about the need to increase the number of H-1B visas available to skilled workers who arrive in America ready to work and contribute to our economy and our culture. Likewise, if foreign students come here to be educated in areas that are important for us to maintain our competitive edge in the global economy, it doesn't make sense to have them benefit from our educational system and then kick them out of the country. We need to increase the number of long-term work visas in order to keep the American economy competitive.
## **End the Visa Lottery**
Each year, tens of thousands of people are given visas to the United States on the basis of luck alone. No test is administered. Those given a visa might be young, they might be old, they might have skills, they might be illiterate. If immigration is going to serve the interests of the country, we have to take luck out of the equation and award visas to the deserving, especially young adults who are seeking to make a fresh start and assimilate into an adopted country. We want the best and the brightest immigrants—they will help to make our nation stronger and more competitive.
## **Activate E-Verify**
_E_ stands for _electronic_ and this national electronic system would make it easy for employers to independently and efficiently verify whether a prospective employee is legal or not. Right now, employers have little to go on: driver's licenses are easily obtained; Social Security cards are often faked. The result is that employers have no choice but to accept what documents they are given. There is no definitive way of verifying their authenticity. Of course, once we solve the problem, employers will have no excuse for hiring unauthorized immigrants. This will be a problem not just for large agribusiness, which hires many migrant and itinerant workers on the fly. I can think of several prominent Republicans and Democrats—Zoë Baird, Linda Chavez, and Meg Whitman—who have hired unauthorized immigrants in their own homes. They broke the law, perhaps without knowing it. So the question is, why is our federal government unable to give employers a reasonable level of confidence that someone is legal or not?
It would be great if immigration reform were simply a matter of putting together a policy paper and seeing it through to law. But that's not really how it works. Because immigration is not merely a security issue and not merely an economic issue, it takes on a special intensity. After all, it touches on the lives of millions of families. And because immigration is closely tied to the legal rights and the culture of tens of millions of Hispanics, any criticism of illegal immigration tends to be misrepresented by the Left as some form of ethnic bigotry on the part of Republicans.
This means Republicans have a special challenge when it comes to talking to millennials about immigration reform. Because the millennial generation is made up of a larger percentage of Hispanics than any previous generation, Republicans must adopt a respectful and compassionate tone.
We should affirm the broadly held American values on immigration and the good that has come from being a nation of immigrants through the constant regeneration of the American dream. And we ought to make the case for effective border security on the basis of the safety of our communities, and the inherent responsibility of the federal government to secure the borders.
This is doable. In fact, it's being done.
We have a rising generation of newly elected Hispanic Republican leaders, including Senator Marco Rubio and border-state governors Brian Sandoval and Susana Martinez, to help lead the way. They can also help us make the case for the importance of assimilation. Marco Rubio has emphasized the responsibilities of immigrants who wish to become Americans: "The most important thing that recent arrivals can do for their children is make them proficient in the English language."
Senator Rubio understands, as do many immigrants who have become active in the Republican Party, that this one thing the Right fixates on—the responsibility of the immigrant to learn English—is in fact the greatest gift an immigrant can receive. Republicans must be prepared to push school systems to quickly wean immigrants off bilingual education and force more and more of them to do their schoolwork in English, the national language of the United States. But immigrants need not give up their native languages or culture even as they assimilate into American culture. And so when those on the Right say "English only," I would say instead, "English first."
While the English language is and will continue to be a unifying thread throughout American culture, we must also avoid being culturally and linguistically alone in a globalized world. I come to this issue with a personal interest and perspective. My grandmother was born in the Arizona Territory, and I've studied Spanish and Spanish literature throughout my school years. I wrote my college thesis in Spanish. Growing up in a western state with a large Latino population, I have always been comfortable hearing Spanish spoken, singing to Spanish-language music, and watching telenovelas on Univision. This familiarity with other languages and cultures can be enormously enriching and should be encouraged.
Republicans can also do something that liberal Democrats rarely do: affirm what it is that most immigrants come here for—to achieve the American dream. Republicans are the champions of entrepreneurs and small-business owners, people who take great risks to achieve a future for themselves and their families. Republicans should articulate how so much of their agenda, based on a pro-growth and pro-innovation philosophy, is vital to the dreams of all immigrants.
Republicans ought to focus in particular on the outrageous bureaucratic delays facing those who want to come to this country legally—who want to play by the rules. Because new immigrants are often fleeing governmental and economic oppression, they are the first to remind us of the incredible opportunities our country provides. They are some of the best examples of American individualism and have been the secret to the regeneration of the American dream throughout our history.
That's why I think Republicans ought to consider that a great majority of those who are in this country illegally have otherwise been law abiding and committed to the hard work and dynamism that American individualism rewards. One day, _after_ the border is successfully secured, we should support efforts for them to emerge from the shadows and be given an opportunity to live in America legally. Conservative critics may call this "amnesty." But let's be honest: it's a far cry from the functional amnesty Ronald Reagan backed in the mid-1980s as part of what we now see as a flawed attempt at immigration reform.
By securing the borders first—expanding the opportunities for legal immigration, and putting in place measures like E-Verify—we can fix our broken system without encouraging new waves of unauthorized immigration. We can restore faith in legal immigration, which has always been a key ingredient of our nation's success—reviving the American dream and the spirit of American individualism through the power of the immigrant's example. Once we establish control of our borders, we can confidently and proudly assert that America remains a nation of immigrants.
# CHAPTER 11
# ISLAMIST SUPREMACY
_A Millennial's Worst Nightmare_
_"The bombers of Manhattan represent fascism with an Islamic face, and there's no point in any euphemism about it."_
—CHRISTOPHER HITCHENS
ON SEPTEMBER 11, 2001, nineteen men hijacked four commercial airplanes and used them as missiles, slamming two of the planes into the World Trade Center towers and one into the Pentagon. The hijackers piloting the fourth plane, United Airlines Flight 93, were thwarted by the heroism of the passengers on board, whose desperate resistance caused the aircraft to crash into an open field in western Pennsylvania.
These events were a defining moment in the lives of millennials and a wake-up call for all of America. In the same way that Pearl Harbor changed the Greatest Generation, and the Kennedy assassination changed baby boomers, 9/11 changed and continues to shape the millennials.
On that fateful morning, the oldest millennials were seniors in high school, while the youngest were just infants. Their lives went from being secure and carefree to uncertain and scary. For the first time in their lives, they saw that their government and their parents were not in control. Evil existed and it needed to be confronted. In the weeks and months afterward, more fear followed, sparked by bomb threats and anthrax exposure. Osama bin Laden became a household name, and to the youngest millennials he was the incarnation of their worst fears.
Millennials, like the rest of the country, understood what would happen next: a war. A war to protect our cities and borders. A war to prevent another 9/11. When America went to war against the Taliban and Al Qaeda in Afghanistan, it went proudly, and millennials saw it in the way our troops were treated, the way the flag was honored, and the way people set aside political grudges to focus on the work ahead. I suspect that it was in this period that the high expectations millennials have for their elected leaders were formed. They came to see that government should be trusted and trustworthy, and that those who give their lives to public service—soldiers, sailors, pilots, firefighters, and police—should be revered. What was for America one of the most challenging moments was also one of its most defining. Our country came through that period stronger.
But that unity of purpose did not linger long. Millennials were inspired to act by 9/11, but they soon soured on America's war in Iraq, which spurred increasing doubts about our prolonged engagement in Afghanistan. Meanwhile, in the absence of another successful attack—despite more than a dozen thwarted incidents—Americans as a whole have grown less concerned about the threat of another 9/11. The intensity is gone. The sense of shared sacrifice has dissipated. A sense of complacency threatens to take its place.
This isn't just a challenge for Republicans trying to win over millennials. This is a problem for all Americans who care about the future security of our nation. If the 9/11 generation no longer understands why we are involved in the conflicts we fight today, we have a problem.
The problem, in my view, is that as a country we never clearly established who it is we were fighting, and what we were fighting for. We looked at those nineteen men on the airplanes as representing a single isolated enemy supported by the Taliban and deployed by Al Qaeda from a remote mountainous corner of Afghanistan. We focused immediately on the tactics they might use: hijacking airplanes and turning them into missiles, deploying "dirty bombs" stuffed with radioactive material, setting off car bombs in major cities, and releasing biological and chemical agents. But we stopped thinking about who these enemies are, what their goals are and why.
Perhaps we did not have the time or imagination to think about the motivating ideology of those nineteen men and the people who planned and financed their operations. We didn't think about how they were the latest foot soldiers in a larger and ongoing movement, a broad effort to push out, through violent force when necessary, Western values from the Middle East, Africa, Asia, and even Europe. We did not recognize that these men were merely the spearhead of the most violent ideology of the twenty-first century so far—an ideology as sweeping and murderous as fascism and communism. We seemed to believe instead that all we needed to focus on were their tactics, as if we could defeat them if we just limited ourselves to three-ounce bottles of liquids in our carry-on luggage.
Initially, we called it the Global War on Terror. But America isn't fighting terror. Terror isn't an enemy. Terror is a tool of warfare, employed to achieve specific goals that other methods cannot. Terror intimidates. It forces civilized nations to spend significant sums on protecting infrastructure, transport systems, and institutions that ordinarily aren't targets of conventional warfare. It is an ancient technique, used by militants in many different times and places, including both left-wing and right-wing radicals at different points in American history. So many groups around the globe use terror that to declare a literal "war on terror" means that America must go to war against groups on every continent, in dozens of nations, including places where we have no strategic or national interests. And ultimately, calling this conflict a "war on terror" makes it impossible to declare victory—if you suffer no terror attacks, it merely means you have matched wits, for the moment, with the terror masters. You haven't eliminated the source of their strength, which is their capacity to constantly dream up new methods of mayhem.
The Bush administration deserves credit for having prevented another 9/11, but it failed to communicate clearly to the American people who it was that we were fighting. Nor have things improved in the years since the Bush administration. Neither John McCain nor Barack Obama used the term _radical Islamism_ in any of the 2008 presidential debates. As soon as President Obama took office, his administration dropped the phrase "Global War on Terror" in favor of an even fuzzier "Overseas Contingency Operations." By avoiding any reference to an enemy of any kind, this terminology made our wars and our sacrifices seem like a technical problem of logistics. It gave no hint of the violent political ideology that was at the heart of 9/11 and, before that, the attacks on the USS _Cole_ in 2000 and our embassies in Tanzania and Kenya in 1998. No wonder people are losing interest in securing victory in Afghanistan. A war that began as a result of the worst attack on American soil has been classified using a phrase that nobody understands and only an Orwellian could love.
We have to go back and start over from square one. We have to think about what it is we are fighting for and whom we are fighting against. Our enemies have been very open about their intentions and motives. There are certain men—this is an enemy that does not consider women to be equals (though it has begun to use women as operatives in rare cases)—who believe that their narrow, intolerant, and violent interpretation of Islam offers the only correct way to be a faithful Muslim.
They view other Muslims who subscribe to less fanatical interpretations of traditional Islamic teachings as apostates. Those Muslims who are secular, or who no longer follow the strictest interpretation of Islamic law, are condemned, sometimes to death. Those who belong to other, non-Muslim religions are considered second-class humans at best.
This political ideology is not drastically different from that of other supremacist movements we have witnessed—the Aryan supremacist movement of Nazism, the racial supremacist movement of the Ku Klux Klan and apartheid, and the countless other ideologies that have depended on the demonization of another group. Our present enemies have latched on to a highly intolerant interpretation of Islam to justify their supremacist ideology. Of course, it's not at all unusual for people of a certain faith to believe that their religion is the one true way—it is common among Hindus, Jews, Catholics, Protestants, and just about every other major religion. What sets our enemies apart today is their belief that those who don't subscribe to their view of Islam should be conquered, subjugated, and if necessary, eliminated.
Our enemies are, in a phrase, Islamist supremacists. Terrorist acts against America and other Western interests are just one tool in their toolbox. In the nations where they live, they operate schools that teach their young to hate members of all other faiths. With rare exceptions, women are treated as the property of their fathers and husbands, and they are required to veil themselves at a young age. Those who accept alternative interpretations of Islamic law and scripture are pushed aside—or worse. Those lands that once were ruled by Islamic leaders centuries ago—Spain, Israel, North Africa, Central Asia, and beyond—are regarded as lands rightfully belonging to a broad Islamic caliphate, and all non-Muslims now residing there are regarded as future subjects. In certain nations that have welcomed hundreds of thousands of Muslims—such as the Netherlands, the United Kingdom, France, and Norway—Islamist supremacists work to undermine civil authorities and laws, and agitate for the recognition of sharia as the sole law applicable to all Muslims, regardless of their desire to live under such laws.
Islamist supremacists not only view Islamic civilization as superior to Western civilization, but they also hold an especially negative view of Western civilization. To Islamist supremacists, the West is a dangerous place filled with temptations for Muslim youth such as alcohol, drugs, sexuality, equality of the sexes, and secularism.
It is not unusual for strictly observant people of _non_ -Islamic faiths to regard the cultural extremes of Western society this way. But while most of the traditional strains of various faiths can accommodate Western values, and seek to promote a voluntary moral culture, Islamist supremacists can't abide the West. In their eyes, the West is decadent and immoral, and through its films, music, and other cultural exports it threatens to erode the morals of young Muslims. To Islamist supremacists, the West's liberated culture actively encourages women to oppose the wishes of fathers and husbands. In the same spirit that white supremacists took offense when a black man spoke to a white woman, Islamist supremacists have created a wall of separation around women, in the name of protecting them against male sexual aggression.
This is obviously not a live-and-let-live ideology, but an aggressive one that seeks to impose itself on others by force. It's a violent ideology based on the notion that one way of living is superior to all others, and therefore should dominate all others. This is the very definition of a supremacist movement.
With all this in mind, it is imperative that we reframe the ongoing conflict in order to more clearly communicate to the millennial generation, and all Americans, who it is exactly we are fighting. There is simply no greater threat to the liberal cultural values nourished on our nation's college campuses than Islamist supremacy. If members of the millennial generation care about women's rights or gay rights, they must understand the present danger and evil of Islamist supremacy.
Additionally, framing the current conflict as a struggle against Islamist supremacy makes it clear that this is not a clash of faiths, or a clash of regions, or a clash of civilizations. It is actually a face-off between two worldviews. One worldview is richly grounded in values of individual freedom, pluralism, and human rights, and encourages Muslims, Christians, Jews, Hindus, Catholics, Buddhists, and members of all other faiths to practice their religion freely and to live together peacefully. The other worldview is a political ideology of intolerance that is fighting for the subordination of all others, including adherents of a modern and pluralistic Islam.
Framing the argument this way will help us make it unambiguously clear that while the leaders of this supremacy movement may call themselves Muslims, they do not represent the religion of Islam or the will of the world's one billion Muslims, most of whom wish to live in peace and do not support this xenophobic and violent ideology.
The Bush and Obama administrations have repeatedly said that America is not at war with Islam. And Republicans need to be vocal about condemning anyone, in America or abroad, who seeks to lump all Muslims together as America's enemies, whether it's a pastor in Florida who burns the Koran, or jailed terrorists in Guantánamo who insist that America hates Muslims. America is not at war with Islam, and Republicans must continue to follow President Bush's example in condemning any expressions of prejudice against Muslims.
Perhaps most galling to Islamist supremacists are those Arab and Muslim leaders who rule countries that have so far not succumbed to Islamist fanaticism. The leaders of Saudi Arabia, Jordan, Bahrain, Lebanon, Kuwait, and even Iraq are regarded as puppets of the West, subject to Western whims and Western culture. And so Islamist supremacists are often most concentrated, and most radical, in nations that are ruled by secular or pro-Western Muslim leaders. For exactly this reason, the United States is highly motivated to see the establishment of representative democratic governments in countries such as Tunisia, Egypt, Libya, and Bahrain. Islamist supremacists are often the most organized actors on the ground, and we fear they might get a foothold in governance as former dictatorships crumble.
This is not an abstract argument. It is rooted in recent human experience. Remember Afghanistan as it was run by the Taliban. This was a country in which Islamist supremacists organized what they considered to be a perfect society. Girls were forbidden to attend school and were attacked when they tried to do so. Women were confined to the home and hidden beneath heavy clothing—and had to keep their faces from view. Flying kites, an old Afghan custom, was banned, as was the playing of music. A cluster of Buddhist statues 165 feet tall and 1,700 years old, recognized as a World Heritage Site, was destroyed by dynamite and tank fire. And thousands of Afghan citizens who dared to question this supremacist rule were murdered.
It should be no surprise that the Taliban gave shelter to Osama bin Laden and Al Qaeda when they were plotting the 9/11 attacks. The mass murder of Western "infidels" is their stock-in-trade. To Islamist supremacists, the West poses a unique threat, greater than that posed by mere nonbelievers in their midst. To them, the West must be resisted, not just with the heart but with the sword.
Millennials understand the dangers of supremacist movements. They have learned about the white supremacists who once dominated the American South. They have studied the rise of Nazism and the genocidal movements in Rwanda and the Balkans, each of which proclaimed that one type of people was better than another and deserved to rule the other. Given that millennials have a strong belief in the value of pluralism and diversity, they recognize that Islamist supremacists represent a special threat to their values and their freedoms. Indeed, Islamist supremacists are the millennial generation's worst nightmare.
The Pew Center for Research tells us that millennials are the most liberal generation in America. And here liberal means _socially liberal_ —"permissive" in their attitudes about sex and sexual orientation, for example. They are, of course, completely comfortable when women participate actively in society as leaders, and in virtually any profession they choose. Millennials have diverse and broad tastes in music and film. They communicate readily with one another, and with their peers in foreign countries, using social media. And by comparison with earlier generations of Americans, they are less likely to be affiliated with organized religion—even if they are privately spiritual. Millennials represent the future of American individualism, which makes them an especially despised target of the political ideology of Islamist supremacists.
But the truth is that Islamist supremacists have done more harm to _Muslims_ through their violence than to any other group. It is not unusual for Muslims who resist Islamist supremacists to be jailed, raped, and forced to watch family members be tortured and murdered.
A cold-eyed realist may say, "Look, it may be terrible what this movement does to its own people, but we can't interfere every time there is injustice in the world." I agree that we cannot and should not try to stop Islamist supremacy through military means wherever it appears in the world. But we must act to contain it and confront it when it threatens the United States.
The reason we cannot abandon Afghanistan today is that the Islamist supremacists there have not been defeated. They have fled into neighboring Pakistan and are hoping to return to Afghanistan. And if we leave Afghanistan before helping to secure a stable society that can defend itself against them, they will return to power. We know precisely what would happen if that were to occur, because we saw the effects of Taliban rule on 9/11.
Likewise, we need to remind millennials that Islamist supremacists are not just cave-dwelling insurgents. Just look at Iran: a nation that was once cosmopolitan, pluralistic, and Western oriented is now an exporter of terrorism and a repressed country whose government (though not its people) is hostile to American values and is actively trying to build nuclear weapons.
Iran's president, Mahmoud Ahmadinejad, declared before the General Assembly of the United Nations that the attacks of 9/11 were organized by the United States "to reverse the declining American economy, and its grip on the Middle East, in order to save the Zionist regime." Ahmadinejad is also a Holocaust denier who has declared that Israel should be "wiped off the face of the earth."
When Ahmadinejad went to Columbia University and was questioned about the lack of gay rights in Iran, he responded that "in Iran we don't have homosexuals like in your country." The audience of millennials could hardly subdue its laughter.
And then there's the state-sanctioned punishment for adultery: stoning. This is the medieval practice of burying a victim partially in the ground, and then taking turns throwing rocks at the victim's head, using just enough force to maximize the pain but without knocking the victim unconscious. There are, at last count, nine women sentenced to death by stoning in Iran.
The way to overcome Islamist supremacists is by empowering young Muslims around the world to think freely. We need to support Iranian millennials like those who rose up in 2009's Green Revolution against a government that had plainly stolen an election, ruined an economy, and put Iran on a path of hostile confrontation with the West. Iran's millennials have grown tired of being oppressed by a political regime led by mullahs and autocrats.
We need to connect America's millennials with those of other nations, and encourage those connections. President Obama's overtures to the Muslim world might just help open up the minds of Muslims who would otherwise fall under the influence of Islamist supremacist thinking. We Republicans may not always agree with what President Obama says, or with the Democratic Party's foreign policy approach. But we are all Americans first, and if we are to emerge victorious in this battle against Islamist supremacists, we have to work together. Millennials look to us to uphold the proud tradition that partisan politics ends at the water's edge.
Finally, we need to champion those Muslims who practice a variation of Islam that respects Western values. I think of people like Irshad Manji, a faithful Muslim who calls on her fellow Muslims around the world to reconnect with the concept of _ijtihad_ , Islam's own tradition of independent reasoning. In her work she calls out the anti-Semitism, homophobia, and sexism of Muslims who have colonized her faith by imposing a narrow, tribal mind-set that violates the Koran's teaching to think critically. Because of her fierce commitment to reconciling faith and freedom, Manji's life is constantly threatened. But her message is welcomed by youth throughout the Muslim world. They have downloaded the Urdu, Farsi, and Arabic translations of her book _The Trouble with Islam Today_ from the Internet—which publishers were too fearful to print in these languages—more than two million times.
I also think of Zainab Al-Suwaij, who was forced to flee Iraq in the 1990s, and who after 9/11 went on to found the American Islamic Congress (AIC) to give a voice to pluralist Muslims on American college campuses. She is a strong advocate of the rights of Muslim women and has devoted her work at the AIC to championing interfaith understanding.
There is Zeba Khan, a writer, social media consultant, and self-described advocate of Muslim-American civic engagement. She is a quintessential millennial who in 2008 launched an online network called Muslim-Americans for Obama to mobilize Muslim-American voters to support President Obama's campaign. While we were on opposite sides of the political aisle then (and probably still are today), I want to see Khan succeed. She is the daughter of devout Muslim parents who emigrated from India to the United States. Her parents insisted that she learn about other faiths, so she and her brother attended Hebrew school, where she studied Hebrew for nine years and learned to read the Torah while also attending her local mosque. Khan is a prime example of the unusual combinations of faith traditions and backgrounds that have found a home in America over the centuries.
Likewise, there is Suhail Khan, a young Muslim Republican whose parents also emigrated from India. He grew up in the western United States, where he practiced his faith with his family while learning about Christianity at the Catholic school he attended. Khan worked in the White House for George W. Bush and is deeply involved in the American conservative movement. Khan is one of thousands of mainstream American Muslims who work to promote Christian-Muslim understanding, in his case at the Institute for Global Engagement. Republicans must harness the knowledge, energy, and dynamism of people like Khan in order to mobilize, as Zeba Kahn did for Obama, a new generation of Muslim-American Republicans.
Republicans should champion the work of Muslims whose religious practice complements America's history of religious pluralism. We must stand against those who use intimidation and thuggery to silence the voices of freethinkers here in America and Europe and the Middle East. We Republicans can show, by our own example, that individual freedoms and liberties need not lead to lives of immorality but rather to lives of discipline, creativity, and dynamism—and to cultures of tolerance and diversity.
We must show that the dream of Islamist supremacists—an end to the rights of women, religious minorities, homosexuals, and others who do not abide by their radicalized interpretations of the Koran—is the nightmare of all civilized people.
Republicans can connect with this new generation by reminding millennials that what they saw on September 11, 2001, represents the Islamist supremacist plan for the future. It was not a _tactic_ that declared war on us that day but a political _ideology_ , one as menacing, as hateful, and as anti-American as Nazism.
Millennials, the most globally oriented, most diverse generation in American history, should understand that their worldview is in the crosshairs of Islamist supremacists. This is a war that none of us—neither Republicans nor Democrats—chose. But it is a war we must win. Otherwise, values that we cherish as Americans—freedom, democracy, and diversity—will perish.
# CHAPTER 12
# AMERICA THE EXCEPTIONAL
_"Our willingness to speak for freedom is no bargaining chip. It's an integral part of our foreign policy."_
—RONALD REAGAN
BY 1919, WHEN Herbert Hoover turned forty-five years old, he had spent his entire adult life outside the United States, working and traveling in the Far East, the Near East, Southeast Asia, Australia, South America, Europe, and Africa. He established residencies in Australia, China, and England. He had visited every continent but Antarctica. He had circumnavigated the globe five times—and this was before the advent of commercial aviation.
He was not a dilettante traveler, casually gliding through countries as a tourist or an art collector, someone merely picking up experiences to use as dinner party fodder back home. He was working. His international travels brought him into direct contact with those who were building industries and managing emerging national governments, or clinging to feudal ones. He met all kinds of citizens in these countries, from average laborers to ruling aristocrats.
What he saw during his travels influenced him greatly. At the beginning of the twentieth century, the world was in upheaval. The old order of monarchies and empires, while still in place, was starting to be challenged. Rapidly modernizing countries such as Japan and Germany were building up their militaries and flexing their muscles. The threat of revolution and anarchy was palpable. The ideas of Karl Marx appeared ready to overturn the ideological tables in Europe. It was a time when the assumptions that underpinned capitalism, democracy, and Western civilization were being called into question.
Hoover experienced this revolutionary upheaval. He observed that in each of the countries where revolution took hold, workers were agitated that they did not reap the benefits of their labor to an extent nearly equal to their contributions to industry. He and my great-grandmother barely escaped China's Boxer Rebellion in 1900. During the following decade, the mines he operated in Russia were seized and nationalized by Lenin's Bolshevik regime. Hoover was asked by his own government to administer massive food relief to combat the starvation and hunger-related disease caused by the First World War and its revolutionary aftermath. He saw that the landed gentry of Europe, those lucky few who hung on to the remnants of the ancient system of feudalism and peerage, were blind to their impending demise.
Hoover had no illusions, as he went about his travels, that he was an exception to the global rules of opportunity. Here was a man, once a frontier orphan of no means, who had become one of the world's wealthiest and most respected individuals. He began to think about why this was, and to wonder whether America itself was the exception—whether the American _system_ was different and, in essential ways, unlike that of any other country.
He looked at countries endowed with greater natural resources, stronger educational traditions, more powerful armies and navies, and deeper cultural heritages—and yet none of them could equal America's economic might and potential at that time. He wondered whether young America, immature America, was benefiting from more than just good fortune. He began to think that perhaps America was on a distinct path because it was unique, because its founding philosophy encouraged individual achievement and placed a premium on equality of opportunity.
After the First World War, Hoover returned to the United States and began to ponder the circumstances that had catapulted the country into its position of global dominance within such a short period of time. He wondered what it was about the American formula that had made his story possible. Like the engineer he was, he began to reverse-engineer America. Mentally he pushed aside all the external features of its national strength and tried to peer inside the machinery, at the hardwiring. He wanted to figure out what the mechanisms were that set America apart from Europe and the rest of the world.
That search led him to craft a commencement address, which eventually became a slender volume called "American Individualism." Instead of looking at America from the top down, as a system of interconnected parts, a vast machine of political philosophy and economic organization, Hoover proceeded from the bottom up. He zeroed in on the smallest unit of society. He focused on how America was guided both culturally and politically by its celebration of the individual. In America, Hoover observed, an individual is just that—a single person, defined only as that person chooses. An individual could come from a background of poverty, be raised by uneducated parents, belong to a church of no particular note, or be born into a race just two generations removed from slavery, yet that individual could advance well beyond the circumstances of his birth. And Herbert Hoover, the orphaned boy of the frontier, the child with no privileges or advantages other than his God-given talents who rose to the greatest heights of wealth and power, understood this as well as anyone.
But Hoover worried that America would be tempted by those proposing a radical redistribution of economic spoils and political power, as the Marxists and socialists of that time prescribed. Upon his return to the United States in 1920, he saw adherents of Marxism and fascism and other ideologies lashing out at America's system. The Socialist Party in America in the late 1910s and early 1920s was regularly garnering between 2.8 percent and 6 percent of the votes in presidential elections.
Hoover wrote "American Individualism" to explain the American system, and to defend it. He wanted to inoculate America against the dangerous temptation to experiment with Europe's radical ideologies. Hoover had witnessed the failure of socialism and communism and the hardship and death it had brought to millions in Europe. And while socialists in America ignored this early evidence of ideological failure, Hoover drew attention to it. Referring to the communist experiment in Soviet Russia, he wrote that "socialism in a nation-wide application has now proved itself with rivers of blood and inconceivable misery to be an economic and spiritual fallacy and has wrecked itself finally upon the rocks of destroyed production and moral degeneracy." He understood the appeal of socialism all too well, and he considered the attempt to implement it in Russia to be useful, in that it showed humanity that socialism's uplifting ideas, when put into practice, led to destruction and misery. And having seen socialism's failure with his own eyes, he was determined to oppose it, especially in America.
It is striking that nearly a hundred years later, "American Individualism" encapsulates much of what Americans still believe about this country today, and why they resisted the temptations of socialism even in the country's darkest economic period, during the Great Depression. Hoover understood that America existed on the strength of an idea, not of national or tribal loyalties. The essence of America could not be captured in a flag or a banner or a song. America represented the collective achievements of its individual people.
Hoover didn't think that Americans were inherently superior to citizens of other countries. He merely felt that Americans were more likely to achieve their fullest potential as individuals because of the freedoms they enjoyed. He understood that America was the first nationwide experiment in economic and social freedom and mobility in history. No nation before had suggested that it was the inalienable right of an individual to pursue happiness and prosperity without fear of disrupting the social order.
Hoover presented the American system as a revolutionary system, a system without the shackles of class or caste. He understood the ambition of the revolutionaries in Europe—to bring about new systems that would offer greater opportunity for more people to share in the nation's wealth. But Hoover believed that the American system of representative democracy had its own built-in mechanisms to make possible such mobility, and that it allowed for gradual social and political reforms, thereby avoiding the need for violent or radical revolution.
It is ironic, even tragic, that my great-grandfather, the first globally minded president of our country, the first man to hold the office who had traveled so widely, is most remembered for having presided over a nation that lost its global confidence, turned inward, shut its borders, cut off its trading partners, and retreated from engagement with the world, a descent into isolation from which it would not emerge until after 1941. The Herbert Hoover of "American Individualism" believed that America was a beacon of light in the story of human freedom. And yet during his presidency and the subsequent twelve years, that light faded, just as he feared it would.
But that light was never fully extinguished. Herbert Hoover was on to something. There really was something extraordinary about the American system. Hoover's "American Individualism" was an early-twentieth-century expression of the idea of American exceptionalism, first articulated in _Democracy in America_ ninety years earlier. To Alexis de Tocqueville, America was "exceptional" because at the time of its founding, it was quite literally the world's _sole exception_ to antiquated governments that concentrated wealth and power in the hands of the few. Hoover's practical experience corroborated Tocqueville's observations, and he recognized that America was the first sustainable example of equality of opportunity, liberty, and social and economic mobility in human civilization. Hoover saw that America had led the world in its commitment to individual freedom, which had created unprecedented wealth and prosperity.
American exceptionalism is not a statement of arrogance or a belief that Americans are inherently better than the citizens of other nations. American exceptionalism is not the notion that America is faultless. And it is certainly not the notion that America has achieved the perfect implementation of its ideals. Rather it asserts that the American _system_ is unique in its emphasis on individual liberty, and that this emphasis has produced extraordinary individuals and extraordinary achievements for humanity.
When Barack Obama was asked in his first year in office whether he believed in American exceptionalism, he said he did but in the same way that Greeks believe in Greek exceptionalism, and that the British believe in British exceptionalism. In other words, while President Obama said he personally believed America is exceptional, he was not prepared to argue that it actually _is_ more exceptional than any other nation—a dodge of the question. Republicans were outraged with his answer and saw it as an example of moral equivalency—every nation is equally exceptional, none more than any other.
Republicans couldn't understand how we had elected a president who confuses the idea of American exceptionalism—which argues that the American system is better not because it's American but because it affords individuals greater freedoms and opportunities than any other system—with the base instincts of American jingoism, which is merely a reflex of the fiercely proud. Republicans like me were incensed that President Obama didn't seem to notice that the results of the American system are unmatched by any other system when it comes to securing personal freedoms and opportunity.
His view that all other nations are our equals—no better but no worse—was blasphemy to Republicans and to many Americans.
But while this statement rightly offended the sensibilities of Republicans and conservatives, President Obama's worldview that other countries are just as exceptional as America seems consistent with the values of the millennial generation. Millennials are the most global generation and the least likely to see significant differences between themselves and their international peers. The impact of globalization among millennials has been a leveling experience. The differences between nations simply fall away, and millennials do not distinguish between them. Millennials are unimpressed with the argument that America is different or special or better, because they've been taught that _everyone_ is special and _everyone_ is exceptional.
According to pollster Frank Luntz, this generation really is "the first to reject 'American exceptionalism,' preferring a 'We're all in this together' philosophy." They are the Facebook generation, with "friends" all over the world. They follow people on other continents via Twitter, and they have access to products from all over the globe. Luntz points out that millennials "aren't as interested in the Olympics as their parents were because they don't really like international competition—the whole country-versus-country thing—US versus Russia, US versus China.... They can't relate to superpower competition on any field—of battle or otherwise." On a personal level they don't think they are better than their peers in other countries, so unless they have had an opportunity to travel, they don't think the American system is better than any other system of government.
Why should they? We haven't made the case for American exceptionalism to them. And because of this, they think American exceptionalism stands for the same arrogant beliefs that President Obama thinks it does—that America is always right, that America should go it alone, and that America doesn't need allies.
I've taken a different view of American exceptionalism than most people my age and younger, thanks as much to my own practical experiences as to any political philosophy. Partly inspired by my great-grandfather, partly by my own itch for independence, and partly because my mother was a flight attendant and I benefited from free travel vouchers, I started my adult life the same way my great-grandfather did: I traveled. Between the ages of eighteen and twenty-five (at which point my United Airlines travel passes expired), I dedicated myself to collecting passport stamps and travel visas, and even had extra pages sewn in to extend the life of my passport. I packed my university years with study-abroad experiences, with extended stays in China, Bolivia, and Mexico. When I graduated from college, I moved to Taiwan, where I landed my first job as an editor and research assistant at a Taiwanese law firm.
I wanted to see and experience an array of cultures, political systems, landscapes, and people. I loved languages and dedicated years of study to Spanish and Mandarin Chinese.
I was conscious of the fact that, in traveling the world at the dawn of a new American century, I was, in my own modest way, following in the footsteps of my great-grandfather and great-grandmother.
My experiences afforded me the ability to compare other political and economic systems with America's and reaffirmed for me that my great-grandfather's observations about American individualism were still true.
In Bolivia, the poorest of South American countries, I saw how impostors throughout their country's history have claimed to offer democracy and more prosperity but have left the country worse off. Bolivia's natural resources have been exploited to enrich the ruling elite and foreign allies, leaving the indigenous people only slightly less impoverished than at the time of the Spanish conquest more than five hundred years ago.
Even the middle class in Bolivia, a tiny share of the population, has limited economic mobility. I lived for seven months in the city of Cochabamba with a middle-class family whose life was quite comfortable compared to that of their fellow countrymen, yet which still presented little opportunity for an improved economic position. I spent one week visiting the countryside and lived alongside the most crushing poverty of the not-yet-developing world. I herded sheep and experienced _campesino_ life with a generous Aymaran family, who, after observing that their daughter and I had formed a friendship, decided that she ought to return to America with me so that she might enjoy a life of enhanced economic possibility.
In China, I saw how even the most creative and ambitious young adults avoided saying anything critical of their government or political system, aware of the watchful and stern authority that silently loomed over them. Yet their cousins in Taiwan, the thriving Chinese democracy on the other side of the Taiwan Strait, could be politically active while enjoying the economic fruits of that island marvel. All of that was possible because of the military protection and diplomatic guarantees of the United States.
I also saw how women around the world edited themselves—not just their words, but their entire personalities—so as not to offend the timeworn sensibilities of their fathers, their brothers, or their husbands. I wondered to myself what these women would be like if they lived in America. Not just how they would dress, or even talk, but how they would think. What would their lives be like? The differences between us could be boiled down to my exceptional luck in having been born under a system of government that has afforded me more freedom and opportunity than most young women around the world will ever know. This is my experience with American exceptionalism.
Most of all, I felt lucky to have been born in a country that afforded me the luxury of this ability to experience and observe, that allowed me to follow my curiosity and inclination to adventure and to test the knowledge I'd acquired from books and in classrooms. I had seen how the American system is exceptional, why it is worth fighting to preserve, and why every generation—the millennials especially—must overcome their doubts about the greatness of the American system. It is only by good fortune that we are born into a country that values individual freedom, social and economic mobility, the rule of law, and human rights. And it is too easy to take these gifts for granted.
Another reason I suspect that millennials are skeptical about American exceptionalism is that in their short lifetimes, America's foreign policy efforts have been defined by significant failures. While the millennial generation was indelibly affected by the attacks of 9/11, their attitudes toward America's assertion of power abroad were shaped by America's blunders in Iraq. And that is clear from the way millennials have voted. The first millennials to vote in a presidential election in 2000 split their votes evenly between George W. Bush and Al Gore. But their decisive break from the Republican Party occurred in the 2004 presidential election, when the defining issue was whether America should withdraw its forces from Iraq. They voted 54 percent to 45 percent for John Kerry, the Democratic nominee who had originally voted in favor of the war but who later came out against it, calling it "the wrong war, in the wrong place, at the wrong time."
The discontent with Iraq spread further in 2006, when Democratic candidates won control of both the House and the Senate, thanks in part to strong millennial turnout. And of course in 2008, millennials voted two to one for Obama, who had opposed the surge in Iraq, over McCain, who had been the first major politician to advocate in favor of it.
Unlike baby boomers during the Vietnam War, millennials do not face the risk of the military draft. There was no self-interest involved in opposing the war, and discontent with the war did not take the form of demonstrations on college campuses. Unlike the Vietnam-era campus protesters that idealized Ho Chi Minh, millennials were in no way sympathetic to Saddam Hussein, Al Qaeda in Iraq, or the multiple Islamist supremacist groups vying to replace Saddam Hussein after his ouster and execution. And millennials were not hostile to our troops, who have enjoyed significant support among all population groups, despite the unpopularity of the Iraq War.
Yet millennials fiercely opposed the war anyway—not in the run-up to the war, but after things started to go badly. They were susceptible to the constant accusations from the Left that Americans had been lied to about weapons of mass destruction. President Bush and the leaders of several nations had argued that, because Saddam Hussein had developed WMD programs in the past and had used them against the Iraqi Kurds in the late 1980s, in a post-9/11 world America could no longer risk allowing a maniacal dictator to pass WMDs on to terrorist groups hostile to the United States.
This was a powerful argument for war, but a faulty one, it turned out, because Saddam didn't have WMDs. Even worse, in the days, weeks, and months after the American military victory, Iraq descended into chaos. From the millennials' point of view, Iraq was a war fought on a false premise—WMDs—and without a clear plan for ensuring lasting victory. A 2006 survey found that a significant majority—60 percent—of millennials surveyed believed that the Iraq War was a "mistake," while nearly a third felt the United States should withdraw from Iraq immediately. They came to view Iraq as an irresponsible and incompetent adventure. This generation is not antiwar, but they oppose war waged without a clear and definable purpose. As a result, the millennial generation soured on Republican leadership in foreign policy.
Against this backdrop, Republicans have a difficult task ahead as they try to restore their credibility with the millennial generation on foreign policy. Nonetheless, I think it can be done—more important, it _must_ be done. Here are five areas Republicans should emphasize in order to win millennials back and set a confident foreign policy agenda for the next generation:
## **The American System Is Exceptional**
America is a force for good in the world, and the American system is worth fighting to defend. When we speak of American exceptionalism, we must be absolutely clear that this isn't a statement of superiority or a belief that Americans are better than other people. There are exceptional _individuals_ in every country, but the American system, by protecting freedom and facilitating social and economic mobility, has allowed its people to reach their fullest potential in greater numbers than is possible in any other system. Along the way it has introduced unprecedented levels of freedom and wealth creation and has increased standards of living for great majorities of people—not just in the United States, but around the world.
I think it's up to us as Republicans to demonstrate to millennials that the idea of American exceptionalism stands for more than waving the flag. We have to make the case that America champions the freedoms and liberties and privileges millennials enjoy. It is the benefits of American exceptionalism that their peers around the world—including those seeking freedom today in the Middle East—desire and deserve.
## **Maintain a Strong Military**
America spends more than any other nation on its military for good reason. The United States must continue to be prepared to defend its freedoms with the strongest military in the world.
We need to remind millennials that as the world's leading power we cannot rely on other nations to defend our freedoms. Many nations of Europe have been able to prune their militaries, in some cases to the bare branches, because for more than fifty years they have been able to count on America to defend their freedom. But who will come to the aid of the United States if it's unable to beat back its enemies? We must be careful not to indulge the impulse to cut military budgets just because they are large—our military budget must always be sizable, since no other nation will fight our wars and few countries will fight for the freedom of others, as America does. The system of American exceptionalism also calls for American sacrifice—military service, a strong defense, an active effort to defend our borders and our friends.
This sacrifice and service to the world has led to enormous shared prosperity. Just look at how the American Navy's securing of the seas has brought about decades of secure trade and wealth creation for the international community. It is in the interest of world prosperity that America remain a global military power. Other nations may not be willing to acknowledge it, or even thank us for it, but there are graveyards all over Europe and Asia that attest to the sacrifices we have made fighting for the freedom and security of others. In return for our sacrifices, as Colin Powell so eloquently reminded the world, all we have ever asked is for enough land to bury our dead.
## **Support Our Friends**
America should be committed to a global strategy that focuses on supporting those countries that share our values. Rather than reach out to hostile and rogue nations such as Iran and North Korea, or aggressive nations like Russia, we must be good to our allies so they do not think America's support is only temporary. In Eastern Europe, we must keep our promise to build missile defenses to protect our friends against the threat of aggression (and not abandon these plans, as President Obama appears willing to do).
Republicans must keep in mind not only our recent struggles but also our recent successes. If not for the Pax Americana, much of the world would be living in the shadow of one form of tyranny or another. Republicans should remind millennials that America has been Israel's strongest friend in its fight for security and peace, and that we continue to provide support to South Korea and the people of Taiwan. We must recall that our friends in Colombia were once prisoners in their own nation, as drug cartels waged a vicious war of control over large swaths of that beautiful country. Thanks to American assistance, Colombia's president at the time, Álvaro Uribe, smashed the cartels and placed the future of Colombia in the hands of the Colombian people.
We should remember that one of the things that makes America exceptional is American friendship—no nation has given so much of itself for its friends, or even for those nations that have rarely supported American interests. We should be proud of that. Not just because it will resonate with millennials. But because it's true.
## **A Confident Foreign Policy Is Also a Generous Foreign Policy**
At the time Herbert Hoover wrote "American Individualism," he was overseeing America's first foray into foreign aid by administering the largest humanitarian food relief effort ever undertaken, to famine-ridden Soviet Russia. The American Relief Administration delivered food and medicine to tens of millions of Soviet citizens and saved millions of lives. America became the benevolent world leader, and in the spirit of the millennial generation's impulse to service, we should remind millennials how America helps the world not only by protecting our friends, but by serving needy and deserving people (even when we don't support their governments, as in the case of Hoover's humanitarian relief to the Soviets). American aid ships to Haiti, and the relief efforts after Asian tsunamis, our efforts to distribute mosquito netting to stop malaria in Africa, and our spearheading of treatments for tuberculosis worldwide—millennials see these endeavors, and they see that America alone does them on a scale that can make the difference. This is American exceptionalism in action.
Although George W. Bush gets low marks from millennials, Republicans should remind them that he led America's effort to save millions of lives in Africa through efforts to fund the distribution of HIV/AIDS medications. "[Bush] has actually done more than any American president for Africa," said British music producer Bob Geldof, one of the most outspoken advocates for aid to Africa. U2's lead singer, Bono, said much the same thing: "250,000 Africans are on anti-viral drugs; they literally owe their lives to America."
## **Free Trade Leads to Free People**
This applies to America's approach to global trade, as well as to the free flow of ideas. Millennials are globally minded, so Republicans should point out that as emerging markets rise to compete for economic dominance, America will welcome them into the global economic marketplace, and will welcome their dynamism, their competition, their collaboration, and their innovations. We can argue that free trade and open markets do more to alleviate poverty than any other public policy. We have to make the connection to millennials—who can be skeptical of free trade because of concerns that foreign workers end up being exploited. They need to see that when we trade with a nation, we are giving that nation a chance at prosperity, while we ourselves benefit from a more globalized marketplace.
Republicans must also argue that American exceptionalism isn't just what we do around the world, but what we do at home as well. We are still a country of extraordinary opportunity. A child born under Soviet Communism can move here with his parents as a young boy and go on to found a company like Google. There is a reason this happens in America and only rarely anywhere else. It is because America does not deny legal newcomers the chance to become productive citizens. In many European countries, you can live your entire life there and yet still be considered a foreigner, simply because your parents came from another place. In America, everyone can belong.
I believe that a generation raised to think globally and act locally will understand that so long as America is free and confident, freedom and prosperity will have a chance in the world. We are, as Madeleine Albright once said, "the world's indispensable nation." I believe that millennials lost faith in the idea of American indispensability not because they lost faith in America, but because they have lost sight of the vital importance of American leadership in the world. But they do want America to be equal to its promise to give every individual an opportunity to dream, to succeed, and to be exceptional. And I think that if they are willing to fight for that, they will find many friends in the Republican Party.
# **ACKNOWLEDGMENTS**
If American individualism is a blend of rugged individualism and a community spirit, then the effort to create this book has epitomized the spirit of its title: to be precise, a rather stubborn individual blessed with a supportive community of friends and colleagues who helped make this book possible.
My editor, Jenna Ciongoli, has been a steady champion from beginning to end—not every author is fortunate enough to have an editor who believes in her book from the moment it begins to take shape. I'd also like to thank the marketing and publicity team at Crown, especially Jennifer Robbins and Sarah Breivogel—and my agent, Ian Kleinert, for discovering this opportunity and helping me out at every step along the way.
In the realm of polishing and editing, I am grateful to Noam Neusner, my former OMB colleague and late-stage collaborator on this project, who helped ensure that I put the right words on the page. I am grateful as well to Bob Asahina for his insightful perspective on how to structure the book. And I am especially grateful to Bert Patenaude, the pristine polisher of words, whose keen eye and industrious pen swooped in at the eleventh hour.
For each of the policy sections of the book, I checked my prescriptions with friends who are experts in their respective fields, among them Stuart Gottlieb, Jen Pollom, Irshad Manji (whose moral courage inspires), and Ayann Hirsi Ali (whose friendship I treasure). Spencer Howard and Tim Walch at the Herbert Hoover Presidential Library and Museum were invaluable in ensuring that my Hoover facts are correct.
Among those of us dedicated to restoring the good reputation of Herbert Hoover, George H. Nash stands out as the preeminent expert on both Hoover and the history of America's modern conservative movement. He is a kindred spirit in his devotion to Herbert Hoover and his dedication is deeply appreciated by the Hoover family. Likewise, film producer Austin Hoyt and historian Bert Patenaude (again) deserve recognition for drawing deeply from the archives at the Hoover Institution for their recent documentary film for PBS, _The Great Famine_ , which recounts the epic story of Hoover's rescue operation in Bolshevik Russia to vanquish the Great Famine.
Ashley Koning will one day make a fine professor, but in the meantime I'm indebted to her for sacrificing her spring break from her doctoral studies in order to fact-check my work. Her prodigious talents as a researcher and her fastidious attention to detail put her in a league of her own. I also owe a debt of gratitude to Sam Abrams of Sarah Lawrence College for finding me Katelyn Bornholdt and Emily MacDonald, whose excellent research helped me immeasurably and who added their refreshing millennial perspectives to the data they supplied. Finally, many thanks to Aileen Hogan for her late-inning notes support.
I would be remiss not to thank David Eisenhower, whom I admire and respect and who first gave me the idea for this book more than ten years ago when I took his class "American Presidency and Communications" at the University of Pennsylvania.
Family and friends make life shine. And so I owe thanks first to my husband, John Avlon—Fipp and LOML—whose devotion and support are manifest throughout these pages and to whom this book is dedicated.
My parents have given me all their love all of my life—and I'm especially happy that my father has enjoyed reading the sections on his grandfather. I'm grateful that my grandmother, for whom I'm named, knew about this book. Her spirit suffuses these pages. My beloved brother and sister-in-law tolerated my absence and stress over the past year, as did my gracious in-laws, John and Dianne Avlon. Lots of love also to my Aunt Lou, whom I will now be free to visit more frequently in her new home in Los Angeles, and to my cousins: Allan and Michelle, Debbie and Jonathan, and Jimmy. And a special hug for my uncle Allan.
I am blessed with an abundance of friends who have supported me this year and steered clear during my intensive writing spells. I can't wait to reconnect with all of you—Lisa and Eric, Jenny and Scott, Libby and Chris, Judy and Bill Casey, Erika and Cameron, Natalie and Mike, Heide and Max, Muffy Lewis, Ivette Fernandez, Melissa Danforth, Christian Zaal, Maury Donahue, Jordan Salcito, Olga Arguello, Sara Dawes, Michael Ahrens, Mike Rixon, BJ Goergan, and my newfound cousin Matthew Mesher.
Likewise, Jen Heller, Lily Appelman, and Penny Rice are lifeline friends who have never flinched, even when this book forced me to shirk baby shower duties!
On a personal and professional level, I want to thank Bill O'Reilly for making me both a Culture Warrior and, to use his word, "a star"; Gretchen Carlson for her friendship; and all my friends at Fox News Channel for their support.
I am also indebted to the wonderful team of policy warriors intimately known as the Blob—Paul Singer, Annie Dickerson, Dan Senor, Michele Packman, Marge Govan, and Terry Kassel—who demonstrate the truth of Margaret Mead's aphorism that a small group of thoughtful, committed people can change the world.
Finally, for their ongoing effort to ensure that the freedom to marry is available to every American, I want to thank Chad Griffin and Adam Umhoffer at AFER, and special thanks to Ken Mehlman for his leadership in this cause. For their courage in breaking down barriers within the conservative movement, I'm grateful to Jimmy LaSalvia and Chris Barron of GOProud.
For more information on Herbert Hoover, please visit the following websites: Herbert Hoover Presidential Library and Museum, www.hoover.nara.gov; the Herbert Hoover Presidential Library Association, www.hooverassociation.org; the Hoover Institution on War, Revolution, and Peace, www.hoover.org.
# **NOTES**
## **Introduction**
_been disappointed:_ "About half of millennials say the president has failed to change the way Washington works, which had been the central promise of his candidacy." <http://pewresearch.org/pubs/1501/millennials-new-survey-generational-personality-upbeat-open-new-ideas-technology-bound>.
_idealistic expectations:_ Pew Research Center for People and the Press (Washington, D.C.), "A Pro-Government, Socially Liberal Generation: Democrats' Edge Among Millennials Slips," 6.
_favorable attitudes:_ Pew poll. Ibid., 24.
_saving more lives:_ George H. Nash, _Reappraising the Right: The Past and Future of American Conservatism_ (Wilmington, Del.: Intercollegiate Studies Institute, 2009), 309.
_"contains the New and Old":_ F. J. Turner to Richard Emmet [Hoover's secretary], January 18, 1923, Commerce Papers, Herbert Hoover Presidential Library, West Branch, Iowa.
## **1. Growing Up Hoover**
_"He certainly has had his hand":_ Lou Henry Hoover to Allan Hoover, September 26, 1931, Hoover Family Papers.
_He was the second son:_ George H. Nash, _The Life of Herbert Hoover: The Engineer, 1874–1914_ (New York: W. W. Norton, 1983), 7.
_He graduated to:_ Herbert Hoover, _The Memoirs of Herbert Hoover: Years of Adventure, 1874–1920_ (New York: Macmillan, 1951), 26–28.
_set sail for China:_ Ibid., 35–36.
_German mail boat:_ Nash, _The Life of Herbert Hoover_ , 117–24.
_By the time he was:_ "The Highest Salaried Man of His Age in the World," _San Francisco Chronicle_ , December 8, 1901.
_Later, at the height:_ Nash, _The Life of Herbert Hoover_ , 568.
_"Twenty million are starving":_ David Burner, _Herbert Hoover: A Public Life_ (New York: Alfred A. Knopf, 1979), 131.
_According to Hoover's biographer:_ George H. Nash, _Reappraising the Right_ (Wilmington, Del.: Intercollegiate Studies Institute, 2009), 251.
_"let the fortune":_ Will Irwin, _Herbert Hoover: A Reminiscent Biography_ (New York: Grosset & Dunlap, 1928), 135.
_"from its records":_ Herbert Hoover's 1959 statement to the Board of Trustees of Stanford University, <http://www.hoover.org/about/mission-statement>.
_encouraged to run:_ Franklin D. Roosevelt, to Hugh Gibson, January 2, 1920, Pre-Commerce Papers, Herbert Hoover Presidential Library, West Branch, Iowa.
_"Secretary of Commerce and":_ This phrase was commonly applied to Hoover in the 1920s, attributed in particular to S. Parker Gilbert, a U.S. agent general for reparations to Germany during the Coolidge administration and previously an undersecretary of the treasury. The attribution was made, without any source cited, by Oswald Garrison Villard, editor of the _Nation_ at the time, in _Prophets True and False_ (New York: Alfred A. Knopf, 1928), 24.
_"every man for himself":_ Herbert Hoover, "American Individualism and Challenge to Liberty" (West Branch, Iowa: Herbert Hoover Presidential Library Association, 1989), 35.
_he articulated his philosophy:_ Ibid., 11–12.
_"few great formulations": New York Times Book Review_ , December 17, 1922, 1.
_"it contains the New":_ F. J. Turner to Richard Emmet [Hoover's secretary], January 18, 1923, Commerce Papers, Herbert Hoover Presidential Library, West Branch, Iowa.
_low-interest loans:_ Herbert Hoover, _The Memoirs of Herbert Hoover: The Cabinet and the Presidency_ (New York: MacMillan, 1952), 126.
_"those were the days":_ Ibid., 126.
_"Rugged Individualism":_ Herbert Hoover, _The New Day: Campaign Speeches of Herbert Hoover, 1928_ (Stanford, Calif.: Stanford University Press), 230.
_He proposed to Congress a $160 million tax cut:_ "Annual Message to Congress on the State of the Union, December 3, 1929," _Public Papers of the Presidents: Herbert Hoover, 1929_ (Washington, D.C.: U.S. Government Printing Office, 1974), 404–36.
_"No one in his place": New York Times_ , March 2, 1930.
_conditions in the Midwest:_ David Burner, _Herbert Hoover: A Public Life_ (New York: Alfred A. Knopf, 1970), 131.
_4.2 percent of the:_ Milton Friedman and Anna Jacobson Schwartz, _A Monetary History of the United States, 1867–1960_ (Princeton, N. J.: Princeton University Press, 1963), 342.
_"raw ambition for power":_ Thomas Fleming, "Channeling George Washington: Presidents Criticizing Presidents," George Mason University's History News Network, March 1, 2010, <http://www.hnn.us/articles/123859.html>.
_"unguarded, discredited, unloved":_ William F. Buckley Jr., _Happy Days Were Here Again: Reflections of a Libertarian Journalist_ (New York: Random House, 1993), 396.
_an injustice so petty:_ Joseph E. Stevens, _Hoover Dam: An American Adventure_ (Norman, Okla.: University of Oklahoma Press, 1988), 173–75.
_"of modern conservative thought":_ Richard Norton Smith, in preface to Herbert Hoover, "American Individualism and Challenge to Liberty" (West Branch, Iowa: Herbert Hoover Presidential Library Association, 1989), vii.
_"cripple or abandon":_ Herbert Hoover, "American Individualism and the Challenge to Liberty," 65.
_Hoover cited several areas:_ Ibid., 124.
_"Hoover depression economics":_ Joseph Biden, CBS interview by Katie Couric, September 22, 2008; reprinted in _Wall Street Journal_ , "Rehabilitating Maligned Hoover," November 4, 2008, <http://online.wsj.com/article/SB122512986151172687.html>.
_"ignorance was bliss":_ Ibid.
_"tough economic times":_ Ibid.
_"Obama's Hoovervilles":_ Matt Bai, "Hooverville Attacks on Obama Offers Previews of Campaign," _New York Times_ , February 12, 2011, A13.
_"Barack 'Hoover' Obama":_ Rush Limbaugh, "Screw this Hussein business. It's Barack 'Hoover' Obama," August 12, 2010, [www.rushlimbaugh.com/home/daily/site_081210/content
/01125106.member.html](http://www.rushlimbaugh.com/home/daily/site_081210/content/01125106.member.html).
_"what was widely believed":_ Thomas Sowell, _Intellectuals and Society_ (New York: Basic Books, 2009), 71.
_"It's a myth":_ Burton Folsom Jr. and Anita Folsom, "Did FDR End the Depression?" _Wall Street Journal_ , April 12, 2010, A17.
## **2. Conservative Tribalism**
_"The term conservatism":_ Milton Friedman, _Capitalism and Freedom: Fortieth Anniversary Edition_ (Chicago: University of Chicago Press, 1962), 6.
_homosexuality should be accepted:_ Pew Research Center, "Religion Among the Millennials: Less Religiously Active Than Older Americans, but Fairly Traditional in Other Ways," February 2010, 18.
_As a generation:_ Cathy Lynn Grossman, "Young Adults 'Less Religious,' Not Necessarily 'More Secular,' " _USA Today_ , February 17, 2010, <http://www.usatoday.com/news/religion/2010-02–17-pewyouth17_ST_N.htm>; Cathy Lynn Grossman, "Survey: 72% of Millennials 'More Spiritual Than Religious,' " _USA Today_ , October 14, 2010, <http://www.usatoday.com/news/religion/2010–04–27–1Amillfaith27_ST_N.htm>; Pew Research Center Forum on Religion and Public Life, "Religion Among the Millennials: Less Religiously Active Than Older Americans, but Fairly Traditional in Other Ways," February 2010, [http://pewforum.org/uploadedFiles/Topics/Demographics
/Age/millennials-report.pdf](http://pewforum.org/uploadedFiles/Topics/Demographics/Age/millennials-report.pdf); "Millennials—A Portrait of Generation Next: Confident. Connected. Open to Change," Pew Research Center, February 2010, <http://pewsocialtrends.org/files/2010/10/millennials-confident-connected-open-to-change.pdf>. See also "Reader's Digest Poll: Millennials Are First Generation with Unique Political Identity; Will Match Seniors in Turnout; Obama Beats McCain by Whopping 22 Points Among 18–29 Year-Olds," _PR Newswire_ , May 6, 2008, <http://www.prnewswire.com/news-releases/young-voters-to-have-unprecedented-impact-on-presidential-election-57162512.html>.
_A disclaimer first:_ Lee Edwards, "The Conservative Consensus: Frank Meyer, Barry Goldwater, and the Politics of Fusionism," _The Heritage Foundation: First Principles Series_ , no. 8, January 22, 2007, 1.
_"Perhaps the most important thing":_ George H. Nash, _Reappraising the Right_ (Wilmington, Del.: Intercollegiate Studies Institute, 2009), 359.
_Buckley's mission statement:_ William F. Buckley Jr., "Our Mission Statement," _National Review_ , November 19, 1955, <http://article.nationalreview.com/346187/our-mission-statement/william-f-buckley-jr>.
_economic libertarians:_ Nash, _Reappraising the Right_ , 321.
_They were not even:_ Friedrich A. Hayek, "Why I Am Not a Conservative," _The Constitution of Liberty_ (Chicago: University of Chicago Press, 1960), 397. Hayek even dedicated an entire essay to the subject of why he refused to accept the label "conservative" for his political and economic philosophy.
_"free minds and free markets":_ "About Reason," _Reason_ , 2011, <http://reason.com/about>.
_They advocated "a revival":_ Nash, _Reappraising the Right_ , 320–21.
_"school prayer":_ Ibid., 325–26.
_"conservagenzia":_ Matthew Dowd, interviewed by author, March 9, 2011; and subsequent e-mail exchanges with author.
_"defiantly nationalist":_ Ibid., 330.
_rethink their opposition:_ Norman Podhoretz, "Neo-Conservatism: A Eulogy," _Commentary Magazine_ , March 1996.
_deserved their support:_ Joyce Huyett Turner and Sam Tanenhaus, "Q & A on William F. Buckley," _NYTimes.com_ , February 27, 2008. <http://artsbeat.blogs.nytimes.com/2008/02/27/qa-with-sam-tanenhaus-on-william-f-buckley/>.
_Crunchy Cons:_ Rod Dreher, _Crunchy Cons: The New Conservative Counterculture and Its Return Roots_ (New York: Three Rivers Press, 2006).
_fusionism:_ Nash, _Reappraising the Right_ , 322.
## **3. Meet the Millennials**
_"Every generation discovers":_ "Concerning Honor in Public Life," a speech by Herbert Hoover broadcast nationwide from the Iowa Centennial Celebration, Des Moines, Iowa, August 30, 1951. See Herbert Hoover, _Addresses on the American Road: 1950–1955_ (Stanford, Calif.: Stanford University Press, 1955), beginning on p. 111.
_the rising generation:_ Pew Research Center, "Millennials—A Portrait of Generation Next: Confident. Connected. Open to Change," Forum on Religion and Public Life, February 2010, 4, <http://pewsocialtrends.org/files/2010/10/millennials-confident-connected-open-to-change.pdf>.
_Sixty-six percent of:_ Pew Research Center, "Young Voters in the 2008 Election," by Scott Keeter, Juliana Horowitz, and Alec Tyson, November 12, 2008, 19, <http://pewresearch.org/pubs/1031/young-voters-in-the-2008-election>.
_In the presidential elections:_ Patrick Fisher, "The Age Gap: Evidence of a Realignment in U.S. Politics," paper presented at the annual meeting of the Western Political Science Association, San Diego, California, March 20, 2008, [http://www.allacademic.com//meta/
p_mla_apa_research_citation/2/3/8/5/1/pages238512/p238512–31.php](http://www.allacademic.com//meta/p_mla_apa_research_citation/2/3/8/5/1/pages238512/p238512–31.php).
_Between the elections of 2008 and 2010:_ Kirk Johnson, "Fewer Voters See Themselves as Democrats," _New York Times_ , September 2, 2010, [http://www.nytimes.com/2010/09/03/us/politics
/03students.html](http://www.nytimes.com/2010/09/03/us/politics/03students.html).
_Forty percent of them are nonwhite:_ Pew Research Center, "Millennials—A Portrait of Generation Next," 9.
_20 percent... at least one immigrant parent:_ Morley Winograd and Michael D. Hais, "The Boomers Had Their Day. Make Way for the Millennials," _Washington Post_ , February 3, 2008, <http://www.washingtonpost.com/wp-dyn/content/article/2008/02/01/AR2008020102826.html>.
_And 93 percent are comfortable:_ Pew Research Center, "Millennials—A Portrait of Generation Next," 78.
_most educated generation:_ Pew Research Center, "Millennials—A Portrait of Generation Next," 1.
_juvenile crime, teen pregnancy:_ Morley Winograd and Michael D. Hais, _Millennial Makeover: MySpace, YouTube, and the Future of American Politics_ (Piscataway, N.J.: Rutgers University Press, 2008), 81.
_They are less conventionally:_ Pew Research Center Forum on Religion and Public Life, "Religion Among the Millennials: Less Religiously Active Than Older Americans, but Fairly Traditional in Other Ways," February 2010, [http://pewforum.org/uploadedFiles/Topics/Demographics
/Age/millennials-report.pdf](http://pewforum.org/uploadedFiles/Topics/Demographics/Age/millennials-report.pdf).
_According to a_ Reader's Digest _poll:_ Carl M. Cannon, "The Facebook Election," _Reader's Digest_ , June 2008.
_created a profile:_ Pew Research Center, "Millennials—A Portrait of Generation Next," 1.
_video of themselves:_ Ibid., 30.
_Ninety-four percent of millennials:_ Ibid., 32–33.
_more than nine thousand books:_ Winograd and Hais, _Millennial Makeover_ , 79.
_Employers report that:_ Morley Safer, "The 'Millennials' Are Coming," _60 Minutes_ , CBS, May 23, 2008, [http://www.cbsnews.com/stories/2007/11/08/
60minutes/main3475200.shtml](http://www.cbsnews.com/stories/2007/11/08/60minutes/main3475200.shtml).
_83 percent... volunteered regularly in high school:_ Lynne C. Lancaster and David Stillman, _The M-Factor: How the Millennial Generation Is Rocking the Workplace_ (New York: HarperCollins, 2010), 96.
_Sixty percent have volunteered:_ Pew Research Center, "Millennials—A Portrait of Generation Next," 83.
_Seasoned pollsters:_ Pew Research Center, "A Pro-Government, Socially Liberal Generation: Democrats' Edge Among Millennials Slips," 6, <http://pewresearch.org/assets/pdf/1497.pdf>, "But, compared with older cohorts, Gen Xers have remained less opposed to active government for more than a decade, suggesting that these attitudes, once formed, tend to persist at least in comparison with other age cohorts."
_"as many raw votes":_ Cannon, "The Facebook Election."
_Millennials cannot abide hyperpartisanship:_ Eric Greenberg and Karl Weber, "The Millennials: America's First Post-Ideological, Post-Partisan, and Post-Political Generation," _The Huffington Post_ , September 14, 2008, <http://www.huffingtonpost.com/eric-greenberg-and-karl-weber/the-millennials-americas_b_126205.html>.
_currently married:_ Pew Research Center, "Millennials—A Portrait of Generation Next," 11.
_About a third (34 percent) are parents:_ Ibid., 17.
_higher share of unwed mothers:_ Ibid., 51.
_devoted to the families:_ Winograd and Hais, _Millennial Makeover_ , 83.
_The oldest of them:_ Pew Research Center, "Millennials—A Portrait of Generation Next," 140.
_45 percent talk on the phone:_ Ibid., 83.
_What's more, 52 percent of millennials:_ Ibid., 2.
_accepted by society:_ Pew Research Center, "Religion Among the Millennials: Less Religiously Active Than Older Americans, but Fairly Traditional in Other Ways," February 2010, 18.
_legalization of same-sex marriage:_ Pew Research Center, "Millennials—A Portrait of Generation Next," 51.
_Nearly 40 percent of them have tattoos:_ Pew Research Center, "Millennials—A Portrait of Generation Next," 57.
_"They've been down":_ quote from Mary Crane, in report by Morley Safer, "The 'Millennials' Are Coming," _60 Minutes_ , CBS, May 23, 2008, [http://www.cbsnews.com/stories/2007/11/08/60minutes/
main3475200.shtml](http://www.cbsnews.com/stories/2007/11/08/60minutes/main3475200.shtml).
_37 percent of people aged:_ Pew Research Center, "Millennials—A Portrait of Generation Next," 44.
_In 2009, only 20 percent:_ Lynne C. Lancaster and David Stillman, _The M-Factor: How the Millennial Generation Is Rocking the Workplace_ (New York: HarperCollins, 2010), 53.
_during the 2010 cycle:_ Carl M. Cannon, "The Facebook Election," _Reader's Digest_ , June 2008.
_On a range of issues:_ Pew Research Center, "Millennials—A Portrait of Generation Next," 63.
_businesses take fair profits:_ Ibid., 74.
_While they tend to favor... affirmative action:_ Ibid., 77.
_While they still favored Democrats:_ Kevin Brennan, "Millennials Missing from the Midterms," _Politics Daily_ , November 3, 2010, <http://www.politicsdaily.com/2010/11/03/millennials-missing-from-the-midterms/>.
_turnout rate dropping:_ Ibid.
_regulation of business:_ Pew Research Center, "Millennials—A Portrait of Generation Next," 70.
_Far fewer eighteen- to twenty-nine-year-olds:_ Kirk Johnson, "Fewer Voters See Themselves as Democrats."
_Republicans would do well:_ David Cameron, keynote speech, Conservative Party Conference, Birmingham, England, October 6, 2010.
_"the spirit of activism":_ Ibid.
## **4. Generational Theft**
_"Blessed are the young":_ "New Deal Agricultural Policies and Some Reforms," a speech by Herbert Hoover sponsored by the Nebraska Republican State Central Committee, Lincoln, Nebraska, January 16, 1936. See Herbert Hoover, _Addresses on the American Road: 1933–1936_ (New York: Scribners, 1936), beginning on p. 101.
_WeCantPayThatTab.org:_ "National Debt: The Quick Facts," _We Can't Pay That Tab_ , accessed March 15, 2011, <http://www.wecantpaythattab.org/debt-posts/quick-facts/>.
_Within his first hundred days:_ John Ensign, "100 Days of Bailouts and Making a Bad Situation Worse," Washington, D.C.: U.S. Senate Republican Policy Committee, April 29, 2009, [http://rpc.senate.gov/public/_files/
042909100DaysofBailoutsandMakingaBadSituationWorse.pdf](http://www.rpc.senate.gov/public/_files/042909100DaysofBailoutsandMakingaBadSituationWorse.pdf).
_His spending spree:_ Mike Allen, "Congress at Work: '$1 Billion an Hour,' " _Politico_ , March 11, 2009, <http://www.politico.com/news/stories/0309/19884.html>.
_We've seen students rioting:_ ABC/AFP, "Riots in London After University Fee Vote," ABC News, December 10, 2010, <http://www.abc.net.au/news/stories/2010/12/10/3089663.htm>.
_In France:_ John Lichfield, "France Braces for Riots as Protests Turn Violent," _The Independent_ , October 19, 2010, [http://www.independent.co.uk/news/world/europe/
france-braces-for-riots-as-protests-turn-violent-2110305.html](http://www.independent.co.uk/news/world/europe/france-braces-for-riots-as-protests-turn-violent-2110305.html).
_In Spain:_ Fiona Govan, "General Strike in Spain to Protest Against Austerity Measures," _Telegraph_ , September 29, 2010, [http://www.telegraph.co.uk/travel/destinations/europe/
spain/8032647/General-strike-in-Spain-to-protest-against-austerity-measures.html](http://www.telegraph.co.uk/travel/destinations/europe/spain/8032647/General-strike-in-Spain-to-protest-against-austerity-measures.html).
_Here are the stakes:_ Paul Ryan, "Roadmap to America's Future: Version 2.0," January 2010, 23, [http://www.roadmap.republicans.budget.house.gov/
UploadedFiles/Roadmap2Final2.pdf](http://www.roadmap.republicans.budget.house.gov/UploadedFiles/Roadmap2Final2.pdf).
_Even today:_ Scott A. Hodge, "Fiscal Facts: Number of Americans Paying Zero Federal Income Tax Grows to 43.4 Million," _Tax Foundation_ , March 30, 2008, <http://www.taxfoundation.org/news/show/1410.html>; Stephen Ohlemacher, "Nearly Half of U.S. Households Escape Fed Income Tax," Associated Press in _Yahoo! Finance_ , April 7, 2010, http://finance.yahoo.com/news/Nearly-half-of-US-households-apf-1105567323.html?x=0&.v=1.
_By the year 2030:_ Richard Wolf, "Social Security Hits First Wave of Boomers," _USA Today_ , October 9, 2007, <http://www.usatoday.com/news/washington/2007-10-08-boomers_N.htm>.
_While President Obama praised:_ Associated Press, "Obama Touts New 'Paygo' Budget Rules," CBSNews.com, February 13, 2010, [http://www.cbsnews.com/stories/2010/02/13/politics/
main6204926.shtml](http://www.cbsnews.com/stories/2010/02/13/politics/main6204926.shtml).
_In the 110th and 111th:_ "Pelosi's PAYGO Ploy: Budgetary Gimmick Provides Cover for Liberals," The Foundry, _The Heritage Foundation_ , October 15, 2010, <http://blog.heritage.org/2010/10/15/pelosis-paygo-ploy-budgetary-gimmick-provides-cover-for-liberals/>.
_Our projected unfunded liability:_ Pamela Villareal, "Social Security and Medicare Projections: 2009," _National Center for Policy Analysis_ , June 11, 2009, <http://www.ncpa.org/pub/ba662>.
_"those under 30 who":_ Andrew Sullivan, "Obama to the Next Generation: Screw You, Suckers," _The Atlantic_ , February 14, 2011, <http://www.theatlantic.com/daily-dish/archive/2011/02/obama-to-the-next-generation-screw-you-suckers/175804/>.
## **5. Freedom Means Freedom for Everyone**
_"It's time America realized":_ Barry Goldwater, quoted in Rebecca Borders and C. C. Dockery, eds., _Beyond the Hill: A Directory of Congress from 1984 to 1993_ (Lanham, MD: University Press of America, Inc., 1995), 104.
_anti-same-sex-marriage initiatives played no:_ Daniel A. Smith, Matthew DeSantis, and Jason Kassel, "Same-Sex Marriage Ballot Measures and the 2004 Presidential Election," _State and Local Government Review_ 38, no. 2 (2006).
_mobilized people_ across _party lines:_ Simon Jackman, "Same-Sex Marriage Ballot Initiatives and Conservative Mobilization in the 2004 Election," presentation at Institute for Research in the Social Sciences, Stanford University, November 9, 2004, http://jackman.stanford.edu/papers/RISSPresentation.pdf.
_analysis of the results in Ohio's:_ Ibid.
_This mythology now holds:_ Matthew Dowd, "When D.C. 'Wisdom' Is Wrong: Be advised not to buy into Washington myths either before or after Election Day," _National Journal_ , September 22, 2010, <http://nationaljournal.com/njmagazine/oo_20100911_3691.php>.
_As many as 39 percent:_ Pew Research Center, "Millennials—A Portrait of Generation Next: Confident. Connected. Open to Change," Forum on Religion and Public Life, February 2010, 102, <http://pewsocialtrends.org/files/2010/10/millennials-confident-connected-open-to-change.pdf>.
_Polling confirms:_ Washington Post–ABC News Poll, February 4–8, 2010 (Horsham, Penn.: TNS), <http://www.washingtonpost.com/wp-srv/politics/polls/postpoll_021010.html>.
_This sea change:_ Gallup, by Lydia Saad, "Americans' Acceptance of Gay Relations Crosses 50% Threshold: Increased acceptance by men driving the change," May 25, 2010, <http://www.gallup.com/poll/135764/americans-acceptance-gay-relations-crosses-threshold.aspx>.
_"Ronald Reagan opposed":_ Matt Lewis, "Are Social Conservatives Losing Clout—or Just the Gay Debate?" _Politics Daily_ , December 10, 2010, <http://www.politicsdaily.com/2010/12/10/are-social-conservatives-losing-clout-or-just-the-gay-debate/>.
_"the conservative movement":_ Lee Edwards, "The Conservative Consensus: Frank Meyer, Barry Goldwater, and the Politics of Fusionism," _The Heritage Foundation: First Principles Series_ , no. 8, January 22, 2007, 2.
_"adopting same-sex marriage": Perry v. Schwarzenegger_ , testimony on page 2803, lines 13–15.
_"This still-revolutionary":_ David Blankenhorn, _The Future of Marriage_ (New York: Encounter Books, 2009), 2.
_The next day:_ Jerry Sanders, "Proud to Testify for Marriage Equality," _The Huffington Post_ , January 22, 2010, <http://www.huffingtonpost.com/jerry-sanders/proud-to-testify-for-marr_b_432890.html>.
_"I hope that everyone":_ Ibid.
## **6. Education Reform: A Civil Rights Win for the Millennial Generation**
_"Education spending will be":_ Milton Friedman, "Our Best Chance for Better Schools," _New York Post_ , February 20, 2002.
_American students now rank:_ fifteenth in reading, nineteenth in math, fourteenth in science, <http://www.nationalmathandscience.org/index.php/staying-competitive/>.
_drop out of high school:_ C. A. Lehr et al., "Essential Tools: Increasing Rates of School Completion" (Minneapolis: National Center on Secondary Education and Transition, 2004), <http://www.ecs.org/html/Document.asp?chouseid=6649>.
_every nine seconds:_ "Every Nine Seconds in America a Student Becomes a Dropout," American Youth Policy Forum, www.aypf.org/publications/EveryNineSeconds.pdf.
_approximately 70 percent:_ Ibid.
_leaving school:_ "High School Dropouts in America," Fact Sheet, Alliance for Excellent Education, updated February 2009, www.all4ed.org/files/GraduationRates_FactSheet.pdf.
_"dropout factories":_ Robert Balfanz and Nettie Legters, "The Graduation Rate Crisis We Know and What Can Be Done About It," Johns Hopkins University Center for Social Organization of Schools, July 12, 2006, http://web.jhu.edu/bin/a/b/Crisis_Commentary.pdf. "Dropout factory" was first defined by Bob Balfanz and Nettie Legters at Johns Hopkins University as a high school in which "the number of seniors is routinely 60% or fewer than the number of freshmen four years earlier."
_from the lowest-income:_ National Center for Education Statistics, U.S. Department of Education, _The Condition of Education 2004_ (Washington, D.C.: U.S. Government Printing Office, 2004), 11.
_"a permanent national":_ "High School Dropout Crisis Threatens U.S. Economic Growth and Competitiveness, Witnesses Tell House Panel," United States House Education and Labor Committee, May 12, 2009, [http://democrats.edworkforce.house.gov/newsroom/2009/
05/high-school-dropout-crisis-thr.shtml](http://www.democrats.edworkforce.house.gov/newsroom/2009/05/high-school-dropout-crisis-thr.shtml).
_"on the order of $103 trillion":_ Karin Zeitvogel, "U.S. Falls to Average in Education Ranking," Agence France-Presse, December 7, 2010.
_We spend between 41 and 50 percent more:_ Institute of Education Sciences, U.S. Department of Education, "Education Expenditures by Country: Contexts of Elementary and Secondary Education," [http://nces.ed.gov/programs/coe/2010/section4/
indicator38.asp](http://www.nces.ed.gov/programs/coe/2010/section4/indicator38.asp).
_the power to shut down:_ June Kronholz, "D.C.'s Braveheart," _Education Next_ 10, no. 1 (Winter 2010), <http://educationnext.org/d-c-s-braveheart/>. Two years after Rhee took over as D.C.'s education chancellor, "scores on district-administered tests are up: 49 percent of elementary school students were reading at grade level, a 21-percentage-point jump in two years, according to test results released in July 2009. Among secondary-school students, 40 percent were at grade level in math, up 13 points" (36).
_more than $1 million in donations:_ Ben Smith, "Teachers union helped unseat Fenty," _Politico_ , September 15, 2010, [http://www.politico.com/blogs/bensmith/0910/
Teachers_union_helped_unseat_Fenty.html](http://www.politico.com/blogs/bensmith/0910/Teachers_union_helped_unseat_Fenty.html).
_bottom 5 percent of teachers:_ Eric. A. Hanushek, "The Economic Value of Higher Teacher Quality," _Economics of Education Review_ , 2011, doi:10.1016/j.econedurev.2010.12.006.
_Although the state spends:_ Jessica Calefati and Jeanette Rundquist, "N.J. School Report Card Data Shows Average Per-Pupil Spending Increased Statewide, Dropped in Urban Districts," _Newark Star-Ledger_ , February 8, 2011, <http://www.nj.com/news/index.ssf/2011/02/nj_per-pupil_spending_increase.html>.
_In a 2010 e-mail:_ Allysia Finley, "Teachers Embrace the Power of Prayer: A New Jersey Teacher's Union Prays for Chris Christie's Death," _Wall Street Journal_ , April 12, 2010, [http://online.wsj.com/article/
SB10001424052702303828304575180160172878050.html](http://online.wsj.com/article/SB10001424052702303828304575180160172878050.html).
_From 1994 to 1999:_ Kurt J. Bauman, _Home Schooling in the United States: Trends and Characteristics_ (Washington, D.C.: U.S. Census Bureau, August 2001).
_KIPP is a national network:_ "KIPP: Schools," _KIPP_ , accessed March 15, 2011, <http://www.kipp.org/schools>.
_President Bush and the Republican Congress:_ Susan Ferrechio, "Democrat inserted provision endangers DC Opportunity Scholarship program," posted February 23, 2009, 1:00 a.m., <http://washingtonexaminer.com/politics/2009/02/democrat-inserted-provision-endangers-dc-opportunity-scholarship-program#ixzz1LyOiHGfV>. 125 _Three months after Katrina:_ Brett Michael Dykes, "New Orleans Schools Stage Impressive Turnaround after Katrina," _Yahoo! News_ , August 27, 2009, [http://news.yahoo.com/s/yblog_upshot/20100827/
us_yblog_upshot/new-orleans-public-schools-stage-impressive-turnaround-five-years-after-katrina](http://news.yahoo.com/s/yblog_upshot/20100827/us_yblog_upshot/new-orleans-public-schools-stage-impressive-turnaround-five-years-after-katrina).
_"Over half of all":_ "The State of Public Education in New Orleans," Cowen Institute for Public Education Initiatives, 2010, <http://www.coweninstitute.com/wp-content/uploads/2010/03/SPENO-2010-Exec-Summ-WEB-22710.pdf>.
_Just follow the money:_ Liz Goodwin, "Despite Fiery Rhetoric, Largest Teachers Union Spending Big for Dems," _Yahoo! News_ , October 6, 2010, [http://news.yahoo.com/s/yblog_upshot/20101006/
el_yblog_upshot/despite-fiery-rhetoric-largest-teachers-union-spending-big-for-dems](http://news.yahoo.com/s/yblog_upshot/20101006/el_yblog_upshot/despite-fiery-rhetoric-largest-teachers-union-spending-big-for-dems).
## **7. A New Republican Feminism**
_"The Independent Girl":_ Lou Henry Hoover Papers, Subject File, School Papers, Lou Henry, High School, Reports and Misc., 1886–1990, Herbert Hoover Presidential Library, West Branch, Iowa.
_"Body Hair":_ "Body Hair: The Final Frontier for Female Liberation," _Ms. Magazine_ , July 1972.
_"postgrievance":_ Barbara Bylenga and Marya Stark, "Have Millennial Women Moved Beyond Feminism?" _BlogHer_ , October 22, 2008, <http://www.blogher.com/have-millennial-women-moved-beyond-feminism-barbara-bylenga-and-marya-stark>.
_"acting white":_ Roddie A. Burris, "Jackson Slams Obama for 'Acting White,' " _The State in Politico_ , September 19, 2007, <http://www.politico.com/news/stories/0907/5902.html>.
_concept of co-parenting:_ Morley Winograd and Michael D. Hais, _Millennial Makeover: My Space, YouTube, and the Future of American Politics_ (Piscataway, N. J.: Rutgers University Press, 2008), 71.
_"Caribou Barbie":_ Maureen Dowd, "Now, Sarah's Folly," _New York Times_ , July 4, 2009, <http://www.nytimes.com/2009/07/05/opinion/05dowd.html>.
_"gang-raped":_ Tracy Miller, "Sandra Bernhard Issues 'Gang Rape' Warning to Sarah Palin," _New York Daily News_ , September 19, 2008, <http://www.nydailynews.com/gossip/2008/09/19/2008–09–19_sandra_bernhard_issues_gang_rape_warning-2.html>.
## **8. The Choice Dilemma**
_"The federal government has no business":_ Barry Goldwater, from Robert Alan Goldberg, _Barry Goldwater_ (New Haven: Yale University Press, 1995), 308.
_78 percent of Republican voters:_ "Trends: Abortion," Gallup, last modified May 2010, <http://www.gallup.com/poll/1576/abortion.aspx>. A Gallup poll conducted May 3–6, 2010, found that 54 percent thought abortion should be "legal only under certain circumstances," and 24 percent thought it should be "legal under any circumstances." This means that 78 percent thought that abortion should be legal at least under certain circumstances. Also see Avalanche Strategic Communications, "Poll Demonstrates That a Big-Tent GOP Ticket Is More Electable," Republican Majority for Choice, August 22, 2008, https://gopchoice.electionmall.name/E-PressRelease/displaycontent.asp?a=5C5A58&z=5D. A poll conducted August 14–16, 2008, by Republican Majority for Choice found that "78% of Republican voters agreed that women should have access to the full range of reproductive options, including education, contraception, motherhood, adoption and abortion." Seventy percent of voters also said "their support for McCain would not be affected if he chose a pro-choice running mate..."
_41 percent of women identify themselves as Democrats:_ Jeffrey M. Jones, "Republicans Face Steep Uphill Climb Among Women," Gallup, May 6, 2009, <http://www.gallup.com/poll/118207/Republicans-Face-Steep-Uphill-Climb-Among-Women.aspx>.
_The procedure:_ "Trends: Abortion," Gallup. In a Gallup poll conducted May 10–13, 2007, the following question was asked: "Now I would like to ask your opinion about a specific abortion procedure known as 'late term' abortion or 'partial birth' abortion, which is sometimes performed on women during the last few months of pregnancy. Do you think that this procedure should be legal or illegal?" Seventy-two percent responded that this should be illegal, up from 68 percent when the question was last polled, in 2003.
_most Americans view abortion as morally wrong:_ Lydia Saad, "Four Moral Issues Sharply Divide Americans," Gallup, May 26, 2010, <http://www.gallup.com/poll/137357/Four-Moral-Issues-Sharply-Divide-Americans.aspx>.
_"54% of self-described pro-life":_ Avalanche Strategic Communications, "August 2008 National Research Inc. Poll" (Washington, D.C.: Republican Majority for Choice, August 2008).
_one of these groups:_ "About Us," Republican National Coalition for Life, accessed March 15, 2011, <http://www.rnclife.org/about/>.
_When the question:_ Lydia Saad, "Abortion Issue Laying Low in 2008 Campaign," Gallup, May 22, 2008, <http://www.gallup.com/poll/107458/abortion-issue-laying-low-2008-campaign.aspx>.
_A Republican Party that would appeal:_ [http://www.rasmussenreports.com/platinum/
political_tracking_crosstabs/february_2011/
crosstabs_abortion_february_14_15_2010](http://www.rasmussenreports.com/platinum/political_tracking_crosstabs/february_2011/crosstabs_abortion_february_14_15_2010).
_Part of that reality:_ Rebecca Wind, "Premarital Sex Is Nearly Universal Among Americans, and Has Been for Decades," Guttmacher Institute, December 19, 2006, <http://www.guttmacher.org/media/nr/2006/12/19/index.html>.
_Forty-seven percent of teenagers:_ Associated Press, "Fewer High School Students Are Having Sex," _MSNBC.com_ , July 13, 2007, <http://www.msnbc.msn.com/id/19733766/#>.
_Most young people:_ Guttmacher Institute, "Facts on American Teens' Sexual and Reproductive Health," <http://www.guttmacher.org/pubs/FB-ATSRH.pdf>.
_those with the genetic mutation:_ Caroline Mansfield, Suellen Hopfer, and Theresa M. Marteau, "Termination Rates After Prenatal Diagnosis of Down Syndrome, Spina Bifida, Anencephaly, and Turner and Klinefelter Syndromes: A Systematic Literature Review," _Prenatal Diagnosis_ 19, no. 9 (1999). Abstract available at: <http://www.ncbi.nlm.nih.gov/pubmed/10521836>.
_Most teenage mothers:_ <http://www.theodora.com/teddy/newyork/teenage.html>. Also see Rebecca A. Maynard, ed., _Kids Having Kids: A Robin Hood Foundation Special Report on the Costs of Adolescent Childbearing_ (New York: Robin Hood Foundation, 1996); and R. H. Haveman, B. Wolfe, and E. Peterson, "Children of Early Childbearers as Young Adults," in Rebecca A. Maynard, ed., _Kids Having Kids: Economic Costs and Social Consequences of Teen Pregnancy_ (Washington, D.C.: Urban Institute Press, 1997), 257–84.
_The teen birth rate:_ Bill Albert, "New Survey: Teens Say Parents Most Influence Their Decisions About Sex," The National Campaign to Prevent Teen and Unplanned Pregnancy, December 21, 2010, <http://www.thenationalcampaign.org/press/press-release.aspx?releaseID=202>.
_"82 percent of":_ Patrik Jonsson, "A Force Behind the Lower Teen Birthrate: MTV's '16 and Pregnant,' " _Christian Science Monitor_ , December 21, 2010, <http://www.csmonitor.com/USA/Society/2010/1221/A-force-behind-the-lower-teen-birthrate-MTV-s-16-and-Pregnant>.
## **9. Conservative Environmentalism**
_"The spiritual uplift":_ Herbert Hoover, National Conference on Outdoor Recreation, Washington, D.C., January 21, 1926, Herbert Hoover Papers—Articles, Addresses, and Public Statements—#546B.
_During his second term:_ Information compiled and edited from research done by the National Geographic Society and The Theodore Roosevelt Association staff. Available at: <http://www.theodoreroosevelt.org/life/conNatlForests.htm> (copyright November 2005).
_a school of modern environmentalism:_ Daniel Engber, "Global Swarming: Is It Time for Americans to Start Cutting Our Baby Emissions?" _Slate_ , September 10, 2007, <http://www.slate.com/id/2173458/>.
_three-quarters of them demand:_ Janis Gaudelli, "The Greenest Generation: The Truth Behind Millennials and the Green Movement," _AdvertisingAge_ , April 29, 2009, <http://adage.com/goodworks/post?article_id=136331>.
Newsweek _once published a cover story:_ Peter Gwynne, "The Cooling World," _Newsweek_ , April 28, 1975, <http://www.denisdutton.com/newsweek_coolingworld.pdf>; "The Little Ice Age," _Windows to the Universe_ , accessed March 15, 2011, <http://windows2universe.org/earth/climate/little_ice_age.html>.
_electricity prices would rise:_ William W. Beach, David W. Kreutzer, Karen A. Campbell, and Ben Lieberman, "Son of Waxman-Markey: More Politics Makes for a More Costly Bill," _The Heritage Foundation_ , May 18, 2009, <http://www.heritage.org/research/reports/2009/05/son-of-waxman-markey-more-politics-makes-for-a-more-costly-bill>.
_Climate scientist:_ Bjorn Lomborg, "Cost-Effective Ways to Address Climate Change," _Washington Post_ , November 17, 2010, <http://www.washingtonpost.com/wp-dyn/content/article/2010/11/16/AR2010111604973.html>.
## **10. A Nation of Immigrants, a Nation of Borders**
_The Republican Party, unfortunately:_ Marco Rubio, from Fox News Sunday interview with Chris Wallace, March 28, 2010.
_10.8 million undocumented immigrants:_ <http://www.reuters.com/article/2010/04/29/us-usa-immigration-idUSTRE63S5TY> 20100429.
_Arizona had 460,000 undocumented immigrants:_ Ibid.
_made 650 arrests a day:_ Ibid.
_state's government loses between:_ "Expensive Aliens: How Much Do Illegal Immigrants Really Cost?" _ABC News Online:_ http://abcnews.go.com/Business/illegal-immigrants-cost-us-100-billion-year-group/story?id=10699317&page=2.
_$113 billion a year:_ "Illegal Immigration a $113 Billion a Year Drain on U.S. Taxpayers: FAIR Releases First-of-Its-Kind Comprehensive Study of Federal, State and Local Costs of Illegal Immigration," July 6, 2010, [http://www.fairus.org/site/News2?page=ewsArticle&id=
23198&security=1601&news_iv_ctrl=1741](http://www.fairus.org/site/News2?page=ewsArticle&id=23198&security=1601&news_iv_ctrl=1741).
_Millennials are the most ethnically diverse:_ Pew Research Center, "Millennials—A Portrait of Generation Next: Confident. Connected. Open to Change," Forum on Religion and Public Life, February 2010, 79, <http://pewsocialtrends.org/files/2010/10/millenials-confident-connected-open-to-change.pdf>.
_"younger people":_ Ibid.
_less apt to say:_ Ibid.
_"much less supportive":_ Ibid.
_After his reelection:_ "Comprehensive Immigration Reform Act of 2007," _SourceWatch_ , last modified December 17, 2008, <http://www.sourcewatch.org/index.php?title=Comprehensive_Immigration_Reform_Act_of_2007>.
_not prepared to support:_ Senator Jon Kyl, "Arizona Immigration Law, State Officials," The Independent Women's Forum and The Georgetown Law Supreme Court Institute (Washington, D.C.: Georgetown University Law Center, May 20, 2010), <http://www.c-spanvideo.org/program/293622–1>.
_Yuma County, Arizona:_ Lauren Gambino, "Failed virtual border fence has politicians pointing to Yuma success," _TucsonSentinel.com_ , Cronkite News Service, posted January 31, 2011, 7:33 p.m., <http://www.tucsonsentinel.com/local/report/013111_border>.
_Some of our liberal:_ Guy Benson, "Democrat Loretta Sanchez Compares US Border Fence to Berlin Wall," _Townhall.com_ , October 6, 2010, [http://townhall.com/tipsheet/guybenson/2010/10/06/
democrat_loretta_sanchez_compares_us_border_
fence_to_berlin_wall](http://townhall.com/tipsheet/guybenson/2010/10/06/democrat_loretta_sanchez_compares_us_border_fence_to_berlin_wall); Stephanie Simon, "Border-Fence Project Hits a Snag: Opposition from Environmentalists, Property Owners Slows Construction of Final Leg," _Wall Street Journal_ , February 4, 2009, <http://online.wsj.com/article/SB123370523066745559.html>.
_"muro de odio":_ Stephanie Simon, "Border-Fence Project Hits a Snag: Opposition from Environmentalists, Property Owners Slows Construction of Final Leg," _Wall Street Journal_ , February 4, 2009, <http://online.wsj.com/article/SB123370523066745559.html>; Bill Addington, " 'Muro de Odio': A Border Wall of Hate and Fear," _Rio Grande Sierran_ , March/April 2009, nmsierraclub.org/sites/default/files/rgsierran_09_03_04.pdf.
_Despite the fact:_ Andrea Nill, "Over 77 Percent of All Arizonans Support Comprehensive Immigration Reform," _Think Progress_ , May 14, 2010, <http://thinkprogress.org/2010/05/14/arizona-poll-immigration/>.
_and 73 percent of all Americans:_ Pew Research Center, "Democrats Divided, but Support Key Provisions: Broad Approval for New Arizona Immigration Law," May 12, 2010, <http://people-press.org/reports/pdf/613.pdf>.
_"The most important":_ Rachel Rose Hartman, "Marco Rubio Defends Spanish Ad, Support for English as Official Language," The Upshot, in _Yahoo! News_ , September 29, 2010, [http://news.yahoo.com/s/yblog_upshot/20100929/
el_yblog_upshot/marco-rubio-defends-spanish-ad-support-for-english-as-official-language](http://news.yahoo.com/s/yblog_upshot/20100929/el_yblog_upshot/marco-rubio-defends-spanish-ad-support-for-english-as-official-language).
## **11. Islamist Supremacy: A Millennial's Worst Nightmare**
_"The bombers of Manhattan":_ Christopher Hitchens, "Against Rationalization," _Nation_ , September 20, 2001, <http://www.thenation.com/article/against-rationalization>. This article appeared in the print edition of the _Nation_ on October 8, 2001.
_Islamist Supremacy:_ Eli Lake, "Study: Iran Indoctrinating Children in Islamic Supremacism," _New York Sun_ , March 19, 2008, <http://www.nysun.com/foreign/study-iran-indoctrinating-children-in-islamic/73162/>; Robert Farley, interview by Eli Lake, "Eli Abandons 'Islamic Fascism,' Defends 'Islamic Supremacism,' " _Bloggingheads.tv_ , posted January 13, 2008, <http://bloggingheads.tv/diavlogs/7939>; James Taranto, " 'Islamic Supremacy': The Solution to a Conundrum of Language and Policy," _Wall Street Journal_ , January 16, 2009, <http://online.wsj.com/article/SB123211637982290301.html>. This term was first used by Eli Lake on _Bloggingheads.tv_ and in his _New York Sun_ article, later borrowed by Taranto in _Wall Street Journal_. I have altered it to "Islamist Supremacy."
_the most liberal generation:_ Pew Research Center, "Millennials—A Portrait of Generation Next," 73.
_Iran's president:_ "Ahmadinejad: Most Blame U.S. Government for 9/11; American Officials Call Iran Leader 'Delusional,' Walk Out of Speech," _MSNBC.com_ , September 23, 2010, <http://www.msnbc.msn.com/id/39331594/ns/world_news/>.
_also a Holocaust denier:_ Ewen MacAskill and Chris McGreal, "Israel Should Be Wiped Off the Map, Says Iran's President," _Guardian_ , October 27, 2005, <http://www.guardian.co.uk/world/2005/oct/27/israel.iran>.
_When Ahmadinejad went:_ "Iran President in NY Campus Row," _BBC News_ , September 25, 2007, <http://news.bbc.co.uk/2/hi/7010962.stm>.
_at last count, nine women:_ Saeed Kamali Dehghan and Ian Black, "Iranians Still Facing Death by Stoning Despite 'Reprieve,' " _Guardian.co.uk_ , July 8, 2010, <http://www.guardian.co.uk/world/2010/jul/08/iran-death-stoning-adultery>.
_partisan politics ends:_ Conrad Black, "A New Isolationism?" _National Review Online_ , November 2, 2009, <http://www.nationalreview.com/articles/228505/new-isolationism/conrad-black>; Tony Karon, "Why Obama Defaulted to Bush Foreign Policy Positions," _Time_ , January 4, 2010, [http://www.time.com/time/world/article/
0,8599,1950827,00.html](http://www.time.com/time/world/article/0,8599,1950827,00.html). "Politics ends at the water's edge" is a phrase associated most often with the Eisenhower and Truman eras. It is still used in politics today, as shown by these two recent articles.
_Zainab Al-Suwaij:_ "American Islamic Congress: Leadership," American Islamic Congress, last modified 2008, <http://www.aicongress.org/about/leadership.html>.
## **12. America the Exceptional**
_"Our willingness to speak":_ Ronald Reagan, "Remarks and a Question-and-Answer Session in Los Angeles at a Meeting with Editors and Broadcasters from Western States," July 1, 1982, http://www.presidency.ucsb.edu/ws/index.php?pid=42695&st=&st1=#ixzz1M5ByzUxj.
_Socialist Party:_ <http://www.u-s-history.com/pages/h890.html>; <http://www.historycentral.com/elections/1920Pop.html>.
_"socialism in a nation-wide":_ Herbert Hoover, "American Individualism and the Challenge to Liberty" (West Branch, Iowa: Herbert Hoover Presidential Library Association, 1989), 46.
_When Barack Obama was asked:_ Michael D. Shear and Scott Wilson, "On European Trip, President Tries to Set a New, Pragmatic Tone," _Washington Post_ , April 5, 2009, <http://www.washingtonpost.com/wp-dyn/content/article/2009/04/04/AR2009040400700.html>; James Fallows, "Obama on Exceptionalism," _Atlantic_ , April 4, 2009, [http://www.theatlantic.com/technology/archive/2009/
04/obama-on-exceptionalism/9874/](http://www.theatlantic.com/technology/archive/2009/04/obama-on-exceptionalism/9874/); Monica Crowley, "American Exceptionalism," commentary in _Washington Times_ , July 1, 2009, <http://www.washingtontimes.com/news/2009/jul/1/american-exceptionalism/>.
_According to pollster:_ Frank I. Luntz, _What Americans Really Want... Really: The Truth About Our Hopes, Dreams, and Fears_ (New York: Hyperion, 2009), 180.
_But their decisive break:_ Carl M. Cannon, "The Facebook Election," _Reader's Digest_ , June 2008.
_"the wrong war":_ William Kristol, "Kerry vs. Kerry: What Does 'the Wrong War in the Wrong Place at the Wrong Time' Mean?" _The Blog_ , in _The Weekly Standard_ , September 7, 2004, [http://www.weeklystandard.com/Content/Public/Articles/
000/000/004/587jxocg.asp](http://www.weeklystandard.com/Content/Public/Articles/000/000/004/587jxocg.asp). This statement by Democratic presidential nominee and Senator John Kerry was widely quoted in the 2004 presidential election campaign.
_From the millennials':_ Cannon, "The Facebook Election."
_A 2006 survey:_ Frank N. Magid Associates, Inc., "The Politics of the Millennial Generation: A New Survey Comparing Political Attitudes Between Generations," _New Politics Institute_ , March 2006, 8, <http://ndn-newpol.civicactions.net/sites/ndn-newpol.civicactions.net/files/MillenialGenerationPolitics.pdf>.
_This generation is:_ Hilary Doe and Zachary Kolodin, "Blueprint for the Millennial America," _The Roosevelt Campus Network_ , 2010, 25, [http://www.rooseveltcampusnetwork.org/chapter/
1875/blueprint-millennial-america](http://www.rooseveltcampusnetwork.org/chapter/1875/blueprint-millennial-america).
_As a result:_ Cannon, "The Facebook Election."
_In return for:_ U.S. Department of State Office of the Spokesman, "Interview of Secretary Colin L. Powell on 'Be Heard: An MTV Global Discussion with Colin Powell' " (Washington, D.C.: U.S. Department of State, February 14, 2002), <http://www.solcomhouse.com/colinpowellmtv.htm>.
_In Eastern Europe:_ "Q&A: US Missile Defence," _BBC News_ , September 20, 2009, <http://news.bbc.co.uk/2/hi/europe/6720153.stm>.
_he led America's:_ Jim Fisher-Thompson, "U.S. Aid to Africa Hits Record Levels: Geldof, Bono Praise Bush Before Group of Eight Summit in Scotland," _America.gov_ , June 27, 2005, [http://www.america.gov/st/washfile-english/2005/June/
200506271748571EJrehsiF0.8724481.html#ixzz1E2ik0cqx](http://www.america.gov/st/washfile-english/2005/June/200506271748571EJrehsiF0.8724481.html#ixzz1E2ik0cqx).
# ABOUT THE AUTHOR
MARGARET HOOVER is a veteran of national political campaigns, Capitol Hill, and the Bush Administration White House. A Fox News contributor, she is committed to modernizing the Republican Party to enable it to connect with a new generation of Americans, even as it remains true to the principles of individual freedom and fiscal conservatism and continues to champion a robust U.S. foreign policy. She serves on the advisory council of the American Foundation for Equal Rights and GOProud, as well as on the boards of the Hoover Institution, the Herbert Hoover Presidential Library Association, and the Belgian American Educational Foundation. Raised in Colorado, Ms. Hoover lives in New York City with her husband.
For more information, or to contact the author, visit <http://MargaretHoover.com>.
| {
"redpajama_set_name": "RedPajamaBook"
} | 543 |
{"url":"https:\/\/www.physicsoverflow.org\/759\/decidability-algorithm-checking-universality-quantum-gate?show=2960","text":"# Decidability\/algorithm for checking universality of a quantum gate set\n\n+ 8 like - 0 dislike\n303 views\n\nGiven a finite set of quantum gates $\\mathcal{G} = \\{G_1, \\dots, G_n\\}$, is it decidable (in computation theoretic sense) whether $\\mathcal{G}$ is a universal gate set? On one hand, \"almost all\" gate sets are universal, on the other, non-universal gate sets are still not well understood (in particular, of course, it is not known whether every non-universal gate set is classically simulatable), so I imagine giving an explicit algorithm for checking universality could be nontrivial.\n\nThis post has been migrated from (A51.SE)\nCan you clarify the question? Joe's answer assumes you have a fixed number of qubits and all gates act on those, but for universality, we often assume gates can act on any subset of qubits. E.g., CNOT + all one-qubit gates are not universal if the one-qubit gates can only act on the first qubit, and CNOT is only from qubit 1 to qubit 2. In the latter case, we might want to extrapolate to many qubits to get universality. In that case, I think the anwer may be unknown.\n\nThis post has been migrated from (A51.SE)\n@DanielGottesman: I agree about the limitations of my answer. Indeed, I believe it is undecidable in the latter case as follows: Take a cellular automata on an infinite lattice of qubits and use it to encode the halting problem (call this update unitary $U_1$). Then take a second lattice with a universal QCA (with update unitary $U_2$). We can define a new unitary $CU_2 = |0\\rangle\\langle0|_H\\otimes I + |1\\rangle\\langle1|\\otimes U_2$, where the subscript $H$ denotes a qubit which is set to $|1\\rangle$ iff the first cellular automata halts.\n\nThis post has been migrated from (A51.SE)\nThus the gate $CU_2 \\times U_1$ is universal if and only if the first Turing machine halts, and is hence undecidable.\n\nThis post has been migrated from (A51.SE)\n\n+ 4 like - 0 dislike\n\nFor the case of Hamiltonians, rather than gates the answer is trivially yes: you simply enumerate the independent elements of the Lie algebra. Since the Lie algebra is a vector space with the addition of the Lie bracket operator. Since the space is finite, it has a finite basis, and which can easily be checked as to whether it is closed or open under the Lie bracket operation. Simply checking the Lie bracket of all pairs of orthogonal operators can be done in time polynomial in the dimensionality of the space, and a suitable operator basis can be found by the Gram-Schmidt method.\n\nFor gates, you don't really have the same option to resort to infinitesimals straight off, and need to construct gates with irrational eigenvalues so that you can arbitrarily well approximate the required infinitesimal generators. I guess that there is a relatively simple way to do this, but it is not immediately obvious to me.\n\nIn any case, taking the log of the gates to obtain a set of operators which generate them when exponentiated and checking whether these generated the full Lie algebra would provide a simple necessary but not sufficient criteria for universality.\n\nThis post has been migrated from (A51.SE)\nanswered Jan 20, 2012 by (3,575 points)\nWhy we should check only pairs?\n\nThis post has been migrated from (A51.SE)\n@AlexV: Because the Lie bracket takes operates on 2 inputs. Every time you produce a new linearly independent operator you produce an orthogonal one and repeat until you get closure.\n\nThis post has been migrated from (A51.SE)\nI meant you should consider $[\\ldots[H_k,H_j],H_l],\\ldots]$, but not only pairs, e.g. see my own paper http:\/\/arxiv.org\/abs\/quant-ph\/0010071\n\nThis post has been migrated from (A51.SE)\n@AlexV: You don't need to. It's a vector space, so a vector is orthogonal to a given subspace if and only if it is orthogonal to any basis for that subspace.\n\nThis post has been migrated from (A51.SE)\nLikely we are talking about different things - which vector space you are talking about? You do not know from very beginning the subalgebra generated by your gates - you need to construct that from given Hamiltonians to check if it whole Lie algebra.\n\nThis post has been migrated from (A51.SE)\nI did not see a work of Daniel Burgarth and Alastair Kay on the particular theme, but in works of other authors most often appear the same $spin(n)$ theme, but it is necessary some time, to check the isomorphism. Anyway, in many classical problems we should consider $n \\to \\infty$ to lost decidability.\n Please use answers only to (at least partly) answer questions. To comment, discuss, or ask for clarification, leave a comment instead. To mask links under text, please type your text, highlight it, and click the \"link\" button. You can then enter your link URL. Please consult the FAQ for as to how to format your post. This is the answer box; if you want to write a comment instead, please use the 'add comment' button. Live preview (may slow down editor)\u00a0\u00a0 Preview Your name to display (optional): Email me at this address if my answer is selected or commented on: Privacy: Your email address will only be used for sending these notifications. Anti-spam verification: If you are a human please identify the position of the character covered by the symbol $\\varnothing$ in the following word:p$\\hbar$ysicsOv$\\varnothing$rflowThen drag the red bullet below over the corresponding character of our banner. When you drop it there, the bullet changes to green (on slow internet connections after a few seconds). To avoid this verification in future, please log in or register.","date":"2020-04-06 05:53:57","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 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\": 0.8270519971847534, \"perplexity\": 422.57482771899714}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2020-16\/segments\/1585371618784.58\/warc\/CC-MAIN-20200406035448-20200406065948-00009.warc.gz\"}"} | null | null |
\section{Introduction}
A large variety of superconducting materials can be theoretically understood within the standard BCS theory proposed by Bardeen, Cooper and Schrieffer \cite{BCSsupra57,tinkham2004introduction}. Within this framework, the metallic electrons of a single, partially filled band are considered to be bound into (Cooper) pairs by a weak attraction, while other bands are discarded based on the premise that they are much more remote in energy than the typical energy scale set by the attractive interaction. Indeed, the attractive interaction between electrons is, within the standard BCS theory, mediated by phonons via the electron phonon coupling. Within the weak-coupling limit, the typical energy scale for superconductivity is then a fraction of the Debye temperature $k_BT_D$ that is itself in the $10...100$ meV range, while the Fermi energy and the typical band gaps are on the order of $\sim 1$ eV \cite{tinkham2004introduction}. In spite of its great success, BCS theory is not capable of explaining all occurrences of superconductivity and finds severe limitations \textit{e.g.} in the case of strongly correlated materials, such as heavy-fermion superconductivity \cite{HF1,reviewheavyfermion} or high-$T_c$ superconductivity \cite{reviewhighTc}, where even the origin of the attractive interaction is still debated.
While the above-mentioned energy-scale consideration has remained unchallenged for a long time, the advent of topological band theory \cite{TBT,cayssolfuchs} and its success in the theoretical description of a plethora of materials \cite{alltopoallmate}, such as topological insulators \cite{TI1,TI2}, topological superconductors \cite{bernevigBook,TSC}, Weyl and Dirac semimetals \cite{WSM}, has shown that the coupling between energy bands is not only governed by energy scales but by more subtle geometric quantities, such as the Berry curvature or the quantum metric. Several recent papers have investigated the role of the latter, namely in the presence of flat bands in which the quantum metric can be the dominant contribution to the superfluid weight \cite{peotta_superfluidity_2015,rossi2021quantum,tormaberneivig,tian2021evidence}. The Berry curvature has been theoretically shown to play a relevant role in a two-body problem that is closely related to the Cooper pair, namely in the physics of excitons. For example, in two-dimensional (2D) semiconducting transition-metal dichalcogenides (TMDC) \cite{TMDC}, excitons -- bound electron-hole pairs -- are formed in the vicinity of the $K$ and $K'$ points of the first Brillouin zone, where the Berry curvature reaches its maximal value \cite{Fuchs2010}. Experimentally, a first hint to the relevance of band-geometric effects came from the failure of the effective hydrogen model, which had been extremely successful before in the theoretical understanding of the measured exciton spectra \cite{exciton1,exciton2}. It was later shown that the Berry curvature affects the exciton spectra, contrary to the one-particle case, because it couples to the electric field that is generated by the attractive interaction between the electron and the hole forming the bound exciton state \cite{BerryExc1,BerryExc2,Trushin_2017,Hichri_2019}. This is a consequence of the intrinsic Dirac character of the low-energy charge carriers in these materials, which are commonly described in terms of a 2D massive Dirac equation \cite{xiao,GoerbigEPL}. Excitons in 2D TMDC and potentially other bound pairs inherit then this Dirac character \cite{Trushin_2016}.
Based on the above-mentioned exciton example, it is therefore natural to consider that the Berry curvature might also affect the formation of the Cooper pair due to the mutual interaction between the two electrons. This is the main motivation of the present theoretical study, where we show that the effective electron-electron interaction is generically weakened when one includes energy terms in the Hamiltonian that take into account the effect of the Berry curvature. We consider conventional BCS-type superconductivity in 2D materials, such as the above-mentioned 2D semiconducting TMDC for a moderate doping range. We emphasize that we do not investigate topological superconductivity \cite{TSC} that arises when one considers the quasiparticle bands, the mutual coupling of which is at the origin of the emergent topological properties. Here, we rather treat the role of the Berry curvature, which affects the formation of Cooper pairs in conventional BCS theory. Within topological band theory, the related wave-vector ($\vec{k}$) dependent Berry connection $\mathcal{A}_n(\vec{k})$ modifies the electrons' positions $\vec{r}$ when the latter are projected by the projectors $P_n$ to the $n$-th band, $\vec{r}\rightarrow P_n \vec{r} P_n=\vec{r}+\mathcal{A}_n(\vec{k})$. This yields a dipole that interacts with the electric field, and this dipolar structure, which the Cooper pair inherits, is at the origin of the weakened Cooper pairing. More precisely, the projection yields two extra terms which affect the electron-electron interaction to the one-body Hamiltonian. One of them is reminiscent of the spin-orbit coupling if one interprets the Berry curvature in terms of a spin, and the second one corresponds to the Darwin term, which arises within a Dirac-fermion treatment of the two bands in the vicinity of the direct gap \cite{FW}. We show that the latter is responsible for a reduced effective BCS coupling constant that results in a smaller superconducting BCS gap, while the former spin-orbit-type term does not play a role in $s$-wave nor other types of pure singlet or triplet pairing.
The paper is organized as follows. In Sec. \ref{sec:1body}, we briefly revisit, along the lines exposed in Ref. \cite{Hichri_2019}, the emergence of corrective terms to the one-body Hamiltonian of a charge projected to a single band. We present two complementary approaches: one based on a generalized version of the Peierls substitution in Sec. \ref{ssec:Peierls} and one based on a treatment within the continuum two-band model of massive Dirac fermions in the vicinity of the direct gap, where the role of the Berry curvature is most prominent. This treatment is the basis of the two-body problem, which we present in Sec. \ref{sec:2body}. After some general considerations (Sec. \ref{ssec:gen}), Sec. \ref{ssec:Cooper} shows how the Cooper pair and its binding energy are modified by the extra terms, while Sec. \ref{sec:BCS} presents the BCS theory of conventional $s$-wave-type superconductivity in the presence of the corrective terms due to the Berry curvature. In the calculations, we consider a Fermi level that is extremely close to the conduction-band bottom, and we discuss then the role of stronger doping on Cooper pairing and BCS superconductivity in Sec. \ref{sec:doping}. In Sec. \ref{sec:beyond}, we briefly discuss how our theoretical picture of superconductivity in the presence of non-zero Berry curvature evolves in other pairing symmetries, be they singlet or triplet. The last section (Sec. \ref{sec:exp}) is devoted to possible experimental implications of our theoretical studies. There, we compare the superconducting gap and the critical temperature in the absence and the presence of the weakened interaction due to the Berry curvature.
\section{One-body Hamiltonian: corrective terms due to the Berry curvature}
\label{sec:1body}
Before discussing the role of possible geometric terms on the superconducting properties of a 2D material, let us briefly revisit the emergence of these terms within a one-particle description. More precisely, we consider a band structure with $N$ bands described by the Bloch Hamiltonian. The Berry curvature may be viewed as the action of virtual interband transitions of electrons that are otherwise restricted to a single band, while there are no true (quantum) transitions in the adiabatic limit. Notice that there are no geometric terms in the Hamiltonian in the absence of a local electric potential $V(\vec{r})$ different from the periodic one that gives rise to the Bloch bands, and the Hamiltonian is then reduced to the bare band dispersion $E_n(\vec{k})$ of the $n$-th band which the electrons are projected to.
In the presence of a local potential $V(\vec{r})$ which acts on our single electron, the simple reduction of the Hamiltonian to the band dispersion is no longer valid -- in the following we consider this potential to be generated by the second electron to which the first one is bound in a Cooper pair, but our arguments are not restricted to this case. Indeed, $V(\vec{r})$ couples directly the different bands and needs thus to be taken into account prior to the adiabatic projection to a single band. This yields extra terms to the Hamiltonian that can be discussed within two complementary approaches that we briefly review in this section. The first one is based on a generalized Peierls substitution \cite{gosselin_menas_berard_mohrbach_2006,PhysRevLett.115.166803,Chang2008BerryCO,Gosselin_2008,Trushin_2017,Hichri_2019}. It yields a corrected (quantum) Hamiltonian that reproduces the semi-classical equations of motion. This approach has the advantage of providing a transparent physical interpretation of the role played by the Berry curvature, namely in the formation of a \textit{dipole-like} term that arises due to the projection to a single band. This approach is similar to the magnetic-field case when the electron motion is restricted to a single Landau level \cite{LLdipole0,LLdipole1}, but it does not provide all corrective terms, even at linear order in the Berry curvature. In order to obtain the missing term, which is analogous to the Darwin term in relativistic quantum mechanics, we interpret the Berry curvature in terms of a two-band model, which describes the band structure locally in reciprocal space in terms of a massive Dirac Hamiltonian.
\subsection{Generalized Peierls substitution: emergence of the Berry dipole}
\label{ssec:Peierls}
Let us first recall how to incorporate the magnetic field to describe the dynamics of an electron in the $n$-th band $E_n(\vec{k})$ via the Peierls substitution (in the absence of a Berry curvature). Because the wave vector $\vec{k}=-i\nabla_{\vec{r}}$ is not a gauge-invariant quantity, it needs to be replaced by its gauge-invariant form
\begin{equation}\label{eq:PeierlsK}
\hbar\vec{k}\longrightarrow\vec{\Pi}=\hbar\vec{k}+e\vec{A}(\vec{r}),
\end{equation}
in terms of the vector potential $\vec{A}(\vec{r})$ which yields the magnetic field, $\vec{B}(\vec{r})=\vec{\nabla}_{\vec{r}}\times\vec{A}(\vec{r})$. We consider, here, electrons of charge $-e$ ($e>0$). From a semi-classical point of view, one obtains the equations of motion
\begin{equation}
\dot{\vec{r}}_n=\vec{v}_n= \frac{1}{\hbar}\nabla_{\vec{k}}E_n \qquad \text{and}\qquad
\hbar\dot{\vec{k}}=-e\vec{v}_n \times \vec{B},
\end{equation}
where $\vec{r}_n$ and $\vec{v}_n$ are the average position and velocity, respectively, of the electron in the $n$-th band. One justification of the Peierls substitution is that the Hamiltonian thus obtained, $H(\vec{\Pi})=E_n(\vec{\Pi})$, yields the same equations of motion if one uses the \textit{quantum} Heisenberg equations of motion
\begin{equation}
i\hbar \dot{\Pi}_j=[\Pi_j,H(\vec{\Pi})],
\end{equation}
with the help of the commutation relations $[\Pi_x,\Pi_y]=-i\hbar^2/l_B^2$, in terms of the magnetic length $l_B=\sqrt{\hbar/eB}$. Indeed, one then obtains
\begin{equation}
\dot{\Pi}_j=-\frac{\hbar}{l_B^2} \epsilon_{jl} \frac{\partial H}{\partial \Pi_l},
\end{equation}
where $\epsilon_{jl}$ is the antisymmetric Levi-Civita tensor. The quantum Hamiltonian $H(\vec{\Pi})$ yields therefore Heisenberg equations of motion that are the same as the semi-classical ones if we identify the (semi-classical) wave vector $\vec{k}$ with the gauge-invariant quantity $\vec{\Pi}/\hbar$, as it is precisely stipulated by the Peierls substitution.
The generalized Peierls substitution follows the same spirit when considering a system with a non-zero Berry curvature in the presence of a spatially varying potential $V(\vec{r})$, thus starting from the band energy $H_n=E_n(\vec{k})+V(\vec{r})$. In this case, the semi-classical equations of motion read \cite{Niu,cayssolfuchs}
\begin{eqnarray}\label{eq:semicl}
\dot{\vec{r}}_n = \vec{v}_n &=& \frac{1}{\hbar}\nabla_{\vec{k}}E_n + \frac{1}{\hbar} \nabla_{\vec{r}}V(\vec{r})\times\vec{\mathcal{B}}_n(\vec{k}) \\
\text{and}\qquad \hbar\dot{\vec{k}} &=& -\nabla_{\vec{r}}V-e\vec{v}_n \times \vec{B},
\end{eqnarray}
where $\vec{\mathcal{B}}_n(\vec{k})=\nabla_{\vec{k}}\times \mathcal{A}_n(\vec{k})$ is the Berry curvature of the $n$-th band in terms of its Berry connection $\mathcal{A}_n(\vec{k})$. Similarly to the case discussed above, one can obtain these equations of motion from a \textit{quantum} Hamiltonian \begin{equation}\label{eq:hamKR}
H(\vec{\Pi},\vec{R})=E_n(\vec{\Pi})+V(\vec{R}),
\end{equation}
where we have replaced not only the wave vector by its gauge-invariant expression (\ref{eq:PeierlsK}) but also the position by its expression projected onto the $n$-th band \cite{Sundaram1999,Niu,cayssolfuchs}
\begin{equation}\label{eq:PeierlsR}
\vec{r}\longrightarrow\vec{R}=\vec{r}+\vec{\mathcal{A}}_n(\vec{k}),
\end{equation}
which involves the Berry connection $\mathcal{A}_n(\vec{k})$. Similarly to the Peierls substitution (\ref{eq:PeierlsK}), the position $\vec{r}$ on the right-hand-side of this expression should be interpreted as a reciprocal-space derivative $\vec{r}=i\nabla_{\vec{k}}$. The replacement (\ref{eq:PeierlsR}) may be viewed as a \textit{generalized Peierls substitution} \cite{gosselin_menas_berard_mohrbach_2006,PhysRevLett.115.166803,Chang2008BerryCO,Gosselin_2008,Hichri_2019}. The semi-classical equations of motion are then retrieved as the Heisenberg equations of motion not only for $\vec{\Pi}$ but also for $\vec{R}=(X,Y)$ on the basis of the Hamiltonian (\ref{eq:hamKR}) and the induced commutation relations $[X,Y]=i\mathcal{B}_n(\vec{k})$ \cite{Hichri_2019}.
Let us now discard the magnetic field, which we have only discussed in order to remind the reader of the Peierls substitution and to justify its generalized form and expand the Hamiltonian (\ref{eq:hamKR}) to lowest order in the Berry connection. This expansion is legitimate as long as the external potential $V(\vec{r})$ varies slowly on a length scale that is set, in orders of magnitude, by the Berry connection and that can be related to an effective Compton length, as we discuss below. The Hamiltonian then becomes
\begin{equation}\label{eq:hamPeierls}
H=E_{n}(\vec{k})+V(\vec{r})+\vec{\mathcal{A}}_n(\vec{k})\cdot\vec{\nabla}_{\vec{r}}V(\vec{r}).
\end{equation}
The last generated term is interesting. First, it can be interpreted as the energy of an electric dipole $-e\vec{\mathcal{A}}_n(\vec{k})$ in an electric field $\vec{E}(\vec{r}) = \nabla V(\vec{r})/e$. We therefore call this term the \textit{Berry dipole term}.
Second, this term can be understood as an effective spin-orbit coupling if we use the \textit{symmetric gauge} for the Berry connection
\begin{equation}
\vec{\mathcal{A}}_n(\vec{k})=\frac{1}{2}\vec{\mathcal{B}}_n(\vec{k})\times\vec{k},
\end{equation}
in which case the corrective term reads
\begin{equation}
\vec{\mathcal{A}}_n(\vec{k})\cdot\vec{\nabla}_{\vec{r}}V(\vec{r})=\frac{1}{2}\Big(\vec{\mathcal{B}}_n(\vec{k})\times\vec{k}\Big)\cdot\vec{\nabla}_{\vec{r}}V(\vec{r}).
\end{equation}
This expression is interesting for the following reason. The Berry curvature is often viewed as the analogue of a magnetic field in reciprocal space, while the extra term in Eq. (\ref{eq:hamPeierls}) has the same form as the spin-orbit coupling term, which arises when one projects the relativistic Dirac equation onto the electron (or positron) branch \cite{greiner2000}. In this analogy, one would however need to identify the Berry curvature with an emergent spin rather than with a magnetic field.
\subsection{Non-relativistic limit of the Dirac equation}
\label{ssec:Dirac}
\begin{figure}[h!]
\label{figbandes}
\centering
\includegraphics[width=0.3\textwidth]{structurebandes2.png}
\caption{Band structure of massive Dirac fermions, with \textit{a priori} two different gaps for the two values of $\xi\sigma$, as one typically encounters in 2D semiconducting TMDC.}
\label{fig:01}
\end{figure}
In many situations the role of the Berry curvature in semiconducting materials can be approached in terms of a massive Dirac equation that describes two coupled bands in the vicinity of a reciprocal-space point, where the band gap is smallest and the Berry curvature has a maximum \cite{DiracBerry,Fuchs2010}. In this picture, coupling to other bands is not \textit{per se} excluded, but we consider that it only gives rise to a negligible contribution to the respective Berry curvatures of the two bands. This situation arises, \textit{e.g.}, in 2D semi-conducting TMDC in which two spin-orbit coupled families of band pairs form a direct gap at the $K$ and $K'$ points. In the vicinity of these points, the two bands are described by the generic Dirac Hamiltonian
\begin{equation}\label{eq:hamDir}
H=
\begin{pmatrix}
\Delta_{\xi\sigma} \sigma_0 & \hbar v_D(\xi\sigma k_x-ik_y) \\
\hbar v_D(\xi\sigma k_x +i k_y) & -\Delta_{\xi\sigma}\sigma_0
\end{pmatrix} + E_{\xi\sigma}^0 + V(\vec{r}),
\end{equation}
where $\xi$ indicates the valley index ($\xi=+$ for the $K$ valley and $\xi=-$ for the $K'$ valley in the case of 2D TMDC, or generally two time-reversal-symmetry related points $\pm \vec{k}_D$) and $\sigma=\pm$ represents the physical spin. In the presence of spin-orbit coupling and time-reversal symmetry, the band gaps $2\Delta_{\xi\sigma}$ of the two valleys are locked and depend only on the product $\xi\sigma$ of the spin and valley index, and so does the shift in energy $E_{\xi\sigma}^0$, which does not play any topological or dynamical role. In the absence of the external potential $V(\vec{r})$, one obtains the four bands
\begin{equation}
\epsilon_{\lambda,\xi\sigma}(\vec{k})=E_{\xi\sigma}^0+\lambda \sqrt{\Delta_{\xi\sigma}^2+\big(\hbar v_Dk)^2},
\end{equation}
which is depicted in Fig. \ref{fig:01}. The index $\lambda $ refers to the conduction ($\lambda=+$) and the valence ($\lambda=-$) bands. Note that there are only four bands since spin and valley are locked -- they enter into the expressions only as the product label $\xi\sigma$ -- as it is required by time-reversal symmetry. The associated Berry curvatures are given by \cite{DiracBerry,Niu}
\begin{equation}\label{eq:Berry}
\vec{\mathcal{B}}_{\lambda,\xi\sigma}(\vec{k})=-\frac{\lambda\xi\sigma}{2}\frac{\lambdabar_{\xi\sigma}^2}{\big(1+\lambdabar_{\xi\sigma}^2k^2\big)^{3/2}}\vec{e}_z\qquad\lambdabar_{\xi\sigma}=\frac{\hbar v_D}{\Delta_{\xi\sigma}},
\end{equation}
where $\vec{e}_z$ denotes the unit vector in the $z$-direction. The last expression $\lambdabar_{\xi\sigma}$ represent the characteristic length scale, which we have already mentioned in the previous subsection and that yields the order of magnitude for the displacement and thus the dipole as a consequence of projection onto a single band. It is inversely proportional to the band gap $\Delta_{\xi\sigma}$ and constitutes a lower bound for all length scales. It is reminiscent of the Compton length in high-energy physics \cite{Compton,greiner2000}. Indeed, if we rewrite the gap in terms of the band masses $m_{\xi\sigma}$, $\Delta_{\xi\sigma}=m_{\xi\sigma}v_D^2$, one retrieves its more familiar form $\lambdabar_{\xi\sigma} =\hbar/m_{\xi\sigma}v_D$. Physically it represents a limiting length below which the Compton effect transforms erratically photons into electron-positron pairs, so that information encoded in the phase of the light field can no longer be used for spectroscopic means. In condensed-matter physics, the interpretation of this length is similar: processes of characteristic length scales below $\lambdabar_{\xi\sigma}$ inevitably yield interband transitions that drive the system out of the regime of validity of the adiabatic approximation, which provided us with the semi-classical equations of motion (\ref{eq:semicl}).
For transport properties, including superconductivity, the most important electrons are those in the vicinity of the Fermi level, which we consider here to be close to the bottom of the conduction band, \textit{i.e.} we consider a moderately doped semiconductor. We can already anticipate that the Berry curvature may play a role as long as the Fermi wave vector $k_F$ satisfies $\lambdabar_{\xi\sigma} k_F\ll 1$ since it vanishes algebraically for $\lambdabar_{\xi\sigma}\rightarrow \infty$ [see Eq. (\ref{eq:Berry})]. We therefore project the Hamiltonian (\ref{eq:hamDir}) onto the conduction-band bottom, $0<\delta E=E-\Delta_{\xi\sigma}-E_{\xi\sigma}^0\ll \Delta_{\xi\sigma}$ (see Fig. \ref{fig:01}), with the help of the Foldy-Wouthuysen transformation to keep track of the electric potential $V(\vec{r})$ \cite{FW}.
This yield the effective one-band Hamiltonian
\begin{eqnarray}\label{eq:hamPauli}
\nonumber
H &\simeq& E_{\xi\sigma}^0+\Delta_{\xi\sigma}+\frac{\hbar^2\vec{k}^2}{2m_D}+ V(\vec{r})\\
&&+\frac{\xi\sigma\lambdabar_{\xi\sigma}^2}{4}\Big(\vec{e}_z\times\vec{k}\Big)\cdot\vec{\nabla}_{\vec{r}}V+\frac{\lambdabar_{\xi\sigma}^2}{8}\vec{\nabla}_{\vec{r}}^2V ,
\end{eqnarray}
which, apart from the last term, is identical to the one (\ref{eq:hamPeierls}) which we have obtained with the help of the generalized Peierls subsitution if we make use of the expression (\ref{eq:Berry}) for the Berry curvature to lowest order in the wave vector and if we redefine the energy with respect to the band bottom. The last term may also be written in terms of the Berry curvature as
\begin{equation}
\frac{\lambdabar_{\xi\sigma}^2}{8}\vec{\nabla}_{\vec{r}}^2V(\vec{r})=\frac{1}{4}\big|\mathcal{B}_{\lambda,\xi\sigma}(0)\big|\vec{\nabla}^2_{\vec{r}}V(\vec{r})
\end{equation}
and corresponds to the Darwin term in high-energy physics. While it does not play any role in the semi-classical equations of motion, it is relevant namely at very short ranges and has been shown to strongly affect \textit{e.g.} the spectra of $s$-state excitions in 2D TMDC \cite{BerryExc1,BerryExc2,Trushin_2017,Hichri_2019}. This is best seen in the case of the 2D Coulomb potential in which case $\nabla^2_{\vec{r}}V=e^2\delta(\vec{r})/\epsilon$, \textit{i.e.} it is relevant for pair wave functions with a non-zero amplitude at the origin ($s$-wave states) such as the BCS wave functions, which we discuss below.
\section{Two-body problem: General case and Cooper pair}
\label{sec:2body}
With the Cooper-pair problem in mind, we now consider how the extra terms discussed within the one-particle picture presented in the preceding section evolves in the case of two electrons at the bottom of the conduction band $\lambda=+$ at the same energy. This choice to consider a Fermi level slightly above the bottom of the conduction band is perfectly arbitrary, but the results obtained in the following sections remain valid for Cooper pairs formed from holes in the valence band. We consider again the spin to be locked to the valley index so that there is only one effective label $\xi\sigma$, which we represent by the valley index ($\xi_1$ for the first electron and $\xi_2$ for the second one) to simplify the notations. Furthermore we consider a two-body potential $V$ that depends only on the relative position of the two electrons $\vec{r}_1-\vec{r}_2$, such as it is the case for the BCS potential.
\subsection{General case}\label{ssec:gen}
Because the two-body interaction potential only depends on the relative distance $\vec{\rho}=\vec{r}_1-\vec{r}_2$ between the electrons, we introduce relative and center-of-mass (CoM) coordinates. Since both electrons have the same mass, we have
\begin{align}
\text{Relative:}\qquad&\vec{\rho}=\vec{r}_1-\vec{r}_2\hspace{2cm}\vec{k}=\frac{\vec{k}_1-\vec{k}_2}{2} \\
\text{CoM:}\qquad&\vec{R}=\frac{\vec{r}_1+\vec{r}_2}{2}\hspace{2cm}\vec{K}=\vec{k}_1+\vec{k}_2,
\end{align}
Separation of the CoM and relative coordinates yields the Hamiltonian
\begin{widetext}
\begin{eqnarray}
\nonumber
H_{2e^-}&=&2\Delta_b+\frac{\hbar^2\vec{K}^2}{4m_D}+\frac{\hbar^2\vec{k}^2}{m_D}+V(\vec{\rho})+\frac{1}{4}\bigg(\vec{\Lambda}^{\xi_1,\xi_2}_{+}\big(\vec{K},\vec{k}\big)\times\vec{K}\bigg)\cdot\vec{\nabla}V(\vec{\rho})+\frac{1}{2}\bigg(\vec{\Lambda}^{\xi_1,\xi_2}_{-}\big(\vec{K},\vec{k}\big)\times\vec{k}\bigg)\cdot\vec{\nabla}V(\vec{\rho}) \\
&&+\frac{1}{2}\big|\mathcal{B}(0)\big|\vec{\nabla}^2V(\vec{\rho})
\qquad\text{with}\quad\vec{\Lambda}^{\xi_1,\xi_2}_\pm\big(\vec{K},\vec{k}\big)=\vec{\mathcal{B}}_{+,\xi_1}\bigg(\frac{1}{2}\vec{K}+\vec{k}\bigg)\pm\vec{\mathcal{B}}_{+,\xi_2}\bigg(\frac{1}{2}\vec{K}-\vec{k}\bigg) \label{eq:ham2body}
\end{eqnarray}
\end{widetext}
within the parabolic approximation, and where we have made use of the Dirac mass $m_D=\Delta_{\xi\sigma}/v_D^2$. Since we no longer consider $k$-space gradients, we omit the index $\vec{r}$ at the gradient $\nabla_{\vec{r}}=\nabla$ from now on. It is interesting to notice that, when moving to CoM/relative coordinates, the Berry dipole term splits into two dipoles acting on the electron pair. One is associated with its \emph{center-of-mass motion} and the \emph{sum} of the two Berry curvatures and the other is associated with its \emph{relative motion} and the \emph{difference} of the two Berry curvatures. To gain further insight into the physical meaning of these two terms, we can calculate the Heisenberg equations of motion
\begin{align}
&\overset{.}{\vec{K}} = \vec{0} \qquad
\hspace{9mm}
\overset{.}{\vec{R}} = \frac{\hbar\vec{K}}{2m_D}+\frac{1}{4\hbar}\vec{\nabla}V(\rho)\times\vec{\Lambda}^{\xi_1,\xi_2}_+(\vec{K},\vec{k}) \\
& \overset{.}{\vec{k}} = -\frac{1}{\hbar}\vec{\nabla}H_{2e^-}\qquad \overset{.}{\vec{\rho}}=2\frac{\hbar\vec{k}}{m_D}+\frac{1}{2\hbar}\vec{\nabla}V(\vec{\rho})\times\vec{\Lambda}^{\xi_1,\xi_2}_-(\vec{K},\vec{k})
\end{align}
The CoM momentum is a conserved quantity, owing to the fact that $H_{2e^-}$ does not depend on $\vec{R}$. We also see that the two dipoles induce two Karplus-Luttinger-type velocities: $\Lambda_+$, which is associated to the CoM dipole, generates a drift velocity of the CoM coordinate, and $\Lambda_-$, which is associated to the relative dipole, yields another drift velocity of the relative coordinate of the Cooper pair.
Before discussing the special case of the Cooper pair, we may already discuss here the relative role of the two quantities $\vec{\Lambda}_+$ and $\vec{\Lambda}_-$ as a function of the two different valleys, \textit{i.e.} in the case of \textit{intra-valley} pairing as compared to \textit{inter-valley} pairing. Indeed, they determine the dipolar moments
\begin{equation}
\vec{d}_{\pm}=-e(\vec{\Lambda}_{\pm}\times \vec{q})/2,
\end{equation}
where $\vec{q}=\vec{K}$ for the CoM dipole (sign $+$) and $\vec{q}=\vec{k}$ for the relative dipole (sign $-$). In the case of intra-valley pairing ($\xi_1=\xi_2$), which corresponds to triplet superconductivity as a consequence of the spin-valley locking, the relative dipole $\vec{d}_-$ is negligible to lowest order in the wave vectors while the CoM dipole is on the order of $\vec{d}_+\sim -e \mathcal{B}_{+,\xi_1}(0)\times \vec{K}$. Their roles are inverted in the case of singlet-type inter-valley pairing, in which case $\vec{d}_+\simeq 0$ while $\vec{d}_-\sim -e \mathcal{B}_{+,\xi_1}(0)\times \vec{k}$.
\subsection{Revisiting the Cooper problem}\label{ssec:Cooper}
We are now in a position to study the effect of the Berry curvature on a Cooper pair, the building block of superconductors. To do so, we revisit the Cooper problem following the lines of Ref. \cite{leonn.cooper1956} and standard textbooks \cite{tinkham2004introduction}. The Hamiltonian we consider here is $H_c= H_{2e^-}(\vec{K}=\vec{0})$, \textit{i.e.} our two-body Hamiltonian (\ref{eq:ham2body}) in the rest frame,
\begin{align}
H_{c}=2\epsilon_+(\vec{k})+V(\vec{\rho})&+\frac{1}{2}\bigg(\vec{\Lambda}^{\xi_1,\xi_2}_{-}\big(\vec{0},\vec{k}\big)\times\vec{k}\bigg)\cdot\vec{\nabla}V(\vec{\rho})\nonumber \\
&+\frac{1}{2}\big|\mathcal{B}(0)\big|\vec{\nabla}^2V(\vec{\rho}),
\end{align}
where $\Lambda_-$ can be rewritten as
\begin{equation}
\vec{\Lambda}_-^{\xi_1,\xi_2}(\vec{0},\vec{k})=-(\xi_1-\xi_2)\frac{\lambdabar_{\xi\sigma}^2}{2\big(1+\lambdabar_{\xi\sigma}^2k^2\big)^{3/2}}\vec{e}_z=\vec{\Lambda}_-^{\xi_1,\xi_2}(\vec{k}).
\end{equation}
As mentioned above, one notices that, for the Berry dipole term to be non-zero, the two electrons of the Cooper pair need to be taken in different valleys and thus with opposite spin, as it is usual for $s$-wave singlet superconductivity. In contrast to this, we have $\Lambda_+^{\xi_1,\xi_2}(0,\vec{k})\propto(\xi_1+\xi_2)$ \textit{i.e.} one needs electrons in the same valley, but even then, the intra-valley CoM dipolar term in the Hamiltonian vanishes unless $\vec{K}\neq 0$. We therefore consider henceforth only the relative dipolar term and the case of inter-valley pairing.
Let us now take a closer look at the wave function of the Cooper pair $\psi(\vec{\rho})$, which is a solution of $H_c\psi(\vec{\rho})=E\psi(\vec{\rho})$. We then decompose $\psi$ and $V$ in a Fourier series
\begin{align}
&\psi(\vec{\rho})=\sum_{\vec{k}}g_{\vec{k}}e^{i\vec{k}\cdot\vec{\rho}},\\
&V(\vec{\rho})=\sum_{\vec{k}\vec{k'}}V_{\vec{k}\vec{k'}}e^{i(\vec{k}-\vec{k'})\cdot\vec{\rho}}\quad.
\end{align}
Following the steps of Ref. \cite{tinkham2004introduction} we find the self-consistent equation
\begin{equation}
\big[E-2\epsilon_+(\vec{k})\big]g_{\vec{k}}=\sum_{\vec{k}'}V^{\text{eff}}_{\vec{k}\vec{k'}}g_{\vec{k'}}
\end{equation}
for the coefficients $g_{\vec{k}}$, in terms of the \textit{effective interaction}
\begin{equation}
\label{inteff}
V^{\text{eff}}_{\vec{k}\vec{k}'}=\Bigg[1+\frac{i}{2}\bigg(\vec{\Lambda}^{\xi_1,\xi_2}_{-}(\vec{k})\times\vec{k}\bigg)\cdot\vec{k}'-\frac{1}{2}\big|\mathcal{B}(0)\big|\big(\vec{k}-\vec{k}'\big)^2\Bigg]V_{\vec{k}\vec{k}'}.
\end{equation}
This equation is one of the main results of our paper. Qualitatively, we see that the two terms appear with opposite signs. The second term stems from the Berry dipole term in Hamiltonian (\ref{eq:ham2body}) and may increase or decrease the interaction potential and thus the strength of the Cooper pairing depending on the sign of $\vec{\Lambda}_-$. As for the last (Darwin) term, it is negative irrespective of the valley index, meaning that it tends to weaken the electron-electron interaction and thus the superconducting phase. On a more practical level, the above expressions tell us that the calculations for the energy of the Cooper pair in the presence of a Berry curvature are the same as in the conventional pairing case \cite{tinkham2004introduction}, but in terms of the effective interaction (\ref{inteff}).
In a second step we need to solve the self-consistency equation
\begin{equation}
\label{Cooper1}
\sum_{\vec{k}}\frac{\langle V^{\text{eff}}_{\vec{k}\vec{k'}}\rangle}{E-2\epsilon_+(\vec{k})}=1,
\end{equation}
where we have defined the average
\begin{equation}
\langle\mathcal{O}(\vec{k'})\rangle=\frac{\sum_{\vec{k'}}\mathcal{O}(\vec{k'})g_{\vec{k'}}}{\sum_{\vec{k'}}g_{\vec{k'}}}
\end{equation}
with respect to the weighting coefficients $g_{\vec{k}}$. The term $\langle V^{\text{eff}}_{\vec{k}\vec{k'}}\rangle$ may be rewritten as
\begin{widetext}
\begin{equation}
\langle V^{\text{eff}}_{\vec{k}\vec{k'}}\rangle=\bigg(1-\frac{1}{2}\big|\mathcal{B}(0)\big|\vec{k}^2\bigg)\langle V_{\vec{k}\vec{k'}}\rangle+\bigg(\big|\mathcal{B}(0)\big|\vec{k}+\frac{i}{2}\vec{\Lambda}_-^{\xi_1,\xi_2}(\vec{k})\times\vec{k}\bigg)\cdot\langle\vec{k'}V_{\vec{k}\vec{k'}}\rangle-\frac{1}{2}\big|\mathcal{B}(0)\big|\langle\vec{k'}^2V_{\vec{k}\vec{k'}}\rangle.
\end{equation}
\end{widetext}
To illustrate the role of the additional terms due to the Berry curvature, let us consider the BCS potential, defined as
\begin{equation}
V_{\vec{k}\vec{k'}}=
\begin{cases}
&-V<0\quad\text{if }\epsilon_F\leq\epsilon_+(\vec{k}),\epsilon_+(\vec{k'})\leq\epsilon_F+\hbar\omega_D \\
&0\quad\text{otherwise},
\end{cases}
\end{equation}
where $\epsilon_F$ is the Fermi energy and $\hbar\omega_D$ the Debye energy. We can compactly rewrite it as
\begin{equation}
V_{\vec{k}\vec{k'}}=-V\mathbbm{1}_{\mathcal{D}}(\vec{k})\mathbbm{1}_{\mathcal{D}}(\vec{k'})
\label{VBCS}
\end{equation}
where $\mathbbm{1}_{\mathcal{D}}$ is the indicator function of the set
\begin{equation}
\label{Dset}
\mathcal{D}=\Big\{\vec{k}\in\mathbb{R}^2\Big|\epsilon_F\leq\epsilon_+(\vec{k})\leq\epsilon_F+\hbar\omega_D\Big\}.
\end{equation}
With this in mind, we write
\begin{equation}
\label{sumPeierls}
\langle\vec{k'}V_{\vec{k}\vec{k'}}\rangle\propto\sum_{\vec{k'}\in\mathcal{D}}\vec{k'}V_{\vec{k}\vec{k'}}g_{\vec{k'}}
\end{equation}
From Eq. (\ref{VBCS}) we see that $V_{\vec{k};-\vec{k'}}=V_{\vec{k}\vec{k'}}$. Moreover, for BCS superconductivity we have $g_{-\vec{k'}}=g_{\vec{k'}}$ so that $\vec{k'}V_{\vec{k}\vec{k'}}g_{\vec{k'}}$ is an odd function of $\vec{k'}$. Because summing an odd function over the set $\mathcal{D}$ gives zero, we have $\langle\vec{k'}V_{\vec{k}\vec{k'}}\rangle=\vec{0}$ so that \emph{the Berry dipole term does not affect the Cooper pair}, which is then solely affected by the Darwin term. Therefore, if we remember the competition between the dipolar and Darwin terms, this suggests that the effect of the Berry curvature is to weaken the Cooper.
As for $\langle V^{\text{eff}}_{\vec{k}\vec{k'}}\rangle$, we are left with
\begin{equation}
\langle V^{\text{eff}}_{\vec{k}\vec{k'}}\rangle=\bigg(1-\frac{1}{2}\big|\mathcal{B}(0)\big|\vec{k}^2\bigg)\langle V_{\vec{k}\vec{k'}}\rangle -\frac{1}{2}\langle \big|\mathcal{B}(0)\big|\vec{k'}^2V_{\vec{k}\vec{k'}}\rangle
\end{equation}
Remember that $V^{\text{eff}}_{\vec{k}\vec{k'}}$ is non-zero only for $\vec{k},\vec{k'}\in\mathcal{D}$, and from the definition of $\mathcal{D}$ we rewrite the energy as $\epsilon_+(\vec{k})=\epsilon_F+\eta_{\vec{k}}\hbar\omega_D$ with $\eta_{\vec{k}}\in[0,1]$.
From this and the expression of $\epsilon_+(\vec{k})$ we obtain
\begin{equation}
\big|\mathcal{B}(0)\big|\vec{k}^2=\frac{\epsilon_F-\Delta_{\xi\sigma}}{\Delta_{\xi\sigma}}+\eta_{\vec{k}}\frac{\hbar\omega_D}{\Delta_{\xi\sigma}}.
