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\section{Introduction}\label{sec.intr}
The subdiffusion transport mechanism in recent years has received much attention for the fact that some physical processes including the electron transport, thermal diffusion, and protein transport, among others, reveal that the underlying stochastic process is the continuous time random walk instead of the Brownian motion \cite{JinLazarovZhouconcise,MetzlerKlafterrandom}.
In this study, we develop robust time-stepping methods for the following $\alpha$th ($\alpha \in (0,1)$) order subdiffusion problem
\begin{equation}\label{I.1}\begin{split}
\begin{cases}
\partial_t^\alpha u(\boldsymbol x,t)-\Delta u(\boldsymbol x,t)=f(\boldsymbol x,t), & (\boldsymbol x,t)\in \Omega\times (0,T], \\
u(\boldsymbol x,t)=0, & \boldsymbol x \in \partial\Omega, ~t\in (0,T],\\
u(\boldsymbol x,0)=v(\boldsymbol x), & \boldsymbol x \in \Omega,
\end{cases}
\end{split}\end{equation}
where the space $\Omega \in \mathbb{R}^d$ $(d=1,2,3)$ is a bounded convex polygonal domain with the boundary denoted by $\partial \Omega$.
The operator $\Delta: D(\Delta) \to L^2(\Omega)$ stands for the Laplacian with $D(\Delta)=H_0^1(\Omega)\cap H^2(\Omega)$, and $f:(0,T]\to L^2(\Omega)$ is a given function.
The initial function $v$, depending on its smoothness, belongs to $D(\Delta)$ or $L^2(\Omega)$.
$\partial_t^\alpha$ is the Caputo fractional operator satisfying $\partial_t^\alpha \phi=D_t^\alpha (\phi-\phi(0))$ for $\alpha \in (0,1)$, where $D_t^\alpha$, known as the Riemann-Liouville fractional operator, is defined by
\[
(D_t^\alpha \phi)(t)=\frac{1}{\Gamma(1-\alpha)}\frac{\mathrm{d}}{\mathrm{d}t}
\int_{0}^{t}\frac{\phi(s)}{(t-s)^\alpha}\mathrm{d}s.
\]
\par
The literature on subdiffusion is vast, for example, the solution regularity exploration can be found in \cite{SakamotoYamamoto}, and some numerical studies were developed in \cite{YinLiuLiZhang1,JinLiZhou1,JinLiZh2,WangWangYin,GaoSun,LiZhaoChen,LiaoMcLeanZhang}, to mention just a few.
See also the overview article \cite{JinLazarovZhouconcise}.
It is well known that the problem (\ref{I.1}) is characterized by the initial singularity of its solution, which frustrates most high-order numerical methods in case the singularity is overlooked.
In \cite{YinLiuLiZhang1}, we proposed a modified $\theta$-method which can preserve the optimal accuracy for $\theta \in (0,\frac{1}{2})$.
As mentioned in \cite{YinLiuLiZhang1}, the case $\theta=\frac{1}{2}$ has deserved much more our attention since the correction terms vanish when $\theta=\frac{1}{2}$, enlightening us that a carefully designed time-stepping method should automatically resolve the singularity.
To sum up, our contribution in this study is twofold:
\begin{itemize}
\item A novel strategy is developed which can transfer known time-stepping methods such as the fractional BDF2 to more robust methods.
\item Rigorous arguments of the optimal error estimates of the transformed fractional BDF2 are provided for the subdiffusion problem (\ref{I.1}).
\end{itemize}
\par
The rest of the article is outlined as follows.
In section \ref{sec.Novel}, a novel strategy is proposed to introduce a shifted parameter $\theta$ into known stepping methods, based on which the fully discrete scheme for (\ref{I.1}) is constructed.
In section \ref{sec.Optimal}, the rigorous error estimates are provided and their correctness is fully validated in section \ref{sec.tests}.
Finally, some concluding remarks are made in section \ref{sec.conc}.
\section{Novel $\theta$-schemes}\label{sec.Novel}
We first propose some general results on constructing high-order accuracy difference formulas for fractional calculus based on generating function (GF) reformulation.
Assume $\varpi_p(\zeta)$ is a GF of the convolution quadrature (CQ) \cite{Lubich} with convergence order $p$, and let $\delta(\zeta)=\sum_{j=1}^{p}\frac{1}{j}(1-\zeta)^j$ denote the GF of backward difference formulas (BDF) with $p\leq 6$.
\begin{lemma}\label{lem.novel.1} (General conversion strategy)
Define $\omega(\zeta)=\varpi_p(\zeta) e^{\theta \delta(\zeta)}$, $\theta \in \mathbb{R}$,
then $\omega(\zeta)$ can generate a $\theta$-method which is convergent of order $p$.
\begin{proof}
The function $e^{\theta \delta(\zeta)}$ is sufficiently differentiable on the unit circle and thus its Fourier coefficients decay faster than, e.g., $O(n^{-k})$ for any positive integer $k$.
Then the asymptotic property of $\omega_n$ is fully determined by $\varpi_n$ which, by the stability in CQ (i.e., $\varpi_n=O(n^{-\alpha-1})$, see Definition 2.1 in \cite{Lubich}), leads to $\omega_n=O(n^{-\alpha-1})$.
Moreover, by the consistency of $\varpi_p(\zeta)$ (see Definition 2.2 in \cite{Lubich})) and the backward difference formulas, i.e.,
\[
\tau^{-\alpha}\varpi_p(e^{-\tau})=1+O(\tau^p),\quad
\tau^{-1}\delta(e^{-\tau})=1+O(\tau^p),
\]
we have
$
\tau^{-\alpha}e^{\theta \tau}\omega(e^{-\tau})
=\tau^{-\alpha}\varpi_p(e^{-\tau})e^{\theta \tau}e^{\theta \delta(e^{-\tau})}=1+O(\tau^p),
$
indicating that $\omega(\zeta)$ is consistent of order $p$ which, combined with $\omega_n=O(n^{-\alpha-1})$, completes the proof of the lemma (see Theorem 1 in \cite{LiuYinunified}).
\end{proof}
\end{lemma}
\begin{remark}\label{rem.1}
Lemma \ref{lem.novel.1} indicates we can approximate $\phi(t_{n-\theta})$ by a discrete convolution as
\begin{equation}\label{novel.1.2}
\sum_{j=0}^{n}\theta_j\phi(t_{n-j}),\quad \text{where $\theta_j$ is generated by~} \sum_{j=0}^{\infty}\theta_j \zeta^j=e^{\theta \delta(\zeta)}.
\end{equation}
\end{remark}
\begin{lemma}\label{lem.novel.2}
Assume $\omega(\zeta)$ takes the form $\big[P(\zeta)\big]^\alpha e^{\theta Q(\zeta)}$ where $P(\zeta)$ and $Q(\zeta)$ are polynomials such that $\omega(\zeta)$ is analytic within the open unit disc, then
\begin{equation}\label{novel.2}
\omega_n=\frac{1}{nP(0)}\bigg[\omega_0 G_{n-1}+\sum_{k=1}^{n-1}\omega_{n-k}\big(G_{k-1}-(n-k)P_k\big)\bigg], \quad n \geq 1,\quad
\omega_0=\big[P(0)\big]^\alpha e^{\theta Q(0)},
\end{equation}
where $G_k$ is the coefficients of $G(\zeta)$ defined by $G(\zeta)=\alpha P'(\zeta)+\theta P(\zeta)Q'(\zeta)$.
\begin{proof}
Take the derivative of $\omega(\zeta)=\big[P(\zeta)\big]^\alpha e^{\theta Q(\zeta)}$ w.r.t $\zeta$ and multiply both sides by $P(\zeta)$ to obtain
\[
P(\zeta)\omega'(\zeta)=\omega(\zeta)G(\zeta).
\]
The formula (\ref{novel.2}) then follows by taking the $n$th coefficient of both sides of the above equality.
\end{proof}
\end{lemma}
\par
It is notable that the algorithm (\ref{novel.2}) is efficient since $G(\zeta)$ and $P(\zeta)$ have finitely many nonzero coefficients, and thus the computing complexity to obtain $\{\omega_j\}_{j=0}^N$ is of $O(N)$.
\par
Denote by $u^n$ the approximation to $u(t_n)$, and introduce the symbols for general functions $\phi$
\begin{equation}\label{novel.2.0}
\phi^{n-\theta}=\sum_{j=0}^{n}\theta_j \phi^{n-j},\quad
D_\tau^{\alpha,n-\theta}\phi=\tau^{-\alpha}\sum_{j=0}^{n}\omega_j \phi^{n-j}
\end{equation}
where $\theta_j$ is defined in (\ref{novel.1.2}) with $\delta(\zeta)=\frac{3}{2}-2\zeta+\frac{1}{2}\zeta^2$, and $\omega_j$ is generated by $\omega(\zeta)=\big[\delta(\zeta)\big]^\alpha e^{\theta\delta(\zeta)}$.
In accordance with Lemma \ref{novel.1.2} (see also Remark \ref{rem.1}), $\phi^{n-\theta}$ and $D_\tau^{\alpha,n-\theta}\phi$ both are of second-order accuracy to their continuous counterparts.
To formulate the fully discrete scheme of the model, define the finite element space as
$
V_h=\{\chi_h \in H_0^1(\Omega): \chi_h |_e \text{ is a linear polynomial function},~ e \in \mathcal{T}_h\}
$
where $\mathcal{T}_h$ is a shape regular, quasi-uniform triangulation of $\Omega$.
\par
Let $P_h:L^2(\Omega)\to V_h$ and $R_h:H_0^1(\Omega)\to V_h$ stand for the $L^2(\Omega)$ and Ritz projection, respectively, and define $\Delta_h: V_h\to V_h$ as the discrete Laplacian.
By replacing $u(t)$ with $w(t)+v$ and $f(t)$ with $g(t)+f(0)$ in (\ref{I.1}), the space semi-discrete scheme then reads
\begin{equation}\label{novel.2.1}
D_t^\alpha w_h(t)-\Delta_h w(t)=g_h(t)+f_h^0+\Delta_h v_h,
\end{equation}
where $g_h:=P_h g$, $f_h^0=P_h f(0)$ and $v_h=R_h v$ if $v\in D(\Delta)$ or $v_h=P_h v$ if $v \in L^2(\Omega)$.
Then the fully discrete scheme can be stated as finding $W_h^n\in V_h$ such that
\begin{equation}\label{novel.3}
D_\tau^{\alpha,n-\theta}W_h-\Delta_h W_h^{n-\theta}=g_h^{n-\theta}+f_h^0+\Delta_h v_h,\quad n\geq 1, \quad \theta \in (-1,1).
\end{equation}
\par
In general cases, the scheme (\ref{novel.3}) can only result first-order convergence rate at positive time due to the initial singularity of the solution.
We propose a corrected scheme, with the motivation explained in the next section, by resorting to a single-step modification:
\begin{equation}\label{novel.4}\begin{split}
D_\tau^{\alpha,1-\theta}W_h-\Delta_h W_h^{1-\theta}=(\theta+3/2)(\Delta_h v_h+f_h^0)
+g_h^{1-\theta},\quad n=1,
\\
D_\tau^{\alpha,n-\theta}W_h-\Delta_h W_h^{n-\theta}=g_h^{n-\theta}+f_h^0+\Delta_h v_h,\quad n\geq 2.
\end{split}\end{equation}
\par
We note that for $\theta=-\frac{1}{2}$, the scheme (\ref{novel.4}) recovers exactly (\ref{novel.3}), indicating that (\ref{novel.3}) can resolve the initial singularity automatically if the problem is discretized at the point $t_{n+\frac{1}{2}}$.
\section{Optimal error estimates}\label{sec.Optimal}
The error estimate is based on solution representation and estimates of some kernels.
Denote by $\widehat{\phi}$ the Laplace transform of $\phi$.
Then, using the Laplace transform and its inverse transform, we obtain
\begin{equation}\label{Optimal.1}\begin{split}
w_h(t)&=-\frac{1}{2\pi{\rm i}}\int_{\Gamma_{\sigma,\epsilon}}e^{zt}\big[K(z)(\Delta_h v_h+f_h(0))+zK(z)\widehat{g_h}(z)\big]\mathrm{d}z,
\end{split}\end{equation}
where $K(z)=-z^{-1}(z^\alpha-\Delta_h)^{-1}$ stands for the kernel function, and the contour (with the direction of an increasing imaginary part) $\Gamma_{\sigma,\epsilon}$ is defined by
\[
\Gamma_{\sigma,\epsilon}:=\{z\in\mathbb{C}:|z|=\epsilon, |\arg z|\leq \sigma\}\cup\{z\in\mathbb{C}: z=re^{\pm{\rm i}\sigma}, r\geq \epsilon\}.
\]
\begin{theorem}\label{thm.1}
For $\alpha \in (0,1)$ and $\theta \in (-1,1)$, there exist $\sigma_0 \in (\pi/2,\pi)$ and $\epsilon_0>0$ both of which are free of $\alpha$ and $\tau$ such that for any $\sigma\in (\pi/2,\sigma_0)$ and any $\epsilon<\epsilon_0$, the solution of (\ref{novel.4}) takes the form
\begin{equation}\label{Optimal.2}\begin{split}
W_h^n=-\frac{1}{2\pi{\rm i}}\int_{\Gamma^\tau_{\sigma,\epsilon}}e^{zt_n}\big[\ell(e^{-z\tau})K(\delta_\tau(e^{-z\tau}))
(\Delta_h v_h+f_h^0)
+\tau\delta_\tau(e^{-z\tau})K(\delta_\tau(e^{-z\tau}))g_h(e^{-z\tau})\big]\mathrm{d}z,
\end{split}\end{equation}
where $\Gamma^\tau_{\sigma,\epsilon}=\{z\in \Gamma_{\sigma,\epsilon}:|\Im(z)|\leq \pi/\tau\}$,
$\delta_\tau(\zeta)=\delta(\zeta)/\tau$ and $\ell(\zeta)=\delta(\zeta)\zeta\big(\frac{1}{1-\zeta}+\theta+\frac{1}{2}\big)e^{-\theta\delta(\zeta)}$.
\begin{proof}
Multiply both sides of (\ref{novel.4}) by $\zeta^n$ and sum the index $n$ from $1$ to $\infty$ to yield
\[
\sum_{n=1}^{\infty}\zeta^n D_\tau^{\alpha,n-\theta}W_h
-\sum_{n=1}^{\infty}\zeta^n\Delta_h W_h^{n-\theta}
=\sum_{n=1}^{\infty}\zeta^n g_h^{n-\theta}
+(f_h^0+\Delta_h v_h)\bigg(\sum_{n=1}^{\infty}\zeta^n+(\theta+1/2)\zeta\bigg),
\]
which, by definitions of symbols in (\ref{novel.2.0}), leads to
\[
\big(\big[\delta_\tau(\zeta)\big]^\alpha
-\Delta_h\big) W_h(\zeta)
=g_h(\zeta)
+(f_h^0+\Delta_h v_h)\kappa(\zeta),
\]
where $\kappa(\zeta)=\zeta\big(\frac{1}{1-\zeta}+\theta+\frac{1}{2}\big)e^{-\theta\delta(\zeta)}$.
By Lemma B.1 in \cite{JinLiZhou1}, for fixed constant $\phi_0\in (\pi/2,\pi)$, there exists $\sigma_0\in (\pi/2,\pi)$ which depends only on $\phi_0$, for any $\sigma\in (\pi/2,\sigma_0)$ and any $\epsilon<\epsilon_0$ where $\epsilon_0$ is small enough, $\delta_\tau(e^{-z\tau})|_{z\in \Gamma_{\sigma,\epsilon}^\tau}\in \Sigma_{\phi_0}:=\{z\in \mathbb{C}:|\arg z|<\phi_0, z\neq 0\}$.
By Cauchy integral formula, we have the expression for $W_h^n$ by
\[
W_h^n=\frac{1}{2\pi{\rm i}}\int_{|\zeta|=\varepsilon}\frac{W_h(\zeta)}{\zeta^{n+1}}\mathrm{d}\zeta
\xlongequal{\zeta=e^{-z\tau}}\frac{\tau}{2\pi{\rm i}}\int_{\Gamma_\varepsilon^\tau}e^{zt_n}W_h(e^{-z\tau})\mathrm{d}z
\]
where $\Gamma^\tau_\varepsilon:=\big\{z=-\frac{1}{\tau}\ln\varepsilon+{\rm i}y: y\in \mathbb{R}, |y|\leq \pi/\tau\big\}$.
Let $\mathcal{L}$ be the region enclosed by contours $\Gamma_{\sigma,\epsilon}^\tau$, $\Gamma_\varepsilon^\tau$, $\Gamma^\tau_{\pm}:=\mathbb{R}\pm {\rm i}\pi/\tau$ (oriented from left to right), one can check $W_h(e^{-z\tau})$ is analytic for $z\in \overline{\mathcal{L}}$.
By using the Cauchy integral formula again, and noting that the integral values along $\Gamma_-^\tau$ and $\Gamma_+^\tau$ are opposite, the result (\ref{Optimal.2}) follows readily by taking $\ell(\zeta)=\tau\delta_\tau(\zeta)\kappa(\zeta)$.
The proof is completed.
\end{proof}
\end{theorem}
\begin{remark}\label{rem.2}
The arguments for Theorem \ref{thm.1} reveal the superiority of our scheme that, on the one hand for arbitrary $\theta$, the transform function $e^{-\theta \delta(\zeta)}|_{\zeta=e^{-z\tau}}$ appeared in $\kappa(\zeta)$ is analytic for $z\in \overline{\mathcal{L}}$, in contrast to the transform function $\frac{1}{1-\theta+\theta\zeta}|_{\zeta=e^{-z\tau}}$ in \cite{YinLiuLiZhang1} which is singular at points $z=\pm \frac{\pi}{\tau}\in \overline{\mathcal{L}}$ when $\theta=\frac{1}{2}$ (in which case, the Crank-Nicolson scheme is excluded).
See also \cite{JinLiZh2,WangWangYin} for similar situations.
Therefore, our scheme or numerical analysis is robust against the shifted parameter $\theta$.
On the other hand, thanks to Lemma \ref{lem.novel.1}, the function $\delta_\tau(\zeta)$ appeared in (\ref{Optimal.2}) is independent of $\alpha$, allowing us to develop robust analysis even for small $\alpha$.
We argue that such kind of robustness is not available for schemes in \cite{JinLiZh2,WangWangYin,YinLiuLiZhang1} as $\delta_\tau(\zeta)$ in those schemes are singular at $\alpha=0$, leading to the blow-up of constants $C$ in their estimates.
See \textit{Example 2} in section \ref{sec.tests}.
\end{remark}
\begin{lemma}\label{lem.optimal.1}
Let $\Gamma^\tau_{\sigma,\epsilon}$ be the contour defined in Theorem \ref{thm.1}.
For given $\theta \in (-1,1)$ and any $z\in \Gamma^\tau_{\sigma,\epsilon}$, there holds
\begin{equation}\label{Optimal.2.1}
|\ell(e^{-z\tau})-1|\leq C\tau^2 |z|^2,
\end{equation}
where $C$ is independent of $\tau, z$, but may dependent on $\theta$.
\begin{proof}
Since $|z|\tau \leq \pi/\sin\sigma<+\infty$, we only need to prove (\ref{Optimal.2.1}) for sufficiently small $|z|\tau$.
By the expansion of $\ell(\zeta)$ at the point $\zeta=1$, we have
$
\ell(\zeta)=1+c(\theta)(1-\zeta)^2+(1-\zeta)^3 r(\zeta),
$
where $r(\zeta)$ is analytic at $\zeta=1$.
One then immediately gets
$
\ell(e^{-z\tau})=1+c(\theta)\tau^2|z|^2+o(\tau^2|z|^2)
$,
which completes the proof of the lemma.
\end{proof}
\end{lemma}
\begin{theorem}\label{thm.2}
Suppose $u_h(t):=w_h(t)+v_h$ is the solution of the space semi-discrete scheme of (\ref{I.1}), and $U_h^n:=W_h^n+v_h$ is the solution of the fully discrete scheme of (\ref{I.1}).
If $f\in W^{1,\infty}(0,T;L^2(\Omega))$ and $\int_{0}^{t}(t-s)^{\alpha-1}\|f''(x)\|\mathrm{d}s\in L^{\infty}(0,T)$ where $\|\cdot\|$ denotes the $L^2$ norm, then
\begin{equation}\label{Optimal.3}
\|U_h^n-u_h(t_n)\|\leq C\tau^2\bigg(\mathcal{R}(t_n,v)
+t_n^{\alpha-2}\|f(0)\|
+t_n^{\alpha-1}\|f'(0)\|
+\int_{0}^{t_n}(t_n-s)^{\alpha-1}\|f^{''}(s)\|\mathrm{d}s
\bigg),
\end{equation}
where $\mathcal{R}(t_n,v)=t_n^{\alpha-2}\|\Delta v\|$ if $v \in D(\Delta)$ and $\mathcal{R}(t_n,v)=t_n^{-2}\|v\|$ if $v \in L^2(\Omega)$.
The constant $C$ is independent of $\tau,\alpha,n,N$ and $f$, but may depend on $\theta$.
\begin{proof}
The arguments for this theorem is essentially based on Lemma \ref{lem.optimal.1} and the following estimates on $\delta_\tau(\zeta)$, which can be found in \cite{JinLiZhou1},
\[
|\delta_\tau(e^{-z\tau})-z|\leq C\tau^2|z|^3,\quad
|\delta_\tau^\alpha(e^{-z\tau})-z^\alpha|\leq C\tau^2|z|^{2+\alpha},\quad
C_1|z|\leq |\delta_\tau(e^{-z\tau})|\leq C_2|z|.
\]
Then, the result (\ref{Optimal.3}) is followed after a lengthy but standard analysis for the contour integral, which is omitted for space reasons.
\end{proof}
\end{theorem}
\begin{remark}
The error $u-u_h$ of the space semi-discrete scheme (\ref{novel.2.1}) has been well studied by researchers which is not our main concern in this article.
Interested readers can refer, e.g., \cite{JinLazarovZhou} for more information.
\end{remark}
\section{Numerical tests}\label{sec.tests}
\textit{Example 1.} Let $T=1$.
Depending on the smoothness of $v$, we consider two cases:
\par
~(i) $f=0$, $v=\sin x \in D(\Delta)$, $\Omega=(0,\pi)$, with the exact solution $u(x,t)=E_\alpha(-t^\alpha)\sin x$;
\par
(ii) $f=0$, $v=\chi_{(0,1/2)}$, $\Omega=(0,1)$;
\par
In Table \ref{tab1} and Table \ref{tab2}, we present the $L^2$ error and convergence rates for different $\alpha$ and $\theta$ for schemes (\ref{novel.3}) and (\ref{novel.4}), respectively.
One observes that the scheme (\ref{novel.4}) with correction terms results in optimal convergence rates while the scheme (\ref{novel.3}) is of first-order accuracy except for $\theta=-0.5$, both of which are in line with our theoretical results.
\begin{table}[htbp]
\centering
\caption{$L^2$ error and convergence rates at time $t=0.5$ of \textit{Example 1} (i).}\label{tab1}
{\scriptsize
{\renewcommand{\arraystretch}{1.4}
\begin{tabular}{crlllllllllll}
\toprule
\multirow{2}{*}{$\alpha$} & \multicolumn{1}{c}{\multirow{2}{*}{$\theta$}} & \multicolumn{5}{c}{Corrected scheme (\ref{novel.4})} & \multicolumn{1}{c}{} & \multicolumn{5}{c}{Standard scheme (\ref{novel.3})} \\ \cline{3-7} \cline{9-13}
& \multicolumn{1}{c}{} & \multicolumn{1}{c}{$\tau=2^{-5}$} & \multicolumn{1}{c}{$\tau=2^{-6}$} & \multicolumn{1}{c}{$\tau=2^{-7}$} & \multicolumn{1}{c}{$\tau=2^{-8}$} & \multicolumn{1}{c}{Rates} & \multicolumn{1}{c}{} & \multicolumn{1}{c}{$\tau=2^{-5}$} & \multicolumn{1}{c}{$\tau=2^{-6}$} & \multicolumn{1}{c}{$\tau=2^{-7}$} & \multicolumn{1}{c}{$\tau=2^{-8}$} & \multicolumn{1}{c}{Rates} \\ \midrule
\multirow{4}{*}{0.1} & -0.9 & 4.33E-06 & 3.10E-06 & 6.92E-07 & 1.62E-07 & 2.09 & & 7.50E-04 & 3.91E-04 & 1.96E-04 & 9.82E-05 & 1.00 \\
& -0.5 & 1.86E-06 & 8.76E-07 & 2.65E-07 & 7.13E-08 & 1.89 & & 1.86E-06 & 8.76E-07 & 2.65E-07 & 7.13E-08 & 1.89 \\
& 0.5 & 1.47E-04 & 3.43E-05 & 8.27E-06 & 2.02E-06 & 2.03 & & 2.02E-03 & 9.97E-04 & 4.95E-04 & 2.47E-04 & 1.01 \\
& 0.9 & 2.53E-04 & 5.78E-05 & 1.38E-05 & 3.37E-06 & 2.03 & & 2.87E-03 & 1.41E-03 & 6.95E-04 & 3.46E-04 & 1.01 \\ \hline
\multirow{4}{*}{0.5} & -0.8 & 1.15E-04 & 2.49E-05 & 5.78E-06 & 1.39E-06 & 2.05 & & 3.15E-03 & 1.60E-03 & 8.04E-04 & 4.03E-04 & 1.00 \\
& -0.5 & 3.86E-05 & 6.97E-06 & 1.44E-06 & 3.24E-07 & 2.15 & & 3.86E-05 & 6.97E-06 & 1.44E-06 & 3.24E-07 & 2.15 \\
& 0 & 2.35E-04 & 5.70E-05 & 1.40E-05 & 3.49E-06 & 2.01 & & 5.49E-03 & 2.72E-03 & 1.35E-03 & 6.74E-04 & 1.00 \\
& 0.6 & 2.35E-04 & 5.70E-05 & 1.40E-05 & 3.49E-06 & 2.01 & & 1.23E-02 & 6.02E-03 & 2.98E-03 & 1.49E-03 & 1.01 \\ \hline
\multirow{4}{*}{0.9} & -0.5 & 2.35E-04 & 5.70E-05 & 1.40E-05 & 3.49E-06 & 2.01 & & 3.05E-04 & 7.23E-05 & 1.76E-05 & 4.35E-06 & 2.02 \\
& -0.2 & 1.28E-04 & 2.95E-05 & 7.10E-06 & 1.74E-06 & 2.03 & & 6.78E-03 & 3.30E-03 & 1.63E-03 & 8.10E-04 & 1.01 \\
& 0.3 & 3.56E-04 & 8.65E-05 & 2.14E-05 & 5.31E-06 & 2.01 & & 1.78E-02 & 8.72E-03 & 4.33E-03 & 2.15E-03 & 1.01 \\
& 0.6 & 7.64E-04 & 1.84E-04 & 4.51E-05 & 1.12E-05 & 2.01 & & 2.44E-02 & 1.20E-02 & 5.95E-03 & 2.96E-03 & 1.01 \\ \bottomrule
\end{tabular}}}
\end{table}
\begin{table}[]
\centering
\caption{$L^2$ error and convergence rates at time $t=0.5$ of \textit{Example 1} (ii).}\label{tab2}
{\scriptsize
{\renewcommand{\arraystretch}{1.4}
\begin{tabular}{crlllllllllll}
\toprule
\multirow{2}{*}{$\alpha$} & \multicolumn{1}{c}{\multirow{2}{*}{$\theta$}} & \multicolumn{5}{c}{Corrected scheme} & \multicolumn{1}{c}{} & \multicolumn{5}{c}{Standard scheme} \\ \cline{3-7} \cline{9-13}
& \multicolumn{1}{c}{} & \multicolumn{1}{c}{$\tau=2^{-5}$} & \multicolumn{1}{c}{$\tau=2^{-6}$} & \multicolumn{1}{c}{$\tau=2^{-7}$} & \multicolumn{1}{c}{$\tau=2^{-8}$} & \multicolumn{1}{c}{Rates} & \multicolumn{1}{c}{} & \multicolumn{1}{c}{$\tau=2^{-5}$} & \multicolumn{1}{c}{$\tau=2^{-6}$} & \multicolumn{1}{c}{$\tau=2^{-7}$} & \multicolumn{1}{c}{$\tau=2^{-8}$} & \multicolumn{1}{c}{Rates} \\ \midrule
\multirow{4}{*}{0.2} & -0.5 & 2.68E-06 & 7.74E-07 & 2.03E-07 & 5.14E-08 & 1.98 & & 2.68E-06 & 7.74E-07 & 2.03E-07 & 5.14E-08 & 1.98 \\
& -0.3 & 7.66E-06 & 1.92E-06 & 4.80E-07 & 1.18E-07 & 2.02 & & 9.41E-05 & 4.69E-05 & 2.28E-05 & 1.07E-05 & 1.09 \\
& 0 & 1.83E-05 & 4.39E-06 & 1.07E-06 & 2.62E-07 & 2.03 & & 2.42E-04 & 1.19E-04 & 5.75E-05 & 2.68E-05 & 1.10 \\
& 0.9 & 7.69E-05 & 1.75E-05 & 4.14E-06 & 9.97E-07 & 2.06 & & 7.07E-04 & 3.40E-04 & 1.63E-04 & 7.56E-05 & 1.11 \\ \hline
\multirow{4}{*}{0.8} & -0.5 & 8.79E-05 & 2.12E-05 & 5.20E-06 & 1.28E-06 & 2.03 & & 8.79E-05 & 2.12E-05 & 5.20E-06 & 1.28E-06 & 2.03 \\
& 0.1 & 1.99E-04 & 4.64E-05 & 1.12E-05 & 2.71E-06 & 2.04 & & 7.59E-04 & 3.95E-04 & 1.95E-04 & 9.18E-05 & 1.09 \\
& 0.5 & 3.28E-04 & 7.47E-05 & 1.77E-05 & 4.27E-06 & 2.05 & & 1.36E-03 & 6.82E-04 & 3.31E-04 & 1.54E-04 & 1.10 \\
& 0.7 & 4.11E-04 & 9.26E-05 & 2.18E-05 & 5.25E-06 & 2.06 & & 1.68E-03 & 8.29E-04 & 3.99E-04 & 1.86E-04 & 1.10 \\ \bottomrule
\end{tabular}}}
\end{table}
\\
\textit{Example 2.} We illustrate the robustness of (\ref{novel.4}) when $\alpha \to 0$.
Let $\Omega=(0,\pi), T=1$ and $u(x,t)=(E_\alpha(-t^\alpha)+t^3)\sin x$ such that $v=\sin x\in D(\Delta)$.
The source term is $f(x,t)=\big(6t^{3-\alpha}/\Gamma(4-\alpha)+t^3\big)\sin x$.
In Fig.\ref{C1} (a), we illustrate the $L^2$ error of the scheme (\ref{novel.4}) for varying $\alpha$ under different $\theta=-0.5,0.1,0.4,0.8$.
Particularly, the cases $\theta=0.1$ and $0.4$ of the scheme in \cite{YinLiuLiZhang1} are also presented.
Obviously, the scheme (\ref{novel.4}) is much more robust when $\alpha \to 0$ than the scheme in \cite{YinLiuLiZhang1}.
\par
It seems weird that in (\ref{novel.1.2}) the term $\phi(t_{n-\theta})$ is approximated by a nonlocal formula with coefficients $\theta_j$ with $j=0,1,\cdots, n$.
We shall argue that $\theta_j$ decays exponentially as plotted in Fig.\ref{C1} (b), and thus we only need the first few $\theta_j$'s.
\begin{figure}[htbp]
\centering
\subfigure[]{
\begin{minipage}[t]{0.48\linewidth}
\centering
\includegraphics[width=1\textwidth]{err_vs_alpha.eps}
\end{minipage
\subfigure[]{
\begin{minipage}[t]{0.48\linewidth}
\centering
\includegraphics[width=1\textwidth]{thetas.eps}
\end{minipage
\centering
\caption{(a) Comparison of $L^2$ error between our scheme and that in \cite{YinLiuLiZhang1} for different $\alpha$. (b) Exponential decay of the weights $|\theta_n|$ defined in (\ref{novel.1.2}).}\label{C1}
\end{figure}
\section{Conclusion}\label{sec.conc}
A general conversion strategy is proposed to develop robust and accurate difference formulas based on known ones by involving a shifted parameter $\theta$.
As a demonstration, the well-known BDF2 is considered and is proved rigorously for the subdiffusion problem (\ref{I.1}) showing that our scheme is robust even for very small $\alpha$ and can resolve the initial singularity of the solution.
\section*{Acknowledgments}
The work of the second author was supported by the NSF of Inner Mongolia 2021BS01003,
the third author was supported in part by Grants NSFC 12061053 and the NSF of Inner Mongolia 2020MS01003,
and the fourth author was supported in part by the grant NSFC 12161063 and the NSF of Inner Mongolia 2021MS01018.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 259 |
Hello Guys , Today , A new Tablet comes into the Market Named Auraslate , Its an Open source Tablet with Android Installed , and lots of Tools and toolkit for hackers inbuilt .
Aura slate may be for you. It's basically an Ice Cream Sandwich-compatible tablet built from the ground up for hax0rz and programmers alike.
There are two models – the 7-inch 726B and the 10-inch 1026 – and the 1026 can run the latest version of Android. You can upload any version you want, however, and even the hardware is open source in that you receive a hardware source disk for about $20 extra.
Auraslate just launched (thanks tipster!) so we'll have to wait and see how popular and useful the product becomes. However, as a tool for developers it seems that the founders' hearts are in the right place.
This is one of the good post.I enjoy a lot by read about your post.This is one of the useful post.Supper. | {
"redpajama_set_name": "RedPajamaC4"
} | 742 |
Ісус Христос може означати:
Ісус Христос — релігійний лідер, засновник і центральна постать християнства
Ісус Христос (в ісламі) — один з найбільших пророків в ісламі
Ісус Христос (Новітні релігійні рухи) — відображення образу Ісуса Христа в Новітніх релігійних рухах
Ісус Христос (в Біблії) — опис життя Ісуса Христа викладений в Біблії
Ісус Христос (в християнстві) — християнське трактування особистості Ісуса Христа
Ісус Христос (в історії) — історична реконструкція життя і вчення Ісуса Христа
Інше:
Ісус Христос — суперзірка — рок-опера 1970 року, що описує останній тиждень земного життя Ісуса Христа
Ісус Христос (Південний парк) — персонаж мультиплікаційного серіалу «Південний парк» | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 7,499 |
Q: KineticJS - Scaling stage to viewport I'm working towards a default resolution of 1366x756.
I would to scale this up and down depending on the viewport.
Similar to the example shown here: http://blogs.msdn.com/b/davrous/archive/2012/04/06/modernizing-your-html5-canvas-games-with-offline-apis-file-apis-css3-amp-hardware-scaling.aspx
I'm kind of confused how I would get this to work in KineticJS as it abstracts away the canvas elements used.
What I would like to achieve is basically:
window.addEventListener("resize", OnResizeCalled, false);
function OnResizeCalled() {
canvas.style.width = window.innerWidth + 'px';
canvas.style.height = window.innerHeight + 'px';
}
I know that this isn't the perfect solution as this would stretch the canvas and not keep aspect ratio.
I'm currently using the following:
stage.setWidth(window.innerWidth);
stage.setHeight(window.innerHeight);
stage.setScale(window.innerWidth / 1366)
It's more or less what I'm looking for, but:
*
*mouse/touch actions aren't scaled to the new proportions.
*not the native scaling which uses the GPU to upscale the canvas object.
Any ideas? Thanks :)
A: On a separate problem I was trying to help solve, the user him/herself (user848039) figured out a solution that dealt with mouse position and KineticJS scaling.
This was the question: kineticjs stage.getAbsoluteMousePosition()?.
Credits go to him, but here was his formula that I modified to meet a more common KineticJS setup:
var mousePos = stage.getMousePosition();
mousePos.x = mousePos.x/stage.getScale().x-stage.getAbsolutePosition().x/stage.getScale().x+stage.getOffset().x;
mousePos.y = mousePos.y/stage.getScale().y-stage.getAbsolutePosition().y/stage.getScale().y+stage.getOffset().y;
Perhaps jsFiddle isn't the best example for this, since you want the stage to be innerWidth and innerHeight. Those measurements seem to play a bit weird in the jsFiddle frame, but if you look at the code, you'll see that I set the stage exactly the same as you mentioned above. With the new calculations for mousePos the mousePos.x and mousePos.y are returning the proper coordinates for your scale.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 2,614 |
The Hotel Hershey®
DestinationsPennsylvania
Supreme Chocolate Almond Tort
Guests at The Hotel Hershey have access to the Hershey Country Club's championship golf courses.
The Hotel Hershey (1933) in Hershey, Pennsylvania, offers its guests access to the Hershey Country Club – known for its historic championship courses, signature service, and abundant amenities – making it a perfect destination for a sweet golf getaway. Golf has been a fixture of Hershey, Pennsylvania since 1909: Before the Hershey Country Club was founded, there was the Hershey Golf Club, a nine-hole course located between the chocolate factory and High Point. Milton S. Hershey, chocolate magnate and philanthropist, had bigger plans for the game in "The Sweetest Place on Earth," and founded the Hershey Country Club in 1930. Mr. Hershey's home, High Point Mansion, served as the clubhouse in the early days of the club. Today, the country club offers several courses for contemporary travelers.
The oldest course at the Hershey Country Club is the par-73 West Course, designed at the founding of the club in 1930 by golf course architect Maurice McCarthy. McCarthy was an active golf course designer in the 1920s and 1930s, primarily working in the mid-Atlantic region. Nearly 40 years later, in 1969, the par-71 East Course was designed by George Fazio. Fazio was a professional golfer with PGA and Open Championship wins under his belt. Then a decade later, he and his nephew, fellow golf professional Tom Fazio, designed Pinehurst No. 6 in Pinehurst, North Carolina. While working for Milton Hershey on the country club's West Course, McCarthy also designed the Hershey Park Golf Club, which is later renamed Parkview Golf Course. This golf course closed in 2005. In 1934, Henry Picard was hired as Head Golf Professional and served until 1941. Nicknamed the "Hershey Hurricane" and "Chocolate Soldier," his on-course skills led to 26 wins on the PGA Tour, including the 1936-1937 Hershey Open, 1938 Masters, and the 1939 PGA Championship. After Picard, legendary professional golfer Ben Hogan – considered one of the greatest players of all time – served as Head Golf Professional until 1951. Of his 63 tournament wins, 52 occurred during his tenure as Hershey's golf professional, including six majors.
Since its founding, numerous national tournaments have been held at the Hershey Country Club and individuals including Arnold Palmer, Jan Stevenson, former Pennsylvania Governor Tom Ridge, have walked its greens. The West Course hosted the Hershey Professional Invitational Golf Tournament in 1933 and 1934, then hosted the Hershey Open sporadically until World War II. In 1940, the West Course hosted the 23rd PGA Championship, where Byron Nelson beat Sam Snead during one of his 11-straight PGA Tour victories. The West Course later hosted the Ladies' PGA Lady Keystone Open between 1978 and 1994. The East Course hosted the Reese's Cup Classic until 2004 (began as the Nike Tour event, then the Nationwide Tour) as well as the 44th PGA Professional National Championship in June 2011, the 51st USGA Women's Senior Amateur Championship in 2012, and the NCAA Division II Men's Golf National Championship in 2013.
Nearby, the club offers an 18-hole putting course at The Hotel Hershey as well as access to the Spring Creek Golf Course, a public nine-hole course designed with junior players in mind. Spring Creek, designed by Maurice McCarthy in 1932, was the nation's first public golf course created for players aged 17 and younger. Originally called the "Juvenile Golf Club," the course allowed adolescent players to golf for a fee of $.35 for nine holes when it first opened. In 1969, the name was changed to highlight Spring Creek, the water hazard that winds throughout the course. In 2006, award-winning 21st century golf course architect Tom Clark oversaw a course renovation that included three new hole designs, restoration of six existing holes, a tee-through-green irrigation system, turf grass replacement throughout the course, and the addition of 220 playing yards.
Optional Push | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 5,332 |
Q: Run multiple test opening mutilple instances of windows application(VB .NET) using one UFT instance Application Under test is developed using .net vb .
Clarification 1 :
is it possible to run multiple tests parellel using one instance of UFT in a one machine ?
Clarification 2 :
Can we open multiple instances of the windows application and run different tests on each instance of the the opened windows application in parellel .
A: This will not workout with windows application. Even the application is minimized the objects will not be able to identify with UFT until the window is activated. Both the criteria will not work as the objects are just like that identified by their properties not by creation time as we use it for browser. Only option is running multiple test cases in multiple machines
A: UFT doesn't support running tests in parallel. Note that UFT simulates users, each tests assumes it controls where the focus is (sometimes by taking control of the mouse and keyboard for typing). If multiple tests are trying to do this you may get unexpected results just like you would if you had two mice connected to your computer and two people tried to do different things at the same time.
Consider the following two tests:
Test A
*
*Click username edit field
*Type "hello"
Test B
*
*Click password edit field
*Type "world"
When run in parallel you may get:
*
*Click password edit field
*Click username edit field
*Type "hello"
*Type "world"
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 131 |
Didymodon wollei är en bladmossart som först beskrevs av Coe Finch Austin, och fick sitt nu gällande namn av Coe Finch Austin 1877. Didymodon wollei ingår i släktet lansmossor, och familjen Pottiaceae. Inga underarter finns listade i Catalogue of Life.
Källor
Lansmossor
wollei | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 8,199 |
Shop for Hayes Auto, offering new & used Hayes Auto listed today from a huge discounts.
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Ke'Bryan Hayes 2016 Bowman Inception Gold Auto #'d 1425. | {
"redpajama_set_name": "RedPajamaC4"
} | 5,190 |
The Diamond Five: Brilliant!
Why some artists achieve the recognition they deserve while other equally talented ones don't depends on many things. But one thing is clear, that obscurity does not always mean lack of talent. Regardless, it is always a pleasure to discover little known but immensely gifted musicians and it is really a great pleasure to hear the reissue of The Diamond Five's Brilliant!.
The Diamond Five, a Dutch quintet led by pianist Cees Slinger, was founded in 1959 and lasted until 1965. They were based at the Sheherazade Club in Amsterdam and were quite popular, playing all over Holland and accompanying expatriate American musicians on their visits to Amsterdam. However, when the club closed its doors due to a shift in popular interest from jazz to rock music, the quintet disbanded. This 1964 recording is their only session available on CD. The music is hard bop on the surface, but is neither formulaic nor a copy of the genres imported from the U.S.
The musicians are quite unique in their style. Slinger plays sparse notes on his solos, utilizing well-placed pauses in the music to create melodic hard bop with hints of more forward-looking styles. The other outstanding soloist is tenor saxophonist Harry Verbeke, whose solos (in contrast to that of the leader) are filled with a multitude of notes played in the modal vein. The others are also quite stellar, the bass and the drums providing a loose bluesy support and horn man Cees Smal adding something unique with the sounds of his different horns, switching between valve trombone, cornet and trumpet.
Two highlights are "Lutuli, by composer Ruud Bos and the final track "Monosyl, composed by Smal. This is a beautiful record and a timely reissue, with crystal clear sound from an extremely talented but sorely under-recognized European group that yet again underscores the universality of jazz.
Johnny's Birthday; Ruined Girl; Lutuli; Lining Up; New Born; Monosyl.
Cees Slinger: piano; Harry Verbeke: tenor sax; Cees Smal: trumpet, cornet and valve trombone; Jacques Schols: bass; Johnny Engels: drums.
Title: Brilliant! | Year Released: 2007 | Record Label: Universal Music France
About The Diamond Five
The Diamond Five CD/LP/Track Review Hrayr Attarian Universal Music France United States Brilliant!
Matthew Alec
saxophone, tenor | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 1,684 |
Q: Pause function while JSON call is running I would like my save button to run the JSON call and then finish it's function, but it finishes the function while running the JSON call, so the variables become initialized as nil.
My DeviceDetailViewController.m
//
// DeviceDetailViewController.m
// Steam Backpack Viewer
//
// Created by Vishwa Iyer on 5/22/14.
// Copyright (c) 2014 MoAppsCo. All rights reserved.
//
#import "DeviceDetailViewController.h"
#import "MasterViewController.h"
#import "ProfileManager.h"
#import "ProfileCommunicator.h"
#import "SteamProfile.h"
#import "DeviceViewController.h"
@interface DeviceDetailViewController () <ProfileManagerDelegate> {
ProfileManager *_manager;
NSArray *profile;
SteamProfile *s;
}
extern NSString *ID;
@end
@implementation DeviceDetailViewController
- (NSManagedObjectContext *)managedObjectContext {
NSManagedObjectContext *context = nil;
id delegate = [[UIApplication sharedApplication] delegate];
if ([delegate performSelector:@selector(managedObjectContext)]) {
context = [delegate managedObjectContext];
}
return context;
}
- (IBAction)cancel:(id)sender {
[self dismissViewControllerAnimated:YES completion:nil];
}
- (IBAction)save:(id)sender {
NSManagedObjectContext *context = [self managedObjectContext];
// Create a new managed object
NSManagedObject *newDevice = [NSEntityDescription insertNewObjectForEntityForName:@"BackpackViewer" inManagedObjectContext:context];
[newDevice setValue:self.steamIDTextField.text forKey:@"steamID"];
ID = [NSString stringWithFormat:@"%@", [newDevice valueForKey:@"steamID"]];
[self startFetchingGroups]; // I would like this JSON call to finish before calling the rest of the function below
[newDevice setValue:s.personaname forKey:@"steamName"];
[newDevice setValue:s.avatar forKey:@"imageURL"];
NSError *error = nil;
// Save the object to persistent store
if (![context save:&error]) {
NSLog(@"Can't Save! %@ %@", error, [error localizedDescription]);
}
[self dismissViewControllerAnimated:YES completion:nil];
}
- (id)initWithNibName:(NSString *)nibNameOrNil bundle:(NSBundle *)nibBundleOrNil
{
self = [super initWithNibName:nibNameOrNil bundle:nibBundleOrNil];
if (self) {
// Custom initialization
}
return self;
}
- (void)viewDidLoad
{
[super viewDidLoad];
_manager = [[ProfileManager alloc] init];
_manager.communicator = [[ProfileCommunicator alloc] init];
_manager.communicator.delegate = _manager;
_manager.delegate = self;
// Do any additional setup after loading the view.
}
- (void)startFetchingGroups
{
[_manager fetchGroups];
}
- (void)didReceieveProfileInfo:(NSArray *)groups
{
//the JSON call finishes here, when the groups are receives from the call. I would then like the rest of the save button method above to run after this runs, so that the s variable (which corresponds to a SteamProfile object) becomes initialized correctly.
profile = groups;
s = [profile objectAtIndex:0];
NSLog(s.personaname);
}
- (void)fetchingGroupsFailedWithError:(NSError *)error
{
NSLog(@"Error %@; %@", error, [error localizedDescription]);
}
- (void)didReceiveMemoryWarning
{
[super didReceiveMemoryWarning];
// Dispose of any resources that can be recreated.
}
@end
A: I think you looking for something like this. Note this syntax could be wrong it is untested. I will leave it to you to read the documentation on function call backs.
@interface MyClass: NSObject
{
void (^_completionHandler)(int someParameter);
}
- (void)startFetchingGroups:(void(^)(int))handler;
@end
@implementation MyClass
- (void)startFetchingGroups:(void(^)(void))handler
{
[_manager fetchGroups];
if (handler) {
handler();
}
}
@end
[var startFetchingGroups:^{
[newDevice setValue:s.personaname forKey:@"steamName"];
[newDevice setValue:s.avatar forKey:@"imageURL"];
NSError *error = nil;
// Save the object to persistent store
if (![context save:&error]) {
NSLog(@"Can't Save! %@ %@", error, [error localizedDescription]);
}
[self dismissViewControllerAnimated:YES completion:nil];
}];
The behavior of when a callback gets called depends on what the _manager fetchGroups actually does. You could also use delegation as some of the people in the comments suggested, and is definatly a clean solution as well.
A: Sorry for ugly formatting. This code does exactly what you want..
#import "DeviceDetailViewController.h"
#import "MasterViewController.h"
#import "ProfileManager.h"
#import "ProfileCommunicator.h"
#import "SteamProfile.h"
#import "DeviceViewController.h"
typedef void(^EmptyBlock_t)();
@interface DeviceDetailViewController () <ProfileManagerDelegate> {
ProfileManager *_manager;
NSArray *profile;
SteamProfile *s;
// 1
// here you define a block, an anonymous function pointer that will be called right after you callback is called..
EmptyBlock_t _blockAfterJSONFetched;
}
extern NSString *ID;
@end
@implementation DeviceDetailViewController
- (NSManagedObjectContext *)managedObjectContext {
NSManagedObjectContext *context = nil;
id delegate = [[UIApplication sharedApplication] delegate];
if ([delegate performSelector:@selector(managedObjectContext)]) {
context = [delegate managedObjectContext];
}
return context;
}
- (IBAction)cancel:(id)sender {
[self dismissViewControllerAnimated:YES completion:nil];
}
- (IBAction)save:(id)sender {
NSManagedObjectContext *context = [self managedObjectContext];
// Create a new managed object
NSManagedObject *newDevice = [NSEntityDescription insertNewObjectForEntityForName:@"BackpackViewer" inManagedObjectContext:context];
[newDevice setValue:self.steamIDTextField.text forKey:@"steamID"];
ID = [NSString stringWithFormat:@"%@", [newDevice valueForKey:@"steamID"]];
[self startFetchingGroups]; // I would like this JSON call to finish before calling the rest of the function below
// 2
// here you assign a value to your block. Notice that all objects inside block are called "retain" automatically. Also they a called "release" when you release the block itself..
_blockAfterJSONFetched=^{
[newDevice setValue:s.personaname forKey:@"steamName"];
[newDevice setValue:s.avatar forKey:@"imageURL"];
NSError *error = nil;
// Save the object to persistent store
if (![context save:&error]) {
NSLog(@"Can't Save! %@ %@", error, [error localizedDescription]);
}
[self dismissViewControllerAnimated:YES completion:nil];
};
}
- (id)initWithNibName:(NSString *)nibNameOrNil bundle:(NSBundle *)nibBundleOrNil
{
self = [super initWithNibName:nibNameOrNil bundle:nibBundleOrNil];
if (self) {
// Custom initialization
}
return self;
}
- (void)viewDidLoad
{
[super viewDidLoad];
_manager = [[ProfileManager alloc] init];
_manager.communicator = [[ProfileCommunicator alloc] init];
_manager.communicator.delegate = _manager;
_manager.delegate = self;
// Do any additional setup after loading the view.
}
- (void)startFetchingGroups
{
[_manager fetchGroups];
}
- (void)didReceieveProfileInfo:(NSArray *)groups
{
//the JSON call finishes here, when the groups are receives from the call. I would then like the rest of the save button method above to run after this runs, so that the s variable (which corresponds to a SteamProfile object) becomes initialized correctly.
profile = groups;
s = [profile objectAtIndex:0];
NSLog(s.personaname);
// 3
// finally after your callback is fired you check if block pointer is not null and if it is you call it as a casual function. Assigning nil in the end is optional..
if(_blockAfterJSONFetched){
_blockAfterJSONFetched();
_blockAfterJSONFetched=nil;
}
}
- (void)fetchingGroupsFailedWithError:(NSError *)error
{
NSLog(@"Error %@; %@", error, [error localizedDescription]);
}
- (void)didReceiveMemoryWarning
{
[super didReceiveMemoryWarning];
// Dispose of any resources that can be recreated.
}
@end
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 6,857 |
Q: What is the use of initializationStatus -> in MobileAds.initialize()? So I know that MobileAds.initialize() is used to reduce latency on the session's first ad request. But the structure as I have seen is MobileAds.initialize(context,app_id), so what is the use of "initializationStatus -> " then ? I have also seen that someone said that "MobileAds.initialize(this, initializationStatus -> {
});" is not working anymore, is that true? Is it OK if I use it?
MobileAds.initialize(this, initializationStatus -> {
});
A: Each GMA SDK update may change some initialization methods.
"is not working anymore, is that true?"
It depends on which version you use. At latest version it will work.
Latest GMA SDK v21.1.0 has only these methods for initialization.
public static void initialize(@NonNull Context context)
public static void initialize(@NonNull Context context,
@NonNull OnInitializationCompleteListener listener)
Also, MobileAds.initialize(context, app_id) was removed from the library, not available anymore in GMA SDK.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 1,746 |
Paulo Gabriel
Is it true? Can't Post
Was the Hobbit trilogy ''done in a hurry'', or this is just a myth created by the ''anti-Hobbit'' to justify their claims that the trilogy is bad (in their view)?
Chen G.
It is and it isn't [In reply to] Can't Post
The Hobbit, like The Lord of the Rings, counts among the most well-planned trilogies in cinema history: having been scripted and shot concurrently - ostensibly a single film spliced into three parts.
Even some of the subplots and original material - which some of the films' detractors fancy to be afterthoughts devised when the expansion to a trilogy was plotted, or a dictate of the studio - were in fact concieved very early in the screenwriting. For instance, the romantic subplot, the Dol Guldur subplot (which goes back to the earliest meetings with Guillermo), Thrain, the Bree prologue, etc...
However, this kind of prolonged production requires a long period of pre-production: a time spent revising the screenplay, manufacturing sets and practical effects, casting actors, storyboarding/previsualsing the camera coverage.
The Hobbit had plenty of pre-production time: about 25 months of it. However, of this period, 18 months were spent under Guillermo Del Toro. When he stepped down, Jackson had to step in and restart the process: he couldn't make Del Toro's film for him - he had to make it his own. However, he only had about six months to do so.
That does show up in the finished film: the over-reliance of CG can be in no small part associated with the lack of time to fabricate practical effects: you can't stall the shoot in order to manufacture a set - you have to do it in pre-production, and being that not enough time was given to it, more CG had to be used.
It also shows in the script: some of the subplots that feel undercooked could have been much better with one or two more revisions. The lack of sufficient time to previsualize the film is perhaps the most felt, when it comes to the more over-the-top action beats. With previsualization, someone will have figured out how preposterous they were earlier down the line, and they'd probably be dialed back more.
Having said all of that, these issues have been grossly overstated. On the whole, it seems to me that Jackson and co-writers Philippa Boyens and Fran Walsh seemed to have had a solid concept of how to shape the trilogy as a whole. He also very wisely divided principal photography into three "blocks", between which more of the practical effects could be produced.
The decision to go to a trilogy was decided upon by Jackson and Co - without even informing the studio - in between Block 2 and 3, and they plotted the three films before presenting the idea to the studio, so that aspect of the trilogy was also planned.
A lot has been made of a "reshoot" period but in actual fact those weren't reshoots - they were pickups. They weren't meant to change the existing footage into something it wasn't - they're part of the way Jackson produces films: he always schedules a pickup period midway through the editing process - because while editing, you always find that you need certain shots (even something as trivial as a reaction shot) that you didn't shoot.
Also, the existence of the extended cuts helped enormously, because the extended cut of An Unexpected Journey came out when the edit of The Desolation of Smaug was shaping up, and with a rough cut of The Battle of the Five Armies already assembled, so Jackson could retroactivelly tweak the film - adding references to the Thrain plotlines, prefiguring Kili's infatuation with Tauriel and inserting Girion into the prologue - to make it all feel all the more pre-planned and organic.
(This post was edited by Chen G. on Dec 27 2018, 8:42pm)
A good summary but you left out [In reply to] Can't Post
The fact that a good chunk of prime pre-production time was spent dealing with a bizarre actors union strike, followed by contentious negotiations between WB and the NZ government that required a special debate and vote in parliament to solve. This took immense time and energy for Peter, and I believe he was also dealing with some health issues at the time. I have never seen him look so ill, run down and utterly exhausted as he did in the snips of video we saw from him then.
That is not a normal set of things to be dealing with during prep for a film, let alone with an already-shortened timetable after an interminable wait for a green light.
(This post was edited by Silverlode on Dec 27 2018, 9:04pm)
Yes, there were other issues [In reply to] Can't Post
Jackson also had an ulcer. There were animal-rights issues and a recasting of the role of Fili early in production, etcetra.
But on the whole, I think Peter Jackson had enough of a clear idea of what each film was going to be, and how to make it that way. He just needed more time with the script, with the effects and with previz especially.
And, once you're behind on pre-production, it cascades and you find yourself behind schedule on post-production, as well, which led to the last-minute, overlong edit of An Unexpected Journey.
2ndBreffest
It's true [In reply to] Can't Post
This thing was plagued with problems and set-backs from the very start and continued all the way to the end, and it is glaringly obvious in the end product. PJ admits this in the behind the scenes materials.
skyofcoffeebeans
Dec 28 2018, 3:28am
Yes [In reply to] Can't Post
Pre-production was cut short when Jackson gained control of the production� Warner would not budge on delaying the project, so he had no choice but to accept an inadequate amount of time to prep. It shows in the final product and is an incontrovertible element of this trilogy's production.
What a privilige [In reply to] Can't Post
I just love the way folks here answer questions, some of them spending a lot of time on thoughtful, lengthy answers, jam packed with a ton of information. I'm learning so much here and feel privileged to be part of this place where so many people are so passionate about the tales we share a mutual love for.Thanks to all who contribute with responses.
"I found it is the small things.....everyday deeds of ordinary folk that keeps the darkness at bay.....simple acts of kindness and love." - Gandalf
And yet [In reply to] Can't Post
It shows in the final product and is an incontrovertible element of this trilogy's production.
But it is also an element that was grossly exaggerated by the films' detractors. In the now infamous Battle of the Five Armies documentary, the part that Jackson recalls "winging" wasn't the entire trilogy or even just that movie: it wasn't even the battle.
Rather, it was only the component of the battle that took place on the open field. Both the battle in the streets of Dale and the majority of Ravenhill were concieved of between block 2 and 3 of principal photography, and shot during early pick-ups.
So they weren't planned from the very outset, but they were were much formed (and indeed shot) before the extended cut of An Unexpected Journey came out.
(This post was edited by Chen G. on Dec 28 2018, 10:38am)
imin
i didn't know that [In reply to] Can't Post
I have watched the documentary and to me it made so much sense and made me feel better about the movies.
I felt like 'ah of course! No wonder the films are pants, they had no time to plan it!'
To me it allowed me to cut PJ some slack, now you're telling me that isn't the case?! I feel i would rather still imagine it wasn't his fault, just doing the best in a bad scenario, fail to plan, plan to fail and all that.
I think (at least for me) the documentary isn't used as a way of being anti hobbit and detracting, to me it actually made me feel sorry for PJ and feel more of an understanding as to why the films are the way they are.
All posts are to be taken as my opinion.
It absolutely is [In reply to] Can't Post
The shortcomings of the films are absolutely the result of a lack of ample time to prepare. Really, that it turned out as well as it did is something of a miracle, unto itself.
But I do think that the films' detractors have grossly exaggerated both the films' shortcomings and how detrimental this lack of ample pre-production time was.
Agreed. [In reply to] Can't Post
Hobbit detractors upset me. Especially when they try to compare book to movie.
PJ's style is to "wing it" [In reply to] Can't Post
PJ did admit that there were times when he felt like he was "winging it," in other words he didn't have as solid of a plan as he would like. He ended up delaying the filming of the battle part of BOT5A for nearly a year, in order to properly plan out the sequences, etc. But then again, PJ also likes to throw ideas out there and see what happens. Remember in "Two Towers" where Arwen was supposed to fight in Helm's Deep? He filmed some scenes, but then obviously decided that wouldn't work and cut those. And in the "Fellowship" BTS, Billy Boyd jokes that he got his lines 15 minutes ago. Anyway, I'll bet there's 20 hrs of film from the Hobbit movies that ended up on the floor. Originally he had Tauriel removing the arrow from Kili's leg in the Laketown scene, and IMO it was better that those scenes were cut from the film, changed to her removing the poison instead. I mean, how could Kili run around Laketown with an arrow sticking out of his leg? And then there's that line about the "jambags," if you can watch the documentary that is HILARIOUS!
Also agree [In reply to] Can't Post
I absolutely LOVE these movies! I admit they are not perfect, but they are FABULOUS!
Mari D.
Well, maybe we can stop feeling upset by and instead ... [In reply to] Can't Post
... try to understand people with other opinions? I'm not very good at this myself but I want to try :-)
I don't understand what's going on on this message board, and I don't know if it's really helping.
Many of the threads and posts these days seem to me like many people are trying to work together with the purpose of convincing the others of how amazing the hobbit movies are.
But the sheer amount of posts at least to me does not have that effect. Restating the same arguments over and over again does not convince me at all. And I guess it's the same the other way round?
- People who love the hobbit movies are equally frustrated and it seems to them that their opinions are not understood by the others?
And so we all keep repeating ... the same things ... and does anything good come out of this?
I wonder if it wouldn't make more sense to just put our opinion on the hobbit movies in our signatures or profiles or something. Rather than state and restate them again and again ...
But then, again, we can also just continue as we do. I for my part will then try to use this dynamic as an excercise in not feeling upset and instead carefully listening, trying to understand, even when I disagree.
[Sorry Paulo Gabriel, this is a long post, it's a reaction to a dynamic I'm observing, which your post reminded me of, not a reaction to your post alone. I hope that's okay.]
(This post was edited by Mari D. on Dec 30 2018, 12:15am)
Yes and no [In reply to] Can't Post
Short answer: yes, the pre-production period after PJ took over as director was rushed but if that really damaged the movies significantly is a matter of opinion.
Chen summed it up well. A good many, if not most, of the major decisions about story and plot, characters and much else were made and the scripts were written before del Toro left, long before the movies were green lit and PJ agreed to direct them at virtually the last minute. But PJ needed these movies to express his own vision, style and design sensibilities and the time to develop what was required was short.
Of course changes had to be made after the shift from two to three movies, such as the creation of a new climax for DoS. That stuff had to be shot during the pickups that are part of PJ's mode of operation.
He also never had time to make any real plan for the actual battle of the five armies and IIRC decided not to proceed with that during principle photography. He took the time he needed to develop that part of the story, which was then shot during an extended pickup period.
As for over-reliance on technology, that too is a matter of opinion. But PJ does love that stuff. He used cutting edge tech for LotR and new cutting edge tech for TH.
I often wonder if the problems they had filming on location for both trilogies with rain, snow, flooding, fog and clouds etc. made PJ choose studio shooting and CGI more often than not. Even the barrel river chase in DoS replaced something PJ had planned for FotR but abandoned when the location set was destroyed by a flooding river.
As has been said, PJ is a chaotic film director who changes things on the day in response to new ideas and really makes his movies in the editing room.
Hamfast Gamgee
Less is more? [In reply to] Can't Post
Sometimes it is a good idea to have a time schedule and stick to it! One can have too much time to think about and alter things sometimes!
There absolutely was a schedule [In reply to] Can't Post
You can't really undertake even just one film of this sort - let alone take on three at the same time - and not have a schedule. What Jackson tends to "wing" it are some of the specifics: variations in blocking, and obviously the incorporation of CG, which is only done after-the-fact. No major turns in the story were made up on the spot.
Thats absolutely okay. [In reply to] Can't Post
I respect your opinion.
entmaiden
Jan 2 2019, 2:35am
Might be a good time [In reply to] Can't Post
to provide a link to Silverlode's excellent Open Letter to the Hobbit Forum.
I'll put a couple paragraphs here, because I can't say it better myself, but whenever I get frustrated, I read the entire post.
Diversity is an idea we tend to praise right up until we realize that it means others will actually be annoyingly different and keep on being annoyingly different, never coming around to our (obviously correct or I wouldn't hold it, stands to reason) point of view. We all want others to love the things we love as much as we love them and in the same way, because then we could share it. But often that doesn't happen. If there is one thing I have learned from being part of this community, it's that many people will love the things I hate, be touched by things that leave me cold, and somehow manage to be supremely oblivious to things, both major and minor, that I love passionately. But everyone is here because they love - the books, the movies, or both.
So may I encourage everyone: While talking about what you love or don't, pay attention to whether you're falling into the habit of telling someone else what and how they should think or be. Because it's not your job to make them be like you, even if you're right, and they won't appreciate it any more than you do when it goes the other way. Nobody here represents a block of opinion, or a monolith of sentiment - we are all individuals and between us we cover pretty much the whole spectrum of opinions on any given question. That's what drives debate and discussions and keeps them interesting, but it need not make them antagonistic. Your fellow fans may have weird tastes, be incomprehensible in their reasoning, and have their priorities all out of order (compared to yours), but they're not your enemy. It's always a good idea to remind ourselves how much common ground we share when differences start to feel like they're filling up all the space.
Lindele
You can certainly [In reply to] Can't Post
perceive that if you want, as many people [who have taken it out of context] have, but that is not the spirit PJ tries to convey at all and it's unfortunate that people have twisted his words (like in that unfortunate YouTube video that was released a few years ago).
Thor 'n' Oakenshield
I should have taken this advice long ago [In reply to] Can't Post
Thank you, Mari D. Very insightful, as always. I, at least, will try to adhere to this. New Year's resolution.
"Torment in the dark was the danger that I feared, and it did not hold me back. But I would not have come, had I known the danger of light and joy. Now I have taken my worst wound in this parting, even if I were to go this night straight to the Dark Lord."
Jan 4 2019, 10:00pm
Isn't this is a discussion board on which people are meant to present often differing opinions? [In reply to] Can't Post
Isn't that why we are all here - to see what other people think about The Hobbit movies? But we needn't take differing opinions to heart.
I love all six movies, though I don't think they are perfect any more than the books are. But many people didn't get what they wanted in either trilogy or in one and not the other, and that's inevitable. Sometimes it's because changes have been made to beloved characters, plots, themes etc. and sometimes it's due to PJ's style as a director. Or other reasons. They are all as valid as the reasons those of us who love the movies have for doing so.
It doesn't matter. The books are still the books, unharmed by the movies. The movies are still the movies, undiminished by any criticism. Though I have loved LotR immensely for fifty years, it and TH are still just books and the movies are just movies.
I consider myself lucky to be one of the legions of people who love the movies and take as much joy from them as I do from the books.
But since it so often seems to me that whenever I try to make some point about my opinions, I end up making some mistake or accidentally leaving out some crucial point of my argument, which usually results in my words backfiring - it just seems easier at this point to say, once, definitively, that I very much enjoy the Hobbit, and let that be that. I know I'm not going to change the opinions of anybody else, and after the seventeenth argument about burping jokes in the movies, or whatnot, I find myself tired of such conversations, so I understand Mari D.'s advice. And I take it to heart. It's not a differing opinion - it's one that I've wanted to act on for quite some time now. I'm not gonna stop talking about other things that are important to me, stuff about the books and the movies. But whether I like the Hobbit? I've said that I do, I don't feel the need to repeat that a hundred times over. I've got it in my signature, also as Mari D. suggested.
And, maybe, when I am able to better phrase my arguments - and not leave out important paragraphs - I will be able to hold my own in one of those battles of opinion over the movies that take place here. But until then, until I have that sort of skill, I will respectfully refer anyone who wants to argue with me about whether the movies are good to my signature, which says it all.
I love The Hobbit. Always will.
Ethel Duath
Hear, hear. :) Hopefully, here. // [In reply to] Can't Post
Honestly not sure that's possible [In reply to] Can't Post
"Upset" is a strong word, maybe annoyed is better. But people have feelings, and it may not be possible to separate the two. Keeping the conversation respectful and the snark to a minimum is just good manners, but I don't know that either will keep people from getting annoyed with each other. As for your dynamic, I've seen it go both ways. We had a fellow for awhile that never missed an opportunity to dump on the movies, no matter what the original post was about. I also remember a few years ago someone started a "movie-haters" thread, the very title invited some relatively hostile responses. In response, I started a "movie-lover thread," and then someone else started a "mixed-feeling" thread, which actually got the most responses. Here's the bottom line: My attempt at explaining my love for these movies, and the reasons thereof (for the however-manieth time), will come across to some people as annoying or even badgering, no matter what language I use. I figured out I just needed to avoid threads that were obviously written by movie-haters for movie-haters. Because there is no point in my re-reading how much someone else hates the movies; it just makes me double-down in my responses so why go down that rabbit hole? | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 532 |
{"url":"https:\/\/www.physicsforums.com\/threads\/earths-rotation-and-atmosphere.825307\/","text":"# Earth's Rotation and Atmosphere\n\n1. Jul 28, 2015\n\n### Iseous\n\nHow does the atmosphere rotate with the Earth on its axis? There are no forces acting on it that would be strong enough to keep it moving with the ground. Gravity acts perpendicular to the direction of rotation, so it would not be able to cause this motion. It would only be able to keep the atmosphere from drifting outside of Earth's gravitational field. Viscosity would not be responsible, or else moving anything through the atmosphere would pull the entire atmosphere above it with it. That is, in order for the atmosphere to be pulled with Earth, each point on the ground would be dragging the atmosphere above it. However, if this were the case, then anything like a car or plane would be able to do the same, but they obviously cannot. And even if the atmosphere somehow got forced into this motion by some other means, viscosity\/friction would have slowed this motion until there was only a small boundary layer near the surface that was affected at all, while the rest would be moving at high speeds relative to the ground. Imagine water flowing through a pipe, or reverse it so the pipe moves relative to the water to make it more similar to the Earth\/atmosphere. If the pipe is moving, then imagine adding pressure to get the water up to the same speed as the pipe, and then remove that pressure, would the water keep moving with the pipe forever like our atmosphere?\n\nFurthermore, even if we were to assume that somehow there was a force keeping the atmosphere relatively stationary to the ground, this would create an atmosphere that does not match what we have around us. Each point on the Earth is not the same distance from the axis of rotation, so the speed of rotation is 0 mph at the poles, and increases to over 1000 mph at the equator. This means the atmosphere would have to be moving at different speeds depending on where it was, which would result in a pressure gradient with highest pressure at the poles and the lowest pressure at the equator. This would cause the air to want to flow from high to low pressure, creating constant winds that went from the poles to the equator. This flow could potentially stabilize as the density increased the closer you got to the equator, but we do not see this density variation either.\n\nLastly, since the atmosphere would have to be at different speeds depending on how far you were from the poles, traveling in an airplane would be largely influenced if the destinations were at very different latitudes. They could either gain or lose 100's of mph, but this does not seem to happen or have to be taken into account.\n\n2. Jul 28, 2015\n\n### Staff: Mentor\n\nThe atmosphere formed with the rest of the Earth and thus has been rotating with it from the very beginning.\n\n3. Jul 28, 2015\n\n### Astronuc\n\nStaff Emeritus\nThere is certainly drag by the earth on the atmosphere. At the earth's surface, the air speed is very low, but aloft the air speed is much higher. Look at the jet stream.\n\nGravity affects the atmosphere through buoyancy effects vis-\u00e0-vis differences in density in regions of air, and by the Coriolis affect. The sun provides plenty of energy to the atmosphere to keep things moving.\n\nAn example of the effect of drag -\n\n4. Jul 28, 2015\n\n### Staff: Mentor\n\nThe atmosphere is stationary with respect to the earth, so there are no pressure gradients. And even if there were, they'd quickly equalize, as just a little air flowing in one direction would result in a pressure change, making the air stop flowing. Obviously, air can't continuously flow from the poles to the equator because of conservation of mass (once the air is gone from the poles, there would be no more air to flow).\n\n5. Jul 28, 2015\n\n### Iseous\n\nThe Earth is rotating about its axis, which means each point on the Earth is not rotating at the same speed since they are at different distances from the axis of rotation. So if it were stationary with the Earth, that does not mean each point on the Earth is moving at the same speed. If you are at the equator, the atmosphere would be moving 1000 mph since the Earth would be rotating that fast at that location. If you are at a pole, the atmosphere would not be rotating since it is on the axis of rotation (or close to it). Thus the atmosphere would be ranging from 0 to 1000 mph from the poles to the equator. So there would be a pressure gradient since each part of the atmosphere would be moving at different speeds to be \"stationary\" with the respective part of Earth it was over.\n\nFurthermore, the very next sentence I said it would probably stabilize as the flow continued, but that would mean the density would increase in order to equalize the pressures. In either case we don't see that. Since there is no flow, that means it had to stabilize, but if it stabilized, then the density of the atmosphere would increase as you approached the equator. That doesn't happen.\n\n6. Jul 28, 2015\n\n### Staff: Mentor\n\nIt would help if you could show us some math. What is the percent change in density that you calculate? The Earth is a pretty big object.\n\n7. Jul 28, 2015\n\n### Staff: Mentor\n\nRight: which means the entire first paragraph of your post was wrong. There is no pressure gradient like you describe because the air is not moving with respect to the earth's surface. Think about it this way: take a parcel of air just above the equator and a parcel of air on the equator. Is the distance between them changing? The reality is that at different latitudes, different points on the earth and in the air are moving parallel to each other. As a result, there are no forces (other than centripetal acceleration, discussed below) caused by this motion. IE, if you are trying to apply Bernoulli's equation, that only works along a streamline, which means a parcel of air must be flowing from one place to another. If the parcel of air at the pole never flows to the equator, then it isn't a streamline and you can't compare the pressures using Bernoulli's equation.\n\nNow, what you do have because of the rotational motion is a centripetal acceleration. That's what makes the earth a slightly flattened sphere. The same effect exists on the atmosphere.\n\nBack to the OP for a sec:\nHave you never felt the wind rushing by from a train or a car passing you? Obviously, they can and do pull the atmosphere with them.\n\nThe atmosphere formed with earth, so there was no need to get it to spin-up to the earth's speed. But even if it hadn't, the drag force you describe and we feel every time there is a breeze would fairly quickly bring the atmosphere to a near halt with respect to the earth's surface.\n\n8. Jul 29, 2015\n\n### Iseous\n\nThe air is not moving relative to the surface, but that's not where the gradient would come from. Imagine moving in a car and rolling your window down. The air inside your car is not moving relative to the car, but it is moving at a different speed than the air outside it. Thus there is air flow when the window goes down. The atmosphere would have to be moving at different speeds relative to itself at different points on Earth, just as different points on the ground are rotating faster depending on their distance from the axis of rotation.\n\nI said the train or car does not pull the entire atmosphere above it and that there would only be a small boundary layer that would go with it. Hence why if you were far away from the car you would not feel the wind rushing by you. So if you were far from the surface of the Earth, the ground would not be pulling it as you suggest.\n\nThe atmosphere was pulled toward the Earth by gravity, so it moves around the sun with it, but that doesn't mean it would have been forced to rotate with it. Those are two different things.\n\n9. Jul 29, 2015\n\n### Staff: Mentor\n\nYour posts are bordering on crackpottery and misinformation. The atmosphere obviously moves with the Earth. Either you post your calculations, or this thread will be closed.\n\n10. Jul 29, 2015\n\n### Iseous\n\nWell, in a simple calculation using Bernoulli's Principle, we have:\nv^2\/2 + gz + p\/rho = constant\n\nv = speed of fluid\ng = gravitational acceleration\nz = altitude\np = pressure\nrho = density\n\nSince we would deal with points at equal altitudes, gz is negligible.\n\nSo, assuming the flow stabilized, all of the pressures would be the same, which is the air pressure around us.\n\nWe could compare two different points, one where the atmosphere was moving at 0 m\/s (pole), and another where the atmosphere was moving at 100 m\/s (up to 465 at equator).\nv1^2\/2 + p\/rho1 = v2^2\/2 + p\/rho2\nv1 = speed of atmosphere at pole (0 m\/s)\nrho1 = density of the atmosphere at pole (sea level)\nv2 = speed of atmosphere at point on Earth that would rotate at 100 m\/s\nrho2 = density of the atmosphere at the point where Earth is rotating at 100 m\/s (sea level)\np = pressure at sea level (101325 Pa)\n\n0^2\/2 + p\/rho1 = 100^2\/2 + p\/rho2\np\/rho1 = 5000 + p\/rho2\nDivide by pressure:\n1\/rho1 = 5000\/101325 + 1\/rho2\nMultiply by rho2:\nrho2\/rho1 = 0.05*rho2 + 1\n\nThus, the ratio of rho2\/rho1 would be greater than 1, meaning the density would increase as you increased in velocity or moved closer to the equator. Using 1.2 kg\/m^3 as rho1, rho2 would be 1.27 kg\/m^3 or about 6% more. This would increase further as you moved to the equator.\n\n11. Jul 29, 2015\n\n### Staff: Mentor\n\nSomeone correct me if I'm wrong, but wouldn't the relative velocity between two points on the Earth's surface be zero? It's like being on a spinning merry-go-round with me in the center and by buddy at the edge. Neither of us are moving relative to one another. Or have I misunderstood how things work in this non-inertial frame?\n\n12. Jul 29, 2015\n\n### Staff: Mentor\n\nThat is not right. Think about how the Coriolis effect sets a hurricane to spinning...\n\n13. Jul 29, 2015\n\n### Staff: Mentor\n\nBernoulli's Principle applies within a flow line, not between flow lines, so it doesn't apply here. The Wikipedia article on Bernouilli's Principle has a section on misapplications of the principle that you might want to look at; you're basically presenting a more sophisticated version of the widespread but bogus classroom demonstration in which blowing across a sheet of paper causes it to rise.\n\nLast edited: Jul 29, 2015\n14. Jul 29, 2015\n\n### Staff: Mentor\n\nIt most certainly does happen and is taken onto account. An aircraft in the northern hemisphere experiences a slight crosswind from the west as it flies south into air that is moving more rapidly to the east. The effect of this crosswind is to increase the eastward component of the aircraft's speed relative to the fixed stars, matching it to the eastward speed of the ground underneath it.\n\n15. Jul 29, 2015\n\n### Staff: Mentor\n\nThat's not the effect the OP is referring to.\nIn a stationary frame at earth's center, the atmosphere is moving at 1000 mph at the equator, but in the rotating frame of earth's surface - not including the wind - the atmosphere is stationary. That means there are no shear (or any other fluid dynamic) forces between adjacent air molecules even though in the stationary frame they rotate around each other.\nThe OP is not referring to the wind as we know it, he's wondering why a plane flying at the equator isn't going 1500 mph in when flying west and barely moving if it flies east.\n\nLast edited: Jul 29, 2015\n16. Jul 29, 2015\n\n### Iseous\n\nIsn't this motion the basis for a centrifuge, however? As it rotates, the more dense fluids get pulled further away from the axis of rotation, just as I am describing. For the merry-go-round, that is an interesting point. I think the rotation makes it a bit confusing, as an outside observer would see one moving and the other not moving, so aren't they moving relative to each other even if they aren't getting further away from each other? That seems odd.\n\nTrue, but does the basic idea still not apply? That a faster flow would result in lower pressure? Or any difference between the two fluids? For instance, imagine a car driving. The air inside the car is not moving relative to a person inside it, but the air outside the car is moving relative to that same person. How would you compare the two different fluids? If you roll down the window, there obviously seems to be a difference, so how would you represent those differences mathematically?\n\nThis is exactly what I am talking about. So as it moves south, the air would be moving more rapidly to the east.\n\nNo I was referring to what Nugatory was talking about. If you stay at the same latitude, you will have the same rotational speed that you started with. If you are at the equator, you would already be moving at 1000 mph with the rotation, so it would be like walking in a plane traveling at 1000 mph; it wouldn't be harder to move forward, back, or side to side because you were already moving with it. So going east or west would not have any effect. It would, however, have an effect if you moved north or south because those points aren't moving as fast as the equator. For instance, a point north of the equator might be rotating at only 900 mph because it is not as far out on the curvature of the Earth (not as far from axis of rotation). Thus, you would have a 100 mph speed difference if you moved to that point.\n\n17. Jul 29, 2015\n\n### Staff: Mentor\n\nI disagree with everything the wiki says about the paper demonstration. I don't think it is relevant here though, because the entire issue of flowing air doesn't exist in the example we are discussing.\n\nEdit: ehh, thinking about it more, the first and key issue is correct, that the static pressure in the jet is equal to the static pressure of the surrounding air. Still don't like the other criticisms though.\n\nLast edited: Jul 29, 2015\n18. Jul 29, 2015\n\n### Staff: Mentor\n\nIn the rotating frame of the earth's surface, there are fictitious forces present because the frame is non-inertial. The key one for this discussion is Coriolis force, as Nugatory said. If every parcel of air on Earth were exactly stationary all the time, then Coriolis force would not come into play, since it is velocity-dependent. But in the real atmosphere, parcels of air do move relative to each other, just from random fluctuation if nothing else, and as soon as they do, Coriolis force comes into play.\n\nYes, it applies. The differences in pressure between air parcels in the atmosphere are a major contributor to weather and wind patterns. But for those differences in pressure to exist, there has to be some relative motion between the air parcels. In your car example, the car provides the relative motion. In the actual atmosphere, the relative motion is ultimately due to random fluctuations in the motion of air molecules, plus differences in density and temperature due to differential heating. See below.\n\nYes, this is the Coriolis force that Nugatory referred to. But, as I said above, there needs to be some initial motion of a parcel of air for it to come into play; if every parcel of air in the atmosphere were exactly stationary with respect to the Earth, then there would be no Coriolis force observed. This idealized case (all parcels of air motionless with respect to the rotating Earth) is what russ_watters is talking about in post #15: in this idealized case, he is correct that there are no shear forces present. If you really don't understand how that idealized case is possible, look up the equations describing a rotating frame, and note how the Coriolis force depends on velocity relative to the rotating frame: an object at rest in the rotating frame experiences no Coriolis force, so the entire atmosphere sitting at rest with respect to the rotating Earth is a possible state.\n\nBut in the real atmosphere, all parcels of air are never motionless with respect to the rotating Earth; there are random fluctuations, plus different parts of the atmosphere are heated differently because of day\/night differences, differences in sun angle, differences in ground temperature, etc., and this differential heating drives convection. And once the air is moving, the Earth's rotation does affect how the air moves (an obvious example is the circulation of air in cyclones, which is driven by the Coriolis force and is different in the Northern and Southern hemispheres). And obviously, if you are in an airplane and you move relative to the rotating Earth, you are introducing motion yourself.\n\n19. Jul 29, 2015\n\n### Staff: Mentor\n\nIseous, your original question asked why the atmosphere rotates with the Earth's surface. That has the simple, two-pronged answer:\nA. It always has.\nB. If it didn't, drag would quickly make it.\n\nBut you don't seem to like that and it looks to me like you are jumping around between several other explanations, hoping one might apply. That makes it hard to follow. So far, we have:\n1. Bernoulli's principle.\n2. Centrifugal\/centripetal force.\n3. Coriolis effect.\n\n#2 and #3 have an impact on the atmosphere (and objects) due to Earth's rotation. #1 does not. But none of these have anything to do with your original question. This jumping-around makes the thread confusing and unfocused. Please try to keep the separate issues separate in your head and be clear that you understand that they are separate when discussing them.\nYes. Earth's rotation causes the earth to be flattened as a result of those forces (#2, above). But this has nothing to do with aerodynamic forces and the rotation (or not) of the atmosphere with Earth. Also, my understanding is that because the Earth is flattened, the gravitational potential is equalized at the Earth's surface. As a result, the atmosphere would not have its own, additional bulge. I'm not certain of that, though.\nA car is not well sealed, so the pressure situation is not clear\/simple. However, if you seal a car air-tight and then set it in motion and measure the absolute pressure inside and outside (for someone stationary), they will be equal, at atmospheric pressure.\n\nWhen you open a window, you create a pathway for air to flow into and out of the car and the orientation of that path determines the pressure change. For a window perpendicular to the direction of motion, no air (net) should be flowing into or out of the car and the pressure should not change (like a \"static port\"). If you angle an air-scoop forward, that directs air into the car and acts like a pitot tube, making the pressure inside rise (up to, potentially, stagnation pressure)\nThe \"crosswind\" is the Jet Stream, it's a weather effect caused by the coriolis effect and regular wind. If the Earth was not heated by the sun and thus had no wind, there would be no jet stream. This is totally different from other things you were discussing.\n\nNote that there is a southern jet stream and it flows in the opposite direction of the northern one.\nThat's the coriolis effect and it has nothing to do with wind\/the atmosphere: it applies in a vacuum, causing a projectile to appear to curve as it moves north or south.\n\n20. Jul 29, 2015\n\n### Staff: Mentor\n\nThat's all fine, as long as we are no longer discussing the question in the first sentence of the OP. It isn't clear to me that the OP recognized that that's not the same issue.\n\n21. Jul 29, 2015\n\n### Staff: Mentor\n\nThis is true, at least as a good approximation. (It would be exactly true if the Earth were a fluid, but since the Earth is solid, its rigidity can resist, to some extent, the gravitational effects that try to make its surface exactly equipotential. This effect is very small, however.)\n\nThis is not true. The gravitational potential as a function of height is different at the equator than at the poles; this effect does not stop at the Earth's surface. To put it another way, the fact that the Earth is an oblate spheroid means that the gravitational potential it produces is not a simple spherically symmetric $1 \/ r$ function; there is an additional term due to the Earth's quadrupole moment, which is different at different latitudes. This affects the shape of the atmosphere.\n\n22. Jul 29, 2015\n\n### Staff: Mentor\n\nYeah, ok, I can see that. I wonder how big an additional oblateness that is. The earth's oblateness is caused by the interaction of gravity and centrifugal force over the entire radius of 6400 km. Any additional oblateness would be due to the additional few tens of km of atmosphere, not the entire radius. So it may be a few orders of magnitude smaller of an effect than the 21km of the surface oblateness.\n\n23. Jul 29, 2015\n\n### Staff: Mentor\n\nThe atmosphere's oblateness is not just caused by the gravity of the atmosphere itself. It's caused by the gravity of the entire Earth being oblate. The oblateness of the Earth's gravity doesn't just stop at the Earth's surface; the variation of potential with distance from the Earth's center does not suddenly start being the same at all latitudes when you get beyond the Earth's surface. That is, if I am, say, 100 km above the North Pole, the difference in potential between that and the surface at the North Pole is not the same as the difference in potential between 100 km above the equator, and the surface at the equator.\n\n24. Jul 31, 2015\n\n### Iseous\n\nI do not understand how B could be true. So the ground is essentially moving eastward, and the atmosphere above each point on the ground is being \"dragged\" with it. If there is a treadmill on the ground, then the atmosphere above is being pulled by each point on the treadmill. But if you turn it on so that the treadmill (ground) is essentially moving faster, the entire atmosphere above does not move faster with it. Even if it were a smaller closed system only a small layer close to it would be pulled with it (boundary layer). Where else would you see a moving or rotating object take a static fluid and get it to move with it over a long distance? So how could drag be the answer? So we are left with A, which is a non-answer. It doesn't actually explain anything.\n\n25. Jul 31, 2015\n\n### Bandersnatch\n\nIt looks like you're under the misapprehension that the only two possible states to consider are drag having no effect at all and drag that is causing the whole column of air to move simultaneously and immediately.\nConsider drag accelerating the layer closest to the ground first, then that layer transferring its kinetic energy to the layer above, and that one to the next one up. The ground keeps adding kinetic energy to the atmosphere until it is all moving at the same angular speed.\n\nThis is the same effect that can be seen when you take a cup of tea and stir it. At first you're only accelerating the volume elements in contact with the spoon, but the rest of the liquid soon starts rotating as well - you don't have to 'hit' every bit of volume of the liquid with the spoon to make it rotate.\nOnce the tea is rotating at the same angular velocity, you again can see the same effect as the walls of the cup begin to slow it down through friction. Again, it is only the layer directly in contact that is decelerated by the cup, yet in a finite time you observe the tea to match the rotation (or lack thereof) of the cup.","date":"2018-07-18 13:13:15","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\": 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\": 0.5654700994491577, \"perplexity\": 393.1284988554276}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2018-30\/segments\/1531676590169.5\/warc\/CC-MAIN-20180718115544-20180718135544-00245.warc.gz\"}"} | null | null |
Search Results for Latest Hindi Movies
Displaying 1 - 15 Latest Hindi Movies
New Hindi Movies - The Current Scenario
Nowadays, directors are able to work with ease while producing movies closest to their direction abilities. Moreover, due to revolution in media and advertisement, it is quite easy to promote new hindi movies and earn a good business.
By: Mac Verma Published in Movies & TV on April 30, 2012
The Trends In Latest Bollywood Movies
The latest Bollywood movies feature excellent stories, editing, direction and cinematography. New talents emerging in every field be it actors, directors, singers, producers etc are not hesitant in trying out new and challenging ideas.
Hindi Movies - The Present Scenario
You get to see technological revolutions in the industry as well. With strong editing techniques and new instruments to work with, today directors find it more convenient to produce new hindi movies. Moreover, today market is more welcoming to hindi cinema and hence it is one of the reasons that tod
By: Lee Daniel Published in Movies & TV on April 08, 2012
What is so Different About The New Hindi Movies?
New Hindi movies have a lot in store for almost everyone. From a whole lot of drama to a bag full of humor, you can have it all here. Moreover, Bollywood is also making great efforts towards coming up with some suitable projects for children.
By: Lee Daniel Published in Movies & TV on March 30, 2012
Latest Bollywood Movies - An Outlook
Finding about Bollywood reviews for latest Bollywood movies is one of the most appealing jobs of every youth. There may not be a comprehensible reason on what makes Bollywood so appealing to them. In fact, there could be several aesthetic reasons on why youths fall in to carry a fatal attraction.
New Hindi Movies - Innovative And Audacious Themes
Filmmaking took a new height with new Hindi movies offering exceptional storylines and high quality cinematography along with excellence in technicalities like animation, special effects and other technical aspects. Some of the movies have even been nominated at the Oscar Awards, though they did ...
By: Mac Verma Published in Movies & TV on March 21, 2012
Latest Bollywood Movies - Creating New Waves And Themes For Audiences
With new and talented actors and directors in the industry, latest Bollywood movies are making a huge impact over international platforms too. At last, new directors are getting a chance in showing their excellence in making innovative films with the backing of some of the top producers in the ..
Bollywood Movie Reviews - Offering Unbiased Reports of Latest Hindi Films
Film lovers today have become smart and thus, they go through the Hindi movie review before booking their tickets. Whether you feel that a movie is worth watching or not, you should read the Bollywood movie reviews, as most of the promos can be very misleading.
The Differing Spice of New Hindi Movies
Formula movies of the 90s were based on a particular track, wherein the hero falls in love with a rich girl, however, the girl's rich family opposes their love and the hero fights them all to get the girl. This was a big limitation of most of the movies produced during that era.
Impact Of Bollywood Movies On Youth
We all know that Bollywood is totally situated in Mumbai and every year's number of Hindi Movies are produced in Bollywood and it is often scrutinize as the biggest film producing center in the world. Bollywood movies are very popular among the Indian people and they relish watching these films.
By: Urvashi Agrawal Published in Movies & TV on July 26, 2016
Far-Fetched Daddies of Bollywood in Our Hindi Movie Video Songs
When it comes to emotions even the Hollywood needs to look up to our film industry. For years now it has skilfully displayed all the human sentiment. Today we look at the Hindi movies video songs that expressed the father and child bond
By: Vinod Kadams Published in Movies & TV on June 22, 2015
Bollywood Movies Then And Now
Hindi new movie started adopting trends somewhat similar to the western movies, as there was rise in popularity of bollywood movies. Western-style scripting formats are followed in today's new Hindi movies. Bollywood industry started producing 800 movies or even more annually.
By: Mac Verma Published in Movies & TV on May 18, 2012
Is Good Subject In Movie A Necessity For Good Hindi Movie Review?
Bollywood industry is one of the oldest industry and has seen many ups and downs. Several critiques have measured the taste of audience in depth to find about the changing trends in latest hindi films. Further, it may be difficult to point to a single era and call it as the best trend of Bollywood.
New Approach to Hindi Films
The Millennium has seen various changes in film making trends as far as the latest Hindi movies are concerned.
By: Kevin Fernandez Published in Movies & TV on June 08, 2011
Bollywood - History of Movie Making
The Hindi film industry or Bollywood's history goes back to 1913, when Dadasaheb Phalke financed and made the silent movie Raja Harishchandra. It was a major hit and served as an impetus to many creative people and technicians to make their own films.
By: Yuyutsu Sen Published in Movies & TV on December 27, 2009 | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 8,595 |
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<div id="topLinks">
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</a>
<a href="https://github.com/spincast/spincast-framework">
<img src="/public/images/github.png" title="Code on Github" />
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<li id="menu_index"><a href="/">Home</a></li>
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Maye Musk, 69, joins CoverGirl's all-star lineup
Jacqueline Laurean Yates
Yahoo Beauty September 27, 2017
Maye Musk is officially a CoverGirl. (Photo: Covergirl)
Major beauty brand CoverGirl seems to be on a bit of speedy hiring spree lately, and following recent news of Issa Rae and Ayesha Curry becoming fresh faces of the brand comes word that Maye Musk, 69, will join its diverse ranks.
The longtime model announced the news Wednesday on Instagram, alongside a strikingly gorgeous photo of her wearing a stylish black turtleneck, leather pants, and a fierce cut of her silver hair, which was pushed back from her face of soft, natural-toned makeup. Musk's caption further explained her excitement: "Who knew, after many years of admiring the gorgeous COVERGIRL models, that I would be one at 69 years of age? It just shows, never give up. Thank you COVERGIRL, for including me in your tribe of diversity. Beauty truly is for women of all ages, and I can't wait to take you all along this amazing journey with me!"
After so many years of wearing makeup and serving as a brand muse for different companies, one might think Musk had grown bored with it all. But according to a statement she released, that couldn't be further from the truth. "With wearing makeup a part of my job as a model, one might think I'd get tired of it, but I absolutely love it," she said. "I've had the benefit of working with a lot of wonderful artists through the years, and it's a fascinating process: You start as a blank canvas and then a transformation occurs, but there's always a very real and honest piece of me there too."
Ukonwa Ojo, CoverGirl's senior vice president, also shared a statement: "This is exactly what COVERGIRL is all about: owning your identity and proudly sharing with the world all the facets that make you, you. Maye is an affirmation of the power and importance of diversity and inclusivity in the world of beauty. She is unstoppable and, together, we're just getting started!"
Musk has appeared on countless magazine covers and walked tons of Fashion Week runways, and now she's adding CoverGirl to her long résumé of accomplishments. Talk about aging gracefully. This lovely lady has got it down to a science.
Ayesha Curry just snagged a gig with a major beauty brand
Issa Rae just landed a major beauty gig
Molly Sims electrocuted herself to look younger: 'Mama wants to look good'
Meredith Videos
Wake Up Now!: "America will never be destroyed from the outside. If we falter and lose our freedoms, it will be because we destroyed ourselves." Abraham Lincoln
Prime Day's best deals are on Amazon devices: Kindle, Echo Dot, Fire TV and more
Amazon Prime Day 2019 - The absolute best 10 deals of the day
Meghan Markle's Gucci Clutch Featured This Sweet Detail, and It Could Be a Callout to Her Pre-Royal Days
News Outlets Call Trump's Attacks on Minority Congresswomen Anything But "Racist"
Source Claims Beyoncé Broke Royal Protocol, As If Royal Protocol Applies to Beyoncé
Lily Aldridge Dyed Her Hair Blonde for the First Time and Looks Unrecognizable | {
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Q: Invoking Mule flows from HTML frontend I am new to Mule and my requirement is to run a mule flow on demand .
Wondering whats the best way to configure this in mule. Is there a way I can build a JSP frontend UI which can then invoke a mule flow directly .
Thanks in Advance
A: Expose your Mule flow with a HTTP inbound endpoint.
From the HTML make a call to the HTTP url which will invoke your mule flow.
A: Yes, there is absolutely a way to invoke Mule flow from JSP front-end (of course you can use any front-end technology you want). All you need to do is to configure a flow with HTTP Listener with appropriate method, including GET, POST, PUT, DELETE and an URI. Now from your JSP page invoke this flow using the URI and appropriate method. For more information on HTTP Listener please go through this link
| {
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Hillary Clinton does not say what she means or mean what she says.
There is an Ends Justify the Means mentality and ultimately, I suspect, the Ends are Me Me Me even more than they are Socialism. Sounding virtuous about wanting to hand out other peoples' hard-earned money�is just her chosen�tool,�while she hangs out with, and collects $ from,�all of her billionaire friends. But she cannot finesse things the way�Slick Willie can.
She doesn't live up to the illusion, does she? She was lightly pressed by pols of modest accomplishment and retreated befuddled behind the "you're just a bunch of big meanies" shield.
I wonder at the sense of inevitability that the press has promoted regarding her ascension. She has even fewer accomplishments to her credit than Kerry and as unlikeable as he was, she is even more so.
As to her "self as the center of the universe" - why, yes, she projects an even higher sense of entitlement than did Kerry. Bubba had a certain "rogue's grace" that provided a sufficient cloak to cover his venality and innate corruption. She lacks that cloak and leaves no doubt that any means will be used to justify her selection of "ends".
She's very beatable. Perhaps more so than Kerry.
Well, she is a girl.
Nonetheless, I dodn't dig the pileon coming from the dims.
Though, I do love the general mayhem.
Good ideas don't always take flight and she was responding to Senator Dodd.
Senatorial banter and chicken feathers is just the process and she just wanna-wannabe one of the boys, me thinks.
Why should a candidate in the primaries with the lead Mrs. Clinton has over her rivals and only a few months to go until the primaries say anything?
So she has picked up her husbands abilities to obfuscate, deny,delay, lie, break the law with impunity and a host of plain ole run of the mill sociopathic markers .... she couldn't care less.
She has a field of squat to pee wannabe alpha males trailing her by double digits late in the game. Her constituents and supporters are rallying to her cause, National Socialism. The MSM already has her in the Oval office and the actual election is over a year away.
Pew shows her beating Rudy but it's absurd this far out.
We haven't even approached pre-crunch time.
Don't open fire until you see the whites of their eyes.
Historically speaking she has practically reverted to the early 1800's "campaign" style where it was considered unseemly for a candidate to leave his front porch or answer questions. They were "available" based on their previous record and were then approached by the party leaders and asked to run.
It wasn't until the election of 1840 when things began to appreciably change and candidates "stumped" for office.
A "rogue's grace"? That is worthy of Shakespeare. | {
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namespace il2cpp
{
namespace debugger
{
struct StackFrameGetValuesCommand : public CustomCommand<StackFrameGetValuesCommand>
{
struct Reply : public CustomReply
{
inline std::vector<Variant> &values()
{
return _values;
}
void WriteContentTo(Buffer &out) const;
private:
std::vector<Variant> _values;
};
CUSTOM_COMMAND(StackFrame, GetValues);
Property<int32_t> frame_id;
Property<Il2CppThread*> thread;
Property<std::vector<int32_t> > positions;
void ReadProperties(Buffer &in)
{
thread.ReadFrom(in);
frame_id.ReadFrom(in);
positions.ReadFrom(in);
}
};
struct StackFrameSetValuesCommand : public CustomCommand<StackFrameSetValuesCommand>
{
CUSTOM_COMMAND_EMPTY_REPLY(StackFrame, SetValues);
Property<int32_t> frame_id;
Property<Il2CppThread*> thread;
Property<int32_t> values_count;
inline Buffer &command_buffer() const
{
return *_command_buffer;
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void ReadProperties(Buffer &in);
~StackFrameSetValuesCommand()
{
delete _command_buffer;
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private:
Buffer *_command_buffer;
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struct StackFrameGetThisCommand : public CustomCommand<StackFrameGetThisCommand>
{
struct Reply : public CustomReply
{
Variant this_object;
void WriteContentTo(Buffer &out) const
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out.WriteVariant(this_object);
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CUSTOM_COMMAND(StackFrame, GetThis);
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Property<Il2CppThread*> thread;
void ReadProperties(Buffer &in)
{
thread.ReadFrom(in);
frame_id.ReadFrom(in);
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EXCEL is responsible for construction services on St. Croix for the EHRVI program in response to Hurricanes Maria and Irma. EXCEL has performed work in the Virgin Islands for 20 years, and we consider the region an extension of our Gulf Coast home.
There are extraordinary challenges within this program, including shipping and logistical hurdles, vehicle shortages, material availability, constraints in local labor, and political pressures. These challenges push our team to be creative, agile, responsive and resolute every day. We work closely with our client and program stakeholders to align our work plan with the program's evolving requirements.
Our approach has been to plan and schedule work orders, crews and materials in coordination with resource availability. Our work plan prioritizes safety, quality and homeowner satisfaction to the highest degree. We balance these priorities and challenges to most effectively and efficiently address the pace and production of the program. | {
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\section{Diagrammatic algebras}
\subsection{Algebras}
In this section, we introduce the algebraic construction that will unify the perspectives appearing here. These algebras have appeared in many previous works of the author and collaborators \cite{Webmerged,WebCB,WebwKLR,KTWWYO,WebGT, WebTGK}:
\begin{definition}
A {\bf KLRW diagram} is a collection of finitely many oriented curves in
$\R\times [0,1]$ whose projection to the second factor is a diffeomorphism. Each curve is either
\begin{itemize}
\item colored red and labeled with the integer $\rankp$ and decorated with finitely many dots;
\item colored black and labeled with $i\in [1,\rankm]$ and decorated with finitely many dots. Let $v_i$ be the number of black strands with label $i$.
\end{itemize}
The diagram must be locally of the form \begin{equation*}
\begin{tikzpicture}
\draw[very thick,postaction={decorate,decoration={markings,
mark=at position .75 with {\arrow[scale=1.3]{<}}}}] (-4,0) +(-1,-1) -- +(1,1);
\draw[very thick,postaction={decorate,decoration={markings,
mark=at position .75 with {\arrow[scale=1.3]{<}}}}](-4,0) +(1,-1) -- +(-1,1);
\draw[very thick,postaction={decorate,decoration={markings,
mark=at position .75 with {\arrow[scale=1.3]{<}}}}](0,0) +(-1,-1) -- +(1,1);
\draw[wei, very thick,postaction={decorate,decoration={markings,
mark=at position .75 with {\arrow[scale=1.3]{<}}}}](0,0) +(1,-1) -- +(-1,1);
\draw[wei,very thick,postaction={decorate,decoration={markings,
mark=at position .75 with {\arrow[scale=1.3]{<}}}}](4,0) +(-1,-1) -- +(1,1);
\draw [very thick,postaction={decorate,decoration={markings,
mark=at position .75 with {\arrow[scale=1.3]{<}}}}](4,0) +(1,-1) -- +(-1,1);
\draw[very thick,postaction={decorate,decoration={markings,
mark=at position .75 with {\arrow[scale=1.3]{<}}}}](8,0) +(0,-1) -- node
[midway,circle,fill=black,inner sep=2pt]{}
+(0,1);
\end{tikzpicture}
\end{equation*}
with each curve oriented in the negative direction. In
particular, no red strands can ever cross. Each curve must
meet both $y=0$ and $y=1$ at distinct points from the other curves.
\end{definition}
Readers familiar with the conventions of \cite{Webmerged, WebTGK}, etc. might be surprised to see dots on red strands as well as black. This corresponds to the ``canonical deformation'' discussed in \cite[\S 2.7]{WebwKLR} or the ``redotting'' of \cite{KSred}.
We'll typically only consider KLRW diagrams up to isotopy. Since
the orientation on a diagram is clear, we typically won't draw it.
We call the lines $y=0,1$ the {\bf bottom} and {\bf top} of the
diagram. Reading across the bottom and top from left to right, we
obtain a sequence $\Bi=(i_1, \dots, i_V)$ of elements of $[1,\rankp]$ labelling both red and black strands, where $V$ is the total number of strands.
\begin{definition}
Given KLRW diagrams $a$ and $b$, their {\bf composition} $ab$ is
given by stacking $a$ on top of $b$ and attempting to join the
bottom of $a$ and top of $b$. If the sequences
from the bottom of $a$ and top of $b$ don't match, then the
composition is not defined and by convention is 0, which is not a
KLRW diagram, just a formal symbol.
\[
ab=
\begin{tikzpicture}[baseline,very thick,yscale=.5]
\draw (-.5,-2) to[out=90,in=-90] node[below,at start]{$i$} (-1,-.8)
to[out=90,in=-90](1,.2) to[out=90,in=-90] node[midway,circle,fill=black,inner sep=2pt]{} (0,2);
\draw (.5,-2) to[out=90,in=-90] node[below,at start]{$j$} (.5,0) to[out=90,in=-90] (1,2);
\draw (1,-2) to[out=90,in=-90] node[below,at start]{$i$} (-1,.8) to[out=90,in=-90] (.5,2);
\draw[wei] (0,-2) to[out=90,in=-90] node[below,at start]{$\la_2$}
(0,0) to[out=90,in=-90]
(-.5,2);
\draw[wei] (-1, -2) to[out=90,in=-90] node[below,at start]{$\la_1$}
(-.5,-1) to[out=90,in=-90] (-1,0) to[out=90,in=-90] (-.5,1)
to[out=90,in=-90] (-1,2);
\end{tikzpicture}\qquad \qquad ba=0
\]
Fix a field $\K$ and let ${\it \doubletilde{\mathbb{T}}}\xspace$ be the formal span over
$\K$ of KLRW diagrams (up to isotopy). The composition law
induces an algebra structure on ${\it \doubletilde{\mathbb{T}}}\xspace$.
\end{definition}
Let $e(\Bi)$ be the unique
crossingless, dotless diagram where the sequence at top and bottom are both $\Bi$.
\begin{definition} The {\bf degree} of a KLRW diagram is the sum over
crossings and dots in the diagram of
\begin{itemize}
\item$-\langle\al_i,\al_j\rangle$ for each crossing of a black strand
labeled $i$ with one labeled $j$;
\item $\langle\al_i,\al_i\rangle=2$ for each dot on a black
strand labeled $i$;
\item $\langle\al_i,\la\rangle=\la^i$ for each crossing of a
black strand labeled $i$ with a red strand labeled $\la$.
\end{itemize}
The degree of diagrams is additive under composition. Thus, the
algebra ${\it \doubletilde{\mathbb{T}}}\xspace$ inherits a grading from this degree function.
\end{definition}
Consider the set $\Omega=\{(i,j) \mid i\in [1,\rankp], j\in [1,v_i]\}$.
Let $\prec$ be a total order on $\Omega$ such that
\begin{equation}
(i,1)\preceq \cdots \preceq (i,v_i). \label{eq:order-1}
\end{equation}
This is equivalent to choosing a word $\Bi =(i_1,\dots, i_{N})$ where where $N=|\Omega|$ and $i_k=i$ for $v_i$ different indices $k$.
We will want to weaken this definition a bit and allow $\preceq$ to be a total preorder (that is, a relation which is transitive and reflexive, but not necessarily anti-symmetric). In this case, we have an induced equivalence relation $(i,k)\approx (j,\ell)$ if $(i,k)\preceq (j,\ell)$ and $(i,k)\succeq (j,\ell)$. We assume that our preorder satisfies the condition that
\begin{equation}
(i,k)\not\approx (j,\ell)\text{ whenever } i\neq j.\label{eq:order-2}
\end{equation}
We can still attach a word $\Bi$ to such a preorder; two equivalent elements give the same letter in the word $\Bi$, so it doesn't matter whether they have a chosen order. We can thus think of a preorder as corresponding to a word in the generators with some subsets where the same letter appears multiple times together grouped together. We can represent this within the word itself by replacing $(i,\dots, i)$ with $i^{(a)}$. Thus, for our purposes $(3,2,2,3,1,3)$ and $(3,2^{(2)},3,1,3)$ are different words with different associated preorders. Every such word has a unique {\bf totalization} satisfying \eqref{eq:order-1}.
\begin{definition}
We say the preorder $\prec$ and word $\Bi$ is $\chi$-parabolic if whenever $\chi_k=\chi_{k+1}$ then $(\rankp,k)\approx (\rankp,k+1)$. In particular, the corresponding appearances of $\rankp$ in $\Bi$ are consecutive.