\end{equation}
Now, for many 2D materials (including any TMDC), the band gap is in the 1eV range (see e.g. Ref. \cite{doi:10.1021/acs.jpclett.5b01686}) while for most crystals $\hbar\omega_D\sim 0.01$eV \cite{LI2012197}. One therefore obtains a ratio $\frac{\hbar\omega_D}{\Delta_b}\sim 0.01$, so that we may neglect the corresponding term and thus make the approximation
\begin{equation}
\label{approxberry} \big|\mathcal{B}(0)\big|k^2\simeq\big|\mathcal{B}(0)\big|k_F^2\qquad \big|\mathcal{B}(0)\big|k'^2\simeq\big|\mathcal{B}(0)\big|k_F^2.
\end{equation}
With this and $\langle\vec{k'}V_{\vec{k}\vec{k'}}\rangle=\vec{0}$, we finally obtain
\begin{equation}
\label{approxint}
\langle V^{\text{eff}}_{\vec{k}\vec{k'}}\rangle=\big(1-\big|\mathcal{B}(0)\big|k_F^2\big)\langle V_{\vec{k}\vec{k'}}\rangle,
\end{equation}
in line with our qualitative argument of a weakening of the electron-electron interaction induced by the Darwin term. With the BCS potential, $\langle V_{\vec{k}\vec{k'}}\rangle=-V$, one finds
\begin{equation}
\sum_{\vec{k}}\frac{1}{E-2\epsilon_+(\vec{k})}=-\frac{1}{\big(1-\big|\mathcal{B}(0)\big|k_F^2\big)V}.
\end{equation}
As usual, the sum over the wave vector may be replaced by an integral over energy with the help of the density of states $\rho(\epsilon)$ and the BCS coupling constant $\lambda= V\rho(\epsilon_F)$. We finally find the binding energy of the Cooper pair
\begin{equation}\label{eq:binding}
E_B=\frac{2\hbar\omega_D}{e^{2/\lambda_{\text{eff}}}-1}\quad\text{with}\quad\lambda_{\text{eff}}=\big(1-\big|\mathcal{B}(0)\big|k_F^2\big)\lambda,
\end{equation}
which is the same as the conventional expression
\begin{equation}
E_B^{\text{BCS}}=\frac{2\hbar\omega_D}{e^{2/\lambda}-1},
\end{equation}
where we have replaced $\lambda$ by an effective (lower) coupling constant. If we set the Berry curvature to zero or if we set the band gap to be infinity, we recover the usual expression, as expected.
To summarize this subsection, we highlight two aspects. First, the effect of the Berry curvature on the Cooper pair reveals itself through a competition between two terms. On the one hand, the Berry dipole term, with its dipolar/spin-orbit form, induces a drift velocity analogous to the Karplus-Luttinger veloctity on the relative position of the electrons of the Cooper pair. It \emph{could} in principle enhance the electron-electron interaction $V_{\vec{k}\vec{k'}}$. On the other hand, the Darwin term yields a negative contribution and thus weakens the effective interaction. Second, the Berry dipole term's contribution to Cooper pairing turns out to be zero for $s$-wave superconductivity, and thus we are only left with a weakened electron-electron interaction due to the Darwin term. This is clearly seen in the expression of the binding energy (\ref{eq:binding}) Indeed, since the interaction $V$ is lowered, so is the BCS coupling $\lambda$, thereby lowering the binding energy of the Cooper pair. In conclusion, the Berry curvature makes the Cooper pairs less bound and thus more easily breakable, \textit{e.g.} by thermal fluctuations. This means that the critical temperature (and the superconducting gap) are lowered as well, as we show explicitely in the following section, where we discuss the action of the Berry-curvature corrective terms in the BCS many-body approach.
\section{BCS Hamiltonian in the presence of Berry curvature}\label{sec:BCS}
In the previous section, we found that the calculations in the electron pair problem with Berry curvature were the same as in its absence, but with an effective interaction. We therefore consider, in this part, the BCS Hamiltonian where we replace the interaction $V_{\vec{k}\vec{k'}}$ with the effective one $V^{\text{eff}}_{\vec{k}\vec{k'}}$ which is given in Eq. (\ref{inteff}) and that accounts for the corrective terms due to the Berry curvature.
\begin{equation}
H=\sum_{\vec{k}\sigma}\xi_{\vec{k}}c^\dagger_{\vec{k}\sigma}c_{\vec{k}\sigma}+\sum_{\vec{k}\vec{k'}}V^{\text{eff}}_{\vec{k}\vec{k'}}c^\dagger_{\vec{k'}\uparrow}c^\dagger_{-\vec{k'}\downarrow}c_{\vec{k}\uparrow}c_{-\vec{k}\downarrow}
\end{equation}
where $\xi_{\vec{k}}=\epsilon_+(\vec{k})-\epsilon_F$, and the bare interaction (in the absence of Berry curvature corrections) is $V_{\vec{k}\vec{k'}}=-V\mathbbm{1}_{\mathcal{D}}(\vec{k})\mathbbm{1}_{\mathcal{D}}(\vec{k'})$ with $\mathcal{D}=\Big\{\vec{k}\in\mathbb{R}^2\Big|\epsilon_F-\hbar\omega_D\leq\epsilon_+(\vec{k})\leq\epsilon_F+\hbar\omega_D\Big\}$. We also keep the same groundstate $|\psi_G\langle$. Since this Hamiltonian has the same form as the original BCS Hamiltonian, the same calculations hold as long as the interaction is not specified. We thus find the textbook gap equation \cite{tinkham2004introduction}
\begin{equation}
\Delta_{\vec{k}}=-\frac{1}{2}\sum_{\vec{k'}}V^{\text{eff}}_{\vec{k}\vec{k'}}\frac{\Delta_{\vec{k'}}}{\sqrt{\Delta_{\vec{k'}}^2+\xi^2_{\vec{k'}}}}\tanh\bigg(\frac{\beta}{2}\sqrt{\Delta_{\vec{k'}}^2+\xi^2_{\vec{k'}}}\bigg)
\end{equation}
with $\Delta_{\vec{k}}=-\sum_{\vec{k'}}V^{\text{eff}}_{\vec{k}\vec{k'}}\langle c^\dagger_{\vec{k'}\uparrow}c^\dagger_{-\vec{k'}\downarrow}\rangle$ and $\beta=(k_BT)^{-1}$. In terms of the auxiliary function
\begin{equation}
\label{kernel}
f_{\beta,\vec{k}}(\vec{k'})=\frac{\Delta_{\vec{k'}}}{\sqrt{\Delta_{\vec{k'}}^2+\xi^2_{\vec{k'}}}}\tanh\bigg(\frac{\beta}{2}\sqrt{\Delta_{\vec{k'}}^2+\xi^2_{\vec{k'}}}\bigg),
\end{equation}
the self-consistent gap equation reads
\begin{widetext}
\begin{equation}
\Delta_{\vec{k}}=-\frac{1}{2}\bigg(1-\frac{1}{2}\big|\mathcal{B}(0)\big|\vec{k}^2\bigg)\sum_{\vec{k'}}V_{\vec{k}\vec{k'}}f_{\beta,\vec{k}}(\vec{k'})-\frac{1}{2}\bigg(\frac{i}{2}\vec{\Lambda}^{\xi_1,\xi_2}_-(\vec{k})\times\vec{k}+\big|\mathcal{B}(0)\big|\vec{k}\bigg)\cdot\sum_{\vec{k'}}\vec{k'}V_{\vec{k}\vec{k'}}f_{\beta,\vec{k}}(\vec{k'})-\frac{1}{2}\sum_{\vec{k'}}\frac{1}{2}\big|\mathcal{B}(0)\big|\vec{k'}^2V_{\vec{k}\vec{k'}}f_{\beta,\vec{k}}(\vec{k'})
\end{equation}
\end{widetext}
One can show that if the bare superconducting gap has a definite parity, then $\Delta_{\vec{k}}$ (so defined through the effective interaction) has the same parity. Therefore for BCS superconductivity we have $\Delta_{-\vec{k}}=\Delta_{\vec{k}}$. From equation (\ref{kernel}), it is then clear that $f_{\beta,\vec{k}}(-\vec{k'})=f_{\beta,\vec{k}}(\vec{k'})$. And since $V_{\vec{k};-\vec{k'}}=V_{\vec{k};\vec{k'}}$, the function $\vec{k'}\longrightarrow\vec{k'}V_{\vec{k}\vec{k'}}f_{\beta,\vec{k}}(\vec{k'})$ is an odd function so that
\begin{equation}
\sum_{\vec{k'}}\vec{k'}V_{\vec{k}\vec{k'}}f_{\beta,\vec{k}}(\vec{k'})=\vec{0},
\end{equation}
and thus the Berry dipole term does again not affect the many-body result, which is consistent with the results obtained in the previous section. We then make the same approximate treatment [see Eqs. (\ref{approxberry}) and (\ref{approxint})] as for the Cooper pair problem and we find
\begin{equation}
\Delta_{\vec{k}}=-\frac{1}{2}\sum_{\vec{k'}}\big(1-\big|\mathcal{B}(0)\big|k_F^2\big)V_{\vec{k}\vec{k'}}f_{\beta,\vec{k}}(\vec{k'}),
\end{equation}
in agreement with our previous result. the Berry curvature reduces the attractive electron-electron interaction due to the Darwin term.
We are now able to calculate the zero-temperature superconducting gap. At $T=0$, the gap equation is
\begin{equation}
\Delta_{\vec{k}}=-\frac{1}{2}\Big(1-\big|\mathcal{B}(0)\big|k_F^2\Big)\sum_{\vec{k'}}V_{\vec{k}\vec{k'}}\frac{\Delta_{\vec{k'}}}{\sqrt{\Delta_{\vec{k'}}^2+\xi_{\vec{k'}}^2}}
\end{equation}
We then use $V_{\vec{k}\vec{k'}}=-V\mathbbm{1}_{\mathcal{D}}(\vec{k})\mathbbm{1}_{\mathcal{D}}(\vec{k'})$ and have
\begin{equation}
\Delta_{\vec{k}}=\mathbbm{1}_{\mathcal{D}}(\vec{k})\frac{1}{2}\Big(1-\big|\mathcal{B}(0)\big|k_F^2\Big)V\sum_{\vec{k}\in\mathcal{D}}\frac{\Delta_{\vec{k'}}}{\sqrt{\Delta_{\vec{k'}}^2+\xi_{\vec{k'}}^2}}
\end{equation}
Thus $\Delta_{\vec{k}}=0$ for $\vec{k}\notin\mathcal{D}$, and then one can show directly that $\Delta_{\vec{k}}=\Delta$ for $\vec{k}\in\mathcal{D}$. The former case is trivially satisfied since if $\vec{k}\notin\mathcal{D}$, the corresponding electron is not subject to the attractive interaction so it cannot condense and participate in a SC state. The latter indicates that the gap is then isotropic for the electrons that are concerned by superconductivity. We may again follow the conventional derivation \cite{tinkham2004introduction} and find the $T=0$ superconducting gap
\begin{equation}
\label{gaplambdaeff}
\Delta(T=0)=\frac{\hbar\omega_D}{\sinh\big(1/\lambda_{\text{eff}}\big)}\quad\text{with}\quad\lambda_{\text{eff}}=\big(1-\big|\mathcal{B}(0)\big|k_F^2\big)\lambda
\end{equation}
with the same effective coupling constant $\lambda_{\text{eff}}$ as that obtained in the previous section [see Eq. (\ref{eq:binding})]. Comparing this to the bare BCS expression
\begin{equation}
\Delta^{\text{BCS}}(T=0)=\frac{\hbar\omega_D}{\sinh\big(1/\lambda\big)}
\end{equation}
we see the same result as in the Cooper pair problem, that is to say a lowering of the BCS coupling constant driven by the Berry curvature thereby lowering the $T=0$ superconducting gap. This is also consistent with what we said about the consequences for the Cooper pairs. Indeed, since the superconducting gap is smaller, so is the energy of the quasiparticles in the superconductor. This makes them more sensitive to variations of energy, e.g. thermal fluctuations. In other words, the superconducting phase is weakened and thus more easily suppressed upon raising temperature.
Similarly, the expression for the critical temperature takes the form \cite{tinkham2004introduction}
\begin{equation}\label{eq:TC}
T_c=2\hbar\omega_D\frac{e^\gamma}{\pi}e^{-1/\lambda_{\text{eff}}}
\end{equation}
and is identical to the standard one except for the fact that the coupling constant needs to be replaced by $\lambda\rightarrow \lambda_{\text{eff}}$ to take into account the extra terms due to the Berry curvature. Here, $\gamma\simeq0.577$ is the Euler-Mascheroni constant, and the approximation is valid if $2T_c\ll\hbar\omega_D/k_B=T_D$, and it is relatively reliable when $2T_c\lesssim T_D$. Notice finally, that the Berry curvature therefore does not affect the universality of the ratio between the superconducting gap and $T_c$ in the weak-coupling limit,
\begin{equation}
\frac{\Delta(T=0)}{k_BT_c}\underset{\lambda\ll1}{=}\frac{\pi}{e^\gamma}\simeq1.76.
\end{equation}
Indeed this ratio is independent of the (effective) coupling constant.
\section{Doping dependence}\label{sec:doping}
Until now, we considered a low-doping limit, in which the Fermi level is close to the bottom of the conduction band. This allowed us to approximate the Berry curvature as $\mathcal{B}(k)\simeq\mathcal{B}(0)$. At larger doping, we first expect a weakening of the inter-band effects since the relevant physics will take place farther away from the other band. We should then expect to recover the usual one-band BCS results as the Fermi energy increases. The main thing to change would be our extra terms. The Berry dipole term does not rely on the low-energy expansion of the Dirac Hamiltonian, and we thus do not need to change it. The Darwin term is different: we have obtained it by expanding the Dirac Hamiltonian in the low-energy/non-relativistic limit. In this limit, the Berry curvature enters as $\big|\mathcal{B}(0)\big|$. Since the physics is controlled by states near the Fermi energy, we change $\big|\mathcal{B}(0)\big|\longrightarrow\big|\mathcal{B}(k_F)\big|$, \textit{i.e.} the most important contribution of the Berry curvature is its value at the Fermi level. The effective coupling constant $\lambda_{\text{eff}}$ takes then the form
\begin{equation}
\label{deforlambda}
\lambda_{\text{eff}}=\big(1-\big|\mathcal{B}(k_F)\big|k_F^2\big)\lambda=\Bigg(1-\frac{\lambdabar_{\xi\sigma}^2k_F^2}{2\big(1+\lambdabar_{\xi\sigma}^2k_F^2\big)^{3/2}}\Bigg)\lambda,
\end{equation}
and we have
\begin{align}
\label{lowdopinglimit}
&\text{Low-doping limit: }\frac{\lambda_{\text{eff}}}{\lambda}\underset{\lambdabar_ck_F\ll1}{\sim}1-\frac{\lambdabar_{\xi\sigma}^2k_F^2}{2} \\
\label{highdopinglimit}
&\text{High-doping limit: }\frac{\lambda_{\text{eff}}}{\lambda}\underset{\lambdabar_{\xi\sigma}k_F\gg1}{\sim}1-\frac{1}{2\lambdabar_ck_F}
\end{align}
for the different limiting cases. As a consistency check, we recover the previous result in the low-doping limit (indeed, $\lambdabar_c^2/2=\big|\mathcal{B}(0)\big|$). In the high-doping limit, the effective coupling constant approaches its bare BCS value as the Fermi level goes to $+\infty$. This is consistent with our expectation of a decreased role of the corrective terms due to the Berry curvature and thus of the inter-band effects in this limit. The doping dependence of the coupling constant (i.e. on $\lambdabar_ck_F$) is depicted in Fig. \ref{dopingdependence}.
\begin{figure}[h!]
\centering
\includegraphics[width=0.36\textwidth]{lambda.png}
\caption{Effective coupling constant $\frac{\lambda_{\text{eff}}}{\lambda}$ as a function of $\lambdabar_{\xi\sigma}k_F$.}
\label{dopingdependence}
\end{figure}
\newline
It is apparent that the effective coupling constant has a minimum that can be shown to occur at $\lambdabar_ck_F=\sqrt{2}$. Therefore the effect of the Berry curvature on conventional BCS type ($s$-wave) superconductivity is expected to be strongest in an intermediate doping regime in which the Fermi wave vector is on the order of the inverse effective Compton length. We then have
\begin{equation}
\min_{\lambdabar_ck_F}\frac{\lambda_{\text{eff}}}{\lambda}=1-\frac{1}{3\sqrt{3}}\simeq81\%
\end{equation}
\textit{i.e.} the maximal reduction is approximately $19\%$. It is interesting to note that while the ratio goes to 1 as the Fermi level goes to $+\infty$, the difference does not go to zero. Indeed,
\begin{equation}
\underset{k_F\rightarrow+\infty}{\lim}\big[\lambda_{\text{eff}}-\lambda\big]=-\frac{AV}{4\pi\Delta_b}
\end{equation}
with $A$ the area of the Brillouin zone. Note that $V$ represents, here, the interaction energy per unit area in reciprocal space so that the quantity $AV$ itself is an energy and the coupling constant is dimensionless. While the reduction of the coupling constant seems rather limited, we must not forget that the critical temperature and the superconducting gap both depend exponentially on this coupling constant, so the effect could be quite substantial.
The central result of this paper is Eq. (\ref{deforlambda}). Indeed, from it ensues most of the results we had so far. Moreover, it could have several uses. First, doping could offer a way to experimentally observe the effects of a Berry curvature on a superconducting phase discussed in this paper. We present some possible paths for an experimental test of Berry-curvature effects on BCS superconductivity in Sec. \ref{sec:exp}. Second, while this specific deformation of the coupling constant may not be true for other types of band structures, these could still exhibit other types of deformations depending on the corrective terms of the one-body problem. If Eq. (\ref{deforlambda}) is true in other types of band structures, it can even be a way to detect the presence of a Berry curvature as well as its $k$-dependence.
\section{Beyond BCS superconductivity}\label{sec:beyond}
Now that we have studied the conventional $s$-wave case, let us see what happens with other types of superconductivity. As in the case for the $s$-wave case (see Sec. \ref{ssec:Cooper}), we first revisit the modified Cooper problem from a more general point of view following Ref. \cite{mineev_samokhin_1999}. We will then study the many-body BCS theory, this time following Refs. \cite{SigristUeda} and \cite{sigrist_2005}.
\subsection{Cooper problem}\label{ssec:Coopergen}
The 2-electron potential may be decomposed in the relative-angular momentum basis as \cite{mineev_samokhin_1999}
\begin{equation}
V_{\vec{k}\vec{k'}}=\sum_{l=0}^{+\infty}V_l(\vec{k},\vec{k'})
\end{equation}
with $V_l(-\vec{k},\vec{k'})=(-1)^lV_l(\vec{k},\vec{k'})=V_l(\vec{k},-\vec{k'})$, and the integer $l$ the angular momentum of the superconducting phase. It is even for singlet pairing and odd for triplet pairing. Let us pick a superconducting phase with fixed $l$, so that the pairing is either singlet or triplet. Then $V_{\vec{k}\vec{k'}}=V_l(\vec{k},\vec{k'})$. The same is done to $g_{\vec{k}}=g_l(\vec{k})$ with also $g_l(-\vec{k})=(-1)^lg_l(\vec{k})$. Equation (\ref{Cooper1}) becomes therefore
\begin{equation}
\sum_{\vec{k}}\frac{\langle V^{\text{eff}}_l(\vec{k},\vec{k'})\rangle_l}{E-2\epsilon_+(\vec{k})}=1,
\end{equation}
and we then proceed in the same way as before, expanding $\langle V^{\text{eff}}_l(\vec{k},\vec{k'})\rangle_l$ and considering
\begin{equation}
\langle\vec{k'}V_l(\vec{k},\vec{k'})\rangle_l\propto\sum_{\vec{k'}}\vec{k'}V_l(\vec{k},\vec{k'})g_l(\vec{k'}).
\end{equation}
If we then take $V_l(\vec{k},\vec{k'})$ to be non-zero only within a thin layer of energy around the Fermi level, with the energy cut-off $\epsilon_l$, and one retrieves Eq. (\ref{Dset}) but with $\mathcal{D}_l=\Big\{\vec{k}\in\mathbb{R}^2\big|\epsilon_F\leq\epsilon_+(\vec{k})\leq\epsilon_F+\epsilon_l\Big\}$. Because of the symmetry
\begin{align}
V_l(\vec{k},-\vec{k'})g_l(-\vec{k'})&=(-1)^lV_l(\vec{k},\vec{k'})(-1)^lg_l(\vec{k'})\nonumber\\
&=V_l(\vec{k},\vec{k'})g_l(\vec{k'}),
\end{align}
the function $\vec{k'}V_l(\vec{k},\vec{k'})g_l(\vec{k'})$ is odd in $\vec{k'}$ so that the sum over the set $\mathcal{D}_l$ yields zero. Since this term carries the Berry dipole term, we can conclude that \emph{the Berry dipole term does not contribute to the energy of the Cooper pair with pure singlet or triplet pairings}. Notice, however, that the Berry dipole term may nevertheless play a significant role in exotic superconductors that mix singlet and triplet pairing, as we sketch out in Sec. \ref{ssec:mixed}.
We then proceed with the same approximation as for the conventional $s$-wave case, which gives the effective interaction
\begin{equation}
\langle V^{\text{eff}}_l(\vec{k},\vec{k'})\rangle_l=\big(1-\big|\mathcal{B}(k_F)\big|k_F^2\big)\langle V_l(\vec{k},\vec{k'})\rangle_l.
\end{equation}
We also take the approach of \cite{mineev_samokhin_1999} and take $V_l(\vec{k},\vec{k'})=V_l(k,k')f(\hat{k},\hat{k'})$ with $V_l(k,k')=-V_l\mathbbm{1}_{\mathcal{D}_l}(\vec{k})\mathbbm{1}_{\mathcal{D}_l}(\vec{k'})$. This approach gives a binding energy $E_{B,l}$ given by
\begin{equation}
E_{B,l}=\frac{2\epsilon_l}{e^{2/\lambda_{\text{eff}}}-1}
\end{equation}
with $\lambda_{\text{eff}}=\big(1-\big|\mathcal{B}(k_F)\big|k_F^2\big)\lambda$, i.e. \emph{the result obtained for the conventional Cooper problem extends to all singlet and triplet pairings}.
\subsection{Many-body problem: generalized BCS theory}\label{ssec:BCSgen}
We now briefly address the many-body problem from a more general point of view, using the generalized BCS theory presented in Refs. \cite{sigrist_2005} and \cite{SigristUeda}. Its Hamiltonian is
\begin{equation}
H=\sum_{\vec{k}}\xi_{\vec{k}}c^\dagger_{\vec{k}\sigma}c_{\vec{k}\sigma}+\frac{1}{2}\sum_{\substack{\sigma_1\sigma_2\\\sigma_3\sigma_4}}\sum_{\vec{k}\vec{k'}}V^{\substack{\sigma_1\sigma_2\\\sigma_3\sigma_4}}_{\text{eff},\vec{k}\vec{k'}}c^\dagger_{\vec{k}\sigma_1}c^\dagger_{-\vec{k}\sigma_2}c_{-\vec{k'}\sigma_3}c_{\vec{k'}\sigma_4},
\end{equation}
with the effective interaction containing the Berry curvature corrections. The mean-field theory of this Hamiltonian gives rise to a $2\times2$ matrix $\widehat{\Delta}_{\vec{k}}$. As in the conventional case, one can prove that the dressed order parameter has the same parity as the bare one. Similarly to the Cooper problem, let us investigate a pairing that is either singlet or triplet. Then the gap equation has the form \cite{SigristUeda}
\begin{equation}
\Delta^{\sigma_1\sigma_2}_{\vec{k}}=-\sum_{\sigma_3\sigma_4}\sum_{\vec{k'}}V^{\substack{\sigma_2\sigma_1\\\sigma_3\sigma_4}}_{\text{eff},\vec{k}\vec{k'}}\mathscr{I}^{\sigma_3\sigma_4}_\beta(\vec{k'}),
\end{equation}
and the expansion of the effective interaction yields
\begin{widetext}
\begin{align}
\label{expandVgen}
&\Delta^{\sigma_1\sigma_2}_{\vec{k}}=-\bigg(1-\frac{1}{2}\big|\mathcal{B}(k_F)\big|\vec{k}^2\bigg)\sum_{\sigma_3\sigma_4}\sum_{\vec{k'}}V^{\substack{\sigma_2\sigma_1\\\sigma_3\sigma_4}}_{\vec{k}\vec{k'}}\mathscr{I}^{\sigma_3\sigma_4}_{\beta}(\vec{k'})-\Bigg(\frac{i}{2}\vec{\Lambda}^{\xi_1,\xi_2}_-(\vec{k})\times\vec{k}+\big|\mathcal{B}(k_F)\big|\vec{k}\Bigg)\cdot\sum_{\sigma_3\sigma_4}\sum_{\vec{k'}}\vec{k'}V^{\substack{\sigma_2\sigma_1\\\sigma_3\sigma_4}}_{\vec{k}\vec{k'}}\mathscr{I}^{\sigma_3\sigma_4}_{\beta}(\vec{k'})\nonumber\\
&-\frac{1}{2}\big|\mathcal{B}(k_F)\big|\sum_{\sigma_3\sigma_4}\sum_{\vec{k'}}\vec{k'}^2V^{\substack{\sigma_2\sigma_1\\\sigma_3\sigma_4}}_{\vec{k}\vec{k'}}\mathscr{I}^{\sigma_3\sigma_4}_{\beta}(\vec{k'}),
\end{align}
\end{widetext}
where the summand of the $\vec{k'}$-linear term is \begin{equation}
\sum_{\sigma_3\sigma_4}\vec{k'}V^{\substack{\sigma_2\sigma_1\\\sigma_3\sigma_4}}_{\vec{k}\vec{k'}}\mathscr{I}^{\sigma_3\sigma_4}_\beta(\vec{k'})
\end{equation}
We study two separate cases now. First, let us consider a unitary pairing, \textit{i.e.} one for which $\widehat{\Delta}_{\vec{k}}\widehat{\Delta}^\dagger_{\vec{k}}\propto\sigma_0$. This entails all singlet pairings and unitary triplet pairings (those without spin polarization). In that case, the kernel $\hat{\mathscr{I}}_{\beta}(\vec{k'})$ is given by \cite{SigristUeda,sigrist_2005}
\begin{equation}
\hat{\mathscr{I}}_\beta(\vec{k'})=\frac{\widehat{\Delta}_{\vec{k'}}}{2E_{\vec{k'}}}\tanh\bigg(\frac{\beta}{2}E_{\vec{k'}}\bigg).
\end{equation}
Since the order parameter generally obeys $\widehat{\Delta}_{-\vec{k}}=-\widehat{\Delta}^\top_{\vec{k}}$ and $E_{-\vec{k}}=E_{\vec{k}}$, we have
\begin{equation}
\mathscr{I}^{\sigma_3\sigma_4}_\beta({-\vec{k'}})=-\mathscr{I}^{\sigma_4\sigma_3}_\beta(\vec{k'}).
\end{equation}
Furthermore, in order to respect the anticommutation relations of the fermionic operators, the \cite{sigrist_2005} interaction must obey $V^{\substack{\sigma_2\sigma_1\\\sigma_3\sigma_4}}_{\vec{k};-\vec{k'}}=-V^{\substack{\sigma_2\sigma_1\\\sigma_4\sigma_3}}_{\vec{k}\vec{k'}}$. With this, we have
\begin{align}
\sum_{\sigma_3\sigma_4}-\vec{k'}V^{\substack{\sigma_2\sigma_1\\\sigma_3\sigma_4}}_{\vec{k};-\vec{k'}}\mathscr{I}^{\sigma_3\sigma_4}_\beta(-\vec{k'})&=-\sum_{\sigma_3\sigma_4}\vec{k'}V^{\substack{\sigma_2\sigma_1\\\sigma_4\sigma_3}}_{\vec{k}\vec{k'}}\mathscr{I}^{\sigma_4\sigma_3}_\beta(\vec{k'})\nonumber\\
&=-\sum_{\sigma_3\sigma_4}\vec{k'}V^{\substack{\sigma_2\sigma_1\\\sigma_3\sigma_4}}_{\vec{k}\vec{k'}}\mathscr{I}^{\sigma_3\sigma_4}_\beta(\vec{k'})
\end{align}
\textit{i.e.} the latter is odd in $\vec{k'}$. If one takes the interaction to be non-zero in a thin layer of energy around the Fermi level with energy cutoff $\epsilon_c$, the sum over the term that is linear in $\vec{k'}$ in Eq. (\ref{expandVgen}) vanishes again, a situation encountered several times in this paper. So the Berry dipole term does not change the gap and critical temperature for unitary pairings. As pointed out in the appendix, the latter is also valid for non-unitary triplet pairings. Therefore, \emph{the Berry dipole term does not change the gap equation for pure singlet and triplet pairings}.
\subsection{Possible situations in which the Berry dipole term may become relevant}\label{ssec:mixed}
In view of the above results, one may then wonder if there is any possible effect of the Berry dipole term on superconductivity. What we proved so far is that it does not change the SC gap or $T_c$ if the parity of the pairing is well defined. So a necessary condition for the Berry dipole term to actually contribute would be a superconducting phase without a fixed parity. We saw in the Cooper problem that the Berry dipole term drops out because the following sum is zero
\begin{equation}
\label{sumVg}
\sum_{\vec{k'}}\vec{k'}V_{\vec{k}\vec{k'}}g_{\vec{k'}}.
\end{equation}
If we decompose the two functions $V_{\vec{k}\vec{k'}}$ and $g_{\vec{k'}}$ in the sum of an even and an odd function
\begin{align}
V_{\vec{k}\vec{k'}}&=V^e_{\vec{k}\vec{k'}}+V^o_{\vec{k}\vec{k'}}\\
g_{\vec{k'}}&=g^e_{\vec{k'}}+g^o_{\vec{k'}}
\end{align}
and interpret the $e$ and $o$ parts respectively as the singlet and triplet parts, we then have
\begin{equation}
V_{\vec{k}\vec{k'}}g_{\vec{k'}}=V^e_{\vec{k}\vec{k'}}g^e_{\vec{k'}}+V^o_{\vec{k}\vec{k'}}g^o_{\vec{k'}}+V^o_{\vec{k}\vec{k'}}g^e_{\vec{k'}}+V^e_{\vec{k}\vec{k'}}g^o_{\vec{k'}}.
\end{equation}
While the first two terms disappear in Eq. (\ref{sumVg}) as they are even functions of $\vec{k'}$, the other two terms do \textit{a priori} not disappear as they are odd functions of $\vec{k'}$. $V^og^e$ may be interpreted as the interactions between triplet pairs in the presence of singlet pairs while $V^eg^o$ is the opposite. These two terms may then be an opportunity for the Berry dipole term to have a non-zero contribution in the superconducting phase, \textit{i.e.} if the latter shows coexistence between singlet and triplet pairs. We would then need a superconducting phase where none of the two dominates. Some materials have been proposed to exhibit two superconducting phases, each with a different parity, such as $\text{CeRh}_2\text{As}_2$ and bilayer-$\text{NbSe}_2$ \cite{doi:10.1126/science.abe7518,möckli2021unconventional,doi:10.1126/science.abe7518}. Notice furthermore that a very recent theoretical study argues that the observed superconducting phase in twisted bilayer graphene \cite{Cao2018} might be due to an admixture of singlet and triplet pairs \cite{Lake2022}, and the Berry dipole term might then be a relevant parameter in the stabilization of this type of superconductivity.
\section{Possible experimental implications of the Berry curvature on 2D BCS superconductivity}\label{sec:exp}
\begin{figure}[htbp]
\centering
\includegraphics[width=\linewidth]{Tcratiodop.png}
\caption{Ratio $T_c/T_c^{\text{BCS}}$ as a function of $\lambdabar_{\xi\sigma}k_F\propto \sqrt{n_{\text{2D}}}$. Here, we have used $AV/2\pi\Delta_{\xi\sigma}\simeq0.2$ for illustration.}.
\label{fig:my_label}
\end{figure}
As shown in Sec. \ref{sec:doping}, the Berry curvature has its strongest effect at Fermi wave vectors that are on the order of the inverse effective (Compton) length $\lambdabar_{\xi\sigma}$. Even if the relative reduction of the coupling constant is on the order of 19\%, one needs to keep in mind that the experimentally measurable superconducting gap and critical temperature depend exponentially on the coupling constant. Indeed, the former is accessible by spectroscopic means, \textit{e.g.} in scanning-tunneling spectroscopy, and the latter within resistive temperature-dependent measurements. Experimentally, it is likely impossible to change the Berry curvature \textit{in situ} because this would require experimental access to the band parameters, such as the direct band gap in 2D TMDC. While one could hope to change it \textit{e.g} under strain, also the phonon spectrum and the electron-phonon coupling would then change, possibly in an uncontrolled manner, thus excluding a direct measurement of the Berry-curvature effect in superconductivity.
However, one may compare the evolution of the Berry-curvature dependent superconducting gap or critical temperature, measured as a function of doping, to the \textit{expected} behavior of these quantities. Direct comparison of the critical temperature $T_c$ in Eq. (\ref{eq:TC}), in terms of the effective coupling constant (\ref{eq:binding}), yields the ratio
\begin{equation}\label{eq:ratioTC}
\frac{T_c}{T_c^{\text{BCS}}}=\exp\Bigg(-\frac{2\pi\Delta_{\xi\sigma}}{AV}\frac{\big|\mathcal{B}(k_F)\big|k_F^2}{\Big(1-\big|\mathcal{B}(k_F)\big|k_F^2\Big)\sqrt{1+\lambdabar_{\xi\sigma}^2k_F^2}}\Bigg),
\end{equation}
where $T_c^{\text{BCS}}$ is the BCS critical temperature in the absence of Berry-curvature terms.
We notice here the clear competition between the Berry curvature (through the gap) and superconductivity (through the attractive interaction $V$.). The ratio (\ref{eq:ratioTC}) is plotted in Fig. \ref{fig:my_label} as a function of the doping-dependent Fermi wave vector, $k_F=\sqrt{(4\pi/g) n_{\text{2D}}}$, in terms of the induced 2D electronic density $n_{\text{2D}}$. The factor $g$ takes into account the degeneracy due to internal degrees of freedom, such as the valley and the spin. Notice that, in 2D TMDC with a prominent spin-orbit coupling, the valley and spin degrees of freedom are generically locked, as mentioned above. One would therefore expect $g=2$ in these materials. This is likely the case in the valence band, with a spin-orbit splitting on the order of $\sim 100$ meV, while it is only in the $\sim 1...10$ meV range in the conduction band. The reduction of the critical temperature is strongest at the minimum, which occurs at $\lambdabar_{\xi\sigma}k_F\simeq1.05$. This corresponds to an electronic density of
\begin{equation}
n_{2\text{D}}=\frac{g}{4\pi}k_F^2\simeq1.1\frac{g}{4\pi}\lambdabar_{\xi\sigma}^{-2}.
\end{equation}
We can then give an approximation of the minimum of the ratio as
\begin{equation}
\min_{k_F}\frac{T_c}{T_c^{\text{BCS}}}\simeq\exp\bigg(-0.15\frac{2\pi\Delta_{\xi\sigma}}{AV}\bigg).
\end{equation}
\section{Conclusions}
In conclusion, we have studied the effect of the Berry curvature on BCS-type superconductors in 2D electronic systems. We have shown that the two-body Hamiltonian for interacting electrons inherits terms that are linear in the Berry curvature and that are inherited from the single-electron band structure. In this case, the Berry curvature, which arises in the adiabatic limit when the electrons are restricted to a single band due to purely virtual transitions to the other bands, is coupled to electric potentials beyond the periodic one, which gives rise to the Bloch bands. While such potentials may arise due to external electric fields, they naturally arise when interactions between the electrons (or holes) are taken into account. Generically, the Berry curvature provides a dipolar structure to the charged pairs, and one of the terms emerging in the two-body Hamiltonian can indeed be interpreted as a dipole in an electric field. A second term emerges in the form of a Darwin term, in which the Berry curvature couples to the Laplacian of the electric potential. This term is best understood within a relativistic treatment of the (massive) Dirac Hamiltonian that mimics the two adjacent bands in a direct-gap semiconductor.
Following the lines of the usual BCS treatment of superconductivity in the weak-coupling limit, we have shown that the latter Darwin term generally lowers the BCS coupling constant. As a consequence, this lowers also the stability of the Cooper pair so that the superconducting gap and critical temperature are decreased. On the contrary, the dipolar term, which potentially has the power to increase superconductivity, does not affect the superconducting properties in an $s$-wave or any pure singlet or triplet superconductor because of their fixed parity. The dipolar term might then play a role in systems where superconducting phases of different parity coexist or where the superconducting order parameter does not have a fixed parity. This path might be explored in future work, but it is beyond the scope of our present paper.
Interestingly, the gap-to-$T_c$ ratio remains the same as in the conventional BCS theory in the weak-coupling limit, that we have considered here. Upon doping, the reduction of BCS superconductivity is strongest when the Fermi wave vector is on the order of the inverse effective Compton length, $k_F\sim\lambdabar_{\xi\sigma}^{-1}$, where the BCS coupling constant is lowered by $19\%$. Indeed, for stronger doping, the Fermi level is situated at wave vectors, where the Berry curvature rapidly tends to zero. Since the superconducting gap and the critical temperature both depend exponentially on the BCS coupling constant, the relatively weak reduction of the coupling constant is more prominent there. Our calculations show that the reduction of the doping-dependent superconducting gap and critical temperature depends then both on the band gap, which determines the value of the Berry curvature, as well as on the effective electron-electron interaction. The experimental measurement of these quantities in 2D materials upon doping might then provide a test of our theoretical studies if compared to the expected evolution predicted by the usual BCS theory in the absence of Berry-curvature corrections.
\section*{Acknowledgements}
We thank J. Meyer and A. Mesaros for valuable discussions.
\bibliographystyle{apsrev4-2}
\section{Introduction}
A large variety of superconducting materials can be theoretically understood within the standard BCS theory proposed by Bardeen, Cooper and Schrieffer \cite{BCSsupra57,tinkham2004introduction}. Within this framework, the metallic electrons of a single, partially filled band are considered to be bound into (Cooper) pairs by a weak attraction, while other bands are discarded based on the premise that they are much more remote in energy than the typical energy scale set by the attractive interaction. Indeed, the attractive interaction between electrons is, within the standard BCS theory, mediated by phonons via the electron phonon coupling. Within the weak-coupling limit, the typical energy scale for superconductivity is then a fraction of the Debye temperature $k_BT_D$ that is itself in the $10...100$ meV range, while the Fermi energy and the typical band gaps are on the order of $\sim 1$ eV \cite{tinkham2004introduction}. In spite of its great success, BCS theory is not capable of explaining all occurrences of superconductivity and finds severe limitations \textit{e.g.} in the case of strongly correlated materials, such as heavy-fermion superconductivity \cite{HF1,reviewheavyfermion} or high-$T_c$ superconductivity \cite{reviewhighTc}, where even the origin of the attractive interaction is still debated.
While the above-mentioned energy-scale consideration has remained unchallenged for a long time, the advent of topological band theory \cite{TBT,cayssolfuchs} and its success in the theoretical description of a plethora of materials \cite{alltopoallmate}, such as topological insulators \cite{TI1,TI2}, topological superconductors \cite{bernevigBook,TSC}, Weyl and Dirac semimetals \cite{WSM}, has shown that the coupling between energy bands is not only governed by energy scales but by more subtle geometric quantities, such as the Berry curvature or the quantum metric. Several recent papers have investigated the role of the latter, namely in the presence of flat bands in which the quantum metric can be the dominant contribution to the superfluid weight \cite{peotta_superfluidity_2015,rossi2021quantum,tormaberneivig,tian2021evidence}. The Berry curvature has been theoretically shown to play a relevant role in a two-body problem that is closely related to the Cooper pair, namely in the physics of excitons. For example, in two-dimensional (2D) semiconducting transition-metal dichalcogenides (TMDC) \cite{TMDC}, excitons -- bound electron-hole pairs -- are formed in the vicinity of the $K$ and $K'$ points of the first Brillouin zone, where the Berry curvature reaches its maximal value \cite{Fuchs2010}. Experimentally, a first hint to the relevance of band-geometric effects came from the failure of the effective hydrogen model, which had been extremely successful before in the theoretical understanding of the measured exciton spectra \cite{exciton1,exciton2}. It was later shown that the Berry curvature affects the exciton spectra, contrary to the one-particle case, because it couples to the electric field that is generated by the attractive interaction between the electron and the hole forming the bound exciton state \cite{BerryExc1,BerryExc2,Trushin_2017,Hichri_2019}. This is a consequence of the intrinsic Dirac character of the low-energy charge carriers in these materials, which are commonly described in terms of a 2D massive Dirac equation \cite{xiao,GoerbigEPL}. Excitons in 2D TMDC and potentially other bound pairs inherit then this Dirac character \cite{Trushin_2016}.
Based on the above-mentioned exciton example, it is therefore natural to consider that the Berry curvature might also affect the formation of the Cooper pair due to the mutual interaction between the two electrons. This is the main motivation of the present theoretical study, where we show that the effective electron-electron interaction is generically weakened when one includes energy terms in the Hamiltonian that take into account the effect of the Berry curvature. We consider conventional BCS-type superconductivity in 2D materials, such as the above-mentioned 2D semiconducting TMDC for a moderate doping range. We emphasize that we do not investigate topological superconductivity \cite{TSC} that arises when one considers the quasiparticle bands, the mutual coupling of which is at the origin of the emergent topological properties. Here, we rather treat the role of the Berry curvature, which affects the formation of Cooper pairs in conventional BCS theory. Within topological band theory, the related wave-vector ($\vec{k}$) dependent Berry connection $\mathcal{A}_n(\vec{k})$ modifies the electrons' positions $\vec{r}$ when the latter are projected by the projectors $P_n$ to the $n$-th band, $\vec{r}\rightarrow P_n \vec{r} P_n=\vec{r}+\mathcal{A}_n(\vec{k})$. This yields a dipole that interacts with the electric field, and this dipolar structure, which the Cooper pair inherits, is at the origin of the weakened Cooper pairing. More precisely, the projection yields two extra terms which affect the electron-electron interaction to the one-body Hamiltonian. One of them is reminiscent of the spin-orbit coupling if one interprets the Berry curvature in terms of a spin, and the second one corresponds to the Darwin term, which arises within a Dirac-fermion treatment of the two bands in the vicinity of the direct gap \cite{FW}. We show that the latter is responsible for a reduced effective BCS coupling constant that results in a smaller superconducting BCS gap, while the former spin-orbit-type term does not play a role in $s$-wave nor other types of pure singlet or triplet pairing.
The paper is organized as follows. In Sec. \ref{sec:1body}, we briefly revisit, along the lines exposed in Ref. \cite{Hichri_2019}, the emergence of corrective terms to the one-body Hamiltonian of a charge projected to a single band. We present two complementary approaches: one based on a generalized version of the Peierls substitution in Sec. \ref{ssec:Peierls} and one based on a treatment within the continuum two-band model of massive Dirac fermions in the vicinity of the direct gap, where the role of the Berry curvature is most prominent. This treatment is the basis of the two-body problem, which we present in Sec. \ref{sec:2body}. After some general considerations (Sec. \ref{ssec:gen}), Sec. \ref{ssec:Cooper} shows how the Cooper pair and its binding energy are modified by the extra terms, while Sec. \ref{sec:BCS} presents the BCS theory of conventional $s$-wave-type superconductivity in the presence of the corrective terms due to the Berry curvature. In the calculations, we consider a Fermi level that is extremely close to the conduction-band bottom, and we discuss then the role of stronger doping on Cooper pairing and BCS superconductivity in Sec. \ref{sec:doping}. In Sec. \ref{sec:beyond}, we briefly discuss how our theoretical picture of superconductivity in the presence of non-zero Berry curvature evolves in other pairing symmetries, be they singlet or triplet. The last section (Sec. \ref{sec:exp}) is devoted to possible experimental implications of our theoretical studies. There, we compare the superconducting gap and the critical temperature in the absence and the presence of the weakened interaction due to the Berry curvature.
\section{One-body Hamiltonian: corrective terms due to the Berry curvature}
\label{sec:1body}
Before discussing the role of possible geometric terms on the superconducting properties of a 2D material, let us briefly revisit the emergence of these terms within a one-particle description. More precisely, we consider a band structure with $N$ bands described by the Bloch Hamiltonian. The Berry curvature may be viewed as the action of virtual interband transitions of electrons that are otherwise restricted to a single band, while there are no true (quantum) transitions in the adiabatic limit. Notice that there are no geometric terms in the Hamiltonian in the absence of a local electric potential $V(\vec{r})$ different from the periodic one that gives rise to the Bloch bands, and the Hamiltonian is then reduced to the bare band dispersion $E_n(\vec{k})$ of the $n$-th band which the electrons are projected to.
In the presence of a local potential $V(\vec{r})$ which acts on our single electron, the simple reduction of the Hamiltonian to the band dispersion is no longer valid -- in the following we consider this potential to be generated by the second electron to which the first one is bound in a Cooper pair, but our arguments are not restricted to this case. Indeed, $V(\vec{r})$ couples directly the different bands and needs thus to be taken into account prior to the adiabatic projection to a single band. This yields extra terms to the Hamiltonian that can be discussed within two complementary approaches that we briefly review in this section. The first one is based on a generalized Peierls substitution \cite{gosselin_menas_berard_mohrbach_2006,PhysRevLett.115.166803,Chang2008BerryCO,Gosselin_2008,Trushin_2017,Hichri_2019}. It yields a corrected (quantum) Hamiltonian that reproduces the semi-classical equations of motion. This approach has the advantage of providing a transparent physical interpretation of the role played by the Berry curvature, namely in the formation of a \textit{dipole-like} term that arises due to the projection to a single band. This approach is similar to the magnetic-field case when the electron motion is restricted to a single Landau level \cite{LLdipole0,LLdipole1}, but it does not provide all corrective terms, even at linear order in the Berry curvature. In order to obtain the missing term, which is analogous to the Darwin term in relativistic quantum mechanics, we interpret the Berry curvature in terms of a two-band model, which describes the band structure locally in reciprocal space in terms of a massive Dirac Hamiltonian.
\subsection{Generalized Peierls substitution: emergence of the Berry dipole}
\label{ssec:Peierls}
Let us first recall how to incorporate the magnetic field to describe the dynamics of an electron in the $n$-th band $E_n(\vec{k})$ via the Peierls substitution (in the absence of a Berry curvature). Because the wave vector $\vec{k}=-i\nabla_{\vec{r}}$ is not a gauge-invariant quantity, it needs to be replaced by its gauge-invariant form
\begin{equation}\label{eq:PeierlsK}
\hbar\vec{k}\longrightarrow\vec{\Pi}=\hbar\vec{k}+e\vec{A}(\vec{r}),
\end{equation}
in terms of the vector potential $\vec{A}(\vec{r})$ which yields the magnetic field, $\vec{B}(\vec{r})=\vec{\nabla}_{\vec{r}}\times\vec{A}(\vec{r})$. We consider, here, electrons of charge $-e$ ($e>0$). From a semi-classical point of view, one obtains the equations of motion
\begin{equation}
\dot{\vec{r}}_n=\vec{v}_n= \frac{1}{\hbar}\nabla_{\vec{k}}E_n \qquad \text{and}\qquad
\hbar\dot{\vec{k}}=-e\vec{v}_n \times \vec{B},
\end{equation}
where $\vec{r}_n$ and $\vec{v}_n$ are the average position and velocity, respectively, of the electron in the $n$-th band. One justification of the Peierls substitution is that the Hamiltonian thus obtained, $H(\vec{\Pi})=E_n(\vec{\Pi})$, yields the same equations of motion if one uses the \textit{quantum} Heisenberg equations of motion
\begin{equation}
i\hbar \dot{\Pi}_j=[\Pi_j,H(\vec{\Pi})],
\end{equation}
with the help of the commutation relations $[\Pi_x,\Pi_y]=-i\hbar^2/l_B^2$, in terms of the magnetic length $l_B=\sqrt{\hbar/eB}$. Indeed, one then obtains
\begin{equation}
\dot{\Pi}_j=-\frac{\hbar}{l_B^2} \epsilon_{jl} \frac{\partial H}{\partial \Pi_l},
\end{equation}
where $\epsilon_{jl}$ is the antisymmetric Levi-Civita tensor. The quantum Hamiltonian $H(\vec{\Pi})$ yields therefore Heisenberg equations of motion that are the same as the semi-classical ones if we identify the (semi-classical) wave vector $\vec{k}$ with the gauge-invariant quantity $\vec{\Pi}/\hbar$, as it is precisely stipulated by the Peierls substitution.
The generalized Peierls substitution follows the same spirit when considering a system with a non-zero Berry curvature in the presence of a spatially varying potential $V(\vec{r})$, thus starting from the band energy $H_n=E_n(\vec{k})+V(\vec{r})$. In this case, the semi-classical equations of motion read \cite{Niu,cayssolfuchs}
\begin{eqnarray}\label{eq:semicl}
\dot{\vec{r}}_n = \vec{v}_n &=& \frac{1}{\hbar}\nabla_{\vec{k}}E_n + \frac{1}{\hbar} \nabla_{\vec{r}}V(\vec{r})\times\vec{\mathcal{B}}_n(\vec{k}) \\
\text{and}\qquad \hbar\dot{\vec{k}} &=& -\nabla_{\vec{r}}V-e\vec{v}_n \times \vec{B},
\end{eqnarray}
where $\vec{\mathcal{B}}_n(\vec{k})=\nabla_{\vec{k}}\times \mathcal{A}_n(\vec{k})$ is the Berry curvature of the $n$-th band in terms of its Berry connection $\mathcal{A}_n(\vec{k})$. Similarly to the case discussed above, one can obtain these equations of motion from a \textit{quantum} Hamiltonian \begin{equation}\label{eq:hamKR}
H(\vec{\Pi},\vec{R})=E_n(\vec{\Pi})+V(\vec{R}),
\end{equation}
where we have replaced not only the wave vector by its gauge-invariant expression (\ref{eq:PeierlsK}) but also the position by its expression projected onto the $n$-th band \cite{Sundaram1999,Niu,cayssolfuchs}
\begin{equation}\label{eq:PeierlsR}
\vec{r}\longrightarrow\vec{R}=\vec{r}+\vec{\mathcal{A}}_n(\vec{k}),
\end{equation}
which involves the Berry connection $\mathcal{A}_n(\vec{k})$. Similarly to the Peierls substitution (\ref{eq:PeierlsK}), the position $\vec{r}$ on the right-hand-side of this expression should be interpreted as a reciprocal-space derivative $\vec{r}=i\nabla_{\vec{k}}$. The replacement (\ref{eq:PeierlsR}) may be viewed as a \textit{generalized Peierls substitution} \cite{gosselin_menas_berard_mohrbach_2006,PhysRevLett.115.166803,Chang2008BerryCO,Gosselin_2008,Hichri_2019}. The semi-classical equations of motion are then retrieved as the Heisenberg equations of motion not only for $\vec{\Pi}$ but also for $\vec{R}=(X,Y)$ on the basis of the Hamiltonian (\ref{eq:hamKR}) and the induced commutation relations $[X,Y]=i\mathcal{B}_n(\vec{k})$ \cite{Hichri_2019}.
Let us now discard the magnetic field, which we have only discussed in order to remind the reader of the Peierls substitution and to justify its generalized form and expand the Hamiltonian (\ref{eq:hamKR}) to lowest order in the Berry connection. This expansion is legitimate as long as the external potential $V(\vec{r})$ varies slowly on a length scale that is set, in orders of magnitude, by the Berry connection and that can be related to an effective Compton length, as we discuss below. The Hamiltonian then becomes
\begin{equation}\label{eq:hamPeierls}
H=E_{n}(\vec{k})+V(\vec{r})+\vec{\mathcal{A}}_n(\vec{k})\cdot\vec{\nabla}_{\vec{r}}V(\vec{r}).
\end{equation}
The last generated term is interesting. First, it can be interpreted as the energy of an electric dipole $-e\vec{\mathcal{A}}_n(\vec{k})$ in an electric field $\vec{E}(\vec{r}) = \nabla V(\vec{r})/e$. We therefore call this term the \textit{Berry dipole term}.
Second, this term can be understood as an effective spin-orbit coupling if we use the \textit{symmetric gauge} for the Berry connection
\begin{equation}
\vec{\mathcal{A}}_n(\vec{k})=\frac{1}{2}\vec{\mathcal{B}}_n(\vec{k})\times\vec{k},
\end{equation}
in which case the corrective term reads
\begin{equation}
\vec{\mathcal{A}}_n(\vec{k})\cdot\vec{\nabla}_{\vec{r}}V(\vec{r})=\frac{1}{2}\Big(\vec{\mathcal{B}}_n(\vec{k})\times\vec{k}\Big)\cdot\vec{\nabla}_{\vec{r}}V(\vec{r}).
\end{equation}
This expression is interesting for the following reason. The Berry curvature is often viewed as the analogue of a magnetic field in reciprocal space, while the extra term in Eq. (\ref{eq:hamPeierls}) has the same form as the spin-orbit coupling term, which arises when one projects the relativistic Dirac equation onto the electron (or positron) branch \cite{greiner2000}. In this analogy, one would however need to identify the Berry curvature with an emergent spin rather than with a magnetic field.
\subsection{Non-relativistic limit of the Dirac equation}
\label{ssec:Dirac}
\begin{figure}[h!]
\label{figbandes}
\centering
\includegraphics[width=0.3\textwidth]{structurebandes2.png}
\caption{Band structure of massive Dirac fermions, with \textit{a priori} two different gaps for the two values of $\xi\sigma$, as one typically encounters in 2D semiconducting TMDC.}
\label{fig:01}
\end{figure}
In many situations the role of the Berry curvature in semiconducting materials can be approached in terms of a massive Dirac equation that describes two coupled bands in the vicinity of a reciprocal-space point, where the band gap is smallest and the Berry curvature has a maximum \cite{DiracBerry,Fuchs2010}. In this picture, coupling to other bands is not \textit{per se} excluded, but we consider that it only gives rise to a negligible contribution to the respective Berry curvatures of the two bands. This situation arises, \textit{e.g.}, in 2D semi-conducting TMDC in which two spin-orbit coupled families of band pairs form a direct gap at the $K$ and $K'$ points. In the vicinity of these points, the two bands are described by the generic Dirac Hamiltonian
\begin{equation}\label{eq:hamDir}
H=
\begin{pmatrix}
\Delta_{\xi\sigma} \sigma_0 & \hbar v_D(\xi\sigma k_x-ik_y) \\
\hbar v_D(\xi\sigma k_x +i k_y) & -\Delta_{\xi\sigma}\sigma_0
\end{pmatrix} + E_{\xi\sigma}^0 + V(\vec{r}),
\end{equation}
where $\xi$ indicates the valley index ($\xi=+$ for the $K$ valley and $\xi=-$ for the $K'$ valley in the case of 2D TMDC, or generally two time-reversal-symmetry related points $\pm \vec{k}_D$) and $\sigma=\pm$ represents the physical spin. In the presence of spin-orbit coupling and time-reversal symmetry, the band gaps $2\Delta_{\xi\sigma}$ of the two valleys are locked and depend only on the product $\xi\sigma$ of the spin and valley index, and so does the shift in energy $E_{\xi\sigma}^0$, which does not play any topological or dynamical role. In the absence of the external potential $V(\vec{r})$, one obtains the four bands
\begin{equation}
\epsilon_{\lambda,\xi\sigma}(\vec{k})=E_{\xi\sigma}^0+\lambda \sqrt{\Delta_{\xi\sigma}^2+\big(\hbar v_Dk)^2},
\end{equation}
which is depicted in Fig. \ref{fig:01}. The index $\lambda $ refers to the conduction ($\lambda=+$) and the valence ($\lambda=-$) bands. Note that there are only four bands since spin and valley are locked -- they enter into the expressions only as the product label $\xi\sigma$ -- as it is required by time-reversal symmetry. The associated Berry curvatures are given by \cite{DiracBerry,Niu}
\begin{equation}\label{eq:Berry}
\vec{\mathcal{B}}_{\lambda,\xi\sigma}(\vec{k})=-\frac{\lambda\xi\sigma}{2}\frac{\lambdabar_{\xi\sigma}^2}{\big(1+\lambdabar_{\xi\sigma}^2k^2\big)^{3/2}}\vec{e}_z\qquad\lambdabar_{\xi\sigma}=\frac{\hbar v_D}{\Delta_{\xi\sigma}},
\end{equation}
where $\vec{e}_z$ denotes the unit vector in the $z$-direction. The last expression $\lambdabar_{\xi\sigma}$ represent the characteristic length scale, which we have already mentioned in the previous subsection and that yields the order of magnitude for the displacement and thus the dipole as a consequence of projection onto a single band. It is inversely proportional to the band gap $\Delta_{\xi\sigma}$ and constitutes a lower bound for all length scales. It is reminiscent of the Compton length in high-energy physics \cite{Compton,greiner2000}. Indeed, if we rewrite the gap in terms of the band masses $m_{\xi\sigma}$, $\Delta_{\xi\sigma}=m_{\xi\sigma}v_D^2$, one retrieves its more familiar form $\lambdabar_{\xi\sigma} =\hbar/m_{\xi\sigma}v_D$. Physically it represents a limiting length below which the Compton effect transforms erratically photons into electron-positron pairs, so that information encoded in the phase of the light field can no longer be used for spectroscopic means. In condensed-matter physics, the interpretation of this length is similar: processes of characteristic length scales below $\lambdabar_{\xi\sigma}$ inevitably yield interband transitions that drive the system out of the regime of validity of the adiabatic approximation, which provided us with the semi-classical equations of motion (\ref{eq:semicl}).
For transport properties, including superconductivity, the most important electrons are those in the vicinity of the Fermi level, which we consider here to be close to the bottom of the conduction band, \textit{i.e.} we consider a moderately doped semiconductor. We can already anticipate that the Berry curvature may play a role as long as the Fermi wave vector $k_F$ satisfies $\lambdabar_{\xi\sigma} k_F\ll 1$ since it vanishes algebraically for $\lambdabar_{\xi\sigma}\rightarrow \infty$ [see Eq. (\ref{eq:Berry})]. We therefore project the Hamiltonian (\ref{eq:hamDir}) onto the conduction-band bottom, $0<\delta E=E-\Delta_{\xi\sigma}-E_{\xi\sigma}^0\ll \Delta_{\xi\sigma}$ (see Fig. \ref{fig:01}), with the help of the Foldy-Wouthuysen transformation to keep track of the electric potential $V(\vec{r})$ \cite{FW}.
This yield the effective one-band Hamiltonian
\begin{eqnarray}\label{eq:hamPauli}
\nonumber
H &\simeq& E_{\xi\sigma}^0+\Delta_{\xi\sigma}+\frac{\hbar^2\vec{k}^2}{2m_D}+ V(\vec{r})\\
&&+\frac{\xi\sigma\lambdabar_{\xi\sigma}^2}{4}\Big(\vec{e}_z\times\vec{k}\Big)\cdot\vec{\nabla}_{\vec{r}}V+\frac{\lambdabar_{\xi\sigma}^2}{8}\vec{\nabla}_{\vec{r}}^2V ,
\end{eqnarray}
which, apart from the last term, is identical to the one (\ref{eq:hamPeierls}) which we have obtained with the help of the generalized Peierls subsitution if we make use of the expression (\ref{eq:Berry}) for the Berry curvature to lowest order in the wave vector and if we redefine the energy with respect to the band bottom. The last term may also be written in terms of the Berry curvature as
\begin{equation}
\frac{\lambdabar_{\xi\sigma}^2}{8}\vec{\nabla}_{\vec{r}}^2V(\vec{r})=\frac{1}{4}\big|\mathcal{B}_{\lambda,\xi\sigma}(0)\big|\vec{\nabla}^2_{\vec{r}}V(\vec{r})
\end{equation}
and corresponds to the Darwin term in high-energy physics. While it does not play any role in the semi-classical equations of motion, it is relevant namely at very short ranges and has been shown to strongly affect \textit{e.g.} the spectra of $s$-state excitions in 2D TMDC \cite{BerryExc1,BerryExc2,Trushin_2017,Hichri_2019}. This is best seen in the case of the 2D Coulomb potential in which case $\nabla^2_{\vec{r}}V=e^2\delta(\vec{r})/\epsilon$, \textit{i.e.} it is relevant for pair wave functions with a non-zero amplitude at the origin ($s$-wave states) such as the BCS wave functions, which we discuss below.
\section{Two-body problem: General case and Cooper pair}
\label{sec:2body}
With the Cooper-pair problem in mind, we now consider how the extra terms discussed within the one-particle picture presented in the preceding section evolves in the case of two electrons at the bottom of the conduction band $\lambda=+$ at the same energy. This choice to consider a Fermi level slightly above the bottom of the conduction band is perfectly arbitrary, but the results obtained in the following sections remain valid for Cooper pairs formed from holes in the valence band. We consider again the spin to be locked to the valley index so that there is only one effective label $\xi\sigma$, which we represent by the valley index ($\xi_1$ for the first electron and $\xi_2$ for the second one) to simplify the notations. Furthermore we consider a two-body potential $V$ that depends only on the relative position of the two electrons $\vec{r}_1-\vec{r}_2$, such as it is the case for the BCS potential.
\subsection{General case}\label{ssec:gen}
Because the two-body interaction potential only depends on the relative distance $\vec{\rho}=\vec{r}_1-\vec{r}_2$ between the electrons, we introduce relative and center-of-mass (CoM) coordinates. Since both electrons have the same mass, we have
\begin{align}
\text{Relative:}\qquad&\vec{\rho}=\vec{r}_1-\vec{r}_2\hspace{2cm}\vec{k}=\frac{\vec{k}_1-\vec{k}_2}{2} \\
\text{CoM:}\qquad&\vec{R}=\frac{\vec{r}_1+\vec{r}_2}{2}\hspace{2cm}\vec{K}=\vec{k}_1+\vec{k}_2,
\end{align}
Separation of the CoM and relative coordinates yields the Hamiltonian
\begin{widetext}
\begin{eqnarray}
\nonumber
H_{2e^-}&=&2\Delta_b+\frac{\hbar^2\vec{K}^2}{4m_D}+\frac{\hbar^2\vec{k}^2}{m_D}+V(\vec{\rho})+\frac{1}{4}\bigg(\vec{\Lambda}^{\xi_1,\xi_2}_{+}\big(\vec{K},\vec{k}\big)\times\vec{K}\bigg)\cdot\vec{\nabla}V(\vec{\rho})+\frac{1}{2}\bigg(\vec{\Lambda}^{\xi_1,\xi_2}_{-}\big(\vec{K},\vec{k}\big)\times\vec{k}\bigg)\cdot\vec{\nabla}V(\vec{\rho}) \\
&&+\frac{1}{2}\big|\mathcal{B}(0)\big|\vec{\nabla}^2V(\vec{\rho})
\qquad\text{with}\quad\vec{\Lambda}^{\xi_1,\xi_2}_\pm\big(\vec{K},\vec{k}\big)=\vec{\mathcal{B}}_{+,\xi_1}\bigg(\frac{1}{2}\vec{K}+\vec{k}\bigg)\pm\vec{\mathcal{B}}_{+,\xi_2}\bigg(\frac{1}{2}\vec{K}-\vec{k}\bigg) \label{eq:ham2body}
\end{eqnarray}
\end{widetext}
within the parabolic approximation, and where we have made use of the Dirac mass $m_D=\Delta_{\xi\sigma}/v_D^2$. Since we no longer consider $k$-space gradients, we omit the index $\vec{r}$ at the gradient $\nabla_{\vec{r}}=\nabla$ from now on. It is interesting to notice that, when moving to CoM/relative coordinates, the Berry dipole term splits into two dipoles acting on the electron pair. One is associated with its \emph{center-of-mass motion} and the \emph{sum} of the two Berry curvatures and the other is associated with its \emph{relative motion} and the \emph{difference} of the two Berry curvatures. To gain further insight into the physical meaning of these two terms, we can calculate the Heisenberg equations of motion
\begin{align}
&\overset{.}{\vec{K}} = \vec{0} \qquad
\hspace{9mm}
\overset{.}{\vec{R}} = \frac{\hbar\vec{K}}{2m_D}+\frac{1}{4\hbar}\vec{\nabla}V(\rho)\times\vec{\Lambda}^{\xi_1,\xi_2}_+(\vec{K},\vec{k}) \\
& \overset{.}{\vec{k}} = -\frac{1}{\hbar}\vec{\nabla}H_{2e^-}\qquad \overset{.}{\vec{\rho}}=2\frac{\hbar\vec{k}}{m_D}+\frac{1}{2\hbar}\vec{\nabla}V(\vec{\rho})\times\vec{\Lambda}^{\xi_1,\xi_2}_-(\vec{K},\vec{k})
\end{align}
The CoM momentum is a conserved quantity, owing to the fact that $H_{2e^-}$ does not depend on $\vec{R}$. We also see that the two dipoles induce two Karplus-Luttinger-type velocities: $\Lambda_+$, which is associated to the CoM dipole, generates a drift velocity of the CoM coordinate, and $\Lambda_-$, which is associated to the relative dipole, yields another drift velocity of the relative coordinate of the Cooper pair.
Before discussing the special case of the Cooper pair, we may already discuss here the relative role of the two quantities $\vec{\Lambda}_+$ and $\vec{\Lambda}_-$ as a function of the two different valleys, \textit{i.e.} in the case of \textit{intra-valley} pairing as compared to \textit{inter-valley} pairing. Indeed, they determine the dipolar moments
\begin{equation}
\vec{d}_{\pm}=-e(\vec{\Lambda}_{\pm}\times \vec{q})/2,
\end{equation}
where $\vec{q}=\vec{K}$ for the CoM dipole (sign $+$) and $\vec{q}=\vec{k}$ for the relative dipole (sign $-$). In the case of intra-valley pairing ($\xi_1=\xi_2$), which corresponds to triplet superconductivity as a consequence of the spin-valley locking, the relative dipole $\vec{d}_-$ is negligible to lowest order in the wave vectors while the CoM dipole is on the order of $\vec{d}_+\sim -e \mathcal{B}_{+,\xi_1}(0)\times \vec{K}$. Their roles are inverted in the case of singlet-type inter-valley pairing, in which case $\vec{d}_+\simeq 0$ while $\vec{d}_-\sim -e \mathcal{B}_{+,\xi_1}(0)\times \vec{k}$.
\subsection{Revisiting the Cooper problem}\label{ssec:Cooper}
We are now in a position to study the effect of the Berry curvature on a Cooper pair, the building block of superconductors. To do so, we revisit the Cooper problem following the lines of Ref. \cite{leonn.cooper1956} and standard textbooks \cite{tinkham2004introduction}. The Hamiltonian we consider here is $H_c= H_{2e^-}(\vec{K}=\vec{0})$, \textit{i.e.} our two-body Hamiltonian (\ref{eq:ham2body}) in the rest frame,
\begin{align}
H_{c}=2\epsilon_+(\vec{k})+V(\vec{\rho})&+\frac{1}{2}\bigg(\vec{\Lambda}^{\xi_1,\xi_2}_{-}\big(\vec{0},\vec{k}\big)\times\vec{k}\bigg)\cdot\vec{\nabla}V(\vec{\rho})\nonumber \\
&+\frac{1}{2}\big|\mathcal{B}(0)\big|\vec{\nabla}^2V(\vec{\rho}),
\end{align}
where $\Lambda_-$ can be rewritten as
\begin{equation}
\vec{\Lambda}_-^{\xi_1,\xi_2}(\vec{0},\vec{k})=-(\xi_1-\xi_2)\frac{\lambdabar_{\xi\sigma}^2}{2\big(1+\lambdabar_{\xi\sigma}^2k^2\big)^{3/2}}\vec{e}_z=\vec{\Lambda}_-^{\xi_1,\xi_2}(\vec{k}).
\end{equation}
As mentioned above, one notices that, for the Berry dipole term to be non-zero, the two electrons of the Cooper pair need to be taken in different valleys and thus with opposite spin, as it is usual for $s$-wave singlet superconductivity. In contrast to this, we have $\Lambda_+^{\xi_1,\xi_2}(0,\vec{k})\propto(\xi_1+\xi_2)$ \textit{i.e.} one needs electrons in the same valley, but even then, the intra-valley CoM dipolar term in the Hamiltonian vanishes unless $\vec{K}\neq 0$. We therefore consider henceforth only the relative dipolar term and the case of inter-valley pairing.
Let us now take a closer look at the wave function of the Cooper pair $\psi(\vec{\rho})$, which is a solution of $H_c\psi(\vec{\rho})=E\psi(\vec{\rho})$. We then decompose $\psi$ and $V$ in a Fourier series
\begin{align}
&\psi(\vec{\rho})=\sum_{\vec{k}}g_{\vec{k}}e^{i\vec{k}\cdot\vec{\rho}},\\
&V(\vec{\rho})=\sum_{\vec{k}\vec{k'}}V_{\vec{k}\vec{k'}}e^{i(\vec{k}-\vec{k'})\cdot\vec{\rho}}\quad.
\end{align}
Following the steps of Ref. \cite{tinkham2004introduction} we find the self-consistent equation
\begin{equation}
\big[E-2\epsilon_+(\vec{k})\big]g_{\vec{k}}=\sum_{\vec{k}'}V^{\text{eff}}_{\vec{k}\vec{k'}}g_{\vec{k'}}
\end{equation}
for the coefficients $g_{\vec{k}}$, in terms of the \textit{effective interaction}
\begin{equation}
\label{inteff}
V^{\text{eff}}_{\vec{k}\vec{k}'}=\Bigg[1+\frac{i}{2}\bigg(\vec{\Lambda}^{\xi_1,\xi_2}_{-}(\vec{k})\times\vec{k}\bigg)\cdot\vec{k}'-\frac{1}{2}\big|\mathcal{B}(0)\big|\big(\vec{k}-\vec{k}'\big)^2\Bigg]V_{\vec{k}\vec{k}'}.
\end{equation}
This equation is one of the main results of our paper. Qualitatively, we see that the two terms appear with opposite signs. The second term stems from the Berry dipole term in Hamiltonian (\ref{eq:ham2body}) and may increase or decrease the interaction potential and thus the strength of the Cooper pairing depending on the sign of $\vec{\Lambda}_-$. As for the last (Darwin) term, it is negative irrespective of the valley index, meaning that it tends to weaken the electron-electron interaction and thus the superconducting phase. On a more practical level, the above expressions tell us that the calculations for the energy of the Cooper pair in the presence of a Berry curvature are the same as in the conventional pairing case \cite{tinkham2004introduction}, but in terms of the effective interaction (\ref{inteff}).
In a second step we need to solve the self-consistency equation
\begin{equation}
\label{Cooper1}
\sum_{\vec{k}}\frac{\langle V^{\text{eff}}_{\vec{k}\vec{k'}}\rangle}{E-2\epsilon_+(\vec{k})}=1,
\end{equation}
where we have defined the average
\begin{equation}
\langle\mathcal{O}(\vec{k'})\rangle=\frac{\sum_{\vec{k'}}\mathcal{O}(\vec{k'})g_{\vec{k'}}}{\sum_{\vec{k'}}g_{\vec{k'}}}
\end{equation}
with respect to the weighting coefficients $g_{\vec{k}}$. The term $\langle V^{\text{eff}}_{\vec{k}\vec{k'}}\rangle$ may be rewritten as
\begin{widetext}
\begin{equation}
\langle V^{\text{eff}}_{\vec{k}\vec{k'}}\rangle=\bigg(1-\frac{1}{2}\big|\mathcal{B}(0)\big|\vec{k}^2\bigg)\langle V_{\vec{k}\vec{k'}}\rangle+\bigg(\big|\mathcal{B}(0)\big|\vec{k}+\frac{i}{2}\vec{\Lambda}_-^{\xi_1,\xi_2}(\vec{k})\times\vec{k}\bigg)\cdot\langle\vec{k'}V_{\vec{k}\vec{k'}}\rangle-\frac{1}{2}\big|\mathcal{B}(0)\big|\langle\vec{k'}^2V_{\vec{k}\vec{k'}}\rangle.
\end{equation}
\end{widetext}
To illustrate the role of the additional terms due to the Berry curvature, let us consider the BCS potential, defined as
\begin{equation}
V_{\vec{k}\vec{k'}}=
\begin{cases}
&-V<0\quad\text{if }\epsilon_F\leq\epsilon_+(\vec{k}),\epsilon_+(\vec{k'})\leq\epsilon_F+\hbar\omega_D \\
&0\quad\text{otherwise},
\end{cases}
\end{equation}
where $\epsilon_F$ is the Fermi energy and $\hbar\omega_D$ the Debye energy. We can compactly rewrite it as
\begin{equation}
V_{\vec{k}\vec{k'}}=-V\mathbbm{1}_{\mathcal{D}}(\vec{k})\mathbbm{1}_{\mathcal{D}}(\vec{k'})
\label{VBCS}
\end{equation}
where $\mathbbm{1}_{\mathcal{D}}$ is the indicator function of the set
\begin{equation}
\label{Dset}
\mathcal{D}=\Big\{\vec{k}\in\mathbb{R}^2\Big|\epsilon_F\leq\epsilon_+(\vec{k})\leq\epsilon_F+\hbar\omega_D\Big\}.
\end{equation}
With this in mind, we write
\begin{equation}
\label{sumPeierls}
\langle\vec{k'}V_{\vec{k}\vec{k'}}\rangle\propto\sum_{\vec{k'}\in\mathcal{D}}\vec{k'}V_{\vec{k}\vec{k'}}g_{\vec{k'}}
\end{equation}
From Eq. (\ref{VBCS}) we see that $V_{\vec{k};-\vec{k'}}=V_{\vec{k}\vec{k'}}$. Moreover, for BCS superconductivity we have $g_{-\vec{k'}}=g_{\vec{k'}}$ so that $\vec{k'}V_{\vec{k}\vec{k'}}g_{\vec{k'}}$ is an odd function of $\vec{k'}$. Because summing an odd function over the set $\mathcal{D}$ gives zero, we have $\langle\vec{k'}V_{\vec{k}\vec{k'}}\rangle=\vec{0}$ so that \emph{the Berry dipole term does not affect the Cooper pair}, which is then solely affected by the Darwin term. Therefore, if we remember the competition between the dipolar and Darwin terms, this suggests that the effect of the Berry curvature is to weaken the Cooper.
As for $\langle V^{\text{eff}}_{\vec{k}\vec{k'}}\rangle$, we are left with
\begin{equation}
\langle V^{\text{eff}}_{\vec{k}\vec{k'}}\rangle=\bigg(1-\frac{1}{2}\big|\mathcal{B}(0)\big|\vec{k}^2\bigg)\langle V_{\vec{k}\vec{k'}}\rangle -\frac{1}{2}\langle \big|\mathcal{B}(0)\big|\vec{k'}^2V_{\vec{k}\vec{k'}}\rangle
\end{equation}
Remember that $V^{\text{eff}}_{\vec{k}\vec{k'}}$ is non-zero only for $\vec{k},\vec{k'}\in\mathcal{D}$, and from the definition of $\mathcal{D}$ we rewrite the energy as $\epsilon_+(\vec{k})=\epsilon_F+\eta_{\vec{k}}\hbar\omega_D$ with $\eta_{\vec{k}}\in[0,1]$.
From this and the expression of $\epsilon_+(\vec{k})$ we obtain
\begin{equation}
\big|\mathcal{B}(0)\big|\vec{k}^2=\frac{\epsilon_F-\Delta_{\xi\sigma}}{\Delta_{\xi\sigma}}+\eta_{\vec{k}}\frac{\hbar\omega_D}{\Delta_{\xi\sigma}}.
\end{equation}
Now, for many 2D materials (including any TMDC), the band gap is in the 1eV range (see e.g. Ref. \cite{doi:10.1021/acs.jpclett.5b01686}) while for most crystals $\hbar\omega_D\sim 0.01$eV \cite{LI2012197}. One therefore obtains a ratio $\frac{\hbar\omega_D}{\Delta_b}\sim 0.01$, so that we may neglect the corresponding term and thus make the approximation
\begin{equation}
\label{approxberry} \big|\mathcal{B}(0)\big|k^2\simeq\big|\mathcal{B}(0)\big|k_F^2\qquad \big|\mathcal{B}(0)\big|k'^2\simeq\big|\mathcal{B}(0)\big|k_F^2.
\end{equation}
With this and $\langle\vec{k'}V_{\vec{k}\vec{k'}}\rangle=\vec{0}$, we finally obtain
\begin{equation}
\label{approxint}
\langle V^{\text{eff}}_{\vec{k}\vec{k'}}\rangle=\big(1-\big|\mathcal{B}(0)\big|k_F^2\big)\langle V_{\vec{k}\vec{k'}}\rangle,
\end{equation}
in line with our qualitative argument of a weakening of the electron-electron interaction induced by the Darwin term. With the BCS potential, $\langle V_{\vec{k}\vec{k'}}\rangle=-V$, one finds
\begin{equation}
\sum_{\vec{k}}\frac{1}{E-2\epsilon_+(\vec{k})}=-\frac{1}{\big(1-\big|\mathcal{B}(0)\big|k_F^2\big)V}.
\end{equation}
As usual, the sum over the wave vector may be replaced by an integral over energy with the help of the density of states $\rho(\epsilon)$ and the BCS coupling constant $\lambda= V\rho(\epsilon_F)$. We finally find the binding energy of the Cooper pair
\begin{equation}\label{eq:binding}
E_B=\frac{2\hbar\omega_D}{e^{2/\lambda_{\text{eff}}}-1}\quad\text{with}\quad\lambda_{\text{eff}}=\big(1-\big|\mathcal{B}(0)\big|k_F^2\big)\lambda,
\end{equation}
which is the same as the conventional expression
\begin{equation}
E_B^{\text{BCS}}=\frac{2\hbar\omega_D}{e^{2/\lambda}-1},
\end{equation}
where we have replaced $\lambda$ by an effective (lower) coupling constant. If we set the Berry curvature to zero or if we set the band gap to be infinity, we recover the usual expression, as expected.
To summarize this subsection, we highlight two aspects. First, the effect of the Berry curvature on the Cooper pair reveals itself through a competition between two terms. On the one hand, the Berry dipole term, with its dipolar/spin-orbit form, induces a drift velocity analogous to the Karplus-Luttinger veloctity on the relative position of the electrons of the Cooper pair. It \emph{could} in principle enhance the electron-electron interaction $V_{\vec{k}\vec{k'}}$. On the other hand, the Darwin term yields a negative contribution and thus weakens the effective interaction. Second, the Berry dipole term's contribution to Cooper pairing turns out to be zero for $s$-wave superconductivity, and thus we are only left with a weakened electron-electron interaction due to the Darwin term. This is clearly seen in the expression of the binding energy (\ref{eq:binding}) Indeed, since the interaction $V$ is lowered, so is the BCS coupling $\lambda$, thereby lowering the binding energy of the Cooper pair. In conclusion, the Berry curvature makes the Cooper pairs less bound and thus more easily breakable, \textit{e.g.} by thermal fluctuations. This means that the critical temperature (and the superconducting gap) are lowered as well, as we show explicitely in the following section, where we discuss the action of the Berry-curvature corrective terms in the BCS many-body approach.
\section{BCS Hamiltonian in the presence of Berry curvature}\label{sec:BCS}
In the previous section, we found that the calculations in the electron pair problem with Berry curvature were the same as in its absence, but with an effective interaction. We therefore consider, in this part, the BCS Hamiltonian where we replace the interaction $V_{\vec{k}\vec{k'}}$ with the effective one $V^{\text{eff}}_{\vec{k}\vec{k'}}$ which is given in Eq. (\ref{inteff}) and that accounts for the corrective terms due to the Berry curvature.
\begin{equation}
H=\sum_{\vec{k}\sigma}\xi_{\vec{k}}c^\dagger_{\vec{k}\sigma}c_{\vec{k}\sigma}+\sum_{\vec{k}\vec{k'}}V^{\text{eff}}_{\vec{k}\vec{k'}}c^\dagger_{\vec{k'}\uparrow}c^\dagger_{-\vec{k'}\downarrow}c_{\vec{k}\uparrow}c_{-\vec{k}\downarrow}
\end{equation}
where $\xi_{\vec{k}}=\epsilon_+(\vec{k})-\epsilon_F$, and the bare interaction (in the absence of Berry curvature corrections) is $V_{\vec{k}\vec{k'}}=-V\mathbbm{1}_{\mathcal{D}}(\vec{k})\mathbbm{1}_{\mathcal{D}}(\vec{k'})$ with $\mathcal{D}=\Big\{\vec{k}\in\mathbb{R}^2\Big|\epsilon_F-\hbar\omega_D\leq\epsilon_+(\vec{k})\leq\epsilon_F+\hbar\omega_D\Big\}$. We also keep the same groundstate $|\psi_G\langle$. Since this Hamiltonian has the same form as the original BCS Hamiltonian, the same calculations hold as long as the interaction is not specified. We thus find the textbook gap equation \cite{tinkham2004introduction}
\begin{equation}
\Delta_{\vec{k}}=-\frac{1}{2}\sum_{\vec{k'}}V^{\text{eff}}_{\vec{k}\vec{k'}}\frac{\Delta_{\vec{k'}}}{\sqrt{\Delta_{\vec{k'}}^2+\xi^2_{\vec{k'}}}}\tanh\bigg(\frac{\beta}{2}\sqrt{\Delta_{\vec{k'}}^2+\xi^2_{\vec{k'}}}\bigg)
\end{equation}
with $\Delta_{\vec{k}}=-\sum_{\vec{k'}}V^{\text{eff}}_{\vec{k}\vec{k'}}\langle c^\dagger_{\vec{k'}\uparrow}c^\dagger_{-\vec{k'}\downarrow}\rangle$ and $\beta=(k_BT)^{-1}$. In terms of the auxiliary function
\begin{equation}
\label{kernel}
f_{\beta,\vec{k}}(\vec{k'})=\frac{\Delta_{\vec{k'}}}{\sqrt{\Delta_{\vec{k'}}^2+\xi^2_{\vec{k'}}}}\tanh\bigg(\frac{\beta}{2}\sqrt{\Delta_{\vec{k'}}^2+\xi^2_{\vec{k'}}}\bigg),
\end{equation}
the self-consistent gap equation reads
\begin{widetext}
\begin{equation}
\Delta_{\vec{k}}=-\frac{1}{2}\bigg(1-\frac{1}{2}\big|\mathcal{B}(0)\big|\vec{k}^2\bigg)\sum_{\vec{k'}}V_{\vec{k}\vec{k'}}f_{\beta,\vec{k}}(\vec{k'})-\frac{1}{2}\bigg(\frac{i}{2}\vec{\Lambda}^{\xi_1,\xi_2}_-(\vec{k})\times\vec{k}+\big|\mathcal{B}(0)\big|\vec{k}\bigg)\cdot\sum_{\vec{k'}}\vec{k'}V_{\vec{k}\vec{k'}}f_{\beta,\vec{k}}(\vec{k'})-\frac{1}{2}\sum_{\vec{k'}}\frac{1}{2}\big|\mathcal{B}(0)\big|\vec{k'}^2V_{\vec{k}\vec{k'}}f_{\beta,\vec{k}}(\vec{k'})
\end{equation}
\end{widetext}
One can show that if the bare superconducting gap has a definite parity, then $\Delta_{\vec{k}}$ (so defined through the effective interaction) has the same parity. Therefore for BCS superconductivity we have $\Delta_{-\vec{k}}=\Delta_{\vec{k}}$. From equation (\ref{kernel}), it is then clear that $f_{\beta,\vec{k}}(-\vec{k'})=f_{\beta,\vec{k}}(\vec{k'})$. And since $V_{\vec{k};-\vec{k'}}=V_{\vec{k};\vec{k'}}$, the function $\vec{k'}\longrightarrow\vec{k'}V_{\vec{k}\vec{k'}}f_{\beta,\vec{k}}(\vec{k'})$ is an odd function so that
\begin{equation}
\sum_{\vec{k'}}\vec{k'}V_{\vec{k}\vec{k'}}f_{\beta,\vec{k}}(\vec{k'})=\vec{0},
\end{equation}
and thus the Berry dipole term does again not affect the many-body result, which is consistent with the results obtained in the previous section. We then make the same approximate treatment [see Eqs. (\ref{approxberry}) and (\ref{approxint})] as for the Cooper pair problem and we find
\begin{equation}
\Delta_{\vec{k}}=-\frac{1}{2}\sum_{\vec{k'}}\big(1-\big|\mathcal{B}(0)\big|k_F^2\big)V_{\vec{k}\vec{k'}}f_{\beta,\vec{k}}(\vec{k'}),
\end{equation}
in agreement with our previous result. the Berry curvature reduces the attractive electron-electron interaction due to the Darwin term.
We are now able to calculate the zero-temperature superconducting gap. At $T=0$, the gap equation is
\begin{equation}
\Delta_{\vec{k}}=-\frac{1}{2}\Big(1-\big|\mathcal{B}(0)\big|k_F^2\Big)\sum_{\vec{k'}}V_{\vec{k}\vec{k'}}\frac{\Delta_{\vec{k'}}}{\sqrt{\Delta_{\vec{k'}}^2+\xi_{\vec{k'}}^2}}
\end{equation}
We then use $V_{\vec{k}\vec{k'}}=-V\mathbbm{1}_{\mathcal{D}}(\vec{k})\mathbbm{1}_{\mathcal{D}}(\vec{k'})$ and have
\begin{equation}
\Delta_{\vec{k}}=\mathbbm{1}_{\mathcal{D}}(\vec{k})\frac{1}{2}\Big(1-\big|\mathcal{B}(0)\big|k_F^2\Big)V\sum_{\vec{k}\in\mathcal{D}}\frac{\Delta_{\vec{k'}}}{\sqrt{\Delta_{\vec{k'}}^2+\xi_{\vec{k'}}^2}}
\end{equation}
Thus $\Delta_{\vec{k}}=0$ for $\vec{k}\notin\mathcal{D}$, and then one can show directly that $\Delta_{\vec{k}}=\Delta$ for $\vec{k}\in\mathcal{D}$. The former case is trivially satisfied since if $\vec{k}\notin\mathcal{D}$, the corresponding electron is not subject to the attractive interaction so it cannot condense and participate in a SC state. The latter indicates that the gap is then isotropic for the electrons that are concerned by superconductivity. We may again follow the conventional derivation \cite{tinkham2004introduction} and find the $T=0$ superconducting gap
\begin{equation}
\label{gaplambdaeff}
\Delta(T=0)=\frac{\hbar\omega_D}{\sinh\big(1/\lambda_{\text{eff}}\big)}\quad\text{with}\quad\lambda_{\text{eff}}=\big(1-\big|\mathcal{B}(0)\big|k_F^2\big)\lambda
\end{equation}
with the same effective coupling constant $\lambda_{\text{eff}}$ as that obtained in the previous section [see Eq. (\ref{eq:binding})]. Comparing this to the bare BCS expression
\begin{equation}
\Delta^{\text{BCS}}(T=0)=\frac{\hbar\omega_D}{\sinh\big(1/\lambda\big)}
\end{equation}
we see the same result as in the Cooper pair problem, that is to say a lowering of the BCS coupling constant driven by the Berry curvature thereby lowering the $T=0$ superconducting gap. This is also consistent with what we said about the consequences for the Cooper pairs. Indeed, since the superconducting gap is smaller, so is the energy of the quasiparticles in the superconductor. This makes them more sensitive to variations of energy, e.g. thermal fluctuations. In other words, the superconducting phase is weakened and thus more easily suppressed upon raising temperature.
Similarly, the expression for the critical temperature takes the form \cite{tinkham2004introduction}
\begin{equation}\label{eq:TC}
T_c=2\hbar\omega_D\frac{e^\gamma}{\pi}e^{-1/\lambda_{\text{eff}}}
\end{equation}
and is identical to the standard one except for the fact that the coupling constant needs to be replaced by $\lambda\rightarrow \lambda_{\text{eff}}$ to take into account the extra terms due to the Berry curvature. Here, $\gamma\simeq0.577$ is the Euler-Mascheroni constant, and the approximation is valid if $2T_c\ll\hbar\omega_D/k_B=T_D$, and it is relatively reliable when $2T_c\lesssim T_D$. Notice finally, that the Berry curvature therefore does not affect the universality of the ratio between the superconducting gap and $T_c$ in the weak-coupling limit,
\begin{equation}
\frac{\Delta(T=0)}{k_BT_c}\underset{\lambda\ll1}{=}\frac{\pi}{e^\gamma}\simeq1.76.
\end{equation}
Indeed this ratio is independent of the (effective) coupling constant.
\section{Doping dependence}\label{sec:doping}
Until now, we considered a low-doping limit, in which the Fermi level is close to the bottom of the conduction band. This allowed us to approximate the Berry curvature as $\mathcal{B}(k)\simeq\mathcal{B}(0)$. At larger doping, we first expect a weakening of the inter-band effects since the relevant physics will take place farther away from the other band. We should then expect to recover the usual one-band BCS results as the Fermi energy increases. The main thing to change would be our extra terms. The Berry dipole term does not rely on the low-energy expansion of the Dirac Hamiltonian, and we thus do not need to change it. The Darwin term is different: we have obtained it by expanding the Dirac Hamiltonian in the low-energy/non-relativistic limit. In this limit, the Berry curvature enters as $\big|\mathcal{B}(0)\big|$. Since the physics is controlled by states near the Fermi energy, we change $\big|\mathcal{B}(0)\big|\longrightarrow\big|\mathcal{B}(k_F)\big|$, \textit{i.e.} the most important contribution of the Berry curvature is its value at the Fermi level. The effective coupling constant $\lambda_{\text{eff}}$ takes then the form
\begin{equation}
\label{deforlambda}
\lambda_{\text{eff}}=\big(1-\big|\mathcal{B}(k_F)\big|k_F^2\big)\lambda=\Bigg(1-\frac{\lambdabar_{\xi\sigma}^2k_F^2}{2\big(1+\lambdabar_{\xi\sigma}^2k_F^2\big)^{3/2}}\Bigg)\lambda,
\end{equation}
and we have
\begin{align}
\label{lowdopinglimit}
&\text{Low-doping limit: }\frac{\lambda_{\text{eff}}}{\lambda}\underset{\lambdabar_ck_F\ll1}{\sim}1-\frac{\lambdabar_{\xi\sigma}^2k_F^2}{2} \\
\label{highdopinglimit}
&\text{High-doping limit: }\frac{\lambda_{\text{eff}}}{\lambda}\underset{\lambdabar_{\xi\sigma}k_F\gg1}{\sim}1-\frac{1}{2\lambdabar_ck_F}
\end{align}
for the different limiting cases. As a consistency check, we recover the previous result in the low-doping limit (indeed, $\lambdabar_c^2/2=\big|\mathcal{B}(0)\big|$). In the high-doping limit, the effective coupling constant approaches its bare BCS value as the Fermi level goes to $+\infty$. This is consistent with our expectation of a decreased role of the corrective terms due to the Berry curvature and thus of the inter-band effects in this limit. The doping dependence of the coupling constant (i.e. on $\lambdabar_ck_F$) is depicted in Fig. \ref{dopingdependence}.
\begin{figure}[h!]
\centering
\includegraphics[width=0.36\textwidth]{lambda.png}
\caption{Effective coupling constant $\frac{\lambda_{\text{eff}}}{\lambda}$ as a function of $\lambdabar_{\xi\sigma}k_F$.}
\label{dopingdependence}
\end{figure}
\newline
It is apparent that the effective coupling constant has a minimum that can be shown to occur at $\lambdabar_ck_F=\sqrt{2}$. Therefore the effect of the Berry curvature on conventional BCS type ($s$-wave) superconductivity is expected to be strongest in an intermediate doping regime in which the Fermi wave vector is on the order of the inverse effective Compton length. We then have
\begin{equation}
\min_{\lambdabar_ck_F}\frac{\lambda_{\text{eff}}}{\lambda}=1-\frac{1}{3\sqrt{3}}\simeq81\%
\end{equation}
\textit{i.e.} the maximal reduction is approximately $19\%$. It is interesting to note that while the ratio goes to 1 as the Fermi level goes to $+\infty$, the difference does not go to zero. Indeed,
\begin{equation}
\underset{k_F\rightarrow+\infty}{\lim}\big[\lambda_{\text{eff}}-\lambda\big]=-\frac{AV}{4\pi\Delta_b}
\end{equation}
with $A$ the area of the Brillouin zone. Note that $V$ represents, here, the interaction energy per unit area in reciprocal space so that the quantity $AV$ itself is an energy and the coupling constant is dimensionless. While the reduction of the coupling constant seems rather limited, we must not forget that the critical temperature and the superconducting gap both depend exponentially on this coupling constant, so the effect could be quite substantial.
The central result of this paper is Eq. (\ref{deforlambda}). Indeed, from it ensues most of the results we had so far. Moreover, it could have several uses. First, doping could offer a way to experimentally observe the effects of a Berry curvature on a superconducting phase discussed in this paper. We present some possible paths for an experimental test of Berry-curvature effects on BCS superconductivity in Sec. \ref{sec:exp}. Second, while this specific deformation of the coupling constant may not be true for other types of band structures, these could still exhibit other types of deformations depending on the corrective terms of the one-body problem. If Eq. (\ref{deforlambda}) is true in other types of band structures, it can even be a way to detect the presence of a Berry curvature as well as its $k$-dependence.
\section{Beyond BCS superconductivity}\label{sec:beyond}
Now that we have studied the conventional $s$-wave case, let us see what happens with other types of superconductivity. As in the case for the $s$-wave case (see Sec. \ref{ssec:Cooper}), we first revisit the modified Cooper problem from a more general point of view following Ref. \cite{mineev_samokhin_1999}. We will then study the many-body BCS theory, this time following Refs. \cite{SigristUeda} and \cite{sigrist_2005}.
\subsection{Cooper problem}\label{ssec:Coopergen}
The 2-electron potential may be decomposed in the relative-angular momentum basis as \cite{mineev_samokhin_1999}
\begin{equation}
V_{\vec{k}\vec{k'}}=\sum_{l=0}^{+\infty}V_l(\vec{k},\vec{k'})
\end{equation}
with $V_l(-\vec{k},\vec{k'})=(-1)^lV_l(\vec{k},\vec{k'})=V_l(\vec{k},-\vec{k'})$, and the integer $l$ the angular momentum of the superconducting phase. It is even for singlet pairing and odd for triplet pairing. Let us pick a superconducting phase with fixed $l$, so that the pairing is either singlet or triplet. Then $V_{\vec{k}\vec{k'}}=V_l(\vec{k},\vec{k'})$. The same is done to $g_{\vec{k}}=g_l(\vec{k})$ with also $g_l(-\vec{k})=(-1)^lg_l(\vec{k})$. Equation (\ref{Cooper1}) becomes therefore
\begin{equation}
\sum_{\vec{k}}\frac{\langle V^{\text{eff}}_l(\vec{k},\vec{k'})\rangle_l}{E-2\epsilon_+(\vec{k})}=1,
\end{equation}
and we then proceed in the same way as before, expanding $\langle V^{\text{eff}}_l(\vec{k},\vec{k'})\rangle_l$ and considering
\begin{equation}
\langle\vec{k'}V_l(\vec{k},\vec{k'})\rangle_l\propto\sum_{\vec{k'}}\vec{k'}V_l(\vec{k},\vec{k'})g_l(\vec{k'}).
\end{equation}
If we then take $V_l(\vec{k},\vec{k'})$ to be non-zero only within a thin layer of energy around the Fermi level, with the energy cut-off $\epsilon_l$, and one retrieves Eq. (\ref{Dset}) but with $\mathcal{D}_l=\Big\{\vec{k}\in\mathbb{R}^2\big|\epsilon_F\leq\epsilon_+(\vec{k})\leq\epsilon_F+\epsilon_l\Big\}$. Because of the symmetry
\begin{align}
V_l(\vec{k},-\vec{k'})g_l(-\vec{k'})&=(-1)^lV_l(\vec{k},\vec{k'})(-1)^lg_l(\vec{k'})\nonumber\\
&=V_l(\vec{k},\vec{k'})g_l(\vec{k'}),
\end{align}
the function $\vec{k'}V_l(\vec{k},\vec{k'})g_l(\vec{k'})$ is odd in $\vec{k'}$ so that the sum over the set $\mathcal{D}_l$ yields zero. Since this term carries the Berry dipole term, we can conclude that \emph{the Berry dipole term does not contribute to the energy of the Cooper pair with pure singlet or triplet pairings}. Notice, however, that the Berry dipole term may nevertheless play a significant role in exotic superconductors that mix singlet and triplet pairing, as we sketch out in Sec. \ref{ssec:mixed}.
We then proceed with the same approximation as for the conventional $s$-wave case, which gives the effective interaction
\begin{equation}
\langle V^{\text{eff}}_l(\vec{k},\vec{k'})\rangle_l=\big(1-\big|\mathcal{B}(k_F)\big|k_F^2\big)\langle V_l(\vec{k},\vec{k'})\rangle_l.
\end{equation}
We also take the approach of \cite{mineev_samokhin_1999} and take $V_l(\vec{k},\vec{k'})=V_l(k,k')f(\hat{k},\hat{k'})$ with $V_l(k,k')=-V_l\mathbbm{1}_{\mathcal{D}_l}(\vec{k})\mathbbm{1}_{\mathcal{D}_l}(\vec{k'})$. This approach gives a binding energy $E_{B,l}$ given by
\begin{equation}
E_{B,l}=\frac{2\epsilon_l}{e^{2/\lambda_{\text{eff}}}-1}
\end{equation}
with $\lambda_{\text{eff}}=\big(1-\big|\mathcal{B}(k_F)\big|k_F^2\big)\lambda$, i.e. \emph{the result obtained for the conventional Cooper problem extends to all singlet and triplet pairings}.
\subsection{Many-body problem: generalized BCS theory}\label{ssec:BCSgen}
We now briefly address the many-body problem from a more general point of view, using the generalized BCS theory presented in Refs. \cite{sigrist_2005} and \cite{SigristUeda}. Its Hamiltonian is
\begin{equation}
H=\sum_{\vec{k}}\xi_{\vec{k}}c^\dagger_{\vec{k}\sigma}c_{\vec{k}\sigma}+\frac{1}{2}\sum_{\substack{\sigma_1\sigma_2\\\sigma_3\sigma_4}}\sum_{\vec{k}\vec{k'}}V^{\substack{\sigma_1\sigma_2\\\sigma_3\sigma_4}}_{\text{eff},\vec{k}\vec{k'}}c^\dagger_{\vec{k}\sigma_1}c^\dagger_{-\vec{k}\sigma_2}c_{-\vec{k'}\sigma_3}c_{\vec{k'}\sigma_4},
\end{equation}
with the effective interaction containing the Berry curvature corrections. The mean-field theory of this Hamiltonian gives rise to a $2\times2$ matrix $\widehat{\Delta}_{\vec{k}}$. As in the conventional case, one can prove that the dressed order parameter has the same parity as the bare one. Similarly to the Cooper problem, let us investigate a pairing that is either singlet or triplet. Then the gap equation has the form \cite{SigristUeda}
\begin{equation}
\Delta^{\sigma_1\sigma_2}_{\vec{k}}=-\sum_{\sigma_3\sigma_4}\sum_{\vec{k'}}V^{\substack{\sigma_2\sigma_1\\\sigma_3\sigma_4}}_{\text{eff},\vec{k}\vec{k'}}\mathscr{I}^{\sigma_3\sigma_4}_\beta(\vec{k'}),
\end{equation}
and the expansion of the effective interaction yields
\begin{widetext}
\begin{align}
\label{expandVgen}
&\Delta^{\sigma_1\sigma_2}_{\vec{k}}=-\bigg(1-\frac{1}{2}\big|\mathcal{B}(k_F)\big|\vec{k}^2\bigg)\sum_{\sigma_3\sigma_4}\sum_{\vec{k'}}V^{\substack{\sigma_2\sigma_1\\\sigma_3\sigma_4}}_{\vec{k}\vec{k'}}\mathscr{I}^{\sigma_3\sigma_4}_{\beta}(\vec{k'})-\Bigg(\frac{i}{2}\vec{\Lambda}^{\xi_1,\xi_2}_-(\vec{k})\times\vec{k}+\big|\mathcal{B}(k_F)\big|\vec{k}\Bigg)\cdot\sum_{\sigma_3\sigma_4}\sum_{\vec{k'}}\vec{k'}V^{\substack{\sigma_2\sigma_1\\\sigma_3\sigma_4}}_{\vec{k}\vec{k'}}\mathscr{I}^{\sigma_3\sigma_4}_{\beta}(\vec{k'})\nonumber\\
&-\frac{1}{2}\big|\mathcal{B}(k_F)\big|\sum_{\sigma_3\sigma_4}\sum_{\vec{k'}}\vec{k'}^2V^{\substack{\sigma_2\sigma_1\\\sigma_3\sigma_4}}_{\vec{k}\vec{k'}}\mathscr{I}^{\sigma_3\sigma_4}_{\beta}(\vec{k'}),
\end{align}
\end{widetext}
where the summand of the $\vec{k'}$-linear term is \begin{equation}
\sum_{\sigma_3\sigma_4}\vec{k'}V^{\substack{\sigma_2\sigma_1\\\sigma_3\sigma_4}}_{\vec{k}\vec{k'}}\mathscr{I}^{\sigma_3\sigma_4}_\beta(\vec{k'})
\end{equation}
We study two separate cases now. First, let us consider a unitary pairing, \textit{i.e.} one for which $\widehat{\Delta}_{\vec{k}}\widehat{\Delta}^\dagger_{\vec{k}}\propto\sigma_0$. This entails all singlet pairings and unitary triplet pairings (those without spin polarization). In that case, the kernel $\hat{\mathscr{I}}_{\beta}(\vec{k'})$ is given by \cite{SigristUeda,sigrist_2005}
\begin{equation}
\hat{\mathscr{I}}_\beta(\vec{k'})=\frac{\widehat{\Delta}_{\vec{k'}}}{2E_{\vec{k'}}}\tanh\bigg(\frac{\beta}{2}E_{\vec{k'}}\bigg).
\end{equation}
Since the order parameter generally obeys $\widehat{\Delta}_{-\vec{k}}=-\widehat{\Delta}^\top_{\vec{k}}$ and $E_{-\vec{k}}=E_{\vec{k}}$, we have
\begin{equation}
\mathscr{I}^{\sigma_3\sigma_4}_\beta({-\vec{k'}})=-\mathscr{I}^{\sigma_4\sigma_3}_\beta(\vec{k'}).
\end{equation}
Furthermore, in order to respect the anticommutation relations of the fermionic operators, the \cite{sigrist_2005} interaction must obey $V^{\substack{\sigma_2\sigma_1\\\sigma_3\sigma_4}}_{\vec{k};-\vec{k'}}=-V^{\substack{\sigma_2\sigma_1\\\sigma_4\sigma_3}}_{\vec{k}\vec{k'}}$. With this, we have
\begin{align}
\sum_{\sigma_3\sigma_4}-\vec{k'}V^{\substack{\sigma_2\sigma_1\\\sigma_3\sigma_4}}_{\vec{k};-\vec{k'}}\mathscr{I}^{\sigma_3\sigma_4}_\beta(-\vec{k'})&=-\sum_{\sigma_3\sigma_4}\vec{k'}V^{\substack{\sigma_2\sigma_1\\\sigma_4\sigma_3}}_{\vec{k}\vec{k'}}\mathscr{I}^{\sigma_4\sigma_3}_\beta(\vec{k'})\nonumber\\
&=-\sum_{\sigma_3\sigma_4}\vec{k'}V^{\substack{\sigma_2\sigma_1\\\sigma_3\sigma_4}}_{\vec{k}\vec{k'}}\mathscr{I}^{\sigma_3\sigma_4}_\beta(\vec{k'})
\end{align}
\textit{i.e.} the latter is odd in $\vec{k'}$. If one takes the interaction to be non-zero in a thin layer of energy around the Fermi level with energy cutoff $\epsilon_c$, the sum over the term that is linear in $\vec{k'}$ in Eq. (\ref{expandVgen}) vanishes again, a situation encountered several times in this paper. So the Berry dipole term does not change the gap and critical temperature for unitary pairings. As pointed out in the appendix, the latter is also valid for non-unitary triplet pairings. Therefore, \emph{the Berry dipole term does not change the gap equation for pure singlet and triplet pairings}.
\subsection{Possible situations in which the Berry dipole term may become relevant}\label{ssec:mixed}
In view of the above results, one may then wonder if there is any possible effect of the Berry dipole term on superconductivity. What we proved so far is that it does not change the SC gap or $T_c$ if the parity of the pairing is well defined. So a necessary condition for the Berry dipole term to actually contribute would be a superconducting phase without a fixed parity. We saw in the Cooper problem that the Berry dipole term drops out because the following sum is zero
\begin{equation}
\label{sumVg}
\sum_{\vec{k'}}\vec{k'}V_{\vec{k}\vec{k'}}g_{\vec{k'}}.
\end{equation}
If we decompose the two functions $V_{\vec{k}\vec{k'}}$ and $g_{\vec{k'}}$ in the sum of an even and an odd function
\begin{align}
V_{\vec{k}\vec{k'}}&=V^e_{\vec{k}\vec{k'}}+V^o_{\vec{k}\vec{k'}}\\
g_{\vec{k'}}&=g^e_{\vec{k'}}+g^o_{\vec{k'}}
\end{align}
and interpret the $e$ and $o$ parts respectively as the singlet and triplet parts, we then have
\begin{equation}
V_{\vec{k}\vec{k'}}g_{\vec{k'}}=V^e_{\vec{k}\vec{k'}}g^e_{\vec{k'}}+V^o_{\vec{k}\vec{k'}}g^o_{\vec{k'}}+V^o_{\vec{k}\vec{k'}}g^e_{\vec{k'}}+V^e_{\vec{k}\vec{k'}}g^o_{\vec{k'}}.
\end{equation}
While the first two terms disappear in Eq. (\ref{sumVg}) as they are even functions of $\vec{k'}$, the other two terms do \textit{a priori} not disappear as they are odd functions of $\vec{k'}$. $V^og^e$ may be interpreted as the interactions between triplet pairs in the presence of singlet pairs while $V^eg^o$ is the opposite. These two terms may then be an opportunity for the Berry dipole term to have a non-zero contribution in the superconducting phase, \textit{i.e.} if the latter shows coexistence between singlet and triplet pairs. We would then need a superconducting phase where none of the two dominates. Some materials have been proposed to exhibit two superconducting phases, each with a different parity, such as $\text{CeRh}_2\text{As}_2$ and bilayer-$\text{NbSe}_2$ \cite{doi:10.1126/science.abe7518,möckli2021unconventional,doi:10.1126/science.abe7518}. Notice furthermore that a very recent theoretical study argues that the observed superconducting phase in twisted bilayer graphene \cite{Cao2018} might be due to an admixture of singlet and triplet pairs \cite{Lake2022}, and the Berry dipole term might then be a relevant parameter in the stabilization of this type of superconductivity.
\section{Possible experimental implications of the Berry curvature on 2D BCS superconductivity}\label{sec:exp}
\begin{figure}[htbp]
\centering
\includegraphics[width=\linewidth]{Tcratiodop.png}
\caption{Ratio $T_c/T_c^{\text{BCS}}$ as a function of $\lambdabar_{\xi\sigma}k_F\propto \sqrt{n_{\text{2D}}}$. Here, we have used $AV/2\pi\Delta_{\xi\sigma}\simeq0.2$ for illustration.}.
\label{fig:my_label}
\end{figure}
As shown in Sec. \ref{sec:doping}, the Berry curvature has its strongest effect at Fermi wave vectors that are on the order of the inverse effective (Compton) length $\lambdabar_{\xi\sigma}$. Even if the relative reduction of the coupling constant is on the order of 19\%, one needs to keep in mind that the experimentally measurable superconducting gap and critical temperature depend exponentially on the coupling constant. Indeed, the former is accessible by spectroscopic means, \textit{e.g.} in scanning-tunneling spectroscopy, and the latter within resistive temperature-dependent measurements. Experimentally, it is likely impossible to change the Berry curvature \textit{in situ} because this would require experimental access to the band parameters, such as the direct band gap in 2D TMDC. While one could hope to change it \textit{e.g} under strain, also the phonon spectrum and the electron-phonon coupling would then change, possibly in an uncontrolled manner, thus excluding a direct measurement of the Berry-curvature effect in superconductivity.
However, one may compare the evolution of the Berry-curvature dependent superconducting gap or critical temperature, measured as a function of doping, to the \textit{expected} behavior of these quantities. Direct comparison of the critical temperature $T_c$ in Eq. (\ref{eq:TC}), in terms of the effective coupling constant (\ref{eq:binding}), yields the ratio
\begin{equation}\label{eq:ratioTC}
\frac{T_c}{T_c^{\text{BCS}}}=\exp\Bigg(-\frac{2\pi\Delta_{\xi\sigma}}{AV}\frac{\big|\mathcal{B}(k_F)\big|k_F^2}{\Big(1-\big|\mathcal{B}(k_F)\big|k_F^2\Big)\sqrt{1+\lambdabar_{\xi\sigma}^2k_F^2}}\Bigg),
\end{equation}
where $T_c^{\text{BCS}}$ is the BCS critical temperature in the absence of Berry-curvature terms.
We notice here the clear competition between the Berry curvature (through the gap) and superconductivity (through the attractive interaction $V$.). The ratio (\ref{eq:ratioTC}) is plotted in Fig. \ref{fig:my_label} as a function of the doping-dependent Fermi wave vector, $k_F=\sqrt{(4\pi/g) n_{\text{2D}}}$, in terms of the induced 2D electronic density $n_{\text{2D}}$. The factor $g$ takes into account the degeneracy due to internal degrees of freedom, such as the valley and the spin. Notice that, in 2D TMDC with a prominent spin-orbit coupling, the valley and spin degrees of freedom are generically locked, as mentioned above. One would therefore expect $g=2$ in these materials. This is likely the case in the valence band, with a spin-orbit splitting on the order of $\sim 100$ meV, while it is only in the $\sim 1...10$ meV range in the conduction band. The reduction of the critical temperature is strongest at the minimum, which occurs at $\lambdabar_{\xi\sigma}k_F\simeq1.05$. This corresponds to an electronic density of
\begin{equation}
n_{2\text{D}}=\frac{g}{4\pi}k_F^2\simeq1.1\frac{g}{4\pi}\lambdabar_{\xi\sigma}^{-2}.
\end{equation}
We can then give an approximation of the minimum of the ratio as
\begin{equation}
\min_{k_F}\frac{T_c}{T_c^{\text{BCS}}}\simeq\exp\bigg(-0.15\frac{2\pi\Delta_{\xi\sigma}}{AV}\bigg).
\end{equation}
\section{Conclusions}
In conclusion, we have studied the effect of the Berry curvature on BCS-type superconductors in 2D electronic systems. We have shown that the two-body Hamiltonian for interacting electrons inherits terms that are linear in the Berry curvature and that are inherited from the single-electron band structure. In this case, the Berry curvature, which arises in the adiabatic limit when the electrons are restricted to a single band due to purely virtual transitions to the other bands, is coupled to electric potentials beyond the periodic one, which gives rise to the Bloch bands. While such potentials may arise due to external electric fields, they naturally arise when interactions between the electrons (or holes) are taken into account. Generically, the Berry curvature provides a dipolar structure to the charged pairs, and one of the terms emerging in the two-body Hamiltonian can indeed be interpreted as a dipole in an electric field. A second term emerges in the form of a Darwin term, in which the Berry curvature couples to the Laplacian of the electric potential. This term is best understood within a relativistic treatment of the (massive) Dirac Hamiltonian that mimics the two adjacent bands in a direct-gap semiconductor.
Following the lines of the usual BCS treatment of superconductivity in the weak-coupling limit, we have shown that the latter Darwin term generally lowers the BCS coupling constant. As a consequence, this lowers also the stability of the Cooper pair so that the superconducting gap and critical temperature are decreased. On the contrary, the dipolar term, which potentially has the power to increase superconductivity, does not affect the superconducting properties in an $s$-wave or any pure singlet or triplet superconductor because of their fixed parity. The dipolar term might then play a role in systems where superconducting phases of different parity coexist or where the superconducting order parameter does not have a fixed parity. This path might be explored in future work, but it is beyond the scope of our present paper.
Interestingly, the gap-to-$T_c$ ratio remains the same as in the conventional BCS theory in the weak-coupling limit, that we have considered here. Upon doping, the reduction of BCS superconductivity is strongest when the Fermi wave vector is on the order of the inverse effective Compton length, $k_F\sim\lambdabar_{\xi\sigma}^{-1}$, where the BCS coupling constant is lowered by $19\%$. Indeed, for stronger doping, the Fermi level is situated at wave vectors, where the Berry curvature rapidly tends to zero. Since the superconducting gap and the critical temperature both depend exponentially on the BCS coupling constant, the relatively weak reduction of the coupling constant is more prominent there. Our calculations show that the reduction of the doping-dependent superconducting gap and critical temperature depends then both on the band gap, which determines the value of the Berry curvature, as well as on the effective electron-electron interaction. The experimental measurement of these quantities in 2D materials upon doping might then provide a test of our theoretical studies if compared to the expected evolution predicted by the usual BCS theory in the absence of Berry-curvature corrections.
\section*{Acknowledgements}
We thank J. Meyer and A. Mesaros for valuable discussions.
\bibliographystyle{apsrev4-2}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 5,288 |
PPECB approved produce is recognised by importers world-wide as meeting the required food safety and quality standards.
The PPECB is a public entity, constituted and mandated in terms of the Perishable Products Export Control Act (PPEC Act), No 9, of 1983 to perform cold chain services. The PPECB also delivers inspection and food safety services assigned by the Department of Agriculture, Forestry and Fisheries (DAFF) under the Agricultural Products Standards Act, No. 119 of 1990.
Through our inspection services we ensure all products meet the minimum food safety and quality requirements of importing countries and further ensure quality is maintained through proper cold chain management until the produce reaches its destination.
International importers wishing to learn more about PPECB's services can contact us here. | {
"redpajama_set_name": "RedPajamaC4"
} | 4,505 |
Return to the Tugela
Nicholas Bratton
Hummingbirds, Hibernation, and Human Medicine: What the World's Tiniest Birds Reveal About Our Future
$25 Ally Price: FREE
« Solving the Plastic-Waste Crisis: Reframing our Relationship with Trash
Penguins as a Catalyst for Change: How a Vulnerable Species Group Can Inspire Global Conservation »
Hummingbirds are incredible animals. Some of the world's tiniest birds, they have among the highest metabolic rates of all vertebrates — they burn through energy so quickly they are almost always a few hours from perishing. But these little creatures have remarkable strategies for survival. For Dr. Anusha Shankar, who has studied hummingbirds from the cloud forests of Ecuador to the deserts of Arizona, it was the hummingbirds' ability to use a hibernation-like state called "torpor" to save energy at night that ignited her interest. In "Hummingbirds, Hibernation, and Human Medicine," Shankar not only shares her fascination with the strategies hummingbirds use to survive, but also how her and other scientists' research on torpor has implications that soar beyond the realm of birds. Such research illuminates possibilities for human medicine, from life-saving techniques to futuristic ambitions such as cryogenics and human hibernation.
Anusha Shankar studies hummingbirds as a Rose Postdoctoral Fellow at the Cornell Lab of Ornithology. She is investigating how hummingbirds can use a hibernation-like state called "torpor" to get cold (50°F) and rewarm safely every night, without damaging organs like their hearts and brains. During her PhD, Shankar captured hummingbird nightlife with infrared video, and before that tracked king cobras and studied giant birds — hornbills — in India. She plans to work longer term in the tropics, with a home base in India.
Shankar is passionate about teaching and mentoring, and has mentored 17 students on her projects in the past few years. She is also a National Geographic Explorer and Young Leader and loves dancing salsa, bachata, and swing, and reading fiction.
Purchase Tickets $25 Ally Price: FREE
Dr. Anusha Shankar
https://www.tickettailor.com/events/hiddencompass/628056/
Alliance Price | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 4,031 |
Justia › US Law › Case Law › Texas Case Law › Texas Court of Criminal Appeals Decisions › 1970 › Martin v. State
Martin v. State
460 S.W.2d 919 (1970)
Leo Edward MARTIN, Appellant, v. The STATE of Texas, Appellee.
No. 43098.
Court of Criminal Appeals of Texas.
Rehearing Denied December 31, 1970.
*920 Gordon McDowell, Dallas, for appellant.
Henry Wade, Dist. Atty., John B. Tolle, W. T. Westmoreland, Jr., Edgar A. Mason and Harry J. Schulz, Asst. Dist. Attys., Houston, and Jim D. Vollers, State's Atty., Austin, for the State.
DOUGLAS, Judge.
The conviction is for robbery with a firearm; the punishment, sixty years.
The sufficiency of the evidence is not challenged.
James T. Clark testified that on April 1, 1968, appellant, armed with a pistol, entered his supermarket in Dallas and demanded his money. All of the cash registers were emptied and appellant put the currency in a bag and after determining that the safe contained only silver, he fled.
Cuba Sheehan, an employee of the store, identified the appellant as the robber.
Appellant testified that he was in Coppell, Texas, on the day in question and that he did not commit the robbery.
He was impeached by his prior criminal record which included convictions for desertion at Fort Leavenworth, Kansas in 1948; for attempted larceny in Oklahoma in 1949; for grand larceny in Arizona in 1954; for violating the Dyer Act in Fort Worth in 1954; for felony theft in Dallas in 1956; for burglary in California in 1958; for two separate offenses of burglary in Dallas in 1960; for burglary in Odessa in 1961. It was shown that he was released from the penitentiary on March 1, 1968, and was convicted for theft of property under the value of $50.00 for an offense which occurred on March 3, 1968, and was assessed punishment of 150 days in the county jail.
In his first ground of error the appellant contends that the trial court erred when it failed to grant motions by both appellant and his counsel to permit said counsel to withdraw from the case. The record reflects that appellant was indigent and that on September 18, 1968, the court appointed Honorable Jon Franks to represent him. The record further shows that the case was passed some six times "generally" and three times at the request of the appellant. The case was tried on April 7, 1969.
On March 14, 1969, counsel filed a motion to withdraw, stating as a reason therefor that appellant was not satisfied with counsel and that they were unable to cooperate and that they did not see eye to eye.
The contention here is that appellant did not agree with and was dissatisfied with his court-appointed counsel. In Jackson v. United States, 258 F. Supp. 175 (U.S.D.C., which was affirmed in 384 F.2d 375 [5th Cir. 1967]), it was held:
"If the defendant does not agree with his counsel, he has a right to present his own contentions; but the sovereign is under no duty to search for counsel until it finds one who will agree with him."
*921 The record reflects nothing prejudicial against the appellant except the evidence against him.
No abuse of discretion has been shown. The first ground of error is overruled.
In his second ground of error, the appellant contends that he was denied effective assistance of counsel because his appointed counsel failed to subpoena two witnesses who, appellant contends, could have testified favorably to his defense of alibi. It is shown, by appellant's own testimony, that appellant did not know where the witnesses were at the time of trial.
Ineffective assistance of counsel is not shown.
The Court fails to find in the record any indication that "the trial was a farce, or a mockery of justice, or was shocking to the conscience of the reviewing court, or the purported representation was only perfunctory, in bad faith, a sham, a pretense, or without adequate opportunity for conference and preparation." Wilson v. State, Tex.Cr.App., 457 S.W.2d 902. See also Williams v. Beto, 354 F.2d 698 (5th Cir. 1965).
Appellant's second ground of error is overruled.
The judgment is affirmed.
Get free summaries of new Texas Court of Criminal Appeals opinions delivered to your inbox! | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 7,640 |
<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 4.01 Transitional//EN" "http://www.w3.org/TR/html4/loose.dtd">
<!--NewPage-->
<HTML>
<HEAD>
<!-- Generated by javadoc (build 1.6.0_20) on Sat Nov 27 21:36:21 PST 2010 -->
<TITLE>
NeighborIndex (JGraphT : a free Java graph library)
</TITLE>
<META NAME="date" CONTENT="2010-11-27">
<LINK REL ="stylesheet" TYPE="text/css" HREF="../../../stylesheet.css" TITLE="Style">
<SCRIPT type="text/javascript">
function windowTitle()
{
if (location.href.indexOf('is-external=true') == -1) {
parent.document.title="NeighborIndex (JGraphT : a free Java graph library)";
}
}
</SCRIPT>
<NOSCRIPT>
</NOSCRIPT>
</HEAD>
<BODY BGCOLOR="white" onload="windowTitle();">
<HR>
<!-- ========= START OF TOP NAVBAR ======= -->
<A NAME="navbar_top"><!-- --></A>
<A HREF="#skip-navbar_top" title="Skip navigation links"></A>
<TABLE BORDER="0" WIDTH="100%" CELLPADDING="1" CELLSPACING="0" SUMMARY="">
<TR>
<TD COLSPAN=2 BGCOLOR="#EEEEFF" CLASS="NavBarCell1">
<A NAME="navbar_top_firstrow"><!-- --></A>
<TABLE BORDER="0" CELLPADDING="0" CELLSPACING="3" SUMMARY="">
<TR ALIGN="center" VALIGN="top">
<TD BGCOLOR="#EEEEFF" CLASS="NavBarCell1"> <A HREF="../../../overview-summary.html"><FONT CLASS="NavBarFont1"><B>Overview</B></FONT></A> </TD>
<TD BGCOLOR="#EEEEFF" CLASS="NavBarCell1"> <A HREF="package-summary.html"><FONT CLASS="NavBarFont1"><B>Package</B></FONT></A> </TD>
<TD BGCOLOR="#FFFFFF" CLASS="NavBarCell1Rev"> <FONT CLASS="NavBarFont1Rev"><B>Class</B></FONT> </TD>
<TD BGCOLOR="#EEEEFF" CLASS="NavBarCell1"> <A HREF="package-tree.html"><FONT CLASS="NavBarFont1"><B>Tree</B></FONT></A> </TD>
<TD BGCOLOR="#EEEEFF" CLASS="NavBarCell1"> <A HREF="../../../deprecated-list.html"><FONT CLASS="NavBarFont1"><B>Deprecated</B></FONT></A> </TD>
<TD BGCOLOR="#EEEEFF" CLASS="NavBarCell1"> <A HREF="../../../index-all.html"><FONT CLASS="NavBarFont1"><B>Index</B></FONT></A> </TD>
<TD BGCOLOR="#EEEEFF" CLASS="NavBarCell1"> <A HREF="../../../help-doc.html"><FONT CLASS="NavBarFont1"><B>Help</B></FONT></A> </TD>
</TR>
</TABLE>
</TD>
<TD ALIGN="right" VALIGN="top" ROWSPAN=3><EM>
</EM>
</TD>
</TR>
<TR>
<TD BGCOLOR="white" CLASS="NavBarCell2"><FONT SIZE="-2">
<A HREF="../../../org/jgrapht/alg/KShortestPaths.html" title="class in org.jgrapht.alg"><B>PREV CLASS</B></A>
<A HREF="../../../org/jgrapht/alg/StrongConnectivityInspector.html" title="class in org.jgrapht.alg"><B>NEXT CLASS</B></A></FONT></TD>
<TD BGCOLOR="white" CLASS="NavBarCell2"><FONT SIZE="-2">
<A HREF="../../../index.html?org/jgrapht/alg/NeighborIndex.html" target="_top"><B>FRAMES</B></A>
<A HREF="NeighborIndex.html" target="_top"><B>NO FRAMES</B></A>
<SCRIPT type="text/javascript">
<!--
if(window==top) {
document.writeln('<A HREF="../../../allclasses-noframe.html"><B>All Classes</B></A>');
}
//-->
</SCRIPT>
<NOSCRIPT>
<A HREF="../../../allclasses-noframe.html"><B>All Classes</B></A>
</NOSCRIPT>
</FONT></TD>
</TR>
<TR>
<TD VALIGN="top" CLASS="NavBarCell3"><FONT SIZE="-2">
SUMMARY: NESTED | FIELD | <A HREF="#constructor_summary">CONSTR</A> | <A HREF="#method_summary">METHOD</A></FONT></TD>
<TD VALIGN="top" CLASS="NavBarCell3"><FONT SIZE="-2">
DETAIL: FIELD | <A HREF="#constructor_detail">CONSTR</A> | <A HREF="#method_detail">METHOD</A></FONT></TD>
</TR>
</TABLE>
<A NAME="skip-navbar_top"></A>
<!-- ========= END OF TOP NAVBAR ========= -->
<HR>
<!-- ======== START OF CLASS DATA ======== -->
<H2>
<FONT SIZE="-1">
org.jgrapht.alg</FONT>
<BR>
Class NeighborIndex<V,E></H2>
<PRE>
java.lang.Object
<IMG SRC="../../../resources/inherit.gif" ALT="extended by "><B>org.jgrapht.alg.NeighborIndex<V,E></B>
</PRE>
<DL>
<DT><B>All Implemented Interfaces:</B> <DD>java.util.EventListener, <A HREF="../../../org/jgrapht/event/GraphListener.html" title="interface in org.jgrapht.event">GraphListener</A><V,E>, <A HREF="../../../org/jgrapht/event/VertexSetListener.html" title="interface in org.jgrapht.event">VertexSetListener</A><V></DD>
</DL>
<HR>
<DL>
<DT><PRE>public class <B>NeighborIndex<V,E></B><DT>extends java.lang.Object<DT>implements <A HREF="../../../org/jgrapht/event/GraphListener.html" title="interface in org.jgrapht.event">GraphListener</A><V,E></DL>
</PRE>
<P>
Maintains a cache of each vertex's neighbors. While lists of neighbors can be
obtained from <A HREF="../../../org/jgrapht/Graphs.html" title="class in org.jgrapht"><CODE>Graphs</CODE></A>, they are re-calculated at each invocation by
walking a vertex's incident edges, which becomes inordinately expensive when
performed often.
<p>Edge direction is ignored when evaluating neighbors; to take edge
direction into account when indexing neighbors, use <A HREF="../../../org/jgrapht/alg/DirectedNeighborIndex.html" title="class in org.jgrapht.alg"><CODE>DirectedNeighborIndex</CODE></A>.
<p>A vertex's neighbors are cached the first time they are asked for (i.e.
the index is built on demand). The index will only be updated automatically
if it is added to the associated graph as a listener. If it is added as a
listener to a graph other than the one it indexes, results are undefined.</p>
<P>
<P>
<DL>
<DT><B>Since:</B></DT>
<DD>Dec 13, 2005</DD>
<DT><B>Author:</B></DT>
<DD>Charles Fry</DD>
</DL>
<HR>
<P>
<!-- ======== CONSTRUCTOR SUMMARY ======== -->
<A NAME="constructor_summary"><!-- --></A>
<TABLE BORDER="1" WIDTH="100%" CELLPADDING="3" CELLSPACING="0" SUMMARY="">
<TR BGCOLOR="#CCCCFF" CLASS="TableHeadingColor">
<TH ALIGN="left" COLSPAN="2"><FONT SIZE="+2">
<B>Constructor Summary</B></FONT></TH>
</TR>
<TR BGCOLOR="white" CLASS="TableRowColor">
<TD><CODE><B><A HREF="../../../org/jgrapht/alg/NeighborIndex.html#NeighborIndex(org.jgrapht.Graph)">NeighborIndex</A></B>(<A HREF="../../../org/jgrapht/Graph.html" title="interface in org.jgrapht">Graph</A><<A HREF="../../../org/jgrapht/alg/NeighborIndex.html" title="type parameter in NeighborIndex">V</A>,<A HREF="../../../org/jgrapht/alg/NeighborIndex.html" title="type parameter in NeighborIndex">E</A>> g)</CODE>
<BR>
Creates a neighbor index for the specified undirected graph.</TD>
</TR>
</TABLE>
<!-- ========== METHOD SUMMARY =========== -->
<A NAME="method_summary"><!-- --></A>
<TABLE BORDER="1" WIDTH="100%" CELLPADDING="3" CELLSPACING="0" SUMMARY="">
<TR BGCOLOR="#CCCCFF" CLASS="TableHeadingColor">
<TH ALIGN="left" COLSPAN="2"><FONT SIZE="+2">
<B>Method Summary</B></FONT></TH>
</TR>
<TR BGCOLOR="white" CLASS="TableRowColor">
<TD ALIGN="right" VALIGN="top" WIDTH="1%"><FONT SIZE="-1">
<CODE> void</CODE></FONT></TD>
<TD><CODE><B><A HREF="../../../org/jgrapht/alg/NeighborIndex.html#edgeAdded(org.jgrapht.event.GraphEdgeChangeEvent)">edgeAdded</A></B>(<A HREF="../../../org/jgrapht/event/GraphEdgeChangeEvent.html" title="class in org.jgrapht.event">GraphEdgeChangeEvent</A><<A HREF="../../../org/jgrapht/alg/NeighborIndex.html" title="type parameter in NeighborIndex">V</A>,<A HREF="../../../org/jgrapht/alg/NeighborIndex.html" title="type parameter in NeighborIndex">E</A>> e)</CODE>
<BR>
Notifies that an edge has been added to the graph.</TD>
</TR>
<TR BGCOLOR="white" CLASS="TableRowColor">
<TD ALIGN="right" VALIGN="top" WIDTH="1%"><FONT SIZE="-1">
<CODE> void</CODE></FONT></TD>
<TD><CODE><B><A HREF="../../../org/jgrapht/alg/NeighborIndex.html#edgeRemoved(org.jgrapht.event.GraphEdgeChangeEvent)">edgeRemoved</A></B>(<A HREF="../../../org/jgrapht/event/GraphEdgeChangeEvent.html" title="class in org.jgrapht.event">GraphEdgeChangeEvent</A><<A HREF="../../../org/jgrapht/alg/NeighborIndex.html" title="type parameter in NeighborIndex">V</A>,<A HREF="../../../org/jgrapht/alg/NeighborIndex.html" title="type parameter in NeighborIndex">E</A>> e)</CODE>
<BR>
Notifies that an edge has been removed from the graph.</TD>
</TR>
<TR BGCOLOR="white" CLASS="TableRowColor">
<TD ALIGN="right" VALIGN="top" WIDTH="1%"><FONT SIZE="-1">
<CODE> java.util.List<<A HREF="../../../org/jgrapht/alg/NeighborIndex.html" title="type parameter in NeighborIndex">V</A>></CODE></FONT></TD>
<TD><CODE><B><A HREF="../../../org/jgrapht/alg/NeighborIndex.html#neighborListOf(V)">neighborListOf</A></B>(<A HREF="../../../org/jgrapht/alg/NeighborIndex.html" title="type parameter in NeighborIndex">V</A> v)</CODE>
<BR>
Returns a list of vertices which are adjacent to a specified vertex.</TD>
</TR>
<TR BGCOLOR="white" CLASS="TableRowColor">
<TD ALIGN="right" VALIGN="top" WIDTH="1%"><FONT SIZE="-1">
<CODE> java.util.Set<<A HREF="../../../org/jgrapht/alg/NeighborIndex.html" title="type parameter in NeighborIndex">V</A>></CODE></FONT></TD>
<TD><CODE><B><A HREF="../../../org/jgrapht/alg/NeighborIndex.html#neighborsOf(V)">neighborsOf</A></B>(<A HREF="../../../org/jgrapht/alg/NeighborIndex.html" title="type parameter in NeighborIndex">V</A> v)</CODE>
<BR>
Returns the set of vertices which are adjacent to a specified vertex.</TD>
</TR>
<TR BGCOLOR="white" CLASS="TableRowColor">
<TD ALIGN="right" VALIGN="top" WIDTH="1%"><FONT SIZE="-1">
<CODE> void</CODE></FONT></TD>
<TD><CODE><B><A HREF="../../../org/jgrapht/alg/NeighborIndex.html#vertexAdded(org.jgrapht.event.GraphVertexChangeEvent)">vertexAdded</A></B>(<A HREF="../../../org/jgrapht/event/GraphVertexChangeEvent.html" title="class in org.jgrapht.event">GraphVertexChangeEvent</A><<A HREF="../../../org/jgrapht/alg/NeighborIndex.html" title="type parameter in NeighborIndex">V</A>> e)</CODE>
<BR>
Notifies that a vertex has been added to the graph.</TD>
</TR>
<TR BGCOLOR="white" CLASS="TableRowColor">
<TD ALIGN="right" VALIGN="top" WIDTH="1%"><FONT SIZE="-1">
<CODE> void</CODE></FONT></TD>
<TD><CODE><B><A HREF="../../../org/jgrapht/alg/NeighborIndex.html#vertexRemoved(org.jgrapht.event.GraphVertexChangeEvent)">vertexRemoved</A></B>(<A HREF="../../../org/jgrapht/event/GraphVertexChangeEvent.html" title="class in org.jgrapht.event">GraphVertexChangeEvent</A><<A HREF="../../../org/jgrapht/alg/NeighborIndex.html" title="type parameter in NeighborIndex">V</A>> e)</CODE>
<BR>
Notifies that a vertex has been removed from the graph.</TD>
</TR>
</TABLE>
<A NAME="methods_inherited_from_class_java.lang.Object"><!-- --></A>
<TABLE BORDER="1" WIDTH="100%" CELLPADDING="3" CELLSPACING="0" SUMMARY="">
<TR BGCOLOR="#EEEEFF" CLASS="TableSubHeadingColor">
<TH ALIGN="left"><B>Methods inherited from class java.lang.Object</B></TH>
</TR>
<TR BGCOLOR="white" CLASS="TableRowColor">
<TD><CODE>clone, equals, finalize, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait</CODE></TD>
</TR>
</TABLE>
<P>
<!-- ========= CONSTRUCTOR DETAIL ======== -->
<A NAME="constructor_detail"><!-- --></A>
<TABLE BORDER="1" WIDTH="100%" CELLPADDING="3" CELLSPACING="0" SUMMARY="">
<TR BGCOLOR="#CCCCFF" CLASS="TableHeadingColor">
<TH ALIGN="left" COLSPAN="1"><FONT SIZE="+2">
<B>Constructor Detail</B></FONT></TH>
</TR>
</TABLE>
<A NAME="NeighborIndex(org.jgrapht.Graph)"><!-- --></A><H3>
NeighborIndex</H3>
<PRE>
public <B>NeighborIndex</B>(<A HREF="../../../org/jgrapht/Graph.html" title="interface in org.jgrapht">Graph</A><<A HREF="../../../org/jgrapht/alg/NeighborIndex.html" title="type parameter in NeighborIndex">V</A>,<A HREF="../../../org/jgrapht/alg/NeighborIndex.html" title="type parameter in NeighborIndex">E</A>> g)</PRE>
<DL>
<DD>Creates a neighbor index for the specified undirected graph.
<P>
<DL>
<DT><B>Parameters:</B><DD><CODE>g</CODE> - the graph for which a neighbor index is to be created.</DL>
</DL>
<!-- ============ METHOD DETAIL ========== -->
<A NAME="method_detail"><!-- --></A>
<TABLE BORDER="1" WIDTH="100%" CELLPADDING="3" CELLSPACING="0" SUMMARY="">
<TR BGCOLOR="#CCCCFF" CLASS="TableHeadingColor">
<TH ALIGN="left" COLSPAN="1"><FONT SIZE="+2">
<B>Method Detail</B></FONT></TH>
</TR>
</TABLE>
<A NAME="neighborsOf(java.lang.Object)"><!-- --></A><A NAME="neighborsOf(V)"><!-- --></A><H3>
neighborsOf</H3>
<PRE>
public java.util.Set<<A HREF="../../../org/jgrapht/alg/NeighborIndex.html" title="type parameter in NeighborIndex">V</A>> <B>neighborsOf</B>(<A HREF="../../../org/jgrapht/alg/NeighborIndex.html" title="type parameter in NeighborIndex">V</A> v)</PRE>
<DL>
<DD>Returns the set of vertices which are adjacent to a specified vertex. The
returned set is backed by the index, and will be updated when the graph
changes as long as the index has been added as a listener to the graph.
<P>
<DD><DL>
</DL>
</DD>
<DD><DL>
<DT><B>Parameters:</B><DD><CODE>v</CODE> - the vertex whose neighbors are desired
<DT><B>Returns:</B><DD>all unique neighbors of the specified vertex</DL>
</DD>
</DL>
<HR>
<A NAME="neighborListOf(java.lang.Object)"><!-- --></A><A NAME="neighborListOf(V)"><!-- --></A><H3>
neighborListOf</H3>
<PRE>
public java.util.List<<A HREF="../../../org/jgrapht/alg/NeighborIndex.html" title="type parameter in NeighborIndex">V</A>> <B>neighborListOf</B>(<A HREF="../../../org/jgrapht/alg/NeighborIndex.html" title="type parameter in NeighborIndex">V</A> v)</PRE>
<DL>
<DD>Returns a list of vertices which are adjacent to a specified vertex. If
the graph is a multigraph, vertices may appear more than once in the
returned list. Because a list of neighbors can not be efficiently
maintained, it is reconstructed on every invocation, by duplicating
entries in the neighbor set. It is thus more efficient to use <A HREF="../../../org/jgrapht/alg/NeighborIndex.html#neighborsOf(V)"><CODE>neighborsOf(Object)</CODE></A> unless duplicate neighbors are important.
<P>
<DD><DL>
</DL>
</DD>
<DD><DL>
<DT><B>Parameters:</B><DD><CODE>v</CODE> - the vertex whose neighbors are desired
<DT><B>Returns:</B><DD>all neighbors of the specified vertex</DL>
</DD>
</DL>
<HR>
<A NAME="edgeAdded(org.jgrapht.event.GraphEdgeChangeEvent)"><!-- --></A><H3>
edgeAdded</H3>
<PRE>
public void <B>edgeAdded</B>(<A HREF="../../../org/jgrapht/event/GraphEdgeChangeEvent.html" title="class in org.jgrapht.event">GraphEdgeChangeEvent</A><<A HREF="../../../org/jgrapht/alg/NeighborIndex.html" title="type parameter in NeighborIndex">V</A>,<A HREF="../../../org/jgrapht/alg/NeighborIndex.html" title="type parameter in NeighborIndex">E</A>> e)</PRE>
<DL>
<DD><B>Description copied from interface: <CODE><A HREF="../../../org/jgrapht/event/GraphListener.html#edgeAdded(org.jgrapht.event.GraphEdgeChangeEvent)">GraphListener</A></CODE></B></DD>
<DD>Notifies that an edge has been added to the graph.
<P>
<DD><DL>
<DT><B>Specified by:</B><DD><CODE><A HREF="../../../org/jgrapht/event/GraphListener.html#edgeAdded(org.jgrapht.event.GraphEdgeChangeEvent)">edgeAdded</A></CODE> in interface <CODE><A HREF="../../../org/jgrapht/event/GraphListener.html" title="interface in org.jgrapht.event">GraphListener</A><<A HREF="../../../org/jgrapht/alg/NeighborIndex.html" title="type parameter in NeighborIndex">V</A>,<A HREF="../../../org/jgrapht/alg/NeighborIndex.html" title="type parameter in NeighborIndex">E</A>></CODE></DL>
</DD>
<DD><DL>
<DT><B>Parameters:</B><DD><CODE>e</CODE> - the edge event.<DT><B>See Also:</B><DD><A HREF="../../../org/jgrapht/event/GraphListener.html#edgeAdded(org.jgrapht.event.GraphEdgeChangeEvent)"><CODE>GraphListener.edgeAdded(GraphEdgeChangeEvent)</CODE></A></DL>
</DD>
</DL>
<HR>
<A NAME="edgeRemoved(org.jgrapht.event.GraphEdgeChangeEvent)"><!-- --></A><H3>
edgeRemoved</H3>
<PRE>
public void <B>edgeRemoved</B>(<A HREF="../../../org/jgrapht/event/GraphEdgeChangeEvent.html" title="class in org.jgrapht.event">GraphEdgeChangeEvent</A><<A HREF="../../../org/jgrapht/alg/NeighborIndex.html" title="type parameter in NeighborIndex">V</A>,<A HREF="../../../org/jgrapht/alg/NeighborIndex.html" title="type parameter in NeighborIndex">E</A>> e)</PRE>
<DL>
<DD><B>Description copied from interface: <CODE><A HREF="../../../org/jgrapht/event/GraphListener.html#edgeRemoved(org.jgrapht.event.GraphEdgeChangeEvent)">GraphListener</A></CODE></B></DD>
<DD>Notifies that an edge has been removed from the graph.
<P>
<DD><DL>
<DT><B>Specified by:</B><DD><CODE><A HREF="../../../org/jgrapht/event/GraphListener.html#edgeRemoved(org.jgrapht.event.GraphEdgeChangeEvent)">edgeRemoved</A></CODE> in interface <CODE><A HREF="../../../org/jgrapht/event/GraphListener.html" title="interface in org.jgrapht.event">GraphListener</A><<A HREF="../../../org/jgrapht/alg/NeighborIndex.html" title="type parameter in NeighborIndex">V</A>,<A HREF="../../../org/jgrapht/alg/NeighborIndex.html" title="type parameter in NeighborIndex">E</A>></CODE></DL>
</DD>
<DD><DL>
<DT><B>Parameters:</B><DD><CODE>e</CODE> - the edge event.<DT><B>See Also:</B><DD><A HREF="../../../org/jgrapht/event/GraphListener.html#edgeRemoved(org.jgrapht.event.GraphEdgeChangeEvent)"><CODE>GraphListener.edgeRemoved(GraphEdgeChangeEvent)</CODE></A></DL>
</DD>
</DL>
<HR>
<A NAME="vertexAdded(org.jgrapht.event.GraphVertexChangeEvent)"><!-- --></A><H3>
vertexAdded</H3>
<PRE>
public void <B>vertexAdded</B>(<A HREF="../../../org/jgrapht/event/GraphVertexChangeEvent.html" title="class in org.jgrapht.event">GraphVertexChangeEvent</A><<A HREF="../../../org/jgrapht/alg/NeighborIndex.html" title="type parameter in NeighborIndex">V</A>> e)</PRE>
<DL>
<DD><B>Description copied from interface: <CODE><A HREF="../../../org/jgrapht/event/VertexSetListener.html#vertexAdded(org.jgrapht.event.GraphVertexChangeEvent)">VertexSetListener</A></CODE></B></DD>
<DD>Notifies that a vertex has been added to the graph.
<P>
<DD><DL>
<DT><B>Specified by:</B><DD><CODE><A HREF="../../../org/jgrapht/event/VertexSetListener.html#vertexAdded(org.jgrapht.event.GraphVertexChangeEvent)">vertexAdded</A></CODE> in interface <CODE><A HREF="../../../org/jgrapht/event/VertexSetListener.html" title="interface in org.jgrapht.event">VertexSetListener</A><<A HREF="../../../org/jgrapht/alg/NeighborIndex.html" title="type parameter in NeighborIndex">V</A>></CODE></DL>
</DD>
<DD><DL>
<DT><B>Parameters:</B><DD><CODE>e</CODE> - the vertex event.<DT><B>See Also:</B><DD><A HREF="../../../org/jgrapht/event/VertexSetListener.html#vertexAdded(org.jgrapht.event.GraphVertexChangeEvent)"><CODE>VertexSetListener.vertexAdded(GraphVertexChangeEvent)</CODE></A></DL>
</DD>
</DL>
<HR>
<A NAME="vertexRemoved(org.jgrapht.event.GraphVertexChangeEvent)"><!-- --></A><H3>
vertexRemoved</H3>
<PRE>
public void <B>vertexRemoved</B>(<A HREF="../../../org/jgrapht/event/GraphVertexChangeEvent.html" title="class in org.jgrapht.event">GraphVertexChangeEvent</A><<A HREF="../../../org/jgrapht/alg/NeighborIndex.html" title="type parameter in NeighborIndex">V</A>> e)</PRE>
<DL>
<DD><B>Description copied from interface: <CODE><A HREF="../../../org/jgrapht/event/VertexSetListener.html#vertexRemoved(org.jgrapht.event.GraphVertexChangeEvent)">VertexSetListener</A></CODE></B></DD>
<DD>Notifies that a vertex has been removed from the graph.