\end{definition}
\begin{definition}
Let $\mathbb{\tilde{T}}$ be the quotient of ${\it \doubletilde{\mathbb{T}}}\xspace$ by
the following local
relations between KLRW diagrams. We draw these below as black, but the same relations apply to red strands (always taken with the label $\rankp$):
\begin{equation*}\subeqn\label{first-QH}
\begin{tikzpicture}[scale=.7,baseline]
\draw[very thick,postaction={decorate,decoration={markings,
mark=at position .2 with {\arrow[scale=1.3]{<}}}}](-4,0) +(-1,-1) -- +(1,1) node[below,at start]
{$i$}; \draw[very thick,postaction={decorate,decoration={markings,
mark=at position .2 with {\arrow[scale=1.3]{<}}}}](-4,0) +(1,-1) -- +(-1,1) node[below,at
start] {$j$}; \fill (-4.5,.5) circle (3pt);
\node at (-2,0){=}; \draw[very thick,postaction={decorate,decoration={markings,
mark=at position .8 with {\arrow[scale=1.3]{<}}}}](0,0) +(-1,-1) -- +(1,1)
node[below,at start] {$i$}; \draw[very thick,postaction={decorate,decoration={markings,
mark=at position .8 with {\arrow[scale=1.3]{<}}}}](0,0) +(1,-1) --
+(-1,1) node[below,at start] {$j$}; \fill (.5,-.5) circle (3pt);
\end{tikzpicture}
\qquad
\begin{tikzpicture}[scale=.7,baseline]
\draw[very thick,postaction={decorate,decoration={markings,
mark=at position .2 with {\arrow[scale=1.3]{<}}}}](-4,0) +(-1,-1) -- +(1,1) node[below,at start]
{$i$}; \draw[very thick,postaction={decorate,decoration={markings,
mark=at position .2 with {\arrow[scale=1.3]{<}}}}](-4,0) +(1,-1) -- +(-1,1) node[below,at
start] {$j$}; \fill (-3.5,.5) circle (3pt);
\node at (-2,0){=}; \draw[very thick,postaction={decorate,decoration={markings,
mark=at position .8 with {\arrow[scale=1.3]{<}}}}](0,0) +(-1,-1) -- +(1,1)
node[below,at start] {$i$}; \draw[very thick,postaction={decorate,decoration={markings,
mark=at position .8 with {\arrow[scale=1.3]{<}}}}](0,0) +(1,-1) --
+(-1,1) node[below,at start] {$j$}; \fill (-.5,-.5) circle (3pt);
\node at (3.5,0){unless $i=j$};
\end{tikzpicture}
\end{equation*}
\begin{equation*}\subeqn\label{nilHecke-1}
\begin{tikzpicture}[scale=.8,baseline]
\draw[very thick,postaction={decorate,decoration={markings,
mark=at position .2 with {\arrow[scale=1.3]{<}}}}](-4,0) +(-1,-1) -- +(1,1) node[below,at start]
{$i$}; \draw[very thick,postaction={decorate,decoration={markings,
mark=at position .2 with {\arrow[scale=1.3]{<}}}}](-4,0) +(1,-1) -- +(-1,1) node[below,at
start] {$i$}; \fill (-4.5,.5) circle (3pt);
\node at (-2,0){$-$}; \draw[very thick,postaction={decorate,decoration={markings,
mark=at position .8 with {\arrow[scale=1.3]{<}}}}](0,0) +(-1,-1) -- +(1,1)
node[below,at start] {$i$}; \draw[very thick,postaction={decorate,decoration={markings,
mark=at position .8 with {\arrow[scale=1.3]{<}}}}](0,0) +(1,-1) --
+(-1,1) node[below,at start] {$i$}; \fill (.5,-.5) circle (3pt);
\node at (1.8,0){$=$};
\end{tikzpicture}\,\,
\begin{tikzpicture}[scale=.8,baseline]
\draw[very thick,postaction={decorate,decoration={markings,
mark=at position .8 with {\arrow[scale=1.3]{<}}}}](-4,0) +(-1,-1) -- +(1,1) node[below,at start]
{$i$}; \draw[very thick,postaction={decorate,decoration={markings,
mark=at position .8 with {\arrow[scale=1.3]{<}}}}](-4,0) +(1,-1) -- +(-1,1) node[below,at
start] {$i$}; \fill (-4.5,-.5) circle (3pt);
\node at (-2,0){$-$}; \draw[very thick,postaction={decorate,decoration={markings,
mark=at position .2 with {\arrow[scale=1.3]{<}}}}](0,0) +(-1,-1) -- +(1,1)
node[below,at start] {$i$}; \draw[very thick,postaction={decorate,decoration={markings,
mark=at position .2 with {\arrow[scale=1.3]{<}}}}](0,0) +(1,-1) --
+(-1,1) node[below,at start] {$i$}; \fill (.5,.5) circle (3pt);
\node at (2,0){$=$}; \draw[very thick,postaction={decorate,decoration={markings,
mark=at position .5 with {\arrow[scale=1.3]{<}}}}](4,0) +(-1,-1) -- +(-1,1)
node[below,at start] {$i$}; \draw[very thick,postaction={decorate,decoration={markings,
mark=at position .5 with {\arrow[scale=1.3]{<}}}}](4,0) +(0,-1) --
+(0,1) node[below,at start] {$i$};
\end{tikzpicture}
\end{equation*}
\begin{equation*}\subeqn\label{black-bigon}
\begin{tikzpicture}[very thick,scale=.8,baseline]
\draw[postaction={decorate,decoration={markings,
mark=at position .55 with {\arrow[scale=1.3]{<}}}}] (-2.8,0) +(0,-1) .. controls (-1.2,0) .. +(0,1)
node[below,at start]{$i$}; \draw[postaction={decorate,decoration={markings,
mark=at position .55 with {\arrow[scale=1.3]{<}}}}] (-1.2,0) +(0,-1) .. controls
(-2.8,0) .. +(0,1) node[below,at start]{$j$};
\end{tikzpicture}=\quad
\begin{cases}
0 & i=j\\
\begin{tikzpicture}[very thick,yscale=.6,xscale=.8,baseline=-3pt]
\draw[postaction={decorate,decoration={markings,
mark=at position .5 with {\arrow[scale=1.3]{<}}}}] (2,0) +(0,-1) -- +(0,1) node[below,at start]{$j$};
\draw[postaction={decorate,decoration={markings,
mark=at position .5 with {\arrow[scale=1.3]{<}}}}] (1,0) +(0,-1) -- +(0,1) node[below,at start]{$i$};
\end{tikzpicture} & i\neq j,j\pm 1\\
\begin{tikzpicture}[very thick,yscale=.6,xscale=.8,baseline=-3pt]
\draw[postaction={decorate,decoration={markings,
mark=at position .8 with {\arrow[scale=1.3]{<}}}}] (2,0) +(0,-1) -- +(0,1) node[below,at start]{$j$};
\draw[postaction={decorate,decoration={markings,
mark=at position .8 with {\arrow[scale=1.3]{<}}}}] (1,0) +(0,-1) -- +(0,1) node[below,at start]{$i$};\fill (2,0) circle (4pt);
\end{tikzpicture}-\begin{tikzpicture}[very thick,yscale=.6,xscale=.8,baseline=-3pt]
\draw[postaction={decorate,decoration={markings,
mark=at position .8 with {\arrow[scale=1.3]{<}}}}] (2,0) +(0,-1) -- +(0,1) node[below,at start]{$j$};
\draw[postaction={decorate,decoration={markings,
mark=at position .8 with {\arrow[scale=1.3]{<}}}}] (1,0) +(0,-1) -- +(0,1) node[below,at start]{$i$};\fill (1,0) circle (4pt);
\end{tikzpicture}& i=j-1\\
\begin{tikzpicture}[very thick,baseline=-3pt,yscale=.6,xscale=.8]
\draw[postaction={decorate,decoration={markings,
mark=at position .8 with {\arrow[scale=1.3]{<}}}}] (2,0) +(0,-1) -- +(0,1) node[below,at start]{$j$};
\draw[postaction={decorate,decoration={markings,
mark=at position .8 with {\arrow[scale=1.3]{<}}}}] (1,0) +(0,-1) -- +(0,1) node[below,at start]{$i$};\fill (1,0) circle (4pt);
\end{tikzpicture}-\begin{tikzpicture}[very thick,yscale=.6,xscale=.8,baseline=-3pt]
\draw[postaction={decorate,decoration={markings,
mark=at position .8 with {\arrow[scale=1.3]{<}}}}] (2,0) +(0,-1) -- +(0,1) node[below,at start]{$j$};
\draw[postaction={decorate,decoration={markings,
mark=at position .8 with {\arrow[scale=1.3]{<}}}}] (1,0) +(0,-1) -- +(0,1) node[below,at start]{$i$};\fill (2,0) circle (4pt);
\end{tikzpicture}& i=j+1
\end{cases}
\end{equation*}
\begin{equation*}\subeqn\label{triple-dumb}
\begin{tikzpicture}[very thick,scale=.8,baseline=-3pt]
\draw[postaction={decorate,decoration={markings,
mark=at position .2 with {\arrow[scale=1.3]{<}}}}] (-2,0) +(1,-1) -- +(-1,1) node[below,at start]{$k$}; \draw[postaction={decorate,decoration={markings,
mark=at position .8 with {\arrow[scale=1.3]{<}}}}]
(-2,0) +(-1,-1) -- +(1,1) node[below,at start]{$i$}; \draw[postaction={decorate,decoration={markings,
mark=at position .5 with {\arrow[scale=1.3]{<}}}}]
(-2,0) +(0,-1) .. controls (-3,0) .. +(0,1) node[below,at
start]{$j$}; \node at (-.5,0) {$-$}; \draw[postaction={decorate,decoration={markings,
mark=at position .8 with {\arrow[scale=1.3]{<}}}}] (1,0) +(1,-1) -- +(-1,1)
node[below,at start]{$k$}; \draw[postaction={decorate,decoration={markings,
mark=at position .2 with {\arrow[scale=1.3]{<}}}}] (1,0) +(-1,-1) -- +(1,1)
node[below,at start]{$i$}; \draw[postaction={decorate,decoration={markings,
mark=at position .5 with {\arrow[scale=1.3]{<}}}}] (1,0) +(0,-1) .. controls
(2,0) .. +(0,1) node[below,at start]{$j$}; \end{tikzpicture}=\quad
\begin{cases}
\begin{tikzpicture}[very thick,yscale=.6,xscale=.8,baseline=-3pt]
\draw[postaction={decorate,decoration={markings,
mark=at position .5 with {\arrow[scale=1.3]{<}}}}] (6.2,0)
+(1,-1) -- +(1,1) node[below,at start]{$k$}; \draw[postaction={decorate,decoration={markings,
mark=at position .5 with {\arrow[scale=1.3]{<}}}}] (6.2,0)
+(-1,-1) -- +(-1,1) node[below,at start]{$i$}; \draw[postaction={decorate,decoration={markings,
mark=at position .5 with {\arrow[scale=1.3]{<}}}}] (6.2,0)
+(0,-1) -- +(0,1) node[below,at
start]{$j$}; \end{tikzpicture}& i=k=j+1\\
-\begin{tikzpicture}[very thick,yscale=.6,xscale=.8,baseline]
\draw[postaction={decorate,decoration={markings,
mark=at position .5 with {\arrow[scale=1.3]{<}}}}] (6.2,0)
+(1,-1) -- +(1,1) node[below,at start]{$k$}; \draw[postaction={decorate,decoration={markings,
mark=at position .5 with {\arrow[scale=1.3]{<}}}}] (6.2,0)
+(-1,-1) -- +(-1,1) node[below,at start]{$i$}; \draw[postaction={decorate,decoration={markings,
mark=at position .5 with {\arrow[scale=1.3]{<}}}}] (6.2,0)
+(0,-1) -- +(0,1) node[below,at
start]{$j$}; \end{tikzpicture}& i=k=j-1\\
0& \text{otherwise}
\end{cases}
\end{equation*}
\end{definition}
Let $e_\chi$ be the idempotent given by summing $e(\Bi)$ for all $\chi$-parabolic total orders satisfying \eqref{eq:order-1}. For a non-total order satisfying \eqref{eq:order-1} and \eqref{eq:order-2}, we associate the ``divided power'' idempotent $e'(\Bi)$ which acts on each equivalence class of consecutive strands by a primitive idempotent in the nilHecke algebra (for example, that introduced in \cite[2.18]{KLMS}).
Let $S_\chi$ be the stabilizer of $\chi$ in $S_n$; this naturally acts on the subalgebra $e_\chi \mathbb{\tilde{T}} e_\chi$ by permuting groups of red strands (or equivalently, dots on those red strands).
\begin{definition}
Let $\mathbb{\tilde{T}}^\chi=(e_\chi \mathbb{\tilde{T}} e_{\chi})^{S_\chi}$ be the invariants of $S_\chi$ acting on the subalgebra $(e_\chi \mathbb{\tilde{T}} e_{\chi})^{S_\chi}$.
\end{definition}
This is a canonical deformation of the algebra $\tilde{T}^{\bla}$ of \cite{Webmerged} attached to the sequence of dominant weights $\bla=(g_1\omega_{\rankm},\dots, g_k\omega_{\rankm})$ where $g_1,\dots, g_k$ are the sizes of the blocks of consecutive equal entries in $\chi$, i.e. $S_\chi=S_{g_1}\times \cdots\times S_{g_k}$.
Note that this algebra breaks up into a sum of subalgebras where we fix the number of strands with each label; as usual, we let $v_i$ denote the number with label $i$. We'll be particularly interested in the case when $v_1\leq v_2\leq \dots \leq v_m$. This is the condition that the corresponding weight of $\mathfrak{sl}_m$ is dominant; in our usual correspondence, it corresponds to the $n$-tuple $\nu$ where $(1,\dots, 1,2,\dots, 2,\dots,)$ where $i$ appears $v_i-v_{i-1}$ times.
This same algebra is considered in \cite[\S 4]{KLSY} in the case $\rankp=2$ and denoted $W(\mathbf{g},v_1)$ (using our $g_*$ and $v_*$ as above).
From its realization as a weighted KLR algebra, the algebra $\mathbb{\tilde{T}}$ inherits a polynomial representation.
\begin{definition}\label{def:poly-rep}
The polynomial representation of $\mathbb{\tilde{T}}$ is the vector space
\begin{equation*}
\poly=\bigoplus_{\Bi}\K[Y_1, \dots, Y_{V}]e({\Bi}),
\end{equation*} with sum running over total orders on $\Omega$ satisfying \eqref{eq:order-1}. The action is given by the rules:
\begin{itemize}
\item $e({\Bi})$ acts by projection to the corresponding summand,
\item a dot on the $k$th strand from the left acts by multiplication by $Y_k$,
\item a crossing of the $k$th and $k+1$st strands with $\Bi$ at the bottom and $\Bi'$ at top acts by
\begin{itemize}
\item If $i_k=i_{k+1}$, the divided difference operator \[fe_{\Bi}\mapsto \frac{f^{(k,k+1)}-f}{Y_{k+1}-Y_k}e_{\Bi'}.\]
\item If $i_k+1=i_{k+1}$, the permutation $(k,k+1)$ followed by a multiplication \[fe_{\Bi}\mapsto (Y_{k+1}-Y_k) f^{(k,k+1)}e_{\Bi'}.\]
\item Otherwise, the permutation $(k,k+1)$ \[fe_{\Bi}\mapsto f^{(k,k+1)}e_{\Bi'}.\]
\end{itemize}
\end{itemize}
The polynomial representation $\poly^\chi$ for $\mathbb{\tilde{T}}^\chi$ is given by $(e_\chi \poly)^{S_\chi}$ where $S_\chi$ acts by permuting red dots as usual.
\end{definition}
\subsection{Violating quotients}
\begin{definition}
We call an idempotent $e(\Bi)$ {\bf violating} if $i_1\neq \rankp$; that is, if $(\rankp,1)$ is not minimal in $\prec$.
Let $\vT^\chi$ be the quotient of $\mathbb{\tilde{T}}^\chi$ by the 2-sided ideal generated by all violating idempotents.
\end{definition}
The algebra $\vT^\chi$ is not precisely the algebra $\bvT^\chi$ defined in \cite{Webmerged}, but a deformation of it which we've considered in several contexts, in particular, in \cite[\S 4]{Webunfurl}. This is flat, since it is a special case of deforming the polynomials defining the KLRW algebra (as discussed in \cite[Prop. 2.23]{WebwKLR}); in the case $\rankp=2$, this is the redotted algebra discussed by Khovanov-Sussan in \cite{KSred}. The algebra $\bvT^\chi$ is the quotient of $\vT^\chi$ by all red dots.
Since the red dots are central, and the polynomial ring is graded local, every gradable simple $\vT^\chi$-module factors through $\bvT^\chi$, and so the Grothendieck group of $\bvT^\chi\mmod$ agrees with the Grothendieck group of $\vT^\chi\wgmod$. From \cite[\S 4]{Webunfurl}, we have:
\begin{theorem}
The categories of $\vT^\chi$-modules and $\bvT^\chi$-modules are categorifications of $\Sym^{\mathbf{g}}(\C^m)$, with the categorical $\mathfrak{sl}_m$-action given by induction and restriction functors changing the number of black strands.
\end{theorem}
\subsection{Ladder bimodules}
In our notation, we identify the dominant weight $\chi$ with a weight of $\mathfrak{sl}_\infty$ by $\mu_{\chi}=\sum_{i=1}^n \epsilon_{\chi_i}$. This is an injective map, but is far from surjective, since it only hits weights where the coefficients of the $\ep_i$'s are positive and sum to $n$ (in the usual parlance, they are level $n$). In particular, \[\al_j^\vee(\mu_{\chi})=\#\{j\mid \chi_j=i+1\}-\#\{j \mid \chi_j=1\}.\]
\begin{definition}
Let $\chi^{+ i}$ denote $\chi$ with an entry $i$ increased to $i+1$ if such a dominant weight exists, and $\chi^{-i}$ denote $\chi$ with an entry $i+1$ decreased to $i$ if such a dominant weight exists. Let $\chi^{\pm i^a}$ be the result of doing this operation $a$ times.
\end{definition}
These operations are uniquely characterized by the fact that:
\begin{lemma}
If they exist, then $\mu_{\chi^{\pm i}}=\mu_{\chi}\pm\al_i$.
\end{lemma}
\begin{proof}
If $\chi^{+ i}$ exists, then for some $k$, we have $i=\chi_k<\chi_{k+1}$. In this case, the dominant weight \[\chi^{+ i}=\chi+\ep_k=(\chi_1,\dots, \chi_k+1,\dots,\chi_n)\] satisfies
\[\al_j^\vee(\chi+\ep_k)=\begin{cases} \al_j^\vee(\mu_{\chi})-1 & j=\chi_k\pm 1,\\
\al_j^\vee(\mu_{\chi})+2 & j=\chi_k,\\
\al_j^\vee(\mu_{\chi}) & \text{otherwise}
\end{cases}
\]
Thus, we have that $\mu_{\chi+\ep_k}=\mu_\chi+\al_{i}$ as desired. The second half of the result follows from the fact that $(\chi^{+ i})^{-i}=\chi$.
\end{proof}
Assume that $i$ appears at least $a$ times in $\chi$. Let $\chi'=\chi^{+i^a}$ be the dominant weight obtained by changing $a$ instances of $i$ to $i+1$. Let $S_{\chi,\chi'}=S_{\chi}\cap S_{\chi'}$.
\begin{definition}
The {\bf ladder bimodules} are the subspace $\mathbb{E}_i^{(a)}=(e_\chi \mathbb{\tilde{T}} e_{\chi'})^{S_{\chi,\chi'}}$ considered as a $\mathbb{\tilde{T}}^{\chi'}$-$\mathbb{\tilde{T}}^\chi$ bimodule, and similarly $\mathbb{F}_i^{(a)}=(e_{\chi'} \mathbb{\tilde{T}} e_\chi)^{S_{\chi,\chi'}}$ simply swaps the roles of $\chi$ and $\chi'$.
\end{definition}
These are generalizations of the ladder bimodules defined in \cite[\S 5.2]{KLSY}.
We draw these by pinching together the red strands in a single equivalence class into a single red strand. Thus, elements of this bimodule look like:
\begin{equation}
\tikz[baseline, yscale=1.2]{\draw[wei] (-2,-1) to[out=90,in=-90] (-2,1);\draw[very thick] (-1.5,-1)to[out=90,in=-90](.5,1);\draw[very thick] (-1,-1)to[out=90,in=-90] node [pos=.7,circle,fill=black,inner sep=2pt] {} (-1.5,1);\draw[very thick] (0,-1)--(0,1);\draw[very thick] (.5,-1)to[out=90,in=-90](1.5,1);
\draw[wei] (-.5,-.85) to[out=90,in=-90] node [pos=.7,circle,fill=red,inner sep=2pt] {} (1,.85) ;\draw[wei] (1,-1)--(1,1); \draw[wei] (-.5,-1)--(-.5,1);\draw[very thick] (1.5,-1)to[out=90,in=-90](-1,1);\draw[wei] (2,-1)--(2,1); }\label{eq:ladder}
\end{equation}
These bimodules have a ``representation'' as well. By a representation of a bimodule $B$ over algebras $A_1$ and $A_2$, we mean a representation of the Morita context:
\[A_B=\begin{bmatrix} A_1 & B \\ 0& A_2\end{bmatrix}\]
with the obvious matrix multiplication. That is a left module $V_i$ of $A_i$, and a bimodule map $B\to \Hom_{\K}(V_2, V_1)$. In our case, we will use the polynomial representations $\poly^\chi$ of $\mathbb{\tilde{T}}^\chi$. Diagrams other than than the split and join of red strands act by the formulas in Definition \ref{def:poly-rep}. The formulas for splitting and joining are the same as in \cite{SWschur}:
\begin{itemize}
\item split corresponds to the inclusion $\poly^\chi \hookrightarrow \poly^{\chi,\chi^{+i^{(a)}}}$, where the latter is the invariants under $S_{\chi,\chi^{+i^{(a)}}}\subset S_{\chi}$.
\item merge corresponds to the divided difference operator $\poly^{\chi,\chi^{+i^{(a)}}}\to \poly^{\chi^{+i^{(a)}}}$ given by
\[f\mapsto \sum_{\sigma\in S_{\chi^{+i^{(a)}}/S_{\chi,\chi^{+i^{(a)}}}}}\frac{\displaystyle f^{\sigma}}{\Delta_{+i^{(a)}}^\sigma},\]
where $\Delta_{+i^{(a)}}$ is the product of $Y_k-Y_\ell$ where $k$ ranges over the red strands with $\la_{n,j}=i+1$, and $\ell$ over the red strands in the ``rung'' of the ladder. This corresponds to the operator in equivariant cohomology which integrates a $P_{\chi,\chi^{+i^{(a)}}}$-equivariant class over $P_{\chi^{+i^{(a)}}}/P_{\chi,\chi^{+i^{(a)}}}$ to give a $P_{\chi^{+i^{(a)}}}$-equivariant class. In terms of the of the nilHecke algebra, this corresponds to the diagram:
\[\tikz[very thick]{\draw (-3,-1) -- (1,1);\draw (-1,-1) -- (3,1);\draw (3,-1) -- (-1,1);\draw (1,-1) -- (-3,1); \node at (1.4,.7){$\cdots$};\node at (-1.4,.7){$\cdots$};\node at (1.4,-.7){$\cdots$};\node at (-1.4,-.7){$\cdots$};}\]
\end{itemize}
\begin{lemma}\label{lem:E-rep}
The formulas above define a representation of the bimodule $\mathbb{E}_i^{(a)}$.
\end{lemma}
\begin{proof}
Here, we use the fact that $\mathbb{E}_i$ by definition is the subbimodule of $e_{\chi^{+i^{(a)}}}\mathbb{\tilde{T}} e_{\chi}$ invariant under $S_{\chi,\chi^{+i^{(a)}}}$. This embedding corresponds to taking a diagram as in \eqref{eq:ladder}, and simply expanding red strands.
\begin{equation*}
\tikz[baseline, yscale=1.2]{\draw[wei] (-2,-1) to[out=90,in=-90] (-2,1);\draw[very thick] (-1.5,-1)to[out=90,in=-90](.5,1);\draw[very thick] (-1,-1)to[out=90,in=-90] node [pos=.7,circle,fill=black,inner sep=2pt] {} (-1.5,1);\draw[very thick] (0,-1)--(0,1);\draw[very thick] (.5,-1)to[out=90,in=-90](2,1);
\draw[wei] (-.25,-1) to[out=90,in=-90] node [pos=.7,circle,fill=red,inner sep=2pt] {} (1.25,1) ;\draw[wei] (1.5,-1)--(1.5,1); \draw[wei] (-.5,-1)--(-.5,1);\draw[very thick] (2,-1)to[out=90,in=-90](-1,1);\draw[wei] (2.5,-1)--(2.5,1); }
\end{equation*}
This does not precisely match the operators above, but it does after we add a crossing of the red strands that joined at the top:
\begin{equation*}
\tikz[baseline, yscale=1.2]{\draw[wei] (-2,-1) to[out=90,in=-90] (-2,1);\draw[very thick] (-1.5,-1)to[out=90,in=-90](.5,1);\draw[very thick] (-1,-1)to[out=90,in=-90] node [pos=.7,circle,fill=black,inner sep=2pt] {} (-1.5,1);\draw[very thick] (0,-1)--(0,1);\draw[very thick] (.5,-1)to[out=90,in=-90](2,1);
\draw[wei] (-.25,-1) to[out=90,in=-90] node [pos=.75,circle,fill=red,inner sep=2pt] {} (1.25,.65) to[out=90,in=-90] (1.5,1) ;\draw[wei] (1.5,-1)to[out=90,in=-90](1.5,.65) to[out=90,in=-90](1.25,1); \draw[wei] (-.5,-1)--(-.5,1);\draw[very thick] (2,-1)to[out=90,in=-90](-1,1);\draw[wei] (2.5,-1)--(2.5,1); }
\end{equation*}
The action of this in the usual polynomial representation of the KLR algebra of $A_{\rankp}$ matches the formulas we have given.
\end{proof}
\section{The classification of orbits}\label{sec:orbits}
\centerline{by Jerry Guan and Ben Webster}
\bigskip
In this appendix, we discuss the classification of orbits of $P_\chi$ on $V$, using the notation introduced earlier.
Recall the definition of a $\chi$-flavored multi-segment from Definition \ref{def:segments}.
A {\bf subsegment} of $\Omega$ is a subset $S$ such that $S=\{ (k,j_k),(k+1,j_{k+1}),\dots, (\ell,j_{\ell})\} $ for a segment $(k,k+1,\dots, \ell)$. A subsegment carries a canonical charge: if $\ell< \rankp$, this requires no new information, and if $\ell=\rankp$, then we use $\chi_{j_{\rankp}}$ as our flavor.
A {\bf segmentation} of the set $\Omega$ is a partition of $\Omega$ into subsegments. Every segmentation gives a choice of flavored multisegment.
\begin{definition}
For each segmentation $\Sigma$ of $\Omega$, there is a canonical map $f_{\Sigma}\colon \C^\Omega\to \C^{\Omega}$ sending $f(b_{k,j_k})=b_{k+1,j_{k+1}}$ when $(k,j_{k})$ and $(k+1,j_{k+1})$ lie in a subsegment together, and $f(b_{k,j})=0$ if there is no element $(k+1,j')$ in the same subsegment.
\end{definition}
\begin{lemma}\label{lem:orbits}
Every $G_\chi$-orbit on $V$ contains $f_{\Sigma}$ for some segmentation $\Sigma$, and $f_{\Sigma}$ and $f_{\Sigma'}$ are in the same orbit if and only if they have the same $\chi$-flavored multisegment. Thus, we have a bijection between $P_\chi$-orbits and $\chi$-flavored multi-segments with the corresponding dimension vector.
\end{lemma}
\begin{proof}
Obviously, if two segmentations $\Sigma, \Sigma'$ correspond to the same $\chi$-flavored multisegment, then there is a permutation $\sigma$ of $\Omega$ with $\sigma(\Sigma)=\Sigma'$ which preserves the 1st index, and such that if $\sigma((\rankp,i))=(\rankp,j)$ then $\chi_i=\chi_j$. The induced linear map $\C^{\Omega}\to \C^\Omega$ lies in $G_\chi$, and shows these are in the same orbit.
Now, we turn to showing that if $\Sigma$ and ${\Sigma'}$ are segmentations where $f_\Sigma$ and $f_{\Sigma'}$ are in the same orbit, then the corresponding $\chi$-flavored multisegment is the same.
We'll prove this by induction on $m$. As before, let $f_{i;j}=f_{i-1}\cdots f_j\colon \C^{v_j}\to \C^{v_i}$. Consider the map $f_{\rankp;1}\colon \C^{v_1}\to \C^{v_{\rankp}}$. The space $\C^{v_{\rankp}}$ has a unique finest partial flag invariant under $P_\chi$. This is of the form $0\subset F_{p_1}\subset \cdots \subset F_{p_\ell}$ where $p_k$ ranges over the integers which appear as values $\chi_i$, and $\dim F_{p_k}/F_{p_{k-1}}=g_k$, the number of indices $i$ such that $p_k=\chi_i$.
Thus, consider the flag \[0\subset \ker f_{\rankp;1}\subset f_{\rankp;1}^{-1}(F_{p_1})\subset f_{\rankp;1}^{-1}(F_{p_2})\subset \cdots\subset f_{\rankp;1}^{-1}(F_{p_\ell})\]
For $f_\Sigma$, obviously, $\dim f_{\rankp;1}^{-1}(F_{p_1})/\ker f_{\rankp;1}$ is the number of segments of the form $(1,\dots, \rankp)$ of charge $p_1$, and $\dim f_{\rankp;1}^{-1}(F_{p_k})/f_{\rankp;1}^{-1}(F_{p_{k-1}})$ is the number of charge $p_k$. Since these dimensions are the same on orbits of $G_\chi$, the segmentations $\Sigma$ and ${\Sigma'}$ must have the same number of segments of the form $(1,\dots, \rankp)$ of each given charge.
Similarly, the rank of the map $f_{\rankm;1}\colon \ker f_{\rankp;1}\subset \C^{v_1}\to \C^{v_{\rankm}}$ is the number of segments of the form $(1,\dots, \rankm),$ and more generally the rank of the map $f_{k;1}\colon \ker f_{k+1;1}\subset \C^{v_1}\to \C^{v_{k}}$ gives the number of segments of the form $(1,\dots, k)$.
This shows each flavored segment containing a 1 appears with the same multiplicity in the multisegments for $\Sigma$ and $\Sigma'$. Thus, if $w,w'\subset \C^{\Omega}$ are the subrepresentations generated by $\C^{v_1}$ under $f_\Sigma$ and $f_{\Sigma'}$, the induced representations on $\C^{\Omega}/W$ and $\C^{\Omega}/W'$ are of the form $f_{\bar{\Sigma}}$ and $f_{\bar{\Sigma}'}$ for $\bar{\Sigma}$ and $\bar{\Sigma}'$ segmentations obtained by throwing out all subsegments containing a $(1,k)$ (and reindexing by decreasing all indices). By induction, $\bar{\Sigma}$ and $\bar{\Sigma}'$ have the same flavored multisegment, so the same is true of $\Sigma$ and $\Sigma'$.
This shows that associated to each flavored multi-segment with the right dimension vector, we have a unique orbit, and these are all distinct. Now, we just need to show that every orbit has this form.
Consider the flag
\begin{equation*}
0\subset\ker f_{2;1}\subset \cdots \subset \ker f_{\rankp;1}\subset f_{\rankp;1}^{-1}(F_{p_1}) \subset f_{\rankp;1}^{-1}(F_{p_2})\subset \cdots\subset f_{\rankp;1}^{-1}(F_{p_\ell})\subset \C^{v_1}.
\end{equation*}
Choose a ordered basis of $\C^{v_1}$ compatible with this flag, and let $g_1\in GL(\C^{v_1})$ be a matrix mapping this basis to $b_{1,1},\dots, b_{1,v_1}.$ By construction, the non-zero images of this set under $f_{k;1}$ form a linearly independent set, which span $\operatorname{image}(f_{k;1})$ compatibly with the intersection with the flag
\begin{equation}
0\subset\ker f_{k+1;k}\subset \cdots \subset \ker f_{\rankp;k}\subset f_{\rankp;k}^{-1}(F_{p_1})\subset \cdots\subset f_{\rankp;k}^{-1}(F_{p_\ell})\subset \C^{v_{k}}.\label{eq:big-flag}
\end{equation}
Thus, we can choose a basis of $\ker f_{3;2}\subset \C^{v_2}$ whose union with the non-zero images $f(g_1^{-1}b_{1,*})$ is still linearly independent; we then in turn extend this in turn to bases of \[\ker f_{4;2}\subset \ker f_{5;2}\subset \dots f_{\rankp;2}^{-1}(F_{p_1})\subset \dots\subset f_{\rankp;2}^{-1}(F_{p_\ell}),\] to obtain a basis containing all non-zero vectors of the form $f_{1}(g_1^{-1}b_{1,*})$ which is compatible with the flag \eqref{eq:big-flag} in the case $k=2$. Let $g_2$ be a matrix mapping this basis to $b_{2,*}$.
Now, apply the same process in $\C^{v_3}$, to construct a basis compatible with the flag \eqref{eq:big-flag} in the case $k=3$, and so on to construct such a basis in $\C^{v_{k+1}}$ compatible with $f_k(g_k^{-1}b_{k,*})$ and the flag \eqref{eq:big-flag} in the general case, with $g_{k+1}$ mapping this to the standard basis.
Note that in $\C^{v_{\rankp}}$, this just gives a basis compatible with $0\subset F_{p_1}\subset \cdots \subset F_{p_\ell}= \C^{v_{\rankp}}$; whereas for previous bases, we have been able to choose any bijection of our basis to the standard basis, we have to assure that $g_{\rankp}\in P_\chi$, which is of course possible since our basis is compatible with the partial flag.
Thus, the bases $\{g_k^{-1}b_{k,*}\}$ satisfy the property that $f$ sends each of these basis vectors to another basis vector or to 0. This shows that $(g_1^{-1},\dots, g_{\rankp}^{-1})\cdot f=f_{\Sigma}$ for a segmentation $\Sigma.$ This completes the proof.
\end{proof}
\begin{lemma}\label{lem:simply-connected}
Each of these orbits is $P_\chi$-equivariantly simply connected.
\end{lemma}
\begin{proof}
To show this, we need only show that the stabilizer of $f_{\Sigma}$ in $P_\chi$ is simply connected.
We can think about this stabilizer as follows. The element $g_1$ acting on $\C^{v_1}$ must preserve the flag \eqref{eq:big-flag}, but otherwise can be chosen freely. This is a connected parabolic subgroup. This choice determines how $g_2$ acts on the image of $f$ in $\C^{v_2}$, but on the complement $U$ to this image (given by the span of the appropriate vectors) we can freely choose an isomorphism of $U$ to itself compatible with the intersection with the flag \eqref{eq:big-flag} (again a connected parabolic), and any map $U \to \operatorname{image}(f)$ compatible with this flag (a contractible set).
Applying this inductively, we obtain that the stabilizer is topologically a product of these parabolics (always connected) with affine spaces. Paying attention to the group structure, we have a product of general linear groups (with ranks given by the multiplicities with which given flavored segments appear) extended by a unipotent radical. Thus, as a quotient the orbit is homotopic to a product of classifying spaces $BGL_{*}$ and in particular is simply connected.
\end{proof}
\section{Gelfand-Tsetlin modules}
Let $U=U(\mathfrak{gl}_n)$. As mentioned in the introduction, we let $\Gamma\subset U(\mathfrak{gl}_n)$ be the {\bf Gelfand-Tsetlin subalgebra} generated by the centers $Z_k=Z(U(\mathfrak{gl}_k))$ where $\mathfrak{gl}_k\subset \mathfrak{gl}_n$ is embedded as the top left corner. In this section, we'll discuss how $U(\mathfrak{gl}_n)$ and its representation theory relate to the algebras $\mathbb{\tilde{T}}^\chi$ in the case where $v_i=i$ for $i=1,\dots, n$. In this case, $\Omega=\{(i,j) \mid 1\leq j\leq i\leq n\}$.
For the dominant weight $\chi$, we have a maximal ideal $\mathfrak{m}_\chi$ of $Z_n $ defined by the kernel of the action on the Verma module with highest weight
\[\chi-\rho=(\chi_1-1,\dots, \chi_n-n).\]
\begin{definition}
A {\bf Gelfand-Tsetlin module} is a finitely generated $U(\mathfrak{gl}_n)$-module on which the action of $\Gamma$ is locally finite.
\end{definition}
Of course, by standard commutative algebra, if $M$ is a Gelfand-Tsetlin module then $M$ breaks up as a direct sum over the maximal ideals in $\MaxSpec(\Gamma)$. Every maximal ideal of $\Gamma$ is generated by maximal ideals in $Z_k$ for each $k$, which we index with an unordered $k$-tuple $\la_k=(\la_{k,1},\dots, \la_{k,k})$; as above, we match this with the maximal ideal in $Z_k$ acts trivially on the Verma module over $U(\mathfrak{gl}_k)$ with highest weight $\la_k-\rho_k=(\la_{k,1}-1,\dots, \la_{k,k}-k).$ Let $\mGT_{\la}\subset \Gamma$ be the corresponding maximal ideal.
Thus, for any Gelfand-Tsetlin module, we have a decomposition \[M=\bigoplus_{\la \in \MaxSpec(\Gamma)}\Wei_\la(M)\hspace{5mm}\text{ where }\hspace{5mm}
\Wei_\la(M)=\{m\in M\mid \mGT_\la^Nm=0 \text{ for } N\gg 0\}.\]
\begin{remark}
To help the reader fix notation in their mind, this means that the maximal ideals that appear in finite dimensional modules with $\chi\in \Z^n$ are given by $\la_k\in \Z^k$ with $\la_n=\chi$ that satisfy
\begin{equation}
\la_{k+1,1}\leq \la_{k,1}< \la_{k+1,2}\leq \la_{k,2}< \cdots <\la_{k+1,k}\leq \la_{k,k}<\la_{k+1,k+1} \label{eq:GT}
\end{equation} for $k=1,\dots, n-1$. Readers will recognize this as the condition that $\la_k-\rho_k$ form a Gelfand-Tsetlin pattern. Under this correspondence, the trivial module gives $\la_k=(1,2,\dots, k).$
\end{remark}
\begin{definition}
Let $\MaxSpec_{\Z}(\Gamma)\subset \MaxSpec(\Gamma)$ be the subset where $\la_{k,i}\in \Z$ for all $(k,i)\in \Omega$, and $\MaxSpec_{\Z,\chi}(\Gamma)\subset\MaxSpec_{\Z}(\Gamma)$ the subset where $\la_{n,i}=\chi_i$ for all $i$. An {\bf integral Gelfand-Tsetlin module} is one where $\Wei_\la(M)=0$ if $\la\notin\MaxSpec_{\Z}(\Gamma)$.
Let $\mathcal{GT}_\chi$ be the category of integral Gelfand-Tsetlin modules over $U(\mathfrak{gl}_n)$ on which the ideal $\mathfrak{m}_\chi\subset Z_n$ acts nilpotently, i.e. those where $\Wei_\la(M)=0$ if $\la\notin\MaxSpec_{\Z,\chi}(\Gamma)$.
\end{definition}
\begin{lemma}\label{lem:GT-action}
Let $V,M$ be $U(\mathfrak{gl}_n)$-modules with $M$ Gelfand-Tsetlin and $V$ finite dimensional. The module $V\otimes M$ is Gelfand-Tsetlin, and if $M\in \mathcal{GT}_\chi$ then $V\otimes M$ is a sum of objects in $\mathcal{GT}_{\chi+\mu}$ for $\mu$ a weight of $V$.
\end{lemma}
\begin{proof}
Since $\Gamma$ is commutative, it is enough to show local finiteness separately under a set of generating subalgebras. That is, it suffices to check that $V\otimes M$ is locally finite under $Z(U(\mathfrak{gl}_k))$ for all $k\leq n$. This follows from \cite[Cor. 2.6(ii)]{BeGe}, which shows that $V\otimes M$ is locally finite under $Z(U(\mathfrak{gl}_k))$ whenever $M$ is locally finite under $Z_k$.
By \cite[Thm. 2.5(ii)]{BeGe}, the set of modules with integral central character is closed under tensor product, so if $M$ is integral Gelfand-Tsetlin, then $V\otimes M$ is as well. Finally, applying \cite[Thm. 2.5(ii)]{BeGe} again implies the statement on characters under $Z_n$.
\end{proof}
In particular, we have functors $E(M)=\C^n\otimes M$ and $F(M)=(\C^n)^*\otimes M$ on the category of Gelfand-Tsetlin modules with integral central character $\chi\in \Z^n$.
Lemma \ref{lem:GT-action} shows that:
\begin{corollary}
If $M\in \mathcal{GT}_\chi$, then $E(M)$ is a sum of objects in $\mathcal{GT}_{\chi^{+i}}$ for $i\in \{\chi_1,\dots, \chi_n\}$ and $F(M)$ is a sum of objects in $\mathcal{GT}_{\chi^{-i}}$ for $i\in \{\chi_1-1,\dots, \chi_n-1\}$
\end{corollary}
Since decomposing according to the spectrum of $Z_n$ is functorial, we can define functors $E_i$ (resp.\ $F_i$) such that for $M\in\mathcal{GT}_{\chi}$, we have that $E_i(M)\subset E(M)$ (resp. $F_i(M)\subset F(M)$) is the unique maximal summand that lies in the category $\mathcal{GT}_{\chi^{\pm i}}$. These are also the generalized weight spaces of the Casimir operator on $V\otimes M$ (this follows, for example, by \cite[Lem. 4.5]{BSW3}). This gives a decomposition of these functors acting on integral Gelfand-Tsetlin modules
\[E=\bigoplus_{i\in \Z} E_i\qquad F=\bigoplus_{i\in \Z} F_i.\]
It is well-known that the functors $E_i$ define categorical actions (for example, see the discussion of category $\mathcal{O}$ in \cite[\S 7.5]{CR04}) on various categories of $\mathfrak{gl}_n$, but due to artifacts of the proofs, it is not obvious that these apply to the category of Gelfand-Tsetlin modules. Recent work of Brundan, Savage and the author shows how to unify these proofs:
\begin{lemma}
The functors $E$ and $F$ define an action of the {\bf level 0 Heisenberg category} of \cite{BruHei} (also called the {\bf affine oriented Brauer category} in \cite{BCNR}). The functors $E_i$ and $F_i$ for $i\in \Z$ give an induced categorical Kac-Moody action of $\mathfrak{sl}_\infty$, and these actions are related by \cite[Th. A]{BSW3}.
\end{lemma}
Note that the formulas of \cite[Th. 4.11]{BSW3} give explicit formulas for the action of $\tU$ in terms of swapping factors in tensor products and the Casimir element, but these formulas are not needed for our purposes.
Let us note another perspective on this result. Recall, that we call a $U$-$U$ bimodule {\bf Harish-Chandra} if it is locally finite for the adjoint action and {\bf pro-Harish-Chandra} if it satisfies this property in the topological sense; obviously the bimodule $V\otimes U$ inducing tensor product has this property. Thus, the functor $E_i\colon \mathcal{GT}_{\chi}\to \mathcal{GT}_{\chi^{+i}}$ is given by tensor product with a pro-Harish-Chandra $U$-$U$ bimodule $\mathscr{E}_i$, formed by completing the left and right actions of $Z_n$ which respect to the appropriate maximal ideal. Thus if we let $\mathsf{HC}$ be the 2-category such that
\begin{itemize}
\item objects are dominant integral weights $\chi=(\chi_1\leq \cdots \leq \chi_n)$,
\item 1-morphisms $\chi\to \chi'$ are sums of pro-Harish-Chandra bimodules where the maximal ideal $\mathfrak{m}_{\chi'}$ acts topologically nilpotently on the left and $\mathfrak{m}_\chi$ acts topologically nilpotently on the right.
\item 2-morphisms are homomorphisms of bimodules.
\end{itemize}
then \cite[Th. A]{BSW3} implies that we have a 2-functor $\tU^*\to \mathsf{HC}$ sending $\EuScript{E}_i\mapsto \mathscr{E}_i$ and similarly with $\mathscr{F}$. Note that Soergel sketches the construction of a functor $\mathsf{Flag}\to \mathsf{HC}$ which can be used to relate this to Khovanov and Lauda's action in the final paragraph of \cite{Soe92}.
\subsection{Presentation of the category \texorpdfstring{$\mathcal{GT}_{\chi}$}{GT}}
A choice of integral $\la\in \MaxSpec_{\Z,\chi}(\Gamma)$ can be used to give a total preorder $\prec$ on the set $\Omega$ as in previous sections for the dimension vector $v_i=i$ for $\rankp=n$. We set $(i,k) \preceq (j,\ell)$ if and only if $\la_{i,k}< \la_{j,\ell}$ or $\la_{i,k}= \la_{j,\ell}$ and $i\geq j$. Note that the resulting preorder satisfies \eqref{eq:order-1} by assumption and \eqref{eq:order-2} since if $(i,k) \approx (j,\ell)$, then we must have $\la_{i,k}=\la_{j,\ell}$ and $i=j$.
\begin{definition}
We let $\Bi(\la)$ define the corresponding word in $[1,m+1]$ and $e'(\la)$ be the corresponding idempotent in $\mathbb{\tilde{T}}^\chi$.
\end{definition}
\begin{example}
The action of $\Gamma$ on the trivial representation gives the weight $\la_{i,k}=k$. Thus, the resulting word is
\[(n,n-1,\dots, 2,1,n, n-1, \dots, 3,2, n,n-1,\dots, 3, \dots n,n-1,n).\]
As mentioned before, every Gelfand-Tsetlin pattern gives an ordering that differs from this one by neutral swaps.
On the other hand, a weight like $\la_3=(1,2,3), \la_2=(4,4), \la_1=(1)$ gives $(3,1,3,3,2^{(2)})$, whereas if $\la_2=(4,5)$, then we have $(3,1,3,3,2,2).$ This is an example of a maximal ideal of the Gelfand-Tsetlin subalgebra that appears in no finite-dimensional module.
\end{example}
Note that \cite[Prop. 5.4]{WebGT} shows that any maximal ideals giving the same preorder on $\Omega$ give isomorphic functors on $\mathcal{GT}_\chi$.
If we change the order by swapping pairs $(i,k) $ and $(j,\ell)$ this is called a {\bf neutral swap}, and more generally neutral swaps don't change the functor given by two maximal ideals. For example, two Gelfand-Tsetlin patterns with the same $\chi$ don't necessarily give the same order, but they differ by neutral swaps, and indeed they give isomorphic functors.
Using the techniques of \cite{WebGT, WebSD, KTWWYO}, we can give an algebraic presentation of the category $\mathcal{GT}_{\chi}$. This is accomplished by presenting the algebra of natural transformations of the functors $\Wei_\la.$
Recall that $\widehat{\mathbb{\tilde{T}}}_\chi$ is the completion of this algebra with respect to its grading. See \cite[\S 2]{WebBKnote} for more discussion of the induced topology (in particular, the continuity of multiplication in it).
\begin{theorem}\label{thm:GT-iso}
We have an isomorphism compatible with multiplication
\[e'(\la') \widehat{\mathbb{\tilde{T}}}_\chi e'(\la)\cong \Hom(\Wei_\la,\Wei_{\la'}).\]
\end{theorem}
This is extremely closely related to \cite[Thm. 5.2]{KTWWYO}, but slightly different, since that result is for Gelfand-Tsetlin modules with honest central character, which corresponds in $\widehat{\mathbb{\tilde{T}}}_\chi $ to setting red dots to 0. Under this isomorphism, the red dots give the nilpotent parts of certain (complicated) elements of the center of $U(\mathfrak{gl}_n).$ In \cite[Lem. 3.3 \& Prop. 3.4]{SilverthorneW}, we give a more direct algebraic proof of this theorem.
\begin{proof}
This follows from \cite[Thm. 4.4]{WebGT}. The space $\Hom(\Wei_\la,\Wei_{\la'})$ is exactly the bimodule ${}_{\la'}\hat{U}_{\la}=\varprojlim U/(\mGT_{\la'}^NU+U\mGT^N_{\la})$, and by \cite[(4.5)]{WebGT}, this is exactly $\Ext(\perv_{\Bi(\la)},\perv_{\Bi(\la')}).$ On the other hand, by Theorem \ref{thm:T-iso}, this is the same as $e'(\la') \widehat{\mathbb{\tilde{T}}}_\chi e'(\la)$. Both of these isomorphisms are compatible with multiplication, so this completes the proof.
\end{proof}
\begin{definition}\label{def:wgmod}
Let $\mathbb{\tilde{T}}^\chi\wgmod$ be the category of finite-dimensional $\mathbb{\tilde{T}}^\chi$-modules $M$ which are weakly gradable, that is, $M$ has a finite filtration with gradable subquotients.
These are precisely the modules that extend to finite dimensional modules over the completion $\widehat{\mathbb{\tilde{T}}}_\chi$ with the discrete topology.
\end{definition}
\begin{corollary}\label{cor:GT-equiv}
We have an equivalence of categories $\boldsymbol{\Theta}\colon \mathcal{GT}_\chi\cong {\mathbb{\tilde{T}}}_\chi\wgmod$ such that $e(\la)\boldsymbol{\Theta}(M)=\Wei_{\la} (M)$ for all integral maximal ideals $\la\in \MaxSpec_{\Z,\chi} (\Gamma)$.
\end{corollary}
\begin{proof}
As $\la$ runs over $\MaxSpec_{\Z,\chi} (\Gamma)$, we only get finitely many different idempotents as $e(\la)$; if we let $\mathsf{S}$ be a set of maximal ideals that contains one representative of each idempotent, then this is {\bf complete} in the sense of \cite[Def. 2.24]{WebGT}. Let $\bar{e}$ be the sum of these idempotents. Then \cite[Def. 2.23]{WebGT} shows that the functor $\oplus_{\la\in \mathsf{S}}\Wei_{\la}$ is an equivalence of categories $\mathcal{GT}_\chi\cong \bar{e}\widehat{\mathbb{\tilde{T}}}_\chi\bar{e}\wgmod$ with the desired propeties. As argued in \cite[Prop. 5.12]{WebGT}, no simple object in $\widehat{\mathbb{\tilde{T}}}_\chi\mmod$ is killed by $\bar{e}$, so $\bar{e}$ defines a Morita equivalence giving the desired result.
Note that if $\chi_i \ll \chi_{i+1}$ whenever $\chi_{i}\neq \chi_{i+1}$, then $\bar{e}$ is the identity in $\mathbb{\tilde{T}}^\chi$ and this last step is not needed. The general case can be deduced from this one using translation functors, but we appeal to \cite[Prop. 5.12]{WebGT} since we have not yet discussed compatibility with translation functors.
\end{proof}
Together with Theorem \ref{thm:T-iso},
this establishes Theorem \ref{thm:main}.
For posterity, let us record how this equivalence records various properties of interest for Gelfand-Tsetlin modules:
\begin{lemma}
A Gelfand-Tsetlin module $M$ satisfies:
\begin{enumerate}
\item $U(\mathfrak{n})\subset U(\mathfrak{gl}_n)$ acts locally finitely if and only if $\Theta(M)$ factors through $\vT^{\chi}_\nu$.
\item the Gelfand-Tsetlin subalgebra acts semi-simply if and only if all dots, red and black, act trivially.
\item the Cartan $\mathfrak{h}$ acts semi-simply if and only if the sum $h_{i,1}$ of all dots on strands with label $i$ acts trivially, for all $i$.
\item the center $Z_n$ acts semi-simply if and only if all positive degree homogeneous polynomials in the red dots act trivially.
\end{enumerate}
\end{lemma}
In particular, the modules over $\mathbb{\tilde{T}}^\chi_\nu$ which factor through the quotient by all red dots and by violating strands is the category $\cO'_\chi$ of modules locally finite under $U(\mathfrak{n})$ and semi-simple for the action of the center with central character $\chi$; such a module is automatically a generalized weight module, but not necessarily an honest weight module, and so not necessarily in category $\cO$.
On the other hand, if we consider the quotient of $\vT^{\chi}_\nu$ by the symmetric polynomials $h_{i,1}$, modules over this algebra are the block $\cO_\chi$ of the usual category $\cO$. Note that we showed in \cite[Cor. 9.10]{Webmerged} that $\bvT^{\nu}_\chi\mmod$ was equivalent to $\cO_\chi$ with the translation functors matched with the induction-restriction action. Thus, this is related to our result using the Morita equivalence of Corollary \ref{cor:chi-nu} between $\bvT^{\nu}_\chi$ and this quotient of $\vT^\chi_\nu$.
\subsection{Translation functors}
Now, let us consider how this perspective interacts with translation functors. This requires studying the bimodule $\C^n\otimes U$ with the right action by right multiplication on the second factor, and the left action via the coproduct. That is, for $v\in \C^n,u\in U, X_1,X_2\in \mathfrak{gl}_n$, we have:
\[X_1\cdot (v\otimes u)\cdot X_2=X_1v\otimes uX_2+v\otimes X_1 u X_2.\]
\begin{lemma}
As a $U$-$U$-bimodule, $\C^n\otimes U$ is isomorphic to the sub-bimodule $U'$ in $U(\mathfrak{gl}_{n+1})$ generated by $e_n$.
\end{lemma}
\begin{proof}
Both these bimodules have an induced adjoint bimodule structure (which exists for any Hopf algebra); under this structure $\C^n\subset \C^n\otimes U$ is a submodule with the usual $\mathfrak{gl}_n$ module structure. Similarly, $e_n$ generates a copy of $\C^{n}$ inside the adjoint module over $\mathfrak{gl}_{n+1}$ restricted to $\mathfrak{gl}_n$. Sending $e_n\mapsto (0,\dots, 0,1)$ induces a unique isomorphism $\psi$ between these subspaces.
In any $U$-$U$ bimodule, a generating subspace for the bimodule action which is closed under the adjoint action is already generating for the right action. The module $\C^n\otimes U$ is freely generated by $\C^n$ as a right $U$ module by definition, and the same is true for $U'$ by the PBW theorem. Thus, $\psi$ canonically extends to a right module homomorphism, which is also a left-module homomorphism since $\psi$ is adjoint equivariant.
\end{proof}
Fix $\chi\in \MaxSpec(Z_n)$, and assume that $\chi^{+i}$ exists. Consider \[\la\in \MaxSpec_{\Z,\chi}(\Gamma)\qquad \la'\in \MaxSpec_{\Z,\chi^{+i}}(\Gamma).\]
\begin{theorem} We have an isomorphism compatible with bimodule structure:
\[e'(\la') \mathbb{E}_i e'(\la)=\varprojlim U'/(\mGT_{\la'}^NU'+U'\mGT_{\la}^N)\]
\end{theorem}
\begin{proof}
Note that $e_n$ commutes with $Z_k$ for $k<n$, embedded in $U(\mathfrak{gl}_{n+1})$ via the standard inclusion. Thus, $e_n$ only has non-zero image if $\la_{i,k}=\la'_{i,k}$ for all $i<n$. We therefore can describe the desired map by specifying the image of $e_n$ in this case.
As in \cite{WebSD, KTWWYO}, we can read this off by carefully studying the polynomial representation of $U(\mathfrak{gl}_{n+1})$, induced by its realization as an orthogonal Gelfand-Zetlin algebra\footnote{Apologies for the inconsistent spelling of \foreignlanguage{russian}{Цетлин. }We use a``Z'' here to match \cite{mazorchukOGZ}.}. This was originally given in \cite{mazorchukOGZ}, though it might be easier to follow the notation in \cite[\S 5.1]{WebGT}. Let $\Gamma_{(n+1)}$ be the Gelfand-Tsetlin subalgebra of $U(\mathfrak{gl}_{n+1})$, which we identify with the space of symmetric polynomials in the alphabets $\{x_{i,1},\dots, x_{i,i}\}$ for $i=1,\dots, i+1$.
Let $\varphi_{i,j}$ be the translation on all polynomials in these alphabets
satisfying \[\varphi_{i,j}(x_{k,\ell})=(x_{k,\ell}+\delta_{ik}\delta_{j\ell})
\varphi_{i,j}.\]
We then have a representation of $\mathfrak{gl}_{n+1}$ where $E_i$ and $F_i$ act by the operators
\[X^\pm_i=\mp\sum_{j=1}^{i}\frac{\displaystyle\prod_{k=1}^{v_{i\pm 1}}
(x_{i,j}-x_{i\pm 1,k})}{\displaystyle\prod_{k\neq j}
(x_{i,j}-x_{i,k})}\varphi_{i,j}^{\pm 1}\]
Thus, $U'$ embeds into the bimodule of operators $\Gamma_{(n+1)}$ when this space is made into a module over $U(\mathfrak{gl}_{n})$ as the subspace generated by the operator
\[X^+_n=-\sum_{j=1}^{n}\frac{\displaystyle\prod_{k=1}^{{n+ 1}}
(x_{n,j}-x_{n+ 1,k})}{\displaystyle\prod_{k\neq j}
(x_{n,j}-x_{n,k})}\varphi_{n,j}\]
Consider $P'=\varprojlim U'/U'\mGT_{\la}^N$. This is a left module over $U$, and clearly pro-Gelfand-Tsetlin, since for any element of $u\in U'$, $\Gamma u\Gamma$ lies in the left $\Gamma$ module generated by finitely many translations (this is a version of the Harish-Chandra property discussed in \cite{FOD}), meaning that $\Gamma u\Gamma\otimes_{\Gamma}\Gamma/\mGT_{\la}^N$ is finite-dimensional so $U'/U'\mGT_{\la}^N$ is Gelfand-Tsetlin. Thus $P'$ decomposes into pieces $P'=\oplus_{i\in \Z} P_i'$ on which $Z_n$ acts by $\chi^{+i}$ for $i$ ranging over the values of $\chi_g$; note that $P_i'=E_i(\varprojlim U/U\mGT_{\la}^N)$.
Consider the image of $X^+_n$ in $P'_i$. If we let $I_i=\{j\mid \la_{n,j}=i\}$, then we have that this image is given by
\[\sum_{j\in I_{i+1}} \frac{\displaystyle\prod_{k=1}^{{n+ 1}}
(x_{n,j}-x_{n+ 1,k})}{\displaystyle\prod_{k\in [1,n]\setminus \{j\}}
(x_{n,j}-x_{n,k})} \varphi_{n,j}\]
Note that if $k\notin I_{i+1}$ then the corresponding factor in the denominator is invertible in the local ring $\varprojlim \Gamma_{n}/\Gamma_{n}\mGT_{\la'}^N$, using the geometric series
\[\frac{1}{x_{n,j}-x_{n,k}}=\frac{1}{\la_{n,j}'-\la'_{n,k}}+\frac{x_{n,k}-\la'_{n,k}-x_{n,j}+\la'_{n,j}}{(\la_{n,j}'-\la'_{n,k})^2}+\frac{(x_{n,k}-\la'_{n,k}-x_{n,j}+\la'_{n,j})^2}{(\la_{n,j}'-\la'_{n,k})^3}+\cdots\]
Thus, we can write the image of $X^+_n$ as a sum of operators of the form \[X_{i}^{(p,s)}=-\sum_{j\in I_{i+1}} \frac{\displaystyle\prod_{k=1}^{{n+ 1}}
(x_{n,j}-x_{n+ 1,k})}{\displaystyle\prod_{p\in I_{i+1}\setminus \{j\}}
(x_{n,j}-x_{n,p})} (x_{n,j}-i-1)^sp(x_{n,1},\dots, x_{n,n})\varphi_{n,j}\] where $p$ is a polynomial symmetric in the alphabets $I_i$.
We can identify $p$ with a polynomial in the red dots at the bottom of the diagram, with the shifted variables $\{x_{n,j}-i\}_{j\in I_i}$ sent to the red dots on the corresponding strand with $i=\chi_j$.
We then send $X_{i}^{(p)}$ to the element of $\mathbb{E}_i$ where we place the polynomial $p$ applied to the red dots at the bottom of the diagram and ``sandwich'' $s$ red dots on the middle strand of the ladder, with all black strands are straight vertical at positions determined by $\la$. That is,
\begin{equation}
X_{i}^{(p,s)}\mapsto\,- \tikz[baseline, yscale=1.2]{\draw[wei] (-2,-1)--(-2,1);\draw[very thick] (-1.5,-1)--(-1.5,1);\draw[very thick] (-1,-1)--(-1,1);\draw[very thick] (0,-1)--(0,1);\draw[very thick] (.5,-1)--(.5,1);
\draw[wei] (-.5,-.35) to[out=90,in=-90] node [red, midway, circle, fill=red, inner sep=3pt,label=above:{$s$}]{} (1,.85);\draw[wei] (1,-1)--(1,1); \draw[wei] (-.5,-1)--(-.5,1);\draw[very thick] (1.5,-1)--(1.5,1);\draw[wei] (2,-1)--(2,1); \node[red,draw=black, fill=white, inner xsep=60pt] at (0,-.6){$p$};} \label{eq:XPS}
\end{equation}
We claim that this defines a map
\[\varprojlim U'/(\mGT_{\la'}^NU'+U'\mGT_{\la}^N)\to e'(\la') \mathbb{E}_i e'(\la)\]
First, let us match the completion $\K[[Y_1,\dots,Y_N]]e'(\la)$ with the completed polynomial ring $\varprojlim \Gamma/\Gamma\mGT_\la^N$ by matching $Y_r$ with $x_{p,j}-\la_{p,j}$ if the $r$th strand from the left is the $j$th (from the left) with label $p$; let us simplify notation by writing $Y_{p,j}:=Y_r$ in this case. Under this isomorphism, $X_i^{(p,s)}$ tranforms to
\[\mathscr{X}_{i}^{(p,s)}=-\sum_{j\in I_{i+1}} \frac{\displaystyle\prod_{k=1}^{{n+ 1}}
(Y_{n,j}+\la_{n,j}-x_{n+ 1,k})}{\displaystyle\prod_{p\in I_{i+1}\setminus \{j\}}
(Y_{n,j}-Y_{n,p})} (Y_{n,j})^sp(Y_{n,1}+\la_{n,1},\dots, Y_{n,n}+\la_{n,1})\varphi_{n,j}\]
This is exactly $C=\prod_{k=1}^{{n+ 1}}
(Y_{n,j}+\la_{n,j}-x_{n+ 1,k})$ times the action in the polynomial representation of $\mathbb{E}_i$ given the diagram on the left hand side of \eqref{eq:XPS} in Lemma \ref{lem:E-rep}. Multiplication by $C$ commutes with the left and right actions on this bimodule, so this shows that the representations of bimodules match.
This shows that \eqref{eq:XPS} defines a homomorphism, which must be injective by the faithfulness of both representations. Obviously, every bare ladder is in the image of this map, so it must be surjective as well.
\end{proof}
\begin{corollary}
The equivalence of Corollary \ref{cor:GT-equiv} intertwines the action of the translation functors $E_i$ on $\mathcal{GT}_\chi$ and the bimodules $\mathbb{E}_i$ on ${\mathbb{\tilde{T}}}_\chi\mmod$.
\end{corollary}
Together with Theorem \ref{thm:E-iso}, this establishes Theorem \ref{thm:action}, completing the main results of the paper.
\section{Introduction}
In this paper, we discuss 3 different perspectives on a category which has shown up in several guises in recent years. We can organize these perspectives as (1) diagrammatic, (2) representation-theoretic and (3) geometric.
For the diagrammatic perspective, work of Khovanov and Lauda \cite{KLI} initiated a great burst of different algebras spanned by diagrams with locally defined relations; while no rubric can contain all of this efflorescence, the author introduced {\bf weighted Khovanov-Lauda-Rouquier algebras} \cite{WebwKLR} which include the algebras discussed in this paper (and many others we will not consider here). The special case of interest to us was considered in recent work of Khovanov-Lauda-Sussan-Yonezawa \cite{KLSY}; in their terminology, this is a {\bf deformed Webster algebra}. In the spirit of compromise, we will follow the terminology suggested by our collaborators in \cite{KTWWYO}, and write {\bf KLRW algebras}.
As suggested by the title, we will focus on the case which is relevant to symmetric Howe duality; specifically, we consider the algebras that categorify $\mathfrak{sl}_\infty$-weight spaces of the symmetric power $U(\mathfrak{n})\otimes \Sym^n(\C^{\infty}\otimes \C^{\rankp})$, where $\mathfrak{n}\subset \mathfrak{sl}_{\rankp}$ is the Lie algebra of strictly upper-triangular matrices. We can identify the $\mathfrak{sl}_\infty$ weights appearing with increasing $n$-tuples $\chi=(\chi_1\leq \chi_2\leq \cdots \leq \chi_n)$, and we denote the algebra categorifying this weight space by $\mathbb{\tilde{T}}^\chi$; one can easily verify that as a $\mathfrak{sl}_m$-module, this weight space can be identified with $U(\mathfrak{n})\otimes \Sym^{g_1}(\C^m)\otimes \cdots \otimes \Sym^{g_k}(\C^m)$, where $g_*$ are multiplicities with which the distinct elements of $\chi$ repeat. The algebra $\mathbb{\tilde{T}}^\chi$ is a deformed version of the algebra $\tilde{T}^\bla$ as introduced in \cite[\S 4]{Webmerged} for this tensor product. In \cite{KLSY}, Khovanov-Lauda-Sussan-Yonezawa consider the case $\rankp=2$ and construct the Howe dual $\mathfrak{sl}_\infty$-action on the categories of $\mathbb{\tilde{T}}^\chi$-modules in this case. One of our goals is to generalize this result to general values of $\rankp$.
We achieve this by considering the other perspectives mentioned above, where the Howe dual actions are consequences of previously constructed $\mathfrak{sl}_\infty$-actions.
From the representation-theoretic perspective, we consider the category of {\bf Gelfand-Tsetlin modules} over $\mathfrak{gl}_n$. Recall that a Gelfand-Tsetlin module over $\mathfrak{gl}_n$ is one on which the center of $U(\mathfrak{gl}_k)$ for all $k\leq n$ acts locally finitely; we will also sometimes want to consider pro-Gelfand-Tsetlin modules, by which we mean topological modules where the action is only topologically locally finite. This category has received a great deal of interest in recent years \cite{FGRnew,RZ18, FGRZVerma} but its objects have remained relatively mysterious. Recent work of the author and collaborators \cite{KTWWYO,WebGT} gave a classification of the simples in a block of this category in general, but the combinatorics of the general case is somewhat complicated; some data on the complexity of the $\mathfrak{sl}_3$ and $\mathfrak{sl}_4$ case are presented in \cite{SilverthorneW}. One of our motivations in this paper is to draw out this structure of this category in the most interesting case, that of an integral character.
We'll show here (based on the techniques in \cite{WebGT}) that the algebras $\mathbb{\tilde{T}}^\chi$ attached to the zero weight space for the action of $\mathfrak{sl}_{\rankp}$ control the category $\mathcal{GT}_\chi$ of the category of integral Gelfand-Tsetlin modules where now we interpret $\chi$ as a central character of $Z_n=Z(U(\mathfrak{gl}_n))$.
These algebras also have a topological interpretation in terms of convolution algebras and perverse sheaves. We can view this as a
generalization of the well-known theorem of Beilinson-Ginzburg-Soergel which shows that the Koszul dual of an integral block $\mathcal{O}_\chi$ of deformed category $\mathcal{O}$ is the category of $P$-equivariant perverse sheaves on $GL_n/B$ where $P=P_\chi$ corresponds to the central character $\chi$ of the block. Our main theorem explains how to extend this to Gelfand-Tsetlin modules.
Consider the vector space $V$ defined by the set of quiver representations on the vector spaces $\C^1\overset{f_1}\to \C^2\overset{f_2}\to \cdots \overset{f_{n-2}}\to\C^{n-1}\overset{f_{n-1}}\to \C^{n}$ divided by the group $G$ that changes bases arbitrarily on $\C^1,\dots, \C^{n-1}$ and on $\C^{n}$ by elements of the group $P\subset GL_n$ (that is, preserving the standard partial flag corresponding to $P$). That is, \newseq
\begin{align*}
\label{eq:OGZ1}\subeqn G&=P\times GL_{{n-1}}\times \cdots \times GL_{1}\\
\label{eq:OGZ2}\subeqn V&=\Hom(\C^1,\C^2)\oplus \cdots \oplus
\Hom(\C^{n-2},\C^{n-1})\oplus \Hom(\C^{n-1},\C^n).
\end{align*}
with $G_0=GL_{{n-1}}\times \cdots \times GL_{1}$.
Note that on an open subset of $V$, the maps $f_i$ are all injective, and the subspaces $$\operatorname{im}(f_{n-1}\cdots f_1)\subset\cdots \subset\operatorname{im}(f_{n-1}f_{n-2})\subset\operatorname{im}(f_{n-1})\subset \C^n$$ give a complete flag. Thus, we can identify this open subset of $V/G_0$ with the flag variety $\Fl=GL_n/B$. In this paper, we will study $G$-equivariant sheaves on $V$ as an enlargement of the category of $P$-equivariant sheaves on $\Fl$.
We consider the usual $G_\chi$-equivariant derived category of $\C$-vector spaces $D^{b}_{G_\chi}(V)$, and its mixed\footnote{Readers expert in Hodge theory and the yoga of weights will rightly object that we are sweeping an enormous amount of detail under the rug here by just saying ``mixed.'' We will discuss this in a bit more detail later, but these techniques will not be used in any serious way; the only important fact we need is that the derived category has a graded lift which is compatible with the Hodge structure on the homology of algebraic varieties. Whether I do this by mixed Hodge modules or by reducing mod $p$ and using the action of the Frobenius is unimportant.} graded lift $D^{b,\operatorname{mix}}_{G_\chi}(V)$.
\begin{itheorem}\label{thm:main}
We have equivalences of categories between:
\begin{enumerate}
\item The category of weakly gradable finite dimensional $\mathbb{\tilde{T}}^\chi$-modules.
\item The category $\mathcal{GT}_\chi$ of integral Gelfand-Tsetlin modules.
\end{enumerate}
Thus, the category $\mathbb{\tilde{T}}^\chi\operatorname{-gmod}$ of all finitely generated graded $\mathbb{\tilde{T}}^\chi$-modules is a graded lift $\widetilde{\mathcal{GT}}_\chi$ of the category of pro-Gelfand-Tsetlin modules.
We also have an equivalence of categories between:
\begin{enumerate}
\item[(1')] The category of linear complexes of projectives over $\mathbb{\tilde{T}}^\chi$.
\item[(3)] The category of $P_\chi$-equivariant perverse sheaves on $V$.
\end{enumerate}
The categories $(1)$ and $(1')$ are in a certain sense Koszul dual, so the same is true of $(2)$ and $(3)$. These equivalences are induced by equivalences of derived categories
\[D^b(\widetilde{\mathcal{GT}}_\chi) \cong D^b(\mathbb{\tilde{T}}^\chi\operatorname{-gmod})\cong D^{b,\operatorname{mix}}_{G_\chi}(V).\]
\end{itheorem}
Furthermore, this equivalence matches two natural actions on these categories. As discussed previously, we can interpret $\chi$ as a weight of $\mathfrak{sl}_\infty$, and $K^0(\mathbb{\tilde{T}}^\chi )$ as a weight space of a $\mathfrak{sl}_\infty$-module. Thus, it's a natural question whether this can be extended to a categorical $\mathfrak{sl}_{\infty}$-action. Not only is this possible, but in fact, the resulting action is one already known in both the representation-theoretic and geometric perspectives.
For Gelfand-Tsetlin modules, this action is by translation functors. The functors of $\mathsf{E}(M)=\C^n\otimes M$ and $\mathsf{F}(M)=(\C^n)^*\otimes M$ act on the category of Gelfand-Tsetlin modules, and define an action of the level 0 Heisenberg category (also called the affine oriented Brauer category in \cite{BCNR}) on the category of all Gelfand-Tsetlin modules. These functors decompose according to how they act on blocks, and by \cite[Th. A]{BSW3}, the summands of this functor define a categorical $\mathfrak{sl_\infty}$ action on the sum of the integral blocks $\mathcal{GT}_\chi$.
On the other hand, as is always true for equivariant sheaves for different subgroups of a single group, the $P_\chi\times G_0$ equivariant derived categories of $V$ for different $\chi$ carry an action by convolution of the derived categories $D^b_{P_{\chi'}\times P_\chi} ( GL_n)$ where these subgroups act by left and right multiplication. This is essentially an action of the category $\mathsf{Perv}$ from \cite[Def. 5]{Webcomparison} with minor notational changes to account for working with $\mathfrak{sl}_\infty.$ There is a 2-functor $\Phi$ from the 2-Kac-Moody algebra for $\mathfrak{sl}_\infty$ to $\mathsf{Perv}$ introduced in \cite{Webcomparison}, uniquely characterized by the property that it agrees with Khovanov and Lauda's original functor from \cite{KLIII} to modules over the cohomology rings of $GL_n/P_\chi$.
Both these actions must have an algebraic description in terms of bimodules over $\mathbb{\tilde{T}}^{\chi'}$ and $\mathbb{\tilde{T}}^{\chi}$. In fact, the resulting bimodules are the {\bf ladder bimodules} defined in \cite{KLSY} in the case of $\mathfrak{sl}_2$. Our comparison of these with the geometric action gives an easy and conceptual proof of the fact that these bimodules induce a categorical $\mathfrak{sl}_\infty$ action in the case not just of $\mathfrak{sl}_2$ (as is shown in \cite{KLSY}), but in general on $\mathfrak{sl}_n$.
\begin{itheorem}\label{thm:action}
The equivalences of Theorem \ref{thm:main} match:
\begin{enumerate}
\item The categorical $\mathfrak{sl}_\infty$-action on $\oplus_\chi \mathbb{\tilde{T}}^\chi\mmod$ defined by ladder bimodules as in \cite{KLSY}.
\item The categorical $\mathfrak{sl}_\infty$-action on $\oplus_\chi \mathcal{GT}_\chi$ defined by translation functors.
\item The categorical $\mathfrak{sl}_\infty$-action on $\oplus_\chi D^{b,\operatorname{mix}}_{G_\chi}(V)$ defined by convolution with sheaves in $\mathsf{Perv}$.
\end{enumerate}
\end{itheorem}
The actions (1) and (3) make sense when we replace \eqref{eq:OGZ1} and \eqref{eq:OGZ2} with more general dimension vectors, as we'll discuss below, but (2) really depends on the identification with the universal enveloping algebra.
\excise{arbitrary dimension vectors, that is for
\[V=\Hom(\C^{v_1},\C^{v_2})\oplus \Hom(\C^{v_2},\C^{v_3})\oplus \cdots \Hom(\C^{v_{m-1}},\C^{v_{m}})\oplus \Hom(\C^{v_k},\C^n)\] for arbitrary $1\leq k\leq m$ and $(v_1,\dots, v_{m})$. This corresponds to the quiver gauge theory
\begin{equation*}
\tikz{
\node[draw, thick, circle, inner sep=5pt,fill=white] (a) at (-6,0) {$v_1$};
\node[draw, thick, circle, inner sep=5pt,fill=white] (b) at (-4,0) {$v_2$};
\node[draw, thick, circle, inner sep=5pt,fill=white] (d) at (0,0) {$v_k$};
\node[draw, thick, circle, inner sep=5pt,fill=white] (f) at (4,0) {$v_m$};
\node[draw, thick,inner sep=8pt] (s) at (0,2){$n$};
\node[inner sep=12pt,fill=white] (c) at (-2,0){$\cdots$};
\node[inner sep=12pt,fill=white] (e) at (2,0){$\cdots$};
\draw[thick] (a) -- (b) ;
\draw[thick] (b) -- (c) ;
\draw[thick] (c) -- (d) ;
\draw[thick] (d) -- (e) ;
\draw[thick] (e) -- (f) ;
\draw[thick] (d) -- (s) ;
}
\end{equation*}}
Theorems \ref{thm:main} and \ref{thm:action} extend essentially without change to the comparison of the equivariant derived category and $\mathbb{\tilde{T}}$, and the match of the categorical $\mathfrak{sl}_\infty$-actions.
The extension of the action (2) will require more effort, though it should possible in some cases where quantum Coulomb branch is a finite W-algebra using Brundan and Kleshchev's definition of translation functors for W-algebras in \cite[\S 4.4]{BKrsy}.
Finally, we discuss the slightly tangled relationship of this construction to previous work relating diagrammatic categories and the representation theory of Lie algebras.
\subsection*{Acknowledgments}
J. G. was supported by NSERC and the University of Waterloo through an Undergraduate Student Research Award. B. W. is supported by an NSERC Discovery Grant. This research was supported in part by Perimeter Institute for Theoretical Physics. Research at Perimeter Institute is supported by the Government of Canada through the Department of Innovation, Science and Economic Development Canada and by the Province of Ontario through the Ministry of Research, Innovation and Science.
\input{algebras}
\input{sheaves}
\input{GT}
\input{appendix}
\section{The geometry of quivers and perverse sheaves}
\subsection{Quiver representations} Throughout, we fix integers $\rankp,n$ and $\chi$ an integral weight $(\chi_1,\dots, \chi_n)\in \Z^n$ such that $\chi_1\leq \cdots \leq \chi_n$. We can think of this as giving a cocharacter into $GL_n$, and let $P_\chi\subset GL_n$ be the parabolic whose Lie algebra is the non-positive weight space for this cocharacter. This is the standard Borel if $\chi_i\neq \chi_j$ for all $i,j$, and in general is block upper-triangular matrices, with blocks corresponding the consecutive $\chi_i$ which are equal.
Fix a dimension vector $\Bv=(v_1,\dots, v_{\rankp-1})\in \Z_{\geq 0}^{\rankp-1}$; by convention, we take $v_{\rankp}=n$. Let
\[V=\Hom(\C^{v_1},\C^{v_2})\oplus \Hom(\C^{v_2},\C^{v_3})\oplus \cdots \oplus \Hom(\C^{v_{\rankp-2}},\C^{v_{\rankp-1}})\oplus \Hom(\C^{v_{\rankp-1}},\C^n)\] with the action of the group $GL_n$, and thus $P_\chi$ by post-composition and the natural action of $G_0=GL_{v_1}\times \cdots \times GL_{v_{\rankm}}$.
In the notation popular with physicists, this corresponds to the following quiver:
\begin{equation*}
\tikz{
\node[draw, thick, circle, inner sep=6pt,fill=white] (a) at (-6,0) {$v_1$};
\node[draw, thick, circle, inner sep=6pt,fill=white] (b) at (-4,0) {$v_2$};
\node[draw, thick, circle, inner sep=2pt,fill=white] (d) at (0,0) {$v_{\rankm}$};
\node[draw, thick,inner sep=12pt] (s) at (2,0){$n$};
\node[inner sep=12pt,fill=white] (c) at (-2,0){$\cdots$};
\draw[thick] (a) -- (b) ;
\draw[thick] (b) -- (c) ;
\draw[thick] (c) -- (d) ;
\draw[thick] (d) -- (s) ;
}
\end{equation*}
Attached to the space $V$ with the action of $G$, we have an equivariant derived category $D^b_G(V)$ as introduced in \cite{BL}; in modern terminology, we would think of this as the derived category of constructible sheaves on the quotient stack $V/G$. There are various other avatars of this category, such as strongly equivariant D-modules, but we will only use a few basic facts about this category, such as the decomposition theorem and the computation of Ext-algebras as Borel-Moore homology familiar from \cite{CG97}.
First, we simply need to classify the orbits of $P_\chi$ in $V$. Recall that a {\bf segment} in $[1,\rankp]$ is a list of consecutive integers $(k,k+1,\dots, \ell)$, and a multi-segment is a multi-set of segments. The {\bf dimension vector} of a segment is the vector $(0,\dots, 1,\dots, 1,\dots, 0)\in \Z^{\rankp}$ with 1 in every position in $[k,\ell]$ and 0 in all others, and the dimension vector of a multi-segment is the sum of those for the constituent segments. That is, it is the vector that records how many times an index $i\in [1,\rankp]$ appears in the constituent segments.
\begin{definition}\label{def:segments}
A flavored segment is a pair consisting of a segment and an integer $\beta\in \Z$. A $\chi$-flavored multi-segment is a multi-segment with a choice of flavoring on each segment with $\ell=\rankp$ (and {\it no} additional information about other segments) such that the flavors of the different segments agree with $\chi$ up to permutation.
We call the segments with $\ell=\rankp$ {\bf flavored} and those with $\ell<\rankp$ {\bf unflavored}.
\end{definition}
\begin{example}
If $n=2$ and $\rankp=2$ and $v_1=1$ then there are two multisegments with the correct dimension vector: $\{(1), (2),(2)\}$ and $\{(1,2), (2)\}$. There is only one way of flavoring $\{(1), (2),(2)\}$, mapping the two copies of $(2)$ to the two coordinates of $\chi$.
On the other hand, for $\{(1,2), (2)\}$, there are two different possible flavors, as long as $\chi_1\neq \chi_2$, depending on the bijection we choose between the sets $\{(1,2), (2)\}$ and $\{\chi_1,\chi_2\}$. If $\chi_1=\chi_2$, then we are back to having a single possible choice of flavor.
\end{example}
These are relevant because of the following fact from the appendix:
\begin{theorem}[Lemmata \ref{lem:orbits} \& \ref{lem:simply-connected}]
The $P_\chi$-orbits in $V$ are in bijection with $\chi$-flavored multi-segments with the corresponding dimension vector. Each of these orbits is equivariantly simply connected.
\end{theorem}
We view the subset $\{(i,1),\cdots,(i,v_i)\}\subset \Omega $ as corresponding to an ordered basis
$\{b_{(i,1)},\dots, b_{(i,v_i)}\}$ of $\C^{v_i}$. Giving this copy of $\C^{v_i}$ degree $i$, we can view
\[\C^\Omega \cong \C^{v_1}\oplus \cdots \oplus \C^{v_{\rankp}}\] as a graded vector space, and we can view a degree 1 map $f\colon \C^\Omega \to \C^\Omega $ as an element of $V$, that is, of quiver representation of $A_{\rankp}$ with dimension vector $(v_1,\dots, v_{\rankp})$.
\subsection{Quiver flag varieties and the equivariant derived category} As before, As before, consider a total preorder $\preceq$ satisfying \eqref{eq:order-1} and \eqref{eq:order-2}. This choice of preorder induces a flag $F^{\prec}_\bullet$, with each subspace given
\[F^{\prec}_{(i,k)}=\operatorname{span}(\{b_{(j,\ell)}\mid (j,\ell)\preceq (i,j)\}.\]
If the preorder is not an order, equivalent elements give the same subspace, so this flag will have some redundancies in it.
We say that a flag on $\C^\Omega $ indexed by the equivalence classes has {\bf type $\preceq$} or {\bf type $\Bi$} if it is conjugate to a flag of this form under $G_0$, and let $\Fl(\Bi)$ be the set of such flags. Note that $G_0$ acts on this space transitively, with the stabilizer of $F^{\prec}_\bullet$ given by a parabolic $P_0$, which only depends on the equivalence of $\preceq$; in particular, for any total order, we get the same Borel $B_0$. Consider the $G_0$-space
\[X(\Bi)=\{ (f,F_\bullet)\in V\times \Fl(\Bi) \mid f(F^{\prec}_{(i,k)})\subset F^{\prec}_{(i,k)}\}.\]
\begin{lemma}
If $\Bi$ is $\chi$-parabolic, then $X(\Bi)$ has an action of $P_\chi$ by the post-composition action on $V$ and the trivial action on $\Fl(\Bi)$. This commutes with the $G_0$-action, inducing a $P_\chi\times G_0$ action for which projection to $V$ is equivariant.
\end{lemma}
\begin{proof}
Assume that $\Bi$ is $\chi$-parabolic. Thus, we have that if $\chi_k=\chi_{k+1}$, then $F^{\prec}_{(\rankp,k+1)}=F^{\prec}_{(\rankp,k)}+\C\cdot b_{\rankp,k+1}$. By degree considerations, $f(b_{\rankp,k+1})=0$, so $f(F^{\prec}_{(\rankp,k)})\subset F^{\prec}_{(\rankp,k)}$, then automatically, we have
\[f(F^{\prec}_{(\rankp,k+1)})=f(F^{\prec}_{(\rankp,k)})\subset F^{\prec}_{(\rankp,k)}\subset F^{\prec}_{(\rankp,k+1)}.\]
Thus, for a $\chi$-parabolic flag, we only need to check that $f(F^{\prec}_{(i,k)})\subset F^{\prec}_{(i,k)}$ for $i<\rankp$ or when $i=\rankp$ and $\chi_k\neq \chi_{k-1}$.
Consider $(f,F_\bullet)\in X(\Bi)$ and $g\in P_\chi$. Consider the map $gf\colon \C^{\Omega} \to \C^{\Omega}$. This is again a quiver representation, which is compatible with the flag $gF^{\prec}_{(i,k)}$. Since $g\in P_\chi$, we have that $gF^{\prec}_{(i,k)}=F^{\prec}_{(i,k)}$ if $i<\rankp$ or if $i=\rankp$ and $\chi_k\neq \chi_{k-1}$. Thus, by our observation above, our original flag is still compatible with $gf$. The fact that this commutes with $G_0$ is clear.
\end{proof}
For each segment $(k,\dots, \ell)$,
let $\Bi_{(k,\dots, \ell)}=(\ell, \ell-1,\dots, k)$ be the word where we list the entries in reverse order.
\begin{definition}
For a $\chi$-flavored multi-segment $\mathbf{Q}$, the corresponding {\bf good word} $\Bi_{\bQ}$ is the result of concatenating \begin{enumerate}
\item the words for the unflavored segments in increasing lexicographic order (with the convention that attaching any suffix makes a word shorter).
\item the words for the flavored segments sorted first by the attached flavor (in increasing order) and then in decreasing order in the usual integers.
\end{enumerate}
\end{definition}
\begin{example}
Let $n=2,\rankp=3$ and consider the multi-segment \[\bQ=\{(1),(2),(2,1), (3,2,1) ,(3,2)\}.\] If $\chi_1>\chi_2$, a flavoring of this multi-segment is a bijection between the sets $\{(3,2,1) ,(3,2)\}$ and $\{\chi_1,\chi_2\}$. In this case, good words for this multi-segment with the flavorings that preserve and reverse the order we've written the sets above are:
\[(1,2,1,2,3,2,1,3,2)\qquad (1,2,1,2,3,2,3,2,1).\]
On the other hand, if $\chi_1=\chi_2$, there is only one possible flavoring with
\[(1,2,1,2,3,3,2,2,1).\]
\end{example}
\begin{lemma}
The good word $\Bi_{\bQ}$ is always $\chi$-parabolic, and the image of $X(\Bi_{\bQ})$ is precisely the closure of the corresponding $G_\chi$-orbit.
\end{lemma}
\begin{proof}
We prove this by induction on the number of segments. Note that since $X(\Bi)$ is irreducible, the same is true of its image in $V$, so its image is the closure of some orbit.
Consider the segment $(k,\dots, \ell)$ which appears first in the good word; if any unflavored segments appear in $\bQ$, then this will be unflavored. Assume for now that $\ell< \rankp$. This portion of the word gives a submodule $M\subset \C^\Omega$, and on an open subset of $X(\Bi)$, this submodule is the unique indecomposible module with this dimension vector. Also, by assumption, the quotient $\C^\Omega/M$ gives a point in $X(\Bi')$, the good word obtained by removing this segment. By induction, on an open subset of $X(\Bi)$, the quotient $\C^\Omega/M$ has the representation type given by $\bQ$ with this segment removed.
By the lexicographic condition $\Ext^1(\C^\Omega/M,M)=0$, so on the open set where both $M$ and $\C^\Omega/M$ have the correct representation type, we have a split extension, and the result follows.
We need to be a bit careful in the case of a flavored word with $\chi_1=\chi_{2}=\cdots =\chi_{p}$; in this case, we don't have a single segment appearing at the bottom, but rather a word of the form
$(\rankp,\dots, \rankp,\rankp-1,\dots, \rankp-1,\dots)$, which again has a corresponding subrepresentation $M$, which by construction satisfies $M\cap \C^n=\operatorname{span}(b_{\rankp,1},\dots, b_{\rankp,p})$. The generic representation with this dimension vector is lies in the orbit given by the corresponding multi-segment, and by induction, the same is true $\C^\Omega/M$. In particular, $\C^{\Omega}/M$ is compatible as desired with the action of $P_{\chi'}$ where $\chi'=(\chi_{p+1},\dots, \chi_n)$. As before, we have $\Ext^1(\C^\Omega/M,M)=0$, so generically on $X(\Bi)$, we have a split extension, and the desired generic representation type.
\end{proof}
Let $\pi^{\Bi}\colon X(\Bi)\to V$ be the projection map, and $\perv_{\Bi}=\pi_*^{\Bi}\C_{X(\Bi)}[\dim X(\Bi)]$.
\begin{lemma}
If $\Bi'$ is the totalization of $\Bi$, then there is a natural map $\phi\colon X(\Bi')\to X(\Bi)$ satisfying $\pi^{\Bi}\circ \phi =\pi^{\Bi'}$. The map $\phi$ is a fiber bundle with fiber given by a product of complete flag varieties.
Thus, $\perv_{\Bi'}$ is a sum of copies of shifts of $\perv_{\Bi}$.
\end{lemma}
\begin{proof}
The map $\phi$ is defined by forgetting the spaces attached to elements which are not maximal in their equivalence class in $\Bi$. This indeed gives an element of $X(\Bi)$ and a point in the fiber over a given point in $X(\Bi)$ is given by choosing a total flag in the subquotient of consecutive spaces in the flag. This has an induced grading, for which it is homogenous of a single degree. Thus, the induced action of $f$ is trivial for degree reasons, and any choice of total flag gives an element of $X(\Bi').$
Since the map $\phi$ is a fiber bundle whose fiber is a smooth projective variety, the pushforward $\phi_*\C_{X(\Bi')}[\dim X(\Bi')]$ is a sum of shifts of local systems by the Hodge theorem. The space $X(\Bi')$ is homotopy equivalent to $G_0/P_0$ which is simply connected, so these local systems are all trivial. This gives the result.
\end{proof}
Recall that for each $G_\chi$-orbit, there is a unique $G_\chi$-equivariant perverse sheaf $\mathbf{IC}_{\bQ}$ which extends the trivial local system on the orbit.
\begin{theorem}\label{thm:all-ICs}
Every simple $G_\chi$-equivariant perverse sheaf on $V$ is of the form $\mathbf{IC}_{\bQ}$ for the orbit of a flavored multi-segment with the trivial local system, and $\mathbf{IC}_{\bQ}$ is a summand of $\perv_{\Bi_{\bQ}}$ up to shift.
\end{theorem}
\begin{proof}
The fact that $\mathbf{IC}_{\bQ}$ is a complete list of simple perverse sheaves follows from the classification of orbits and their equivariant 1-connectedness.
Since $\pi^{\Bi}$ is a proper $G_\chi$-equivariant map with $X(\Bi)$ smooth, the decomposition theorem shows that $\perv_{\Bi}$ is a Verdier self-dual direct sum of shifts of perverse sheaves supported on the orbit closure $\overline{\mathbb{O}_{\bQ}}$.
At least one of these summands must have support precisely equal to $\overline{\mathbb{O}_{\bQ}}$, and $\mathbf{IC}_{\bQ}$ is the only such option. Thus, it must appear as a summand up to shift.
\end{proof}
Let a {\bf mixed structure} on a complex of sheaves $\mathcal{M}\in D^b_{G_\chi}(V)$ be a mixed Hodge structure on the D-module of which $\mathcal{M}$ is the solution sheaf. The pushforward $\perv_{\Bi}$ has a canonical mixed Hodge structure which is pure of weight 0, since $\pi^{\Bi}$ is proper. Let $D^{b,\operatorname{mix}}_{G_\chi}(V)$ be the derived category of complexes of sheaves with a mixed structure with a generated by Tate twists of $\perv_{\Bi}$. This is a graded lift of $D^b_{G_\chi}(V)$, with grading shift given by Tate twist.
Let $\perv_{\chi}=\oplus_{\Bi}\perv_{\Bi}$ where $\Bi$ ranges over totalizations of $\chi$-parabolic words and $X_{\chi}=\sqcup_{\Bi} X(\Bi)$. By \cite{CG97}, we have that
\[A_\chi=H^{BM,G_\chi}(X_{\chi}\times_V X_{\chi})\cong \Ext^\bullet(\perv_{\chi},\perv_{\chi})\] as formal dg-algebras. On the other hand, by \cite[Thm. 4.5]{WebwKLR}, we have that
\begin{theorem}\label{thm:T-iso}
For any preorders $\Bi,\Bj$ satisfying \eqref{eq:order-1} and \eqref{eq:order-2}, we have an isomorphism
\[\Ext^\bullet(\perv_{\Bi},\perv_{\Bj})\cong e'(\Bi)\mathbb{\tilde{T}}^\chi e'(\Bj)\]
matching convolution product with multiplication in $\mathbb{\tilde{T}}^\chi$, the red dots with the cohomology ring $H^*(BP_\chi)$ and the black dots with Chern classes of the tautological bundles on $X(\Bi)$.
In particular, we have an isomorphism $A_\chi\cong \mathbb{\tilde{T}}^\chi$, and the induced Hodge structure on $A_\chi$ is pure of weight 0.
\end{theorem}
It might seem strange that we used only honest orders, not preorders; this is needed to match the usual definition of $\mathbb{\tilde{T}}^\chi$. If we incorporated preorders, the result would be a larger, Morita equivalent algebra which contained thick strands as in the extended graphical calculus of \cite{KLMS,SWschur}. This is much more difficult to give a nice presentation of.
These results combine to show that the algebra $\mathbb{\tilde{T}}^\chi$ controls the derived category $D^b_{G_\chi}(V)$:
\begin{theorem}\label{thm:generator}
The object $\perv_{\chi}$ is a compact generator of $D^b_{G_{\chi}}(V)$. Thus, the functor $\Ext(\perv_{\chi},-)$ induces equivalences:
\[D^b_{G_{\chi}}(V)\cong \mathbb{\tilde{T}}^\chi\operatorname{-dg-mod}\qquad D^{b,\operatorname{mix}}_{G_{\chi}}(V)\cong D^b(\mathbb{\tilde{T}}^\chi\operatorname{-gmod}).\]
The functor of forgetting mixed structure corresponds to the functor considering a complex of graded modules as a dg-module by collapsing gradings.
\end{theorem}
\begin{proof}
Any finite complex with constructible cohomology is compact as an object in the derived category, so this includes $\perv_{\chi}$.
Clearly, by the $t$-structure properties, the object $\perv_{\chi}$ is a classical generator, and thus a generator of the derived category. By dg-Morita theory, we thus have the desired equivalences given by $\Ext(\perv_{\chi},-)$. The equivalence in the mixed setting is to dg-modules where the Hodge structure equips them with a second grading on which the differential has degree 0. By taking the difference of the dg-grading and the Hodge gradings, one obtains a homological grading is preserved by $\mathbb{\tilde{T}}^\chi$, which we can then use to think of our module as an object in $D^b(\mathbb{\tilde{T}}^\chi\operatorname{-gmod})$.
\end{proof}
\subsection{The categorical action}
Consider the 2-category $\mathsf{Perv}$ whose
\begin{itemize}
\item objects are dominant integral weights $\chi=(\chi_1\leq \cdots \leq \chi_n)$,
\item 1-morphisms $\chi\to \chi'$ are sums of shifts of semi-simple $P_{\chi'}\times P_{\chi}$-equivariant perverse sheaves on $GL_n$, or equivalently, $GL_n$-equivariant perverse sheaves on $GL_n/P_{\chi'}\times GL_n/P_{\chi'}$ with composition given by convolution,
\item 2-morphisms in the equivariant derived category.
\end{itemize}
As discussed in \cite[Thm. 6]{Webcomparison}, this equivalent to the category $\mathsf{Flag}$ whose
\begin{itemize}
\item objects are dominant integral weights $\chi=(\chi_1\leq \cdots \leq \chi_n)$,
\item 1-morphisms $\chi\to \chi'$ are sums of singular Soergel bimodules over the rings $H^*(BP_{\chi'})\times H^*(BP_{\chi})$.
\item 2-morphisms are degree 0 homomorphisms of bimodules.
\end{itemize}
via the functor that takes hypercohomology of a sum of shifts of equivariant perverse sheaves. For technical reasons, we add to these categories an object $\emptyset$, which has Hom with any other object given by the zero category.
Consider the fiber product \[P_{\chi'}\setminus GL_n/P_{\chi}\times_{BP_{\chi}}V/G_{\chi}=\frac{GL_{n}\times V}{P_{\chi'}\times G_{\chi}}.\] where the action is via $(p',p)\cdot (g,v)=(p'gp^{-1},pv)$. This is, of course, equipped with an action map \[a\colon P_{\chi'}\setminus GL_n/P_{\chi}\times_{BP_{\chi}}V/G_{\chi}\to V/G_{\chi'}, \] and as usual, we define convolution of sheaves $\mathcal{F}\in D^b_{P_{\chi'}\times P_\chi}(GL_n)$ and $\mathcal{G}\in D^b_{G_\chi}(V)$ as $\mathcal{F}\star \mathcal{G}=a_* (\mathcal{F}\boxtimes \mathcal{G}).$
\begin{lemma}
Convolution of sheaves induces a representation of $\mathsf{Perv}$ in $\mathsf{Cat}$ sending $\chi\mapsto D^b_{G_\chi}(V)$.
\end{lemma}
The category $\mathsf{Flag}$ carries a well-known categorical action defined by Khovanov and Lauda \cite[\S 6]{KLIII}.
We define $\mathcal{U}$ to be the 2-category with
\begin{itemize}
\item objects are integral weights of $\mathfrak{sl}_\infty$, that is, finite sums of the unit vectors $\ep_k$ for $k\in \Z$.
\item 1-morphisms are generated by
\[\eE_i\colon \mu \to \mu+\al_i=\mu+\ep_{i+1}-\ep_{i}\qquad \eF_i\colon \mu \to \mu-\al_i=\mu-\ep_{i+1}+\ep_{i}\]
\item 2-morphisms are given by certain string diagrams modulo the relations in \cite{Brundandef}.
\end{itemize}
As discussed in \cite[\S 2.2]{Webcomparison}, we thus have a 2-functor $\Phi_{\mathsf{P}}\colon \tU\to \mathsf{Perv}$ sending $\mu_\chi \to \chi$ for weights of the correct form, and to $\emptyset$ otherwise.
If $\chi'=\chi^{+i^a}$, then consider the action of $P_{\chi}\times P_{\chi'}$-orbits on $G$ are in bijection with orbits of double cosets for $(S_\chi',S_\chi)$. In particular, there is a unique closed orbit, given by the product $P_{\chi',\chi}=P_{\chi'}P_{\chi}$. The cohomology $H^*_{P_{\chi'}\times P_{\chi}}(P_{\chi',\chi})$ of this orbit gives the action of \cite[\S 6]{KLIII}. As discussed below the proof of Theorem 6 in \cite[\S 2.2]{Webcomparison}, we have that:
\begin{lemma}
The image $\Phi_{\mathsf{P}}(\eE_i^{(a)})$ is the Verdier-self-dual shift of the constant sheaf $\C_{\chi',\chi}$ on $P_{\chi',\chi}$. Similarly $\Phi_{\mathsf{P}}(\eF_i^{(a)})$ is the same sheaf with the role of the factors switched.
\end{lemma}
For us, the important new ingredient here is that since we have an algebraic manifestation of the category $D^b_{P_\chi}(V)$ given by the algebra $\mathbb{\tilde{T}}^\chi$, the action of the 2-category $\tU$ has a similar manifestation.
For every 1-morphism $\mathcal{F}\colon \chi\to \chi'$, we can define a bimodule
\[B_{\mathcal{F}}=\Ext_{D^b_{G_{\chi'}}(V)}(\perv_{\chi'}, \mathcal{F}\star \perv_{\chi}).\]
By Theorem \ref{thm:generator}, we have a commutative diagram \[\tikz[->,thick]{
\matrix[row sep=18mm,column sep=35mm,ampersand replacement=\&]{
\node (d) {$D^{b,\operatorname{mix}}_{G_\chi}(V)$}; \& \node (e)
{$ D^{b,\operatorname{mix}}_{G_{\chi'}}(V)$}; \\
\node (a) {$D^b(\cT^\chi\operatorname{-gmod})$}; \& \node (b)
{$D^b(\cT^{\chi'}\operatorname{-gmod})$}; \\
};
\draw (a) -- (b) node[below,midway]{$B_{\mathcal{F}}\Lotimes_{A_\chi}-$};
\draw (d) -- (a) node[left,midway]{$\Ext_{D^b_{G_{\chi}}(V)}(\perv_{\chi}, -)$} ;
\draw (e) -- (b) node[right,midway]{$\Ext_{D^b_{G_{\chi'}}(V)}(\perv_{\chi'}, -)$};
\draw (d) -- (e) node[above,midway]{$\mathcal{F}\star$};
}\]
\begin{lemma}
For any sum of shifts of semi-simple perverse sheaves $\mathcal{F}$ in $D^b_{P_{\chi'}\times P_{\chi}}(GL_n)$, the bimodule $B_{\mathcal{F}}$ is sweet, that is, it is projective as a left module over $\cT^{\chi'}$ and as a right module over $\cT^\chi$.
\end{lemma}
\begin{proof}
Since the anti-automorphism of taking inverse switches left and right module structures, it's enough to prove this for the left module structure.
Note that for any flavored multi-segment $\bQ$ for $\chi'$, we have that $\mathbf{IC}_{\bQ}$ is a summand of $\perv_{\chi'}$ by Theorem \ref{thm:all-ICs}. Thus, $\Ext_{D^b_{G_{\chi'}}(V)}(\perv_{\chi'}, \mathbf{IC}_{\bQ})$ is a projective as a left module over $\cT^{\chi'}$; in fact it is of the form $\cT^{\chi'} e_{\bQ}$ for an idempotent projecting to $\mathbf{IC}_{\bQ}$ as a summand of $\perv_{\chi'}$.
Since $\mathcal{F}$ is semi-simple, by the Decomposition theorem, the complex $\mathcal{F}\star \perv_{\chi}$ is a sum of shifts of simple perverse sheaves in $D^b_{G_{\chi'}}(V)$. Thus, $B_{\mathcal{F}}=\Ext_{D^b_{G_{\chi'}}(V)}(\perv_{\chi'}, \mathcal{F}\star \perv_{\chi})$ is a sum of projective left $\cT^{\chi'}$-modules, and thus projective.
\end{proof}
It follows immediately that the functor of tensor product with $B_{\mathcal{F}}$ is exact and:
\begin{lemma}\label{lem:BFG}
$B_{\mathcal{F}\star \mathcal{G}}\cong B_{\mathcal{F}}\Lotimes B_{\mathcal{G}}=B_{\mathcal{F}}\otimes B_{\mathcal{G}}$.
\end{lemma}
\begin{theorem}\label{thm:E-iso}
If $\chi'=\chi^{+i^a}$, then we have that $B_{\C_{\chi',\chi}}=\mathbb{E}_i^{(a)}$ and $B_{\C_{\chi,\chi'}}=\mathbb{F}_i^{(a)}$.
\end{theorem}
\begin{proof}
Of course, we only need to check one idempotent at a time. That is, let $\Bi$ be a $\chi$-parabolic word, and $\Bi'$ a $\chi'$ parabolic. We need only show that
\[e(\Bi')\mathbb{E}_i^{(a)}e(\Bi)\cong e(\Bi')B_{\C_{\chi',\chi}}e(\Bi) =\Ext_{D^b_{G_\chi}(V)}(\perv_{\Bi'}, \C_{\chi',\chi}\star \perv_{\Bi}).\]
Note that this isomorphism is only correct up to grading shift due to the need to shift $\C_{\chi',\chi}$ to make it Verdier-self-dual.
Since $\C_{\chi',\chi}\star-$ is the composition of restriction of $\perv_\Bi$ to the $P_\chi\cap P_{\chi'}$-equivariant derived category, with the induction to the $P_{\chi'}$-equivariant derived category, we can use adjunction to show that
\begin{equation}\label{eq:adjunction}
\Ext_{D^b_{G_\chi}(V)}(\perv_{\Bi'}, \C_{\chi',\chi}\star \perv_{\Bi})\cong \Ext_{D^b_{G_{\chi}\cap G_{\chi'}}(V)}(\perv_{\Bi'},\perv_{\Bi})\cong\Ext_{D^b_{B\times G_0}(V)}(\perv_{\Bi'},\perv_{\Bi})^{S_{\chi',\chi}}
\end{equation}
Of course, by Theorem \ref{thm:T-iso} in the case where $P_\chi=B$, we have:
\begin{equation}\label{eq:B-case}
\Ext_{D^b_{B\times G_0}(V)}(\perv_{\Bi'},\perv_{\Bi})\cong e(\Bi')\mathbb{\tilde{T}} e(\Bi).
\end{equation}
Thus, combining \eqref{eq:adjunction} and \eqref{eq:B-case}, we find that:
\[e(\Bi')\mathbb{E}_i^{(a)}e(\Bi)=(e(\Bi')\mathbb{\tilde{T}} e(\Bi))^{S_{\chi',\chi}}=\Ext_{D^b_{B\times G_0}(V)}(\perv_{\Bi'},\perv_{\Bi})^{S_{\chi',\chi}}=e(\Bi')B_{\C_{\chi',\chi}}e(\Bi).\qedhere\]
\end{proof}
Combining Lemma \ref{lem:BFG} and Theorem \ref{thm:E-iso}, we find that:
\begin{corollary}
We have a representation of the category $\tU$ sending $\mu_\chi \mapsto \mathbb{\tilde{T}}^\chi\mmod$ where any 1-morphism $u$ acts by tensor product with the bimodule $B_{\Phi_{\mathsf{P}}(u)}$, and in particular, $\eE_i^{(a)},\eF_i^{(a)}$ act by the bimodules $\mathbb{E}_i^{(a)},\mathbb{F}_i^{(a)}$.
\end{corollary}
This generalizes \cite[Thm. 9.1]{KLSY}, which is the special case where $\rankp=2$, and avoids the long computation required by the direct proof given in that paper.