<P>
<DD><DL>
<DT><B>Specified by:</B><DD><CODE><A HREF="../../../org/jgrapht/event/VertexSetListener.html#vertexRemoved(org.jgrapht.event.GraphVertexChangeEvent)">vertexRemoved</A></CODE> in interface <CODE><A HREF="../../../org/jgrapht/event/VertexSetListener.html" title="interface in org.jgrapht.event">VertexSetListener</A><<A HREF="../../../org/jgrapht/alg/NeighborIndex.html" title="type parameter in NeighborIndex">V</A>></CODE></DL>
</DD>
<DD><DL>
<DT><B>Parameters:</B><DD><CODE>e</CODE> - the vertex event.<DT><B>See Also:</B><DD><A HREF="../../../org/jgrapht/event/VertexSetListener.html#vertexRemoved(org.jgrapht.event.GraphVertexChangeEvent)"><CODE>VertexSetListener.vertexRemoved(GraphVertexChangeEvent)</CODE></A></DL>
</DD>
</DL>
<!-- ========= END OF CLASS DATA ========= -->
<HR>
<!-- ======= START OF BOTTOM NAVBAR ====== -->
<A NAME="navbar_bottom"><!-- --></A>
<A HREF="#skip-navbar_bottom" title="Skip navigation links"></A>
<TABLE BORDER="0" WIDTH="100%" CELLPADDING="1" CELLSPACING="0" SUMMARY="">
<TR>
<TD COLSPAN=2 BGCOLOR="#EEEEFF" CLASS="NavBarCell1">
<A NAME="navbar_bottom_firstrow"><!-- --></A>
<TABLE BORDER="0" CELLPADDING="0" CELLSPACING="3" SUMMARY="">
<TR ALIGN="center" VALIGN="top">
<TD BGCOLOR="#EEEEFF" CLASS="NavBarCell1"> <A HREF="../../../overview-summary.html"><FONT CLASS="NavBarFont1"><B>Overview</B></FONT></A> </TD>
<TD BGCOLOR="#EEEEFF" CLASS="NavBarCell1"> <A HREF="package-summary.html"><FONT CLASS="NavBarFont1"><B>Package</B></FONT></A> </TD>
<TD BGCOLOR="#FFFFFF" CLASS="NavBarCell1Rev"> <FONT CLASS="NavBarFont1Rev"><B>Class</B></FONT> </TD>
<TD BGCOLOR="#EEEEFF" CLASS="NavBarCell1"> <A HREF="package-tree.html"><FONT CLASS="NavBarFont1"><B>Tree</B></FONT></A> </TD>
<TD BGCOLOR="#EEEEFF" CLASS="NavBarCell1"> <A HREF="../../../deprecated-list.html"><FONT CLASS="NavBarFont1"><B>Deprecated</B></FONT></A> </TD>
<TD BGCOLOR="#EEEEFF" CLASS="NavBarCell1"> <A HREF="../../../index-all.html"><FONT CLASS="NavBarFont1"><B>Index</B></FONT></A> </TD>
<TD BGCOLOR="#EEEEFF" CLASS="NavBarCell1"> <A HREF="../../../help-doc.html"><FONT CLASS="NavBarFont1"><B>Help</B></FONT></A> </TD>
</TR>
</TABLE>
</TD>
<TD ALIGN="right" VALIGN="top" ROWSPAN=3><EM>
</EM>
</TD>
</TR>
<TR>
<TD BGCOLOR="white" CLASS="NavBarCell2"><FONT SIZE="-2">
<A HREF="../../../org/jgrapht/alg/KShortestPaths.html" title="class in org.jgrapht.alg"><B>PREV CLASS</B></A>
<A HREF="../../../org/jgrapht/alg/StrongConnectivityInspector.html" title="class in org.jgrapht.alg"><B>NEXT CLASS</B></A></FONT></TD>
<TD BGCOLOR="white" CLASS="NavBarCell2"><FONT SIZE="-2">
<A HREF="../../../index.html?org/jgrapht/alg/NeighborIndex.html" target="_top"><B>FRAMES</B></A>
<A HREF="NeighborIndex.html" target="_top"><B>NO FRAMES</B></A>
<SCRIPT type="text/javascript">
<!--
if(window==top) {
document.writeln('<A HREF="../../../allclasses-noframe.html"><B>All Classes</B></A>');
}
//-->
</SCRIPT>
<NOSCRIPT>
<A HREF="../../../allclasses-noframe.html"><B>All Classes</B></A>
</NOSCRIPT>
</FONT></TD>
</TR>
<TR>
<TD VALIGN="top" CLASS="NavBarCell3"><FONT SIZE="-2">
SUMMARY: NESTED | FIELD | <A HREF="#constructor_summary">CONSTR</A> | <A HREF="#method_summary">METHOD</A></FONT></TD>
<TD VALIGN="top" CLASS="NavBarCell3"><FONT SIZE="-2">
DETAIL: FIELD | <A HREF="#constructor_detail">CONSTR</A> | <A HREF="#method_detail">METHOD</A></FONT></TD>
</TR>
</TABLE>
<A NAME="skip-navbar_bottom"></A>
<!-- ======== END OF BOTTOM NAVBAR ======= -->
<HR>
</BODY>
</HTML>
| {
"redpajama_set_name": "RedPajamaGithub"
} | 4,246 |
The school had a Mass that was around 40 minutes long.
The Mass started in the morning at 8:00 a.m. The Bishop was present along with other Catholic students, teachers, the principal and some other guests. One of the highlighted events was that the Bishop gave a blessing to our new-constructed chapel. In the 1st Reading, 2nd Reading, and Gospel, everyone listened well. In the Homily, the Bishop was speaking and he was talking about our school and Catholic schools. After the Mass ended, all of the students and teachers took photos with the Bishop.
All of the people there were very happy that the Bishop was present. After all, you don't get to see the Bishop everyday. This is another wonderful event that everyone in CDSJ5 experienced.
On November 1st will be the start of the "Month of Souls". The definition of the month of souls is the time when we as Catholics offer our prayers to the Holy Souls. Recently, students have paper cranes as a memorial activity. Through this activity students thanked and remembered their past relatives, pupils were reminded of the attitudes towards life. These crafts are then given to Art teachers and will be hung up in the school lobby.
"All Saints' Day" will be on November 1st, the start of the "Month of Souls". The day after that will be "All Souls Day" on November 2nd.These meaningful days are similar to Chinese Holidays, such as "Lunar New Year, The Ching Ming Festival and The Chung Yeung Festival". During these Chinese holidays, people worship their ancestors, which is fairly similar to the events in the "Month of Souls".
During RE (Religious Education) classes have written down some words for their ancestors and are then given to the teacher to check their work.
The "Month of Souls" is a month where you can take the time to worship or thank your ancestors for what you wish to say.
The Mass was on the 7th of September, beginning at 9:30 in the morning. Some schools like CDSJ5 partook in this special event. This took place in Se Cathedral.
In the beginning of the Mass, the choir would sing and everyone had to stand up. The First Reading began and everyone listened. The Second Reading was a few minutes later. Then the Homily begins. The Bishop talked about the two readings and how to always move forward.
The people kneeled and prayed to God. The people would then go and take the Eucharist. After that, everyone would pray some more while the choir was singing. The Mass ended shortly afterwards. All the schools took pictures to remember this event.
The Mass was a refreshing breeze to all the Catholic students who attended. Though it was just like any ordinary Mass, the amount of students who attended made it an experience for every individual person who attended. | {
"redpajama_set_name": "RedPajamaC4"
} | 3,078 |
if ( Centi && !Centi.plugins ) Centi.plugins = [];
try {
Centi.plugins.push(
function(centi){
this.target = centi;
this.prefix = 'THREE';
this.shortPrefix = 'T';
this.target.renderer = null;
this.target.scene = null;
this.target.cam = null;
this.destroy = function(){
this.target.renderer = null;
this.target.scene = null;
this.target.cam = null;
this.target = null;
};
this.preSetup = function(){
var obj = this.target;
if ( obj.b3d ) {
obj.ctx = null;
obj.scene = new THREE.Scene();
} else {
obj.renderer = null;
obj.scene = null;
obj.cam = null;
}
};
this.postSetup = function(){
var obj = this.target;
if ( obj.b3d ) {
obj.size(obj.w, obj.h);
try {
console.log(obj.canvas.width, obj.canvas.height);
obj.renderer = new THREE.WebGLRenderer({canvas:obj.canvas, preserveDrawingBuffer: true});
} catch(e){
obj.renderer = new THREE.CanvasRenderer({canvas:obj.canvas});
}
obj.cam = new THREE.PerspectiveCamera( 60, obj.w/obj.h, 1, 100000 );
obj.cam.position.set( 0, 0, 367 );
}
};
this.preUpdate = function(){
};
this.postUpdate = function(){
var obj = this.target;
if ( obj.b3d && obj.renderer ) obj.renderer.render(obj.scene, obj.cam);
};
}
);
} catch(e){} | {
"redpajama_set_name": "RedPajamaGithub"
} | 742 |
EVENTING RIDERS ASSOCIATION
ERA of GB
Novice Masters
Rider Working Groups
LRBHT Hall of Fame
ERAs Work
ERA Team
OFFICIAL PARTNERS OF THE 2021 BLENHEIM PALACE INTERNATIONAL HORSE TRIALS
The Jockey Club is delighted to announce seven prestigious partnerships have been agreed for the 2021 Blenheim Palace International Horse Trials, which take place between Thursday 16th and Sunday 19th September.
The four-day event is being organised for the first time this year by The Jockey Club and takes place against the spectacular backdrop of Blenheim Palace in Oxfordshire, the seat of the Dukes of Marlborough and birthplace of Sir Winston Churchill.
Pol Roger - Official Champagne Partner
Pol Roger has been producing exceptional champagne for over 170 years. To this day the house remains small, family-owned, fiercely independent and unrivalled in its reputation for quality. Indeed, it is this quality which brought about Sir Winston Churchill's lifelong attachment to Pol Roger. Churchill, who famously insisted of Pol Roger "In defeat I need it, in victory I deserve it", was born at Blenheim Palace, hence the partnership with the Jockey Club for the 2021 Blenheim Palace International Horse Trials is particularly fitting. The Champagne Pol Roger Lodge will be the ultimate environment in which to unwind and enjoy a glass of Pol Roger; an oasis from the adrenaline-filled Cross-Country course and fantastic shopping opportunities.
Bentley - Official Luxury Automotive Partner
In 1919, it was W.O. Bentley's vision that kick-started this iconic marque. He set out to make 'a fast car, a good car, the best in its class.' From this flash of inspiration he created an extraordinary company that attracted visionaries, adventurers, entrepreneurs and heroes of motorsport from around the world. People with spirit, people with innovation, people with imagination. Bentley, which celebrated its 100thanniversary in 2019, has been involved at The Jockey Club since 2015.
Boodles - Official Jewellery Partner
Boodles is the leading British fine jeweller whose reputation has been built on the quality of their innovative design, quality and exceptional customer relationships and service. Boodles have nine shops - its flagship stores are on London's Bond Street and in Liverpool. There are three Boodles in the North West of the UK, one in Dublin, two further shops in London, a concession in Harrods Fine Jewellery Hall, and one in the Savoy Hotel. The company was founded as Boodle and Dunthorne in Liverpool in 1798 and has remained a family company. Boodles' relationship with The Jockey Club is longstanding, with the company having sponsored at the Cheltenham Festival since 2014.
Ben & Jerry's - Official Ice Cream Partner
Unilever ice cream has partnered with the event and will activate with the Ben & Jerry's brand. This unconventional and quirky ice cream brand will bring a new dimension to the renamed Ben & Jerry's Arena featuring fun activities across the four days of competition and no doubt a welcome source of dessert for those customers that will be staying on the Campsite! Other opportunities to indulge in their wonderful products will be available extensively across the show grounds.
Mark Todd Collection - Official Clothing Partner
Working with the legendary dual Olympic Gold medal-winning eventer Sir Mark Todd, the Mark Todd Collection is one of the UK's leading brands offering a comprehensive range of high-quality equestrian products. Since launching 25 years ago the brand has over 2,000 products and has become synonymous with quality and craftmanship. The collection will be showcased at the event, including an exclusive Blenheim Palace International Horse Trials range to celebrate the partnership.
Horse & Country – Official Broadcast Partner
Horse & Country is the leading international sports network for the passionate and active equestrian community. Headquartered in London, it is available globally via connected TVs, mobile and web, and on leading digital and pay-TV platforms in the US, UK, Ireland, Germany, Austria, the Netherlands, Sweden, and Australia. Horse & Country's programming line-up includes live coverage from leading sporting competitions in all equestrian disciplines, as well as training and learning shows, documentaries and entertainment.
During the Blenheim Palace International Horse Trials, members of Horse & Country's streaming service, H&C+, will be able to watch all of the sport from all three phases. There will also be a range of free content including the trot up and daily round-up show After Hours (Thursday - Sunday evenings), presented by Alice Fox-Pitt and Chris Hughes.
Ford - Official E-Car Partner
Ford UK, will showcase the Ford Go Electric roadshow during the Blenheim International Horse Trials. This interactive exhibition has been designed to demystify and answer any questions about switching to electric cars and vans and help banish range anxiety. Visitors will learn how to plug in and charge, the benefits of living with an electrified vehicle and what vehicle is best suited to your personal and/or Mach-W and the plug-in hybrid Kuga as well as find out all there is to know about the science of electrification technology and reducing CO2 emissions and motor running costs.
Ian Renton, regional managing director of The Jockey Club, commented: "Our planning for the 2021 Blenheim International Horse Trials is very much a case of full steam ahead and we are delighted to be able to announce a number of our Official Partners today.
"From the likes of Bentley and Boodles, with whom we already have longstanding and successful partnerships, through to renowned and respected names such as Pol Roger and Ben and Jerry's, our team has worked hard to form partnerships with brands which share our vision of taking the Blenheim International Horse Trials to a new level."
About The Blenheim Place International Horse Trials
The Blenheim Palace International Horse Trials is one of the crown jewels of the eventing calendar and takes place over four days in September each year. The COVID-19 pandemic forced the cancellation of the 2020 event and this year's competition will be subject to Government protocols and safety guidelines in place at the time.
The fixture includes FEI-accredited and internationally-recognised competitions; a CCI4*-L (four-star Long) and CCI4*-S for eight and nine year old horses (four-star Short). The 2019 CCI4*-L competition was won by one of Britain's leading event riders, Piggy March, with her then-10 year old gelding, Brookfield Inocent. The 2019 CCI4*-S was won by Australian Chris Burton and his nine year old, Clever Louis.
For all background information on the 2021 Blenheim Palace International Horse Trials, please visit https://bpiht.co.uk/.
JOIN THE EVENTING RIDERS COMMUNITY TODAY:
SUPPORTING BRANDS:
Images courtesy of Equuis Photography/Jason Bax, Eventing Images/Tim Wilkinson & Tilly Berendt
Website by EquiConsulting | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 1,156 |
{"url":"http:\/\/www.cs.cornell.edu\/courses\/cs4780\/2015fa\/web\/lecturenotes\/lecturenote05.html","text":"## Bayes Classifier and Naive Bayes\n\nIdea: Estimate $\\hat{P}(y | \\vec{x})$ from the data, then use the Bayes Classifier on $\\hat{P}(y|\\vec{x})$.\n\nSo how can we estimate $\\hat{P}(y | \\vec{x})$?\n\nOne way to do this would be to use the MLE method. Assuming that $y$ is discrete, $$\\hat{P}(y|\\vec{x}) = \\frac{\\sum_{i=1}^{n} I(\\vec{x}_i = \\vec{x} \\wedge \\vec{y}_i = y)}{ \\sum_{i=1}^{n} I(\\vec{x}_i = \\vec{x})}$$\nFrom the above diagram, it is clear that, using the MLE method, we can estimate $\\hat{P}(y|\\vec{x})$ as $$\\hat{P}(y|\\vec{x}) = \\frac{|C|}{|B|}$$ But there is a big problem with this method.\n\nProblem: The MLE estimate is only good if there are many training vectors with the same identical features as $\\vec{x}$!\n\nIn high dimensional spaces (or with continuous $\\vec{x}$), this never happens! So $|B| \\rightarrow 0$ and $|C| \\rightarrow 0$.\n\nTo get around this issue, we can make a 'naive' assumption.\n\n### Naive Bayes\n\nWe can approach dilemma with a simple trick, and an additional assumption. The trick part is to estimate $P(y)$ and $P(\\vec{x} | y)$ instead, since, by Bayes rule, $$P(y | \\vec{x}) = \\frac{P(\\vec{x} | y)P(y)}{P(\\vec{x})}$$ Estimating $P(y)$ is easy. If $Y$ takes on discrete binary values, for example, this just becomes coin tossing. Also, recall from Estimating Probabilities from Data that estimating $P(y)$ and $P(\\vec{x} | y)$ is called genreative learning.\n\nEstimating $P(\\vec{x}|y)$, however, is not easy! (you will explore this in the written homework). The additional assumption that we make is the Naive Bayes assumption.\n\nNaive Bayes Assumption: $$P(\\vec{x} | y) = \\prod_{\\alpha = 1}^{d} P([\\vec{x}]_\\alpha | y)$$ i.e. Feature values are independent given the label! This is a very bold assumption. As a quick example, suppose that you want to predict the label that indicates if you have the flu or not ($Y \\in \\{\\text{yes flu}, \\text{no flu}\\}$) And you want to predict this using binary features on caughing, fever. Under the naive assumption, if I know that a person has the flu, then knowing that the person is also caughing does not change the probability that the person has a fever. It is clear that this is not the case in reality.\n\nBut, for now, let's assume that it holds.\n\nThen the Bayes Classifier can be defined as \\begin{align} h(\\vec{x}) &= argmax_y P(y | \\vec{x}) \\\\ &= argmax_y \\frac{P(\\vec{x} | y)P(y)}{P(\\vec{x})} \\\\ &= argmax_y P(\\vec{x} | y) P(y) && \\text{($P(\\vec{x})$ does not depend on $y$)} \\\\ &= argmax_y \\prod_{\\alpha=1}^{d} P([\\vec{x}]_\\alpha | y) P(y) && \\text{(By the naive assumption)}\\\\ &= argmax_y \\sum_{\\alpha = 1}^{d} log(P([\\vec{x}]_\\alpha | y)) + log(P(y)) && \\text{(As log is a monotonic function)} \\end{align} Estimating $log(P([\\vec{x}]_\\alpha | y))$ is easy as we only need to consider one dimension. And estimating $P(y)$ is not affected by the assumption, but it is still easy.\n\n### Estimating $P([\\vec{x}]_\\alpha | y)$\n\nNow that we know how we can use our assumption to make estimation of $P(y|\\vec{x})$ tractable, let's move on and look at how we can actually apply our method to different problems. There are 3 notable cases in which we can use our naive Bayes classifier.\n\n#### Case #1: Categorical features\n\nFeatures: $$[\\vec{x}]_\\alpha \\in \\{f_1, f_2, \\cdots, f_{K_\\alpha}\\}$$ Each feature $\\alpha$ falls into one of $K_\\alpha$ categories. (Note that the case with binary features is just a specific case of this)\n\nModel: $$P([\\vec{x}]_{\\alpha} = j | y) = [p_{j}^{y}]_{\\alpha} \\\\ \\text{ where } \\sum_{j=1}^{k} [p_{j}^{y}]_{\\alpha} = 1$$ where $[p_{j}^{y}]_{\\alpha}$ is the probability of feature $\\alpha$ having the value $j$, given that the label is $y$. And the constraint indicates that $[\\vec{x}]_{\\alpha}$ must have one of $\\{1, \\cdots, k\\}$.\n\nEstimator: \\begin{align} [p_{j}^{y}]_{\\alpha} &= \\frac{\\sum_{i=1}^{n} I(y_i = y \\wedge [\\vec{x}_i]_\\alpha = j) + l}{\\sum_{i=1}^{n} I(y_i = y) + lk} \\end{align} where $l$ is the smoothing constant.\n\n#### Case #2: Multinomial features\n\nFeatures: \\begin{align} [\\vec{x}]_\\alpha \\in \\{0, 1, 2, \\cdots \\} && \\text{(Each feature $\\alpha$ represents a count)} \\end{align} An example of this could be the count of a specific word $\\alpha$ in a document.\n\nModel: \\begin{align} P([\\vec{x}]_1 = j_1,\\dots,[\\vec{x}]_d = j_d | y) = \\frac{m!}{j_1!\\dots j_d!}([p^{y}]_{1})^{j_1}\\dots ([p^{y}]_{d})^{j_d} && \\text{and} && \\sum_{\\alpha = 1}^{d} [p^y]_\\alpha = 1 && \\textrm{where $m=\\sum_{\\alpha=1}^n j_\\alpha$} \\end{align} So, for example, if there are $d$ number of words in the vocabulary, each word $\\alpha$ has some probability mass of being spam of ham. For both cases, $y = \\text{spam}$ and $y = \\text{ham}$, the probabilities sum to 1.\n\nEstimator: \\begin{align} [p^y]_\\alpha = \\frac{\\sum_{i = 1}^{n} I(y_i = y)[\\vec{x}]_\\alpha + l}{\\sum_{i=1}^{n}\\sum_{\\beta = 1}^{d} [\\vec{x}]_{\\beta} I(y_i = y) + dl } \\end{align}\n\n#### Case #3: Continuous features (Gaussian Naive Bayes)\n\nFeatures: \\begin{align} [\\vec{x}]_\\alpha \\in \\mathbb{R} && \\text{(each feature takes on a real value)} \\end{align}\n\nModel: \\begin{align} P([\\vec{x}]_\\alpha | y) = \\mathcal{N}([\\vec{\\mu}_y]_\\alpha, [\\vec{\\sigma}_{y}^{2}]_\\alpha) \\end{align} Note that the model specified above is based on our assumption about the data - that each feature $\\alpha$ comes from a class-conditional Gaussian distribution. Other specification could be used as well.\n\nEstimator: \\begin{align} [\\vec{\\mu}_y]_\\alpha &\\leftarrow \\frac{1}{n_y} \\sum_{i = 1}^{n} I(y_i = y) [\\vec{x}]_\\alpha && \\text{where $n_y = \\sum_{i=1}^{n} I(y_i = y)$} \\\\ [\\vec{\\sigma}_y^2]_\\alpha &\\leftarrow \\frac{1}{n_y} \\sum_{i=1}^{n} I(y_i = y)([\\vec{x}_i]_\\alpha - [\\vec{\\mu}_y]_\\alpha)^2 \\end{align}\n\n### Naive Bayes is a linear classifier\n\nSuppose that $y_i \\in \\{-1, +1\\}$ and features are multinomial ($P([\\vec{x}]_\\alpha | y) =$)\n\nLet us define: \\begin{align} [\\vec{w}]_\\alpha &= log(p_{\\alpha}^{+1}) - log(p_{\\alpha}^{-1}) \\\\ b &= log(P(Y = +1)) - log(P(Y = -1)) \\end{align} If we use the above to do classification, we can compute for $\\vec{w}^T \\cdot \\vec{x} + b$\n\nSimplifying this further leads to \\begin{align} \\vec{w}^T \\cdot \\vec{x} + b > 0 &\\Longleftrightarrow \\sum_{\\alpha = 1}^{d} [\\vec{x}]_\\alpha (log(p_{\\alpha}^{+1}) - log(p_{\\alpha}^{-1})) + log(P(Y = +1)) - log(P(Y = -1)) > 0 \\\\ &\\Longleftrightarrow \\frac{\\prod_{\\alpha = 1}^{d} P([\\vec{x}]_\\alpha | Y = +1)P( Y = +1)}{\\prod_{\\alpha =1}^{d}P([\\vec{x}]_\\alpha | Y = -1)P(Y = -1))} > 0 \\\\ &\\Longleftrightarrow P(Y = +1 | \\vec{x}) > P(Y = -1 | \\vec{x}) && \\text{(By our naive Bayes assumption)} \\\\ &\\Longleftrightarrow h(\\vec{x}) = +1 && \\text{(By definition of $h(\\vec{x})$)} \\end{align}\n\n### Some extra notes related to lecture on 9\/21\/2015\n\nI think there is some confusion about Naive Bayes with the multinomial distribution. Let me explain it one more time in a different way.\n\nThe multinomial view: Let us assume we have d possible words in our dictionary. We have a data instance (e.g. an email) consisting of m words. Each of the $d$ words has a probability for spam $p_1,\\dots,p_d$ and for ham $q_1,\\dots,q_d$ (with $\\sum_{\\alpha=1}^d p_\\alpha=1=\\sum_{\\alpha=1}^d q_\\alpha$.) We also have probabilities that an email is spam $\\pi_s$ and that it is ham $\\pi_h$ with $\\pi_s+\\pi_h=1$.\n\nIf you want to generate an email, you follow the following procedure. First you decide with probability $\\pi_s$ if you want to write a spam email or a ham email. Then you pick the corresponding probability distribution and randomly pick $m$ words. $w_1,\\dots,w_m$. Each of these words is picked independently. So if you generate a spam email, the probability that the first term is word $\\alpha$ is $p_\\alpha$. Let us write $w_i=\\alpha$ to denote that the $i^{th}$ word in the email was word $\\alpha$, where $\\alpha\\in\\{1,\\dots,d\\}$.\n\nWhat is the probability that a particular email was generated that way, with the spam label (so we used $p_\\alpha$ instead of $q_\\alpha$)? Keep in mind that in the bag of words representation word order does not matter and we are therefore mostly concerned with how many times each word appeared. Let these counts be $n_1,\\dots, n_d$ to denote the occurrences of word 1, word 2, ... word d respectively (with $\\sum_{\\alpha=1}^d n_\\alpha=m$). To represent your email as a vector, you write $\\vec x=[n_1,\\dots,n_d]^\\top$.\n\nThe multinomial distribution then specifies that the likelihood of the email is \\begin{align} P(\\vec x | Y=spam)=\\frac{m!}{n_1!\\times \\dots \\times n_d!}p_1^{n_1}\\times \\dots \\times p_d^{n_d} \\end{align} We don't actually need the exact likelihood, as we can decide the label by evaluating $\\frac{P(Y=spam|\\vec x)}{P(Y=ham|\\vec x)}$. (If this fraction is greater than one the email is classified as spam otherwise ham.) In this fraction all normalization constants disappear as they are the same for spam and ham. It is therefore enough to evaluate $$P(\\vec x | Y=spam)\\propto p_1^{n_1}\\times \\dots \\times p_d^{n_d}$$ This is exactly what we used in class. You notice, that $m$ disappeared. So, to use naive bayes on document data we do not have to make any assumptions about the length of the document. We simply check if this particular document, no matter what its length is, contains words that are more likely to be generated with the spam generating process or with the ham generating process.\n\nSo the ratio we finally compute is: \\begin{align} \\frac{P(Y=spam| \\vec{x})}{P(Y=ham|\\vec x)}=\\frac{P(\\vec x|Y=spam)P(Y=spam)}{P(\\vec x|Y=ham)P(Y=ham)}=\\frac{p_1^{n_1}\\times\\dots\\times p_d^{n_d}\\times\\pi_s}{q_1^{n_1}\\times\\dots\\times q_d^{n_d}\\times\\pi_h} \\end{align}\n\nHow do we get the probabilities $p_\\alpha$?\n\nWell, $p_\\alpha$ is the probability that word $\\alpha$ is chosen to generate a spam email. So we can estimate it as the fraction of words in spam emails that are word $\\alpha$: $$p_\\alpha=\\frac{\\textrm{no. of times word \\alpha appears in spam emails in our training data}}{\\textrm{no. of words in all spam emails in our training data}}$$ -Kilian","date":"2017-11-20 00:18:24","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\": 12, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.9982469081878662, \"perplexity\": 913.995511370483}, \"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-2017-47\/segments\/1510934805881.65\/warc\/CC-MAIN-20171119234824-20171120014824-00496.warc.gz\"}"} | null | null |
using Lucene.Net.Index;
using Lucene.Net.Util;
namespace Lucene.Net.Codecs.Memory
{
/// <summary>
/// Tests FSTOrdPulsing41PostingsFormat
/// </summary>
public class TestFSTOrdPulsing41PostingsFormat : BasePostingsFormatTestCase
{
private readonly Codec codec = TestUtil.AlwaysPostingsFormat(new FSTOrdPulsing41PostingsFormat());
protected override Codec GetCodec()
{
return codec;
}
}
} | {
"redpajama_set_name": "RedPajamaGithub"
} | 107 |
Skip to MenuSkip to Content
Home / Blog / Top Resources
Top resources for Canadian nonprofits and charities.
Bloomberg Businessweek: "How a Charity Superstar Innovated Its Way to Political Scandal" – with coverage of the WE Charity scandal
Today an article appeared in Bloomberg Businessweek entitled "How a Charity Superstar Innovated Its Way to Political Scandal: The Kielburger brothers…
New course – Running and Maintaining a Federal Non-Profit Corporation under the CNCA – An Introduction
We have recently prepared a new course on Running and Maintaining a Federal Non-Profit Corporation under the CNCA – An Introduction. Here is…
Canadian charities can have relationships with non-charities but need to have adequate separation
We recently did a course entitled Multiple Corporate Structures for Canadian For-Profits, NPOs, + Charities to Enhance Flexibility and Impact.…
Happy Holidays from the OECD – a very interesting report on different countries and how they treat philanthropy and charities
An interesting report from the OECD on charities in many different countries recently was published. It was given a really exciting title namely…
New Ontario Business Registry will one day become active and you should sign up now to get reminders and information
The Ontario Ministry of Government Services is launching a new Ontario Business Registry at some point over the next year or two. This Registry will…
Another version of CRA's guidance on advancement of religion released in an access to information request
We have been blogging about CRA's draft guidance "Advancement of religion and charitable registration" in 2009, 2017 and 2018. Essentially CRA has…
Why does CRA ask about charitable "activities" and not just about charitable "purposes"
In a recent CRA document, we obtained that was released through access to information, it provides a good explanation of why CRA cares about…
Mark Blumberg's testimony to the Standing Committee on Access to Information, Privacy and Ethics (ETHI) on CSSG/WE Charity Scandal
On Friday, December 11, 2020 I gave testimony to the Standing Committee on Access to Information, Privacy and Ethics (ETHI) on the CSSG and We…
CRA updates and releases new charity guidances
CRA released several new updates and guidances for registered charities on Friday. This includes a new guidance on the advancement of education…
Some are calling for changes to Canadian charity rules for foreign activities
I wish I did not have to write this article about the Canadian rules around international philanthropy and foreign activities but unfortunately, a…
CRA's New Education Guidance – CG-030, Advancement of education and charitable registration
CRA issued a few new guidances today, including a guidance on the advancement of education: CG-030, Advancement of education and charitable…
CRA releases foreign activity guidance after multi-year delays as well as guidance on intermediaries working in Canada
CRA released today the updated guidance CG-002, Canadian registered charities carrying on activities outside Canada. Essentially, this guidance…
Capacity building, education, and online courses for the Canadian charity and non-profit sector
Our firm has been using online tools for over 10 years – primarily webinars. In 2018, we moved to providing paid online courses dealing with charity…
The Charity Report provides a profile of the Blumbergs' Snapshot of the Canadian Charity Sector
We were pleasantly surprised to see that The Charity Report "your independent source of news in the charity sector" recently published an article…
Blumbergs Canadian Charity Sector Snapshot 2018
Here is our Blumbergs Canadian Charity Sector Snapshot 2018.We recently reviewed the T3010 Registered Charity Information Return database for 2018…
Blumbergs' Canadian Charity Law Institute 2020 – online
Every year Blumbergs has its Blumbergs' Canadian Charity Law Institute. This year we will be doing a "mini-institute" – very virtual, much shorter…
Upcoming Webinars for the Canadian Charity Law Association
The Canadian Charity Law Association is delivering some upcoming webinars. Registration is free but space is limited. Topics are subject to change.…
The WE Charity scandal and its impact on the Canadian charity sector?
A lot has happened since we originally published this article on September 4th so here is an updated version below. The WE Charity…
ONCA update but still don't know when ONCA will be brought into force
[Update - the motion to extend the period within ONCA can be brought into force was carried on September 21, 2020.] As we have noted before,…
What will be the impact of the WE Charity scandal on the Canadian charity sector?
The WE Charity scandal has now been going on for over 2 months. A few weeks ago the Minister of Finance resigned and parliament was prorogued. Some…
We just launched a new course – Membership of Non-Profits and Charities in Canada
There is so much confusion on memberships for Non-Profits and Charities in Canada that we have just launched a 2-hour course on the topic.Here is…
Transparency in the Canadian Charitable Sector and the T3010 Registered Charity Information Return Online Course
Every year we do a 3-hour course on transparency and the T3010. This year there has been a lot of interest in transparency around charities. On…
WE Charity filings with Corporations Canada
WE Charity is a Federal Corporation under the Canada Not-for-profit Corporations Act ("CNCA"). The CNCA is corporate legislation that applies to…
Mark Blumberg's written submission to the Finance Committee on transparency in the non-profit and charity sector
Here is my Written Submission for the Pre-Budget Consultations in Advance of the Upcoming Federal Budget on Transparency and Accountability in the…
Do you require legal advice with respect to Canadian or Ontario non-profits or charities?
Mark Blumberg is a partner at the law firm of Blumberg Segal LLP in Toronto and works almost exclusively in the areas of non-profit and charity law.
mark@blumbergs.ca416.361.1982Download vCard
Blumberg Segal LLP
Barristers & Solicitors
M4P 1E4 Canada
Charity Law List
Join Blumbergs' non-profit and charities newsletter.
View recent issue: December 2020
The material on this site is for general information and it is not offered as legal advice or opinion. Canadian Charity Law is a project of Blumberg Segal LLP, a law firm based in Toronto, ON. If you would like to discuss your specific legal concerns please contact us.
© 2007 – 2021 Blumberg Segal LLP. Disclaimer, Copyright and Privacy Notice.
Join Charity Law List!
Receive Blumbergs' FREE monthly newsletter with the latest information on Canadian charity law and compliance information. | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 6,425 |
{"url":"https:\/\/www.vedantu.com\/question-answer\/if-the-quadratic-equation-px2-2sqrt-5-px-+-15-0-class-10-maths-cbse-5ee7213347f3231af290ec69","text":"Question\n\n# If the quadratic equation $p{x^2} - 2\\sqrt 5 px + 15 = 0$ has two equal roots then find the value of p.\n\nVerified\n146.4k+ views\nHint: Here we go through the properties of the quadratic equation as we know when the roots of the quadratic equation are equal then their discriminant must be equal to zero. So we equate discriminant of this equation equal to zero for finding the value of p.\n\nWe know that if in quadratic equation $a{x^2} + bx + c = 0$ when the two roots are equal then its discriminant is equal to zero I.e. ${b^2} - 4ac = 0$.\nNow in the question the given quadratic equation is $p{x^2} - 2\\sqrt 5 px + 15 = 0$.\nBy equating it with the general quadratic equation we get a=p, b$= - 2\\sqrt 5 p$ and c=15.\nNow we will calculate its discriminant by formula ${b^2} - 4ac = 0$.\n$\\Rightarrow {\\left( { - 2\\sqrt 5 p} \\right)^2} - 4 \\times p \\times 15 = 0 \\\\ \\Rightarrow 20{p^2} - 60p = 0 \\\\$\n$\\Rightarrow 20p(p - 3) = 0$\n$p \\ne 0$ As it makes a coefficient of ${x^2} = 0$.","date":"2021-10-28 10:25:59","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.9298645257949829, \"perplexity\": 162.1683845668563}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.3, \"absolute_threshold\": 20, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2021-43\/segments\/1634323588284.71\/warc\/CC-MAIN-20211028100619-20211028130619-00006.warc.gz\"}"} | null | null |
Q: Markdown: how to split a cell in a table G'evening.
I'm doing a page for a lab work report at the GitHub and I have to do a table similar to this using "Markdown":
the table
How can I split a cell into 8 cells like it is done in the second (2) row of that table?
A: GitHub does not support this feature using pure markdown syntax. It does, however, support HTML tables as explained in this answer. You would have to use the colspan attribute on the <td> tag to tell how many columns that cell should span. Don't see it as splitting a cell, but more like merging multiple cells.
It all boils down to whatever markdown processor you are using. Some do support markdown syntax for table cells to span multiple columns, and on the other hand, not all markdown processors support HTML syntax.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 2,319 |
Legerdemain ($0.15) Price History from major stores - Tempest - MTGPrice.com Values for Ebay, Amazon and hobby stores!
Flavor Text: Squee tucked the warm ball in his pocket and slipped a pebble in its place. ?Glok,' he mumbled, and hid. | {
"redpajama_set_name": "RedPajamaC4"
} | 4,337 |
The German Agency for Technical Cooperation (GIZ) provided support to the Kenya health sector to improve access to and demand for good quality sexual and reproductive health services for the Kenyan population. In particular, it aimed to increase the provision of, and quality of, age specific sexual and reproductive health services for adults and adolescents.
On behalf of GIZ, Options provided a programme of technical support to Kenya's Ministry of Health Division of Reproductive Health focused on policy formulation and strategic planning; performance monitoring; grant management and fund disbursement; capacity strengthening; resource mobilisation and coordination and operational research.
Options also provided capacity building support through technical assistance to faith based and civil society organisations and their networks in the areas of reproductive and adolescent sexual health. The programme of support was demand-led and based on the priority needs of organisations in order to increase effective provision of affordable quality health care. | {
"redpajama_set_name": "RedPajamaC4"
} | 8,101 |
package ua_parser;
import java.util.ArrayList;
import java.util.List;
import java.util.Map;
import java.util.regex.Matcher;
import java.util.regex.Pattern;
/**
* Device parser using ua-parser regexes. Extracts device information from user agent strings.
*
* @author Steve Jiang (@sjiang) <gh at iamsteve com>
*/
public class DeviceParser {
List<DevicePattern> patterns;
public DeviceParser(List<DevicePattern> patterns) {
this.patterns = patterns;
}
public Device parse(String agentString) {
if (agentString == null) {
return null;
}
String device = null;
for (DevicePattern p : patterns) {
if ((device = p.match(agentString)) != null) {
break;
}
}
if (device == null) device = "Other";
return new Device(device);
}
public static DeviceParser fromList(List<Map<String,String>> configList) {
List<DevicePattern> configPatterns = new ArrayList<DevicePattern>();
for (Map<String,String> configMap : configList) {
configPatterns.add(DeviceParser.patternFromMap(configMap));
}
return new DeviceParser(configPatterns);
}
protected static DevicePattern patternFromMap(Map<String, String> configMap) {
String regex = configMap.get("regex");
if (regex == null) {
throw new IllegalArgumentException("Device is missing regex");
}
Pattern pattern = "i".equals(configMap.get("regex_flag")) // no ohter flags used (by now)
? Pattern.compile(regex, Pattern.CASE_INSENSITIVE) : Pattern.compile(regex);
return new DevicePattern(pattern, configMap.get("device_replacement"));
}
protected static class DevicePattern {
private static final Pattern SUBSTITUTIONS_PATTERN = Pattern.compile("\\$\\d");
private final Pattern pattern;
private final String deviceReplacement;
public DevicePattern(Pattern pattern, String deviceReplacement) {
this.pattern = pattern;
this.deviceReplacement = deviceReplacement;
}
public String match(String agentString) {
Matcher matcher = pattern.matcher(agentString);
if (!matcher.find()) {
return null;
}
String device = null;
if (deviceReplacement != null) {
if (deviceReplacement.contains("$")) {
device = deviceReplacement;
for (String substitution : getSubstitutions(deviceReplacement)) {
int i = Integer.valueOf(substitution.substring(1));
String replacement = matcher.groupCount() >= i && matcher.group(i) != null
? Matcher.quoteReplacement(matcher.group(i)) : "";
device = device.replaceFirst("\\" + substitution, replacement);
}
device = device.trim();
} else {
device = deviceReplacement;
}
} else if (matcher.groupCount() >= 1) {
device = matcher.group(1);
}
return device;
}
private List<String> getSubstitutions(String deviceReplacement) {
Matcher matcher = SUBSTITUTIONS_PATTERN.matcher(deviceReplacement);
List<String> substitutions = new ArrayList<String>();
while (matcher.find()) {
substitutions.add(matcher.group());
}
return substitutions;
}
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 9,489 |
AllCelebritiesEntertainmentLifestyle
Brock Lesnar Wiki, Age, Weight, Height, Wife, Net Worth
Brock Lesnar is a popular name in wrestling. Every wrestling fan knows him well. He is known as the Beast in wrestling. That is because he is a man with skills in the field of wrestling. He became so much popular at such a young age, and he is the only and youngest wrestler who won the WWE World Championship. It shows how much skill does he has and how destructive he is.
Early Life of Brock Lesnar
One of the most popular wrestlers, Brock Lesnar, was born on 12th July 1977 in a place known as Webster South Dakota.
Brock is the son of Richard Lesnar and Stephanie. However, Brock has three siblings, and he is of German descent too. Although he is recognized and holds an American nationality, he actually belongs to German ethnicity. At the same time, he is a follower of Christianity.
Brock went to Webster High School at an early age. On there, he played football and also took part in amateur wrestling tournaments. After completing high school, he joined the Bismarck State College, and on there, he participated in the National Junior College Athletic Association heavyweight wrestling championship and won the championship as well. He didn't complete his graduation because he got the wrestling scholarship at the University of Minnesota for his amazing wrestling skills.
Brock is a married man, and his wife is Rena Greek, who is professionally a Sable. They got married in 2006. They have two sons. At the same time, Brock has twins from his former fiancé named Nicole McClain.
Brock chose wrestling as his career, but he also did some other things too:
Brock Lesnar started his career as a wrestler by signing with the World Wrestling Federation WWF in 2000. After he signed in, he was sent to OVW that stands for Ohio Valley Wrestling. On there, he met with Paul Heyman, who is a spokesman, friend, and manager as well. However, he won the Southern Tag Team Championship three times along with his partner Shelton Benjamin.
Brock is one of the skilled wrestlers, and he won the WWE Championship four times in his career. He also won the King of the Ring in 2002. Besides, he also has the title of the Royal Rumble in the year 2003. Later, he won the IWGP Heavy Weight Championship in the New Japan Pro-Wrestling. Now, during his time at Inoki Genome Federation, he won the IWGP Heavy Weight Championship as well. Moreover, he was the winner of the UFC Heavyweight Championship at the Ultimate Fighting Championship.
Unknown Facts about Brock Lesnar
There are some unknown facts about Brock Lesnar that you surely don't know, and those are as follows:
Nickname – Most people don't know that Brock used to have a funny nickname when he was a kid. It's a vegetable nickname called "Broccoli."
Color Blind – Brock is color blind, and he can't differentiate between various colors.
National Guard – This WWE star joined the National Guard when he was only 17 years old.
First Match – The first match that he won in WWE was against Jeff Hardy.
Republican – Most people don't know, but he is a republican conservative.
Brock Lesnar is a name of terror in wrestling since he destroys his opponents and is known for his attacks. He is a very skilled wrestler and the highest earned wrestler in the world. Now, there are some rumors going on that he is going to retire soon.
Best Real Estate Books for Beginners
Free Download Call of Duty Mobile with 4 Easy Steps
Grand Theft Auto: Vice City | Free Download for Your PC with Simple Steps
Grand Theft Auto: San Andreas | Free Download for Your PC with Simple Steps | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 5,382 |
Gone Soft
By Aidan Crilly | Nov 3 2017
Aidan Crilly explores an exciting Dublin label Despite gaining some traction over recent years, for the most part Irish hip-hop remains a relatively underground genre. Soft Boy Records is one of the leading Dublin-based independent labels providing a platform to elevate local hip-hop artists, as well as hosting its own events and radio shows.The label was founded in 2015 by Kean Kavanagh and Kevin Smith (Kojaque). The two first met while working in the Gaeltacht in Galway, and their mutual love for hip-hop, soul, and jazz resulted in a strong friendship and musical partnership. Since then Soft Boy has grown in size and reputation, now hosting a sonically eclectic roster of acts including Gaptoof, Henry Earnest, Matt Finnegan, Peter Brien, Five to Two, as well as the two founders.Kojaque has emerged as Soft Boy's most prominent artist. One of the label's early successes came in 2015 with the striking music video for 'Midnight Flower,' which features Kojaque rapping the song with his head submerged in a tank of water. Kojaque claims that this was inspired by watching a tadpole drown in air, while being held over a mug of water, "I wanted to know what that would feel like to be a tadpole sized human held over a mug of air," he told District Magazine. 'Midnight Flower' quickly went viral, and was featured on websites like Vice and Joe.ie.Kojaque's festival performances, particularly at Electric Picnic, have gathered him, and Soft Boy, further recognition. In 2016, he was described by the Irish Times as "pure fire," and the same paper awarded him four stars for this year's set. In September, the Rubberbandits announced that "talented bastard Kojaque" would be supporting them for their Vicar Street show.The influential Nialler9 has shown plenty of support for Soft Boy, having featured Kojaque's 'Wificode,' producer Henry Earnest's 'Good Day,' and the label's compilation tape, Soft World, on his website. Recent releases from the label include 'Latinvader' from jazz outfit Five to Two, and 'Can I Lie With U' from Belfast producer Peter Brien. The first release from his Tandem EP will be dropping in December.Soft Boy works tirelessly to promote home-grown hip hop, collaborating with District Magazine and the RnB Club to host events. Recently these have included 'Soft Sundaes' in the Bernard Shaw, and a sold-out night at Yamamori Tengu on 20th October.With Irish hip-hop becoming increasingly popular, there is certainly a lot of room for this independent record label to grow. With more events and more music coming, the Soft Boys are a crowd to which we should pay close attention. | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 3,172 |
Q: How to mock string variable from a service file in component spec.ts using jasmine How to mock up the screenName value in the component spec.ts file.? Though I mocked the value for screenName in spec file it shows as undefined value only
sample.service.ts
screenName: string;
sampleComponent.ts
setScreen(){
if (sampleService.screenName === 'CREATE'){
// do this
} else if (sampleService.screenName === 'View') {
// do this
} else {
// do this
}
samplecomponent.spec.ts
it('sample Screen as Create', () => {
limitsService = jasmine.createSpyObj('SampleService', ['screenName']);
limitsService.screenName = 'CREATE';
component.setScreen();
expect(component.setScreen).toBeTruthy();
});
A: Override the service using useValue in the providers array and then directly use component.sampleService.screenName = 'View' to mock the values
beforeEach(() => {
TestBed.configureTestingModule({
...
providers: [{ provide: SampleService, useValue: {} }],
});
...
});
it('sample Screen as Create', () => {
component.sampleService.screenName = 'View';
component.setScreen();
expect(component.setScreen).toBeTruthy();
});
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 4,371 |
{"url":"https:\/\/mmb.irbbarcelona.org\/MCDNA\/help\/analysis\/circular","text":"# MC DNA Help - Analysis Circular\n\nCircular\n\n## Circular\n\nThis analysis is available when tool \u201cCircular MC DNA\u201d is chosen. The analysis parameters for circular DNA are Twist, Writhe and Radius of gyration. Twist Tw reflects the number of helical turns ($Tw = \\sum_{i=1}^{N - 1} tw_i \/ 360$; N is the length of the sequence, $tw_i$ is the value for Twist in degree of base-pair step i.) and writhe Wr is the number of times the double helix crosses over on itself (supercoils). The relaxed structure for the circle is defined as the structure with Wr = 0 and twist values are the values of the relaxed twist state. Thus the total linking number $Lk_0$ of the relaxed circle is $Lk_0 = Tw_0$. To induce additional stress the twist value of each base pair step of the circle can be changed which results in new value of Tw. Over- or under-twisting of the relaxed structure results in a different linking number $\\Delta Lk = Lk - Lk_0 = Tw - Tw_0$ and thus a different starting structure with $\\Delta Lk \\ne 0$. $\\Delta Lk$ can only take integer numbers. $\\Delta Lk = Tw + Wr$ will stay constant throughout the whole simulation, however Tw will change throughout the simulation due to the Monte Carlo moves and Wr becomes non-zero. The values of Tw and Wr of the final structure of the simulation are plotted.\n\nAnother parameter to analyze the compactness of the circle is the radius of gyration $R_g$. We define the position $r_i$ of base-pair I as the middle between the C6 and C8 atom. The radius of gyration $R_g$ is then calculated as follows:\n\n$R_g = \\sqrt{ \\frac{1}{N} \\sum_{i=1}^N (r_i - r_{mean})^2}$\n\n$r_{mean}$ is the mean position of the base-pairs and N is the total number of base-pairs.\n\n### For \u201cStructure Flexibility Analysis\u201d\n\nThe values of Twist, Writhe and Radius of gyration are given for the relaxed circular structure.\n\n### For \u201cTrajectory Flexibility Analysis\u201d\n\nRadius of gyration (in nm), Twist (in turns) and Writhe (in turns) are plotted against the index of the snapshot of the trajectory.","date":"2021-09-16 19:32:08","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 2, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.8781641721725464, \"perplexity\": 946.490734352901}, \"config\": {\"markdown_headings\": true, \"markdown_code\": false, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2021-39\/segments\/1631780053717.37\/warc\/CC-MAIN-20210916174455-20210916204455-00621.warc.gz\"}"} | null | null |
Aap Ki Parchhaiyan Lyrics movie songs lyrics & videos: The music of Aap Ki Parchhaiyan Lyrics is composed by Madan mohan, and the songs are sung by Lata mangeshkar, Mohammad rafi, Asha bhosle.
Love Aap Ki Parchhaiyan Lyrics Songs? Comment and say something!! | {
"redpajama_set_name": "RedPajamaC4"
} | 4,459 |
Michael Dalton McLean (2 December 1880 – 12 August 1958) was elected a Conservative member of the House of Commons of Canada. He was born in Nova Scotia and became a foreman.
McLean won the Kootenay East riding in the July 1930 general election, but resigned on 7 August 1930 to open the seat for Henry Herbert Stevens whom Prime Minister R. B. Bennett appointed Minister of Trade and Commerce. McLean accepted an unspecified federal appointment.
References
External links
1880 births
1958 deaths
Members of the House of Commons of Canada from British Columbia
Conservative Party of Canada (1867–1942) MPs | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 7,402 |
\section{Formulation of the problem}
\section{Formulation of the problem}\label{S1}
We study the generalized Navier--Stokes--Fourier system
\begin{subequations}\label{bl}
\begin{align}
\label{i1}
\partial_t \vc{v} + \operatorname{div}(\vc{v} \otimes \vc{v} ) - \operatorname{div} \S + \nabla \pi &= 0 \\
\label{i2}
\operatorname{div} \vc{v} &= 0\\
\label{i3}
\partial_t(c_\nu\vartheta)+\operatorname{div}(c_\nu\vartheta\vc{v}) + \operatorname{div} \vc{q} &= \S:\mathbf{D}{\vc{v}}
\end{align}
\end{subequations}
in $Q:=(0,T) \times \Omega\subset(0, +\infty)\times {\mathbb R}^3$ with $\Omega$ a bounded domain. The system~\eqref{i1}--\eqref{i3} is completed by the following boundary conditions
\begin{equation}
\vc{v}=0,
\ \
\vartheta = \vartheta_{b} \qquad \mbox{ on } (0, T)\times\partial\Omega,
\end{equation}
where the function $\vartheta_{b}=\vartheta_b(x)$ is a nontrivial function of the position, and the following initial conditions
\begin{equation}\label{condizioni-iniziali}
\vc{v} = \vc{v}_0, \ \
\vartheta = \vartheta_0 \qquad \mbox{ in } \{0\}\times\Omega.
\end{equation}
Here $\vc{v}: Q\to {\mathbb R}^3$ denotes the velocity field, $\mathbf{D}\vc{v}:= (\nabla\vc{v} + (\nabla\vc{v})^{t})/2$ is the symmetric part of the velocity gradient $\nabla \vc{v}$, $\pi:Q\to {\mathbb R}$ is the pressure, $\vartheta:Q\to {\mathbb R}$ is the temperature; $\S: Q\to {\mathbb R}^{3\times 3}_{\rm sym}$ denotes the viscous part of the Cauchy stress tensor and $\vc{q}:Q \to {\mathbb R}^3$ is the heat flux.
Concerning the material parameters, the constant $c_\nu>0$ in \eqref{i3} denotes the heat capacity and, for simplicity and without losing the generality, we set $c_\nu\equiv 1$ in what follows. The heat flux $\vc{q}$ is represented by the Fourier law
\begin{equation}\label{Fourier}
\vc{q} = - \kappa(\vartheta) \nabla \vartheta
\end{equation}
with the heat conductivity $\kappa: {\mathbb R} \to (0, +\infty)$ being a continuous function of the temperature satisfying, for all $\vartheta \in (0, +\infty)$ and for some $0<\underline{\kappa}, \overline{\kappa} <+\infty,$
\begin{align}\label{k}
0<\underline{\kappa}\leq \kappa(\vartheta) \leq \overline{\kappa}<+\infty.
\end{align}
We assume that $\S=\S^*(\vartheta, \mathbf{D}\vc{v})$, where $\S^*:(0,\infty)\times {\mathbb R}^{3\times 3}_{\rm sym} \to {\mathbb R}^{3\times 3}_{\rm sym}$ is a continuous mapping fulfilling for some $p\ge 11/5$, some $0< \underline{\nu}, \overline{\nu}<+\infty$ and for all $\vartheta\in {\mathbb R}_+$, $\mathbf{D}_1,\mathbf{D}_2 \in {\mathbb R}^{3\times 3}_{\rm sym}$
\begin{subequations}\label{nu}
\begin{align}
(\S^*(\vartheta, \mathbf{D}_1)-\S^*(\vartheta, \mathbf{D}_2)): (\mathbf{D}_1-\mathbf{D}_2)&\ge 0,\label{nu1}\\
\S^*(\vartheta,\mathbf{D}_1):\mathbf{D}_1\ge \underline{\nu}|\mathbf{D}_1|^p - \overline{\nu}, \quad |\S^*(\vartheta, \mathbf{D}_1)|&\le \overline{\nu}(1+|\mathbf{D}_1|)^{p-1}, \quad \S^*(\vartheta, 0)=0.\label{nu2}
\end{align}
\end{subequations}
Note that the prototypic relation $\S\sim \nu(\vartheta) |\mathbf{D}\vc{v}|^{p-2}\mathbf{D}\vc{v}$ falls into the class \eqref{nu}.
The key motivation of the paper is the following stability result: to show that $(\vc{v}, \vartheta)\to (0,\hat\vartheta)$ in a suitable topology as $t\to \infty$, where $\hat\vartheta$ is the unique solution to
\begin{equation}\label{thetahat}
\begin{split}
-\operatorname{div}(\kappa(\hat\vartheta)\nabla\hat\vartheta)&=0 \ \mbox{ in $\Omega$},\qquad \hat\vartheta=\vartheta_b \ \mbox{ on $\partial\Omega$}.
\end{split}
\end{equation}
For linear models of the form $\S=\nu(\vartheta)\mathbf{D} \vc{v}$, such result was already proven in~\cite{DosPruRaj} provided that the solution $(\vc{v}, \theta)$ is smooth. However, the existence of smooth (or sufficiently regular) solution is not known and the only available results focus only on weak solutions. In three dimensional setting, the existence of a weak solution was firstly proved for $p\ge 11/5$ in \cite{Consiglieri}, see also the related work \cite{FrMaRu10}. Later, for $p\in (9/5,11/5)$ and slightly different boundary conditions, the existence of a weak solution was established in
\cite{BuMaRa09}, see also \cite{MaZa18} for more complicated model, with one proviso. The identity \eqref{i3} was replaced by the inequality, which in terms of the entropy
\begin{equation}\label{entropy2}
\eta = \log\vartheta
\end{equation}
can be rewritten into the so-called entropy inequality
\begin{equation}
\label{entropy}
\partial_t \eta + \operatorname{div}(\eta \vc{v}) + \operatorname{div}\left(\frac{\vc q}{\vartheta}\right) \geq \frac{1}{\vartheta} \S:\mathbf{D}\vc{v} + \kappa(\vartheta)\frac{|\nabla\vartheta|^2}{\vartheta^2}.
\end{equation}
Unfortunately, such a result does not allow us to use the procedure developed in \cite{DosPruRaj} and to prove a stability result. Therefore, we need to change the methods and results used in \cite{Consiglieri,BuMaRa09,BulFeiMal} significantly. Thus, our main goal is to prove the existence of a weak solution, which satisfies \eqref{entropy} with the \emph{equality sign} and also to show that the temperature is continuous with respect to time into the topology of $L^1(\Omega)$, which is the natural function space. Then, inspired by \cite{DosPruRaj}, we know that one can renormalize \eqref{entropy} by a properly chosen set of functions. Indeed, in the standard approach of renormalization, based just on \eqref{i3}, one needs (due to the commutator lemma) that $\vartheta \in L^{p'}(Q)$. Unfortunately, this is true only for $p>\frac52$. Therefore, we introduce \eqref{entropy} with equality sign and then to renormalize it, one just requires $\eta\in L^{p'}(Q)$, which is true for any $p>1$. Our result can be understood as a starting point and the corner stone of further rigorous stability results in various models of incompressible heat conducting fluids. Furthermore, it seems that we identify the proper notion of solution for which we are able to show its existence and also, in future, to show the convergence to equilibrium.
\section{Rigorous statement of the main result}
We use in what follows the standard notation for the Lebesgue, the Sobolev and the Bochner spaces and endow them with standard norms. The symbol $C^{\infty}_0$ is reserved for smooth compactly supported functions and the function spaces related to the incompressible setting are denoted by $W^{1,p}_{0, \operatorname{div}}:=\{\vc{v} \in W^{1,p}_{0}(\Omega; {\mathbb R}^3);\; \operatorname{div} \vc{v} =0\}$ and $L^2_{0,\operatorname{div}}$ denotes the closure of $W^{1,2}_{0,\operatorname{div}}$ in $L^2$ topology. Duality pairing between $W^{1,p}_{0, \operatorname{div}}$ and their duals is denoted $\langle\cdot,\cdot\rangle$.
Next, we prescribe the assumptions on data. Recall, we assume that $\vc{q}$ and $\S^*$ satisfy \eqref{Fourier}--\eqref{nu}. For initial and boundary data we consider
\begin{equation}\label{data}
\vc{v}_0\in L^2_{0,\operatorname{div}}, \ \vartheta_0\in L^1(\Omega), \ \hat\vartheta \in W^{1,2}(\Omega)\cap L^{\infty}(\Omega),
\end{equation}
where $\hat\vartheta$ is the solution to \eqref{thetahat}. Hence, we transferred all assumptions on the boundary behaviour of $\vartheta_b$ to the uniquely defined $\hat\vartheta$. Finally, we suppose that
\begin{equation}\label{min}
\mu:=\min\left\{\essinf_{x\in \Omega}\hat\vartheta(x), \; \essinf_{x\in \Omega}\vartheta_0(x)\right\}>0.
\end{equation}
The main result of the paper is following.
\begin{theorem}[Existence of a solution fulfilling entropy equality]\label{thm}
Let $\Omega\subset {\mathbb R}^3$ be a bounded domain with Lipschitz boundary. Assume that $\S^*$ and $\kappa$ satisfy \eqref{k}--\eqref{nu} with $p\geq 11/5$. Then for any data $\vc{v}_0, \ \vartheta_0, \hat\vartheta$ fulfilling \eqref{data}--\eqref{min} there exists a quadruplet $(\vc{v}, \S, \vartheta, \eta)$ fulfilling
\begin{align}
\vc{v}&\in C([0, T]; L^2_{0,\operatorname{div}})\cap L^p(0, T; W_{0, {\rm div}}^{1, p}),\\
\partial_t\vc{v} &\in L^{p'}(0, T; (W_{0, {\rm div}}^{1, p})^*), \ \S\in L^{p'}(Q; {\mathbb R}^{3\times 3}),\\
\vartheta &\in C([0, T]; L^1(\Omega)), \ (\vartheta)^\alpha \in L^2(0, T; W^{1,2}(\Omega)) &&\mbox{ for any } \alpha\in (0, {1}/{2}),\\
\vartheta&\in L^r(Q) &&\mbox{ for any } r\in [1, 5/3), \\
\vartheta - \hat\vartheta &\in L^s(0, T; W_0^{1, s}(\Omega)) &&\mbox{ for any } s\in [1, 5/4),\\
\eta &\in L^2(0,T; W^{1,2}(\Omega))\cap L^q(Q) &&\mbox{ for any } q\in [1, +\infty),
\end{align}
and satisfying \eqref{bl} and \eqref{entropy} in the following sense:
\begin{itemize}
\item[] {\bf Momentum equation:} The Cauchy stress is of the form $\S=\S^*(\vartheta, \mathbf{D}\vc{v})$ a.e. in $Q$, the initial datum fulfils $\vc{v}(0)=\vc{v}_0$ and for all $\vc{w}\in L^p(0, T; W_{0, {\rm div}}^{1,p})$
\begin{equation}\label{T1}
\begin{aligned}
\int_0^T{\langle\partial_t\vc{v}, \vc{w}\rangle}\,{\rm d} t + \int_0^T\intO{\S:\mathbf{D}\vc{w}}\,{\rm d} t = \int_0^T\intO{(\vc{v}\otimes\vc{v}): \mathbf{D}\vc{w}}\,{\rm d} t ;
\end{aligned}
\end{equation}
\item[]{\bf Internal energy balance:} Temperature satisfies the minimum principle $\vartheta\geq \mu \mbox{ a.e. in } Q$, the initial condition fulfils $\vartheta(0)=\vartheta_0$ and for all $\varphi\in C^\infty_0((-\infty, T) \times \Omega)$
\begin{equation}
\begin{aligned}\label{T2}
-\int_0^T\intO{\vartheta \partial_t\varphi}\,{\rm d} t - \int_0^T\intO{\vartheta\vc{v}\cdot\nabla\varphi}\,{\rm d} t + \int_0^T\intO{\kappa(\vartheta)\nabla\vartheta\cdot\nabla\varphi}\,{\rm d} t \\
= \int_0^T\intO{\S:\mathbf{D}\vc{v}\, \varphi}\,{\rm d} t + \intO{\vartheta_0\varphi(0)};
\end{aligned}
\end{equation}
\item[] {\bf Entropy equation:} Entropy is given as $\eta =\ln \vartheta \mbox{ a.e. in } Q$, $\eta_0:= \ln \vartheta_0$ and for all $\varphi \in C_0^\infty((-\infty, T)\times\Omega)$
\begin{equation}\label{entropy-limit}
\begin{aligned}
-\intTO{\eta\partial_t\varphi} - \intTO{\eta\,\vc{v}\cdot \nabla\varphi} + \intTO{\kappa(\vartheta)\nabla\eta\cdot\nabla\varphi} \\
= \intTO{\frac{1}{\vartheta}\,\S:\mathbf{D}{\vc{v}}\,\varphi} + \intTO{\kappa(\vartheta)\frac{|\nabla\vartheta|^2}{(\vartheta)^2}\,\varphi} + \intO{\eta_0\, \varphi(0)}.\end{aligned}
\end{equation}
\end{itemize}
\end{theorem}
\section{Proof of Theorem~\ref{thm}}
The existence proof relies on the methods developed in \cite{BuMaRa09} and a large part of the proof is identical. Therefore, we omit unnecessary details and focus mainly on the new aspects of the proof, i.e., on the proof of entropy equality~\eqref{entropy-limit}.
Hence, following \cite{BuMaRa09} (compare also with \cite{Consiglieri}, where a different approach is used), we introduce a basis $\{ \vc{w}_j\}_{j=1}^\infty$ of the space $W^{3,2}(\Omega; {\mathbb R}^3)\cap W^{1,p}_{0,\operatorname{div}}$ that is orthonormal in $L^2_{0,\operatorname{div}}$, see \cite[Appendix A.4]{MaBook}. Next, for given initial conditions $\vc{v}_0$ and $\vartheta_0$, we denote $\vc{v}^{n}_0$ the projection of $\vc{v}_0$ onto the subspace $[\vc{w}_1,\dots, \vc{w}_n]$, and $\vartheta^n_0\in L^2(\Omega)$, fulfilling $\vartheta^n_0\ge \mu$ a.e. in $\Omega$, is the regularization of $\vartheta_0$ such that
\begin{align}
\vc{v}_0^{n}&\to \vc{v}_0 \ &&\mbox{ strongly in } L^2_{0,\operatorname{div}} \mbox{ as } n\to +\infty,\\
\vartheta^n_0 &\to \vartheta_0\ &&\mbox{ strongly in } L^1(\Omega) \mbox{ as }n\to +\infty.\label{thet-zero
\end{align}
Then, for every $n\in \mathbb{N}$, we can find a triple $(\vc{v}^n, \vartheta^n, \S^n)$, such that
$\vc{v}^n\in W^{1,2}((0,T), W^{3,2}(\Omega;{\mathbb R}^3)\cap W^{1,p}_{0,\operatorname{div}})$, $\theta^n\in L^\infty((0, T); L^2(\Omega))\cap L^2((0, T); W^{1,2}(\Omega))\cap W^{1,2}((0, T); (W^{1,2}_0(\Omega))^*)$ and $\S^n \in L^{\infty}(Q)$ and such that for a.e. $t\in(0,T)$
\begin{align}
\label{ode11}
&\intO{[\partial_t\vc{v}^{n} \cdot \vc{w}_j - (\vc{v}^{n} \otimes \vc{v}^{n}): \nabla \vc{w}_j + \S^{n}: \mathbf{D}\vc{w}_j]} = 0 \qquad \mbox{ for all } j=1,\dots,n,
\end{align}
and for all $\psi\in L^2(0,T;W_0^{1,2}(\Omega))$
\begin{align}
\label{ode22}
\begin{aligned}
&\int_0^T\dual{\partial_t\vartheta^{n}}{\psi}\,{\rm d} t +\intTO{[-\vartheta^{n} \vc{v}^{n} \cdot \nabla \psi + \kappa(\vartheta^n)\nabla\vartheta^{n}\cdot \nabla \psi]} \\
&= \int_0^T\intO{\S^{n}: \mathbf{D}\vc{v}^{n} \psi} \,{\rm d} t .
\end{aligned}
\end{align}
In addition, $\vc{v}^n$ and $\S^n$ are given by
\begin{equation}
\vc{v}^n= \sum_{i=1}^n c^{n}_i(t)\vc{w}_i(x) \quad \mbox{ and } \quad \S^n= \S^*(\vartheta^n, \mathbf{D} \vc{v}^n),
\end{equation}
the initial conditions $\vc{v}^n(0, \cdot)=\vc{v}^n_0$, $\vartheta^n(0, \cdot)=\vartheta_0^n$ are satisfied, $\vartheta^n$ attains the boundary conditions, i.e., $\vartheta^n_{|\partial\Omega}=\hat\vartheta$ and fulfills the minimum principle, i.e., $\vartheta^n \ge \mu$ a.e. in $Q$.
Then, following e.g. \cite{BuMaRa09} and defining $\tilde\vartheta^n:=\vartheta^n - \hat\vartheta$, it is rather standard to deduce the following $n$-independent a~priori estimates valid for all $r\in [1,5/3)$, $s\in [1, 5/4)$ and $\alpha \in (0,1/2)$
\begin{align}
\label{apr:vn}
\nm{\vc{v}^{n}}_{L^\infty(0,T; L^2_{0,\operatorname{div}})}+\nm{\vc{v}^{n}}_{L^p(0,T; W^{1,p}_{0,\operatorname{div}})}+\nm{\S^n}_{L^{p'}(Q)}+\nm{\vc{v}^{n}}_{L^{\frac{5p}3}(Q)}&\leq C
\\
\label{timen}
\nm{\tilde\vartheta^n}_{L^\infty(0,T;L^1(\Omega))}+\nm{\partial_t \vartheta^n}_{L^1(0, T; (W^{1, 10}_0(\Omega))^*)}+\nm{\partial_t\vc{v}^n}_{L^{p'}(0, T; (W^{1,p}_{0,\operatorname{div}})^*)} &\leq C,\\
\label{eq:re3}
\nm{(\theta^n)^\alpha}_{L^2(0,T;W^{1,2}(\Omega))}+\nm{\tilde\vartheta^n}_{L^r(Q)}+\nm{\tilde\vartheta^n}_{L^s(0, T; W_0^{1,s}(\Omega))}&\leq C(\alpha, r, s)
\end{align}
This is the starting point of the proof. It is rather sketchy, but it does not contain any essentially new information. However, it is not the case for what follows and therefore we prove it in full details.
\subsection{Limit in momentum and energy equations as \texorpdfstring{$n\to+\infty$}{n}}
By virtue of the established uniform estimates \eqref{apr:vn}--\eqref{eq:re3} and employing the Aubin--Lions compactness lemma, we can extract a subsequence that we do not relabel and we can find $(\vc{v}, \vartheta, \S)$ such that
\begin{align}
\vc{v}^n &\rightharpoonup^* \vc{v} && \mbox{ weakly-* in } L^\infty(0, T; L^2_{0,\operatorname{div}}), \label{Linfty2}\\
\vc{v}^n &\rightharpoonup \vc{v} &&\mbox{ weakly in } L^p(0, T; W^{1,p}_{0,\operatorname{div}})\cap W^{1,p'}(0,T; (W^{1,p}_{0,\operatorname{div}})^*),\label{Lp}\\
\S^n &\rightharpoonup \S &&\mbox{ weakly in } L^{p'}(Q; {\mathbb R}^3),\label{Lp-S}\\
\vc{v}^n&\to\vc{v} &&\mbox{ strongly in }L^q(Q; {\mathbb R}^3) \mbox{ for any $q\in [1, {5p}/{3})$, and a.e. in $Q$} \label{v-strong}\\
(\vartheta^n)^\alpha&\rightharpoonup (\vartheta)^\alpha &&\mbox{ weakly in } L^2(0, T; W^{1,2}(\Omega)) \mbox{ for any } \alpha\in (0, {1}/{2}),\label{thet-weak}\\
\vartheta^n &\to \vartheta &&\mbox{ strongly in $L^r(Q)$ for any $r\in [1, 5/3)$, and a.e. in $Q$},\label{thet-strong}\\
\vartheta^n -\hat\vartheta &\rightharpoonup \vartheta -\hat\vartheta && \mbox{ weakly in } L^s(0, T; W_0^{1, s}(\Omega)) \mbox{ for any } s\in [1, 5/4).\label{grad-thet-weak}
\end{align}
In addition, using \eqref{thet-strong} and the first part of the uniform estimate \eqref{timen}, we have that
\begin{equation}
\begin{split}\label{theinf1}
\vartheta\in L^{\infty}(0,T; L^1(\Omega)).
\end{split}
\end{equation}
Now we are in position to analyze the limit of the formulation \eqref{ode11}-\eqref{ode22}. Indeed,
by virtue of the convergence results \eqref{Lp}-\eqref{v-strong}, we may follow \cite{BleMalRaj} to take the limit in the formulation \eqref{ode11} to deduce that for all $\vc{w}\in L^p(0, T; W^{1,p}_{0,\operatorname{div}})$
\begin{equation}
\begin{split}\label{AF1}
\int_0^T{\langle\partial_t\vc{v}, \vc{w}\rangle}\,{\rm d} t + \int_0^T\intO{\S:\mathbf{D}\vc{w}}\,{\rm d} t = \int_0^T\intO{(\vc{v}\otimes\vc{v}): \mathbf{D}\vc{w}}\,{\rm d} t .
\end{split}
\end{equation}
In addition, we have $\vc{v}\in C([0, T]; L^2_{0,\operatorname{div}})$ and $\vc{v}(0, \cdot)=\vc{v}_0$. To complete the proof of \eqref{T1}, it remains to show
\begin{equation}\label{Minty1}
\S= \S^*(\vartheta, \mathbf{D}\vc{v}) \qquad \mbox{ a.e. in } Q.
\end{equation}
Next, thanks to \eqref{v-strong}--\eqref{grad-thet-weak}, one may follow a standard procedure (see e.g. \cite{BulFeiMal}) and let $n\to \infty$ in \eqref{ode22} to show that for all $\psi\in C^\infty_0((-\infty, T)\times \Omega)$ there holds (compare with \eqref{T2})
\begin{equation}
\begin{split}
-\int_0^T&\intO{\vartheta \partial_t\psi}\,{\rm d} t - \int_0^T\intO{\vartheta\vc{v}\cdot\nabla\psi}\,{\rm d} t + \int_0^T\intO{\kappa(\vartheta)\nabla\vartheta\cdot\nabla\psi}\,{\rm d} t \\
&= \int_0^T\intO{\S:\mathbf{D}\vc{v}\, \psi}\,{\rm d} t + \intO{\vartheta_0\psi(0)} \label{3.18}
\end{split}
\end{equation}
provided that we show
\begin{equation}
\S^n:\mathbf{D}{\vc{v}}^n \rightharpoonup \S:\mathbf{D}{\vc{v}} \mbox{ weakly in } L^1(Q).\label{Sn}
\end{equation}
Obviously, we also have $\vartheta\geq \mu$ a.e. in $Q$. It remains to show \eqref{Minty1} and \eqref{Sn}. Note that both of them are consequences of the Minty method. Indeed, it follows from \eqref{ode11} that (using the fact that $\operatorname{div} \vc{v}^n=0$ and \eqref{Lp})
$$
\begin{aligned}
&\limsup_{n\to \infty}\int_0^T \intO{\S^n\cdot \mathbf{D} \vc{v}^n}\,{\rm d} t =\limsup_{n\to \infty} -\int_0^T \langle \partial_t \vc{v}^n, \vc{v}^n \rangle \,{\rm d} t \\
&=\limsup_{n\to \infty}\left( -\int_0^T \langle \partial_t (\vc{v}^n-\vc{v}), \vc{v}^n-\vc{v} \rangle \,{\rm d} t -\int_0^T \langle \partial_t \vc{v}, \vc{v}^n-\vc{v} \rangle \,{\rm d} t -\int_0^T \langle \partial_t \vc{v}^n, \vc{v} \rangle \,{\rm d} t \right)\\
&\le
\frac12 (\|\vc{v}_0\|_2^2- \|\vc{v}(T)\|_2^2)=\int_0^T \intO{\S : \mathbf{D} \vc{v}} \,{\rm d} t ,
\end{aligned}
$$
where the last identity follows from \eqref{AF1} with setting $\vc{w}:=\vc{v}$. Consequently, using this estimate, the monotonicity and the growth assumption \eqref{nu}, the strong convergence result \eqref{thet-strong}, the weak convergence \eqref{Lp} and the Lebesgue dominated convergence theorem, we deduce that for all $\overline{\mathbf{D}}\in L^{p}(Q; {\mathbb R}^{3\times 3})$ there holds
\begin{equation}\label{Mintyc}
\begin{split}
0&\le \limsup_{n\to \infty} \int_0^T \intO{(\S^n-\S^*(\vartheta^n,\overline{\mathbf{D}})):(\mathbf{D}\vc{v}^n -\overline{\mathbf{D}}) }\,{\rm d} t \\
&\le \int_0^T \intO{(\S-\S^*(\vartheta,\overline{\mathbf{D}})):(\mathbf{D}\vc{v} -\overline{\mathbf{D}}) }\,{\rm d} t .
\end{split}
\end{equation}
The classical Minty method then leads to \eqref{Minty1}. Moreover, setting $\overline{\mathbf{D}}:=\mathbf{D}\vc{v}$ in \eqref{Mintyc}, we have
\begin{equation}\label{Mintyd}
\begin{split}
&\limsup_{n\to \infty} \int_0^T \intO{\left|(\S^n-\S^*(\vartheta^n,\mathbf{D}\vc{v})):(\mathbf{D}\vc{v}^n -\mathbf{D}\vc{v})\right| }\,{\rm d} t =0
\end{split}
\end{equation}
Hence,
\begin{align}\label{Mintye}
(\S^n-\S^*(\vartheta^n,\mathbf{D}\vc{v})):(\mathbf{D}\vc{v}^n -\mathbf{D}\vc{v}) &\to 0 &&\textrm{ strongly in } L^1(Q).
\end{align}
Since
\begin{align*}
\S^*(\vartheta^n,\mathbf{D}\vc{v}):(\mathbf{D}\vc{v}^n -\mathbf{D}\vc{v}) &\rightharpoonup 0 &&\textrm{ weakly in } L^1(Q),
\end{align*}
which follows from \eqref{Lp}, \eqref{thet-strong} and \eqref{nu2}, we see that \eqref{Mintye} directly implies \eqref{Sn}.
\subsection{Limit in entropy equation as \texorpdfstring{$n\to+\infty$}{n}}
In this section we show the validity of \eqref{entropy-limit}. To see this, we first set $\psi:=\varphi/\vartheta^n$ in \eqref{ode22} with arbitrary $\varphi \in C_0^\infty((-\infty, T)\times\Omega)$ to derive the following identity for approximated entropy $\eta^n:=\ln \vartheta^n$
\begin{equation}
\begin{split}\label{entropy-weak}
&-\intTO{\eta^n\partial_t\varphi+\eta^n\,\vc{v}^n\cdot \nabla\varphi-\kappa(\vartheta^n)\nabla\eta^n \cdot\nabla\varphi}\\
&= \intTO{\frac{\S^n:\mathbf{D}{\vc{v}}^n}{\vartheta^n}\,\varphi+\kappa(\vartheta^n)\frac{|\nabla\vartheta^n|^2}{(\vartheta^n)^2}\,\varphi} +\intO{\eta^n_0\, \varphi(0)},
\end{split}
\end{equation}
where we set $\eta^n_0:=\ln \vartheta^n_0$. Next, we want to let $n\to \infty$. The identification of the limit in the terms on the left hand side is rather standard and follows from the convergence results \eqref{Lp}, \eqref{v-strong}, \eqref{thet-strong} and \eqref{grad-thet-weak}. Similarly, to pass to the limit in the first term on the right hand side of \eqref{entropy-weak} is straightforward thanks to \eqref{thet-strong}, \eqref{Sn}, the Egorov and Dunford-Pettis theorems. Also the limit passage in the last term is obvious. The most problematic term is however the second term on the right hand side since $\kappa(\vartheta^n)|\nabla\vartheta^n|^2/(\vartheta^n)^2$ is uniformly bounded only in $L^1((0, T)\times\Omega)$ and so we cannot even a~priori extract an $L^1$ weakly convergent subsequence. We overcome this in two steps. First, the point-wise convergence of $\nabla\vartheta^n$ is shown and then the strong convergence of $\kappa(\vartheta^n)|\nabla\eta^n|^2$ in $L^1((0, T)\times\Omega)$ is deduced.
\subsubsection{Almost everywhere convergence of \texorpdfstring{$\nabla\vartheta^n$}{n}}
\newcommand{\thet^n_K}{\vartheta^n_K}
\newcommand{\thet^m_K}{\vartheta^m_K}
\newcommand{w^{m,n}}{w^{m,n}}
\newcommand{\mathcal F}{\mathcal F}
\newcommand{\mathcal G}{\mathcal G}
\newcommand{w^{m,n}_{\delta}}{w^{m,n}_{\delta}}
We start this part with definition of auxiliary cut-off functions. For arbitrary $k >0$, we define
\begin{equation}\label{def:tk}
\mathcal{T}_k(z) := {\rm sign}(z) \min\{|z|,k\}.
\end{equation}
Its primitive function attaining zero at zero is denoted $\mathcal G_k$, i.e. $\mathcal G_k'=\mathcal{T}_k$, $\mathcal G_k(0)=0$. Note that $|\mathcal G_k(s)|\leq k|s|$ for all $s\in{\mathbb R}$. Next, we also introduce a mollification of $\mathcal{T}_k$. For arbitrary $\delta\in (0, k)$ (typically $\delta \ll 1$), we denote by $\mathcal{T}_{k, \delta}\in C^2(\mathbb{R})$ a mollification of $\mathcal T_k$, which is given by a convolution with a symmetric, positive kernel of radius $\delta$. Such a mollification then has the following properties
\begin{align*}
&\mathcal{T}_{k, \delta}(z)= \mathcal{T}_k(z) \qquad \mbox{ if } |z|\leq k-\delta \mbox{ or } |z|\geq k+\delta, \qquad | \mathcal{T}^{\,''}_{k, \delta}|\leq C\delta^{-1}\\
&0\leq \mathcal{T}^{\,'}_{k, \delta}\leq 1, \quad \mathcal{T}^{\,''}_{k, \delta}\leq 0, \quad \mathcal{T}_{k, \delta}\leq \mathcal{T}_k\quad \mbox{on $(0,+\infty)$.}
\end{align*}
We fix $m,n,k\in\mathbb N$, $k>2\nm{\hat\vartheta}_{L^\infty(\partial\Omega)}$ and $\varepsilon, \delta>0$, $\varepsilon<k$ and define $ w^{m,n}_{\delta}=\mathcal{T}_{k+\varepsilon, \delta}(\vartheta^n)-\mathcal{T}_{k, \delta}(\vartheta^m)$. We set $\psi:=\mathcal{T}^{\,'}_{k+\varepsilon, \delta}(\vartheta^n)\mathcal{T}_\varepsilon(w^{m, n}_{\delta})$ in \eqref{ode22} for $\vartheta^n$ and $\psi:=\mathcal{T}^{\,'}_{k, \delta}(\vartheta^m)\mathcal{T}_\varepsilon(w^{m, n}_{\delta})$ in \eqref{ode22} for $\vartheta^m$. Note that it is allowed since $\mathcal{T}^{\,'}_{k+\varepsilon, \delta}(\vartheta^n)\mathcal{T}_\varepsilon(w^{m, n}_{\delta})$ and also $\mathcal{T}^{\,'}_{k, \delta}(\vartheta^m)\mathcal{T}_\varepsilon(w^{m, n}_{\delta})$ belong to $L^2((0, T); W_0^{1,2}(\Omega))$. Then, we subtract the so obtained equations to get
\begin{equation}\label{Tep}
\begin{split}
&\intTO{\kappa(\vartheta^n)\nabla(w^{m,n}_{\delta})\cdot \nabla \mathcal{T}_\varepsilon(w^{m,n}_{\delta})}\\
=&\intTO{\bigl(\kappa(\vartheta^n)-\kappa(\vartheta^m)\bigr)\nabla \mathcal{T}_{k, \delta}(\vartheta^m)\cdot\nabla\mathcal{T}_\varepsilon(w^{m,n}_{\delta})} -\int_0^T \langle \partial_t w^{m,n}_{\delta}, \mathcal{T}_\varepsilon(w^{m,n}_{\delta})\rangle \,{\rm d} t \\
&\phantom{=} + \intTO{ G^{m,n} \mathcal{T}_\varepsilon(w^{m,n}_{\delta})
\left[\mathcal{T}_{k+\varepsilon, \delta}(\vartheta^n) \vc{v}^n - \mathcal{T}_{k, \delta}(\vartheta^m) \vc{v}^m\right] \cdot \nabla \mathcal{T}_\varepsilon(w^{m,n}_\delta)},
\end{split}
\end{equation}
where we denoted
\begin{equation*}
\begin{split} G^{m,n}:&=\left[\mathcal{T}^{\,'}_{k+\varepsilon, \delta}(\vartheta^n) \,\S^n:\mathbf{D}\vc{v}^n - \mathcal{T}^{\,'}_{k, \delta}(\vartheta^m) \,\S^m:\mathbf{D}\vc{v}^m\right]\\
&-\left[ \kappa(\vartheta^n) |\nabla \vartheta^n|^2 \mathcal{T}^{\,''}_{k+\varepsilon, \delta}(\vartheta^n)- \kappa(\vartheta^m)|\nabla \vartheta^m|^2 \mathcal{T}^{\,''}_{k, \delta}(\vartheta^m)\right].
\end{split}
\end{equation*}
Our first goal is to let $\delta \to 0_+$ in \eqref{Tep}. Note that such convergence procedure is very standard in all terms except the term $G^{m,n}$ involving the second derivative of $\mathcal{T}_{\cdot,\delta}$. To get a proper $\delta$-independent bound, we consider $M\ge \|\hat\vartheta\|_{L^{\infty}(\partial \Omega)}$ and $\delta \in (0,M/2)$. Since $\vartheta^n=\hat\vartheta$ on $\partial \Omega$, we can deduce that $\psi:=1-\mathcal{T}^{\,'}_{M, \delta}(\vartheta^n)\in L^2(0,T;W^{1,2}_0(\Omega))$ and therefore it can be used in \eqref{ode22}. Such choice then leads to
\begin{equation}
\begin{split}\label{T22}
&\intTO{ \kappa(\vartheta^n)|\mathcal{T}^{\,''}_{M, \delta}(\vartheta^n)| |\nabla\vartheta^n|^2} =\intO{(\vartheta^n_0-\mathcal{T}_{M, \delta}(\vartheta^n_0))} \\
&-\intO{(\vartheta^n-\mathcal{T}_{M, \delta}(\vartheta^n))(T)} + \intTO{(1-\mathcal{T}^{\,'}_{M, \delta}(\vartheta^n)) \S^n:\mathbf{D}{\vc{v}}^n }\\
&\leq \int_{\{\vartheta^n_0>M/2\}}\!\!\!\!\! \vartheta^n_0\,{\rm d} {x} + \int_{\{|\vartheta^n|>M/2\}}\!\!\!\!\!\S^n:\mathbf{D}{\vc{v}}^n \,{\rm d} {x}\,{\rm d} t \le C,
\end{split}
\end{equation}
where we exploited that $\operatorname{div} \vc{v}^n=0$, used the properties of $\mathcal{T}_{M, \delta}$ (in particular the concavity) and \eqref{apr:vn} and \eqref{thet-zero}. In a very similar manner we can estimate the term with time derivative in \eqref{Tep}. Since $\vartheta^n$ and $\vartheta^m$ belong for fix $n,m$ to $C([0,T]; L^2(\Omega))$, we have (using the nonnegativity of $\mathcal G_{\varepsilon}$ as well as the estimate $\mathcal G_{\varepsilon}(s)\le s\varepsilon$)
\begin{equation}\label{dert}
\begin{split}
-\int_0^T \langle \partial_t w^{m,n}_{\delta}, \mathcal{T}_\varepsilon(w^{m,n}_{\delta})\rangle\,{\rm d} t =
- \intTO{\partial_t \mathcal G_\varepsilon(w^{m,n}_{\delta})}\le \intO{\mathcal G_\varepsilon(w^{m,n}_{\delta}(0))}\leq C\varepsilon.
\end{split}
\end{equation}
Hence, we can apply estimates \eqref{T22} and \eqref{dert} and then let $\delta \to 0_+$ in the remaining terms of \eqref{Tep} to deduce (recall $k\ge 2\|\hat\vartheta\|_{L^{\infty}(\partial \Omega)}$)
\begin{equation}\label{Tep2}
\begin{split}
&\underline{\kappa}\intTO{|\nabla\mathcal{T}_\varepsilon(w^{m,n})|^2}\le \intTO{\bigl(\kappa(\vartheta^n)-\kappa(\vartheta^m)\bigr)\nabla \mathcal{T}_{k}(\vartheta^m)\cdot \nabla\mathcal{T}_\varepsilon(w^{m,n})}\\
&\phantom{=} + \intTO
\left[\mathcal{T}_{k+\varepsilon}(\vartheta^n) \vc{v}^n - \mathcal{T}_{k}(\vartheta^m) \vc{v}^m\right] \cdot \nabla \mathcal{T}_\varepsilon(w^{m,n})}+C\varepsilon,
\end{split}
\end{equation}
where $w^{m,n}:=\mathcal{T}_{k+\varepsilon}(\vartheta^n) - \mathcal{T}_k(\vartheta^m)$. Next goal is to let $n,m \to \infty$ and finally $\varepsilon \to 0_+$. We start with the second term on the right hand side. Defining $w_{\varepsilon}:=\mathcal{T}_{k+\varepsilon}(\vartheta) - \mathcal{T}_k(\vartheta)$ and using \eqref{v-strong}--\eqref{grad-thet-weak}, we deduce
\begin{equation}\label{lim}
\begin{aligned}
\lim_{\varepsilon \to 0_+} &\limsup_{n\to \infty} \limsup_{m\to \infty} \intTO{\left[\mathcal{T}_{k+\varepsilon}(\vartheta^n) \vc{v}^n - \mathcal{T}_{k}(\vartheta^m) \vc{v}^m\right] \cdot \nabla \mathcal{T}_\varepsilon(w^{m,n})}\\
&=\lim_{\varepsilon \to 0_+} \intTO{\left[\mathcal{T}_{k+\varepsilon}(\vartheta) \vc{v} - \mathcal{T}_{k}(\vartheta) \vc{v}\right] \cdot \nabla \mathcal{T}_\varepsilon(w_{\varepsilon})}=0.
\end{aligned}
\end{equation}
The remaining term in \eqref{Tep2} is estimated with the help of \eqref{eq:re3} and the H\"{o}lder inequality as follows (note that $\{|w^{m,n}|<\epsilon\}\cap\{\vartheta^m\leq k\}=\{|\vartheta^n-\vartheta^m|<\epsilon\}\cap\{\vartheta^m\leq k\}$)
\begin{equation}
\begin{split}
&\intTO{\bigl(\kappa(\vartheta^n)-\kappa(\vartheta^m)\bigr)\nabla \mathcal{T}_{k}(\vartheta^m)\cdot \nabla\mathcal{T}_\varepsilon(w^{m,n})}\\
&\leq 2\intTO{|\kappa(\vartheta^n)-\kappa(\vartheta^m)|(|\nabla \vartheta^n|^2 + |\nabla \vartheta^m|^2)\chi_{\{|w^{m,n}|<\varepsilon\}\cap \{\vartheta^m\le k\}}}\\
&\le C(k)\||\kappa(\vartheta^n)-\kappa(\vartheta^m)|\chi_{\{|w^{m,n}|<\varepsilon\}\cap \{\vartheta^m\le k\}}\|_{L^{\infty}(Q)}\\
&\le C(k)\||\kappa(\vartheta^n)-\kappa(\vartheta^m)|\chi_{\{|\vartheta^n-\vartheta^m|<\varepsilon\}\cap \{\vartheta^m\le k\}}\|_{L^{\infty}(Q)}\\
&\le C(k)\sup_{l,s\in [\mu,2k]; \, |s-l|\le \varepsilon}|\kappa(l)-\kappa(s)| \to 0
\end{split}\label{small}
\end{equation}
as $\varepsilon \to 0_+$ thanks to the uniform continuity of $\kappa$ on $[\mu,2k]$.
Hence, using \eqref{lim}--\eqref{small} in \eqref{Tep2}, have
\begin{equation*
\begin{split}
\lim_{\varepsilon \to 0_+} \limsup_{n\to \infty} \limsup_{m\to \infty} \intTO{|\nabla\mathcal{T}_\varepsilon(w^{m,n})|^2}=0,
\end{split}
\end{equation*}
which, thanks to the weak lower semicontinuity of the norm in $L^2(Q)$, gives
\begin{equation}\label{Tep3}
\begin{split}
\lim_{\varepsilon \to 0_+} \limsup_{n\to \infty} \intTO{|\nabla\mathcal{T}_\varepsilon(\mathcal{T}_{k+\varepsilon}(\vartheta^n)-\mathcal{T}_k(\vartheta))|^2}=0.
\end{split}
\end{equation}
Finally, we use the H\"{o}lder inequality, the uniform bound \eqref{eq:re3} and the Tschebyschev inequality to get
$$
\begin{aligned}
&\intTO{|\nabla \vartheta^n -\nabla \vartheta|}\le \intTO{|\nabla \mathcal{T}_{\varepsilon}(\vartheta^n -\vartheta)|}+ \intTO{|\nabla \vartheta^n -\nabla \vartheta|\chi_{\{|\vartheta^n-\vartheta|>\varepsilon\}}}\\
&\le \intTO{|\nabla \mathcal{T}_{\varepsilon}(\mathcal{T}_{k+\varepsilon}(\vartheta^n)-\mathcal{T}_k(\vartheta))|}+\intTO{|\nabla \vartheta^n -\nabla \vartheta|\chi_{\{|\vartheta^n-\vartheta|>\varepsilon\}\cup \{|\vartheta^n|+|\vartheta|>k/2\}}}\\
&\le C\left(\intTO{|\nabla \mathcal{T}_{\varepsilon}(\mathcal{T}_{k+\varepsilon}(\vartheta^n)-\mathcal{T}_k(\vartheta))|^2}\right)^{\frac12}\\
&\quad +C\|\nabla \vartheta^n -\nabla \vartheta\|_{L^{\frac98}(Q)}\left(|\{|\vartheta^n-\vartheta|>\varepsilon\}|^{\frac19}+|\{|\vartheta^n|+|\vartheta|>k/2\}|^{\frac19}\right)\\
&\le C\left(\intTO{|\nabla \mathcal{T}_{\varepsilon}(\mathcal{T}_{k+\varepsilon}(\vartheta^n)-\mathcal{T}_k(\vartheta))|^2}\right)^{\frac12}+\frac{C\|\vartheta^n-\vartheta\|^{\frac19}_{L^1(Q)}}{\varepsilon^{\frac{1}{9}}}+\frac{C}{k^{\frac{1}{9}}}.
\end{aligned}
$$
Therefore, the convergence result \eqref{thet-strong} leads to
$$
\begin{aligned}
&\lim_{n\to \infty}\intTO{|\nabla \vartheta^n -\nabla \vartheta|}\le C\left(\limsup_{n\to \infty}\intTO{|\nabla \mathcal{T}_{\varepsilon}(\mathcal{T}_{k+\varepsilon}(\vartheta^n)-\mathcal{T}_k(\vartheta))|^2}\right)^{\frac12}+\frac{C}{k^{\frac{1}{9}}}.
\end{aligned}
$$
As the left hand side is independent of $\varepsilon$ and $k$, we may let first $\varepsilon \to 0_+$ and use \eqref{Tep3} to eliminate the first term on the right hand side and then let $k\to\infty$ to handle the second term on the right hand side and thus to observe that
\begin{equation}
\nabla \vartheta^n \to \nabla \vartheta \qquad \textrm{strongly in } L^1(Q)\label{NS1}
\end{equation}
and consequently (for a subsequence)
\begin{equation}\label{ae-nabla}
\nabla \vartheta^n \to \nabla \vartheta \qquad \mbox{ a.e. in } Q.
\end{equation}
\let\tktn\relax
\subsubsection{Strong convergence of \texorpdfstring{${\kappa(\vartheta^n)|\nabla \vartheta^n|^2}/{(\vartheta^n)^2}$}{n} in \texorpdfstring{$L^1$}{L1}-norm}
We start the proof of the claim by showing a strong convergence of $\nabla \mathcal{T}_k(\vartheta^n-\hat\vartheta)$ in $L^2(Q)$ for arbitrary $k$. To prove such a result, we want to set $\psi:=\mathcal{T}_k(\vartheta^n-\hat\vartheta)$ in \eqref{ode22}, let $n \to \infty$ and compare the limit with \eqref{T2} tested by $\varphi:= \mathcal{T}_k(\vartheta-\hat\vartheta)$. However, such test functions are not allowed in general and therefore we must proceed more carefully. We fix an arbitrary $T^*\in (0,T)$, which is the Lebesgue point of $\vartheta(t)$ as a function in $L^1(0,T;L^1(\Omega))$.
Using the fact that $\operatorname{div} \vc{v}^n=0$, we see that
\begin{equation}\label{thetab-n}
\begin{split}
&\intO{\nabla \vartheta^n \cdot \vc{v}^n \, \mathcal{T}_k(\vartheta^n -\hat\vartheta)
=\intO{\nabla \hat\vartheta \cdot \vc{v}^n \, \mathcal{T}_k(\vartheta^n -\hat\vartheta)}.
\end{split}
\end{equation}
Then, we set $\psi:=\mathcal{T}_k(\vartheta^n-\hat\vartheta) \chi_{[0,\tau]}$ with $\tau\in (T^*,T)$ arbitrary in \eqref{ode22}. We deduce (since $\vartheta^n \in C([0,T]; L^2(\Omega))$ and $\hat\vartheta$ is independent of time) by using \eqref{thetab-n} that
\begin{equation}\label{MBa}
\begin{split}
&\int_0^{\tau }\intO{\kappa(\vartheta^n)\nabla (\vartheta^n-\hat\vartheta) \cdot \nabla [\mathcal{T}_k(\vartheta^n -\hat\vartheta)]}\,{\rm d} t = - \intO{\mathcal{G}_k(\vartheta^n(\tau) -\hat\vartheta)-\mathcal{G}_k(\vartheta^n_0 -\hat\vartheta)}\\
& + \int_0^{\tau}\intO{-\nabla \hat\vartheta \cdot \vc{v}^n \, \mathcal{T}_k(\vartheta^n -\hat\vartheta)-\kappa(\vartheta^n)\nabla \hat\vartheta \cdot \nabla [\mathcal{T}_k(\vartheta^n -\hat\vartheta)]+\mathcal{T}_k(\vartheta^n -\hat\vartheta)\, \S^n:\mathbf{D}{\vc{v}^n} } \,{\rm d} t .
\end{split}
\end{equation}
Then, it follows from \eqref{v-strong}, \eqref{thet-strong}, \eqref{thet-zero}, \eqref{Sn} and \eqref{MBa} that for $\delta\in(0,T-T^*)$
\begin{equation}\label{MB}
\begin{split}
&\limsup_{n\to \infty}\int_0^{T^*}\intO{\kappa(\vartheta^n)\nabla \vartheta^n \cdot \nabla [\mathcal{T}_k(\vartheta^n -\hat\vartheta)]}\,{\rm d} t \\
&=\limsup_{n\to \infty}\int_0^{T^*}\intO{\kappa(\vartheta^n)\nabla [\vartheta^n-\hat\vartheta] \cdot \nabla [\mathcal{T}_k(\vartheta^n -\hat\vartheta)]+\kappa(\vartheta^n)\nabla \hat\vartheta \cdot \nabla [\mathcal{T}_k(\vartheta^n -\hat\vartheta)]}\,{\rm d} t \\
&\le \limsup_{n\to \infty} \fint_{T^*}^{T^*+\delta}\int_0^{\tau}\intO{\kappa(\vartheta^n)\nabla [\vartheta^n-\hat\vartheta] \cdot \nabla [\mathcal{T}_k(\vartheta^n -\hat\vartheta)]}\,{\rm d} t \,{\rm d} \tau\\
&\qquad\qquad +\int_0^{T^*}\intO{\kappa(\vartheta)\nabla \hat\vartheta \cdot \nabla [\mathcal{T}_k(\vartheta -\hat\vartheta)]}\,{\rm d} t \\
&\overset{\eqref{MBa}}{\to} -\intO{\mathcal{G}_k(\vartheta(T^*) -\hat\vartheta)-\mathcal{G}_k(\vartheta_0 -\hat\vartheta)}
+ \int_0^{T^*}\intO{(\S:\mathbf{D}{\vc{v}}-\nabla \hat\vartheta \cdot \vc{v})\mathcal{T}_k(\vartheta -\hat\vartheta) } \,{\rm d} t
\end{split}
\end{equation}
as $\delta \to 0_+$.
Now, the aim is to identify the final expression in \eqref{MB}. We want to set $\psi:=\mathcal{T}_k(\vartheta-\hat\vartheta)$ in \eqref{3.18}. Unfortunately, it is not possible due to low regularity of $\vartheta$. Therefore, we must proceed differently. In what follows, we always consider $M,k,\varepsilon,\delta>0$ such that $M>k+1+\|\hat\vartheta\|_{\infty}$ and $\delta\in(0,1)$. Due to the smoothness of $\vartheta^n$, we can set $\psi(\tau,x):=\mathcal{T}'_{M, \delta}(\vartheta^n(\tau,x)) \chi_{(t,t+\varepsilon)}(\tau)\varphi(x)$, $\tau\in(0,T)$, $x\in\Omega$ in \eqref{ode22} to deduce that for all $\varphi\in W^{1,2}_0(\Omega) \cap L^{\infty}(\Omega)$ and all $t\in (0,T^*)$, $\varepsilon\in(0,T-T^*)$ we have
\begin{equation*}
\begin{split}
&\intO{\partial_t \fint_t^{t +\varepsilon} \mathcal{T}_{M, \delta}(\vartheta^n) \,{\rm d} \tau \varphi}
+ \intO{\fint_t^{t +\varepsilon} \nabla [\mathcal{T}_{M, \delta}(\vartheta^n)]\cdot \vc{v}^n \,{\rm d} \tau \varphi}\\
&+ \intO{ \fint_t^{t +\varepsilon} \kappa(\vartheta^n)\nabla \mathcal{T}_{M, \delta}(\vartheta^n)\,{\rm d} \tau \cdot \nabla \varphi}
= \intO{\fint_t^{t +\varepsilon} \mathcal{T}^{\,'}_{M, \delta}(\vartheta^n) \, \S^n:\mathbf{D}{\vc{v}}^n\,{\rm d} \tau \varphi}\\
&+ \intO{\fint_t^{t +\varepsilon} \kappa(\vartheta^n) \mathcal{T}^{\,''}_{M, \delta}(\vartheta^n) |\nabla\vartheta^n|^2\,{\rm d} \tau\varphi}.
\end{split}
\end{equation*}
Then, we fix a measurable set $J\subset(0,T^*)$ and integrate over $t\in J$. Using the convergence results \eqref{v-strong}--\eqref{thet-strong} and \eqref{Sn} next the density of step functions in $L^2(0,T^*;W^{1,2}_0(\Omega))\cap L^\infty(0,T;L^\infty(\Omega))$, we obtain that for all $\varphi\in L^{2}(0,T; W^{1,2}_0(\Omega))\cap L^{\infty}(Q)$,
\begin{equation}\label{eq:dvakrize}
\begin{split}
&\int_0^{T^*}\intO{\partial_t \vartheta^{M,\delta}_{\varepsilon} \varphi}\,{\rm d} t
+ \int_0^{T^*}\intO{\fint_t^{t +\varepsilon} \nabla [\mathcal{T}_{M, \delta}(\vartheta)]\cdot \vc{v} \,{\rm d} \tau \varphi}\,{\rm d} t \\
&+ \int_0^{T^*}\intO{ \fint_t^{t +\varepsilon} \kappa(\vartheta)\nabla \mathcal{T}_{M, \delta}(\vartheta)\,{\rm d} \tau \cdot \nabla \varphi}\,{\rm d} t
= \int_0^{T^*}\intO{\fint_t^{t +\varepsilon} \mathcal{T}^{\,'}_{M, \delta}(\vartheta) \, \S:\mathbf{D}{\vc{v}}\,{\rm d} \tau \varphi}\,{\rm d} t \\
&+ \limsup_{n\to \infty} \int_0^{T^*}\intO{\fint_t^{t +\varepsilon} \kappa(\vartheta^n) \mathcal{T}^{\,''}_{M, \delta}(\vartheta^n) |\nabla\vartheta^n|^2\,{\rm d} \tau\varphi}\,{\rm d} t ,
\end{split}
\end{equation}
where we denoted $\vartheta^{M,\delta}_{\varepsilon}(t,x):=\fint_t^{t +\varepsilon} \mathcal{T}_{M, \delta}(\vartheta(\tau, x)) \,{\rm d} \tau$. Note that for every $\varepsilon>0$ the function $\vartheta^{M,\delta}_{\varepsilon}$ belongs to $W^{1,\infty}(0,T; L^{\infty}(\Omega))$ and so the integral with time derivative is well defined.
From \eqref{T22}, \eqref{thet-zero}, \eqref{thet-strong} and \eqref{Sn} we get
\begin{equation}\label{eq:nextest}
\begin{split}
&\limsup_{n\to\infty}\intTO{ \kappa(\vartheta^n)|\mathcal{T}^{\,''}_{M, \delta}(\vartheta^n)| |\nabla\vartheta^n|^2} \leq \intO{(\vartheta_0-\mathcal{T}_{M, \delta}(\vartheta_0))} \\
&\quad + \intTO{(1-\mathcal{T}^{\,'}_{M, \delta}(\vartheta)) \S:\mathbf{D}{\vc{v}} }
\leq \int_{\{\vartheta_0>M/2\}}\!\!\!\!\! \vartheta_0\,{\rm d} {x} + \int_{\{|\vartheta|>M/2\}}\!\!\!\!\!\S:\mathbf{D}{\vc{v}} \,{\rm d} {x}\,{\rm d} t .
\end{split}
\end{equation}
Setting $\varphi=\mathcal{T}_k( \vartheta^{M,\delta}_{\varepsilon} - \hat\vartheta )$ in \eqref{eq:dvakrize} and using \eqref{eq:nextest} we obtain
\begin{equation*}
\begin{split}
&\int_0^{T^*}\intO{ \fint_t^{t +\varepsilon} \kappa(\vartheta)\nabla \mathcal{T}_{M, \delta}(\vartheta)\,{\rm d} \tau \cdot \nabla \mathcal{T}_k \left(\vartheta^{M,\delta}_{\varepsilon} - \hat\vartheta\right)}\,{\rm d} t \\
&\ge-\intO{\mathcal{G}_k\left( \vartheta^{M,\delta}_{\varepsilon}(T^*)- \hat\vartheta\right)-\mathcal{G}_k\left( \vartheta^{M,\delta}_{\varepsilon}(0)- \hat\vartheta\right)}\\
& + \int_0^{T^*}\intO{\left(\fint_t^{t +\varepsilon} \mathcal{T}^{\,'}_{M, \delta}(\vartheta) \, \S:\mathbf{D}{\vc{v}}-\nabla [\mathcal{T}_{M, \delta}(\vartheta)]\cdot \vc{v} \,{\rm d} \tau \right) \mathcal{T}_k \left( \vartheta^{M,\delta}_{\varepsilon} - \hat\vartheta\right)}\,{\rm d} t \\
&-k\left(\int_{\{\vartheta_0>M/2\}}\!\!\!\!\! \vartheta_0\,{\rm d} {x} + \int_{\{\vartheta>M/2\}}\!\!\!\!\!\S:\mathbf{D}{\vc{v}} \,{\rm d} {x}\,{\rm d} t \right).
\end{split}
\end{equation*}
Since $T^*$ is the Lebesgue point of $\vartheta$ and since the initial condition is attained (see e.g. \cite{BuMaRa09, Consiglieri}) we can let $\varepsilon\to 0_+$ and obtain
\begin{equation}
\begin{split}
&\int_0^{T^*}\intO{ \kappa(\vartheta)\nabla \mathcal{T}_{M, \delta}(\vartheta) \cdot \nabla \mathcal{T}_k \left(\mathcal{T}_{M, \delta}(\vartheta) - \hat\vartheta\right)}\,{\rm d} t \\
&\ge-\intO{\mathcal{G}_k\left( \mathcal{T}_{M, \delta}(\vartheta)(T^*)- \hat\vartheta\right)-\mathcal{G}_k\left( \mathcal{T}_{M, \delta}(\vartheta_0)- \hat\vartheta\right)}\\
& + \int_0^{T^*}\intO{\left(\mathcal{T}^{\,'}_{M, \delta}(\vartheta) \, \S:\mathbf{D}{\vc{v}}-\nabla [\mathcal{T}_{M, \delta}(\vartheta)]\cdot \vc{v} \right) \mathcal{T}_k \left( \mathcal{T}_{M, \delta}(\vartheta) - \hat\vartheta\right)}\,{\rm d} t \\
&-k\left(\int_{\{\vartheta_0>M/2\}}\!\!\!\!\! \vartheta_0\,{\rm d} {x} + \int_{\{\vartheta>M/2\}}\!\!\!\!\!\S:\mathbf{D}{\vc{v}} \,{\rm d} {x}\,{\rm d} t \right).
\end{split}\label{ep}
\end{equation}
For $M> k+1 + \|\hat\vartheta\|_{\infty}$, we have $\mathcal{T}_k ( \mathcal{T}_{M, \delta}(\vartheta) - \hat\vartheta) = \mathcal{T}_k ( \vartheta - \hat\vartheta)$, thus one can let $M\to \infty$ in \eqref{ep}. Using integration by parts and the fact that $\operatorname{div} \vc{v}=0$ similarly as in \eqref{thetab-n} one obtains
\begin{equation*}
\begin{split}
&\int_0^{T^*}\intO{ \kappa(\vartheta)\nabla \vartheta \cdot \nabla \mathcal{T}_k (\vartheta - \hat\vartheta)}\,{\rm d} t \\
&\ge-\intO{\mathcal{G}_k (\vartheta(T^*)- \hat\vartheta)-\mathcal{G}_k(\vartheta_0- \hat\vartheta)} + \int_0^{T^*}\intO{( \S:\mathbf{D}{\vc{v}}-\nabla \hat\vartheta\cdot \vc{v} ) \mathcal{T}_k ( \vartheta - \hat\vartheta)}\,{\rm d} t .
\end{split
\end{equation*}
Comparing the result with \eqref{MB}, we see that
\begin{equation}\label{GradTk-L2}
\begin{split}
\limsup_{n\to +\infty}
\int_0^{T^*}\intO{\kappa(\vartheta^n)\nabla \vartheta^n \cdot \nabla \mathcal{T}_k(\vartheta^n -\hat\vartheta)}\,{\rm d} t
\le \int_0^{T^*}\intO{ \kappa(\vartheta) \nabla \vartheta \cdot \nabla \mathcal{T}_k (\vartheta - \hat\vartheta )}\,{\rm d} t ,
\end{split}
\end{equation}
which is the corner stone for the strong convergence. Indeed, it follows from \eqref{thet-weak}--\eqref{thet-strong} and from \eqref{GradTk-L2} that (using also the Vitali convergence theorem and \eqref{k})
\begin{equation}\label{goal}
\begin{split}
&\limsup_{n\to +\infty} \int_0^{T^*}\intO{\kappa(\vartheta^n)|\nabla \mathcal{T}_k(\vartheta^n -\hat\vartheta)|^2}\,{\rm d} t \le \int_0^{T^*}\intO{ \kappa(\vartheta) |\nabla \mathcal{T}_k (\vartheta - \hat\vartheta )|^2}\,{\rm d} t .
\end{split}
\end{equation}
Since
\begin{align}\label{weakGradTk}
\sqrt{\kappa(\vartheta^n)}\nabla \mathcal{T}_k(\vartheta^n -\hat\vartheta) &\rightharpoonup \sqrt{\kappa(\vartheta)}\nabla \mathcal{T}_k(\vartheta -\hat\vartheta) &&\mbox{ weakly in } L^2(Q),\\
\intertext{the inequality \eqref{goal}, weak lower semicontinuity of $L^2$ norm and uniform convexity of $L^2$ imply}
\label{strong-Tk}
\sqrt{\kappa(\vartheta^n)}\nabla \mathcal{T}_k(\vartheta^n -\hat\vartheta) &\to \sqrt{\kappa(\vartheta)}\nabla \mathcal{T}_k(\vartheta -\hat\vartheta) &&\mbox{ strongly in } L^2((0, T^*)\times\Omega).
\end{align}
Finally, we want to show that
\begin{align}\label{grad-eta}
\sqrt{\kappa(\vartheta^n)}\frac{\nabla \vartheta^n}{\vartheta^n} \to \sqrt{\kappa(\vartheta)} \frac{\nabla \vartheta}{\vartheta} &&\mbox{ strongly in } L^2((0, T^*)\times\Omega).
\end{align}
We have
\begin{align}\label{weak-nablathet-thet}
\sqrt{\kappa(\vartheta^n)}\frac{\nabla \vartheta^n}{\vartheta^n} \rightharpoonup \sqrt{\kappa(\vartheta)}\frac{\nabla \vartheta}{\vartheta} &&\mbox{ weakly in } L^2((0, T^*)\times\Omega).
\end{align}
because of the uniform boundedness in $L^2((0, T^*)\times\Omega).$
To show convergence of norms, we write
\begin{equation}\label{nablathet-thet}
\begin{split}
& \int_0^{T^*}\intO{ \kappa(\vartheta^n)\frac{|\nabla \vartheta^n|^2}{(\vartheta^n)^2}}\,{\rm d} t \\
&= \int_0^{T^*}\intO{\kappa(\vartheta^n)\frac{\nabla \vartheta^n}{\vartheta^n}\cdot\frac{\nabla (\vartheta^n - \hat\vartheta)}{\vartheta^n}}\,{\rm d} t + \int_0^{T^*}\intO{\kappa(\vartheta^n)\frac{\nabla \vartheta^n}{\vartheta^n}\cdot\frac{\nabla \hat\vartheta}{\vartheta^n}}\,{\rm d} t \\
&= \int_0^{T^*}\intO{\kappa(\vartheta^n)\frac{\nabla \vartheta^n}{\vartheta^n}\cdot\frac{\nabla \mathcal{T}_k(\vartheta^n - \hat\vartheta)}{\vartheta^n}}\,{\rm d} t + \int_{\{|\vartheta^n-\hat\vartheta|>k\}}\!\!\!\!\! \kappa(\vartheta^n) \frac{\nabla \vartheta^n}{\vartheta^n}\cdot\frac{\nabla (\vartheta^n - \hat\vartheta)}{\vartheta^n} \,{\rm d} {x}\,{\rm d} t \\&+
\int_0^{T^*}\intO{\kappa(\vartheta^n)\frac{\nabla \vartheta^n}{\vartheta^n}\cdot\frac{\nabla \hat\vartheta}{\vartheta^n}}\,{\rm d} t .
\end{split}
\end{equation}
Next, we let $n\to \infty$ on the right-hand side of \eqref{nablathet-thet}. First, by the weak convergence \eqref{weak-nablathet-thet}, the strong convergence results \eqref{thet-strong} and \eqref{strong-Tk}, the Vitali theorem and the minimum principle $\vartheta^n\ge \mu$
\begin{equation*}
\lim_{n\to + \infty}\int_0^{T^*}\!\!\!\intO{\kappa(\vartheta^n)\frac{\nabla \vartheta^n}{\vartheta^n}\cdot\frac{\nabla \mathcal{T}_k(\vartheta^n - \hat\vartheta)}{\vartheta^n}}\,{\rm d} t = \int_0^{T^*}\!\!\!\intO{\kappa(\vartheta)\frac{\nabla \vartheta}{\vartheta}\cdot\frac{\nabla \mathcal{T}_k(\vartheta - \hat\vartheta)}{\vartheta}}\,{\rm d} t
\end{equation*}
and
\begin{equation} \lim_{n\to+\infty} \int_0^{T^*}\intO{\kappa(\vartheta^n)\frac{\nabla \vartheta^n}{\vartheta^n}\cdot\frac{\nabla \hat\vartheta}{\vartheta^n}}\,{\rm d} t = \int_0^{T^*}\intO{\kappa(\vartheta)\frac{\nabla \vartheta}{\vartheta}\cdot\frac{\nabla \hat\vartheta}{\vartheta}}\,{\rm d} t .
\end{equation}
In addition, assuming that $k>\|\hat\vartheta\|_{\infty}$, we see that $\vartheta^n\geq k$ on the set $\{|\vartheta^n-\hat\vartheta|>k\}$. Consequently, with the help of the Young inequality, we deduce that for any $\lambda\in(0,1)$
\begin{equation}\label{limit-k}
\begin{split}
\left|\int_{\{|\vartheta^n-\hat\vartheta|>k\}}\!\!\!\!\! \kappa(\vartheta^n)\frac{\nabla \vartheta^n}{\vartheta^n}\cdot\frac{\nabla (\vartheta^n - \hat\vartheta)}{\vartheta^n} \,{\rm d} {x}\,{\rm d} t \right|& \leq 2\overline{\kappa}\intTO{\frac{|\nabla \vartheta^n|^2+ |\nabla\hat\vartheta|^2}{(\vartheta^n)^{1+\lambda} k^{1-\lambda} }}\leq \frac{C(\lambda)}{k^{1-\lambda}},
\end{split}
\end{equation}
where $C$ is independent of $n$ and $k$ thanks to the uniform estimate in \eqref{eq:re3} and the minimum principle $\vartheta^n\ge \mu$.
Equality \eqref{nablathet-thet} and inequality \eqref{limit-k} remain valid also if we replace $\vartheta^n$ with $\vartheta$.
Consequently, it follow
\begin{equation}\label{conv-norms}
\limsup_{n\to \infty}\int_0^{T^*}\intO{\kappa(\vartheta^n)\frac{|\nabla \vartheta^n|^2}{(\vartheta^n)^2}}\,{\rm d} t \le \int_0^{T^*}\intO{\kappa(\vartheta)\frac{|\nabla \vartheta|^2}{\vartheta^2}}\,{\rm d} t +\frac{C(\lambda)}{k^{1-\lambda}}.
\end{equation}
Since $k>\|\hat\vartheta\|_{\infty}$ is arbitrary, \eqref{conv-norms} gives convergence of norms which combined with \eqref{weak-nablathet-thet} implies the strong convergence \eqref{grad-eta}. Finally, let us recall that $T^*$ can be chosen arbitrarily close to $T$. In addition, we can a~priori construct the solution on the time interval $(0,2T)$. Consequently, $T^*$ can be chosen bigger than $T$ and so we obtain \eqref{grad-eta} with $T^*$ replaced by $T$.
\subsubsection{Limit in the entropy equality}
Let us analyze the limit of \eqref{entropy-weak} as $n\to+\infty$ for each term.
The strong convergence of $\vartheta^n$ and $\nabla\vartheta^n$, see \eqref{thet-strong} and \eqref{NS1}, implies that $\eta^n\to\eta$ and $\nabla\eta^n\to\nabla\eta$ a.e. in $Q$. Thanks to the definition of $\eta^n$, and the a~priori bound \eqref{eq:re3}, we obtain a uniform bound for $\nm{\eta^n}_{L^r(Q)}$ for any $r\in(1,+\infty)$ and for $\nm{\nabla\eta^n}_{L^2(Q)}$. These facts, the Vitali convergence theorem and \eqref{thet-zero} allow us, up to subsequence, easily pass to the limit on the left hand side of~\eqref{entropy-weak}.
To pass to the limit also in terms on the right hand side, we use \eqref{Lp}, \eqref{Lp-S}, \eqref{thet-strong}, \eqref{Mintye} and \eqref{grad-eta} combined with the Lebesgue dominated convergence theorem and together with the minimum principle for $\vartheta^n$.
Consequently, $\eta$ satisfies entropy equation~\eqref{entropy-limit}.
\subsection{Continuity of \texorpdfstring{$\vartheta$}{theta} in time}
Finally, we focus on the attainment of initial conditions and continuity with respect to time variable. Concerning the velocity field, we can recall \eqref{Lp} and by standard parabolic interpolation, we observe $\vc{v} \in C([0,T]; L^2_{0,\operatorname{div}})$. The fact that $\vc{v}(0)=\vc{v}_0$ is then proven analogously as for Navier--Stokes equations, see e.g. \cite{MaBook,FrMaRu10}.
Now, we focus on the temperature. For the attainment of the initial condition, one can follow \cite{Consiglieri,BuMaRa09,FrMaRu10}. Thus we present here only the proof of continuity of $\vartheta$ with respect to time into the $L^1$ topology. The key idea is the following. We investigate a function (here $M>\max\{\|\hat\vartheta\|_{L^{\infty}(\Omega)},2,2\mu\}$)
$$
g(\vartheta-\hat\vartheta):= \sign (\vartheta - \hat\vartheta) \sqrt{\mathcal{G}_M(\vartheta - \hat\vartheta)}
$$
and show first that for all $\psi \in L^2(\Omega)$ the mapping $t\mapsto \intO{g(\vartheta(t,x)-\hat\vartheta(t,x))\psi(x)}$ is continuous on $[0,T]$. Second, we show that the mapping $t\mapsto \intO{g^2(\vartheta(t,x)-\hat\vartheta(t,x))}$ is continuous on $[0,T]$. As a direct consequence of these two properties, we obtain that
\begin{equation}
g(\vartheta-\hat\vartheta)\in C([0,T]; L^2(\Omega)).
\label{C2}
\end{equation}
Since,
$$
g'(\vartheta - \hat\vartheta)=\frac{|T_M(\vartheta - \hat\vartheta)|}{2\sqrt{\mathcal{G}_M(\vartheta - \hat\vartheta)}}= \left\{
\begin{aligned}
&\frac{1}{\sqrt{2}},&&\textrm{if }\vartheta\le M+\hat\vartheta\\
&\frac{M}{2\sqrt{M(\vartheta - \hat\vartheta)-\frac{M^2}{2}}}&&\textrm{if }\vartheta\ge M+\hat\vartheta
\end{aligned}
\right\} \ge \frac{\sqrt\mu}{\sqrt{2\vartheta}},
$$
we have a trivial estimate
$$
|\vartheta_1-\vartheta_2|\le \sqrt{2}(\sqrt{\vartheta_1}+\sqrt{\vartheta_2})|g(\vartheta_1 - \hat\vartheta)-g(\vartheta_2 - \hat\vartheta)|.
$$
Consequently, using the H\"{o}lder inequality and \eqref{theinf1}, we observe
$$
\begin{aligned}
\|\vartheta(t_1)-\vartheta(t_2)\|_{L^1(\Omega)} &\le \sqrt{2}\|\sqrt{\vartheta(t_1)}+\sqrt{\vartheta(t_2)}\|_{L^2(\Omega)} \|g(\vartheta(t_1) - \hat\vartheta)-g(\vartheta(t_2) - \hat\vartheta)\|_{L^2(\Omega)}\\
&\le C \|g(\vartheta(t_1) - \hat\vartheta)-g(\vartheta(t_2) - \hat\vartheta)\|_{L^2(\Omega)}.
\end{aligned}
$$
Thus, we see that \eqref{C2} implies $\vartheta \in C([0,T];L^1(\Omega))$.
It remains to show \eqref{C2}. First, we set $\psi:=g'(\vartheta^n-\hat\vartheta)\chi_{[0,\tau]} \varphi$ with $\varphi \in C^1_0(\Omega)$ arbitrary in \eqref{ode22}. Using integration by parts, we have
\begin{equation*}
\begin{aligned}
&\intO{(g(\vartheta^n(\tau)-\hat\vartheta)-g(\vartheta^n(0)-\hat\vartheta))\varphi}= -\int_0^{\tau}\intO{g'(\vartheta^n-\hat\vartheta) \varphi \vc{v}^{n} \nabla \hat\vartheta}\,{\rm d} t \\
& + \int_0^{\tau}\intO{\left(\vc{v}^{n} (\vartheta^{n}-\hat\vartheta)-\kappa(\vartheta^n)\nabla \vartheta^n\right) \cdot (\varphi g''(\vartheta^n-\hat\vartheta) \nabla (\vartheta^n-\hat\vartheta) +g'(\vartheta^n-\hat\vartheta) \nabla \varphi)}\,{\rm d} t \\
& + \int_0^{\tau}\intO{\S^{n}: \mathbf{D}\vc{v}^{n} g'(\vartheta^n-\hat\vartheta) \varphi} \,{\rm d} t .
\end{aligned}
\end{equation*}
Next, we let $n\to \infty$ in the above identity. Using the uniform estimates \eqref{apr:vn}--\eqref{eq:re3}, the properties of $g$, the convergence results \eqref{Lp}, \eqref{thet-strong}, \eqref{ae-nabla}, \eqref{weak-nablathet-thet} and the Vitali convergence theorem, we can easily identify the limits in the first two terms on the right hand side. For the last term on the right hand side, we also use \eqref{Sn}. In addition, for almost all $\tau \in (0,T)$ we can also identify the limit on the left hand side
\begin{equation}
\begin{aligned}
\label{ode22n}
&\intO{(g(\vartheta(\tau)-\hat\vartheta)-g(\vartheta_0-\hat\vartheta))\varphi}= -\int_0^{\tau}\intO{g'(\vartheta-\hat\vartheta) \varphi \vc{v} \nabla \hat\vartheta}\,{\rm d} t \\
& + \int_0^{\tau}\intO{\left(\vc{v} (\vartheta-\hat\vartheta)-\kappa(\vartheta)\nabla \vartheta\right) \cdot (\varphi g''(\vartheta-\hat\vartheta) \nabla (\vartheta-\hat\vartheta) +g'(\vartheta-\hat\vartheta) \nabla \varphi)}\,{\rm d} t \\
& + \int_0^{\tau}\intO{\S: \mathbf{D}\vc{v} g'(\vartheta-\hat\vartheta) \varphi} \,{\rm d} t .
\end{aligned}
\end{equation}
Since the right hand side is a continuous function of $\tau \in [0,T]$, we can redefine $g(\vartheta-\hat\vartheta)$ on zero subset of $[0,T]$ to get
\begin{equation}\label{weak-cont1}
\left(\intO{g(\vartheta)\varphi}\right) \in C([0,T]) \qquad \textrm{ for all }\varphi \in C_0^1(\Omega).
\end{equation}
Since $g\in L^{\infty}(0,T; L^2(\Omega))$ and $C_0^1(\Omega)$ is dense in $L^2(\Omega)$, \eqref{weak-cont1} implies that
\begin{equation}\label{WC2}
\left(\intO{g(\vartheta)\varphi}\right) \in C([0,T]) \qquad \textrm{ for all }\varphi \in L^2(\Omega).
\end{equation}
Finally, we pass in \eqref{MBa} to the limit as $n\to+\infty$ similarly as in \eqref{MB} using also \eqref{weak-nablathet-thet}
\begin{equation}\label{MBaa}
\begin{split}
&\intO{\mathcal{G}_M(\vartheta(\tau) -\hat\vartheta)-\mathcal{G}_M(\vartheta_0 -\hat\vartheta)}=\int_0^{\tau}\intO{-\kappa(\vartheta)\nabla \vartheta \cdot \nabla [\mathcal{T}_M(\vartheta -\hat\vartheta)]} \,{\rm d} t \\
& + \int_0^{\tau}\intO{-\nabla \hat\vartheta \cdot \vc{v} \, \mathcal{T}_M(\vartheta -\hat\vartheta)+\mathcal{T}_M(\vartheta -\hat\vartheta)\, \S:\mathbf{D}{\vc{v}} } \,{\rm d} t =:\int_0^{\tau}\intO{h}\,{\rm d} t ,
\end{split}
\end{equation}
where $h\in L^{1}(Q)$. Hence, we see that (it can be continuously extended)
$$
\intO{\mathcal{G}_M(\vartheta(\tau) -\hat\vartheta)} \in C([0,T]).
$$
Since $(g(\theta-\hat\vartheta))^2= \mathcal{G}_M(\vartheta -\hat\vartheta)$, the above relation combined with \eqref{WC2} implies \eqref{C2}. The proof is\footnote{In fact in above procedure we somehow extended $g$ and $\mathcal{G}_M$ also to a possible non-Lebesgue points. This can be done more carefully. Namely, one can consider $\fint_t^{t+\delta} g(\tau)\,{\rm d} \tau$ and $\fint_{t}^{t+\delta}\mathcal{G}_M(\vartheta(\tau)-\hat\vartheta)$. These are surely continuous with respect to $t\in [0,T]$. Then thanks to \eqref{WC2} and \eqref{MBaa}, we see that
$$
\fint_t^{t+\delta} g(\tau)\,{\rm d} \tau \to g \quad \textrm{strongly in } C([0,T]; L^2(\Omega)).
$$
} complete. \qed
\section*{Acknowledgment}
{A.~Abbatiello has been supported by the ERC-STG Grant n. 759229 HiCoS ``Higher Co-dimension Singularities: Minimal Surfaces and the Thin Obstacle Problem" and is member of the Italian National Group for the Mathematical Physics (GNFM) of the Italian National Institute of the High Mathematics (INdAM). M. Bul\'{\i}\v{c}ek and P. Kaplick\'{y} acknowledge the support of the project No. 20-11027X financed by Czech Science Foundation (GA\v{C}R). M. Bul\'{\i}\v{c}ek is member of the Jind\v{r}ich Ne\v{c}as Center for Mathematical Modelling.}
\bibliographystyle{amsplain}
\providecommand{\bysame}{\leavevmode\hbox to3em{\hrulefill}\thinspace}
\providecommand{\MR}{\relax\ifhmode\unskip\space\fi MR }
\providecommand{\MRhref}[2]{%
\href{http://www.ams.org/mathscinet-getitem?mr=#1}{#2}
}
\providecommand{\href}[2]{#2}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 7,978 |
<?php
class sfDoctrineiCalStoragePluginConfiguration extends sfPluginConfiguration
{
/**
* @see sfPluginConfiguration
*/
public function initialize() {
$this->dispatcher->connect('doctrine.configure_connection', array($this, 'configureDoctrineConnection'));
}
public function configureDoctrineConnection(sfEvent $event)
{
$connection = $event['connection'];
$connection->setAttribute(Doctrine_Core::ATTR_VALIDATE, Doctrine_Core::VALIDATE_ALL);
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 3,337 |
This past August, we had one of the most difficult conventions to date. Someone who is not a brother, nor a gentleman, pulled off a victory after lying to the commuter railroads, and many other railroads as well. How do we move forward without being divisive, yet still inform our locals? Does it make us just as bad for decrying the actions of this liar or is the liar the divisive one? OUR PRESIDENT, W. Dan Pickett, will speak for himself and all of us as well.
I am dismayed to learn of the cancellation of my retirement party, and I would like an explanation. It has been a long past practice to throw a celebration of career for each Grand Lodge Officer and each Member of the Grand Board of Trustees that retire. Additionally, the date is always of the honoree's choosing, and they are consulted before making any arrangements. Brothers, December 15 was not an arbitrary date, it was chosen specifically so I and my family could attend. That date worked with my calendar and that of my family, and we have all been making plans around holding that date.
I have done my best to stay out of the day-to-day business of the Brotherhood since my retirement on September 30, and all I have heard from my Brothers at other unions and within our own, not at Grand Lodge, is that Mr. Mason has had a vendetta, working to undo my legacy and tenure, piece-by-piece. Cancelling my retirement party and forcing Mr. Boles and Ms. Lasky to clean up his mess is beyond egregious. I also understand that he has decided Cynthia Haley will plan it, whenever he deems it will be scheduled. To my understanding, that date is March 29, and I am unavailable. I, respectfully, request that my party happen on December 15, or another date that I am consulted on prior to signing a contract for the venue, and that Ms. Lasky under the direction of Secretary-Treasurer Boles be the team that plans this party, just as they worked with me on all other functions of this nature, in recent memory.
Lastly, I have heard from members, and I want to express my profound disappointment that my farewell to the members, who I served as President for 26 years and as an officer far more than that, was not run in the third quarter journal. Because Mr. Mason decided to run roughshod over a journal I had approved in September, and was held for some unknown reason, our members did not get the most valuable piece of information, the political endorsements. This election was important for working people. Our members, who have come to count on the political endorsements from our organization, were not in receipt of this information prior to the election, and many races are being decided by close margins. It is a disappointment, to say the least.
I am profoundly disappointed in what has transpired in the last 6 weeks, I hope something is done soon to "right the ship" for our members. BRS has a long, proud history as leaders in the rail industry, rail labor, and labor as a whole and that does not seem to be a primary focus and concern of this new President, and I request that you, Brothers, help him bring that into view. | {
"redpajama_set_name": "RedPajamaC4"
} | 1,688 |
{"url":"https:\/\/stacks.math.columbia.edu\/tag\/01MC","text":"Lemma 27.8.8. Let $S$ be a graded ring. The scheme $\\text{Proj}(S)$ is separated.\n\nProof. We have to show that the canonical morphism $\\text{Proj}(S) \\to \\mathop{\\mathrm{Spec}}(\\mathbf{Z})$ is separated. We will use Schemes, Lemma 26.21.7. Thus it suffices to show given any pair of standard opens $D_{+}(f)$ and $D_{+}(g)$ that $D_{+}(f) \\cap D_{+}(g) = D_{+}(fg)$ is affine (clear) and that the ring map\n\n$S_{(f)} \\otimes _{\\mathbf{Z}} S_{(g)} \\longrightarrow S_{(fg)}$\n\nis surjective. Any element $s$ in $S_{(fg)}$ is of the form $s = h\/(f^ ng^ m)$ with $h \\in S$ homogeneous of degree $n\\deg (f) + m\\deg (g)$. We may multiply $h$ by a suitable monomial $f^ ig^ j$ and assume that $n = n' \\deg (g)$, and $m = m' \\deg (f)$. Then we can rewrite $s$ as $s = h\/f^{(n' + m')\\deg (g)} \\cdot f^{m'\\deg (g)}\/g^{m'\\deg (f)}$. So $s$ is indeed in the image of the displayed arrow. $\\square$\n\n## Comments (0)\n\nThere are also:\n\n\u2022 10 comment(s) on Section 27.8: Proj of a graded ring\n\n## Post a comment\n\nYour email address will not be published. Required fields are marked.\n\nIn your comment you can use Markdown and LaTeX style mathematics (enclose it like $\\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).\n\nUnfortunately JavaScript is disabled in your browser, so the comment preview function will not work.\n\nAll contributions are licensed under the GNU Free Documentation License.\n\nIn order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 01MC. Beware of the difference between the letter\u00a0'O' and the digit\u00a0'0'.","date":"2021-07-25 10:37:31","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 2, \"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\": 2, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.8298658728599548, \"perplexity\": 431.48026570627985}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": false}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2021-31\/segments\/1627046151641.83\/warc\/CC-MAIN-20210725080735-20210725110735-00606.warc.gz\"}"} | null | null |
Case IH is a global leader in agricultural equipment and has powered agriculture for more than 175 years.
Kubota Tractor Corporation is a leading marketer and distributor of under 120-h.p. tractors in the United States, offering more than 80 tractor models.
JCB has developed and implemented the newest technology in their industry-changing machines which makes you more productive.
Great Plains is a leader in the agricultural implement business and has earned a reputation of satisfying customers needs by providing excellant products and service.
Geringhoff is more than just a brand of harvesting equipment, it's the name that represents the confidence of farming professionals. Efficient, reliable, responsive and legendary.
Kinze is a market leader in innovative, durable and efficient planters and grain carts.
Kuhn North America offers a full range of tillage, feeding and bedding equipment. With fifty-five years of building experience and innovation, Kuhn delivers results you can count on.
Unverferth is a leading manufacturer and marketer of tillage equipment, pull-type sprayers, hay, manure and grain handling equipment, and agricultural dual and specialty wheels.
Hesston Hay equipment. Setting new standards in performance and capacity.
J. & M. Manufacturing Co., Inc. has been a leading manufacturer of grain handling equipment for over 50 years and has consistently provided innovative, dependable, and high quality products.
MacDon Industries Ltd. is a manufacturer of harvesting equipment specializing in the production of pull-type and self-propelled windrowers and specialty and pick-up headers for combines for world markets.
Brillion Farm Equipment manufacturers a full line of tillage, seedbed preparation and seeding equipment made for today's top producers. Brillion equipment is designed to be durable, long-lasting and productive.
Meyer Manufacturing Corporation is a leading producer of farm equipment products including manure spreaders, front unload forage boxes, front/rear unload forage boxes, rear unload dump boxes, mineral feeders, bunk feeders, farm wagons and bale boxes.
Lanco Manufacturing is located in Bird-In-Hand, Lancaster County, PA – the heart of the Pennsylvania Dutch Country. At Lanco Manufacturing, "Building Excellence" is more than just a saying. Lanco is committed to providing equipment to make farming easier and more efficient.
Thunder Creek Equipment offers fuel and service trailers that efficiently deliver diesel fuel, diesel exhaust fluid, tools and parts directly to off-road equipment. It is built to travel safely at highway speeds, handle DEF without contamination and withstand rugged off-road conditions.
Crary manufactures harvesting attachments that are innovative and designed to improve harvesting! Farmers count on Crary for faster ground speeds, increased yield and improved efficiency.
Yetter is recognized as the leader in designing planter attachments, precision fertilizer placement tools, and bulk seed handling products for over 83 years.
Soil-Max products have revolutionized surface drainage and land improvement management around the globe. They produce consistent and quality results on all soil types, making them invaluable tools for any farmer.
CropCare manufactures innovative and dependable agricultural and vegetable equipment for farms, commercial applications, lawns, wildlife food plots, and more.
KMC Over 50 Years of Dependable Equipment for Progressive Farming.
Creek View Manufacturing Our commitment is to offer the highest quality, In-House Poultry Litter Management Equipment.
Parma Company is a leading manufacturer of specialized agricultural equipment for the farming and livestock industries, including Arena Groomers and more.
Walinga PTO Grain Vacs. The only system you'll ever need. Gives one-person total grain-handling capability! Do it all with the Walinga Agri-Vac. | {
"redpajama_set_name": "RedPajamaC4"
} | 1,825 |
\section{Introduction}
\label{sec:intro}
The `MHV rules' approach proposed in Ref.~\cite{CSW1}, has led to the
establishment of
a new and powerful framework for computing large classes of
previously unknown tree-level and one-loop
scattering amplitudes in gauge theories, in a compact form, and
without appealing to Feynman diagrams.
In this paper, we apply the MHV rules to study the singular limits of QCD
amplitudes when $n$ partons (gluons and massless quarks) are simultaneously
collinear. This continues the program started in our earlier
work~\cite{Birthwright:2005ak} where MHV rules were used to derive
multi-collinear limits of amplitudes involving only gluons. Understanding the
infrared singular behaviour of tree-level QCD amplitudes is a prerequisite for
computing infrared-finite cross sections at fixed order in perturbation
theory. In general, when one or more final state particles are either soft or
collinear, the amplitudes factorise. The first factor in this product is a
scattering amplitude that depends only on the remaining hard partons in the
process (including any hard partons constructed from an ensemble of unresolved
partons). The second factor is the splitting amplitude, it contains all of the
singularities due to the unresolved particles. One of the best known examples
of this type of factorisation is the limit of tree amplitudes when two
particles are collinear. This factorisation is universal and can be
generalised to more particles~\cite{Gehrmann-DeRidder:dblunres,Campbell:dblunres,Catani:NNLOcollfact,
Catani:IRtreeNNLO,delduca} and any number of
loops~\cite{Kosower:allorderfact}.
One of the main points of our approach \cite{Birthwright:2005ak}
is that, in order to derive all required splitting
functions we do not need to know the full amplitude. Out of the complete set of
MHV-diagrams contributing to the full amplitude, only a subset will contribute
in the multi-collinear limit. This subset includes only those MHV-diagrams
where {\em all} of the internal propagators go on-shell in the multi-collinear
limit.
Moreover, the functions multiplying these singular propagators
in the splitting amplitude
are constrained by the MHV rules to take a purely holomorphic form: they are
functions which depend only on the holomorphic spinor products,
$\langle i ~j \rangle$, of the right-handed (undotted) spinors
and not on the anti-holomorphic ones $[i~j]$.
This points towards a simple twistor space picture for the multi-collinear limits,
in terms of a degree-one curve in twistor space.
The MHV rules approach also enables us to calculate infinite
sequences of splitting amplitudes -- with fixed numbers of negative helicity partons
and arbitrary numbers of positive helicity ones, or vice versa.
The basic building blocks
of the MHV rules approach~\cite{CSW1}
are the colour-ordered
$n$-point vertices which are connected by scalar propagators. These MHV
vertices are off-shell continuations of the maximally helicity-violating (MHV)
$n$-gluon scattering amplitudes of Parke and Taylor~\cite{ParkeTaylor,BG}. They
contain precisely two negative helicity gluons.
Written in terms of spinor inner
products~\cite{SpinorHelicity}, they are composed entirely of the holomorphic
products $\spa{i}.{j}$,
rather than their anti-holomorphic partners $\spb{i}.{j}$,
\begin{equation}
A_n(1^+,\ldots,p^-,\ldots,q^-,\ldots,n^+) =
\frac{\spa{p}.{q}^4}{ \spa1.2 \spa2.3 \cdots \spa{n-1,}.{n} \spa{n}.{1} },
\label{MHV}
\end{equation}
where we introduce the common notation
$\spa{p_i}.{p_j}=\spa{i}.{j}$ and $\spb{p_i}.{p_j}=\spb{i}.{j}$.
By connecting MHV vertices, amplitudes involving more
negative helicity gluons can be built up.
The MHV rules for gluons~\cite{CSW1} have been extended to amplitudes with
fermions~\cite{GK}. New compact results for tree-level
gauge-theory results for non-MHV amplitudes involving arbitrary numbers of
gluons~\cite{Zhu,KosowerNMHV,BBK}, and fermions~\cite{GK,GGK,Wu2} have been
derived. They have been applied to processes involving external Higgs
bosons~\cite{DGK,BGK} and electroweak bosons~\cite{BFKM}.
MHV rules have also been shown to work at one-loop level for
supersymmetric theories~\cite{BST}.
Building on the earlier work of Bern, Dixon, Dunbar and Kosower~\cite{BDDK1,BDDK2},
there has been a remarkable progress in computing cut-constructible multi-leg
loop amplitudes in ${\cal N}=4$~\cite{CSW2,BST,Cachazo,
BCF1,BDDK7,BCF3,BDKNMHV}
and ${\cal N}=1$~\cite{QuigleyRozali,BBST1,BBDD,BBDP1,Britto:2005ha}
supersymmetric gauge theories. Encouraging
progress has also been made using MHV rules for non-supersymmetric loop
amplitudes~\cite{BBST2,BBDP2}.
Remarkably, the expressions obtained for the infrared singular parts of
${\cal N}=4$ one-loop amplitudes (which are known to be proportional to
tree-level results) were found to produce even more compact expressions for
gluonic tree amplitudes~\cite{BDKNMHV,RSV}. This observation led to the BCF
recursion relations~\cite{BCF4,BCFW} of Britto, Cachazo, Feng and Witten
as well as extremely
compact six-parton amplitudes~\cite{BCF4,LW1,LW2}. These tree-level BCF recursion relations
for massless particles have recently been generalised in two ways.
In Refs.~\cite{BDKrec,BDKrec2} a new version of recursion relations was adopted to calculate
all finite one-loop amplitudes in non-supersymmetric QCD.
At the same time, Ref.~\cite{Badger:2005zh} generalised BCF recursion relations to
include massive particles at tree level.
A comprehensive list of references
and a more detailed discussion of recent developments
can be found in the recent review \cite{Cachazo:2005ga}.
This progress has been stimulated by the original proposal of Witten
in \cite{Witten1} of a weak-to-weak coupling duality between a perturbative
${\cal N}=4$ gauge theory and a topological string theory in twistor space.
\medskip
The factorisation properties of amplitudes in the infrared
play several roles in developing higher order
perturbative predictions for observable quantities. First, a detailed
knowledge of the structure of unresolved emission enables phase space
integrations to be organised such that the infrared singularities due to soft
or collinear emission can be analytically
subtracted at NLO~\cite{Giele:1992vf,Frixione:1996ms,Catani:1997vz}
or at NNLO~\cite{Gehrmann-DeRidder:antenna}. Second, they
enable large logarithmic corrections to be identified and resummed.
Third, the collinear limit plays a crucial role in the unitarity-based method
for loop calculations~\cite{BDDK1,BDDK2,Bern:1996db,Bern:1996je}.
In general, to compute a cross section at N$^n$LO, one requires detailed
knowledge of the infrared factorisation functions describing the unresolved
configurations for $n$-particles at tree-level, $(n-1)$-particles at one-loop
etc. The universal behaviour in the double collinear limit is well known at
tree-level (see for example Refs.~\cite{Altarelli:1977zs,Bassetto:1983ik}),
one-loop~\cite{Bern:1995ix,BDDK1,Bern:split1gluon,
Kosower:split1,Bern:split1QCD,Catani:2000pi} and at
two-loops~\cite{Bern:2lsplit,Badger:2lsplit}. Similarly, the triple collinear
limit has been studied at
tree-level~\cite{Gehrmann-DeRidder:dblunres,Campbell:dblunres,Catani:NNLOcollfact,Catani:IRtreeNNLO} and,
in the case of distinct quarks, at one-loop~\cite{Catani:2003}. Finally, the
tree-level quadruple gluon collinear limit was derived in Ref.~\cite{delduca,Birthwright:2005ak}.
\medskip
Our paper is organised as follows. In Section~{\bf \ref{sec:col}}, we
briefly review the colour ordered formalism that underpins the MHV rules.
The relevant MHV vertices are given in Section~{\bf \ref{sec:mhv}}.
Section~{\bf \ref{sec:limit1}} describes the procedure for taking the
collinear limit while the analytic structure of the splitting functions is
discussed in Section~{\bf \ref{sec:limit2}}.
We write down general collinear
factorization formulae in Section~{\bf \ref{sec:general-results}}, which
are valid for specific numbers of negative helicity partons and an
arbitrary number of positive helicity partons.
These results involve quarks and gluons in the collinear set and are
complementary to the multi-gluon splitting functions derived in
Ref.~\cite{Birthwright:2005ak}.
Specific explicit results
for the collinear limits of up to three collinear partons
are given in Sec.~{\bf
\ref{sec:specific-results}}.
Our findings are
summarized in Sec.~{\bf \ref{sec:conclusion}}.
\section{Colour-ordered amplitudes}
\label{sec:col}
Tree-level multi-particle amplitudes can be decomposed into
colour-ordered partial amplitudes.
For gluons only, this decomposition is given by
\begin{equation}
{\cal A}_n(\{p_i,\lambda_i,a_i\}) =
i g^{n-2}
\sum_{\sigma \in S_n/Z_n} {\rm Tr}(T^{a_{\sigma(1)}}\cdots T^{a_{\sigma(n)}})\,
A_n(\sigma(1^{\lambda_1},\ldots,n^{\lambda_n}))\,.
\label{TreeColourDecomposition}
\end{equation}
Here $S_n/Z_n$ is the group of non-cyclic permutations on $n$
symbols, and $j^{\lambda_j}$ labels the momentum $p_j$ and helicity
$\lambda_j$ of the $j^{\rm th}$ gluon, which carries the adjoint
representation index $a_i$. The $T^{a_i}$ are fundamental
representation SU$(N_c)$ colour matrices, normalized so that
${\rm Tr}(T^a T^b) = \delta^{ab}$. The strong coupling constant is
$\alpha_s=g^2/(4\pi)$.
Note that the MHV rules method of Ref.~\cite{CSW1} is used to evaluate only the
purely kinematic amplitudes $A_n.$
Full amplitudes are then determined uniquely from the kinematic part $A_n$,
and the known expressions for the colour traces.
For processes involving a quark-antiquark pair and an arbitrary number of
gluons, the colour
decomposition is given by
\begin{eqnarray}
\lefteqn{
{\cal A}_n(\{p_i,\lambda_i,a_i\},\{p_j,\lambda_j,i_j\}) }\\
&&=
i g^{n-2}
\sum_{\sigma \in S_{n-2}} (T^{a_{\sigma(2)}}\cdots T^{a_{\sigma(n-1)}})_{i_1i_n}\,
A_n(1_q^{\lambda_1},\sigma(2^{\lambda_2},\ldots,{(n-1)}^{\lambda_{n-1}}),
n_{\bar q}^{\lambda_n})\,,\nonumber
\label{TreeColorDecomposition}
\end{eqnarray}
where $S_{n-2}$ is the set of permutations of $(n-2)$ gluons
and the fermions carry the fundamental colour labels $i_1$ and $i_n$.
By current conservation, the quark and antiquark helicities are related such
that $\lambda_1 = -\lambda_n \equiv \lambda$ where $\lambda = \pm \frac{1}{2}$.
When an additional photon with momentum $P_{\gamma}$ is emitted,
the amplitudes have the following form,
\begin{eqnarray}
\lefteqn{
{\cal A}_n(\{p_i,\lambda_i,a_i\},\{p_j,\lambda_j,i_j\},P_\gamma) }\\
&&=
i e g^{n-2}
\sum_{\sigma \in S_{n-2}} (T^{a_{\sigma(2)}}\cdots T^{a_{\sigma(n-1)}})_{i_1i_n}\,
\tilde{A}_n(1_q^{\lambda_1},\sigma(2^{\lambda_2},\ldots,{(n-1)}^{\lambda_{n-1}}),
n_{\bar q}^{\lambda_n};P_{\gamma})\,,\nonumber
\label{qqgammacoldecomp}
\end{eqnarray}
where $e$ is the electric charge of the quark.
When there are two quark-antiquark pairs the
tree-level amplitude can be decomposed into colour ordered
amplitudes as,
\begin{eqnarray}
&&\mathcal{A}_n(\{p_i,\lambda_i,a_i\},\{p_j,\lambda_j,i_j\})
= i g^{n-2} \sum_k^{n-4}\sum_{\sigma\in S_k}\sum_{\rho\in S_l} \bigg\{ \nonumber\\
&\phantom{-}&(T^{a_{\sigma(1)}}\cdots T^{a_{\sigma(k)}})_{i_1 i_n}
(T^{a_{\rho(1)}}\cdots T^{a_{\rho(l)}})_{i_{s+1} i_{s}} \nonumber\\
&&\times
A_n(1_q^{\lambda},\sigma(1),\ldots,\sigma(k)),s^{-\lambda^\prime}_{\bar{Q}};
(s+1)^{\lambda^\prime}_{Q},\rho(1),\ldots,\rho(l),n^{-\lambda}_{\bar{q}}) \nonumber\\
&-&\frac{1}{N}(T^{a_{\sigma(1)}}\cdots T^{a_{\sigma(k)}})_{i_1 i_s} (T^{a_{\rho(1)}}\cdots T^{a_{\rho(l)}})_{
i_{s+1}i_n} \nonumber\\
&&\times\tilde{A}_n(1_q^{\lambda},\sigma(1),\ldots,\sigma(k),s^{-\lambda}_{\bar{q}};
(s+1)^{\lambda^\prime}_{Q},\rho(1),\ldots,\rho(l),n_{\bar Q}^{-\lambda^\prime})\bigg\}
\label{eq:qqQQcoldecomp}
\end{eqnarray}
where $S_k$ and $S_l$ are permutation groups such that $k+l=n-4$ and
represent the possible ways of distributing the gluons in a colour ordered way between
the quarks. For $i=j=0$, $(T^{a_i}\ldots T^{a_j})_{kl}$ reduces to $\delta_{kl}$.
We see that the two amplitudes $A_n$ and $\tilde{A}_n$ correspond to
different ways of connecting the fundamental colour charges.
For the $A$ amplitudes, there is a colour line connecting $q$ and $\bar Q$ and a second
line connecting $Q$ and $\bar q$, while for
the QED-like $\tilde A$ amplitudes the colour lines connect $q$ to $\bar q$ and $Q$ to
$\bar Q$. Any number of gluons may be radiated from each colour
line.
As before, by current conservation, the quark and antiquark helicities are related such
that $\lambda_q = -\lambda_{\bar q} \equiv \lambda$
and $\lambda_Q = -\lambda_{\bar Q} \equiv \lambda^\prime$
where $\lambda, ~\lambda^\prime = \pm \frac{1}{2}$.
\section{MHV amplitudes}
\label{sec:mhv}
The colour ordered $n$-gluon MHV amplitude is given by
\begin{equation}
\label{eq:gluon}
A_n(1^+,\ldots,m_1^-,\ldots,m_2^-,\ldots,n^+) = \frac{\aab{m_1}{m_2}^4}{\prod_{i=1}^{n}\aab{i}{i+1}},
\end{equation}
while the two-quark multi-gluon MHV amplitudes are,
\begin{eqnarray}
A_n(1^{\lambda}_q,\ldots,m^-,\ldots,n^{-\lambda}_{\bar{q}})
&=& \frac{\spa{m}.{1}^{2-2\lambda}\spa{m}.{n}^{2+2\lambda}}{\prod_{l=1}^n\spa{l}.{l+1}}
\label{eq:2qMHV}.
\end{eqnarray}
Here the helicity of the quark is denoted by $\lambda = \pm \frac{1}{2}$ while $\ldots$ denotes an arbitrary number
of positive helicity gluons.
Amplitudes for a quark-antiquark pair, many gluons and a photon are given by,
\begin{eqnarray}
\label{eq:2qMHVphoton}
\tilde{A}( 1_q^\lambda,\ldots,n_{\bar q}^{-\lambda};P_{\gamma}^-)
&=& \frac{\spa{P}.{1}^{2-2\lambda} \spa{P}.{n}^{2+2\lambda}}
{\spa{P}.{1}\spa{1}.{2}\cdots \spa{n}.{P}}, \\
\tilde{A}(1_q^\lambda,\ldots,m^-,\ldots,n_{\bar q}^{-\lambda};P_{\gamma}^+) &=&
\frac{\spa{m}.{1}^{2-2\lambda} \spa{m}.{n}^{2+2\lambda}}
{\spa{P}.{1}\spa{1}.{2}\cdots \spa{n}.{P}}.
\end{eqnarray}
In the four-quark case, there are four MHV amplitudes where two of the fermions have
negative helicity and two have positive helicity for each colour structure. For each
helicity configuration we can write,
\begin{eqnarray}
\label{eq:4qmhv1}
A_n(1^+_q,\ldots,s^-_{\bar{Q}},(s+1)^+_{Q},\ldots,n^-_{\bar{q}}) &=& \frac{\spa{1}.{s}\spa{s}.{n}^2\spa{n}.{s+1}}{\prod_{l=1}^n \spa{l}.{l+1}} ,\\
\label{eq:4qmhv2}
A_n(1^+_q,\ldots,s^+_{\bar{Q}},(s+1)^-_{Q},\ldots,n^-_{\bar{q}}) &=& \frac{\spa{1}.{s}\spa{n}.{s+1}^3}{\prod_{l=1}^n \spa{l}.{l+1}} ,\\
\label{eq:4qmhv3}
A_n(1^-_q,\ldots,s^+_{\bar{Q}},(s+1)^-_{Q},\ldots,n^+_{\bar{q}}) &=&
\frac{\spa{1}.{s}\spa{1}.{s+1}^2\spa{n}.{s+1}}{\prod_{l=1}^n \spa{l}.{l+1}} ,\\
\label{eq:4qmhv4}
A_n(1^-_q,\ldots,s^-_{\bar{Q}},(s+1)^+_{Q},\ldots,n^+_{\bar{q}}) &=& \frac{\spa{1}.{s}^3\spa{n}.{s+1}}{\prod_{l=1}^n \spa{l}.{l+1}},
\end{eqnarray}
with the other colour ordering given by,
\begin{eqnarray}
\label{eq:4qmhv5}
\tilde{A}_n(1^+_q,\ldots,s^-_{\bar{q}},(s+1)^+_{Q},\ldots,n^-_{\bar{Q}}) &=& \frac{\spa{1}.{n}\spa{n}.{s}^2\spa{s}.{s+1}}{\prod_{l=1}^n \spa{l}.{l+1}} ,\\
\label{eq:4qmhv6}
\tilde{A}_n(1^+_q,\ldots,s^-_{\bar{q}},(s+1)^-_{Q},\ldots,n^+_{\bar{Q}}) &=& \frac{\spa{1}.{n}\spa{s}.{s+1}^3}{\prod_{l=1}^n \spa{l}.{l+1}} ,\\
\label{eq:4qmhv7}
\tilde{A}_n(1^-_q,\ldots,s^+_{\bar{q}},(s+1)^-_{Q},\ldots,n^+_{\bar{Q}})
&=& \frac{\spa{1}.{n}\spa{1}.{s+1}^2\spa{s}.{s+1}}{\prod_{l=1}^n \spa{l}.{l+1}} ,\\
\label{eq:4qmhv8}
\tilde{A}_n(1^-_q,\ldots,s^+_{\bar{q}},(s+1)^+_{Q},\ldots,n^-_{\bar{Q}}) &=& \frac{\spa{1}.{n}^3\spa{s}.{s+1}}{\prod_{l=1}^n \spa{l}.{l+1}}.
\label{eq:4qMHVamps2}
\end{eqnarray}
The $\overline{\text{MHV}}$~amplitudes are related by parity and can be obtained by conjugating
the MHV expressions,
\begin{equation}
A_n(1^{\lambda_1},\ldots,n^{\lambda_n}) =
(-1)^n\left(A_n(1^{-\lambda_1},\ldots,n^{-\lambda_n})\right)^*,
\end{equation}
and similarly for the $\tilde{A}$ amplitudes.
\section{Collinear limits}
\label{sec:limit1}
To find the splitting functions we work with the colour stripped amplitudes.
For these colour ordered amplitudes, it is known that when the
collinear particles are not adjacent there is no collinear
divergence~\cite{delduca}. Therefore, without loss of generality, we can take
particles $1 \dots n$ collinear.
The multiple collinear limit is approached when the momenta $p_1,
\dots, p_n$ become parallel. This implies that all the
particle subenergies $s_{ij}=(p_i+p_j)^2$, with $i,j=1,\dots,n$, are
simultaneously small. We thus introduce a pair of
light-like momenta $P^\nu$
and $\xi^\nu$ ($P^2=0, \xi^2=0$), and we write
\begin{equation}
(p_1 + \dots + p_n)^\nu = P^\nu
+ \frac{s_{1,n} \; \xi^\nu}{2 \, \xi \cdot P} \;, \quad
s_{i,j} = (p_i + \dots + p_j)^2 \;,
\end{equation}
where $s_{1,n}$ is the total invariant mass of the system of collinear
partons. In the collinear limit, the vector $P^\nu$
denotes the collinear direction, and the individual collinear momenta are
$p_i^\nu \to z_i P^\nu$. Here the longitudinal-momentum
fractions $z_i$ are given by
\begin{equation}
z_i = \frac{\xi \cdot p_i}{\xi \cdot P}
\end{equation}
and fulfil the constraint $\sum_{i=1}^m z_i =1$.
To be definite, in the rest of the paper we work
in the time-like region so that
($s_{ij} > 0, \; 1>z_i > 0$).
\begin{figure}[htbp]
\centering
\psfrag{-P-l}{$-P^{-\lambda}$}
\psfrag{Pl}{$P^{\lambda}$}
\psfrag{pn+1}{$(n+1)$}
\psfrag{pn}{$n$}
\psfrag{pN}{$N$}
\psfrag{p1}{$1$}
\includegraphics[width=12cm]{limit.eps}
\caption{Factorisation of an $N$-point colour ordered amplitude with gluons $p_1,\ldots,p_n$ collinear
into splitting function for $P \to 1, \ldots, n$
multiplied by an $(N-n+1)$-point amplitude.}
\label{fig:limit}
\end{figure}
As illustrated in Fig.~\ref{fig:limit},
in the multi-collinear limit an $N$-particle colour ordered tree amplitude
factorises
and can be written as
\begin{eqnarray}
\label{eq:factorise}
A_N(1^{\lambda_1},\ldots,N^{\lambda_N}) &\to&
\mathrm{split}({{1}^{\lambda_{1}},\ldots,n^{\lambda_n} \to P^\lambda})
\times
A_{N-n+1}((n+1)^{\lambda_{n+1}},\ldots ,N^{\lambda_N},P^\lambda).\nonumber \\
\end{eqnarray}
This labelling of the splitting amplitude
$\mathrm{split}({1^{\lambda_1},\ldots,n^{\lambda_n}\to P^\lambda})$ differs from the
usual definition because we use the momentum and helicity that
participates in the resultant amplitude $P^\lambda$ rather than $-P^{-\lambda}$.
With this choice, it is easier to see how the helicity is conserved
in the splitting, i.e. helicity $\lambda^1,\ldots,\lambda^n$ is replaced by
$\lambda$. Since eq.~(\ref{eq:factorise}) applies for all $N$, we can
use it to derive the splitting amplitude by systematically choosing $N = 3+n$.
In this case, we always factorise onto a four-point amplitude.
\section{Analytic structure of splitting amplitudes}
\label{sec:limit2}
The MHV rules of Ref.~\cite{CSW1}
were developed for calculating
purely gluonic amplitudes at tree level and extended to
amplitudes involving fermions in Ref.~\cite{GK}. In this approach
all non-MHV $N$-particle
amplitudes (including $\overline{\rm MHV}$) are expressed
as sums of tree diagrams in an effective scalar perturbation theory.
The vertices in this theory are the MHV amplitudes of Eq.~(\ref{MHV})
continued off-shell and connected by scalar
propagators $1/q^2$.
Following~\cite{Birthwright:2005ak}, we classify
collinear limits according to the difference between the number of negative
helicity particles before taking the collinear limit, and the number after,
$\Delta M$.
Splitting amplitudes are calculated using the factorisation formula
eq.~(\ref{eq:factorise}). To facilitate the calculation, it makes sense to factorise
onto hard amplitudes with the simplest analytic structure.
Hence, in the MHV-rules formalism
we will always factorise
onto MHV amplitudes which are listed in section \ref{sec:mhv}.
In this case we find that $\Delta M$ of the splitting amplitude
satisfies the relation,
\begin{equation}
\Delta M + 2\, =\, N_{-}
\end{equation}
where $2$ is the number of negative helicities in the hard MHV amplitude, and
$N_{-}$ is the total number of negative helicities in the full amplitude.
$\Delta M$ determines the
order of MHV diagram~\cite{CSW1} for the full amplitude $A_N$
\begin{eqnarray}
\Delta M=0 \quad \Rightarrow \quad
&&1^+,2^+,3^+, \ldots ,n^+ \rightarrow P^+ \nonumber \qquad A_N={\rm MHV}\\
&&1^-,2^+,3^+, \ldots ,n^+ \rightarrow P^-\nonumber\\
& &\nonumber\\
\Delta M=1 \quad \Rightarrow \quad
&&1^-,2^+,3^+, \ldots ,n^+ \rightarrow P^+ \nonumber \qquad A_N={\rm NMHV}\\
&&1^-,2^-,3^+, \ldots ,n^+ \rightarrow P^- \label{mmp} \nonumber\\
& &\nonumber\\
\Delta M=2 \quad \Rightarrow \quad
&&1^-,2^-,3^+, \ldots ,n^+ \rightarrow P^+\nonumber \qquad A_N={\rm NNMHV}\\
&&1^-,2^-,3^-, \ldots ,n^+ \rightarrow P^-\nonumber\\
\end{eqnarray}
and so on for all $\Delta M>2$ cases.
If we choose to use $\overline{\text{MHV}}$\ rules, we extract the splitting function
by factorising onto $\overline{\text{MHV}}$\ amplitudes.
Splitting amplitudes are then classified by
the difference in the number of positive helicity particles,
$\Delta P$, and similar observations apply.
In general, any splitting amplitude can be obtained from either MHV or
$\overline{\text{MHV}}$\ rules.
A simple power counting argument~\cite{Birthwright:2005ak} gives
\begin{equation}
\label{powerct}
\mathrm{split} \, \propto \,
\frac{1}{\spb{\ }.{\ }^{\Delta M} \spa{\ }.{\ }^{\Delta P}} \ .
\end{equation}
For an MHV-rules diagram to contribute to $ \Delta M \neq 0$ collinear limits,
it
must contain anti-holomorphic spinor products $\spb{i}.{j}$ of
collinear momenta. However, because on-shell MHV vertices are
entirely holomorphic,
within the MHV rules there are only two potential sources of the
anti-holomorphic spinor products. One source is scalar propagators
$1/s_{ij}=1/\spa{i}.{j} \spb{j}.{i}$ which connect MHV vertices. The
second source is the off-shell continuation of the corresponding connected
legs in the MHV vertices. Each off-shell continued leg of momentum $P$ gives
rise to a factor $\langle i P \rangle \propto \langle i | P | \eta]$ which
amounts to anti-holomorphic factors of the form $[j \eta]$.
When the reference
spinors $\eta_{\dot\alpha}$ are kept general, the
$\eta$-dependence must cancel and therefore
the off-shell continuation cannot give rise
to an overall factor of $\spb{i}.{j}$.
This implies that within the MHV rules, the anti-holomorphic
spinor products
in \eqref{powerct} arise solely from the internal propagators.
Since $\Delta M= v_{MHV}-1$,\footnote{In principle,
$\Delta M= v_{MHV}-v_{MHV}^\prime$ where $v_{MHV}^\prime$ is the number of
MHV vertices remaining in the factored amplitude. The splitting function is
independent of $v_{MHV}^\prime$, and because we systematically choose
to factor directly onto a single MHV vertex, we set $v_{MHV}^\prime = 1$.}
where $v_{MHV}$ is the number of MHV vertices
in the diagram, the total number of internal propagators is $\Delta M$,
in agreement with \eqref{powerct}.
Similarly, in the $\overline{\text{MHV}}$\ approach, the holomorphic products would arise solely
from internal propagators whose total number in $\overline{\text{MHV}}$\ diagrams is
$\Delta P$.
More precisely, it follows that all
splitting amplitudes can be recast as
\begin{eqnarray}
\mathrm{split} & = & \sum \, \frac{1}{\prod_{{i,j}=1}^{\Delta M} s_{i,j}} \,\, f(\spa{ }.{ })
\label{54}\\
& = & \sum \, \frac{1}{\prod_{{i,j}=1}^{\Delta P} s_{i,j}} \,\, \tilde{f}(\spb{}.{})
\label{55}
\end{eqnarray}
where the first expression follows from the MHV rules representation,
and the second expression -- from the $\overline{\text{MHV}}$\ formalism.
Here the summations are over all inequivalent choices of $\Delta M$
($\Delta P $) products of vanishing kinematic
invariants $s_{i,j}$ which corresponds to
different MHV ($\overline{\text{MHV}}$) rules diagrams.
The
coefficient functions $f$ depend only on holomorphic spinor products, while
the $\overline{\text{MHV}}$\ coefficients
$\tilde{f}$ are purely anti-holomorphic. Moreover, $f$ and $\tilde f$ have
dimensions,
\begin{equation}
f \, \propto\, \frac{1}{\spa{\ }.{\ }^{\Delta P -\Delta M}} \ , \qquad
\tilde{f} \, \propto\, \frac{1}{\spb{\ }.{\ }^{\Delta M -\Delta P}} \ .
\end{equation}
The fact that $f$ ($\tilde{f}$) is purely (anti)-holomorphic suggests a simple
twistor-space interpretation. All splitting functions can be represented as
sums over the corresponding poles in $s$ with the coefficients being supported
on a single degree-one curve in (anti)-twistor space. This pure (anti)-holomorphic
representation of multi-collinear limits is specific to the MHV ($\overline{\text{MHV}}$) formalism
and is lost in the usual Feynman-diagram-type approaches as in Ref.~\cite{delduca},
or in the BCF recursive
approach, as shown in \cite{Birthwright:2005ak}.
We further note that MHV rules for collinear limits are substantially
simpler than the rules
for the full amplitudes. Collinear splitting functions follow from a
{\it subset} of the MHV rules diagrams
\cite{Birthwright:2005ak}. The subset is determined by requiring that all internal
propagators are on-shell in the multi-collinear limit.\footnote{This is dictated
by eq.~\eqref{54} in the MHV formalism (or eq.~\eqref{55} for $\overline{\text{MHV}}$\ rules).}
This is a powerful
constraint on the types of the contributing diagrams and it simplifies
taking the collinear limit dramatically.
As mentioned earlier, each splitting amplitude can be calculated in both
the MHV and
in the $\overline{\text{MHV}}$\ approaches.
In practice, eqs.~\eqref{powerct}-\eqref{55} imply that
the MHV approach is simpler if $\Delta M < \Delta P$,
while the $\overline{\text{MHV}}$\ approach is more
compact in the opposite case, $\Delta P < \Delta M$.
In most of what follows we will concentrate on the splitting amplitudes
with $\Delta M \le \Delta P$ and will follow the MHV rules. The remaining
amplitudes with $\Delta P < \Delta M$ are obtained from
these by complex conjugation.
\subsection{ An example}
When $\Delta M = \Delta P$ both MHV and $\overline{\text{MHV}}$\ rules are expected
to yield results of similar complexity.
As an example, let us consider a triple collinear splitting with $\Delta M = \Delta P = 1$.
In full generality, the MHV ($\overline{\text{MHV}}$) rules approach should generate a maximum of three terms
corresponding to simple poles in $s_{1,2}$, $s_{2,3}$ and $s_{1,3}\equiv (p_1+p_2+p_3)^2$.
For the specific splitting
$1_q^-,2_{\bar Q}^+,3_Q^-\to P_q^-$, the MHV rules approach yields,
\begin{eqnarray}
\label{eq:qpQpQm1}
\mathrm{split}(1_q^+,2_{\bar Q}^+,3_Q^- \to P_q^+)&=&
-{\frac {\spa{2}.{3}z_{{2}}}{s_{{2,3}} \left( \spa{1}.{2}\sqrt {z_{{2}
}}+\spa{1}.{3}\sqrt {z_{{3}}} \right) \left( z_{{2}}+z_{{3}} \right)
}}\nonumber \\&&+{\frac { \left( \spa{1}.{3}\sqrt {z_{{1}}}+\spa{2}.{3}\sqrt {z_{{2}
}} \right) ^{2}}{s_{{1,3}} \left( \spa{1}.{2}\sqrt {z_{{2}}}+\spa{1}.{
3}\sqrt {z_{{3}}} \right) \spa{2}.{3}}}\ ,
\end{eqnarray}
while the $\overline{\text{MHV}}$\ rules approach finds,
\begin{eqnarray}
\label{eq:qpQpQm2}
\mathrm{split}(1_q^+,2_{\bar Q}^+,3_Q^- \to P_q^+)&=&
{\frac {z_{{1}}z_{{3}}\spb{2}.{3}}{s_{{2,3}}
\left( \spb{1}.{2}\sqrt {z_{{2}}}+\spb{1}.{3}\sqrt {z_{{3}}} \right)
\left( z_{{2}}+z_{{3}} \right)
}}\nonumber \\&&
-{\frac { \spb{1}.{2}^2}{s_{{1,3}} \left( \spb{1}.{2}\sqrt {z_{{2}}}+\spb{1}.{
3}\sqrt {z_{{3}}} \right) \spb{2}.{3}}}\ .
\end{eqnarray}
As expected, the $s_{12}$ pole is absent because there is no $q \bar Q$ collinear limit.
By taking the limit of a Feynman diagram calculation, Ref.~\cite{delduca}
finds,
\begin{eqnarray}
\label{eq:qpQpQm3}
\mathrm{split}(1_q^+,2_{\bar Q}^+,3_Q^- \to P_q^+)&=&
\frac {\sqrt{z_1z_2z_3}}{s_{2,3}(z_2+z_3)}
+\frac { \spb{1}.{2}
\left( \spa{1}.{3}\sqrt {z_1}+\spa{2}.{3}\sqrt {z_2} \right)}{s_{1,3}s_{2,3}}\ .
\end{eqnarray}
Results \eqref{eq:qpQpQm1}, \eqref{eq:qpQpQm2} and \eqref{eq:qpQpQm3}
are for the same amplitude and
all three expressions agree numerically. But the analytic form of these specific
representations is different.
In agreement with eqs.~\eqref{54}-\eqref{55},
the functions accompanying the $1/s$ poles, are holomorphic in the
MHV result \eqref{eq:qpQpQm1},
are anti-holomorphic in the $\overline{\text{MHV}}$\ expression \eqref{eq:qpQpQm2},
while the Feynman diagram result \eqref{eq:qpQpQm3} contains a
mixture of holomorphic and anti-holomorphic terms.
(In this case, it happens to give a more compact result.)
In general, the limit of an amplitude computed using the BCF recursion relations
will also provide a mixed holomorphic/anti-holomorphic splitting function
(as discussed in sections {\bf 4.2.2} and {\bf 4.2.3} of Ref.~\cite{Birthwright:2005ak}).
In this specific case, taking the collinear limit of the compact
expression for the appropriate six-parton amplitude given in Ref.~\cite{LW2} exactly
reproduces the MHV result of eq.~\eqref{eq:qpQpQm2}.
\section{General results}
\label{sec:general-results}
In this section we give the results for the multiple collinear limit
of quarks and gluons.
We categorise the results according to the number of quarks
involved in the limit.
In each case, we give the general results for
collinear limits with $\Delta M =\, 0\, , \, 1$ and involving
an arbitrary number of positive helicity particles.
Limits of the type
$\mathrm{split}({1^+,\ldots,n^+\to P^+})$ and
$\mathrm{split}({1^-,2^+,\ldots,n^+ \to P^-})$ can contribute to the $\Delta M=0$,
and these collinear splitting functions
are straightforward to derive directly from the simple MHV vertex.
For the remaining splitting functions, it is useful
to introduce the more compact notation
\begin{eqnarray}
\mathrm{split}(1^+,\ldots , m_1^-, \ldots, m_2^-, \ldots,m_r^-,\ldots ,
n^+ \to P^\pm)={\mathrm{Split}}_{\pm}(m_1,\ldots,m_r)\ .
\end{eqnarray}
\begin{figure}[t!]
\centering
\psfrag{i+1+}{$~$}
\psfrag{a}{$(a)$}
\psfrag{b}{$(b)$}
\psfrag{j+1+}{$~$}
\psfrag{j+}{$j+$}
\psfrag{i+}{{\small $i+$}}
\psfrag{m1-}{$m_1^-$}
\psfrag{m2-}{$m_2^-$}
\begin{center}
\includegraphics[width=12cm]{m1m2.eps}
\end{center}
\caption{MHV topologies contributing to (a) ${\mathrm{Split}}_+(m_1)$ and (b) ${\mathrm{Split}}_-(m_1,m_2)$. Negative helicity particles are indicated
by solid lines, while
arbitrary numbers of positive helicity particles emitted from each vertex are shown as dotted arcs.
All particles that are not in the collinear set must be emitted from the left-hand vertex.}
\label{fig:Split}
\end{figure}
For $\Delta M = 1$, there are two possible types of splitting function,
${\mathrm{Split}}_{+}(m_1)$ and ${\mathrm{Split}}_{-}(m_1,\ldots,m_r)$.
The possible MHV topologies contributing to these splitting functions are illustrated in
Fig.~\ref{fig:Split}. Only negative helicity particles are
shown. In the collinear limit, the propagator goes on-shell.
Any MHV diagram with a hard particles emitted from both vertices produces an off-shell
propagator.
This means that {\em only} particles from the collinear set are allowed to
couple to the right-hand vertex. All hard partons couple to the left-hand vertex.
Throughout we adopt the notation of Ref.~\cite{Birthwright:2005ak}.
In order that the limits can be read directly from the MHV diagrams,
we make the following substitutions.
If $a$ is a
particle from the collinear set,
$b$ is a particle which is not in the collinear set, and
$q$ is the sum of the collinear momenta from $i+1$ to $j$, then
\begin{eqnarray}
\spa{a}.{q}&\rightarrow&\, \spb{P}.{\eta}\sum_{l=i+1}^{j}\spa{a}.{l}\sqrt{z_l} \, \equiv\,
\spb{P}.{\eta}\Del{i}.{j}.{a}, \label{aqqq}\\
\spa{b}.{q}&\rightarrow&\, \spb{P}.{\eta}\spa{b}.{P}\sum_{l=i+1}^{j} z_l ,\label{Xq}\\
\spa{b}.{a}&\rightarrow&\, \spa{b}.{P} \sqrt{z_a} \ .\label{Xa}
\end{eqnarray}
The $\Delta$ is defined as
\begin{equation}
\label{eq:7}
\Del{i}.{j}.{a} = \sum_{l=i+1}^{j}\spa{a}.{l}\sqrt{z_l} \ ,
\end{equation}
noting that
the
boundary terms involving either $\spa{0}.{1}$ or $\spa{n}.{n+1}$,
are given by,
\begin{eqnarray}
\frac{\spa{n}.{n+1}}{\Del{i}.{n}.{n+1}} &\rightarrow&\,
-\frac{\sqrt{z_n}}{\sum_{l=i+1}^{n}z_l}, \label{jn}\\
\frac{\spa{0}.{1}}{\Del{0}.{j}.{0}} &\rightarrow&\,
\frac{\sqrt{z_1}}{\sum_{k=1}^{j}z_k}.\label{inp3}
\end{eqnarray}
We also introduce
\begin{eqnarray}
\label{D}
D(i,j,q_{i+1,j})&=&\frac{q_{i+1,j}^2}{\spa{i,}.{i+1} \spa{j,}.{j+1}}
\Del{i}.{j}.{i}\Del{i}.{j}.{i+1}\Del{i}.{j}.{j}\Del{i}.{j}.{j+1}\ .\nonumber \\
\end{eqnarray}
\subsection{One quark in the collinear set: $q(ng) \to q$}
\subsubsection{$\Delta M = 0$}
This is the simplest case which is read directly off the single MHV vertex.
For positive helicity quarks, we
use the two-quark MHV amplitude of Eq.~(\ref{eq:2qMHV}) and find,
\begin{eqnarray}
\mathrm{split}(1_q^+ ,\ldots, n^+ \to P_q^+)= \frac{\sqrt{z_1}}{\sqrt{z_1 z_n} \prod_{l=1}^{n-1} \spa{l,}.{l+1} } \ .
\end{eqnarray}
For negative helicity quarks,
\begin{eqnarray}
\mathrm{split}(1_q^- ,\ldots, n^+ \to P_q^-)= \frac{\sqrt{z_1}^3}{\sqrt{z_1 z_n} \prod_{l=1}^{n-1} \spa{l,}.{l+1} } \ .
\end{eqnarray}
Note that helicity conservation ensures that the helicity of $P$ is the same as that of $q$.
It is often convenient to combine results for quarks of helicity $\lambda = \pm
\frac{1}{2}$ such that,
\begin{eqnarray}
\mathrm{split}(1_q^\lambda ,\ldots, n^+ \to P_q^\lambda)=
\frac{\sqrt{z_1}^{2-2\lambda}}{\sqrt{z_1 z_n} \prod_{l=1}^{n-1} \spa{l,}.{l+1} } \ .
\end{eqnarray}
Using parity we find,
\begin{eqnarray}
\mathrm{split}(1_q^+ ,\ldots, n^- \to P_q^+)= \frac{(-1)^{n-1}\sqrt{z_1}^{2+2\lambda}}{\sqrt{z_1 z_n} \prod_{l=1}^{n-1} \spb{l,}.{l+1} } \ .
\end{eqnarray}
The amplitudes where an antiquark is collinear with several gluons are obtained by charge conjugation.
\subsubsection{$\Delta M = 1$}
Because of helicity conservation, $\Delta M=1$ implies that a single gluon has negative helicity.
When the quark has positive helicity, then the
MHV diagrams contributing in the collinear limit correspond to topology (a) of Fig.~\ref{fig:Split}.
There are two types of diagram -- one class where the quark is emitted from the right-hand vertex
(and the propagating particle
is a quark) and one class mediated by gluon exchange where the quark is emitted from the left-hand vertex.
We find,
\begin{eqnarray}
\label{eq:qgg1}
\lefteqn{ \mathrm{split}(1_q^+,\ldots,m^-,\ldots,n^+\to P_q^+) =
\frac{1}{\sqrt{z_1 z_n} \prod_{l=1}^{n-1} \spa{l,}.{l+1} }}\nonumber\\
&&\hspace{2cm}\times\bigg[
-\sum_{j=m}^{n}\frac{\Delta^3(0,j;m)
\aab{1}{m}}{D(0,j,q_{1,j})}\left (\sum_{k=1}^{j}z_k\right ) + \sum_{i=1}^{m-1} \sum_{j=m}^{n}
\frac{\Delta^4(i,j;m)}{D(i,j,q_{i+1,j})}\sqrt{z_1} \bigg].\nonumber \\
\end{eqnarray}
In the same manner,
for negative helicity quarks, the allowed MHV diagrams correspond to the first and second
topologies shown in Fig.~\ref{fig:Split}(b),
\begin{eqnarray}
\label{eq:qgg2}
\lefteqn{ \mathrm{split}(1_q^-,\ldots,m^-,\ldots,n^+\to P_q^-)=
\frac{1}{\sqrt{z_1 z_n} \prod_{l=1}^{n-1} \spa{l,}.{l+1} }}\nonumber \\
&&\hspace{2cm}\times\bigg[
-\sum_{j=m}^{n}\frac{\Delta(0,j;m)
\aab{1}{m}^3}{D(0,j,q_{1,j})}\left (\sum_{k=1}^{j}z_k \right)^3+ \sum_{i=1}^{m-1} \sum_{j=m}^{n}
\frac{\Delta^4(i,j;m)}{D(i,j,q_{i+1,j})}\sqrt{z_1}^3 \bigg].\nonumber \\
\end{eqnarray}
\subsection{Two quarks in the collinear set: $(ng)\bar qq \to g$ }
In this collinear limit, the $\bar q q$ pair is in the adjoint representation
and effectively acts as a gluon.
\subsubsection{$\Delta M = 0$}
This is the simplest case which is read directly off the single MHV vertex.
Unlike the previous case, here we start with a two-quark MHV amplitude and factorise onto
a gluonic MHV amplitude. Alternatively, we could start with a four-quark amplitude and
factorise onto a two-quark amplitude.
For quarks with helicity $\lambda = \pm\frac{1}{2}$, we find,
\begin{eqnarray}
\mathrm{split}(1^+ ,\ldots,s^{-\lambda}_{\bar q},(s+1)^\lambda_q,\ldots, n^+ \to P^-)=
\frac{\sqrt{z_s}^{2+2\lambda}\sqrt{z_{s+1}}^{2-2\lambda}}{\sqrt{z_1 z_n} \prod_{l=1}^{n-1} \spa{l,}.{l+1} } \ .
\end{eqnarray}
\subsubsection{$\Delta M = 1$}
For amplitudes of the ${\mathrm{Split}}_+(m_1)$-type, we find
\begin{eqnarray}
\label{eq:gqq}
\lefteqn{
\mathrm{split}(1^+ ,\ldots,s^{-\lambda}_{\bar q},(s+1)^\lambda_q,\ldots, n^+ \to P^+)= }\nonumber \\
&&
\frac{1}{\sqrt{z_1 z_n} \prod_{l=1}^{n-1} \spa{l,}.{l+1} }
\sum_{i=0}^{s-1}\sum_{j=s+1}^{n}
\frac{\Delta^{2+2\lambda}(i,j;s) \Delta^{2-2\lambda}(i,j;s+1)}{D(i,j,q_{i+1,j})} .
\end{eqnarray}
There are four diagrams contributing to splitting functions of
${\mathrm{Split}}_-(m_1,m_2)$ type\footnote{Diagrams where both the negative helicity fermion and gluon couple to the
right-hand vertex in Fig.~\ref{fig:Split}(b) can be mediated by either fermion or gluon exchange.},
\begin{eqnarray}
\lefteqn{
\mathrm{split}(1^+,\ldots, s^{-\lambda}_{\bar q},(s+1)^\lambda_q,\ldots, m^-,\ldots,n^+ \to P^-)=
\frac{1}{\sqrt{z_1 z_n} \prod_{l=1}^{n-1} \spa{l,}.{l+1} }}\nonumber \\
&&\hspace{5cm}\times\bigg[
\sum_{i=0}^{s-1}\sum_{j=m}^{n}
\frac{\aab{s}{m}^{2+2\lambda}\aab{s+1}{m}^{2-2\lambda}}{D(i,j,q_{i+1,j})}
\left(\sum_{k=i+1}^{j}z_k\right)^4 \nonumber\\
&&\hspace{5cm}+
\sum_{i=0}^{s-1}\sum_{j=s+1}^{m-1}\frac{\Delta^{2+2\lambda}(i,j;s)
\Delta^{2-2\lambda}(i,j;s+1)}{D(i,j,q_{i+1,j})} z_m^2 \nonumber\\
&&\hspace{5cm}- \sum_{j=m}^{n}
\frac{\Delta^{2+2\lambda}(s,j;m)\aab{s+1}{m}^{2-2\lambda}}
{D(s,j,q_{s+1,j})}\sqrt{z_s}^{2+\lambda}
\left(\sum_{k=s+1}^{j} z_k \right)^{2-2\lambda} \nonumber\\
&&\hspace{5cm}+ \sum_{i=s+1}^{m-1}\sum_{j=m}^{n} \frac{\Delta^4(i,j;m)}{D(i,j,q_{i+1,j})}
\sqrt{z_s}^{2+2\lambda}\sqrt{z_{s+1}}^{2-2\lambda} \bigg].
\end{eqnarray}
Splitting functions of the type
$\mathrm{split}(1^+,\ldots, m^-, \ldots, s^{-\lambda}_{\bar q},(s+1)^\lambda_q,\ldots,n^+ \to P^-)$ are obtained
by line reversal. These results for the two-quark sector are sufficient to calculate all
splitting amplitudes for up to four partons.
\subsection{Two quarks in the collinear set: $q(ng)\bar q \to \gamma$}
In this collinear limit, the $q\ldots \bar q $ system
forms a colour singlet and effectively acts as a photon.
\begin{figure}[t!]
\centering
\psfrag{n+1}{$n+1$}
\psfrag{n}{$n^+$}
\psfrag{N}{$N$}
\psfrag{1}{$1^-$}
\psfrag{m}{$m^-$}
\begin{center}
\includegraphics[height=5cm]{2qtilde.eps}
\end{center}
\caption{MHV topologies contributing to the two
quark collinear limit of the type
$\widetilde{\mathrm{split}}(1_q^-,\ldots,m^-,\ldots,n_{\bar q}^+\to P_\gamma^-)$.
Quarks of type $Q$ ($q$) are shown as green(red)-dotdashed lines and negative
helicity gluons as black solid lines. The negative helicity photon is shown
as a blue dashed line.
}
\label{fig:2qtilde}
\end{figure}
\subsubsection{$\Delta M = 0$}
In this limit the four-quark $\tilde A$ MHV amplitudes of
eqs.~(\ref{eq:4qmhv5})--(\ref{eq:4qmhv8}) factorise directly onto
the two-quark+photon amplitudes of (\ref{eq:2qMHVphoton}).
We find that,
\begin{eqnarray}
\widetilde{\mathrm{split}}(1_q^\lambda,\ldots,n_{\bar q}^{-\lambda}\to P_\gamma^-) &=&
\frac{z_1^{\frac{1}{2}-\lambda}z_n^{\frac{1}{2}+\lambda}}
{\spa{1}.{2}\cdots\spa{n-1}.{n}}.
\end{eqnarray}
\subsubsection{$\Delta M = 1$}
For amplitudes of the ${\mathrm{Split}}_+(m_1)$ type, there is a single MHV diagram and
we find
\begin{eqnarray}
\widetilde{\mathrm{split}}(1_q^-,\ldots,n_{\bar q}^+\to P_\gamma^+) &=&
\frac{1}{\sqrt{z_1 z_n} \prod_{l=1}^{n-1} \spa{l,}.{l+1} }
\times \frac{\Del{0}.{n}.{1}^3 \Del{0}.{n}.{n} }{D(0,n;q_{1,n})}. \nonumber
\\
\end{eqnarray}
As in the previous case, there are four diagrams shown in Fig.~\ref{fig:2qtilde}
contributing
to splitting functions of
${\mathrm{Split}}_-(m_1,m_2)$ type such that,
\begin{eqnarray}
\lefteqn{\widetilde{\mathrm{split}}(1_q^-,\ldots,m^-,\ldots,n_{\bar q}^+\to P_\gamma^-) =
\frac{1}{\sqrt{z_1 z_n} \prod_{l=1}^{n-1} \spa{l,}.{l+1} }}\nonumber \\
&&\hspace{2cm}\times\bigg[
\sum_{i=1}^{m-1} \sum_{j=m}^{n-1} \frac{\Del{i}.{j}.{m}^4
\sqrt{z_1}^3\sqrt{z_n}}{D(i,j;q_{i+1,j})}-\sum_{i=1}^{m-1}
\frac{\spa{n}.{m}\Del{i}.{n}.{m}^3}{D(i,n;q_{i+1,n})}\sqrt{z_1}^3
\left(\sum_{k=i+1}^{n} z_k\right) \nonumber\\
&&\hspace{2cm}-\sum_{j=m}^{n-1}
\frac{\spa{1}.{m}^3\Del{0}.{j}.{m}}{D(0,j;q_{1,j})}\sqrt{z_n}
\left(\sum_{k=1}^{j} z_k\right)^3 +
\frac{\spa{1}.{m}^3\spa{n}.{m} }{D(0,n;q_{1,n})}
\bigg ].
\end{eqnarray}
\subsection{Three quarks in the collinear set: $q (ng) \bar Q Q \to q$}
In this configuration, the $\bar Q$ is adjacent with $Q$
and therefore the vertices in the MHV rules
include the
four-quark amplitudes of eqs.~(\ref{eq:4qmhv1})--(\ref{eq:4qmhv4}).
The factorised amplitude is a two-quark MHV as given in eq.~(\ref{eq:2qMHV})
Furthermore, since the helicity of $q$ is conserved and the helicities
of $Q$ and $\bar Q$ are opposite, there are no $\Delta M=0$ splitting functions.
\subsubsection{$\Delta M = 1$}
For $\Delta M = 1$, the two diagrams (with quark and gluons exchanged)
of ${\mathrm{Split}}_+(m_1)$-type yield,
\begin{eqnarray}
\lefteqn{
\mathrm{split}(1_q^+,\ldots, s^{-}_{\bar Q},(s+1)^+_Q,\ldots,n^+ \to P_q^+)=
\frac{1}{\sqrt{z_1 z_n} \prod_{l=1}^{n-1} \spa{l,}.{l+1}
}\hspace{2cm}\phantom{~}.}\nonumber \\
&&\hspace{4cm}\times\bigg[
\sum_{i=1}^{s-1} \sum_{j=s+1}^n
\frac{\Delta(i,j;s+1)\Delta^3(i,j;s)}{D(i,j,q_{i+1,j})} \sqrt{z_1} \nonumber \\
&&\hspace{4cm}-\sum_{j=s+1}^n \frac{\aab{1}{s} \Delta^2(0,j;s)\Delta(0,j;s+1)}{D(0,j,q_{1,j})} \left(\sum_{k=1}^jz_k\right)\bigg],\\
\lefteqn{
\mathrm{split}(1_q^+,\ldots, s^{+}_{\bar Q},(s+1)^-_Q,\ldots,n^+ \to P_q^+)=
\frac{1}{\sqrt{z_1 z_n} \prod_{l=1}^{n-1} \spa{l,}.{l+1} }}\nonumber \\
&&\hspace{4cm}\times\bigg[
\sum_{i=1}^{s-1} \sum_{j=s+1}^n
\frac{\Delta^3(i,j;s+1)\Delta(i,j;s)}{D(i,j,q_{i+1,j})} \sqrt{z_1} \nonumber \\
&&\hspace{4cm}-\sum_{j=s+1}^n \frac{\aab{1}{s} \Delta^3(0,j;s+1)}{D(0,j,q_{1,j})} \left(\sum_{k=1}^jz_k\right)\bigg].
\end{eqnarray}
Similarly, the two diagrams of ${\mathrm{Split}}_-(m_1,m_2)$-type yield,
\begin{eqnarray}
\lefteqn{
\mathrm{split}(1_q^-,\ldots, s^{-}_{\bar Q},(s+1)^+_Q,\ldots,n^+ \to P_q^-)=
\frac{1}{\sqrt{z_1 z_n} \prod_{l=1}^{n-1} \spa{l,}.{l+1} }}\nonumber \\
&&\hspace{4cm}\times\bigg[
\sum_{i=1}^{s-1} \sum_{j=s+1}^n
\frac{\Delta(i,j;s+1)\Delta^3(i,j;s)}{D(i,j,q_{i+1,j})} \sqrt{z_1}^3 \nonumber \\
&&\hspace{4cm}-\sum_{j=s+1}^n \frac{\aab{1}{s}^3 \Delta(0,j;s+1)}{D(0,j,q_{1,j})}
\left( \sum_{k=1}^jz_k \right)^3\bigg],\\
\lefteqn{
\mathrm{split}(1_q^-,\ldots, s^{+}_{\bar Q},(s+1)^-_Q,\ldots,n^+ \to P_q^-)=
\frac{1}{\sqrt{z_1 z_n} \prod_{l=1}^{n-1} \spa{l,}.{l+1} }}\nonumber \\
&&\hspace{4cm}\times\bigg[
\sum_{i=1}^{s-1} \sum_{j=s+1}^n
\frac{\Delta^3(i,j;s+1)\Delta(i,j;s)}{D(i,j,q_{i+1,j})} \sqrt{z_1}^3 \nonumber\\
&&\hspace{4cm}-\sum_{j=s+1}^n \frac{\aab{1}{s}\aab{1}{s+1}^2 \Delta(0,j;s+1)}{D(0,j,q_{1,j})}
\left( \sum_{k=1}^jz_k \right)^3 \bigg].
\end{eqnarray}
\subsection{Three quarks in the collinear set: $q (ng) \bar q Q \to Q$}
Here the relevant vertices in the MHV rules include
the $\tilde A$
four-quark amplitudes of eqs.~(\ref{eq:4qmhv5})--(\ref{eq:4qmhv8})
and the factorised amplitude is a two-quark MHV as given in eq.~(\ref{eq:2qMHV})
As in the previous case, the quark helicities are constrained such that
there are no $\Delta M=0$ splitting functions.
\subsubsection{$\Delta M = 1$}
There are two diagrams for both ${\mathrm{Split}}_+(m_1)$- and ${\mathrm{Split}}_-(m_1,m_2)$-types
and we find,
\begin{eqnarray}
\lefteqn{
\widetilde{\mathrm{split}}(1_q^+,\ldots, s^{-}_{\bar q},(s+1)^+_Q,\ldots,n^+ \to P_Q^+)=
\frac{1}{\sqrt{z_1 z_n} \prod_{l=1}^{n-1} \spa{l,}.{l+1} }}\nonumber \\
&&\hspace{4cm}\times\bigg[
\frac{\Delta(0,s;1)\Delta^3(0,s;s)}{D(0,s,q_{1,s})} \sqrt{z_{s+1}} \nonumber \\
&&\hspace{4cm}+ \sum_{j=s+1}^n \frac{\aab{s}{s+1} \Delta^2(0,j;s)\Delta(0,j;1)}{D(0,j,q_{1,j})} \left(\sum_{k=1}^jz_k\right)\bigg],\\
\lefteqn{
\widetilde{\mathrm{split}}(1_q^-,\ldots, s^{+}_{\bar q},(s+1)^+_Q,\ldots,n^+ \to P_Q^+)=
\frac{1}{\sqrt{z_1 z_n} \prod_{l=1}^{n-1} \spa{l,}.{l+1} }}\nonumber \\
&&\hspace{4cm}\times\bigg[
\frac{\Delta^3(0,s;1)\Delta(0,s;s)}{D(0,s,q_{1,s})} \sqrt{z_{s+1}} \nonumber \\
&&\hspace{4cm}+ \sum_{j=s+1}^n \frac{\aab{s}{s+1} \Delta^3(0,j;1)}{D(0,j,q_{1,j})} \left(\sum_{k=1}^jz_k\right)\bigg],
\end{eqnarray}
and,
\begin{eqnarray}
\lefteqn{
\widetilde{\mathrm{split}}(1_q^+,\ldots, s^{-}_{\bar q},(s+1)^-_Q,\ldots,n^+ \to P_Q^-)=
\frac{1}{\sqrt{z_1 z_n} \prod_{l=1}^{n-1} \spa{l,}.{l+1} }}\nonumber \\
&&\hspace{4cm}\times\bigg[
\frac{\Delta^3(0,s;1)\Delta(0,s;s)}{D(0,s,q_{1,s})} \sqrt{z_{s+1}}^3 \nonumber \\
&&\hspace{4cm}+ \sum_{j=s+1}^n \frac{\aab{s}{s+1}^3 \Delta(0,j;1)}{D(0,j,q_{1,j})} \left(\sum_{k=1}^jz_k\right)^3\bigg],\\
\lefteqn{
\widetilde{\mathrm{split}}(1_q^-,\ldots, s^{+}_{\bar q},(s+1)^-_Q,\ldots,n^+ \to P_Q^-)=
\frac{1}{\sqrt{z_1 z_n} \prod_{l=1}^{n-1} \spa{l,}.{l+1} }}\nonumber \\
&&\hspace{4cm}\times\bigg[
\frac{\Delta^3(0,s;1)\Delta(0,s;s)}{D(0,s,q_{1,s})} \sqrt{z_{s+1}}^3 \nonumber \\
&&\hspace{4cm}+ \sum_{j=s+1}^n \frac{\aab{s}{s+1} \aab{1}{s+1}^2\Delta(0,j;1)}{D(0,j,q_{1,j})} \left(\sum_{k=1}^jz_k\right)^3
\bigg].
\end{eqnarray}
\subsection{Four quarks in the collinear set: $\bar Q Q (ng) \bar q q \to g$}
This limit is associated with the four-quark $A$-type colour ordered amplitude and
is obtained by factoring onto a gluonic MHV.
\subsubsection{$\Delta M = 1$}
Because of helicity
conservation for the quarks, $\Delta M = 0$ is forbidden. Furthermore, at least two
negative helicity quarks participate in the scattering so that $\Delta M = 1$
splittings must be of the ${\mathrm{Split}}_-(m_1,m_2)$-type.
The five contributing diagrams
are shown in Fig.~\ref{fig:4q}.
\begin{figure}[t!]
\centering
\psfrag{s+1}{$s+1$}
\psfrag{s}{$s$}
\psfrag{t+1}{$t+1$}
\psfrag{t}{$t$}
\begin{center}
\includegraphics[width=12cm]{4q.eps}
\end{center}
\caption{MHV topologies contributing to the four quark collinear limit of the type
$\mathrm{split}(1^+,\ldots,s_{\bar Q}^\lambda, (s+1)_Q^{-\lambda}, \ldots t_{\bar q}^{\lambda^\prime},
(t+1)_q^{-\lambda^\prime}, \ldots, n^+ \to P^-)$.
Quarks of type $Q$ ($q$) are shown as green(red)-dotdashed lines and negative
helicity gluons as black solid lines.
}
\label{fig:4q}
\end{figure}
Explicit evaluation of the four independent helicity configurations yields,
\begin{eqnarray}
\lefteqn{
\mathrm{split}(1^+,\ldots,s_{\bar Q}^+, (s+1)_Q^-, \ldots t_{\bar q}^-,
(t+1)_q^+, \ldots, n^+ \to P^-)=
\frac{1}{\sqrt{z_1 z_n} \prod_{l=1}^{n-1} \spa{l,}.{l+1} }}\nonumber \\
&&\hspace{4cm}\times\bigg[
\sum_{i=0}^{s-1}\sum_{j=s+1}^{t-1}
\frac{\Delta(i,j;s)\Delta^3(i,j;s+1)}{D(i,j,q_{i+1,j})}
\sqrt{z_t}^3\sqrt{z_{t+1}} \nonumber\\
&&\hspace{4cm}+ \sum_{i=s+1}^{t-1}\sum_{j=t+1}^n
\frac{\Delta^3(i,j;t)\Delta(i,j;t+1)}{D(i,j,q_{i+1,j})}
\sqrt{z_s}\sqrt{z_{s+1}}^3 \nonumber\\
&&\hspace{4cm}+ \sum_{i=0}^{s-1} \sum_{j=t+1}^n
\frac{\aab{s+1}{t}^3\aab{s}{t+1}}{D(i,j,q_{i+1,j})}\left(\sum_{k=i+1}^j z_k\right)^4 \nonumber\\
&&\hspace{4cm}- \sum_{j=t+1}^n
\frac{\Delta(s,j;t+1)\aab{s+1}{t}^3}{D(s,j,q_{s+1,j})}\sqrt{z_{s}}
\left(\sum_{k=s+1}^j z_k\right)^3 \nonumber\\
&&\hspace{4cm}+ \sum_{i=0}^{s-1}
\frac{\Delta(i,t;s)\aab{s+1}{t}^3}{D(i,t,q_{i+1,t})}\sqrt{z_{t+1}}
\left( \sum_{k=i+1}^{t} z_k \right)^3 \bigg],
\\
\lefteqn{
\mathrm{split}(1^+,\ldots,s_{\bar Q}^-, (s+1)_Q^+, \ldots t_{\bar q}^-,
(t+1)_q^+, \ldots, n^+ \to P^-)=
\frac{1}{\sqrt{z_1 z_n} \prod_{l=1}^{n-1} \spa{l,}.{l+1} }}\nonumber \\
&&\hspace{4cm}\times\bigg[
\sum_{i=0}^{s-1}\sum_{j=s+1}^{t-1}
\frac{\Delta^3(i,j;s)\Delta(i,j;s+1)}{D(i,j,q_{i+1,j})}
\sqrt{z_t}^3\sqrt{z_{t+1}} \nonumber\\
&&\hspace{4cm}+ \sum_{i=s+1}^{t-1}\sum_{j=t+1}^n
\frac{\Delta^3(i,j;t)\Delta(i,j;t+1)}{D(i,j,q_{i+1,j})}
\sqrt{z_s}^3\sqrt{z_{s+1}} \nonumber\\
&&\hspace{4cm}+ \sum_{i=0}^{s-1} \sum_{j=t+1}^n
\frac{\aab{s+1}{t}\aab{s}{t+1}\aab{t}{s}^2}{D(i,j,q_{i+1,j})}\left(\sum_{k=i+1}^j z_k\right)^4 \nonumber\\
&&\hspace{4cm}- \sum_{j=t+1}^n
\frac{\Delta(s,j;t+1)\Delta^2(s,j;t)\aab{s+1}{t}}{D(s,j,q_{s+1,j})}\sqrt{z_{s}}^3
\left(\sum_{k=s+1}^j z_k\right) \nonumber\\
&&\hspace{4cm}+ \sum_{i=0}^{s-1}
\frac{\Delta(i,t;s)\aab{s+1}{t}\aab{t}{s}^2}{D(i,t,q_{i+1,t})}\sqrt{z_{t+1}}
\left( \sum_{k=i+1}^{t} z_k \right)^3 \bigg],
\\
\lefteqn{
\mathrm{split}(1^+,\ldots,s_{\bar Q}^+, (s+1)_Q^-, \ldots t_{\bar q}^+,
(t+1)_q^-, \ldots, n^+ \to P^-)=
\frac{1}{\sqrt{z_1 z_n} \prod_{l=1}^{n-1} \spa{l,}.{l+1} }}\nonumber \\
&&\hspace{4cm}\times\bigg[
\sum_{i=0}^{s-1}\sum_{j=s+1}^{t-1}
\frac{\Delta(i,j;s)\Delta^3(i,j;s+1)}{D(i,j,q_{i+1,j})}
\sqrt{z_t}\sqrt{z_{t+1}}^3 \nonumber\\
&&\hspace{4cm}+ \sum_{i=s+1}^{t-1}\sum_{j=t+1}^n
\frac{\Delta(i,j;t)\Delta^3(i,j;t+1)}{D(i,j,q_{i+1,j})}
\sqrt{z_s}\sqrt{z_{s+1}}^3 \nonumber\\
&&\hspace{4cm}+ \sum_{i=0}^{s-1} \sum_{j=t+1}^n
\frac{\aab{s+1}{t}\aab{s}{t+1}\aab{s+1}{t+1}^2}{D(i,j,q_{i+1,j})}\left(\sum_{k=i+1}^j z_k\right)^4 \nonumber\\
&&\hspace{4cm}- \sum_{j=t+1}^n
\frac{\Delta(s,j;t+1)\aab{s+1}{t}\aab{s+1}{t+1}^2}{D(s,j,q_{s+1,j})}\sqrt{z_{s}}
\left(\sum_{k=s+1}^j z_k\right)^3 \nonumber\\
&&\hspace{4cm}+ \sum_{i=0}^{s-1}
\frac{\Delta(i,t;s)\Delta^2(i,t;s+1)\aab{s+1}{t}}{D(i,t,q_{i+1,t})}\sqrt{z_{t+1}}^3
\left( \sum_{k=i+1}^{t} z_k \right)\bigg],\nonumber \\
\\
\lefteqn{
\mathrm{split}(1^+,\ldots,s_{\bar Q}^-, (s+1)_Q^+, \ldots t_{\bar q}^+,
(t+1)_q^-, \ldots, n^+ \to P^-)=
\frac{1}{\sqrt{z_1 z_n} \prod_{l=1}^{n-1} \spa{l,}.{l+1} }}\nonumber \\
&&\hspace{4cm}\times\bigg[
\sum_{i=0}^{s-1}\sum_{j=s+1}^{t-1}
\frac{\Delta^3(i,j;s)\Delta(i,j;s+1)}{D(i,j,q_{i+1,j})}
\sqrt{z_t}\sqrt{z_{t+1}}^3 \nonumber\\
&&\hspace{4cm}+ \sum_{i=s+1}^{t-1}\sum_{j=t+1}^n
\frac{\Delta(i,j;t)\Delta^3(i,j;t+1)}{D(i,j,q_{i+1,j})}
\sqrt{z_s}^3\sqrt{z_{s+1}} \nonumber\\
&&\hspace{4cm}+ \sum_{i=0}^{s-1} \sum_{j=t+1}^n
\frac{\aab{s+1}{t}\aab{s}{t+1}^3}{D(i,j,q_{i+1,j})}\left(\sum_{k=i+1}^j z_k\right)^4 \nonumber\\
&&\hspace{4cm}- \sum_{j=t+1}^n
\frac{\Delta^3(s,j;t+1)\aab{s+1}{t}}{D(s,j,q_{s+1,j})}\sqrt{z_{s}}^3
\left(\sum_{k=s+1}^j z_k\right) \nonumber\\
&&\hspace{4cm}+ \sum_{i=0}^{s-1}
\frac{\Delta^3(i,t;s)\aab{s+1}{t}}{D(i,t,q_{i+1,t})}\sqrt{z_{t+1}}^3
\left( \sum_{k=i+1}^{t} z_k \right) \bigg].
\end{eqnarray}
\subsection{Four quarks in the collinear set: $\bar Q q (ng) \bar q Q \to g$}
This limit is associated with the four-quark $\tilde A$-type colour ordered
amplitude and
is obtained by factoring onto a gluonic MHV.
\subsubsection{$\Delta M = 1$}
As in the previous case, helicity
conservation for the quarks, ensures that $\Delta M = 0$ is forbidden and
that ${\mathrm{Split}}_+(m_1)$-type $\Delta M = 1$ splittings are absent.
The four contributing diagrams of ${\mathrm{Split}}_-(m_1,m_2)$-type
are shown in Fig.~\ref{fig:4qtilde}.
\begin{figure}[t!]
\centering
\psfrag{s+1}{$s+1$}
\psfrag{s}{$s$}
\psfrag{t+1}{$t+1$}
\psfrag{t}{$t$}
\begin{center}
\includegraphics[height=5cm]{4qtilde.eps}
\end{center}
\caption{MHV topologies contributing to the four quark collinear limit of the type
$\widetilde{\mathrm{split}}(1^+,\ldots,s_{\bar Q}^\lambda, (s+1)_q^{-\lambda^\prime}, \ldots t_{\bar q}^{\lambda^\prime},
(t+1)_Q^{-\lambda}, \ldots, n^+ \to P^-)$.
Quarks of type $Q$ ($q$) are shown as green(red)-dotdashed lines and negative
helicity gluons as black solid lines.
}
\label{fig:4qtilde}
\end{figure}
The four independent helicity configurations are given by,
\begin{eqnarray}
\lefteqn{
\widetilde{\mathrm{split}}(1^+,\ldots,s_{\bar Q}^+, (s+1)_q^+, \ldots t_{\bar q}^-,
(t+1)_Q^-, \ldots, n^+ \to P^-)=
\frac{1}{\sqrt{z_1 z_n} \prod_{l=1}^{n-1} \spa{l,}.{l+1} }}\nonumber \\
&&\hspace{5cm}\times\bigg[
\sum_{i=0}^{s-1} \sum_{j=t+1}^{n}
\frac{\aab{t+1}{t}^3\aab{s}{s+1}}{D(i,j,q_{i+1,j})} \left( \sum_{k=i+1}^j z_k\right)^4
\nonumber\\
&&\hspace{5cm}- \sum_{j=t+1}^n \frac{\Delta(s,j;s+1)\aab{t+1}{t}^3}{D(s,j,q_{s+1,j})}
\sqrt{z_s} \left( \sum_{k=s+1}^j z_k \right)^3 \nonumber\\
&&\hspace{5cm}- \sum_{i=0}^{s-1} \frac{\Delta^3(i,t;t)\aab{s}{s+1}}{D(i,t,q_{i+1,t})}
\sqrt{z_{t+1}}^3 \left( \sum_{k=i+1}^t z_k \right) \nonumber\\
&&\hspace{5cm}+ \frac{\Delta^3(s,t;t)\Delta(s,t;s+1)}{D(s,t,q_{s+1,t})}\sqrt{z_s}\sqrt{z_{t+1}}^3\bigg],
\\
\lefteqn{
\widetilde{\mathrm{split}}(1^+,\ldots,s_{\bar Q}^+, (s+1)_q^-, \ldots t_{\bar q}^+,
(t+1)_Q^-, \ldots, n^+ \to P^-)=
\frac{1}{\sqrt{z_1 z_n} \prod_{l=1}^{n-1} \spa{l,}.{l+1} }}\nonumber \\
&&\hspace{5cm}\times\bigg[
\sum_{i=0}^{s-1} \sum_{j=t+1}^{n}
\frac{\aab{t+1}{t}\aab{s}{s+1}\aab{s+1}{t+1}^2}{D(i,j,q_{i+1,j})} \left( \sum_{k=i+1}^j z_k\right)^4
\nonumber\\
&&\hspace{5cm}- \sum_{j=t+1}^n \frac{\Delta(s,j;s+1)\aab{t+1}{t}\aab{s+1}{t+1}^2}{D(s,j,q_{s+1,j})}
\sqrt{z_s} \left( \sum_{k=s+1}^j z_k \right)^3 \nonumber\\
&&\hspace{5cm}- \sum_{i=0}^{s-1} \frac{\Delta(i,t;t)\Delta^2(i,t;s+1)\aab{s}{s+1}}{D(i,t,q_{i+1,t})}
\sqrt{z_{t+1}}^3 \left( \sum_{k=i+1}^t z_k \right) \nonumber\\
&&\hspace{5cm}+ \frac{\Delta(s,t;t)\Delta^3(s,t;s+1)}{D(s,t,q_{s+1,t})}\sqrt{z_s}\sqrt{z_{t+1}}^3\bigg],
\\
\lefteqn{
\widetilde{\mathrm{split}}(1^+,\ldots,s_{\bar Q}^-, (s+1)_q^+, \ldots t_{\bar q}^-,
(t+1)_Q^+, \ldots, n^+ \to P^-)=
\frac{1}{\sqrt{z_1 z_n} \prod_{l=1}^{n-1} \spa{l,}.{l+1} }}\nonumber \\
&&\hspace{5cm}\times\bigg[
\sum_{i=0}^{s-1} \sum_{j=t+1}^{n}
\frac{\aab{t+1}{t}\aab{s}{s+1}\aab{t}{s}^2}{D(i,j,q_{i+1,j})} \left( \sum_{k=i+1}^j z_k\right)^4
\nonumber\\
&&\hspace{5cm}- \sum_{j=t+1}^n \frac{\Delta(s,j;s+1)\Delta^2(s,j;t)\aab{t+1}{t}}{D(s,j,q_{s+1,j})}
\sqrt{z_s}^3 \left( \sum_{k=s+1}^j z_k \right) \nonumber\\
&&\hspace{5cm}- \sum_{i=0}^{s-1} \frac{\Delta(i,t;t)\aab{s}{s+1}\aab{t}{s}^2}{D(i,t,q_{i+1,t})}
\sqrt{z_{t+1}} \left( \sum_{k=i+1}^t z_k \right)^3 \nonumber \\
&&\hspace{5cm}+ \frac{\Delta^3(s,t;t)\Delta(s,t;s+1)}{D(s,t,q_{s+1,t})}\sqrt{z_s}^3\sqrt{z_{t+1}}\bigg],
\\
\lefteqn{
\widetilde{\mathrm{split}}(1^+,\ldots,s_{\bar Q}^-, (s+1)_q^-, \ldots t_{\bar q}^+,
(t+1)_Q^+, \ldots, n^+ \to P^-)=
\frac{1}{\sqrt{z_1 z_n} \prod_{l=1}^{n-1} \spa{l,}.{l+1} }}\nonumber \\
&&\hspace{5cm}\times\bigg[
\sum_{i=0}^{s-1} \sum_{j=t+1}^{n}
\frac{\aab{t+1}{t}\aab{s}{s+1}^3}{D(i,j,q_{i+1,j})} \left( \sum_{k=i+1}^j z_k\right)^4
\nonumber\\
&&\hspace{5cm}- \sum_{j=t+1}^n \frac{\Delta^3(s,j;s+1)\aab{t+1}{t}}{D(s,j,q_{s+1,j})}
\sqrt{z_s}^3 \left( \sum_{k=s+1}^j z_k \right) \nonumber\\
&&\hspace{5cm}- \sum_{i=0}^{s-1} \frac{\Delta(i,t;t)\aab{s}{s+1}^3}{D(i,t,q_{i+1,t})}
\sqrt{z_{t+1}} \left( \sum_{k=i+1}^t z_k \right)^3 \nonumber\\
&&\hspace{5cm}+ \frac{\Delta(s,t;t)\Delta^3(s,t;s+1)}{D(s,t,q_{s+1,t})}\sqrt{z_s}^3\sqrt{z_{t+1}}\bigg].
\end{eqnarray}
\section{Selected results for triple collinear limits}
\label{sec:specific-results}
To illustrate our general results for multi-collinear limits,
in this section we list some of the
triple-collinear splitting functions. The
$\Delta M = 0$ splitting amplitudes are obtained directly from
MHV amplitudes and we do not list them here. Explicit results are given in
Section~\ref{sec:general-results}.
For the $\Delta M = 1$ (and therefore $\Delta P = 1$)
amplitudes, there are two types of splitting function corresponding to
${\mathrm{Split}}_{+}(m_1)$ and ${\mathrm{Split}}_{-}(m_1,m_2)$. In the specific case of
three collinear particles, these are related by parity thereby reducing the
number of independent amplitudes to at most
two for each splitting. Here, we
list only the most compact form of the amplitudes.
\subsection{$q g g \to q$}
There are only two independent $\Delta M = 1$ splitting amplitudes,
which can be obtained by setting $m=n=3$ in
Eqs.~(\ref{eq:qgg1}) and (\ref{eq:qgg2}). Explicitly we find,
\begin{eqnarray}
\mathrm{split}(1_q^+,2^+,3^- \to P_q^+)&=&
-{\frac {\spa{2}.{3}{z_{{2}}}^{3/2}}
{\sqrt {z_{{3}}} s_{{2,3}}
\left( \spa{2}.{1}\sqrt{z_{{2}}}+\spa{3}.{1}\sqrt {z_{{3}}} \right)
\left( z_{{2}}+z_{{3}} \right) }}\nonumber \\
&&-{\frac {\spa{1}.{3}
\left( \spa{1}.{3}\sqrt {z_{{1}}}+\spa{2}.{3}\sqrt {z_{{2}}} \right) ^{2}}
{\spa{1}.{2}\spa{2}.{3}s_{{1,3}}
\left( \spa{2}.{1}\sqrt {z_{{2}}}+\spa{3}.{1}\sqrt {z_{{3}}} \right) }} ,\\
\mathrm{split}(1_q^-,2^+,3^-\to P_q-)&=&
-{\frac {z_{{1}}\spa{2}.{3}{z_{{2}}}^{3/2}}
{\sqrt {z_{{3}}}s_{{2,3}}
\left( \spa{2}.{1}\sqrt {z_{{2}}}+\spa{3}.{1}\sqrt {z_{{3}}} \right)
\left( z_{{2}}+z_{{3}} \right) }}\nonumber \\
&&-{\frac { \spa{1}.{3}^{3}}
{\spa{1}.{2}\spa{2}.{3}s_{{1,3}}
\left( \spa{2}.{1}\sqrt {z_{{2}}}+\spa{3}.{1}\sqrt {z_{{3}}} \right)
}}.
\end{eqnarray}
All others can be obtained by parity and charge conjugation.
These expressions numerically agree with the splitting functions
given in \cite{delduca}\footnote{Note that there is a small
typographical error
in Eq. (5.56) of Ref.~\cite{delduca}. $s_{23}$ should be replaced by $s_{12}$
in the last term of the equation for ${\rm split}_-^{q\to
qgg}(k_1^+,k_2^+,k_3^-)$},
\begin{eqnarray}
\mathrm{split}(1_q^+,2^+,3^- \to P_q^+)&=&-\frac{1}{s_{1,2}s_{2,3}}\nonumber \\
&\times&
\bigg[
\frac{\spb{1}.{2}(\spa{1}.{3}\sqrt{z_1}+\spa{2}.{3}\sqrt{z_2})^2
(\spb{1}.{2}\sqrt{z_1}+\spb{3}.{2}\sqrt{z_3})}{s_{1,3}}\nonumber \\
&&
+\frac{\sqrt{z_2}(z_1+z_2)
\spb{1}.{2}(\spa{1}.{3}\sqrt{z_1}+\spa{2}.{3}\sqrt{z_2})}{\sqrt{z_3}}\nonumber \\
&&+\frac{\sqrt{z_1}z_2s_{1,2}}{(z_2+z_3)}\bigg], \\
\mathrm{split}(1_q^-,2^+,3^-\to P_q-)&=&-\frac{1}{s_{1,2}s_{2,3}}\nonumber \\
&\times&
\bigg[
\frac{\spb{1}.{3}(\spa{1}.{2}\sqrt{z_1}+\spa{3}.{2}\sqrt{z_3})^2
(\spb{1}.{3}\sqrt{z_1}+\spb{2}.{3}\sqrt{z_2})}{s_{1,3}}\nonumber \\
&&
+\frac{\sqrt{z_2}(z_1+z_2)
\spb{1}.{3}(\spa{1}.{2}\sqrt{z_1}+\spa{3}.{2}\sqrt{z_3})}{\sqrt{z_3}}\nonumber \\
&&+\frac{\sqrt{z_1}z_2s_{1,2}}{(z_2+z_3)}+\sqrt{z_2}\spa{2}.{3}\spb{1}.{3}\bigg].
\end{eqnarray}
We see that the two sets of results have the same types of
singularity structure as $z_3 \to 0$ and $z_1 \to 1$ corresponding to the soft
and double soft gluon limits.
\subsection{$\bar q q g \to g$}
There are again only two independent $\Delta M = 1$ amplitudes,
both of which can be obtained from Eq.~(\ref{eq:gqq})
by setting $s=2$ and $\lambda=\pm\frac{1}{2}$.
We find,
\begin{eqnarray}
\label{eq:qggppm}
\mathrm{split}(1^+,2_{\bar q}^+,3_q^-\to P^+)&=&-
{\frac { \left( \spa{1}.{2}\sqrt {z_{{1}}}+\spa{3}.{2}\sqrt {z_{{3}}}
\right) \left( \spa{1}.{3}\sqrt {z_{{1}}}+\spa{2}.{3}\sqrt {z_{{2}}}
\right) ^{2}}{\spa{1}.{2}\spa{2}.{3}s_{{1,3}} \left( \spa{2}.{1}
\sqrt {z_{{2}}}+\spa{3}.{1}\sqrt {z_{{3}}} \right) }}\nonumber\\
&&+{\frac {z_{{2}}\spa{2}.
{3}}{\sqrt {z_{{1}}}s_{{2,3}} \left( \spa{2}.{1}\sqrt {z_{{2}}}
+\spa{3}.{1}\sqrt {z_{{3}}} \right) \left( z_{{2}}+z_{{3}} \right) }}
,\\
\label{eq:qggpmp}
\mathrm{split}(1^+,2_{\bar q}^-,3_{q}^+\to P^+)&=&
-{\frac {
\left( \spa{1}.{2}\sqrt {z_{{1}}}+\spa{3}.{2}\sqrt {z_{{3}}} \right)^{3}}
{\spa{1}.{2}\spa{2}.{3}s_{{1,3}}
\left( \spa{2}.{1}\sqrt {z_{{2}}}+\spa{3}.{1}\sqrt {z_{{3}}} \right)
}}\nonumber \\&&
+{\frac {z_{{3}}\spa{2}.{3}}{\sqrt {z_{{1}}}s_{{2,3}}
\left( \spa{2}.{1}\sqrt {z_{{2}}}+\spa{3}.{1}\sqrt {z_{{3}}} \right)
\left( z_{{2}}+z_{{3}} \right) }}.
\end{eqnarray}
All others can be obtained via parity and charge
conjugation.
Eq.~(\ref{eq:qggppm}) numerically agrees with
the analogous
expression given in Ref.~\cite{delduca}.
We were not able to find agreement between Eq.~(\ref{eq:qggpmp})
and the expression for
${\rm split}_-^{g \to g \bar q q}(k_1^+,k_2^-,k_3^+)$ in Eq.~(5.53)
of Ref.~\cite{delduca}.
\subsection{$q\bar Q Q \to q$ and $q\bar q Q \to Q$}
In this special case both colour structures lead to the same splitting
amplitude. There is only one
independent $\Delta M = 1$ helicity configurations.
It is given in Eq.~(\ref{eq:qpQpQm1}).
\section{Conclusion}
\label{sec:conclusion}
In this paper we have considered the collinear limit of multi-parton QCD
amplitudes at tree level. We have used the new MHV rules for constructing
colour ordered amplitudes from MHV vertices to generalise our previous results
for gluon-only splitting functions. Our main results are general formulae for
timelike splitting functions involving up to two negative helicity partons and
an arbitrary number of positive helicity partons. We anticipate that the
expressions presented here will be useful in
developing higher order perturbative
predictions for observable quantities, such as jet cross sections at the LHC
or in examining the high energy limit of QCD.
A key point of our approach is that in the collinear limit only a
subset of MHV rules diagrams contribute - those where every propagator invariant
$s$ goes on-shell in the multi-collinear limit.
We observe that the splitting functions have a simple structure,
and can be written as
sums over the corresponding poles in $s$ multiplied
by a coefficient that is either entirely composed of
holomorphic spinor products $\spa{i}.{j}$ or entirely composed of
anti-holomorphic spinor products $\spb{i}.{j}$.
This implies that the coefficients are
supported on a single degree-one curve in (anti)-twistor space.
\bigskip
\section{Acknowledgements}
EWNG and VVK acknowledge the support of PPARC through
Senior Fellowships and TGB acknowledges the award of a PPARC studentship.
\bibliographystyle{JHEP-2}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 7,847 |
Nederlandse Amateur Vereniging voor Raket-Onderzoek
Dutch amateur association for rocket research
NAVRO launch days
CanSat launch days
Fly to the Sky launch days
Other launch days
NAVRO's media presence
Model rocketry
High Power Rocketry
Amateur rocketry
NAVRO rockets
Altical
Motor research
Kalinidex motor tests
Kalinitrox motor tests
Other motor tests
TNO classification
NAVRO motors
NAVRO history
All amateur rockets built by the NAVRO as a club project and some private projects have received a designation beginning with "N". This page describes all the rockets, which received this designation. By clicking on the image next to the description of the rocket you will be taken to the picture page of that rocket.
After the first club projects N1 and N2, the next type of rockets were of the Midget B design with the Kalinitrox K600 motor. The Midget C was an improved version using a more powerful Kalinitrox K1800/K2000 motor. The second series were the Hercules'. The Hercules A1 is redesigned and improved version of the original Hercules A. It was easier to build, but most of all was reduced a quarter in weight.
The next series of NAVRO rockets is the Titan. This type of rocket has a diamete of 135 mm. Significantly lager than the 90 mm of the Hercules. Purpose of the Titan project is perfecting the NDU parachuting system (nose-down nose-up also see N23) and eventualy reach an altitude of 7000 m with a Kalinidex motor with 15.000 Ns.
Most NAVRO amateur rockets are launched at our own NAVRO Lanceerdagen (NAVRO Launch Days) at ASK 't Harde, but in the earlier nineties we also launched at several French launch campaigns.
N1 "Pluvius Tubus"
Name: N1 "Pluvius Tubus"
Type: N1
Motor: Bambi
Launch: Mourmelon, 28 August 1990
The N1 is the first amateur rocket built by the NAVRO. "Pluvius Tubus" is latin for "Rain-pipe". It was launched at the National French Launching Campaign. After 0.7 seconds the French Bambi motor that was used, destroyed the rocket. This was a great disappointment, but it was one of the reasons to develop our own rocket motors.
The French government provided rockets motors to amateurs free of charge under supervision of the CNES (the French NASA), to stop amateurs making their own rocket motors. Unfortunately these are developed from military rocket motors and are smokeless. As user you don't have any control on the quality of the motor.
N2 "Vindicta Pluvii Tubi"
Name: N2 "Vindicta Pluvii Tubi"
Motor: Isard
The N2 was very similar to the N1 and even reused surviving parts of the N1. Unlike the N1, the N2 flew successful. Not only it landed almost unscratched, the NAVRO also won the Prix Joseph Mercier, a safety award. "Vindicta Pluvii Tubi" means quite appropriately "Revenge of the Rain-pipe".
Name: N3
Type: Midget A
Motor: K600
Launch: NLC2, ASK 't Harde, 26 September 1992
The N3 was the first rocket to fly with our own K600 Kalinitrox composite rocket motor. It was a small rocket, built just to test the motor in flight. The flight was a success, as far as we could know at the time. The N3 had no recovery transmitter and it wasn't found that day. This flight was also the first flight of a composite rocket motor developed in The Netherlands.
A year later at our next launch the military presented us the rocket they had found when they cleared the shooting range. We were very pleased with the return of the N3 and since it is proudly on display.
Type: Midget B
Launch: NLD1, ASK 't Harde, 6 August 1993
After success of the N3 and its K600 motor, the N4 was a more advanced rocket and was the prototype of the Midget B/C rockets. It had a recovery transmitter and a timer for the parachute, but no further electronics. A new feature for us was that the parachute was behind an hatch. Such a hatch opens when the rocket parachutes, instead of a detachable nose cone. The flight was a success.
N5 "Partiarius"
Name: N5 "Partiarius"
Launch: WWLC 1993, Bourges, 29 August
The N5 was the private project of Vincent Kouer with some assistance of other NAVRO members. He had obtained an aluminium tube, which had to fly. The tube was said to be a part of a Fokker 100's fuel pipe (The Fokker 100 is a 100 man passenger aircraft that was built by Fokker Aircraft, which is now bankrupt...).
The N5 was designed for the K600 motor. The flight was a success, except for the recovery transmitter, which did not function. The N5 was found anyway. The N5 reached a maximum speed of 221m/s and an altitude of 1800 metres.
Launch: NLD2, ASK 't Harde, 27 May 1994
The N6 was built to test if the recovery transmitter would survive a crash. Kees Jan Groenendijk put much effort in building the rocket. The rocket had no parachute. As you might have guessed, the transmitter did not survive the crash. The N6 was never found and is now entertaining worms.
The N7 was a rocket made by three youth members of the NAVRO. The purpose of this project was to introduce amateur rocketry to NAVRO youth members, who had been building model rockets for some years. The rocket was built with assistance of experienced members. The launch was a success and the rocket was recovered almost intact. Even the parachute hatch was found.
Launch: WWLC 1993, Bourges, 29 August 1993
The N8 was our first rocket, which had an onboard computer. This computer was based on the 80C552 processor and was designed by one of our members. It could measure acceleration, rotation, onboard temperature and could trigger the separation of the parachute hatch. A month earlier the N8 had flown as the N4 and was repaired and rebuild for the Bourges launch. The main difference was that the N8 was longer to accommodate more electronics. The flight was a success and the N8 rotated around its vertical axis once each 8 seconds.
The N9 was our first rocket with a two-stage parachute system and its main purpose was to test it. After reaching the top of the flight a (small) drogue chute was deployed, and only at a few hundred metres high the (big) main chute was deployed. This way the rocket lands closer to the launching site. The flight was a success.
Name: N10
Type: Midget C
Motor: K1800
Launch: NLD3, ASK 't Harde, 19 August 1994
The N10 was the first flight of the K1800 motor, later upgraded to the K2000 motor. The rocket itself was the modified N9, which was launched three months earlier. The flight was successful.
During the winter of 1994/1995 the K1800 was upgraded to the K2000, giving the N11 more thrust than the N10 of a year earlier. The N11 main purposes however, were its experiments. Using an improved version of the N8's electronics, the measured values were: atmospheric pressure, rotation, acceleration, temperature on top of the nose cone, as well as the temperature of the rockets surrounding air and a Doppler-measurement. A barometric altimeter was also present. All data was recorded to be downloaded into the computer when the rocket was recovered. However, the rocket did not parachute and has not been found to this day. The probable cause was pyro-technical. Since then our rockets have had multiple pyro-technical systems.
Type: Hercules A
The N12 was the prototype of the Hercules rocket. Apart from being prototype, the N12 was supposed to fly with two camera's, one facing down and one facing up. The images were transmitted live to our command centre and recorded. The camera's could not transmit simultaneously, so a switch ensured the most interesting moments were transmitted. However the electronic switch did not work, so the N12 flew using only its downward looking camera. The electronics were similar to the electronics of the N8. Parachuting wasn't perfect again, because the N12 landed on its drogue chute due to a miscalculation. Fortunately all internal parts, like camera's and electronics were intact.
The N13 was like the N5 a private project of Vincent Kouer, who was fascinated by speed and the availability of a more powerful motor now had to build a rocket which could break the sound barrier. The N13 was essentially a motor with on top of it a small compartment with simple electronics and a parachute. The N13's speed was measured with a Doppler measurement. It broke the sound barrier and parachuted correctly, but the rocket was not recovered that day. However, in 2003 it was found.
The N14 has flown before as the N7, but had been enlarged and given a new paint job. It was supposed to fly some electronic projects, but they were never finished. After the N14 was launched it never parachuted, so somewhere in the clouds their still must be flying a black rocket. If you see it please report it to the NAVRO.
Launch: NLD6, ASK 't Harde, 23 April 1996
The N15 was a copy of the N12. The small differences were the single downwards faced camera and the sound recording and atmospheric pressure measurement experiments. The most important difference was, that all data was transmitted to ground control. The resulting values were close to what was calculated. The N15 was featured on national television on the "Klokhuis" show.
Again we had troubles parachuting, but it wasn't our fault. The cable of the main parachute broke, because it wasn't nylon. The nylon was bought at a DIY store as nylon. When we informed the DIY store, they sued their supplier.
Type: Hercules A1
The N16 was built as a redesigned and improved Hercules A1. The N16 was launched successful and parachuted as planned.
Name: N17, Tintin rocket
Type: Tintin rocket
Motor: 1 AeroTech I284-W
2 AeroTech H180-W
This Tintin rocket was built by Chiel Klein. The N17 is an exception in the N-series, as it is in fact an High Power Rocket, which normally do not get a N-registration, but this one was very special and the first HPR rocket launched at a NLD. The rocket was completely built from scratch and 1290mm (ca. 4 feet) in length! It had a great flight, which you could follow all the way. In some documents the rocket was also allocated the N18 number, to add to the confusion (see below).
N18 (1997)
Name: N18 (1997)
The N18 (1997) was the refurbished N16. Again we were victim of Murphy's Laws as we had another parachute failure. This time the drogue chute didn't pull out the main parachute. Later the N18 (1997) was dug out and nothing more than small fragments of plastic and metal were found. All the electronics boards were wiped clean. This rocket is called N18 (1997) on the website, to distinguish it from the more successful similar named N18 of 1999.
After the year of cancelled NLD's we finally were to launch again in 1999. During that year we forgot we already had allocated the N18 number, so the new rocket was also called N18. On the website this N18 is called just N18. The new N18 was again equipped with a camera, this time facing upwards to shoot images of the deployment of the parachutes. Unfortunately the antenna was badly targeted, so we had some interference in the images, but most of the parachute deployments could be seen. It happens fast, only in a few frames! This time the parachuting was done correctly, much to our relieve. The altimeter was the commercial IA-X96 Cambridge Accelerometer of Emmanuel Avionics.
Launch: NLD10, ASK 't Harde, 13 August 1999
The N19 was the refurbished N18 without the camera. Again the IA-X96 Cambridge Accelerometer was used. The flight was successful, but landed in a tree.
Launch: NLD11, ASK 't Harde, 7 April 2000
The faulty IA-X96 Cambridge Accelerometer was replaced by the R-DAS flight computer of AED Electronics, which is based on the same plans as the N8's electronics, but is further developed and more advanced. The N20's flight was a success, but the N20 pointed to the wind far more than expected. Also the K2000 rocket motor seemed not to have the thrust is was thought to have. It landed virtually unscratched. The N20 had flown earlier as the N18 and N19.
Launch: NLD12, ASK 't Harde, 20 October 2000
The N21 is the same rocket as the N20, and thus as the N18 and N19. The flight was successful. The purpose of this flight was to measure the thrust of our K2000 rocket motor in flight. According to the R-DAS it reached an altitude of 1500 metre (5000 feet).
Motor: AeroTech K550-W
The proven airframe, which started life as the N22, would fly again. This time for his last flight (so we thought). The rocket had a different motor, an AeroTech K550-W, which has the same performance as the proposed K2000 replacement. Also the R-DAS was expanded with a GPS module. The rocket had a good flight, this time leaving the tower with enough surplus speed. It reached an altitude of 1600 metres.
Launch: NLD16, ASK 't Harde, 6 September 2002
The N23 used the well tested old airframe, which was original the N18. It was its sixth flight, this time with an AeroTech K550-W. The R-DAS was again supplemented with the GPS module and the NAVRO radio beacon. New this flight was a new NDU parachute configuration (nose-down nose-up). The N23 was to descend nose down on the drogue chute and then using a pyrotechnic charge the rocket would tumble and descend tail down on the main chute. This elaborated system worked perfectly in flight and the N23 made a flawless flight.
Motor: AeroTech K1100-T
The N24 is essentially the N23 with a new paint job. Its seventh flight was the last flight of the rocket that was first launched as the N18. The electronics included R-DAS with GPS and the NAVRO radio beacon. An other experiment is the NUND parachute system (see N23). The rocket had a nice lift off, but but crashed as it went ballistic. The reason was that the safety plug was forgotten and thus the parachuting system didn't operate. The remains were crushed very badly.
The N25 project started life with an eight years old K600 motor we had in storage and wanted to use in a rocket. We also had the reasonable complete 1993 vintage N8 hanging on our ceiling. So those two were matched. On the rocket we replaced the parachute hatch, the parachute hatch release mechanism, the electronics, the lower body tube and the fins. Both the parachute hatch and two of the fins were reclaimed from old N8 parts. The electronics compromised a transmitter and an R-DAS. Like the N8, we also measured rotation.
The launch and flight of the N25 were perfect and it went up in a straight line. It landed almost unscratched. It reached an altitude of 1260m and it rotated around its axis once every 1.8 seconds.
Type: Titan A
Motor: 70 mm Kalinidex motor from Mark Uitendaal
Launch: Cansat launchday, ASK 't Harde, 7 june 2013
This rocket with number N26 is the first Titan and was launched 7 June 2013. For the propulsion we used the 70 mm Kalinidex motor from Mark Uitendaal. With this 2400 Ns motor the rocket reached an altitude of 1200 m and parachuted partially successful with the NDU system. The main we used, a 15 years old parachute with a diameter of 1,80 m, was not up to the opening forces witch partly tore the parachute. For the flight electronics we used the RDAS.
Type: Titan B
Launch: NLD39, ASK 't Harde, 20 september 2013
This Titan rocket was launched on September 20, 2013. Revised electronics and RDAS of the N26 where used. The differens was that this time we used the 80 mm Kalinidex motor (4500 Ns) from Mark. In addittion, the rocket is now provided with two cameras. One camera (camera A) produces images in the direction of the nose cone and the other (camera B) produced images downwards to th fins. The NDU parachute system used an slower opening ballute as main parachute. The rocket reach an altitude of 2500 m and parachited now successful. Camera A produced stunning images including the parachtuting. Unfortunately data from Camera B could not be read.
Type: Titan C
Motor: NAVRO Kalinidex K90.6.5700
Launch: NLD42, ASK 't Harde, 1 May 2015
This Titan rocket was launched on 1 May 2015. For the flight the Titan C was used which was used as the N27. The difference with the N27 was that now a K90.6 type engine was used with a total impulse of 5700 Ns. This K90.6.5700 was the result of the engine test KSB2014-T003 / 004 witch where held in 2014. The rocket reached an altitude of 2500 m and parachuted successful. This time two small video cameras were mounted on the rocket, one facing up and the other down. Unfortunately, the camera looking up didn't make recordings, the other cameras made beautiful recordings of the flight.
Launch: NLD44, ASK 't Harde, 20 may 2016
The N29 was launched on May 20, 2016. For this flight the Titan-C was used witch was already used as N27. The electronics frame for this flight was extended with a sensor package. The aim was to test different sensors for use in the new NAVRO flight computer.
The rocket flew again with K90.6.5700 rocket engine. Now, however, it was decided not to recover the hatch. This time both cameras recorded beautiful images of, among others, the parachuting of the rocket. The rocket reached an altitude of 2800 m (according to then new sensors, RDAS > 2500m). Parachuting unfortunately was only partly successful and the rocket landed hanging on the drogue ballutte. By this relatively hard landing, the nose cone and parachute section of the rocket were destroyed. The recordings of the parachuting camera where carefully studied. Also there where different theories developed that might explain why the main ballutte remained unopened. A satisfactory explanation was not found.
Type: Hercules B
The N30 was a Hercules B type rocket where the engine has the same diameter as the rocket. Therefore the lower part of the rocket is completely motor and the fins are clamed on the motor. The purpose of this launch was to test te NUND parachuting system at higher speeds. Because of this we used a small drougue shute. Unfortunatelely the drougue chute was to small and therfore the rocket did not turn before the deployment of the main parachute. As a result, the main parachute has not been pulled out of the parachute compartment. The N30 has therefore suffered great damage during the landing.
Launch: NLD46, ASK 't Harde, 19 April 2017
After the launch of the N30, it was decided to recreate the same rocket. Of course the parachuting system was adjusted. The N31 was again launched with a self-made Kalinidex K90.4.3800 motor. The launch went beautifully, but after the rocket disappeared in the clouds, no one has seen it anymore. We assume that the rocket has crashed. Unfortunately, nothing has been recovered so we have no idea what went wrong. | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 8,215 |
BeMyGuest closes Series A, launches its own lab
By Martin Cowen | July 19, 2017
Asian tours, activities and attractions specialist BeMyGuest has closed its Series A funding round and launched BeMyGuest Labs.
While the amount has not been disclosed, BeMyGuest confirmed that it has now raised a total of S$11.5 million ($8.5 million). Its Crunchbase entry however values the Series A at S$8.5 million ($6.2 million).
Last year the business decided to focus on B2B and says that it is on track to reach S$100 million ($72 million) of sales through existing and new partnerships. It provides content to the tours and activities channels of many leading OTAs in the region, such as Ctrip and Yatra.
In total it works with some 500 distribution partners had has access to more than 4,500 suppliers and 25,000 products covering 900 destinations.
The Series A round involves a number of investors including Raffles Venture Partners, a government investment agency and Singapore's biggest travel agent and tour operator, Chan Brothers.
BeMyGuest launched its own agent marketplace for travel agents to access inventory and also feeds into Travelport's Rooms & More.
The Lab project is intended to help suppliers and distributors access its tools and services, ranging from straightforward AP integrations to e-ticketing and revenue management.
According to Phocuswright data referenced in the announcement, tours and activities in Asia was a $45 billion market in 2016 with just over 10% transacted online.
Related reading:Ctrip gives tours and activities channel a major boost (Oct 15) | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 2,377 |
Q: PHP set class property to retain its value on every call I know this is a simple question, sorry, but I'm still learning OOP concepts.
I'd like to have a class property that should retain its value during the execution and I need to get and set that value by different methods from within the same class.
My code:
class incarico extends globale {
static $contatore;
// delete the product
function delete(){
$query = "DELETE FROM " . $this->table_name . " WHERE id = ?";
$stmt = $this->conn->prepare($query);
$stmt->bindParam(1, $this->id);
// on every delete I need to get the total number of records
if($result = $stmt->execute()){
self::$contatore = $this->conn->query("select count(*) from ". $this->table_name)->fetchColumn();
return true;
}
else{
return false;
}
}
function generaProtocollo () {
// here I need the $contatore value
error_log(self::$contatore);
return $annocorrente . self::$contatore;
}
}
When I delete the record with
$incarico = new incarico($db);
$incarico->delete()
The value $contatore is correctly set, when I (after) call:
$incarico = new incarico($db);
$incarico->protocollo = $incarico->generaProtocollo();
The value is empty.
What am I doing wrong?
Thank you,
Alex
A: Try this code:
class Incarico extends Globale
{
private $contatore;
// delete the product
public function delete()
{
$query = "DELETE FROM " . $this->table_name . " WHERE id = ?";
$stmt = $this->conn->prepare($query);
$stmt->bindParam(1, $this->id);
return $stmt->execute();
}
public function generaProtocollo ()
{
return $annocorrente . $this->contatore;
}
public function countRows()
{
return $this->contatore = $this->conn->query("select count(*) from ". $this->table_name)->fetchColumn();
}
}
What you need to do is:
*
*Write code in English it really helps and code is more readable.
*Use PSR standards
*You need to use the class not like Global don't think in global way, try to change your thinking to DI and object that has its state which is not global.
*I didn't write it in the class but it's good habit to use getters and setters
*I don't think that counting rows should be in method named delete. You should probably create the other method that count rows
usage:
$obj = new Incranico();
$obj->delete();
echo $obj->countRows();
$obj->delete();
echo $obj->countRows();
I don't really know what are your needs but also delete method could use param which is $id, it would be more clear that way.
A: Could be that your $stmt->execute() or query() returned false. Note that $annocorrente is also never set.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 9,077 |
Blog Posts > Ending Environmental Regulations Won't Save Central Appalachian Coal
Sean O'Leary Senior Policy Analyst
Ending Environmental Regulations Won't Save Central Appalachian Coal
Last month, using new data from the Energy Information Administration (EIA), Ted showed that West Virginia coal production is headed for a steep decline in the coming years, led by falling production in Central Appalachia.
According to the EIA, "Appalachian coal production declines substantially from current levels, as coal produced from the extensively mined, higher cost reserves of Central Appalachia is supplanted by lower cost coal from other supply regions," saying that Appalachian coal is being priced out of the market by cheaper, easier to mine coal from other parts of the country.
But the rhetoric around coal in the state from politicians, the coal industry, and prominent media personalities suggests that EPA regulations are to blame for the coming decline of coal in West Virginia, as part of the so-called War on Coal.
But what if the coal industry got its way? What if we could guarantee that the EPA would never limit greenhouse gas emissions, allowing utilities to burn as much coal as they want? Would coal production in Central Appalachia reverse its decline? The answer is no.
The EIA has modeled this exact scenario, with its "No greenhouse gas concern" case scenario, which projects coal production assuming no greenhouse gas limiting policies are enacted and the market never anticipates any. And according to the projections, even with a guarantee of no carbon tax, no emission limits, no cap and trade system, coal production in Central Appalachia will continue to decline, just as much as in the current baseline projection.
Under both the reference case, and the no greenhouse gas policy scenario, coal production in Central Appalachia is projected to decline by around 62% between 2009 and 2020.
The same is true of the mercury emission standards enacted in December. While starting a firestorm on controversy in the state, projections from the EIA show a decline in Central Appalachian Coal production with or without the mercury standards. The projections in the 2012 early release AEO were made before the emissions standards were finalized, and were not incorporated into the projections. The final 2012 AEO did include the new rules in the projections, but once again, the effect of the regulations on Central Appalachian coal production was negligible. (Check here for a full list of changes from the 2012 early release to the 2012 official release).
The reality is that even without greenhouse gas or mercury regulations, coal production in Central Appalachia is going to dramatically decline. Repealing environmental regulations won't make the remaining coal seams in West Virginia any thicker or easier to mine, and it won't stop power plants from converting to natural gas. To ignore this reality, and to act as if stopping the EPA will save the coal industry in West Virginia, is shortsighted and dangerous to the state's future.
Categories: Blog, Energy & Environment | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 6,857 |
\section{Introduction}
Topic models \cite{blei2012probabilistic,DBLP:journals/ipm/Chen17} underpin many successful applications within the field of Natural Language Processing (NLP). Variants of topic models have been proposed for different tasks including content analysis of e-petitions\cite{DBLP:journals/ipm/Hagen18}, topic-associated sentiment analysis \cite{lin2009joint}, event extraction from social media \cite{zhou2014simple,zhou2016jointly, zhou2017unsupervised} and product aspect mining \cite{DBLP:journals/ipm/XiaoJLZS18}. However, topic models typically rely on mean-field variational inference\cite{asuncion2009smoothing} or collapsed Gibbs sampling for model learning. A small change to the modeling assumption requires the re-derivation of the whole inference algorithm, which is mathematically arduous and time consuming.
In recent years, word embeddings (such as Word2vec\cite{mikolov2013efficient}, GloVe\cite{pennington2014glove}, fastText\cite{joulin2016bag,bojanowski2016enriching} and probabilistic fastText\cite{athiwaratkun2018probabilistic}) have gained an increasing interest thanks to their improved efficiency in representing words as continuous vectors in a low-dimensional space. The resulting embeddings encode numerous semantic relations (similarity or analogies) and are helpful for NLP tasks\cite{DBLP:journals/ipm/Fernandez-Reyes18,DBLP:journals/ipm/HsuLCS18}. But the traditional topic models could not generate such word-level semantic representations.
{\color{black}To overcome the limitation that traditional topic model often need sophisticated inference algorithm}, Neural Variational Document Model (NVDM) \cite{miao2016neural} was devised based on the Variational Auto-Encoder (VAE)\cite{kingma2013auto} and used a hidden layer to reconstruct the document by generating the words independently. However, the usage of gaussian prior over topics in NVDM may lead to incoherent and similar topics being generated. On the contrary, Srivastava \cite{srivastava2017autoencoding} proposed LDA-VAE, a neural topic model based on the VAE, in which the logistic normal distribution was employed as the prior over topics for topic generation. To further enhance the quality of the generated topic, Srivastava replaced the mixture assumption with a weighted product of experts at the word-level and proposed the ProdLDA. {\color{black} But both the LDA-VAE and the ProdLDA were not able to produce word-level semantic representations. Besides, the logistic normal prior used in LDA-VAE and ProdLDA also could not capture the multiplicity topical aspects in a document and result in generating bad topics. }
{\color{black}To overcome the limitations that the traditional topic models often need sophisticated inference algorithm and the exist neural based topic models could not generate coherent topic words.} In this paper, we propose the Adversarial-neural Topic Model (ATM) based on adversarial training. The principle idea is to use a generator network to learn the projection function between the document-topic distribution and the document-word distribution. Instead of providing an analytic approximation, as in traditional topic models, the ATM uses a discriminator network to recognize if the input document is real or fake and its output signal could help the generator to construct a more realistic document from a random noise drawn from a dirichlet distribution. Due to the flexibility of neural networks, the generator is capable of learning complicated non-linear distributions. And the supervision provided by the discriminator in the adversarial training phase will help the generator to capture the semantic patterns embedded in the latent topics. Besides, the connection weights between the embedding layer and the word distribution layer of the generator also encodes the semantic information and naturally provides distributed representations of words as side product.
{\color{black} The objectives of our work in this paper are, more succinctly, as follows:}
{\color{black}
\begin{enumerate}
\item Traditional topic models based on gibbs sampling or variational inference often need sophisticated inference algorithms and obtain incoherent topics. We are interested in devising a novel neural-based topic model which could mine coherent topics from text corpora automatically in an unsupervised manner. To this end, based on the Generative Adversarial Net, we propose the ATM model which could extract the coherent topics among text corpus.
\item From a practical perspective, we would like to devise a neural-based topic model which could be transplanted to other task easily with limited modification. For this purpose, we modify the topic generation process of the proposed ATM and employ it for open domain event extraction task \cite{zhou2015unsupervised}, experiments on news articles corpus shows that the proposed model is able to extract meaningful events and also verifies the portability of ATM.
\end{enumerate}
}
The practical significance of this work is that the proposed approach (ATM) could generate more coherent topics than the state-of-the-art
topic modeling approaches. Meanwhile, it could also produce semantic representations for each word in the vocabulary as side product, which is currently not supported by the compared models. Besides, the proposed ATM could be easily ported to other NLP task (such as open domain event extraction) with limited modification. The rest of the paper is organized as follows. Section 2 reviews the related literature on neural topic models and generative adversarial nets. In Section 3, we provide the details of the proposed Adversarial-neural Topic Model. Section 4 will introduce our evaluation corpora and our obtained experimental results. Finally, the paper is concluded in Section 5 with suggestions for further work.
\section{Related Work}
Our work is related to two lines of research, neural-based topic modeling and the Generative Adversarial Nets. Thus, we will next briefly introduce the related work in two domain separately.
\subsection*{Neural-based Topic Modeling}
To overcome the difficult exact inference of topic models based on directed graph, Hinton \cite{hinton2009replicated} modified the Restricted Boltzmann Machines and proposed a replicated softmax model (called RSM). Inspired by the variational autoencoder, Miao \cite{miao2016neural} used the multivariate gaussian as the prior distribution of latent space and proposed the Neural Variational Document Model (NVDM) for text modeling. More recently, to deal with the inappropriate gaussian prior of topic distributions in NVDM, Srivastava \cite{srivastava2017autoencoding} proposed the LDA-VAE which approximated the dirichlet prior using a logistic normal distribution, and the usage of logistic normal prior could help to generate more coherent and diverse topics. Srivastava \cite{srivastava2017autoencoding} replaced the mixture assumption with a weighted product of experts at the word-level and proposed the ProdLDA which further improved topic coherence.
\subsection*{Generative Adversarial Nets}
As a neural-based generative model, the Generative Adversarial Nets \cite{goodfellow2014generative} have been extensively researched from both theoretical and practical aspects.
Theoretically, \cite{nowozin2016f} used the Fenchel conjugate to define the F-divergence and proposed the F-GAN to generalize its optimization objective. To precisely measure the distance between two high dimensional distributions, \cite{arjovsky2017wasserstein} defined the Earth Mover's Distance (Wasserstein distance) and gave a computational method based on the weight clipping mechanism. Along this line, \cite{gulrajani2017improved} improved the Wasserstein GAN by adding a gradient penalty loss and promoted the stability of adversarial training.
In practical applications, {\color{black} GAN-based models have been extensively researched in computer vision community, especially in image generation scenario. To incorporate the conditional information, Mirza \cite{mirza2014conditional} employed the random noise together with label as input and proposed the Conditional-GAN to generate image under the {\color{black}supervision} of the annotated label. The deep convolutional neural network were employed as the generator and the discriminator in \cite{radford2015unsupervised} to improve the quality of generated image. And \cite{ledig2016photo} also used the GAN-basd approch to generate super-resolution image.} {\color{black}On the other hand, many variants of GAN have been developed for NLP tasks. Such as text generation, a hot research area in NLP.} The sequence generative adversarial network (SeqGAN) proposed in \cite{yu2017seqgan} incorporated a policy gradient strategy to optimize the generation process. Based on the policy gradient, Lin \cite{lin2017adversarial} proposed the RankGAN to capture the rich structures of language by ranking and analysing a collection of human-written and machine-written sentences. To overcome the mode collapse when dealing with discrete data, Fedus \cite{fedus2018maskgan} proposed the MaskGAN which used an actor-critic conditional GAN to fill in missing text conditioned on the surrounding context. Along this line, Wang \cite{wang2018sentigan} employed multiple generator network (each for one sentiment) and proposed the SentiGAN to generate texts of different sentiment labels. {\color{black}Hu \cite{hu2017toward} incorporated the VAE into GAN framework for text generation. Besides, \cite{miyato2016adversarial,li2018learning} improved the performance of semi-supervised text classification using adversarial training. Zeng \cite{zeng2018adversarial} designed GAN-based models for distance supervision relation extraction. Wang \cite{wang-lee-2018-learning} incorporated the generative adversarial net into a encoder-decoder framework and proposed a GAN based model for text summarization. Yang \cite{NIPS2018_7959} employed the target domain language model into GAN framework to transfer style of text.}
{\color{black}Despite many successful applications using GAN-based approaches, none of these approaches tackles the topic modeling problem. We propose the first GAN-based topic model called ATM, which differs from the existing approaches to neural topic modeling in the following aspects: (1) Unlike the NVDM and the LDA-VAE which use either multivariate gaussian prior or logistic-normal prior for latent topics, ATM uses the dirichlet prior instead. It makes sure that ATM could provide K-dimensional noise and each capture certain semantic patterns in the text corpus; (2) Unlike most GAN-based text generation approaches, a generator network is employed by ATM to learn the projection function between the document-topic distribution and the document-word distribution, which essentially captures the semantic patterns among latent topics rather than generating text sequences; (3) Unlike the traditional topic model, ATM is able to generate meaningful word-level semantic representations as a side product.}
\begin{figure*}[!h]
\centering
\includegraphics[
width=1\textwidth,
keepaspectratio]
{VisioATMmodelrevisedcut.pdf}
\caption{The framework of the Adversarial-neural Topic Model (ATM).}
\label{fig:atm_framework}
\end{figure*}
\section{Adversarial-neural Topic Model}
We propose the Advesarial-neural Topic Model (ATM) as shown in Figure \ref{fig:atm_framework}.
The proposed ATM contains three main components: (1) the document sampling module shown at the top of Figure \ref{fig:atm_framework}, which defines the representation mapping function and samples a real document $d_{r}\in\mathbb{R}^{V}$ from an input text corpus; (2) the generator $G$ takes a topic distribution $\vec\theta$ sampled from a dirichlet prior as input and generates the corresponding fake document $d_{f}$; (3) the discriminator $D$ takes $d_{f}$ and $d_{r}$ as input and discriminates the fake document from the real ones, whose output is subsequently used as a learning signal to update the parameters of $G$ and $D$. We explain the design and function of each of these modules in more details below.
\subsection{Representation Mapping}
Each document $d$ is represented by a normalized $V$-dimensional vector weighted by TF-IDF. More concretely:
\begin{align*}
tf_{i,d} & =\frac{n_{i,d}}{\sum_{v}n_{v,d}} \\
idf_{i} &=\log \frac{|C|}{|C_{i}|}\\
tf\textrm{-}idf_{i,d} &=tf_{i,d}\times idf_{i}\\
d_{r}^{i} &=\frac{tf\textrm{-}idf_{i,d}}{\sum_{v}tf\textrm{-}idf_{v,d}}
\end{align*}
where $V$ is the vocabulary size, $n_{i,d}$ denotes the number of times the $i$-th word appears in document $d$, $|C|$ denotes the total number of documents in the corpus, and $|C_{i}|$ is the number of documents containing the $i$-th word. With this representation, each document in the corpus could be regarded as a multinomial distribution over $V$ words, and each dimension reflects the semantic coherence between the $i$-th word and the document $d$.
\subsection{Network Architecture}
The $G$ network contains three layers, the $K$-dimensional document-topic distribution layer, the $S$-dimensional embedding layer and the $V$-dimensional document-word distribution layer as shown in Figure \ref{fig:atm_framework}. First, the $G$ network takes a randomly sampled topic distribution $\vec\theta$ as input and transforms it into a document-word distribution. To model the multinomial property of the document-topic distribution, $\vec\theta$ is drawn from $Dir(\vec\theta|\vec\alpha)$:
\begin{align}
p(\vec\theta|\vec \alpha)= &Dir(\vec\theta|\vec\alpha)\triangleq\frac{1}{\triangle\left(\vec\alpha\right)}\prod_{k=1}^{K}\theta_{k}^{\alpha_{k}-1}
\end{align}
where $\vec\alpha$ is the hyper-parameter of the dirichlet distribution, $\triangle(\vec\alpha)=\frac{\prod_{k=1}^{K}\Gamma(\alpha_{k})}{\Gamma(\sum_{k=1}^{K}\alpha_{k})}$, $K$ is the number of topics, $\theta_{k}\in[0,1]$ denotes the proportion of topic $k$ in the document and $\sum_{k=1}^{K}\theta_{k}=1$.
Then, $G$ projects $\vec\theta$ into the $S$-dimensional (set to 100 in experiments) semantic space through the embedding layer based on equations :
\begin{align}
&\vec a_{s}=\max ((W_{s}\vec\theta+\vec b_{s}),leak*(W_{s}\vec\theta+\vec b_{s}))\\
&\quad\quad\quad\quad\quad \vec o_{s}=BN(\vec a_{s})
\end{align}
where $W_{s}\in \mathbb{R}^{S\times K}$ is the weight matrix and $\vec b_{s}$ represents the bias term of the embedding layer, $\vec a_{s}$ is the state vector activated by the LeakyReLU function parameterized with $leak$, $BN$ denotes batch normalization and $\vec o_{s}$ is the output of the embedding layer.
Finally, $G$ transforms $\vec o_{s}$ to a $V$-dimensional multinomial distribution $d_{f}$ using :
\begin{align}
&\vec h_{w}=W_{w}\vec o_{s}+\vec b_{w}\\
&o_{w}^{i}=\frac{\exp (h_{w}^{i})}{\sum_{v=1}^{V}\exp (h_{w}^{v})}
\end{align}
where $W_{w}\in \mathbb{R}^{V\times S}$ learns the semantic word embeddings and $\vec b_{w}$ represents the bias term, $\vec h_{w}$ is the state vector and $o_{w}^{i}$ denotes the probability of $i$-th word in $d_{f}$.
Likewise, we design the discriminator as a three layer fully connected network. The $D$ network employs the $d_{f}$ and the $d_{r}$ as input and outputs a scalar as shown in Figure \ref{fig:atm_framework}. A higher $D_{out}$ means that the discriminator is prone to consider the input data as a real document and vice versa.
\subsection{Training}
The fake document $d_{f}$ and the real document $d_{r}$ shown in Figure \ref{fig:atm_framework} could be viewed as the random sample from two $V$-dimensional dirichlet distribution $\mathbb{P}_{g}$ and $\mathbb{P}_{r}$. And the training objective of ATM is to let the generated distribution $\mathbb{P}_{g}$ approximate the real data distribution $\mathbb{P}_{r}$ as much as possible. Thus, the choice of divergence that measures the distance between two distributions is crucial for effective training of ATM.
The original GAN \cite{goodfellow2014generative} used the Jensen-Shannon divergence as the optimization objective. However, \cite{arjovsky2017wasserstein} argued that the divergences which GANs typically minimize are potentially not continuous with respect to the generator's parameters, leading to mode collapse and training difficulty. They proposed instead using the Earth-Mover's distance (also called Wasserstein-1) which is defined as the minimum cost of transporting mass in order to transform the distribution $\mathbb{P}_{g}$ into the distribution $\mathbb{P}_{r}$. Further, \cite{gulrajani2017improved} improved the Wassertein-1 with a gradient penalty strategy which performed more stable. We follow their work and define the objective of ATM as:
\begin{align}
&L_{d} = \underset{d_{f}\sim \mathbb{P}_{g}}{\mathbb{E}}\left[D(d_{f})\right]-\underset{d_{r}\sim \mathbb{P}_{r}}{\mathbb{E}}[D(d_{r})]\\
&\quad L_{gp}= \underset{\hat{d}\sim\mathbb{P}_{\hat{d}}}{\mathbb{E}}[(\parallel\nabla_{\hat{d}}D(\hat{d})\parallel_2 -1)^{2}]\\
&\quad\quad\quad\quad L = L_{d}+\lambda L_{gp}
\end{align}
\begin{algorithm}[!h]
\renewcommand{\algorithmicrequire}{\textbf{Input:}}
\renewcommand{\algorithmicensure}{\textbf{Output:}}
\caption{Training procedure for ATM}
\label{alg:1}
\begin{algorithmic}[1]
\REQUIRE $K$, $\lambda$, $n_{d}$, $m$, $\alpha_{1}$, $\beta_{1}$, $\beta_{2}$
\ENSURE the trained generator network $G$.
\STATE Initial $D$ parameters $\omega_{d}$ and $G$ parameter $\omega_{g}$
\WHILE{$\omega_{g}$ has not converged}
\FOR{$t=1,...,n_{d}$}
\FOR{$j=1,...,m$}
\STATE Sample $d_{r}\sim \mathbb{P}_{r}$,
\STATE Sample a random $\vec\theta\sim Dir(\vec\theta|\vec\alpha)$
\STATE Sample a random number $\epsilon\sim U[0,1]$
\STATE $d_{f}\leftarrow G(\vec\theta)$
\STATE $\hat{d}\leftarrow \epsilon d_{r}+(1-\epsilon) d_{f}$
\STATE {\color{black}$L_{d}^{(j)}=D(d_{f})-D(d_{r})$}
\STATE $L_{gp}^{(j)}=(\parallel \nabla_{\hat{d}}D(\hat{d}) \parallel-1)^{2}$
\STATE $L^{(j)}\leftarrow L_{d}^{(j)}+\lambda L_{gp}^{(j)}$
\ENDFOR
\STATE $\omega_{d}\leftarrow Adam(\nabla_{\omega_{d}}\frac{1}{m}\sum_{j=1}^{m}L^{(j)},\omega_{d},p_{a}) $
\ENDFOR
\STATE Sample $m$ noise $\left\{ \vec\theta^{(j)}\sim Dir(\vec\theta|\vec\alpha) \right\}$
\STATE $\omega_{g}\leftarrow Adam(\nabla_{\omega_{g}}\frac{-1}{m}\sum_{j=1}^{m}D(G(\vec\theta^{(j)})),\omega_{g},p_{a})$
\ENDWHILE
\end{algorithmic}
\end{algorithm}
where $L_{d}$ and $L_{gp}$ denote the loss of discriminator $D$ and the gradient penalty, respectively, $\lambda$ is the gradient penalty coefficient, $\hat{d}$ could be obtained by sampling uniformly along a straight line between a real document $d_{r}$ and a generated document $d_{f}$, and $\mathbb{P}_{\hat{d}}$ is the distribution from which $\hat{d}$ is sampled.
{\color{black}In each training step, the same number of $d_r$ and $d_f$ samples are fed into the Discriminator and the distance between $\mathbb{P}_{g}$ and $\mathbb{P}_{r}$ is estimated using Eqs. 6-8. Thus, G and D networks could be updated to minimize the distance between $\mathbb{P}_{g}$ and $\mathbb{P}_{r}$.
} Based on the model structure and the optimization objective described above, the training procedure for ATM is given in Algorithm 1. Here, $n_{d}$ denotes the number of discriminator iterations per generator iteration, $m$ represents the batch size, $\alpha_{1}$ is the learning rate, $\beta_{1}$ and $\beta_{2}$ are other hyper-parameters of Adam optimizer \cite{kingma2014adam}, and $p_{a}$ denotes $\left\{\alpha_{1},\beta_{1},\beta_{2}\right\}$. We use the default values of $\lambda=10$, $n_{d}=5$, $m=512$. Moreover, the $\alpha_{1}$, $\beta_{1}$ and $\beta_{2}$ are set to 0.0001, 0 and 0.9 respectively.
\subsection{Topic Generation}
The trained generator $G$ learns the projection function between the document-topic distribution and the document-word distribution. That is, given a topic distribution $\vec\theta_{d}$ for a document $d$, $G$ is able to generate the corresponding word distribution.
To generate the word distribution of each topic, we use $\vec ts_{(k)}$, a $K$-dimensional vector, as the one-hot encoding of the $k$-th topic. For example, $\vec ts_{(1)} = [1, 0, 0, 0, 0]^{\intercal}$ in the five topic number setting. We could then obtain the word distribution $\vec \phi_{k}$ for topic $k$ using:
\begin{equation}
\vec \phi_{k}=G(\vec ts_{(k)})
\end{equation}
\section{Experiments}
We evaluate our proposed ATM on two tasks, topic extraction and open domain event extraction. We first describe the datasets and the baseline approaches, and then present the topic coherence evaluation results for the topic extraction task. Finally, we discuss the results of using ATM for open domain event extraction to validate the feasibility of applying ATM for tasks other than topic modeling.
\subsection{Experimental Setup}
Two publicly accessible datasets, Grolier\footnote{https://cs.nyu.edu/$\sim$roweis/data/ \label{grolier}} and NYtimes\footnote{http://archive.ics.uci.edu/ml/datasets/Bag+of+Words \label{nytimes}} datasets, are used for topic coherence evaluation, and an event dataset built based on the Global Database of Events, Language, and Tone (GDELT)\footnote{http://data.gdeltproject.org/events/index.html \label{event}} is used for event extraction. Details are summarized below:
\begin{itemize}
{\color{black}
\item \emph{Grolier dataset}\textsuperscript{\ref{grolier}} is built from Grolier Multimedia Encyclopedia, and its content covers almost all the fields in the world, such as sports, economics, politics and etc. It contains 29,762 documents and is a benchmark text corpora in topic modeling.
\item \emph{NYtimes dataset}\textsuperscript{\ref{nytimes}} is a collection of newswire articles written and published by New York Times between January 1, 1987 and June 19, 2007 with article metadata provided by the New York Times Newsroom. This corpus also has a wide range of topics in real world, such as politics and entertainment.
\item \emph{Event dataset}. This dataset is the subset of GDELT which is released by Google. we crawl the Database\textsuperscript{\ref{event}} and built the event dataset by selecting the articles published on the first day of May in 2014. It contains many real events occurred at that day, such as MH370 and Indian Election.
}
\end{itemize}
\begin{table}[h]
\centering
\caption{The statistics of datasets.}
\label{tbs:statistic}
\begin{tabular}{lrr}
\hline
{\bfseries Dataset}& {\bfseries \#Document}&{\bfseries \#Words} \\
\hline
Grolier& 29,762& 15,276\\
NYtimes & 99,992&12,604\\
Event&20,199& 9,346\\
\hline
\end{tabular}
\end{table}
We choose the following five models as the baselines:
\begin{itemize}
{\color{black}
\item \textbf{LDA}~\cite{blei2003latent}, is a topic model that generates topics based on word the co-occurrence patterns from documents. With the usage of dirichlet prior topic distribution and word distribution, LDA could capture the multiplicity topic aspects from document collections in an unsupervised manner. We implement the LDA model and set the dirichlet prior of the document-topic distribution $\alpha=50/K$ and the dirichlet prior of the topic-word distributions $\beta=0.01$, following what have been suggested in \cite{griffiths2004finding}.
\item \textbf{NVDM}~\cite{miao2016neural}, is an neural based approach which models topics using variational auto-encoder. In NVDM, multivariate gaussian distribution is used as prior distribution of the latent space, and it is trained under the supervision of evidence lower bound (ELBO). We use the original implementation\footnote{https://github.com/ysmiao/nvdm}.
\item \textbf{LDA-VAE}~\cite{srivastava2017autoencoding}, is a neural topic model based on variational auto-encoder. To obtain readable topics, LDA-VAE substitute multivariate gaussian with a logistic normal distribution as the prior of the latent space. In this paper, the original implementation\footnote{https://github.com/akashgit/autoencoding\_vi\_for\_topic\_models\label{vae}} of LDA-VAE is employed to obtain the compared results.
\item \textbf{ProdLDA}~\cite{srivastava2017autoencoding}, is a variant of LDA-VAE which also uses logistic normal as the prior of the latent space. Beside, it assumes that the distribution over individual words is a product of experts rather than the mixture model used in LDA. The original implementation is used in this paper.
\item \textbf{LEM} \cite{zhou2014simple}, is a bayesian modeling approach for open domain event extraction. It treats an event as a latent variable and models the generation of an event as a joint distribution of its individual event elements (organization , location , person , keyword)\footnote{means organization, location, person and keywords. \label{olpk}}. We implement the algorithm with the default configuration.
}
\end{itemize}
For the NYtimes dataset, we random select 100,000 articles and remove the low frequent words. For the Event dataset, we use the Stanford Named Entity Recognizer\footnote{https://nlp.stanford.edu/software/CRF-NER.html\label{ner}} \cite{finkel2005incorporating} for identifying the named entities (Location, Organization and Person). In addition, we remove common stopwords and only keep the recognized name entities and the tokens which are verbs, nouns, or adjectives from these event documents. The statistics of the processed corpora are shown in Table \ref{tbs:statistic}.
\begin{table*}[!ht]
\centering
\small
\label{table:stu}
\caption{Average topic coherence on Grolier and NYtimes corpus with five topic settings [20, 30, 50, 75, 100].}
\label{tbs:average}
\begin{tabular}{l|l|cccccc}
\hline
{\bfseries Dataset}&{\bfseries Model}& {\bfseries C\_P}&{\bfseries C\_A}&{\bfseries NPMI}&\bfseries UCI& \bfseries UMass \\
\hline
\multirow{5}{*}{Grolier}&NVDM&-0.187746 &0.145684 &-0.061911 &-2.114927 &-4.291624\\
&LDA-VAE&-0.220548 &0.150469 &-0.065378 &-2.479750 &-4.755522\\
&ProdLDA&-0.037436 &0.173391 &-0.019347 &-1.639878 &-4.542689\\
&LDA&0.190845 &0.200942 &0.049753 &-0.050336 &-2.918612\\
&ATM&\textbf{0.210448} &\textbf{0.218898} &\textbf{0.058167} &\textbf{0.105086} &\textbf{-2.765081}\\
\hline
\multirow{5}{*}{NYtimes}&NVDM&-0.413086 &0.134154 &-0.143711 &-4.307269 &-5.931614\\
&LDA-VAE&-0.157560 &0.148221 &-0.061418 &-2.420816 &-4.640276\\
&ProdLDA&-0.003455 &0.196395 &-0.028223 &-1.917367 &-4.193377\\
&LDA&0.308336 &0.212750 & 0.077278 &0.516503 &-2.420221\\
&ATM&\textbf{0.356771} &\textbf{0.237524}&\textbf{0.089874} &\textbf{0.658218} &\textbf{-2.324093}\\
\hline
\end{tabular}
\end{table*}
\begin{figure}[!ht]
\centering
\includegraphics[
width=\textwidth,
keepaspectratio]
{measuresbarcut.pdf}
\caption{Average topic coherence on Grolier and NYtimes with five topic settings [20, 30, 50, 75, 100] among topics whose coherence values are ranked at the top 50\%, 70\%, 90\% and 100\% positions.}
\label{fig:avg_bar}
\end{figure}
\subsection{Topic Coherence Evaluation}
\begin{figure}[h!]
\centering
\includegraphics[
width=\textwidth,
keepaspectratio]
{measurecurvecut.pdf}
\caption{Average topic coherence (100\%) on Grolier and NYtimes datasets vs. different topic setting [20, 30, 50, 75, 100].}
\label{fig:avg_curve}
\end{figure}
\begin{figure}[!b]
\centering
\includegraphics[
width=\textwidth,
keepaspectratio]
{topic_example.pdf}
\caption{visualization of the topic words from the six selected topics.}
\label{fig:topic_visualization}
\end{figure}
\begin{table*}[h]
\centering
\footnotesize
\label{table:stu}
\caption{Topic examples of all the models, italics means out-of-topic.}
\label{tbs:example_topics}
\scalebox{0.84}{
\begin{tabular}{c|l}
\hline
{\bfseries Model}& \multicolumn{1}{c}{\bfseries Topics}\\
\hline
\multirow{5}{*}{ATM}&\textbf{jet flight airline hour plane passenger trip plan travel pilot} \\
& \textbf{stock market companies money investor technology fund investment company business}\\
& \textbf{music song musical album jazz band record recording mp3 composer}\\
& \textbf{voter vote poll republican race primary percent election campaign democratic}\\
& \textbf{film movie actor director award movies character theater production play}\\
\hline
\multirow{5}{*}{LDA} & \textbf{flight plane} \emph{ship} \emph{crew} \textbf{air pilot hour} \emph{boat} \textbf{passenger airport}\\
& \textbf{stock market} \emph{percent} \textbf{investor analyst} \emph{quarter} \textbf{investment shares share fund}\\
& \textbf{music song band sound record artist album show musical rock}\\
& \textbf{voter vote poll election campaign primary candidates republican race party}\\
& \textbf{film movie character play actor director movies} \emph{minutes} \textbf{theater} \emph{cast}\\
\hline
\multirow{5}{*}{ProdLDA}& \emph{wireless} \textbf{customer} \emph{telecommunication} \textbf{airlines} \emph{broadband satellites phones subscriber} \textbf{airline} \emph{provider}\\
& \textbf{brokerage securities broker lender buyer transaction investor investment stock borrower}\\
& \textbf{musical album} \emph{playwright} \textbf{composer} \emph{choreographer} \emph{onstage} \textbf{songwriter song guitarist repertory}\\
& \textbf{voter vote votes election electoral polling poll presidential primaries} \emph{turnout}\\
& \textbf{film comedy} \emph{beginitalic} \emph{enditalic} \emph{sci} \textbf{filmmaker cinematic filmmaking movie starring}\\
\hline
\multirow{5}{*}{LDA-VAE}& \textbf{passenger destination traveler} \emph{fares} \emph{booking} \textbf{airlines luggage routes} \emph{rider} \emph{excursion}\\
& \textbf{acquisition shareholder merge} \emph{takeover} \emph{acquire} \textbf{merger} \emph{consolidated} \textbf{stockholder} \emph{suitor} \emph{consolidation}\\
& \textbf{soloist operatic composer} \emph{repertory} \textbf{troupe} \emph{choreographer} \emph{choreography} \textbf{sung} \emph{dances} \textbf{recital}\\
& \textbf{balloting nominating election elect incumbent victor primaries} \emph{contested} \textbf{electoral vote}\\
& \textbf{moviegoer studios filmmaker movies film filming} \emph{vh1} \textbf{studio stardom} \emph{rapper}\\
\hline
\multirow{5}{*}{NVDM}& \emph{nesting instructor ranchers wingspan veteran} \textbf{fly} \emph{manager} \textbf{pilot} \emph{ecosystems} \textbf{flight}\\
& \textbf{company billion companies} \emph{production} \emph{equipment} \emph{processed processing producer manufacturing products}\\
& \textbf{conducting conductor instrumental} \emph{interval staff} \textbf{discography} \emph{knighted radioactive charge} \textbf{director}\\
& \emph{degrees} \textbf{national party} \emph{billion} \textbf{nations} \emph{decrease university exceed disorder nuclear}\\
& \emph{bay} \textbf{film} \emph{indian french company} \textbf{novel} \emph{dec lake explorer travels}\\
\hline
\end{tabular}
}
\end{table*}
Typically topic models are evaluated based on the likelihood of held-out documents. However, as pointed out in \cite{chang2009reading}, higher likelihood of held-out document does not necessarily correspond to human judgement of topic coherence. In this subsection, we follow \cite{roder2015exploring} and choose five coherence metrics to evaluate the topics generated by models. They are C\_P (a metric based on a sliding window, a one-preceding segmentation of the given words and the confirmation measure of Fitelson's coherence), C\_A (a metric based on a context window, a pairwise comparison of the given words and an indirect confirmation measure that uses normalized pointwise mutual information and the cosine similarity), UCI (a metric based on a sliding window and the pointwise mutual information of all word pairs of the given topics), NPMI (an enhanced version of UCI using the normalized pointwise mutual information) and UMass \cite{mimno2011optimizing} (a metric based on document cooccurrence counts, a one-preceding segmentation and a logarithmic conditional probability as confirmation measure). For all these five metrics, higher value implies more coherent topic. In our evaluation, we choose the top 10 words to represent each topic and compute the topic coherence using the Palmetto library\footnote{https://github.com/dice-group/Palmetto}.
To compare the performance of the proposed approach, experiments are conducted on Grolier and NYtimes with five topic number settings [20, 30, 50, 75, 100]. The average coherence values are listed in Table \ref{tbs:average} and each value is computed by averaging the average topic coherences (all the topics are used) over five topic number settings. Besides, we calculate the average topic coherence among topics whose coherence values are ranked at the top 50\%, 70\%, 90\%, 100\% positions. For example, to calculate the average UCI coherence of ATM @ 70\%, we first compute the average UCI coherence with the select topics whose UCI values are ranked at the top 70\% positions for each topic number setting, and then average the five averaged coherence values. The corresponding results are shown in Figure \ref{fig:avg_bar}. It can be observed from Figure \ref{fig:avg_bar} that the proposed model outperforms the LDA, NVDM, LDA-VAE and ProdLDA in general. {\color{black}This maybe caused by following factors: i) ATM models the multinomial distribution over topics using a Dirichlet prior, which is more proper than the Gaussian prior (used in NVDM) and logistic normal prior (used in LDA-VAE and ProdLDA). The usage of the Dirichlet prior in ATM make it could capture the multiplicity topic aspects from document collection and further obtain more coherent topics. 2.) The strong representation ability of the neural network makes the ATM could fit the true data distribution better than the traditional topic model and generate more coherent topics.}
\begin{table*}[!h]
\centering
\footnotesize
\label{table:stu}
\caption{The event examples extracted by ATM and LEM.}
\label{tbs:example_event}
\begin{tabular}{p{1.0cm}|c|l}
\hline
\multicolumn{1}{c|}{{\bfseries Events}}& \multicolumn{1}{c|}{\bfseries Method} &\multicolumn{1}{c}{\bfseries Representative Words}\\
\hline
\multirow{8}{*}{\makecell[c]{MH370}}&
\multirow{4}{*}{\makecell[cc]{ATM}}&org: \textbf{air airlines} ministry transport international \\
&&loc: \textbf{malaysia} beijing france vietnam dubai \\
&&per: \textbf{hishammuddin hussein} najib kerry lee \\
&&key: \textbf{search flight aircraft air plane} \\
\cline{2-3}
&
\multirow{4}{*}{\makecell[tc]{LEM}}&org: \textbf{airlines air} international transport government \\
&&loc: \textbf{malaysia} south korea beijing us \\
&&per: \textbf{hussein hishammuddin} fitch long park \\
&&key: \textbf{flight airlines plane} preliminary \textbf{search} \\
\hline
\multirow{8}{*}{\makecell[tc]{Saudi\\ MERS}}&
\multirow{4}{*}{\makecell[tc]{ ATM }}&org: \textbf{community} ministry \textbf{saudi} \textbf{healthcare} government \\
&&loc: \textbf{saudi} ontario iran canada jeddah \\
&&per: \textbf{president} obama jordan kerry walker \\
&&key: \textbf{health hospital patients disease medical} \\
\cline{2-3} &
\multirow{4}{*}{\makecell[tc]{ LEM}}&org: \textbf{saudi} jordan army eastern state \\
&&loc: east \textbf{saudi} jordan egypt israel \\
&&per: jordan \textbf{president} frank rob geldof \\
&&key: east middle \textbf{respiratory syndrome health} \\
\hline
\multirow{8}{*}{\makecell[tc]{Pakistan\\ vs. India}}&
\multirow{4}{*}{\makecell[tc]{ATM}} & org: \textbf{army kashmir} sharif taliban afghanistan \\
&&loc: \textbf{pakistan kashmir india} afghanistan islamabad \\
&&per: \textbf{sharif} kerry khan president lovell \\
&&key: \textbf{army peace chief region} province \\
\cline{2-3} &
\multirow{4}{*}{\makecell[tc]{LEM}}&org: \textbf{army kashmir} sharif government congress \\
&&loc: \textbf{pakistan kashmir} islamabad \textbf{india} delhi \\
&&per: \textbf{sharif} tsvangirai morgan dube biti \\
&&key: \textbf{army chief} vein news \textbf{peace} \\
\hline
\multirow{8}{*}{\makecell[tc]{Indian\\ Election}}&
\multirow{4}{*}{\makecell[tc]{ATM}}&org: \textbf{bjp party congress singh gandhi} \\
&&loc: \textbf{gujarat india varanasi delhi} seemandhra \\
&&per: \textbf{modi singh} gandhi naidu khan \\
&&key: \textbf{congress election candidate minister leader} \\
\cline{2-3} &
\multirow{4}{*}{\makecell[tc]{ LEM}}&org: \textbf{bjp congress party commission delhi} \\
&&loc: \textbf{delhi gujarat} modis \textbf{varanasi india} \\
&&per: \textbf{modi} gandhi \textbf{singh} modis president \\
&&key: \textbf{prime candidate election ministerial congress} \\
\hline
\multirow{8}{*}{\makecell[tc]{Taksim\\ Clash}}&
\multirow{4}{*}{\makecell[tc]{ ATM}}&org: \textbf{police} city \textbf{government} erdogan \textbf{union} \\
&&loc: \textbf{taksim istanbul} city \textbf{turkey union} \\
&&per: \textbf{erdogan} park walker quinn hall \\
&&key: \textbf{square protesters tear demonstrators street} \\
\cline{2-3} &
\multirow{4}{*}{\makecell[tc]{ LEM}} & org: \textbf{police} international \textbf{labor} central greenpeace \\
&&loc: \textbf{istanbul taksim turkey} rotterdam \textbf{union} \\
&&per: mark \textbf{erdogan} geldof park hall \\
&&key: \textbf{protesters square} international \textbf{gas water} \\
\hline
\end{tabular}
\end{table*}
To explore how topic coherence results vary with different topic numbers, we show in Figure \ref{fig:avg_curve} the average topic coherence of two datasets vs. different topic number settings. We can observe that ATM achieves better results compared to other baselines most of the time with 20, 30, 50 or 75 topics. However, when the topic number is 100, the performance gap between ATM and LDA diminishes and in some cases (e.g., C\_P and C\_A for the Grolier dataset), ATM gives slightly worse results compared to LDA, though it still largely outperforms all the other baselines. This might attribute to the increased network complexity due to the larger topic number setting.
From the above topic coherence evaluation results, it is clear that ATM is able to extract more coherence topics compared to baselines. To verify this qualitatively, we show examples of topics from all the models in Table \ref{tbs:example_topics}. These topics correspond to \emph{`airline'}, \emph{`trade'}, \emph{`music'}, \emph{`election'} and \emph{`film'} respectively. Words that do not seem to belong to its corresponding topic are highlighted in italic. It can be observed that the number of less semantically relevant words somewhat correlates with the coherence results observed earlier in Table \ref{tbs:average} and Figure \ref{fig:avg_bar}.
Unlike traditional topic models, the proposed ATM could learn the semantic embeddings of words apart from generating coherent topics. The weights matrix $W_{w}\in \mathbb{R}^{V\times S}$ contains the word-level semantic information, and each row could be viewed as the corresponding word embedding. Thus, we select the topic words of six topics from a 50-topic run on the NYtimes corpus and use the Principal Component Analysis (PCA) to project their word embeddings into a two-dimensional space. The visualization of these topic words is shown as Figure \ref{fig:topic_visualization}. We can clearly see that the words related to the \emph{`trade'} topic are grouped at the lower right corner, and the topic words of \emph{`religious'} are displayed at the top region. Besides, the words related to the topics \emph{`music'} and \emph{`film'} are close to each other, which is not surprising, since these topics are closely related.
\subsection{Open Domain Event Extraction}
To further prove the feasibility of porting ATM to tasks other than topic modeling, we apply it for open domain event extraction. For this task, an event is represented in a structured form as $<org, loc, per, key>$\textsuperscript{\ref {olpk}} \cite{zhou2015unsupervised}, with each of the elements in the quadruples represented by a list of words.
We use the pre-identified named entities\textsuperscript{\ref{ner}}, verbs, nouns and adjectives to construct the word set of organization, location, person and keywords. When using ATM for event extraction, these four word sets and the event-specific word distribution are used to generate the related topics. For example, the organization topic of an event could be obtained by sorting the words in the organization word set based on the corresponding probabilities in the event-specific word distribution learned by ATM. Table \ref{tbs:example_event} shows the example events extracted by ATM and LEM where the relevant words are highlighted in bold. It can be observed that ATM performs comparably with LEM. However, while LEM required the model-specific inference algorithm to be derived, ATM did not need any modification of its network architecture or parameter estimation procedure.
To validate the correctness of the extracted events, we retrieve the title of articles using the event-related words from ATM and obtain the following results:
\begin{itemize}
\item \emph{Missing Malaysia Airlines flight MH370: Government report suggests official search for plane did not begin until four hours after disappearance.}
\item \emph{Saudi Arabia finds 26 more cases of MERS, Egypt reports first sufferer}.
\item \emph{India's defence experts and politicos condemn Pak Army Chief's Kashmir statement}.
\item \emph{Top BJP leaders, Rajnath Singh, MM Joshi, Sushma Swaraj to campaign for Narendra Modi in Varanasi}.
\item \emph{Turkey May Day protests hit by tear gas near Taksim Square - Panorama}.
\end{itemize}
It is clear that the retrieved titles indeed correspond well with the extracted events by ATM.