\subsection{Restriction to the flag variety}
\label{sec:restr-flag-vari}
Assume now that $v_1\leq v_2\leq \cdots \leq v_n$. In this case, the stack $V/G_0$ contains an open subset where the compositions $f_{i;1}=f_i\cdots f_1$ are injective for all $i$. On this open subset, \[\operatorname{image}(f_1)\subset \operatorname{image}(f_{2;1})\subset \cdots \subset \operatorname{image}(f_{n;1})\subset \C^n\] gives a partial flag. As before, we let $\nu=(1,\dots, 1,2, \dots, 2,\dots)\in \Z^n$ be the unique increasing sequence with $i$ occuring $v_i-v_{i-1}$ times. We can then describe our space of partial flags as exactly $GL_n/P_{\nu}.$ Thus we obtain a functor
\[\Res\colon D^{b,\operatorname{mix}}_{G_\chi}(V)\to D^{b,\operatorname{mix}}_{P_\chi}(GL_n/P_{\nu})\]
by restriction to this open set. This is, of course, compatible with the natural action of $\Perv$.
\begin{theorem}
We have an equivalence of categories \[\beta=\Ext_{D^b_{P_{\chi}}(GL_n/P_\nu)}(\Res(\perv_{\chi}), -)\colon D^{b,\operatorname{mix}}_{P_\chi}(GL_n/P_\nu)\cong D^b(\vT^\chi_\nu\gmod)\] such that we have a commutative diagram:
\[\tikz[->,thick]{
\matrix[row sep=18mm,column sep=30mm,ampersand replacement=\&]{
\node (d) {$D^{b,\operatorname{mix}}_{G_\chi}(V)$}; \& \node (e)
{$ D^{b,\operatorname{mix}}_{P_{\chi}}(G/P_{\nu})$}; \\
\node (a) {$D^b(\mathbb{\tilde{T}}^\chi\operatorname{-gmod})$}; \& \node (b)
{$D^b(\vT^{\chi}\operatorname{-gmod})$}; \\
};
\draw (a) -- (b) node[below,midway]{$\vT^{\chi}\Lotimes_{\mathbb{\tilde{T}}^\chi}-$};
\draw (d) -- (a) node[left,midway]{$\Ext_{D^b_{G_{\chi}}(V)}(\perv_{\chi}, -)$} ;
\draw (e) -- (b) node[right,midway]{$\beta$};
\draw (d) -- (e) node[above,midway]{$\Res$};
}\]
\end{theorem}
\begin{proof}
This essentially a restatement of \cite[Cor. 5.4]{Webqui}, but let us be a bit more careful about the details of this point. We have a map $\mathbb{\tilde{T}}^\chi\to \Ext^*(\Res(\perv_{\chi}), \Res(\perv_{\chi}))$ induced by the functor $\Res$.
If $\Bi$ is violating, then $(i_1,1)\prec (\rankp,1)$. For any point in $\Fl(\Bi)$, by assumption, we have $f_{\rankm;i_1}F_{\preceq(i_1,1)}=0$, and so one of $f_k$ is not injective. This shows that $\Res(\mathscr{F}_{\Bi})=0$.
On the other hand, if $\Res(\mathbf{IC}_{\bQ})=0$, then $\bQ$ must be a multi-segment that includes a segment without $\rankp$, so the corresponding good word $\Bi_{\bQ}$ is violating. Thus, $\Res(\mathbf{IC}_{\bQ})=0$ if and only if $\mathbf{IC}_{\bQ}$ is a summand of $\mathscr{F}_{\Bi}$.
The usual formalism of recollement shows that $D^{b,\operatorname{mix}}_{P_\chi}(GL_n/P_\nu)$ (after suitable enhancement) is a dg-quotient of $D^{b,\operatorname{mix}}_{G_\chi}(V)$ by the subcategory generated by $\mathscr{F}_{\Bi}$ for $\Bi$ violating.
Since $\mathbb{\tilde{T}}^\chi$ is quasi-isomorphic to $\Ext^*(\perv_{\chi}, \perv_{\chi})$ as a formal dg-algebra, taking dg-quotient by the projectives $\mathbb{\tilde{T}}^\chi e(\Bi)$ for $\Bi$ violating has the effect of simply modding out by the corresponding 2-sided ideal. Thus, the map above induces an isomorphism
\[\vT^\chi=\Ext^*_{D^{b,\operatorname{mix}}_{P_\chi}(GL_n/P_\nu)}(\Res(\perv_{\chi}), \Res(\perv_{\chi})).\] By dg-Morita theory, this completes the proof.
\end{proof}
Of course, we can also interpret $D^b_{P_{\chi}}(GL_n/P_\nu)$ as a Hom-category in the equivalent 2-categories $\Flag\cong \Perv$, and this identification intertwines the left regular action on $\Perv$ with the action we've discussed on $D^b_{P_{\chi}}(GL_n/P_\nu)$.
To fix this equivalence algebraically, we need to describe the image of the identity. This corresponds to the constant sheaf on $P_\nu/P_\nu\subset GL_n/P_{\nu}$. Consider the flavored multi-segment where we take the segment $(i,\dots,m)$ exactly $v_i-v_{i-1}$ times, and flavor these with $i$.
The corresponding word is
\[ \Bi_{\nu}=(m^{(v_1)},(m-1)^{(v_1)},\dots, 1^{(v_1)},m^{(v_2-v_1)},(m-1)^{(v_2-v_1)},\dots, 2^{(v_2-v_1)},\dots).\]
\begin{lemma}
We have an isomorphism $\Res(\mathscr{F}_{\Bi_\nu})\cong \C_{P_{\nu}/P_{\nu}}$.
\end{lemma}
\begin{proof}
Consider the space $\C^{v_{i}}$. If $f$ is injective on $\C^{v_j}$ for all $j<m$, then $f^{m-i}(\C^{v_{i}})$ is a $v_i$-dimensional subspace in $\C^n$. Furthermore, this subspace must lie in the span of $b_{m,j}$ with $(m,j)\prec (i,v_i),$ since $F_{\preceq (i,v_i)}$ is a submodule containing $\C^{v_i}.$ In the word $\Bi_{\nu}$, we have $(m,1) \preceq \cdots \preceq (m,v_i)\prec (i,v_i)\prec (m,v_{i+1}).$ This shows that the image of $f^{m-i}$ must be exactly the $v_i$ dimensional space in the standard flag; this completes the proof.
\end{proof}
\begin{corollary}\label{cor:Soergel-T}
The category of graded projective $\vT^\chi_\nu$-modules is equivalent to the category $\Hom_{\Flag}(\chi,\nu)$ of singular Soergel bimodules for the singularity $(\chi,\nu)$. This equivalence is characterized by
\begin{enumerate}
\item sending the diagonal bimodule in $\Hom_{\Flag}(\nu,\nu)$ to the module $\vT^\nu_\nu e(\Bi_{\nu})$,
\item intertwining the action of $\Flag\cong \Perv$ by ladder bimodules with the left regular action on singular Soergel bimodules.
\item intertwining the action of $\Flag$ by induction and restriction functors with the right regular action on singular Soergel bimodules.
\end{enumerate}
\end{corollary}
This generalizes the main theorem of \cite{KSred}. Of course, the category of Soergel bimodules has a ``reversing functor'' where one swaps the left and right actions; this is an equivalence $\Hom_{\Flag}(\chi,\nu)\cong \Hom_{\Flag}(\nu,\chi)$ which is anti-monoidal. Applying Corollary \ref{cor:Soergel-T}, we find that:
\begin{corollary}\label{cor:chi-nu}
We have a Morita equivalence between $\vT^\chi_\nu$ and $\vT^\nu_\chi$ which swaps the action of $\Flag$ via ladder bimodules with the action via induction and restriction functors.
\end{corollary}
Readers of an artistic bent are encouraged to attempt drawing the bimodule realizing this equivalence; the author wishes them luck in this endeavor.
| {
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{"url":"http:\/\/mathhelpforum.com\/algebra\/49877-simplify.html","text":"Math Help - simplify\n\n1. simplify\n\n1-m\n----\nm\n----\nm-1\/m\n\n$\\frac{\\dfrac{1-m}{m}} {m - \\dfrac{1}{m}}$\nMultiply top and bottom by $m:$\n. . $\\frac{{\\color{blue}m}\\left(\\dfrac{1-m}{m}\\right)} {{\\color{blue}m}\\left(m - \\dfrac{1}{m}\\right)} \\;=\\;\\frac{1-m}{m^2-1} \\;=\\;\\frac{-(m-1)}{(m-1)(m+1)} \\;=\\;\\frac{-1}{m+1}$","date":"2014-09-03 01:45:36","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 3, \"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\": 0.512492299079895, \"perplexity\": 3257.223672861317}, \"config\": {\"markdown_headings\": false, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2014-35\/segments\/1409535923940.4\/warc\/CC-MAIN-20140909040754-00454-ip-10-180-136-8.ec2.internal.warc.gz\"}"} | null | null |
Investigation in the Canary Islands reveals the threat of electrocution to birds like the endangered Egyptian Vulture
Investigating mortality of birds © Agentes de Medio Ambiente de Canarias
Environmental Agents across the Canary Islands (Agentes de Medio Ambiente de Canarias) have investigated the deaths of 247 birds caused by collision or electrocution with power lines between mid-2016 to the end of 2020. Victims also included the Egyptian Vulture, an endemic subspecies, known as 'guirre', living in the three easternmost islands — Fuerteventura, Lanzarote and Alegranza.
Results of the investigation
Of these 247 cases of mortality, 145 correspond to the province of Las Palmas (Lanzarote 71, Fuerteventura 37 and Gran Canaria 37) and 102 to Santa Cruz de Tenerife (Tenerife 79, La Gomera 20, La Palma 2 and El Hierro 1). These numbers, however, maybe be even higher. According to SEO-BirdLife studies carried out in Lanzarote and Fuerteventura, for every 100 birds found, it is estimated that the number of incidents amounts to approximately 3,000 birds. So, the estimated impact could reach 7,000 birds throughout the archipelago.
As for the type of incident, the agents have recorded a total of 123 possible electrocutions, 78 possible collisions and 7 cases that could not be determined. The electrocution of 8 Egyptian Vulture cases, stands out, whose recovery plan aims to tackle the threat of power lines. With a current estimate of only 80 pairs in Fuerteventura and 9 in Lanzarote, the finding of 8 dead individuals presumably electrocuted is devastating.
Egyptian Vulture conservation in the Canary Islands
The Egyptian Vulture was widespread across the archipelago in the early 20th century, but its numbers drastically declined, and in 1998 only about 21 breeding pairs remained at Fuerteventura. Thanks to targeted conservation actions ever since, the population of the Canary Egyptian Vulture (Neophron percnopterus majorensis), one of the most endangered raptors in Europe, is recovering.
In 1998, an intensive long-term monitoring programme carried out by the Estación Biológica de Doñana-CSIC detected the main threats the species was facing. These include human activities such as collision with power lines and illegal poisoning. Between 2004 to 2008, a LIFE conservation project carried out education campaigns to help minimise illegal poisoning and advocated for the modification of power lines to reduce the risk of collision to mitigate these threats. Conservationists have been monitoring the species for the last 20 years, and their analysis indicated that since the project, the survival of Egyptian Vultures has increased, especially for adult and subadult birds. Also, the population almost quadrupled between 1998 and 2020, from 21 to over 80 breeding pairs living in Fuerteventura, Lanzarote and Alegranza according to the latest monitoring carried out by the Egyptian Vulture LIFE Project.
Egyptian Vultures in the Canary Islands and Spain
Egyptian Vulture/illustrative © C. Bougain
All European vulture species live in Spain, Europe's vulture stronghold. The Egyptian Vulture has two subspecies in Spain; the Peninsular subspecies is currently listed in the Spanish Catalogue of Threatened Species in the category of Vulnerable and the Canary Islands subspecies in the category of Endangered. Periodic censuses are carried out for these two subspecies by the Autonomous Communities in compliance with the Law on Natural Heritage and Biodiversity. Spain is home to the largest European population of Egyptian Vultures and is a global stronghold for the species, with SEO BirdLife's national Egyptian Vulture census 2018 estimating 1,490-1,567 pairs.
The Canary Egyptian Vulture is the only vulture species that exists on the Canary Islands. It is essential to continue to tackle poisoning, electrocution, collisions with power lines and wind turbines, and to undertake conservation policies that address the abandonment of fields and the extensive decline of livestock to safeguard the species.
Source: Info Norte Digital
tagPlaceholderTags: egyptianvulture, Electrocution, Collision , 2021-05 | {
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La sindrome di Ogilvie è la dilatazione acuta del colon in assenza di ostruzione meccanica.
La pseudo-ostruzione del colon è caratterizzata da massiccia dilatazione dell'intestino cieco (diametro > 10 cm) e del colon destro sulla radiografia addominale. È un tipo di megacolon, a volte indicato come "megacolon acuto", per distinguerlo dal megacolon tossico.
La condizione porta il nome del chirurgo britannico Sir William Heneage Ogilvie (1887-1971), che per primo la riferì nel 1948.
Segni e sintomi
Di solito il paziente ha distensione addominale, dolore e alterazioni dei movimenti intestinali. Ci possono anche essere nausea e vomito.
Causa
La sindrome di Ogilvie può verificarsi dopo un intervento chirurgico, in particolare dopo l'intervento di bypass delle arterie coronarie, la sostituzione totale dell'articolazione o per la somministrazione di farmaci che disturbano la motilità del colon (es. anticolinergico, oppiaceo o analgesico) e contribuiscono allo sviluppo di questa condizione.
Fisiopatologia
Il meccanismo esatto non è noto. La spiegazione probabile è lo squilibrio nella regolazione dell'attività motoria del colon da parte del sistema nervoso autonomo. È stato postulato che la riattivazione di varicella zoster virus (che causa varicella e herpes zoster) nei gangli enterici può essere una causa della sindrome di Ogilvie.
Il megacolon acuto si sviluppa a causa dell'anormale motilità intestinale. La normale motilità del colon richiede l'integrazione di influenze miogeniche, neurologiche e ormonali. Il sistema nervoso enterico è indipendente ma è collegato al sistema nervoso centrale dal simpatico e dal sistema nervoso parasimpatico. I bersagli del neurone enterico sono cellule muscolari, cellule secretorie, cellule endocrine, microvasculature e cellule infiammatorie. I neuroni nei plessi enterici sono stimolati dal bolo, che distende l'intestino e stimola la superficie della mucosa, portando al rilascio di fattori che stimolano gli interneuroni. Gli interneuroni stimolati trasmettono segnali eccitatori prossimalmente, che causano contrazioni e segnali inibitori distalmente, e questi a loro volta causano rilassamento. Questi segnali sono trasmessi dai neurotrasmettitori, acetilcolina e serotonina, tra gli altri.
Il megacolon acuto può anche portare a necrosi ischemica in segmenti intestinali massicciamente dilatati. Questo è spiegato dalla legge di Pascal e dalla legge di Laplace. Il principio di Pascal afferma che un cambiamento di pressione in qualsiasi punto in un fluido chiuso a riposo viene trasmesso non distrutto a tutti i punti nel fluido; la pressione su tutte le parti del lume è uguale. La legge di Laplace afferma che:
dove = tensione della parete, = pressione, = raggio, = spessore della parete.
Poiché la tensione della parete è proporzionale al raggio, un segmento intestinale dilatato ha una tensione della parete maggiore rispetto a un segmento non dilatato; se la dilatazione e la tensione sono sufficientemente grandi, il flusso sanguigno può essere ostruito e si verificherà l'ischemia dell'intestino.
Diagnosi
La diagnosi inizia con esame fisico, osservazione e colloquio del paziente. L'imaging per diagnosticare la dilatazione del colon coinvolge una serie di radiografie o ostruzioni addominali (torace PA, addome eretto e immagini dell'addome supino). Se è necessario ulteriore imaging, è possibile ordinare CT.
Trattamento
Solitamente si risolve con la terapia conservativa fermando l'ingestione orale, cioè niente per bocca e un sondino nasogastrico, ma potrebbe richiedere la decompressione colonoscopica che ha successo nel 70% dei casi. Uno studio pubblicato su The New England Journal of Medicine ha dimostrato che la neostigmina è un potente mezzo farmacologico per decomprimere il colon. Secondo l'American Society for Gastrointestinal Endoscopy (ASGE), la sua somministrazione dovrebbe essere considerata prima della decompressione colonscopica. L'uso di neostigmina non è privo di rischi poiché può indurre bradiaritmie e broncospasmi. Pertanto, l'atropina deve essere a portata immediata quando si usa questa terapia.
Prognosi
È un disturbo medico grave e il tasso di mortalità può arrivare al 30%. L'alto tasso di mortalità è probabilmente dovuto al suo manifestarsi in pazienti già critici piuttosto che alla mortalità della sindrome di per sé, sebbene possa presentarsi anche in individui altrimenti sani (specialmente se il disturbo è stato indotto da agenti farmacologici). Il megacolon indotto da farmaci (cioè da Clozapina) è stato associato a mortalità del 27,5%.
Note
Altri progetti
Malattie dell'intestino tenue e crasso | {
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After a disappointing loss last week to Southwestern, the Brahmas rebounded in a big way on the road against the Glendale Vaqueros.
Pierce never trailed, keeping its foot on the gas to a convincing 27-16 victory. Quarterback Sean Smith dazzled, completing 24-37 passes for 303 yards and two touchdowns.
Both touchdown passes were to receiver Xavier Ubosi, who had five catches for 90 yards, including a 53-yard touchdown haul to put the Brahmas up 20-9 toward the end of the third quarter.
Head coach Jason Sabolic offered high praise to the offense, crediting the team's success to Pierce's aerial attack.
The defense and special teams also played a huge role in the team's win, as the team lead the Vaqueros in almost all statistical categories.
Defensively, Pierce dominated counting for two interceptions, five sacks, and two forced fumbles.
Special teams set the tone early, as the Brahmas forced a fumble during the Vaqueros first kickoff return, leading to an early 3-0 lead.
Next Vaqueros drive, Pierce special teams would come up again, blocking a 28-yard field goal attempt.
After the block, the Brahma offense fired on all cylinders behind the legs of running back Calvin Howard, who had two runs during the drive for over 20 yards and included 3-yard rushing touchdown.
The defense shut down Glendale for the remainder of the first half, 13-0.
Glendale finally put points on the scoreboard during the third quarter after Smith was sacked in his own end-zone for a safety.
Smith would respond by tossing a long 53-yard touchdown pass to Ubosi.
In the fourth quarter Ubosi caught another touchdown off of a beautiful 15-yard pass from Smith to put Pierce up 27-9.
Penalties plagued the Brahmas for the third game in a row as the team was flagged multiple times throughout the night.
The biggest penalties came during a Glendale drive in the middle of the fourth which led to a touchdown run by running Travis Custis.
Pierce would then hold on to keep the score 27-16 to end the game.
Glendale's head coach John Rome said Pierce showed up, and that Glendale stopped themselves.
"The same mistakes we've been making we need to correct," Rome said. | {
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Perception is reality, particularly in the world of pre- construction condominium sales. When there is nothing to show but floor plans and blueprints, success hinges on your ability to convey the developer's vision and pique the interest of your intended buyer through sophisticated branding.
An integral part of branding is Visualization – photorealistic renderings that capture a project's every detail from interiors, exteriors, lighting and views to finishes, furnishings, amenities and décor.
More and more, developers are engaging celebrated architects to create buildings that are remarkable in their own right. Digital or 3D rendering is the best, and often the only, way to present these complex designs to the public to drive critical pre-construction sales. LGD Communications offers the most sophisticated visualization services available in the realm of residential real estate. | {
"redpajama_set_name": "RedPajamaC4"
} | 4,918 |
Emma Elina Kimiläinen-Liuski (Helsinki, 1989. július 8. –) finn autóversenyző.
Pályafutása
Kimiläinen 2005-ben kezdte versenyzői pályafutását, amikor is az észak-európai Formula Ford bajnokságban vett részt. Első szezonjában az ötödik helyen végzett, egy évvel később pedig az összetett második helyén zárt, kevesebb győzelme miatt alulmaradva a bajnok Sami Isohellával szemben. 2007-ben a svéd Radical Sportscars elnevezésű formulaautós-sorozatban állt rajthoz.
Az Audi 2008-ban szerződtette a Formula Mastersben versenyző csapatába. Első idényében 10. lett az összetettben. majd egy évvel később az Audi pénzügyi nehézségei miatt saját maga finanszírozta, hogy a Formula Palmer Audi elnevezésű bajnokságban rajthoz állhasson. Itt összességében négy dobogós helyezést gyűjtve az ötödik helyen végzett a bajnoki pontversenyben. 2009-ben családot alapított, és szüneteltette pályafutását.
2014-ben a skandináv túraautó-bajnokság mezőnyében állt rajthoz a PWR Racing színeiben. Nettan Lindgren-Jansson 1999-es szereplése óta ő lett az első nő a sorozatban. A következő években is ebben a bajnokságban indult, igaz a 2016-os szezon nagy részét ki kellett hagynia, miután a a Skövde-i nyitófordulóban súlyos balesetet és nyaksérülést szenvedett. 2017-ben a svéd ThunderCar elnevezésű bajnokságban vett részt és bár tervezte, hogy nevez a 2018-as Electric GT sportautós bajnokságba, ezt nem sikerült megvalósítania.
2019-ben az újonnan alapított, és kizárólag nőket versenyeztető W Seriesben indult. A Hockenheimringen tartott első versenyen hiába szerepelt nagyszerűen a kvalifikáción, a versenyen ütközött Megan Gilkesszel, és kiújuló nyaksérülése miatt a küövetkező két fordulót ki kellett hagynia. Visszatérése után Assenben Grand Chelemet ért el, azaz a pole pozícióból indulva, a verseny leggyorsabb körét megfutva nyerte meg a holland futamot. A szezon végén ötödik helyen zárt az összetettben.
Személye körüli botrányok
2020-ban egy podcast-show-ban úgy nyilatkozott, hogy 2009-ben azért nem lett az Indy Lights egyik csapatának versenyzője, mert a csapat főtámogatója, amely egy neves férfimagazin volt, topless fotókat kért volna cserébe a szerződésért, amit ő visszautasított. Ugyanebben a műsorban célzásokat tett arra is, hogy pályafutása során többször érte negatív megkülönböztetés az autósportban, azért mert nő.
Eredményei
Karrier összefoglaló
Teljes Skandináv túraautó-bajnokság eredménysorozata
Teljes W Series eredménylistája
Jegyzetek
További információk
Emma Kimiläinen, Driver Database
Hivatalos honlapja
Finn autóversenyzők
1989-ben született személyek
Élő személyek
Finn nők | {
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Ramūnas Grikevičius (i internationella sammanhang Ramunas Grikevicius), född 1963 i Panevėžys, är en litauisk konstnär.
Källor
Litauiska konstnärer
Konstnärer under 1900-talet
Konstnärer under 2000-talet
Födda 1963
Levande personer
Män
Personer från Panevėžys | {
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Imagine having three children—two who are drama queens, and one who is calm and mature, as if there were another grownup in the family.
The first two might represent recent United Kingdom and French elections that sparked outsize moves in markets. Meanwhile, Germany's upcoming ballot plays the part of the "strong and stable" child—to borrow a Tory campaign slogan that failed to resonate with U.K. voters.
German Chancellor Angela Merkel looks set to cruise to her fourth term, and the main question has involved which parties will team up with her Christian Democrats to form a governing coalition. Yet investors are still keeping an eye on the Sept. 24 federal election in Europe's largest economy, says Seema Shah, a London-based strategist for Principal Global Investors. It could provide investing opportunities, even though it doesn't look like a big market mover, she says.
The June vote in Europe's second-biggest economy was a different story, as U.K. Prime Minister Theresa May's gamble to secure a larger Conservative majority backfired, sparking a slump in the pound for several sessions. That came a year after the Brits voted to leave the European Union, shaking up markets worldwide. And voters in France, the No. 3 economy, triggered a global rally for stocks in April, when they gave now-President Emmanuel Macron a victory in the election's first round.
"In the past year, the elections you've had in Europe—as well as in the U.S.—have gotten a lot of market attention, with at least before-the-election speculation that there could be a major change in policies," Shah says. On the other hand, neither Merkel's Christian Democrats nor the Martin Schulz–led Social Democrats, the party polling a distant second, appear likely to alter the status quo in foreign or domestic policies.
However, there could be a buying opportunity in European stocks if Germany's election campaign, described as "bland" and "boring" so far by the media, somehow starts to put traders on edge in its final week. "If you do see any widening in spreads on Bund yields, or if there is some market concern growing about the German election, that's an opportunity to get in, because this should not have a big impact," Shah says.
Investors should buy the dips in European equities because more gains are coming, thanks to a favorable backdrop, she argues. The Stoxx Europe 600, Germany's DAX, and other European indexes have been holding below 2017 peaks hit in May and June, but fresh highs will arrive in due course, says Shah. She predicts that the European Central Bank will gradually taper its bond-buying program, a key stimulus effort, over nine to 12 months, with that process starting early next year. She forecasts no interest-rate hikes until 2019 and sees ECB President Mario Draghi possibly talking down the rallying euro, which has been a headwind for shares of European exporters.
"So putting that together, it's a kind of slow, steady improvement for Europe. That's the best kind of circumstances that you want for equity markets," says Shah.
Popular U.S.-listed plays for betting broadly on European stocks include the iShares MSCI Eurozone exchange-traded fund (ticker: EZU) and the iShares MSCI Germany ETF (EWG), which trade around 14 to 15 times predicted forward-year earnings, well below the SPDR S&P 500 ETF's (SPY) multiple of 19.
Many strategists have been upbeat on European stocks' prospects this year. While the Stoxx Europe 600 has fallen about 4% from its mid-May peak, it's still up more than 5% for the year—and bulls aren't throwing in the towel.
The index looks set to rise by year's end to 400, according to a Barclays team of European equity strategists led by Dennis Jose. That implies an advance of 5% from the gauge's recent print around 381.
The "primary culprit" for the decline from French election highs has been the euro rally, but that also creates buying opportunities, write Jose and his colleagues in a recent note. Domestically oriented companies such as bank and transportation stocks "stand out as key winners from euro strength," and they should start to outperform, they note. On their list of domestically focused firms worth considering: German lender Commerzbank (CBK.Germany), mail and courier service Deutsche Post (DPW.Germany), and airline Deutsche Lufthansa (LHA.Germany).
The German election could buoy European stocks by simply getting out of the way.
IN EUROPEAN MARKETS last week, the main indexes mostly gained, with the DAX rising nearly 2% and stretching its 2017 gain to about 9%. The FTSE 100 bucked the positive trend, hurt by a rally for the Brexit-battered pound that came after the Bank of England signaled that it's preparing to raise rates within months to rein in U.K. inflation.
German elections aren't likely to drive a big move in improving markets. | {
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Originally from Northern California, Atarah grew up in a small town amongst the giant redwood trees. She first fell in love with photography through the photographs of her childhood taken by her mother, immediately drawn to the nostalgic nature of the medium. After graduating with a BFA and honors from The Brooks Institute of Photography in Santa Barbara, Atarah moved to Brooklyn, NY to continue working within fashion and fine art. Over the years Atarah's work has developed a signature style of a bright etherial look with a modern edge. For Atarah, photography has always been a form of self exploration, so you'll often find her using personal experiences as narratives to drive emotion into her work. She's described her creative approach as efforts to create what she calls "eye pillows" - calm soft moments where the viewer may rest their eyes and be transported into her world.
"As if One Is" by Atarah Atkinson
"Prickly" by Atarah Atkinson
"Access" by Atarah Atkinson
"XXX" by Atarah Atkinson
"The Vase" by Atarah Atkinson
"Pastel Shadows" by Atarah Atkinson
"In the Course of Time" by Atarah Atkinson
The Ceiling by Atarah Atkinson
The Pink Stairs by Atarah Atkinson | {
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Q: java avoid Objects pass by reference I'm facing problem with object cloning in java.
public class TestClass {
private int count;
private List<Integer> list;
public final int getCount() {
return count;
}
public final void setCount(int count) {
this.count = count;
}
public final List<Integer> getList() {
return list;
}
public final void setList(List<Integer> list) {
this.list = list;
}
}
public class Main {
public static void main(String[] args) {
TestClass obj = new TestClass();
obj.setCount(5);
List<Integer> temp = new ArrayList<Integer>();
temp.add(1);
temp.add(2);
obj.setList(temp);
TestClass newObj= obj;
System.out.println(newObj.getList().size());
obj.getList().add(3);
System.out.println(newObj.getList().size());
}
}
The output I'm getting here is 2,3. But my desired output is 2,2 but since java assign reference of "obj" to newObj. Is there anyway to avoid this? I have seen that serialize the object and deserialize it will give brand new reference to "newObj". But is there anyother efficient ways?
A: First, you aren't actually cloning which in java means you use the object.clone() method. clone makes a shallow copy of an object. You just made a 2nd reference to the same object.
In your code, calling
obj.getList().add(3);
Is the same as
newObj.getList().add(3);
Which is why it is printing the way it is.
You need to make a deep copy of the object so not only is the instance of TestClass different, but all the fields are different too.
The easiest way to do this is to make a copy method and/or a copy constructor
public class TestClass{
public TestClass(){
}
//copy constructor
public TestClass( TestClass copy){
this.list = new ArrayList<Integer>(copy.list);
this.count = copy.count;
}
...
}
Then to use it :
TestClass newObj= new TestClass(obj);
| {
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Royal Mothers, Part IV: What On Earth Did Pierre André Mean?
[Theresa]: Transcribing the manuscript was daunting for all of us, and I think the students got a lesson in professorial humility watching both Monica and I discuss and dispute the meaning and significance of the words on the page. Students are accustomed to thinking of their professors as the source of knowledge, not uncertainty. No matter how much I tell them that I don't know everything, I think they just nod and say, "Yeah, sure." They know that I'm the one who determines their grades, and so I must know everything right? Yeah, sure. My Latin was rusty but my paleography was good. Still, I was more familiar with a few fifteenth-century Catalan royal secretaries who rarely abbreviated and used words that didn't require a dictionary of medieval medical terminology.
Paris, Bibliothèque Nationale de France, MS lat. 6992, f. 87vb (detail), Pomum aureum, Practica, chapter 2. This is an example of the kinds of abbreviations the students had to learn. Click me for the translation of this text ("It ought to be noted [. . . ]."
Above all, they learned one more valuable lesson in research: It takes time. A lot of time. And it takes a fine-toothed comb, a penchant for persistence, and a love of linguistic nuance. Time was more precious than everything else because Seattle University runs on a quarter system and we were already at the halfway mark when we began the transcription. Monica and I based our schedule on prior transcriptions, so we honestly thought we were in fine shape. But we were wrong.
On the first day of a collaborative transcription, when we were all on a Skype call, it took nearly two hours to get through one sentence. One. In the prologue, where the text was flowery and filled with boilerplate phrasings such as "I am not worthy" and "My esteemed lord." One sentence. Clearly, we needed to move faster, but how? The answer came from a student.
Students are crafty in terms of technology, and the answer was "Google Drive" (the application formerly known as GoogleDocs). Bingo! If you've never done this before, let me tell you that working collaboratively on Google Drive is brilliant. At first, we transcribed as a group with each of us typing along. This was fun, a bit like having a mystery hand in the room, with letters and words taking shape before your very eyes. Each person's edits were highlighted in a different color, with a date and time stamp. But this got out of control quickly, with overtyping and speed editing making it hard to see where we were. So we decided that each of us would take chunks of the text to transcribe, post it, and with any luck, we'd knit the whole thing together in a jiffy.
Yeah, right. We got about 2/3 of the way through by the end of the term. By that time, we could all agree on one thing: the manuscript was a working copy, not a presentation copy.
A numerical charm to be written on a tile and placed above the woman's right foot, as an aid to childbirth. Paris, Bibliothèque Nationale de France, MS lat. 6992, f. 88va (detail).
One student, with a keen eye and ear for codicology, made a clear argument for this in the final paper and noted that the disparity between what the original text was intended for and what this manuscript offers is significant to a broader understanding of medieval medicine. Comparing this text to André's other work will shed light on his thinking, how it developed over time, and how he might have revised it for various patrons. Here we see a fairly polished text, but one that was hardly finished, giving us a glimpse into the seams and the stitching of a medieval physician's handiwork.
As if it wasn't hard enough to transcribe the heavily abbreviated handwriting in the manuscript of Pomum aureum, we then had to turn to translating it. We did some of this while we were transcribing, just to be sure the jumble of letters made sense. And as we did, our students got increasingly excited about some vexing and problematic sections of the text. Here are some of the issues they lost some sleep over.
We were stumped by the word "virtualitas," a medieval Latin term that made a few perplexed classicists scratch their heads in wonder at the novelty. Two students took up the challenge. They agreed that it must have something to do with the necessity to balance what André called the male and female "instruments" in the creation of the child in order to ensure that the combined "essences" of the parents are passed down to the child. But one student, whose interest is in political history, translated it as "virtuality," a word that suggests that André and his contemporaries believed that not only physical but also moral characteristics were passed through generations, through both the male and female seed. "[E]ven if there is disproportion," which the student argued refers to physical disproportionality, and thus that the "virtuality" of the offspring will still pass down as result of the combination of the mother and father's seeds.
One other student, a philosophy major, was intrigued by Aristotle's natural philosophy. This student argued that André plausibly establishes the eternal nature of the cosmos, but had a tough time with the question, "Why do children look like their mothers?" This question was linked to Aristotle's hylomorphic ("form-matter") account of generation, particularly so when André states that "when this proportion draws near their own [the parents'] potential essence" [cum sit propinquior ipsorum virtualitas]. Here, the philosophy student differs with the student of political history, and translates "virtualitas" as "potential essence," turning to both Aristotle and Galen, noting that the essence—the "esse" or being—of an object was derived from its four causes, with form the most important. "Form," for Aristotle, was the abstract idea of a person or thing; existing beings were instantiated in "matter," but "form" was what stamped a raw material substrate with its unique characteristics. Form contains a set of potentials to act towards the good, which results in the full actualization of a human being, and this is closely tied to virtue. The key to the medieval meaning of "virtualitas" thus is "virtue," which is logically connected to one's physical appearance. Ergo, good kings were kings that looked and acted like a king should. When André asserts that when all three qualities are correctly balanced, the child will draw close to the parent's "virtualitas" not only looking more like one than the other, but receive the same formal qualities, including the virtues, of that parent as well.
Both of these interpretations of "virtualitas" have serious implications for our understanding of medieval inheritance, and of special interest to those of us who study queens and queenship, a woman's right to inherit a realm and rule it in her own right. More specifically, this transfer of moral or character traits through bloodlines has political implications for the royalty of Navarre, whose political culture permitted that a ruling monarch could be either a queen or a king. And it has a lot to tell us about the troubled reign of King Enrique IV of Castile, derogatorily known as "el impotente" (the impotent man) because after two marriages he had only one child, a daughter. His death resulted in a decade-long contest for the throne between a daughter allegedly not his own and a step-sister who eventually won the war and ruled as Isabel I, "the Catholic."
One student took up the question of André's understanding of biological sex and the cultural construction of notions of masculinity and femininity in the fourth chapter of theorica on the signs of masculinity and femininity in pregnancy. This section outlines the first two of twelve signs of the sex of a child during pregnancy and provides valuable insight into the more intricate nature of gender norms in medieval Europe. Because of the physical subject of medical theory, the descriptions of gender that result from the text are also physical in nature. The student argued that medical descriptions of sex influence, or perhaps serve as the basis for, the social norms which establish physical ideals for masculinity and femininity.
[Monica]: The topic that grabbed the attention of three students concerned what André meant by the term aborsus. As we mentioned earlier, the issue of how to translate this word had already raised questions for them when they were reading the Trotula. Pierre André used the word frequently to describe fetal death, both miscarriage (unplanned loss of a fetus or embryo) and, more rarely, abortion (deliberate termination of a pregnancy). But he also used the term, surprisingly, to refer to situations where no conception had yet happened.
One of the first things the students did was to look up the original meaning of the Latin verb aborior. Yes, it can mean to miscarry; but it also means to disappear, pass away, set (as in the sun), or to fail. It was this last meaning, it seemed, that André had in mind when, under the chapter heading "quot sint aborsus et qualiter regenda sit pregnans ne aborciat" ("how many are the kinds of aborsus and how the pregnant woman ought to be governed so that she does not abort"), he gathered his beliefs on the range of impediments to producing viable offspring. In reading through "the six causes of aborsus due to the male partner and twenty-three due to the female," it became clear to us that André used the concept of aborsus to cover all circumstances that impeded generation, not simply those that caused the loss of an embryo or fetus already conceived.
Paris, Bibliothèque Nationale de France, MS lat. 6992, f. 85rb (detail): Pomum aureum, Practica, cap. 1. The annotation in the margin reads "Note the 20 causes of aborsus."
For example, among the six causes of aborsus he attributed to males was excessive coitus that produced "insufficiently cooked" semen, or problems in the size or shape of his generative organs. For women, the causes were more what we would recognize as conditions that might terminate an established pregnancy, though situations such as ulcers in the uterus might have impeded conception in the first place. Totally unambiguous, however, was the sixth cause: "si usa est aliquibus provocantibus aborsum que nominari non debent propter abusores" ("if she uses any substances provoking abortion/miscarriage which ought not be named because of abusers"). However, this may not have been as stern as a condemnation as it sounds, since André had admitted that a woman might inadvertently use substances that, in other circumstances, were totally innocuous, such as a fumigation of myrrh. And aborsus caused by thunder was, of course, entirely beyond the woman's control.
[Theresa]: One student compared modern medical definitions of losing a child that consider fetal death before twenty weeks of gestation as a miscarriage; after that, it is considered a stillbirth. The student noted that the way we talk about fetal death changes based on audience, that among medical professionals terms are straightforward, but when talking to a mother or a family the physician will be more figurative, using gentle phrases such as "losing a child."
The transcription is still up there on Google Drive, waiting for us to finish it. But that's a task for Monica and me. And we'll keep you posted.
3 thoughts on "Royal Mothers, Part IV: What On Earth Did Pierre André Mean?"
Yvonne said:
This has been a very interesting and useful series of posts—thank you!
Have you tried using the T-PEN transcription tool from St Louis University? I've not used it myself for group collaborations, but I know that it does have that capability and I've found it quite a stable tool for image annotation and transcription.
Monica Green said:
Hi Yvonne, thanks for your comment about T-PEN. I am in fact already affiliated with the folks at the Center for Digital Theology at Saint Louis University, working on another project: http://www.slu.edu/department-of-theology-home/center-for-digital-theology/projects/tradamus/case-testers. However, we decided not to use the T-PEN program for this course just because we already had the students on such a steep learning curve (codicology and paleography and medical history) and we didn't want to add something else they had to learn. GoogleDrive had the advantage of not asking them to use any skills besides basic Word functions. I agree that T-PEN is the way to go for professional transcription, and I would certainly use it when teaching at the graduate level.
Brenda M Cook said:
Brilliant project. This is what scholarship should be all about. Breaking new ground, not reworking old topics. | {
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</table>
<a name="details" id="details"></a><h2 class="groupheader">Detailed Description</h2>
<p>Experimental features not specified by GLSL specification. </p>
<p>Experimental extensions are useful functions and types, but the development of their API and functionality is not necessarily stable. They can change substantially between versions. Backwards compatibility is not much of an issue for them.</p>
<p>Even if it's highly unrecommended, it's possible to include all the extensions at once by including <<a class="el" href="a00377.html" title="Core features (Dependence) ">glm/ext.hpp</a>>. Otherwise, each extension needs to be included a specific file. </p>
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| {
"redpajama_set_name": "RedPajamaGithub"
} | 1,825 |
tofu and tempe : all the same?? » COMMON HOUSE-PLANT AGAINST AIRPOLLUTION [ NASA RESEARCH]
Family • Liliaceae
Spider plant
Chlorophytum comosum (Thunb.) Jacques
RIBBON PLANT
Dian lan
Scientific names Common names
Chlorophytum comosum (Thunb.) Jacques Airplane plant (Engl.)
Ribbon plant (Engl.)
Spider ivy (Engl.)
Spider plant (Engl.)
Dian lan (Chin.)
A tufted grass-like perennial herb growing to a height of 60 cm. Leaves are narrow-linear, blunt at the tip, up to 2 cm wide, recurved, glossy, solid green. The variegated form may be pale green with white longitudinal stripes. Flowering racemes are long, pendulous. Flowers are small and white.
Native to South Africa.
Recently introduced to the Philippines.
Suited for use as groundcover.
Propagated by division of rhizomes and from plantlets.
Parts utilized
Properties and constituents
• Study isolated three new spirostanol pentaglycosides and four known saponins.
• No reported folkloric use in the Philippines.
• In Chinese traditional medicine, used for treating bronchitis, fractures and burns.
• Steroidal Saponins / Antitumor-Promoter Activity: Study isolated three new spirostanol pentaglycosides and four known saponins. The saponins were examined for inhibitory activity on tumor promoter-induced phospholipid metabolism of HeLa cells.
• Antiproliferative: The antiproliferative effects of a n-butanol extract from C comosum was tested in vitro against four human cell lines. Results showed the extract to have antiproliferative effects and apoptosis in human cell lines.
• Indoor Air Purifier: According to a NASA study, spider plants absorb 96 percent of carbon monoxide in a controlled environment within a 24-hour period, making it one of the most effective air purifier in its research. | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 5,273 |
Q: converting doc to html and pdf How can we convert a doc file into an html and a pdf file using c# in a web application?
I am making a web application and i want to convert a doc file into html and pdf as soon as the user clicks on the desired button
A: If you want to convert a word doc to pure html, you will need to run a function that strips all of the garbage characters from your word document.. you would be recreating your own basic CMS.
If you want to convert a word doc to PDF, you will need to run Microsoft Office on a Microsoft server and it gets quite complex and expensive to buy licences for using Office on the server.
If you want to simply upload a word doc to a server, you can display it online using Google Doc Viewer without any conversion. Here is an short article on how to embed a document on a website. All that is required is a short amount of code:
<iframe src="http://docs.google.com/gview?url=http://infolab.stanford.edu/pub/papers/google.pdf&embedded=true" style="width:600px; height:500px;" frameborder="0"></iframe>
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 3,094 |
Application Essay Help Online - A Savior For Prospective Students?
Finding a writer for your application essay is the same as finding a writer for any other essay you may have done. The quickest way to find essay writing assistance online is to find a college social forum, a college writing forum, or any variety of review sites. A general search and assist with this. Let's look at the general process of getting help with your essay.
Granted that you've finally found a place to take care of that for you, you'll have to fill out the order form on their site to get everything set.
On the order form, you'll setup how many pages you need, what kind of essay, the time window it's needed in, the academic level of the writer handling the paper, and if you need proofreading. The other things that you'll setup are notes to the writer and the subject and topic you'll want covered in the paper. Lastly, you'll want to choose the formatting style. Most of selections—except for the text box for notes—are in the form of drop down boxes and checkboxes.
It's also a good idea to use the order form as a means to get a quote on how much your essay will cost.
Pricing on these sites use a combination of number of pages, the type essay you want done, the level of writer, and the time window it needs to be done in. The paper can get pricey depending on the type of paper, the time window's tightness, the academic level of the writer to handle the paper, and the number of pages for this essay.
The level of writer is the most questionable part of the pricing. You don't know if the levels of writing are any different from each other at all. They could be, but you're not going to shell out money on three or four different levels of writers, give them a certain number of pages to do, and read over their work to see the difference. It's a waste of money and you could be wasting money on the uncertainty of how much more different these writers are from each other and if the selectable academic level makes a difference.
Take all of this in mind when getting your paper done online. | {
"redpajama_set_name": "RedPajamaC4"
} | 3,942 |
Q: Why don't environment variables work in $(shell) commands? export HELLO=Hello,world
all:
@echo $(HELLO)
@echo $(shell echo $$HELLO)
@echo `echo $$HELLO`
outputs:
Hello,world
Hello,world
Why is there a difference between backtick and $(shell), and is there a way to pass the environment variables to $(shell) invocations?
I'm trying to use pkg-config in a cross-compilation environment, so I need to set $PKG_CONFIG_SYSROOT. I can use backticks, but it's executed once for every .o file. As per Computing Makefile variable on assignment, I need to use PKG_CFLAGS := $(shell pkg-config $(PACKAGES)), but I can't pass in the required environment variable to make that work properly.
Tested on GNU Make 4.0.
A: Congratulations you have hit a make bug: $(shell) doesn't honor export but this is undocumented?
There's a comment in the code (that precede's the filing of the ticket and is quoted in the ticket) which indicates that there are complicated situations where this cannot work correctly and so it appears that it just isn't done.
I can think of two ways to get the $(shell) environment to have the variables you want to set manually available.
*
*Set them in the $(shell) context explicitly.
PCVAR:=$(shell PKG_CONFIG_SYSROOT=$(make-level-variable-PKG_CONFIG_SYSROOT) pkg-config ...)
*Set them in the make processes environment so the $(shell) environment inherits them normally.
$ PKG_CONFIG_SYSROOT=/some/path make
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 1,543 |
Q: Rsyslog doesn't process log queue from disk We're running rsyslogd on CentOS 6.
We have log forwarding to a central server but the connection was broken due to missing certificates. Now we have processing working again but we have almost 1G of fwdLog1.00000xxx files.
Even after restarting rsyslog multiple times these queue files are not processed. Any ideas on how to force processing of these queue files?
Thanks
These are our settings.
$ActionFileDefaultTemplate RSYSLOG_FileFormat
$ActionSendStreamDriverMode 1 # run driver in TLS-only mode
$ActionSendStreamDriverAuthMode x509/name
$ActionSendStreamDriverPermittedPeer *.domain.com
$ActionQueueMaxDiskSpace 1g # 1gb space limit (use as much as possible)
$ActionQueueFileName fwdLog1 # unique name prefix for spool files
$ActionQueueSaveOnShutdown on # save messages to disk on shutdown
$ActionQueueType LinkedList # run asynchronously
$ActionResumeRetryCount -1 # infinite retries if host is down
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 7,150 |
Q: How to fix ParseError: not well-formed (invalid token): line 1, column 0 in Python I'm using a piece of code that is supposed to be able to detect road damage from images by some trained models. This is part of the code that checks statistical information of the dataset to calculate the number of total images and labels. There is an error with the xml.etree
from xml.etree import ElementTree
from xml.dom import minidom
import collections
import os
import matplotlib.pyplot as plt
import matplotlib as matplot
import seaborn as sns
%matplotlib inline
cls_names = []
total_images = 0
for gov in govs:
file_list = os.listdir(base_path + gov + '/Annotations/')
for file in file_list:
total_images = total_images + 1
if file =='.DS_Store':
pass
else:
infile_xml = open(base_path + gov + '/Annotations/' +file)
tree = ElementTree.parse(infile_xml)
root = tree.getroot()
for obj in root.iter('object'):
cls_name = obj.find('name').text
cls_names.append(cls_name)
print("total")
print("# of images:" + str(total_images))
print("# of labels:" + str(len(cls_names)))
I expect the number of images and number of labels to show
A: The exception indicates that one of the files you're trying to load isn't well formed XML. Try surrounding the ElementTree.parse() section in a try...except block, and printing the filename so you can see which file has the problem.
Update:
from xml.etree import ElementTree, ParseError
from xml.dom import minidom
import collections
import os
import matplotlib.pyplot as plt
import matplotlib as matplot
import seaborn as sns
cls_names = []
total_images = 0
for gov in govs:
file_list = os.listdir(base_path + gov + '/Annotations/')
for file in file_list:
total_images = total_images + 1
if file =='.DS_Store':
pass
else:
try:
infile_xml = open(base_path + gov + '/Annotations/' +file)
tree = ElementTree.parse(infile_xml)
root = tree.getroot()
for obj in root.iter('object'):
cls_name = obj.find('name').text
cls_names.append(cls_name)
except ParseError:
print("Parse error with %s", file)
print("total")
print("# of images:" + str(total_images))
print("# of labels:" + str(len(cls_names)))
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 568 |
\section{Introduction}
\label{sec:intro}
The popular, exuberant, and efficient Deep Neural Networks (DNNs) techniques provide a DNN-based routine for the Fixed-Point (FP) method to handle optimization problems in computer vision \cite{bai2022flow,chang2022obj,fung2022jfb}, decision-making \cite{heaton2021game} and other domains, achieve adorable performance due to existing physical advantages (like GPU computing) or human experiences (like tuning or networks structure settings).
Conventional DNN-FP methods that directly unroll the FP via Convolutional Neural Networks (CNNs) or Recurrent Neural Networks (RNNs) are trained by Back-Propagation (BP) \cite{chen2016tnrd,zhang2017dncnn} or Back-Propagation Through Time (BPTT) \cite{rumelhart1985bptt,werbos1990bptt}. These methods have limited representation ability \cite{chen2016tnrd,qiao2017tnlrd,zhang2017dncnn} and suffer from high memory consumption \cite{gruslys2016memory}, and the gradient vanishing/exploding issues \cite{bengio1994bpttVE,jozefowicz2015empirical,cho2014gru,hochreiter1997lstm} as the number of unrolled depth increases. While Deep EQuilibrium (DEQ) \cite{bai2019deq,bai2020msdeq} methods unroll FP with implicit depth and seek the equilibrium point of FP, they are trained by computing the inversion of the Jacobian of loss w.r.t. the equilibrium point. To alleviate the heavy computation burden of inversion of the Jacobian, an inexact gradient is proposed. However, it yields an undesirable solution.
With the great success of Transformer based models in Natural Language Processing (NLP) \cite{vaswani2017MHSA,lan2019albert,devlin2018bert,brown2020GPT3,shoeybi2019megatron,raffel2020t5} and Computer Vision (CV) \cite{dosovitskiy2021vit,bao2022beit,he2022mae,liu2021swin,liang2021swinir,chen2021ipt,li2021edt,zamir2022restormer}, it has been shown that Transformer-based models are suitable for model the sequential relation with a powerful representation. Motivated by this fact, we propose to unroll the FP and approximate each unrolled process via Transformer called FPformer.
Nevertheless, Transformer-based methods increase the consumption of memory and computation.
To handle this issue, we investigate the parameter sharing \cite{lan2019albert} in FPformer, called FPRformer. In this setting, the successive blocks in Transformers share the parameters, resulting in fewer trainable parameters and maintaining the unrolled iteration times.
Based on our analysis in Section \ref{sec:method_fpaformer}, we further apply Anderson acceleration \cite{anderson1965aa,toth2015aac} to FPRformer via a simplified ConvGRU \cite{cho2014gru,teed2020raft} module to enlarge the unrolled iterations, called FPAformer.
To verify the effectiveness of the proposed FPformer and its variants, we apply it to image restoration task sets as a general image restoration framework. In order to fully exploit the capability of the Transformer, we train the FPformer, FPRformer, and FPAformer using self-supervised pre-training and supervised fine-tuning, widely used in NLP and high-level vision tasks. In the self-supervised pre-training, fixed-point finding for solving image restoration problems becomes a natural interpretation for general image restoration, serving as the self-supervised pre-training problem. We use 161 tasks from 4 categories of image restoration problems to pre-train the proposed FPformer, FPRformer, and FPAformer. Namely, the image restoration tasks are Gaussian denoising in grayscale and color space (noise levels ranging from 0 to 75), single image super-resolution (scale factors are 2, 3, 4, and 8), and JPEG deblocking (quality factors are 10, 20, 30, 40 and 50). During the supervised fine-tuning, the pre-trained FPformer, FPRformer, and FPAformer are further fine-tuned for a specific comparison scenario, e.g., Gaussian denoising in color space with noise level $\sigma=25$.
Using self-supervised pre-training and supervised fine-tuning, the proposed FPformer, FPRformer, and FPAformer achieve competitive performance with state-of-the-art image restoration methods and better efficiency, as shown in Figure \ref{fig:top}, providing a promising way to introduce Transformer in low-level vision tasks.
\begin{figure*}[ht]
\centering
\includegraphics[width=0.8\textwidth]{arch.png}
\caption{The architecture of the proposed FPformer, FPRformer, and FPAformer.\textbf{CDN} and \textbf{GDN} mean Color Gaussian Denoising and Grayscale Gaussian Denoising, respectively. \textbf{JPEG} stands for image JPEG deblocking. \textbf{SISR} means Single Image Super-Resolution.}
\label{fig:arch}
\end{figure*}
\section{Related works}
\label{sec:related}
\subsection{Fixed-Point via DNNs}
The fixed-point is formulated as
\begin{equation}\label{eq:fp}
z^* = \mathcal{F}(z^*).
\end{equation}
The fixed-point finding in Algorithm \ref{alg:fpf}, details in supplementary materials, generates a series of $\{{z_t}\}_{t=1}^{T}$ by successively applying the contraction mapping $\mathcal{F}(.)$, given a initial point $z_0$.
When we focus on the states $z_t$, we can simplify the Algorithm \ref{alg:fpf} as
\begin{equation}\label{eq:unroll1}
z_0 \stackrel{\mathcal{F}(.)}{\longrightarrow} z_1 \longrightarrow \cdots \longrightarrow z_{T-1} \stackrel{\mathcal{F}(.)}\longrightarrow z_T.
\end{equation}
Conventional DNN-FP methods that directly unroll the FP via Convolutional Neural Networks (CNNs) or Recurrent Neural Networks (RNNs), i.e., parameterizing $\mathcal{F}(.)$ with $\mathcal{F}_{\theta}$. These methods either have limited representation ability \cite{chen2016tnrd,qiao2017tnlrd,zhang2017dncnn} or suffer from high memory consumption \cite{gruslys2016memory} and the gradient vanishing/exploding issues \cite{bengio1994bpttVE,jozefowicz2015empirical,cho2014gru,hochreiter1997lstm} as the number of unrolled iterations increases. While Deep EQuilibrium (DEQ) \cite{bai2019deq} methods unroll the FP with implicit depth and seek the equilibrium point of the FP, they are trained by computing the inversion of the Jacobian of loss w.r.t. the equilibrium point. DEQ and its variants \cite{fung2022jfb} suffer from high computation and inexact gradient to achieve an unacceptable solution. In \cite{gilton2021deq}, DEQ is applied in solving inverse problems in imaging, where $\mathcal{F}(.)$ is a specific proximal operator and is further parameterized via $\theta$.