\section{Conclusions}
\label{sec:length}
We have proposed a novel topic modeling approach based on adversarial training. The proposed approach, ATM, models the topics with Dirichlet prior and employs the generator network to learn the semantic patterns among latent topics. Apart from automatically generating latent topics from a text corpus, it could also produce word-level semantic representations as a side product. The experimental comparison with the state-of-the-art methods show that ATM achieves improved topical coherence results. Moreover, the feasibility of porting ATM for tasks other than topic modeling has been verified for open domain event extraction. {\color{black}In the future, we want to incorporate the sequential information contained in texts into GAN based topic modeling approaches and devise a topic driven sentence generation model. And an extension to cope with the data sparsity in short text is also our future work. Besides, another direction we are interested in exploring is to develop dynamic and correlated topic models based on adversarial training.}
\section{Acknowledgements}
We would like to thank anonymous reviewers for their valuable comments and helpful suggestions. {\color{black}This work was funded by the National Key Research and Development Program of China (2016YFC1306704), the National Natural Science Foundation of China (61772132), the Natural Science Foundation of Jiangsu Province of China (BK20161430).}
\bibliographystyle{unsrt}
\clearpage
\pagebreak
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 9,954 |
cmake_host_system_information(HOSTNAME)
| {
"redpajama_set_name": "RedPajamaGithub"
} | 9,538 |
{"url":"http:\/\/www.byond.com\/forum\/?post=147555","text":"ID:147555 \u00a0 Apr\u00a02\u00a02004, 9:48\u00a0pm I'm making a mess of commands that work through the browser or a popup window (It's just so much niftier), but I can't seem to make the \\icon[] instruction work.\n <-> Apr\u00a02\u00a02004, 9:50\u00a0pm The \\icon macro doesn't work in the browser. Instead you need to use browse_rsc() to send an image, and use the filename you defined for it as the src attribute for your img tag. Lummox JR\n <-> Apr\u00a02\u00a02004, 9:51\u00a0pm In response to Lummox JR Ah. Yet another simple solution that slipped my mind.\n <-> Apr\u00a02\u00a02004, 10:09\u00a0pm In response to Enigmaster2002 Enigmaster2002 wrote: Ah. Yet another simple solution that slipped my mind. Well, it's not immediately obvious. To that end I've added a note to reference entry that will appear in a future update. Lummox JR\n <-> Apr\u00a02\u00a02004, 10:18\u00a0pm (Edited on\u00a0Apr\u00a02\u00a02004, 10:49\u00a0pm) In response to Lummox JR Arg, not as simple as I thought. What I'm doing is listing all of a certain mob in the world and I'm including their icon, but the problem is that there's going to be many many instances of one mob with completely different icons.\n <-> Apr\u00a02\u00a02004, 10:23\u00a0pm In response to Enigmaster2002 Your code doesn't seem to use browse_rsc() at all, which you need to send an icon to the client before using browse(). In browse() just use whatever filename you chose in browse_rsc() as the filename for the image. Lummox JR\n <-> Apr\u00a02\u00a02004, 10:33\u00a0pm (Edited on\u00a0Apr\u00a02\u00a02004, 10:50\u00a0pm) In response to Lummox JR Okay, I'm getting somewhere, I tried using browse_rsc and I forgot the instruction to output to src the first time... anyways... I can cache the icon, but I can't figure out how to store the icon_state; I'm getting the first state in its' specified icon. Also, is there a way to remove these files after they've been intitialized and displayed? I don't want to distribute people's icons to my moderators.\n <-> Apr\u00a02\u00a02004, 10:42\u00a0pm In response to Enigmaster2002 Enigmaster2002 wrote: Okay, I'm getting somewhere, I tried using browse_rsc and I forgot the instruction to output to src the first time... anyways... I can cache the icon, but I can't figure out how to store the icon_state; I'm getting the first state in its' specified icon. You're doing well so far. To specify an icon state you need to use new\/icon() in the first argument to browse_rsc(). This is okay even though an \/icon datum isn't the same as a regular icon, which normally does matter for something. browse_rsc(new\/icon('myicon.dmi', \"mystate\"), \"myicon.png\") I usually name my images .png for the browser, because that's what they become when exported. But the browser recognizes them regardless. Note that the result is not animated. Also, is there a way to remove these files after they've been intitialized and displayed? I don't want to distribute people's icons to my moderators. This isn't a huge concern; it's only as much a worry as taking screen shots, which anybody can do. Lummox JR\n <-> Apr\u00a02\u00a02004, 10:48\u00a0pm In response to Lummox JR Success! Thanks Lummox, I haven't a clue where I'd get without you.\n <-> Apr\u00a03\u00a02004, 3:36\u00a0pm (Edited on\u00a0Apr\u00a03\u00a02004, 5:34\u00a0pm) In response to Enigmaster2002 Hmm, it appears I need to delete these files in the cache regardless; if I were to use this verb, it lists all the custom-made cars. I delete two of them, and now two of the images in the cache are obsolete, because browse_rsc() doesn't do anything if the specified file is already in the user's cache.\n <-> Apr\u00a03\u00a02004, 4:43\u00a0pm In response to Enigmaster2002 Eeeek! Long unbreaking lines on forum.... arrrgh... *dies* =) You can split those up onto multiple lines using slashes, you know, like this: html_doc+=\"all the HTML stuff, etc. goes here; but now this line is getting a bit long, so.... \\ we jump down onto another line. All the blank space before \"we\" will be cut off.\" Much friendlier. Anyway, to your problem - I believe that if the file already exists in the cache, but the file your sending is different, DS will detect that and replace the old cached file with the new one. I might be wrong on that though. A way to avoid the issue altogether is to get a unique ID number for each car image. For example, in my latest project the NPC dialogue images are called \"talk\\ref[src].dmi\" - where src is the NPC mob. So there'll be one \"talk[whatever].dmi\" file in the cache for each NPC the player has talked to, and each one will be unique to each NPC mob because I've used the \\ref macro to get a unique reference to the mob.\n <-> Apr\u00a03\u00a02004, 5:46\u00a0pm In response to Crispy Obsolete image files are still being used. for(var\/mob\/Car\/Custom\/C in world) src << browse_rsc(new\/icon(C.icon, \"[C.icon_state]\"), \"\\ref[C]car.png\") html_doc += \"\n[C.name]\n\" \/\/etc etc \n <-> Apr\u00a03\u00a02004, 6:35\u00a0pm In response to Enigmaster2002 Hmm... really? Not good. Try basing the name of the file on the icon state, that should work until you change the icon state itself.\n <-> Apr\u00a03\u00a02004, 7:04\u00a0pm In response to Crispy I also add a string based on world.realtime to the filename. As long as people don't look at a different icon with the same name within 6 seconds or so, it will have a new name and use the new icon.\n <-> Apr\u00a04\u00a02004, 5:45\u00a0am In response to Crispy Crispy wrote: Eeeek! Long unbreaking lines on forum.... arrrgh... *dies* =) You can split those up onto multiple lines using slashes, you know, like this: html_doc+=\"all the HTML stuff, etc. goes here; but now this line is getting a bit long, so.... \\ > we jump down onto another line. All the blank space before \"we\" will be cut off.\" Much friendlier. I my self am a fan of, html={\" \"} That way, I can write the html just as if it were in a txt file. ^_^\n <-> Apr\u00a04\u00a02004, 9:36\u00a0am In response to Goku72 Goku72 wrote: I my self am a fan of, html={\" \"} That way, I can write the html just as if it were in a txt file. ^_^ I use the {\"\"} format so that I don't have to escape quotes in the code, but I also end each line with \\ so that it ignores the carraige return and leading whitespace on the next line for more compact HTML files. It's not much, but each and every character that isn't tieing up bandwidth is a triumph in my book. :)\n <-> Apr\u00a05\u00a02004, 2:51\u00a0am In response to Shadowdarke Shadowdarke wrote: It's not much, but each and every character that isn't tieing up bandwidth is a triumph in my book. :) You, strange, strange individual...=P\n <-> Apr\u00a05\u00a02004, 7:31\u00a0am In response to Crispy Crispy wrote: Anyway, to your problem - I believe that if the file already exists in the cache, but the file your sending is different, DS will detect that and replace the old cached file with the new one. I might be wrong on that though. You're not; that's correct. Lummox JR","date":"2015-04-25 19:51:48","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 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\": 0.49242115020751953, \"perplexity\": 2104.9006331488486}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 20, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2015-18\/segments\/1429246651471.95\/warc\/CC-MAIN-20150417045731-00132-ip-10-235-10-82.ec2.internal.warc.gz\"}"} | null | null |
{"url":"https:\/\/deamentiaemundi.wordpress.com\/2014\/11\/18\/multiple-precision-in-javascript-fast-division-ii\/","text":"# Multiple Precision in JavaScript: Fast Division\u00a0II\n\n## Barrett Division\n\nThe Barrett division uses Barrett\u2019s correction algorithm (Barrett reduction) meant for polynomial division, which are in this case the big-integers. It does work for fractions of the form $N \\leq 2D$ only so it needs some precomputations for other fractions.\n\nIt basically works by computing a low-precision reciprocal of the denominator and using slices of it with slices of the numerator to build the result.\n\nBigint.prototype.barrettDivision = function(bint) {\nvar m = this.highBit() + 1;\nvar n = bint.highBit() + 1;\nvar mu, largemu, start, q, r, mask, digit, c;\nif (m < n) {\nreturn [new Bigint(0), this.copy()];\n} else if (m <= 2 * n) {\nmu = bint.inverse(m - n);\nreturn this.barretDivisionCorrection(bint, mu);\n} else {\n\/\/ do school-division with big-digits of a size chosen such that\n\/\/ the condition N<=2*D holds.\n\n\/\/ Overall mu, gets splitted later\nlargemu = bint.inverse(n);\n\/\/ choose the startpoint as an integer multiple of n\nstart = Math.floor(m \/ n) * n;\nq = new Bigint(0);\n\/\/ first part of the new numerator\nr = this.rShift(start);\nwhile (start > 0) {\nstart -= n;\n\/\/ Snip a large digit from the LSB side of the original numerator\n\/\/ make room for it in the new numerator\nr.lShiftInplace(n);\n\/\/ put the digit there\n\/\/ get the right amount of mu (still under the condition N<=2*D)\nmu = largemu.rShiftRounded(2 * n - (r.highBit() + 1));\n\/\/ correct the result\nc = r.barretDivisionCorrection(bint, mu);\n\/\/ make room for the quotient-part\nq.lShiftInplace(n);\n\/\/ put it there\n\/\/ the remainder is the new numerator\nr = c[1];\n}\nreturn [q, r];\n}\n};\n\n\nThe reciprocal gets calculated with some rounds of Newton-Raphson root-finding with the precision of the denominator. This already gives a hint that this algorthm works best for fractions of the form $N \\leq kD$ with $k \\ggg 2$ and a sufficiently large denominator.\nSome tests resulted in a breakeven to the Burnikel-Ziegler algorithm described earlier at $N = 10 D$ with a bit length of about 50k for the denominator but not only might your mileage vary but it is also different with different relations and sizes.\n\nvar BARRETT_NEWTON_CUTOFF = 100\nBigint.prototype.inverse = function(n) {\nvar m = this.highBit() + 1;\nvar giantsteps;\nvar steps, gs, gs0, i;\nvar r, s, t, u, w, a, b;\n\/\/ truncated division\nif (n <= BARRETT_NEWTON_CUTOFF) {\nvar ret = new Bigint(1);\nret.lShiftInplace(2 * n);\nreturn ret.div(this.rShiftRounded(m - n));\n}\n\/\/ some rounds of Newton-Raphson\ngiantsteps = computeGiantsteps(MP_DIGIT_BIT >> 1, n, 2);\nsteps = giantsteps.length;\nr = new Bigint(1);\nr.lShiftInplace(2 * giantsteps[0]);\nr = r.div(this.rShiftRounded(m - giantsteps[0]));\ngs0 = giantsteps[0];\nfor (i = 0; i < steps; i++) {\ngs = giantsteps[i];\na = r.lShift(giantsteps[i] - gs0 + 1);\nb = r.sqr().mul(this.rShift(m - giantsteps[i])).rShift(2 * gs0);\nr = a.sub(b);\ngs0 = giantsteps[i];\n}\nreturn r;\n};\n\n\nThe function computeGiantsteps(start, end, stepsize) is a small function to calulate the precision needed for the respective iteration rounds.\n\n\/\/ Computes iteration steps for e.g. Newton-Raphson\n\/\/ \"stepsize\" is the length of the steps and a multiplicator.\n\/\/ For example stepsize=2 for quadratic convergences (Newton), stepsize=3\n\/\/ for cubic ones (Housholder), etc.\n\/\/ Yep, just like the similarily named Python function\nfunction computeGiantsteps(start, end, stepsize) {\nvar ret = [ end ],\ni = 1;\nif (arguments.length != 3) {\nreturn MP_VAL;\n}\nwhile (true) {\nif (ret[ ret.length - 1 ] <= start * stepsize) {\nbreak;\n}\nret[ i++ ] = Math.floor(ret[ret.length - 1] \/ stepsize) + 2;\n}\nreturn ret.reverse();\n}\n\n\nThe actual Barrett reduction is astonishingly small. It is just:\n\nBigint.prototype.barretDivisionCorrection = function(b, mu) {\nvar m = this.highBit() + 1;\nvar n = b.highBit() + 1;\n\nvar digit = this.rShift(n - 1);\nvar q = digit.mul(mu).rShift(m - n + 1);\nvar r = this.sub(b.mul(q));\n\nwhile (r.sign < 0 || r.cmp(b) != MP_LT) {\nif (r.sign < 0) {\nq.decr();\n} else {\nr = r.sub(b);\nq.incr();\n}\n}\nreturn [q, r];\n};\n\n\nThis works quite well if the reciprocal is not too much off but there are exceptions. Assuming that I did not make an error in the implementation teh following numbers made with the function Bigint.random() from my Bigint implementation\n\nvar C = new Bigint(0);\nvar D = new Bigint(0);\nvar N = 100;\nC.random(100 * 26 * N, 123);\nD.random(40 * 26 * N , 124);\n\n\nit will \u201chang\u201d in the loop trying to correct a ~5,300 limbs large result with a 4,000 limbs \u201csmall\u201d denominator by way of stepwise subtraction.\nFunnily, it works with\n\nvar N = 100;\nC.random(99 * 26 * N, 123);\nD.random(40 * 26 * N , 124);\n\n\nand\n\nvar N = 100;\nC.random(101 * 26 * N, 123);\nD.random(40 * 26 * N , 124);\n\n\nI was not able to find another example of this behaviour but if you know the reason: please let me know!","date":"2017-10-18 12:59:38","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\": 4, \"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\": 0.4014776349067688, \"perplexity\": 8552.455556279578}, \"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-2017-43\/segments\/1508187822966.64\/warc\/CC-MAIN-20171018123747-20171018143747-00170.warc.gz\"}"} | null | null |
Below is a playlist of the Fragment in Android Tutorials for Absolute Beginners. Fragment is one of the most useful feature in Android in every Android API. In one Activity, you can have multiple Fragments. If you have Android Phone like Samsung Note 8, LG, or OPPO or other manufactures, you will can a swipe menu from the left of your screen. That is called Navigation Drawer. And the bottom icons at the below of the screen, it is called Bottom Navigation Bar. And the icons tabs at the top below the toolbar is called Tabbed Activity bar. These features all need to use Fragment. If you wanna use Fragment in your real world application projects to impress your client, here is the full list compiled from my YouTube Channel Oum Saokosal. | {
"redpajama_set_name": "RedPajamaC4"
} | 733 |
@interface ALPreferenceViewController : NSViewController {
IBOutlet NSTextField* hostField;
IBOutlet NSTextField* portField;
IBOutlet NSSecureTextField* passwordField;
}
@property (retain) NSString* versionString;
@property (retain) NSString* configuredHost;
@property (retain) NSString* configuredPassword;
- (IBAction)didEnterHostname:(id)sender;
- (IBAction)didEnterPassword:(id)sender;
+ (void)updatePassword:(NSString*)newPassword forHost:(NSString*)host;
+ (NSString*)retrievePasswordForHost:(NSString*)host;
+ (NSString*)messageForStatusCode:(OSStatus)status;
@end
| {
"redpajama_set_name": "RedPajamaGithub"
} | 7,341 |
\section{Introduction} \label{ch:intro}
Stock price predictability has been an important research topic in both academia and industry since it reflects our economic and social organization and the stock market plays an important role in the world economy.
Although the dynamic nature of our economic activity makes it harder to predict future stock prices, significant efforts are made to explain the dynamism.
From this perspective, stock markets have often been modeled as a complex, evolutionary, and nonlinear dynamical system~\cite{cont2001empirical,hommes2001financial,borges2010efficient}.
Due to the dynamic nature of our economic activity, machine learning is an increasingly popular tool with some success in predicting stock prices~\cite{henrique2019literature,nakagawa2020ric,imajo2021deep}.
This is because many machine learning methods can automatically capture nonlinear relationships between relevant factors from the input data~\cite{cavalcante2016computational,chen2018integrating}.
One promising method among them is the Trader-Company~(TC) method, a metaheuristic stock prediction model that mimics the roles of an actual financial institute and traders within it~\cite{ito2021trader}.
The TC method consists of two components, the predictor called a Trader and the aggregation algorithm called a Company.
The TC method considers the dynamism of the stock market and has both high predictive power and interpretability.
Stock prediction methods based on machine learning, including the TC method, have been concentrating on estimating and improving point predictions.
However, point predictions in the absence of uncertainty estimates lack credibility quantification and raise concerns about safety.
Considering the significant consequences of decision-making in financial practice, quantifying the uncertainty of predictions is proving to be a key step in putting machine learning models into practice~\cite{jaeger2021understanding,philps2021interpretable}.
For example, most trades (80\%) are automated~\cite{URL} and the algorithmic tradings based on the machine learning method have played a crucial role in financial markets.
The algorithmic tradings focused on the large investment universe of stocks and sampled data at very high frequencies (intraday or tick by tick).
In such an environment with a large amount of data, it is important for practitioners to quantify the uncertainty of predictions.
Therefore, the challenge in this paper is to make an investment strategy with high predictive power and can quantify the uncertainty of predictions.
To formalize our discussion of the uncertainty of predictions, we will rely on probabilistic modeling. Probabilistic modeling, which can provide probabilistic predictions and uncertainty estimations simultaneously, has been a fundamental tool in machine learning and related fields~\cite{bishop2006pattern}.
Most of these studies rely on a Bayesian framework, and their applications to complex models such as neural networks and decision tree models have been actively studied. Among them, one standard approach is directly estimating the distribution of predictions. However, it has been pointed out that these methods tend to make predictions biased toward one specific mode~\cite{Fort2019,malinin2021uncertainty}.
Another approach is ensemble-based uncertainty estimation, which focuses on the dispersion of predictions~\cite{pmlr-v48-gal16,NIPS2017_9ef2ed4b,malinin2021uncertainty}.
These methods are experimentally confirmed to be more robust to dataset shift than methods that explicitly learn distributions~\cite{NIPS2017_9ef2ed4b,malinin2021uncertainty}. That is effective in predicting a dynamic environment, such as financial markets.
Based on the above studies, we propose a novel approach called the Uncertainty Aware Trader-Company~(UTC) method.
The core idea of this approach is to combine the strengths of both frameworks by merging the TC method with the
probabilistic modeling framework.
We expect to retain the predictive power and interpretability of the TC method while capturing the uncertainty.
To be more concrete, we propose the method of estimating the prediction's uncertainty from the Traders' output.
\if 0
We separately estimate the uncertainty of the prediction into inter-trader, which represents the uncertainty on weighting, and intra-trader variance, which represents the uncertainty of effectiveness of the whole traders' strategies.
The inter-trader variance corresponds to the uncertainty on weighting given a trader's strategy, while the intra-trader variance addresses uncertainty about the effectiveness of the whole traders' strategies.
\fi
We estimate the variance by the two-stage algorithm: estimation by the Trader and estimation by the Company.
The Trader estimates the uncertainty on weighting given a trader's strategy while the Company estimates the uncertainty about the effectiveness of the whole traders' strategies.
We theoretically show that our uncertainty estimation reflects the posterior variance of predictive return given the past return.
Also, we prove that the predictive return of the UTC method is identical to that of the original TC method under some assumptions in the prior distribution of traders. That means our method does not introduce additional biases.
We conduct a comprehensive evaluation of our approach based on the synthetic and real market datasets.
Our evaluation of the synthetic datasets demonstrates that the UTC method can detect situations where the uncertainty increases and the prediction is difficult.
We also confirmed that our method can detect abrupt changes of data generating distributions.
Furthermore, experiments using actual data show that the investment strategy based on our UTC method gains stable returns while suppressing risks compared to existing investment strategies.
The remainder of this paper is organized as follows.
Section 2 describes our problem formulation and TC method briefly. Section 3 presents our UTC method and theoretical properties. Section 4 performs experiments. Section 5 reviews the related work, and Section 6 is the conclusion.
\section{Preliminary} \label{sec:prelim}
In this section, we formulate our problem and then provide the overview of the TC method~\cite{ito2021trader}.
\subsection{Problem Formulation} \label{sec:problem}
Our problem is to forecast future returns of stocks based on their historical observations.
Let $\price[i][t]$ be the price of stock $i$ at time $t$ where $1 \le i \le S$ denotes the index of stocks and $0 \le t \le T$ denotes the time index.
We use the logarithmic returns of stock prices as input features of models; we denote the one period ahead return of stock $i$ by
\begin{align}\label{logret}
\ret[i][t] := \log (\price[i][t] / \price[i][t-1]) \approx \frac{\price[i][t]-\price[i][t-1]}{\price[i][t-1]}.
\end{align}
Then we define the returns of stock $i$ and returns of multiple stocks $1 \le i \le j \le S$ from time period $u$ to $v~(u \leq v)$ by
\begin{align}\label{vecret}
&\vecret[i][u:v]:=(\ret[i][u],\cdots,\ret[i][v]),\\
&\vecret[i:j][u:v]:=(\ret[i][u:v],\cdots,\ret[j][u:v])
\end{align}
We can formulate our main problem as follows.
\begin{problem}[one-period-ahead prediction]\label{main_problem}
We sequentially observe the returns $\ret[i][t]$ ($1 \le i \le S$) at every time $0 \le t \le T-1$.
We predict the one-period-ahead return $\ret[i][t+1]$ and estimate its predictive uncertainty $\sigma_i[t+1]$ based on the past $t$ returns $\vecret[1:S][0:t]$.
That is, the one-period-ahead return and uncertainty prediction can be written as
\begin{equation}\label{predret}
\predret[i][t+1], \hat{\sigma}_i[t+1] = f_{t}(\vecret[1:S][0:t])
\end{equation}
for some function $f_t$.
The purpose of this study is to find $f_t$ whose output $\predret[i][t+1]$ approximates the true return $\ret[i][t+1]$ and predictive standard deviation $\hat{\sigma}_i[t+1]$ approximates the estimation error between $\predret[i][t+1]$ and $\ret[i][t+1]$ well
\end{problem}
\subsection{Trader Company Method}\label{sec:tc}
This section introduces the Trader-Company method briefly. The TC method consists of two main components, \textit{Traders} and \textit{Companies}. A Trader predicts the returns using a simple model expressing realistic trading strategies, while a Company combines strategies from multiple Traders into a single prediction.
A Company applies an evolutionary algorithm that mimics the role of financial institutes as employers of traders.
During training, a Company generates promising new candidates for Traders and deletes poorly performing ones.
We provide more detailed definitions and training algorithms for the TC method.
\begin{table}
\caption{Notation.}
\label{tab:notation}
\begin{center}
\begin{tabular}{clc}
\hline
\textbf{Notation} & \multicolumn{1}{c}{\textbf{Meaning}} & \textbf{Def.} \\
\hline
$\price[i][t]$ &
\begin{tabular}{l}
stock price of stock $i$ at time $t$\\
where $1\le i \le S, 0\le t \le T$
\end{tabular}& $\S$ \ref{sec:problem} \\
$\ret[i][t]$ & logarithmic return of $i$ at $t$& \eqref{logret}\\
$\vecret[i][u:v]$ & $(\ret[i][u],\cdots,\ret[i][v])$ & \eqref{vecret} \\
$\vecret[i:j][u:v]$ & $(\ret[i][u:v],\cdots,\ret[j][u:v])$ & \eqref{vecret} \\
$\predret[i][t+1],\hat{\sigma}_i[t+1]$ & predicted value and standard deviation of $\ret[i][t]$& \eqref{predret}\\
\begin{tabular}{l}$\numterm,P_j,Q_j,D_j$\\$F_j,A_j,O_j$\end{tabular}
& hyper-parameters of Traders & \eqref{TCtraders}\\
\hline
\end{tabular}
\end{center}
\end{table}
\subsubsection{Traders - Simple Prediction Module} \label{subsec:TCtraders}
\begin{definition}
A Trader is a predictor of one period ahead returns defined as follows.
Let $\numterm$ be the number of terms in the prediction formula.
For each $1 \le j \le \numterm$, we define $P_j,Q_j$ as the indices of the stock to use, $D_j,F_j$ as the delay parameters, $O_j$ as the binary operator, $A_j$ as the activation function, and $w_j$ as the weight of the $j$-th term.
Then, the Trader predicts the return value $\ret[i][t+1]$ at time $t + 1$ by the formula
\begin{align}\label{TCtraders}
f_{\theta,w}(\vecret[1:S][0:t])= \sum_{j=1}^\numterm w_j A_j(O_j(\ret[P_j][t-D_j],\ret[Q_j][t-F_j])).
\end{align}
where $\theta$ is the parameters of the Trader:
$$\theta := \{\numterm,\{P_j,Q_j,D_j,F_j,O_j,A_j\}_{j=1}^\numterm\}.$$
\end{definition}
For activation functions $A_j$, the TC method used standard activation functions used in deep learning, such as the identity function, hyperbolic tangent function, hyperbolic sine function, and Rectified Linear Unit (ReLU~\cite{nair2010rectified}).
For the binary operators $O_j$, we use several arithmetic binary operators (e.g., $x + y$, $x - y$, and $x \times y$), the coordinate projection, $(x, y) \mapsto x$, the max/min functions, and the comparison function $(x > y) = \mathop{\rm sign}\limits(x - y)$.
The formula \eqref{TCtraders} is interpretable in that it has a similar form to typical human-generated trading strategies ~\cite{Kakushadze2018}.
Second, the Trader model has sufficient expressive power.
The Trader has various binary operators as fundamental units, which allows it to represent any binary operations commonly used in practical trading strategies. Besides, the model also
encompasses the linear models since we can choose the projection operator $(x, y) \mapsto x$ as $O_j$.
The Trader is optimized to maximize the cumulative return of its strategy.
\begin{align}\label{traderloss}
\mathop{\rm arg~max}\limits_{\theta,w} \sum_{u}\mathop{\rm sign}\limits(f_{\theta,w}(\vecret[1:S][0:u]))\cdot \ret[i][u+1]
\end{align}
Since the parameter $\theta$ is a discrete variable, standard optimization methods with derivatives are difficult to apply.
Therefore, the TC method introduces an evolutionary algorithm driven by Company models.
\subsubsection{Companies - Optimization and Aggregation Module}\label{sec:TCcompany}
\begin{algorithm}[ht]
\caption{Educate algorithm of Company in TC}
\algrenewcommand\algorithmicrequire{\textbf{Input:}}
\algrenewcommand\algorithmicensure{\textbf{Output:}}
\label{TCcompanyeducate}
\begin{algorithmic}[1]
\Require $\vecret[1:S][0:t]$:stock returns before $t$
\Require Traders. $N$ : the number of Traders. $Q$: ratio.
\Ensure Traders
\Function{CompanyEducate}{}
\State{$R_n\Leftarrow R(f_{\theta_n,w_n},\vecret[1:S][0:t],\ret[i][0:t+1])$}
\Comment{Trader's return \eqref{traderloss}}
\State{$R^* \Leftarrow$ bottom $Q$ percentile of $\{R_n \}$}
\For{$n \in \{m| R_m \le R^* \}$}
\Comment{for all bad traders}
\State{Update $w_i$ in (\ref{TCtraders}) by least squares method}
\EndFor\\
\Return{Traders}
\EndFunction
\end{algorithmic}
\end{algorithm}
\begin{algorithm}[ht]
\caption{Prune-and-Generate algorithm of Company}
\algrenewcommand\algorithmicrequire{\textbf{Input:}}
\algrenewcommand\algorithmicensure{\textbf{Output:}}
\label{TCcompanygen}
\begin{algorithmic}[1]
\Require $\vecret[1:S][0:t]$:stock returns before $t$, F: \# of fit times
\Require $N$: the number of Predictors. $Q$: ratio.
\Ensure $N'$ Predictors
\State{$\theta_n,w_n \sim $ Uniform Distribution}
\For{$k=1,\cdots ,F$}
\State{$R_n \Leftarrow R(f_\Theta,\vecret[1:S][0:t],\ret[i][0:t+1])$}
\Comment{Trader's return \eqref{traderloss}}
\State{$R^* \Leftarrow$ bottom $Q$-percentile of $\{ R_n \}$}
\State{$\{(\theta_j,w_j)\}_j \Leftarrow \{(\theta_n,w_j) | R_n \ge R^* \}$}
\Comment{Pruning}
\State{$\{(\theta_j,w_j)\}_{j=1}^{N'} \sim$ GM fitted to $\{(\theta_j,w_j)\}_j$ *}
\Comment{Generation}
\EndFor\\
\Return{$N'$ Predictors with $\{(\theta_j,w_j)\}_{j=1}^{N'}$}
\end{algorithmic}
* If the parameter is an integer, we round it off.
\end{algorithm}
In this framework, a Company maintains $N$ Traders that act as weak learners or feature extractors and aggregate them. Given $N$ Traders specified by parameters $\theta_1, \ldots,\theta_N$,
$w_1, \ldots, w_N$ and the past observations of stock returns $\vecret[1:S][0:t]$, a Company predicts the one-period-ahead return by
\[
\hat{r}[t + 1] =
\mathrm{Aggregate}(f_{\theta_1,w_1}, \ldots, f_{\theta_n,w_n}).
\]
Here, we employed the simple averaging
$$\frac{1}{N} \sum_{n = 1}^N f_{\theta_n,w_n}(\vecret[1:S][0:t])$$
for $\mathrm{Aggregate}$ function.
The Company should maintain the average quality as well as the diversity of the Traders' strategies to achieve low training errors whilst avoiding overfitting.
For this purpose, the TC method introduced the Educate algorithm (Algorithm \ref{TCcompanyeducate}) and the Prune-and-Generate algorithm (Algorithm \ref{TCcompanygen}), which update the weights and formulae of Traders, respectively.
\begin{description}
\item[\textbf{Educating Traders}:] ~\\
Recall that a Trader \eqref{TCtraders} is a linear combination of $M$ mathematical formula.
We update the weights $\{ w_j \}_j $ by the least-squares method.
\item[\textbf{Pruning Traders and generating new candidates}:]~\\
Since the discrete parameters such as stock indices are difficult to optimize, they update these parameters by Algorithm \ref{TCcompanygen}.
First, we evaluate the (cumulative) returns of the Traders and remove them with relatively low returns.
Then, we generate new Traders by randomly fluctuating the existing Traders with good performances, i.e., we fit continuous Gaussian mixture distribution, draw new parameters from it and discretize the generated samples for discrete indices.
\end{description}
\begin{figure*}[t]
\begin{center}
\includegraphics[width=2.0\columnwidth]{figure/UTC_process.png}
\caption{Overview of prediction procedure in our Uncertainty Aware Trader-Company Method}
\label{fig:UTCprocess}
\end{center}
\end{figure*}
\section{Uncertainty Aware Trader-Company Method}
In this section, we present a \textit{Uncertainty Aware Trader-Company Method}, which extends the Trader-Company method to consider the uncertainty.
As with the original TC method, our method consists of two main components, \textit{Traders} and \textit{Companies}.
We propose extensions in three major aspects to exploit the uncertainty in the Trader's prediction and the diversification of Traders' prediction.
We show the overview of our method in Figure \ref{fig:UTCprocess}.
\subsection{Trader} \label{sec:trader}
We basically follow the definition of traders in TC methods in section \ref{subsec:TCtraders}.
The difference from the TC method is that we assume each parameter $\theta_n$ and $w_n$ is randomly chosen from the empirical distribution $\Theta$.
As mentioned in the introduction, few solid strategies will always be effective due to the volatile and uncertain behavior of the financial market. It is important to estimate the reliability of the strategies in such an environment. Therefore, instead of learning a deterministic strategy, we learn a probability distribution over strategies to estimate the uncertainty of the prediction.
In other words, our goal is to find a distribution $\Theta$ which approximates the posterior distribution.
Next, we will discuss how the Trader estimates the predictive uncertainty.
Since we assumed the weights of the Trader are chosen from the empirical distribution $\Theta$, the output of the Trader is also probabilistic.
Here, we assumed that the distribution of weights follows a multivariate normal distribution with a mean of $m$ and the covariance of $\Sigma$.
We can calculate the mean and variance for the Trader's output as follows.
\begin{align}\label{btctrader}
& \mu_n = m^\top z_n,~~\sigma_n^2 = z_n^\top \Sigma z_n
\end{align}
Here, $z_n$ is the signal defined as
\begin{align} \label{eq:signal}
z_n[j] = A_j(O_j(\ret[P_j][t-D_j],\ret[Q_j][t-F_j])).
\end{align}
The variance of Trader's prediction estimated here is used to predict the return and estimate the variance of the Company prediction.
Next, we will discuss how the Company optimizes the distribution $\Theta$.
\subsection{Company}\label{sec:company}
We basically employ the same framework as the TC method; our Company maintains $N$ traders that act as weak learners or feature extractors and aggregate them.
However, each Trader estimates not only the expected value but the variance of the predicted return.
First, we introduce how to estimate the predictive return and its uncertainty based on these outputs.
Given $N$ Traders
with parameters $\theta_1, \ldots, \theta_n$ and the past observations of stock returns $\vecret[1:S][0:t]$, a Company predicts the one-period-ahead return $\hat{r}[t + 1]$.
The original TC method applies the most naive aggregating method, simply averaging.
If we apply this, we could calculate the mean and variance of the prediction by
\begin{align}
&\mu = \frac{1}{N}\sum_{n \in \{1,2,\cdots,N\}} \mu_n,~~\sigma = \sqrt{\sigma^2_{intra}+\sigma^2_{inter}}, \nonumber\\
& \sigma^2_{intra} = \frac{1}{N}\sum_{n \in \{1,2,\cdots,N\}} (\mu_n - \mu)^2,\nonumber \\ & \sigma^2_{inter} = \frac{1}{N}\sum_{n \in \{1,2,\cdots,N\}} \sigma_n^2. \label{eq:variance}
\end{align}
For clarity, this procedure is presented in Algorithm \ref{companypredict}.
\begin{algorithm}[ht]
\caption{Prediction and uncertainty estimation algorithm of Company in UTC}
\label{companypredict}
\algrenewcommand\algorithmicrequire{\textbf{Input:}}
\algrenewcommand\algorithmicensure{\textbf{Output:}}
\begin{algorithmic}[1]
\Require $\vecret[1:S][0:t]$: stock returns before $t$, $\{(\theta_n, w_n)\}_{n=1}^N$ : Traders
\Ensure $\mu[i][t+1]$: predicted return of stock $i$ at $t$, $\sigma^2[i][t+1]$: estimated variance of return of stock $i$ at $t$
\Function{CompanyPrediction}{}
\For{$n = 1,\cdots,N$}
\State{$\mu_n$, $\sigma_n \Leftarrow f_{\theta_n,w_n}(\vecret[1:S][0:t])$
}
\Comment{Prediction by Trader \eqref{btctrader}}
\EndFor
\State{$\mu = \frac{1}{N}\sum_{t \in \{1,2,\cdots,N\}} \mu_t$}
\State{$\sigma^2_{intra} = \frac{1}{N}\sum_{t \in \{1,2,\cdots,N\}} (\mu_t - \mu)^2$}
\State{$\sigma^2_{inter} = \frac{1}{N}\sum_{t \in \{1,2,\cdots,N\}} \sigma_t^2$}
\State{$\sigma = \sqrt{\sigma^2_{intra}+\sigma^2_{inter}}$}\\
\Return {$\mu, \sigma$}
\EndFunction
\end{algorithmic}
\end{algorithm}
Next, we explain how the Company optimizes Traders.
First, with respect to strategy selection, we perform an iterative process of pruning and generating, similar to the TC method.
Then, we optimize the weight distribution for each Trader in the UTC method.
We use MAP estimation since its solution can be analytically obtained because we assumed a simple normal distribution for the weights.
It helps reduce the computational cost of the UTC method because this optimization process is repeated for many traders.
\begin{algorithm}[ht]
\caption{Educate algorithm of Company in UTC}
\algrenewcommand\algorithmicrequire{\textbf{Input:}}
\algrenewcommand\algorithmicensure{\textbf{Output:}}
\label{companyeducate}
\begin{algorithmic}[1]
\Require $\vecret[1:S][0:t]$:stock returns before $t$
\Require Traders. $N$ : the number of Traders. $Q$: ratio.
\Ensure Traders
\Function{CompanyEducate}{}
\State{$R_n\Leftarrow R(f_{\Theta_n},\vecret[1:S][0:t],\ret[i][0:t+1])$}
\Comment{Trader's return \eqref{traderloss}}
\State{$R^* \Leftarrow$ bottom $Q$ percentile of $\{R_n \}$}
\For{$n \in \{m| R_m \le R^* \}$}
\Comment{for all bad traders}
\State{Update traders' weight distributions by MAP estimation}
\EndFor\\
\Return{Traders}
\EndFunction
\end{algorithmic}
\end{algorithm}
Overall, they train the TC and UTC models as follows.
\begin{enumerate}
\item Educate a fixed proportion of poorly performing Traders by Algorithm \ref{TCcompanyeducate} in TC and Algorithm \ref{companyeducate} in UTC.
\item Replace a fixed proportion of poorly performing Traders with random new Traders by Algorithm \ref{TCcompanygen}.
\item If the aggregation function $\mathrm{Aggregate}$ has trainable parameters, update them using the data $\vecret[1:S][t_1: t_2]$ and any optimization algorithm.
\item Predict future returns by averaging in TC and Algorithm \ref{companypredict} in UTC.
\end{enumerate}
\subsection{Theoretical Properties}\label{sec:comp-tc}
Here we discuss the following properties of our UTC method.
\begin{description}
\item[Proposition\ref{prop1}:]~\\ The UTC method can estimate the uncertainty~(posterior variance) of the prediction.
\item[Proposition\ref{prop2}:]~\\The UTC method does not introduce additional biases from the TC method.
\end{description}
Traders and their empirical distribution are optimized to earn returns under the training data.
Therefore, the empirical distribution of the Traders is expected to approximate the true posterior distribution given training data.
The following proposition shows that the variance of our method approximates the posterior variance given the data under these assumptions.
\begin{proposition}[Posterior Variance Estimation]\label{prop1}
If the empirical distribution of the Trader $q(D)$ trained by UTC approximates the posterior distribution of the Trader $p(\theta|D)$ well, then the posterior variance of the return $\mathbb{V}_{p(y|x,D)}[y]$ can be approximated by the variance calculated by equation (\ref{eq:variance}).
\end{proposition}
\begin{proof}
Since we optimize the weight of the Trader by MAP estimation in Educate step, the Trader's output satisfies
$$\mu_\theta(x) \approx \mathbb{E}_{p(y|x,\theta,D)}[y],\sigma^2_\theta(x) \approx \mathbb{V}_{p(y|x,\theta,D)}[y].$$
Furthermore, since the Trader $\{\theta_n\}$ is sampled from the distribution $
q(D) \approx p(y|x,D)$,
\begin{align*}
& \mathbb{V}_{p(y|x,D)}[\mu_\theta(x)] \approx \frac{1}{N} \sum_{n} (\mu_n - \mu)^2 \\
& \mathbb{E}_{p(y|x,D)}[\sigma^2_\theta(x)] \approx \frac{1}{N} \sum_{n} \sigma_n^2.
\end{align*}
From above,
\begin{align*}
&\mathbb{V}_{p(y|x,D)}[y] \\
&= \mathbb{V}_{p(\theta|D)}\left[\mathbb{E}_{p(y|x,\theta,D)}[y]\right] + \mathbb{E}_{p(\theta|D)}\left[\mathbb{V}_{p(y|x,\theta,D)}[y]\right] \\
&\approx \mathbb{V}_{p(y|x,D)}[\mu_\theta(x)] + \mathbb{E}_{p(y|x,D)}[\sigma^2_\theta(x)]\\
&\approx \frac{1}{N} \sum_{n} (\mu_n - \mu)^2 + \frac{1}{N} \sum_{n} \sigma_n^2.
\end{align*}
\end{proof}
\if 0
Next, we show that our method has at least as same predictive power as the TC method.
That is, we show that the output of our UTC method is consistent with that of the TC method when the parameters are set to satisfy the assumptions of the following proposition.
First, we show two lemmas about the consistency of the Traders and Educate algorithm between the UTC and the TC.
\fi
Next, we show that our method does not introduce additional biases from the TC method.
That is, we show that the expected value of the output of our UTC method is identical with that of the TC method when the parameters are set to satisfy the assumptions of the following proposition.
First, we show two lemmas about the unbiasedness of the Traders and Educate algorithm of the UTC.
\begin{lemma}[Unbiasedness of Prediction from TC] \label{lemma:prediction}
Let $C$ be the Company trained by the UTC method and $C'$ be the Company trained by the TC method.
Let $\mathbb{E}[y]$ be the expected return of $C$ and $y'$ be the predicted return of $C'$.
If the parameters of each Traders $\{\theta_n\}_{n \in C}$ and $\{\theta_n\}_{n \in C'}$ is identical and expected weights of Traders $\{ \mathbb{E}[w_n] \}_{n \in C}$ is identical with the weights of Traders $\{ w'_n \}_{n \in C'}$, then, $\mathbb{E}[y] = y'$.
\end{lemma}
\begin{proof}
Let $\{z_n\}_{n \in C}$ and $\{z'_n\}_{n \in C'}$ be the signal of each Traders calculated by \eqref{eq:signal}.
Since, the parameters except weights is same, the signals satisfy $z_n = z'_n $.
From the definition of Trader, the predicted return by TC $\{y_n\}_{n \in C'}$ and our expected return $\{\mu_n\}_{n \in C}$ is calculated as
\begin{align}
& \mathbb{E}[y_n] = \mathbb{E}[w_n^\top z_n] = \mathbb{E}[w_n]^\top z_n, \nonumber\\
& y'_n = w^\top z_n'. \nonumber
\end{align}
Therefore, $\mathbb{E}[y_n] = y'_n$.
If we use a simple averaging in Aggregate as well as TC method, then
$$\mathbb{E}[y] = \frac{1}{N} \sum_{n} \mathbb{E}[y_n] = \frac{1}{N} \sum_{n} y'_n = y'$$.
\end{proof}
\begin{lemma}[Unbiasedness of Educate Algorithm] \label{lemma:educate}
Let $X,Y$ be the training data and $T$ be the Trader in UTC method and whose parameter be $\theta_{T}$ and weight distribution be $w$, $T'$ be the Trader in TC method and whose parameter be $\theta_{T'}$ and weight be $w'$, If the parameters satisfy $\theta_{T} = \theta_{T'}$, then, the distribution of weights in UTC satisfies $\mathbb{E}[w] = w'$.
\end{lemma}
\begin{proof}
Let $Z$ and $Z'$ be the signal of each Trader.
Since, the parameters are identical, the signals satisfy $Z = Z' $.
We update the Trader's weight distribution to maximize the posterior distribution of weights given the data.
Suppose the prior distribution of weights is given as
$$ w \sim \mathcal{N}(0,\sigma_0^2), T \sim \mathcal{N}(Y,\sigma^2). $$
Then, the posterior distribution is
$$ p(w|T,Z) \propto \exp\left( \frac{-\|Zw-T\|^2}{2\sigma^2} - \frac{1}{2\sigma_0^2} w^\top w\right).$$
On the other hand, since the TC method updated weight by least square method, their weight is $$w' = (Z^\top Z + \lambda I)^{-1}Z^\top Y $$.
If we set the parameter of prior to satisfy $\lambda = \frac{\sigma^2}{\sigma_0^2}$, both weights satisfy $\mathbb{E}[w] = w'$.
\end{proof}
Finally, we prove the unbiasedness of the Company's prediction between UTC and TC when the parameters are set to satisfy the assumptions of these lemmas.
\begin{proposition}[Unbiasedness of Training and Prediction from TC]\label{prop2}
If the hyper-parameters of Company $C$ trained by the UTC method and Company $C'$ trained by the TC method satisfy the assumption of Lemma \ref{lemma:prediction} and \ref{lemma:educate}, the expected return of $C$ equals to the expected return of $C'$.
\end{proposition}
\begin{proof}
From Lemma \ref{lemma:educate}, The Traders' weights are consistent with respect to the expected value in the Educate process.
On the other hand, the TC method and our method are executed in the same way for the Prune-and-Generate algorithm. \\
Therefore, the distribution of Traders' parameters $\Theta$ in our method is identical to that of Traders' parameters $\Theta'$ in the TC method. Also, Traders' weights are also identical with respect to expected values. Therefore, from Lemma \ref{lemma:prediction}, the expected return of $C$ equals to the expected return of $C'$.
\end{proof}
\section{Experiment}
\subsection{Analysis on Synthetic Data} \label{sec:synthetic}
\subsubsection{Simple Nonlinear Case}
We used the following artificial data as a simple example of nonlinear time series.
\begin{align}
& y_{0}(t) = 0.5 y_{0}(t-1) - 0.5y_{0}(t-1)y_{1}(t-1) \\
& ~~~~~~~~~+ 0.1\min(y_{0}(t-1),y_{1}(t-1)) + \varepsilon_{0}(t) \nonumber \\
& y_{1}(t) = -0.2 y_{1}(t-1) + 0.8y_{0}(t-1) \\
& ~~~~~~~~~+ 0.5\max(y_{0}(t-1),y_{1}(t-1)) + \varepsilon_{1}(t) \nonumber
\end{align}
where $\varepsilon_{0}(t),\varepsilon_{1}(t)$ are generated by independently and identical normal distribution $N(0,0.1)$.
We used a simple Vector AutoRegression~(VAR~\cite{hamilton2020time}) model with lag 1 as a baseline.
Makridakis et al.\cite{makridakis2018statistical} demonstrated that traditional statistical methods such as the VAR model are more accurate than machine learning ones and suitable for the baseline of time series prediction tasks.
We used 1800 samples for training and used 200 samples for the test.
\begin{figure}[tb]
\begin{minipage}[b]{0.5\textwidth}
\centering
\includegraphics[keepaspectratio, scale=0.5]{figure/Test2_2.png}
\end{minipage}
\begin{minipage}[b]{0.5\textwidth}
\centering
\includegraphics[keepaspectratio, scale=0.5]{figure/Cov_test2.png}
\end{minipage}
\caption{Prediction error of synthetic data}
\label{fig:synthetic}
\end{figure}
The upper panel of Figure \ref{fig:synthetic} shows the comparison of prediction error~(Absolute Error) between UTC and VAR.
The lower panel shows the correlation diagram between the predictive standard deviation and the prediction error.
We can confirm that our UTC method makes better predictions than the VAR model because it captures nonlinear features of synthetic data.
Also, we can observe that the estimated standard deviation is larger when the prediction error is large, i.e., prediction is difficult.
That means our UTC method can capture such uncertainty.
\subsubsection{Dataset Shift Case}
Next, we evaluated the performance of the UTC method in the dataset shift case using the artificial data defined as follows.
\begin{align*}
y_2(t) = \left\{ \begin{array}{ll}
y_0(t-1)+y_1(t-1) + \epsilon_2(t) & (t < 200) \\
y_0(t-1)-y_1(t-1) + \epsilon_2(t) & (t \geq 200)
\end{array} \right.
\end{align*}
where $\varepsilon_{2}(t)$ are generated by independently and identical normal distribution $N(0,0.1)$.
To verify the adaptability of our method to dataset shift, we sequentially updated the model using the last 100 data as training data.
Figure \ref{fig:datashift} shows the result.
In this experiment, the error increases as data distribution are shifted (Blue region). Along with this, the estimated variance is also increasing. Furthermore, as the environment shifts to the original static environment, the error and the estimated variance decrease again.
This means that our method detects the decrease in reliability of prediction due to the dataset shift.
\begin{figure}[tb]
\begin{minipage}[b]{0.5\textwidth}
\centering
\includegraphics[keepaspectratio, scale=0.3]{figure/Test3_u2.png}
\end{minipage}
\caption{Prediction error and predictive standard error in dataset shift}
\label{fig:datashift}
\end{figure}
\subsection{Experiments on Real Datasets}\label{sec:exp}
\subsubsection{Dataset}
We tested two different settings using real market data, the TOPIX100 index.
The TOPIX100 Index is a market capitalization-weighted index of large-cap Japanese stocks, consisting of the top 100 stocks with particularly high market capitalization and liquidity (trading value) among the stocks included in the TOPIX index.
We performed daily trading at the close and hourly intraday trading.
In the daily trading setting, we used daily closing prices of the constituents of the TOPIX100 index from January 4, 2010, to January 31, 2022.
We used the data form from January 4, 2010, to December 30, 2017, for training and from January 4, 2018, to January 31, 2022, for the test.
In the intraday trading setting, we used hourly closing prices of the constituents of the TOPIX100 index from January 4, 2016, to June 30, 2021.
We used the data form from January 4, 2016, to December 30, 2019, for training and from January 4, 2020, to June 30, 2021, for the test.
\subsubsection{Experimental Settings}
The following settings are the same as \cite{ito2021trader}.
We introduced a time window $w>0$ and a trading execution lag $l>0$ as in \cite{ito2021trader}.
Throughout experiments, we used $w= 10$ and $l = 1$.
We trained models using observations $\vecret[1:S][t-l-w:t-l]$ and predict returns at every time $t$.
Recall $r_i[t]$ be the return of stock $i$ ($i \in \{ 1, \ldots, S \}$) at time $t$.
Here, the time between $t$ and $t+1$ represents 1 day in the case of daily trading, and 1 hour in the case of intraday trading.
To evaluate the effectiveness of our UTC method, we performed the following virtual trading strategy.
We buy the stock if the predicted label for the time $t$ (indicating whether the price of the time $t+1$ would rise or fall) is positive and sell it otherwise.
We define the strategy's return at time $t$ as
$$R[t] = \mathop{\rm mean}\limits(\hat{b}_i[t]\times r_i[t])$$
where $\hat{r}_i[t]$ is its prediction from each method, $\hat{b}_i[t] = \mathrm{sign}(\hat{r}_i[t])$ and $\mathop{\rm mean}\limits(\cdot)$ represents the average value of the function over its input.
We compared our UTC methods with the following strategies:
\begin{itemize}
\item Market: Simply buy all stocks in TOPIX100 index equally~\cite{demiguel2009optimal} i.e., $R[t] = \mathop{\rm mean}\limits(r_i[t])$.
\item VAR: $\hat{r}_i[t]$ is estimated by the VAR model using historical observations.
We used the VAR(1) model selected by the AIC for lags from 1 to 10.
\item RF: $\hat{r}_i[t]$ is estimated by Random Forest model with Scikit-learn package~\cite{pedregosa2011scikit}.
We set "n\_estimators" to 100, "min\_samples\_split" to 10, "min\_samples\_leaf" to 4, "max\_features" to sqrt, "max\_depth" to 60.
We determined these hyper-parameters by cross-validation.
\item TC: $\hat{r}_i[t]$ is estimated by the original TC method.
\item UTC: Let $\sigma_i[t]$ be its prediction uncertainty from UTC method. Define $\hat{b}_i[t] = \mathrm{sign}(\hat{r}_i[t] \times I_{A})$ where $I$ is an indicator function and $A$ is an event when $\sigma_i[t]$ is higher than the threshold calculated from past predictive variance.
Then $R[t] = \mathop{\rm mean}\limits(\hat{b}_i[t]r_i[t])$.
\end{itemize}
Table \ref{tab:hyperparamters} lists the hyper-parameters used in both the TC and UTC methods.
\begin{table}[tb]
\caption{Hyper-parameters used in TC and UTC in the experiment}
\label{tab:hyperparamters}
\centering
\begin{tabular}{ccc}
\hline
\textbf{Parameter} & \multicolumn{1}{c}{\textbf{Value}} & \textbf{Def} \\
$\numterm$ & $\{1,\cdots,10\}$ & Definition \eqref{TCtraders}\\
$D_j,F_j$ & $\{0,\cdots,10\}$ & Definition \eqref{TCtraders}\\
$A_j(x) $ & $\{x,\mathrm{tanh}(x),\mathrm{exp}(x),\mathrm{sign}(x),\mathrm{ReLU}(x)\}$ & Definition \eqref{TCtraders}\\
$O_j(x,y)$ & \begin{tabular}{@{}c@{}}$\{x+y, x-y , xy, x, y,\max(x,y)$, \\ $ \min(x,y),x>y,x<y,\textrm{Corr}(x,y) \}$\end{tabular} & Definition \eqref{TCtraders}\\
$N$ & $200$ & Algorithm \ref{companypredict} \\
Aggregate & Simple Averaging & Algorithm \ref{companypredict} \\
$Q$ & $0.1$ & Algorithm \ref{TCcompanygen} \\
\hline
\end{tabular}
\end{table}
\subsubsection{Performance Measures}
To evaluate the performances of each strategy, we adopted the four metrics widely used in financial experiment\cite{brandt2010portfolio} as follows.
$T_Y$ represents the number of periods of trading in one year.
We used $T_Y = 250$ in the daily setting, $T_Y = 1250$ in the hourly setting.
\begin{itemize}
\item \textbf{Annualized Return(AR)}:
We define the Annualized Return \textbf{(AR)} as
\begin{align}
\mathrm{AR} := T_Y\times \mathop{\rm mean}\limits(R[t])
\end{align}
This measure represents the profitability of the strategy.
\item \textbf{Annualized Risk(RISK)}:We define the Annualized Risk \textbf{(RISK)} as
\begin{align}
\mathrm{RISK} := \sqrt{T_Y} \times \sqrt{\sum_{t=1}^T(R[t]-\mathop{\rm mean}\limits(R[t]))^2/(T-1)}.
\end{align}
\item \textbf{Sharpe Ratio (SR)}: The Sharpe ratio \cite{sharpe1964capital}, or the Return/Risk ratio (R/R) is the return adjusted by its risk.
That is,
\begin{align}
\mathrm{SR} := \mathrm{AR} / \mathrm{RISK}.
\end{align}
\item \textbf{Maximum DrawDown (MDD)}: The Maximum DrawDown is defined as the largest drop from an extremum~\cite{magdon2004maximum}:
\begin{align}
\mathrm{MDD} := \max_{1 \leq t \leq T}\max_{t < s \leq T}(1 - C[t]/C[s])
\end{align}
where $C[t] := \sum_{j=1}^t R[j]$ is the cumulative return.
\item \textbf{Cramer Ratio(CR)}: The CR are adjusted returns by its MDD~\cite{young1991calmar}. CR is more sensitive to drawdown events that occur less frequently:
\begin{align}
\mathrm{CR} := \mathrm{AR} / \mathrm{MDD}
\end{align}
\end{itemize}
Larger AR, SR, and CR are better, while smaller RISK and MDD are better.
\subsubsection{Result}
\begin{table}[t]
\caption{Performance Comparison in the daily trading setting: Average results of 5 repeated experiments}
\label{tab:jpn}
\begin{center}
\begin{tabular}{cccccc}\hline\hline
& UTC & TC & Market & RF & VAR \\ \hline
AR & 5.42 & \textbf{9.03} & 8.56 & 7.80 & 8.13 \\
RISK & \textbf{4.45} & 9.05 & 17.90 & 8.37 & 16.90 \\
SR & \textbf{1.22} & 1.00 & 0.48 & 0.93 & -0.48\\
MDD & \textbf{-6.43} & -13.99 & -28.66 & -12.12 & -27.03\\
CR & \textbf{0.86} & 0.64 & 0.30 & 0.64 & 0.30 \\
\hline
\end{tabular}
\end{center}
\end{table}
\begin{figure}[tb]
\centering
\includegraphics[width=\columnwidth]{figure/TPX100_3.png}
\caption{Cumulative daily returns on JPN market}
\label{fig:offline_jpn}
\end{figure}
Table \ref{tab:jpn} shows the comparison between our proposed method (UTC) and other baselines in the daily trading. We ran the experiment 5 times and averaged the result.
Figure \ref{fig:offline_jpn} shows the transition of cumulative return $C[t]$ of each method.
As previous studies have shown, the TC method was confirmed to earn higher returns than other methods in both daily and intraday trading settings.
However, TC also had a high risk because it does not consider uncertainty.
On the other hand, our method invests only in stocks with low predictive uncertainty, thereby achieving stable returns while limiting risk. Therefore, our method achieved the highest SR and CR, risk-adjusted performance evaluation metrics.
In particular, during the COVID-19 shock in the market in the first half of 2020, our method successfully captured market uncertainties and limited investments during those periods and significantly suppressed the sharp fall.
Table \ref{tab:jpn_hour} shows the comparison between our proposed method (UTC) and other baselines in the intraday trading. We ran the experiment 5 times and averaged the result.
We also show the transition of cumulative return $C[t]$ of each method in Figure \ref{fig:offline_jpn_hour}.
As in the daily case, our UTC method also achieved the highest SR and CR in the intraday case.
\begin{table}[t]
\caption{Performance Comparison in the hourly trading setting: Average results of 5 repeated experiments}
\label{tab:jpn_hour}
\begin{center}
\begin{tabular}{cccccc}\hline\hline
& UTC & TC & Market & RF & VAR \\ \hline
AR & 9.60 & \textbf{11.9} & 7.75 & 2.70 & 4.11 \\
RISK & 6.42 & 8.42 & 21.2 & 11.62 & \textbf{5.06} \\
SR & \textbf{1.52} & 1.46 & 0.37 & 0.23 & 0.81\\
MDD & -6.95 & -9.83 & -30.96 & -15.87 & \textbf{-4.57}\\
CR & \textbf{1.52} & 1.37 & 0.25 & 0.17 & 0.90 \\
\hline
\end{tabular}
\end{center}
\end{table}
%
\begin{figure}[t]
\centering
\includegraphics[width=\columnwidth]{figure/TPX100_h.png}
\caption{Cumulative hourly returns on JPN market}
\label{fig:offline_jpn_hour}
\end{figure}
\section{Related Work}
\subsection{Stock Return Prediction in Finance}
Predicting future stock prices has been actively researched for a long time but is still a highly challenging task~\cite{chen2018integrating}.
Stock price prediction can be broadly divided into two types: fundamental analysis and technical analysis\cite{atsalakis2009,sedighi2019novel}.
Fundamental analysis focuses on fundamental information of the corporation, such as a company's revenues and expenses, yearly growth rate, and other information contained in financial statements.
On the other hand, the technical analysis predicts using market data such as historical stock price and volume data.
The methods of the former analysis perform a regression analysis using cross-sectional data of fundamental information.
Such strategies aim to build a portfolio for investing as a subset of a large bucket of stocks. These approaches are frequently applied to a practical quantitative investment strategy~\cite{nakagawa2020ric}.
One of the most significant interests in a cross-sectional analysis lies in finding ``factors'' that have strong predictive powers to the stock return of portfolio strategy~\cite{fama1992}.
This argument inspires a lot of subsequent studies that propose more sophisticated versions of factors~\cite{harvey2016}.
The methods of the latter strategy analyze past stock prices as time-series data~\cite{atsalakis2009} and are applied to a practical trading strategy that focuses on a particular stock.
The autoregressive integrated moving average (ARIMA) model, Vector autoregression (VAR), and generalized autoregressive conditional heteroscedasticity (GARCH~\cite{bollerslev1986generalized}) model are basic benchmarks often used in financial time series prediction. These linear models take into account uncertainties as error terms, and some studies have also modeled their distributions~\cite{Ledolter1979}.
\subsection{Application of Machine Learning to Stock Return Prediction}
With the introduction of artificial intelligence and machine learning, these techniques have received increased attention in stock prediction studies\cite{atsalakis2009,cavalcante2016computational}.
Unlike traditional time series methods, these methods can handle the nonlinear, noisy, and complex data of the stock market, leading to more effective predictions~\cite{chen2018integrating,ito2021trader}.
Among them, Trader-Company~(TC) method achieves state-of-the-art performance in this field~\cite{ito2021trader}.
The TC method takes into account the dynamism of the stock market and has both high predictive power and interpretability.
The TC method is based on a meta-heuristic approach, which has been extensively applied to stock prediction~\cite{soler2017survey,sedighi2019novel}.
Another promising approach is the application of neural-network-based models such as CNNs and RNNs.
They have been proposed as innovative and advantageous alternatives to traditional methods~\cite{hu2021,Lai2018,ijcai2020-640,imajo2021deep}.
These studies showed promising performance, but unfortunately, they lack a perspective on the uncertainty of the prediction.
In other words, all of these methods are only focused on point prediction. This is a major practical problem because of the possibility of significant losses in investment strategies.
As noted in the previous subsection, research in the field of finance has incorporated uncertainty, but in terms of adapting machine learning to finance, there remains room for improvement. Among them, several applications of Bayesian neural networks have been proposed to deal with the issue. These studies show their effectiveness in highly volatile and uncertain markets while COVID-19 is reported~\cite{chandra2021}.
Inspired by these studies, this study introduced the uncertainty estimation technique to the TC method to capture uncertainty.
\subsection{Uncertainty Estimation with Weight Parameterization}
Most methods for uncertainty estimation have been proposed under a Bayesian framework. Bayesian estimation requires some approximation to estimate the uncertainty unless the linear model since exact computation is usually intractable.
One basic approach to the approximation is to set parameters corresponding to the mean and variance in the model and use these to approximate the posterior distribution.
In linear regression, Bayesian regression follows this approach, and methods that follow this approach have been proposed for other standard models, such as neural networks~\cite{pmlr-v37-blundell15} and decision trees~\cite{pmlr-v119-duan20a}.
The former assumes the Gaussian distribution for weights in neural networks, and the latter assumes the outputs follow Gaussian distributions and predict their mean and variance. Both works train the model to minimize the KL divergence from the posterior distribution. Our method introduces these approaches in the estimation of the uncertainty in the Trader. This approach is computationally efficient, but it has been pointed out that these methods tend to make predictions biased toward one specific mode ~\cite{Fort2019,malinin2021uncertainty}.
\subsection{Uncertainty Estimation with Ensemble}
Another approach is ensemble-based, which approximates the posterior distribution using multiple outputs. For a neural network, Gal and Ghahramani proposed Monte Carlo Dropout~\cite{pmlr-v48-gal16}, which applies dropout during inference and estimates uncertainty. This method can be interpreted as using multiple sub-networks that share the parameters.
Deep Ensemble~\cite{NIPS2017_9ef2ed4b,d'angelo2021repulsive} trains several neural networks with different initialization and estimates the predictive uncertainty by the variance of their outputs. Learning independent networks makes them less likely to be biased toward one particular mode and enhances their robustness~\cite{fort2020deep}.
For tree-based models, initial work trained using MCMC~\cite{NIPS2006_1706f191} and the combination with GBDT~\cite{malinin2021uncertainty} shows better performance in terms of both prediction and uncertainty estimation.
These methods are experimentally confirmed to be more robust to dataset shift than methods that explicitly learn distributions\cite{NIPS2017_9ef2ed4b,malinin2021uncertainty}. These characteristics are effective in non-stational environments such as financial markets.
Our method introduces these approaches for the Company to capture the uncertainty due to the dataset shift.
\section{Conclusion}
We proposed a novel approach called the Uncertainty Aware Trader-Company Method~(UTC), which extends the TC method with probabilistic modeling framework.
The UTC method can keep the predictive power and interpretability of the TC method while capturing uncertainty.
Our contributions are as follows:
\begin{itemize}
\item We showed that the UTC method could estimate the prediction's uncertainty (the posterior variance) under the assumption that the empirical distribution is a good approximation of the posterior distribution.
\item We also proved that the UTC method does not introduce additional biases from the TC method.
\item We confirmed on synthetic data that the UTC method could detect the situations where the prediction is difficult or the data distribution is abruptly changed.
\item We demonstrated on real market data that the UTC method could achieve higher risk-adjusted returns than baselines.
\end{itemize}
For the direction of further study, we may consider uncertainty when recruiting or firing Traders in the Prune-and-Generate algorithm.
Furthermore, our method assumed the distribution of the parameters is the Gaussian mixture distribution. We may consider modeling them in a non-parametric way.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 9,617 |
James H. Friend, Esq.
James H. Friend, Counsel, founder, president and CEO of Friend Development Group, LLC, headquartered in New York City, has more than 25 years of deep and broad experience in the hospitality and real estate industries as a developer, investor, operator, financier, asset manager and advisor. In addition, Jim currently serves as Chairman of the Board of Condor Hospitality Trust, Inc., which has more than 60 hotels in 20 states throughout the U. S. Jim has worked in several foreign jurisdictions and speaks fluent Spanish. He has extensive experience in the drafting and negotiation of land acquisition agreements, management agreements, franchise agreements, construction agreements, architectural agreements and partnerships agreements.
jfriend@osbornlaw.com
T: 212 576 2670 | F: 212 686 4023
7 Penn Plaza, Suite 914
Jim has specialized for several decades in hotel and real estate development and acquisition both in the U. S. and overseas, working in dozens of urban and suburban markets and in numerous property types. He has many years of active, hands-on development experience and has developed many significant ground-up and adaptive re-use properties as well as significant renovations of existing properties. Friend Development has purchased, repositioned and re-branded numerous hotels and worked on dozens of other hotel and other real estate projects. The firm has partnered with and advised NYSE companies, major institutions, REIT's, high net worth family offices, banks and privately held companies.
Has secured tens of millions of dollars of tax abatements, industrial development bond financing and other types of government development incentives for many real estate projects
Has arranged financing for more than $700 million of hotel and real estate transactions, including hotels, assisted living, retail and mixed-use projects and has drawn on his investment banking, legal and development background to bring to fruition complicated real estate projects which include:
the adaptive re-use of a 50-year old hospital into a Hyatt Place Hotel;
the environmental remediation of a former gas station to develop a Hampton Inn & Suites;
the conversion of a former office building in Texas to a 300-room hotel;
the lengthy and highly complex negotiation of a 99-year leasehold within a million sf shopping center;
transfer of air rights between two New York City land parcels to develop a major Hilton-branded hotel;
negotiations with the Department of Transportation to permit a guest drop-off in the heart of downtown Brooklyn;
negotiation of numerous complex land purchase agreements; and much more.
Professor | New York University | Jonathan M. Tisch Center of Hospitality
The Rocky Road Toward the Rule of Law in China: 1979-2000, Forward to Symposium: China Revisited: Examining the Rule of Law After Twenty Years, Northwestern Journal of International Law & Business, 1999: Vol. 20, No. 3.
Suing a Foreign Government Under the United States Antitrust Laws: The Need for Clarification of the Commercial Activity Exception to the Foreign Immunities Act of 1976. Northwestern Journal of International Law & Business, 1979:
Vol. 1, No. 1.
Condor Hospitality Trust, Inc. (Chairman)
Brooklyn Chamber of Commerce
Brooklyn Downtown Partnership
Richard Tucker Music Foundation (Treasurer)
Stanford Alumni Association
Stanford New York Alumni Board (Chairman)
Stanford New York Arts Council (Chairman)
Northwestern University, School of Law | J.D., 1979
Stanford University | 1975 | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 9,686 |
package learning.java.lang;
import org.junit.jupiter.api.Test;
import static org.hamcrest.CoreMatchers.*;
import static org.hamcrest.MatcherAssert.assertThat;
public class IntegerTest {
@Test
public void test() {
assertThat(3 / 2, is(1));
assertThat(3 / 2 * 10, is(10));
assertThat(3.0 / 2.0 * 10.0, is(15.0));
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 6,213 |
/* global describe it expect beforeEach afterEach jasmine spyOn */
var $ = window.jQuery
describe('MultivariateTest', function () {
'use strict'
var GOVUK = window.GOVUK
beforeEach(function () {
GOVUK.cookie = jasmine.createSpy('GOVUK.cookie')
GOVUK.analytics = {setDimension: function () {}, trackEvent: function () {}}
spyOn(GOVUK.analytics, 'setDimension')
spyOn(GOVUK.analytics, 'trackEvent')
})
afterEach(function () {
delete GOVUK.analytics
})
describe('#run', function () {
it('should pick a random cohort on first run', function () {
GOVUK.cookie.and.returnValue(null)
var fooSpy = jasmine.createSpy('fooSpy')
var barSpy = jasmine.createSpy('barSpy')
new GOVUK.MultivariateTest({ // eslint-disable-line no-new
name: 'stuff',
cohorts: {
foo: {callback: fooSpy},
bar: {callback: barSpy}
}
})
expect(GOVUK.cookie.calls.count()).toEqual(2)
expect(GOVUK.cookie.calls.argsFor(1)[0]).toEqual('multivariatetest_cohort_stuff')
if (GOVUK.cookie.calls.argsFor(1)[1] === 'foo') {
expect(fooSpy).toHaveBeenCalled()
} else {
expect(barSpy).toHaveBeenCalled()
}
})
it('should use an existing cohort choice on subsequent runs', function () {
GOVUK.cookie.and.returnValue('foo')
var fooSpy = jasmine.createSpy('fooSpy')
var barSpy = jasmine.createSpy('barSpy')
new GOVUK.MultivariateTest({ // eslint-disable-line no-new
name: 'stuff',
cohorts: {
foo: {callback: fooSpy},
bar: {callback: barSpy}
}
})
expect(fooSpy).toHaveBeenCalled()
})
it('should set a custom var with the name and cohort if one is defined', function () {
GOVUK.cookie.and.returnValue('foo')
new GOVUK.MultivariateTest({ // eslint-disable-line no-new
name: 'stuff',
cohorts: {
foo: {},
bar: {}
},
customDimensionIndex: 2
})
expect(GOVUK.analytics.setDimension).toHaveBeenCalledWith(
2,
'multivariatetest_cohort_stuff__foo'
)
})
it('should be able to set multiple custom vars with the name and cohort if one is defined as an array', function () {
GOVUK.cookie.and.returnValue('foo')
new GOVUK.MultivariateTest({ // eslint-disable-line no-new
name: 'stuff',
cohorts: {
foo: {},
bar: {}
},
customDimensionIndex: [2, 3]
})
expect(GOVUK.analytics.setDimension).toHaveBeenCalledWith(
2,
'multivariatetest_cohort_stuff__foo'
)
expect(GOVUK.analytics.setDimension).toHaveBeenCalledWith(
3,
'multivariatetest_cohort_stuff__foo'
)
})
it('should trigger an event to track that the test has been run', function () {
GOVUK.cookie.and.returnValue('foo')
new GOVUK.MultivariateTest({ // eslint-disable-line no-new
name: 'stuff',
cohorts: {
foo: {},
bar: {}
}
})
expect(GOVUK.analytics.trackEvent).toHaveBeenCalledWith(
'multivariatetest_cohort_stuff',
'run',
{nonInteraction: true}
)
})
it('should set html for a cohort', function () {
GOVUK.cookie.and.returnValue('foo')
var $el = $('<div>')
new GOVUK.MultivariateTest({ // eslint-disable-line no-new
name: 'stuff',
el: $el,
cohorts: {
foo: {html: 'foo'},
bar: {html: 'bar'}
}
})
expect($el.html()).toEqual('foo')
})
it('should call the callback for a cohort', function () {
var fooSpy = jasmine.createSpy('fooSpy')
var barSpy = jasmine.createSpy('barSpy')
GOVUK.cookie.and.returnValue('bar')
var $el = $('<div>')
new GOVUK.MultivariateTest({ // eslint-disable-line no-new
name: 'stuff',
el: $el,
cohorts: {
foo: {callback: fooSpy},
bar: {callback: barSpy}
}
})
expect(barSpy).toHaveBeenCalled()
})
it('should call the callback for a cohort if it is a string', function () {
GOVUK.cookie.and.returnValue('foo')
var test = new GOVUK.MultivariateTest({
name: 'stuff',
cohorts: {
foo: {callback: 'fooCallback'},
bar: {}
},
runImmediately: false
})
test.fooCallback = jasmine.createSpy('fooCallback')
test.run()
expect(test.fooCallback).toHaveBeenCalled()
})
it("should assign 30 if cookieDuration isn't defined", function () {
GOVUK.cookie.and.returnValue('foo')
var test = new GOVUK.MultivariateTest({
name: 'cookie_duration_test',
cohorts: {
foo: {callback: function () {}}
}
})
expect(test.cookieDuration).toEqual(30)
})
it("should assign the user's cookie duration, when cookieDuration is defined", function () {
GOVUK.cookie.and.returnValue('foo')
var test = new GOVUK.MultivariateTest({
name: 'cookie_duration_test',
cookieDuration: 14,
cohorts: {
foo: {callback: function () {}}
}
})
expect(test.cookieDuration).toEqual(14)
})
it('should assign a new random cohort if the assigned cohort does not exist', function () {
var fooSpy = jasmine.createSpy('fooSpy')
var barSpy = jasmine.createSpy('barSpy')
GOVUK.cookie.and.returnValue('baz')
new GOVUK.MultivariateTest({ // eslint-disable-line no-new
name: 'stuff',
cohorts: {
foo: {callback: fooSpy},
bar: {callback: barSpy}
}
})
if (GOVUK.cookie.calls.argsFor(1)[1] === 'foo') {
expect(fooSpy).toHaveBeenCalled()
} else {
expect(barSpy).toHaveBeenCalled()
}
})
})
describe('#weightedCohortNames', function () {
it('should return the weighted names of the cohorts when no weights are defined', function () {
var test = new GOVUK.MultivariateTest({
name: 'stuff',
cohorts: {foo: {}, bar: {}, baz: {}}
})
expect(test.weightedCohortNames()).toEqual(['foo', 'bar', 'baz'])
})
it('should return the weighted names of the cohorts when weights are defined', function () {
var test = new GOVUK.MultivariateTest({
name: 'stuff',
cohorts: {foo: { weight: 2 }, bar: { weight: 1 }, baz: { weight: 3 }}
})
expect(test.weightedCohortNames()).toEqual(['foo', 'foo', 'bar', 'baz', 'baz', 'baz'])
})
it('should return the weighted names of the cohorts using default weighting', function () {
var test = new GOVUK.MultivariateTest({
name: 'stuff',
defaultWeight: 2,
cohorts: {foo: {}, bar: {}, baz: {}}
})
expect(test.weightedCohortNames()).toEqual(['foo', 'foo', 'bar', 'bar', 'baz', 'baz'])
})
it('should return the weighted names of the cohorts using default weighting or defined weighting', function () {
var test = new GOVUK.MultivariateTest({
name: 'stuff',
defaultWeight: 2,
cohorts: {foo: {}, bar: { weight: 1 }, baz: {}}
})
expect(test.weightedCohortNames()).toEqual(['foo', 'foo', 'bar', 'baz', 'baz'])
})
})
describe('#chooseRandomCohort', function () {
it('should choose a random cohort', function () {
var test = new GOVUK.MultivariateTest({
name: 'stuff',
cohorts: {foo: {}, bar: {}}
})
expect(['foo', 'bar']).toContain(test.chooseRandomCohort())
})
})
describe('Google Content Experiment Integration', function () {
beforeEach(function () {
window.ga = function () {}
spyOn(window, 'ga')
})
it('should report the experiment data to Google', function () {
new GOVUK.MultivariateTest({ // eslint-disable-line no-new
name: 'stuff',
contentExperimentId: 'asdfsadasdfa',
cohorts: {foo: {variantId: 0, weight: 1}, bar: {variantId: 1, weight: 0}}
})
expect(window.ga.calls.first().args).toEqual(['set', 'expId', 'asdfsadasdfa'])
expect(window.ga.calls.mostRecent().args).toEqual(['set', 'expVar', 0])
expect(GOVUK.analytics.trackEvent).toHaveBeenCalledWith(
'multivariatetest_cohort_stuff',
'run',
{nonInteraction: true}
)
})
})
})
| {
"redpajama_set_name": "RedPajamaGithub"
} | 8 |
\section{Introduction and summary}
The static dielectric susceptibility in the materials like SrTiO3 and KTaO3
does not show any divergence down to zero Kelvin, it rather saturates at a
very high value $(\mathcal{O}(10^4))$. Ferroelectric transitions in materials
which are similar in structure, for example in BaTiO3, are of displacive type
and are well described by softening of a zone center transverse optical mode.
This scenario, that is, absence of transition and high dielectric susceptibility,
has been attributed to smallness of the gap in the corresponding optical branch,
smallness of the optical gap makes quantum fluctuations relevant in these materials.
These materials are, therefore, called quantum paraelectrics\cite{Muller}. There
is a revival of interest, particularly regarding the nature of phase transition at
quantum critical point in these materials \cite{Millis, we, Coleman}. The effects
of pressure\cite{expt-pressure} and impurities on the dielectric susceptibilities
have been well studied experimentally. One recent experimental\cite{Taniguchi} report indicates that these materials show phase separation near QCP. A general consensus
arising form these experiments is that application of hydrostatic pressure
moves them away from criticality and the possibility of transition is suppressed.
One needs to apply some kind of \emph{negative pressure} to induce phase transitions in these materials. One way to apply negative pressure is to put non-polar impurities
which creates local pressure deficiencies and thus induce phase transitions. Remarkably when experimentally\cite{expt-impurity} one finds $T_c \sim (n-n_c)^{\frac{1}{2}}$
(where n is the average impurity concentration and $n_c$ is the critical value), the theoretically estimated\cite{we} transition temperature for pressure induced transition goes as $T_c \sim (p+p_c)^{\frac{1}{2}}$ (where $p$ is hydrostatic pressure
and $p_c$ is the critical value). This motivates us to develop a description,
suitable for properties of pressure induced phase transition, which can be used to understand the newly found phase separation in ferroelectric transition near
ferroelectric quantum critical point. In the following, using a microscopic theory
in the paraelectric phase, we first show that $T_c \sim (p+p_c)^{\frac{1}{2}}$
within some reasonable approximation. Then we use this scaling behavior of $T_c$
in a Landau functional to discuss ferroelectric transition in such systems.
\section{Mean Field Analysis}
Application of hydrostatic pressure (also the effect of non-polar impurity) will couple to optical mode via strain. In this case our starting Hamiltonian takes the form
\begin{eqnarray}
H&=& \int dq \frac{1}{2} \left[ \frac{1}{2} p_q ^2 +
\frac{1}{2} \left( \omega_0^2
- v \delta \cos qa \right) u_q u_{-q} \right] \nonumber \\
&+& \frac{1}{4} \lambda \int \Pi_i dq_i u_{q_1} u_{q_2}
u_{q_3} u_{-q_1 -q_2 - q_3} + \frac{1}{2} K \int dq \epsilon^2 (q)~\nonumber\\
&+&g \int dk~dq~ \epsilon(k)~ u_q u_{k-q} - p\int dq \delta(q)\epsilon(q)~
\label{H}
\end{eqnarray}
Here terms in the parentheses describe optical phonons in harmonic approximation,
the quartic term describes the anharmonic coupling between these phonons. Last
three terms describe the result of application of pressure to lowest possible order.
The strain field $\epsilon (q) $ couples to unit cell displacement related to optic
mode with amplitude $g$, the harmonic acoustic phonons have force constant
$K$ and in the last term the coupling of the hydrostatic pressure $p$ to the
static strain has unit strength. The details of this hamiltonian are described
in our earlier work\cite{we}. On Integrating out the strain field, we get an
effective Hamiltonian of the form
\bea
H &=& \int dq \left[ \frac{1}{2} p_q ^2 +
\frac{1}{2} \left( \omega_0^2 -v+
gp+ v \delta q^2a^2 \right) u_q u_{-q} \right]
\nonumber\\ && + \frac{1}{4} \lambda_R \int \Pi_i dq_i u_{q_1} u_{q_2}
u_{q_3} u_{-q_1 -q_2 - q_3}
\end{eqnarray}
where the normalized quartic coupling
\be
\lambda_R = (\lambda -\frac{2g^2}{K}).
\ee
Now with $ p_q = \dot {u}_q = - \imath \omega u_q $ in the kinetic energy term, and within the quasi harmonic approximation, for the quartic term, i.e.
\be
\sum_l u_l^4 \approx 6 N (\sigma +\frac{1}{2} \langle u \rangle^2) \sum_{ q_1 } u_{q_1} u_{- q_1}
\ee
the Hamiltonian is given by,
\be
H = \frac{1}{2} \sum_q (\omega_q^2 - \omega^2) u_q u_{-q}.
\ee
Here $\omega_q$ is renormalized optical frequency in the paraelectric phase where
$<u>$ is identically zero and is given by,
\bea
\omega^2_q &=& \omega_0^2 +gp + v\delta cosqa + 3 \lambda_R \sigma \nonumber\\
&\simeq& \omega_0^2 - v+gp + v \delta a^2 q^2 + 3 \lambda_R \sigma.
\label{re-frequency}
\end{eqnarray}
and
\be
\sigma = \sum_{q} \langle T u_q (0) u_{-q} (0^+) \rangle
\label{sigma}
\ee
is the mean square fluctuations of displacement in optical mode.
The susceptibility, which is related to $<u^2 > $, is
essentially the phonon propagator,
\be
\chi(q,n)= -\frac{1}{ (\imath \omega_n)^2 - \omega_q^2 },~~
\omega_n= 2n\pi T.
\ee
With $\omega_q$ already defined we have a self consistent equation for the fluctuation in the optical mode,
\be
\sigma =\sum_{q} \frac{1}{2\omega_q} \coth \left( \frac{\omega_q}{2T} \right)
\label{self-consist}
\ee
Which in its asymptotic forms reduces to,
\bea
\sigma &\sim& \int \frac{Td^3q}{ \omega_0^2 - v +gp
+ v \delta q^2 + 3 \lambda_R \sigma }
\label{highT}
\end{eqnarray}
in the high temperature limit and to
\bea
\sigma &\sim& \int \frac{d^3q}{\sqrt{\omega_0^2 -v+gp + v \delta q^2 + 3
\lambda_R \sigma }}
\label{lowT}
\end{eqnarray}
in the low temperature limit.
Defining the parameters
\be
\Delta =\frac{( \omega_0^2 -v )}{3\lambda} {\rm ~and~} p_c = \frac{3K\Delta\lambda}{g},
\ee
which give vicinity to the quantum critical point and the critical pressure respectively, we can write
\be
\omega^2(q) = 3\Delta \lambda (1+ p/p_c)
- v \delta q^2a^2/2
+ 3\lambda_R \sigma.
\label{omega-final}
\ee
In absence of coupling to strain, $\Delta$ completely determines the quantum phase diagram. with $\Delta = 0$
giving the quantum critical point.
Up to this point the result is just the renormalization of the factor $\Delta$ to $\Delta(1+p/p_c)$ and it becomes an experimentally controllable parameter. To find out the dependence of $T_c$ on pressure, one needs to find out temperature dependence of $\sigma$ at certain value of $p$ and then $T_c$ can be found out from the equation $\omega^2(q,T_c )= 0$.
Near quantum critical point a non-self consistent calculation, with a temperature dependent momentum cut off
($q_{max}\sim T$) gives $\sigma \sim T^2$ which thereby imply $T_c\sim (1+p/p_c)^{\frac{1}{2}}$.
\section{Landau Expansion}
Now using this scaling behavior for $T_c$, we switch over to a Landau description for the ordered phase, that is, we
write the variational Gibbs free energy functional as,
\be
F(P, \epsilon : p, T) = \frac{\alpha}{2}(T - T_c(\epsilon))P^2 +
\frac{\lambda}{4}P^4 + \frac{1}{2}K\epsilon^2 - p\epsilon.
\label{F-Gibbs}
\ee
Here $\alpha, \lambda \ge 0$, and $K$ is the stiffness constant for strain and
$p$ is external hydrostatic pressure. $P$ and $\epsilon$ are the average values
of polarization and strain respectively. In
this scheme we can proceed up to the tricritical point along a second order line,
i.e. the coefficient of quartic term is positive and truncation of free energy
functional at quartic term is sufficient. A similar function was used recently
by Gehring \cite{Gehring} to describe various aspects of pressure induced quantum
phase transition in itinerant magnets. Starting from equation (\ref{H}), if we
consider only mean field behavior of both the order parameter and strain, we get $<\epsilon(q)> = \frac{p}{K}\delta(q)$ and $T_c \sim (1+p/p_c)$. It is to be noted that the behavior near QCP can be understood only beyond the Landau scheme
and a proper account of local quantum fluctuations as well as the spatial variations of order parameter in the free energy functional is needed. With introduction of one more variational parameter ($\sigma$), the mean square fluctuations defined in the equation (\ref{sigma}) and elimination of it through minimization of free energy, will lead to the self consistent equation in the disordered phase. To a good approximation we neglect fluctuations in strain (when $K\gg 0$) and considering it's mean field value $\frac{p}{K}\delta(q)$, it leads to $T_c \sim (1+p/p_c)^{\frac{1}{2}}$.
The reason for such behavior in the ordered phase is that in the self consistent calculation there $\sigma$
will be replaced by $\sigma + <u>^2$ in the expression for renormalized frequency (\ref{re-frequency}).
Here $<u>$ is the mean average displacement and the polarization ($P$) is proportional to it. Classically $<u> \sim (T_c(p))^{\frac{1}{2}}$, hence its contribution retains the form of equation (\ref{omega-final}) same except modifying the numerical factor in the right hand side. In quantum domain there is possibility that $<u> \sim T_c^{\nu}$. As long as $\nu \geq \frac{1}{2}$, this assumption is valid.
One can also question the validity of the linear expansion in temperature around $T_c$ when $T_c\rightarrow 0$. In a suitable description for the phase transition at finite temperature near QCP, a higher power of $T$ should appear. Here we focus only on QCP at zero temperature. Now we solve the coupled equation obtained from the minimization of free energy with respective to these parameters which gives their physical values. Minimization of free energy with respective to polarization gives
\be
\frac{\partial F(P, \epsilon : p, T)}{\partial P} = 0 = P \left[ \alpha(T - T_c(\epsilon)) + \lambda P^2 \right].
\ee
We need one more equation to eliminate $\epsilon$ from the above equation and this equation is obtained from the minimization of free energy with respective to strain as follows,
\be
\frac{\partial F(P, \epsilon : p, T)}{\partial \epsilon} = 0 = -\frac{\alpha}{2}\frac{\partial T_c(\epsilon)}{\partial \epsilon}P^2 + K\epsilon -p
\ee
One needs to solve coupled equation (\ref{re-frequency}) and (\ref{self-consist}) to get physically observable quantities.\\
{\bf Paraelectric Phase:} This phase is characterized by zero average value of polarization and leads to the solutions
\be P = 0,~~ \epsilon = p/K~~ and ~~F_0(p,T) = -\frac{p^2}{2K}
\ee
{\bf Ferroelectric Phase:} In this phase average polarization takes non zero value and interplay between strain and polarization becomes interesting. Here,
\bea
P &=& \pm\sqrt{\frac{\alpha(T_c(\epsilon)-T)}{\lambda}}, \nonumber \\
\epsilon - \frac{p}{K}
&=& \frac{\alpha^2}{2\lambda K}(T_c(\epsilon)-T)\frac{\partial T_c(\epsilon)}{\partial\epsilon} \nonumber \\ & \equiv & g(\epsilon) ~~\text{(say)}
\label{strain-pol}
\end{eqnarray}
and on substituting this expression the free energy in the polar phase becomes
\bea
F_{pol}(p,T) &=& F_0 (p,T) - \frac{\alpha^2} {4\lambda} \left( T_c(\epsilon_0)-T \right) ^2 + \frac{Kg^2(\epsilon_0)}{2}\nonumber\\
&=& F_0 (p,T) - \frac{\alpha^2}{4\lambda} \left( T_c(\epsilon_0)-T \right) ^2 \nonumber\\
&\times& \left[ 1-\frac{\alpha^2}{2\lambda K}(\frac{\partial T_c(\epsilon)}{\partial\epsilon}|_{\epsilon_0})^2\right]
\label{F-pol}
\end{eqnarray}
Where $\epsilon_0$ is a solution of equation (\ref{strain-pol}). The second term in the right hand side of the equation (\ref{F-pol}) vanishes at $T_c$ and also below $T_c$ if
\be
1-\frac{\alpha^2}{2\lambda K}(\frac{\partial T_c(\epsilon)}{\partial\epsilon}|_{\epsilon_0})^2 = 0
\ee
Clearly the above equation is the condition for tricriticality.
Thus the condition to achieve first order transition is given by
\be
F_{pol}(p,T) \geq F_0 (p,T) \Rightarrow \frac{\alpha^2}{2K\lambda}
\left( \frac{\partial T_c}{\partial \epsilon}\right) ^2 \geq 1
\ee
In terms of pressure this relation becomes
\be
\frac{\alpha^2}{2K\lambda}
\left( \frac{\partial T_c}{\partial p}\frac{\partial p}{\partial\epsilon} \right) ^2 \geq 1\Rightarrow \frac{\alpha^2 K}{2\lambda}
\left( \frac{\partial T_c}{\partial p} \right) ^2 \geq 1
\ee
The point in p-T phase diagram satisfying the equality marks the transition between second and first order is called ``tricritical point''. In our estimation of $T_c(\epsilon)$ or $T_c(p)$ (estimated microscopically and supported experimentally) is taken as
\be
T_c(p) = \left( 1-p/p_0 \right)^{\xi}
\ee
where $\xi = \frac{1}{2} \leq 1$, which tells that at quantum critical point the transition is inevitably first order. \\
\section{Discussions}
The above mentioned analysis is a qualitative answer to the recently found phase separation behavior near ferroelectric quantum critical point and its absence when one is away from the quantum critical point. Our analysis shows that the exponent $\xi$ which determines the scaling behavior of $T_c$ with pressure near quantum critical point, determines whether
phase separation is there or not. The fact that $\xi\le 1$ is clearly a consequence of
the large quantum fluctuations near QCP. Near QCP system becomes very much sensitive to any perturbation and is manifested by the divergence of ${\partial T_c}/{\partial p})_{p_c}$. These tells that in the proposed form of free energy in equation (\ref{F-Gibbs}) some amount of fluctuation effects are included and the first orderness and the occurrence of phase separation is essentially fluctuation induced.
Here we have estimated exponent $\xi$ in a qualitative manner. Correct estimation needs to take higher order correlations into account and needs more involved calculations. For the present purpose only relevant aspect about $\xi$ is, whether it is $\le 1$ or not.
As long as one have $\xi\le 1$ (from theory or experiments) the whole discussion is true.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 4,845 |
Japan Carrier Forces Wheelchair Man to Crawl onto Plane. Required to crawl for safety reasons. A low-priced Japanese airline on Wednesday (June 26th) apologized after having forced a man in a wheelchair to crawl up the stairs leading to his plane.
The scene took place in early June on the island of Amami-Oshima in southern Japan. Hideto Kijima, 44, was returning from vacation when a Vanilla Air employee informed him that the safety rules prohibited him from wearing it. According to the Asahi newspaper, the man had to hoist himself with the strength of his arms up to the 17 steps to embark aboard the plane bound for Osaka.
According to the Asahi newspaper, the man had to hoist himself with the strength of his arms up to the 17 steps to embark aboard the plane bound for Osaka.
Hideto Kijima, who travels frequently, said he was "surprised" when the staff exposed him to the problem. "I wondered if the employees at the airport did not find it incorrect," he told a Japanese TV station.
"We are sorry to have subjected this test," said a spokesman for Vanilla Air (ANA Holdings), contacted by AFP. Since this unfortunate episode, the company says it has set up at the airport a lifting device for people with disabilities, equipment used by airlines.
In April, US airline United Airlines had been heavily criticized after the violent deportation by the Chicago airport police of the passenger of an overbooked flight, a filmed scene that had raised a wave of international outrage.
Web Designers Cannot Perform Web Development-Is it True? | {
"redpajama_set_name": "RedPajamaC4"
} | 1,741 |
Zbiornik zaporowy Bukovec (słow. Vodná nádrž Bukovec) – sztuczny zbiornik wodny w południowo-wschodniej Słowacji, powstały przez przegrodzenie rzeki Ida zaporą wodną usytuowaną powyżej wsi Bukovec.
Położenie
Zbiornik znajduje się w południowej części Gór Wołowskich, w Rudawach Spiskich. Utworzono go na rzece Ida, która stanowi w tym miejscu granicę pomiędzy dwoma mniejszymi jednostkami wyróżnianymi w tej grupie górskiej: Pasmem Kojszowskiej Hali (na północy) oraz Pasmem Holički (na południu). Administracyjnie znajduje się w granicach powiatu Koszyce-okolice w kraju koszyckim.
Historia
Zbiornik i zapora budowane były w latach 1968–1976, jako kolejne źródło wody do celów komunalnych, w obliczu szybkiego rozrostu Koszyc i całej koszyckiej aglomeracji. Dodatkowo stanowi element stabilizujący przepływ rzeki Idy.
Charakterystyka
Zbiornik tworzy zapora ziemna typu ciężkiego. Maksymalna wysokość korony zapory ponad terenem wynosi 56 m, a długość zbiornika 2,8 km. Pojemność zbiornika to maksymalnie 21,4 milionów m³ wody. Przy maksymalnym poziomie wody w zbiorniku lustro wody leży na wysokości 416,75 m n.p.m.
W zależności od lokalnych uwarunkowań (jakości wody) czerpnia wody umożliwia pobór wody z czterech różnych poziomów. Maksymalny możliwy pobór wody pitnej wynosi 700 litrów/s. W rzeczywistości obecnie średni pobór wynosi 120–140 l/s.
Znaczenie przyrodnicze
Zbiornik jest cennym biotopem ptactwa wodnego, natomiast jego brzegi wraz z porastającymi je zespołami roślinnymi – płazów i gadów.
Znaczenie turystyczne
Ze względu na przeznaczenie (źródło wody pitnej) zbiornik nie jest przeznaczony do jakiegokolwiek wykorzystania rekreacyjnego, a jego brzegi objęte są strefą ochrony sanitarnej.
Bibliografia
wg
Linki zewnętrzne
Bukovec, Zbiornik zaporowy
Kraj koszycki | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 7,781 |
Stratus Building Solutions of New Orleans Honored with IFA Franchisee of the Year Award
Fox 8 WVUE New Orleans
Stratus Building Solutions Regional Master Franchise Owner Michael J. Seiler was presented IFA's Annual Franchisee of the Year Award in Washington, DC this month.
New Orleans, LA (PRWEB) September 27, 2017
Stratus Building Solutions Regional Master Franchise of New Orleans owner Michael J. Seiler was honored Tuesday, September 12th by the International Franchise Association and the Stratus Building Solutions Corporate Team with its annual Franchisee of the Year Award. Seiler was recognized during the IFA's Franchise Action Network Annual Meeting for his first year as a Stratus Master Franchise in the New Orleans Metropolitan.
"IFA is proud to recognize Michael with the Franchisee of the Year award," said IFA President & CEO, Robert Cresanti, CFE. "The franchising community is a critical component of the U.S. economy in large part due to the dedication of exemplary individuals like Michael who have helped grow the industry and continue to give back to their communities."
Celebrating his office's year anniversary in October, Seiler is on pace for a record breaking first year; alongside his drive, passion and loyalty to the Stratus system and the steadfast belief to help Stratus franchise owners in the New Orleans market, Seiler's personal attention to customer service has been attributed to his outstanding success and recognition for the award.
"We were honored to present this year's Franchisee of the Year award to Michael Seiler. In less than one-year Seiler and his team have demonstrated that through hard work, dedication, support to his Franchisees and outstanding customer service he is able to exceed all expectations," states Afshin Cangarlu, CEO of Stratus Building Solutions Corporate, "His commitment to drive value and benefits to his customers will enable Stratus of New Orleans to successfully grow year after year."
"The New Orleans team all have the same vision of helping our franchise owners better their families' lives by building successful small businesses. I am proud to bring it back to New Orleans and present it to the team because it's through their commitment and collective efforts we are being recognized," said Seiler after receiving the award in DC on Tuesday.
Stratus Building Solutions of New Orleans regional master franchise office currently provides support to 19 janitorial service franchisees who provide services to nearly 60 customers across the New Orleans metro. Stratus stands out as a commercial cleaning service provider as the only franchise to offer Green Cleaning services with proprietary, Green Seal Certified line of cleaning chemicals.
About Stratus Building Solutions
Stratus Building Solutions is an international franchise company in the commercial cleaning industry, founded in 2006 and headquartered in Los Angeles, CA. Stratus was developed to provide environmentally friendly commercial cleaning services driven by dedicated, entrepreneurial, small business owners and regional support offices. Stratus has over 1,400 unit franchisees in 31 major cities across the United States. Stratus is setting new standards in the building services and maintenance franchise industry by being the first to offer green janitorial with their proprietary, Green Seal Certified line of cleaning chemicals. | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 3,577 |
package integration
import (
"fmt"
"log"
"math/rand"
"os"
"strconv"
"testing"
"github.com/coreos/etcd/client"
"github.com/coreos/etcd/pkg/testutil"
"golang.org/x/net/context"
)
func init() {
// open microsecond-level time log for integration test debugging
log.SetFlags(log.Ltime | log.Lmicroseconds | log.Lshortfile)
if t := os.Getenv("ETCD_ELECTION_TIMEOUT_TICKS"); t != "" {
if i, err := strconv.ParseInt(t, 10, 64); err == nil {
electionTicks = int(i)
}
}
}
func TestClusterOf1(t *testing.T) { testCluster(t, 1) }
func TestClusterOf3(t *testing.T) { testCluster(t, 3) }
func testCluster(t *testing.T, size int) {
defer testutil.AfterTest(t)
c := NewCluster(t, size)
c.Launch(t)
defer c.Terminate(t)
clusterMustProgress(t, c.Members)
}
func TestTLSClusterOf3(t *testing.T) {
defer testutil.AfterTest(t)
c := NewClusterByConfig(t, &ClusterConfig{Size: 3, PeerTLS: &testTLSInfo})
c.Launch(t)
defer c.Terminate(t)
clusterMustProgress(t, c.Members)
}
func TestClusterOf1UsingDiscovery(t *testing.T) { testClusterUsingDiscovery(t, 1) }
func TestClusterOf3UsingDiscovery(t *testing.T) { testClusterUsingDiscovery(t, 3) }
func testClusterUsingDiscovery(t *testing.T, size int) {
defer testutil.AfterTest(t)
dc := NewCluster(t, 1)
dc.Launch(t)
defer dc.Terminate(t)
// init discovery token space
dcc := mustNewHTTPClient(t, dc.URLs(), nil)
dkapi := client.NewKeysAPI(dcc)
ctx, cancel := context.WithTimeout(context.Background(), requestTimeout)
if _, err := dkapi.Create(ctx, "/_config/size", fmt.Sprintf("%d", size)); err != nil {
t.Fatal(err)
}
cancel()
c := NewClusterByConfig(
t,
&ClusterConfig{Size: size, DiscoveryURL: dc.URL(0) + "/v2/keys"},
)
c.Launch(t)
defer c.Terminate(t)
clusterMustProgress(t, c.Members)
}
func TestTLSClusterOf3UsingDiscovery(t *testing.T) {
defer testutil.AfterTest(t)
dc := NewCluster(t, 1)
dc.Launch(t)
defer dc.Terminate(t)
// init discovery token space
dcc := mustNewHTTPClient(t, dc.URLs(), nil)
dkapi := client.NewKeysAPI(dcc)
ctx, cancel := context.WithTimeout(context.Background(), requestTimeout)
if _, err := dkapi.Create(ctx, "/_config/size", fmt.Sprintf("%d", 3)); err != nil {
t.Fatal(err)
}
cancel()
c := NewClusterByConfig(t,
&ClusterConfig{
Size: 3,
PeerTLS: &testTLSInfo,
DiscoveryURL: dc.URL(0) + "/v2/keys"},
)
c.Launch(t)
defer c.Terminate(t)
clusterMustProgress(t, c.Members)
}
func TestDoubleClusterSizeOf1(t *testing.T) { testDoubleClusterSize(t, 1) }
func TestDoubleClusterSizeOf3(t *testing.T) { testDoubleClusterSize(t, 3) }
func testDoubleClusterSize(t *testing.T, size int) {
defer testutil.AfterTest(t)
c := NewCluster(t, size)
c.Launch(t)
defer c.Terminate(t)
for i := 0; i < size; i++ {
c.AddMember(t)
}
clusterMustProgress(t, c.Members)
}
func TestDoubleTLSClusterSizeOf3(t *testing.T) {
defer testutil.AfterTest(t)
c := NewClusterByConfig(t, &ClusterConfig{Size: 3, PeerTLS: &testTLSInfo})
c.Launch(t)
defer c.Terminate(t)
for i := 0; i < 3; i++ {
c.AddMember(t)
}
clusterMustProgress(t, c.Members)
}
func TestDecreaseClusterSizeOf3(t *testing.T) { testDecreaseClusterSize(t, 3) }
func TestDecreaseClusterSizeOf5(t *testing.T) { testDecreaseClusterSize(t, 5) }
func testDecreaseClusterSize(t *testing.T, size int) {
defer testutil.AfterTest(t)
c := NewCluster(t, size)
c.Launch(t)
defer c.Terminate(t)
// TODO: remove the last but one member
for i := 0; i < size-1; i++ {
id := c.Members[len(c.Members)-1].s.ID()
c.RemoveMember(t, uint64(id))
c.waitLeader(t, c.Members)
}
clusterMustProgress(t, c.Members)
}
func TestForceNewCluster(t *testing.T) {
c := NewCluster(t, 3)
c.Launch(t)
cc := mustNewHTTPClient(t, []string{c.Members[0].URL()}, nil)
kapi := client.NewKeysAPI(cc)
ctx, cancel := context.WithTimeout(context.Background(), requestTimeout)
resp, err := kapi.Create(ctx, "/foo", "bar")
if err != nil {
t.Fatalf("unexpected create error: %v", err)
}
cancel()
// ensure create has been applied in this machine
ctx, cancel = context.WithTimeout(context.Background(), requestTimeout)
if _, err = kapi.Watcher("/foo", &client.WatcherOptions{AfterIndex: resp.Node.ModifiedIndex - 1}).Next(ctx); err != nil {
t.Fatalf("unexpected watch error: %v", err)
}
cancel()
c.Members[0].Stop(t)
c.Members[1].Terminate(t)
c.Members[2].Terminate(t)
c.Members[0].ForceNewCluster = true
err = c.Members[0].Restart(t)
if err != nil {
t.Fatalf("unexpected ForceRestart error: %v", err)
}
defer c.Members[0].Terminate(t)
c.waitLeader(t, c.Members[:1])
// use new http client to init new connection
cc = mustNewHTTPClient(t, []string{c.Members[0].URL()}, nil)
kapi = client.NewKeysAPI(cc)
// ensure force restart keep the old data, and new cluster can make progress
ctx, cancel = context.WithTimeout(context.Background(), requestTimeout)
if _, err := kapi.Watcher("/foo", &client.WatcherOptions{AfterIndex: resp.Node.ModifiedIndex - 1}).Next(ctx); err != nil {
t.Fatalf("unexpected watch error: %v", err)
}
cancel()
clusterMustProgress(t, c.Members[:1])
}
func TestAddMemberAfterClusterFullRotation(t *testing.T) {
defer testutil.AfterTest(t)
c := NewCluster(t, 3)
c.Launch(t)
defer c.Terminate(t)
// remove all the previous three members and add in three new members.
for i := 0; i < 3; i++ {
c.RemoveMember(t, uint64(c.Members[0].s.ID()))
c.waitLeader(t, c.Members)
c.AddMember(t)
c.waitLeader(t, c.Members)
}
c.AddMember(t)
c.waitLeader(t, c.Members)
clusterMustProgress(t, c.Members)
}
// Ensure we can remove a member then add a new one back immediately.
func TestIssue2681(t *testing.T) {
defer testutil.AfterTest(t)
c := NewCluster(t, 5)
c.Launch(t)
defer c.Terminate(t)
c.RemoveMember(t, uint64(c.Members[4].s.ID()))
c.waitLeader(t, c.Members)
c.AddMember(t)
c.waitLeader(t, c.Members)
clusterMustProgress(t, c.Members)
}
// Ensure we can remove a member after a snapshot then add a new one back.
func TestIssue2746(t *testing.T) {
defer testutil.AfterTest(t)
c := NewCluster(t, 5)
for _, m := range c.Members {
m.SnapCount = 10
}
c.Launch(t)
defer c.Terminate(t)
// force a snapshot
for i := 0; i < 20; i++ {
clusterMustProgress(t, c.Members)
}
c.RemoveMember(t, uint64(c.Members[4].s.ID()))
c.waitLeader(t, c.Members)
c.AddMember(t)
c.waitLeader(t, c.Members)
clusterMustProgress(t, c.Members)
}
// Ensure etcd will not panic when removing a just started member.
func TestIssue2904(t *testing.T) {
defer testutil.AfterTest(t)
// start 1-member cluster to ensure member 0 is the leader of the cluster.
c := NewCluster(t, 1)
c.Launch(t)
defer c.Terminate(t)
c.AddMember(t)
c.Members[1].Stop(t)
// send remove member-1 request to the cluster.
cc := mustNewHTTPClient(t, c.URLs(), nil)
ma := client.NewMembersAPI(cc)
ctx, cancel := context.WithTimeout(context.Background(), requestTimeout)
// the proposal is not committed because member 1 is stopped, but the
// proposal is appended to leader's raft log.
ma.Remove(ctx, c.Members[1].s.ID().String())
cancel()
// restart member, and expect it to send UpdateAttributes request.
// the log in the leader is like this:
// [..., remove 1, ..., update attr 1, ...]
c.Members[1].Restart(t)
// when the member comes back, it ack the proposal to remove itself,
// and apply it.
<-c.Members[1].s.StopNotify()
// terminate removed member
c.Members[1].Terminate(t)
c.Members = c.Members[:1]
// wait member to be removed.
c.waitMembersMatch(t, c.HTTPMembers())
}
// TestIssue3699 tests minority failure during cluster configuration; it was
// deadlocking.
func TestIssue3699(t *testing.T) {
// start a cluster of 3 nodes a, b, c
defer testutil.AfterTest(t)
c := NewCluster(t, 3)
c.Launch(t)
defer c.Terminate(t)
// make node a unavailable
c.Members[0].Stop(t)
<-c.Members[0].s.StopNotify()
// add node d
c.AddMember(t)
// electing node d as leader makes node a unable to participate
leaderID := c.waitLeader(t, c.Members)
for leaderID != 3 {
c.Members[leaderID].Stop(t)
<-c.Members[leaderID].s.StopNotify()
c.Members[leaderID].Restart(t)
leaderID = c.waitLeader(t, c.Members)
}
// bring back node a
// node a will remain useless as long as d is the leader.
err := c.Members[0].Restart(t)
select {
case <-c.Members[0].s.StopNotify():
t.Fatalf("should not be stopped")
default:
}
// must waitLeader so goroutines don't leak on terminate
leaderID = c.waitLeader(t, c.Members)
// try to participate in cluster
cc := mustNewHTTPClient(t, []string{c.URL(0)}, c.cfg.ClientTLS)
kapi := client.NewKeysAPI(cc)
ctx, cancel := context.WithTimeout(context.Background(), requestTimeout)
_, err = kapi.Set(ctx, "/foo", "bar", nil)
cancel()
if err != nil {
t.Fatalf("unexpected error on Set (%v)", err)
}
}
// clusterMustProgress ensures that cluster can make progress. It creates
// a random key first, and check the new key could be got from all client urls
// of the cluster.
func clusterMustProgress(t *testing.T, membs []*member) {
cc := mustNewHTTPClient(t, []string{membs[0].URL()}, nil)
kapi := client.NewKeysAPI(cc)
ctx, cancel := context.WithTimeout(context.Background(), requestTimeout)
key := fmt.Sprintf("foo%d", rand.Int())
resp, err := kapi.Create(ctx, "/"+key, "bar")
if err != nil {
t.Fatalf("create on %s error: %v", membs[0].URL(), err)
}
cancel()
for i, m := range membs {
u := m.URL()
mcc := mustNewHTTPClient(t, []string{u}, nil)
mkapi := client.NewKeysAPI(mcc)
mctx, mcancel := context.WithTimeout(context.Background(), requestTimeout)
if _, err := mkapi.Watcher(key, &client.WatcherOptions{AfterIndex: resp.Node.ModifiedIndex - 1}).Next(mctx); err != nil {
t.Fatalf("#%d: watch on %s error: %v", i, u, err)
}
mcancel()
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 6,584 |
Q: C# Adding a tabpage to a listView I have a window form application, and in that I have a ListView called lstView.
How do I add a tabpage to that through code.
thanks in advance.
A: here is your code
private void Form1_Load(object sender, EventArgs e)
{
TabControl tbl = new TabControl();
tbl.TabPages.Add("page1");
lstView.Controls.Add(tbl);
}
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 2,954 |
Q: Sql Server pivot table not grouping result set I have table with values as follows -
EMP_CODE | LEAVENAME | APP_TYPE | LEAVE_DATE | ACT_DAYS
--------------------------------------------------------
ST006 | CL | P | 2012-01-03 | 1.0
ST006 | CL | P | 2012-01-18 | 1.0
ST006 | SL | P | 2012-01-27 | 1.0
ST002 | CL | P | 2012-01-04 | 1.0
ST002 | CL | P | 2012-01-12 | 1.0
ST002 | SL | P | 2012-01-27 | 1.0
OCO038 | CL | P | 2012-01-27 | 1.0
HO188 | CL | P | 2012-01-09 | 1.0
HO188 | CL | P | 2012-01-30 | 1.0
HO085 | CL | P | 2012-01-19 | 1.0
HO085 | SL | P | 2012-01-23 | 1.0
I have written this query to sum all leave types as columns for each employee. Each employee must have only one row.
SELECT EMP_CODE,[CL],[LWP],[PL],[SL] FROM LEAVE_DETAIL L
PIVOT (SUM(ACT_DAYS) FOR LEAVENAME IN ([CL],[LWP],[PL],[SL]))
AS PVT ORDER BY EMP_CODE;
But this query is not giving me the expected output. There are more than one row for each employee which is not what I want.
The following table show the expected output -
EMP_CODE | CL | SL |
---------|------|-----|
ST006 | 2.0 | 1.0 |
ST002 | 2.0 | 1.0 |
OCO038 | 1.0 | 0.0 |
HO188 | 2.0 | 0.0 |
HO085 | 1.0 | 1.0 |
Please help.
A: you can try as follow. You can repalce your table as I tested temp table same as your table.
create table #tmp_emp (EMP_CODE varchar(10),LEAVENAME char(2), APP_TYPE char(1),LEAVE_DATE datetime,ACT_DAYS decimal(2,1))
insert into #tmp_emp values ('ST006','CL','P ','2012-01-03','1.0');
insert into #tmp_emp values ('ST006','CL','P ','2012-01-18','1.0');
insert into #tmp_emp values ('ST006','SL','P ','2012-01-27','1.0');
insert into #tmp_emp values ('ST002','CL','P ','2012-01-04','1.0');
insert into #tmp_emp values ('ST002','CL','P ','2012-01-12','1.0');
insert into #tmp_emp values ('ST002','SL','P ','2012-01-27','1.0');
insert into #tmp_emp values ('OCO038','CL','P ','2012-01-27','1.0');
insert into #tmp_emp values ('HO188','CL','P ','2012-01-09','1.0');
insert into #tmp_emp values ('HO188','CL','P ','2012-01-30','1.0');
insert into #tmp_emp values ('HO085','CL','P ','2012-01-19','1.0');
insert into #tmp_emp values ('HO085','SL','P ','2012-01-23','1.0');
SELECT EMP_CODE,[CL],[LWP],[PL],[SL]
FROM
(
select EMP_CODE, LEAVENAME, sum(ACT_DAYS) ACT_DAYS
from #tmp_emp
group by EMP_CODE, LEAVENAME
) L
PIVOT (SUM(ACT_DAYS) FOR LEAVENAME IN ([CL],[LWP],[PL],[SL]))
AS PVT ORDER BY EMP_CODE;
A: You don't even need the group by in the query. Because what a pivot does is that it "group by" on the other columns. The key to this solution is the inner select. I think it is not a god idé to first do a sum with group by and then apply a sum and a group by again.
SELECT
EMP_CODE,
[CL],
[LWP],
[PL],
[SL]
FROM
(
SELECT
EMP_CODE,
LEAVENAME,
ACT_DAYS
FROM
@tmp_emp
) L
PIVOT
(
SUM(ACT_DAYS)
FOR LEAVENAME IN ([CL],[LWP],[PL],[SL])
)
AS PVT
ORDER BY EMP_CODE
This will get you the same result.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 5,371 |
La voix humaine est l'ensemble des sons produits par le frottement de l'air des poumons sur les replis du larynx de l'être humain. La voix inclut la parole et le chuchotement, le cri, le rire et le chant.
L'étude des sons produits par la voix humaine s'appelle la phonétique. C'est une des branches de la linguistique. Dans le domaine de la médecine, l'étude de la physiologie et de la pathologie de la voix et les soins de santé qui lui sont apportés sont du ressort de la phoniatrie. L'orthophonie s'occupe de la rééducation fonctionnelle de la voix.
Fonction
La voix est un instrument à la fois dans sa fonction physico-acoustique de communication avec l'autre, et dans sa fonction psychologique ou de lien à l'autre et à soi-même, donc dans sa fonction pragmatique et identitaire.
Physiologie
Les organes qui assurent la production vocale constituent lappareil phonatoire ou appareil de la phonation. Cet ensemble convertit le souffle en sons en plusieurs étapes : la soufflerie alimentée par l'appareil respiratoire (contrôlé et régulé par les appuis et le système musculaire abdomino-diaphragmatique) ; l'émission par l'appareil de production sonore constitué par le larynx, cavité mobile à l'extrémité de la trachée où se trouvent les cordes vocales dont la vibration module très rapidement le débit (et la pression) d'air, tandis que les muscles laryngés permettent d'ajuster la hauteur du son et dans une certaine mesure le timbre (contenu harmonique) ; l'amplification assurée par des résonateurs (pharynx, cavité buccale, les cavités crâniennes, la masse osseuse) ; l'articulation grâce au bout de la langue, du voile du palais, des gencives supérieures, des incisives supérieures et des deux lèvres. Les cavités buccales et nasales altèrent le timbre de la voix et un peu la hauteur : elles favorisent certaines fréquences au détriment d'autres en fonction de leur forme et de leurs dimensions, et réagissent avec la cavité laryngée.
Les types de voix et les cordes vocales
Les hommes et les femmes ont des cordes vocales de tailles différentes ; les hommes adultes ont généralement une voix plus grave et des cordes vocales plus longues, entre . Celles des femmes se situent entre .
Des facteurs génétiques sont à l'origine de la différence de taille des cordes vocales au sein d'un même sexe, ce qui a donné lieu au classement des voix de chanteurs par tessiture. On trouvera par exemple chez les hommes les basses, barytons et ténors alors qu'on aura chez les femmes les contraltos, mezzo-sopranos et sopranos. La taille des cordes vocales n'est pas la seule cause de différence entre les voix d'hommes et de femmes. La cavité résonnante (trachée, bouche, pharynx…) est généralement plus grande chez les hommes, ce qui favorise aussi les sons graves indépendamment des cordes vocales elles-mêmes. De même la saillie antérieure du cartilage thyroïde qui entoure les cordes vocales a un angle de 90° chez les hommes et de 120° chez les femmes.
Physiologie et timbre vocal
La voix de chaque humain est unique, du fait de la forme et de la taille non seulement de ses cordes vocales, mais aussi du reste du corps de la personne. Les humains peuvent relâcher ou resserrer leurs cordes vocales, ou changer leur épaisseur, ainsi que changer la pression d'air transférée. La forme de la poitrine et du cou, la position de la langue, et la tension de nombreux muscles peuvent être altérées en produisant un effet sur la hauteur, le volume et le timbre du son produit. Le son résonne aussi en différentes parties du corps ; la taille et la structure osseuse d'un individu peuvent affecter sa voix.
Les chanteurs peuvent apprendre à travailler sur la respiration, le positionnement de la gorge et l'ouverture de la bouche afin de produire des registres différents. Un changement des cavités de résonance pourra donner une « voix de poitrine » ou au contraire une « voix de tête ».
Mécanismes vocaux
On distingue quatre façons d'émettre des sons vocaux, dits mécanismes :
le , dit Fry ou : essentiellement accessible aux hommes, il est parfois utilisé en voix parlée sur le son « euh » et ressemble à un gargarisme sans eau ;
le , dit voix de poitrine : c'est le mécanisme le plus fréquent pour un homme. Il permet de produire des sons de fréquence fondamentale comprise entre ;
le , dit voix de tête : c'est le mécanisme le plus fréquent pour une femme. Il permet de produire des sons de fréquence fondamentale comprise entre ;
le , dit voix de sifflet : c'est une voix détimbrée, comparable à une sirène ou un crissement de craie.
Souvent, en début d'apprentissage du chant, le passage de la voix de poitrine à la voix de tête pose problème. Un peu d'entraînement permet en général de franchir cet obstacle sans cassure de la voix grâce à la voix mixte.
Registres vocaux
Les registres vocaux, appelés aussi tessitures vocales, sont souvent confondus avec les mécanismes vocaux. On en dénombre généralement huit (cinq pour les hommes et trois pour les femmes) en chant lyrique : basse, baryton-basse, baryton, ténor, contre-ténor, contralto (ou simplement alto), mezzo-soprano et soprano. Chaque pupitre présente également des subdivisions.
Un registre vocal est l'ensemble des fréquences émises avec une résonance identique, c'est-à-dire la partie de l'étendue vocale dans laquelle le chanteur émet des hauteurs avec un timbre à peu près identique. Les artistes en chant lyrique utilisent plusieurs registres vocaux en modifiant le son émis par différents résonateurs de leur corps (cavité orale, fosses nasales...) afin d'homogénéiser le timbre de leur voix. Par exemple, le ténor lyrique peut monter jusqu'au contre-ut soit un do4 en mécanisme de poitrine, bien que cette note soit extrêmement haute et corresponde plutôt à une tessiture vocale de contreténor en voix de tête. De même une soprano lyrique peut monter aussi jusqu'au contre-ut, soit un do5 en voix de tête, mais la voix de sifflet est normalement de rigueur tant la note semble difficile à atteindre.
Phonétique de la voix parlée
La phonétique est la discipline scientifique qui étudie les sons produits par la voix en vue de la communication verbale.
Voyelles
Le triangle vocalique ou le trapèze vocalique représente les voyelles suivant deux axes, selon qu'elles sont ouvertes ou fermées à la verticale, antérieures ou postérieures à l'horizontale. Ainsi, en bas du triangle se situe le son [a] qui est le plus ouvert. Vers le haut et la gauche se trouvent les sons [è] et [é] notés [ɛ] et [e] jusqu'au [i] qui est fermé et le plus antérieur. Vers le haut et la droite, les sons [au], [o], [ou] notés [ɔ], [o], [u] qui vont vers une voyelle fermée et postérieure. Entre ces deux lignes, les sons [œ], [e], [u] notés [œ], [ə], [y]. En plus de ces voyelles orales, il existe les voyelles nasales [an], [in], [un], [on] notées [ɑ̃], [ẽ], [œ̃], [ɔ̃] où le voile du palais s'ouvre laissant passer l'air dans les fosses nasales.
Les formants sont une amplification de certaines fréquences grâce aux résonateurs. Les valeurs de ces fréquences varient en fonction de la voyelle suivant si elle est ouverte ou fermée, antérieure ou postérieure. Les formants sont en quelque sorte la signature d'une voyelle. On met en évidence en général les deux premiers formants notés F1 et F2. Les voyelles ouvertes comme les sons proches du [a] sont celles où la fréquence du premier formant est la plus élevée. Les voyelles antérieures comme le [i] sont celles où la fréquence du deuxième formant est la plus élevée. Le triangle vocalique des formants permet de visualiser ces deux premiers formants pour toutes les voyelles avec un sens inversé pour les deux axes, de sorte que l'emplacement des voyelles sur ce graphique ressemble un peu au schéma du triangle vocalique.
Consonnes
La prononciation des consonnes est indispensable pour comprendre les paroles. Toutes les consonnes peuvent être classées en deux catégories : voisées ou sourdes. De même, il existe les consonnes fricatives ou occlusives. Les consonnes voisées font vibrer les cordes vocales : le [b], le [d], le [j], le [g], le [l], le [m], le [n], le [v], le [z]. Les consonnes fricatives peuvent se prolonger par friction de l'air : le [s], le [f], le [j], le [r], le [v], le [z], le [ch], le th anglais. Les consonnes occlusives résultent d'un relâchement instantané : le [b], le [d], le [g], le [k], le [l], le [m], le [n], le [p], le [t]. Chaque consonne sourde est couplée à une consonne voisée : p-b, f-v, t-d, s-z, ch-j, k-g.
Voix chantée
Le son de la voix humaine est riche, ce qui signifie qu'il a une composition spectrale et une dynamique complexes. Une note tenue s'analyse en de multiples fréquences variant en permanence. Sur un sonagramme, on peut voir surtout la fréquence fondamentale et ses partiels harmoniques, les amplitudes décroissant en général avec le rang. Des fréquences apparaissent aussi dans l'aigu, avec une forte intensité ; ce sont les formants. Comparée à la voix chantée des femmes à l'unisson normal, celle des hommes présente une fréquence fondamentale une octave plus bas. Cette différence du son de la voix trouve son origine dans les modifications morphologiques de l'appareil vocal à l'époque de la mue. Elle permet une grande richesse harmonique dans les chorales polyphoniques entre les voix graves des basses et les voix aiguës des sopranos. Elle explique peut-être aussi pourquoi, en soliste, les registres de ténor et d'alto sont souvent utilisés, du moins pour les chansons contemporaines.
Pathologies de la voix
La voix humaine peut être atteinte de nombreux dysfonctionnements ; parmi eux, les défauts de prononciation et les lésions aux cordes vocales. Le fait de parler trop longtemps ou trop fort peut amener à une fatigue des organes de la parole. Les extinctions de voix peuvent aussi être le résultat d'une maladie infectieuse. Une extinction de voix qui dure plus de deux semaines est le symptôme d'un problème de fond et doit faire l'objet d'une consultation en ORL.
Bilan vocal
Les phoniatres évaluent la dégradation de la voix à l'écoute selon cinq critères, appréciés grâce à l'expérience des thérapeutes :
G () degré d'atteinte globale la voix,
R () voix rauque ou éraillée,
B () son voilé,
A () manque de puissance,
S () timbre « forcé ».
Le praticien peut établir avec peu d'appareillage un phonétogramme. La personne suivie chante une dizaine de notes bien réparties, le praticien mesure le niveau atteint à une distance spécifiée avec un sonomètre. L'examen peut servir pour mesurer les progrès lors d'une rééducation vocale. Le sonagramme, aujourd'hui réalisable avec un ordinateur de bureau, peut aider à caractériser les problèmes audibles. L'examen audiométrique est le complément nécessaire des examens phoniatriques
Les examens physiques peuvent inclurent l'électroglottographie, inventée en 1957. Elle consiste à mesurer la transmission d'un courant électrique à haute fréquence à travers le larynx pendant l'émission d'un son. Sa variation est liée aux mouvements d'ouverture et de fermeture des cordes vocales .
L'examen des cordes vocales en vibration par un endoscope, avec lumière stroboscopique ou non, est plus invasif, mais donne des informations précises sur leur état .
Affections anatomiques
Des lésions de l'appareil vocal peuvent causer une dysphonie. Certaines sont congénitales, comme le glotidis, sillon qui sépare les cordes vocales, compensé par un effort musculaire ; d'autres sont traumatiques ou irritatives : accidents, séquelles d'une intubation trachéale, tabac, maladies laryngées, reflux gastro-œsophagien, surmenage vocal .
Les nodules, polypes, kystes, cancers des cordes vocales provoquent une dysphonie. La microchirurgie par endoscopie intervient pour traiter ces lésions.
Le nodule de la corde vocale est une petite tuméfaction qu'on observe fréquemment chez les professionnels de la voix. Il provoque une baisse de la tonicité vocale et un timbre voilé et perturbe l'attaque des sons chantés. S'ils sont récents, le repos vocal et la rééducation orthophonique les fait disparaître en quelques mois ; s'ils sont anciens, il faut avoir recours à la microchirurgie .
Chez l'enfant, il existe parfois une insuffisance vélo-pharyngée, un défaut d'occlusion du voile du palais qui gêne la prononciation des voyelles orales. Elle est causée la plupart du temps par la fente palatine, une malformation qui survient avant la naissance.
Affections psychosomatiques
La voix, élément de la personnalité, reflète ses caractéristiques fondamentales. Chaque élément de la performance vocale peut s'interpréter comme définition de soi : parler fort ou bas, aigu ou grave.
Voir aussi
Bibliographie
.
.
Articles connexes
Phonétique
Formant
Prothèse phonatoire
Reconnaissance automatique de la parole
Synthèse vocale
Voix (instrument)
Chant
Liens externes
.
Notes et références
Orthophonie | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 4,704 |
I'm currently in flight on my way back to the great state of Texas! We loved Disney World, but 7 days in the parks wore us flat out! I'll be online tomorrow afternoon, playing catch up!
How to Get Rid of Junk eMail!
Hang Tight…something worth reading is coming! | {
"redpajama_set_name": "RedPajamaC4"
} | 4,001 |
using System.IO;
using Umbraco.Core.Configuration;
namespace Umbraco.Core.IO
{
public class SystemFiles
{
public static string TinyMceConfig => SystemDirectories.Config + "/tinyMceConfig.config";
// TODO: Kill this off we don't have umbraco.config XML cache we now have NuCache
public static string GetContentCacheXml(IGlobalSettings globalSettings)
{
return Path.Combine(globalSettings.LocalTempPath, "umbraco.config");
}
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 6,711 |
\section{Introduction}
Large hierarchical classification with more than 10000 categories is a challenging task (\cite{partalas2015lshtc}). Traditionally, hierarchical classification can be done by training a classifier on the flattened labels (\cite{babbar2013flat}) or by training a classifier at each hierarchical node (\cite{silla2011survey}), whereby each hierarchical node is a decision maker of which subsequent node to route to. However, the second method scales inefficiently with the number of categories ($> 10000$ categories). Models such as hierarchical-SVM (\cite{vural2004hierarchical}) becomes difficult to train when there are 10000 SVMs in the entire hierarchy. Therefore for large number of categories, the hierarchy tree is flattened to produce single labels. While training becomes easier, the data then loses prior information about the labels and their structural relationships.
In this paper, we model the large hierarchical labels with layers of neurons directly. Unlike the traditional structural modeling with classifier at each node, here we represent each label in the hierarchy simply as a neuron.
\section{Model}
HiNet has different procedures for training and inference. During training, as illustrated in Figure \ref{fig:training}, the model is forced to learn MAP (Maximum a Posteriori) hypothesis over predictions at different hierarchical levels independently. Since the hierarchical layers contain shared information as child node is conditioned on the parent node, we employ a combined cost function over errors across different levels. A combined cost allows travelling of information across levels which is equivalent to transfer learning between levels.
During inference, after predicting the posterior distribution at each level of the hierarchy, we employ a greedy downpour algorithm to efficiently infer the MAP (Maximum a Posteriori) hierarchical trace (Figure \ref{fig:inference}) from the posterior predictions at each level (Figure \ref{fig:training}).
\subsection{Hierarchy as Layers of Neurons}
\begin{figure}
\hspace{2cm}
\begin{subfigure}{0.3\textwidth}
\includegraphics[width=4cm]{./images/dag3.png}
\caption{DAG}
\label{fig:dag}
\end{subfigure}
\hspace{1cm}
\begin{subfigure}{0.3\textwidth}
\includegraphics[width=4cm]{./images/tree3.png}
\caption{Tree}
\label{fig:tree}
\end{subfigure}
\caption{Neural network for modeling hierarchical relationships. Figure \ref{fig:dag} shows a DAG (Directed Acyclic Graph) where a child neuron is possible to have more than one parents versus Figure \ref{fig:tree} showing a tree where each child neuron only belongs to one parent. The path will end in a stop neuron (red neuron).}
\label{fig:mask}
\vspace{-0.5cm}
\end{figure}
Using layers of neurons to model hierarchy is very efficient and flexible. It can be easily used to model a Directed Acyclic Graph (DAG) (Figure \ref{fig:dag}) or a Tree (Figure \ref{fig:tree}) by masking out the unnecessary connections. Unlike node-based architecture (\cite{silla2011survey,vens2008decision, dumais2000hierarchical}), whereby each node is an object with pointers to its child and parent, and takes up large memory, neural network models the connections as compact matrix which takes up much less memory. In order to model hierarchies of different length, we append a stop neuron (red neuron in Figure \ref{fig:mask}) at each layer. So a top-down path will end when it reaches the stop neuron.
\subsection{Training}
\begin{figure}[t]
\centering
\centering
\includegraphics[width=8cm]{images/training.png}
\label{fig:training}
\caption{The above figure illustrates the architecture during training.}
\label{fig:training}
\end{figure}
Figure \ref{fig:training} shows the model for transfer learning with combined cost function. Given an input feature $\mathbf{X}$ with multiple levels of outputs $\{\mathbf{y}^{(1)}, \mathbf{y}^{(2)}, \dots, \mathbf{y}^{(n)}\}$, where the outputs may have inter-level dependencies $p(\mathbf{y}^{(k)}|\mathbf{y}^{(1:k-1)}, \mathbf{y}^{(k+1:n)})$. For each output level from network $f_{\theta_k}(\mathbf{X}) = \mathbf{y}^{(k)}$ and its corresponding label $ \tilde{\mathbf{y}}^{(k)}$. The combined cost is defined as $E = \sum_{k}^n\Big(\tilde{\mathbf{y}}^{(k)} -f_{\theta_k}(\mathbf{X})\Big)^2$ allows the parameters $\theta_k$ from different levels to exchange knowledge.
\subsection{Inference}
\begin{figure}[h]
\centering
\centering
\includegraphics[width=7cm]{images/inference2.png}
\caption{Inference: Downpour Algorithm}
\caption{The above figure illustrates the downpour algorithm (Algorithm \ref{algo:inference}) for deriving the MAP trace.}
\label{fig:inference}
\end{figure}
\paragraph{Downpour Algorithm}
\begin{algorithm}[h]
Given output posteriors $\mathbf{y}^{(l)} = \{y_a^{(l)}\}$ at each layer $l \in \{1, \dots, K\}$
Define $T^{(l)}_i = [\mbox{ }]$ as MAP trace of neuron $i$ at level $l$
$T^{(1)}_i = [i]$
\For{\mbox{layer} $l=2$ to $k$}{
Find MAP parent $A = \arg\max_a y_b^{(l)} y_a^{(l-1)}$
$T_b^{(l)} = T_A^{(l-1)}.append(b)$
Update $y_b^{(l)}$ = $\max_a y_b^{(l)} y_a^{(l-1)}$
%
}
Find MAP level for stop neuron $s$
$L = \arg\max_l y_s^{(l)}$
\Return $T_s^{(L)}$
\vspace{0.2cm}
\hrule
\vspace{0.3cm}
\caption{Downpour algorithm for inferencing the MAP trace.}
\label{algo:inference}
\end{algorithm}
During inference, as illustrated in Figure \ref{fig:inference}, the model will output a normalized probability distribution at each level $\{\mathbf{y}^{(1)}, \mathbf{y}^{(2)}, \dots, \mathbf{y}^{(k)}\}$, where $\mathbf{y}^{(k)} = \{y_{a_{k}}^{(k)}\}$ is a vector with indexes $a_k \in \{1,\dots, n\}$, where $n$ is the size of layer $k$, and $\sum_{a_k}y_{a_k}^{(k)}=1$. From the second level onwards, we include a stop neuron (red neuron) which is used for stopping the hierarchical trace. The path of the trace from top down ends in a stop neuron (red neuron). Define the MAP trace up to level $k$ which ends at stop neuron $s$ as $T^{(k)}_{a_k=s} = \arg\max\limits_{a_{1:k-1}} p(a_1, a_2, \dots, a_{k-1}, a_k=s)$. The objective of the downpour in finding the MAP from the hierarchy is equivalent to finding the maximum MAP trace out of all MAP traces that ends in a stop neuron from different levels which is $T^{(L)}_{a_L=s}$ where $L = \arg\max_k T^{(k)}_{a_k=s}$.
The probability of MAP trace at level $k$ can be derived greedily from the MAP trace at level $k-1$ as
\begin{equation}
p(T_{a_k}^{(k)})=\max_{a_{k-1}}p(a_k|a_{k-1})p(T_{a_{k-1}}^{(k-1)})
\label{eqn:greedy}
\end{equation}
From Equation \ref{eqn:greedy}, we can derive Theorems \ref{thm:general}-\ref{thm:downpour} which prove that Downpour Algorithm will always yield the MAP trace of $T^{(L)}_{a_L=s} = \max_n p(T_{a_n=s}^{(n)}) \geq p(a_1, a_2,\dots,a_m=s)$ $\forall m$.
\begin{theorem}
\label{thm:general}
For a greedy downpour that ends at a stop neuron at level $n$ with MAP trace $T_{a_n}^{(n)}$, then $p(S_m) \leq p(T_{a_n}^{(n)})$ for every sequence $S_m=\{a_{1}, a_{2}, \dots, a_n, \dots, a_m\}$ of $m \geq n$ that pass through $a_n$ or ends at $a_n$. \textit{Refer to Appendix A for proof.}
\end{theorem}
\begin{theorem}
\label{thm:longer}
For a MAP trace $T_{a_n=s}^{(n)}$ that ends at stop neuron $s$ at level $n$ such that $p(T_{a_n=s}^{(n)}) \geq p(T_{a_n}^{(n)})$ $\forall a_n \neq s$, then $p(S_m) \leq p(T_{a_n=s}^{(n)})$ for every sequence $S_m=\{a_{1}, a_{2}, \dots, a_m\}$ of $m>n$. \textit{Refer to Appendix A for proof.}
\end{theorem}
\begin{theorem}
\label{thm:downpour}
The maximum of the MAP traces that end in a stop neuron from each level is $T_{a_k=s}^{(L)}$ where $L=\arg\max_k p(T_{a_k=s}^{(k)})$ is the MAP for the hierarchy, which means $p(T_{a_k=s}^{(L)}) \geq p(S_m)$ $\forall m$. \textit{Refer to Appendix A for proof.}
\end{theorem}
\section{Results and Conclusion}
\begin{table}[h]
\begin{center}
\begin{tabular}{|c|c|c|}
\hline
& DMOZ (Tree) & No. of Params \\ \hline
Flatten Network & 39.2 & $O(kn^h)$ \\ \hline
HiNet & 41.4 & $O(kn + hn^2)$ \\ \hline
\end{tabular}
\end{center}
\caption{Accuracy on the DMOZ dataset (\cite{partalas2015lshtc}) with 11947 classes. $k$: dimension of the first feature layer connected to the first hierarchical layer. $n$: dimension of each hierarchical layer. $h$: height of the hierarchy. For a fully dense hierarchy, the total number of classes is $n^h$.
}
\end{table}
We compared HiNet with a Flatten Network which have the same architecture except the output layer for HiNet is hierarchical as illustrated in Figure \ref{fig:training} and flatten for Flatten Network. The number of outputs for Flatten Network corresponds to the number of classes in the dataset. From the results, HiNet out-performs Flatten Network for both a Tree hierarchical dataset with much lesser parameters. We see that the number of parameters in the classification layer for Flatten Network is exponential to the maximum length of the trace. Thus for very deep hierarchies, the number of parameters in Flatten Network will be exponentially large while HiNet is always polynomial. This makes HiNet not only better architecture in terms of accuracy but also way more efficient in parameters space.
\newpage
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 5,748 |
\section{Introduction}
\label{sec:Introduction}
In this paper we study geometric numerical integration algorithms for
non-autonomous systems. By classifying the appropriate Lie sub-algebra
of vector fields, the standard framework for backward error analysis
can be used to explain the superior qualitative behaviour of
geometric methods based on the splitting approach.
The current section continues with a brief review of
the general framework for geometric methods, mainly following
the approach by Reich~\cite{Re1999}.
In Section~\ref{sec:linear_systems} we study geometric integration
of linear systems with non-constant periodic coefficients.
A numerical example from classical rotor dynamics is given.
Conclusions are given in Section~\ref{sec:conclusions}.
We adopt the following notation. $\P$ denotes a phase space manifold of dimension
$n$, with local coordinates $\vect{x}=(x_1,\ldots,x_n)$. In the case when $\P$
is a linear space we also use $\set{P}$.
Further, $\mathfrak{X}(\P)$ denotes the
linear space of vector fields on~$\P$.
The flow of $X\in\mathfrak{X}(\P)$ is denoted $\varphi^t_X$, where
$t$ is the time parameter.
The Lie derivative
along $X$ is denoted~$\LieD{X}$.
If~$X,Y\in\mathfrak{X}(\P)$ then the vector field commutator~$[X,Y]_\mathfrak{X}=\LieD{X}Y$
supplies~$\mathfrak{X}(\P)$ with an infinite dimensional Lie algebra structure.
Its corresponding Lie group is
the set~$\mathrm{Diff}(\P)$ of diffeomorphisms on~$\P$, with composition
as group operation.
\begin{remark}
More precisely, the group $\mathrm{Diff}(\P)$ has the structure
of a \emph{Fréchet Lie group}. See~\cite{Ha1982,Sc2004} for
issues concerning infinite dimensional Lie groups.
\end{remark}
As usual, the general linear group of $n\times n$--matrices is denoted
$\GL(n)$ and its corresponding Lie algebra
$\gl(n)$. We use $[A,B]_\gl$ for the matrix commutator~$AB-BA$.
If $\set{V}$ is a metric linear space, then the linear space of
smooth periodic functions $\mathbb{R}\to\set{V}$
with period $2\pi/\Omega$ is denoted $\mathcal{C}_\Omega(\set{V})$. Notice that
this space is closed under differentiation, i.e., if $f \in \mathcal{C}_\Omega(\set{V})$
then it also holds that $f' \in \mathcal{C}_\Omega(\set{V})$.
\subsection{Geometric Integration and Backward Error Analysis}
\label{sub:geometric_integration_and_backward_error_analysis}
Let $\mathfrak{X}_S(\P)$ be a sub-algebra of $\mathfrak{X}(\P)$, i.e.,
a linear sub-space which is closed under the commutator.
Its corresponding sub-group of $\mathrm{Diff}(\P)$ is denoted
$\mathrm{Diff}_S(\P)$.
Let $X\in\mathfrak{X}_S(\P)$ be a vector field which is to be integrated
numerically. Assume that $X$ can be splitted as a sum
of explicitly integrable vector field also belonging to
$\mathfrak{X}_S(\P)$. That is,
$X=Y+Z$ where $Y,Z \in\mathfrak{X}_S(\P)$ and
$\varphi^t_{Y},\varphi^t_{Z}$ can be computed explicitly.
By various compositions, various numerical integration
schemes for $\varphi^t_X$ are obtained.
The most classical example is $\Phi_h = \varphi_Y^{h/2}\circ
\varphi_Z^{h\vphantom{/2}}\circ\varphi_Y^{h/2}$, which yields a second
order symmetric method ($h$~is the step-size parameter of the method).
Since $\varphi^t_Y,\varphi^t_Z\in \mathrm{Diff}_S(\P)$, and since $\mathrm{Diff}_S(\P)$
is closed under composition (since it is the group operation),
it holds that $\Phi_h \in \mathrm{Diff}_S(\P)$.
Thus, the splitting approach yields structure preserving methods,
which is a key property.
Backward error analysis for structure preserving integrators
deals with the question of finding a modified vector field $\tilde{X}\in\mathfrak{X}_S(\P)$
such that $\Phi_h = \varphi_{\tilde{X}}^h$. In conjunction with perturbation
theory, such an analysis can be used to study the dynamical properties of~$\Phi_h$.
For splitting methods, backward error analysis is particularly simple as
the modified vector field, at least formally,
is obtained from the Baker--Campbell--Hausdorff (BCH) formula.
For details on this framework we refer to Reich~\cite{Re1999}.
\section{Linear Systems}
\label{sec:linear_systems}
In this section we study non-autonomous systems on a linear phase space $\set{P}$
with global coordinates $\vect{x}=(x_1,\ldots,x_n)$. More precisely, let
$\Ggroup$ be a Lie sub-group of $\GL(n)$ and $\Galg$ its corresponding
Lie sub-algebra. We consider systems of the form
\begin{equation} \label{eq:linear_systems}
\dot{\vect{x}} = A(t)\vect{x} + f(t)
\end{equation}
where $A \in \mathcal{C}_\Omega(\Galg)$ and $f \in \mathcal{C}_\Omega(\set{P})$ is a smooth vector valued
periodic function with period~$T=2\pi/\Omega$. Our objective is to construct
geometric integrators for~\eqref{eq:linear_systems}. Of course,
since the system is linear, there is a closed form formula
for its solution. However, in engineering applications, e.g.\ finite element analysis,
the system is typically very large so computing the exponential matrix, which is
necessary for the exact solution, is not computationally efficient.
Also, it might not be possible
to analytically integrate $f$~and~$A$ over~$t$,
which is necessary for the exact solution.
In order to study dynamical systems of the form~\eqref{eq:linear_systems}
in the framework of geometric integration, we need, first of all, to
extend the phase space to $\overline{\set{P}}=\set{P}\times\mathbb{R}$ to include the time
variable in the dynamics. Coordinates on $\overline{\set{P}}$ are now
given by $(\vect{x},t)$ and the new independent variable is denoted~$\tau$
(in practice we always have $t(\tau)=\tau$).
Further, we need to find a Lie sub-algebra of $\mathfrak{X}(\overline{\set{P}})$
which captures the form~\eqref{eq:linear_systems}. For this purpose, consider the
set of vector field on $\overline{\set{P}}$ given by
\begin{equation} \label{eq:linear_sub_algebra}
\mathfrak{L}_{\Omega}(\set{P},\Galg) = \big\{
X \in \mathfrak{X}(\overline{\set{P}}) \; \big| \;
X(\vect{x},t) = (A(t) x + f(t),\alpha), \; A\in\mathcal{C}_{\Omega}(\Galg),\; f\in\mathcal{C}_\Omega(\set{P}),\;\alpha\in\mathbb{R}
\big\} \; .
\end{equation}
We now continue with some results concerning properties of~$\mathfrak{L}_{\Omega}(\set{P},\Galg)$.
The first result states that it actually \emph{is} a Lie sub-algebra.
\begin{proposition} \label{prop:linear_lie_algebra}
The set of vector fields $\mathfrak{L}_{\Omega}(\set{P},\Galg)$ is a Lie sub-algebra
of $\mathfrak{X}(\overline{\set{P}})$.
\end{proposition}
\begin{proof}
We need to check that $\mathfrak{L}_{\Omega}(\set{P},\Galg)$ is closed under vector operations and under
the Lie bracket. That is, $X,Y\in \mathfrak{L}_{\Omega}(\set{P},\Galg)$ should imply $a X + b Y \in \mathfrak{L}_{\Omega}(\set{P},\Galg)$ for
$a,b\in\mathbb{R}$ and $[X,Y]_\mathfrak{X} \in \mathfrak{L}_{\Omega}(\set{P},\Galg)$.
With $X(\vect{x},t)=(A(t)\vect{x}+f(t),\alpha)$ and
$Y(\vect{x},t)=(B(t)\vect{x}+g(t),\beta)$ we get $(aX+bY)(\vect{x},t)=((a A+b B)(t)\vect{x}+(a f+b g)(t),a \alpha+b \beta)$
which is of the desired form. Further,
\begin{multline*}
[X,Y]_{\mathfrak{X}} = \begin{pmatrix} A(t) & A'(t)\vect{x}+f'(t) \\ 0 & 0 \end{pmatrix}
\begin{pmatrix} B(t)\vect{x} + g(t) \\ \beta
\end{pmatrix}
-
\begin{pmatrix} B(t) & B'(t)\vect{x}+g'(t) \\ 0 & 0 \end{pmatrix}
\begin{pmatrix} A\vect{x} + f(t) \\ \alpha
\end{pmatrix}
\\
= \begin{pmatrix}
(A(t)B(t)-B(t)A(t)+\beta A'(t)-\alpha B'(t))\vect{x} + A(t)g(t) - B(t)f(t) + \beta f'(t) - \alpha g'(t) \\ 0
\end{pmatrix}
\end{multline*}
which is of the desired form since $AB-BA+\beta A'-\alpha B'=
[A,B]_\GL+\beta A' + \alpha B' \in \mathcal{C}_{\Omega}(\Galg)$ and
$(Ag - Bf + \beta f' - \alpha g') \in \mathcal{C}_\Omega(\set{P})$.
\end{proof}
From the proof above we immediately obtain the following corollary.
\begin{corollary}
The set $\mathfrak{l}_\Omega=\mathcal{C}_{\Omega}(\Galg)\times\mathcal{C}_\Omega(\set{P})\times\mathbb{R}$ equipped
with the induced vector operation
\[
a(A,f,\alpha)+b(B,g,\beta)=(aA+bB,af+bg,a\alpha+b\beta) , \quad a,b\in\mathbb{R}
\]
and with the bracket operation
\[
[(A,f,\alpha),(B,g,\beta)]_{\mathfrak{L}} = ([A,B]_\GL+\beta A' + \alpha B',Ag-Bf+\beta f'-\alpha g',0)
\]
is a Lie algebra which is isomorphic to $\mathfrak{L}_{\Omega}(\set{P},\Galg)$ with isomorphism
$\mathfrak{l}_\Omega \ni (A,f,\alpha) \mapsto (A\vect{x}+f,\alpha) \in \mathfrak{L}_{\Omega}(\set{P},\Galg)$.
\end{corollary}
Since $\mathcal{C}_\Omega(\set{P})$ and $\mathcal{C}_{\Omega}(\Galg)$ are infinite dimensional
it follows that $\mathfrak{l}_\Omega$, and therefore
also $\mathfrak{L}_{\Omega}(\set{P},\Galg)$, is infinite dimensional. However, a finite dimensional sub-space of
$\mathcal{C}_\Omega(\set{P})$ is given by
\begin{equation} \label{eq:finite_frequency}
\mathcal{C}_{\Omega,k}(\set{P}) = \big\{ f \in \mathcal{C}_\Omega(\set{P}) \;\big| \;
f(t) = \vect{a}_0 + \sum_{i=1}^k \vect{a}_i \cos(i \Omega t) +
\vect{b}_i \sin( i \Omega t), \; \vect{a}_i,\vect{b}_i\in\set{P} \big\}
\end{equation}
which is the sub-space of $\mathcal{C}_\Omega(\set{P})$ with angular frequencies bounded by~$k\Omega$.
Notice that the dimension of $\mathcal{C}_{\Omega,k}(\set{P})$ is $(2k+1)n$ and that
$\mathcal{C}_{\Omega,\infty}(\set{P})=\mathcal{C}_{\Omega}(\set{P})$ and $\mathcal{C}_{\Omega,0}(\set{P})=\set{P}$.
Further, $\mathcal{C}_{\Omega,k}(\set{P})$
is closed under differentiation. Clearly, these results also holds for
the corresponding sub-space $\mathcal{C}_{\Omega,l}(\Galg)$ of $\mathcal{C}_{\Omega}(\Galg)$,
except that the dimension is given by $(2l+1)\dim{\Galg}$ instead.
By replacing $\mathcal{C}_{\Omega}(\set{P})$
with $\mathcal{C}_{\Omega,k}(\set{P})$ and $\mathcal{C}_{\Omega}(\Galg)$ with $\mathcal{C}_{\Omega,l}(\Galg)$
we get the sub-spaces $\mathfrak{l}_{\Omega,k,l} = \mathcal{C}_{\Omega,l}(\Galg)\times\mathcal{C}_{\Omega,m}(\set{P})\times\mathbb{R}$
of~$\mathfrak{l}_\Omega$.
In general $\mathfrak{l}_{\Omega,k,l}$ is \emph{not} a sub-algebra, due to the fact that
$A,B\in\mathcal{C}_{\Omega,l}(\Galg)$ does not in general imply $AB\in\mathcal{C}_{\Omega,l}(\Galg)$. However,
in some special cases the implication holds true.
\begin{proposition} \label{prop:finite_lie_algebra}
The sub-spaces $\mathfrak{l}_{\Omega,k,0} = \Galg\times\mathcal{C}_{\Omega,k}(\set{P})\times\mathbb{R}$
and $\mathfrak{l}_{\Omega,k,\infty} = \mathcal{C}_{\Omega}(\Galg)\times\mathcal{C}_{\Omega,k}(\set{P})\times\mathbb{R}$ are
Lie sub-algebras of~$\mathfrak{l}_\Omega$. Further, $\mathfrak{l}_{\Omega,k,0}$ is finite dimensional
with dimension $\dim\Galg + (2k+1)n + 1$.
\end{proposition}
Clearly, $\mathfrak{l}_{\Omega,k,0}$ and $\mathfrak{l}_{\Omega,k,\infty}$ induces corresponding Lie
sub-algebras~$\mathfrak{L}_{\Omega,k,0}(\set{P},\Galg)$ and~$\mathfrak{L}_{\Omega,k,\infty}(\set{P},\Galg)$ of~$\mathfrak{L}_{\Omega}(\set{P},\Galg)$.
\subsection{Geometric Integration}
\label{sub:geometric_integration}
In this section we describe an approach for geometric integration of
of systems of the form~\eqref{eq:linear_systems}. The approach is based on splitting.
To this extent we write~\eqref{eq:linear_systems} as an extended system
\begin{equation} \label{eq:linear_system_extended}
\frac{\, \mathrm{d} }{\, \mathrm{d} \tau} \begin{pmatrix} \vect{x} \\ t \end{pmatrix} =
\begin{pmatrix} A(t)\vect{x} + f(t) \\ 1 \end{pmatrix} \equiv X(\vect{x},t) \; .
\end{equation}
It is clear that $X\in\mathfrak{L}_{\Omega}(\set{P},\Galg)$. Since $\mathfrak{L}_{\Omega}(\set{P},\Galg)$ is a Lie sub-algebra
of $\mathfrak{X}(\overline{\set{P}})$ it corresponds to a Lie sub-group
$\mathrm{Diff}_\Omega$ of $\mathrm{Diff}(\overline{\set{P}})$. Geometric integration
of~\eqref{eq:linear_system_extended} now means to find a one-step integration
algorithm~$\Phi_h \in \mathrm{Diff}_\Omega$.
There are of several ways to obtain such integrators.
One of the simplest, but yet most powerful ways, is to use a splitting
approach. That is, to split the vector field~$X$ as a sum of
two vector fields each of them of the form~\eqref{eq:linear_system_extended}.
\newpage
\subsection{Example: Linear Rotor Dynamics}
\label{sub:example_linear_rotor_dynamics}
\begin{wrapfigure}[12]{r}[0ex]{0.25\textwidth}
\centering
\raisebox{-29ex}[0ex][0ex]{\includegraphics[width=0.17\textwidth]{SimpleRotor.pdf}}
\end{wrapfigure}
This example is the simplest possible rotor dynamical problem. It consists of
a disc attached to a shaft which is rotating with constant angular velocity~$\Omega$.
The shaft is held by a bearing, which is modelled as a linear spring with
stiffness~$k$. (See figure.)
The disc is slightly unbalanced, i.e., its centre of mass does not align with rotational
axis. This implies a time-dependent periodic centrifugal force acting on the rotor.
The phase space for this system is given by $\set{P}=\mathbb{R}^4$,
with coordinates~$\vect{x}=(q_1,q_2,p_1,p_2)$, which is the horizontal and vertical position
of the shaft in a plane perpendicular to the axis of rotation, and their corresponding momenta.
The equations of motion are of the form~\eqref{eq:linear_system_extended} with
\begin{equation*} \label{eq:gov_simple_rotor}
A = \begin{pmatrix}
0& 0 & m^{-1}& 0\\
0&0 & 0& m^{-1} \\
-k &0 & 0& 0\\
0 & -k &0& 0
\end{pmatrix}
\qquad \text{and} \qquad
f(t) = \varepsilon \Omega^2 \begin{pmatrix}
0 \\ 0 \\ -\cos(\Omega t) \\ \sin(\Omega t)
\end{pmatrix}
\end{equation*}
where $m$ is the total mass and $\varepsilon$ is the magnitude of the unbalance.
It holds that $A^T J + J A = 0$, so $A$ is an element in the canonical symplectic
Lie sub-algebra of~$\gl(4)$, i.e., we have~$\Galg=\spg(4)$.
Further, since $A$ is independent of $t$, and $f$ only contains a single
frequency, the appropriate Lie sub-algebra of~$\mathfrak{X}(\mathbb{R}^4)$ is
$\mathfrak{L}_{\Omega,1,0}(\mathbb{R}^4,\spg(4))$, which is finite dimensional.
The eigenvalues of $A$ are $\pm \i \sqrt{k/m}$. Thus if $\Omega$ is close to
a multiple of the eigen frequency $\omega=\sqrt{k/m}$ of the system
starts to resonate. In this example we investigate how well various numerical
integrators capture that behaviour, both qualitatively and quantitatively.
For the data given in Table~\ref{tab:data} the problem is
numerically integrated with four different methods, two which
are geometric and two which are not.
\begin{center}
\begin{tabular}{lc}
\textbf{Method} & \textbf{Geometric?} \\
\hline\hline
Implicit midpoint & Yes \\
Splitting method & Yes \\
Heun's method & No \\
Implicit extrapolation method & No
\end{tabular}
\end{center}
\begin{table} \label{tab:data}
\centering
\begin{tabular}{l|rl}
& \textbf{Value} & \textbf{Unit} \\
\hline\hline
$m$ & 1 & \unit{kg} \\
$k$ & 1 & \unit{N/m} \\
$\Omega$ & 1.02 & \unit{rad/s} \\
$\varepsilon$ & 0.1 & $\unit{m \cdot kg}$ \\
$\vect{x}_0$ & (0,0,0,0) & \unit{(m,m,m/s,m/s)}
\end{tabular}
\caption{Data used in the simulations of the rotor dynamical problem.}
\end{table}
The results of the $x_1$--variable are shown in Figure~\ref{fig:SimpleRotor_xplot}.
Notice that the geometric integrators captures the resonance phenomena in a qualitatively
correct way, whereas the non-geometric methods does not show the correct behaviour.
\begin{figure}
\centering
\includegraphics[width=0.99\textwidth]{SimpleRotor_xplot_exact.pdf} \\
\includegraphics[width=0.99\textwidth]{SimpleRotor_xplot_met1.pdf} \\
\includegraphics[width=0.99\textwidth]{SimpleRotor_xplot_met2.pdf} \\
\includegraphics[width=0.99\textwidth]{SimpleRotor_xplot_met3.pdf} \\
\includegraphics[width=0.99\textwidth]{SimpleRotor_xplot_met4.pdf} \\
Time $t$
\caption{Results for $q_1$ variable for the simple linear rotor example.
\emph{From top:} (1) exact solution, (2) implicit midpoint, geometric,
(3) Störmer--Verlet, geometric, (4) implicit Runge--Kutta, non-geometric,
(5) explicit Runge-Kutta, non-geometric. All methods are second order accurate.
Notice the superior qualitative and
quantitative behaviour of the geometric methods.}
\label{fig:SimpleRotor_xplot}
\end{figure}
\section{Conclusions}
\label{sec:conclusions}
A structural analysis of non-autonomous systems has been carried out using the framework
of Lie sub-algebras of the Lie algebra of vector fields. As a direct application,
backward error analysis results are obtained for this class of problems.
Numerical examples of a classical rotor dynamical problem show that the geometric
methods preserving the structure of the problem indeed are favourable over
non-geometric dito.
\paragraph{Acknowledgement}
\label{par:acknowledgement}
\onSKF{
The author is grateful to Dag Fritzson, Claus Führer and Gustaf Söderlind
for helpful discussions.
The author would also like to thank SKF for the support given.
}{
The authors are grateful to Claus Führer and Gustaf Söderlind
for fruitful discussions.
The work is supported by SKF and by the Swedish Research Council
under contract \textsf{VR-621-2006-5737}.
}
\bibliographystyle{abbrv}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 4,304 |
\section{\label{intro}Introduction}
Recent applications of tools from Statistical Physics have brought
about new perspectives to theoretical Neuroscience. On the one hand,
networks of simplified neuron models seem to capture essential
features of collective neuronal
dynamics~\cite{lewis00,Copelli02,Copelli05a,Copelli05b,Haldeman05},
very often being also amenable to analytical calculations via mean
field approximations or Fokker-Planck
equations~\cite{Furtado06,Kinouchi06a,Doiron06}. On the other hand,
experimental data from real neural networks have yielded extremely
interesting results when analyzed within the framework of complex
networks, often revealing the small-world character for structural
(i.e. anatomical) connectivity in different spatial
scales~\cite{Watts98,Amaral00,Sakata05}, as well as scale-free
characteristics for functional connectivity both in {\it
in-vitro}~\cite{Beggs03} and in fMRI~\cite{Eguiluz05, Chialvo04} data
(see~\cite{Sporns04a} for a recent review).
In the context of modelling, one intriguing question in Neuroscience
regards the stunning ability that brains have to cope with sensory
stimuli that vary over many orders of
magnitude~\cite{Kinouchi06a}. The experimental evidence supporting
this claim has been accumulating for about a century in the
Psychophysics literature: the perception of a given stimulus grows
with a power law of the stimulus intensity (Stevens law), with an
exponent (Stevens exponent) which is typically $<1$, implying
low-stimulus amplification and large dynamic range~\cite{Stevens}.
This is in stark contrast with the poor performance of single neurons:
as a function of the stimulus intensity, the mean firing rate of
sensory neurons experimentally shows the linear saturating shape that
one expects for general excitable systems, so their responses
consistently have a small dynamic range, typically about one or two
decades only~\cite{Firestein93, Rospars00, Rospars03, Deans02}. How
can these two experimental results be reconciled?
A solution which has been proposed for this apparent paradox involves
a collective phenomenon. The idea is that if excitable elements with
small dynamic range are coupled, signal propagation in the network
amplifies the average activity, as compared to that of an isolated
node. This collectively leads to a significant enhancement of dynamic
range, thus providing a possible solution to a problem faced by
biological as well as artificial sensors: how to code for several
orders of magnitude of stimulus intensity, starting from narrow-coding
elements~\cite{Copelli02, Copelli05a, Copelli05b, Furtado06,
Kinouchi06a}.
The reasoning underlying the enhancement of dynamic range is very
general and applies to essentially any network topology. Consider the
limit of very weak stimulus, where each excitable element has a small
probability of being excited. By coupling the elements, a single
stimulus event will be amplified to neighboring sites, which will
further amplify it, and so forth. If the coupling strength is small,
this excitable wave will eventually die out, but the overall network
activity (the response to the stimulus) will still be larger than that
of the isolated element that originally received the stimulus. The
larger the coupling strength, the larger the amplification, and so the
sensitivity and the dynamic range of the response curve initially
increase with coupling. There is, however, a critical value of the
coupling above which self-sustained activity becomes stable. Above
this (typically second order) nonequilibrium phase transition, the
response of the network for weak stimulus is hindered, because it is
masked by the self-sustained activity of the network. This gets worse
and worse as the coupling increases, so above criticality the dynamic
range of the response curve decreases with increasing coupling
strength. Therefore, the dynamic range is optimal at
criticality~\cite{Kinouchi06a}.
The above mechanism has been tested in regular~\cite{Copelli02,
Copelli05a, Copelli05b, Furtado06} as well as
random~\cite{Kinouchi06a} networks of excitable elements. The maximum
enhancement in dynamic range is about 100\% in one-dimensional
networks~\cite{Furtado06} and 50\% in random
graphs~\cite{Kinouchi06a}. It is not {\it a priori\/} clear how the
performance depends on the network topology and, in particular, which
one gives the best results. Given the potential applications of the
mechanism to artificial sensors, as well as the relevance to
Neuroscience, in this paper we study the performance of a scale-free
topology in the dynamic range problem. We show that a particular class
of scale-free networks, those with no loops, yield the best
performance known so far. We investigate the role of loops on the
dynamic range by introducing a slightly modified version of the
Barab\'asi-Albert scale-free model where we can now tune the amount of
loops in the network.
The paper is organized as follows. In section~\ref{model} we introduce
the model and give a precise definition of the dynamic range. Results
are discussed in section~\ref{results} for the standard
Barab\'asi-Albert scale-free model (\ref{ba}) as well as for a
slightly modified ``loop-diluted'' version that we introduce
(\ref{ld}). Our concluding remarks are presented in
section~\ref{conclusions}.
\section{\label{model}The model}
We consider a variant of the Greenberg-Hastings cellular
automaton~\cite{Greenberg78}, which is one of the simplest models of
an excitable system and can be used in large-scale simulations. In the
model, each excitable node $i=1,\ldots,N$ could represent either a
neuron, an active dendritic patch or even sub-cellular excitable
processes. Each node can be in one of $n$ states: $x_i=0$ is the
quiescent state (e.g. a polarized neuron), $x_i=1$ is the excited
state (e.g. a spiking neuron) and $x_i=2,\ldots,n-1$ are refractory
states (e.g. a hyperpolarized neuron). Once a site is excited
($x_i=1$), it deterministically goes through the next $n-2$ refractory
states, after which it jumps to the quiescent state $x_i=0$ (the
automaton is therefore cyclic~\cite{Marro99}). Each node is
independently excited by a stochastic external source, which mimics
the effect of an stimulus on the lattice. We model the arrival of a
suprathreshold stimulus by a Poisson process with rate $r$: at each
time step $\tau$ an attempt to stimulate a site occurs with
probability
\begin{equation}
\lambda = 1-\exp(-r\tau)
\end{equation}
(we adopt the arbitrary time scale of $\tau=1$~ms, which is the
characteristic time scale of a neuronal spike). We refer to the rate
$r$ as the stimulus intensity. In order to become excited in time
$t+\tau$ a given site has to be in state $0$ at time $t$. There are
two different ways by which a site can be excited: by the continuous
stimulation of the external source (with probability $\lambda$ per
time step) or by stimulus propagation from its excited neighbors. Thus
the probability that a quiescent site $i$ is excited in the next time
step is
\begin{equation}
P_i(t+\tau) = 1 - (1-\lambda)\prod_{j=1}^{k_i}(1-p_{ij})\delta(x_j(t),1)\; ,
\end{equation}
where $\delta(a,b)$ is the Kronecker delta, $k_i$ is the number of
neighbors (connectivity or degree) of site $i$ and $p_{ij}$ is the
probability that excitation from site $j$ gets transmitted to site
$i$. There is quenched disorder in the coupling: the probabilities
$p_{ij}$ are initially drawn from a uniform distribution in
$[0,2\sigma/K]$ if $2\sigma/K<1$, or $[2\sigma/K-1,1]$ if
$2\sigma/K>1$, where $K = \left<k\right>$ is the mean connectivity of
the network and $\sigma$ is the coupling parameter (for simplicity, we
consider the case of bidirectional coupling $p_{ij}=p_{ji}$). Note
that, in a mean field approximation, $\sigma$ coincides with the
branching ratio, defined as the average of the number of descendant
excitations divided by the number of ancestor excitations of each
site. Such mean field approximation provides a quite satisfactory
agreement with simulation results for random graph topologies (as
expected) and accurately predicts a phase transition at
$\sigma_c=1$~\cite{Kinouchi06a,Haldeman05}.
The mean firing rate of the network is defined as $F\equiv
T^{-1}\sum_t^T \rho_t$, where $\rho_t \equiv N^{-1}\sum_{i}^N
\delta(x_i(t),1)$ is the instantaneous density of active (excited)
sites and $T$ is a given time window for measurements (we have used
$N=10^4$, $T=10^4$ steps and $n=5$ states in most simulations). We
refer to $F(r)$ as the {\em response function\/} (or transfer
function) of the network. It typically shows the sigmoidal shape in a
log-linear scale exemplified in Fig.~\ref{fig:response}(a), with a
baseline activity $F_0 \equiv \lim_{r\to 0} F(r)$ and saturation at
$F_{max}\equiv \lim_{r\to\infty} F(r)$. The dynamic range $\Delta$ of
the response function is defined as the width (measured in dB) in
stimulus intensity $r$ which can be ``appropriately coded'' by $F$. In
the biological literature, this is usually operationalized as
follows~\cite{Firestein93,Rospars00}: by letting $F_x \equiv F_0 +
x(F_{max} - F_0)$, where $0 \leq x \leq 1$,
and $r_x$ be the corresponding stimulus intensity, ($F(r_x)=F_x$, see
triangles in Fig.~\ref{fig:response}(a) for an example), the dynamic
range is defined as
\begin{equation}
\Delta = 10\log_{10}\left(\frac{r_{0.9}}{r_{0.1}} \right)\; ,
\end{equation}
therefore excluding stimuli whose response is just above baseline
($r<r_{0.1}$) or too close to saturation ($r>r_{0.9}$). For an
isolated Greenberg-Hastings excitable node, one can easily show that
the dynamic range is $\Delta \lesssim
19$~dB~\cite{Copelli05b,Furtado06}.
\section{\label{results}Results}
\subsection{\label{ba}Barab\'asi-Albert networks}
We consider scale-free networks~\cite{Barabasi99Sci} of such excitable
elements. Several investigations show that distinct systems such as
World-Wide Web~\cite{Barabasi99Sci}, scientific~\cite{Newman01e},
metapopulation dynamics~\cite{Moreno02,Vuorinen04} and biochemical
networks~\cite{Jeong00metabolic,Jeong01lethality} self-organize into a
scale-free configuration \cite{Albert02}, which means that the
probability $P_{k}$ that a given node has $k$ edges follows a
power-law distribution like
\begin{equation}\label{scale}
P_{k} \propto k^{-\gamma}.
\end{equation}
Measurements in real systems estimate $\gamma$ in the range $[2,3]$.
Eq. (\ref{scale}) basically means that poorly-connected nodes are most
frequent in the network than well-connected nodes (hubs).
To establish scale-free networks, we follow the standard algorithm by
Barab\'asi and Albert (BA)~\cite{Barabasi99Sci}, which regards
preferential attachment and growth as mechanisms for the emergence of
the scale-free character. In this algorithm, the resulting networks
display connectivity distribution according to $P_{k} \propto k^{-3}$.
The parameters of the BA model are the number of nodes $N$ and $m$,
which corresponds to the number of links that a newly introduced node
adds to the network. These $m$ links are most probably attached to
those nodes with an already large number of edges.
\begin{figure}
\centerline{\includegraphics[width=0.95\columnwidth]{figuraresponsecurves.ps}}
\caption{\label{fig:response}(a) Response functions for BA scale-free
networks with $m=10$: mean firing rate $F$ versus stimulus rate
$r$. Different curves denote different values of the branching
parameter $\sigma$: from bottom to top, $\sigma=0.1,
0.3,\ldots,1.9$. Filled circles: $\sigma=0.5$, which is close to the
critical value. The horizontal lines exemplify how the dynamic range
$\Delta$ is calculated for $\sigma=1.9$ (filled triangles). (b)
Log-log version of (a). The dashed line shows an exponent
$1/2$. Inset: Self-sustained activity $F_0$ versus $\sigma$,
illustrating the phase transition close to $\sigma_c\simeq 0.5$.}
\end{figure}
\begin{figure}
\centerline{\includegraphics[width=0.95\columnwidth]{DeltaVsSigma.eps}}
\caption{\label{fig:deltasigma}Dynamic range $\Delta$ versus branching
parameter $\sigma$ for distinct network topologies: BA scale-free
networks (open circles) and Erd{\H o}s-R\'enyi random graphs (filled
circles) with approximately the same mean connectivity. Inset:
Response function for BA scale-free networks with $m=1$ and $\sigma=2$
for $N=10^4$ (filled circles) and $N=3\times 10^4$ (open
circles). The solid line shows a slope $\simeq 0.07$.}
\end{figure}
Figure~\ref{fig:response} shows the results for $m=10$. For small
values of $\sigma$, the response function $F(r)$ increases linearly
for weak stimulus. This linearity can be easily interpreted: each
stimulus arrival generates an excitable wave that will have a finite
lifetime and will die before another wave is created. For stronger
stimulus (larger $r$) linearity breaks down, since there is
interaction among waves, which partially annihilate each other. For
very large $r$, the firing rates reach a saturation value which scales
with the inverse of the refractory period, $F_{max} =
1/n$~\cite{Copelli02, Copelli05a, Copelli05b}. As the value of
$\sigma$ increases, so does the lifetime of an excitable wave, leading
to larger amplification of weak stimuli and a corresponding
enhancement of dynamic range (see Fig.~\ref{fig:deltasigma} for
$\sigma \lesssim 0.5$). When $\sigma=\sigma_c$, the lifetime of the
excitable waves effectively diverges and the system undergoes a second
order phase transition (notice the change in the exponent in the
filled circles of Fig.~\ref{fig:response}(b)). For $\sigma>\sigma_c$,
any perturbation in the network will lead to a stable self sustained
activity, $F_0>0$ (inset of Fig.~\ref{fig:response}(b)) which, as
explained in section~\ref{intro}, leads to smaller values of the
dynamic range as the coupling increases~\cite{Kinouchi06a} (see
Fig.~\ref{fig:deltasigma} for $\sigma \gtrsim 0.5$).
One observes that, differently from random graph
topologies~\cite{Kinouchi06a}, the transition for scale-free excitable
networks occurs at $\sigma_c< 1$. We speculate that this is due to the
hubs, which have a local branching ratio $\sigma_i = \sum_j^{k_i}
p_{ij}$ larger than unit even for $\sigma<1$ and could therefore
facilitate the transition. It is also interesting to note that
deviations from mean field behavior have been predicted for the
contact process (CP) in a scale-free
network~\cite{Castellano06}. Apart from the refractory period and the
disorder, the CP is similar to the model we study here (in the sense
that it has a unique absorbing state with no symmetries). In
Fig.~\ref{fig:response}, however, the response exponent at criticality
(defined by $F(r;\sigma_c) \sim r^{1/\delta_{h}}$) seems to be
compatible with the mean field value $1/\delta_h =
1/2$~\cite{Marro99}.
Results in Fig.~\ref{fig:response} are typical, similar curves are
obtained for any $m>1$. The performance of these scale-free networks
in enhancing the dynamic range is poor: while the dynamic range of
isolated excitable elements is $\Delta(\sigma=0)\simeq 16.7$~dB, the
network (optimal) performance at criticality is only $\Delta(\sigma_c)
\simeq 20.8$~dB, an enhancement of less than a decade. This is
slightly worse than the enhancement produced by random networks with
equivalent size and average connectivity, as can be seen in the curves
$\Delta(\sigma)$ of Fig.~\ref{fig:deltasigma}.
The case $m=1$, however, is particularly interesting. Notice that in
this situation the network is still scale-free, but does not comprise
any loop in its structure and consequently has a tree-like
pattern. This condition, together with the deterministic nature of
each excitable node after excitation, prevents the phase transition to
self-sustained activity from occurring~\cite{lewis00}, a fact that has
also been observed in one-dimensional excitable
networks~\cite{Furtado06}. In these conditions the only transition
occurs at $\sigma=\sigma_{max} = K$ ($=2m$ for scale-free networks),
whereby propagation of excitable waves becomes deterministic
(ballistic). Therefore low-stimulus amplification increases steadily
with $\sigma$, but in the absence of self-sustained activity
($F_0=0$). This allows the dynamic range to increase monotonically
with $\sigma$, reaching values near 50~dB, which is the largest value
obtained so far in excitable network models.
\subsection{\label{ld}Loop-diluted model}
As we observe a remarkable difference between the dynamic range of
scale-free networks with $m=1$ and other values of $m$, and the former
has a typical feature (non-existence of loops) which is not present in
$m>1$ topologies, we are interested in investigating the role of loops
in the response functions of the networks. For this purpose, we
propose a variant of the BA model which is referred to as loop-diluted
model. In the model each new node is added according to the usual
preferential attachment rule, but can have $m=1$ or $m=2$ links
according to the probability distribution
\begin{equation}
P(m)=(1-p)\delta_{m,1}+p\delta_{m,2}\; ,
\end{equation}
where $p$ is the probability of having two edges. So $p$ adjusts the
amount of loops in the network and the case $p=0$ recovers the
structure with no loops. The mean degree is now $K = 2\left< m\right>
= 2(1+p)$.
\begin{figure}[t]
\centerline{\includegraphics[width=0.95\columnwidth]{motifs.eps}}
\caption{\label{fig:motif} Occurrences of 3- and 4-site patterns in
the loop-diluted scale-free networks (filled symbols) and their
randomized counterparts (open symbols). (a) Number of triangles
versus $p$ and (b) Number of squares versus $p$. Points (bars)
represent the mean (standard deviation) over 5 realizations for
$N=1000$. Patterns are sequentially numbered (1-8) for further
reference in the text.}
\end{figure}
Notice that, since all sites belong to a single giant component, the
average number of loops created by each newly added site is bounded
from below by $p$. Each new two-edged node can give rise to loops of
any size, but the relationship between parameter $p$ and the number of
loops becomes already apparent in a simple 3-site motif
analysis~\cite{Milo02}. Figure~\ref{fig:motif}(a) shows the mean
number of triangles as a function of $p$ (calculated by the free
software available at {\tt www.weizman.ac.il/mcb/UriAlon}). As
expected, this is a monotonically increasing function, which
nevertheless stays well below $Np$, hinting that most loops comprise
more than three sites. The same qualitative scenario is observed when
we plot the number of squares (Fig.~\ref{fig:motif}(b)), which are
more abundant than triangles. In both cases, we notice that for
equally sized randomized graphs (which preserve the degrees of every
node~\cite{Milo02}), the numbers of triangles and squares are
considerably larger. This means that triangles and squares are
actually {\em anti-motifs\/} in the loop-diluted
model~\cite{Milo04}. In fact, this is true for all patterns that
contain loops (numbered 2, 5, 6 and 8 in Fig.~\ref{fig:motif}), except
for pattern 7, which cannot occur according to the growth rules of the
model. The occurrences of patterns 1, 3 and 4 in the loop-diluted
model and in the randomized networks are statistically
indistinguishable (therefore they are neither motifs nor anti-motifs
-- data not shown).
\begin{figure}[t]
\centerline{\includegraphics[width=0.95\columnwidth]{pDeltaVsSigma.eps}}
\caption{\label{fig:figp} Dynamic range $\Delta$ versus branching
parameter $\sigma$ for the loop-diluted model and different values of
the probability $p$ that a newly added node has two incoming
edges. Inset: maximum value of the dynamic range as a function of
$p$.}
\end{figure}
Figure \ref{fig:figp} displays the dynamic range $\Delta$ as a
function of the coupling $\sigma$ for some values of $p$. From the
figure, we clearly notice that the insertion of loops by increasing
the probability $p$ has a striking effect on the dynamic range. This
effect is already mensurable for values of $p$ such that $pN \sim
1$. The peak value of the dynamic range $\Delta_{max}(p)\equiv
\max_{\sigma}\Delta(\sigma;p)=\Delta(\sigma_c(p);p)$ seems to decrease
logarithmically with $p$.
\section{\label{conclusions}Concluding remarks}
Recently, some investigations have addressed the enlargement in
average activity of excitable elements by coupling these entities and
so giving form to a new larger and more sensitive unit. Although this
collective phenomenon has been widely accepted, little is known about
the way the arrangement of connections among the excitable elements
acts physically on the system dynamics. The creation of more robust
and functional units from smaller units (whose pattern of interactions
is a determining aspect) is of course not exclusive of
Neuroscience. For instance, the hypercycle, a catalytic feedback
network whereby each element helps the replication of the next one in
a regulatory cycle closing on itself, has been pondered as an
alternative resolution for the information crisis in prebiotic
evolution~\cite{Eigen78,Campos00}. We believe that all the recent
contributions on this issue have a bearing on a more general context,
that is, the understanding of the interplay between system dynamics
and the underlying interaction network topologies. We hope that our
contribution gives a small step in this direction, when we corroborate
that the amount of loops in network structure could be a key
topological feature.
We have presented simulation results for the transfer function of
excitable scale-free networks. The behavior of the dynamic range
$\Delta$ as a function of the coupling $\sigma$ shows the general
behavior predicted in Ref.~\cite{Kinouchi06a}: in the subcritical
regime ($\sigma<\sigma_c$) $\Delta(\sigma)$ increases, while in the
supercritical regime ($\sigma>\sigma_c$) $\Delta(\sigma)$
decreases. The maximum value is obtained at criticality, but for
scale-free networks with $m>1$ this result is even smaller than that for
a random graph.
For $m=1$ the phase transition to self-sustained activity disappears,
and the dynamic range increases steadily, reaching its maximum value
when excitable waves become deterministic. This suggests that the
presence of loops in the network could be a relevant feature in
determining the dynamic range of its transfer function. We have
introduced a simple extension to the BA scale-free model which allows
one to interpolate between $m=1$ and $m=2$, showing that dynamic range
decreases as the density of loops increases. This reinforces the need
to study other topologies with tree structure, which abound in
biological structures.
It remains at present unclear whether the response exponent for $m>1$
is indeed compatible with the mean field universality class (even
though this seems to be supported by recent simulation results in
Ref.~\cite{Wu07}, which independently addressed a similar
problem). Also, for $m=1$ at maximum coupling, the response function
seems to be governed by a power law with a much smaller exponent (see
inset of Fig.~\ref{fig:deltasigma}) which might not belong to the
directed percolation universality class. We believe that a detailed
study of the critical exponents of excitable scale-free networks is
still lacking and should be dealt with in the future.
MC and PRAC are supported by Conselho Nacional de Desenvolvimento
Cient\'{\i}fico e Tecnol\'ogico (CNPq), FACEPE and special program
PRONEX.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 9,733 |
//
// TFUser.h
// 101-百思不得姐
//
// Created by 陶飞 on 15/6/4.
// Copyright © 2015年 taofei. All rights reserved.
//
#import <Foundation/Foundation.h>
@interface TFUser : NSObject
/** 用户名 */
@property (nonatomic,copy) NSString *username;
/** 性别 */
@property (nonatomic,copy) NSString *sex;
/** 头像 */
@property (nonatomic,copy) NSString *profile_image;
@end
| {
"redpajama_set_name": "RedPajamaGithub"
} | 9,384 |
\section{Introduction}
Exactly solvable models play a very useful role in various fields
of physics. They help improving our understanding of
physical processes and allow us gain more insight into
complicated phenomena that take place in nature~\cite{1}. Needless to
recall the usefulness of exactly solvable models such as
the harmonic oscillator, the nuclear shell model and the Ising
model, to mention but a few. From a practical point of view, exactly
solvable models serve as a very convenient tool for testing the
accuracy of numerical algorithms, often used in the study of
problems that cannot be analytically solved due to the complexity of
the systems under investigation.
In nature, quantum systems are influenced by their surrounding
environment through, in general, complicated coupling interactions,
leading them to lose their coherence~\cite{2}. This refers to as the
decoherence process~\cite{3,4,5}. Moreover, quantum systems exhibit
properties that do not have classical analogous~\cite{6}. Of great
interest is entanglement, the main ingredient for quantum
teleportation and quantum computation~\cite{7,8,9,10,11,12}. Over
the last years, many proposals have been made for the implementation
of quantum information processing. Solid state systems are very
promising~\cite{13,14} and have been the subject of many
investigations. In particular, decoherence and entanglement of
qubits coupled to spin environments~\cite{15} attracted much
attention~\cite{16,17}. Thus new exactly solvable models describing
the dynamics of qubits in spin baths are highly welcome. Recently,
the spin star configuration, initially proposed by Bose, has been
extensively investigated~\cite{18,19,20,21,22}. An exact treatment
of the dynamics of two qubits coupled to common spin star bath via
$XY$ interactions is presented in~\cite{23,24}. In this paper we
propose to investigate analytically the dynamics when the two qubits
interact with separate spin star baths.
The paper is organized as follows. In section~\ref{sec2} the model
Hamiltonian is introduced. In section~\ref{sec3} we present a
detailed derivation of the time evolution operator and we
investigate the dynamics of the qubits at finite $N$ for some
particular initial conditions. In section~\ref{sec4} we study the
thermodynamic limit, in which the sizes of the spin environments
become infinite. Section~\ref{sec5} is devoted to the second-order
master equation. We end the paper with a short summary.
\section{Model\label{sec2}}
The system under study consists of two two-level systems (
{ e.g.,} spin-$\frac{1}{2}$ particles) each of which is
embedded in its own spin star
environment composed of $N$ spins-$\frac{1}{2}$. The central
particles interact with each other through a Ising interaction; the
corresponding coupling constant is equal to $4\delta$, where the
factor 4 is introduced for later convenience. We shall assume that
each qubit couples to its environment via Heisenberg $XY$
interaction whose coupling constant is $\alpha$, which is, in turn,
scaled by $N^{1/2}$ in order to ensure good thermodynamic behavior.
The spin baths will be denoted by $B_1$ and $B_2$. The Hamiltonian
for the composite system has the form
\begin{equation}
H=H_0+H_{S_1B_1}+H_{S_2B_2},
\end{equation}
where
\begin{equation}
H_0=4\delta S^{1}_z S^{2}_z,\end{equation} and
\begin{equation}
H_{S_iB_i}=\frac{\alpha}{\sqrt{N}} (S_+^{i}\sum_{k=1}^N S^{ik}_-+
S_-^{i}\sum_{k=1}^N S^{ik}_+), \quad (i=1,2).
\end{equation}
Here $\vec{S^1}$ and $\vec{S^2}$ denote the spin operators
corresponding to the central qubits, whereas $\vec{S^{ik}}$ denotes
the spin operator corresponding to the $k^{th}$ particle within the
$i^{th}$ environment. Introducing the total spin operators
$\vec{J}=\sum_{k=1}^N\vec{S^{1k}}$ and $\vec{\mathcal
J}=\sum_{k=1}^N\vec{S^{2k}}$ of the environments $B_1$ and $B_2$,
respectively, one can rewrite the full Hamiltonian as
\begin{equation}
H=4\delta
S^1_zS^2_z+\frac{\alpha}{\sqrt{N}}(S^1_+J_-+S^1_-J_++S^2_+\mathcal{J}_-
+S^2_-\mathcal{J}_+)\label{fullh}.\end{equation}
The dynamics of the two-qubit system is fully described by its
density matrix $\rho(t)$ obtained, as usual, by tracing the
time-dependent total density matrix $\rho_{\rm tot}(t)$, describing
the composite system, with respect to the environmental degrees of
freedom, namely,
\begin{eqnarray}
\rho(t)&=&{\rm tr}_{B_1+B_2}[\rho_{\rm tot}(t)]\nonumber \\
&=& {\rm tr}_{B_1+B_2}\Bigl[\mathbf U(t)\rho_{\rm tot}(0)\mathbf
U^\dag(t)\Bigl],
\end{eqnarray}
where $\mathbf U(t)$ and $\rho_{\rm tot}(0)$ designate the time
evolution operator and the initial total density matrix,
respectively.
At $t=0$ the central qubits are assumed to be uncoupled with the
environments; the latter are assumed to be at infinite temperature.
This means that the initial total density density matrix can be
written as
\begin{equation}
\rho_{\rm tot}(0)=\rho(0)\otimes\frac{\mathbf 1 }{2^N}\otimes
\frac{\mathbf 1 }{2^N}.\end{equation} Here $\rho(0)$ is the initial
density matrix of the two-qubit system, and $\mathbf 1$ is the unit
matrix on the space $\mathbb C^{2\otimes N}$. The former can be
written as $\rho(0)=\sum_{k,\ell,}
\rho_{k\ell}^0|\chi_k\rangle\langle\chi_\ell|$, with
$|\chi_\ell\rangle\in\{|--\rangle,|-+\rangle,|+-\rangle,|++\rangle\}$
for $\ell=\overline{1,4}$. Similarly, we introduce the basis state
vectors $|j,m\rangle$ of $\mathbb C^{2\otimes N}$, such that
$\kappa\le j \le N/2$ ($\kappa=0$ for $N$ even and $\kappa=1/2$ for
$N$ odd), and $-j\le m\le j$. The time-dependent reduced density
matrix can be expressed as
\begin{equation}
\rho(t)=2^{-2N}\sum_{k,\ell}\rho^0_{k\ell}\sum_{j,m}\sum_{r,s}\nu(N,j)\nu(N,r)\langle
j,r,m,s|\mathbf U(t)|\chi_k\rangle\langle \chi_\ell|\mathbf
U^\dag(t)|j,r,m,s\rangle,\end{equation} where
$|j,r,m,s\rangle=|j,m\rangle\otimes|r,s\rangle$, and
$\nu(N,j)=\binom{N}{N/2-j}-\binom{N}{N/2-j-1}$~\cite{25}. Hence, our
task reduces to finding the exact form of the matrix elements of the
time evolution operator ${\mathbf U}(t)=\exp(-iHt)$ ($\hbar=1$).
This will be the subject of the next section.
\section{Derivation of the exact form of the time evolution operator\label{sec3}}
The time evolution operator can be expanded as
\begin{equation} \mathbf
U(t)=\sum\limits_{n=0}^{\infty} \frac{{(-1)^n
t^{2n}}}{(2n)!}(H)^{2n}-i\sum\limits_{n=0}^{\infty}\frac{(-1)^n
t^{2n+1}}{(2n+1)!}( H)^{2n+1} \label{taylor}.
\end{equation}
In order to derive analytical expressions for even and odd powers of
the total Hamiltonian $H$ let us notice that $H_0$ anticommutes with
$H_{S_1B_1}+H_{S_2B_2}$, that is,
\begin{equation}
[H_0,H_{S_1B_1}+H_{S_2B_2}]_+=0.
\end{equation}
This can easily be shown using the following properties for
spin-$\frac{1}{2}$ operators: $S_z S_\pm=\pm S_\pm$, and $S_\pm
S_z=\mp S_\pm$. Moreover, it is easily seen that
$H_0^{2n}\equiv\delta^{2n}$, which simply implies that for $n \ge0$,
\begin{eqnarray}
H^{2n}&=&\sum\limits_{\ell=0}^{n}\binom{n}{\ell}(H_{S_1B_1}+H_{S_2B_2})^{2\ell}\delta^{2(n-\ell)}\label{binom}.
\end{eqnarray}
In the standard
basis of $\mathbb C^2\otimes \mathbb C^2$, it can be shown that
powers of $H_{S_1B_1}$ and $H_{S_2B_2}$ are given by
\begin{eqnarray}
H_{S_1B_1}^{2k}&=&\Bigl(\frac{\alpha}{\sqrt{N}}\Bigl)^{2k}\begin{pmatrix}(J_+J_-)^k&&0&&0&&0\\0&&(J_+J_-)^k&&0&&0\\
0&&0&&(J_-J_+)^k&&0\\0&&0&&0&&(J_-J_+)^k\end{pmatrix},\\
H_{S_1B_1}^{2k+1}&=&\Bigl(\frac{\alpha}{\sqrt{N}}\Bigl)^{2k+1}\begin{pmatrix}0&&0&&J_+(J_-J_+)^k&&0\\
0&&0&&0&&J_+(J_-J_+)^k\\J_-(J_+J_-)^k&&0&&0&&0\\0&&J_-(J_+J_-)^k&&0&&0\end{pmatrix},\\
H_{S_2B_2}^{2k}&=&\Bigl(\frac{\alpha}{\sqrt{N}}\Bigl)^{2k}\begin{pmatrix}(\mathcal J_+\mathcal J_-)^k&&0&&0&&0\\0&&(\mathcal J_-\mathcal J_+)^k&&0&&0\\
0&&0&&(\mathcal J_+\mathcal J_-)^k&&0\\0&&0&&0&&(\mathcal J_-\mathcal J_+)^k\end{pmatrix},\\
H_{S_2B_2}^{2k+1}&=&\Bigl(\frac{\alpha}{\sqrt{N}}\Bigl)^{2k+1}\begin{pmatrix}0&&\mathcal J_+(\mathcal J_-\mathcal J_+)^k&&0&&0\\
\mathcal J_-(\mathcal J_+\mathcal J_-)^k&&0&&0&&0\\0&&0&&0&&\mathcal
J_+(\mathcal J_-\mathcal J_+)^k\\0&&0&& \mathcal J_-(\mathcal
J_+\mathcal J_-)^k&&0\end{pmatrix}.
\end{eqnarray}
It follows that
\begin{eqnarray}
(H_{S_1B_1}+
H_{S_2B_2})^{2\ell}&=&\sum\limits_{k=0}^{\ell}\binom{2\ell}{2k}H_{S_1B_1}^{2k}H_{S_2B_2}^{2(\ell-k)}+
\sum\limits_{k=0}^{\ell-1}\binom{2\ell}{2k+1}H_{S_1B_1}^{2k+1}H_{S_2B_2}^{2(\ell-k)-1}\nonumber
\\
&=&\Bigl(\frac{\alpha}{\sqrt{N}}\Big)^{2\ell}\Biggl[\sum\limits_{k=0}^{\ell}\binom{2\ell}{2k}
D_{\ell k} +\sum\limits_{k=0}^{\ell-1}\binom{2\ell}{2k+1}L_{\ell
k}\Biggl].
\end{eqnarray}
where
\begin{eqnarray}
D_{\ell k}=\mathit{diag}\Bigl[
(J_+J_-)^k(\mathcal J_+\mathcal
J_-)^{\ell-k}, (J_+J_-)^k(\mathcal J_-\mathcal
J_+)^{\ell-k},\nonumber \\ (J_-J_+)^k(\mathcal J_+ \mathcal
J_-)^{\ell-k},(J_-J_+)^k(\mathcal J_-\mathcal
J_+)^{\ell-k}\Bigl]\end{eqnarray} and
\begin{eqnarray}
L_{\ell k}= \mathit{antidiag}\Bigl[ J_+\mathcal J_+
(J_-J_+)^k(\mathcal J_-\mathcal J_+)^{\ell-k-1}, J_+\mathcal J_-
(J_-J_+)^k(\mathcal J_+\mathcal J_-)^{\ell-k-1},\nonumber \\
J_-\mathcal J_+(J_+J_-)^k(\mathcal J_- \mathcal
J_+)^{\ell-k-1},J_-\mathcal J_+ (J_+J_-)^k(\mathcal J_+\mathcal
J_-)^{\ell-k-1}\Bigl].\end{eqnarray}
Using the fact that
\begin{eqnarray} \sum\limits_{k=0}^{\ell}
\binom{2\ell}{2k} x^k y^{\ell-k}&=&\frac{1}{2} \Bigl[( \sqrt{x} +
\sqrt{y})^{2\ell}
+ (\sqrt{x} - \sqrt{y})^{2 \ell}\Bigl],\\
\sum\limits_{k=0}^{\ell-1}
\binom{2\ell}{2k+1} x^k y^{\ell-k-1}&=& \frac{1}{2\sqrt{x y}}\Bigl[(
\sqrt{x} + \sqrt{y})^{2 \ell} - (\sqrt{x} - \sqrt{y})^{2
\ell}\Bigl],\end{eqnarray} one obtains
\begin{eqnarray}
&&(H_{S_1B_1}+H_{S_2B_2})^{2\ell}=\Bigl(\frac{\alpha}{\sqrt{N}}\Bigl)^{2\ell}\nonumber
\\&& \times \begin{pmatrix}F_1^+&&0&&0&&J_+\mathcal J_+
\frac{F_4^-}{\sqrt{J_-J_+\mathcal J_-\mathcal
J_+}}\\0&&F_2^+&&J_+\mathcal J_- \frac{F_3^-}{\sqrt{J_-J_+\mathcal
J_+\mathcal J_-}}&&0\\0&&J_-\mathcal J_+
\frac{F_2^-}{\sqrt{J_+J_-\mathcal J_-\mathcal
J_+}}&&F_3^+&&0\\J_-\mathcal J_- \frac{F_1^-}{\sqrt{J_+J_-\mathcal
J_+\mathcal
J_-}}&&0&&0&&F^+_4\end{pmatrix},\label{powereven}\end{eqnarray}
where
\begin{eqnarray}
F^\pm_1=\frac{1}{2}\Bigl[\Bigl(\sqrt{J_+J_-}+\sqrt{\mathcal
J_+\mathcal J_-}\Bigl)^{2\ell}\pm\Bigl(\sqrt{J_+J_-}-\sqrt{\mathcal
J_+\mathcal J_-}\Bigl)^{2\ell}\Bigl],\\
F^\pm_2=\frac{1}{2}\Bigl[\Bigl(\sqrt{J_+J_-}+\sqrt{\mathcal
J_-\mathcal J_+}\Bigl)^{2\ell}\pm\Bigl(\sqrt{J_+J_-}-\sqrt{\mathcal
J_-\mathcal J_+}\Bigl)^{2\ell}\Bigl],\\
F^\pm_3=\frac{1}{2}\Bigl[\Bigl(\sqrt{J_-J_+}+\sqrt{\mathcal
J_+\mathcal J_-}\Bigl)^{2\ell}\pm\Bigl(\sqrt{J_-J_+}-\sqrt{\mathcal
J_+\mathcal J_-}\Bigl)^{2\ell}\Bigl],\\
F^\pm_4=\frac{1}{2}\Bigl[\Bigl(\sqrt{J_-J_+}+\sqrt{\mathcal
J_-\mathcal J_+}\Bigl)^{2\ell}\pm\Bigl(\sqrt{J_-J_+}-\sqrt{\mathcal
J_-\mathcal J_+}\Bigl)^{2\ell}\Bigl].
\end{eqnarray}
Inserting equation~(\ref{powereven}) into equation~(\ref{binom}),
yields
\begin{eqnarray}
&& H^{2n}=\frac{1}{2}\nonumber\\
&&\times \begin{pmatrix}(\mathcal M_1^+)^n+(\mathcal M_1^-)^n &&0&&0&&J_+\mathcal
J_+ \frac{(\mathcal M_4^+)^n-(\mathcal
M_4^-)^n}{\sqrt{J_-J_+\mathcal J_-\mathcal J_+}}\\0&&(\mathcal
M_2^+)^n+(\mathcal M_2^-)^n&&J_+\mathcal J_- \frac{(\mathcal
M_3^+)^n-(\mathcal M_3^-)^n}{\sqrt{J_-J_+\mathcal J_+\mathcal
J_-}}&&0\\0&&J_-\mathcal J_+ \frac{(\mathcal M_2^+)^n-(\mathcal
M_2^-)^n}{\sqrt{J_+J_-\mathcal J_-\mathcal J_+}}&&(\mathcal
M_3^+)^n+(\mathcal M^-_3)^n&&0\\J_-\mathcal J_- \frac{(\mathcal
M_1^+)^n-(\mathcal M_1^-)^n}{\sqrt{J_+J_-\mathcal J_+\mathcal
J_-}}&&0&&0&&(\mathcal M^+_4)^n+(\mathcal
M^-_4)^n\end{pmatrix},\label{power1}\end{eqnarray} where
\begin{eqnarray}
\mathcal
M_1^\pm=\delta^2+\frac{\alpha^2}{N}\Bigl(\sqrt{J_+J_-}\pm\sqrt{\mathcal
J_+\mathcal J_-}\Bigl)^2,\\
\mathcal
M_2^\pm=\delta^2+\frac{\alpha^2}{N}\Bigl(\sqrt{J_+J_-}\pm\sqrt{\mathcal
J_-\mathcal J_+}\Bigl)^2,\\
\mathcal
M_3^\pm=\delta^2+\frac{\alpha^2}{N}\Bigl(\sqrt{J_-J_+}\pm\sqrt{\mathcal
J_+\mathcal J_-}\Bigl)^2,\\
\mathcal
M_4^\pm=\delta^2+\frac{\alpha^2}{N}\Bigl(\sqrt{J_-J_+}\pm\sqrt{\mathcal
J_-\mathcal J_+}\Bigl)^2.\end{eqnarray} The above operators satisfy
\begin{eqnarray}
&&M_{1,2}^\pm J_+=J_+M_{3,4}^\pm, \qquad M_{1,2}^\pm \mathcal
J_+=\mathcal J_+M_{3,4}^\pm, \\ && M_1^\pm J_+\mathcal
J_+=J_+\mathcal J_+ M_4^\pm, \qquad M_2^\pm J_+\mathcal
J_-=J_+\mathcal J_-M_3^\pm.
\end{eqnarray}
Furthermore, one can show that the matrix elements of $H^{2n+1}$
are given by
\begin{align}
(H^{2n+1})_{11}&=\frac{1}{2}\delta[(\mathcal M_1^+)^n+(\mathcal M_1^-)^n],\label{powerodd1} \\
(H^{2n+1})_{12}&=\mathcal J_+\frac{\alpha}{2\sqrt{N \mathcal
J_-\mathcal J_+}}[(\sqrt{\mathcal J_-\mathcal
J_+}+\sqrt{J_+J_-})(\mathcal M_2^+)^n \\&+(\sqrt{\mathcal
J_-\mathcal J_+}-\sqrt{J_+J_-})(\mathcal M_2^-)^n], \\
(H^{2n+1})_{13}&=\ J_+\frac{\alpha}{2\sqrt{N J_-
J_+}}[(\sqrt{\mathcal J_+\mathcal J_-}+\sqrt{J_-J_+})(\mathcal
M_3^+)^n \\&+(\sqrt{ J_-J_+}-\sqrt{\mathcal J_+\mathcal
J_-})(\mathcal M_3^-)^n], \\
(U^{2n+1})_{14}&=(\delta/2) J_+\mathcal J_+ \frac{(\mathcal
M_4^+)^n-(\mathcal M_4^-)^n}{\sqrt{J_-J_+\mathcal J_-\mathcal J_+}}
,\end{align}
\begin{eqnarray}
(H^{2n+1})_{21}&=&\mathcal
J_-\frac{\alpha}{2\sqrt{N \mathcal J_+\mathcal J_-}}[(\sqrt{\mathcal
J_+\mathcal J_-}+\sqrt{J_+J_-})(\mathcal M_1^+)^n
\\&+&(\sqrt{\mathcal
J_+\mathcal J_-}-\sqrt{J_+J_-})(\mathcal M_1^-)^n], \\
(H^{2n+1})_{22}&=&-\frac{1}{2}\delta[(\mathcal M_2^+)^n+(\mathcal M_2^-)^n], \\
(H^{2n+1})_{23}&=&-(\delta/2) J_+\mathcal J_- \frac{(\mathcal
M_3^+)^n-(\mathcal M_3^-)^n}{\sqrt{J_-J_+\mathcal J_+\mathcal
J_-}}, \\
(H^{2n+1})_{24}&=&J_+\frac{\alpha/2}{\sqrt{N J_-
J_+}}[(\sqrt{\mathcal J_-\mathcal J_+}+\sqrt{ J_-
J_+}) (\mathcal M_4^+)^n \\&+&(\sqrt{J_- J_+}-\sqrt{\mathcal J_-\mathcal
J_+})(\mathcal M_4^-)^{n}], \end{eqnarray}
\begin{eqnarray} (H^{2n+1})_{31}&=&
J_-\frac{\alpha/2}{\sqrt{N J_+J_-}}[(\sqrt{J_+J_-}+\sqrt{\mathcal
J_+ \mathcal J_-})( \mathcal M_1^+)^n
\\&+&(\sqrt{J_+J_-}-\sqrt{\mathcal J_+ \mathcal
J_-})(\mathcal M_1^-)^n], \\
(H^{2n+1})_{32}&=&-(\delta/2)J_-\mathcal J_+\frac{(\mathcal M_2^+)^n-(\mathcal M_1^-)^n}{\sqrt{J_+J_-\mathcal
J_-\mathcal
J_+}}, \\
(H^{2n+1})_{33}&=&-\frac{1}{2}\delta[(\mathcal M_3^+)^n+(\mathcal M_3^-)^n], \\
(H^{2n+1})_{34}&=&\mathcal J_+\frac{\alpha/2}{\sqrt{N \mathcal
J_-\mathcal J_+}}[(\sqrt{\mathcal J_-\mathcal J_+}+\sqrt{J_-
J_+})(\mathcal M_4^+)^n \\&+&(\sqrt{\mathcal J_-\mathcal
J_+}-\sqrt{J_- J_+})( \mathcal M_4^-)^n], \end{eqnarray}
\begin{eqnarray}
(H^{2n+1})_{41}&=&(\delta/2)J_-\mathcal J_-\frac{(\mathcal
M_1^+)^n-(\mathcal M_1^-)^n}{\sqrt{J_+J_-\mathcal
J_+\mathcal
J_-}}, \\
(H^{2n+1})_{42}&=&J_-\frac{\alpha/2}{\sqrt{N J_+
J_-}}[(\sqrt{\mathcal J_-\mathcal J_+}+\sqrt{ J_+
J_-}) (\mathcal M_2^+)^n \\&+&(\sqrt{J_+ J_-}-\sqrt{\mathcal J_-\mathcal
J_+})(\mathcal M_2^-)^{n}], \\
(H^{2n+1})_{43}&=&\mathcal J_-\frac{\alpha/2}{\sqrt{N \mathcal
J_+\mathcal J_-}}[(\sqrt{\mathcal J_+\mathcal J_-}+\sqrt{J_-
J_+})(\mathcal M_3^+)^n \\&+&(\sqrt{\mathcal J_+\mathcal
J_-}-\sqrt{J_- J_+})( \mathcal M_3^-)^n], \\
(H^{2n+1})_{44}&=&\frac{1}{2}\delta[(\mathcal M_4^+)^n+(\mathcal
M_4^-)^n].\label{powerodd2}
\end{eqnarray}
Having in hand the explicit expressions of powers of the total
Hamiltonian, it can easily be verified that the elements of the time
evolution operator, obtained by inserting equations~(\ref{power1})
and~(\ref{powerodd1})-(\ref{powerodd2}) into
equation~(\ref{taylor}), are given by
\begin{align}
U_{11}(t)=&\frac{1}{2}\Bigl\{\cos\Bigl(t \sqrt{ \mathcal
M_1^+}\Bigl)+\cos\Bigl(t\sqrt{ \mathcal
M_1^-}\Bigl)-i\delta\Bigl[\frac{\sin\Bigl(t\sqrt{ \mathcal
M_1^+}\Bigl)}{\sqrt{ \mathcal M_1^+}}+\frac{\sin\Bigl(t \sqrt{
\mathcal M_1^-}\Bigl)}{ \sqrt{ \mathcal M_1^-}}\Bigl]\Bigl\},\\
U_{21}(t)=&-\mathcal J_-\frac{i\alpha/2}{\sqrt{N\mathcal J_+\mathcal
J_-}}\Bigl\{\frac{\sqrt{J_+J_-}+\sqrt{\mathcal J_+ \mathcal
J_-}}{\sqrt{ \mathcal M_1^+}}\sin\Bigl(t\sqrt{ \mathcal
M_1^+}\Bigl)\nonumber\\&-\frac{\sqrt{J_+J_-}-\sqrt{\mathcal J_+
\mathcal J_-}}{\sqrt{ \mathcal M_1^-}}\sin\Bigl(t \sqrt{ \mathcal
M_1^-}\Bigl)\Bigl\},\\
U_{31}(t)=&- J_-\frac{i\alpha/2}{\sqrt{N
J_+J_-}}\Bigl\{\frac{\sqrt{J_+J_-}+\sqrt{\mathcal J_+ \mathcal
J_-}}{\sqrt{ \mathcal M_1^+}}\sin\Bigl(t\sqrt{ \mathcal
M_1^+}\Bigl)\nonumber\\&+\frac{\sqrt{J_+J_-}-\sqrt{\mathcal J_+
\mathcal J_-}}{\sqrt{ \mathcal M_1^-}}\sin\Bigl(t \sqrt{ \mathcal
M_1^-}\Bigl)\Bigl\},\end{align} \begin{align} U_{41}(t)=&J_-\mathcal
J_-\frac{1}{2\sqrt{J_+J_-\mathcal J_+\mathcal
J_-}}\Bigl\{\cos\Bigl(t \sqrt{ \mathcal
M_1^+}\Bigl)-\cos\Bigl(t\sqrt{ \mathcal
M_1^-}\Bigl)\nonumber\\&-i\delta\Bigl[\frac{\sin\Bigl(t\sqrt{
\mathcal M_1^+}\Bigl)}{\sqrt{ \mathcal M_1^+}}-\frac{\sin\Bigl(t
\sqrt{ \mathcal M_1^-}\Bigl)}{ \sqrt{ \mathcal
M_1^-}}\Bigl]\Bigl\},\\ U_{22}(t)=&\frac{1}{2}\Bigl\{\cos\Bigl(t
\sqrt{ \mathcal M_2^+}\Bigl)+\cos\Bigl(t\sqrt{ \mathcal
M_1^-}\Bigl)+i\delta\Bigl[\frac{\sin\Bigl(t\sqrt{ \mathcal
M_2^+}\Bigl)}{\sqrt{ \mathcal M_2^+}}+\frac{\sin\Bigl(t \sqrt{
\mathcal M_2^-}\Bigl)}{ \sqrt{ \mathcal M_2^-}}\Bigl]\Bigl\},\\
U_{12}(t)=&-\mathcal J_+\frac{i\alpha/2}{\sqrt{N\mathcal J_-\mathcal
J_+}}\Bigl\{\frac{\sqrt{J_+J_-}+\sqrt{\mathcal J_- \mathcal
J_+}}{\sqrt{ \mathcal M_2^+}}\sin\Bigl(t\sqrt{ \mathcal
M_2^+}\Bigl)\nonumber\\&-\frac{\sqrt{J_+J_-}-\sqrt{\mathcal J_-
\mathcal J_+}}{\sqrt{ \mathcal M_2^-}}\sin\Bigl(t \sqrt{ \mathcal
M_2^-}\Bigl)\Bigl\},\\
U_{32}(t)=&J_-\mathcal J_+\frac{1}{2\sqrt{J_+J_-\mathcal J_-\mathcal
J_+}}\Bigl\{\cos\Bigl(t \sqrt{ \mathcal
M_2^+}\Bigl)-\cos\Bigl(t\sqrt{ \mathcal
M_2^-}\Bigl)\nonumber\\&+i\delta\Bigl[\frac{\sin\Bigl(t\sqrt{
\mathcal M_2^+}\Bigl)}{\sqrt{ \mathcal M_2^+}}-\frac{\sin\Bigl(t
\sqrt{ \mathcal M_2^-}\Bigl)}{ \sqrt{ \mathcal
M_2^-}}\Bigl]\Bigl\},\\
U_{42}(t)=&- J_-\frac{i\alpha/2}{\sqrt{N
J_+J_-}}\Bigl\{\frac{\sqrt{J_+J_-}+\sqrt{\mathcal J_- \mathcal
J_+}}{\sqrt{ \mathcal M_2^+}}\sin\Bigl(t\sqrt{ \mathcal
M_2^+}\Bigl)\nonumber\\&+\frac{\sqrt{J_+J_-}-\sqrt{\mathcal J_-
\mathcal J_+}}{\sqrt{ \mathcal M_2^-}}\sin\Bigl(t \sqrt{ \mathcal
M_2^-}\Bigl)\Bigl\},\\
U_{33}(t)=&\frac{1}{2}\Bigl\{\cos\Bigl(t
\sqrt{ \mathcal M_3^+}\Bigl)+\cos\Bigl(t\sqrt{ \mathcal
M_3^-}\Bigl)+i\delta\Bigl[\frac{\sin\Bigl(t\sqrt{ \mathcal
M_3^+}\Bigl)}{\sqrt{ \mathcal M_3^+}}+\frac{\sin\Bigl(t \sqrt{
\mathcal M_3^-}\Bigl)}{ \sqrt{ \mathcal M_3^-}}\Bigl]\Bigl\},\\
U_{13}(t)=&- J_+\frac{i\alpha/2}{\sqrt{N J_-
J_+}}\Bigl\{\frac{\sqrt{\mathcal J_+\mathcal J_-}+\sqrt{ J_-
J_+}}{\sqrt{ \mathcal M_3^+}}\sin\Bigl(t\sqrt{ \mathcal
M_3^+}\Bigl)\nonumber\\&-\frac{\sqrt{\mathcal J_+\mathcal
J_-}-\sqrt{ J_- J_+}}{\sqrt{ \mathcal M_3^-}}\sin\Bigl(t \sqrt{
\mathcal
M_3^-}\Bigl)\Bigl\},\\
U_{23}(t)=&J_+\mathcal J_- \frac{1}{2\sqrt{J_- J_+ \mathcal
J_+\mathcal J_- }}\Bigl\{\cos\Bigl(t \sqrt{ \mathcal
M_3^+}\Bigl)-\cos\Bigl(t\sqrt{ \mathcal
M_3^-}\Bigl)\nonumber\\&+i\delta\Bigl[\frac{\sin\Bigl(t\sqrt{
\mathcal M_3^+}\Bigl)}{\sqrt{ \mathcal M_3^+}}-\frac{\sin\Bigl(t
\sqrt{ \mathcal M_3^-}\Bigl)}{ \sqrt{ \mathcal
M_3^-}}\Bigl]\Bigl\},\\
U_{43}(t)=&- \mathcal J_-\frac{i\alpha/2}{\sqrt{N
\mathcal J_+\mathcal J_-}}\Bigl\{\frac{\sqrt{\mathcal J_+\mathcal
J_-}+\sqrt{ J_- J_+}}{\sqrt{ \mathcal M_3^+}}\sin\Bigl(t\sqrt{
\mathcal M_3^+}\Bigl)\nonumber\\&+\frac{\sqrt{\mathcal J_+\mathcal
J_-}-\sqrt{ J_- J_+}}{\sqrt{ \mathcal M_3^-}}\sin\Bigl(t \sqrt{
\mathcal M_3^-}\Bigl)\Bigl\},\end{align}
\begin{align}
U_{44}(t)=&\frac{1}{2}\Bigl\{\cos\Bigl(t \sqrt{ \mathcal
M_4^+}\Bigl)+\cos\Bigl(t\sqrt{ \mathcal
M_4^-}\Bigl)-i\delta\Bigl[\frac{\sin\Bigl(t\sqrt{ \mathcal
M_4^+}\Bigl)}{\sqrt{ \mathcal M_4^+}}+\frac{\sin\Bigl(t \sqrt{
\mathcal M_4^-}\Bigl)}{ \sqrt{ \mathcal M_4^-}}\Bigl]\Bigl\},\\
U_{24}(t)=&- J_+\frac{i\alpha/2}{\sqrt{N J_-
J_+}}\Bigl\{\frac{\sqrt{\mathcal J_-\mathcal J_+}+\sqrt{ J_-
J_+}}{\sqrt{ \mathcal M_4^+}}\sin\Bigl(t\sqrt{ \mathcal
M_4^+}\Bigl)\nonumber\\&-\frac{\sqrt{\mathcal J_-\mathcal
J_+}-\sqrt{J_- J_+}}{\sqrt{ \mathcal M_4^-}}\sin\Bigl(t \sqrt{
\mathcal
M_4^-}\Bigl)\Bigl\},\\
U_{34}(t)=&-\mathcal J_+\frac{i\alpha/2}{\sqrt{N \mathcal
J_-\mathcal J_+}}\Bigl\{\frac{\sqrt{\mathcal J_-\mathcal
J_+}+\sqrt{J_- J_+}}{\sqrt{ \mathcal M_4^+}}\sin\Bigl(t\sqrt{
\mathcal M_4^+}\Bigl)\nonumber\\&+\frac{\sqrt{\mathcal J_-\mathcal
J_+}-\sqrt{J_- J_+}}{\sqrt{ \mathcal M_4^-}}\sin\Bigl(t \sqrt{
\mathcal
M_4^-}\Bigl)\Bigl\},\\
U_{14}(t)=&J_+\mathcal J_+\frac{1}{2\sqrt{J_- J_+\mathcal
J_-\mathcal J_+}}\Bigl\{\cos\Bigl(t \sqrt{ \mathcal
M_4^+}\Bigl)-\cos\Bigl(t\sqrt{ \mathcal
M_4^-}\Bigl)\nonumber\\&-i\delta\Bigl[\frac{\sin\Bigl(t\sqrt{
\mathcal M_4^+}\Bigl)}{\sqrt{ \mathcal M_4^+}}-\frac{\sin\Bigl(t
\sqrt{ \mathcal M_4^-}\Bigl)}{ \sqrt{ \mathcal M_4^-}}\Bigl]\Bigl\}.
\end{align}
It should be noted that the above components of the operator
$\mathbf U(t)$ can also be derived by solving the Schr\"{o}dinger
equation~\cite{22}
\begin{equation}
i\frac{d \mathbf U(t)}{dt}=H{\mathbf U}(t).\end{equation} For
instance, we have
\begin{align}
i \frac{d U_{11}(t)}{dt}=\delta
U_{11}(t)+\frac{\alpha}{\sqrt{N}}\mathcal J_+ U_{21}(t)+
\frac{\alpha}{\sqrt{N}}J_+U_{31}(t)\label{case2eq1},\\
i \frac{d U_{21}(t)}{dt}=\frac{\alpha}{\sqrt{N}}\mathcal J_-
U_{11}(t)-\delta U_{21}(t)+
\frac{\alpha}{\sqrt{N}}J_+U_{41}(t),\\
i \frac{d U_{31}(t)}{dt}=\frac{\alpha}{\sqrt{N}} J_-U_{11}(t)-\delta
U_{31}(t)+
\frac{\alpha}{\sqrt{N}}\mathcal J_+ U_{41}(t),\\
i \frac{d U_{41}(t)}{dt}=\frac{\alpha}{\sqrt{N}} J_- U_{21}(t)+
\frac{\alpha}{\sqrt{N}}\mathcal J_-U_{31}(t)+\delta
U_{41}(t)\label{case2eq4}.\end{align} This set of differential
equation can be solved by introducing the following transformations:
\begin{align}
U_{11}(t)&\rightarrow e^{-i\delta t}U_{11}(t),\\
U_{21}(t)&\rightarrow e^{-i\delta t}\mathcal J_- U_{21}(t),\\
U_{31}(t)&\rightarrow e^{-i\delta t}J_-U_{31}(t), \\
U_{41}(t)&\rightarrow e^{-i\delta t}J_-\otimes \mathcal
J_-U_{41}(t).
\end{align}
The resulting differential equations involve diagonal terms; they
can be solved by taking into account the initial conditions:
\begin{equation}
U_{ij}(0)=\left\{
\begin{array}{ll}
{\mathbf 1} &\text{for} \quad \hbox{$i=j$,} \\
0 & \text{for} \quad\hbox{$i\neq j$.}
\end{array}
\right.
\end{equation}
Following the same procedure, it is possible to derive the remaining matrix elements of the time evolution operator.
There exist many measures for entanglement. Here we shall use the concurrence, defined
by~\cite{26}
\begin{equation}
C(\rho)=\max\{0,2\max[\sqrt{\lambda_i}]-\sum_{i=1}^4\sqrt{\lambda_i}\},\end{equation}
where the quantities $\lambda_i$ are the eigenvalues of the
operator
$\rho(t)(\sigma_y\otimes\sigma_y)\rho(t)^*(\sigma_y\otimes\sigma_y)$.
The above measure is equal to one for maximally entangled states,
and is equal to zero for separable states. The purity
\begin{equation} P(t)={\rm tr}\rho(t)^2\end{equation} can be used
to quantify the decoherence of the central system; it is equal to
$\tfrac{1}{4}$ for maximally mixed states, and one for pure states.
It turns out that the density matrices corresponding to the
initial product states $|\epsilon_1\epsilon_2\rangle$, where
$\epsilon_i\equiv\pm$, are always diagonal. Furthermore, the
numerical simulation shows that if the qubits are prepared in one of
the above states, they remain unentangled regardless of the values
of $N$ and $\delta$. The purity decays less with the increase of
$\delta$.
\begin{figure}[htba]
{\centering
\resizebox*{0.65\textwidth}{!}{\includegraphics{fig1.eps}}
\par}
\caption{\label{figure1} The evolution in time of the concurrence
(solid curve) and the purity (dashed curve) corresponding to the
singlet state for $\delta=\alpha$ and $N=10$.}
\end{figure}
\begin{figure}[htba]
{\centering
\resizebox*{0.65\textwidth}{!}{\includegraphics{fig2.eps}}
\par}
\caption{\label{figure2} The evolution in time of the concurrence
(solid curve) and the purity (dashed curve) corresponding to the
singlet state for $\delta=4\alpha$ and $N=10$.}
\end{figure}
The matrix elements of the reduced density matrices corresponding to
the states $\tfrac{1}{\sqrt{2}}(|-+\rangle\pm|+-\rangle)$ and
$\tfrac{1}{\sqrt{2}}(|++\rangle\pm|--\rangle)$ are shown in the
Appendix. The evolution in time of the concurrence and the purity
corresponding to the above maximally entangled states is
practically the same; we only present the results obtained for the
singlet state. It is found that, for fixed $\delta$, the concurrence
and the purity saturate as the number of spins increases. This
naturally suggests the investigation of the case $N\to\infty$ (see
the next section). For small values of the coupling constant
$\delta$, the concurrence decays from its initial maximum value
$C_{\mathrm{max}}=1$, then vanishes at a certain moment of time
(i.e. entanglement sudden death~\cite{27}). At long times, and
sufficiently large $N$ and $\delta$, the purity and the concurrence
converge to certain asymptotic values, which increase with the
increase of the strength of interaction. Here it should be noted
that, in contrast to
the case of common spin bath, the
singlet state is not decoherence free. This was expected because the
latter state is not eigenvector of the Hamiltonian $H$.
Nevertheless, we find that decoherence can be reduced
with strong coupling between the qubits, in agreement with~\cite{22}. Finally let us remark that,
although we only have considered infinite temperature, we can
ensure that for long-range antiferromagnetic Heisenberg interactions within the baths,
low temperatures will have the same effect
on decoherence and entanglement of the qubits as strong coupling
constants.
\section{Thermodynamic limit\label{sec4}}
In the thermodynamic limit, $N\to\infty$, the operators $\sqrt{J_\pm
J_\mp/N}$ converge to the positive real random variable $r$ whose
probability density function is given by
\begin{equation}
r\mapsto f(r)=4 r {\rm e}^{-2 r^2}, \quad r \ge
0.\label{prod5}\end{equation} Indeed, it has been shown
in~\cite{22,23} that the operator $J_+/\sqrt{N}$ converges to the
complex normal random variable $z$ with the probability density
function
\begin{equation}
z\mapsto \frac{2}{\pi} \mathrm e^{-2|z|^2}.
\end{equation}
Expressing
$z$ in terms of the polar coordinates $r$ and $\phi$, i.e.,
$z=r \mathrm e^{i \phi}$, simply gives $|z|^2=r^2$. Then
integrating the corresponding probability density function over the
variable $\phi$ from 0 to $2 \pi$ yields
\begin{eqnarray}
dP(r)=f(r)dr&=&\frac{2}{\pi} \int\limits_0^{2\pi} d\phi \ r \ dr \mathrm e^{-2
r^2}\nonumber\\
&=&4 r \mathrm e^{-2
r^2} \ dr,
\end{eqnarray}
from which~(\ref{prod5}) follows.
Hence we can ascertain that
\begin{equation}
\lim\limits_{N\to\infty} 2^{-2N}{\rm tr}_{B_1+B_2}
\Omega\Bigl(\sqrt{J_\pm J_\mp/N},\sqrt{\mathcal J_\pm\mathcal
J_\mp/N}\Bigl)=16\int\limits^{\infty}_{0}\int\limits^{\infty}_{0} r\
s\ \mathrm e^{-2(r^2+s^2)}\Omega(r,s) dr
ds,\label{sumint}\end{equation} where $\Omega(r,s)$ is some
complex-valued function for which the integrals in the right-hand
side of equation~(\ref{sumint}) converge.
Using the above result, one can express the nonzero elements of the reduced density matrix
corresponding to the initial state
$\frac{1}{\sqrt{2}}(|-+\rangle-|+-\rangle)$, in the thermodynamic
limit, as
\begin{eqnarray}
\rho_{11}(t)&=&\rho_{44}(t)=\frac{1}{4}[\Lambda_+(t)+\Lambda_-(t)],\\
\rho_{22}(t)&=&\rho_{33}(t)=\frac{1}{4}[\Upsilon_+(t)+\Upsilon_-(t)+\Xi_+(t)+\Xi_-(t)],\\
\rho_{23}(t)&=&-\frac{1}{8}[\Upsilon_+(t)+\Upsilon_-(t)+\Xi_+(t)+\Xi_-(t)+2\Psi(t)],
\end{eqnarray}
where ( we set $\alpha=1$ for the sake of shortness)
\begin{eqnarray}
\Lambda_\pm(t)&=&16\int\limits_0^\infty\int\limits_0^\infty r s \
\mathrm e^{-2(r^2+s^2)} \frac{(r\pm s)^2}{\delta^2+(r\pm
s)^2}\sin^2\Bigl(t \sqrt{\delta^2+(r\pm s)^2}\Bigl)dr ds,\label{f1}\\
\Upsilon_\pm(t)&=&16\int\limits_0^\infty\int\limits_0^\infty r s \
\mathrm e^{-2(r^2+s^2)} \cos^2\Bigl(t \sqrt{\delta^2+(r\pm
s)^2}\Bigl)dr ds,\label{f2}\\
\Xi_\pm(t)&=& 16\int\limits_0^\infty\int\limits_0^\infty r s \
\mathrm e^{-2(r^2+s^2)} \frac{\delta^2}{\delta^2+(r\pm
s)^2}\sin^2\Bigl(t
\sqrt{\delta^2+(r\pm s)^2}\Bigl)dr ds,\label{f3}\\
\Psi(t)&=&16\int\limits_0^\infty\int\limits_0^\infty r s \
\mathrm e^{-2(r^2+s^2)} \Bigl\{\cos\Bigl(t \sqrt{\delta^2+(r+
s)^2}\Bigl) \cos\Bigl(t \sqrt{\delta^2+(r- s)^2}\Bigl)\nonumber
\\&+&\delta^2 \frac{\sin\Bigl(t
\sqrt{\delta^2+(r+ s)^2}\Bigl)}{\delta^2+(r+ s)^2}\frac{\sin\Bigl(t
\sqrt{\delta^2+(r- s)^2}\Bigl)}{\delta^2+(r- s)^2}\Bigl\} dr
ds\label{f4}.
\end{eqnarray}
Unfortunately the above functions cannot be evaluated analytically;
one should make recourse to numerical integration. This task can be
significantly simplified by transforming the double integration into
single one, which is much easier to carry out. To do that notice
that the analysis of the expressions of the functions
$\Lambda_\pm(t)$, $\Upsilon_\pm(t)$, and $\Xi_\pm(t)$ leads to the
evaluation of the probability density functions $Q(\mu)$ and
$R(\eta)$ corresponding, respectively, to the random variables
$\mu=r+s$ and $\eta=r-s$ (see~\cite{28} for a similar situation).
Let us begin with the variable $\mu$; its probability density
function is simply given by the convolution of $f(r)$ with itself:
\begin{equation}
Q(\mu)=16 \int\limits_{0}^\mu (\mu-r) r \mathrm e^{-2(\mu-r)^2-2
r^2} dr .\end{equation} Note that the upper limit of the integration
over $r$ is $\mu$ because the quantity $\mu-r$ should be positive.
The evaluation of the integral is somewhat lengthy, but elementary;
one finds that
\begin{equation}
Q(\mu)=[2 \mu-\sqrt{\pi}\mathrm e^{\mu^2}(1-2\mu^2) {\rm
erf}(\mu)]\mathrm e^{-2\mu^2},\label{probfun1}\end{equation} where
${\rm erf}(x)$ designates the error function~\cite{29}.
Now consider the variable $\eta=r-s$. One should be careful when
using the definition of the convolution,
since, in this case, $\eta$ belongs to the interval
$]-\infty,\infty[ $. We have to distinguish between two cases,
namely, $\eta\ge 0$ and $\eta\le 0$. In the first case $r\in
[0,\infty[$, and hence
\begin{eqnarray}
R(\eta\ge0)&=&16 \int\limits_0^\infty (\eta+r) r \mathrm
e^{-2(r+s)^2-2r^2} dr\nonumber \\
&=&\frac{1}{2} \{2\eta +\sqrt{\pi} \mathrm e^{\eta^2} (1 - 2
\eta^2) [1-{\rm erf}(\eta)]\}\mathrm e^{-2 \eta^2}.\label{pp1}
\end{eqnarray}
When $\eta\le 0$, then $r\in[-\eta,\infty[$, which implies that
\begin{eqnarray}
R(\eta\le 0)&=&16 \int\limits_{-\eta}^\infty (\eta+r) r \mathrm
e^{-2(r+s)^2-2r^2} dr\nonumber \\
&=&\frac{1}{2} \{-2\eta +\sqrt{\pi} \mathrm e^{\eta^2} (1 - 2
\eta^2) [1+{\rm erf}(\eta)]\}\mathrm e^{-2 \eta^2}.\label{pp2}
\end{eqnarray}
Combining (\ref{pp1}) and (\ref{pp2}), we obtain the following
expression for the probability density function of $\eta$ over the
real line:
\begin{eqnarray}
R(\eta)=\frac{1}{2} \{2|\eta| +\sqrt{\pi} \mathrm e^{\eta^2} (1 - 2
\eta^2) [1-{\rm erf}(|\eta|)]\}\mathrm e^{-2 \eta^2}\label{profun2}.
\end{eqnarray}
The above functions are depicted in figures~\ref{figure3}
and~\ref{figure4}. Clearly, $R(\eta)$ is an even function of its
argument; it takes its maximum value at the origin, that is,
$\max\{R(\eta)\}=R(0)=0.886227$. The maximum value of $Q(\mu)$
occurs at $\mu_0=1.14209$, such that
$\max\{Q(\mu)\}=Q(\mu_0)=0.859664$.
\begin{figure}[htba]
{\centering
\resizebox*{0.60\textwidth}{!}{\includegraphics{fig3.eps}}
\par}
\caption{\label{figure3} The probability density function
$Q(\mu)$.}
\end{figure}
\begin{figure}[htba]
{\centering
\resizebox*{0.60\textwidth}{!}{\includegraphics{fig4.eps}}
\par}
\caption{\label{figure4} The probability density function
$R(\eta)$.}
\end{figure}
\vline
As a simple application let us prove the following:
\begin{theorem}
The moments around origin of
the random variables $\mu$ and $\eta$ are given by:
\begin{align}
\langle \mu^{2n} \rangle&=\frac{n!}{2^n}\Bigl[1+2^{n+1} n\
_2F_1\Bigl( 1 + n,\frac{1}{2}; \frac{3}{2}; -1\Bigl)\Bigl],\label{mom1}\\
\langle \mu^{2n+1}
\rangle&=\frac{\Gamma(\tfrac{3}{2}+1)}{2^n}\Bigl[\tfrac{1}{\sqrt{2}}+2^{n}
(2n+1)\ _2F_1\Bigl( \frac{3}{2} + n,\frac{1}{2}; \frac{3}{2};
-1\Bigl)\Bigl],\\
\langle \eta^{2n}
\rangle&=\langle \mu^{2n}
\rangle-n\sqrt{\pi}\Gamma\Bigl(\frac{1}{2}+n\Bigl),\\
\langle \eta^{2n+1} \rangle&=0,\label{mom4}
\end{align}
where $\Gamma(x)$, and $_2F_1(a,b;c;d)$ denote the Gamma and the
hypergeometric functions, respectively.
\end{theorem}
\begin{proof} Relation~(\ref{mom4})
is obvious since the function $R(\eta)$ is even. Let us
prove~(\ref{mom1}). We have that
\begin{eqnarray}
\langle \mu^{2n} \rangle&=&\int\limits_0^\infty \mu^{2n}Q(\mu)\
d\mu\nonumber\\
&=&2I_{n+1}-I_n+2 Y_n,\end{eqnarray}
where
\begin{eqnarray}
I_n&=&\int\limits_0^\infty\sqrt{\pi} \mu^{2n} \mathrm e^{-
\mu^2}{\rm erf}(\mu)\ d\mu,\label{funi}\\
Y_n&=&\int\limits_0^\infty \mu^{2n+1} \mathrm e^{-2 \mu^2}\ d\mu.
\end{eqnarray}
To calculate $Y_n$ and $I_{n}$, introduce the functions of the real
variable $x>0$:
\begin{equation}
Y_n(x)=\int\limits_0^\infty\mu^{2n+1}\mathrm
e^{-\mu^2(1+\tfrac{1}{x})} d\mu,
\end{equation}
\begin{equation}
I_n(x)=\int\limits_0^\infty\sqrt{\pi} \mu^{2n} \mathrm e^{-
\mu^2/x}{\rm erf}(\mu)\ d\mu. \label{newfun}\end{equation} The first
integral can be easily evaluated:
\begin{eqnarray}
\nonumber\\
Y_n(x)&=&\frac{1}{2}\Bigl(\tfrac{x}{1+x}\Bigl)^{n+1}
\int\limits_0^\infty\chi^{n}\mathrm e^{-\chi}d\chi=
\frac{n!}{2}\Bigl(\tfrac{x}{1+x}\Bigl)^{n+1}\label{firsteq}.\end{eqnarray}
The second integral satisfies
\begin{equation}
\frac{d I_n(x)}{dx}=\frac{1}{x^2}
I_{n+1}(x).\label{difffn}\end{equation} Integrating by parts the RHS
of (\ref{newfun})
with respect to $\mu$, and using~(\ref{firsteq}), yield
\begin{equation}
I_{n+1}(x)=\frac{x(2n+1)}{2}I_n(x)+\frac{x
n!}{2}\Bigl(\frac{x}{x+1}\Bigl)^{n+1}.\label{redn}\end{equation}
Here we have used the fact that ${\rm erf}(x)^{'} =2 \mathrm
e^{-x^2}/\sqrt{\pi}$.
Let $I_n(x)=n! x^{n+1} g_n(x)$. Then
from~(\ref{redn}) we have
\begin{equation}
2(n+1)
g_{n+1}(x)=(2n+1)g_n(x)+\frac{1}{(x+1)^{n+1}}\label{dign}.\end{equation}
On the other hand equation~(\ref{difffn}) implies that
\begin{equation}
x \frac{d g_n(x)}{dx}+(n+1)g_n(x)=(n+1)g_{n+1}(x).\end{equation}
Combining the last two equations yields the following first order
differential equation for the function $g_n(x)$:
\begin{equation}
2 x \frac{d
g_n(x)}{dx}+g_n(x)-\frac{1}{(x+1)^{n+1}}=0\label{difff}.\end{equation}
Differentiating both sides of~(\ref{difff}), and again
using~(\ref{dign}), we obtain \begin{equation}
\Bigl[\frac{d^2}{dx^2}+\Bigl(\frac{3}{2x}+\frac{n+1}{x+1}\Big)\frac{d}{dx}+\frac{n+1}{2x(x+1)}\Bigl]g_n(x)=0.\end{equation}
By setting $y=-x$, and $h_n(y)=g_n(-x)$, we obtain \begin{equation}
\Bigl[\frac{d^2}{dy^2}+\Bigl(\frac{3}{2y}+\frac{n+1}{y-1}\Big)\frac{d}{dy}+\frac{n+1}{2y(y-1)}\Bigl]h_n(y)=0,
\end{equation}
which should be compared with the hypergeometric equation
\begin{equation}
\Bigl[\frac{d^2}{dy^2}+\Bigl(\frac{c}{y}+\frac{1+a+b-c}{y-1}\Big)\frac{d}{dy}+\frac{ab}{y(y-1)}\Bigl]\
_2F_1(a,b;c;y)=0.
\end{equation}
Thus
\begin{equation*}a=n+1,\quad b=\tfrac{1}{2}, \quad
c=\tfrac{3}{2}.\end{equation*} It follows that
\begin{equation}
I_n(x)=n! x^{n+1} \
_2F_1(n+1,\tfrac{1}{2};\tfrac{3}{2};-x).\end{equation} Putting $x=1$
yields
\begin{equation}
I_n=n! \ _2F_1(n+1,\tfrac{1}{2};\tfrac{3}{2};-1), \qquad
Y_n=\frac{n!}{2^{n+2}}.\end{equation}
Also, using~(\ref{redn}), we obtain
\begin{equation}2I_{n+1}=(2n+1)n! \
_2F_1(n+1,\tfrac{1}{2};\tfrac{3}{2};-1)+\frac{n!}{2^{n+1}},\end{equation}
from which (\ref{mom1}) readily follows. The other moments can be
evaluated with a similar method. \hfill
\medskip
\end{proof}
The functions~(\ref{f1})-(\ref{f3}) can
easily be expressed in terms of the functions $Q(\mu)$ and
$R(\eta)$. For example, we have:
\begin{eqnarray}
\Lambda_+(t)&=&\int_0^\infty
Q(\mu)\frac{\mu^2}{\delta^2+\mu^2}\sin^2\Bigl(t\sqrt{\delta^2+\mu^2}\Bigl)
d\mu,\\
\Lambda_-(t)&=&\int_{-\infty}^\infty
R(\mu)\frac{\mu^2}{\delta^2+\mu^2}\sin^2\Bigl(t\sqrt{\delta^2+\mu^2}\Bigl)
d\mu.
\end{eqnarray}
It should be noted that in contrast to $r$ and $s$, the random
variables $\eta$ and $\mu$ are not independent. The function
$\Psi(t)$ can not be further simplified, and should be evaluated
using the double integration over the variables $r$ and $s$.
Nevertheless, using the Riemann-Lebesgue lemma, we can infer that
\begin{equation}
\lim\limits_{t\to\infty}\Psi(t)=\Psi(\infty)=0.\end{equation} In a
similar way, the remaining functions tend asymptotically to:
\begin{eqnarray}
\Lambda_+(\infty)&=&\frac{1}{2}\int_0^\infty
Q(\mu)\frac{\mu^2}{\delta^2+\mu^2}
d\mu,\\
\Lambda_-(\infty)&=&\frac{1}{2}\int_{-\infty}^\infty
R(\mu)\frac{\mu^2}{\delta^2+\mu^2} d\mu,\\
\Upsilon_\pm(\infty)&=&\frac{1}{2},\\
\Xi_+(\infty)&=&\frac{1}{2}\int_0^\infty
Q(\mu)\frac{\delta^2}{\delta^2+\mu^2} d\mu,\\
\Xi_-(\infty)&=&\frac{1}{2}\int_{-\infty}^\infty
R(\mu)\frac{\delta^2}{\delta^2+\mu^2} d\mu.
\end{eqnarray}
Notice that \begin{equation}
\Lambda_\pm(\infty)+\Xi_\pm(\infty)=\frac{1}{2},
\end{equation}
independently of the values of $\delta$. It follows that the
asymptotic density matrix can be expressed as
\begin{equation}
\rho(\infty)=\begin{pmatrix} \frac{\Pi}{4}&&0&&0&&0\\
0&& \frac{2-\Pi}{4}&&-\frac{2-\Pi}{8}&&0\\
0&&-\frac{2-\Pi}{8}&&\frac{2-\Pi}{4}&&0\\0&&0&&0&&\frac{\Pi}{4}\end{pmatrix},\end{equation}
where
\begin{equation}
\Pi=\Lambda_+(\infty)+\Lambda_-(\infty).\end{equation} It is easily
seen that
\begin{equation}
\lim\limits_{\delta\to 0}\Xi_\pm(\infty)=0, \quad
\lim\limits_{\delta\to 0}\Lambda_\pm(\infty)=\frac{1}{2}.
\end{equation}
The corresponding asymptotic reduced density matrix reads
\begin{equation}
\rho(\infty)_{\delta=0}=\begin{pmatrix} \frac{1}{4}&&0&&0&&0\\
0&& \frac{1}{4}&&-\frac{1}{8}&&0\\
0&&-\frac{1}{8}&&\frac{1}{4}&&0\\0&&0&&0&&\frac{1}{4}\end{pmatrix},\end{equation}
which has a concurrence identically equal to zero.
On the contrary, in the limit of strong coupling between the central
qubits,
\begin{equation}
\lim\limits_{\delta\to \infty}\Xi_\pm(\infty)=\frac{1}{2}, \quad
\lim\limits_{\delta\to \infty}\Lambda_\pm(\infty)=0.\end{equation}
Consequently,
\begin{equation}
\rho(\infty)_{\delta=\infty}=\begin{pmatrix} 0&&0&&0&&0\\
0&& \frac{1}{2}&&-\frac{1}{4}&&0\\
0&&-\frac{1}{4}&&\frac{1}{2}&&0\\0&&0&&0&&0\end{pmatrix}.\end{equation}
A straightforward calculation shows that
\begin{equation}
\lim\limits_{\delta\to\infty}C(\rho(\infty))=\frac{1}{2}.\end{equation}
In general, since $0\le \mu^2/(\mu^2+\delta^2)\le 1$, then
\begin{eqnarray}
0\le \Pi&=\frac{1}{2} \int_0^\infty
Q(\mu)\frac{\mu^2}{\delta^2+\mu^2} d\mu+ \frac{1}{2}
\int_{-\infty}^\infty R(\mu)\frac{\mu^2}{\delta^2+\mu^2} d\mu
\nonumber
\\
&\le \frac{1}{2}\int_0^\infty Q(\mu) d\mu+\frac{1}{2}
\int_{-\infty}^\infty R(\mu) d\mu=1.
\end{eqnarray}
This allows us to find the following explicit form of the asymptotic
value of the concurrence:
\begin{equation}
C(\infty)=\max\Bigl\{0,\frac{2-3\Pi}{4}\Bigl\}.
\end{equation}
The latter can also be rewritten as:
\begin{equation}
C(\infty)=\left\{
\begin{array}{ll}
\frac{2-3\Pi}{4} &\text{for} \quad \hbox{$0 \le \Pi \le \frac{2}{3} $,} \\
0 & \text{for} \quad\hbox{$ \frac{2}{3}\le \Pi \le 1 $.}
\end{array}
\right.
\end{equation}
The variation of the asymptotic
concurrence as a function of $\delta$ is shown in
figure~\ref{figure5}. It can be seen that $C(\infty)$ remains zero
up to a critical value $\delta_c$ after which it increases, to tend
asymptotically to $\frac{1}{2}$. The value of $\delta_c$ can be
evaluated numerically:
\begin{figure}[htba]
{\centering
\resizebox*{0.65\textwidth}{!}{\includegraphics{fig5.eps}}
\par}
\caption{\label{figure5} The variation of $C(\infty)$ as a
function of the coupling constant $\delta$. The inset shows the
critical point $\delta_c$.}
\end{figure}
\begin{equation}\delta_c= 0.342842, \qquad \Pi|_{\delta=\delta_c}=0.666667.\end{equation}
At the critical point, the density
matrix reads
\begin{equation}
\rho_c(\infty)=\begin{pmatrix} \frac{1}{6}&&0&&0&&0\\
0&& \frac{1}{3}&&-\frac{1}{6}&&0\\
0&&-\frac{1}{6}&&\frac{1}{3}&&0\\0&&0&&0&&\frac{1}{6}\end{pmatrix}.\end{equation}
\section{Second-order master equation\label{sec5}}
Under Born Approximation, the second-order master equation yields
the following set of integro-differential equations:
\begin{align}
\dot{\tilde{\rho}}_{11}(t)&=-\alpha^2\int\limits_0^t\Big(2\tilde{\rho}_{11}(s)-\tilde{\rho}_{22}(s)
-\tilde{\rho}_{33}(s)\Bigr)\cos[2\delta(t-s)]\ ds,\label{master1}\\
\dot{\tilde{\rho}}_{12}(t)&=-\alpha^2\int\limits_0^t\Big(2\tilde{\rho}_{12}(s)\mathrm
e^{2i\delta(t-s)}-\tilde{\rho}_{34}(s) \mathrm e^{2i\delta(t+s)}\Bigr)\ ds,\label{master2}\\
\dot{\tilde{\rho}}_{13}(t)&=-\alpha^2\int\limits_0^t\Big(2\tilde{\rho}_{13}(s)\mathrm
e^{2i\delta(t-s)}-\tilde{\rho}_{24}(s) \mathrm e^{2i\delta(t+s)}\Bigr)\ ds,\label{master3}\\
\dot{\tilde{\rho}}_{14}(t)&=-\alpha^2\int\limits_0^t
2\tilde{\rho}_{13}(s) \cos[2\delta(t-s)]\ ds,\label{master4}\\
\dot{\tilde{\rho}}_{22}(t)&=-\alpha^2\int\limits_0^t\Big(2\tilde{\rho}_{22}(s)-\tilde{\rho}_{11}(s)
-\tilde{\rho}_{44}(s)\Bigr)\cos[2\delta(t-s)]\ ds,\label{master5}
\end{align}
\begin{align}
\dot{\tilde{\rho}}_{23}(t)&=-\alpha^2\int\limits_0^t
2\tilde{\rho}_{23}(s) \cos[2\delta(t-s)]\ ds,\label{master6}\\
\dot{\tilde{\rho}}_{24}(t)&=-\alpha^2\int\limits_0^t\Big(2\tilde{\rho}_{24}(s)\mathrm
e^{2i\delta(s-t)}-\tilde{\rho}_{13}(s) \mathrm e^{-2i\delta(t+s)}\Bigr)\ ds,\label{master7}\\
\dot{\tilde{\rho}}_{33}(t)&=-\alpha^2\int\limits_0^t\Big(2\tilde{\rho}_{33}(s)-\tilde{\rho}_{11}(s)
-\tilde{\rho}_{44}(s)\Bigr)\cos[2\delta(t-s)]\ ds,\label{master8}\\
\dot{\tilde{\rho}}_{34}(t)&=-\alpha^2\int\limits_0^t\Big(2\tilde{\rho}_{34}(s)\mathrm
e^{2i\delta(s-t)}-\tilde{\rho}_{12}(s) \mathrm e^{-2i\delta(t+s)}\Bigr)\ ds,\label{master9}\\
\dot{\tilde{\rho}}_{44}(t)&=-\alpha^2\int\limits_0^t\Big(2\tilde{\rho}_{44}(s)-\tilde{\rho}_{22}(s)
-\tilde{\rho}_{33}(s)\Bigr)\cos[2\delta(t-s)]\ ds\label{master10}.
\end{align}
Some of the above equations can be solved under a time-local
approximation for which the matrix elements $\tilde{\rho}_{ij}(s)$
are replaced by $\tilde{\rho}_{ij}(t)$. One can find that ($\delta$
and $t$ given in units of $\alpha^{-1}$ and $\alpha$ respectively)
\begin{align}
\tilde{\rho}_{11}(t)&=\frac{1}{4}\Biggl\{1+\Bigl[-1+2(\rho_{11}^0+\rho_{44}^0)\Bigr]\exp\Bigl\{\frac{1}{\delta^2}[\cos(2
\delta t)-1]\Bigl\}\nonumber \\&\qquad +2(\rho_{11}^0-\rho_{44}^0)\exp\Bigl\{\frac{1}{2\delta^2}[\cos(2 \delta t)-1]\Bigr\}\Biggr\},\label{ms11} \\
\tilde{\rho}_{22}(t)&=\frac{1}{4}\Biggl\{1+\Bigl[-1+2(\rho_{22}^0+\rho_{33}^0)\Bigr]\exp\Bigl\{\frac{1}{\delta^2}[\cos(2
\delta t)-1]\Bigl\}\nonumber \\&\qquad +2(\rho_{22}^0-\rho_{33}^0)\exp\Bigl\{\frac{1}{2\delta^2}[\cos(2 \delta t)-1]\Bigr\}\Biggr\}, \\
\tilde{\rho}_{33}(t)&=\frac{1}{4}\Biggl\{1+\Bigl[-1+2(\rho_{33}^0+\rho_{22}^0)\Bigr]\exp\Bigl\{\frac{1}{\delta^2}[\cos(2
\delta t)-1]\Bigl\}\nonumber \\&\qquad
+2(\rho_{33}^0-\rho_{22}^0)\exp\Bigl\{\frac{1}{2\delta^2}[\cos(2
\delta t)-1]\Bigr\}\Biggr\},
\end{align}
\begin{align}
\tilde{\rho}_{44}(t)&=\frac{1}{4}\Biggl\{1+\Bigl[-1+2(\rho_{44}^0+\rho_{11}^0)\Bigr]\exp\Bigl\{\frac{1}{\delta^2}[\cos(2
\delta t)-1]\Bigl\}\nonumber \\&\qquad +2(\rho_{44}^0-\rho_{11}^0)\exp\Bigl\{\frac{1}{2\delta^2}[\cos(2 \delta t)-1]\Bigr\}\Biggr\}, \\
\tilde{\rho}_{14}(t)&=\rho_{14}^0\exp\Bigl\{\frac{1}{\delta^2}[\cos(2
\delta t)-1]\Bigr\},\\
\tilde{\rho}_{23}(t)&=\rho_{23}^0\exp\Bigl\{\frac{1}{\delta^2}[\cos(2
\delta t)-1]\Bigr\}\label{ms23}.
\end{align}
These solutions describe approximately the dynamics at short times.
In fact, the smaller the coupling constant $\delta$, the better
these solutions are.
\begin{figure}[htba]
{\centering
\resizebox*{0.65\textwidth}{!}{\includegraphics{fig6.eps}}
\par}
\caption{\label{figure5} The variation in time of the the matrix
element $\rho_{11}(t)$ corresponding to the singlet state. The solid
curve represents the exact solution, and the dashed curve represents
the approximate solution~(\ref{ms11}). The parameters are $N=10$ and
$\delta=\alpha$.}
\end{figure}
Note that when $\delta=0$ ( i.e. nonlocal dynamics), then
\begin{equation}
\exp\{\frac{1}{n\delta^2}[\cos(2 \delta t)-1]\Bigr\}\to \mathrm
e^{-2 t^2/n}, \qquad n=1,2. \end{equation} Thus the second
order time-local master equation shows that the nonlocal dynamics,
or, in general, the short time behavior follow a Gaussian decay law.
Note that the solutions corresponding to the diagonal elements
reproduce their asymptotic limit, namely,
$\rho_{ii}(\infty)=\frac{1}{4}$. However, those corresponding to the off-diagonal elements fail to reproduce
the steady state, since, for
example, equation~(\ref{ms23}) implies that $\rho_{23}(t)\to 0$. To
end our discussion let us remark that
equations~(\ref{master2}), (\ref{master3}), (\ref{master7}) and
(\ref{master9}) can be analytically solved only when $\delta=0$.
For instance (see figure~\ref{figure7}),
\begin{align}
\rho_{12}(t)=\frac{1}{2}\Bigl[(\rho_{12}^0+\rho_{34}^0)\mathrm
e^{-t^2/2}+(\rho_{12}^0-\rho_{34}^0)\mathrm
e^{-3t^2/2}\Bigr].\label{ms12}
\end{align}
\begin{figure}[htba]
{\centering
\resizebox*{0.65\textwidth}{!}{\includegraphics{fig7.eps}}
\par}
\caption{\label{figure7} The variation in time of the the matrix
element $\rho_{12}(t)$ corresponding to the singlet state. The solid
curve represents the exact solution, and the dashed curve represents
the approximate solution~(\ref{ms12}). The parameters are $N=10$ and
$\delta=0$.}
\end{figure}
\section{Summary}
In summary we have investigated the dynamics of two qubits coupled
to separate spin star environment via Heisenberg $XY$
interactions. We have derived the exact form of the time evolution
operator and calculated the matrix elements of the reduced density
operator. The analysis of the evolution in time of the concurrence
and the purity shows that decoherence can be minimized by allowing
the central qubits to strongly interact with each other. The
short-time behavior, studied by deriving the second-order master
equation, is found to be Gaussian.
The next step may consist in considering more central qubits, and
investigate whether the above results still hold.
\section*{Appendix
Using trace properties of the lowering and raising operators, it can
be shown that the nonzero matrix elements corresponding to the
initial maximally entangled states
$\frac{1}{\sqrt{2}}(|-+\rangle\pm|+-\rangle)$ are explicitly given
by:
\begin{eqnarray}
\rho_{11}(t)&=&2^{-(2N+1)}\mathrm{tr}_{B_1+B_2}\Bigl\{U_{12}(t)U^\dag_{12}(t)+ U_{13}(t)U_{13}^\dag(t)\Bigl\},\\
\rho_{22}(t)&=&2^{-(2N+1)}\mathrm{tr}_{B_1+B_2}\Bigl\{U_{22}(t)U_{22}^\dag(t)+ U_{23}(t)U_{23}^\dag(t)\Bigl\},\\
\rho_{23}(t)&=&\pm 2^{-(2N+1)}\mathrm{tr}_{B_1+B_2}\Bigl\{U_{22}(t)U_{33}^\dag(t)\Bigl\},\\
\rho_{33}(t)&=&2^{-(2N+1)}\mathrm{tr}_{B_1+B_2}\Bigl\{U_{32}(t)U_{32}^\dag(t)+ U_{33}(t)U_{33}^\dag(t)\Bigl\},\\
\rho_{44}(t)&=&2^{-(2N+1)}\mathrm{tr}_{B_1+B_2}\Bigl\{U_{42}(t)U_{42}^\dag(t)+
U_{43}(t)U_{43}^\dag(t)\Bigl\}.
\end{eqnarray}
Those associated with the initial state
$\frac{1}{\sqrt{2}}(|--\rangle\pm|++\rangle)$ read:
\begin{eqnarray}
\rho_{11}(t)&=&2^{-(2N+1)}\mathrm{tr}_{B_1+B_2}\Bigl\{U_{11}(t)U_{11}^\dag(t)+ U_{14}(t)U_{14}^\dag(t)\Bigl\},\\
\rho_{22}(t)&=&2^{-(2N+1)}\mathrm{tr}_{B_1+B_2}\Bigl\{U_{21}(t)U_{21}^\dag(t)+ U_{24}(t)U_{24}^\dag(t)\Bigl\},\\
\rho_{14}(t)&=&\pm2^{-(2N+1)}\mathrm{tr}_{B_1+B_2}\Bigl\{U_{11}(t)U_{44}^\dag(t)\Bigl\},\\
\rho_{33}(t)&=&2^{-(2N+1)}\mathrm{tr}_{B_1+B_2}\Bigl\{U_{31}(t)U_{31}^\dag(t)+ U_{34}(t)U_{34}^\dag(t)\Bigl\},\\
\rho_{44}(t)&=&2^{-(2N+1)}\mathrm{tr}_{B_1+B_2}\Bigl\{U_{41}(t)U_{41}^\dag(t)+
U_{44}(t)U_{44}^\dag(t)\Bigl\}.
\end{eqnarray}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 6,599 |
class CreateRecipients < ActiveRecord::Migration
def change
create_table :recipients do |t|
t.integer :giver_id
t.string :name
t.integer :age
t.string :gender
t.string :relationship
t.timestamps
end
end
end
| {
"redpajama_set_name": "RedPajamaGithub"
} | 2,129 |
The 6th Regiment, Canadian Mounted Rifles, CEF, was a mounted infantry unit of the Canadian Expeditionary Force in the First World War.
History
The regiment was formed on March 15, 1915, at Amherst, Nova Scotia . It recruited in Nova Scotia, New Brunswick, and Prince Edward Island. It sailed to England in July 1915, and after training arrived in France on October 22, 1915. It served in the field as infantry until December 1915.
On January 1, 1916, the six regiments of Canadian Mounted Rifles were converted to infantry and reorganized into the four battalions of the 8th Canadian Infantry Brigade. The personnel of the 6th Regiment were absorbed into the 4th and 5th Battalions, CMR.
Perpetuation
The 6th Regiment was perpetuated by the King's Canadian Hussars, which was converted to an artillery unit in 1939, and from 1960 by the 8th Canadian Hussars (Princess Louise's).
Battle honours
France and Flanders, 1915
See also
List of mounted regiments in the Canadian Expeditionary Force
References
Mounted Regiments of the Canadian Expeditionary Force
Mounted rifle regiments of Canada
Military units and formations disestablished in 1916
8th Canadian Hussars (Princess Louise's) | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 1,111 |
Looking for that "ahhhh" feeling?
Tired of cleaning up the dog hair?
Want to keep the home clean for the new baby?
Just can't seem to get past surface cleaning to the deep stuff?
Need a dependable maid who does the job right?
We make the entire process of having your home cleaned effortless! | {
"redpajama_set_name": "RedPajamaC4"
} | 1,981 |
Russia & Coronavirus: Business Updates
March 1, 2021 Posted by Russia Briefing
Россия и Коронавирус: Бизнес Обновления
Last updated March 1.
Глобальные вспышки вируса Covid-19: координация ваших бизнес-операций из дома. Читать здесь
Current Russia Coronavirus Updates
Russia confirmed 11,571 new coronavirus cases and 333 deaths today overnight from Sunday. Around 4 million Russians have been vaccinated against the coronavirus so far, according to Russia's Health Ministry. The country has confirmed 4,257,650 cases of coronavirus and 86,455 deaths.
Russia confirmed 12,604 new coronavirus cases and 337 deaths overnight from Sunday, bringing the total number of cases to 4,177,330 with 83,630 deaths.
Russia confirmed 13,233 new coronavirus cases and 459 deaths overnight from Monday, making a total of 4,099,323 cases of coronavirus and 80,979 deaths.
Russia confirmed 15,089 new coronavirus cases and 507 deaths overnight, with a total of 4,042,837 cases of coronavirus and 79,194 deaths since the pandemic began. The overall trend of infections continues to show a sharp decline.
Russia confirmed 15,916 new coronavirus cases and 407 deaths overnight from Sunday, with a total of 3,983,197 cases of coronavirus and 77,068 deaths.
Russia confirmed 16,688 new coronavirus cases and 527 deaths overnight, bringing a total of Russia has confirmed 3,934,606 cases of coronavirus and 75,732 deaths. The overall infection trend has been de-creasing since the beginning of the year. Russia will resume flights with Armenia and Azerbaijan starting February 15, while rail connections to and from Minsk will resume from February 8.
Russia confirmed 17,648 new coronavirus cases and 437 deaths overnight from Sunday, with a total of 3,868,087 confirmed cases of coronavirus and 73,619 deaths.
Cases have been trending down since the beginning of the year.
Russia confirmed 19,238 new coronavirus cases and 534 deaths overnight from Thursday, meaning there has been a total of 3,813,048 cases of coronavirus and 72,185 deaths.
All Russian universities will be allowed to resume in-person classes starting February 8, according to a decree issued by Russia's Ministry of Science and Higher Education today.Russia has resumed flights with Finland, India, Qatar and Vietnam from January 27 and will resume flights with Greece and Singapore from February 8. Russian authorities on Tuesday said mass production of its second coronavirus vaccine would begin next month and that a third homemade jab is currently in the process of undergoing registration approvals.
Russia confirmed 18,241 new coronavirus cases and 564 deaths overnight, for a pandemic total of 3,756,931 cases of coronavirus and 70,482 deaths. Hungary, Malaysia, Mexico and the UAE have all placed orders for Russia's Sputnik V vaccine.
Russia confirmed 21,887 new coronavirus cases and 612 deaths overnight, and now has confirmed 3,655,839 cases of coronavirus and 67,832 deaths.
Russia confirmed 22,857 new coronavirus cases and 471 deaths overnight from Sunday, resulting in a Russian total of 3,591,066 cases of coronavirus and 66,037 deaths.Russia's republic of Bashkortostan, between the Volga River and Ural Mountains, is set to become the first region to introduce immunity passports for residents with Covid-19 antibodies starting February, Bashkortostan Governor Radiy Khabirov has stated.
Russia confirmed 24,715 new coronavirus cases and 555 deaths overnight, bringing a total confirmed 3,520,531 cases of coronavirus and 64,495 deaths.
Russia confirmed 23,315 new coronavirus cases and 436 deaths overnight from Sunday, providing a total of 3,425,269 cases of coronavirus and 62,273 deaths.
Russia confirmed 23,541 new coronavirus cases and 506 deaths overnight, bringing the total to 3,332,142 cases of coronavirus and 60,457 deaths.
Russia confirmed 23,351 new coronavirus cases and 482 deaths overnight from Sunday. Russia has now confirmed 3,260,138 cases of coronavirus and 58,988 deaths.
Russia confirmed 27,747 new coronavirus cases and 593 deaths overnight from Wednesday. In total, Russia has confirmed 3,159,297 cases of coronavirus and 57,019 deaths.
Russia confirmed 27,787 new coronavirus cases and 487 deaths overnight from Sunday, and now has a total of 3,078,035 cases of coronavirus and 55,265 deaths.
Russia confirmed 28,776 new coronavirus cases and 561 deaths overnight from Tuesday, bringing the total number of reported cases to 2,906,503 cases of coronavirus and 51,912 deaths.
Russia confirmed 28,552 new coronavirus cases and 611 deaths overnight from Thursday, bringing a total of 2,791,220 cases of coronavirus and 49,762 deaths. Moscow residents working in the industrial sector and transportation as well as journalists will be eligible to receive Russia's Sputnik V shot starting this coming Monday.
Russia confirmed 26,509 new coronavirus cases and 596 deaths overnight from Wednesday, and has recorded a total of 2,734,454 cases of coronavirus and 48,564 deaths.
Russia confirmed 27,328 new coronavirus cases and 450 deaths overnight from Monday. This brings Russia's total to 2,681,256 cases of coronavirus and 47,391 deaths.
Moscow municipal workers as well as residents working in the culture and service sectors have been receiving Russia's Sputnik V vaccine since yesterday.
Russia confirmed 27,927 coronavirus cases and 562 deaths overnight from Thursday, and now has a total of 2,569,126 cases of coronavirus and 45,280 deaths.
Russia has confirmed 26,190 coronavirus cases and 559 deaths overnight in the past 24 hours, bringing the total number of cases in the country since the pandemic began to 2,541,199 cases of coronavirus and 44,718 deaths. Recipients of Russia's Sputnik V coronavirus vaccine should abstain from alcohol for three days after each of the required two injections, according to the head of the Gamaleya Research Center for Epidemiology and Microbiology.
Russia confirmed 28,142 coronavirus cases and 456 deaths overnight from yesterday (Sunday December 6). This brings the total number of cases in Russia to 2,488,912 cases of coronavirus with 43,597 deaths. The maximum price of Russia's Sputnik V coronavirus vaccine will be set at 1,942 rubles (US$26), the state-run Interfax news agency reported on Saturday. Russian citizens will be able to receive the homegrown vaccine for free.
Russia confirmed 27,403 coronavirus cases and 569 deaths overnight from Thursday (yesterday). Moscow will start vaccinating at-risk groups against the coronavirus this weekend, Mayor Sergei Sobyanin said yesterday. Online booking for health workers, teachers and social service workers deemed at higher risk of severe infection has opened today.
The total number of Covid-19 cases in Russia is 2,402,949, resulting in 42,176 deaths.
Russia confirmed 26,338 new coronavirus cases and 368 deaths overnight to today (Tuesday), and now has a total confirmed 2,295,654 cases of coronavirus and 39,895 deaths. Universities in Moscow and St. Petersburg will switch students to remote learning until early February to prevent the spread of coronavirus.
Russia has confirmed a record-breaking new daily infection rate of 25,487 coronavirus cases and 524 deaths overnight from Thursday (November 26). Moscow has further extended measures aimed at slowing the spread of the coronavirus, and these will be in place during the Orthodox Christmas and New Year period until January 15. These include a self-isolation order for all residents aged 65 and older and the requirement for 30% of all businesses' employees to work remotely. Russia currently has a total of 2,187,990 cases of coronavirus and 38,062 deaths.
Russia confirmed 23,675 Covid-19 cases and a new record of 507 deaths overnight (Wednesday), and now has a total number of 2,162,503 of Covid-19 cases, and recorded 37,538 deaths from the disease.
Russia confirmed a new daily high of 24,318 new coronavirus cases and 461 deaths in the past 24 hours. In total, Russia has confirmed 2,039,926 cases of coronavirus and 35,311 deaths.
Russia confirmed a new one-day record of 21,983 coronavirus cases overnight from yesterday (Thursday November 12). It now has a total of 1,880,551 cases of coronavirus and recorded 32,443 deaths from the virus.
Russia confirmed 19,851 new coronavirus cases yesterday (Tuesday), ending a five-day streak of new cases surpassing 20,000. To date Russia has confirmed 1,836,960 cases of coronavirus and 31,593 deaths. Moscow will order all bars, restaurants, clubs and other nighttime establishments to close between 11 p.m. and 6 a.m. in order to slow the spread of coronavirus. The order will be valid from November 13 through until January 15, 2021, Mayor Sergei Sobyanin stated. Students at city universities and colleges in Moscow will be switched to remote learning, Sobyanin said.
Russia has confirmed 18,257 new coronavirus cases overnight, and 238 additional deaths. The national total since the pandemic began is now 1,655,038 cases and 28,473 deaths.
Russia announced 18,283 Covid-19 cases today (Friday),bringing its official number of cases to 1,599,976 and setting a new one-day record for new daily infections. In the past 24 hours, 355 people have died.
Russia has confirmed another 16,550 Covid-19 cases on Tuesday, bringing its official number of cases to 1,547,774. A record 320 people have died in the past 24 hours. A nationwide mask-wearing mandate starts from tomorrow (Wednesday October 28).
Russia has confirmed 17,340 new Covid-19 cases in the past 24 hours to Friday 23rd October, bringing its official number of cases to 1,480,646 and setting a new record for one-day infections. A total of 283 people have died in the past 24 hours from a national total of 25,525 since the pandemic began.A device that tests for the coronavirus based on the sound of a cough could appear at Russian airports in the next few months, according to Kremlin spokesman Dmitry Peskov.Senior Russian healthcare professionals have stated that they expect Russia's number of daily Covid-19 infections to peak at 20,000 before stabilizing within the next two weeks.
Russia reported 15,099 new Covid-19 cases yesterday (Sunday), bringing its official number of cases to 1,399,334. This is a rise of over three times the daily new infections of 4,828 new cases being reported two months ago.
Russia confirmed 13,592 new Covid-19 cases today (Monday), bringing its official number of cases to 1,312,310.
Russia has confirmed 11,615 new Covid-19 cases Tuesday, bringing the country's official number of cases to 1,237,504. Tuesday's increase is just 41 fewer than the country's highest daily total of 11,656 recorded on May 11. Moscow reopened two temporary hospitals for coronavirus patients yesterday.
Russia announced 9,412 new Covid-19 cases today (Friday), bringing the country's official number of cases to 1,194,643 as the number of new infections across the country continues to rise.
Russia has confirmed 1,176,286 cases of coronavirus and 20,722 deaths. 8,481 new Covid-19 cases were reported in the past 24 hours as strong signs of a second wave increase.
Moscow has asked resident businesses to reinstate work-from-home measures and for elderly residents to avoid going outside from next Monday as the city sees a new uptick in coronavirus cases, Mayor Sergei Sobyanin said today. Moscow, with a population of 12.7 million is one of the largest cities in Europe, and confirmed more than 1,000 Covid-19 cases for the first time in three months yesterday. Hospitalizations in the Russian capital have gone up by 30% in the past week, according to officials.
Overall, as at today, Russia has confirmed 1,128,836 cases of coronavirus and 19,948 deaths, and confirmed 6,595 new Covid-19 cases yesterday.
Russia has confirmed 1,122,241 cases of coronavirus and 19,79 deaths, with 6,431 new Covid-19 cases reported overnight to Wednesday (12th). The number of new infections across the country continues to rise.
Russia has confirmed 1,103,399 cases of coronavirus and 19,418 deaths. There were 6,148 new Covid-19 cases reported Sunday (20th).
Russia has confirmed 1,041,007 cases of coronavirus and 18,135 deaths, with 5,218 new coronavirus infections on Wednesday.
Russia's confirmed 5,099 new coronavirus infections on Tuesday, bringing the country's official number of cases to 1,035,789. This is the fifth consecutive day that more than 5,000 new infections have been recorded.
Russia has confirmed 1,025,505 cases of coronavirus and 17,820 deaths, with a tick up of 5,195 new coronavirus infections overnight on Sunday.
Russia has confirmed 1,005,000 cases of coronavirus and 17,414 deaths. The country confirmed another 4,952 new coronavirus infections on Wednesday, an uptick on recent infection rates sparking fears a second wave may be emerging.
Russia has confirmed 975,576 cases of Covid-19 and 16,804 deaths. There were 4,711 new coronavirus infections in the past 24 hours. This means Russia is expected to reach 1 million infections sometime next week. A second wave is also presumed likely, with parents reluctant to purchase new school clothes for the new academic year (1st September) as many feel children will be locked down again from home schooling by October.
Russia has confirmed 961,493 cases of coronavirus and 16,448 deaths. The country confirmed 4,744 new coronavirus infections Monday, bringing the country's official number of cases to 961,493. However we are hearing unconfirmed yet credible reports from Moscow that patients testing positive for Covid-19 are then being reassessed and being noted as negative.
Russia has confirmed 937,321 cases of coronavirus and 15,989 deaths. 4,828 new coronavirus infections were announced over the 24 hours from Wednesday to Thursday (today).
Russia today has confirmed 932,493 cases of coronavirus and 15,872 deaths. There were 4,748 new coronavirus infections in the past 24 hours.
Russia has confirmed 912,823 cases of coronavirus and 15,498 deaths. Russia had 5,065 new coronavirus infections in the 24 hours to Friday 14.
Russia has confirmed 902,701 cases of coronavirus and 15,260 deaths, and confirmed another 5,102 new coronavirus infections within the past 24 hours.
Russia has confirmed 897,599 cases of coronavirus and 15,131 deaths. The country confirmed 4,945 new coronavirus infections in the 24 hours to Tuesday. Russia has registered the world's first coronavirus vaccine today.
Russia has confirmed 877,135 cases of coronavirus and 14,725 deaths. The country confirmed 5,241 new coronavirus infections overnight to Friday.
Russia has confirmed 871,894 cases of coronavirus and 14,606 deaths. The country confirmed 5,267 new coronavirus infections in the 24 hours from Wednesday (yesterday) morning.
Russia has confirmed 823,515 cases of coronavirus and 13,504 deaths, confirming another 5,395 new coronavirus infections on Tuesday. The European Union won't open its borders to Russian citizens until at least mid-August, according to an EU member states' delegation. The EU is revising its "white list" of permitted travelers every two weeks. International flights to Britain, Turkey and Tanzania commence from August 1st.
Russia has confirmed 818,120 cases of coronavirus and 13,354 deaths. New daily infections are running at about the 5,500 mark, with very small signs of a downwards trend.
Russia confirmed 5,940 new coronavirus infections overnight, bringing the country's official number of cases to 777,486. Daily infection rates are slowly decreasing.
Russia has confirmed 739,947 cases of coronavirus and 11,614 deaths. New daily infection rates are slowly decreasing and amounted to 6,200 overnight. Russian immigration has announced that there will be checks on arriving international passengers when borders reopen, those with no symptoms will not face self-isolation and will be free to move around.The country is considering the resumption of international flights
Russia has confirmed 727,162 cases of coronavirus and 11,335 deaths as at Monday July 13. New daily infections continue to be around the 6,500 mark and have been the past three weeks.
Russia has confirmed 700,792 cases of coronavirus and 10,667 deaths. Daily new infections have continued to be about the 6,500 mark for the past two weeks with no sign of these declining at yet.
Russia has confirmed 681,251 cases of coronavirus and 10,161 deaths. New daily infections remain about the 6,500 level.
Russia has reported 641,156 cases of coronavirus infection reported as at June 30th and 9,166 deaths. Current daily infection rates are averaging about 6,700.
There have been 641,156 cases of coronavirus infection reported in Russia so far and 9,166 deaths. Daily new infection rates remain around 6,500 to 7,000.
There have been 606,881 cases of coronavirus infection reported in Russia so far and 8,513 deaths. New daily infection rates, which have reached as high as 12,000 are now at slightly below the 7,500 mark for the first time since April.
Russia confirmed 7,728 new coronavirus infections on Saturday, bringing the country's official number of cases to 584,680. It is the third most affected country globally after the United States and Brazil.
There have been 553,301 cases of coronavirus infection reported in Russia so far and 7,478 deaths.
There have been 502,436 cases of coronavirus infection reported in Russia so far and 6,532 deaths. There were 8,779 new coronavirus infections Thursday, meaning the virus is still spreading.
Meanwhile, Moscow has ended its self-isolation orders — including its digital pass system and schedule system for taking walks outside. The city also released a schedule of which restrictions will be lifted over the next few weeks.
Almost 60% more people in Moscow died last month (May) than the city's average toll for the past three years, the city health department stated as questions continue to swirl around Russia's low coronavirus death figures and the continuing high daily infection rate.
There have been 493,657 cases of coronavirus infection reported in Russia so far and 6,358 deaths. Restrictions on staff going to work have been lifted.
Russia confirmed 8,985 new coronavirus infections today (Monday), bringing the country's official number of cases to 476,658.
The Mayor of Moscow, Sergei Sobyanin has stated that the city is expected to lift restrictions imposed during the Covid-19 outbreak from July 1st. Russia has reported 441,108 cases and 5,384 deaths.
There have been 414,878 cases of coronavirus infections reported in Russia so far and 4,855 deaths.
There have been 405,843 cases of coronavirus infections reported in Russia so far and 4,693 deaths. Moscow is starting to reopen from today, with all non-food shops and some service sector businesses available for business. The city will also test lifting restrictions on walks outside using a schedule system for apartment buildings and all parks except for Zaryadye Park will re-open.
There have been 362,342 cases of coronavirus infections reported in Russia so far and 3,807 deaths. Media reports suggest the rate of infections is now slowing, although Russian President Putin has warned of a potential resurgence in the Autumn.
Russia's Coronavirus infection and death rates are starting to flatten out but still remain serious. There have been 335,882 cases of coronavirus infections reported in Russia as at May 25 and 3,388 deaths.
There are now 252,245 cases of coronavirus infections reported in Russia, and 2,305 deaths. We are hearing unofficial comments from reputable sources within Russia that the continuing outbreak in the country is expected to last until the Autumn.
Russia confirmed 10,899 new coronavirus infections on May 12th, bringing the country's official number of cases to 232,243. Russia is now the third most-affected country in terms of infections and has the world's second-fastest rate of new infections behind the United States.
Despite reporting more than 10,000 new cases a day for over a week, President Vladimir Putin on Monday announced that a "non-working" period in place for six weeks would be lifted from Tuesday. A lockdown in Moscow, the epicenter of the crisis in Russia, remains in place until the end of May, however some restrictions are being lifted.
Some 500,000 employees of companies involved in industry and construction are now permitted to return to work, though authorities have now made it mandatory to wear masks and gloves in shops and on public transport.
There is some good news: Russia's reported mortality rate is much lower compared to other European countries hit hard by the pandemic, with 107 new deaths and 2,116 dead from the coronavirus as of Tuesday. Authorities say this is because Russia moved quickly to close its borders and isolate those at risk, convert hospitals to treating virus patients and launch a vast campaign to test and quarantine those infected.
Russia confirmed 11,231 new coronavirus infections Thursday, bringing the country's official number of cases to 177,160 and marking a new one-day record rise in infections. Russia is now the fifth most-affected country in terms of infections, surpassing Germany and France on Thursday.
Russian Covid-19 cases surged by 10,559 over the past 24 hours, bringing the national count to 165,929, the country's coronavirus crisis response centre has announced.
It is the fourth consecutive day that cases have risen by more than 10,000. The crisis response centre also reported 86 new coronavirus deaths, meaning the death toll in Russia has reached 1,537.
Coronavirus cases in Russia continue to rise, with 99,399 cases of coronavirus infections reported in Russia so far and 972 deaths. Just over 50% of all cases are in Moscow. The country has 'indefinately' extended a ban on foreign nationals entering Russia.
There have been 93,558 cases of coronavirus infections reported in Russia so far and 867 deaths. President Vladimir Putin has extended the national "non-working" month through until May 11, warning that Russia has yet to see the peak of its coronavirus outbreak.
There have been 87,147 cases of coronavirus infections reported in Russia so far and 794 deaths. Russia has now surpassed China to become the world's ninth most-affected country amid the pandemic. Meanwhile, the head of Russia's consumer health watchdog has stated that self-isolation should be in force until at least the middle of May.
There are now 62,773 cases of coronavirus infections reported in Russia, and 555 deaths. 21 Russian regions have requested permission to produce digital travel passes a week after coronavirus-hit Moscow enacted its system to enforce lockdown measures and slow the deadly outbreak, the Communications and Press Ministry has stated. These are issued to individuals who show no signs of infection and who have been verified as virus free after passing the appropriate medical tests.
There are now 62,773 cases of coronavirus infections reported in Russia, and 555 deaths.
There have been 52,763 cases of coronavirus infections reported in Russia so far. This updated map from the Moscow Times illustrates the spread.
Moscow is in complete lockdown, citizens must obtain a digital ID pass to be seen on the streets.
There have been 47,121 cases of coronavirus infections reported in Russia as at April 20th, and 405 deaths. Aeroflot, the Russian airline, has suspended all international flight ticket bookings until August 1st, while the May 9th annual parade to celebrate the WWII victory of Nazism – to which US President Donald Trump had been invited – has been cancelled.
Over 2 million Russians have now been screened for the virus.
There have been 24,490 cases of coronavirus infections reported in Russia so far and 198 deaths.
Russia now has 18,328 cases of Covid-19 and 148 deaths. Additional lock down measures are in place until April 19 with all nonessential business and activity suspended. Grocery stores and pharmacies will stay open, and only the government, hospitals, protective gear manufacturers and the defense sector will continue to work. All construction and maintenance work is suspended, as are car-sharing services. Moscow will introduce a digital pass system to allow residents to leave their homes this week. Traffic police have also been deployed at all city entry points to control movement and check all drivers reasons for entering the Russian capital during the lock down.
There are now 10,131 cases of coronavirus infections reported in Russia to date and 76 deaths. Russia is expected to reach the peak of its coronavirus outbreak in 10-14 days. Veronika Skvortsova, the head of the Federal Biomedical Agency has predicted that the country's number of infections will start to fall in early to mid-June.
There have been 7,497 cases of coronavirus infections reported in Russia so far and 58 deaths. The country confirmed 1,154 new coronavirus infections on Tuesday, bringing the country's official number of cases up to 7,497 and marking a new record one-day increase in infections. President Vladimir Putin has announced an extension of the nationwide "non-working week" until April 30 after the country registered a sharp increase in coronavirus cases on Thursday. Speaking in a televised address, he added that he would delegate the decisionmaking power on anti-coronavirus measures to regional authorities given the regional differences in infection rates.
Russia has 771 new coronavirus infections on Thursday morning, bringing the country's official number of cases up to 3,548 and marking a record one-day increase in infections. Thirty people in Russia have been killed by the virus.
Cases have risen to 2,777, with 24 fatalities. Moscow authorities have developed a QR code system to allow residents to leave their homes as well as a smartphone app to monitor coronavirus patients' movement in self-isolation, says the city's IT chief.
There have been 2,337 cases of coronavirus infections reported in Russia so far and 17 deaths.
Cases in Russia have now risen to 1,836.
Cases in Russia have risen to 1,534, with over 1,000 of these in Moscow. The capital is now in lockdown and the entire country has now closed all borders. Other cities are also considering a complete lockdown for residents, including Murmansk. The Russian Church has also advised worshippers to stay at home.
Russia has confirmed 196 new Covid-19 infections on Thursday, bringing the country's official number of cases up to 1,036. The total number of deaths has risen to four.
Sberbank and VTB with backing from the Central Bank will launch a pilot business loan program that offers six-month, 0% interest loans to businesses to help them pay employee salaries during the coronavirus crisis.
Cases in Russia have risen to 840. The country has now suspended all international flights and ordered the closure of all shops except for food and medical supplies.
The number of cases in Russia has nearly tripled overnight to Wednesday 25th with a huge increase to 658. This is because Moscow changed the way it counts its coronavirus cases, with patients considered coronavirus-positive now upon a single positive test rather than having samples sent to a lab in Novosibirsk for further verification.
Russia cases have risen to 495. The majority of these have occurred from Russians flying in from Europe. The country has now suspended flights incoming from most EU nations.
Russia confirmed 71 new coronavirus infections today (Monday), bringing the country's official number of cases up to 438.
Russian cases leap to 367 by Sunday.
Cases in Russia now rise to 199. The Russian government has announced a "high alert" status for all 85 of its regions, requiring the entire country to take anti-coronavirus measures including banning large gatherings, moving schools to online classes and encouraging working from home.
First Covid-19 death reported, an elderly lady in Moscow with serious underlying existing illnesses.
A map of the current situation in Russia can be seen below, courtesy of the Moscow Times.
The number of infections in Russia has seen a sizable increase to 147. Of these, 98 are in Moscow, 10 in St.Petersburg and the remainder single cases across the country.
Cases rise in 114.
Covid-19 cases rise to 93. Foreign nationals are barred from entering Russia until 1st May excepting diplomats and special cases – please check on latest restrictions before planning any travel. All major sporting events have been cancelled.
Russia has now closed all its borders excepting those with Azerbaijan and Finland.
Russia has closed its border with Belarus.
63 cases now confirmed with additional incidents in Kemorovo.
Russian Railways announced it would stop international passenger trains from Moscow to Berlin and Paris, in addition to suspending train connections to and from Ukraine, Moldova and Latvia.
Russian cases rise to 59, closes its borders with Poland and Norway.
Russian cases reported now up to 34. There are 18 cases in Moscow, with others in St. Petersburg, Kaliningrad, Belgorod, Lipetsk, Nizhny Novgorod, Tyumen, Krasnodar and Chita
About 100 Chinese students in Moscow will be deported from Russia because they violated their self-quarantine orders.
Russia infections rise to 28, all had recently visited Italy.Russia will suspend most flights to and from Italy, Germany, France and Spain over the coronavirus outbreak, commencing from Friday 13th March, Russia's coronavirus crisis center has said. Russia will also stop issuing tourist visas to Italian citizens to prevent further spreading of the virus until further notice.
1,000 people placed under quarantine after tracking movements of a returning, infected Russian from Italy. Moscow to build and have ready a US$135 million Covid-19 specific hospital on edge of Moscow.
Moscow has banned large events of more than 5,000 people for the next month, until April 10th in a move to prevent the spread of the virus, Mayor Sergei Sobyanin said in a decree. The ban applies to "sports, entertainment, public and other mass events."
20 cases confirmed, latest all Russian citizens who recently returned from Italy. Moscow plans for city shut down "if required", St. Petersburg students quarantined in dormitories.
The annual St.Petersburg International Economic Forum (SPIEF) due to be held in June has been postponed until 2021 as a result of the Covid-19 outbreak, the organisers Roscongress have confirmed.
The Moscow City Mayor has issued a "High Alert" and stated that individuals, and employers employing individuals who have entered Russia from China, South Korea, Italy, Iran, Germany & Spain must inform the authorities of their whereabouts and self quarantine for 14 days. It is understood that the UK, Switzerland, Norway and United States are about to be included on this list. Russian airlines who have sold tickets on these routes will offer full refunds for any cancellations.
The Moscow hotline for assistance with cases of suspected Covid-19 is +7 495 870 4509.
An imported Coronvirus case has been reported, a returning Russian national in Moscow who had visited Italy.
An interesting article about Facemask production and distribution in St.Petersburg from Fontanka here.
Second Chinese patient recovers and is discharged from hospital in Chita. No additional cases reported.
Seoul reports that the North Korean city of Sinuiju, close to the border with China has multiple deaths, and that infections in North Korea could escalate. The North Korean border with Russia has been closed.
China Factory re-opening schedule, map of national dates here. Russian businesses needing updates or assistance on the situation in China may email us at russia@dezshira.com
Many businesses in Asia are running out of supplies from China due to the problems. Russian businesses with Chinese suppliers need to know when China factories will return to work. Some factories return next Monday, 9th February, others are still in lock down. If you need clarifications on the factory opening situation in China, email us at russia@dezshira.com and we will liaise with our China colleagues to obtain answers.
Sourcing and supply manufacturers in China may not be able to meet delivery schedules to Russia. This could mean Force Majeure clauses need to be implemented. Our article Coronavirus In China: Applicability Of Force Majeure provides advise and protocols to follow.
Russia working with China to develop vaccine, screening of passengers stepped up at airports.
Sochi Investment Forum, due to be held in March, has been postponed.
Country to deport all foreign nationals found with Coronavirus.
Russia closes complete border with China, suspends all rail services except direct Moscow-Beijing Trans-Siberian route.
Two patients are coronavirus positive in the Baikal and Tyumen regions.
Russia has barred Chinese tour groups from entering the country.
Far East Russia, Mongolia, and North Korea have closed their borders with China.
Russia Briefing is written by Dezan Shira & Associates. The firm has 28 offices throughout Eurasia, including China, Russia, India, and the ASEAN nations, assisting foreign investors into the Eurasian region. Through our membership of the Leading Edge Alliance, we also have partner firms throughout Africa and have numerous Russian and African clients. For enquiries please contact us at russia@dezshira.com or visit us at www.dezshira.com
« Russia's Export To Asia Opportunity
Opora Rossii Recommends Huge Tax Discounts For Russian SME's » | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 4,428 |
\section{\NoCaseChange{Doublon-holon condensate on three-colorable lattices}}
We consider a half-filled fermionic Hubbard model under external driving, focusing on the triangular, the Kagome, and the Bethe lattice, see Fig.~\ref{diagram}.
All these lattices are 3-colorable (we use the color labels $R$, $G$, $B$) and consist of connected $RGB$ chains or loops, i.e., triangular motifs.
The Hamiltonian reads
\begin{align}
H=-t_0\sum_{\langle ij\rangle\sigma}e^{i\varphi_{ij}}c^\dag_{i\sigma}c_{j\sigma}+U\sum_{i} n_{i\uparrow}n_{i\downarrow} + g H_{\rm dr},
\label{model}
\end{align}
where $c_{i\sigma}$ is the electron annihilation operator at site $i$ with spin $\sigma$, $\langle ij\rangle$ denotes nearest-neighbor pairs, and $t_0$ and $U>0$ are hopping and interaction parameters. The Peierls phases $\varphi_{ij}$ represent an artificial gauge field implemented through a periodic (Floquet) modulation \cite{struck2012,hauke2012}. As will be demonstrated below, the gauge field can be used to tune the superconducting (SC) condensate's stability, but the twisted condensate also exists for $\varphi=0$ on the triangular and Bethe lattices.
To realize a long-lived photodoped state, we assume that a driving term $gH_{\rm dr}$ with an overall amplitude $g$ generates a nonthermal population of doublons and holons. One example of
$H_{\rm dr}$ is a resonant optical excitation between the Hubbard bands, as widely adopted in experiments. Another example is the coupling of the system to two separate fermion baths (electrodes), as explained in Methods, and widely used in theoretical studies to emulate the photoexcitation protocol.
We will focus on the strong interaction regime with a weak driving $g\ll t_0\ll U$ which is nearly resonant with the Mott gap.
In this regime one generically finds a stationary nonequilibrium state whose properties are independent of the details of the driving. The system is Mott insulating in equilibrium, and the photodoped carriers have a long lifetime due to the large Mott gap \cite{sensarma2010,eckstein2011,mitrano2014}. An effective description of the photodoped state can be obtained from a $1/U$ expansion. As $g\ll t_0$, the driving term does not affect $H_{\rm eff}$ to leading order, but it can control the doublon density.
Analogous to the doped Mott insulators at equilibrium, the effective physics of the photodoped state is well described by a generalized $t$-$J$ model \cite{li2020,kaneko2020,murakami2022} $H_{\rm eff}=H_t+H_J+H_{dh}$ with hopping $H_t=-t_0\sum_{\langle ij\rangle\sigma}e^{i\varphi_{ij}}[n_{i\bar{\sigma}}c^\dag_{i\sigma}c_{j\sigma}n_{j\bar{\sigma}}+\bar{n}_{i\bar{\sigma}}c^\dag_{i\sigma}c_{j\sigma}\bar{n}_{j\bar{\sigma}}]+\text{h.c.}$ and spin exchange $H_J=\sum_{\langle ij\rangle}J_{\rm ex}\boldsymbol{S}_i\cdot \boldsymbol{S}_j$, where $J_{\rm ex}=4t_0^2/U$.
We have defined $\bar{n}_{i\bar{\sigma}}=1-n_{i,-\sigma}$. The doublon-holon interaction term reads
\begin{align}
H_{dh}=\frac{J_{\perp}}{2}\sum_{\langle ij\rangle}(e^{2i\varphi_{ij}}\phi^+_i\phi^-_j+\text{h.c.})+J_{z}\sum_{\langle ij\rangle}\phi^z_i\phi^z_j,
\label{Heff}
\end{align}
where the pairing operators $\phi^+_i=(\phi^-_i)^\dag=c^\dag_{i\uparrow}c^\dag_{i\downarrow}$ and $\phi^z_i=(n_i-1)/2$ span a pseudospin $\mathfrak{su}(2)$ algebra similar to that of spin $\bm S_i$. The original model \eqref{model} yields $J_{\perp}=-J_{z}=J_{\rm ex}$.
The first term in Eq.~\eqref{Heff}
originates from a doublon-holon exchange process, illustrated in Fig.~\ref{diagram}(a), which favors a doublon-holon condensation with $\langle \phi^+_i\rangle\ne0$. In solids, the second term is generically renormalized by the intersite Coulomb repulsion which suppresses charge segregation. We will focus in the following on uniform phases with $\langle n_i\rangle=1$ ($\langle \phi^z_i\rangle=0$).
In pump-probe experiments on Mott insulating solids, a strong pump pulse is often applied for a short duration to create a quasi-stationary photodoped state, which can also be described by the above Hamiltonian $H_{\rm eff}$ with $g=0$. In either set-up, the effective theory for the nonequilibrium state of the Hubbard model is a generalized $t$-$J$ model of $n_d$ doublons and holons and $n_s=1-2n_d$ unpaired electrons per site.
In this prethermal phase, the positive exchange amplitude $J_{\perp}$ tends to impose a staggered phase twist for the doublon-holon condensate ($\eta$--pairing) \cite{yang1989}, but this alternating SC order is generically impossible on a frustrated non-bipartite lattice. Instead, we consider a $120^\circ$ twisted pairing for the three-colorable lattices, defined by $\langle\phi^+_{i \in R}\rangle=e^{- i2\pi/3}\langle\phi^+_{i \in G}\rangle=e^{- i4\pi/3}\langle\phi^+_{i \in B}\rangle=\phi_0$, which spontaneously breaks time-reversal and inversion symmetry. To understand the energetics of the condensate, we can examine the mean-field energy for the order $\langle\phi^+_i\rangle=\phi_0e^{i\bm q\cdot\bm r_i}$ with momentum $\bm q$, given by $\langle H_{dh}\rangle/N_{\rm site}=|\phi_0|^2\epsilon({\bm q})$ per site, see Supplemental Note 1. In the following, we restrict ourselves to the case $\varphi_{ij}=\varphi$ along each bond of an $R\to G\to B$ cycle. For the triangular lattice, the above $120^\circ$ order is of momentum $\bm q =\frac{2\pi}{3}\bm b_1-\frac{2\pi}{3}\bm b_2$ with reciprocal lattice vectors $\bm b_1,\bm b_2$, and corresponds to one of the two chiral minima of the energy dispersion. This minimum can be further
stabilized
by an artificial Peierls phase $0<\varphi<\pi/3$, with $\varphi=\pi/6$ realizing the most stable condensate. The above discussion applies equally to the opposite chirality with a reversed phase twist. The Kagome lattice has three sites in a unit cell, giving rise to three bands in $\epsilon(\bm q)$. For $J_{\perp}>0$ the lowest-lying band for $J_{\perp}>0$ is flat for $\varphi=0$, which implies
that no ordering pattern is singled out as energetically most favorable.
The artificial gauge field $\varphi_{ij}$ can however distort the flat band and favor a certain $120^\circ$ twisted order. As we shall show later in the paper, the order with ${\bm q}=0$ illustrated in Fig.~\ref{diagram}(c) can be stabilized by an artificial gauge field generated by circularly polarized light.
It remains to be shown that the $120^\circ$ condensate is stable against quantum fluctuations. We first consider the maximum photodoping situation ($n_d=0.5$) where all sites are either doubly occupied or empty. In this case $H_{\rm eff}=H_{dh}$ corresponds to an XXZ model of pseudospin ${\bm \phi}$. With spatial homogeneity assumed, it is known that the $120^\circ$ condensate is generically stabilized on the triangular lattice. Furthermore, the $120^\circ$ condensates constitute the exact ground-state manifold if $J_{z}=J_{\perp}\cos(2\pi/3+\varphi)$ ($J_{z}=-J_{\perp}/2$ for $\varphi=0$, which may be realized with an NN Coulomb repulsion) for both the triangular and Kagome lattices considered here, see Supplemental Note 1 for a proof following the idea of Ref.~\citenum{changlani2018}. Away from maximum photodoping, the twisted $120^\circ$ condensate is challenged by the presence of singly occupied sites, and
the additional
terms in $H_{\rm eff}$, such as electron hopping, while the condensate should survive at least for $1/2-n_d\ll \mathcal{O}(t_0/U)$. We will numerically confirm that this condensate is in fact stable in an extended parameter regime away from the maximum photodoping limit $n_d=1/2$.
\section{\NoCaseChange{Optical response of the $120^\circ$ chiral condensate}}
The twisted $120^\circ$ condensate embodies a spatially varying phase twist
and thus carries a persistent current even in the absence of an external field. With an external vector potential $A_{ij}$ along bond $\langle ij\rangle$, the doublon-holon current contribution along the cycle $R\to G\to B$ is
$\mathcal{J}^{dh}_{ij}(A_{ij})=\delta H_{dh}/\delta A_{ij}\approx2\mathcal{J}^{dh}_0\sin(2\pi/3+2\varphi+2A_{ij})$ on the mean-field level, where $\mathcal{J}^{dh}_0=-2eJ_{1}|\phi_0|^2$ with the elementary charge $e$. A persistent current $\mathcal{J}^{dh}_{ij}(0)=2\mathcal{J}^{dh}_0\sin(2\pi/3+2\varphi)$ flows even when $A_{ij}=0$, see arrows in Fig.~\ref{diagram}. Indeed, the phase-twisted condensate can be viewed as a frustrated array of Josephson junctions \cite{theron1994}.
As usual, a macroscopic superconducting current emerges when the condensate undergoes a uniform electric pulse $\bm E(t)$, which generates a vector potential $\bm A(t)=-\int^t ds\bm E(s)$. However, the breaking of the time-reversal and inversion symmetries allows for a nonlinear and anisotropic supercurrent response, of the general form
$\mathcal{J}^a= D^{ab}A_b+T^{abc}A_b A_c+\cdots$ with indices $a,b,c=x/y$ and Einstein convention. This is in contrast to conventional superconductors, where the fully symmetric tensor $T^{abc}$ vanishes due to unbroken inversion or time-reversal symmetries, which
imply
$\bm J(-\bm A)=-\bm J(\bm A)$, and where the London equation $\bm J\propto \bm A$ usually holds.
In particular, the chiral condensate allows for a nonlinear transverse current response perpendicular to $\bm A$, which constitutes a characteristic signature of the symmetry breaking. For the triangular lattice in Fig.~\ref{diagram}(b), the three-fold dihedral symmetry ($D_3$) imposes $T^{xyy}=T^{xxx}=0$, but allows for nonzero entries obeying $T^{xxy}=-T^{yyy}$. The value of $T^{xxy}$ is determined by evaluating the gauge-invariant supercurrent density $\mathcal{J}^a=\frac{1}{S}\langle\frac{\delta H_{dh}}{\delta{A}_a}\rangle$ with $a=x,y$ and the total area $S$. At the mean-field level, one obtains $D^{ab}=\frac{\phi_0^2}{S_{\rm u.c.}}\frac{\partial^2 \epsilon(\bm q+\bm A)}{\partial A_a\partial A_b}$ and $T^{abc}=\frac{\phi_0^2}{2S_{\rm u.c.}}\frac{\partial^3 \epsilon(\bm q+\bm A)}{\partial A_a\partial A_b\partial A_c}$, with the unit-cell area $S_{\rm u.c.}$, see Supplemental Note 2. Here we consider $\bm A$ along the $x$--direction. The second-order response emerges due to the trigonal warping $\partial^3\epsilon/\partial k_x\partial k_x\partial k_y$ of the energy dispersion near the condensation minimum.
The current density is illustrated in Fig.~\ref{diagram}(d) and depends strongly on the artificial gauge field $\varphi$. In particular, the transverse current vanishes for $\varphi=\pi/6$.
The nonlinear and anisotropic response
is phenomenologically similar to the second-order anomalous Hall effect \cite{sodemann2015,nagaosa2017} with inversion symmetry breaking, but in contrast to these works, it appears in the context of a SC response and requires the breaking of time-reversal symmetry. It is intriguing to note that if one applies a continuous-wave sinusoidal driving along $x$ ($E_x(t)= E_0\sin(\omega t)$), the transverse current oscillates at frequency $2\omega$. This Hall-like response and second-harmonic generation could serve as a smoking gun of the chiral order and can be tested with a four-point measurement.
\section{\NoCaseChange{Numerical determination of the phase diagram}}
In the remainder of the article, we will explore the phase diagram of the chiral condensate on three different frustrated lattices: the Bethe, the triangular, and the Kagome lattices. To provide a concrete example of the $120^\circ$ twisted order, we first consider a numerically solvable model in the thermodynamic limit, the driven Hubbard model on the Bethe lattice with hopping $t_0/\sqrt{z}$ where the coordination number is taken to $z\to\infty$. In this case, $t_0$ is taken as the unit of energy. While this lattice has no loops, the driven system can support the twisted condensate and, most importantly,
can be
solved using nonequilibrium dynamical mean-field theory (DMFT) in the strong interaction regime \cite{georges1996, aoki2014}.
We drive the system by coupling it to two fermion baths with a semielliptic density of states with temperature $T_b$ and different chemical potentials $\pm \mu_b$. As discussed above, the details of the driving do not matter as long as $g\ll t_0$. The baths can be exactly integrated out and incorporated into the DMFT iterations through a hybridization density of states $D_{\pm}(\epsilon)=\Gamma\sqrt{1-(\epsilon\pm U/2)^2/W^2}$ with $\Gamma = g^2/W$ and half-bandwidth $W$. The parameters $\pm \mu_b$ and $T_b$ are varied to control the doublon and holon distribution in the Hubbard system \cite{li2021}.
We show the spectral function $A=-\frac{1}{\pi}\operatorname{Im}G^r$ and the occupation $A^<=\frac{1}{2\pi}\operatorname{Im}G^<$ for a typical driven state in Fig.~\ref{spec}(a). The external driving creates two separate Fermi surfaces in the lower and upper Hubbard bands around $\omega=\pm\mu_b$, indicating the presence of excess doublons and holons. The two Fermi energies correspond to the energy cost or gain associated with the replacement of a doublon or holon by an unpaired electron. Even though the system is highly excited, the Fermi edges can be rather sharp, with low effective temperatures $T_{\rm eff}$ defined by fitting the distribution function $f(\omega)=A^<(\omega)/A(\omega)$ separately near the two Fermi levels. This confirms that the system can be described by a constrained quasi-equilibrium with excess doublons and holons of density $n_d$ and temperature $T_{\rm eff}$, as already shown in previous works \cite{li2020,li2021,murakami2022}. The order parameter $\phi=\langle\phi^+\rangle$ is sampled for various bath parameters, which allows to generate the phase diagram for the $120^\circ$ condensate shown in Fig.~\ref{spec}(b). The system exhibits a transition to the $120^\circ$-ordered phase beyond a critical double occupancy $n^c_d$, indicated by blue dots (see Supplemental Note 3), and is enhanced as $\varphi$ increases. A persistent current $\mathcal{J}$ flows along $R\to G\to B$, whose magnitude is plotted in panel (c). In experiments, carefully designed protocols can minimze the entropy production and thus achieve low effective temperatures \cite{werner2019}. In our calculations, we can tune the effective temperature of the doublons and holons by changing bath parameters, and then extrapolate the phase boundary to $T_{\rm eff}=0$. This procedure yields the upper bound of the critical doublon number $n^c_d\approx 0.4$ at $\varphi=0$ (black diamond in (b)).
The above results for the Bethe lattice confirm the existence of the $120^\circ$ SC condensate in a wide parameter range, which extends to $\varphi=0$ and at least down to $n_d\approx 0.4$, and also the validity of the effective theory~\eqref{Heff} at finite $U$. It remains to be investigated if this intriguing state can be realized on an ultrafast timescale. To address this question, we consider a real-time entropy-cooling protocol \cite{werner2019,werner2019prb}, which allows to generate cold photodoped states. Here, the system is coupled to two narrow bands with width $W=0.1$ centered at $\omega=\pm6.0$ and identical chemical potentials $\mu_b=0$, while the system-bath coupling $v(t)$ oscillates fast to excite photocarriers, see Methods.