In the fixed-point finding method, the iteration process in Algorithm \ref{alg:fpf} requires quite a lot of iterations, i.e., a large $T$, to reach a feasible equilibrium point $z^*$. One should also note that the choice of hyper-parameters of $\epsilon$ and $T$ is vitally important to achieving a good performance using $\mathcal{F}_{\theta}$, whose contraction property is not well-guaranteed. When applying DNN-FP methods, repeating the modern DNN a couple of times is computation-consuming and time-consuming. For example, DEQ and its variants still consume GPU memory as large as modern DNN, and even more computational consumption to perform the fixed-point finding algorithm to achieve a reasonable performance.
Anderson acceleration (\cite{anderson1965aa,toth2015aac}, AA) is proposed to accelerate the fixed-point finding utilizing the previous $m$ states $\{z_{t-m+i}\}_{i=1}^{m}$ to estimate the next state $z_{t+1}$, as shown in Algorithm \ref{alg:fpaa} in supplementary materials.
In \cite{bai2021ndeq}, Anderson acceleration is integrated into DEQ and parameterized via NNs. In order to explicitly combine the preview $m$ states, the proposed AA module exploits a bottle-neck-like architecture to produces real value weights. Thus, the preview $m$ states are needed to be buffered.
To save the storage costs, in \cite{bai2021ndeq}, the bottle-neck-like network that benefits NLP but degrades image processing because images are represented in 2d-data and have vivid context information in spatial domains while natural languages only need 1d-data.
\subsection{Image Restoration}
Conventional image restoration methods can be used to recover various image degradation problems by minimizing the following energy function,
\begin{equation}\label{eq:ebm}
\mathcal{E}(u,f)=\mathcal{D}(u,f)+\lambda \mathcal{R}(u),
\end{equation}
where $\mathcal{D}(u,f)$ is the data term related to one specific image restoration problem, $f$ is the degraded input image, and $u$ is the restored image. Taking Gaussian denoising as an example, $\mathcal{D}(u,f)=\frac{1}{2{\sigma}^2} \| u -f\| ^2$, where $\sigma$ is the noise level for a specific Gaussian denoising problem. $\mathcal{R}(u)$ is the regularization term known as the image prior model \cite{rudin1992tv,buades2005nlmeans,dabov2007bm3d,danielyan2011bm3d,gu2014wnnm,chen2016tnrd}. Empirically, one can get a minimizer of Equation \ref{eq:ebm} via gradient descent. It can be reformulated into a fixed-point finding diagram when $\mathcal{I}-\nabla \mathcal{E}$ is a contraction mapping.
Benefiting from machine learning methods, the above hand-crafted methods can be further boosted. For example, the diffusion models in \cite{chen2016tnrd,qiao2017tnlrd} are the learned counterparts of their conventional ones. With the rapid development of CNNs and Transformers, the restoration methods are learned in a data-driven manner and provide very impressive performance in image denoising \cite{zhang2021drunet,zhang2017dncnn}, single image super-resolution \cite{wang2018esrgan,zamir2021MPRNet}, JPEG deblocking \cite{zhang2020rdn,zhang2021drunet,fu2021dejpeg}, image deblurring \cite{li2018bid,tao2018srd}, et. al.
Most of CNNs based image restoration methods can be regarded as learning the mapping between the degraded input image $f$ and its corresponding ground-truth image $u_{gt}$. Therefore, we can summarize that these CNNs based methods are parameterized in the very first iteration in Equation \ref{eq:unroll1}, i.e., $z_1=\mathcal{F}_{\theta}(z_0)$. $z_0$ is mapped from the degraded image $f$ via $S_{\phi}$. $z_T \big|_{T=1}$ is mapped to restored image via $U_{\psi}$. Note that, both $S_{\phi}$ and $U_{\psi}$ are identical mapping in \cite{zhang2017dncnn}, and convolution operators in \cite{liang2021swinir}. Due to large parameters and low training efficiency, approximating each step $z_t$ using CNNs is not a common practice. On the contrary, TNRD \cite{chen2016tnrd} and its variants\cite{qiao2017tnlrd} are trained to approximate the unrolling of Equation \ref{eq:unroll1} in fixed iterations. In each iteration, the contraction mapping $\mathcal{F}_{\theta_t}$ is a learned diffusion model parameterized by $\theta_t$. However, the representation ability of TNRD is quite limited. So, we need an effective way to approximate each step $z_{t}$ in Algorithm \ref{alg:fpf}.
\subsection{Vision Transformer}
Multi-Head Self Attention (MHSA) introduced in \cite{vaswani2017MHSA} has been widely used in natural language processing \cite{devlin2018bert,brown2020GPT3,shoeybi2019megatron,raffel2020t5}. In \cite{dosovitskiy2021vit}, MHSA was adopted in computer vision, resulting in vision transformers. And since then, vision transformers \cite{dosovitskiy2021vit,bao2022beit,liu2021swin,he2022mae} trained by self-supervised pre-training and supervised fine-tuning have achieved better performance than CNN models in high-level vision.
In low-level vision represented by image restoration problems, vision transformer architecture is adapted \cite{liang2021swinir,chen2021ipt,li2021edt,zamir2022restormer,tu2022maxim} and also benefits the performance compared with the CNN counterparts \cite{zhang2017dncnn,zhang2021drunet}. These works trained transformer models via supervised training for one specific task and even one specific task setting. In \cite{liang2021swinir,zamir2022restormer}, the proposed methods were trained for Gaussian denoising with different noise levels, for single image super-resolution with different scales, respectively. In \cite{chen2021ipt}, IPT was trained for a couple of image restoration tasks with an auxiliary token, i.e., task-specific token. In \cite{zamir2022restormer}, Restormer is proposed and provides a way to efficiently train for high-resolution image restoration problems. Therefore, the self-supervised pre-training for image restoration is worth discussing to achieve an efficient way to utilize a vision transformer.
\section{Methodology}\label{sec:method}
In this section, we first describe the proposed FPformer approximating the unrolling of the FP. To reduce the memory consumption, we utilize parameter sharing between the successive blocks in FPformer, called FPRformer. To boost the restoration performance, we design a module inspired by the Anderson acceleration algorithm, called FPAformer.
\subsection{Unrolling FP with Transformers: FPformer}\label{sec:method_fpformer}
As described above, DEQ and its variants benefit from the implicitly infinite iterations to conduct fixed-point finding with a large $T$ and small $\epsilon$; they need a huge computational consumption to reach the FP when $\mathcal{F}$ is approximated via modern DNNs. Directly learning the unrolling of the fixed-point finding in fixed iterations, e.g., DnCNN with $T=1$ and TNRD with a larger $T$ less than 10, it is limited by the unrolled times $T$ or the representation ability $\mathcal{F}_{\theta_t}$. We need an effective way to enlarge the unrolled times and strengthen the representation ability.
Having witnessed the success of NLP and CV, the Transformer-based models are suitable for modeling the sequential relation with a powerful representation. Motivated by this fact, we propose to unroll the FP and approximate each unrolled process via Transformer called FPformer.
To this end, we resort to Transformers. To efficiently capture global information, we use Residual Swin Transformer Block (RSTB) blocks as proposed in \cite{liang2021swinir} to learn each contraction mapping in Equation \ref{eq:unroll1}, $\mathcal{F}_{\theta_t}, t=1,\cdots,T$.
\begin{equation}\label{eq:unroll2}
z_0 \stackrel{F_{\theta_1}}{\longrightarrow} z_1 \longrightarrow \cdots \longrightarrow z_{T-1} \stackrel{F_{\theta_{T}}}\longrightarrow z_T.
\end{equation}
The resulting architecture for image restoration is called FPformer and is shown in Figure~\ref{fig:arch} and Equation \ref{eq:unroll2}.
Naturally, the fixed-point finding in Equation \ref{eq:unroll1} of minimizing Equation \ref{eq:ebm} is agnostic to the image restoration task. It just tries to recover a degraded image $f$ to a restored image $u_T$, which is close to the ground-truth clean image $u_{gt}$. Therefore, to fully explore the capability of the Transformer, we trained the FPformer using multiple image restoration problems.
The training of FPformer is formulated as
\begin{align}\label{eq:train_fpformer}
\begin{aligned}
& \mathop{\min}_{\Theta} \sum_{s \in S} \mathcal{L} (u_{gt}^s, u_T^s) \\
z_0^s &= S_{\phi}(f_{task}^s) \\
z_T^s & =\mathcal{F}_{\theta_{T}}(\cdots (\mathcal{F}_{\theta_1}(z_0^s)))= \text{FPformer}_{\Theta}(z_0^s)\\
u_T^s &= U_{\psi}(z_T^s)
\end{aligned}
\end{align}
where $\Theta=\{\theta_1,\cdots,\theta_{T}\}$ is the parameters in FPformer, $\mathcal{L}$ is the loss function that measures the difference between the ground-truth image $u_{gt}^s$ and the restored image $u_T^s$. $u_{gt}^s, f_{task}^s$ are generated using a specific sample in dataset $S$, details are described in Section \ref{sec:exp}. Following \cite{liang2021swinir}, both $S_{\phi}$ and $U_{\psi}$ are convolution operators. To be specific, $S_{\phi}$ represents the degraded image $f_{task}\in \mathbb{R}^{B\times H \times W \times 3}$ as the first state $z_0 \in \mathbb{R}^{B\times H \times W \times C}$. $B$ is the size of the minibatch, $H$ and $W$ are the height and width of the image (or patch), and $C$ is the channel number of features. $U_{\psi}$ maps the last state $z_T \in \mathbb{R}^{B\times H \times W \times C}$ to the restored image $u_T\in \mathbb{R}^{B\times H \times W \times 3}$.
In FPformer, both $S_{\phi}$ and $U_{\psi}$ are shared among image restoration problems and specific tasking settings. To handle different upscales in single image super-resolution, instead of upscaling the features via upsampling blocks, we upscaled the downscaled images with the corresponding scaling factor when preparing the minibatch. Details are discussed in Section \ref{sec:exp}. In SwinIR\cite{liang2021swinir}, $S_{\phi}$ and $U_{\psi}$ are task-specific convolutional operators. In single image super-resolution, dedicated upsampling modules are used. In IPT \cite{chen2021ipt}, $S_{\phi}$ and $U_{\psi}$ are also task-specific, only Transformer blocks are shared. In Restormer \cite{zamir2022restormer}, the degraded and restored image have the same resolution. It may explain why single image super-resolution is not discussed. In summary, FPformer can be treated as a general image restoration solver. Detail performance is discussed in Section \ref{sec:exp}.
\subsection{Sharing parameters: FPRformer}\label{sec:method_fprformer}
In ALBERT \cite{lan2019albert}, sharing parameters among Transformer blocks results in fewer amounts of models parameters and smaller model sizes. In this context, one can maintain the depth of the Transformers while regularizing the whole Transformers, resulting in a small model while providing a competitive performance.
Inspired by this idea, we enforce successive $N_j$ blocks in the $T$ blocks of FPformer to share parameters, coined as FPRformer. Therefore, we have $\sum_{j=1}^{R} N_j = T$, where $R$ is the number of unique RSTB blocks; $N_j$ is the recurrent times of $j_{th}$ unique RSTB block. The amounts of the parameters are about $R/T$ times less compared with FPformer with $T$ RSTB blocks. Note that FPformer can be regarded as a special case of FPRformer with $R=T$ and $N_j = 1, \forall j \in \{1, \cdots, R\}$.
We trained FPRformer with $R=2$, $R=3$ and $R=T$, separately. The details of this ablation study are shown in Section \ref{sec:exp_abla}.
\subsection{Anderson Acceleration: FPAformer}\label{sec:method_fpaformer}
We present a theorem to characterize the performance of \eqref{eq:unroll2}, whose details can be found in supplementary materials.
\begin{theorem}\label{th0}[Informal]
If the trained model $(\theta_t)_{1\leq t\leq T}$ can fit $\mathcal{F}$ well. Let $z^*$ be the fixed point of $\mathcal{F}$, we have
\begin{equation}
\|z_T-z^*\|=\mathcal{O}\Big(\rho^T+\frac{\delta}{1-\rho}\Big)
\end{equation}
for some fixed $0<\rho<1$ and $\delta\geq 0$ reflecting how the model fits (a smaller $\delta$ indicate better fitting).
\end{theorem}
Based on Theorem \ref{th0}, we can immediately get two claims.
\begin{itemize}
\item As $T$ increases, the bound of $\|z_T-z^*\|$ gets small. That indicates when we use a larger $T$, the unrolling yields better results.
\item When $T=\frac{\ln\frac{1-\rho}{\delta}}{\ln\frac{1}{\rho}}$, it holds $\|z_T-z^*\|=\mathcal{O}(\frac{\delta}{1-\rho})$. That means as $T$ is fixed as some integer, the performance is only then determined by how the model fits.
\end{itemize}
Following the above two claims, we conclude that the performance of FPRformer may lag behind that of FPformer because of its parameter-sharing setting. To boost the performance of FPRformer while enjoying the parameter sharing, we design a module analog to the Anderson acceleration to explicitly enlarge the iteration times and get $\delta$ smaller.
As described in Algorithm \ref{alg:fpaa}, the Anderson acceleration algorithm accelerates the FP using the previous states. It is complex to directly translate the Algorithm \ref{alg:fpaa}, especially lines 5-7, into CNNs or RNNs. It is because the forward and backward computation involves the previous outcome of FP and causes nested dependency. This will take more GPU memory and cause more complicated computational graphs which harms the GPU performance greatly. To this end, we simplify the computation in Algorithm \ref{alg:fpaa} into a recurrent module depending on the current state $z_t$ and hidden state $h_t$. Let hidden state $h_t$ to maintain and summarize the previous $m$ states, $\{\mathcal{G}_{t-m_t+1}, \cdots, \mathcal{G}_{t}\}$. In this way, the simplified Anderson acceleration of the fixed-point finding algorithm is formulated as
\begin{align}\label{eq:aafunc}
\begin{aligned}
\hat{z}_{t+1} &= \mathcal{F}_{\theta} (z_t), \\
z_{t+1}, h_{t+1} &=\mathcal{H}_{\mu}(z_t, \hat{z}_{t+1}, h_t),
\end{aligned}
\end{align}
where $\mathcal{H}_{\mu}$ is parameterized by $\mu$. Line 6 of Anderson acceleration algorithm \ref{alg:fpaa} determines the weights of the previous states. Then the weights combine these states to update the next state, as shown in Line 7 of \ref{alg:fpaa}. In our simplified version (\ref{eq:aafunc}), the weights calculation and combination can be summarized as GRU \cite{cho2014gru}. Follow \cite{teed2020raft}, we adopt ConvGRU module to learning $\mathcal{H}_{\mu}$. The proposed simplified ConvGRU module is as follows,
\begin{align}\label{eq:convgru}
\small
\begin{aligned}
\mathcal{G}_t&= \mathbf{Conv}(\hat{z}_{t+1} -z_t ), \\
r_h&= \sigma(\mathbf{Conv}(\mathcal{G}_t)+\mathbf{Conv}(h_t )), \\
r_z&= \sigma(\mathbf{Conv}(r_h )), \\
h_{t+1}&= \mathbf{Norm}((1-r_h ) \odot h_t+r_h \odot \mathbf{Conv}(\mathcal{G}_t)), \\
z_{t+1}&= \mathbf{Norm}((1-r_z ) \odot \hat{z}_{t+1} + r_z \odot \mathbf{Conv}(\mathcal{G}_t)),
\end{aligned}
\end{align}
To analog the $m$ previous states setting in Algorithm \ref{alg:fpaa}, the hidden states $h_t\in \mathbb{R}^{B\times H \times W \times mC}$ is $m$ times larger than $z_t\in \mathbb{R}^{B\times H \times W \times C}$. $\mathbf{Conv}$ is 2d convolutional layer. $\sigma(.)$ is the sigmoid function. $\mathbf{Norm}$ is the layer normalization \cite{ba2016ln}. Layer normalization is commonly used in Transformers and benefits the convergence of the training. Therefore, we add $\mathbf{Norm}$ on the output of $h_{t+1}$ and $x_{t+1}$. $\odot$ is the element-wise product. $h_0$ is initialized as all zero values.
\section{Experiments}\label{sec:exp}
\subsection{Experimental Setup}
\noindent\textbf{Image restoration tasks in training.}
We use commonly used image restoration tasks, e.g., color and gray Gaussian denoising, single image super-resolution (SISR), and image JPEG deblocking in training. \textbf{For color and gray Gaussian denoising}, we obtain noisy images by adding additive white Gaussian noises with noise level $\sigma$ ranging from 0 to 75. \textbf{For SISR}, we downscale and upscale\footnote{We use code from \url{https://github.com/fatheral/matlab_imresize/}.} images with scale 2, 3, 4, 8. Note that instead of upscaling the features $z_t$ via upsampling blocks in FPformer, we upscaled the downscaled images $f$ with the corresponding scaling factor. So, the resulting images $u_T$ are in the same spatial resolution as the ground-truth images $u_{gt}$. \textbf{For image JPEG deblocking}, we generate low-quality images using a JPEG encoder with quality factor $q$=10, 20, 30, 40, 50. Therefore, we train FPformer for 161 tasks from 4 categories of image restoration problems simultaneously. FPRformer and FPAformer are trained in the same setting.
\noindent\textbf{Training datasets.}\label{sec:exp_dataset}
Following \cite{liang2021swinir,wang2018esrgan}, we train FPformer, FPRformer and FPAformer in the above image restoration tasks using random cropped patches from 800 images in DIV2K \cite{agustsson2017div2k}, 2560 images in Flickr2K \cite{timofte2017flickr2k}, 300 images in BSD500 \cite{arbelaez2010bsd500} and all images in WED \cite{ma2016wed}. For DIV2K, we use the first 800 images. For Flickr2K, we use the first 2560 images. For BSD500, we have the whole 300 images in the trainset. For WED, we use the whole dataset.
\noindent\textbf{Pre-Training.}
All training of FPformer, FPRformer, and FPAformer is run on a server with 8 NVIDIA GeForce V100 GPUs. The batch size is 16. The patch sizes are $48 \times 48$, $72 \times 72$, $120 \times 120$ (window size is $8 \times 8$). The RSTB block is set as follows. In FPformer, the number of RSTB blocks is 9 ($T=9$). In FPRformer and FPAformer, the influences of $T$ are discussed in Section \ref{sec:exp_abla}. In each RSTB block, the number of Swin Transformer Layer is 6; the channel number $C$ is 240. The head number of MHSA is 8.
When preparing the training minibatch, we first sample clean images from the above training datasets, crop them into patches (with the above patch size, e.g. $48 \times 48$), and augment these patches. Then we randomly choose an image restoration problem from color and gray Gaussian denoising, SISR, and image JPEG deblocking for each augmented patch. For the chosen problem, we randomly choose the task setting, i.e., the noise level $\sigma$, the scale, or the quality factor $q$. Then we apply the chosen image degradation and setting to each augmented patch. The resulting degraded patches and the augmented clean patches consist of training pairs.
Note that the degraded patches in training pairs are generated on-the-fly instead of generating degraded images offline as \cite{liang2021swinir}. We augment the training images using color space convert augmentation, flipping, rotating and other data augmentation methods as \cite{liang2021swinir}.
The learning rate is halved at [500K, 750K, 900K, 950K, 1000K], the initial learning rate is 2e-4. We use Adam \cite{kingma2014adam} optimizer with $\beta_1 = 0.9$ and $\beta_2 = 0.99$. The training loss $\mathcal{L} (u_{gt}, z_{T} )$ in Equation \ref{eq:train_fpformer} is the Charbonnier loss \cite{charbonnier1994Charbonnier}.
\noindent\textbf{Fine-Tuning.}
When the above pre-training of FPformer, FPRformer, and FPAformer is done, each of them is capable to restore the degraded images from those image restoration tasks. To boost the performance for a specific image restoration problem or task setting, we further finetuned FPformer, FPRformer, and FPAformer. The initial learning rate and schedule are discussed in Section \ref{sec:exp_abla}.
\noindent\textbf{Evaluation.} We pad the image in testing so that the image size is a multiple of the window size. We compare the proposed FPformer, FPRformer, and FPAformer with the previous state-of-the-art methods in color and gray Gaussian denoising, SISR, and image JPEG deblocking. The performance metrics are PSNR and SSIM \cite{liang2021swinir}. The evaluation details are shown in \ref{sec:exp_sota}.
\begin{figure*}[tbph!]
\centering
\includegraphics[width=0.9\textwidth]{srx4.png}
\caption{The $l_2$ distances of input $z_t$ and output $z_{t+1}$ for each layer in SRx4. From left to right, each column FPformer, FPRformer, and FPAformer. The first row is for pre-trained FPformer and its variants. The second row is for fine-tuned models for SRx4.} \label{fig:abla_dist}
\end{figure*}
\subsection{Ablation Studies}\label{sec:exp_abla}
In ablation studies, we investigate the influence of key hyper-parameters on the performance of FPformer, FPRformer, and FPAformer.
The RSTB block is set as above. We compare the performance on Set5 \cite{bevilacqua2012set5} in SISR with scale 4 (SRx4).
\noindent\textbf{The recurrent times $T$ and patch size.}
We trained FPRformer with $R=2$ and $N_1=1$, $N_2=T-1$ given $T$ (6, 8, 10, and 14). We trained FPformer with the default RSTB setting for the different patch sizes listed above. The result is summarized in Table \ref{tab:abla_fprformer}.
The number of parameters in FPformer with different patch sizes is the same, about 27.8M. The number of parameters in FPRformer is 6.5M. Increasing the patch size in training is able to extend the image context, which in turn benefits the performance of FPformer and its variants. Therefore, we believe the performance of FPformer and its variants will continue to increase with larger patch size, as progressive training in \cite{zamir2022restormer}. Due to the limited GPU memory, we were not able to train FPformer and its variants with patch size larger than $120 \times 120$.
\begin{table}[bt!]
\caption{The influence of $T$ and patch size. The evaluation metric is PSNR.}
\label{tab:abla_fprformer}
\begin{center}
{\begin{tabular}{c|c|c|c|c|c}
\hline
patch & \multirow{2}*{FPformer} & \multicolumn{4}{c}{FPRformer} \\
\cline{3-6}
size & & $T=6$ & $T=8$ & $T=10$ & $T=14$ \\
\hline
48 & 32.52 & 32.37 & 32.46 & 32.46 & 32.49 \\
72 & 32.68 & 32.54 & 32.62 & 32.64 & 32.61 \\
120 & 32.78 & 32.64 & 32.76 & 32.78 & 32.78 \\
\hline
\end{tabular}}
\end{center}
\end{table}
As the increase of $T$, the performance of FPformer, FPRformer, and FPAformer increases, as shown in Table \ref{tab:abla_fprformer} and Table \ref{tab:abla_fpaformer}. The performance of FPAformer improves as $T$ increases, even achieving better performance than FPformer.
One can find that the performance of FPRformer and FPAformer is on-par with FPformer. The parameter number of FPRformer and FPAformer is reduced to $23.4\%$ and $30.2\%$ of FPformer's, respectively.
\noindent\textbf{The number of previous states $m$ in $\mathcal{H}_{\mu}$.}
We trained FPAformer with different kernel sizes, $ks$, and various numbers of previous states $m$. The result is summarized in the upper table of \ref{tab:abla_fpaformer}. When $ks=1$, as the increase of $m$ the performance peak is presented in $m=3$. $m$ is set to analog the number of previous states used in Anderson acceleration. Too large or too small $m$ harms the performance. When $m=3$, as the increase of $ks$, the performance is degraded. This indicates that spatial fusion via a larger convolutional kernel is not as important as temporal fusion via larger $m$.
\begin{table}[tpb]
\caption{For a given patch size 48 and 120, $R=2$. In the first super block, $T=10$, the influence of $m$ and kernel size (ks) in $\mathcal{H}_{\mu}$. In the second super block, $ks=1$, $m=3$, the influence of $T$ in FPAformer. The evaluation metric is PSNR.}
\label{tab:abla_fpaformer}
\begin{center}
{\begin{tabular}{c|c|c|c|c|c}
\hline
patch & \multirow{3}*{FPformer} & \multicolumn{4}{c}{FPAformer ($T=10$)} \\
\cline{3-6}
size & & $ks=1$ & $ks=1$ & $ks=1$ & $ks=3$ \\
& & $m=1$ & $m=3$ & $m=5$ & $m=3$ \\
\hline
48 & 32.52 & 32.47 & 32.52 & 32.48 & 32.45 \\
120 & 32.78 & - & 32.82 & - & - \\
\hline
\end{tabular}}
{\begin{tabular}{c|c|c|c|c|c}
\hline
patch & \multirow{3}*{FPRformer} & \multicolumn{4}{c}{FPAformer} \\
size & & \multicolumn{4}{c}{$ks=1$,$m=3$} \\
\cline{3-6}
& & $T=2$ & $T=6$ & $T=10$ & $T=14$ \\
\hline
48 & 32.46 & 32.06 & 32.47 & 32.52 & 32.52 \\
120 & 32.78 & - & 32.70 & 32.82 & - \\
\hline
\end{tabular}}
\end{center}
\end{table}
\noindent\textbf{Contraction mapping.}
Parameter sharing in FPRformer and FPAformer benefits the memory footprint of the models. It is natural to ask why FPRformer and FPAformer exploit fewer parameters while providing competitive performance. And what else can we learn from this setting?
As shown in Figure \ref{fig:abla_dist}, the $l_2$ distances of input $z_t$ and output $z_{t+1}$ for each layer of FPformer becomes larger. On the contrary, those of FPRformer and FPAformer are narrowing down but do not get close to zero. Meanwhile, the minimum $l_2$ distance of FPAformer is smaller than FPRformer. Intuitively, both FPRformer and FPAformer seem to be contractive. It may indicate that FPRformer and FPAformer behave like seeking the fixed-point of image restoration problems. More curves in Gaussian denoising and image JPEG deblocking are shown in the supplementary materials.
We further investigate the behavior when repeating FPRformer with another iteration $T=15$, larger than their training setting, $T=10$. The result is shown in Figure \ref{fig:abla_multiout}. We ran FPRformer in color Gaussian denoising ranging from 5 to 75 with step 5. These noises were added to images in Set5. The denoising performance for each $\sigma$ was averaged over images in Set5. Figure \ref{fig:abla_multiout} indicates that the peak performance is achieved around $T=10 \pm 2$. As iteration exceeds $T=10$, the performance almost stays unchanged. Considering both phenomena, in FPRformer and FPAformer, one can shorten $T$ to balance between the image restoration performance and the inference speed.
The ablation studies reveal that the ability to provide almost the same performance while sharing parameters among blocks comes from the increase of $T$. As $T$ gets larger, the distance in the final $T_{th}$ layer gets smaller, the performance gets better. It seems that FPRformer is seeking the contraction point in the feature space. As mentioned in \cite{bai2019deq}, deep equilibrium models are implicitly seeking the equilibrium point of Equation \ref{eq:unroll1}. On the contrary, FPformer, FPRformer, and FPAformer are trained to find the equilibrium point of Equation \ref{eq:unroll1} with $T$ steps, each of them learned via Transformer blocks. Meanwhile, our proposed methods avoid the computation or approximation of the inverse Jacobian in \cite{bai2019deq}.
\begin{figure}[tbp!]
\centering
\includegraphics[width=0.4\textwidth]{abla-multiout.png}
\caption{The behavior when repeating FPRformer with another iterations $T=15$, larger than their training setting, $T=10$.} \label{fig:abla_multiout}
\end{figure}
\begin{table*}[hbt!]
\caption{Comparison with state-of-the-art methods for single image super-resolution on benchmark datasets in terms of average PSNR/SSIM. The best and second-best results are highlighted and underlined, respectively. $\dagger$ means Fine-tuned for a specific task. $*$ means Pre-trained.}
\label{tab:sr}
\begin{center}
\begin{threeparttable}
{
\begin{tabular}{c|c|c|c|c|c|c||c|c}
\hline
\multirow{3}*{Dataset} & \multirow{3}*{scale} & \multicolumn{7}{c}{Methods} \\
\cline{3-9}
& & NLSA & IPT & SwinIR & FPformer\tnote{$\dagger$} & FPAformer\tnote{$\dagger$} & FPformer\tnote{*} & FPAformer\tnote{*} \\
& & \cite{mei2021nlsa} & \cite{chen2021ipt} & \cite{liang2021swinir} & (ours) & (ours) & (ours) & (ours) \\
\hline
Set5 & 2 & 38.34/0.9618 & 38.37/- & \textbf{38.42/0.9623} & \textbf{38.42}/ \underline{0.9620} & \underline{38.39}/0.9618 & 38.30/0.9616 & 38.28/0.9614 \\
\cite{bevilacqua2012set5}& 3 & 34.85/0.9306 & 34.81/- & \underline{34.97}/\textbf{0.9318} & \underline{34.97/0.9317} & \textbf{34.98}/0.9314 & 34.92/0.9311 & 34.89/0.9308 \\
& 4 & 32.59/0.9000 & 32.64/- & \textbf{32.92/0.9044} & \textbf{32.92}/\underline{0.9033} & \textbf{32.92}/0.9031 & 32.78/0.9024 & 32.82/0.9017\\
\hline
\hline
Set14 & 2 & 34.08/0.9231 & 34.43/- & \underline{34.46/0.9250} & 34.43/0.9241 & \textbf{34.49/0.9251} & 34.08/0.9224 & 34.08/0.9224 \\
\cite{zeyde2010set14} & 3 & 30.70/0.8485 & 30.85/- & 30.93/\textbf{0.8534} & \underline{30.94/0.8523} & \textbf{30.99}/0.8520 & 30.85/0.8511 & 30.81/0.8503 \\
& 4 & 28.87/0.7891 & 29.01/- & \underline{29.09}/\textbf{0.7950} & \textbf{29.10}/\underline{0.7939} & 29.07/0.7928 & 29.03/0.7922 & 29.00/0.7909 \\
\hline
\hline
Manga109 & 2 & 39.59/0.9789 & -/- & 39.92/\textbf{0.9797} & \underline{39.93/0.9795} & \textbf{39.95}/\underline{0.9795} & 39.70/0.9791 & 39.66/0.9789 \\
\cite{matsui2017manga109} & 3 & 34.57/0.9508 & -/- & 35.12/\textbf{0.9537} & \underline{35.15/0.9532} & \textbf{35.22}/\underline{0.9532} & 34.90/0.9522 & 34.96/0.9520 \\
& 4 & 31.27/0.9184 & -/- & 32.03/\textbf{0.9260} & \underline{32.05/0.9251} & \textbf{32.09}/0.9247 & 31.80/0.9227 & 31.87/0.9229 \\
\hline
\end{tabular}}
\end{threeparttable}
\end{center}
\end{table*}
\begin{table*}[bth!]
\caption{Comparison with state-of-the-art methods for grayscale and color image denoising on benchmark datasets in terms of average PSNR. The best and second-best results are highlighted and underlined, respectively. From left to right, the first 7 methods are task-specific trained models. The last 4 methods are task-agnostic models. $\dagger$ means Fine-tuned for a specific task. $*$ means Pre-trained.}
\label{tab:denoise}
\begin{center}
\begin{threeparttable}
{
\begin{tabular}{c|c|c|c|c|c|c|c|c||c|c|c|c}
\hline
\multirow{3}*{Dataset} & \multirow{3}*{$\sigma$} & \multicolumn{11}{c}{Methods} \\
\cline{3-13}
& & BM3D & TNRD & DnCNN & IPT & SwinIR & FPformer\tnote{$\dagger$} & FPAformer\tnote{$\dagger$} & DRUNet & ResTormer & FPformer\tnote{*} & FPAformer\tnote{*} \\
& & \cite{dabov2007bm3d} & \cite{chen2016tnrd} & \cite{zhang2017dncnn} & \cite{chen2021ipt} & \cite{liang2021swinir} & (ours) & (ours) & \cite{zhang2021drunet} & \cite{zamir2022restormer} & (ours) & (ours)\\
\hline
BSD68 & 15 & 31.08 & 31.43 & 31.73 & - & \textbf{31.97} & \underline{31.95} & 31.93 & \underline{31.91} & \textbf{31.95} & \underline{31.91} & 31.89\\
\cite{martin2001bsd68}& 25 & 28.57 & 28.95 & 29.23 & - & \textbf{29.50} & \underline{29.48} & 29.47 & \underline{29.48} & \textbf{29.51} & 29.45 & 29.43 \\
& 50 & 25.60 & 26.01 & 26.23 & - & \textbf{26.58} & \underline{26.56} & 26.55 & \underline{26.59} & \textbf{26.62} & 26.52 & 26.52 \\
\hline
\hline
CBSD68 & 15 & 33.52 & - & 33.90 & - & \textbf{34.42} & \underline{34.40} & 34.37 & 34.30 & \textbf{34.39} & \underline{34.34} & 34.31\\
\cite{martin2001bsd68}& 25 & 30.71 & - & 31.24 & - & \textbf{31.78} & \underline{31.76} & 31.74 & 31.69 & \textbf{31.78} & \underline{31.72} & 31.69\\
& 50 & 27.38 & - & 27.95 & 28.39 & \textbf{28.56} & \underline{28.54} & 28.53 & \underline{28.51} & \textbf{28.59} & \underline{28.51} & 28.49\\
\hline
Kodak24 & 15 & 34.28 & - & 34.60 & - & \textbf{35.34} & \textbf{35.34} & \underline{35.30} & \underline{35.31} & \textbf{35.44} & 35.27 & 35.22 \\
\cite{franzen1999koard24}& 25 & 32.15 & - & 32.14 & - & \textbf{32.89} & \underline{32.88} & 32.86 & \underline{32.89} & \textbf{33.02} & 32.83 & 32.79\\
& 50 & 28.46 & - & 28.95 & 29.64 & \textbf{29.79} & \textbf{29.79} & \underline{29.77} & \underline{29.86} & \textbf{30.00} & 29.75 & 29.72 \\
\hline
\end{tabular}}
\end{threeparttable}
\end{center}
\end{table*}
\noindent\textbf{The learning rate schedule in fine-tuning.}
\begin{figure}[tbp!]
\centering
\includegraphics[width=0.4\textwidth]{ft-SRx4-120-5.jpg}
\caption{The learning rate schedule in fine-tuning. Fine-tuning is performed on SRx4, and results are summarized from Set5 with PSNR and SSIM.}
\label{fig:abla_lr}
\end{figure}
After pre-training, one can further fine-tune FPformer, FPRformer, and FPAformer for a specific image restoration task. Meanwhile, fine-tuning provides an effective way to train, instead of training each task a model from scratch. We fine-tuned the pre-trained FPformer (patch size is 48) using a small learning rate, e.g. 5e-5, for another 10w steps. The performance improves from 32.52dB to 32.57$\sim$32.59dB. The improvement is quite marginal. We also fine-tuned the pre-trained FPformer (patch size is 48) with a large learning rate of 2e-4 used in pre-training. The learning rate is halved at [5K, 105K, 185K, 245K, 285K] in the additional steps. The performance improves from 32.52dB to 32.70dB.
In fine-tuning the pre-trained FPformer with patch size 120, we performed three learning rate schedules starting at 2e-4. The results are summarized in Figure \ref{fig:abla_lr}. 5-5-3-2 schedule means the learning rate is halved at [5K, 55K, 105K, 135K, 155K]. It is a quick fine-tuning strategy, the improvement is about 0.1dB. In the 10-8-6-4 schedule, the learning rate is halved at [5K, 105K, 185K, 245K, 285K]. The fine-tuned models are on-par with SwinIR. In the 20-16-12-8 schedule, the learning rate is halved at [5K, 205K, 365K, 485K, 565K]. It takes about half a step for pre-training. The performance exceeds SwinIR 0.6dB to 32.98dB. It seems that pre-training provides a better initialization. Large initial learning rate and long fine-tuning benefit the performance as well. To balance performance and fine-tuning time consumption, we use a 10-8-6-4 schedule to fine-tune FPformer, FPRformer, and FPAformer. In this schedule, the fine-tuning runtime is about one-quarter of pre-training. Meanwhile, color and grayscale Gaussian denoising are joint fine-tuned. Fine-tuning these pre-trained models for 13 comparison tasks, takes $26.9\%$ time training each task-specific model from scratch. Pre-train + fine-tune training strategy provides an effective training of Vision Transformer-based image restoration models, saving energy.
\subsection{Comparison with state-of-the-art}\label{sec:exp_sota}
In the following comparison, we choose FPformer with patch size 120. FPRformer\footnote{Due to the limited space, the performance of FPRformer is listed in the supplementary materials.} is with $T=10$ and patch size 120. FPAformer is with $T=10$, patch size 120, $ks=1$, $M=3$. The pre-trained and fine-tuned FPformer, FPRformer, and FPAformer are compared with other methods in SISR (x2, x3, and x4), color and grayscale Gaussian denoising ($\sigma$=15, 25, 50) and image JPEG deblocking ($q$=10, 20, 30, 40)\footnote{Due to the limited space, the comparison results for image JPEG deblocking are listed in the supplementary materials.}.
\noindent\textbf{Comparison with DEQ and JFB.}
We adopted and trained JFB and DEQ framework in \cite{fung2022jfb} for single image super-resolution with scale 2. The $\mathcal{F}_{\theta}$ was built on the network (38.7M parameters) for CIFAR10 in \cite{fung2022jfb}. We finetuned key hyper-parameters, i.e., $T$ and $\epsilon$. Comparison is conducted on Set5 in terms of PSNR and SSIM. JFB achieves 33.67/0.9303 (PSNR/SSIM), and DEQ achieves 38.15/0.9608. As shown in Table \ref{tab:sr}, JFB and DEQ are behind others a lot.
\noindent\textbf{SISR.}
We test comparison methods on Set5 \cite{bevilacqua2012set5}, Set14 \cite{zeyde2010set14} and Manga109 \cite{matsui2017manga109} for SISR with scale 2, 3 and 4. Following \cite{liang2021swinir}, we report PSNR and SSIM on the Y channel of the YCbCr space, as summarized in Table \ref{tab:sr}. FPformer$^\dagger$ outperforms IPT (115.5M parameters) and is on-par with SwinIR. FPAformer$^\dagger$ uses fewer model parameters and is on-par with SwinIR on Set5, outperforming SwinIR about 0.02dB on Set14 and 0.06dB on Manga109.
\noindent\textbf{Image denoising.}
We test comparison methods on BSD68 \cite{martin2001bsd68} for grayscale denoising with noise levels 15, 25, and 50. We compare color denoising with noise levels 15, 25, and 50 on CBSD68 \cite{martin2001bsd68} and Kodak24 \cite{franzen1999koard24}. Following \cite{liang2021swinir}, we report the PSNR on the RGB channel and Y channel for color and grayscale denoising, respectively. The experimental results are summarized in Table \ref{tab:denoise}.
In task-specific methods, the first 7 methods in Table \ref{tab:denoise}, FPformer$^\dagger$ and FPAformer$^\dagger$ outperform IPT and are on-par with SwinIR. In task-agnostic methods, the last 4 methods in Table \ref{tab:denoise}, FPformer$^*$ and FPAformer$^*$ use fewer model parameters and provide competitive performance with DRUNet (32.7M parameters) and Restormer (25.3M parameters). The number of model parameters of FPAformer is only $7.3\%$ of those in IPT, $25.7\%$ of DRUNet, $33.2\%$ of Restormer.
\section{Conclusion}
In this work, we propose to learn the unroll of fixed-point via Transformer based models, called FPformer. By sharing parameters, we achieved a lightweight model, FPRformer. A module is proposed to analog the Anderson acceleration to boost the performance of FPRformer, called FPAformer. To fully exploit the capability of Transformer, we apply the proposed model to image restoration, using self-supervised pre-training and supervised fine-tuning. The proposed FPformer, FPRformer, and FPAformer use fewer parameters and achieve competitive performance with state-of-the-art image restoration methods and better training efficiency. FPRformer and FPAformer use only 23.21\% and 29.82\% parameters used in SwinIR models, respectively. To train these comparison models, we use only 26.9\% time used for training from scratch.
\section*{Acknowledgments}
This work was supported by the National Natural Science Foundation of China under the Grant No.61902415.
\bibliographystyle{IEEEtran}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 9,258 |
\section{Introduction}
Object No. 433 in the H-alpha-emission objects catalog by \cite{1977ApJS...33..459S} was recorded exhibiting highly variable radio emissions for the first time in 1978. Since then the SS 433 system has been extensively observed across the electromagnetic spectrum. SS 433 is a binary system located at about $5.5\pm0.2$ kpc from Earth with an accreting black hole of $9\mathrm{M_{\odot}}$ orbitting a $30\mathrm{M_\odot}$ A type supergiant Wolf-Rayet star with an orbital period of 13.082 days \cite{2004ApJ...616L.159B,2002sf2a.conf..317F}. The relativistic, bipolar, hadronic jets spew material with velocity of 0.26c and precess with a period of 162.5 days in cones of half-opening angle $\theta \approx 20^\circ$\cite{2005A&A...437..561C}. The jets are inclined at an angle of $\sim79^\circ$ with the line of sight. The high kinetic luminosity of the jets $L_k\sim10^{39} \mathrm{erg \ s^{-1}}$ \cite{1998AJ....116.1842D} indicates exceptional power output, possibly contributing a reasonable fraction of the Galactic cosmic-ray flux\cite{2002A&A...390..751H}. Models had suggested the cosmic microwave background (CMB) photons could be inverse-Compton upscattered by electrons at the interaction region between the eastern jet and interstellar medium thereby producing TeV gamma rays \cite{1998NewAR..42..579A}. Another model by \cite{2010IJMPD..19..749B} suggests that the interaction region between the western jet and W50 nebula serves as a possible site for GeV-TeV emission arising from relativistic Bremsstrahlung emission. A detection of gamma rays in either eastern or western jet interaction region would provides a unique laboratory to study relativistic shock acceleration with well-known input parameters in addition to confirming the first TeV gamma-ray binary with a microquasar.
\section{VERITAS Observations and Analysis}
The Very Energetic Radiation Imaging Telescope Array Sytem (VERITAS) is an array 4 telescopes that uses the imaging atmospheric-Cherenkov technique to observe gamma rays in the 85 GeV to $>30$ TeV energy range. The telescopes, located at the Fred Lawrence Whipple Observatory (FLWO) in southern Arizona ($31^{\circ}40$' N, $110^{\circ}57$' W, 1.3 km a.s.l.), each have a 12-m diameter reflector focusing light onto a 499 pixel photomultiplier tube (PMT) camera, giving it a $3.5^{\circ}$ field of view. For detailed description and characterization of the instrument refer to \cite{2015ICRC...34..771P}.
The VERITAS observations of SS 433 used in this work were taken between September 2009 and July 2012. This data corresponds to the configuration of the instrument which is after the relocation of one the telescopes to make the array more symmetric and prior to the upgrade of the PMTs in the camera. For a detailed characterization of the instrument during this epoch see \cite{2010HEAD...11.3904P}. Located in the galactic plane, SS 433 is only a few degrees away from the extended TeV source MGRO J1908+06 (R.A. 19h 07m 54s Dec. $+06^{\circ}16'07''$) \cite{2014ApJ...787..166A}. To maximise exposure on both of these objects, the camera was pointed at a central location between them, thereby accommodating both SS 433 and MGRO J1908+06 within the field of view. A total of $\sim70$ hours of quality-selected data (live time) were obtained, all with the SS 433 central black hole within $1.5^{\circ}$ from the camera center. Table \ref{veritasobs} shows a summary of the VERITAS observations of the SS 433 region.
\begin{table}[h]
\centering
\caption{VERITAS observations of SS 433
\label{veritasobs}
\vspace{\baselineskip}
\begin{center}
\begin{tabular}{|c | c | c | c | c |}
\hline \hline
Year & Exposure [h] & No. of Runs & Camera Offset $(^{\circ})$ & Mean Elevation $(^{\circ})$\\
\hline
2009 & 9.33 & 27 & 0.7-1.0 & 59 \\
2010 & 10.5 & 37 & 0.7-0.8 & 60 \\
2011 & 26.11 & 88 & 0.05-1.1 & 61.2 \\
2012 & 25.23 & 79 & 0.7-1.5 & 62.4 \\
\hline
Total & 71.17 & 231 & 0.05-1.5 & 61 \\
\hline \hline
\end{tabular}
\end{center}
\end{table}
The SS 433 system is extended in the sky over a $2^{\circ}\times2^{\circ}$ region. Four locations were selected as regions of interest (ROIs) for this work. These are the central location of the black hole, two locations e1 and e2 in the eastern jet interaction region and, one location w2 in the western jet interaction region. These ROIs were selected from a previous X-ray spectral analysis study using ROSAT/ASCA data, as they were deemed possible sites for VHE emission (see \cite{1997ApJ...483..868S,2009AA...497..325B}).
Each of the pre-selected ROIs were searched for point source emission. The data was analyzed using a standard analysis package, implementing a specialized Boosted Decision Tree (BDT) technique \cite{2017ICRC...M,2017APh....89....1K}. According to the model outlined in \cite{2008MNRAS.387.1745R}, the precessing jet should cause phase-based flux variations from SS 433. Setting $\phi=0$ at JD 2443507.47 when the accretion disc is maximally open\footnote{In this orientation of the disc the gamma rays emitted in the innermost regions would escape without travelling through the thick extended and disc undergoing $\gamma\gamma$ absorption} to the observer and using a precession period of 162.5 days, the data are divided into five phase bins of width $\Delta\phi=0.2$ \cite{2013MNRAS.436.2004C}. Each ROI is also searched for TeV emission during each of the five individual phases.
\section{Results}
No significant emission is found from the location of the black hole or any of the three jet and interstellar medium interaction regions w2, e1 or e2. The VERITAS skymap of the SS 433 region is shown in Figure \ref{ss433_skymap}. The location of the ROIs are marked in white. The skymap is overlayed with X-ray emission contours in black from ROSAT/ASCA \cite{1997ApJ...483..868S}. The green radio contours from VLA \cite{1998AJ....116.1842D} are also shown on the VERITAS skymap. There is weak evidence of emission ($\sim4\sigma$ above the background not accounting for the number of statistical trials conducted in this analysis) from e1, one of the ROIs in the eastern jet interaction region, but as the significance is below $5\sigma$, no positive detection is claimed.
\begin{figure}[t]
\centerline{\includegraphics[scale = 0.5]{ss433_contours.png}}
\caption{ VERITAS significance skymap of the SS 433 region. The 4 regions of interest, SS 433 black hole, w2, e1 and e2 are marked in white. Contours in green are from radio observations \cite{1998AJ....116.1842D}. Contours in black are X-ray observations from ROSAT/ASCA \cite{1997ApJ...483..868S}. Bright extended emission from MGRO J1908+06 is seen on the top right of the skymap located within the cyan circle \cite{2014ApJ...787..166A}, this region is excluded from the background estimation of the analysis.}%
\label{ss433_skymap}
\end{figure}
A plot of the model for flux variation due to the jets precession adapted from \cite{2008MNRAS.387.1745R} is overlaid with integral flux upper limits from the four ROIs, and shown with respect to the phase-based flux variation model in Figure \ref{reynoso_ul}. For each of the phase bins, 99\% confidence level flux upper limits above 600 GeV are also calculated for the four ROIs, a summary of all the upper limits is also presented in Table \ref{ultable}.
\begin{figure}[t]
\centerline{\includegraphics[scale = 0.4]{reynoso_model_results.png}}
\caption{ VERITAS 99\% confidence level flux upper limits $>600$ GeV for the 4 regions of interest, SS 433 black hole in red, w2 in green, e1 in magenta and e2 in cyan. These upper limits are overlayed on a predicted model of phased emission adapted from \cite{2008MNRAS.387.1745R}. The doted line shows prediction from 2005 of current generation Cherenkov Telescope sensitivity. The modelled contribution to gamma-ray flux $>100$ GeV from the two jets are shown separately, where the solid line represents the approaching or western jet and the dotted line represents the recessing or eastern jet}%
\label{reynoso_ul}
\end{figure}
\section{Discussion}
Analysis of 10.4 hours of data recorded between September 2007 and July 2008 which is not included in this analysis, found a $4.9\sigma$ excess at w2 (not accounting for statistical trials), a location in the interaction region between the western jet and the surrounding interstellar medium \cite{2010PhDT.......228G}. This motivated further observations by VERITAS during the 2009-2012 period. The additional observations presented in this work did not detect any statistically significant VHE emission from w2 or any of the other selected regions in the SS 433 system, namely the position of the black hole and the two locations e1 and e2, which are situated in the interaction region of the eastern jet and the surrounding interstellar medium. Out of the total $\sim70$ h of VERITAS data, nearly 26 h are taken in an unfavorable phase when the star obscures the disc and jet. In this orientation the high density of matter from the star and the surrounding W50 nebula subjects gamma rays to a high degree of absorption by mechanisms like $\gamma\gamma$ interactions with ambient soft photons and by $\gamma N$ (where $N$ represents nucleons) interactions with disc and stellar matter. For details of various possible absorption mechanisms see \cite{2008APh....28..565R}.
X-ray emission detected from the inner and outer lobes of SS 433 may be of non-thermal origin as suggested in \cite{1994PASJ...46L.109Y,1997ApJ...483..868S}. Recently, gamma-ray emission from the direction of SS 433 was detected by \textit{Fermi}-LAT revealing a very peculiar spectral energy distribution of the source with a distinct maximum at 250 MeV and extending only up to 800 MeV \cite{2015ApJ...807L...8B}. Although it is not yet clear whether the detected emission is associated with SS 433 the lack of other plausible counterparts in the region supports this hypothesis. If this is the case, the cutoff in the GeV spectrum implies that the maximum energies to which electrons and protons are accelerated are just a few GeV \cite{2015ApJ...807L...8B}. Thus, TeV emission from the source is not expected. However, such low maximum energies of accelerated particles are in contradiction with the non-thermal interpretation of the X-ray emission which requires electrons with energies in the order of 10-100 TeV \cite{1994PASJ...46L.109Y,1997ApJ...483..868S}. It is also unclear where the particles responsible for the HE gamma-ray emission are accelerated. The poor angular resolution of \textit{Fermi}-LAT at these energies ($\geq1.5^{\circ}$ at energies of about 300 MeV) \cite{2015ApJ...807L...8B} does not allow it to resolve the origin of the emission. However, the lack of flux variability suggests that the emission is generated in outer regions far from the binary system, since otherwise it would be subject to strong phase dependent absorption by photo-hadronic interactions with disk and stellar matter, which would show evidence of precessional modulation \cite{2008APh....28..565R}. If this is the case, possible, detectable very-high-energy emission from inner regions of jets cannot be ruled out.
\begin{table}[h]
\center
\caption{VERITAS upper limits of the SS 433 binary system }
\label{ultable}
\vspace{\baselineskip}
\begin{tabular}{ |c|c|c|c|c|c| }
\hline \hline & & & & & 99\% Flux UL \footnotemark\\
Phase & $N_{ON}$ & $N_{OFF}$ & $\sigma$ & Live time (min) & ($>600$ GeV)\\ & & & & & $[ \mathrm{cm^{-2} \ s^{-1}}]$\\
\hline
\multicolumn{6}{|c|}{\cellcolor{red!50}Upper limits of the SS 433 Black hole position} \\
\hline
0.1-0.3 & 52 & 586 & 0.2 & 1113.7 & 4.12$\times 10^{-13}$\\
0.3-0.5 & 62 & 731 & -0.1 & 1515.9 & 3.39$\times 10^{-13}$ \\
0.5-0.7 & 35 & 357 & 0.7 & 630.5 & 7.95$\times 10^{-13}$ \\
0.7-0.9 & 22 & 231 & 0.3 & 414.8& 7.90$\times 10^{-13}$\\
0.9-1.1 & 25 & 282 & 0.0 & 577.1 & 5.70$\times 10^{-13}$\\
\hline
All & 196 & 2195 & 0.4 & 4272.1 & 2.29$\times 10^{-13}$ \\
\hline
\multicolumn{6}{|c|}{\cellcolor{green!50}Upper limits of western lobe (w2)} \\
0.1-0.3 & 57 & 821 & 0.6 & 1113.7 & 4.36$\times 10^{-13}$ \\
0.3-0.5 & 69 & 1080 & 0.0 & 1515.9 & 3.16$\times 10^{-13}$\\
0.5-0.7 & 42 & 546 & 1.1 & 630.5 & 8.07$\times 10^{-13}$ \\
0.7-0.9 & 31 & 325 & 2.0 & 414.8 & 1.17$\times 10^{-12}$\\
0.9-1.1 & 26 & 463 & -0.6 & 577.1 & 4.05$\times 10^{-13}$\\
\hline
All & 225 & 3245 & 1.2 & 4272.1 & 2.66$\times 10^{-13}$\\
\hline
\multicolumn{6}{|c|}{\cellcolor{magenta!50}Upper limits of eastern lobe (e1)} \\
0.1-0.3 & 49 & 563 & 1.1 & 1113.7 & 6.22$\times 10^{-13}$\\
0.3-0.5 & 68 & 583 & 2.6 & 1505.8 & 8.54$\times 10^{-13}$\\
0.5-0.7 & 36 & 278 & 2.5 & 630.5 & 1.45$\times 10^{-12}$\\
0.7-0.9 & 23 & 218 & 1.6 & 414.8 & 1.40$\times 10^{-12}$\\
0.9-1.1 & 23 & 280 & 0.5 & 577.1& 7.97$\times 10^{-13}$\\
\hline
All & 199 & 1925 & 3.7 & 4262.1& 5.68$\times 10^{-13}$ \\
\hline
\multicolumn{6}{|c|}{\cellcolor{cyan!50}Upper limits of eastern lobe (e2)} \\
\hline
0.1-0.3 & 39 & 427 & -0.1 & 1113.7 & 4.47$\times 10^{-13}$ \\
0.3-0.5 & 61 & 435 & 3.0 & 1445.7 & 1.03$\times 10^{-12}$\\
0.5-0.7 & 21 & 231 & 0.2 & 630.5 & 7.79$\times 10^{-13}$\\
0.7-0.9 & 15 & 172 & -0.1 & 414.8 & 9.23$\times 10^{-13}$\\
0.9-1.1 & 17 & 172 & 0.4 & 577.1 & 7.77$\times 10^{-13}$\\
\hline
All & 153 & 1442 & 1.8 & 4202.0 & 4.21$\times 10^{-13}$\\
\hline \hline
\end{tabular}
\end{table}
\section{Conclusion}
A targeted point source search was applied to various locations in the SS 433 system. VERITAS did not detect any significant TeV emission from these predefined regions which could imply that SS 433 may be not be intrinsically as luminous above 600 GeV as was previously thought or if gamma rays are produced in the inner regions of the system, they are absorbed significantly by the surrounding W50 nebula. The possibility of extended emission from the jet and interstellar medium interaction regions, as suggested by the $\sim4\sigma$ excess from e1 region in the eastern jet, and a study exploring the skymap with extended source analysis methods is currently being done. \\
\noindent\textbf{Acknowledgments} \\ \footnotetext{All 99\% confidence level upper limits calculated using Rolke metod \cite{2001NIMPA.458..745R}}
The author would like to thank Gloria Dubner and Samar Safi-Harb for providing the radio and X-ray contours. This research is supported by grants from the U.S. Department of Energy Office of Science, the U.S. National Science Foundation and the Smithsonian Institution, and by NSERC in Canada. We acknowledge the excellent work of the technical support staff at the Fred Lawrence Whipple Observatory and at the collaborating institutions in the construction and operation of the instrument.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 1,315 |
package net.sansa_stack.query.tests
/**
* A single test case for SPARQL query evaluation.