As shown in Fig.~\ref{spec}(d), a nonzero $120^\circ$ SC order parameter quickly emerges at $\varphi=0$ under the driving $v(t)$, in the presence of a small symmetry-breaking field $h=0.001$ coupling to $\phi^+_i$. In the case of transition metal compounds, our unit of time is of the order of femtoseconds. Within the time range accessible in our numerical simulations, the order appears to decay very slowly after the driving is switched off. In addition, we have also simulated a system perturbed by a short pulse of the symmetry-breaking field $h(t)$, see Fig.~\ref{spec}(e). In this case, the $120^\circ$ order continues to grow after the pulse, which strongly suggests a spontaneous symmetry breaking.
We now use exact diagonalization to directly treat the generalized $t$-$J$ model \eqref{Heff} on a triangular lattice, to address the existence of the condensate in 2D systems at $T_{\rm eff}=0$. We study an $N_{\rm site}=12$ cluster with periodic boundary conditions, and calculate the pairing structure factor at the $120^\circ$ point, namely $S(120)=\sum_{ij}\theta_i\theta^*_j\langle \phi^+_i\phi^-_j\rangle/N_{\rm site}$ with $\theta_{i\in R}=e^{i2\pi/3}\theta_{i\in G}=e^{i4\pi/3}\theta_{i\in B}=1$ corresponding to the ``color". $S(120)$ quantifies the total $120^\circ$ order. The result is shown in Fig.~\ref{ed}(a), and
suggests
the stability of the $120^\circ$ condensate away from $n_d=1/2$ down to $\varphi=0$, in line with the observation for the Bethe lattice from DMFT. The critical doublon number is $n^c_d(\varphi=0)\sim 0.37$.
\section{\NoCaseChange{Realization on the Kagome lattice}}
Finally, we comment on how to realize the $120^\circ$ SC order on the Kagome lattice. The generalized $t$-$J$ model in the maximum photodoping limit ($n_d=0.5$) is equivalent to an $XXZ$ spin model, which exhibits a complex phase diagram on the Kagome lattice. As discussed above, the key task is to induce an artificial gauge field which favors a specific $120^\circ$ ordering pattern,
for example the $\bm q =0$ three-coloring of the lattice shown
in Fig.~\ref{diagram}(c). In the case of solid-state systems, we consider applying a strong laser with circular polarization, similar to the setup in Ref.~\cite{claassen2017}, with a vector potential $\bm A(t)=\mathcal{A}_0(\cos(\Omega t), -\sin(\Omega t))$.
The field couples through a time-dependent Peierls phase $\exp(i(\bm r_{i}-\bm r_j)\cdot \bm A(t))$ to the bond $\langle ij\rangle$. In the high-frequency limit ($\Omega\gg t_0$, off-resonant with $U$), the hopping parameters are renormalized, leading to a complex nearest-neighbor (NN) hopping $t_{\rm R}=|t_{\rm R}|e^{i\varphi(\mathcal{A})}$.
This renormalized hopping appears to favor the ${\bm q} = 0$ condensate shown in Fig.~\ref{diagram}(c). The next-nearest neighbor hopping which is also created and would favor a uniform order is usually very small, see the Supplemental Note 4.
We study the resulting effective model on a 12-site cluster with exact diagonalization,
defining $\mathcal{A}=\mathcal{A}_0 a/2$ with lattice constant $a$. The results are shown in Fig.~\ref{ed}(b), and clearly indicate the appearance of the (${\bm q} = 0$) $120^\circ$ SC condensate for $\varphi(\mathcal{A})\gtrsim0.067$ and $n_d\gtrsim 0.3$. The dependence of $S(120)$ on $n_d$ is qualitatively similar for both the triangular and Kagome lattices above the transition. In contrast, the uniform order (indicated by the dashed line) is always suppressed. It is worth noting that other orders, such as an $\sqrt{3}\times\sqrt{3}$ type $120^\circ$ order cannot be studied with the small cluster shown in Fig.~\ref{diagram}.
The
two orders share the identical mean-field energy at $\varphi=0$. However, a nonzero $|\varphi|<\pi/3$ stabilizes the ${\bm q} = 0$ order considered here, see Supplemental Note 4. The $\sqrt{3}\times\sqrt{3}$ type order can however be stabilized with a different gauge field, see Supplemental Note 4. Finally we note that other terms, which are generated by the Floquet driving but ignored here, can alter the phase diagram at the quantitative level.
\section{\NoCaseChange{Conclusion}}
Our work established a new type of chiral superconductivity in photodoped Mott insulators, which breaks the time-reversal and inversion symmetries through a spatially twisted order parameter, namely the $120^\circ$ condensate. This condensate originates from a positive doublon-holon exchange amplitude $J_{\perp}>0$, which is intrinsically related to the nonequilibrium nature of the photodoped states and contrasts with the negative exchange in equilibrium BEC-like pairing induced by charge attraction \cite{micnas1990}. The exchange processes furthermore generate a doublon-holon interaction (the $J_{z}$ term), which favors charge segregation. This effect is ignored here, since it should be suppressed by the inter-site Coulomb repulsion in solids. Also, if the ordered phase is created by an ultrafast uniform excitation, we can assume that the state remains homogeneous on the femtoseconds timescale. In the presence of a light-induced artificial gauge field, the order can be further enhanced and even stabilized on the Kagome lattice. The persistent loop current and the nonlinear transverse superconducting current are characteristic signatures of the chiral $120^\circ$ condensate and allow to realize a second harmonic generation. The phenomenology contrasts with the conventional description of the SC electromagnetic response based on the linear London equation $\bm j\propto \bm A$, where even-order responses are excluded by time-reversal and inversion symmetry.
In experiments, the photodoped state can be realized by applying femtosecond laser pulses to condensed matter systems \cite{stojchevska2014}, or via the
tilting of
optical lattices \cite{greif2011}. When low entropies are maintained, both protocols allow to create long-lived photodoped doublons and holons in the presence of a large Mott gap \cite{sensarma2010,eckstein2011,mitrano2014}, which quickly relax to a prethermal regime characterized by the generalized $t$-$J$ physics. Since the condensate exists down to $\varphi=0$, it can be relevant to the photodoped SC states in correlated materials of triangular lattice geometry, such as $\kappa$--(BEDT--TTF)$_2$Cu[N(CN)$_2$]Br \cite{buzzi2020}.
Finally, doped Mott insulators have a very rich phase diagram, and the competition between the chiral condensate and other long-lived and hidden phases, especially away from maximum photodoping, is an interesting topic for further investigations.
\bibliographystyle{naturemag}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 582 |
\section{Introduction}
In the last 15 years, the paradigm of Answer Set Programming (ASP)~\cite{MT99,Nie99,BET11} has experienced a boost in practical tools and applications that has come in parallel with a series of significant results in its theoretical foundations. Focusing on the latter, a long way has been traversed since the original definition of the \emph{stable models} semantics~\cite{GL88} for normal logic programs, until the current situation where stable models constitute a complete non-monotonic approach for arbitrary theories in the syntax of First Order Logic~\cite{PV04,FLL07}. An important breakthrough that undoubtfully contributed to this evolution was the characterization of stable models in terms of \emph{Equilibrium Logic}~\cite{Pea96,Pea06}, allowing a full coverage of arbitrary propositional theories and inspiring a new definition of program reduct for that syntax~\cite{Fer05}. Equilibrium Logic is defined in terms of a model minimisation criterion for an intermediate logic called the \emph{Logic of Here-and-There} (HT) first introduced in~\cite{Hey30} and, shortly after, reappeared in~\cite{God32} as G\"odel's three-valued logic $G_3$. In~\cite{LPV01} it was shown that equivalence in HT was a necessary and sufficient condition for the property of \emph{strong equivalence}, that is, that two programs yield the same stable/equilibrium models regardless of the context in which they may be included. After that, many theoretical results have followed from the use of Equilibrium Logic and HT, such as the study of variants of strong equivalence~\cite{PV04b,Wol08} or the series of papers considering different forms of strongly equivalent transformations~\cite{CPV05,CF07,CPV07}. Besides, Equilibrium Logic allowed the already mentioned extension to first order syntax~\cite{PV04}, engendering an extensive literature, as well as many other extensions such as the inclusion of a strong negation operator~\cite{OP05} or new formalisms such as \emph{Partial Equilibrium Logic}~\cite{COPV07b}, \emph{Temporal Equilibrium Logic}~\cite{ACD+13} or, more recently, \emph{Infinitary Equilibrium Logic}~\cite{HLPV14}.
All these contributions provide results about HT or Equilibrium Logic that are proved with meta-logical textual descriptions. These proofs lack a common formal basis on which meta-properties of HT and Equilibrium Logic can be mathematically or even automatically checked. Another interesting observation is that many of these theoretical results in the literature use the concept of \emph{sets of models} of different types: classical models, HT models, equilibrium models, etc. It is, therefore, natural to wonder whether a formal treatment of sets of interpretations could help in the development of fundamental results for Equilibrium Logic and ASP.
In this paper we explore the idea of characterising HT (or $G_3$) and Equilibrium Logic using the concept of \emph{denotation} of a formula. Given a formula $\alpha$, its denotation $\den\alpha$ collects the set of $G_3$ models of $\alpha$ and can be described as a compositional function, that is, the denotation of a formula is a function of the denotations of its subformulas. Since their introduction by~\cite{SS71}, denotations constitute a common device for defining the semantics of programming languages, although their use for non-classical logics is also frequent -- a prominent case, for instance, is the semantics of $\mu$-Calculus~\cite{Koz83}. The use of denotational semantics in Logic Programming is not so common: in the case of Prolog we can mention~\cite{NF89} but for ASP, to the best of our knowledge, no attempt has previously been made.
Here we explain how the denotational semantics actually constitutes an alternative description of HT/$G_3$ and provides several interesting features. We define some elementary operations on sets of interpretations and the ordering relation used in the equilibrium models minimisation. Using those elementary set operations and analysing structural properties of the denotation of formulas, we derive some expressivity results for $G_3$ such as, for instance, that conjunction is not expressible in terms of implication, falsum and disjunction. More importantly, we are able to capture the equilibrium models of a formula as a set expression constituting a subset of $\den\alpha$. This allows us to study properties of equilibrium models by using formal results from set theory, something that in many cases is more compact than an informal proof in natural language and, moreover, has allowed us to use a theorem prover for a semi-automated verification (see the sequel~\cite{CMMSE15} of the current paper).
As an application of the denotational semantics, we provide several definitions (in terms of denotations) for entailment relations foundl in the literature, and further introduce a new one called \emph{strong entailment}. We say that $\alpha$ strongly entails $\beta$ when the equilibrium models of $\alpha \wedge \gamma$ are also equilibrium models of $\beta \wedge \gamma$ for any context $\gamma$. This obviously captures one of the directions of strong equivalence. We also provide the corresponding denotational characterisation for this new strong entailment and give an example of a sufficient condition that can be applied in some cases.
The rest of the paper is organised as follows. In Section~\ref{sec:g3} we provide the basic definitions of G\"odel's $G_3$ logic that, as explained, is an equivalent formulation of HT. In Section~\ref{sec:sets} we describe several useful operators on sets of interpretations that we then use in Section~\ref{sec:zero} to define the denotational semantics for $G_3$ and for equilibrium models. After describing some applications of this semantics, Section~\ref{sec:entail} defines different types of entailments and, in particular, presents the idea of strong entailment together with its denotational charaterisation and some examples. Finally, Section~\ref{sec:conc} concludes the paper. Most proofs have been collected in the on line appendix (Appendix A).
\section{G\"odel's three-valued logic $G_3$ and equilibrium models}
\label{sec:g3}
We describe next the characterisation of Equilibrium Logic in terms of G\"odel's three-valued logic -- for further details on multi-valued characterisations of Equilibrium Logic, see~\cite{Pea06}, section 2.4. \\
We start from a finite set of atoms $\Sigma$ called the \emph{propositional signature}. A \emph{formula} $\alpha$ is defined by the grammar:
\[
\alpha ::= \bot \mid p \mid \alpha_1 \wedge \alpha_2 \mid \alpha_1 \vee \alpha_2 \mid \alpha_1 \rightarrow \alpha_2
\]
\noindent where $\alpha_1$ and $\alpha_2$ are formulas in their turn and $p \in \Sigma$ is any atom. We define the derived operators $\neg \alpha \mathbin{\stackrel{\mathrm{def}}{=}} \alpha \rightarrow \bot$ and $\top \mathbin{\stackrel{\mathrm{def}}{=}} \neg \bot$. By ${\cal L}_{\Sigma}$ we denote the language of all well-formed formulas for signature $\Sigma$ and just write ${\cal L}$ when the signature is clear from the context.
A \emph{partial} (or \emph{three-valued}) interpretation is a mapping $v: \Sigma \rightarrow \{0,1,2\}$ assigning $0$ (false), $2$ (true) or $1$ (undefined) to each atom $p$ in the signature $\Sigma$. A partial interpretation $v$ is said to be \emph{classical} (or \emph{total}) if $v(p)\neq 1$ for every atom $p$. We write ${\cal I}$ and ${\cal I}_{\, c}$ to stand for the set of all partial and total interpretations, respectively (fixing signature $\Sigma$). Note that ${\cal I}_{\, c} \subseteq {\cal I}$.
For brevity, we will sometimes represent interpretations by (underlined) strings of digits from $\{0,1,2\}$ corresponding to the atom values, assuming the alphabetical ordering in the signature. Thus, for instance, if $\Sigma=\{p,q,r\}$, the interpretation $v=\str{102}$ stands for $v(p)=1$, $v(q)=0$ and $v(r)=2$.
Given any partial interpretation $v \in {\cal I}$ we define a classical interpretation $v_t \in {\cal I}_{\, c}$ as:
\begin{eqnarray*}
v_t(p) & \mathbin{\stackrel{\mathrm{def}}{=}} & \left\{
\begin{array}{r@{\ \ }l}
2 & \hbox{if } v(p)=1 \\
v(p) & \hbox{otherwise}
\end{array}
\right.
\end{eqnarray*}
\noindent In other words, $v_t$ is the result of transforming all $1$'s in $v$ into $2$'s. For instance, given $v'=\str{1021}$ for signature $\Sigma=\{p,q,r,s\}$, then $v'_t=\str{2022}$.
\begin{definition}[Valuation of formulas]\label{def:val}
Given a partial interpretation $v \in {\cal I}$ we define a corresponding \emph{valuation of formulas}, a function also named $v$ (by abuse of notation) of type $v: {\cal L} \rightarrow \{0,1,2\}$ and defined as:
\[
\begin{array}{c@{\hspace{30pt}}c}
\begin{array}{rcl}
v(\alpha \wedge \beta) & \mathbin{\stackrel{\mathrm{def}}{=}} & \min(v(\alpha), v(\beta))\\[5pt]
v(\alpha \vee \beta) & \mathbin{\stackrel{\mathrm{def}}{=}} & \max(v(\alpha), v(\beta))
\end{array}
&
\begin{array}{rcl}
v(\bot) & \mathbin{\stackrel{\mathrm{def}}{=}} & 0 \\[5pt]
v(\alpha \rightarrow \beta) & \mathbin{\stackrel{\mathrm{def}}{=}} & \left\{
\begin{array}{r@{\ \ }l}
2 & \hbox{if } v(\alpha) \leq v(\beta) \\
v(\beta) & \hbox{otherwise} \hspace{14pt} \Box
\end{array}
\right.
\end{array}
\end{array}
\]
\end{definition}
From the definition of negation, it is easy to see that $v(\neg \alpha)=2$ iff $v(\alpha)=0$, and $v(\neg \alpha)=0$ otherwise.
We say that $v$ \emph{satisfies} $\alpha$ when $v(\alpha)=2$. We say that $v$ is a \emph{model} of a theory $\Gamma$ iff $v$ satisfies all the formulas in $\Gamma$.
\begin{example}\label{ex:1}
As an example, looking at the table for implication, the models of the formula:
\begin{eqnarray}
\neg p \rightarrow q \label{f1}
\end{eqnarray}
\noindent are those where $v(\neg p)=0$ or $v(q)=2$ or both $v(\neg p)=v(q)=1$. The latter is impossible since the evaluation of negation never returns $1$, whereas $v(\neg p)=0$ means $v(p)\neq 0$. Therefore, we get $v(p) \neq 0$ or $v(q)=2$ leading to the following 7 models $\str{10}, \str{11}, \str{12}, \str{20}, \str{21}, \str{22}, \str{02}$. ~\hfill$\Box$
\end{example}
Given two 3-valued interpretations $u,v$, we say that $u \leq v$ when, for any atom $p \in \Sigma$, the following two conditions hold: $u(p) \leq v(p)$; and $u(p)=0$ implies $v(p)=0$. As usual, we write $u < v$ to stand for both $u\leq v$ and $u\neq v$. An equivalent, and perhaps simpler, way of understanding $u \leq v$ is that we can get $v$ by switching some $1$'s in $u$ into $2$'s. This immediately means that classical interpretations are $\leq$-maximal, because they contain no $1$'s. Moreover, since $u_t$ is the result of switching \emph{all} $1$'s in $u$ into $2$'s, we easily conclude $u \leq u_t$ for any $u$. As an example of how $\leq$ works, among models of \eqref{f1}, we can check that $\str{10} < \str{20}$ and that $\str{11}$, $\str{12}$ and $\str{21}$ are strictly smaller than $\str{22}$. On the other hand, for instance, $\str{10}$, $\str{02}$ or $\str{12}$ are all pairwise incomparable.
Once we have defined an ordering relation among interpretations, we can define the concept of \emph{equilibrium model} as a $\leq$-minimal model that is also classical.
\begin{definition}[Equilibrium model]
A classical interpretation $v \in {\cal I}_{\, c}$ is an \emph{equilibrium model} of a theory $\Gamma$ iff it is a $\leq$-minimal model of $\Gamma$.~\hfill$\Box$
\end{definition}
Back to the example \eqref{f1}, from the 7 models we obtained, only three of them $\str{20}, \str{22}$ and $\str{02}$ are classical (they do not contain $1$'s). However, as we saw, $\str{20}$ is not $\leq$-minimal since $\str{10}<\str{20}$ and the same happens with $\str{22}$, since $\str{11}, \str{12}, \str{21}$ are strictly smaller too. The only $\leq$-minimal classical model is $\str{02}$, that is, $p$ false and $q$ true, which becomes the unique equilibrium model of \eqref{f1}. Equilibrium models coincide with the most general definition of stable models, for the syntax of arbitrary (propositional) formulas~\cite{Fer05}. Indeed, we can check that model $\str{02}$ coincides with the only stable model of the ASP rule $(q \leftarrow \text{not} \ p)$ which is the usual rewriting of formula \eqref{f1} in ASP syntax.
\section{Sets of interpretations}\label{sec:sets}
In this section we will introduce some useful operations on sets of interpretations. Some of them depend on the partial ordering relation $\leq$. Given a set of interpretations $S\subseteq {\cal I}$ we will define the operations:
\[
\begin{array}{rcl@{\hspace{50pt}}rcl}
\non{S} & \mathbin{\stackrel{\mathrm{def}}{=}} & {\cal I} \setminus S &
S \downarrow & \mathbin{\stackrel{\mathrm{def}}{=}} & \{u \in {\cal I} \ : \ \hbox{there exists } v \in S, v \geq u\} \\
S_c & \mathbin{\stackrel{\mathrm{def}}{=}} & {\cal I}_{\, c} \cap S &
S \uparrow & \mathbin{\stackrel{\mathrm{def}}{=}} & \{u \in {\cal I} \ : \ \hbox{there exists } v \in S, v \leq u\}
\end{array}
\]
To avoid too many parentheses, we will assume that $\downarrow$, $\uparrow$ and $c$ have more priority than standard set operations $\cup$, $\cap$ and $\setminus$. As usual, we can also express set difference $S \setminus S'$ as $S \cap \non{S'}$. We can easily check that the $c$ operation distributes over $\cap$ and $\cup$, whereas $\downarrow$ and $\uparrow$ distribute over $\cup$. For intersection, we can only prove that:
\begin{proposition}\label{prop:distr} \zlabel{prop:distr}
For any pair $S, S'$ of sets of interpretations: \\
$(S \cap S')\uparrow \ \ \subseteq \ S\uparrow \cap \ S'\uparrow$ and $(S \cap S')\downarrow \ \ \subseteq \ S\downarrow \cap \ S'\downarrow$.~\hfill$\Box$
\end{proposition}
\noindent In the general case, the other direction does not hold. As a simple example, for signature $\Sigma=\{p,q\}$, take $S=\{\str{12}\}$ and $S'=\{\str{21}\}$. Then $S\uparrow = \{\str{12},\str{22}\}$ and $S'\uparrow =\{\str{21},\str{22}\}$ and thus $S\uparrow \cap \ S'\uparrow=\{\str{22}\}$ but $(S\cap S')\uparrow = \emptyset$.
With these new operators we can formally express that $v_t$ is the only classical interpretation greater than or equal to $v$ in the following way:
\begin{proposition}\label{prop:vt} \zlabel{prop:vt}
For any $v \in {\cal I}$ it holds that $\{v\} \uparrow_c \ = \{ v_t \}$. ~\hfill$\Box$
\end{proposition}
\begin{corollary}\label{cor:vt} \zlabel{cor:vt}
For any $S \subseteq {\cal I}$ and for any interpretation $v$ we have: $v \in S_c \downarrow$ iff $v_t \in S$.~\hfill$\Box$
\end{corollary}
A particularly interesting type of sets of interpretations are those $S$ satisfying that, for any $v \in S$, we also have $v_t \in S$. When this happens, we say that $S$ is \emph{total-closed} or \emph{classically closed}. As we will see, there is a one-to-one correspondence between a total-closed set of interpretations and a set of models for some (set of equivalent) formula(s). The definition of total-closed set can be formally captured as follows:
\begin{proposition}\label{prop:totalc} \zlabel{prop:totalc}
The following three assertions are equivalent:\\
(i) \ $S$ is total-closed
\hspace{45pt}
(ii) \ $S\subseteq S_c \downarrow$
\hspace{45pt}
(iii) \ $S \uparrow_c = S_c$.~\hfill$\Box$
\end{proposition}
\begin{lemma}\label{lem:cm} \zlabel{lem:cm}
For any set of interpretations $S$, it holds that $(\non{S})_c \downarrow \subseteq \non{(S_c \downarrow)}$.~\hfill$\Box$
\end{lemma}
From this, together with Proposition \ref{prop:totalc} (ii) we immediately conclude
\begin{proposition}\label{prop:cm} \zlabel{prop:cm}
For any total-closed set of interpretations $S$, it holds that $(\non{S})_c \downarrow \subseteq \non{S}$.~\hfill$\Box$
\end{proposition}
\noindent When $S$ is a total-closed set of models, this proposition asserts that any interpretation below a classical countermodel is also a countermodel. In fact, Proposition~\ref{prop:cm} corresponds to what~\cite{CF07} defined as \emph{total-closed set of countermodels} $\non{S}$.
\section{Denotational semantics}
\label{sec:zero}
In this section we consider a denotational semantics for $G_3$ and for equilibrium models. Rather than saying when an interpretation $v$ is a model of a formula $\varphi$, the main idea is to capture the \emph{whole set of models} of $\varphi$ as a set of interpretations we will denote by $\den{\varphi}$. As we explain next, this set can be completely defined by structural induction without actually resorting to the valuation of formulas.
\begin{definition}[Denotation]
The \emph{denotation} of a formula $\varphi$, written $\den{\varphi}$, is recursively defined as follows
\[
\begin{array}{rcl@{\hspace{30pt}}rcl}
\den{\bot} & \mathbin{\stackrel{\mathrm{def}}{=}} & \emptyset &
\den{\alpha \wedge \beta} & \mathbin{\stackrel{\mathrm{def}}{=}} & \den{\alpha} \cap \den{\beta}\\
\den{p} & \mathbin{\stackrel{\mathrm{def}}{=}} & \{v \in {\cal I} \; : \; v(p)=2 \} &
\den{\alpha \vee \beta} & \mathbin{\stackrel{\mathrm{def}}{=}} & \den{\alpha} \cup \den{\beta} \\
\den{ \alpha \rightarrow \beta} & \mathbin{\stackrel{\mathrm{def}}{=}} & \big( \non{\den{\alpha}} \cup \den{\beta} \big) \cap \big( \ \non{\den{\alpha}} \cup \den{\beta} \ \big)_c\downarrow
\end{array}
\]
\noindent where $p \in \Sigma$ is an atom, and $\alpha, \beta \in {\cal L}$ are formulas in their turn.~\hfill$\Box$
\end{definition}
We say that a formula $\alpha$ is a \emph{tautology} iff $\den\alpha={\cal I}$ and that the formula is \emph{inconsistent} iff $\den\alpha=\emptyset$. The following theorem shows that this definition actually captures the set of models of $\alpha$, i.e., the set of interpretations that make $v(\alpha)=2$ using $G_3$ valuations of formulas (Definition~\ref{def:val}). Moreover, it also proves that $v_t \in \den{\alpha}$ is equivalent to $v(\alpha)\neq 0$.
\begin{theorem}\label{th:den-g3} \zlabel{th:den-g3}
Let $v \in {\cal I}$ be a partial interpretation and $\alpha \in {\cal L}$ a formula. Then:
\begin{enumerate}
\item[\rm (i)] $v(\alpha)=2$ in $G_3$ iff $v \in \den\alpha$.
\item[\rm (ii)] $v(\alpha) \neq 0$ in $G_3$ iff $v_t \in \den\alpha$.~\hfill$\Box$
\end{enumerate}
\end{theorem}
\noindent As $v(\alpha)=2$ implies $v(\alpha)\neq 0$, then $v\in \den\alpha$ implies $v_t \in \den\alpha$ and thus:
\begin{corollary}\label{cor:vt1}
For any $\alpha \in {\cal L}$, $\den\alpha$ is total-closed.~\hfill$\Box$
\end{corollary}
\noindent In fact, this relation between models of a formula and total-closed sets of interpretations also holds in the other direction, that is, for any total-closed set of interpretations $S$, there always exists\footnote{This was proved in Theorem 2 from~\cite{CF07} using the dual concept of total-closed set of countermodels.} a formula $\alpha$ such that $\den\alpha = S$.
When compared to denotational semantics for other formalisms, it is clear that the denotation of implication is the most representative characteristic of $G_3$. Defining its denotation provides a powerful tool for studying fundamental properties of this logic. For instance, we can derive the denotation for negation as $\den{\neg \alpha}=\den{\alpha \rightarrow \bot}= \non{\den{\alpha}} \cap \non{\den{\alpha}}_c \downarrow = \non{\den{\alpha}}_c \downarrow$ where the last step follows from Proposition~\ref{prop:cm}. With this correspondence and Corollary~\ref{cor:vt} we conclude that $v \in \den{\neg \alpha} $ iff $v_t \in \non{\den{\alpha}}$, that is, $v$ is a model of $\neg \alpha$ iff $v_t$ is a classical countermodel of $\alpha$. Another application example of the denotation of implication is, for instance, this simple proof of the Deduction Theorem for $G_3$.
\begin{theorem}
For any pair of formulas $\alpha, \beta$: $\den\alpha \subseteq \den\beta$ iff $\den{\alpha \rightarrow \beta} ={\cal I}$. Moreover, $\den\alpha=\den\beta$ iff $\den{\alpha \leftrightarrow \beta} ={\cal I}$.
\end{theorem}
\begin{proof}
For the result with implication, from left to right, assume $\den\alpha \subseteq \den\beta$. Then, $\big( \non{\den{\alpha}} \cup \den{\beta} \big) = {\cal I}$ and so, $\den{\alpha \rightarrow \beta}={\cal I} \cap {\cal I}_c\downarrow={\cal I}$. For right to left, if $\den{\alpha \rightarrow \beta} ={\cal I}$, take any $v \in \den\alpha$. As $v \in \den{\alpha \rightarrow \beta} \subseteq \big( \non{\den{\alpha}} \cup \den{\beta} \big)$ we conclude $v \in \den\beta$. For the double implication, simply note that $\den\alpha=\den\beta$ now means $\den{\alpha \rightarrow \beta}=\den{\beta \rightarrow \alpha}={\cal I}$. Therefore, $\den{\alpha \leftrightarrow \beta}=\den{\alpha \rightarrow \beta} \cap \den{\beta \rightarrow \alpha}={\cal I}\cap{\cal I}={\cal I}$.
\end{proof}
\noindent This denotation of implication is an intersection of two sets. We can also alternatively capture implication as a union of sets:
\begin{proposition}\label{prop:vt2} \zlabel{prop:vt2}
For any $\alpha, \beta \in {\cal L}$, it follows that:
\begin{eqnarray*}
\hspace{68pt} \den{ \alpha \rightarrow \beta} & = & \overline{\den\alpha}_c \downarrow \ \cup \ ( \overline{\den{\alpha}} \cap \den\beta_c \downarrow) \ \cup \ \den\beta
\hspace{68pt}\Box
\end{eqnarray*}
\end{proposition}
From this alternative representation of implication and the fact that $\non{\den{\alpha}}_c\downarrow \ \subseteq \non{\den\alpha}$ (from Proposition~\ref{prop:cm}) we immediately conclude $\den\alpha \cap \den{\alpha \rightarrow \beta} = \den\alpha \cap \den\beta$. In other words, we have trivially proved that $\den{\alpha \wedge (\alpha \rightarrow \beta)}=\den{\alpha \wedge \beta}$ in $G_3$.
\subsection{Expressiveness of operators}
As an application of the denotational semantics, we will study the expressiveness of the set of propositional operators usually provided as a basis for $G_3$: $\{\wedge,\vee,\rightarrow,\bot\}$. In Intuitionistic Logic, it is well-known that we cannot represent any of these operators in terms of the others. In $G_3$, however, it is also known
that $\vee$ can be represented in terms of $\wedge$ and $\rightarrow$. In particular:
\begin{theorem}\label{th:or} \zlabel{th:or}
For any $\Sigma$, the system ${\cal L}_{\Sigma}\{\bot, \wedge, \rightarrow\}$ is complete because given any pair of formulas $\alpha, \beta$ for $\Sigma$, it holds that:
$\den{ \alpha \vee \beta} = \den{(\alpha \rightarrow \beta) \rightarrow \beta} \cap \den{(\beta \rightarrow \alpha) \rightarrow \alpha}$. ~\hfill$\Box$
\end{theorem}
Now, one may wonder whether $\rightarrow$ or $\wedge$ can be expressed in terms of the rest of operators. However, we prove next that this is not the case.
\begin{lemma}\label{lem:imply} \zlabel{lem:imply}
Let $\Sigma=\{p_1,\dots,p_n\}$ and let $\gamma \in {\cal L}_{\Sigma}\{\bot, \wedge, \vee\}$. Then $\den\gamma \subseteq \bigcup^n_{i=1} \den{p_i}$. ~\hfill$\Box$
\end{lemma}
\begin{theorem}\label{th:imp} \zlabel{th:imp}
If $\{p_1,p_2\}\subseteq \Sigma$ then $p_1 \rightarrow p_2$ cannot be equivalently represented in ${\cal L}_{\Sigma}\{\bot, \vee, \wedge\}$.~\hfill$\Box$
\end{theorem}
\noindent This result is not surprising since we can further observe that the denotations for $\wedge$ and $\vee$, respectively the intersection and the union, are monotonic with respect to set inclusion, whereas $\den{\alpha \rightarrow \beta}$ is monotonic for the consequent and anti-monotonic for the antecedent (see Proposition~\zref{prop:mon} in the online appendix).
We will show next that conjunction cannot be expressed in terms of $\vee, \rightarrow, \bot$. To this aim, we begin proving the following lemma.
\begin{lemma}\label{lem:pq} \zlabel{lem:pq}
Let $\Sigma$ be of the form $\Sigma=\{p,q,\dots\}$ and let $\gamma \in {\cal L}_{\Sigma}\{\bot, \vee, \rightarrow\}$, then for any subformula $\delta$ of $\gamma$ and any $v \in \den\delta$ of the form $v=\str{22\dots}$ (i.e. making both atoms true), there exists some $u \in \den{\delta}$ such that $u<v$ and $u$ coincides with $v$ in all atoms excepting $p,q$.~\hfill$\Box$
\end{lemma}
\begin{theorem}\label{th:wedge} \zlabel{th:wedge}
If $\{p_1,p_2\}\subseteq \Sigma$ then $p_1 \wedge p_2$ cannot be equivalently represented in ${\cal L}_{\Sigma}\{\bot, \vee, \rightarrow\}$.~\hfill$\Box$
\end{theorem}
\subsection{Denotation of equilibrium models}
We can use the denotational semantics to capture equilibrium models as follows.
\begin{theorem}\label{th:1eq} \zlabel{th:1eq}
A classical interpretation $v \in {\cal I}_{\, c}$ is an equilibrium model of $\alpha$ iff it satisfies the fixpoint condition $\den{\alpha} \cap \{v\}\downarrow \ = \{v\}$.~\hfill$\Box$
\end{theorem}
The set of equilibrium models can also be captured as the denotation below.
\begin{theorem}\label{th:eq} \zlabel{th:eq}
The set of equilibrium models of $\alpha$, denoted as $\den{\alpha}_e$, corresponds to the expression:
\[
\hspace{110pt}
\den{\alpha}_e \mathbin{\stackrel{\mathrm{def}}{=}} \den{\alpha}_c \ \setminus \ (\den{\alpha} \setminus {\cal I}_{\, c} ) \uparrow \hspace{110pt} \Box
\]
\end{theorem}
As an application of Theorem~\ref{th:eq}, we have used it to obtain the following characterisation of equilibrium models of a disjunction:
\begin{proposition}\label{prop:or}
For any pair of formulas $\alpha$ and $\beta$:
\begin{eqnarray*}
\den{\alpha \vee \beta}_e & = &
\big( \den{\alpha}_e \setminus \den{\beta}_c \big)
\ \cup \
\big( \den{\beta}_e \setminus \den{\alpha}_c \big)
\ \cup \
\big( \den{\alpha}_e \cap \den{\beta}_e \big)
\end{eqnarray*}
\end{proposition}
\begin{proof}
We begin applying some basic set operations:
\begin{eqnarray*}
\den{\alpha \vee \beta}_e & = & \den{\alpha \vee \beta}_c \ \setminus \ (\den{\alpha \vee \beta} \setminus {\cal I}_{\, c} ) \uparrow \\
& = &\big( \den{\alpha}_c \cup \den{\beta}_c \big)\ \setminus \ (\ (\den{\alpha}\setminus {\cal I}_{\, c}) \cup (\den{\beta} \setminus {\cal I}_{\, c}) \ ) \uparrow \\
& = &\big( \den{\alpha}_c \cup \den{\beta}_c \big)\ \setminus \ ( \ (\den{\alpha}\setminus {\cal I}_{\, c})\uparrow \cup \ (\den{\beta} \setminus {\cal I}_{\, c})\uparrow \ ) \\
& = &\big( \den{\alpha}_c \cup \den{\beta}_c \big)\ \cap \ \non{(\den{\alpha}\setminus {\cal I}_{\, c})\uparrow} \cap \ \non{(\den{\beta} \setminus {\cal I}_{\, c})\uparrow} \\
& = &\den{\alpha}_c \cap \ \non{(\den{\alpha}\setminus {\cal I}_{\, c})\uparrow} \cap \ \non{(\den{\beta} \setminus {\cal I}_{\, c})\uparrow} \\
& & \cup \ \den{\beta}_c \cap \ \non{(\den{\alpha}\setminus {\cal I}_{\, c})\uparrow} \cap \ \non{(\den{\beta} \setminus {\cal I}_{\, c})\uparrow} \\
& = &\underbrace{\den{\alpha}_e \cap \ \non{(\den{\beta} \setminus {\cal I}_{\, c})\uparrow}}_{\gamma_1} \ \ \cup \ \ \underbrace{\den{\beta}_e \cap \ \non{(\den{\alpha}\setminus {\cal I}_{\, c})\uparrow}}_{\gamma_2}
\end{eqnarray*}
Since $\den{\alpha}_e \subseteq {\cal I}_c = (\den{\beta}\cup \non{\den{\beta}})_c = \den{\beta}_c \cup \non{\den\beta}_c$ we can rewrite $\gamma_1$ as follows:
\begin{eqnarray*}
\gamma_1 & = & \den{\alpha}_e \cap \big( \den{\beta}_c \cup \non{\den\beta}_c \big) \cap \ \non{(\den{\beta} \setminus {\cal I}_{\, c})\uparrow} \nonumber \\
& = & \den{\alpha}_e \cap \underbrace{\den{\beta}_c \cap \ \non{(\den{\beta} \setminus {\cal I}_{\, c})\uparrow}}_{\den{\beta}_e} \ \cup \ \den{\alpha}_e \cap \non{\den{\beta}}_c \cap \ \non{(\den{\beta} \setminus {\cal I}_{\, c})\uparrow} \nonumber \\
& = & \den{\alpha}_e \cap \den{\beta}_e \ \cup \ \den{\alpha}_e \cap \non{\den{\beta}}_c \cap \ \non{(\den{\beta} \setminus {\cal I}_{\, c})\uparrow}
\end{eqnarray*}
Now, we will prove that $\non{\den{\beta}}_c \subseteq \non{(\den{\beta} \setminus {\cal I}_{\, c}) \uparrow}$ and so, we can remove the latter in $\gamma_1$. To this aim, we will show that $\non{\den{\beta}}_c \cap (\den{\beta} \setminus {\cal I}_{\, c}) \uparrow =\emptyset$. First, note that $\non{\den{\beta}}_c \cap (\den{\beta} \setminus {\cal I}_{\, c}) \uparrow = \non{\den{\beta}}_c \cap (\den{\beta} \setminus {\cal I}_{\, c}) \uparrow_c$. Then $(\den{\beta} \setminus {\cal I}_{\, c}) \uparrow_c = (\den{\beta} \cap \non{{\cal I}_{\, c}}) \uparrow_c \subseteq \den{\beta}\uparrow_c \cap\ \non{{\cal I}_{\, c}} \uparrow_c \subseteq \den{\beta}\uparrow_c = \den{\beta}_c$ where, in the last step, we have used Proposition~\ref{prop:totalc} (iii). Finally, as $\non{\den{\beta}}_c \cap \den{\beta}_c = \emptyset$, we conclude $\non{\den{\beta}}_c \cap (\den{\beta} \setminus {\cal I}_{\, c}) \uparrow_c = \non{\den{\beta}}_c \cap (\den{\beta} \setminus {\cal I}_{\, c}) \uparrow= \emptyset$ too.
Therefore, we can further simplify the expression we obtained for $\gamma_1$ as:
\begin{eqnarray*}
\gamma_1 & = & \den{\alpha}_e \cap \den{\beta}_e \ \cup \ \den{\alpha}_e \cap \non{\den{\beta}}_c \ = \
\den{\alpha}_e \cap \den{\beta}_e \ \cup \ \big(\den{\alpha}_e \setminus \den{\beta}_c\big)
\end{eqnarray*}
Finally, making a similar reasoning for $\gamma_2$ we get $\gamma_2= \den{\alpha}_e \cap \den{\beta}_e \cup \big(\den{\beta}_e \setminus \den{\alpha}_c\big)$ and the result in the enunciate follows from $\den{\alpha\vee \beta}_e = \gamma_1 \cup \gamma_2$.
\end{proof}
\noindent In other words, equilibrium models of $\alpha \vee \beta$ consists of three possibilities: (1) common equilibrium models of $\alpha$ and $\beta$; (2) equilibrium models of $\alpha$ that are not classical models of $\beta$; and (3), vice versa, equilibrium models of $\beta$ that are not classical models of $\alpha$. Note that $\den{\alpha}_e \cap \den{\beta}_e \subseteq \den{\alpha \vee \beta}_e \subseteq \den{\alpha}_e \cup \den{\beta}_e$. As an example, consider the disjunction $p \vee (\neg p \rightarrow q)$ with $\alpha=p$ and $\beta=(\neg p \rightarrow q)$. The equilibrium models of each disjunct are $\den{p}_e=\{\str{20}\}$ and $\den{\neg p \rightarrow q}_e = \{\str{02}\}$, respectively. Obviously, $\alpha$ and $\beta$ have no common equilibrium model. Interpretation $\str{02}$ is an equilibrium model of $\beta$ and is not classical model of $\alpha$, and thus, it is an equilibrium model of $\alpha \vee \beta$. However, $\str{20}$ is both an equilibrium model of $\alpha$ and a classical model of $\beta$, and so it is disregarded. As a result, $\den{p \vee (\neg p \rightarrow q)}_e=\{\str{02}\}$.
As another example, take $r \vee (\neg p \rightarrow q)$. In this case, $\den{r}_e=\{\str{002}\}$ and $\den{\neg p \rightarrow q}_e=\{\str{020}\}$. Since each equilibrium model of one disjunct is not a classical model of the other disjunct, $\den{r \vee (\neg p \rightarrow q)}_e=\den{r}_e \cup \den{\neg p \rightarrow q}_e = \{\str{002},\str{020}\}$.
\section{Entailment relations}\label{sec:entail}
Logical entailment is usually defined by saying that the models of a formula (or a theory) are a subset of models of another formula (the entailed consequence). In our setting, we may consider different sets of models of a same formula $\alpha$: $\den\alpha$, $\den{\alpha}_c$ and $\den{\alpha}_e$. Therefore, it is not so strange that we can find different types of entailments for ASP in the literature. We summarize some of them in the following definition.
\begin{definition}
Given two formulas $\alpha, \beta$ we say that:
\begin{tabular}{rlcl}
$\alpha$ \emph{entails} $\beta$ (in $G_3$), & written $\alpha \models \beta$, & iff & $\den{\alpha} \subseteq \den{\beta}$\\
$\alpha$ \emph{classically entails} $\beta$, & written $\alpha \models_c \beta$, & iff & $\den{\alpha}_c \subseteq \den{\beta}_c$\\
$\alpha$ \emph{skeptically entails} $\beta$, & written $\alpha \models_{sk} \beta$, & iff & $\den{\alpha}_e \subseteq \den{\beta}_c$\\
$\alpha$ \emph{credulously entails} $\beta$, & written $\alpha \models_{cr} \beta$, & iff & $\den{\alpha}_e \cap \den{\beta}_c \neq \emptyset$\\
$\alpha$ \emph{weakly entails} $\beta$, & written $\alpha \models_{e} \beta$ & iff & $\den{\alpha}_{e} \subseteq \den{\beta}_{e}$\\
$\alpha$ \emph{strongly entails} $\beta$, & written $\alpha \models_{s} \beta$, & iff & for any formula $\gamma$, \\
& & & $\alpha \wedge \gamma \models_{e} \beta \wedge \gamma$, that is,\\
& & & $\den{\alpha \wedge \gamma}_e \subseteq \den{\beta \wedge \gamma}_e$ ~\hfill$\Box$
\end{tabular}
\end{definition}
The first two relations, $\models$ and $\models_c$, correspond to logical entailments in the monotonic logics of $G_3$ and classical propositional calculus, respectively. Obviously, $G_3$ entailment implies classical entailment (remember that $S_c=S\cap {\cal I}_c$). The next two entailments, $\models_{sk}$ and $\models_{cr}$ are typically used for non-monotonic queries where $\alpha$ is assumed to be a program and $\beta$ some query in classical logic. In this way, $\beta$ is a skeptical (resp. credulous) consequence of $\alpha$ if any (resp. some) equilibrium model of $\alpha$ is a classical model of $\beta$. In~\cite{Pea06}, an \emph{equilibrium entailment}, $\alpha \ |\!\!\!\sim \beta$, is defined as $\alpha \models_{sk} \beta$ when $\den\alpha \neq {\cal I}$ and $\den{\alpha}_e \neq \emptyset$, and $\alpha \models_c \beta$ otherwise.
The direct entailment between two programs would correspond to $\models_e$ which we have called here \emph{weak entailment}. The idea is that $\alpha \models_e \beta$ means that the equilibrium models of program $\alpha$ are also equilibrium models of $\beta$. An operational reading of this entailment is that, in order to obtain equilibrium models for $\beta$, we can try solving $\alpha$ and, if a solution for the latter is found, it will also be a solution to the original program. If this same relation holds \emph{for any context} $\gamma$, i.e., we can replace $\beta$ by $\alpha$ inside some larger program and the solutions of the result are still solutions for the original program, then we talk about \emph{strong entailment}.
To the best of our knowledge, the strong entailment relation has not been studied in the literature although its induced equivalence relation, \emph{strong equivalence}~\cite{LPV01}, is well-known and was, in fact, one of the main motivations that originated the interest in ASP for $G_3$ and equilibrium logic. It is obvious that strong entailment implies weak entailment (it suffices with taking $\gamma=\top$). Using the previous entailment relations, we can define several equivalence relations by considering entailment in both directions. As a result, we get the following derived characterisations:
\begin{definition}
Given two formulas $\alpha, \beta$ we say that:
\begin{tabular}{rl@{\!}c@{\!}l}
$\alpha$ is \emph{equivalent} to $\beta$ (in $G_3$), & written $\alpha \equiv \beta$, & iff & $\den{\alpha}=\den{\beta}$\\
$\alpha$ is \emph{classically equivalent} to $\beta$, & written $\alpha \equiv_c \beta$, & iff & $\den{\alpha}_c = \den{\beta}_c$\\
$\alpha$ is \emph{weakly equivalent} to $\beta$, & written $\alpha \equiv_{e} \beta$ & iff & $\den{\alpha}_{e} = \den{\beta}_{e}$\\
$\alpha$ is \emph{strongly equivalent} to $\beta$, & written $\alpha \equiv_{s} \beta$, & iff & for any formula $\gamma$, \\
& & & $\den{\alpha \wedge \gamma}_e = \den{\beta \wedge \gamma}_e$. ~\hfill$\Box$
\end{tabular}
\end{definition}
Note how $\alpha \equiv_s \beta$ iff both $\alpha \models_s \beta$ and $\beta \models_s \alpha$. The following result is a rephrasing of the main theorem in~\cite{LPV01}.
\begin{theorem}[From Theorem~1 in~\cite{LPV01}]\label{th:se}
Two formulas $\alpha, \beta$ are strongly equivalent iff they are equivalent in $G_3$ (or HT). In other words: $\alpha \equiv_s \beta$ iff $\alpha \equiv \beta$.~\hfill$\Box$
\end{theorem}
It is, therefore, natural to wonder whether this relation also holds for entailment, that is, whether strong entailment $\alpha \models_s \beta$ also corresponds to entailment\footnote{As a matter of fact, other authors~\cite{DSTW08,SL14} have implicitly or explicitly used HT entailment (i.e. our $G_3$ relation $\models$) as one of the two directions of strong equivalence without considering that there could exist a difference between $\models$ and $\models_s$ as we defined here.} in $G_3$, $\alpha \models \beta$. However, it is easy to see that these two relations \emph{are different}. As a counterexample, let $\alpha=(p \vee q)$ and $\beta=(\neg p \rightarrow q)$ from Example~\ref{ex:1}. We can easily check that $\alpha \models \beta$: indeed, $\den{\alpha}=\{\str{20},\str{02},\str{21},\str{12},\str{22}\}$ $\subseteq \den{\beta}$ as we saw in Example~\ref{ex:1}. However, the interpretation $\str{20}$ ($p$ true and $q$ false) is an equilibrium model of $\alpha$ which is not equilibrium model of $\beta$. Thus, $\alpha \not\models_e \beta$ and so $\alpha \not\models_s \beta$ either, since weak entailment is obviously a necessary condition for strong entailment.
Fortunately, strong entailment can be compactly captured using the denotational semantics, as we prove next. We begin proving an auxiliary result.
\begin{lemma}
\label{one_aux_lemma} \zlabel{one_aux_lemma}
Given any $v \in {\cal I}$, let $\gamma_v$ be the formula: $$\gamma_{v} \mathbin{\stackrel{\mathrm{def}}{=}} \bigwedge_{v(p)=2} p$$
Then, for any formula $\alpha$ and any $v \in \den{\alpha}_c$, we have $v \in \den{\alpha \wedge \gamma_v}_e$.~\hfill$\Box$
\end{lemma}
\begin{theorem}\label{th:sentail}
$\alpha \models_s \beta$ iff the following two conditions hold:
\begin{enumerate}
\item[(i)] $\alpha \models_c \beta$
\item[(ii)] $ \den{\alpha}_c \downarrow \cap \, \den{\beta} \subseteq \den{\alpha}$
\end{enumerate}
\end{theorem}
\begin{proof}
We are going to start proving that the two conditions are sufficient for $\alpha \models_{s} \beta$. Let us take any formula $\gamma$ and any $v \in \den{\alpha \wedge \gamma}_e$. Then both $v \in \den{\gamma}_c$ and $v \in \den{\alpha}_c \subseteq \den{\beta}_c$. Thus, $v \in \den{\beta \wedge \gamma}_c$. Suppose we had some $u < v$, $u \in \den{\beta \wedge \gamma}$. Then, $u \in \den{\alpha}_c \downarrow$ because $u < v \in \den{\alpha}_c$. But then, $u \in \den{\alpha}_c \downarrow \cap \, \den{\beta} \subseteq \den{\alpha}$, and as $u \in \den{\gamma}$ too, we would get that $v$ is not in equilibrium for $\alpha \wedge \gamma$, reaching a contradiction.
For proving that the two conditions are necessary, suppose $\alpha \models_s \beta$. For (i), take $v \in \den{\alpha}_c$. Since $v \in \den{\alpha \wedge \gamma_v}_e$ because of Lemma \ref{one_aux_lemma}, it follows that $v \in \den{\beta \wedge \gamma_v}_e \subseteq \den{\beta}_c$.
For (ii), take some $u \in \den{\alpha}_c \downarrow \cap \, \den\beta$ and assume $u \not \in \den{\alpha}$. Since $u \in\den{\alpha}_c \downarrow$ and $u \not\in\den\alpha$, we conclude $u_t \in \den{\alpha}_c$ and $u<u_t$. Consider the formula:
$$\gamma:= \gamma_{u} \wedge \bigwedge_{u(p)=u(q)=1} p \rightarrow q$$
\noindent so that, obviously, $u \in \den\gamma$. We are going to show that $u_t \in \den{\alpha \wedge \gamma}_e$ but $u_t \not \in \den{\beta \wedge \gamma}_e$ something that contradicts strong entailment. We begin observing that $u_t \not \in \den{\beta \wedge \gamma}_e$ because $u \in \den\gamma \cap \den\beta = \den{\gamma \wedge \beta}$ but $u<u_t$ so $u_t$ is not in equilibrium. Now, to show $u_t \in \den{\alpha \wedge \gamma}_e$, it is easy to see that $u_t \in \den{\alpha \wedge \gamma}_c$, since we had $u_t \in \den{\alpha}_c$ and $u \in \den\gamma$ implies $u_t\in \den{\gamma}_c$. To see that $u_t$ is in equilibrium, take any $w \in \den{\alpha \wedge \gamma}$ such that $w < u_t$. Now, notice that $w_t=u_t$, but $w \in \den\gamma \subseteq \den{\gamma_u}$ and, thus, the only possibility is that $w \geq u$. Moreover $w > u$ because $w\in\den\alpha$ while $u \not\in \den\alpha$. From $u<w<u_t(=w_t)$ we conclude that there exists some atom $p$, $u(p)=1$ and $w(p)=2$, and some atom $q$, $w(q)=1$ and $w_t(q)=u_t(q)=2$. But then, $w(q)=1$ implies $u(q)=1$ too and we get $u(p)=u(q)=1$ so that implication $p \rightarrow q$ occurs in the conjunction in $\gamma$. However, $w(p)=2$ and $w(q)=1$ means that $w$ is not a model of $p\rightarrow q$, which contradicts the assumption $w \in\den{\alpha \wedge \gamma}$.
\end{proof}
The proof to show that (ii) is a necessary condition for strong entailment relies on showing that, if it does not hold, we can build a formula $\gamma$ (a logic program) for which $\alpha \wedge \gamma \not\models_e \beta \wedge \gamma$. In fact, this part of the proof is not new: it reproduces the logic program built in the proof for Theorem 1 in~\cite{LPV01} for strong equivalence. However, \cite{LPV01} did not explicitly consider the concept of strong entailment, nor its characterisation in terms of sets of models, as provided here in Theorem~\ref{th:sentail}.
Once Theorem~\ref{th:sentail} is separated as an independent result, we can easily provide an immediate proof of Theorem~\ref{th:se}. Combining both entailment directions of $\alpha \equiv_s \beta$ amounts now to satisfying the three conditions:
\begin{enumerate}
\item[(i)] $\den{\alpha}_c=\den{\beta}_c$
\item[(ii)] $\den{\alpha}_c \downarrow \cap \, \den{\beta} \subseteq \den{\alpha}$
\item[(iii)] $\den{\beta}_c \downarrow \cap \, \den{\alpha} \subseteq \den{\beta}$
\end{enumerate}
\noindent but as $\den\beta \subseteq \den{\beta}_c \downarrow = \den{\alpha}_c \downarrow$ and $\den\alpha \subseteq \den{\alpha}_c \downarrow = \den{\beta}_c \downarrow$ we eventually get: (i) $\den{\alpha}_c=\den{\beta}_c$; (ii) $\den{\beta} \subseteq \den{\alpha}$; and (iii) $\den{\alpha} \subseteq \den{\beta}$. But these, altogether, are equivalent to $\den\alpha = \den\beta$.
To conclude this section, we consider an application of Theorem~\ref{th:sentail}, providing a sufficient condition for strong entailment that may be useful in some cases. Suppose that, apart from condition (i) of Theorem~\ref{th:sentail}, we further had $\beta \models \alpha$. Then, condition (ii) would become trivial since $\den{\alpha}_c \downarrow \cap \, \den{\beta} \subseteq \den{\beta}$ and $\den{\beta} \subseteq \den\alpha$. Therefore:
\begin{corollary}\label{cor:ex}
If $\alpha \models_c \beta$ and $\beta \models \alpha$ then $\alpha \models_s \beta$.~\hfill$\Box$
\end{corollary}
As an example, suppose we have a program $\Pi=\beta \wedge \gamma$ containing the disjunction $\beta=p \vee q$, typically used, for instance, to generate a choice between $p$ and $q$ in ASP. This formula is classically equivalent to $\alpha=(\neg p \rightarrow q) \wedge (\neg q \rightarrow p)$ which is also a common way for generating choices in ASP that does not use disjunction. Unfortunately, it is well-known that, in the general case $\alpha$ and $\beta$ are not strongly equivalent. For instance, if $\Pi=\beta \wedge (p\rightarrow q) \wedge (q \rightarrow p)$ we get the equilibrium model $\str{22}$ ($p$ and $q$ true) whereas for $\Pi'=\alpha \wedge (p\rightarrow q) \wedge (q \rightarrow p)$ we get no equilibrium model. However, $\beta \models \alpha$ in $G_3$ and, by Corollary~\ref{cor:ex}, if we replace $\beta$ by $\alpha$ in $\Pi$, any equilibrium (or stable) model we obtain in the new program will also be an equilibrium model of the original one (although, perhaps, we may lose equilibrium models with the replacement). Moreover, we can also replace $\beta=p \vee q$ by $\alpha'=\neg p \rightarrow q$ or by $\alpha''=\neg q \rightarrow p$ and the same property will still hold.
\section{Conclusions}\label{sec:conc}
We have introduced an alternative formulation of equilibrium models and its monotonic basis, Here-and-There (or, more precisely, G\"odel's three-valued logic $G_3$) that assigns a set of models (called a denotation) to each formula. This semantics, the main contribution of the paper, allows describing $G_3$, classical and equilibrium models using several compact set operations. Using denotations, we have proved again some already known fundamental results for $G_3$ or Equilibrium Logic to show that much textual effort usually done in the literature can be rephrased in terms of formal equivalences on sets of interpretations that, in many cases, even amount to simple properties from standard set theory. On the other hand, as side contributions or applications of this semantics, we have also obtained some additional fundamental results. For instance, we have proved that, while disjunction in $G_3$ is definable in terms of the other connectives, conjunction is a basic operation and cannot be derived from disjunction and implication. We have also shown that the equilibrium models of a disjunction can be obtained in a compositional way, in terms of the equilibrium and classical models of the disjuncts. Finally, we have defined (and characterised in denotational terms) a new type of entailment we called \emph{strong entailment}: a formula strongly entails another formula if the latter can be replaced by the former in any context while keeping a subset of the original equilibrium models.
A recent outcome of our current work is~\cite{CMMSE15} focused on the formulation of the denotational semantics using a theorem prover so that most of the meta-theorems for Equilibrium Logic and $G_3$ in this paper have been automatically checked using the PVS theorem prover~\cite{PVS92}. Future work includes the reformulation in denotational terms of different classes of models that are known to charaterise syntactic subclasses of logic programming~\cite{FINK11,FINK13} and the extension to the infinitary and first order versions of Equilibrium Logic. Finally, it would also be interesting to explore how the new definition of strong entailment can be applied in belief update or even inductive learning for ASP.
\newpage
\bibliographystyle{acmtrans}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 7,601 |
\section{Introduction} AT LINE 202
\documentclass[11pt,reqno,tbtags]{amsart}
\usepackage{amsthm,amssymb,amsfonts,amsmath}
\usepackage{amscd}
\usepackage{mathrsfs}
\usepackage[numbers, sort&compress]{natbib}
\usepackage{subfigure, graphicx}
\usepackage{vmargin}
\usepackage{setspace}
\usepackage{paralist}
\usepackage[pagebackref=false]{hyperref}
\usepackage{color}
\usepackage[normalem]{ulem}
\usepackage{stmaryrd}
\usepackage[refpage,noprefix,intoc]{nomencl}
\usepackage{setspace}
\usepackage{cancel}
\usepackage{tikz}
\usetikzlibrary{patterns}
\usepackage{sidecap}
\usepackage[font=small, labelfont=bf, labelsep=period]{caption}
\providecommand{\R}{}
\providecommand{\Z}{}
\providecommand{\N}{}
\providecommand{\C}{}
\providecommand{\Q}{}
\providecommand{\G}{}
\providecommand{\Lt}{}
\renewcommand{\R}{\mathbb{R}}
\renewcommand{\Z}{\mathbb{Z}}
\renewcommand{\N}{{\mathbb N}}
\renewcommand{\C}{\mathbb{C}}
\renewcommand{\Q}{\mathbb{Q}}
\renewcommand{\G}{\mathbb{G}}
\renewcommand{\Lt}{\mathbb{L}}
\newcommand{\E}[1]{{\mathbf E}\left[#1\right]}
\newcommand{{\mathbf E}}{{\mathbf E}}
\newcommand{\V}[1]{{\mathbf{Var}}\left\{#1\right\}}
\newcommand{{\mathbf{Var}}}{{\mathbf{Var}}}
\newcommand{\p}[1]{{\mathbf P}\left\{#1\right\}}
\newcommand{\psub}[2]{{\mathbf P}_{#1}\left\{#2\right\}}
\newcommand{\psup}[2]{{\mathbf P}^{#1}\left\{#2\right\}}
\newcommand{\I}[1]{{\mathbf 1}_{[#1]}}
\newcommand{\set}[1]{\left\{ #1 \right\}}
\newcommand{\Cprob}[2]{\mathbf{P}\set{\left. #1 \; \right| \; #2}}
\newcommand{\probC}[2]{\mathbf{P}\set{#1 \; \left| \; #2 \right. }}
\newcommand{\phat}[1]{\ensuremath{\hat{\mathbf P}}\left\{#1\right\}}
\newcommand{\Cexp}[2]{\mathbf{E}\set{\left. #1 \; \right| \; #2}}
\newcommand{\expC}[2]{\mathbf{E}\set{#1 \; \left| \; #2 \right. }}
\newcommand{\Ehat}[1]{\ensuremath{\hat{\mathbf E}}\left[#1\right]}
\newcommand{\ensuremath{\hat{\mathbf E}}}{\ensuremath{\hat{\mathbf E}}}
\newcommand{\Esup}[2]{{\mathbf E^{#1}}\left\{#2\right\}}
\newcommand{\esup}[1]{{\mathbf E^{#1}}}
\newcommand{\Esub}[2]{{\mathbf E_{#1}}\left\{#2\right\}}
\newcommand{\esub}[1]{{\mathbf E_{#1}}}
\newcommand\cA{\mathcal A}
\newcommand\cB{\mathcal B}
\newcommand\cC{\mathcal C}
\newcommand\cD{\mathcal D}
\newcommand\cE{\mathcal E}
\newcommand\cF{\mathcal F}
\newcommand\cG{\mathcal G}
\newcommand\cH{\mathcal H}
\newcommand\cI{\mathcal I}
\newcommand\cJ{\mathcal J}
\newcommand\cK{\mathcal K}
\newcommand\cL{{\mathcal L}}
\newcommand\cM{\mathcal M}
\newcommand\cN{\mathcal N}
\newcommand\cO{\mathcal O}
\newcommand\cP{\mathcal P}
\newcommand\cQ{\mathcal Q}
\newcommand\cR{{\mathcal R}}
\newcommand\cS{{\mathcal S}}
\newcommand\cT{{\mathcal T}}
\newcommand\cU{{\mathcal U}}
\newcommand\cV{\mathcal V}
\newcommand\cW{\mathcal W}
\newcommand\cX{{\mathcal X}}
\newcommand\cY{{\mathcal Y}}
\newcommand\cZ{{\mathcal Z}}
\newcommand{\bA}{\mathbf{A}}
\newcommand{\bB}{\mathbf{B}}
\newcommand{\bC}{\mathbf{C}}
\newcommand{\bD}{\mathbf{D}}
\newcommand{\bE}{\mathbf{E}}
\newcommand{\bF}{\mathbf{F}}
\newcommand{\bG}{\mathbf{G}}
\newcommand{\bH}{\mathbf{H}}
\newcommand{\bI}{\mathbf{I}}
\newcommand{\bJ}{\mathbf{J}}
\newcommand{\bK}{\mathbf{K}}
\newcommand{\bL}{\mathbf{L}}
\newcommand{\bM}{\mathbf{M}}
\newcommand{\bN}{\mathbf{N}}
\newcommand{\bO}{\mathbf{O}}
\newcommand{\bP}{\mathbf{P}}
\newcommand{\bQ}{\mathbf{Q}}
\newcommand{\bR}{\mathbf{R}}
\newcommand{\bS}{\mathbf{S}}
\newcommand{\bT}{\mathbf{T}}
\newcommand{\bU}{\mathbf{U}}
\newcommand{\bV}{\mathbf{V}}
\newcommand{\bW}{\mathbf{W}}
\newcommand{\bX}{\mathbf{X}}
\newcommand{\bY}{\mathbf{Y}}
\newcommand{\mathbf{Z}}{\mathbf{Z}}
\newcommand{\ensuremath{\stackrel{\mathrm{d}}{=}}}{\ensuremath{\stackrel{\mathrm{d}}{=}}}
\newcommand{\ensuremath{\stackrel{\mathrm{d}}{\rightarrow}}}{\ensuremath{\stackrel{\mathrm{d}}{\rightarrow}}}
\newcommand{\ensuremath{\stackrel{\mathrm{a.s.}}{\rightarrow}}}{\ensuremath{\stackrel{\mathrm{a.s.}}{\rightarrow}}}
\newcommand{\ensuremath{\stackrel{\mathrm{a.s.}}{=}}}{\ensuremath{\stackrel{\mathrm{a.s.}}{=}}}
\newcommand{\pran}[1]{\left(#1\right)}
\providecommand{\eps}{}
\renewcommand{\eps}{\epsilon}
\providecommand{\ora}[1]{}
\renewcommand{\ora}[1]{\overrightarrow{#1}}
\newcommand\urladdrx[1]{{\urladdr{\def~{{\tiny$\sim$}}#1}}}
\DeclareRobustCommand{\SkipTocEntry}[5]{}
\newtheorem{thm}{Theorem}
\newtheorem{lem}[thm]{Lemma}
\newtheorem{prop}[thm]{Proposition}
\newtheorem{cor}[thm]{Corollary}
\newtheorem{dfn}[thm]{Definition}
\newtheorem{definition}[thm]{Definition}
\newtheorem{conj}{Conjecture}
\newtheorem{ex}{Exercise}[section]
\newtheorem{fact}[thm]{Fact}
\newtheorem{claim}[thm]{Claim}
\newtheorem{cla}[thm]{Claim}
\renewcommand{\nomname}{List of notation and terminology}
\renewcommand*{\pagedeclaration}[1]{\unskip\dotfill\hyperpage{#1}}
\makenomenclature
\setlength{\nomitemsep}{-\parsep}
\graphicspath{ {./images/} }
\usepackage{wrapfig}
\newcommand{\ensuremath{v}}{\ensuremath{v}}
\newcommand{\ensuremath{e}}{\ensuremath{e}}
\newcommand{\ensuremath{r}}{\ensuremath{r}}
\newcommand{\ensuremath{\mathrm{d}}}{\ensuremath{\mathrm{d}}}
\newcommand{\ensuremath{\mathrm{wid}}}{\ensuremath{\mathrm{wid}}}
\newcommand{\ensuremath{\mathrm{ht}}}{\ensuremath{\mathrm{ht}}}
\providecommand{\deg}{}
\renewcommand{\deg}{\ensuremath{d}}
\renewcommand{\ensuremath{\mathrm{d}}}{\ensuremath{\mathrm{d}}}
\newcommand{\ensuremath{\mathscr{T}}}{\ensuremath{\mathscr{T}}}
\newcommand{\ensuremath{\mathrm{n}}}{\ensuremath{\mathrm{n}}}
\newcommand{\ensuremath{p}}{\ensuremath{p}}
\newcommand{\ensuremath{S}}{\ensuremath{S}}
\newcommand{\ensuremath{\mathbf{d}}}{\ensuremath{\mathbf{d}}}
\newcommand{\ensuremath{\mathrm{v}}}{\ensuremath{\mathrm{v}}}
\newcommand{\ensuremath{\mathrm{oe}}}{\ensuremath{\mathrm{oe}}}
\newcommand{\ensuremath{\mathrm{w}}}{\ensuremath{\mathrm{w}}}
\newcommand{\ensuremath{\mathcal{T}}}{\ensuremath{\mathcal{T}}}
\numberwithin{equation}{section}
\numberwithin{thm}{section}
\usepackage{framed}
\renewenvironment{leftbar}[1][\hsize]
{%
\def\FrameCommand
{%
{\color{red}\vrule width 3pt}%
\hspace{0pt
\fboxsep=\FrameSep\colorbox{white}%
}%
\MakeFramed{\hsize#1\advance\hsize-\width\FrameRestore}%
}
{\endMakeFramed}
\newcommand\LAB{\marginal{LAB}}
\definecolor{ccel}{rgb}{0.36, 0.54, 0.66}
\newcommand{\cel}[1]{\textcolor{ccel}{#1}}
\newcommand{\celt}[1]{\textcolor{ccel}{\sout{#1}}}
\begin{document}
\setstretch{1.2}
\title{Universal height and width bounds for random trees}
\address{Department of Mathematics and Statistics, McGill University, Montr\'eal, Canada}
\author{Louigi Addario-Berry}
\email{louigi.addario@mcgill.ca}
\author{Anna Brandenberger}
\email{anna.brandenberger@mail.mcgill.ca}
\author{Jad Hamdan}
\email{jad.hamdan@mail.mcgill.ca}
\author{C\'eline Kerriou}
\email{celine.kerriou@mail.mcgill.ca}
\date{September 25, 2021; revised April 25, 2022}
\keywords{Random trees, Galton--Watson trees, Bienaym\'e trees, simply generated trees, width, height.}
\subjclass[2010]{60C05,60J80,05C05}
\begin{abstract}
We prove non-asymptotic stretched exponential tail bounds on the height of a randomly sampled node in a random combinatorial tree, which we use to prove bounds on the heights and widths of random trees from a variety of models. Our results allow us to prove a conjecture and settle an open problem of Janson~\cite{MR2908619}, and nearly prove another conjecture and settle another open problem from the same work (up to a polylogarithmic factor).
The key tool for our work is an equivalence in law between the degrees along the path to a random node in a random tree with given degree statistics, and a random truncation of a size-biased ordering of the degrees of such a tree. We also exploit a Poissonization trick introduced by Camarri and Pitman \cite{MR1741774} in the context of inhomogeneous continuum random trees, which we adapt to the setting of random trees with fixed degrees.
Finally, we propose and justify a change to the conventions of branching process nomenclature: the name ``Galton--Watson trees'' should be permanently retired by the community, and replaced with the name ``Bienaym\'e trees''.
\end{abstract}
\maketitle
\section{\bf Introduction}\label{sec:intro}
This paper concerns the height and width of random plane trees, and applications of bounds thereof to the study of random simply generated trees and to the family trees of branching processes. Our results in particular allow us to settle two conjectures from \cite{MR2908619}, and to nearly settle two others.
By a plane tree, we mean a finite rooted tree $t=(\ensuremath{v}(t),\ensuremath{e}(t))$ in which the set of children of each node is endowed with a left-to-right order. The root of $t$ is denoted $\ensuremath{r}(t)$. The {\em degree} of a node $v \in \ensuremath{v}(t)$, denoted $\deg_t(v)$, is its number of children in $t$, so leaves have degree $0$ and all other nodes have strictly positive degree.
The {\em degree statistics} of $t$ is the sequence $\ensuremath{\mathrm{n}}_t=(\ensuremath{\mathrm{n}}_t(c),c \ge 0)$, where $\ensuremath{\mathrm{n}}_t(c) = |\{v \in \ensuremath{v}(t): \deg_t(v)=c\}|$ is the number of nodes of $t$ with $c$ children.
Note that
\[
|v(t)|
= \sum_{c \ge 0} \ensuremath{\mathrm{n}}_t(c)
=1+\sum_{c \ge 0} c\ensuremath{\mathrm{n}}_t(c)
=1+|e(t)| \, .
\]
A sequence $\ensuremath{\mathrm{n}}=(\ensuremath{\mathrm{n}}(c),c \ge 0)$ is the degree statistics of some tree if and only if $\sum_{c \ge 0} \ensuremath{\mathrm{n}}(c)=1+\sum_{c \ge 0} c\ensuremath{\mathrm{n}}(c)$. For such sequences, we write $\ensuremath{\mathscr{T}}_{\ensuremath{\mathrm{n}}}$ for the set of plane trees with degree statistics $\ensuremath{\mathrm{n}}$, and write
\[
\ensuremath{\mathscr{T}}_{\ensuremath{\mathrm{n}}}^{\bullet}
= \{(t,v): t \in \ensuremath{\mathscr{T}}_{\ensuremath{\mathrm{n}}}, v \in v(t)\}.
\]
A {\em marked tree} is a pair $(t,v)$ where $t$ is a plane tree and $v \in v(t)$; so the elements of $\ensuremath{\mathscr{T}}_{\ensuremath{\mathrm{n}}}^{\bullet}$ are precisely the marked trees with degree statistics $\ensuremath{\mathrm{n}}$.
For a node $v \in \ensuremath{v}(t)$, the {\em height} of $v$, denoted $|v|$, is the graph distance from $v$ to $\ensuremath{r}(t)$. The height of $t$, denoted $\ensuremath{\mathrm{ht}}(t)$, is $\max(|v|: v \in \ensuremath{v}(t))$. The {\em width} of $t$ at level $k$, denoted $\ensuremath{\mathrm{wid}}(t,k)$, is $|\{v \in t: |v|=k\}|$, and the width of $t$, denoted $\ensuremath{\mathrm{wid}}(t)$, is $\max(\ensuremath{\mathrm{wid}}(t,k),k \ge 0)$.
Given a a sequence $\ensuremath{\mathrm{n}}=(\ensuremath{\mathrm{n}}(c),c \ge 0)$ of non-negative real numbers, for $p>0$, we write $|\ensuremath{\mathrm{n}}|_p = (\sum_{c \ge 0} c^p \ensuremath{\mathrm{n}}(c))^{1/p}$. Note that for a plane tree $t$, we have $|\ensuremath{\mathrm{n}}_t|_1+1 = |t|$.
We prove the following non-asymptotic tail bounds on the height of a randomly sampled node in a random plane tree with given degree statistics. For a finite set $S$ we write $X \in_u S$ to mean that $X$ is a uniformly random element of the set $S$.
\begin{thm}
\label{thm:main}
Fix degree statistics $\ensuremath{\mathrm{n}}=(\ensuremath{\mathrm{n}}(c),c \ge 0)$ and let $(T,V) \in_u \ensuremath{\mathscr{T}}_\ensuremath{\mathrm{n}}^{\bullet}$. Then for all $\beta > 17^{3/2}$,
\[
\p{|V| > \beta \frac{|\ensuremath{\mathrm{n}}|_1}{(|\ensuremath{\mathrm{n}}|_2^2-\ensuremath{\mathrm{n}}(1))^{1/2}}}
\le
\exp\pran{-\frac{\beta^{1/3}}{3}\frac{|\ensuremath{\mathrm{n}}|_1}{(|\ensuremath{\mathrm{n}}|_2^2-\ensuremath{\mathrm{n}}(1))^{1/2}}} + 2\exp\pran{-\frac{\beta^{2/3}}{24}}\, ,
\]
and if $\ensuremath{\mathrm{n}}(1)=0$ then for all $\ell \ge 1$,
\[
\p{|V| \ge \ell}
\le
\exp\pran{-\frac{\ell^2}{2|\ensuremath{\mathrm{n}}|_1}}\, .
\]
\end{thm}
A related bound, recently proved by Marzouk \cite[Proposition 5]{marzouk}, strengthens the first of the two bounds stated in the preceding theorem, up to constant factors.
These bounds, interesting in their own right, also have several consequences for the family trees of branching processes, which are summarized in our other main theorems, below. In order to state the theorems, we need a little more terminology.
Given a tree $t$, write $t^{\le k}$ for the subtree of $t$ consisting of all nodes $u \in \ensuremath{v}(t)$ with $|u| \le k$.
Let $\mu$ be a probability distribution with support $\N$ (by which we mean that $\mu(\N)=1$). By a {\em Bienaym\'e tree with offspring distribution $\mu$}, we mean the family tree $T$ of a branching process with offspring distribution $\mu$.\footnote{We propose this name as an alternative to the term ``Galton--Watson tree'' since (a) Bienaym\'e's introduction of such trees both predates that of Galton and Watson and is more mathematically correct - see \cite{bienayme} and \cite[Pages 83-86]{cournot}; and (b) we prefer not to honour the founder of eugenics, Francis Galton, by continuing to attach his name to these mathematical objects. (Galton wrote: ``We greatly want a brief word to express
the science of improving stock, which is by no means confined to questions of judicious
mating, but which, especially in the case of man, takes cognisance of all influences that
tend in however remote a degree to give to the more suitable races or strains of blood a
better chance of prevailing speedily over the less suitable than they otherwise would have
had. The word eugenics would sufficiently express the idea'' \cite[Page 25]{galton83}.)}
The law of $T$ is uniquely determined by the property that for any plane tree $t$ of height at most $k$,
\[
\p{T^{\le k}=t} = \prod_{v \in t^{\le k-1}} \mu(d_t(v))\, .
\]
In the preceding formula and below, we write $\mu(k) = \mu(\{k\})$ for readability.
For $n \in \N$, if $\p{|\ensuremath{v}(T)|=n}>0$ then we define a Bienaym\'e tree conditioned to have size $n$ in the natural way: this is a random tree $T_n$ such that for any plane tree $t$ with $n$ vertices,
\[
\p{T_n=t} = \probC{T=t}{|\ensuremath{v}(T)|=n}\, .
\]
Finally, for a measure $\mu$ on $\R$, for $p > 0$ we write $|\mu|_p := (\int_\R |x|^p\mu(\mathrm{d}x))^{1/p}$. This agrees with the above notation $|\ensuremath{\mathrm{n}}|_p$ for sequences $\ensuremath{\mathrm{n}}=(\ensuremath{\mathrm{n}}(c),c \ge 0)$, by interpreting the sequence as the discrete measure assigning mass $\ensuremath{\mathrm{n}}(c)$ to each non-negative integer $c$.
\begin{thm}\label{thm:main2}
Fix a probability distribution $\mu$ with support $\N$, with $|\mu|_1 \le 1$ and $|\mu|_2 = \infty$. For $n \in \N$, let $T_n$ be a Bienaym\'e tree with offspring distribution $\mu$, conditioned to have size $n$, and let $V_n$ be a uniformly random node of $T_n$. Then $\ensuremath{\mathrm{wid}}(T_n)/n^{1/2} \to \infty$, $|V_n|/n^{1/2} \to 0$, and $\ensuremath{\mathrm{ht}}(T_n)/(n^{1/2}\log^3 n) \to 0$. All convergence results hold both in probability and in expectation, as $n \to \infty$.
\end{thm}
\begin{thm}\label{thm:main3}
Fix a probability distribution $\mu$ with support $\N$, with $|\mu|_1 <1$ and with $\sum_{c \ge 0} e^{tc}\mu(c)=\infty$ for all $t$. For $n \in \N$ let $T_n$ be a Bienaym\'e tree with offspring distribution $\mu$, conditioned to have size $n$, and let $V_n$ be a uniformly random node of $T_n$. Then $\ensuremath{\mathrm{wid}}(T_n)/n^{1/2} \to \infty$, $|V_n|/n^{1/2} \to 0$, and $\ensuremath{\mathrm{ht}}(T_n)/(n^{1/2}\log^3 n) \to 0$. All convergence results hold both in probability and in expectation, as $n \to \infty$.
\end{thm}
The results of Theorems~\ref{thm:main2} and~\ref{thm:main3} are close analogues of Conjectures 21.5 and 21.6 and Problems 21.7 and 21.8 from~\cite{MR2908619}, but those conjectures are stated for the slightly more general model of {\em simply generated trees}. In Section~\ref{sec:simply_generated} we define simply generated trees, state the aforementioned conjectures and problems precisely, and explain how to use Theorem~\ref{thm:main} to prove Conjecture~21.6 and solve Problem~21.8 from \cite{MR2908619}, and to nearly prove Conjecture 21.5 and nearly solve Problem~21.7 from the same paper.\footnote{``Nearly prove'' and ``nearly solve'' rather than ``prove'' and ``solve'' due to the presence of a $\log^{3} n$ factor in two of our bounds.} The key fact about simply generated trees is that, like conditioned Bienaym\'e trees, they are uniformly random conditional on their degree statistics, which allows us to apply Theorem~\ref{thm:main} to them.
We also prove height and width bounds for conditioned Bienaym\'e trees which hold without any assumptions on the offspring distribution at all, aside from the requirement that the resulting family trees have both leaves and branch points. \begin{thm}\label{thm:main4}
There exists a constant $C > 0$ such that the following holds. Fix a probability distribution $\mu$ with support $\N$ with $\mu(0)+\mu(1)<1$. For $n \in \N$, let $T_n$ be a Bienaym\'e tree with offspring distribution $\mu$, conditioned to have size $n$, and let $V_n$ be a uniformly random node of $T_n$.
Then ${\mathbf E} |V_n| \le Cn^{1/2}/(1-\mu(0)-\mu(1))^{1/2}$,
\[
\E{
\ensuremath{\mathrm{wid}}(T_n)} \ge \frac{(1-\mu(0)-\mu(1))^{1/2}}{C} n^{1/2} \quad \mbox{ and }\quad
\E{\ensuremath{\mathrm{ht}}(T_n)}\le C\frac{n^{1/2} \log^{3} n}{(1-\mu(0)-\mu(1))^{1/2}}.
\]
\end{thm}
\subsection{Discussion}
There is a substantial amount of past work on the heights and widths of random Bienaym\'e trees and random combinatorial trees \cite{MR2956056,MR3077536,MR3916103,MR3651047,MR3335012}, and bounds on these quantities, particularly the height, often feature in scaling limit theorems for random trees and associated objects \cite{ss21,MR4132643,MR3947331,MR3551197,MR3188597}. The works \cite{MR2956056,MR3077536,MR3916103} all bound the height via the study of the {\em depth-first exploration process} of the tree. This technique gives bounds which are frequently tight, up to constant factors, for trees whose offspring distributions are sufficiently light tailed (e.g. with finite variance). However, it does not appear well-suited to studying trees with heavy-tailed degrees (in which case the depth-first queue length is a poor proxy for the height).
For critical conditioned Bienaym\'e trees with finite variance ($|\mu|_1=1$, $|\mu|_2<\infty$), sub-Gaussian tail bounds for $n^{-1/2}\ensuremath{\mathrm{ht}}(T_n)$ and $n^{-1/2}\ensuremath{\mathrm{wid}}(T_n)$ are known \cite{MR3077536}. However, the authors of that paper state that they ``are not aware of any results [for the height and width] that hold for arbitrary offspring distributions.'' As far as the authors of the current paper are aware, this is still the case, and this paper is the first work to provide such results.
We do not expect that the stretched exponential tail bound of Theorem~\ref{thm:main} is tight. However, it is not completely clear what form an optimal bound ought to take. We now record some observations which limit how quickly the optimal bounds can decay, to help provide a sense of the potential complexities. These observations in particular show that the exponents $1/3$ and $2/3$ in Theorem~\ref{thm:main} can not be replaced by any values strictly greater than $1$, which means that one can not hope for sub-Gaussian tail bounds like those we prove for degree statistics with $\ensuremath{\mathrm{n}}(1)=0$ to hold in general. (The computations underlying the observations are not fully spelled out here but are not too complicated.)
First, fix $\alpha \in (1,2)$, and suppose that $|\mu|_1=1$ and $\mu(k,\infty) =(1+o(1))c k^{-\alpha}$ as $k \to \infty$, so $\mu$ is a critical offspring distribution in the domain of attraction of an $\alpha$-stable law.
In this setting, it is known \cite[Theorem 1.5]{MR3634265} that
\begin{equation}\label{eq:alpha_stable_bd}
\p{\ensuremath{\mathrm{ht}}(T_n) \ge c n^{1-1/\alpha}} \asymp c^{1+\alpha/2}\exp(- (\alpha-1)^{1/(\alpha-1)}c^{\alpha})
\end{equation}
as first $n\to\infty$, then $c \to \infty$.
Such a tree $T_n$ will typically have $\Theta(n k^{-\alpha-1})$ nodes of degree $k$ for $k \le n^{1/\alpha}$ and no nodes of degree much larger than $n^{1/\alpha}$, and so will satisfy $|\ensuremath{\mathrm{n}}_{T_n}|_2^2 \asymp n^{2/\alpha}$. Since $\alpha$ can be taken arbitrarily close to $1$, comparing the upper bound from Theorem~\ref{thm:main} with (\ref{eq:alpha_stable_bd}) shows that the exponent $2/3$ in Theorem~\ref{thm:main} can not be replaced with anything strictly greater than one.
Second, consider degree statistics of the form $\ensuremath{\mathrm{n}}=(k,k,\ldots,0,1,0,\ldots)$, corresponding to a tree with $n=2k+1$ nodes, with a single node of degree $k$, and $k$ nodes each of degrees $0$ and $1$. For such degree statistics, $|\ensuremath{\mathrm{n}}|_1/(|\ensuremath{\mathrm{n}}|_2^2-\ensuremath{\mathrm{n}}(1))^{1/2} = \Theta(1)$.
Moreover, it is not hard to see that with high probability a random tree with these degree statistics has height $\Theta(\log n)$, so there is $\delta >0$ such that the probability that a randomly sampled node has height at least $\delta\log n$ is at least $(\delta \log n)/n$. Combining these observations shows that neither the exponent $1/3$ nor the exponent $2/3$ in the first bound in Theorem~\ref{thm:main} can in general be replaced with anything greater than $1$. (Marzouk's result \cite[Proposition 5]{marzouk} shows that one {\em can} essentially replace both constants $1/3$ and $2/3$ by $1$; the above observations show that this is then best possible.)
We conclude the discussion with a word about Theorem~\ref{thm:main4}.
The dependence on $\mu(0)$ and $\mu(1)$ in that theorem is necessary; if $\mu(0)+\mu(1)=1$ then with probability one $T_n$ is a path with $n$ vertices, which has width $1$ and height $n-1$. Moreover, the form of the dependence in the theorem is essentially optimal. To see this, suppose that $\mu(1)=1-\eps$ and $\mu(0)=\mu(2)=\eps/2$.
Then with high probability $T_n$ will have
$(1+o(1))(1-\eps)n$ vertices with exactly one child. Let $\hat{T}_n$ be the tree obtained from $T_n$ by suppressing all vertices with exactly one child, so that $\hat{T}_n$ has only nodes with $0$ or $2$ children, and $T_n$ can be recovered from $\hat{T}_n$ by subdividing edges. Then $\hat{T}_n$ has size $(1+o(1))\eps n$ with high probability, and is a uniform binary tree conditional on its size, so has height $\Theta((\eps n)^{1/2})$ and width $\Theta((\eps n)^{1/2})$ in probability. Each edge of $\hat{T}_n$ is subdivided $\Theta(\eps^{-1})$ times on average in $T_n$, from which it is easy to believe (and not too hard to prove) that $T_n$ has height $\Theta((n/\eps)^{1/2})=\Theta((n/(1-\mu(0)-\mu(1)))^{1/2})$ and width $\Theta((\eps n)^{1/2}) = \Theta(((1-\mu(0)-\mu(1))n)^{1/2})$ in probability and in expectation.
\subsection{Notation}
For a sequence $(r_n,n \ge 1)$ of real numbers, we write $r_n=\ensuremath{\mathrm{oe}}(1)$ if there exists $c > 0$ such that $r_n \le e^{-cn}$ for all $n$ sufficiently large.
Given a sequence of events $(E_n,n \ge 1)$,
we say that $E_n$ occurs {\em with very low probability} (and that $E_n^c$ occurs {\em with very high probability}) if $\p{E_n}= \ensuremath{\mathrm{oe}}(1)$.
\section{\bf An overview of the proofs.}
\subsection{\bf A sampler for the height of the marked node}
Fix degree statistics $\ensuremath{\mathrm{n}}$, and let $(T,V) \in_u \ensuremath{\mathscr{T}}_{\ensuremath{\mathrm{n}}}^{\bullet}$. The tool which unlocks all the results of the paper is a sampling procedure which generates a random variable with the same law as $|V|$. To describe the sampling procedure, some notation is needed.
Given degree statistics $\ensuremath{\mathrm{n}}=(\ensuremath{\mathrm{n}}(c),c \ge 0)$, we say a random vector $D=(D_1,\ldots,D_n)$ is a {\em size-biasing of $\ensuremath{\mathrm{n}}$} if $n =\sum_{c \ge 0} \ensuremath{\mathrm{n}}(c) $ and for any sequence $\ensuremath{\mathrm{d}}=(d_1,\ldots,d_n)$ such that $|\{i \in [n]: d_i=c\}|=\ensuremath{\mathrm{n}}(c)$ for all $c$, the following holds. For each $1 \le k\le n$,
\begin{align}\label{eq:size_biasing}
\probC{D_k=d_k}{(D_1,\ldots,D_{k-1})=(d_1,\ldots,d_{k-1})} & = \frac{d_k(\ensuremath{\mathrm{n}}(d_k) - w((d_1,...,d_{k-1}),d_k))}{|\ensuremath{\mathrm{n}}|_1-d_1-\ldots-d_{k-1}}\, ,
\end{align}
where $w((d_1,...,d_{k-1}), d) = |\{i\in [k-1]:d_i = d\}|$ for $d\geq 0$. The fraction is interpreted as equal to $1$ if $d_1+\ldots+d_{k-1}=|\ensuremath{\mathrm{n}}|_1$. The definition implies that $|\{1 \le k \le n: D_k=c\}|=\ensuremath{\mathrm{n}}(c)$ almost surely, for all $c \ge 0$. It also implies that the final $\ensuremath{\mathrm{n}}(0)\ge 1$ entries of $D$ all equal $0$, and in particular $D_n=0$.
\begin{prop}\label{prop:sampler}
Fix degree statistics $\ensuremath{\mathrm{n}}$ and write $n=|\ensuremath{\mathrm{n}}|_1+1$. Let $(T,V) \in_u \ensuremath{\mathscr{T}}_{\ensuremath{\mathrm{n}}}^{\bullet}$, and let $D=(D_1,\ldots,D_n)$ be a size-biasing of $\ensuremath{\mathrm{n}}$.
Next let $(U_1,\ldots,U_n)$ be independent Uniform$[0,1]$ random variables, independent of $D$. For $i \in [n]$ let
\[
A_i =
\begin{cases}
1 & \mbox{ if }U_i \le\frac{1+\sum_{j=1}^{i-1} (D_j-1)}{n+1-i}\\
0 & \mbox{ otherwise}.
\end{cases}
\]
Finally, let $M = \min(i: A_i = 1)$. Then $|V|\ensuremath{\stackrel{\mathrm{d}}{=}} M-1$.
\end{prop}
In the above proposition, note that since $D_n=0$, when $i=n$ we have
\[
\frac{1+ \sum_{j=1}^{i-1} (D_j-1)}{n+1-i}
= 1+\sum_{j=1}^{n-1} (D_j-1) = 1,
\]
so $A_n=1$ and thus $M \le n$.
Theorem~\ref{thm:main} is an essentially immediate consequence of Proposition~\ref{prop:sampler} together with the next result.
\begin{thm}
\label{thm:main_mod}
Fix degree statistics $\ensuremath{\mathrm{n}}$, write $n=|\ensuremath{\mathrm{n}}|_1 + 1$, and let $D=(D_1,\ldots,D_n)$ be a size-biasing of $\ensuremath{\mathrm{n}}$.
Next let $(U_1,\ldots,U_n)$ be independent Uniform$[0,1]$ random variables, independent of $D$. For $i \in [n-1]$ let
\[
B_i =
\begin{cases}
1 & \mbox{ if }U_i \le\frac{\sum_{j=1}^{i-1} (D_j-1)}{n-i}\\
0 & \mbox{ otherwise}.
\end{cases}
\]
Finally, let $\sigma = \inf(i: B_i = 1)$. If
$\ensuremath{\mathrm{n}}(1)>0$ then for all $\beta > 17^{3/2}$,
\[
\p{\sigma > \beta \frac{|\ensuremath{\mathrm{n}}|_1}{(|\ensuremath{\mathrm{n}}|_2^2-\ensuremath{\mathrm{n}}(1))^{1/2}}}
\le
\exp\pran{-\frac{\beta^{1/3}}{3}\frac{|\ensuremath{\mathrm{n}}|_1}{(|\ensuremath{\mathrm{n}}|_2^2-\ensuremath{\mathrm{n}}(1))^{1/2}}} + 2\exp\pran{-\frac{\beta^{2/3}}{24}}\, ,
\]
and if $\ensuremath{\mathrm{n}}(1)=0$ then for all $\ell \ge 1$,
\[
\p{\sigma \ge \ell} \le e^{-(\ell-1)^2/(2|\ensuremath{\mathrm{n}}|_1)}\, .
\]
\end{thm}
Note that in Theorem~\ref{thm:main_mod}, we have $B_i=1$ iff $U_i \le \tfrac{\sum_{j=1}^{i-1} (D_j-1)}{n-i}$, whereas in Proposition~\ref{prop:sampler}, we have $A_i=1$ iff $U_i \le \tfrac{1+\sum_{j=1}^{i-1} (D_j-1)}{n+1-i}$.
Since $\tfrac{a+1}{b+1} \ge \tfrac{a}{b}$ whenever $\tfrac{a}{b} < 1$, it follows that the random variable $\sigma=\inf(i: B_i=1)$ stochastically dominates the random variable $M=\min(i:A_i=1)$. Thus, upper tail bounds for $\sigma$ automatically apply to $M$. Since $|V|\ensuremath{\stackrel{\mathrm{d}}{=}} M-1$, in view of this observation, the bounds of Theorem~\ref{thm:main} follow immediately from those of Theorem~\ref{thm:main_mod}.
\subsection{\bf Moving from random marked trees to Bienaym\'e trees}
Once Theorem~\ref{thm:main} is established, the primary work in proving the other results of the paper is to understand the degree statistics of conditioned Bienaym\'e trees, under various assumptions on their offspring distributions. We prove the following bounds.
\begin{prop}\label{prop:dstats_infinite_variance}
Fix a probability distribution $\mu$ with support $\N$ with $|\mu|_1 \le 1$ and $|\mu|_2 = \infty$. For $n \in \N$ let $T_n$ be a Bienaym\'e tree with offspring distribution $\mu$, conditioned to have size $n$, and let $\ensuremath{\mathrm{n}}_{T_n}$ be the degree statistics of $T_n$. Then for any $C > 0$, with very high probability $|\ensuremath{\mathrm{n}}_{T_n}|_2^2 \ge C |\ensuremath{\mathrm{n}}_{T_n}|_1$.
\end{prop}
\begin{prop}\label{prop:dstats_zero_radius}
Fix a probability distribution $\mu$ with support $\N$, with $|\mu|_1 < 1$ and with $\sum_{c \ge 0} e^{tc}\mu(c)=\infty$ for all $t>0$. For $n \in \N$ let $T_n$ be a Bienaym\'e tree with offspring distribution $\mu$, conditioned to have size $n$, and let $\ensuremath{\mathrm{n}}_{T_n}$ be the degree statistics of $T_n$. Then for any $C > 0$, with very high probability $|\ensuremath{\mathrm{n}}_{T_n}|_2^2 \ge C |\ensuremath{\mathrm{n}}_{T_n}|_1 $.
\end{prop}
\begin{prop}\label{prop:dstats_finite_variance}
Fix a probability distribution $\mu$ with support $\N$ and with $\mu(0) +\mu(1) < 1$. For $n \in \N$ let $T_n$ be a Bienaym\'e tree with offspring distribution $\mu$, conditioned to have size $n$, and let $\ensuremath{\mathrm{n}}_{T_n}$ be the degree statistics of $T_n$. Then for any $\eps > 0$, with very high probability $|\ensuremath{\mathrm{n}}_{T_n}|_2^2-\ensuremath{\mathrm{n}}_{T_n}(1)\ge |\ensuremath{\mathrm{n}}_{T_n}|_1\cdot 4(1-\mu(0)-\mu(1)-\eps)$.
\end{prop}
In the remainder of this section, we explain how Theorems~\ref{thm:main2},~\ref{thm:main3} and~\ref{thm:main4} follow from Propositions~\ref{prop:dstats_infinite_variance},~\ref{prop:dstats_zero_radius} and~\ref{prop:dstats_finite_variance} together with the bound from Theorem~\ref{thm:main}.
\begin{proof}[Proof of Theorem~\ref{thm:main2}]
For $n \ge 1$, let $V_n$ be a uniformly random node of $T_n$.
Next, fix $\eps > 0$ small, and let $C=C(\eps)=1+\eps^{-4}$. Then let $E_n$ be the event that
$|\ensuremath{\mathrm{n}}_{T_n}|_2^2 \ge C|\ensuremath{\mathrm{n}}_{T_n}|_1$.
Now fix any degree statistics $\ensuremath{\mathrm{n}}$ with $\p{\ensuremath{\mathrm{n}}_{T_n}=\ensuremath{\mathrm{n}}} > 0$. Conditionally given that $\ensuremath{\mathrm{n}}_{T_n}=\ensuremath{\mathrm{n}}$, then $(T_n,V_n) \in_u \ensuremath{\mathscr{T}}_\ensuremath{\mathrm{n}}^{\bullet}$. Thus, if $|\ensuremath{\mathrm{n}}|_2^2 \ge C|\ensuremath{\mathrm{n}}|_1$, then $|\ensuremath{\mathrm{n}}|_2^2-\ensuremath{\mathrm{n}}(1) \ge (C-1)|\ensuremath{\mathrm{n}}|_1=|\ensuremath{\mathrm{n}}|_1/\eps^4$,
so for all $\beta \ge 17^{3/2}$ we have
\begin{align}
\probC{|V_n| \ge \beta\eps^2 |\ensuremath{\mathrm{n}}|_1^{1/2}}{\ensuremath{\mathrm{n}}_{T_n}=\ensuremath{\mathrm{n}}}
& \le
\probC{|V_n| \ge \beta\frac{|\ensuremath{\mathrm{n}}|_1}{(|\ensuremath{\mathrm{n}}|_2^2-\ensuremath{\mathrm{n}}(1))^{1/2}}}{\ensuremath{\mathrm{n}}_{T_n}=\ensuremath{\mathrm{n}}}
\nonumber\\
& \le \exp\pran{-\frac{\beta^{1/3}}{3}} + 2\exp\pran{-\frac{\beta^{2/3}}{24}}\, ,\label{eq:vnht_cond_bd}
\end{align}
where for the second equality we have used the bound from Theorem~\ref{thm:main} together with the fact that $|\ensuremath{\mathrm{n}}|_2 \le |\ensuremath{\mathrm{n}}|_1$ so $\tfrac{|\ensuremath{\mathrm{n}}|_1}{(|\ensuremath{\mathrm{n}}|_2^2-\ensuremath{\mathrm{n}}(1))^{1/2}} \ge 1$.
We now use the bound
\begin{align}\label{eq:vht_expbd}
\E{|V_n| \I{|V_n| \ge \beta\eps^2|\ensuremath{\mathrm{n}}|_1^{1/2}}}
& \le n\p{E_n^c} +
\sup_{\ensuremath{\mathrm{n}}:|\ensuremath{\mathrm{n}}|_2^2 \ge C|\ensuremath{\mathrm{n}}|_1} \expC{|V_n| \I{|V_n| \ge \beta\eps^2 |\ensuremath{\mathrm{n}}|_1^{1/2}}}{\ensuremath{\mathrm{n}}_{T_n}=\ensuremath{\mathrm{n}}}\, .
\end{align}
The first term is $o(1)$ since $E_n$ occurs with very high probability by Proposition~\ref{prop:dstats_infinite_variance}. To bound the second term we use that for any non-negative integer random variable $X$ and any $z > 0$, we have
\[
\E{X\I{X \ge \beta z}} \le \lfloor \beta z\rfloor + z\int_{\beta}^\infty \p{X \ge xz}\mathrm{d}x\,
\le z\pran{\beta + \int_\beta^\infty \p{X \ge xz}\mathrm{d}x}\, .
\]
Using the bound from \eqref{eq:vnht_cond_bd}, it follows that
\begin{align*}
& \sup_{\ensuremath{\mathrm{n}}:|\ensuremath{\mathrm{n}}|_2^2 \ge C|\ensuremath{\mathrm{n}}|_1} \expC{|V_n| \I{|V_n| \ge \beta\eps^2 |\ensuremath{\mathrm{n}}|_1^{1/2}}} {\ensuremath{\mathrm{n}}_{T_n}=\ensuremath{\mathrm{n}}}\\
& \le \eps^2|\ensuremath{\mathrm{n}}|_1^{1/2} \pran{\beta +
\int_\beta^\infty
\exp\pran{-\frac{x^{1/3}}{3}}\mathrm{d}x
+ \int_\beta^\infty 2\exp\pran{-\frac{x^{2/3}}{24}}\mathrm{d}x\,} .
\end{align*}
Taking $\beta=1/\eps$, for $\eps$ small the first integral is $O(e^{-\eps^{-1/3}/3})$ and the second is $O(e^{-\eps^{-2/3}/24})$, so \eqref{eq:vht_expbd} gives that
\[
\E{|V_n| \I{|V_n| \ge \eps|\ensuremath{\mathrm{n}}|_1^{1/2}}}
\le \eps|\ensuremath{\mathrm{n}}|_1^{1/2} + O(\eps^2 e^{-\eps^{-1/3}/3}|\ensuremath{\mathrm{n}}|_1^{1/2})+ O(\eps^2 e^{-\eps^{-2/3}/24}|\ensuremath{\mathrm{n}}|_1^{1/2})\,.
\]
Since $\eps^2 e^{-\eps^{-1/3}/3}=O(\eps)$ and $\eps^2 e^{-\eps^{-2/3}/24} = O(\eps)$ for $\eps > 0$ small, this implies that
\[
{\mathbf E}{|V_n|} = O(\eps |\ensuremath{\mathrm{n}}|_1^{1/2}),
\]
and so ${\mathbf E}|V_n| = o(|\ensuremath{\mathrm{n}}|_1^{1/2})$ as $\eps>0$ can be chosen arbitrarily small. Since $|\ensuremath{\mathrm{n}}|_1 = (n-1)$, it follows that $n^{-1/2}|V_n| \to 0$ in probability and in expectation.
Next, for any $\eps > 0$, with $C=1+\eps^{-4}$ as above, taking $\beta = (6\log n)^{3}$ in \eqref{eq:vnht_cond_bd}, we obtain that
\begin{align*}
& \p{|V_n| \ge (6 \log n)^3\eps^2(n-1)^{1/2}}\\
&\le \ensuremath{\mathrm{oe}}(1) + \sup_{\ensuremath{\mathrm{n}}:|\ensuremath{\mathrm{n}}|_2^2 \ge C|\ensuremath{\mathrm{n}}|_1}
\probC{|V_n| \ge (6 \log n)^3\eps^2(n-1)^{1/2}}{\ensuremath{\mathrm{n}}_{T_n}=\ensuremath{\mathrm{n}}}\\
& \le \ensuremath{\mathrm{oe}}(1) + \exp\pran{-2 \log n} + 2\exp\pran{-\frac{3 \log^{2} n}{2}}\,\\
& = O\pran{\frac{1}{n^2}}\, .
\end{align*}
On the other hand, since $V_n$ is a uniformly random node of $T_n$, for any positive integer $h$ we have $\probC{|V_n|\ge h}{\ensuremath{\mathrm{ht}}(T_n)\ge h} \ge 1/n$, so $\p{\ensuremath{\mathrm{ht}}(T_n)\ge h}\le n\p{|V_n| \ge h}$ and thus
\[
\p{\ensuremath{\mathrm{ht}}(T_n) \ge (6 \log n)^3\eps^2(n-1)^{1/2}}=O\pran{\frac{1}{n}}.
\]
Since this holds for any $\eps > 0$, it follows that $\ensuremath{\mathrm{ht}}(T_n)/((n-1)^{1/2}\log^3 n) \to 0$ in probability, and since
\begin{align*}
\E{\ensuremath{\mathrm{ht}}{ ( T_n ) }} & \le (6 \log n)^3\eps^2(n-1)^{1/2} +
n\p{\ensuremath{\mathrm{ht}}{ ( T_n ) } \ge (6 \log n)^3\eps^2(n-1)^{1/2}} \\
& \le (6 \log n)^3\eps^2(n-1)^{1/2}+ O(1)\, ,
\end{align*}
again for any $\eps > 0$, also $\ensuremath{\mathrm{ht}}(T_n)/( (n-1)^{1/2}\log^3 n) \to 0$ in expectation.
Finally, fix $\eps > 0$. For all $n$ large enough that ${\mathbf E}|V_n| \le \eps^2 n^{1/2}$, by Markov's inequality,
\[
\p{|V_{n}| \ge \eps n^{1/2}} \le \eps.
\]
On the other hand, since $V_n$ is a uniformly random node of $T_n$,
\[
\p{|V_{n}| \ge \eps n^{1/2}}
\ge
\frac{1}{2} \p{ |\{u \in T_n: |u| \ge \eps n^{1/2}\}| \ge \frac{n}{2}},
\]
so
\[
\p{ |\{u \in T_n: |u| \ge \eps n^{1/2}\}| \ge \frac{n}{2}} \le 2\eps.
\]
Further, if $\{u \in T_n: |u| \ge \eps n^{1/2}\}| <n/2$ then there are more than $n/2$ nodes in the first $\eps n^{1/2}$ levels of the tree, so $\ensuremath{\mathrm{wid}}(T_n) \ge n^{1/2}/(2\eps)$. It follows that
\[
\p{\ensuremath{\mathrm{wid}}(T_n) \ge \frac{n^{1/2}}{2\eps}} \ge 1-2\eps\, ;
\]
since $\eps > 0$ was arbitrary, it follows that $n^{-1/2}\ensuremath{\mathrm{wid}}(T_n) \to \infty$ in probability and in expectation.
\end{proof}
Theorem~\ref{thm:main3} follows from Proposition~\ref{prop:dstats_zero_radius} in exactly the same way as Theorem~\ref{thm:main2} follows from Proposition~\ref{prop:dstats_infinite_variance}, so we omit the details. The proof of Theorem~\ref{thm:main4} from Proposition~\ref{prop:dstats_finite_variance} is quite similar but not identical, so we do provide a (somewhat terser) explanation.
\begin{proof}[Proof of Theorem~\ref{thm:main4}]
Fix any degree statistics $\ensuremath{\mathrm{n}}$ and let $(T,V) \in_u \ensuremath{\mathscr{T}}_{\ensuremath{\mathrm{n}}}^{\bullet}$. Then integrating the tail bound from Theorem~\ref{thm:main} over $\beta \ge 17^{3/2}$, in essentially the same way as in the proof of Theorem~\ref{thm:main2}, it follows that
\begin{equation}\label{eq:universal_sample_bd}
{\mathbf E}|V| \le C \frac{|\ensuremath{\mathrm{n}}|_1}{(|\ensuremath{\mathrm{n}}|_2^2 - \ensuremath{\mathrm{n}}(1))^{1/2}}\, ,
\end{equation}
where $C>0$ is a universal constant.
Since $(|\ensuremath{\mathrm{n}}|_2^2-\ensuremath{\mathrm{n}}(1))^{1/2} \le |\ensuremath{\mathrm{n}}|_2 \le |\ensuremath{\mathrm{n}}|_1=n-1$, using the tail bound from Theorem~\ref{thm:main} with $\beta = (6 \log n)^3$,
we also obtain that
\[
\p{|V| \ge (6 \log n)^3 \frac{|\ensuremath{\mathrm{n}}|_1}{(|\ensuremath{\mathrm{n}}|_2^2 - \ensuremath{\mathrm{n}}(1))^{1/2}}}
\le \exp(-2\log n) + 2\exp\pran{- \frac{3\log^{2} n}{2}}\le \frac{1}{n},
\]
the last bound holding whenever $|\ensuremath{\mathrm{n}}|_1$ is sufficiently large. Since
\[
\p{\ensuremath{\mathrm{ht}}(T)\ge h}\le n \p{|V| \ge h}
\]
and $\ensuremath{\mathrm{ht}}(T)\le n-1$, it follows that
\begin{equation}\label{eq:uniform_ht_bd}
\E{\ensuremath{\mathrm{ht}}(T)} \le C\log^{3}n \frac{|\ensuremath{\mathrm{n}}|_1}{(|\ensuremath{\mathrm{n}}|_2^2 - \ensuremath{\mathrm{n}}(1))^{1/2}}\, ,
\end{equation}
with $C>0$ again a universal constant.
Now let $T_n$ be a Bienaym\'e tree with offspring distribution $\mu$ as in the statement of Theorem~\ref{thm:main4}, and let $V_n$ be a uniformly random node of $T_n$.
Let $\eps = (1-\mu(0)-\mu(1))/2$ and let $E_n$ be the event that $|\ensuremath{\mathrm{n}}_{T_n}|_2^2-\ensuremath{\mathrm{n}}_{T_n}(1)\ge |\ensuremath{\mathrm{n}}_{T_n}|_1\cdot 4(1-\mu(0)-\mu(1)-\eps) = 2|\ensuremath{\mathrm{n}}_{T_n}|_1(1-\mu(0)-\mu(1))$. By Proposition~\ref{prop:dstats_finite_variance}, $E_n$ occurs with very high probability.
On the event $E_n$ we have
\[
\frac{|\ensuremath{\mathrm{n}}_{T_n}|_1}{(|\ensuremath{\mathrm{n}}_{T_n}|_2^2 - \ensuremath{\mathrm{n}}_{T_n}(1))^{1/2}}
\le \frac{n^{1/2}}{2^{1/2}(1-\mu(0)-\mu(1))^{1/2}}\, ,
\]
so by (\ref{eq:universal_sample_bd}) we have
\begin{align*}
{\mathbf E}{|V_n|} & \le n\p{E_n^c}+ \mathop{\sup_{\ensuremath{\mathrm{n}}:|\ensuremath{\mathrm{n}}|_1=n +1}}_{|\ensuremath{\mathrm{n}}_{T_n}|_2^2-\ensuremath{\mathrm{n}}_{T_n}(1)\ge2|\ensuremath{\mathrm{n}}_{T_n}|_1(1-\mu(0)-\mu(1))}
\expC{|V_n|}{\ensuremath{\mathrm{n}}_{T_n}=\ensuremath{\mathrm{n}}}\\
& \le o(1)+C\frac{n^{1/2}}{2^{1/2}(1-\mu(0)-\mu(1))^{1/2}}\, ,
\end{align*}
where in the second inequality we have used that $\p{E_n^c}=\ensuremath{\mathrm{oe}}(1)$. The lower bound on $\E{\ensuremath{\mathrm{wid}}(T_n)}$ follows from this upper bound on ${\mathbf E}|V_n|$ just as in the proof of Theorem~\ref{thm:main2}. Finally,
\begin{align*}
\E{\ensuremath{\mathrm{ht}}(T_n)} & \le n\p{E_n^c} + \mathop{\sup_{\ensuremath{\mathrm{n}}:|\ensuremath{\mathrm{n}}|_1=n+1}}_{|\ensuremath{\mathrm{n}}_{T_n}|_2^2-\ensuremath{\mathrm{n}}_{T_n}(1)\ge2|\ensuremath{\mathrm{n}}_{T_n}|_1(1-\mu(0)-\mu(1))}
\expC{\ensuremath{\mathrm{ht}}(T_n)}{\ensuremath{\mathrm{n}}_{T_n}=\ensuremath{\mathrm{n}}} \\
& \le o(1) + C\log^3 n \frac{n^{1/2}}{2^{1/2}(1-\mu(0)-\mu(1))^{1/2}}\, ,
\end{align*}
the second bound holding by (\ref{eq:uniform_ht_bd}) and since $\p{E_n^c}=\ensuremath{\mathrm{oe}}(1)$. This establishes the requisite bound on $\E{\ensuremath{\mathrm{ht}}(T_n)}$, and completes the proof.
\end{proof}
\section{\bf Proof of Proposition~\ref{prop:sampler}}
We begin with some combinatorial definitions and facts which we will require for the proof. A {\em forest} is an ordered sequence $f=(t_1,\ldots,t_a)$ of plane trees.
The degree statistics of $f$ is the sequence $\ensuremath{\mathrm{n}}_f=(\ensuremath{\mathrm{n}}_f(c),c \ge 0)$ where $\ensuremath{\mathrm{n}}_f(c)$ is the number of nodes of $f$ with $c$ children.
Fix integers $1 \le a \le n$ and let $\ensuremath{\mathrm{n}}=(\ensuremath{\mathrm{n}}(c),c\ge 0)$ be a sequence of non-negative integers with $\sum_{c \ge 0} \ensuremath{\mathrm{n}}(c) = n$ and $\sum_{c \ge 0} c\ensuremath{\mathrm{n}}(c)=n-a$. Any forest with degree statistics $\ensuremath{\mathrm{n}}$ has $n$ nodes and is composed of $a$ trees.
We write $\ensuremath{\mathscr{T}}_{\ensuremath{\mathrm{n}}}$ to denote the set of forests with degree statistics $\ensuremath{\mathrm{n}}$.
A single tree $t$ can be interpreted as a forest $f=(t)$, which makes this notation agree with and extend the previously introduced notation $\ensuremath{\mathscr{T}}_\ensuremath{\mathrm{n}}$ for the set of trees with given degree statistics (in which case $a=1$, i.e., $|\ensuremath{\mathrm{n}}|_1 = n-1$).
By \cite[Exercise 6.2.1]{MR2245368}, it holds that
\begin{equation}\label{eq:dstat_count}
|\ensuremath{\mathscr{T}}_{\ensuremath{\mathrm{n}}}| = \frac{a}{n} {n \choose \ensuremath{\mathrm{n}}(c),c \ge 0} = \frac{a}{n} \frac{n!}{\prod_{c \ge 0} \ensuremath{\mathrm{n}}(c)!} \, .
\end{equation}
Next write $\ensuremath{\mathscr{T}}_\ensuremath{\mathrm{n}}^\bullet$ for the set of forests with degree statistics $\ensuremath{\mathrm{n}}$ with a marked node, and $\ensuremath{\mathscr{T}}_\ensuremath{\mathrm{n}}^{(1)}$ for the subset of $\ensuremath{\mathscr{T}}_{\ensuremath{\mathrm{n}}}^\bullet$ where the mark is in the first tree:
\begin{align*}
\ensuremath{\mathscr{T}}_\ensuremath{\mathrm{n}}^{\bullet} & = \{(f,v):f=(t_1,\ldots,t_a) \in \ensuremath{\mathscr{T}}_\ensuremath{\mathrm{n}},v \mbox{ is a node of }f\},\\
\ensuremath{\mathscr{T}}_\ensuremath{\mathrm{n}}^{(1)} & = \{(f,v):f=(t_1,\ldots,t_a) \in \ensuremath{\mathscr{T}}_\ensuremath{\mathrm{n}},v \in t_1\}\, .
\end{align*}
For any forest $f \in \ensuremath{\mathscr{T}}_\ensuremath{\mathrm{n}}$ there are $n$ ways to choose a node to mark, so $|\ensuremath{\mathscr{T}}_\ensuremath{\mathrm{n}}^{\bullet}|=n|\ensuremath{\mathscr{T}}_\ensuremath{\mathrm{n}}|$. Moreover, there is a natural $a$-to-$1$ correspondence between $\ensuremath{\mathscr{T}}_\ensuremath{\mathrm{n}}^{\bullet}$ and $\ensuremath{\mathscr{T}}_\ensuremath{\mathrm{n}}^{(1)}$: if $(f,v) \in \ensuremath{\mathscr{T}}_\ensuremath{\mathrm{n}}^{\bullet}$ with $f=(t_1,\ldots,t_a)$ and $v \in t_i$, then
\[
((t_i,t_{i+1},\ldots,t_a,t_1,\ldots,t_{i-1}),v) \in \ensuremath{\mathscr{T}}_\ensuremath{\mathrm{n}}^{(1)}.
\]
It follows that
\begin{equation}\label{eq:marked_forest_count}
|\ensuremath{\mathscr{T}}_\ensuremath{\mathrm{n}}^{(1)}| = \frac{n}{a}|\ensuremath{\mathscr{T}}_{\ensuremath{\mathrm{n}}}| = {n \choose \ensuremath{\mathrm{n}}(c),c \ge 0} \, .
\end{equation}
\begin{figure}[htb]
\color{black}
\centering
\tikzset{every picture/.style={line width=0.75pt}}
\begin{tikzpicture}[x=0.75pt,y=0.75pt,yscale=-1,xscale=1]
\draw [color={rgb, 255:red, 0; green, 0; blue, 0 } ,draw opacity=1 ] (332.48,96.71) -- (332.48,123.99) ;
\draw (352.93,42.16) -- (363.15,69.43) -- (342.7,69.43) -- cycle ;
\draw [color={rgb, 255:red, 208; green, 2; blue, 27 } ,draw opacity=1, line width = 0.6mm ] (332.61,15.17) -- (352.93,42.16) ;
\draw (312.02,42.16) -- (322.25,69.43) -- (301.8,69.43) -- cycle ;
\draw [color={rgb, 255:red, 208; green, 2; blue, 27 } ,draw opacity=1, line width = 0.6mm ][fill={rgb, 255:red, 208; green, 2; blue, 27 } ,fill opacity=1 ] (332.48,14.87) -- (312.02,42.16) ;
\draw (352.79,96.43) -- (363.02,123.7) -- (342.57,123.7) -- cycle ;
\draw [color={rgb, 255:red, 208; green, 2; blue, 27 } ,draw opacity=1, line width = 0.6mm ] (332.48,69.93) -- (352.79,96.93) ;
\draw (312.02,124.01) -- (322.25,151.28) -- (301.8,151.28) -- cycle ;
\draw [color={rgb, 255:red, 0; green, 0; blue, 0 } ,draw opacity=1 ] (332.48,96.71) -- (312.02,124.01) ;
\draw (288.16,42.14) -- (298.39,69.41) -- (277.94,69.41) -- cycle ;
\draw [color={rgb, 255:red, 208; green, 2; blue, 27 } ,draw opacity=1, line width = 0.6mm ] (332.61,15.17) -- (288.16,42.14) ;
\draw (376.79,42.16) -- (387.02,69.43) -- (366.56,69.43) -- cycle ;
\draw [color={rgb, 255:red, 208; green, 2; blue, 27 } ,draw opacity=1, line width = 0.6mm ] (332.48,14.87) -- (376.79,42.16) ;
\draw (376.79,96.73) -- (387.02,123.99) -- (366.56,123.99) -- cycle ;
\draw [color={rgb, 255:red, 208; green, 2; blue, 27 } ,draw opacity=1, line width = 0.6mm ] (332.48,69.93) -- (376.79,97.23) ;
\draw (288.03,123.69) -- (298.25,150.96) -- (277.8,150.96) -- cycle ;
\draw [color={rgb, 255:red, 0; green, 0; blue, 0 } ,draw opacity=1 ] (332.48,96.71) -- (288.03,123.69) ;
\draw (332.48,178.56) -- (342.7,205.83) -- (322.25,205.83) -- cycle ;
\draw [color={rgb, 255:red, 208; green, 2; blue, 27 } ,draw opacity=1, line width = 0.6mm ] (332.48,15.37) -- (332.48,69.93) ;
\draw [shift={(332.48,69.93)}, rotate = 90] [color={rgb, 255:red, 208; green, 2; blue, 27 } ,draw opacity=1][fill={rgb, 255:red, 208; green, 2; blue, 27 } ,fill opacity=1 ][line width=0.75] (0, 0) circle [x radius= 2.01, y radius= 2.01] ;
\draw [shift={(332.48,42.65)}, rotate = 90] [color={rgb, 255:red, 208; green, 2; blue, 27 } ,draw opacity=1 ][fill={rgb, 255:red, 208; green, 2; blue, 27 } ,fill opacity=1 ][line width=0.75] (0, 0) circle [x radius= 2.01, y radius= 2.01] ;
\draw [shift={(332.48,15.37)}, rotate = 90] [color={rgb, 255:red, 208; green, 2; blue, 27 } ,draw opacity=1, line width = 0.6mm ][fill={rgb, 255:red, 208; green, 2; blue, 27 } ,fill opacity=1 ][line width=0.75] (0, 0) circle [x radius= 2.01, y radius= 2.01] ;
\draw [color={rgb, 255:red, 208; green, 2; blue, 27 } ,draw opacity=1 ][fill={rgb, 255:red, 208; green, 2; blue, 27 } ,fill opacity=1 ] (310.07,42.16) .. controls (310.07,41.08) and (310.94,40.2) .. (312.02,40.2) .. controls (313.1,40.2) and (313.98,41.08) .. (313.98,42.16) .. controls (313.98,43.24) and (313.1,44.12) .. (312.02,44.12) .. controls (310.94,44.12) and (310.07,43.24) .. (310.07,42.16) -- cycle ;
\draw [color={rgb, 255:red, 208; green, 2; blue, 27 } ,draw opacity=1 ][fill={rgb, 255:red, 208; green, 2; blue, 27 } ,fill opacity=1 ] (350.97,42.16) .. controls (350.97,41.08) and (351.85,40.2) .. (352.93,40.2) .. controls (354.01,40.2) and (354.89,41.08) .. (354.89,42.16) .. controls (354.89,43.24) and (354.01,44.12) .. (352.93,44.12) .. controls (351.85,44.12) and (350.97,43.24) .. (350.97,42.16) -- cycle ;
\draw [color={rgb, 255:red, 208; green, 2; blue, 27 } ,draw opacity=1 ][fill={rgb, 255:red, 208; green, 2; blue, 27 } ,fill opacity=1 ] (374.83,42.16) .. controls (374.83,41.08) and (375.71,40.2) .. (376.79,40.2) .. controls (377.87,40.2) and (378.75,41.08) .. (378.75,42.16) .. controls (378.75,43.24) and (377.87,44.12) .. (376.79,44.12) .. controls (375.71,44.12) and (374.83,43.24) .. (374.83,42.16) -- cycle ;
\draw [color={rgb, 255:red, 208; green, 2; blue, 27 } ,draw opacity=1 ][fill={rgb, 255:red, 208; green, 2; blue, 27 } ,fill opacity=1 ] (350.83,96.43) .. controls (350.83,95.35) and (351.71,94.47) .. (352.79,94.47) .. controls (353.87,94.47) and (354.75,95.35) .. (354.75,96.43) .. controls (354.75,97.51) and (353.87,98.39) .. (352.79,98.39) .. controls (351.71,98.39) and (350.83,97.51) .. (350.83,96.43) -- cycle ;
\draw [color={rgb, 255:red, 208; green, 2; blue, 27 } ,draw opacity=1 ][fill={rgb, 255:red, 208; green, 2; blue, 27 } ,fill opacity=1 ] (374.83,96.73) .. controls (374.83,95.64) and (375.71,94.77) .. (376.79,94.77) .. controls (377.87,94.77) and (378.75,95.64) .. (378.75,96.73) .. controls (378.75,97.81) and (377.87,98.68) .. (376.79,98.68) .. controls (375.71,98.68) and (374.83,97.81) .. (374.83,96.73) -- cycle ;
\draw [color={rgb, 255:red, 0; green, 0; blue, 0 } ,draw opacity=1 ] (332.48,123.99) -- (332.48,178.56) ;
\draw [shift={(332.48,178.56)}, rotate = 90] [color={rgb, 255:red, 0; green, 0; blue, 0 } ,draw opacity=1 ][fill={rgb, 255:red, 0; green, 0; blue, 0 } ,fill opacity=1 ][line width=0.75] (0, 0) circle [x radius= 2.01, y radius= 2.01] ;
\draw [shift={(332.48,151.28)}, rotate = 90] [color={rgb, 255:red, 0; green, 0; blue, 0 } ,draw opacity=1 ][fill={rgb, 255:red, 0; green, 0; blue, 0 } ,fill opacity=1 ][line width=0.75] (0, 0) circle [x radius= 2.01, y radius= 2.01] ;
\draw [shift={(332.48,123.99)}, rotate = 90] [color={rgb, 255:red, 0; green, 0; blue, 0 } ,draw opacity=1 ][fill={rgb, 255:red, 0; green, 0; blue, 0 } ,fill opacity=1 ][line width=0.75] (0, 0) circle [x radius= 2.01, y radius= 2.01] ;
\draw [color={rgb, 255:red, 0; green, 0; blue, 0 } ,draw opacity=1 ][fill={rgb, 255:red, 0; green, 0; blue, 0 } ,fill opacity=1 ] (286.07,123.69) .. controls (286.07,122.61) and (286.94,121.73) .. (288.03,121.73) .. controls (289.11,121.73) and (289.98,122.61) .. (289.98,123.69) .. controls (289.98,124.77) and (289.11,125.65) .. (288.03,125.65) .. controls (286.94,125.65) and (286.07,124.77) .. (286.07,123.69) -- cycle ;
\draw [color={rgb, 255:red, 0; green, 0; blue, 0 } ,draw opacity=1 ][fill={rgb, 255:red, 0; green, 0; blue, 0 } ,fill opacity=1 ] (310.07,124.01) .. controls (310.07,122.92) and (310.94,122.05) .. (312.02,122.05) .. controls (313.1,122.05) and (313.98,122.92) .. (313.98,124.01) .. controls (313.98,125.09) and (313.1,125.96) .. (312.02,125.96) .. controls (310.94,125.96) and (310.07,125.09) .. (310.07,124.01) -- cycle ;
\draw [color={rgb, 255:red, 208; green, 2; blue, 27 } ,draw opacity=1 ][fill={rgb, 255:red, 208; green, 2; blue, 27 } ,fill opacity=1 ] (286.2,42.14) .. controls (286.2,41.06) and (287.08,40.18) .. (288.16,40.18) .. controls (289.24,40.18) and (290.12,41.06) .. (290.12,42.14) .. controls (290.12,43.22) and (289.24,44.1) .. (288.16,44.1) .. controls (287.08,44.1) and (286.2,43.22) .. (286.2,42.14) -- cycle ;
\draw [color={rgb, 255:red, 208; green, 2; blue, 27 } ,draw opacity=1 ] (394,100) .. controls (398.67,100) and (401,97.67) .. (401,93) -- (401,65.88) .. controls (401,59.21) and (403.33,55.88) .. (408,55.88) .. controls (403.33,55.88) and (401,52.55) .. (401,45.88)(401,48.88) -- (401,18.75) .. controls (401,14.08) and (398.67,11.75) .. (394,11.75) ;
\draw (180.68,42.06) -- (190.9,69.33) -- (170.45,69.33) -- cycle ;
\draw [color={rgb, 255:red, 74; green, 144; blue, 226 } ,draw opacity=1, line width = 0.6mm] (160.76,15.07) -- (181.08,42.06) ;
\draw (139.77,42.06) -- (150,69.33) -- (129.55,69.33) -- cycle ;
\draw [color={rgb, 255:red, 74; green, 144; blue, 226 } ,draw opacity=1, line width = 0.6mm] (160.63,14.77) -- (140.17,42.06) ;
\draw (180.54,96.33) -- (190.77,123.6) -- (170.32,123.6) -- cycle ;
\draw [color={rgb, 255:red, 74; green, 144; blue, 226 } ,draw opacity=1, line width = 0.6mm ] (160.23,69.83) -- (180.54,96.83) ;
\draw (139.77,123.91) -- (150,151.17) -- (129.55,151.17) -- cycle ;
\draw [color={rgb, 255:red, 74; green, 144; blue, 226 } ,draw opacity=1, line width = 0.6mm ] (160.23,97.11) -- (139.77,124.41) ;
\draw (115.91,42.04) -- (126.14,69.31) -- (105.69,69.31) -- cycle ;
\draw [color={rgb, 255:red, 74; green, 144; blue, 226 } ,draw opacity=1, line width = 0.6mm ] (160.36,15.07) -- (115.91,42.04) ;
\draw (204.54,42.06) -- (214.77,69.33) -- (194.31,69.33) -- cycle ;
\draw [color={rgb, 255:red, 74; green, 144; blue, 226 } ,draw opacity=1 , line width = 0.6mm] (160.63,14.77) -- (204.94,42.06) ;
\draw (204.54,96.62) -- (214.77,123.89) -- (194.31,123.89) -- cycle ;
\draw [color={rgb, 255:red, 74; green, 144; blue, 226 } ,draw opacity=1, line width = 0.6mm ] (160.23,69.83) -- (204.54,97.12) ;
\draw (115.78,123.59) -- (126,150.86) -- (105.55,150.86) -- cycle ;
\draw [color={rgb, 255:red, 74; green, 144; blue, 226 } ,draw opacity=1, line width = 0.6mm ] (160.23,97.11) -- (115.78,124.09) ;
\draw (160.23,178.46) -- (170.45,205.73) -- (150,205.73) -- cycle ;
\draw (183.68,178.46) -- (193.9,205.73) -- (173.45,205.73) -- cycle ;
\draw [color={rgb, 255:red, 74; green, 144; blue, 226 } ,draw opacity=1, line width = 0.6mm ] (160.23,151.67) -- (183.68,178.96) ;
\draw [color={rgb, 255:red, 74; green, 144; blue, 226 } ,draw opacity=1, line width = 0.6mm ][fill={rgb, 255:red, 74; green, 144; blue, 226 } ,fill opacity=1 ] (160.63,14.77) -- (160.63,69.33) ;
\draw [shift={(160.63,69.33)}, rotate = 90] [color={rgb, 255:red, 74; green, 144; blue, 226 } ,draw opacity=1 ][fill={rgb, 255:red, 74; green, 144; blue, 226 } ,fill opacity=1 ][line width = 0.75pt] (0, 0) circle [x radius= 2.01, y radius= 2.01] ;
\draw [shift={(160.63,42.05)}, rotate = 90] [color={rgb, 255:red, 74; green, 144; blue, 226 } , draw opacity=1 ][fill={rgb, 255:red, 74; green, 144; blue, 226 } ,fill opacity=1 ][line width = 0.75pt] (0, 0) circle [x radius= 2.01, y radius= 2.01] ;
\draw [shift={(160.63,14.77)}, rotate = 90] [color={rgb, 255:red, 74; green, 144; blue, 226 } ,draw opacity=1][fill={rgb, 255:red, 74; green, 144; blue, 226 } ,fill opacity=1 ][line width = 0.75pt] (0, 0) circle [x radius= 2.01, y radius= 2.01] ;
\draw [color={rgb, 255:red, 74; green, 144; blue, 226 } ,draw opacity=1 ][fill={rgb, 255:red, 74; green, 144; blue, 226 } ,fill opacity=1 ] (137.82,42.06) .. controls (137.82,40.98) and (138.69,40.1) .. (139.77,40.1) .. controls (140.85,40.1) and (141.73,40.98) .. (141.73,42.06) .. controls (141.73,43.14) and (140.85,44.02) .. (139.77,44.02) .. controls (138.69,44.02) and (137.82,43.14) .. (137.82,42.06) -- cycle ;
\draw [color={rgb, 255:red, 74; green, 144; blue, 226 } ,draw opacity=1 ][fill={rgb, 255:red, 74; green, 144; blue, 226 } ,fill opacity=1 ] (178.72,42.06) .. controls (178.72,40.98) and (179.6,40.1) .. (180.68,40.1) .. controls (181.76,40.1) and (182.64,40.98) .. (182.64,42.06) .. controls (182.64,43.14) and (181.76,44.02) .. (180.68,44.02) .. controls (179.6,44.02) and (178.72,43.14) .. (178.72,42.06) -- cycle ;
\draw [color={rgb, 255:red, 74; green, 144; blue, 226 } ,draw opacity=1 ][fill={rgb, 255:red, 74; green, 144; blue, 226 } ,fill opacity=1 ] (202.58,42.06) .. controls (202.58,40.98) and (203.46,40.1) .. (204.54,40.1) .. controls (205.62,40.1) and (206.5,40.98) .. (206.5,42.06) .. controls (206.5,43.14) and (205.62,44.02) .. (204.54,44.02) .. controls (203.46,44.02) and (202.58,43.14) .. (202.58,42.06) -- cycle ;
\draw [color={rgb, 255:red, 74; green, 144; blue, 226 } ,draw opacity=1 ][fill={rgb, 255:red, 74; green, 144; blue, 226 } ,fill opacity=1 ] (178.58,96.33) .. controls (178.58,95.25) and (179.46,94.37) .. (180.54,94.37) .. controls (181.62,94.37) and (182.5,95.25) .. (182.5,96.33) .. controls (182.5,97.41) and (181.62,98.29) .. (180.54,98.29) .. controls (179.46,98.29) and (178.58,97.41) .. (178.58,96.33) -- cycle ;
\draw [color={rgb, 255:red, 74; green, 144; blue, 226 } ,draw opacity=1 ][fill={rgb, 255:red, 74; green, 144; blue, 226 } ,fill opacity=1 ] (202.58,96.62) .. controls (202.58,95.54) and (203.46,94.67) .. (204.54,94.67) .. controls (205.62,94.67) and (206.5,95.54) .. (206.5,96.62) .. controls (206.5,97.71) and (205.62,98.58) .. (204.54,98.58) .. controls (203.46,98.58) and (202.58,97.71) .. (202.58,96.62) -- cycle ;
\draw [color={rgb, 255:red, 74; green, 144; blue, 226 } ,draw opacity=1, line width = 0.6mm ] (160.23,69.83) -- (160.23,124.39) ;
\draw [shift={(160.23,124.39)}, rotate = 90] [color={rgb, 255:red, 74; green, 144; blue, 226 } ,draw opacity=1 ][fill={rgb, 255:red, 74; green, 144; blue, 226 } ,fill opacity=1 ][line width=0.75] (0, 0) circle [x radius= 2.01, y radius= 2.01] ;
\draw [shift={(160.23,97.11)}, rotate = 90] [color={rgb, 255:red, 74; green, 144; blue, 226 } ,draw opacity=1 ][fill={rgb, 255:red, 74; green, 144; blue, 226 } ,fill opacity=1 ][line width=0.75] (0, 0) circle [x radius= 2.01, y radius= 2.01] ;
\draw [shift={(160.23,69.83)}, rotate = 90] [color={rgb, 255:red, 74; green, 144; blue, 226 } ,draw opacity=1 ][fill={rgb, 255:red, 74; green, 144; blue, 226 } ,fill opacity=1 ][line width=0.75] (0, 0) circle [x radius= 2.01, y radius= 2.01] ;
\draw [color={rgb, 255:red, 74; green, 144; blue, 226 } ,draw opacity=1, line width = 0.6mm ] (160.23,124.39) -- (160.23,178.96) ;
\draw [shift={(160.23,178.96)}, rotate = 90] [color={rgb, 255:red, 74; green, 144; blue, 226 } ,draw opacity=1 ][fill={rgb, 255:red, 74; green, 144; blue, 226 } ,fill opacity=1 ][line width=0.75] (0, 0) circle [x radius= 2.01, y radius= 2.01] ;
\draw [shift={(160.23,151.67)}, rotate = 90] [color={rgb, 255:red, 74; green, 144; blue, 226 } ,draw opacity=1 ][fill={rgb, 255:red, 74; green, 144; blue, 226 } ,fill opacity=1 ][line width=0.75] (0, 0) circle [x radius= 2.01, y radius= 2.01] ;
\draw [shift={(160.23,124.39)}, rotate = 90] [color={rgb, 255:red, 74; green, 144; blue, 226 } ,draw opacity=1 ][fill={rgb, 255:red, 74; green, 144; blue, 226 } ,fill opacity=1 ][line width=0.75] (0, 0) circle [x radius= 2.01, y radius= 2.01] ;
\draw [color={rgb, 255:red, 74; green, 144; blue, 226 } ,draw opacity=1 ][fill={rgb, 255:red, 74; green, 144; blue, 226 } ,fill opacity=1 ] (113.82,123.59) .. controls (113.82,122.51) and (114.69,121.63) .. (115.78,121.63) .. controls (116.86,121.63) and (117.73,122.51) .. (117.73,123.59) .. controls (117.73,124.67) and (116.86,125.55) .. (115.78,125.55) .. controls (114.69,125.55) and (113.82,124.67) .. (113.82,123.59) -- cycle ;
\draw [color={rgb, 255:red, 74; green, 144; blue, 226 } ,draw opacity=1 ][fill={rgb, 255:red, 74; green, 144; blue, 226 } ,fill opacity=1 ] (137.82,123.91) .. controls (137.82,122.82) and (138.69,121.95) .. (139.77,121.95) .. controls (140.85,121.95) and (141.73,122.82) .. (141.73,123.91) .. controls (141.73,124.99) and (140.85,125.86) .. (139.77,125.86) .. controls (138.69,125.86) and (137.82,124.99) .. (137.82,123.91) -- cycle ;
\draw [color={rgb, 255:red, 74; green, 144; blue, 226 } ,draw opacity=1 ][fill={rgb, 255:red, 74; green, 144; blue, 226 } ,fill opacity=1 ] (181.72,178.46) .. controls (181.72,177.37) and (182.6,176.5) .. (183.68,176.5) .. controls (184.76,176.5) and (185.64,177.37) .. (185.64,178.46) .. controls (185.64,179.54) and (184.76,180.41) .. (183.68,180.41) .. controls (182.6,180.41) and (181.72,179.54) .. (181.72,178.46) -- cycle ;
\draw [color={rgb, 255:red, 74; green, 144; blue, 226 } ,draw opacity=1 ][fill={rgb, 255:red, 74; green, 144; blue, 226 } ,fill opacity=1 ] (113.95,42.04) .. controls (113.95,40.96) and (114.83,40.08) .. (115.91,40.08) .. controls (116.99,40.08) and (117.87,40.96) .. (117.87,42.04) .. controls (117.87,43.12) and (116.99,44) .. (115.91,44) .. controls (114.83,44) and (113.95,43.12) .. (113.95,42.04) -- cycle ;
\draw [color={rgb, 255:red, 74; green, 144; blue, 226 } ,draw opacity=1 ] (99.25,12.5) .. controls (94.58,12.5) and (92.25,14.83) .. (92.25,19.5) -- (92.25,86.38) .. controls (92.25,93.05) and (89.92,96.38) .. (85.25,96.38) .. controls (89.92,96.38) and (92.25,99.71) .. (92.25,106.38)(92.25,103.38) -- (92.25,173.25) .. controls (92.25,177.92) and (94.58,180.25) .. (99.25,180.25) ;
\draw (355.93,178.56) -- (366.15,205.83) -- (345.7,205.83) -- cycle ;
\draw [color={rgb, 255:red, 0; green, 0; blue, 0 } ,draw opacity=1 ] (332.48,151.28) -- (355.93,178.56) ;
\draw [color={rgb, 255:red, 0; green, 0; blue, 0 } ,draw opacity=1 ][fill={rgb, 255:red, 0; green, 0; blue, 0 } ,fill opacity=1 ] (353.97,178.56) .. controls (353.97,177.48) and (354.85,176.6) .. (355.93,176.6) .. controls (357.01,176.6) and (357.89,177.48) .. (357.89,178.56) .. controls (357.89,179.64) and (357.01,180.52) .. (355.93,180.52) .. controls (354.85,180.52) and (353.97,179.64) .. (353.97,178.56) -- cycle ;
\draw [color={rgb, 255:red, 208; green, 2; blue, 27 } ,draw opacity=1 ][fill={rgb, 255:red, 208; green, 2; blue, 27 } ,fill opacity=1 ] (330.38,96.71) .. controls (330.38,95.56) and (331.32,94.62) .. (332.48,94.62) .. controls (333.63,94.62) and (334.57,95.56) .. (334.57,96.71) .. controls (334.57,97.87) and (333.63,98.81) .. (332.48,98.81) .. controls (331.32,98.81) and (330.38,97.87) .. (330.38,96.71) -- cycle ;
\draw [color={rgb, 255:red, 208; green, 2; blue, 27 } ,draw opacity=1, line width = 0.6mm ] (332.48,69.93) -- (332.48,97.21) ;
\draw (354.32,9.85) node [align=left] {\begin{minipage}[lt]{19.97pt}\setlength\topsep{0pt}
$\displaystyle r( t)$
\end{minipage}};
\draw (434.5,57.6) node [color={rgb, 255:red, 208; green, 2; blue, 27 } ,opacity=1 ] [align=left] {\begin{minipage}[lt]{33.67pt}\setlength\topsep{0pt}
$\displaystyle S_{k}( t,v)$
\end{minipage}};
\draw (322.57,179) node [align=left] {\begin{minipage}[lt]{10.45pt}\setlength\topsep{0pt}
$\displaystyle v$
\end{minipage}};
\draw (182.07,9.75) node [align=left] {\begin{minipage}[lt]{19.97pt}\setlength\topsep{0pt}
$\displaystyle r( t)$
\end{minipage}};
\draw (62,97) node [color={rgb, 255:red, 74; green, 144; blue, 226 } ,opacity=1 ] [align=left] {\begin{minipage}[lt]{33.67pt}\setlength\topsep{0pt}
$\displaystyle S( t,v)$
\end{minipage}};
\draw (150.32,178.9) node [align=left] {\begin{minipage}[lt]{10.45pt}\setlength\topsep{0pt}
$\displaystyle v$
\end{minipage}};
\draw (318.07,94) node [align=left] {\begin{minipage}[lt]{10.45pt}\setlength\topsep{0pt}
$\displaystyle v^{k}$
\end{minipage}};
\end{tikzpicture}
\caption{A visualization of a spine $S(t, v)$ and of a $k$-spine $S_k(t,v)$, for $k=3$.}\label{fig:spine}
\end{figure}
Some of the definitions of the coming paragraph are illustrated in Figure~\ref{fig:spine}.
For nodes $x,y$ of a tree $t$, we write $x \prec y$ if $x$ is an ancestor of $y$ in $t$, and for a node $z\in v(t) \setminus \{r(t)\}$, we denote its parent by $p(z)$.
Given a marked tree $(t,v)$, the {\em spine} $\ensuremath{S}(t,v)$ of $(t,v)$ is the subtree of $t$ with vertices $\{w: p(w) \prec v\} \cup \{r(t)\}$.
For $0 \le k \le |v|$, write $v^k$ for the unique ancestor of $v$ with $|v^k|=k$; so $v^0=r(t)$ and $v^{|v|}=v$. If $k \le |v|$ then
the {\em $k$-spine} $\ensuremath{S}_k(t,v)$ of $(t,v)$ is the subtree of $t$ with vertices $\{w: p(w) \prec v^k\}\cup \{r(t)\}$, and the {\em marked $k$-spine} of $(t,v)$ is the marked tree $(\ensuremath{S}_k(t,v) ,v^k)$.
The \textit{spinal degree sequence} of $\ensuremath{S}_k(t,v)$ is $(d_t(v^0),\ldots,d_t(v^{k-1}))$.
Given a sequence $\ensuremath{\mathrm{d}}=(d_0,\ldots,d_{k-1})$ of non-negative integers and degree statistics $\ensuremath{\mathrm{n}}=(\ensuremath{\mathrm{n}}(c),c \ge 0)$ satisfying $\sum_{c \ge 0} c\ensuremath{\mathrm{n}}(c) = \sum_{c \ge 0} \ensuremath{\mathrm{n}}(c) - 1$, write
\[
\ensuremath{\mathscr{T}}_\ensuremath{\mathrm{n}}^\bullet(\ensuremath{\mathrm{d}}) = \{(t,v) \in \ensuremath{\mathscr{T}}_\ensuremath{\mathrm{n}}^\bullet: |v| \ge k, (\deg_t(v^0),\ldots,\deg_t(v^{k-1}))=\ensuremath{\mathrm{d}}\}\, .
\]
Using these definitions, we establish the following combinatorial result, whose probabilistic corollary is then used to prove Proposition~\ref{prop:sampler}. (This result is closely related to the backbone decomposition of trees given in \cite[Proposition 4 (a)]{MR3188597} in order to prove convergence of large random trees with fixed degrees to the Brownian continuum random tree after rescaling, under suitable assumptions on the degree sequences.)
\begin{prop}\label{prop:tnd_count}
Fix degree statistics $\ensuremath{\mathrm{n}}=(\ensuremath{\mathrm{n}}(c),c \ge 0)$ with $\sum_{c \ge 0} \ensuremath{\mathrm{n}}(c)=n$ and with $\sum_{c \ge 0} c\ensuremath{\mathrm{n}}(c)=n-1$.
For any sequence $\ensuremath{\mathrm{d}}=(d_0,\ldots,d_{k-1})$ of non-negative integers, let $w(\ensuremath{\mathrm{d}}) = (w(\ensuremath{\mathrm{d}},c), c\geq 0)$ where $w(\ensuremath{\mathrm{d}},c) = |\{0 \le i \le k-1: d_i = c\}|$ is as in \eqref{eq:size_biasing}. If $w(\ensuremath{\mathrm{d}},c) \le \ensuremath{\mathrm{n}}(c)$ for all $c \ge 0$, then
\[
|\ensuremath{\mathscr{T}}_\ensuremath{\mathrm{n}}^\bullet(\ensuremath{\mathrm{d}})| = \pran{\prod_{i=0}^{k-1} d_i} \cdot\left|\ensuremath{\mathscr{T}}_{\ensuremath{\mathrm{n}}-w(\ensuremath{\mathrm{d}})}^{(1)}\right| = \prod_{i=0}^{k-1} d_i \cdot {n-k \choose \ensuremath{\mathrm{n}}(c)-w(\ensuremath{\mathrm{d}},c),c \ge 0}\, .
\]
\end{prop}
\begin{proof}
To describe an element $(t,v)$ of $\ensuremath{\mathscr{T}}_\ensuremath{\mathrm{n}}^\bullet(\ensuremath{\mathrm{d}})$, it is necessary and sufficient to specify the marked $k$-spine $(\ensuremath{S}_k(t,v) ,v^k)$, the subtrees of $t$ rooted at the leaves of $\ensuremath{S}_k(t,v)$, and the identity of the mark $v$, which must lie within the subtree of $\ensuremath{S}_k(t,v)$ rooted at $v^k$.
The number of marked $k$-spines with spinal degree sequence $\ensuremath{\mathrm{d}}$ is $\prod_{i=0}^{k-1} d_i$, since to specify such a tree it is necessary and sufficient to indicate which of the $d_i$ children of $v^i$ is $v^{i+1}$ for each $0 \le i \le k-1$. The subtrees rooted at the leaves of $\ensuremath{S}_k(t,v)$ form a rooted forest with degree statistics $\ensuremath{\mathrm{n}}-w(\ensuremath{\mathrm{d}})$, with a marked vertex in a specific tree; by (\ref{eq:marked_forest_count}) the number of such marked forests is
\[
{n-k \choose \ensuremath{\mathrm{n}}(c)-w(\ensuremath{\mathrm{d}},c),c \ge 0}.
\]
The result follows.