* @param uri the URI of the test case resource
* @param name the name of the test case
* @param description an (optional) description of the test case
* @param queryFile the path to the file containing the query to evaluate
* @param dataFile the path to the file containing the data on which the query will be evaluated
* @param resultsFile the path to the file containg the result of the query evaluation, i.e. the target solution
*
* @author Lorenz Buehmann
*/
case class SPARQLQueryEvaluationTest(uri: String,
name: String,
description: String,
queryFile: String,
dataFile: String,
resultsFile: Option[String])
| {
"redpajama_set_name": "RedPajamaGithub"
} | 5,798 |
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This entry was posted in Miscellaneous on September 3, 2018 by admin. | {
"redpajama_set_name": "RedPajamaC4"
} | 6,425 |
Discover the heart of Perth's nightlife, dining and cultural scene from our DoubleTree by Hilton Perth Northbridge hotel. We are a short walk from the city center and many of Perth's top attractions. Discover the 'new heart of Perth' at Yagan Square, just steps from our hotel. This new hot spot offers, premier dining, shopping and entertainment. Take a short 20-minute leisurely stroll to Elizabeth Quay on the Swan River, the place to see and be seen with its island playground, BHP Billiton Water Park, public artwork, promenades, open spaces and dining options.
Perth Arena is a popular venue for basketball games and live concerts. Explore the more than 17,000 works of art at the Art Gallery of Western Australia or the many exhibits at Perth Museum. Take in a performance at the State Theatre Centre and enjoy many of the festival events in Northbridge throughout the year.
James St Bar & Kitchen
City Center 1.0 KM S
Elizabeth Quay 1.2 KM S
Kings Park 3.1 KM SW
Optus Stadium 8.2 KM E
Perth Arena 1.0 KM W
Perth Cultural Centre 0.3 KM E
Yagan Square 0.5 KM N
Sight Seeing Tours 0.1 KM
BHP Billiton 1.2 KM NE
Chevron 1.5 KM N
Chicago, Bridge & Iron 1.2 KM N
Commonwealth Bank 1.1 KM N
McKinsey & Co 1.1 KM N
South32 0.85 KM N
Wesfarmers 1.1 KM N
In and Around Town
Yagan Square
A pedestrian precinct with a permanent food market, native gardens, play areas, cafés, restaurants and Aboriginal art.
Explore Yagan Square
Just a 5 minute walk to the Murray and Hay Street Malls with department stores, international brands and quirky local boutiques.
Be entertained at the Perth Cultural Centre where arts, culture, knowledge and community come together. | {
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Q: In what situations should I use Entity SQL? I was wondering whether there are situation that's more advisable to use ESQL?
Generally, I find ESQL frustrating (specially with all the special cases where you need to fully qualify an entity type) & probably anything done using ESQL can be done through SQL or LINQ.
But I was wondering of situations where ESQL is a better fit for a solution or would have a competitive edge over using SQL or LINQ (either easier/faster to code, or better performance, etc.)
So, what's the compromise here? When is it better to use each one of the three approaches in querying over EF4?
A: I came across these pretty similar questions (which didn't show up when I typed mine, earlier) on stack over flow, I guess they have deeper discussions (though T-SQL is not mentioned a lot, but it's kinda pretty obvious though):
*
*Linq to Entities vs ESQL - Stack
Overflow
*Entity framework entity sql vs linq
to entities
Note that the two questions are a bit "old" so you might validate some of data based on your current understanding of EF4
A: I find that ESQL to be good for edge cases, for example:
*
*Where it's just really hard to express something in LINQ.
*For building very dynamic searches.
*If you want to use a database specific function that is exposed by the provider you are using.
Also, if you know Entity SQL, you will be able to express QueryViews and Model-Defined Queries.
A: Just as what julie mentioned. Esql is required to write model defined functions which in return can be used through LINQ to Entities Queries.
Most cased you'll be using LINQ to Entities. One another case where you can't use LINQ TO Entities us when you want to build queries with store functions, either buit in or UDF. In this case esql is your only way in EF1 but in EF4 you can use its features to expose those functions to be used in LINQ.
About performance, esql is performing better. but you might prefer productivity using Linq over performance gain using esql.
| {
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Akhal-Teke
Despite Ashgabat's fervent diplomacy, one ingredient is missing from its dream project across Afghanistan. Our weekly briefing.
It was not enough that the president earlier this month subjected minions and diplomats to his latest contribution to world culture – a poem titled "Long live peaceful life!"
Artists and officials responsible for Turkmenistan's creative industry were last week compelled to gather and sing praise to this work's allegedly glorious merits. As TDH state news agency has it in a report on the December 15 conference, leading figures in the arts and culture scene unanimously expressed gratitude for Gurbanguly Berdymukhamedov's policies of "peace-loving, humanness, and mutually beneficial cooperation with all the peoples of the planet."
The poem has now also been put to music and was performed at the conference. Several people in the audience appeared to have memorized the verses already, although many others merely moved their lips indistinctly.
This phase of what might come in future to be looked back at as late Berdymukhamedovism is chock-full of transparent attempts at legacy-making. The president is desperate for international recognition of his contributions to regional stability. Afghanistan is offering ample opportunity to put this stated idealism into practice.
Speaking at a council of foreign ministers of the Organization of Islamic Cooperation in Pakistan on December 19, Turkmen Deputy Foreign Minister Vepa Khadzhiyev reminded listeners that his government has already been actively providing their neighbor to the east with aid in this extremely difficult time.
"Humanitarian convoys are dispatched on a regular basis. The last was organized on December 14. Humanitarian cargo sent to Afghanistan includes oil products, foodstuffs, and children's clothing," the Foreign Ministry said in a statement summarizing Khadzhiyev's speech.
Images of a handful of trucks bearing the sign "Humanitarian aid to the Afghan people from Turkmenistan's president" being distributed to grateful aid recipients were shown on state TV news on December 15.
There is a more pragmatic, hard-nosed dimension to all this, though. Ashgabat is eager that its vision for a series of Afghanistan-related, energy, transportation and telecommunications infrastructure projects not come to nothing.
Turkmen Foreign Minister Rashid Meredov met in Islamabad on December 20 with Pakistan's President Arif Alvi to remind him about the need to "intensify collective action" on implementation of the Turkmenistan-Afghanistan-Pakistan-India, or TAPI, natural gas pipeline project, the TAP high-voltage power transmission line project and a third concurrent project to lay fiber-optic communication cables alongside the power lines.
Ashgabat rarely does muscular diplomacy, but there is a sense here that the Turkmens are growing more than a little exasperated at Pakistan's foot-dragging on TAPI. Only last month, a senior Pakistani government official noted that work on TAPI had been halted pending stabilization of the situation in Afghanistan. And it's not just TAPI. "At the moment no one is working on this project in Afghanistan, or any other projects either," Pakistan's Economic Affairs Minister Omar Ayub Khan told Russia's TASS news agency.
Meredov delivered the same TAPI talking points a day earlier, on December 19, at the third edition of the India-Central Asia Dialogue conference held in India's capital, New Delhi. The matter was once more raised in Meredov's tête-à-tête with his Indian counterpart, Subrahmanyam Jaishankar.
The frenetic nature of Meredov's travels attest to just how much emphasis Ashgabat is placing on this agenda. He met with Pakistan's foreign minister in Islamabad on December 18. He traveled to India the next day. And then he returned to Pakistan the day after that to meet the president.
The perennial stumbling block is, however, to quote the title of an Abba song "money, money, money."
The Asian Development Bank has previously signaled it may be good for USD 1 billion or so, although that was long before the Taliban took over in Afghanistan. Turkmenistan took out a USD 700 million loan in October 2016 from the Jeddah-based Islamic Development Bank to fund work on TAPI, but there is no knowing whether that has already been spent, and if so, on what. And members of the TAPI Pipeline Company Limited consortium – the four countries plus the ADB, in other words – in 2016 pledged to collectively invest USD 200 million into preparatory work. But seeing as the final cost of making the pipeline a reality may end up somewhere in the neighborhood of USD 10 billion, these are laughable figures.
A new outfit created in October may, possibly, be intended to cover over some cracks. The Turkmen Investment Company was established as a joint venture between the State Bank for Foreign Economic Affairs, or Turkmenvnesheconombank, as it is better known, and the Abu Dhabi Fund for Development. On December 17, Berdymukhamedov signed a degree formalizing Turkmenvnesheconombank's control over 50 percent of the equity capital in the Turkmen Investment Company – a maneuver implying that money has been disbursed and put on the table. There are no details about how much money is involved or what ends it has been set aside for, though, so any suggestion the fund could be used for the purposes of implementing TAPI may well be idle speculation.
One figure that has been made public is how much the OPEC Fund for International Development has lent Turkmenistan for the development of a marine merchant fleet. The USD 45 million worth of credit extended by the fund will be used to build three ships for the transportation of rail, passengers and dry cargo, according to Russia's TASS news agency.
As he turned to domestic matters during the December 17 Cabinet meeting, Berdymukhamedov called upon his son, Serdar, who is also the deputy prime minister with the portfolio for economic affairs, to run checks on the regions to ensure bazaars keep their wares affordable as the New Year beckons.
"Before the New Year, visit markets and shops, familiarize yourself with the prices of goods," the president told his son and heir apparent.
Amsterdam-based Turkmen.news has relayed some grumbling on this front. The outlet's sources have claimed that government inspectors in the Dashoguz province, which Serdar Berdymukhamedov was scheduled to visit on December 21, have flooded the bazaars demanding that butchers sell their meat at 35 manat per kilo (around USD 1.25 at black market rates), instead of the real market price of 60 manat. Traders have reportedly responded by refusing to come to work until the president's son wraps up his visit.
There may finally be some recognition of reality occurring among officials in charge of agriculture at least. At the Cabinet meeting, Esenmyrad Orazgeldiyev, the deputy prime minister with the portfolio for agriculture, informed the president that the volume of land being set aside for growing cotton is being reduced to make more room for potatoes, other vegetables, and melons, as well as for the increased production of silkworm cocoons. Fully 40,000 hectares will be freed up for those purposes, almost doubling the 55,550 hectares now available, while 580,000 hectares of agricultural land will still be used for cotton.
Officials are pretending this change is being made possible by a surge in productivity in the cotton fields, but it is actually because Turkmenistan has been having to spend lavishly on potato imports to compensate for shortages. So much for import-substitution. | {
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{"url":"https:\/\/physics.stackexchange.com\/questions\/527465\/dynamic-or-kinematic-viscosity\/527534","text":"# Dynamic or Kinematic viscosity\n\nUtilising the formula $$\\eta = \\frac { \\left( 2\\left( { p }_{ s }-{ p }_{ l } \\right) g{ r }^{ 2 } \\right) }{ 9v }$$\n\nWhere $$p_s$$ is density of a sphere with radius $$r$$ traveling at velocity $$v$$ through a liquid $$l$$. I would like to know whether this equation is utilised to find kinematic or dynamic viscosity.\n\nThanks in advanced for the help.\n\nIt is quite easy to figure out using dimensional analysis. The SI unit of the dynamic viscosity commonly denoted $$\\eta$$ or $$\\mu$$ is the pascal-second (also called poiseuille named after Jean-L\u00e9onard-Marie Poiseuille). Therefore its dimension is\n\n$$\\left[\\eta\\right] = \\mathsf{M}\\mathsf{L}^{-1}\\mathsf{T}^{-1}.$$\n\nSince the kinematic viscosity $$\\nu$$ is a diffusivity coefficient\n\n$$\\left[\\nu\\right] = \\mathsf{L}^2\\mathsf{T}^{-1}.$$\n\n$$\\eta \\propto \\frac{(\\rho_\\mathrm{s}-\\rho_\\mathrm{l})gr^2}{v} \\Longrightarrow \\left[\\eta\\right] = \\frac{[\\rho][g][r]^2}{[v]} = \\frac{(\\mathsf{M}\\mathsf{L}^{-3})(\\mathsf{L}\\mathsf{T}^{-2})(\\mathsf{L})^2} {\\mathsf{L}\\mathsf{T}^{-1}} = \\mathsf{M}\\mathsf{L}^{-1}\\mathsf{T}^{-1}$$","date":"2020-06-01 14:29:28","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 11, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.9287241101264954, \"perplexity\": 194.08830027886515}, \"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-2020-24\/segments\/1590347417746.33\/warc\/CC-MAIN-20200601113849-20200601143849-00044.warc.gz\"}"} | null | null |
CALL FOR A FREE CONSULTATION: (855) 529-7761 //
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CASE RESULT: Suspended Sentence — Charged with 9 felony counts including Possession of Controlled Substance, Armed with Firearm, Receiving Stolen Property, possible life in prison. Result: Suspended Sentence
CASE RESULT: Deferred Sentence — Charged with Possession of Child Pornography. Result: Negotiated plea, 4 years Deferred Sentence.
CASE RESULT: Time Served — Charged with Conspiracy to Distribute Meth, Mandatory 10 years to life in prison. Result: Time Served, 4 years Supervised Release. $1,500 fine.
CASE RESULT: Not Guilty On All Counts— Charged with CONSPIRACY TO COMMIT THEFT FROM A FEDERALLY FUNDED PROGRAM AND THEFT, FACING: 30 YEARS
CASE RESULT: Probation Only— Charged with POSSESSION OF CHILD PORNOGRAPHY, FACING: 4 YEARS IN PRISON
CASE RESULT: Suspended Sentence— Charged with First Degree Burglary with Enhancement for Committing Violent Felony, mandatory 27 years to life in prison, $750k in fines. Result: Suspended Sentence. 3 years probation. No restitution to the victim.
CASE RESULT: No Jail Time — Charged with Traveling to Meet a Minor for Sex, up to 15 years in prison. Charges dropped to Illegal Use of Two Way Communication Device. Result: No sex offender registration, three years probation.
CASE RESULT: HUNG JURY ON ALL COUNTS — Charged with CRIMINAL SEXUAL PENETRATION AND CRIMINAL SEXUAL CONTACT OF A MINOR UNDER 13, FACING: 54 YEARS IN PRISON
CASE RESULT: Probation — Charged with 6 COUNTS OF SEXUAL ASSAULT ON A CHILD, FACING: 25 YEARS IN PRISON
CASE RESULT: No Jail Time — Charged with 6 counts of Sexual Assault of a Child Under 17. Result: 5 years of deferred adjudication on count 1. Counts 2-6 dismissed.
CASE RESULT: 120 in County Jail, was Booked & Released — Charged with multiple counts of Fraud and Theft, with 6 prior arrests for theft and 5 prior arrests for fraud. Up to 75 years prison sentence and hundreds of thousands in fines. Result: 120 days in County Jail, Booked and Released
CASE RESULT: HUNG JURY ON ALL COUNTS AT TRIAL — HEALTHCARE FRAUD AND PAYING KICKBACKS FOR PATIENT REFERRALS, FACING: OVER 40 YEARS.
CASE RESULT: Not Guilty — Charged with RAPE OF AN ADULT, FACING: 30 YEARS AND LIFETIME REGISTRATION
CASE RESULT: Deferred Sentence — Charged with Possession of Child Pornogrpahy. Result: Negotiated plea, 4 years Deferred Sentence.
Bankruptcy Fraud
Ponzi & Pyramid Schemes
Child Enticement / Solicitation
Date Rape
Failing to Register
Lewd & Lascivious Conduct
Child Molestation / Sexual Abuse
Criminal & Immigration
Tax Fraud and Evasion
Weapons Trafficking
Early Release for Prisoners
HAVE YOU BEEN ACCUSED OF:
Failing to Register as a Sex Offender
Failing to register as a sex offender can have serious consequences, including imprisonment in some states. Many sexual related crimes require a convicted person to register as a sex offender. Whenever the registered sex offender moves, he or she typically must re-register in the most current jurisdiction.
Failure to comply with sex offender registration rules prescribed in a particular jurisdiction will lead to additional criminal charges and, if convicted, further penalties. Failing to maintain up to date sex offender registration only compounds your problems by adding further criminal penalties.
Los Angeles sex crimes attorney knows that failing to register as a sex offender often stems from the defendant's misunderstanding or a total lack of knowledge of their registration requirements. For example, a defendant may have completed his or her sex offender registration requirements in one particular state, but by relocating to another state, they are again required to register in their residing state. The defendant may assume the new state has the same registration rules or that sex offender registration is automatically transferred when moving.
RECENT WINS IN SEX CASES:
Experienced kids tattoo artist gets charged with 2 counts of child molestation at hotel where he was working. Client was facing Life in Prison. After substantial work by our team including experts and other defense resources, we were able to negotiate an amazing result of Probation and immediate release from Jail!
Decorated Air Force officer charged with Rape! Client was facing 30 years in Prison and would have been registered as a sex offender. Our client would have lost everything from a distinguished 20-year career in the military to all his benefits and any future jobs. He even would have been limited on where he could live. Our team of attorneys put together a solid and aggressive defense, with experts and private investigators along with other critical resources to prove our client's innocence. We took the case to trial and we won! Not Guilty on all counts!
MORE RECENT TOP WINS!
DEFENSE ATTORNEY FOR FAILING TO REGISTER AS A SEX OFFENDER!
Top California criminal defense lawyers know how to defend a sex offender who failed to register or keep his/her registration current. However, you must find an attorney who is familiar with these situations and knows how to deal with them. Our attorneys often run up against these problems where the laws of one jurisdiction conflict with those of other states.
Requirements vary from state to state. It is important to seek lawyers familiar with the various registration requirements. Obtaining attorneys is just as important when accused with Failure to Register as a Sex Offender it was when defending the original case.
People convicted of sexual or violent crimes often do not understand the registration requirements of sex or violent offender registries. They often knowingly compound their problems with additional criminal charges by failing to follow registration rules.
An experienced attorney with a background in these cases and sexual and violent criminal defense can often dramatically reduce the penalties from failing to register as a sex offender.
LibertyBell Law Group was founded by a group of criminal defense attorneys with a track record of getting the best pleas and winning cases. Our lawyers grew frustrated with the fact that thousands of people still suffer life-altering consequences because they do not have the ability to mount an adequate defense.
Our criminal defense attorneys united as a team to level the playing field. We hand-picked only the foremost top lawyers in sex offender registration defense. Our attorneys have extensive experience in sensitive and highly challenging sex crime cases.
Our lawyers are legal veterans with all of the resources they need to get the best ending to your case. Our powerful legal force of sex offender defense attorneys in California are available and affordable.
CALL OUR CALIFORNIA CRIMINAL ATTORNEYS NOW
You can now draw on the combined expertise and resources of many top criminal defense attorneys with one goal — getting the best possible result for your case. We stand ready to defend you both inside and outside the courtroom.
Early intervention is the key to our mutual success. Call us now at 855-LAW-PRO1 (855-529-7761) for a free consultation with one of our sex crime lawyers.
Gina Tennen
Expert legal strategist.
Experienced in both prosecution and defense.
Relentless fighter for her clients with results to prove it.
Received the National Top 100 Trial Lawyers Award
Featured in Time Magazine's "Leaders of Criminal Defense" in the 100 Most Influential People edition.
Executive Managing Attorney Gina Tennen is a renowned leading advocate of personal liberties. Her meticulous trial preparation and exceptional ability to pursue every possible defense has earned her the admiration of clients and respect from prosecutors, judges, and other criminal attorneys all over the nation.
Lawyer Tennen began her career as a Senior Law Clerk for one of the toughest District Attorney's offices in the country where she gained remarkable experience in juvenile crimes, elder abuse, parole hearings, child abuse, sex and other serious crimes. Her experience at the District Attorney's Office armed her knowledge and case strategy in exploiting the other side's weaknesses for the client's advantage and building a rock solid defense. Criminal attorney Tennen's experience on both sides and brilliant maneuvering throughout is evidenced in her winning track record. In fact, even before earning her law degree, she was instrumental in getting excellent results on several criminal cases outshining her peers.
A consummate strategist and top criminal attorney, lawyer Tennen works tirelessly on many high profile criminal cases. She is known as a relentless fighter who never gives up on any case. Whether the case is hers or handled by another criminal defense attorney from LibertyBell Law Group, no matter how small the case may seem she asserts that every client deserves the right to the best legal representation.
It is her deep belief and what she has built her entire career and lifelong dedication to that there is always hope for every client and every avenue should be pursued regardless of the time and energy it takes. For criminal attorney, Gina Tennen, devoting the finest criminal defense for clients is a habit and talent that you must apply and do all the time.
You demand the best defense. Take hold of your life and speak to our defense attorneys by calling 855-LAW-PRO1 (855-529-7761) now.
Founder Gina Tennen and all the defense attorneys at LibertyBell Law Group uphold that when the bell rings, it rings for life and liberty for all.
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I hired Gina and her team for my son who was facing 10 years for something they said he did when he was 17 years old. Gina was like a lioness protecting her cub. She did not stop. She fought hard and was able to defend the charges. We are so grateful to her and her team. They gave us our son. He recovered his life!
– L. W.
Best defense lawyers in the country.
I hired this law firm in 2014 for my son who was facing 16 years in prison for a felony robbery. Lina Helou answered all our questions. I was extremely distraught regarding what my son was facing but Lina comforted me and told me that it would be okay. And thank goodness LibertyBell saved my son from going to jail and they were able to obtain only 3 years of probation. I could not have hired a better law firm for my son and if you want the best defense call LibertyBell Law Group. Thank God for this great group of lawyers.
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She took her time to review all of my evidence and proved my innocence.
My father had found this group through his company and coworkers. I was assigned Gina Tennen and I cannot explain in just a review on how gratefull I am to be working with her. My case was an accusation of unlawful sex. This had looked extremely bad on my part since I was in a fraternity and she was in a sorority. During these times with Greek life, there are too many bad stigmas surrounding these organizations. Gina had believed in me and knew that I was innocent and that these accusations just were not true. She took her time to review all of my evidence and proved my innocence. I totally recommend this law group because they are willing to put in the work to help your situation.
– C. D.
Gina always kept me fully aware of how the case was proceeding.
When I first heard I was under a criminal investigation, I initially thought I would try and resolve this in the best way possible, without legal representation, as we were all interested in a "just" outcome. I quickly discovered that was not the case and contacted Gina at Liberty Bell. She clearly described the expectations and the choices available to me. She was always available to me, and when I called, if she couldn't talk with me at that time, she would quickly get back to me. Gina always kept me fully aware of how the case was proceeding. Her fee's were stipulated up front, as well as what those fees covered and what they did not.
– S. B.
Attorney Gina Tennen was on point from day one.
Attorney Gina Tennen was on point from day one putting pressure on the sherrif's office to get things handled. With Child molestation accusatinos being thrown around Gina and Attorney Castro were great at keeping the pressure on and had no charges filled. The DCFS indications were appealed and we won with flying colors. Great service, great work, and will use again for all legal matters.
– A. B.
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© 2019 LibertyBell Law Group | Marketing by Claim Your Space | {
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Q: In JPA and Spring can a persistence unit be created on the fly? So apparently we have the need to create persistence units on the fly. Basically we have this web service and a bunch of identical schemas with identical domain classes. We want to be able to pass a query to the web service where the context path matches a schema. The first time that the service is queried then pass in that schema name and create the persistence unit on the fly and then using it every time thereafter and repeating the process every time a request is made of the service for a schema that has not yet been created.
Is this possible using Spring and JPA given all the building that has to be done at start up for normal PU creation? Is this brilliant idea scalable?
A: So if your database schemas are predefined I do not quite get why you would want to create persistence units "on the fly".
The only example - and quite contrived one, at that - I can think of where that may make sense is if the number of schemas is rather large and you do not expect all of them to be necessary during your service lifetime. Even then, all you save is some memory.
So, unless I'm missing something here (and if I am, please clarify what is it you're looking to achieve by creating persistence units "on the fly"), I would suggest you pre-define all your persistence units. You can then create or inject appropriate EntityManagerFactory instances by specifying persistence unit name as parameter.
| {
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New Price! Central Mapleton Hill location!-stroll to Pearl St Mall,Ideal Market and Mt Sanitas open space trails.Nice&quite part of the complex with sweet Sanitas view.Newer kitchen, carpet, wood flring,bathroom, & in unit washer and dryer!!Just move in and enjoy. End your summer hikes the right way, on a lounge chair by the complex's pool and landscaped patio area.New roof.Seller to pay special assessment rebuild of parking garage (in process)$9k-unit gets underground parking and storage space! | {
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Q: Primefaces - Ajax commandButton I have a problem , ajax the way it is in the low code does not call the exampleController.include( )
Someone can tell me what I'm doing wrong ?
<h:form id="exampleForm">
<p:panel toggleable="true" id="pgInfo">
<h:panelGrid columns="2" cellpadding="2">
<p:inputMask value="" id="val"/>
</h:panelGrid>
<h:panelGrid columns="3" cellpadding="3">
<p:commandButton icon="ui-icon-disk"
value="Add"
actionListener="#{exampleController.include()}"
ajax="true" update="dataTable" process="pgInfo"/>
</h:panelGrid>
</p:panel>
<p:panel toggleable="true" id="panelTable" >
<h:panelGrid columns="1" cellpadding="1">
<p:dataTable id="dataTable">
...
</p:dataTable>
</h:panelGrid>
</p:panel>
</h:form>
| {
"redpajama_set_name": "RedPajamaStackExchange"
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Q: Flutter Bloc - Register Handler issue I am looking for some help with my code. I want to create some swappable cards and for that I have created blocs. However, when I finished and tried to launch it, following message appears :
Bad state: add(LoadMissionsEvent) was called without a registered event handler.
Make sure to register a handler via on((event, emit) {...})
class MyApp extends StatelessWidget {
const MyApp({
Key key,
}) : super(key: key);
// final StreamChatClient client;
// final Channel channel;
@override
Widget build(BuildContext context) {
return MultiBlocProvider(
providers: [
BlocProvider(
create: (_) => SwipeBloc()
..add(LoadMissionsEvent(
missionDescriptions:
MissionDescription.missionsDescriptions)))
],
child: MaterialApp(
debugShowCheckedModeBanner: false,
title: "Student App",
home: MissionChoice(),
localizationsDelegates: [GlobalMaterialLocalizations.delegate],
supportedLocales: [Locale('en'), Locale('fr')],
// Scaffold(
// backgroundColor: d_green,
// ),
));
}
}
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 259 |
Clement Paston ( – 1597) of Oxnead Hall in Norfolk, was an English sea-captain and served as a Member of Parliament for Norfolk in 1563. Three of his brothers also served as Members of Parliament. Having married a wealthy widow he rebuilt Oxnead Hall, inherited from his father.
Origins
He was the fifth but fourth surviving son of Sir William Paston ( – 1554) of Caister and Oxnead, by his wife Bridget Heydon, a daughter of Sir Henry Heydon of Baconsthorpe, Norfolk, MP. His brothers included Erasmus Paston, MP, John Paston, MP, and Sir Thomas Paston, MP. The Paston family originated at the manor of Paston in Norfolk. Clement Paston is said by Lloyd (1665) to have served the King of France in the time of King Henry VII (1485-1509), but the inscription on his monument, which gives the date of his death as 1597, says: "Twice forty years he lived and somewhat more", fixing the date of his birth at about 1515.
Career
He is first mentioned in 1544 as "one of the pensioners" and a fitting man to command a king's ship. In 1545 he commanded the Pelican of Danzig, of three hundred tons, in the fleet under John Dudley, Viscount Lisle. In 1546, still, presumably, in the Pelican, he captured a French galley, having on board the Baron St. Blanchard, who appears to have been coming to England on some informal embassy from the King of France. The galley was probably the Mermaid, which was added to the English navy, but no documents describing the capture survive. It was afterwards debated whether the galley was "good prize", and whether St. Blanchard ought to pay ransom, for which Paston demanded five thousand crowns, with two thousand more for maintenance. At the request of the king, on giving his bond for the money, the baron was released, and he returned to France with his servants, "two horses, and twelve mastiff dogs". Afterwards he pleaded that he was under compulsion at the time, and that the bond was worthless, nor does it appear that the money was paid. Paston, however, kept the plunder of the galley, of which a gold cup, with two snakes forming the handles, was in 1829 still in the possession of his descendants. Lloyd's statement that Paston captured the Admiral of France and received thirty thousand crowns for his ransom is as incorrect as that "he was the first that made the English navy terrible". At the Battle of Pinkie in 1547, Paston was wounded and left for dead. It is said that he was the captor of Thomas Wyatt in 1554, which is contrary to evidence, and that he commanded the fleet at Havre in 1562, which is fiction. In 1570 he was a magistrate of Norfolk, and a commissioner for the trial and execution of traitors, and in 1587, though a deputy-lieutenant of the county, he was suspected of being lukewarm in the interests of religion. In 1588 he was Sheriff of Norfolk. He died on 18 February 1597, and was buried in the church of Oxnead, where a "stately marble tomb" testifies that
Princes he served four,
In peace and war,
As fortune did command,
Sometimes by sea
And sometimes on the shore.
Marriage
At some time after 1567 he married Alice Packington, a daughter of Humphrey Pakington of London and the widow of Richard Lambert of London, but appears to have had no children. He left the bulk of his property to his wife, with remainder to his nephew Sir William Paston.
Sources
Blomefield and Parkins's History of Norfolk, vi. 487;
Chambers's History of Norfolk, p. 211, 959;
The account in Lloyd's State Worthies is untrustworthy;
State Papers of Henry VIII (1830, &c.), i. 811, 866, 894, xi. 329;
Acts of the Privy Council (Dasent), 1542–7 pp. 514, 566, 1547–50 p. 447;
State Papers of Henry VIII (in the Public Record Office), vols. xvi–xix.
References
Bibliography
Mimardière, A. M. "Paston (1981). "Clement (by 1523-98), of Oxnead, Norfolk". In Hassler, P. W. The History of Parliament: the House of Commons 1558-1603. London: Boydell & Brewer. n.p.
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\section{Introduction}\label{sec:1}
This paper is dedicated to the comparison
of multivariate probability
distributions with respect to extreme portfolio losses.
A new notion of stochastic ordering
named \emph{asymptotic portfolio loss order} ($\mathrel{\preceq_{\mathrm{apl}}}$) is introduced.
Specially designed for the ordering of stochastic risk models
with respect to extreme portfolio losses,
this notion allows to compare the inherent extreme portfolio risks
associated with different model parameters such as correlations,
other kinds of dependence coefficients, or diffusion parameters.
\par
In a recent paper of \cite{Mainik/Rueschendorf:2010} the notion of
\emph{extreme risk index} has been introduced in the framework of
multivariate regular variation. This index, denoted by $\gamma_\xi$,
is a functional of the vector $\xi$ of portfolio weights and of the
characteristics of the multivariate regular variation of $X$ given
by the tail index $\alpha$ and the spectral measure $\Psi$.
It measures the sensitivity of the portfolio loss
to extremal
events and characterizes the probability distribution of extreme losses.
In particular, it serves to determine the optimal portfolio diversification
with respect to extreme losses.
Within the framework of multivariate regular variation
the notion of asymptotic portfolio loss ordering introduced in this paper
is tightly related to model comparison in terms of
the extreme risk index $\gamma_\xi$.
Thus this paper can be seen as a supplement of the previous one,
allowing to order multivariate risk models with respect to
their extremal portfolio loss behaviour.
\par
In Section \ref{sec:2} of the present paper we introduce the
asymptotic portfolio loss order $\mathrel{\preceq_{\mathrm{apl}}}$ and highlight some relationships
to further well-known ordering notions.
It turns out that even strong dependence and convexity
orders do not imply the asymptotic portfolio loss order in general.
We present counter-examples, based on the the inversion of diversification
effects in models with infinite loss expectations.
Another example of particular interest discussed here is given
by the elliptical distributions.
In this model family we establish a precise criterion for the asymptotic
portfolio loss order, which perfectly accords with the classical
results upon other well-known order relations.
Section \ref{sec:3} is devoted to multivariate regularly varying models.
We discuss the relationship between the asymptotic portfolio loss order
and the comparison of the extreme risk index and characterize $\mathrel{\preceq_{\mathrm{apl}}}$
in terms of a suitable ordering of the canonical spectral measures.
These findings allow to establish sufficient conditions for
$\mathrel{\preceq_{\mathrm{apl}}}$ in terms of spectral measures, which can be verified
by analytical or numerical methods.
In particular, we characterize the dependence structures that yield
the best and the worst possible diversification effects for a multivariate
regularly varying risk vector $X$ in $\Rplus^{d}$ with tail index $\alpha$.
For $\alpha\ge1$ the best case is given by the asymptotic independence and
the worst case is the asymptotic comonotonicity.
The result for $\alpha\le 1$ is exactly the opposite
(cf.\ Theorem~\ref{thm:3.8} and Corollary~\ref{cor:3.10}).
Restricting $X$ to $\Rplus^{d}$ means that $X$ represents only the losses, whereas
the gains are modelled separately.
This modelling approach is particularly suitable for applications in
insurance, operational risk, and credit risk.
If $X$ represents both losses and gains, these results remain valid if the
extremal behaviour of the gains is weaker than that of the losses, so
that there is no loss-gain compensation for extremal events.
In Section \ref{sec:4} we discuss the interconnections between $\mathrel{\preceq_{\mathrm{apl}}}$
or ordered canonical spectral measures and other well-known
notions of stochastic ordering.
Ordering of canonical spectral measures allows to conclude $\mathrel{\preceq_{\mathrm{apl}}}$
from the (directionally)
convex or the supermodular order.
It is not obvious how to obtain this implication in a general setting.
Finally, in Section~\ref{sec:5} we present a series of examples
with graphics illustrating the numerical results upon the ordering
of spectral measures.
The relationship to spectral measures provides a useful numerical tool to
establish $\mathrel{\preceq_{\mathrm{apl}}}$ in practical applications.
\section{Asymptotic portfolio loss ordering}\label{sec:2}
To compare stochastic risk models with respect to extreme portfolio losses,
we introduce the asymptotic portfolio loss order $\mathrel{\preceq_{\mathrm{apl}}}$.
This order relation is designed
for the analysis of the asymptotic
diversification effects and the identification of models that generate
portfolio risks with stronger extremal behaviour.
\par
Before stating the definition, some basic notation is needed.
Focusing on risks, let $X$ be a \emph{random loss vector} with values in
$\R^{d}$, i.e., let positive values of the components $X^{(i)}$, $i=1,\ldots,d$,
represent losses and let negative values of $X^{(i)}$ represent gains
of some risky assets.
Following the intuition of diversifying a unit capital over several assets,
we restrict the set of portfolios to the unit simplex in $\R^{d}$:
\[
\Simp^d := \cubrfl{\xi\in\Rplus^{d}: \sum_{i=1}^{d}\xi_i=1 }
\ldotp
\]
The portfolio loss resulting from a random vector $X$ and the portfolio $\xi$
is given by the scalar product of $\xi$ and $X$.
In the sequel it will be denoted by $\xi^{\top} X$.
\begin{definition}
Let $X$ and $Y$ be $d$-dimensional random vectors.
Then $X$ is called smaller than $Y$ in
\emph{asymptotic portfolio loss order}, $X\mathrel{\preceq_{\mathrm{apl}}} Y$,
if
\begin{equation}\label{eq:2.1}
\forall\xi\in\Simp^d
\quad
\limsup_{t\to\infty}
\frac{\mathrm{P}\cubr{\xi^{\top} X> t}}{\mathrm{P}\cubr{\xi^{\top} Y\ge t}} \le 1
\ldotp
\end{equation}
Here, $\frac00$ is defined to be 1.
\end{definition}
\begin{remark}\label{rem:2.1}
\begin{enumerate}[(a)]
\item
Although designed for random vectors, $\mathrel{\preceq_{\mathrm{apl}}}$ is also defined for
random variables. In this case, the portfolio set has only one element,
$\Sigma^1=\cubr{1}$.
\item\label{item:rem:2.1.b}
It is obvious that $\mathrel{\preceq_{\mathrm{apl}}}$ is invariant under componentwise rescaling.
Let $vx$ denote the componentwise product of $v,x\in \R^{d}$:
\begin{equation}\label{eq:apl.2}
vx:= (v^{(i)} x^{(i)},\dots,v^{(d)} x^{(d)}),
\end{equation}
Then it is easy to see that $ X\mathrel{\preceq_{\mathrm{apl}}} Y$ implies $vX \mathrel{\preceq_{\mathrm{apl}}} vY$ for
all $v\in\Rplus^{d}$.
Hence condition~\eqref{eq:2.1} can be equivalently stated for $\xi\in\Rplus^{d}$.
\end{enumerate}
\end{remark}
\par
The ordering statement $X\mathrel{\preceq_{\mathrm{apl}}} Y$ means that for all portfolios
$\xi\in\Simp^d$
the portfolio loss $\xi^{\top} X$ is asymptotically smaller $\xi^{\top} Y$.
Thus $\mathrel{\preceq_{\mathrm{apl}}}$ concerns only the extreme portfolio losses.
In consequence, this order relation is weaker than the (usual)
stochastic ordering $\mathrel{\preceq_{\mathrm{st}}}$ of the portfolio losses:
\begin{equation}\label{eq:2.2}
\xi^{\top} X \mathrel{\preceq_{\mathrm{st}}} \xi^{\top} Y \text{ for all } \xi\in\Simp^d \text{ implies } X\mathrel{\preceq_{\mathrm{apl}}} Y.
\end{equation}
Here, for real random variables $U$, $V$ the \emph{stochastic ordering}
$U \mathrel{\preceq_{\mathrm{st}}} V$ is defined by
\begin{equation}\label{eq:2.3a}
\forall t\in\R\quad \mathrm{P}\cubr{U>t}\le\mathrm{P}\cubr{V>t}.
\end{equation}
\par
Some related, well-known stochastic orderings
\citep[cf.][]{Mueller/Stoyan:2002,Shaked/Shanthikumar:1997}
are collected in the following list. Remind that $f:\R^{d}\to \R$
is called \emph{supermodular} if
\begin{equation}\label{eq:2.3b}
\forall x,y\in\R^{d}
\quad
f(x\wedge y)+f(x\vee y)\ge f(x)+f(y)
\ldotp
\end{equation}
\begin{definition}\label{def:2.2}
Let $X$, $Y$ be random vectors in $\R^{d}$. Then $X$ is said to be smaller than $Y$ in
\begin{enumerate}[(a)]
\item \emph{(increasing) convex order},
$X \mathrel{\preceq_{\mathrm{cx}}} Y$ ($X\mathrel{\preceq_{\mathrm{icx}}} Y$), if $\mathrm{E} f(X) \le \mathrm{E} f(Y)$ for all (increasing) convex functions $f:\R^{d}\mapsto \R$ such that the expectations exist;
\item
\emph{linear convex order}, $X \mathrel{\preceq_{\mathrm{lcx}}} Y$, if
$\xi^{\top} X \mathrel{\preceq_{\mathrm{cx}}} \xi^{\top} Y$ for all $\xi\in\R^{d}$;
\item
\emph{positive linear convex order}, $X \mathrel{\preceq_{\mathrm{plcx}}} Y$,
if $\xi^{\top} X \mathrel{\preceq_{\mathrm{cx}}} \xi^{\top} Y$ for all $\xi\in\Rplus^{d}$;
\item \emph{supermodular order} $X\mathrel{\preceq_{\mathrm{sm}}} Y$, if
$\mathrm{E} f(X) \le \mathrm{E} f(Y)$ for all supermodular functions $f:\R^{d}\to\R$ such
that the expectations exist;
\item
\emph{directionally convex order}, $X\mathrel{\preceq_{\mathrm{dcx}}} Y$, if
$\mathrm{E} f(X) \le \mathrm{E} f(Y)$ for all directionally convex, i.e., supermodular and componentwise convex functions
$f:\R^{d}\to\R$ such that the expectations exist.
\end{enumerate}
\end{definition}
\par
The stochastic orderings listed in Definition \ref{def:2.2} are useful
for describing the risk induced by larger diffusion (convex risk) as well as
the risk induced by positive dependence
(supermodular and directionally convex).
The following implications are known to hold generally for random
vectors $X$, $Y$ in $\R^{d}$:
\begin{enumerate}[(a)]
\item $(X\mathrel{\preceq_{\mathrm{sm}}} Y)_{\phantom{icx}\kern-2ex} \Rightarrow
(X\mathrel{\preceq_{\mathrm{dcx}}} Y)_{\phantom{l}\kern-.5ex} \Rightarrow (X\mathrel{\preceq_{\mathrm{plcx}}} Y)$
\item $(X\mathrel{\preceq_{\mathrm{cx}}} Y)_{\phantom{ism}\kern-2ex} \Rightarrow %
(X\mathrel{\preceq_{\mathrm{lcx}}} Y)_{\phantom{d}\kern-.5ex} \Rightarrow (X\mathrel{\preceq_{\mathrm{plcx}}} Y)$
\item $(X\mathrel{\preceq_{\mathrm{icx}}} Y)_{\phantom{sm}\kern-2ex} \Rightarrow %
(X\mathrel{\preceq_{\mathrm{plcx}}} Y)$
\end{enumerate}
\begin{remark}\label{rem:apl.1}
\begin{enumerate}[(a)]
\item\label{item:apl.1}
It is easy to see that the usual stochastic order $\mathrel{\preceq_{\mathrm{st}}}$ implies
$\mathrel{\preceq_{\mathrm{apl}}}$ in the univariate case.
\item\label{item:apl.2}
In spite of being strong risk comparison orders, the order relations
outlined in Definition~\ref{def:2.2} do not imply $\mathrel{\preceq_{\mathrm{apl}}}$ in general.
For instance, it is known that the comonotonic dependence structure is
the worst case with respect to the strong supermodular ordering $\mathrel{\preceq_{\mathrm{sm}}}$,
whereas it is not necessarily the worst case with respect to $\mathrel{\preceq_{\mathrm{apl}}}$
(cf.\ Examples~\ref{ex:6} and \ref{ex:2}).
\end{enumerate}
\end{remark}
\par
The following proposition helps to establish sufficient criteria
for $\mathrel{\preceq_{\mathrm{apl}}}$ in the univariate case.
To obtain multivariate results,
it can be separately applied to each portfolio loss $\xi^{\top} X$ for
$\xi\in\Simp^d$.
\par
\begin{proposition}\label{prop:2.3}
Let $R_1$, $R_2\ge 0$ be real random variables and let $V$ be a real random variable independent of $R_i$, $i=1,2$.
\begin{enumerate}[(a)]
\item
\label{item:prop2.3b}
If $R_1\mathrel{\preceq_{\mathrm{apl}}} R_2$ and $V < K$
for some constant $K$, then
\begin{equation}\label{eq:2.7}
R_1V\mathrel{\preceq_{\mathrm{apl}}} R_2V
\ldotp
\end{equation}
\item \label{item:prop2.3a}
If $R_1\mathrel{\preceq_{\mathrm{st}}} R_2$, then
\begin{equation}
\label{eq:2.5}
\robr{R_1V}_{+} \mathrel{\preceq_{\mathrm{st}}} \robr{R_2V}_{+}
\quad
\text{and}
\quad
\robr{R_2V}_{-} \mathrel{\preceq_{\mathrm{st}}} \robr{R_1V}_{-}
\ldotp
\end{equation}
In addition, if $V$ and $R_i$ are integrable and $EV \ge 0$, then
\begin{equation}\label{eq:2.6}
R_1V \mathrel{\preceq_{\mathrm{icx}}} R_2V
\ldotp
\end{equation}
Moreover, if $EV=0$, then $R_1V \mathrel{\preceq_{\mathrm{cx}}} R_2V$.
\end{enumerate}
\end{proposition}
\par
\begin{myproof}
\par Part~(\ref{item:prop2.3b}).
Since $R_1V \mathrel{\preceq_{\mathrm{apl}}} R_2V$ is trivial for $V \le 0$, we
assume that $\mathrm{P}\cubr{V>0}>0$.
Hence $V\le K$ implies for all $t>0$
\begin{align}
\mathrm{P}\cubrfl{R_1V>t}
&= \nonumber
\int_{(0,K)} \mathrm{P}\cubrfl{R_1>{t}/{v}} \mathrm{d}\mathrm{P}^V(v) \\
&=\label{eq:2.10b}
\int_{(0,K)} f\robrfl{{t}/{v}} \mathrm{P}\cubrfl{R_2>{t}/{v}} \mathrm{d}\mathrm{P}^V(v),
\end{align}
where
\[
f(z):=\frac{\mathrm{P}\cubr{R_1>z}}{\mathrm{P}\cubr{R_2>z}}
\ldotp
\]
An obvious consequence of \eqref{eq:2.10b} is the inequality
\begin{equation}
\mathrm{P}\cubr{R_1V > t}
\le\label{eq:2.10c}
\sup \cubrfl{ f(z): z > {t}/{K}} \cdot \mathrm{P}\cubrfl{R_2V > t}
\end{equation}
Since $R_1\mathrel{\preceq_{\mathrm{apl}}} R_2$ is equivalent to $\limsup_{z\to\infty} f(z)\le 1$,
we obtain
\[
\limsup_{t\to\infty}\frac{\mathrm{P}\cubr{R_1V>t}}{\mathrm{P}\cubr{R_2V>t}}\le 1
\ldotp
\]
\par
Part (\ref{item:prop2.3a}).
By the well-known coupling principle for the stochastic ordering $\mathrel{\preceq_{\mathrm{st}}}$
we may assume without loss of generality that $R_1 \le R_2$
pointwise on the underlying probability space.
This implies
\[
\mathrm{P}\cubr{R_1 V > t} \le \mathrm{P}\cubr{R_2 V > t}
,\quad t \ge 0,
\]
and, similarly,
\[
\mathrm{P}\cubr{R_1 V \le t} \le \mathrm{P}\cubr{R_2 V \le t}
,\quad t \le 0
\ldotp
\]
In consequence we obtain~\eqref{eq:2.5}.
\par
From the proof of \eqref{eq:2.5}
it follows that the distribution functions
of the products $R_iV$, $i=1,2$, satisfy the cut criterion of Karlin--Novikov
(cf.\ \citealp{Shaked/Shanthikumar:1994}, Theorem 2.A.17 and
\citealp{Mueller/Stoyan:2002}, Theorem 1.5.17)
Hence we obtain
\begin{equation}\label{eq:2.9}
R_1V \mathrel{\preceq_{\mathrm{icx}}} R_2V
\ldotp
\end{equation}
If $EV=0$, then $E\sqbr{R_1V}=E\sqbr{R_2V}$ and therefore
\begin{equation}\label{eq:2.10a}
R_1V\mathrel{\preceq_{\mathrm{cx}}} R_2V
\ldotp
\end{equation}
\end{myproof}
\begin{remark}\label{rem:2.3}
\begin{enumerate}[(a)]
\item %
Note that
\eqref{eq:2.5}
implies (without assuming the existence of moments)
that $\robr{R_2V}_{+} \mathrel{\preceq_{\mathrm{decx}}} \robr{R_1V}_{+}$
where $\mathrel{\preceq_{\mathrm{decx}}}$ denotes the \emph{decreasing convex order}.
Similarly one obtains
$\robr{R_2V}_{-} \mathrel{\preceq_{\mathrm{icx}}} \robr{R_1V}_{-}$
\item
If $f(t):={\mathrm{P}\cubr{R_1>t}}/{\mathrm{P}\cubr{R_2>t}} \le C < \infty$ and
$R_1\mathrel{\preceq_{\mathrm{apl}}} R_2$, then $R_1V\mathrel{\preceq_{\mathrm{apl}}} R_2V$.
\item
A related problem is the ordering of products $RV_i$ for $R\ge 0$ with
$V_1$ and $V_2$ independent of $R$.
In the special case when $R$ is \emph{regularly varying} with \emph{tail index}
$\alpha>0$, i.e.,
\begin{equation}\label{eq:apl.5}
\lim_{t\to\infty}
\frac{\mathrm{P}\cubr{R>tx}}{\mathrm{P}\cubr{R>t}}
=x^{-\alpha}
,\quad
x>0,
\end{equation}
exact criteria for $\mathrel{\preceq_{\mathrm{apl}}}$ can be obtained from Breiman's Theorem
\citep[cf.][Proposition 7.5]{Resnick:2007}.
If $\mathrm{E} \robr{V_i}_{+}^{\alpha+\varepsilon} <\infty$ for $i=1,2$ and
some $\varepsilon>0$, then
\[
\lim_{t\to\infty}\frac{\mathrm{P}\cubr{RV_i>t}}{\mathrm{P}\cubr{R>t}}
=
E\sqbrfl{\robr{V_i}_{+}^{\alpha}}
\ldotp
\]
This yields
\[
\lim_{t\to\infty}
\frac{\mathrm{P}\cubr{RV_1>t}}{\mathrm{P}\cubr{RV_2>t}}
=
\frac
{\mathrm{E}\sqbrfl{\robr{V_1}_{+}^{\alpha}}}
{\mathrm{E}\sqbrfl{\robr{V_2}_{+}^{\alpha}}}
\ldotp
\]
\end{enumerate}
\end{remark}
An important class of stochastic models with various applications are
\emph{elliptical distributions},
which are natural generalizations of multivariate
normal distributions.
A random vector $X\in\R^{d}$ is called elliptically distributed,
if there exist $\mu\in\R^{d}$ and a $d\times d$ matrix $A$ such that
$X$ has a representation of the form
\begin{equation}\label{eq:2.11}
X \mathrel{\stackrel{\mathrm{d}}{=}} \mu + RAU,
\end{equation}
where $U$ is uniformly distributed on the Euclidean unit sphere $\Sbb^d_2$,
\[
\Sbb^d_2=\cubrfl{x\in\R^{d} : \norm{x}_2 =1},
\]
and $R$ is a non-negative random variable independent of $U$.
By definition we have
\begin{equation}\label{eq:2.12}
E\norm{X}_2^2 <\infty \Leftrightarrow E R^2<\infty,
\end{equation}
and in this case
\begin{equation}\label{eq:2.13}
\mathrm{Cov}(X)=\mathrm{Var}(R) A A^{\top}
\ldotp
\end{equation}
The matrix $C:= A A^{\top}$ is unique except for a constant factor and
is also called the \emph{generalized covariance matrix} of $X$.
We denote the elliptical distribution constructed
according to~\eqref{eq:2.11} by $\Ecal(\mu,C,F_R)$,
where $F_R$ is the distribution of $R$.
\par
A classical stochastic ordering result going back to
\cite{Anderson:1955} and \cite{Fefferman/Jodeit/Perlman:1972}
\citep[cf.][p.~70]{Tong:1980}
says that \emph{positive semidefinite ordering}
of the generalized covariance matrices $C_1 \mathrel{\preceq_{\mathrm{psd}}} C_2$, defined as
\begin{equation}
\label{eq:2.13a}
\forall \xi\in\R^{d} \quad \xi^{\top} C_1 \xi \le \xi^{\top} C_2 \xi,
\end{equation}
implies symmetric convex ordering if
the location parameter $\mu$ and the distribution $F_R$ of the radial
factor are fixed:
\begin{equation}\label{eq:2.14}
\Ecal(\mu, C_1,F_R) \mathrel{\preceq_{\mathrm{symmcx}}} \Ecal(\mu, C_2, F_R)
\ldotp
\end{equation}
It is also known that for elliptical random vectors $X\sim \Ecal(\mu,C,F_R)$
the multivariate distribution function
$F(x):=\mathrm{P}\cubr{X_1\le x_1,\dots,X_d\le x_d}$ is increasing in $C_{i,j}$ for $i\not=j$, where $C=(C_{i,j})$ \citep[see, e.g.,][Theorem 2.21]{Joe:1997}.
\par
The following result is concerned
with the asymptotic portfolio loss ordering $\mathrel{\preceq_{\mathrm{apl}}}$ for elliptical
distributions.
\begin{theorem}\label{theo:2.4}
Let $X\mathrel{\stackrel{\mathrm{d}}{=}} \mu_1+R_1A_1U$, $Y\mathrel{\stackrel{\mathrm{d}}{=}}\mu_2+R_2A_2U$ be elliptically distributed
with generalized covariances $C_i:=A_iA_i^{\top}$. If
\begin{equation}\label{eq:2.15}
\mu_1\le \mu_2, \enskip
R_1\mathrel{\preceq_{\mathrm{apl}}} R_2,
\end{equation}
and
\begin{equation}\label{eq:2.15a}
\forall \xi\in\Simp^d \quad \xi^{\top} C_1\xi \le \xi^{\top} C_2 \xi,
\end{equation}
then
\begin{equation}\label{eq:2.16}
X\mathrel{\preceq_{\mathrm{apl}}} Y.