\end{proof}
For the next corollary we introduce the falling factorial notation $(m)_b = m(m-1)\cdot \ldots\cdot (m-b+1)$.
\begin{cor}\label{cor:dseq_formula}
Fix degree statistics $\ensuremath{\mathrm{n}}=(\ensuremath{\mathrm{n}}(c),c \ge 0)$ with $\sum_{c \ge 0} \ensuremath{\mathrm{n}}(c)=n$ and with $\sum_{c \ge 0} c\ensuremath{\mathrm{n}}(c)=n-1$.
For any sequence $\ensuremath{\mathrm{d}}=(d_0,\ldots,d_{k-1})$ of non-negative integers with $w(\ensuremath{\mathrm{d}}, c)\le \ensuremath{\mathrm{n}}(c)$ for all $c\ge 0$, if $(T,V) \in_u \ensuremath{\mathscr{T}}_\ensuremath{\mathrm{n}}^\bullet$ then we have
\[
\p{\big(\deg_t(V^0),\ldots,\deg_t(V^{k-1})\big)=\ensuremath{\mathrm{d}},|V| \ge k}
=
\frac{1}{(n)_k} \cdot \prod_{i=0}^{k-1} d_i\cdot \prod_{c \ge 0} (\ensuremath{\mathrm{n}}(c))_{w(\ensuremath{\mathrm{d}},c)}\, .
\]
\end{cor}
\begin{proof}
Since $(T,V) \in_u \ensuremath{\mathscr{T}}_\ensuremath{\mathrm{n}}^\bullet$, this probability is simply $|\ensuremath{\mathscr{T}}_\ensuremath{\mathrm{n}}^\bullet(\ensuremath{\mathrm{d}})|/|\ensuremath{\mathscr{T}}_\ensuremath{\mathrm{n}}^\bullet|$, and the result follows from the formula for $|\ensuremath{\mathscr{T}}_\ensuremath{\mathrm{n}}^\bullet(\ensuremath{\mathrm{d}})|$ given in Proposition~\ref{prop:tnd_count}.
\end{proof}
\begin{proof}[Proof of Proposition~\ref{prop:sampler}]
Write $n=\sum_{c \ge 0}\ensuremath{\mathrm{n}}(c)$.
It suffices to show that for any sequence of non-negative integers $\ensuremath{\mathrm{d}}=(d_1,\ldots,d_{k})$ with $w(\ensuremath{\mathrm{d}}, c)\le \ensuremath{\mathrm{n}}(c)$ for all $c\ge 0$,
\begin{equation}\label{eq:sampler_toprove}
\p{(D_1,\ldots,D_k)=\ensuremath{\mathrm{d}},M \ge k+1}
=\frac{1}{(n)_k}\cdot\prod_{i=1}^{k} d_i \cdot \prod_{c \ge 0} (\ensuremath{\mathrm{n}}(c))_{w(\ensuremath{\mathrm{d}},c)}\, ,
\end{equation}
since then by summing over $\ensuremath{\mathrm{d}}$ in this equation and in Corollary~\ref{cor:dseq_formula} (note the shift by $1$ of the indices) it follows that $\p{|V| \ge k} = \p{M \ge k+1}$, which establishes the distributional identity.
By the defining equation (\ref{eq:size_biasing}) for the size-biased sequence $(D_1,\ldots,D_n)$ and since $|\ensuremath{\mathrm{n}}|_1=n-1$, we have
\begin{align}
\p{(D_1,\ldots,D_k)=\ensuremath{\mathrm{d}}}
& =
\prod_{i=1}^k \probC{D_i=d_i}{(D_1,\ldots,D_{i-1})=(d_1,\ldots,d_{i-1})}\nonumber\\
& = \prod_{i=1}^k
\frac{d_{i}(\ensuremath{\mathrm{n}}(d_i)-w((d_1,\ldots,d_{i-1}),d_i))}{n-1-d_1-\ldots-d_{i-1}}.
\label{eq:d1todk_ident}
\end{align}
On the event that $(D_1,\ldots,D_{k})=(d_1,\ldots,d_k)$, we have $M \ge k+1$ if and only if
\[
U_i > \frac{1+\sum_{j=1}^{i-1} (d_{j}-1)}{n+1-i}
\]
for each $1 \le i \le k$.
Since
\[
n+1-i-\Big(1+\sum_{j=1}^{i-1} (d_j-1)\Big)
= n-1-d_1-\ldots-d_{i-1}\, ,
\]
it follows that
\begin{align*}
\probC{M \ge k+1}{(D_1,\ldots,D_k)=\ensuremath{\mathrm{d}}}
& = \prod_{i=1}^{k} \p{U_i > \frac{1+\sum_{j=1}^{i-1} (d_{j}-1)}{n+1-i}}
\\
& = \prod_{i=1}^{k} \frac{n-1-d_1-\ldots-d_{i-1}}{n+1-i}\, ,
\end{align*}
so by (\ref{eq:d1todk_ident}),
\begin{align*}
\p{(D_1,\ldots,D_k)=\ensuremath{\mathrm{d}},M \ge k+1}
& =
\prod_{i=1}^k
\frac{d_{i}(\ensuremath{\mathrm{n}}(d_i)-w((d_1,\ldots,d_{i-1}), d_i)}{n+1-i} \\
& = \frac{1}{(n)_k} \prod_{i=1}^k d_i (\ensuremath{\mathrm{n}}(d_i)-w((d_1,\ldots,d_{i-1}), d_i).
\end{align*}
Equation (\ref{eq:sampler_toprove}) follows since
\[
\prod_{i=1}^k (\ensuremath{\mathrm{n}}(d_i)-w((d_1,\ldots,d_{i-1}),d_i)) = \prod_{c \ge 0} (\ensuremath{\mathrm{n}}(c))_{w(\ensuremath{\mathrm{d}},c)}\, . \qedhere
\]
\end{proof}
\section{\bf Proof of Theorem~\ref{thm:main_mod}}
Fix degree statistics $\ensuremath{\mathrm{n}}=(\ensuremath{\mathrm{n}}(c),c \ge 0)$ with $\sum_{c \ge 0} \ensuremath{\mathrm{n}}(c) = n$ and $|\ensuremath{\mathrm{n}}|_1=\sum_{c \ge 0} c\ensuremath{\mathrm{n}}(c)=n-1$.
To prove the theorem, we construct a size-biasing of $\ensuremath{\mathrm{n}}$ using a Poissonization trick similar to one introduced in \cite{MR1741774} in the context of inhomogeneous continuum random trees. Several of the definitions of the next two paragraphs are illustrated in Figure~\ref{fig:poisson}.
Let $(d_1,\ldots,d_n)$ be such that
$d_i = c$ if and only if $\sum_{b=0}^{c-1} \ensuremath{\mathrm{n}}(b) < i \le \sum_{b=0}^c \ensuremath{\mathrm{n}}(b)$, so that $(d_1,\ldots,d_n)$ contains $\ensuremath{\mathrm{n}}(c)$ entries with value $c$ for each $c \ge 0$. Let $l_1=0$ and for $1 \le i \le n$, let
$l_{i+1} = l_i + d_i/(n-1)$, and define $I_i = [l_i,l_{i+1})$. The intervals $I_1,\ldots,I_n$ are disjoint and partition $[0,1)$.
Next let $\bN$ be a homogeneous Poisson process on $[0,\infty) \times[0,1)$, with atoms $((S_\ell,U_\ell),\ell \ge 1)$ listed in increasing order of arrival time (so $0 < S_1 < S_2 < \ldots$). Then $(S_\ell,\ell \ge 1)$ is a rate-one Poisson process on $[0,\infty)$, and the random variables $(U_\ell,\ell \ge 1)$ are independent Uniform$[0,1]$, independent of $(S_\ell,\ell \ge 1)$.
For each $\ell \ge 1$, let $J(\ell)$ be the index of the interval containing the point $U_\ell$, so $U_\ell \in I_{J(\ell)}$. Let $M(1) = 1$, and for $\ell \ge 1$ let
\[
M(\ell+1) = \inf\{k> M(\ell): U_k \not\in I_{J(M(1))}\cup\ldots\cup I_{J(M(\ell))}\}\, ,
\]
so $i \in \{M(\ell),\ell \ge 1\}$ precisely if $U_i \not\in I_1\cup\ldots \cup I_{i-1}$.
Then for any vector $(j(1),\ldots,j(\ell))$ of distinct elements of $[n]$ and any increasing sequence of integers $(m(1),\ldots,m(\ell))$ with $m(1)=1$, we have
\begin{align*}
& \probC{J(m(\ell))=j(\ell),M(\ell)=m(\ell)}{J(m(k))=j(k)\mbox{ and }M(k)=m(k)\mbox{ for all }1 \le k \le \ell-1}\\
& =
\p{U_{m(\ell)} \in I_{j(\ell)}\mbox{ and } U_j \in \bigcup_{k=1}^{\ell-1} I_{j(k)}\mbox{ for all }j \in \{m(\ell-1)+1,\ldots,m(\ell)-1\}}\\
& =
\frac{d_{j(\ell)}}{n-1} \cdot \pran{\frac{\sum_{k=1}^{\ell-1} d_{j(k)}}{n-1}}^{m(\ell)-m(\ell-1)-1}.
\end{align*}
Summing over all possible values for $(M(1),\ldots,M(\ell))$, it follows that if $\sum_{k=1}^{\ell-1}d_{j(k)}< n-1$ then
\[
\probC{J(M(\ell))=j(\ell)}{J(M(k))=j(k)\mbox{ for all } 1 \le k \le \ell-1} = \frac{d_{j(\ell)}}{n-1-d_{j(1)}-\ldots-d_{j(\ell-1)}}\, .
\]
This implies that, writing $n'=n-\ensuremath{\mathrm{n}}(0)$,
the sequence $(D(1),\ldots,D(n))$ defined by
\[
D(\ell) = \begin{cases}
d_{J(M(\ell))} & \mbox{ if }1 \le \ell \le n'\\
0 & \mbox{ otherwise}
\end{cases}
\]
is a size-biasing of $\ensuremath{\mathrm{n}}$, since
\begin{align*}
& \p{D(l) = d\mid (D(1),...,D(l-1)) = (d_1,...,d_{l-1})} \\
& = \probC{d_{J(M(l))} = d}{(D(1),...,D(l-1)) = (d_1,...,d_{l-1})}\\
& = \sum_{i\in [n]} \probC{J(M(l))= i,\, d_i = d }{(D(1),...,D(l-1)) = (d_1,...,d_{l-1})}\\
& = \frac{d(\ensuremath{\mathrm{n}}(d) - w((d_1,...,d_{l-1}),d)}{n - 1 - d_1 - ... - d_{l-1}},
\end{align*}
where we recall that $w((d_1,...,d_{l-1}),d) = |\{i\in [l-1]: d_i = d\}|$.
When parsing the definitions of the coming paragraph, Figure~\ref{fig:poisson} will again be useful. For each $1 \le i \le n$, let $r_i = l_i + \max(0,(d_i-1)/(n-1)) \le l_{i+1}$. For $\ell \ge 1$ let $C_\ell = \bigcup_{k=1}^\ell [r_{J(k)},l_{J(k)+1})$, so for each interval $I_i$ which contains at least one point from $U_1,\ldots,U_\ell$, the region $C_\ell$ contains a sub-interval of $I_i$ of length $1/(n-1)$.
Let
\[
\tau = \min\pran{\ell\ge 1: U_\ell \in \bigcup_{k=1}^{\ell-1} I_{J(k)}\setminus C_{\ell-1}} =
\min\pran{\ell\ge 1: U_\ell \in \bigcup_{k=1}^{\ell-1} [l_{J(k)},r_{J(k)})}\, .
\]
By the definition of the indices $(M(\ell),\ell \ge 1)$, we have $\tau \not \in \{M(\ell),\ell \ge 1)\}$, since the points $(U_{M(\ell)},\ell \ge 1)$ are precisely those which, on their arrival, land in a previously empty interval, whereas $U_\tau$ falls in a subinterval of an interval which already contains one of $U_1,\ldots,U_{\tau-1}$.
\begin{figure}[hbt]
\centering
\tikzset{
pattern size/.store in=\mcSize,
pattern size = 5pt,
pattern thickness/.store in=\mcThickness,
pattern thickness = 0.3pt,
pattern radius/.store in=\mcRadius,
pattern radius = 1pt}
\makeatletter
\pgfutil@ifundefined{pgf@pattern@name@_jy889d05m}{
\pgfdeclarepatternformonly[\mcThickness,\mcSize]{_jy889d05m}
{\pgfqpoint{0pt}{0pt}}
{\pgfpoint{\mcSize+\mcThickness}{\mcSize+\mcThickness}}
{\pgfpoint{\mcSize}{\mcSize}}
{
\pgfsetcolor{\tikz@pattern@color}
\pgfsetlinewidth{\mcThickness}
\pgfpathmoveto{\pgfqpoint{0pt}{0pt}}
\pgfpathlineto{\pgfpoint{\mcSize+\mcThickness}{\mcSize+\mcThickness}}
\pgfusepath{stroke}
}}
\makeatother
\tikzset{
pattern size/.store in=\mcSize,
pattern size = 5pt,
pattern thickness/.store in=\mcThickness,
pattern thickness = 0.3pt,
pattern radius/.store in=\mcRadius,
pattern radius = 1pt}
\makeatletter
\pgfutil@ifundefined{pgf@pattern@name@_perv8pg8o}{
\pgfdeclarepatternformonly[\mcThickness,\mcSize]{_perv8pg8o}
{\pgfqpoint{0pt}{0pt}}
{\pgfpoint{\mcSize+\mcThickness}{\mcSize+\mcThickness}}
{\pgfpoint{\mcSize}{\mcSize}}
{
\pgfsetcolor{\tikz@pattern@color}
\pgfsetlinewidth{\mcThickness}
\pgfpathmoveto{\pgfqpoint{0pt}{0pt}}
\pgfpathlineto{\pgfpoint{\mcSize+\mcThickness}{\mcSize+\mcThickness}}
\pgfusepath{stroke}
}}
\makeatother
\tikzset{
pattern size/.store in=\mcSize,
pattern size = 5pt,
pattern thickness/.store in=\mcThickness,
pattern thickness = 0.3pt,
pattern radius/.store in=\mcRadius,
pattern radius = 1pt}
\makeatletter
\pgfutil@ifundefined{pgf@pattern@name@_tm4jdcxlk}{
\pgfdeclarepatternformonly[\mcThickness,\mcSize]{_tm4jdcxlk}
{\pgfqpoint{0pt}{0pt}}
{\pgfpoint{\mcSize+\mcThickness}{\mcSize+\mcThickness}}
{\pgfpoint{\mcSize}{\mcSize}}
{
\pgfsetcolor{\tikz@pattern@color}
\pgfsetlinewidth{\mcThickness}
\pgfpathmoveto{\pgfqpoint{0pt}{0pt}}
\pgfpathlineto{\pgfpoint{\mcSize+\mcThickness}{\mcSize+\mcThickness}}
\pgfusepath{stroke}
}}
\makeatother
\tikzset{
pattern size/.store in=\mcSize,
pattern size = 5pt,
pattern thickness/.store in=\mcThickness,
pattern thickness = 0.3pt,
pattern radius/.store in=\mcRadius,
pattern radius = 1pt}
\makeatletter
\pgfutil@ifundefined{pgf@pattern@name@_nlv3f6ai7}{
\pgfdeclarepatternformonly[\mcThickness,\mcSize]{_nlv3f6ai7}
{\pgfqpoint{0pt}{0pt}}
{\pgfpoint{\mcSize+\mcThickness}{\mcSize+\mcThickness}}
{\pgfpoint{\mcSize}{\mcSize}}
{
\pgfsetcolor{\tikz@pattern@color}
\pgfsetlinewidth{\mcThickness}
\pgfpathmoveto{\pgfqpoint{0pt}{0pt}}
\pgfpathlineto{\pgfpoint{\mcSize+\mcThickness}{\mcSize+\mcThickness}}
\pgfusepath{stroke}
}}
\makeatother
\tikzset{every picture/.style={line width=0.75pt}}
\begin{tikzpicture}[x=0.75pt,y=0.75pt,yscale=-1,xscale=1]
\draw [color={rgb, 255:red, 66; green, 118; blue, 179 } ,draw opacity=1 ][pattern=_jy889d05m,pattern size=6pt,pattern thickness=0.75pt,pattern radius=0pt, pattern color={rgb, 255:red, 74; green, 144; blue, 226}] (137.84,145.03) -- (507.83,145.03) -- (507.83,153.02) -- (137.84,153.02) -- cycle ;
\draw [color={rgb, 255:red, 0; green, 0; blue, 0 } ,draw opacity=1 ] (87.72,52.85) -- (508.82,52.85) ;
\draw [color={rgb, 255:red, 74; green, 144; blue, 226 } ,draw opacity=1 ] (92.28,182.75) -- (507.45,182.75) ;
\draw [color={rgb, 255:red, 74; green, 144; blue, 226 } ,draw opacity=1 ] (92.28,145.03) -- (507.83,145.03) ;
\draw [color={rgb, 255:red, 74; green, 144; blue, 226 } ,draw opacity=1 ] (92.28,118.1) -- (507.83,118.1) ;
\draw [color={rgb, 255:red, 74; green, 144; blue, 226 } ,draw opacity=1 ] (92.18,95.27) -- (507.54,95.27) ;
\draw [color={rgb, 255:red, 74; green, 144; blue, 226 } ,draw opacity=1 ] (92.28,74.6) -- (507.83,74.6) ;
\draw [color={rgb, 255:red, 139; green, 87; blue, 42 } ,draw opacity=1 ] [dash pattern={on 4.5pt off 4.5pt}] (137.2,171.17) -- (137.2,233.39) ;
\draw [color={rgb, 255:red, 139; green, 87; blue, 42 } ,draw opacity=1 ] [dash pattern={on 4.5pt off 4.5pt}] (181.8,131.47) -- (181.8,233.39) ;
\draw [color={rgb, 255:red, 139; green, 87; blue, 42 } ,draw opacity=1 ] [dash pattern={on 4.5pt off 4.5pt}] (137.2,171.17) -- (94.79,171.17) ;
\draw [color={rgb, 255:red, 139; green, 87; blue, 42 } ,draw opacity=1 ] [dash pattern={on 4.5pt off 4.5pt}] (181.8,131.47) -- (104.57,131.47) ;
\draw [color={rgb, 255:red, 0; green, 0; blue, 0 } ,draw opacity=1 ][fill={rgb, 255:red, 0; green, 0; blue, 0 } ,fill opacity=1 ] (134.48,171.17) .. controls (134.48,169.67) and (135.7,168.45) .. (137.2,168.45) .. controls (138.71,168.45) and (139.92,169.67) .. (139.92,171.17) .. controls (139.92,172.67) and (138.71,173.89) .. (137.2,173.89) .. controls (135.7,173.89) and (134.48,172.67) .. (134.48,171.17) -- cycle ;
\draw [color={rgb, 255:red, 0; green, 0; blue, 0 } ,draw opacity=1 ][fill={rgb, 255:red, 0; green, 0; blue, 0 } ,fill opacity=1 ] (179.08,131.47) .. controls (179.08,129.97) and (180.3,128.76) .. (181.8,128.76) .. controls (183.3,128.76) and (184.52,129.97) .. (184.52,131.47) .. controls (184.52,132.98) and (183.3,134.19) .. (181.8,134.19) .. controls (180.3,134.19) and (179.08,132.98) .. (179.08,131.47) -- cycle ;
\draw [color={rgb, 255:red, 0; green, 0; blue, 0 } ,draw opacity=1 ][fill={rgb, 255:red, 0; green, 0; blue, 0 } ,fill opacity=1 ] (266.09,204.89) .. controls (266.09,203.39) and (267.31,202.17) .. (268.81,202.17) .. controls (270.31,202.17) and (271.53,203.39) .. (271.53,204.89) .. controls (271.53,206.39) and (270.31,207.61) .. (268.81,207.61) .. controls (267.31,207.61) and (266.09,206.39) .. (266.09,204.89) -- cycle ;
\draw [color={rgb, 255:red, 0; green, 0; blue, 0 } ,draw opacity=1 ][fill={rgb, 255:red, 0; green, 0; blue, 0 } ,fill opacity=1 ] (309.59,66.81) .. controls (309.59,65.3) and (310.81,64.09) .. (312.31,64.09) .. controls (313.81,64.09) and (315.03,65.3) .. (315.03,66.81) .. controls (315.03,68.31) and (313.81,69.52) .. (312.31,69.52) .. controls (310.81,69.52) and (309.59,68.31) .. (309.59,66.81) -- cycle ;
\draw [color={rgb, 255:red, 0; green, 0; blue, 0 } ,draw opacity=1 ][fill={rgb, 255:red, 0; green, 0; blue, 0 } ,fill opacity=1 ] (397.15,109.22) .. controls (397.15,107.72) and (398.37,106.5) .. (399.87,106.5) .. controls (401.37,106.5) and (402.59,107.72) .. (402.59,109.22) .. controls (402.59,110.72) and (401.37,111.94) .. (399.87,111.94) .. controls (398.37,111.94) and (397.15,110.72) .. (397.15,109.22) -- cycle ;
\draw [color={rgb, 255:red, 0; green, 0; blue, 0 } ,draw opacity=1 ][fill={rgb, 255:red, 0; green, 0; blue, 0 } ,fill opacity=1 ] (435.76,109.18) .. controls (435.76,107.68) and (436.98,106.46) .. (438.48,106.46) .. controls (439.98,106.46) and (441.2,107.68) .. (441.2,109.18) .. controls (441.2,110.68) and (439.98,111.9) .. (438.48,111.9) .. controls (436.98,111.9) and (435.76,110.68) .. (435.76,109.18) -- cycle ;
\draw [color={rgb, 255:red, 139; green, 87; blue, 42 } ,draw opacity=1 ] [dash pattern={on 4.5pt off 4.5pt}] (486.33,85.25) -- (486.33,232.31) ;
\draw [color={rgb, 255:red, 139; green, 87; blue, 42 } ,draw opacity=1 ] [dash pattern={on 4.5pt off 4.5pt}] (486.33,87.97) -- (96.42,87.97) ;
\draw [color={rgb, 255:red, 0; green, 0; blue, 0 } ,draw opacity=1 ][fill={rgb, 255:red, 0; green, 0; blue, 0 } ,fill opacity=1 ] (483.61,87.97) .. controls (483.61,86.47) and (484.83,85.25) .. (486.33,85.25) .. controls (487.84,85.25) and (489.05,86.47) .. (489.05,87.97) .. controls (489.05,89.47) and (487.84,90.69) .. (486.33,90.69) .. controls (484.83,90.69) and (483.61,89.47) .. (483.61,87.97) -- cycle ;
\draw [color={rgb, 255:red, 139; green, 87; blue, 42 } ,draw opacity=1 ] (268.81,221.42) -- (268.81,233.39) ;
\draw [color={rgb, 255:red, 139; green, 87; blue, 42 } ,draw opacity=1 ] (181.8,221.43) -- (181.8,233.39) ;
\draw [color={rgb, 255:red, 139; green, 87; blue, 42 } ,draw opacity=1 ] (137.2,221.43) -- (137.2,233.39) ;
\draw [color={rgb, 255:red, 139; green, 87; blue, 42 } ,draw opacity=1 ] (486.33,221.43) -- (486.33,233.39) ;
\draw [color={rgb, 255:red, 139; green, 87; blue, 42 } ,draw opacity=1 ] (104.03,171.17) -- (92.61,171.17) ;
\draw [color={rgb, 255:red, 139; green, 87; blue, 42 } ,draw opacity=1 ] (103.49,131.47) -- (92.07,131.47) ;
\draw [color={rgb, 255:red, 139; green, 87; blue, 42 } ,draw opacity=1 ] (103.49,87.97) -- (92.07,87.97) ;
\draw [color={rgb, 255:red, 66; green, 118; blue, 179 } ,draw opacity=1 ][fill={rgb, 255:red, 74; green, 144; blue, 226 } ,fill opacity=0.16 ] (181.8,118.1) -- (507.83,118.1) -- (507.83,126.08) -- (181.8,126.08) -- cycle ;
\draw (98.59,38.71) -- (98.59,238.83) ;
\draw (87.72,227.19) -- (520.77,227.19) ;
\draw [shift={(522.77,227.19)}, rotate = 180] [color={rgb, 255:red, 0; green, 0; blue, 0 } ][line width=0.75] (10.93,-4.9) .. controls (6.95,-2.3) and (3.31,-0.67) .. (0,0) .. controls (3.31,0.67) and (6.95,2.3) .. (10.93,4.9) ;
\draw [color={rgb, 255:red, 74; green, 144; blue, 226 } ,draw opacity=1 ][fill={rgb, 255:red, 74; green, 144; blue, 226 } ,fill opacity=0.16 ] (399.87,95.27) -- (507.54,95.27) -- (507.54,103.25) -- (399.87,103.25) -- cycle ;
\draw [color={rgb, 255:red, 66; green, 118; blue, 179 } ,draw opacity=1 ][fill={rgb, 255:red, 74; green, 144; blue, 226 } ,fill opacity=0.16 ] (312.49,52.85) -- (508.82,52.85) -- (508.82,60.83) -- (312.49,60.83) -- cycle ;
\draw [color={rgb, 255:red, 74; green, 144; blue, 226 } ,draw opacity=1 ][fill={rgb, 255:red, 74; green, 144; blue, 226 } ,fill opacity=0.16 ] (486.33,74.6) -- (507.83,74.6) -- (507.83,82.59) -- (486.33,82.59) -- cycle ;
\draw [color={rgb, 255:red, 74; green, 144; blue, 226 } ,draw opacity=1 ][fill={rgb, 255:red, 74; green, 144; blue, 226 } ,fill opacity=0.16 ] (137.84,145.03) -- (507.83,145.03) -- (507.83,153.02) -- (137.84,153.02) -- cycle ;
\draw [color={rgb, 255:red, 66; green, 118; blue, 179 } ,draw opacity=1 ][fill={rgb, 255:red, 74; green, 144; blue, 226 } ,fill opacity=0.16 ] (269.06,182.73) -- (507.45,182.73) -- (507.45,190.71) -- (269.06,190.71) -- cycle ;
\draw [color={rgb, 255:red, 74; green, 144; blue, 226 } ,draw opacity=1 ][pattern=_perv8pg8o,pattern size=6pt,pattern thickness=0.75pt,pattern radius=0pt, pattern color={rgb, 255:red, 74; green, 144; blue, 226}] (181.8,118.1) -- (507.83,118.1) -- (507.83,126.08) -- (181.8,126.08) -- cycle ;
\draw [color={rgb, 255:red, 74; green, 144; blue, 226 } ,draw opacity=1 ][pattern=_tm4jdcxlk,pattern size=6pt,pattern thickness=0.75pt,pattern radius=0pt, pattern color={rgb, 255:red, 74; green, 144; blue, 226}] (312.49,52.85) -- (508.82,52.85) -- (508.82,60.83) -- (312.49,60.83) -- cycle ;
\draw [color={rgb, 255:red, 74; green, 144; blue, 226 } ,draw opacity=1 ][pattern=_nlv3f6ai7,pattern size=6pt,pattern thickness=0.75pt,pattern radius=0pt, pattern color={rgb, 255:red, 74; green, 144; blue, 226}] (269.06,182.73) -- (507.45,182.73) -- (507.45,190.71) -- (269.06,190.71) -- cycle ;
\draw [color={rgb, 255:red, 208; green, 2; blue, 27 } ,draw opacity=1 ] [dash pattern={on 4.5pt off 4.5pt}] (368.87,53.96) -- (368.87,163.56) -- (368.87,233.39) ;
\draw [color={rgb, 255:red, 208; green, 2; blue, 27 } ,draw opacity=1 ][fill={rgb, 255:red, 208; green, 2; blue, 27 } ,fill opacity=1 ] (366.15,161.38) .. controls (366.15,159.88) and (367.37,158.67) .. (368.87,158.67) .. controls (370.37,158.67) and (371.59,159.88) .. (371.59,161.38) .. controls (371.59,162.89) and (370.37,164.1) .. (368.87,164.1) .. controls (367.37,164.1) and (366.15,162.89) .. (366.15,161.38) -- cycle ;
\draw [color={rgb, 255:red, 139; green, 87; blue, 42 } ,draw opacity=1 ] [dash pattern={on 4.5pt off 4.5pt}] (224.21,149.42) -- (224.21,227.04) ;
\draw [color={rgb, 255:red, 139; green, 87; blue, 42 } ,draw opacity=1 ] [dash pattern={on 4.5pt off 4.5pt}] (224.21,149.42) -- (100,149.42) ;
\draw [color={rgb, 255:red, 0; green, 0; blue, 0 } ,draw opacity=1 ][fill={rgb, 255:red, 0; green, 0; blue, 0 } ,fill opacity=1 ] (221.5,149.42) .. controls (221.5,147.92) and (222.71,146.7) .. (224.21,146.7) .. controls (225.72,146.7) and (226.93,147.92) .. (226.93,149.42) .. controls (226.93,150.92) and (225.72,152.14) .. (224.21,152.14) .. controls (222.71,152.14) and (221.5,150.92) .. (221.5,149.42) -- cycle ;
\draw [color={rgb, 255:red, 139; green, 87; blue, 42 } ,draw opacity=1 ] (224.21,221.04) -- (224.21,233.01) ;
\draw [color={rgb, 255:red, 139; green, 87; blue, 42 } ,draw opacity=1 ] (104,149.42) -- (92.58,149.42) ;
\draw [color={rgb, 255:red, 74; green, 144; blue, 226 } ,draw opacity=1 ] (269.06,190.75) -- (534.33,190.75) ;
\draw (525.83,227.06) node [anchor=north west][inner sep=0.75pt] {$t$};
\draw (66.67,215.1) node [anchor=north west][inner sep=0.75pt] [color={rgb, 255:red, 0; green, 0; blue, 0 } ,opacity=1 ] {$0$};
\draw (66.67,41.08) node [anchor=north west][inner sep=0.75pt] [color={rgb, 255:red, 0; green, 0; blue, 0 } ,opacity=1 ] {$1$};
\draw (125.79,232.63) node [anchor=north west][inner sep=0.75pt] [font=\normalsize,color={rgb, 255:red, 139; green, 87; blue, 42 } ,opacity=1 ] {$S_{1}$};
\draw (171.3,232.63) node [anchor=north west][inner sep=0.75pt] [font=\normalsize,color={rgb, 255:red, 139; green, 87; blue, 42 } ,opacity=1 ] {$S_{2}$};
\draw (53.96,161.76) node [anchor=north west][inner sep=0.75pt] [font=\small,color={rgb, 255:red, 139; green, 87; blue, 42 } ,opacity=1 ] {$U_{1}$};
\draw (53.96,120.52) node [anchor=north west][inner sep=0.75pt] [font=\small,color={rgb, 255:red, 139; green, 87; blue, 42 } ,opacity=1 ] {$U_{2}$};
\draw (54.1,78.01) node [anchor=north west][inner sep=0.75pt] [font=\small,color={rgb, 255:red, 139; green, 87; blue, 42 } ,opacity=1 ] {$U_{N}$};
\draw (378.13,232.63) node [anchor=north west][inner sep=0.75pt] [font=\normalsize,color={rgb, 255:red, 139; green, 87; blue, 42 } ,opacity=1 ] {$\ \ \ \ \ \ \ \ \cdots \ \ \ \ \ \ \ \ S_{N}$};
\draw (516.99,51.91) node [anchor=north west][inner sep=0.75pt] [font=\normalsize,color={rgb, 255:red, 74; green, 144; blue, 226 } ,opacity=1 ] {$I_{6}$};
\draw (516.99,74.75) node [anchor=north west][inner sep=0.75pt] [font=\normalsize,color={rgb, 255:red, 74; green, 144; blue, 226 } ,opacity=1 ] {$I_{5}$};
\draw (515.9,96.5) node [anchor=north west][inner sep=0.75pt] [font=\normalsize,color={rgb, 255:red, 74; green, 144; blue, 226 } ,opacity=1 ] {$I_{4}$};
\draw (515.9,120.43) node [anchor=north west][inner sep=0.75pt] [font=\normalsize,color={rgb, 255:red, 74; green, 144; blue, 226 } ,opacity=1 ] {$I_{3}$};
\draw (515.9,151.97) node [anchor=north west][inner sep=0.75pt] [font=\normalsize,color={rgb, 255:red, 74; green, 144; blue, 226 } ,opacity=1 ] {$I_{2}$};
\draw (515.9,193.3) node [anchor=north west][inner sep=0.75pt] [font=\normalsize,color={rgb, 255:red, 74; green, 144; blue, 226 } ,opacity=1 ] {$I_{1}$};
\draw (258.31,232.63) node [anchor=north west][inner sep=0.75pt] [font=\normalsize,color={rgb, 255:red, 139; green, 87; blue, 42 } ,opacity=1 ] {$S_{4}
\ \ \ \ \ \ \cdots$};
\draw (79.31,176.64) node [anchor=north west][inner sep=0.75pt] [font=\scriptsize,color={rgb, 255:red, 74; green, 144; blue, 226 } ,opacity=1 ] {$l_{2}$};
\draw (79.31,139.66) node [anchor=north west][inner sep=0.75pt] [font=\scriptsize,color={rgb, 255:red, 74; green, 144; blue, 226 } ,opacity=1 ] {$l_{3}$};
\draw (79.31,112.83) node [anchor=north west][inner sep=0.75pt] [font=\scriptsize,color={rgb, 255:red, 74; green, 144; blue, 226 } ,opacity=1 ] {$l_{4}$};
\draw (78.95,89.63) node [anchor=north west][inner sep=0.75pt] [font=\scriptsize,color={rgb, 255:red, 74; green, 144; blue, 226 } ,opacity=1 ] {$l_{5}$};
\draw (80.04,68.96) node [anchor=north west][inner sep=0.75pt] [font=\scriptsize,color={rgb, 255:red, 74; green, 144; blue, 226 } ,opacity=1 ] {$l_{6}$};
\draw (361.05,232.59) node [anchor=north west][inner sep=0.75pt] [font=\normalsize,color={rgb, 255:red, 208; green, 2; blue, 27 } ,opacity=1 ] {$S_{\tau }$};
\draw (213.31,232.63) node [anchor=north west][inner sep=0.75pt] [font=\normalsize,color={rgb, 255:red, 139; green, 87; blue, 42 } ,opacity=1 ] {$S_{3}$};
\draw (53.96,137.76) node [anchor=north west][inner sep=0.75pt] [font=\small,color={rgb, 255:red, 139; green, 87; blue, 42 } ,opacity=1 ] {$U_{3}$};
\draw (535.98,181.64) node [anchor=north west][inner sep=0.75pt] [font=\scriptsize,color={rgb, 255:red, 74; green, 144; blue, 226 } ,opacity=1 ] {$r_{1} =l_{2} -1/( n-1)$};
\end{tikzpicture}
\caption{
The black dots represent the atoms $((S_i,U_i),i \ge 1)$ of the Poisson process $\mathbf{N}$.
The union of the striped blue regions is the ``forbidden'' region up to the stopping time $S_\tau$; the projection of the striped blue regions onto the $y$-axis is $C_{\tau-1}$.}
\label{fig:poisson}
\end{figure}
Fix any vector $(j(1),\ldots,j(\ell))$ of distinct elements of $[n]$ and any increasing sequence of integers $(m(1),\ldots,m(\ell-1))$ with $m(1)=1$. Suppose that $\tau > M(\ell-1)$, that $M(k)=m(k)$ and that $J(m(k))=j(k)$ for all $1 \le k < \ell$. Then $\tau \le M(\ell)$ (and so $\tau < M(\ell)$) precisely if the first point among $(U_m,m > m(\ell-1))$ which does {\em not} fall into $C_{m(\ell-1)} = \bigcup_{k=1}^{\ell-1} [r_{j(k)},l_{j(k)+1})$, {\em does} belong to the set
\[
\bigcup_{k=1}^{\ell-1} I_{j(k)} \setminus C_{m(\ell-1)}
=
\bigcup_{k=1}^{\ell-1} [l_{j(k)},r_{j(k)}).
\]
Since $r_{j(k)}-\ell_{j(k)} = (d_{j(k)}-1)/(n-1)$, writing $U$ for a Uniform$[0,1]$ random variable, it follows that
\begin{align*}
& \probC{\tau < M(\ell)}{\tau > M(\ell-1),M(k)=m(k)\mbox{ and }J(m(k))=j(k)\mbox{ for all }1 \le k < \ell}\\
& = \probC{U \in \bigcup_{k=1}^{\ell-1} [l_{j(k)},r_{j(k)})}{U \not\in \bigcup_{k=1}^{\ell-1} [r_{j(k)},l_{j(k)+1})} \\
& =
\frac{\sum_{k=1}^{\ell-1} (d_{j(k)}-1)/(n-1)}{1-((\ell-1)/(n-1))} = \frac{\sum_{k=1}^{\ell -1}(d_{j(k)}-1)}{n-\ell}\, .
\end{align*}
Since the final expression depends on $(j(1),\ldots,j(\ell-1))$ and $(m(1),\ldots,m(\ell-1))$ only through the values $(d_{j(1)},\ldots,d_{j(\ell-1)})$, we have
\begin{align}\label{eq:tau_condprob}
& \probC{\tau \le M(\ell)}{D(1),\ldots,D(\ell-1),\tau > M(\ell-1)} \nonumber\\
& = \probC{\tau < M(\ell)}{D(1),\ldots,D(\ell-1),\tau > M(\ell-1)} \nonumber\\
& = \frac{\sum_{k=1}^{\ell-1} (D(k)-1)}{n-\ell}.
\end{align}
Writing $\sigma=\sup(k\ge 1: \tau > M(k))$, the preceding identity implies that $\sigma$ has the same distribution as the random variable from Theorem~\ref{thm:main_mod} with the same name. Since $M(k) \ge k$, it follows that $\sigma \le \sup(k \ge 1: \tau > k) = \tau-1$, so for all $\ell \ge 1$,
\begin{equation}\label{eq:sigma_tau_ident}
\p{\sigma \ge \ell} = \p{\tau > M(\ell)} \le \p{\tau > \ell}\, .
\end{equation}
The second bound of Theorem~\ref{thm:main_mod} now follows quite straightforwardly: if $\ensuremath{\mathrm{n}}(1)=0$ then $[l_i,r_{i+1}) = (d_i-1)/(n-1) \ge 1/(n-1)$ whenever $d_i \ne 0$. Therefore, for any $\ell \ge 2$, if $\tau > M(\ell-1)$ then $D(k) \ge 2$ for all $1 \le k \le \ell-1$, so by (\ref{eq:tau_condprob}),
\[
\probC{\tau \le M(\ell)}{\tau > M(\ell-1)} \ge \frac{\ell-1}{n-\ell}.
\]
It follows by induction that
\[
\p{\sigma\ge \ell}=\p{\tau > M(\ell)} \le \prod_{k=1}^{\ell-1}
\pran{1-\frac{k}{n-k}} \le e^{-(\ell-1)^2/(2(n-1))}\, ,
\]
proving the second inequality in Theorem~\ref{thm:main_mod}.
We now turn to proving the first inequality in Theorem~\ref{thm:main_mod}; this bound is an immediate consequence of the next proposition.
\begin{prop}\label{prop:main_mod}
Write $\ensuremath{\mathrm{v}}=\sum_{i:d_i \ge 2} d_i^2/(n-1)$.
Then for all $\beta \ge 17^{3/2}$,
\[
\p{\tau > \beta \pran{\frac{n-1}{\ensuremath{\mathrm{v}}}}^{1/2}}
\le
\exp\pran{-\frac{1}{3}\pran{\frac{\beta^{2/3}(n-1)}{\ensuremath{\mathrm{v}}}}^{1/2}} + 2\exp\pran{-\frac{\beta^{2/3}}{24}}\, .
\]
\end{prop}
The first bound of Theorem~\ref{thm:main_mod} follows from the proposition since $\sum_{i:d_i \ge 2} d_i^2 = |\ensuremath{\mathrm{n}}|_2^2 - \ensuremath{\mathrm{n}}(1)$ and $|\ensuremath{\mathrm{n}}|_1=(n-1)$, so $\tfrac{n-1}{\ensuremath{\mathrm{v}}} = \tfrac{|\ensuremath{\mathrm{n}}|_1^2}{|\ensuremath{\mathrm{n}}|_2^2 - \ensuremath{\mathrm{n}}(1)}$.
The proposition's proof is where the Poisson process setup comes into its own. Write $\bN(t) = \bN([0,t]\times[0,1])$ for the number of points of $\bN$ arriving by time $t$, and let $\bN_i(t) = \bN([0,t]\times[l_i,r_i))$ be the number of points arriving in the interval $[l_i,r_i)$ by time $t$.
Note that for all $i \in [n]$, if $\bN_i(t) \ge 2$ then $\tau \le \bN(t)$. Thus, letting $T =\inf\{t \ge 0: \max_{i \in [n]}\bN_i(t) \ge 2\}$, we have $\tau \le \bN(T)$. It follows that for all $h \in \N$, if $\bN(t) \le h$ and $T \le t$ then $\tau \le h$, so
\begin{equation}\label{eq:generic_poissonization_bound}
\p{\tau > h} \le \inf_{t \ge 0} \Big(\p{\bN(t) > h} + \p{T > t}\Big)\, .
\end{equation}
We control the first of these probabilities using standard Poisson tail estimates. The second requires a little more work. The random variables $(\bN_i(t),i \in [n])$ are independent and $\bN_i(t)$ is Poisson$(t(r_i-l_i))$-distributed.
Note that $r_i-l_i=0$ when $d_i \le 1$.
Writing $p_i = d_i/(2(n-1))$, we have $r_i-l_i \ge p_i$ whenever $d_i \ge 2$, so
\begin{equation}\label{eq:poisson_upper}
\p{T>t}
= \prod_{i:d_i \ge 2} \p{\bN_i(t) \le 1}
\le\prod_{i:d_i \ge 2} \p{\mathrm{Poisson}(p_it)\le 1}
= \prod_{i: d_i \ge 2} (1+p_it)e^{-p_it}\, .
\end{equation}
Combining this with (\ref{eq:generic_poissonization_bound}) and the tail bound
$\p{\mathrm{Poisson}(t)> h}
=e^{-t((h/t)\log(h/t) - h/t + 1)}$, which holds for $h \ge t$ and can be found in, e.g., \cite[Page 23]{MR3185193}, we obtain that
\begin{equation}\label{eq:poissonization_bd}
\p{\tau > h}
\le \inf_{t \le h}
\pran{
e^{-t((h/t)\log(h/t)- h/t + 1)} + \prod_{i: d_i \ge 2} (1+p_it)e^{-p_it}\,
}.
\end{equation}
We next focus on proving bounds for the second term on the right-hand side of (\ref{eq:poissonization_bd}); our approach is based on that of Lemma~9 in \cite{MR1741774}.
\begin{lem}\label{lem:gtd_bd}
Write $\ensuremath{\mathbf{d}}=(d_1,\ldots,d_n)$ and
let $g(t,\ensuremath{\mathbf{d}}) = \prod_{i: d_i \ge 2} (1+p_it)e^{-p_i t}$, where $p_i = d_i/(2(n-1))$. Also write $p_{\max} = \max_{i \in [n]} p_i$ and $d_{\max}=\max_{i \in [n]} d_i$. Then
for all $0 \le t < 1/p_{\max}=2(n-1)/d_{\max}$,
\[
\log g(t,\ensuremath{\mathbf{d}}) = \sum_{k \ge 2} \frac{(-1)^{k+1}}{k}\sum_{i: d_i \ge 2}\pran{\frac{d_it}{2(n-1)}}^k\, ,
\]
and with $\ensuremath{\mathrm{v}}=\sum_{i:d_i \ge 2} d_i^2/(n-1)$, we have
\[
\left|
\log g(t,\ensuremath{\mathbf{d}})+\frac{\ensuremath{\mathrm{v}} t^2}{8(n-1)}
\right|
\le \frac{d_{\max} t}{6(n-1)-3d_{\max}t}\cdot \frac{\ensuremath{\mathrm{v}} t^2}{4(n-1)} \, .
\]
\end{lem}
\begin{proof}
First, note that
\[
\sum_{k \ge 2} \frac{1}{k} \sum_{i:d_i \ge 2} (p_i t)^k < \infty
\]
for $0 \le t < 1/p_{\max}$, since the inner sum has finitely many summands, each of which decreases geometrically in $k$. By a Taylor expansion of $\log(1+x)$ around $x=0$ and Tonelli's theorem, it follows that
\begin{align*}
\log g(t,\ensuremath{\mathbf{d}})
& = \sum_{i: d_i \ge 2} (\log(1+p_i t)-p_i t)\\
& = \sum_{i: d_i \ge 2} \sum_{k\ge 1} \frac{(-1)^{k+1}}{k} (p_it)^k -
\sum_{i: d_i \ge 2}p_i t\\
& = \sum_{k \ge 2} \frac{(-1)^{k+1}}{k}\sum_{i: d_i \ge 2} (p_it)^k\, .
\end{align*}
Next, note that
\[
\sum_{i: d_i \ge 2} p_i^k \le p_{\max}^{k-2}\sum_{i:d_i \ge 2} p_i^2,
\]
so
\[
\sum_{k \ge 3}\sum_{i: d_i \ge 2} \frac{(p_it)^k}{k}
\le \pran{\sum_{i:d_i \ge 2} (p_it)^2}
\sum_{k \ge 3} \frac{(p_{\max}t)^{k-2}}{k}
\le \pran{\sum_{i:d_i \ge 2} (p_it)^2}
\cdot \frac{p_{\max} t}{3(1-p_{\max}t)}\,
\]
and thus
\[
\left|\log g(t,\ensuremath{\mathbf{d}}) + \sum_{i:d_i \ge 2} \frac{(p_it)^2}{2}\right| \le
\pran{\sum_{i:d_i \ge 2} (p_it)^2}
\cdot \frac{p_{\max} t}{3(1-p_{\max}t)}\, .
\]
Using that $p_i = d_i/(2(n-1))$ and $p_{\max}=d_{\max}/(2(n-1))$ and $\ensuremath{\mathrm{v}}=\sum_{i:d_i \ge 2} d_i^2/(n-1)$, this is precisely the bound claimed in the lemma; this completes the proof.
\end{proof}
\begin{cor}\label{cor:gtd_bd}
Write $\ensuremath{\mathrm{v}}=\sum_{i:d_i \ge 2} d_i^2/(n-1)$. Then for all $0 \le t \le (n-1)/d_{\max}$,
\[
g(t,\ensuremath{\mathbf{d}}) \le \exp\pran{\frac{-\ensuremath{\mathrm{v}} t^2}{24(n-1)}}.
\]
\end{cor}
\begin{proof}
For $t \le (n-1)/d_{\max}$ we have
\[
\frac{d_{\max} t}{6(n-1)-3d_{\max}t}
\cdot
\frac{\ensuremath{\mathrm{v}} t^2}{4(n-1)}
\le \frac{\ensuremath{\mathrm{v}} t^2}{12(n-1)},
\]
and the bound in the lemma then gives
\[
\log g(t,\ensuremath{\mathbf{d}})
\le
-\frac{\ensuremath{\mathrm{v}} t^2}{8(n-1)}
+ \frac{\ensuremath{\mathrm{v}} t^2}{12(n-1)}
=-\frac{\ensuremath{\mathrm{v}} t^2}{24(n-1)}\, .\qedhere
\]
\end{proof}
\begin{proof}[Proof of Proposition~\ref{prop:main_mod}]
First, if $d_{\max}=1$ then $\ensuremath{\mathrm{v}}=0$ and the lemma asserts a non-negative upper bound on $\p{\tau > \infty}$, so clearly holds. We thus assume that $d_{\max} > 1$ for the rest of the proof.
Fix $C > 2$. If $d_{\max} \le (\sum_{i:d_i \ge 2} d_i^2)^{1/2}/C=\tfrac{((n-1)\ensuremath{\mathrm{v}})^{1/2}}{C}$, then taking $t = \tfrac{C(n-1)^{1/2}}{\ensuremath{\mathrm{v}}^{1/2}} \le \tfrac{n-1}{d_{\max}}$, by
Lemma~\ref{lem:gtd_bd} we have
\begin{align*}
g(t,\ensuremath{\mathbf{d}}) \le \exp\pran{-\frac{\ensuremath{\mathrm{v}} t^2}{24(n-1)}} = \exp\pran{-\frac{C^2}{24}}\, .
\end{align*}
Taking $h=2t=\tfrac{2C(n-1)^{1/2}}{\ensuremath{\mathrm{v}}^{1/2}}$, and noting that
\[
t\pran{\frac{h}{t}\log\frac{h}{t}-\frac{h}{t} + 1}
= t(2\log 2 - 1)> \frac{t}{3}\, ,
\]
by (\ref{eq:poissonization_bd}) we obtain that
\begin{align}\label{eq:dsmall_taubd}
\p{\tau > 2C \frac{(n-1)^{1/2}}{\ensuremath{\mathrm{v}}^{1/2}}}
& \le \exp\pran{-\frac{C}{3}\frac{(n-1)^{1/2}}{\ensuremath{\mathrm{v}}^{1/2}}} + \exp\pran{-\frac{C^2}{24}}\, .
\end{align}
Now suppose that $d_{\max} > (\sum_{i:d_i \ge 2} d_i^2)^{1/2}/C=\tfrac{((n-1)\ensuremath{\mathrm{v}})^{1/2}}{C}$. By construction the entries of $(d_1,\ldots,d_n)$ are non-decreasing, so $d_n=d_{\max}$. For any positive real $K\ge 2$, if at least two of the points $U_1,\ldots,U_{\lfloor K\rfloor}$ lie in the interval $[l_n,r_n)$ then $\tau \le K$, so
\begin{align*}
\p{\tau > K}
&\le \p{|\{k \in [\lfloor K \rfloor]: U_k \in [l_n,r_n)\}|\le 1}\\
& = \p{\mathrm{Bin}(\lfloor K \rfloor,r_n-l_n) \in \{0,1\}}\\
& = (1-(r_n-l_n))^{\lfloor K \rfloor-1}(1-(r_n-l_n)+\lfloor K \rfloor(r_n-l_n))\, .
\end{align*}
Since $d_{\max} \ge 2$ we have $r_n-l_n = (d_{\max}-1)/(n-1) \ge d_{\max}/(2(n-1))$, so using that $1-x \le e^{-x}$ it follows that for $K \ge 2C(\tfrac{n-1}{\ensuremath{\mathrm{v}}})^{1/2}\ge 4$,
\begin{align}
\p{\tau > K}
& \le \pran{1+\frac{(\lfloor K \rfloor-1)d_{\max}}{n-1}}\cdot\exp\pran{-\frac{(\lfloor K \rfloor-1) d_{\max}}{2(n-1)}}\, \nonumber\\
& \le \frac{2Kd_{\max}}{n-1}\exp\pran{-\frac{Kd_{\max}}{4(n-1)}}\, ,\label{eq:tau_bd_d_big}
\end{align}
where for the second inequality we have used the lower bound on $K$ to deduce that
$\lfloor K\rfloor -1 > K/2$ and that
$\tfrac{(\lfloor K \rfloor-1)d_{\max}}{n-1} > \tfrac{K}{2}\tfrac{d_{\max}}{n-1} \ge 2$ and so $1 + \tfrac{(\lfloor K \rfloor-1)d_{\max}}{n-1} < \tfrac{2Kd_{\max}}{n-1}$.
Taking $K=xC(\tfrac{n-1}{\ensuremath{\mathrm{v}}})^{1/2}$ for $x \ge 4$, the lower bound on $d_{\max}$ implies that $Kd_{\max}/(n-1) \ge x$; since $2xe^{-x/4}$ is decreasing for $x \ge 4$, the bound (\ref{eq:tau_bd_d_big}) then implies that
\begin{align}\label{eq:dbig_taubd}
\p{\tau \ge xC\frac{(n-1)^{1/2}}{\ensuremath{\mathrm{v}}^{1/2}}}
\le 2xe^{-x/4}\, .
\end{align}
To finish the proof, we combine (\ref{eq:dsmall_taubd}) and (\ref{eq:dbig_taubd}) to get a bound which does not depend on the value of $d_{\max}$. Take $\beta \ge 17^{3/2}$, let $C=\beta^{1/3}>2$ and $x=\beta^{2/3}\ge 4$. Then $2C \le \beta$ and $xC=\beta$. Whatever the value of $d_{\max}$, one of (\ref{eq:dsmall_taubd}) and (\ref{eq:dbig_taubd}) applies, so we obtain that
\begin{align*}
\p{\tau \ge \beta\frac{(n-1)^{1/2}}{\ensuremath{\mathrm{v}}^{1/2}}}
& = \p{\tau \ge xC\frac{(n-1)^{1/2}}{\ensuremath{\mathrm{v}}^{1/2}}}\\
& \le
\exp\pran{-\frac{C}{3}\frac{(n-1)^{1/2}}{\ensuremath{\mathrm{v}}^{1/2}}} + \exp\pran{-\frac{C^2}{24}}
+ 2x e^{-x/4}\\
& =
\exp\pran{-\frac{\beta^{1/3}}{3}\frac{(n-1)^{1/2}}{\ensuremath{\mathrm{v}}^{1/2}}}
+
\exp\pran{-\frac{\beta^{2/3}}{24}}
+
2\beta^{2/3}\exp\pran{-\frac{\beta^{2/3}}{4}}\, .
\end{align*}
Finally, it is straightforward to check that $e^{-y/24}+2ye^{-y/4} \le 2e^{-y/24}$ for $y \ge 17$, which combined with the previous inequality yields the first bound of the proposition.
\end{proof}
\section{\bf Proofs of the conjectures from \cite{MR2908619} and of Propositions~
\ref{prop:dstats_infinite_variance},~\ref{prop:dstats_zero_radius} and~\ref{prop:dstats_finite_variance}}
\label{sec:simply_generated}
The sort of random trees considered by Janson~\cite{MR2908619} are called {\em simply generated trees}; they are defined as follows. Fix non-negative real weights $\ensuremath{\mathrm{w}}=(w_k,k \ge 0)$ with $w_0>0$. For a finite plane tree $t$, the weight of $t$ is
\[
\ensuremath{\mathrm{w}}(t) = \prod_{v \in \ensuremath{v}(t)} w_{\deg_t(v)}\, .
\]
For positive integers $n$ write
\[
Z_n=Z_n(\ensuremath{\mathrm{w}}) = \sum_{\mathrm{plane~trees}~t: |\ensuremath{v}(t)|=n} \ensuremath{\mathrm{w}}(t) \, ,
\]
and when $Z_n > 0$ define a random tree $\ensuremath{\mathcal{T}}_n=\ensuremath{\mathcal{T}}_n(\ensuremath{\mathrm{w}})$ by
\[
\p{\ensuremath{\mathcal{T}}_n=t} = \frac{\ensuremath{\mathrm{w}}(t)}{Z_n}\,
\]
for plane trees $t$ with $|\ensuremath{v}(t)|=n$. Then $\ensuremath{\mathcal{T}}_n$ is called a {\em simply generated tree of size $n$ with weight sequence $\ensuremath{\mathrm{w}}$}.
If $\sum_{k \ge 0} w_k=1$ then
$\cT_n(\ensuremath{\mathrm{w}})$ is distributed as a Bienaym\'e tree with offspring distribution $\ensuremath{\mathrm{w}}$ conditioned to have $n$ vertices.
Write $\Phi(z) = \Phi_{\ensuremath{\mathrm{w}}}(z) = \sum_{k \ge 0} w_k z^k$ for the generating function of the sequence $\ensuremath{\mathrm{w}}$, and $\rho=\rho_{\ensuremath{\mathrm{w}}}$ for the radius of convergence of $\Phi$. For $t > 0$ such that $\Phi(t) < \infty$, define
\[
\Psi(t) = \Psi_{\ensuremath{\mathrm{w}}}(t)=\frac{t\Phi'(t)}{\Phi(t)} = \frac{\sum_{k\ge 0} kw_kt^k}{\sum_{k \ge 0} w_kt^k}\, .
\]
If $\Phi(\rho)=\infty$ then also define
\[
\Psi(\rho)=\Psi_\ensuremath{\mathrm{w}}(\rho) = \lim_{t \uparrow \rho} \Psi(t)\, ;
\]
the function $\Psi$ is strictly increasing on $[0,\rho)$ by \cite[Lemma~3.1(i)]{MR2908619}, so this limit exists. In all cases, write $\nu=\nu(\ensuremath{\mathrm{w}})=\Psi_\ensuremath{\mathrm{w}}(\rho)$. Note that $\Psi(t) \in (0,\infty]$ for all $t > 0$, so $\nu=0$ if and only if $\rho=0$.
The questions from \cite{MR2908619} that we address in this paper concern exclusively weight sequences with $\nu \le 1$, and we assume this is the case from now on. We define $\sigma^2 = \rho\Psi'(\rho)$; this is a slight simplification of the definition from \cite[Theorem~7.1]{MR2908619}, made possible by the assumption that $\nu \le 1$.
The following conjecture summarizes Conjectures~21.5 and~21.6 and Problems~21.7 and~21.8 from \cite{MR2908619}.
\begin{conj}[\cite{MR2908619}]\label{conj_svante}
Let $\ensuremath{\mathrm{w}}=(w_k,k \ge 0)$ be a weight sequence with $w_0>0$ and with $w_k>0$ for some $k \ge 2$, and for $n \ge 0$ with $Z_n(\ensuremath{\mathrm{w}})>0$ let $\ensuremath{\mathcal{T}}_n$ be a simply generated tree of size $n$ with weight sequence $\ensuremath{\mathrm{w}}$.
\begin{enumerate}
\item If $\nu=1$ and $\sigma^2 = \infty$ then $\ensuremath{\mathrm{ht}}(\ensuremath{\mathcal{T}}_n)/n^{1/2} \to 0$ in probability.
\item If $\nu=1$ and $\sigma^2 = \infty$ then $\ensuremath{\mathrm{wid}}(\ensuremath{\mathcal{T}}_n)/n^{1/2} \to \infty$ in probability.
\item If $\nu < 1$ then $\ensuremath{\mathrm{ht}}(\ensuremath{\mathcal{T}}_n)/n^{1/2} \to 0$ in probability.
\item If $\nu < 1$ then $\ensuremath{\mathrm{wid}}(\ensuremath{\mathcal{T}}_n)/n^{1/2} \to \infty$ in probability.
\end{enumerate}
In all four statements, the convergence is as $n \to \infty$ along integers $n$ such that $Z_n(\ensuremath{\mathrm{w}})>0$.
\end{conj}
The results of this work establish points (2) and (4) of this conjecture, and establish (1) and (3) up to polylogarithmic factors. To make these deductions, we rely on the following result from \cite{MR2908619} about the typical degree statistics of simply generated trees.
\begin{thm}[\cite{MR2908619}]\label{thm:dstats_simply_generated}
Let $\ensuremath{\mathrm{w}}=(w_k,k \ge 0)$ be a weight sequence with $w_0>0$ and with $w_k>0$ for some $k \ge 2$. Whenever $Z_n(\ensuremath{\mathrm{w}})>0$ let $\ensuremath{\mathcal{T}}_n$ be a simply generated tree with weight sequence $\ensuremath{\mathrm{w}}$ and size $n$.
Assume that $\nu=\nu(\ensuremath{\mathrm{w}})\in (0,1]$.
Writing $\rho=\rho_{\ensuremath{\mathrm{w}}}\in (0,\infty]$, for $k \ge 0$ let
\begin{equation}\label{eq:pi_def}
\pi(k) := \frac{w_k \rho^k}{\Phi(\rho)}\, .
\end{equation}
Then $\pi=(\pi(k),k \ge 0)$ is a probability distribution, with expectation $\nu$ and variance $\sigma^2 = \rho\Psi'(\rho)$, and the degree statistics $\ensuremath{\mathrm{n}}_{\ensuremath{\mathcal{T}}_n}$ satisfy that for every integer $k \ge 0$ and real $\eps > 0$,
\[
\p{\left|\frac{\ensuremath{\mathrm{n}}_{\ensuremath{\mathcal{T}}_n}(k)}{n} - \pi(k)\right| > \eps } = \ensuremath{\mathrm{oe}}(1)\, .
\]
\end{thm}
This theorem is essentially a special case of \cite[Theorem 11.4]{MR2908619}. The error bounds stated above are not made explicit in the statement of that theorem, but are recorded in the course of its proof (see \cite[page 164]{MR2908619}).
We also require a version of Theorem~\ref{thm:dstats_simply_generated} which addresses the case that $\nu=\rho=0$.
Before stating this result, note that if $\rho_{\ensuremath{\mathrm{w}}}=0$ then the probability distribution $\pi$ defined by \eqref{eq:pi_def} has
$\pi(0)=1$ and $\pi(k)=0$ for $k > 0$.
\begin{thm}\label{thm:nu_zero}
Let $\ensuremath{\mathrm{w}}=(w_k,k \ge 0)$ be a weight sequence with $w_0>0$ and with $w_k>0$ for some $k \ge 2$. Whenever $Z_n(\ensuremath{\mathrm{w}})>0$ let $\ensuremath{\mathcal{T}}_n$ be a simply generated tree with weight sequence $\ensuremath{\mathrm{w}}$ and size $n$.
Suppose that $\rho_\ensuremath{\mathrm{w}}=0$. Then the degree statistics $\ensuremath{\mathrm{n}}_{\ensuremath{\mathcal{T}}_n}$ satisfy that for every real $\eps > 0$,
\[
\p{\frac{\ensuremath{\mathrm{n}}_{\ensuremath{\mathcal{T}}_n}(0)}{n} < 1- \eps} = \ensuremath{\mathrm{oe}}(1)\, .
\]
\end{thm}
This theorem asserts that when the radius of convergence of $\Phi$ is zero, with very high probability $\ensuremath{\mathcal{T}}_n$ has $n-o(n)$ leaves.
\begin{proof}
Fix $\delta > 0$ and integer $L>2$. We claim that
\begin{equation}\label{eq:toprove}
\frac{\p{\sum_{1 \le c \le L} \ensuremath{\mathrm{n}}_{\ensuremath{\mathcal{T}}_n}(c) \ge 2\delta n}}{\p{\sum_{1 \le c \le L} \ensuremath{\mathrm{n}}_{\ensuremath{\mathcal{T}}_n}(c) \le \delta n}} = \ensuremath{\mathrm{oe}}(1)\, .
\end{equation}
Since $\sum_{c \ge 1} c\ensuremath{\mathrm{n}}_{\ensuremath{\mathcal{T}}_n}(c) =n-1$,
we deterministically have $\sum_{c > L} \ensuremath{\mathrm{n}}_{\ensuremath{\mathcal{T}}_n}(c) \le (n-1)/L$,
so the bound \eqref{eq:toprove} implies that
\[
\p{\sum_{c \ge 1} \ensuremath{\mathrm{n}}_{\ensuremath{\mathcal{T}}_n}(c) \le 2\delta n + \frac{n-1}{L}} \ge
\p{\sum_{1 \le c \le L} \ensuremath{\mathrm{n}}_{\ensuremath{\mathcal{T}}_n}(c) \le 2\delta n} = 1-\ensuremath{\mathrm{oe}}(1).
\]
Since $\delta > 0$ and $L>2$ are arbitrary, this proves the theorem. It thus remains to prove~\eqref{eq:toprove}.
It's convenient in what follows to assume that $w_0=1$. (We can achieve this by multiplying all weights by $w_0^{-1}$; this does not change the distribution of $\ensuremath{\mathcal{T}}_n$.)
Now fix $K$ large enough that $K^\delta > 2(L+1)$.
Note that if $\limsup_{k \to \infty}(\log w_k)/k = r < \infty$, then $\rho_\ensuremath{\mathrm{w}} \ge 1/r$; since we assume $\rho_\ensuremath{\mathrm{w}}=0$, it follows that $\limsup_{k \to \infty}(\log w_k)/k=\infty$,
so we may further choose an integer $M > 2L$ such that
\[
w_M \ge
\max
\pran{(K w_c)^{M/c}, 0 < c \le L}.
\]
In what follows, given a sequence $\ensuremath{\mathrm{n}}$ which is the degree statistics of a tree (so $\sum_{c \ge 0}\ensuremath{\mathrm{n}}(c)=|\ensuremath{\mathrm{n}}|_1+1$), it's useful to write
$\ensuremath{\mathrm{w}}(\ensuremath{\mathrm{n}}) := \prod_{c \ge 0} w_c^{\ensuremath{\mathrm{n}}(c)}$ --- so if $t$ is a tree with degree statistics $\ensuremath{\mathrm{n}}$ then $\ensuremath{\mathrm{w}}(t)=\ensuremath{\mathrm{w}}(\ensuremath{\mathrm{n}})$.
Now, for such a sequence $\ensuremath{\mathrm{n}}$, form degree statistics $\hat{\ensuremath{\mathrm{n}}}$ as follows.
For $c \in \N$ let $m(c) = \lfloor \ensuremath{\mathrm{n}}(c)/M\rfloor$. Then for $i \in \N$ define
\[
\hat{\ensuremath{\mathrm{n}}}(i) =
\begin{cases}
\ensuremath{\mathrm{n}}(0) + \sum_{0 < c \le L} (M-c)m(c) & \mbox{ if } i=0\\
\ensuremath{\mathrm{n}}(i)-Mm(i) &\mbox{ if }0 < i \le L \\
\ensuremath{\mathrm{n}}(i)+\sum_{0 < c \le L} cm(c) &\mbox{ if }i=M\\
\ensuremath{\mathrm{n}}(i) &\mbox{ otherwise.}
\end{cases}
\]
Then $|\hat{\ensuremath{\mathrm{n}}}|_1=|\ensuremath{\mathrm{n}}|_1$ and $\sum_{c \ge 0} c\hat{\ensuremath{\mathrm{n}}}(c) = \sum_{c \ge 0} \ensuremath{\mathrm{n}}(c)$, so $\hat{\ensuremath{\mathrm{n}}}$ is again the degree statistics of a tree with $|\ensuremath{\mathrm{n}}|_1+1$ vertices.
Since $w_0=1$, we also have
\begin{align*}
\frac{\ensuremath{\mathrm{w}}(\hat{\ensuremath{\mathrm{n}}})}{\ensuremath{\mathrm{w}}(\ensuremath{\mathrm{n}})}
& =
(w_M)^{\sum_{0 < c \le L} cm(c)}
\cdot
\prod_{0 < c \le L} (w_c)^{-Mm(c)}
=
\prod_{0 < c \le L}
\pran{\frac{w_M^c}{w_c^M}}^{m(c)}
\end{align*}
Since $w_M \ge (Kw_c)^{M/c}$, this yields that
\[
\frac{\ensuremath{\mathrm{w}}(\hat{\ensuremath{\mathrm{n}}})}{\ensuremath{\mathrm{w}}(\ensuremath{\mathrm{n}})}
\ge \prod_{0 < c\le L}
K^{Mm(c)} = K^{\sum_{0 < c \le L} M\lfloor \ensuremath{\mathrm{n}}(c)/M\rfloor}\, .
\]
Now write $n = \sum_{c \ge 0} \ensuremath{\mathrm{n}}(c)=\sum_{c \ge 0} \hat{\ensuremath{\mathrm{n}}}(c)$.
Note that for all $i\in\N$ we have $\ensuremath{\mathrm{n}}(i)-Mm(i) \le M-1$, so
$\sum_{0 < c \le L} \hat{\ensuremath{\mathrm{n}}}(c) \le (M-1)L$, and thus if $n \ge (M-1)L/\delta$ then $\sum_{0 < c \le L} \hat{\ensuremath{\mathrm{n}}}(c) \le \delta n$.
For such $n$, if $\sum_{0 < c \le L} \ensuremath{\mathrm{n}}(c) \ge 2\delta n$, then we also have
$\sum_{0 < c \le L} M \lfloor \ensuremath{\mathrm{n}}(c)/M\rfloor \ge (\sum_{0 < c \le L} \ensuremath{\mathrm{n}}(c))-(M-1)L \ge \delta n$, so it follows from the previous lower bound on $\ensuremath{\mathrm{w}}(\hat{\ensuremath{\mathrm{n}}})/\ensuremath{\mathrm{w}}(\ensuremath{\mathrm{n}})$ that
\begin{equation}\label{eq:weight_gain_is_good}
\frac{\ensuremath{\mathrm{w}}(\hat{\ensuremath{\mathrm{n}}})}{\ensuremath{\mathrm{w}}(\ensuremath{\mathrm{n}})} \ge K^{\delta n}\, .
\end{equation}
To use this bound to complete the proof, it remains to control (a) the number of degree statistics $\ensuremath{\mathrm{n}}$ that can give rise to a given degree statistics $\hat{\ensuremath{\mathrm{n}}}$, and (b), for a given pair of degree statistics $\ensuremath{\mathrm{n}}$ and $\hat{\ensuremath{\mathrm{n}}}$, the relative numbers of trees with these degree statistics.
To control (a),
fix a sequence $\ensuremath{\mathrm{n}}'$ which is the degree statistics of a tree of size $n$. Then for any degree statistics $\ensuremath{\mathrm{n}}$ with $\hat{\ensuremath{\mathrm{n}}}=\ensuremath{\mathrm{n}}'$, there are non-negative integers $m_1,\ldots,m_L$ with $\sum_{0 < c \le L} cm_c < n$
such that $\ensuremath{\mathrm{n}}'(c)=\ensuremath{\mathrm{n}}(c)-Mm_c$ for $0 < c \le L$. Moreover, $\ensuremath{\mathrm{n}}$ may be reconstructed from $\ensuremath{\mathrm{n}}'$ and the values $m_1,\ldots,m_L$. It follows that
\[
|\{\text{Degree statistics }\ensuremath{\mathrm{n}}: \hat{\ensuremath{\mathrm{n}}}=\ensuremath{\mathrm{n}}'\}| \le
\Big|\Big\{(m_1,\ldots,m_L)\in \N^L: \sum_{0 < c < L} cm_c < n\Big\}\Big| < n^L\, .
\]
To control (b), note that if $\ensuremath{\mathrm{n}}$ is the degree statistics of a tree of size $n$, then by the formula \eqref{eq:dstat_count} for the number of trees with given degree statistics, we have
\begin{align*}
\frac{|\ensuremath{\mathscr{T}}_{\ensuremath{\mathrm{n}}}|}{|\ensuremath{\mathscr{T}}_{\hat{\ensuremath{\mathrm{n}}}}|}
& =
\prod_{c \ge 0} \frac{\hat{\ensuremath{\mathrm{n}}}(c)!}{\ensuremath{\mathrm{n}}(c)!}\\
& =
\frac{(\ensuremath{\mathrm{n}}(0)+\sum_{c=1}^L (M-c)m(c))!}{\ensuremath{\mathrm{n}}(0)!}
\cdot
\frac{(\ensuremath{\mathrm{n}}(M)+\sum_{c=1}^L cm(c))!}{\ensuremath{\mathrm{n}}(M)!}
\cdot
\prod_{c=1}^L \frac{(\ensuremath{\mathrm{n}}(c)-Mm(c))!}{\ensuremath{\mathrm{n}}(c)!}\, .
\end{align*}
Since $M > 2L > 2$, we have $\ensuremath{\mathrm{n}}(0)> \ensuremath{\mathrm{n}}(M)$ and $\sum_{c=1}^L (M-c)m(c) \ge \sum_{c=1}^L cm(c)$. Thus
$(\ensuremath{\mathrm{n}}(0)+\sum_{c=1}^L (M-c)m(c))! > (\ensuremath{\mathrm{n}}(M)+\sum_{c=1}^L cm(c))!$ and so
\[
\frac{(\ensuremath{\mathrm{n}}(0)+\sum_{c=1}^L (M-c)m(c))!}{\ensuremath{\mathrm{n}}(0)!}
\cdot
\frac{(\ensuremath{\mathrm{n}}(M)+\sum_{c=1}^L cm(c))!}{\ensuremath{\mathrm{n}}(M)!}
\le \frac{(\ensuremath{\mathrm{n}}(0)+\sum_{c=1}^L Mm(c))!}{\ensuremath{\mathrm{n}}(0)!}.
\]
It follows that
\[
\frac{|\ensuremath{\mathscr{T}}_{\ensuremath{\mathrm{n}}}|}{|\ensuremath{\mathscr{T}}_{\hat{\ensuremath{\mathrm{n}}}}|}
\le
\frac{(\ensuremath{\mathrm{n}}(0)+\sum_{c=1}^L Mm(c))!}{\ensuremath{\mathrm{n}}(0)!}
\cdot
\prod_{c=1}^L \frac{(\ensuremath{\mathrm{n}}(c)-Mm(c))!}{\ensuremath{\mathrm{n}}(c)!}\, .
\]
Since $\ensuremath{\mathrm{n}}(c)-Mm(c) \le M-1$ and
$Mm(c)\le \ensuremath{\mathrm{n}}(c)$ for all $c \in \N$, this yields the bound
\begin{equation}\label{eq:set_size_relation}
\frac{|\ensuremath{\mathscr{T}}_{\ensuremath{\mathrm{n}}}|}{|\ensuremath{\mathscr{T}}_{\hat{\ensuremath{\mathrm{n}}}}|}
\le
((M-1)!)^L \frac{(\sum_{c=0}^L \ensuremath{\mathrm{n}}(c))!}{\prod_{c=0}^L\ensuremath{\mathrm{n}}(c)!}
\le ((M-1)!)^L (L+1)^n\, ,
\end{equation}
where in the last inequality we have used the fact that the final fraction is a multinomial coefficient and that $\sum_{c=0}^L \ensuremath{\mathrm{n}}(c) \le \sum_{c \ge 0} \ensuremath{\mathrm{n}}(c)=n$.
To conclude, write $N_n$ (resp.\ $\hat{N}_n$) for the set of degree statistics $\ensuremath{\mathrm{n}}$ with $\sum_{c \ge 0}\ensuremath{\mathrm{n}}(c)=n=|\ensuremath{\mathrm{n}}|_1+1$ and such that
$\sum_{1 \le c \le L} \ensuremath{\mathrm{n}}(c) \ge 2\delta n$ (resp.\ such that $\sum_{1 \le c \le L} \ensuremath{\mathrm{n}}(c) \le \delta n$), and note that if $\ensuremath{\mathrm{n}} \in N_n$ then $\hat{\ensuremath{\mathrm{n}}} \in \hat{N}_n$ provided that $n \ge (M-1)L/\delta$.
We have
\begin{align*}
\p{\sum_{1 \le c \le L} \ensuremath{\mathrm{n}}_{\ensuremath{\mathcal{T}}_n}(c) \ge 2\delta n}
& = \sum_{\ensuremath{\mathrm{n}} \in N_n} \p{\ensuremath{\mathrm{n}}_{\ensuremath{\mathcal{T}}_n}=\ensuremath{\mathrm{n}}}\\
& =
\sum_{\ensuremath{\mathrm{n}} \in N_n}
\sum_{t \in \ensuremath{\mathscr{T}}_{\ensuremath{\mathrm{n}}}}
\p{\ensuremath{\mathcal{T}}_n=t}\\
& =
\sum_{\ensuremath{\mathrm{n}} \in N_n}
|\ensuremath{\mathscr{T}}_\ensuremath{\mathrm{n}}| \frac{\ensuremath{\mathrm{w}}(\ensuremath{\mathrm{n}})}{Z_n(\ensuremath{\mathrm{w}})} \\
& \le
\sum_{\ensuremath{\mathrm{n}} \in N_n}
((M-1)!)^L (L+1)^n |\ensuremath{\mathscr{T}}_{\hat{\ensuremath{\mathrm{n}}}}|
\frac{\ensuremath{\mathrm{w}}(\hat{\ensuremath{\mathrm{n}}})}{Z_n(\ensuremath{\mathrm{w}})}\frac{1}{K^{\delta n}}\,,
\end{align*}
where we have used \eqref{eq:weight_gain_is_good} and \eqref{eq:set_size_relation} for the final bound.
For each $\ensuremath{\mathrm{n}}' \in \hat{N}_n$,
there are at most $n^L$ sequences $\ensuremath{\mathrm{n}} \in N_n$ with $\hat{\ensuremath{\mathrm{n}}}=\ensuremath{\mathrm{n}}'$, so for $n \ge (M-1)L/\delta$ the above bound yields
\begin{align*}
\p{\sum_{1 \le c \le L} \ensuremath{\mathrm{n}}_{\ensuremath{\mathcal{T}}_n}(c) \ge 2\delta n}
& \le n^L((M-1)!)^L ((L+1)/K^\delta)^n \sum_{\ensuremath{\mathrm{n}}' \in \hat{N}_n}
|\ensuremath{\mathscr{T}}_{\ensuremath{\mathrm{n}}'}|
\frac{\ensuremath{\mathrm{w}}(\ensuremath{\mathrm{n}}')}{Z_n(\ensuremath{\mathrm{w}})} \\
& = (n(M-1)!)^L ((L+1)/K^\delta)^n \
\p{\sum_{1 \le c \le L} \ensuremath{\mathrm{n}}_{\ensuremath{\mathcal{T}}_n}(c) \le \delta n}\, .
\end{align*}
Since $K^\delta > 2(L+1)$, the term $(n(M-1)!)^L ((L+1)/K^\delta)^n$ tends to zero as $n \to \infty$, which establishes \eqref{eq:toprove} and completes the proof.
\end{proof}
The next corollary is the key takeaway from Theorems~\ref{thm:dstats_simply_generated} and~\ref{thm:nu_zero}, for the purposes of this work.
\begin{cor}\label{cor:dstats_simply_generated}
In the setting of Theorems~\ref{thm:dstats_simply_generated} and~\ref{thm:nu_zero}, if $\nu < 1$, or if $\nu=1$ and $\sigma^2=\infty$, then for any $C > 0$, with very high probability
\[
|\ensuremath{\mathrm{n}}_{\ensuremath{\mathcal{T}}_n}|_2^2 \ge C|\ensuremath{\mathrm{n}}_{\ensuremath{\mathcal{T}}_n}|_1.
\]
\end{cor}
In the same way that Theorems~\ref{thm:main2} and~\ref{thm:main3} follow from Propositions~\ref{prop:dstats_infinite_variance} and~\ref{prop:dstats_zero_radius}, Corollary~\ref{cor:dstats_simply_generated} implies that if $\nu < 1$, or if $\nu=1$ and $\sigma^2=\infty$, then
$\ensuremath{\mathrm{ht}}(\ensuremath{\mathcal{T}}_n)/(n^{1/2}\log^3 n) \to 0$ and $\ensuremath{\mathrm{wid}}(\ensuremath{\mathcal{T}}_n)/n^{1/2} \to \infty$ in probability. This proves the second and fourth points of the above conjecture and nearly proves the first and third points, up to the polylogarithmic factors.
\begin{proof}[Proof of Corollary~\ref{cor:dstats_simply_generated}]
We begin by noting that if the support of $\ensuremath{\mathrm{w}}$ is finite then $\rho=\infty$ and thus $\nu = \Psi(\infty) = \max(k:w_k>0) > 1$.
We now argue in three cases. First, suppose that $0<\nu<1$. In this case the support of $\ensuremath{\mathrm{w}}$ is infinite, so for any fixed $K\in \N$, $\sum_{k\leq K}k\pi(k) < \sum_{k\geq 0} k\pi(k) = \nu$.
It follows by Theorem~\ref{thm:dstats_simply_generated} that with very high probability $\sum_{k\leq K} k\ensuremath{\mathrm{n}}_{\ensuremath{\mathcal{T}}_n}(k) \leq \nu n-1$. Since
\[
\sum_{k> K} k\ensuremath{\mathrm{n}}_{\ensuremath{\mathcal{T}}_n}(k)
=
|\ensuremath{\mathrm{n}}_{\ensuremath{\mathcal{T}}_n}|_1 -
\sum_{k\leq K} k\ensuremath{\mathrm{n}}_{\ensuremath{\mathcal{T}}_n}(k)
= n-1-
\sum_{k\leq K} k\ensuremath{\mathrm{n}}_{\ensuremath{\mathcal{T}}_n}(k)\, ,
\]
this implies that with very high probability
\begin{equation*}
\sum_{k> K} k\ensuremath{\mathrm{n}}_{\ensuremath{\mathcal{T}}_n}(k)
\geq n(1-\nu).
\end{equation*}
Now let $C>0$ arbitrary and fix $K\in \N$ such that $K(1-\nu) >C$. Then
since
$|\ensuremath{\mathrm{n}}_{\ensuremath{\mathcal{T}}_n}|_2^2 \geq \sum_{k> K} k^2\ensuremath{\mathrm{n}}_{\ensuremath{\mathcal{T}}_n}(k) > K \sum_{k> K} k\ensuremath{\mathrm{n}}_{\ensuremath{\mathcal{T}}_n}(k)$, we have
\begin{align*}
\p{|\ensuremath{\mathrm{n}}_{\ensuremath{\mathcal{T}}_n}|_2^2 < C|\ensuremath{\mathrm{n}}_{\ensuremath{\mathcal{T}}_n}|_1} &\leq \p{ K \sum_{k> K} k\ensuremath{\mathrm{n}}_{\ensuremath{\mathcal{T}}_n}(k) < Cn} \\
& = \p{ K \sum_{k> K} k\ensuremath{\mathrm{n}}_{\ensuremath{\mathcal{T}}_n}(k) < Cn,\, \sum_{k> K} k\ensuremath{\mathrm{n}}_{\ensuremath{\mathcal{T}}_n}(k) < n(1-\nu)} = \ensuremath{\mathrm{oe}}(1)\, .
\end{align*}
This establishes the corollary in the case that $\nu \in (0,1)$.
Next, suppose $\nu = 1$ and $\sigma^2 = \infty$. Let $C>0$ arbitrary. Then since we have $\lim_{K \to \infty} \sum_{k=0}^K k^2 \pi(k) = \infty$, there exists $K \in \N$ such that $\sum_{k=0}^K k^2 \pi(k) \geq 2C$. Noting that $|\ensuremath{\mathrm{n}}_{\ensuremath{\mathcal{T}}_n}|_1 = n-1$, we can write
\begin{align*}
\p{|\ensuremath{\mathrm{n}}_{\ensuremath{\mathcal{T}}_n}|_2^2 < C|\ensuremath{\mathrm{n}}_{\ensuremath{\mathcal{T}}_n}|_1} &\leq \p{|\ensuremath{\mathrm{n}}_{\ensuremath{\mathcal{T}}_n}|_2^2 < Cn} \\
&\leq \p{ \sum_{k=0}^K k^2 \frac{\ensuremath{\mathrm{n}}_{\ensuremath{\mathcal{T}}_n} (k)}{n} < C} \, .
\end{align*}
Then, taking $\eps < 6C/{(K(K+1)(2K+1)})$ and applying Theorem~\ref{thm:dstats_simply_generated}, we have
\begin{equation*}
\p{|\ensuremath{\mathrm{n}}_{\ensuremath{\mathcal{T}}_n}|_2^2 < C|\ensuremath{\mathrm{n}}_{\ensuremath{\mathcal{T}}_n}|_1} \leq
\p{\sum_{k=0}^K k^2 \frac{\ensuremath{\mathrm{n}}_{\ensuremath{\mathcal{T}}_n} (k)}{n} < C , \bigcap_{k=0}^K \left\{\left|\frac{\ensuremath{\mathrm{n}}_{\ensuremath{\mathcal{T}}_n}(k)}{n} - \pi(k)\right| \leq \eps\right\} } + \ensuremath{\mathrm{oe}}(1) \, .
\end{equation*}
However, if $|\tfrac{\ensuremath{\mathrm{n}}_{\ensuremath{\mathcal{T}}_n}(k)}{n} - \pi(k)| \leq \eps$ for all $0 \le k \le K$ then
\begin{equation*}
\sum_{k=0}^K k^2 \frac{\ensuremath{\mathrm{n}}_{\ensuremath{\mathcal{T}}_n}(k)}{n} \geq \sum_{k=0}^K k^2 (\pi(k) - \eps) \geq 2C - \eps \sum_{k=0}^K k^2 > C \, ,
\end{equation*}
so the intersection of events in the probability on the right-hand side is empty and thus this probability is zero. Therefore $\p{|\ensuremath{\mathrm{n}}_{\ensuremath{\mathcal{T}}_n}|_2^2 < C|\ensuremath{\mathrm{n}}_{\ensuremath{\mathcal{T}}_n}|_1}=\ensuremath{\mathrm{oe}}(1)$, as required.
Finally, suppose that $\nu=0$, and fix $C \in \N$. By Theorem~\ref{thm:nu_zero}, we have
\[
\p{\sum_{k=1}^C \ensuremath{\mathrm{n}}_{\ensuremath{\mathcal{T}}_{n}}(k) > \frac{n}{2C^2}}
\le
\p{\frac{\ensuremath{\mathrm{n}}_{\ensuremath{\mathcal{T}}_{n}}(0)}{n} < 1-\frac{1}{2C^2}} = \ensuremath{\mathrm{oe}}(1).
\]
However, if $\sum_{k=1}^C \ensuremath{\mathrm{n}}_{\ensuremath{\mathcal{T}}_{n}}(k) < n/(2C^{2})$ then $\sum_{k=1}^C k\ensuremath{\mathrm{n}}_{\ensuremath{\mathcal{T}}_{n}}(k) < n/(2C)$, so
\[
\sum_{k > C}k\ensuremath{\mathrm{n}}_{\ensuremath{\mathcal{T}}_{n}}(k)
=
n-1 -\sum_{k \le C} k\ensuremath{\mathrm{n}}_{\ensuremath{\mathcal{T}}_{n}}(k)
\ge
n-1-\frac{n}{2C}\, ,
\]
and thus
\[
\sum_{k \ge 1}k^2\ensuremath{\mathrm{n}}_{\ensuremath{\mathcal{T}}_{n}}(k)
\ge
(C+1)\sum_{k > C}k\ensuremath{\mathrm{n}}_{\ensuremath{\mathcal{T}}_{n}}(k)
\ge (C+1)(n-1)-\frac{n(C+1)}{2C} > C(n-1)=C|\ensuremath{\mathrm{n}}_{\ensuremath{\mathcal{T}}_n}|_1\, ,
\]
the last inequality holding for $n$ large provided $C >1$. This shows that $|\ensuremath{\mathrm{n}}_{\ensuremath{\mathcal{T}}_n}|_2^2>C|\ensuremath{\mathrm{n}}_{\ensuremath{\mathcal{T}}_n}|_1$ with very high probability, and completes the proof.
\end{proof}
\begin{proof}[Proofs of Propositions~
\ref{prop:dstats_infinite_variance} and~\ref{prop:dstats_zero_radius}]
Define a weight sequence $\ensuremath{\mathrm{w}}$ by $\ensuremath{\mathrm{w}}_k = \mu(k)$. Then $\ensuremath{\mathcal{T}}_n=\ensuremath{\mathcal{T}}_n(\ensuremath{\mathrm{w}})$ is distributed as a Bienaym\'e tree with offspring distribution $\mu$, conditioned to have size $n$.
Now suppose that $\mu$ satisfies the assumptions of either Proposition~\ref{prop:dstats_infinite_variance} or Proposition~\ref{prop:dstats_zero_radius}.
Then either $|\mu|_2^2=\infty$ or $\sum_{k \ge 0} e^{tk}\mu(k) =\infty$ for all $t > 0$. In either case, $\ensuremath{\mathrm{w}}$ has radius of convergence $\rho$ equal to $1$. Thus $\nu = \Psi(\rho) = \sum_{k \ge 0} kw_k = |\mu|_1$, and
either $\nu < 1$ or else $\nu=1$ and $\sigma^2 = \rho\Psi'(\rho)=\Psi'(1) = \sum_{k \ge 0} k^2 w_k-(\sum_{k \ge 0} kw_k)^2 = |\mu|_2^2-|\mu|_1=\infty$, whence the (common) conclusion of the propositions follows from Corollary~\ref{cor:dstats_simply_generated}.
\end{proof}
\begin{proof}[Proof of Proposition~\ref{prop:dstats_finite_variance}]
Again define a weight sequence $\ensuremath{\mathrm{w}}$ by $\ensuremath{\mathrm{w}}_k = \mu(k)$. Then $\ensuremath{\mathcal{T}}_n=\ensuremath{\mathcal{T}}_n(\ensuremath{\mathrm{w}})$ is distributed as a Bienaym\'e tree with offspring distribution $\mu$, conditioned to have size $n$.
Fix $\eps > 0$ and let $E_n(k)$ be the event that $\ensuremath{\mathrm{n}}_{\ensuremath{\mathcal{T}}_n}(k) \ge n(\mu(k)-\eps/2^k)$.
By Theorem~\ref{thm:dstats_simply_generated}, $E_n(k)$ happens with very high probability. Now fix $K\ge 2$ large enough that $\mu(K,\infty)< \eps/2$. If $E_n(k)$ occurs for each $2 \le k \le K$ then
\[
|\ensuremath{\mathrm{n}}_{\ensuremath{\mathcal{T}}_n}|_2^2 \ge \ensuremath{\mathrm{n}}_{\ensuremath{\mathcal{T}}_n}(1) + \sum_{k=2}^K k^2 n (\mu(k)-\eps/2^k) \ge \ensuremath{\mathrm{n}}_{\ensuremath{\mathcal{T}}_n}(1) + 4 n \Big( \sum_{k=2}^K \mu(k) - \sum_{k=2}^K \eps / 2^k \Big) \, ,
\]
in which case, since $|\ensuremath{\mathrm{n}}_{\ensuremath{\mathcal{T}}_n}|_1 = n-1$, we have
\[
|\ensuremath{\mathrm{n}}_{\ensuremath{\mathcal{T}}_n}|_2^2 - \ensuremath{\mathrm{n}}_{\ensuremath{\mathcal{T}}_n}(1) \ge 4 n \Big(1 - \mu(0) - \mu(1) - \mu(K, \infty) - \sum_{k=2}^K \frac{\eps}{2^k} \Big) > |\ensuremath{\mathrm{n}}_{\ensuremath{\mathcal{T}}_n}|_1 \cdot 4 (1 - \mu(0) - \mu(1) - \eps)\,.
\]
Since $\bigcap_{k=2}^K E_n(k)$ occurs with very high probability, the result follows.
\end{proof}
\addtocontents{toc}{\SkipTocEntry}
\section*{\bf Acknowledgements}
During this work LAB was partially supported by NSERC Discovery Grant 643473. The authors also thank Serte Donderwinkel for useful discussions, and for pointing out a gap in one of the proofs in an early version of the paper (as well as how to fix it); and two referees, whose careful reading substantially improved the paper.
\small
\bibliographystyle{plainnat}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 3,879 |
Russian Navy
Mindstorm
mnztr
hoom
verkhoturye51
Singular_Transform
GunshipDemocracy
sepheronx
Backinblack
artjomh
Morpheus Eberhardt
Stealthflanker
Poseidon carrier Submarines
Location : Fort Evil, Serbia
Re: Poseidon carrier Submarines
PapaDragon Mon Jan 18, 2021 2:02 am
kvs wrote: Worrying about the optics of nuclear contamination during WWIII is to put it mildly, detached from reality. Nobody will giving a
flying fcuk about the environment when faced with nuclear annihilation.
Not a single person here said anything about nuclear contamination
Poseidon is expensive complicated extravaganza when dealing with surface ships
Missiles are more than sufficient, if you want guaranteed results just add nuclear warhead
Isos Mon Jan 18, 2021 2:07 am
PapaDragon wrote:
64 nuclear armed Tsirkon... at least the world would end with a beautifull firework . I doubt Mars life had such plaisure when they ended.
kvs Mon Jan 18, 2021 4:57 am
So Russia now must use conventional torpedoes to take out carrier groups? WTF retarded drivel is this.
Location : Melbourne, Australia
Big_Gazza Mon Jan 18, 2021 7:08 am
Isos wrote: Actually wire guided torpedoes are powered by the submarine until the wire breaks then they run for the final attack on batteries.
They are controlled via the signals in the wire, but not powered. The electrical power required to drive a torpedo at high speeds thru water is considerable, and there is no way to conduct sufficient current through the long thin wire that are spooled out behind a torpedo in flight.
Location : Brasilia
Kiko Sat Feb 13, 2021 6:00 pm
Measure of power: why does Russia need the world's longest submarine
The nuclear submarine Belgorod is one of the most ambitious shipbuilding projects in the country
Dmitry Boltenkov, Alexey Ramm
Special-purpose submarine "Belgorod" is preparing to go to sea for the first time. Sources in the military department told Izvestia that the world's longest submarine has a high degree of factory readiness. Preliminary tests of units and mechanisms have shown their serviceability and technical reliability. The nuclear-powered submarine is over 180 meters long. This is 11 meters longer than that of the largest nuclear submarine Typhoon, and 12 meters longer than that of the American Ohio-class submarine. The domestic submarine is one of the most ambitious projects in Russian shipbuilding. But how much does the Navy need it today? After all, the Navy, albeit with a delay, began to receive the Yasen multipurpose nuclear submarines and the Borey strategic missile carriers. The military historian Dmitry Boltenkov dealt with the issue especially for Izvestia.
The history of the giant
The nuclear-powered ship "Belgorod" was created to combat enemy surface groupings. It was supposed to carry 20 Granit cruise missiles on board. It was planned that the submarine will become the 14th in the family of boats of Project 949 and 949A.
Although the nuclear submarine was officially laid down after the collapse of the Soviet Union, on July 24, 1992, in fact, its construction began earlier. That is why Belgorod can be considered the last submarine of the departed superpower. In 1997, the construction of the submarine and two more of its "sisterships" was stopped due to the difficult economic situation. Subsequently, the metal structures of the two ships laid down after the "Belgorod" went to the creation of submarine missile cruisers of the "Borey" type.
After the tragedy with the nuclear submarine "Kursk" in August 2000, the question of completing the "Belgorod" to replace the deceased submarine was considered for some time. However, in the end, they decided to build a nuclear-powered ship for other purposes. The process of developing new types of weapons is slow. The systems that Vladimir Putin told the Federal Assembly about their appearance in 2018 in his message to the Federal Assembly have been created for a long time. And one can understand why in the early 2000s it was decided to use the Belgorod as a carrier of a variety of promising underwater systems, including combat ones.
On December 20, 2012, the construction of the nuclear submarine resumed, but already according to the new project 09852. On April 23, 2019, the submarine was taken out of the boathouse for completion afloat and conducting various tests. The transfer of the submarine to the Russian Navy is scheduled for 2021.
During the work, the boat underwent major changes. In particular, its hull was increased by 30 m due to the insertion of new compartments. Thus, "Belgorod" became the longest submarine in the world. New compartments were needed to accommodate a variety of underwater vehicles, as well as to release them into the marine environment and receive them back.
Tsunami underwater
The nuclear submarine Belgorod will become the first regular carrier of a new strategic deterrent weapon - the Poseidon super torpedo. In fact, it is a Doomsday weapon. With its help, a retaliatory nuclear strike can be delivered on the territory of a potential adversary if he attacks first.
The highest population density and the largest cities and industrial centers are located near the coastline. A nuclear explosion of enormous power near the enemy's territory, in addition to the damaging factors themselves, will also cause a tsunami.
It should be noted that the Poseidon system is already being tested on the Sarov submarine, but it can also be used autonomously. The construction of the Belgorod submarine is proceeding at a faster pace, especially against the background of delays in the schedules of other nuclear submarines, in particular, the attack submarine Kazan.
And other robots
In addition to Poseidon, Russia has created or is creating many more underwater robotic systems, in particular the autonomous unmanned vehicle Harpsichord-2R-PM. It is capable of conducting research and prospecting work at a depth of up to 6 thousand meters. For example, it can find sensors of underwater observation systems located at the bottom of the sea. Curiously, in the publicly available images of Belgorod there is always a hangar for this product.
For testing anti-submarine operations and, most importantly, for camouflaging and protecting submarines in Russia, the "Surrogate" apparatus was created. It is a submarine simulator with reproduction of its acoustic and magnetic fields. Such a technique will divert the enemy's attention to false targets during the conduct of hostilities. It can be assumed that the Belgorod will also carry it on board.
"Harmony" and "Losharik"
The development of the Harmony underwater observation system is underway in Russia. This is a complex of quickly deployed target detection sensors on the seabed. In peacetime, the system will be able to monitor areas of mining, primarily oil and gas. In particular, "Harmony" will be able to find in time foreign objects that can damage the production facilities.
In wartime, "Harmony" will be deployed to create the so-called defensive bastions. These are sea areas completely isolated for the enemy, where strategic submarine cruisers and boats with Poseidon drones on board will be on duty.
In addition, Belgorod is the carrier of the so-called nuclear deep-water stations, such as AS-31 Losharik. These miniature boats are designed for a wide range of underwater technical operations at great depths (according to some data, up to 6 thousand meters). In particular, for searching for lost military property, detecting or installing various underwater sensors, connecting or destroying communication cables.
The above systems are far from the only ones that are created in the interests of the Russian Navy. They are simply the most famous. On the development and adoption of attack submarines equipped with missiles and torpedoes, information has not yet appeared. However, this does not mean that they are not being developed.
The cruiser "Belgorod" has a wide range of underwater vehicles and robots as a carrier. The use of a variety of systems in the course of combat operations will ensure control over vast sea areas and will minimize the loss of warships and submarines. It will also give the possibility of guaranteed destruction of both the enemy's combat assets and his territory. Therefore, we can safely say that the commissioning of "Belgorod" as a whole brings the concept of "naval war" to a new level.
https://yandex.ru/turbo/iz.ru/s/1124101/dmitrii-boltenkov-aleksei-ramm/mera-sil-zachem-rossii-nuzhna-samaia-dlinnaia-v-mire-submarina?promo=navbar&utm_referrer=https%3A%2F%2Fzen.yandex.com
GarryB, dino00, d_taddei2, Big_Gazza, slasher, DerWolf, zardof and Hole like this post
PapaDragon Fri Feb 26, 2021 1:12 am
Two tubes for Poseidon
George1, dino00, Big_Gazza, LMFS and The_Observer like this post
Isos Fri Feb 26, 2021 1:55 am
They put weapons on a "research" vessel. That's weired.
We saw that such vessels conducts dangerous missions and adding two micro nuk power plants on board isn't making it safe.
They should have used a borei for that IMO.
Isos wrote: ...They should have used a borei for that IMO.
They are, it's called Khabarovsk-class
GarryB Fri Feb 26, 2021 4:44 am
Hahahaha... research sub.... almost as much a joke as deep submergence rescue vessel...
Would be like having a sun rescue vessel that can fly to the outer layers of the sun to rescue astronauts that might get caught there...
The thing is that no one who goes below about 400m can be rescued... and the depths the deep submergence rescue vessels operate at no one would be alive to rescue...
Research Sub and DSRV are cover terms for special ops cable cutters and cable buggers... (ie people who put bugs on cables not people who bugger cables...)
Location : England
JohninMK Mon Mar 01, 2021 6:54 pm
Interesting size comparison between the two torpedoes
H I Sutton
@CovertShores
Major new refresh of the #Russian Navy Belgorod #Submarine article http://hisutton.com/Belgorod-Class-Submarine.html
Not quite in time for #SubSunday.
The internal arrangement to accommodate the ginormous Poseidon weapons is still speculative but we are getting there.
Big_Gazza and PapaDragon like this post
PapaDragon Mon Mar 01, 2021 8:52 pm
JohninMK wrote: Interesting size comparison between the two torpedoes
I took the liberty of adding full size image there, John
Big_Gazza likes this post
owais.usmani Tue Mar 02, 2021 10:29 am
https://tass.ru/armiya-i-opk/10810809
MOSCOW, March 2. / TASS /. Tests of the special-purpose nuclear submarine (SPN) "Belgorod" of project 09852 will begin in May. This was reported to TASS by a source in the military-industrial complex.
"The state tests of Belgorod will begin in May after the White Sea breaks open from ice. They are in no way tied to the tests of the deep-sea nuclear Poseidon," the agency's interlocutor said.
TASS does not have a comment from the Sevmash press service (the enterprise where Belgorod is being built is part of the United Shipbuilding Corporation) on this score.
The submarine SPN "Belgorod" was launched on April 23, 2019. Initially, it was planned to transfer it to the fleet in 2020. According to reports, this did not happen due to the non-completion of the test program, including those related to the coronavirus pandemic.
George1, dino00 and Big_Gazza like this post
George1 Mon Mar 22, 2021 5:57 pm
The Ministry of Defense is considering the issue of placing the first carrier of nuclear "Poseidons"
Today, 12: 19
The Ministry of Defense is working on the issue of placing the first regular carrier of Poseidon submarine drones, a special-purpose submarine of Project 09851 Khabarovsk. Reportedly "News" with reference to the military department, Kamchatka is called one of the possible deployment options.
The Russian Navy is considering Vilyuchinsk in Kamchatka as one of the main options for the placement of the Poseidon carrier of the Khabarovsk nuclear submarine, which will become part of the Pacific Ocean fleet... Several factors speak in favor of Kamchatka: by the end of the year, all the necessary infrastructure for basing new generation submarines will be put into operation in Vilyuchinsk, as well as the sea depths in this region allow submarines to conduct covert deployment.
To date, as the newspaper emphasizes, the final decision on the placement of "Khabarovsk" has not been made. The submarine should be launched this year, the exact date is still unknown.
Project 09851 Khabarovsk nuclear submarine will be a regular carrier of unmanned underwater vehicles and, according to unconfirmed reports, will be able to carry at least six Poseidons on board. The Khabarovsk project was developed at the Rubin Central Design Bureau (CDB), the submarine itself was laid down at Sevmash in July 2014. The technical details of the project were not disclosed.
The Poseidon strategic submarine is designed to engage a variety of targets, including carrier groups and coastal fortifications. The apparatus is equipped with a nuclear power plant. It is capable of diving to a depth of over 1 km and has an unlimited range. It can be armed with a 2 megaton nuclear warhead.
It should be noted that earlier it was reported that the Ministry of Defense plans to adopt up to 32 Poseidon underwater uninhabited vehicles, with the prospect of building four underwater carriers under them. According to the plans of the military department, two submarines with drones should be deployed in the Northern and Pacific fleets.
https://en.topwar.ru/181136-minoborony-prorabatyvaet-vopros-razmeschenija-pervogo-nositelja-jadernyh-posejdonov.html
dino00 and Big_Gazza like this post
George1 Tue Apr 06, 2021 3:08 pm
Belgorod nuclear submarine carrier with Poseidon nuke drones to serve in Pacific — source
The sub will be able to perform missions in any location of the World Ocean, according to the top brass
MOSCOW, April 6. /TASS/. The Belgorod special-purpose nuclear submarine will serve in the Pacific Ocean after passing state trials and commissioning, a source close to Defense Ministry told TASS.
"According to preliminary information, the Belgorod submarine will serve in the Pacific Ocean after commissioning. However, it would be able to perform missions in any location of the World Ocean," he said.
TASS does not have any commentary by Sevmash, the Belgorod manufacturer, on this point.
Earlier, a TASS source revealed that the Belgorod would enter state trials in May 2021. Another source disclosed that, in addition to the Poseidon nuclear drones, the submarine will carry the AS-15 deep-sea nuclear station.
The Project 09852 Poseidon submarine - the first ever carrier of Poseidon nuclear underwater drones - was launched on April 23, 2019. The submarine was initially scheduled for commissioning in 2020. According to TASS information, this did not pan out because of the coronavirus pandemic.
Under the current armament program, three special-purpose submarines will be built before 2027.
https://tass.com/defense/1274461
GarryB, dino00, Big_Gazza and PapaDragon like this post
owais.usmani Fri Jun 25, 2021 7:55 pm
https://www.navalnews.com/naval-news/2021/06/russias-gigantic-submarine-belgorod-sails-for-the-first-time/
The Russian Navy's latest 'Special Mission' submarine, K-329 Belgorod, has put to sea for the first time. Open source intelligence seen by Naval News indicates that the submarine left Severodvinsk on June 25 2021. This represents the start of sea trials which are critical to the delivery of the new boat to the Russian Navy.
https://www.instagram.com/p/CQjFyxiDHiW
Above link contains a picture and a video of Belgorod making her transit out to White Sea.
PapaDragon Fri Jun 25, 2021 8:49 pm
owais.usmani wrote: ...https://www.instagram.com/p/CQjFyxiDHiW
And here be some pictures:
Also official scale model from IMDS-2021:
GarryB, Arrow, Big_Gazza, LMFS, Hole and lancelot like this post
Arrow Sat Jun 26, 2021 2:01 pm
It looks beautiful. I wonder if Belgotod will serve in the Russian navy or in GUGI. In addition to carrying the Poseidons, it has many other interesting missions.
ALAMO Sat Jun 26, 2021 4:09 pm
It serves in the rank of GUGI. There is even a batch for it.
Arrow and LMFS like this post
PapaDragon Sun Jul 25, 2021 12:58 am
Fresh photos of Belgorod, two very large Poseidon tube visible, could be more below:
Big_Gazza, zardof, LMFS and Mir like this post
Big_Gazza Sun Jul 25, 2021 3:33 am
Its remarkable how similar the Belgorod is to the 955A like Kynaz Vladimir. Belgorod has no outward similarily to the 949A, and if we didn't have prior knowledge I suspect we would never guess her lineage (except for her rudder and towed sonar array arrangement). It looks like they have repurposed her pressure hulls, reactor, propulsion and major equipment, and repacked them in a completely new outer casing, forwards weapon compartment, sail and mothership interface (and no doubt some other goodies). No wonder the rework took so long.
Arrow Sun Jul 25, 2021 10:30 am
Project 09851 is also based probably on the 955 project. I wonder what the situation with project 09853 looks like.Probably one under construction.
Belgorod actually looks quite different than the 949A. This is a completely new submarine. It probably has quite new equipment, a new reactor and power plant from the 955A submarine. So it is a completely new 4th generation submarine new class.
PapaDragon Mon Jul 26, 2021 12:36 pm
GarryB, George1, Big_Gazza and LMFS like this post
PapaDragon Wed Aug 04, 2021 11:55 pm
Belgorod went to another round of trials
GarryB, George1, Arrow, dino00, Big_Gazza, Hole and Mir like this post
JohninMK Wed Sep 01, 2021 1:29 pm
Satellite images from Maxar taken earlier this month and provided to USNI News confirm that a special purpose ship, Akademik Aleksandrov, is using the facility. And the vessel appears to have a Poseidon round, or related surrogate load, aboard.
The facility is on the Northern shore of the Northern Dvina River on the edge White Sea. Work on the new quay started in 2018 and was substantially completed in 2020. Akademik Aleksandrov has been observed there in July and August. There is also a large building that was recently built adjacent to the new quay and may also be related to Poseidon operations. This specific quay appears directly connected to Poseidon testing activities, according to the imagery.
https://news.usni.org/2021/08/31/new-satellite-images-hint-how-russian-navy-could-use-massive-nuclear-torpedos
Arrow, Big_Gazza and zardof like this post
PapaDragon Thu Sep 30, 2021 12:02 am
Fresh photo of Belgorod, big torpedo tube visible
George1, dino00, Big_Gazza, zepia and zardof like this post | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 88 |
Q: Gifting teenagers access to "the real deal" in technology competitions I run a CTF for teenagers as part of a local LAN party. Our goal is to "nerdsnipe" teenagers in our local area into developing their technical skills, as well as expose the interest to more kids.
As a part of this, I want to give them something Technology-related as a prize. My hope was to be able to give them credits on Google Cloud or similar, so they can afford to run their own server for a while and play around on it(without the pitfalls of self-hosting at home). The main narrative behind the technical crew of the LAN is "you join us to play with gear your parents won't buy for you(servers, cisco switches, expensive routers, etc)", and I want the prize to follow this.
It appears none of the services I am used to - Google Cloud, AWS, Linode, DigitalOcean, offer something like this. I recognize that you can get a lot of free credits for many of these services through educational packages as well.
I am currently at a loss for options, so I am asking here for tips. Are there any ideas I've overlooked? What kind of "real life" technologies, approachable for teenagers, are possible to gift?
Thanks
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 2,478 |
<?xml version="1.0" encoding="utf-8"?>
<RelativeLayout xmlns:android="http://schemas.android.com/apk/res/android"
xmlns:android1="http://schemas.android.com/apk/res/android"
android1:id="@+id/marginlayout"
android1:layout_width="match_parent"
android1:layout_height="wrap_content" >
<RelativeLayout
android1:id="@+id/nav_item_layout"
android1:layout_width="match_parent"
android1:layout_height="wrap_content"
android1:layout_marginLeft="15dp"
android1:background="@drawable/navigation_section2" >
<ImageView
android1:id="@+id/nav_item_icon"
android1:layout_width="20dp"
android1:layout_height="20dp"
android1:layout_alignParentLeft="true"
android1:layout_alignParentTop="true"
android1:layout_centerInParent="true"
android1:layout_centerVertical="true"
android1:contentDescription="@string/dummy"
android1:scaleType="fitCenter"
android1:src="@drawable/simulator" />
<TextView
android1:id="@+id/nav_item_text"
android1:layout_width="wrap_content"
android1:layout_height="match_parent"
android1:layout_alignBottom="@+id/nav_item_icon"
android1:layout_alignParentRight="true"
android1:layout_alignParentTop="true"
android1:layout_centerInParent="true"
android1:layout_centerVertical="true"
android1:layout_toRightOf="@+id/nav_item_icon"
android1:paddingLeft="5dp"
android1:text="Test"
android1:textAppearance="?android:attr/textAppearanceSmall" />
</RelativeLayout>
</RelativeLayout>
| {
"redpajama_set_name": "RedPajamaGithub"
} | 6,410 |
\section{Introduction and main result}
Let $\Omega \subset {\mathbb R}^2$ be a bounded, simply connected domain with regular boundary.
We keep this assumption in the entire paper.\\
Let ${\bf F}(x) = (F_1,F_2) = (-x_2/2, x_1/2)$---a standard choice for a vector potential generating a unit magnetic field: ${\operatorname{curl}\,} {\bf F} = 1$.
We consider ${\mathcal H}(B)$, the self-adjoint operator associated with the closed, symmetric quadratic form,
\begin{align*}
W^{1,2}(\Omega) \ni u \mapsto Q_B(u)=\int_{\Omega} |(-i\nabla + B {\bf F})u|^2\,dx.
\end{align*}
We will use the notation $p_{{\bf A}} = (-i\nabla + {\bf A})$. Then,
more explicitly, ${\mathcal H}(B)$ is the differential operator $p_{B {\bf F}}^2$ with domain $\{ u \in W^{2,2}(\Omega) \, \Big|\, \nu \cdot p_{B {\bf F}}u |_{\partial \Omega} = 0 \}$, where $\nu$ is the unit interior normal to $\partial \Omega$.
We choose and fix a smooth parametrization $\gamma: \frac{|\partial \Omega|}{2 \pi} {\mathbb S}^1\mapsto \partial \Omega$ of the boundary. We may assume that $|\gamma'(s)|=1$ for all $s$. We will further parametrize $\frac{|\partial \Omega|}{2 \pi}{\mathbb S}^1$ by $[-|\partial \Omega|/2, |\partial \Omega|/2]$ with periodicity being tacitly understood.
For a point $p = \gamma(s) \in \partial \Omega$ we define $k(p)$---also denoted by $k(s)$---to be the curvature of the boundary at the point $\gamma(s)$, i.e.
$$
\gamma''(s) = k(s) \nu(s),
$$
where $\nu(s)$ is the interior normal (to the boundary) vector at the point $\gamma(s)$. The maximum of $k$ will play an important role, we define therefore,
$k_{\rm max} := \max_s \{ k(s) \}$.
Define $\lambda_1(B)= \inf \operatorname{Spec} {\mathcal H}(B)$ to be the lowest eigenvalue of ${\mathcal H}(B)$. The diamagnetic inequality tells us that
$$
\lambda_1(B) \geq \lambda_1(0),
$$
for all $B\geq 0$.\\
One may ask whether the more general inequality
$$
0< B_1 < B_2 \quad \Rightarrow \quad \lambda_1(B_1) \leq \lambda_1(B_2),
$$
which one can consider as a strong form of diamagnetism, holds (see \cite{Erd1},
\cite{Erd2}
and \cite{LoTha}).
In this paper we prove that strong diamagnetism holds for sufficiently large $B$.
\begin{thm}\label{thm:Derivative}~\\
The one sided derivatives,
$$
\lambda_{1,+}'(B) = \lim_{\epsilon \rightarrow 0_{+}} \frac{\lambda_1(B+\epsilon) - \lambda_1(B)}{\epsilon},
\quad\quad\lambda_{1,-}'(B) = \lim_{\epsilon \rightarrow 0_{+}} \frac{\lambda_1(B) - \lambda_1(B-\epsilon)}{\epsilon}
$$
exist for all $B>0$ and $\lambda_{1,+}'(B)$ satisfies
\begin{align}
\label{eq:LimitDerivative}
\liminf_{B \rightarrow \infty} \lambda_{1,+}'(B) > 0.
\end{align}
Furthermore, there exists a universal constant $\Theta_0 > 0$ such that if $\Omega$ is not a disc, then the limit actually exists and satisfies,
\begin{align}
\label{eq:PreciseNotDisc}
\lim_{B \rightarrow \infty} \lambda_{1,-}'(B) =
\lim_{B \rightarrow \infty} \lambda_{1,+}'(B) = \Theta_0.
\end{align}
If $\Omega$ is a disc, then
\begin{align*}
&\limsup_{B \rightarrow \infty} \lambda_{1,+}'(B) > \Theta_0, \\
&0 < \liminf_{B \rightarrow \infty} \lambda_{1,+}'(B) < \Theta_0.
\end{align*}
In particular, in any case, there exists $B_0>0$ such that $\lambda_1(B)$ is strictly increasing on $[B_0, \infty)$.
\end{thm}
Results similar to \eqref{eq:LimitDerivative} have been proved recently in \cite{FournaisHelffer3} under extra assumptions. First of all (in \cite{FournaisHelffer2}) a complete asymptotics of $\lambda_1(B)$ was derived for $\Omega$ satisfying a certain `generic' assumption, i.e. that the boundary curvature only has a finite number of maxima, all being non-degenerate. This complete asymptotics was then used to obtain \eqref{eq:LimitDerivative}.
The most prominent domain excluded in this approach is the disc---where the curvature is constant. However, \cite{FournaisHelffer3} includes a special analysis of the disc proving that Theorem~\ref{thm:Derivative} remains true in that case.
What remained was the study of all the other `non-generic' cases. Also it seemed desirable to be able to establish Theorem~\ref{thm:Derivative}
without using the existence of a complete asymptotic expansion, since such expansions are difficult to obtain and their structure depends heavily on the different kinds of maxima of the boundary curvature. In this paper we realize such a strategy. It turns out that for all domains, {\it except the disc}, one can modify the approach from \cite{FournaisHelffer3} to obtain \eqref{eq:LimitDerivative} with only very limited knowledge on the asymptotic behavior of $\lambda_1(B)$. For the disc one can use the special symmetry (separation of variables) of the domain to conclude.
Thus the structure of the proof of Theorem~\ref{thm:Derivative} is as follows. The statements for the disc follow from the analysis in \cite{FournaisHelffer3} which will not be repeated.
Thus we only consider the case where $\Omega$ is not a disc.
If $\Omega$ is not a disc then there exists a part of the boundary where the ground state will be very small. Thus one can choose a gauge such that $|\widehat {\bf A} \psi| \ll 1$ (for large $B$ and in the $L^2$-sense), where $\widehat {\bf A}$ is the vector field ${\bf F}$ in the new gauge. This new input to the proof in \cite{FournaisHelffer3} allows us to differentiate the leading order asymptotics for $\lambda_1(B)$.
Notice that if $\Omega$ is not a disc, then it satisfies the following assumption~:
\begin{assumption}\label{assump:Notdisc}~\\
If we denote by $\Pi$ the set of maxima for the curvature, i.e.
$$
\Pi = \{ p \in \partial \Omega \,\big | \,k(p) = k_{\rm max} \},
$$
then $$
\Pi \neq \pa \Omega\;.
$$
\end{assumption}
Finally, we will prove in Section \ref{s3} (Theorem \ref{thm:HPimproved})
that all the natural definitions of
the third critical field appearing in the theory of superconductivity coincide
without any other geometric assumption than regularity and simply connectedness.\\
\section{The analysis of the diamagnetism}
Two universal constants $\Theta_0, C_1$ will play an important role in this paper, as in any investigation of the magnetic Neumann Laplacian. For detailed information about these constants, one can refer to \cite{He-Mo}.
For the second constant $C_1$, we only use the fact that it is strictly positive.
The first, $\Theta_0$ can be defined as the ground state energy of the magnetic Neumann Laplacian with unit magnetic field in the case of the half-plane,
$$
\Theta_0 := \lambda_1(B=1), \quad\quad \text{ for } \quad\quad \Omega = {\mathbb R}^2_{+}.
$$
The numerical value of $\Theta_0$ can be calculated with precision ($\Theta_0 \approx 0. 59$), however for our purposes the following (easily established) rigorous bounds
$$
0< \Theta_0 < 1,
$$
suffice.
We recall the following general, leading order asymptotics of $\lambda_1(B)$ proved in \cite{He-Mo}.
\begin{prop}~\\
As $B \rightarrow + \infty$, then
\begin{align}
\label{eq:leading}
\lambda_1(B) = \Theta_0 B + o(B)\,.
\end{align}
\end{prop}
If a state $u$ is localized away from the boundary, i.e. $u \in C_0^{\infty}(\Omega)$, we have the following standard inequality
$$
\langle u \,,\, {\mathcal H}(B) u \rangle \geq B \| u \|_{L^2(\Omega)}^2\,,
$$
where, for $v$, $w$ in $L^2(\Omega)$,
$\langle v\,,\,w\rangle$ denotes the $L^2$ scalar product of $v$ and $w$.\\
Using that $\Theta_0<1$ it is therefore a standard consequence of \eqref{eq:leading} (for the proof see \cite{He-Mo}) that ground states are exponentially localized near the boundary.
\begin{lemma}[Normal Agmon estimates]\label{lem:NormalAgmon}~\\
There exists $\alpha, M,C>0$ such that if $B\geq 1$ and $\psi_1(\,\cdot\,;B)$ is a ground state of ${\mathcal H}(B)$ then
\begin{align}
\int_{\Omega} e^{2\alpha \sqrt{B} {\operatorname{dist}}(x,\partial \Omega)} &\big\{
|\psi_1(x;B) |^2 + \frac{1}{B} | p_{B{\bf F}} \psi_1(\,\cdot\,;B) |^2 \big\}\,dx\nonumber\\
&\leq C \int_{\{ \sqrt{B} {\operatorname{dist}}(x,\partial \Omega) \leq M\}}|\psi_1(x;B) |^2
\,dx\, .
\end{align}
In particular, for all $N>0$,
\begin{align}
\int {\operatorname{dist}}(x,\partial \Omega)^N |\psi_1(x;B) |^2 \,dx = {\mathcal O}(B^{-N/2}).
\end{align}
\end{lemma}
>From \cite[Proposition 10.5]{He-Mo} we also get the following (stronger than \eqref{eq:leading}) result,
\begin{prop}\label{prop:LowerPotential}~\\
Let $\Theta_0, C_1$ be the usual universal constants and define, for $C>0$
$$
U_B(x) = \begin{cases} B, & {\operatorname{dist}}(x,\partial \Omega) \geq 2 B^{-1/6},\\
\Theta_0 B - C_1 k(s)\sqrt{B} - C B^{1/3},& {\operatorname{dist}}(x,\partial \Omega) \leq 2 B^{-1/6}.
\end{cases}
$$
Then, if $B\geq 1$ and $C$ is sufficiently big, we have for all $\psi \in W^{2,2}(\Omega)$,
$$
\langle \psi \,,\, {\mathcal H}(B) \psi \rangle
\geq \int_{\Omega} U_B(x) |\psi(x)|^2\,dx.
$$
\end{prop}
Proposition~\ref{prop:LowerPotential} and a corresponding improved upper bound (also proved in \cite{He-Mo}),
\begin{align}
\label{eq:BetterAsymp}
\lambda_1(B) = \Theta_0 B - C_1 k_{\rm max} \sqrt{B} + o(\sqrt{B}),
\end{align}
imply, by suitable Agmon estimates, that ground states have to be localized near the set $\Pi$. We actually only need the following very weak version of this localization.
\begin{lemma}\label{lem:localisation}~\\
Let $\epsilon_0 >0$. Then, for all $N>0$, there exists $C>0$ such that if
$\psi_1(\,\cdot\,;B)$ is a ground state for ${\mathcal H}(B)$, then
$$
\int_{\{{\operatorname{dist}}(x,\Pi) \geq \epsilon_0\}} |\psi_1(x;B) |^2 \,dx\leq C \; B^{-N}\;.
$$
\end{lemma}
We now introduce adapted coordinates near the boundary. Define, for $t_0>0$
\begin{align*}
& \Phi : \frac{|\partial \Omega|}{2 \pi} {\mathbb S}^1\times (0,t_0) \rightarrow \Omega
& \Phi(s,t) = \gamma(s) + t \nu(s).
\end{align*}
For $t_0$ sufficiently small we have that ${\operatorname{dist}}(\Phi(s,t), \partial \Omega) = t$ and that
$\Phi$ is a diffeomorphism with image $\{ x \in \Omega \,| \, {\operatorname{dist}}(x, \partial \Omega) < t_0 \}$.
Furthermore, the Jacobian satisfies $|D\Phi| = 1-tk(s)$.
\begin{lemma}\label{lem:gauge}~\\
Let us define for $\epsilon \leq \min( t_0/2, |\partial \Omega|/2)$ and $s_0\in \pa \Omega$
$$
\Omega(\epsilon,s_0) := \{ x = \Phi(s,t) \, \big | \, t \leq \epsilon, |s-s_0| \geq \epsilon \}.
$$
Then there exists $\phi \in C^{\infty}(\Omega)$ such that
$\widehat {\bf A} = {\bf F}+\nabla \phi$ satisfies
$$
|\widehat {\bf A}(x) | \leq C\, {\operatorname{dist}}(x, \partial \Omega),
$$
for $x \in \Omega(\epsilon,s_0)$.
\end{lemma}
\begin{proof}~\\
Let $\widetilde{\bf A}=(\widetilde{A}_1, \widetilde{A}_2)$ be the magnetic $1$-form pulled back to $(s,t)$ coordinates,
$$
F_1 dx + F_2 dy = \widetilde{A}_1ds+ \widetilde{A}_2 dt.
$$
Taking the exterior derivative, and using $dx \wedge dy = |D\Phi| ds \wedge dt$, we find
$$
{\operatorname{curl}\,}_{s,t}\widetilde{\bf A} = \partial_s \widetilde{A}_2 - \partial_t \widetilde{A}_1 = (1-tk(s)).
$$
Since $\{(s,t) \, | \, t \leq \epsilon, |s-s_0| \geq \epsilon \}$ is simply connected there exists a function $\widetilde{\phi} \in C^{\infty}(\Phi^{-1}(\Omega(\epsilon,s_0)))$ such that
$$
\widetilde{\bf A} + \nabla_{s,t} \widetilde{\phi} = (t - t^2k(s)/2,0).
$$
Let $\chi \in C^{\infty}(\overline{\Omega})$,
\begin{align*}
&\chi = 1 \quad\text{ on } \quad \{x \, | \, t \leq \epsilon, |s-s_0| \geq \epsilon \}, \\
&\chi =0 \quad\text{ on } \quad \{x \, | \, {\operatorname{dist}}(x, \partial \Omega) \geq 2 \epsilon \mbox{ or } |s-s_0| \leq \epsilon/2 \} ,
\end{align*}
and define $\phi(x) = \widetilde{\phi}(\Phi^{-1}(x)) \chi(x)$. Then $\phi$ solves the problem.
\end{proof}
\begin{proof}[Proof of Theorem~\ref{thm:Derivative}]
$\,$\\
Let $\overline\phi \in C^{\infty}(\overline{\Omega})$ be such that ${\bf \overline F}:= {\bf F} + \nabla \overline\phi$ satisfies ${\bf \overline F} \cdot \nu = 0$ on $\partial \Omega$. The existence of such a $ \overline \phi$ is easy to prove.
Define $\overline{ {\mathcal H}}(B)$ to be the self-adjoint operator associated to the closed quadratic form
$$
W^{1,2}(\Omega) \ni u\mapsto \int_\Omega|-i\nabla u + B {\bf \overline F} u|^2 dx\,.
$$
Then $\overline{ {\mathcal H}}(B)$ and ${\mathcal H}(B)$ are unitarily equivalent and so they have the same spectrum. Furthermore, the domain of $\overline{ {\mathcal H}}(B)$ is
$$
{\mathcal D}(\overline{ {\mathcal H}}(B)) = \{ u \in W^{2,2}(\Omega) \,:\, \nu \cdot \nabla u \,\big|_{\partial \Omega} = 0 \},
$$
in particular, ${\mathcal D}(\overline{ {\mathcal H}}(B))$ is independent of $B$.
Applying analytic perturbation theory to $\overline{ {\mathcal H}}(B)$ we get the
existence of $\lambda_{1,+}'(B), \lambda_{1,-}'(B)$.
We recall that Theorem~\ref{thm:Derivative} was proved already in \cite{FournaisHelffer2}
in the case of the disk, so it remains to consider the case where $\Omega$ is not the disc. Thus $\Omega$ satisfies Assumption~\ref{assump:Notdisc}.
Therefore, there exist $s_0 \in [-|\partial \Omega|/2, |\partial \Omega|/2]$ and $0 < \epsilon_0 < \min( t_0/2, |\partial \Omega|/4)$ such that
$$
[s_0 - 2\epsilon_0, s_0 + 2\epsilon_0] \cap \Pi = \emptyset.
$$
Let $\widehat {\bf A} $ be the vector potential defined in
Lemma~\ref{lem:gauge}, $\widehat Q_B$ the quadratic form
$$
W^{1,2}(\Omega) \ni u\mapsto
\widehat Q_B(u)= \int_\Omega|-i\nabla u + B \widehat {\bf A}u|^2 dx\,,
$$
and
$\widehat {\mathcal H}(B)$ be the associated operator.
Then $\widehat {\mathcal H}(B)$ and $\overline{\mathcal H}(B)$ are unitarily
equivalent: $\widehat {\mathcal H}(B)=e^{iB\phi}\overline{\mathcal H}(B)e^{-iB\phi}$, for some $\phi$ independent of $B$.
By analytic perturbation theory applied to $\overline{\mathcal H}(B)$ there exists an analytic branch of eigenfunctions,
$$
\overline{\mathcal H}(\beta) \overline{\psi}_{1,+}(\,\cdot\,;\beta) = \lambda_1(\beta) \overline{\psi}_{1,+}(\,\cdot\,;\beta),
$$
for $\beta \in [B, B+\epsilon)$, some $\epsilon>0$, with $\| \overline{\psi}_{1,+}(\beta) \| = 1$.\\
With $\psi^+_1(\,\cdot\,;\beta):=e^{i\beta\phi} \overline{\psi}_{1,+}(\,\cdot\,;\beta)$ being the corresponding eigenfunctions of
$\widehat{\mathcal H}(\beta)$, we get
\begin{align}
\label{eq:FormDeriv}
\lambda_{1,+}'(B) &= \frac{d}{d\beta} \widehat Q_{\beta}(\psi^+_1(\beta)) \big|_{\beta=B}\nonumber\\
&= \langle \widehat {\bf A} \psi^+_1(\,\cdot\,;B) \,,\, p_{B
\widehat {\bf A}}
\psi^+_1(\,\cdot\,;B)\rangle + \langle p_{B
\widehat {\bf A}} \psi^+_1(\,\cdot\,;B) \,,\, \widehat {\bf A}
\psi^+_1(\,\cdot\,;B)\rangle \nonumber\\
&\quad+ 2 \Re \{ \widehat Q_B(v, \psi^+_1(B)) \},
\end{align}
where $v = \frac{d}{d\beta} \psi^+_1(\beta) \big|_{\beta=B}$.
The last term in \eqref{eq:FormDeriv} vanishes because $\psi^+_1$ is a normalized eigenfunction of $\widehat {\mathcal H}$, and therefore,
\begin{equation}
\lambda_{1,+}'(B) = \langle \widehat {\bf A} \psi^+_1(\,\cdot\,;B) \,,\, p_{B
\widehat {\bf A}}
\psi^+_1(\,\cdot\,;B)\rangle + \langle p_{B
\widehat {\bf A}} \psi^+_1(\,\cdot\,;B) \,,\, \widehat {\bf A}
\psi^+_1(\,\cdot\,;B)\rangle \;.
\end{equation}
We now obtain for any $\beta >0$,
\begin{align}
\lambda_{1,+}'(B) & = \frac{\widehat{Q}_{B+\beta}(\psi^+_1(\,\cdot\,;B)) - \widehat{Q}_B (\psi^+_1(\,\cdot\,;B))}{\beta} - \beta \int_{\Omega} \vert \widehat {\bf A} \vert ^2 \, |\psi^+_1(x;B)|^2\,dx \nonumber\\
&\geq \frac{\lambda_1(B+\beta) - \lambda_1(B)}{\beta} - \beta \int_{\Omega} \vert \widehat {\bf A} \vert ^2 \, |\psi^+_1(x;B)|^2\,dx\,.
\end{align}
By Lemma~\ref{lem:gauge} we can estimate
\begin{align}
\int_{\Omega} \vert \widehat {\bf A} \vert ^2\; |\psi^+_1(x;B)|^2\,dx
&\leq C \int_{\Omega} {\operatorname{dist}}(x,\partial \Omega)^2 |\psi^+_1(x;B)|^2\,dx \nonumber\\
&\quad+
\|\widehat {\bf A} \|_{\infty}^2 \int_{\Omega \setminus \Omega(\epsilon_0,s_0)} |\psi^+_1(x;B)|^2\,dx.
\end{align}
Combining Lemmas~\ref{lem:NormalAgmon} and ~\ref{lem:localisation} we therefore find the existence of a constant $C>0$ such that~:
\begin{align}
\int_{\Omega} \vert \widehat {\bf A} \vert ^2\, |\psi^+_1(x;B)|^2\,dx \leq C\, B^{-1}\,.
\end{align}
We now choose $\beta = \eta\, B$, where $\eta>0$ is arbitrary. By the weak asymptotics \eqref{eq:leading} for $\lambda_1(B)$, we therefore find~:
\begin{align}
\liminf_{B \rightarrow \infty} \lambda_{1,+}'(B) \geq \Theta_0 - \eta\, C\,.
\end{align}
Since $\eta$ was arbitrary this implies
\begin{align}
\liminf_{B \rightarrow \infty} \lambda_{1,+}'(B) \geq \Theta_0\,.
\end{align}
Applying the same argument to the derivative from the left, $\lambda_{1,-}'(B)$, we get (the inequality gets turned since $\beta<0$)
\begin{align}
\limsup_{B \rightarrow \infty} \lambda_{1,-}'(B) \leq \Theta_0.
\end{align}
Since, by perturbation theory, $\lambda_{1,+}'(B) \leq \lambda_{1,-}'(B)$ for all $B$, we get \eqref{eq:PreciseNotDisc}.
\end{proof}
\section{Application to superconductivity}\label{s3}
As appeared
from the works of Bernoff-Sternberg~\cite{BeSt}, Del Pino-Felmer-Stern\-berg \cite{PiFeSt}, Lu-Pan~\cite{LuPa1, LuPa2, LuPa3}, and Helffer-Pan~\cite{He-Pan}, the determination of
the lowest eigenvalues of the magnetic Schr\"{o}dinger operator is
crucial for a detailed description of the nucleation of
superconductivity (on the boundary) for superconductors of Type II and
for accurate estimates of the critical field $H_{C_3}$.
In this section we will clarify the relation between the different definitions
of critical fields considered in the mathematical or physical literature
and all supposed to describe the same quantity.
This is a continuation and an improvement of \cite{FournaisHelffer3}~:
we will be indeed able to eliminate all the geometric assumptions of that paper.\\
We recall that the Ginzburg-Landau functional is given by
\begin{multline}
\label{eq:GL_F}
{\mathcal E}[\psi,{\bf A}] = {\mathcal
E}_{\kappa,H}[\psi,{\bf A}] =
\int_{\Omega} \Big\{ |p_{\kappa H {\bf A}}\psi|^2
- \kappa^2|\psi|^2
+\frac{\kappa^2}{2}|\psi|^4 \\
+ \kappa^2 H^2
|{\operatorname{curl}\,} {\bf A} - 1|^2\Big\}\,dx\;,
\end{multline}
with
$ (\psi, {\bf A}) \in W^{1,2}(\Omega;{\mathbb C})\times
W^{1,2}(\Omega;{\mathbb R}^2)$.
We fix the choice of gauge by imposing that
\begin{align}
\label{eq:gauge}
{\operatorname{div}\,} {\bf A} &= 0 \quad \text{ in } \Omega\;, & {\bf A} \cdot \nu = 0 \quad \text{ on } \partial \Omega\;.
\end{align}
We recall that the domains $\Omega$ are assumed to be smooth, bounded and simply-connected and refer the reader to \cite{Bo2},\cite{BoDa} and \cite{BoFo} for the analysis of the case with corners.
By variation around a minimum for ${\mathcal E}_{\kappa,H}$ we find that minimizers $(\psi, {\bf A})$ satisfy the Ginzburg-Landau equations,
\begin{subequations}
\label{eq:GL}
\begin{align}
\left.\begin{array}{c}
p_{\kappa H {\bf A}}^2\psi =
\kappa^2(1-|\psi|^2)\psi \\
\label{eq:equationA}
{\operatorname{curl}\,}^2 {\bf A} =-\tfrac{i}{2\kappa H}(\overline{\psi} \nabla
\psi - \psi \nabla \overline{\psi}) - |\psi|^2 {\bf A}
\end{array}\right\} &\quad \text{ in } \quad \Omega \, ;\\
\left. \begin{array}{c}
(p_{\kappa H {\bf A}} \psi) \cdot \nu = 0 \\
{\operatorname{curl}\,} {\bf A} - 1 = 0
\end{array} \right\} &\quad \text{ on } \quad \partial\Omega \, ,
\end{align}
\end{subequations}
with
$$
{\operatorname{curl}\,}^2 {\bf A} =
(\partial_{x_2}({\operatorname{curl}\,} {\bf A}),-\partial_{x_1}({\operatorname{curl}\,} {\bf
A})) \, .
$$
It is known that, for given values of the parameters $\kappa, H$, the functional ${\mathcal E}$ has (possibly non-unique) minimizers. However, after some analysis of the functional, one finds (see \cite{Giorgi-Phillips} for details)
that, for any $\kappa >0$, there exists $H(\kappa)$ such that if $H>H(\kappa)$ then $(0,{\bf F}_\Omega)$ is the only minimizer of ${\mathcal E}_{\kappa,H}$ (up to change of gauge).\\
Here we choose ${\bf F}_\Omega$ as the unique solution in $\Omega$
of
${\operatorname{curl}\,} {\bf F}_\Omega =1$ satisfying \eqref{eq:gauge}.
Following Lu and Pan \cite{LuPa1}, one can therefore first define
\begin{align}
\underline{H}_{C_3}(\kappa) = \inf\{ H>0 \;:\; (0, {\bf F}_\Omega) \text{ is a minimizer of } {\mathcal E}_{\kappa,H}\}\;.
\end{align}
In the physical interpretation of a minimizer $(\psi,{\bf A})$, $|\psi(x)|$ is a measure of the superconducting properties of the material near the point $x$. Therefore, $\underline{H}_{C_3}(\kappa)$ is the value of the external magnetic field, $H$, at which the material loses its superconductivity completely.
Actually, as already used implicitly in \cite{LuPa1}
and more explicitly in \cite{FournaisHelffer3}, we should also introduce an upper critical field,
$\underline{H}_{C_3}(\kappa)\leq \overline{H}_{C_3}(\kappa)$, by
\begin{align}
\overline{H}_{C_3}(\kappa) &= \inf\{ H>0 \;:\; \text{for all } H'>H,
(0, {\bf F}_\Omega) \text{ is the only minimizer of } {\mathcal E}_{\kappa,H'}\} \;.
\end{align}
$\,$
The physical idea of a sharp transition from the superconducting to the normal state, requires the different
definitions of the critical field to coincide.
Most works analyzing $\underline{H}_{C_3}$ relate (more or less implicitly)
these {\it global} critical fields to {\it local} ones given purely
in terms of spectral data of the magnetic Schr\"{o}dinger operator ${\mathcal H}(B)$,
i.e. in terms of a {\it linear} problem. The local fields are defined as follows.
\begin{align}
\label{eq:SpecLocalDef}
\overline{H}_{C_3}^{\rm loc}(\kappa) &= \inf\{ H>0 \;:\; \text{ for all } H'>H, \quad \lambda_1(\kappa H') \geq \kappa^2 \} \;, \nonumber \\
\underline{H}_{C_3}^{\rm loc}(\kappa) &= \inf\{ H>0 \;:\; \lambda_1(\kappa H) \geq \kappa^2 \}\;.
\end{align}
The difference between $\overline{H}_{C_3}^{\rm loc}(\kappa)$ and
$\underline{H}_{C_3}^{\rm loc}(\kappa)$---and also between
$\overline{H}_{C_3}(\kappa)$ and $\underline{H}_{C_3}(\kappa)$---can be retraced to the general non-existence of an inverse to the function $B \mapsto \lambda_1(B)$, i.e. to lack of strict monotonicity of $\lambda_1$.
In the previous section, we have solved this monotonicity question
and we now explain, following mainly \cite{FournaisHelffer3}, how
this permits to close the discussion about this `third' critical field
in the high $\kappa$ regime.
The next theorem, which is proved in \cite{FournaisHelffer3}, is typical of Type II materials, in the sense that it is only valid for large values of $\kappa$.
\begin{thm}
\label{thm:Identical}~\\
There exists a constant $\kappa_0>0$ such that, for $\kappa > \kappa_0$, we have
\begin{align}
\underline{H}_{C_3}(\kappa) = \underline{H}_{C_3}^{{\rm loc}}(\kappa)\;,
\quad\quad
\overline{H}_{C_3}(\kappa) = \overline{H}_{C_3}^{{\rm loc}}(\kappa)\;.
\end{align}
\end{thm}
$\,$
On the other hand, we have from Theorem \ref{thm:Derivative}~:
\begin{prop}
\label{prop:InverseFunction}~\\
There exists $\kappa_0$ such that, if $\kappa \geq \kappa_0$, then the equation for $H$:
\begin{align}
\label{eq:FormalField}
\lambda_1(\kappa H) = \kappa^2\;,
\end{align}
has a unique solution $H(\kappa)$.
\end{prop}
In other words, for large $\kappa$, the upper and lower {\it local} fields, defined in \eqref{eq:SpecLocalDef}, coincide.
We define, for $\kappa \geq \kappa_0$, the local critical field $H_{C_3}^{{\rm loc}}(\kappa)$ to be the solution given by Proposition~\ref{prop:InverseFunction}, i.e.
\begin{align}
\label{eq:Hnobar}
\lambda_1(\kappa H_{C_3}^{{\rm loc}}(\kappa)) = \kappa^2\;.
\end{align}
Using Proposition~\ref{prop:InverseFunction} we can identify the lower and upper local fields and therefore find the following result.
\begin{thm}
\label{thm:HPimproved}~\\
Suppose $\Omega$ is smooth, bounded and simply connected.
There exists $\kappa_0 >0$ such that, when $\kappa>\kappa_0$, then
\begin{align}
H_{C_3}^{{\rm loc}}(\kappa) = \underline{H}_{C_3}(\kappa) =
\overline{H}_{C_3}(\kappa)\;.
\end{align}
\end{thm}
\begin{remark}~\\
This result was established in \cite{FournaisHelffer3} under the additional
assumption that $\Omega$ was either a disk or a domain whose boundary
has only a finite number of points of maximal curvature (with in addition
some non degeneracy condition).
\end{remark}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 1,864 |
Pequea Elementary School is asking for your help to provide a few supplies for your child throughout the school year. Please contact the school office if you're unable to purchase the supplies for your student, or if you're interested in providing supplies for another family.
Place a piece of tape around one of the the earbud strings and write your child's name on the tape.
Place a piece of tape around one of the the earbud/headphone strings and write your child's name on the tape.
Place the earbuds in a clear plastic bag labeled with your student's name.
You will need a backpack to carry things to and from school each day. Please bring a pencil box to keep your materials inside your desk. Scissors and a pencil sharpener are recommended supplies, but not mandatory.
The Penn Manor School District provides a TI – 30X IIS Texas Instruments calculator for students in grades 4 through 6. However, the teachers recommend that students purchase their own own scientific calculator that will be used in grades 4 through 6. In seventh grade, all Marticville Middle School students are required to purchase his or her own scientific TI – 30X IIS Texas Instruments calculator. Just think, the earlier you buy your child their own scientific calculator, the more use for the money!
The calculator you purchase MUST be a TI – 30X IIS Texas Instruments calculator so all the students are learning the same functions (see image). The calculator can be found for about $15.00 and in multiple colors at K-Mart, Target, Wal-Mart, Radio Shack, Office Max, Staples, and other office supply stores. | {
"redpajama_set_name": "RedPajamaC4"
} | 2,470 |
/********************************
* EZ-TC Plugin by jd *
* Makes adding trainer cards EZ *
********************************/
'use strict';
const fs = require('fs');
const serialize = require('node-serialize');
let trainerCards = {};
function loadTrainerCards() {
try {
trainerCards = serialize.unserialize(fs.readFileSync('config/trainercards.json', 'utf8'));
Object.assign(CommandParser.commands, trainerCards);
} catch (e) {}
}
setTimeout(function load() {
loadTrainerCards();
}, 1000);
function saveTrainerCards() {
fs.writeFileSync('config/trainercards.json', serialize.serialize(trainerCards));
Object.assign(CommandParser.commands, trainerCards);
}
exports.commands = {
eztc: 'trainercard',
trainercards: 'trainercard',
tc: 'trainercard',
trainercard: function (target, room, user) {
if (!target) target = 'help';
let parts = target.split(',');
for (let u in parts) parts[u] = parts[u].trim();
switch (parts[0]) {
case 'add':
if (!this.can('trainercard')) return false;
if (!parts[2]) return this.sendReply("Usage: /trainercard add, [command name], [html]");
let commandName = toId(parts[1]);
if (CommandParser.commands[commandName]) return this.sendReply("/trainercards - The command \"" + commandName + "\" already exists.");
let html = parts.splice(2, parts.length).join(',');
trainerCards[commandName] = new Function('target', 'room', 'user', "if (!room.disableTrainerCards) if (!this.runBroadcast()) return; this.sendReplyBox('" + html.replace(/'/g, "\\'") + "');"); // eslint-disable-line no-new-func
saveTrainerCards();
this.sendReply("The trainer card \"" + commandName + "\" has been added.");
this.logModCommand(user.name + " added the trainer card " + commandName);
break;
case 'rem':
case 'del':
case 'delete':
case 'remove':
if (!this.can('trainercard')) return false;
if (!parts[1]) return this.sendReply("Usage: /trainercard remove, [command name]");
let command = toId(parts[1]);
if (!trainerCards[command]) return this.sendReply("/trainercards - The command \"" + command + "\" does not exist, or was added manually.");
delete CommandParser.commands[command];
delete trainerCards[command];
saveTrainerCards();
this.sendReply("The trainer card \"" + command + "\" has been removed.");
this.logModCommand(user.name + " removed the trainer card " + command);
break;
case 'list':
if (!this.can('trainercard')) return false;
let output = "<b>There's a total of " + Object.keys(trainerCards).length + " trainer cards added with this command:</b><br />";
for (let tc in trainerCards) {
output += tc + "<br />";
}
this.sendReplyBox(output);
break;
case 'off':
if (!this.can('roommod', null, room)) return false;
if (room.disableTrainerCards) return this.sendReply("Broadcasting trainer cards is already disabled in this room.");
room.disableTrainerCards = true;
room.chatRoomData.disableTrainerCards = true;
Rooms.global.writeChatRoomData();
this.privateModCommand("(" + user.name + " has disabled broadcasting trainer cards in this room.)");
break;
case 'on':
if (!this.can('roommod', null, room)) return false;
if (!room.disableTrainerCards) return this.sendReply("Broadcasing trainer cards is already enabled in this room.");
delete room.disableTrainerCards;
delete room.chatRoomData.disableTrainerCards;
Rooms.global.writeChatRoomData();
this.privateModCommand("(" + user.name + " has enabled broadcasting trainer cards in this room.)");
break;
default:
case 'info':
case 'help':
if (!this.runBroadcast()) return;
this.sendReplyBox(
"EZ-TC Commands:<br />" +
"/trainercard add, [command name], [html] - Adds a trainer card.<br />" +
"/trainercard remove, [command name] - Removes a trainer card.<br />" +
"/trainercard list - Shows a list of all trainer cards added with this command.<br />" +
"/trainercard off - Disables broadcasting trainer cards in the current room.<br />" +
"/trainercard on - Enables broadcasting trainer cards in the current room.<br />" +
"/trainercard help - Shows this help command.<br />" +
"<a href=\"https://gist.github.com/jd4564/399934fce2e9a5ae29ad\">EZ-TC Plugin by jd</a>"
);
}
},
};
| {
"redpajama_set_name": "RedPajamaGithub"
} | 3,284 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.