\end{equation}
\end{theorem}
\par
\begin{myproof}
It suffices to show that $\xi^{\top} Y \mathrel{\preceq_{\mathrm{apl}}} \xi^{\top} Y$ for an
arbitrary portfolio $\xi\in\Simp^d$.
Furthermore, without loss of generality we can assume $\mu_1=\mu_2=0$.
For $i=1,2$ and $\xi\in\Simp^d$ denote
\[
a_i = a_i(\xi):= \robrfl{\xi ^{\top} C_i \xi}^{1/2}
\]
and
\[
v_i = v_i(\xi) := \frac{\xi^{\top} A_i}{a_i}
\ldotp
\]
Then, by definition of elliptical distributions, we have
\begin{equation}\label{eq:2.17}
\xi^{\top} X \mathrel{\stackrel{\mathrm{d}}{=}} R_1 a_1 v_1 U
\quad\text{and}\quad
\xi^{\top} Y \mathrel{\stackrel{\mathrm{d}}{=}} R_2 a_2 v_2 U
\ldotp
\end{equation}
Since the vectors $v_i=v_i(\xi)$ have unit length by construction,
the random variables $v_i U$ are orthogonal projections of $U\sim \mathrm{unif}(S_2^d)$
on vectors of unit length.
Symmetry arguments yield that the distribution of $v_i U$ is independent of
$v_i$ and that $v_iU \mathrel{\stackrel{\mathrm{d}}{=}} (1,0,\ldots,0)^{\top} U=U^{(1)}$.
\par
Thus we have
\[
\xi^{\top} X \mathrel{\stackrel{\mathrm{d}}{=}} a_1 R_1 V
\quad\text{and}\quad
\xi^{\top} Y\mathrel{\stackrel{\mathrm{d}}{=}} a_2 R_2 V
\]
with $V:=U^{(1)}$.
By assumption we have $a_1\le a_2$ and $R_1\mathrel{\preceq_{\mathrm{apl}}} R_2$.
Applying Proposition \ref{prop:2.3}(\ref{item:prop2.3b})
we obtain $\xi^{\top} X\mathrel{\preceq_{\mathrm{apl}}} \xi^{\top} Y$.
\end{myproof}
\par
\begin{remark}\label{rem:2.6}
\begin{enumerate}[(a)]
\item\label{item:rem:2.6.a}
It should be noted that condition~\eqref{eq:2.15a} is indeed weaker than
\eqref{eq:2.13a}.
Let $-1 < \rho_1 < \rho_2 <1$ and consider covariance matrices
\[
C_i:=
\robrfl{
\begin{array}{cc}
1 & \rho_i\\
\rho_i & 1
\end{array}
}
,\quad
i=1,2
\ldotp
\]
Straightforward calculations show that $C_i$ satisfy~\eqref{eq:2.15a}, but
not~\eqref{eq:2.13a}.
\item
For subexponentially distributed
$R_i$ the assumption $\mu_1\le \mu_2$ in \eqref{eq:2.15} can be omitted.
\end{enumerate}
\end{remark}
\section{Multivariate regular variation: $\mathrel{\preceq_{\mathrm{apl}}}$ in terms of spectral measures}
\label{sec:3}
\par
This section is concerned with the characterization of the asymptotic
portfolio loss order $\mathrel{\preceq_{\mathrm{apl}}}$ in the framework of multivariate regular
variation. The results obtained here highlight the influence of the tail
index $\alpha$ and the spectral measure $\Psi$ on $\mathrel{\preceq_{\mathrm{apl}}}$,
with primary focus put on dependence structures captured by $\Psi$.
It is shown that $\mathrel{\preceq_{\mathrm{apl}}}$ corresponds to a family of order relations
on the set of canonical spectral measures and that these order relations
are intimately related to the extreme risk index $\gamma_\xi$ introduced
in \citet{Mainik/Rueschendorf:2010} and \citet{Mainik:2010}.
\par
The main result of this section is stated in Theorem~\ref{theo:3.4},
providing criteria for $X \mathrel{\preceq_{\mathrm{apl}}} Y$ in terms of componentwise ordering
$X^{(i)} \mathrel{\preceq_{\mathrm{apl}}} Y^{(i)}$ for $i=1,\ldots,d$
and ordering of canonical spectral measures.
A particular consequence of these criteria is the
characterization of the dependence structures that
yield the best and the worst possible diversification effects for
random vectors in $\Rplus^{d}$
(cf.\ Theorem~\ref{thm:3.8} and Corollary~\ref{cor:3.10}).
Another application concerns elliptical distributions. Combining
Theorem~\ref{theo:3.4} with results on $\mathrel{\preceq_{\mathrm{apl}}}$ obtained in
Theorem~\ref{theo:2.4}, we obtain ordering of the corresponding
canonical spectral measures.
\par
Recall the notions of regular variation.
In the univariate case it can be defined separately for the lower
and the upper tail of a random variable via~\eqref{eq:apl.5}.
A random vector $X$ taking values in $\R^{d}$ is called
\emph{multivariate regularly varying} with tail index $\alpha\in(0,\infty)$
if there exist a sequence $a_n\to\infty$ and a (non-zero) Radon measure $\nu$ on
the Borel $\sigma$-field $\mathcal{B}\robr{[-\infty,\infty]^d\setminus\cubr{0}}$
such that $\nu\robr{[-\infty,\infty]^d \setminus \R^{d}}=0$ and,
as $n\to\infty$,
\begin{equation}
\label{eq:29}
\index{$\nu$}
n \mathrm{P}^{\,a_n^{-1} X} \stackrel{\mathrm{v}}\rightarrow \nu
\text{ on }\mathcal{B}\robr{[-\infty,\infty]^d\setminus\cubr{0}},
\end{equation}
where $\stackrel{\mathrm{v}}\rightarrow$ denotes the
\emph{vague convergence} of Radon measures
and $\mathrm{P}^{\,a_n^{-1} X}$ is the probability distribution of
$a_n^{-1} X$.
\par
It should be noted that random vectors with non-negative components
yield limit measures $\nu$ that are concentrated on
$[0,\infty]^d\setminus\cubr{0}$.
Therefore multivariate regular variation in this special case can also
be defined by vague convergence on $\mathcal{B}([0,\infty]^d\setminus\cubr{0})$.
\par
Many popular distribution models are multivariate regularly varying.
In particular, according to \citet{Hult/Lindskog:2002},
multivariate regular variation of an elliptical distribution
$\Ecal\robr{\mu,C,F_R}$ is equivalent to the regular variation of the
radial factor $R$ and the tail index $\alpha$ is inherited from $R$.
Other popular examples are obtained by endowing regularly varying margins
$X^{(i)}$ with an appropriate copula
\citet[cf.][]{Wuethrich:2003, Alink/Loewe/Wuethrich:2004, Barbe/Fougeres/Genest:2006}
\par
For a full account of technical details related to the notion of
multivariate regular variation, vague convergence, and
the Borel $\sigma$-fields on the punctured spaces
$[-\infty,\infty]^d\setminus\cubr{0}$ and $[0,\infty]^d\setminus\cubr{0}$
the reader is referred to \citet{Resnick:2007}.
\par
It is well known that the limit measure $\nu$ obtained in~\eqref{eq:29}
is unique except for a constant factor, has a singularity in the origin
in the sense that
$\nu\robr{(-\varepsilon,\varepsilon)^d}=\infty$ for any $\varepsilon>0$,
and exhibits the scaling property
\begin{equation}
\label{eq:30}
\nu(tA)=t^{-\alpha}\nu(A)
\end{equation}
for all sets $A\in\mathcal{B}\robrfl{[-\infty,\infty]^d\setminus\cubr{0}}$ that
are bounded away from $0$.
\par
It is also well known that~\eqref{eq:29} implies that the random variable
$\norm{X}$ with an arbitrary norm $\norm{\cdot}$ on $\R^{d}$ is
univariate regularly varying with tail index $\alpha$.
Moreover,
the sequence $a_n$ can always be chosen as
\begin{equation}
\label{eq:181}
a_n:=F_{\norm{X}}^{\leftarrow}(1-1/n),
\end{equation}
where $F_{\norm{X}}^{\leftarrow}$ is the quantile function of
$\norm{X}$. The resulting limit measure $\nu$
is normalized on the set $A_{\norm{\cdot}}:=\cubr{x\in\R^{d}: \norm{x}>1}$ by
\begin{equation}
\label{eq:182}
\nu\robrfl{A_{\norm{\cdot}}}=1
\ldotp
\end{equation}
\par
Thus, after normalizing $\nu$ by~\eqref{eq:182},
the scaling relation~\eqref{eq:30} yields an equivalent rewriting of
the multivariate regular variation condition~\eqref{eq:29}
in terms of weak convergence:
\begin{equation}
\label{eq:34}
\mathcal{L}\cubrfl{t^{-1} X\,|\,\norm{X}>t}
\stackrel{\mathrm{w}}{\rightarrow}
\nu|_{A_{\norm{\cdot}}}
\text{ on }
\mathcal{B}\robrfl{A_{\norm{\cdot}}}
\end{equation}
for $t\to\infty$,
where $\nu|_{A_{\norm{\cdot}}}$ is the restriction of $\nu$ to the set $A_{\norm{\cdot}}$.
\par
Additionally to~\eqref{eq:29}
it is assumed that the limit measure $\nu$ is
non-degen\-erate
in the
following sense:
\begin{equation}
\label{eq:4}
\nu\robrfl{\cubrfl{x\in\R^{d}: \absfl{x^{(i)}}> 1}} >0
,\quad i=1,\ldots,d
\ldotp
\end{equation}
This assumption ensures that
all asset losses $X^{(i)}$ are relevant for the extremes of the portfolio loss
$\xi^{\top} X$. If~\eqref{eq:4} is satisfied in the upper tail region, i.e., if
\begin{equation}
\label{eq:4a}
\nu\robrfl{\cubrfl{x\in\R^{d}: x^{(i)}> 1}} >0
,\quad i=1,\ldots,d,
\end{equation}
then $\nu$ also characterizes the asymptotic distribution
of the componentwise maxima
$M_n:=\robr{M^{(1)},\ldots,M^{(d)}}$ with
$M^{(i)}:=\max\cubr{X_1^{(i)},\ldots,X_n^{(i)}}$
by the limit relation
\begin{equation}
\label{eq:164}
\mathrm{P}\cubrfl{a_n^{-1} M_n\in[-\infty,x]} \stackrel{\mathrm{w}}{\rightarrow}
\exp\robrfl{-\nu\robrfl{[-\infty,\infty]^d\setminus[-\infty,x]}}
\end{equation}
for $x\in(0,\infty]^d$.
Therefore $\nu$ is called
\emph{exponent measure}.
For more details concerning the asymptotic distributions of maxima
the reader is referred to~\citet{Resnick:1987}
and \citet{de_Haan/Ferreira:2006}.
\par
Another consequence of the scaling property~\eqref{eq:30} is the
product representation of $\nu$ in polar coordinates
\[
(r,s):=\tau(x):=(\norm{x},\norm{x}^{-1} x)
\]
with respect to an arbitrary norm $\norm{\cdot}$ on $\R^{d}$.
The induced
measure $\nu^\tau:=\nu\circ\tau^{-1}$ necessarily satisfies
\begin{equation}
\label{eq:28}
\nu^\tau=c\cdot\rho_\alpha\otimes\Psi
\end{equation}
with the constant factor
\[
c=\nu\robrfl{A_{\norm{\cdot}}}
>0,
\]
the measure $\rho_\alpha$ on $(0,\infty]$ defined by
\begin{equation}
\label{eq:176}
\rho_\alpha((x,\infty]):=x^{-\alpha},
\quad
x\in(0,\infty],
\end{equation}
and a probability measure $\Psi$ on the unit sphere $\Sbb^d_{\norm{\cdot}}$
with respect to $\norm{\cdot}$,
\[
\Sbb^d_{\norm{\cdot}}:=\cubrfl{s\in\R^{d} : \norm{s} = 1}
\ldotp
\]
The measure $\Psi$ is called
\emph{spectral measure}
of $\nu$ or $X$.
Since the term \enquote{spectral measure} is already used in other areas,
$\Psi$ is also referred to as
\emph{angular measure}.
In the special case of $\Rplus^{d}$-valued random vectors $X$ it
may be convenient to reduce the domain of $\Psi$ to
$\Sbb^d_{\norm{\cdot}}\cap\Rplus^{d}$.
\par
Although the domain of the spectral measure $\Psi$ depends on the
norm $\norm{\cdot}$ underlying the polar coordinates, the
representation~\eqref{eq:28} is norm-independent in the following sense:
if~\eqref{eq:28} holds for some norm $\norm{\cdot}$, then it also holds for
any other norm $\norm{\cdot}_\diamond$ that is equivalent to $\norm{\cdot}$.
The tail index $\alpha$ is the same and the spectral measure $\Psi_\diamond$
on the unit sphere $\Sbb^d_\diamond$ corresponding to $\norm{\cdot}_\diamond$
is obtained from $\Psi$ by the following transformation:
\[
\Psi_\diamond=\Psi^T,\quad T(s):=\norm{s}_\diamond^{-1} s
\ldotp
\]
\par
Finally, it should be noted that multivariate regular variation of
the loss vector $X$ is intimately related with the univariate regular variation
of portfolio losses $\xi^{\top} X$.
As shown in \citet{Basrak/Mikosch/Davis:2002},
multivariate regular variation of $X$
implies existence of a portfolio vector $\xi_0\in\R^{d}$ such that $\xi_0 ^{\top} X$
is regularly varying with tail index $\alpha$ and any
portfolio loss $\xi^{\top} X$ satisfies
\begin{equation}
\label{eq:192}
\lim_{t\to\infty}
\frac{\mathrm{P}\cubrfl{\xi^{\top} X >t}}{\mathrm{P}\cubrfl{\xi_0^{\top} X >t}}
=c(\xi,\xi_0)
\in [0,\infty)
\ldotp
\end{equation}
This means that all portfolio losses $\xi^{\top} X$ are either regularly
varying with tail index $\alpha$ or asymptotically negligible
compared to $\xi_0^{\top} X$.
\par
Moreover, it is also worth a remark
that for $\Rplus^{d}$-valued random vectors $X$
the converse implication is true in the sense that~\eqref{eq:192}
and univariate regular variation of $\xi_0^{\top} X$
imply multivariate regular variation of the random vector $X$.
This sort of Cram\'er-Wold theorem was established in
\citet{Basrak/Mikosch/Davis:2002} and \citet{Boman/Lindskog:2009}.
\par
Under the assumption of multivariate regular variation of $X$
the \emph{extreme risk index} $\gamma_\xi = \gamma_\xi(X)$
is defined as
\begin{equation}\label{eq:3.2}
\gamma_\xi(X)=\lim_{t\to\infty} \frac{\mathrm{P}\cubr{\xi^{\top} X>t}}{\mathrm{P}\cubr{\norm{X}_1>t}}.
\end{equation}
In \citet{Mainik/Rueschendorf:2010} the random vector $X$ is restricted to
$\Rplus^{d}$ and the portfolio vector $\xi$ is restricted to $\Simp^d$.
The general case with $X$ in $\R^{d}$ and possible negative portfolio
weights, i.e., short positions, is considered in \citet{Mainik:2010}.
Normalizing the exponent measure $\nu$ by~\eqref{eq:182},
one obtains
\begin{equation}\label{eq:3.1}
\gamma_\xi(X)=\nu\robrfl{\cubrfl{x\in\R^{d}: \xi^{\top} x > 1}}
\ldotp
\end{equation}
Rewriting this representation in terms of the spectral measure $\Psi$
and the tail index $\alpha$ yields
\begin{equation}\label{eq:apl.1}
\gamma_\xi
=
\int_{\Sbb^d_1}\robrfl{\xi^{\top} s}_{+}^{\alpha} \mathrm{d} \Psi(s)
\ldotp
\end{equation}
Denoting the integrand by $f_{\xi,\alpha}$, we will write this representation
as $\gamma_\xi=\Psif_{\xi,\alpha}$.
\par
The extreme risk index $\gamma_\xi(X)$ allows to compare the risk of different
portfolios. It is easy to see that \eqref{eq:3.2} implies
\begin{equation}\label{eq:3.4}
\lim_{t\to\infty} \frac{\mathrm{P}\cubr{\xi_1^{\top} X>t}}{\mathrm{P}\cubr{\xi_2^{\top} X>t}} = \frac{\gamma_{\xi_1}(X)}{\gamma_{\xi_2}(X)}.
\end{equation}
Thus, by construction,
ordering of the extreme risk index $\gamma_\xi$ is related to the
asymptotic portfolio loss order $\mathrel{\preceq_{\mathrm{apl}}}$.
\par
However, designed for the comparison of different portfolio risks within one
model, the extreme risk index $\gamma_\xi$ cannot be directly applied
to the comparison of different models.
The major problem is the standardization by $\mathrm{P}\cubr{\norm{X}_1>t}$ in
\eqref{eq:3.2}. Indeed, since $\mathrm{P}\cubr{\norm{X}_1>t}$ also depends on the
spectral measure $\Psi_X$ of $X$, criteria for $\mathrel{\preceq_{\mathrm{apl}}}$ in terms of
$\gamma_\xi$ demand the specification of the limit
\[
\lim_{t\to\infty}
\frac{\mathrm{P}\cubr{\norm{X}_1>t}}{\mathrm{P}\cubr{\norm{Y}_1>t}}
\ldotp
\]
\par
Another technical issue arises from the invariance of $\mathrel{\preceq_{\mathrm{apl}}}$ under
componentwise rescalings. Since the spectral measure $\Psi$ does not exhibit
this property, ordering of spectral measures needs additional normalization
of margins that makes it consistent with $\mathrel{\preceq_{\mathrm{apl}}}$. To solve these problems,
we use an alternative representation of $\gamma_\xi$ in terms of the
so-called canonical spectral measure $\Psi^\ast$,
which has standardized marginal weights.
\par
This representation is closely related to the asymptotic risk aggregation
coefficient discussed by \cite{Barbe/Fougeres/Genest:2006}.
Furthermore, the link between the canonical spectral measure and
extreme value copulas
allows to transfer ordering results for copulas into the $\mathrel{\preceq_{\mathrm{apl}}}$
setting. These results are presented in Section~\ref{sec:4}.
\par
To reduce the problem to the essentials,
we start with the observation that $\mathrel{\preceq_{\mathrm{apl}}}$ is trivial for
multivariate regularly varying random vectors with different
tail indices and non-degenerate portfolio losses.
\par
\begin{proposition}\label{prop:3.1}
Let $X$ and $Y$ be multivariate regularly varying on $\R^{d}$ and assume that $\gamma_\xi(Y)>0$ for all $\xi\in\Simp^d$.
\begin{enumerate}[(a)]\label{item:prop.3.1a}
\item If
\begin{equation}\label{eq:3.5}
\lim_{t\to\infty} \frac{\mathrm{P}\cubr{\norm{X}_1>t}}{\mathrm{P}\cubr{\norm{Y}_1>t}} = 0,
\end{equation}
then $X \mathrel{\preceq_{\mathrm{apl}}} Y$.
\vspace{0.5em
\item If $\alpha_X>\alpha_Y$, then $X \mathrel{\preceq_{\mathrm{apl}}} Y$.
\end{enumerate}
\end{proposition}
\par
\begin{myproof}
\begin{enumerate}[(a)]
\item %
Using relation \eqref{eq:3.2} we obtain
\begin{align*}
\hspace{2em}&\hspace{-2em}
\limsup_{t\to\infty}
\frac{\mathrm{P}\cubrfl{\xi^{\top} X > t}}{\mathrm{P}\cubrfl{\xi^{\top} Y > t}}\\
&=
\limsup_{t\to\infty}
\robrfl{
\frac{\mathrm{P}\cubrfl{\xi^{\top} X > t}}{\mathrm{P}\cubrfl{\norm{X}_1>t}}
\cdot
\frac{\mathrm{P}\cubrfl{\norm{Y}_1>t}}{\mathrm{P}\cubrfl{\xi^{\top} Y > t}}
\cdot
\frac{\mathrm{P}\cubrfl{\norm{X}_1 > t}}{\mathrm{P}\cubrfl{\norm{Y}_1>t}}
}\\
&=
\frac{\gamma_\xi(X)}{\gamma_\xi(Y)}
\cdot
\limsup_{t\to\infty}\frac{\mathrm{P}\cubrfl{\norm{X}_1 > t}}{\mathrm{P}\cubrfl{\norm{Y}_1>t}}\\
&=0
\ldotp
\end{align*}
\item %
Recall that multivariate regular variation of $X$ implies regular variation of $\norm{X}_1$ with tail index $\alpha_X$. Analogously, $\norm{Y}_1$ is regularly varying with tail index $\alpha_Y$. Finally, $\alpha_X > \alpha_Y$ yields \eqref{eq:3.5} and by~(\ref{item:prop.3.1a}) we obtain $X\mathrel{\preceq_{\mathrm{apl}}} Y$.
\qed
\end{enumerate}
\end{myproof}
\par
Thus the primary setting for studying the influence of dependence
structures on the ordering of extreme portfolio losses is the case of
random variables $X$ and $Y$ with equal tail indices:
\[
\alpha_X=\alpha_Y=:\alpha
\ldotp
\]
In the framework of multivariate regular variation, asymptotic dependence in the tail region
is characterized by the spectral measure $\Psi$ or its canonical version $\Psi^\ast$.
The \emph{canonical exponent measure} $\nu^{\ast}$ of $X$ is obtained from the exponent
measure $\nu$ as
\[
\nu^{\ast}=\nu\circ T
\]
with the transformation $T:\R^{d}\to\R^{d}$ defined by
\begin{equation}
\quad
T(x)
:=\label{eq:3.7}
\robrfl{T_\alpha\robrfl{\nu\robr{B_1}\cdot x^{(1)}},\ldots, T_\alpha\robrfl{\nu\robr{B_d}\cdot x^{(d)}}},
\end{equation}
where
\begin{equation}
T_\alpha(t)
:=\label{eq:3.8}
\robrfl{t_{+}^{1/\alpha} - t_{-}^{1/\alpha}} \text{ and }
B_i := \cubrfl{x\in\R^{d}: \absfl{x^{(i)}} > 1}
\ldotp
\end{equation}
Furthermore, $\nu^{\ast}$ exhibits the scaling property
\[
\nu^{\ast}(tA)=t^{-1}\nu^{\ast}(A),
\quad t>0,
\]
and, analogously to~\eqref{eq:28}, has a product structure in polar
coordinates:
\begin{equation}\label{eq:apl.3}
\nu^{\ast}\circ\tau^{-1} = \rho_1 \otimes \Psi^\ast,
\end{equation}
The measure $\Psi^\ast$ is the \emph{canonical spectral measure} of $X$.
\par
Since $\mathrel{\preceq_{\mathrm{apl}}}$ and $\Psi^\ast$ are invariant under componentwise rescalings,
the canonical spectral measure $\Psi^\ast$ is more suitable for the
characterization of $\mathrel{\preceq_{\mathrm{apl}}}$.
The following lemma provides a representation of the extreme risk index
$\gamma_\xi$ in terms of $\Psi^\ast$. Note that the formulation makes use of
the componentwise product notation~\eqref{eq:apl.2}.
\par
\begin{proposition}
\label{prop:3.2}
Let $X$ be multivariate regularly varying on $\R^{d}$ with tail index
$\alpha\in(0,\infty)$.
If $X$ satisfies the non-degeneracy condition~\eqref{eq:4},
then
\begin{equation}\label{eq:3.9}
\gamma_\xi(X)=\int_{\Sbb^d_1}g_{\xi,\alpha}\robrfl{v s} \, \mathrm{d} \Psi^\ast(s),
\end{equation}
where $\Psi^\ast$ denotes the canonical spectral measure of $X$,
the rescaling vector $v=\robr{v^{(1)},\ldots,v^{(d)}}$
is defined by
\begin{equation}\label{eq:3.10}
v^{(i)}:=\robr{\gamma_{\ei}(X)+\gamma_{-\ei}(X)},
\end{equation}
and the function $g_{\xi,\alpha}:\R^{d}\to\R$ is defined as
\begin{equation}\label{eq:3.11}
g_{\xi,\alpha}(x):=%
\robrfl{\sum_{i=1}^d\xi^{(i)}\cdot\robrfl{\robrfl{x^{(i)}}_{+}^{1/\alpha} - \robrfl{x^{(i)}}_{-}^{1/\alpha}}}_{+}^{\alpha}
\ldotp
\end{equation}
\end{proposition}
\par
\begin{myproof}
Denote $A_{\xi,1}:=\{x\in\R^{d}: \xi^{\top} x\ge 1\}$. Then, by definition of $\nu^{\ast}$,
\begin{align}
\gamma_\xi(X)
&=\nonumber
\nu(A_{\xi,1})\\
&=\nonumber
\nu^{\ast}\robr{T^{-1}(A_{\xi,1})}\\
&=\nonumber
\nu^{\ast}\cubrfl{x\in\R^{d}: T(x) \in A_{\xi,1}}\\
&=\label{eq:3.12}
\int_{\Sbb^d_1}\int_{(0,\infty)}
1\cubrfl{\xi^{\top} T(rs) > 1} \, \mathrm{d} \rho_1(r) \, \mathrm{d} \Psi^\ast(s)
\ldotp
\end{align}
It is easy to see that~\eqref{eq:3.8} implies
$T_\alpha(rt)=r^{1/\alpha}T_\alpha(t)$ for $r>0$ and $t\in\R$.
Consequently, \eqref{eq:3.7} yields
\begin{equation}
\label{eq:3.13}
T(rx) = r^{1/\alpha} T(x)
\end{equation}
for $r>0$ and $x\in\R^{d}$.
Applying~\eqref{eq:3.13} to~\eqref{eq:3.12}, one obtains
\begin{align}
\gamma_\xi(X)
&=\nonumber
\int_{\Sbb^d_1}\int_{(0,\infty)}
1\cubrfl{r^{1/\alpha} \xi^{\top} T(s) > 1} \, \mathrm{d} \rho_1(r) \, \mathrm{d} \Psi^\ast(s)\\
&=\nonumber
\int_{\Sbb^d_1}\int_{(0,\infty)}
1\cubrfl{\xi^{\top} T(s) > 0}
1\cubrfl{r > \robrfl{\xi^{\top} T(s)}^{-\alpha}} \, \mathrm{d} \rho_1(r) \, \mathrm{d} \Psi^\ast(s)\\
&=\nonumber
\int_{\Sbb^d_1}
1\cubrfl{\xi^{\top} T(s) > 0} \robrfl{\xi^{\top} T(s)}^{\alpha} \, \mathrm{d} \Psi^\ast(s)\\
&=\label{eq:3.14}
\int_{\Sbb^d_1}
\robrfl{\xi^{\top} T(s)}^{\alpha}_{+} \, \mathrm{d} \Psi^\ast(s)
\ldotp
\end{align}
Finally, consider the sets $B_i$ defined in~\eqref{eq:3.8}.
It is easy to see that
\begin{equation*}
\nu(B_i)
=
\gamma_{\ei}(X)+\gamma_{-\ei}(X) = v^{(i)}
\ldotp
\end{equation*}
Hence
\begin{align*}
\robrfl{\xi^{\top} T(s)}_{+}^{\alpha}
&=
\robrfl{
\sum_{i=1}^d\xi^{(i)}\cdot\robrfl{T_\alpha\robrfl{v^{(i)} s^{(i)}}}
} _{+}^{\alpha}\\
&=
g_{\xi,\alpha}\robrfl{vs}
\ldotp
\qed
\end{align*}
\end{myproof}
As already mentioned above, $\mathrel{\preceq_{\mathrm{apl}}}$ and $\Psi^\ast$ are invariant under
rescaling of components.
Consequently, characterization of $\mathrel{\preceq_{\mathrm{apl}}}$ can be reduced to the case when the marginal weights $v^{(i)}=\gamma_{\ei}(X)+\gamma_{-\ei}(X)$ in~\eqref{eq:3.9} are standardized by
\begin{equation}
\label{eq:3.15}
\forall i,j\in\cubr{1,\ldots,d}
\quad
\lim_{t\to\infty}
\frac{\mathrm{P}\cubr{\abs{X^{(i)}}> t}}{\mathrm{P}\cubr{\abs{X^{(j)}}>t}}
=1
\ldotp
\end{equation}
This condition will be referred to as the
\emph{balanced tails condition}.
The following result shows that this condition significantly simplifies the representation~\eqref{eq:3.9}.
\par
\begin{proposition}
\label{prop:3.3}
Suppose that $X$ is multivariate regularly varying on $\R^{d}$ with tail index
$\alpha\in(0,\infty)$.
\begin{enumerate}[(a)]
\item
\label{item:L39.3}%
If $X$ has balanced tails in the sense of~\eqref{eq:3.15}, then
\begin{equation}
\label{eq:3.17}
\frac{\gamma_\xi(X)}{\gamma_{e_1}(X) + \gamma_{-e_1}(X)} = \Psi^\ast g_{\xi,\alpha}
\ldotp
\end{equation}
\item
\label{item:L39.1}%
The non-degeneracy condition~\eqref{eq:4} is equivalent to
the existence of a vector $w\in(0,\infty)^d$
such that $wX$ has balanced tails.
\item
\label{item:L39.2}%
The extreme risk index $\gamma_\xi$ of the rescaled vector $wX$ obtained
in part~(\ref{item:L39.1}) satisfies
\begin{equation}
\label{eq:3.18}
\frac{\gamma_\xi(wX)}{\gamma_{e_1}(wX) + \gamma_{-e_1}(wX)} = \Psiast_X g_{\xi,\alpha}
\ldotp
\end{equation}
\end{enumerate}
\end{proposition}
\par
\begin{myproof}
Part~(\ref{item:L39.3}).
Consider the integrand $g_{\xi,\alpha}(vs)$ in the representation~\eqref{eq:3.9}:
\[
g_{\xi,\alpha}(vs)
=
\robrfl{\sum_{i=1}^{d} \xi^{(i)} \cdot
\robrfl{\robrfl{v^{(i)} s^{(i)}}_{+}^{1/\alpha} - \robrfl{v^{(i)} s^{(i)}}_{-}^{1/\alpha}}
}_{+}^{\alpha}
\ldotp
\]
The balanced tails condition~\eqref{eq:3.15} implies that $X$ is
non-degenerate in the sense of~\eqref{eq:4}.
Furthermore, all weights $v^{(i)}$ in the representation~\eqref{eq:3.9}
are equal:
\begin{align*}
1
&=
\lim_{t\to\infty}
\frac{\mathrm{P}\cubrfl{\absfl{X^{(i)}}>t} / \mathrm{P}\cubrfl{\normfl{X}_1 >t}}
{\mathrm{P}\cubrfl{\absfl{X^{(j)}}>t} / \mathrm{P}\cubrfl{\normfl{X}_1 >t}}
=
\frac{\gamma_{\ei}(X) + \gamma_{-\ei}(X)}{\gamma_{\ej}(X) + \gamma_{-\ej}(X)}\\
&=
\frac{v^{(i)}}{v^{(j)}}
,\quad i,j\in\cubr{1,\ldots,d}
\ldotp
\end{align*}
Hence $g_{\xi,\alpha}(vs)$ simplifies to
\begin{align*}
g_{\xi,\alpha}(vs)
&=
v^{(1)} g_{\xi,\alpha}(s)\\
&=
\robrfl{\gamma_{e_1}(X) + \gamma_{-e_1}(X)} g_{\xi,\alpha}(s)
\ldotp
\end{align*}
\par
Part~(\ref{item:L39.1}).
Suppose that $X$ satisfies~\eqref{eq:4}. Then the sets $B_i$ defined in~\eqref{eq:3.8} satisfy $\nu(B_i)>0$ for $i=1,\ldots,d$. Consequently, the random variables $\abs{X^{(i)}}$ are regularly varying with tail index $\alpha$. Denoting
\begin{equation}
\label{eq:3.19}
w^{(i)}:=\robr{\nu(B_i)}^{-1/\alpha},
\end{equation}
one obtains
\begin{align*}
\lim_{t\to\infty}\frac{\mathrm{P}\cubrfl{\absfl{w^{(i)} X^{(i)}} > t}}{\mathrm{P}\cubrfl{\norm{X}_1>t}}
&=
\lim_{t\to\infty}
\robrfl{
\frac{\mathrm{P}\cubrfl{\absfl{X^{(i)}} > t/w^{(i)}}}{\mathrm{P}\cubrfl{\absfl{X^{(i)}} > t}}
\cdot
\frac{\mathrm{P}\cubrfl{\absfl{X^{(i)}} > t}}{\mathrm{P}\cubrfl{\norm{X}_1>t}}
}\\
&= \robrfl{w^{(i)}}^{\alpha}\cdot\nu(B_i)\\
&= 1
\end{align*}
for $i=1,\ldots,d$. Hence, for any $i,j\in\cubr{1,\ldots,d}$,
\begin{align*}
\lim_{t\to\infty}
\frac{\mathrm{P}\cubrfl{\absfl{w^{(i)} X^{(i)}} > t}}{\mathrm{P}\cubrfl{\absfl{w^{(j)} X^{(j)}} > t}}
&=
1
\ldotp
\end{align*}
\par
To prove the inverse implication, suppose that $Z:=wX$ has balanced tails
for some $w\in(0,\infty)^d$. Then the the exponent measure $\nu$ of $X$
satisfies
\begin{align*}
\frac{\nu(B_i)}{\nu(B_1)}
&=
\lim_{t\to\infty}\frac{\mathrm{P}\cubrfl{\absfl{X^{(i)}}>t}}{\mathrm{P}\cubrfl{\absfl{X^{(1)}} > t}}\\
&=
\lim_{t\to\infty}\frac{\mathrm{P}\cubrfl{\absfl{Z^{(i)}}>w^{(i)} t}}{\mathrm{P}\cubrfl{\absfl{Z^{(1)}} > w^{(1)} t}}\\
&=
\lim_{t\to\infty}\robrfl{
\frac{\mathrm{P}\cubrfl{\absfl{Z^{(i)}}> w^{(i)} t}}{\mathrm{P}\cubrfl{\absfl{Z^{(i)}} >t}}
\cdot
\frac{\mathrm{P}\cubrfl{\absfl{Z^{(1)}} >t}}{\mathrm{P}\cubrfl{\absfl{Z^{(1)}} > w^{(1)} t}}
\cdot
\frac{\mathrm{P}\cubrfl{\absfl{Z^{(i)}} >t}}{\mathrm{P}\cubrfl{\absfl{Z^{(1)}} >t}}
}\\
&=
\robrfl{\frac{w^{(i)}}{w^{(1)}}}^{-\alpha}
\in(0,\infty)
,\quad i\in\cubrfl{1,\ldots,d}
\ldotp
\end{align*}
Since multivariate regular variation of $X$ implies $\nu(B_j)>0$ for at least
one index $j\in\cubr{1,\ldots,d}$, this yields $\nu(B_i)>0$ for all $i$.
\par
Part~(\ref{item:L39.2}). This is an immediate consequence of
part (\ref{item:L39.3}) and the invariance of canonical spectral measures
under componentwise rescaling.
\end{myproof}
\par
Representation~\eqref{eq:3.17} suggests that ordering of
the normalized extreme risk indices $\gamma_\xi/(\gamma_{e_1} + \gamma_{-e_1})$
in the balanced tails setting can be considered
as an \emph{integral order relation}
for canonical spectral measures with respect to the function class
\begin{equation}\label{eq:3.20}
\Gcal_{\alpha}:=\cubrfl{g_{\xi,\alpha}:\xi\in\Simp^d}
\ldotp
\end{equation}
This justifies the following definition.
\par
\begin{definition}
\label{def:3.4}
Let $\Psi^\ast$ and $\Phi^\ast$ be canonical spectral measures on $\Sbb^d_1$
and let $\alpha>0$.
Then the order relation $\Psi^\ast \mathrel{\preceq_{\Gcalalpha}} \Phi^\ast$ is defined by
\begin{equation}\label{eq:3.21}
\forall g\in\Gcal_{\alpha}
\quad
\Psi^\ast g \le\Phi^\ast g
\ldotp
\end{equation}
\end{definition}
\par
\begin{remark}
\label{rem:3.1}
\begin{enumerate}[(a)]
\item
\label{item:r14.1}%
For $\alpha=1$ and spectral measures on $\Simp^d$ the extreme risk index
$\gamma_\xi(X)$ is linear in $\xi$ \citep[cf.][Lemma~3.2]{Mainik/Rueschendorf:2010}.
Consequently, $\mathrel{\preceq_{\Gcalalpha}}$ is indifferent in this case,
i.e.,
any $\Psi^\ast$ and $\Phi^\ast$ on $\mathcal{B}\robr{\Simp^d}$ satisfy
\begin{equation}
\label{eq:3.22}
\Psi^\ast \mathrel{\preceq}_{\mathcal{G},1} \Phi^\ast
\quad\text{and}\quad
\Phi^\ast \mathrel{\preceq}_{\mathcal{G},1} \Psi^\ast
\ldotp
\end{equation}
\item
\label{item:r14.2}%
The order relation $\mathrel{\preceq_{\Gcalalpha}}$ is mixing invariant
in the sense that
uniform ordering of two parametric families
$\cubr{\Psi^\ast_\theta:\theta\in\Theta}$ and
$\cubr{\Phi^\ast_\theta:\theta \in\Theta}$,
\[
\forall \theta\in\Theta
\quad
\Psi^\ast_\theta\mathrel{\preceq_{\Gcalalpha}}\Phi^\ast_\theta
,
\]
implies
\[
\int_\Theta\Psi^\ast_\theta \, \mathrm{d} \mu(\theta)
\mathrel{\preceq_{\Gcalalpha}}
\int_\Theta\Phi^\ast_\theta \, \mathrm{d} \mu(\theta)
\]
for any probability measure $\mu$ on $\Theta$.
\end{enumerate}
\end{remark}
\par
The following theorem states that $\mathrel{\preceq_{\mathrm{apl}}}$ is in a certain sense
equivalent to the ordering of canonical spectral measures
and allows to reduce the verification of $\mathrel{\preceq_{\mathrm{apl}}}$ to the verification
of $\mathrel{\preceq_{\Gcalalpha}}$.
Some exemplary applications are given in Section~\ref{sec:5}.
Furthermore, given explicit representations of spectral measures or their
canonical versions,
this result allows to verify $\mathrel{\preceq_{\mathrm{apl}}}$ numerically, which is very
useful in practice.
\par
\begin{theorem}
\label{theo:3.4}
Let $X$ and $Y$ be multivariate regularly varying random vectors on $\R^{d}$ with tail index $\alpha\in(0,\infty)$ and canonical spectral measures $\Psiast_X$ and $\Psiast_Y$. Further, suppose that $X$ and $Y$ satisfy the balanced tails condition~\eqref{eq:3.15}.
\begin{enumerate}[(a)]
\item
\label{item:t4.1}%
If $\absfl{X^{(1)}} \mathrel{\preceq_{\mathrm{apl}}} \absfl{Y^{(1)}}$,
then $\Psiast_X \mathrel{\preceq_{\Gcalalpha}} \Psiast_Y$ implies $X \mathrel{\preceq_{\mathrm{apl}}} Y$.
\vspace{0.5em
\item
\label{item:t4.2}%
If
$\absfl{X^{(1)}} \mathrel{\preceq_{\mathrm{apl}}} \absfl{Y^{(1)}}$
and
$\absfl{Y^{(1)}} \mathrel{\preceq_{\mathrm{apl}}} \absfl{X^{(1)}}$,
then
$\Psiast_X \mathrel{\preceq_{\Gcalalpha}} \Psiast_Y$ is equivalent to $X \mathrel{\preceq_{\mathrm{apl}}} Y$.
\end{enumerate}
\end{theorem}
\par
\begin{myproof}
(\ref{item:t4.1})
Since $X$ has balanced tails, Proposition \ref{prop:3.3}(\ref{item:L39.3}) yields
\begin{align*}
\lim_{t\to\infty}
\frac{\mathrm{P}\cubrfl{\xi^{\top} X >t}}{\mathrm{P}\cubrfl{\absfl{X^{(1)}}>t}}
&=
\lim_{t\to\infty}
\robrfl{
\frac{\mathrm{P}\cubrfl{\xi^{\top} X >t}}{\mathrm{P}\cubrfl{\norm{X}_1>t}}
\cdot
\frac{\mathrm{P}\cubrfl{\norm{X}_1>t}}{\mathrm{P}\cubrfl{\absfl{X^{(1)}}>t}}
}\\
&=
\frac{\gamma_\xi(X)}{\gamma_{e_1}(X) + \gamma_{-e_1}(X)}\\
&=
\Psiast_X g_{\xi,\alpha}
\ldotp
\end{align*}
Analogously one obtains
\[
\lim_{t\to\infty}
\frac{\mathrm{P}\cubrfl{\xi^{\top} Y >t}}{\mathrm{P}\cubrfl{\absfl{Y^{(1)}}>t}}
=
\Psiast_Y g_{\xi,\alpha}
\ldotp
\]
Moreover, $\Psiast_X \mathrel{\preceq_{\Gcalalpha}} \Psiast_Y$ implies
\begin{equation}
\label{eq:3.23}
\frac{\PsiastXg_{\xi,\alpha}}{\PsiastYg_{\xi,\alpha}}
\le
1
\ldotp
\end{equation}
Consequently,
\begin{align}
\hspace{2em}&\hspace{-2em}\nonumber
\limsup_{t\to\infty}
\frac{\mathrm{P}\cubrfl{\xi^{\top} X >t}}{\mathrm{P}\cubrfl{\xi^{\top} Y >t}}\\
&=\nonumber
\limsup_{t\to\infty}\robrfl{
\frac{\mathrm{P}\cubrfl{\xi^{\top} X >t}}{\mathrm{P}\cubrfl{\absfl{X^{(1)}}>t}}
\cdot
\frac{\mathrm{P}\cubrfl{\absfl{Y^{(1)}}>t}}{\mathrm{P}\cubrfl{\xi^{\top} Y >t}}
\cdot
\frac{\mathrm{P}\cubrfl{\absfl{X^{(1)}}>t}}{\mathrm{P}\cubrfl{\absfl{Y^{(1)}}>t}}
}\\
&=\label{eq:3.24}
\frac{\PsiastXg_{\xi,\alpha}}{\PsiastYg_{\xi,\alpha}}
\cdot
\limsup_{t\to\infty}\frac{\mathrm{P}\cubrfl{\absfl{X^{(i)}}>t}}{\mathrm{P}\cubrfl{\absfl{Y^{(i)}}>t}}\\
&\le\nonumber
1
\end{align}
due to~\eqref{eq:3.23} and $\abs{X^{(i)}} \mathrel{\preceq_{\mathrm{apl}}} \abs{Y^{(i)}}$.
\par
\medskip
\noindent %
(\ref{item:t4.2})
By part~(\ref{item:t4.1}), it suffices to show that
$X \mathrel{\preceq_{\mathrm{apl}}} Y$ implies $\Psiast_X \mathrel{\preceq_{\Gcalalpha}} \Psiast_Y$.
By assumption $\abs{X^{(1)}}$ and $\abs{Y^{(1)}}$ have asymptotically equivalent tails,
\[
\lim_{t\to\infty}
\frac{\mathrm{P}\cubrfl{\absfl{X^{(1)}} >t}}{\mathrm{P}\cubrfl{\absfl{Y^{(1)}} >t}}
=
1
\ldotp
\]
Thus~\eqref{eq:3.24} yields
\[
\frac{\PsiastXg_{\xi,\alpha}}{\PsiastYg_{\xi,\alpha}}
=
\limsup_{t\to\infty}
\frac{\mathrm{P}\cubrfl{\xi^{\top} X >t}}{\mathrm{P}\cubrfl{\xi^{\top} Y >t}}
\]
and $X \mathrel{\preceq_{\mathrm{apl}}} Y$ implies $\Psiast_X \mathrel{\preceq_{\Gcalalpha}} \Psiast_Y$.
\end{myproof}
\par
The following result answers the question for dependence structures
corresponding to the best and the worst possible diversification effects
for multivariate regularly varying random vectors in $\Rplus^{d}$.
According to Theorem~\ref{theo:3.4}, it suffices to find the upper and
the lower elements with respect to $\mathrel{\preceq_{\Gcalalpha}}$ in the set
of all canonical spectral measures on $\Simp^d$.
It turns out that for $\alpha > 1$
the best diversification effects are obtained in case of asymptotic
independence, i.e., the $\mathrel{\preceq_{\Gcalalpha}}$-maximal element is given by
\begin{equation}\label{eq:apl.8}
\Psi^\ast_0 := \sum_{i=1}^{d} \Dirac{e_i},
\end{equation}
whereas the worst diversification effects are obtained in case of
the asymptotic comonotonicity, represented by
\begin{equation}\label{eq:apl.9}
\Psi^\ast_1 := d \cdot \Dirac{(1/d,\ldots,1/d)}
\ldotp
\end{equation}
For $\alpha < 1$ the situation is inverse.
\begin{theorem}\label{thm:3.8}
Let $\Psi^\ast$ be an arbitrary canonical spectral measure on $\Simp^d$ and let
$\Psi^\ast_0$ and $\Psi^\ast_1$ be defined according to~\eqref{eq:apl.8}
and~\eqref{eq:apl.9}. Then
\begin{enumerate}[(a)]
\item\label{item:thm:3.8.a}
$\Psi^\ast_0 \mathrel{\preceq_{\Gcalalpha}} \Psi^\ast \mathrel{\preceq_{\Gcalalpha}} \Psi^\ast_1$
for $\alpha \ge 1$.
\vspace{0.5em}
\item\label{item:thm:3.8.b}
$\Psi^\ast_1 \mathrel{\preceq_{\Gcalalpha}} \Psi^\ast \mathrel{\preceq_{\Gcalalpha}} \Psi^\ast_0$
for $\alpha \in (0,1]$.
\end{enumerate}
\end{theorem}
\begin{proof}
Let $X$ be multivariate regularly varying on $\Rplus^{d}$ with canonical spectral
measure $\Psi^\ast$. Without loss of generality we can assume that $X$
satisfies the balanced tails condition~\eqref{eq:3.15}.
Then, according to~\eqref{eq:3.17}, we have
\begin{equation}
\label{eq:apl.6}
\Psiastg_{\xi,\alpha}=\frac{\gamma_\xi(X)}{\gamma_{e_1}(X)}
\ldotp
\end{equation}
Furthermore,
we have $\Psi^\ast g_{e_i,\alpha} = 1$ for $i=1,\ldots,d$.
Recall that the mapping $\xi\mapsto\gamma_\xi$ is convex for $\alpha \ge 1$
\citep[cf.][Lemma~3.2]{Mainik/Rueschendorf:2010}.
Due to~\eqref{eq:apl.6} this behaviour is inherited by the mapping
$\xi\mapsto\Psiastg_{\xi,\alpha}$.
Thus for $\alpha \ge 1$
we have $\Psiastg_{\xi,\alpha} \le 1 = \Psi^\ast_1g_{\xi,\alpha}$ for all $\xi\in\Simp^d$,
which exactly means $\Psi^\ast \mathrel{\preceq_{\Gcalalpha}} \Psi^\ast_1$ for $\alpha \ge 1$.
\par
To complete the proof of part~(\ref{item:thm:3.8.a}), note that
the normalization of canonical spectral measures yields
\begin{equation}\label{eq:apl.7}
\forall\xi\in\Simp^d
\quad
\Psi^\ast_0g_{\xi,\alpha}
=
\sum_{i=1}^{d}\robrfl{\xi^{(i)}}^{\alpha}
=
\int_{\Simp^d} \sum_{i=1}^{d} \robrfl{\xi^{(i)}}^{\alpha} s ^{(i)}
\,\Psi^\ast(\mathrm{d} s)
\end{equation}
Comparing the integrand on the right side of \eqref{eq:apl.7} with
the function $g_{\xi,\alpha}(s)=\robr{\xi^{\top} s^{1/\alpha}}^{\alpha}$,
we see that
\[
\sum_{i=1}^{d} \robrfl{\xi^{(i)}}^{\alpha} s^{(i)}
=
g_{\xi,\alpha}(s)
\cdot
\sum_{i=1}^{d} z_i^{\alpha}
\]
with
\[
z_i := \frac{\xi^{(i)} \cdot\robrfl{s^{(i)}}^{1/\alpha}}
{\xi^{\top} s^{1/\alpha}}
\ldotp
\]
Thus it suffices to demonstrate that $\sum_{i=1}^{d} z_i^{\alpha} \le 1$,
which follows from $z_i\in[0,1]$, $z_i^{\alpha} \le z_i$ for $\alpha \ge 1$, and
$\sum_{i=1}^{d} z_i=1$.
\par
The inverse result for $\alpha\in(0,1]$ stated in~(\ref{item:thm:3.8.b})
follows from
the concavity of the mapping $\xi\mapsto\Psiastg_{\xi,\alpha}$
and the inequality $z_i^{\alpha} \ge z_i$.
\end{proof}
\par
Due to Theorem~\ref{theo:3.4}, an analogue of the foregoing
result for $\mathrel{\preceq_{\mathrm{apl}}}$ is straightforward.
\begin{corollary}\label{cor:3.10}
Let $X$ be multivariate regularly varying in $\Rplus^{d}$ with tail index
$\alpha\in(0,\infty)$ and identically
distributed margins $X^{(i)}\sim F$, $i=1,\ldots,d$.
Further, let $Y$ be a random vector with independent margins
$Y ^{(i)}\sim F$, and let $Z$ be a random vector with totally dependent
margins $Z^{(i)}=Z^{(1)}$ $\mathrm{P}$-a.s.\ and $Z^{(1)}\sim F$.
Then
\begin{enumerate}[(a)]
\item
$Y \mathrel{\preceq_{\mathrm{apl}}} X \mathrel{\preceq_{\mathrm{apl}}} Z$ for $\alpha \ge 1$
\item
$Z \mathrel{\preceq_{\mathrm{apl}}} X \mathrel{\preceq_{\mathrm{apl}}} Y$ for $\alpha \in (0,1]$.
\end{enumerate}
\end{corollary}
\par
\begin{remark}
The strict assumptions of Corollary~\ref{cor:3.10} are chosen for clearness and simplicity.
The independence of $Y^{(i)}$ and the total dependence of $Z^{(i)}$
are needed only in the tail region, i.e., it suffices for $Y$ and $Z$
to be multivariate regularly varying with canonical spectral measures
$\Psi^\ast_0$ and $\Psi^\ast_1$, respectively.
Furthermore, the assumption of identically distributed margins
can be replaced by equivalent tails:
\[
1=
\lim_{t\to\infty}\frac{\mathrm{P}\cubrfl{Y^{(i)} >t}}{\mathrm{P}\cubrfl{X^{(i)} >t}}
=
\lim_{t\to\infty}\frac{\mathrm{P}\cubrfl{Z^{(i)} >t}}{\mathrm{P}\cubrfl{X^{(i)} >t}}
,\quad i=1,\ldots,d
\ldotp
\]
Finally, the non-negativity of $X^{(i)}$, $Y^{(i)}$, and $Z^{(i)}$
is needed only in the asymptotic sense. The ordering results remain true
if the spectral measure of $X$ is restricted to the unit simplex $\Simp^d$.
\end{remark}
\par
Combining Theorem~\ref{theo:3.4} with Theorem~\ref{theo:2.4}, one obtains an
ordering result for the canonical spectral measures of multivariate regularly
varying elliptical distributions.
The notation $\Psi^\ast=\Psi^\ast(\alpha,C)$ is justified by the fact that
spectral measures of elliptical distributions depend only on the tail
index $\alpha$ and the generalized covariance matrix $C$.
An explicit representation of spectral densities for bivariate elliptical
distributions was obtained by \citet{Hult/Lindskog:2002}.
Alternative representations that are valid for all dimensions $d\ge2$
are given in \citet{Mainik:2010}, Lemma~2.8.
\par
\begin{proposition}
\label{prop:3.7}
Let $C$ and $D$ be $d$-dimensional covariance matrices satisfying
\begin{equation}
\label{eq:3.25}
C_{i,i} = D_{i,i} > 0
,\quad
i = 1,\ldots,d,
\end{equation}
and
\begin{equation}
\label{eq:3.26}
\forall \xi\in\Simp^d
\quad
\xi^{\top} C \xi \le \xi^{\top} D \xi
\ldotp
\end{equation}
Then
\[
\forall \alpha>0
\quad
\Psi^\ast\robr{\alpha,C} \mathrel{\preceq_{\Gcalalpha}} \Psi^\ast\robr{\alpha,D}
\ldotp
\]
\end{proposition}
\par
\begin{myproof}
Fix $\alpha\in(0,\infty)$ and consider random vectors
\[
X\mathrel{\stackrel{\mathrm{d}}{=}} R A U
,\quad
Y\mathrel{\stackrel{\mathrm{d}}{=}} R B U,
\]
where $A$ and $B$ are square roots of the matrices $C$ and $D$
in~\eqref{eq:3.26}, i.e.,
\[
C=A A^{\top}
,\quad
D=B B^{\top},
\]
and $R$ is an arbitrary regularly varying non-negative
random variable with tail index $\alpha$.
\par
As a consequence of Theorem \ref{theo:2.4} one obtains $X \mathrel{\preceq_{\mathrm{apl}}} Y$.
Furthermore, invariance of $\mathrel{\preceq_{\mathrm{apl}}}$ under componentwise rescaling
yields $wX \mathrel{\preceq_{\mathrm{apl}}} wY$ for $w=\robr{w^{(1)},\ldots,w^{(d)}}$ with
\[
w^{(i)}:={C_{i,i}}^{-1/2}={D_{i,i}}^{-1/2}
,\quad
i=1,\ldots,d
\ldotp
\]
Moreover, as a particular consequence of arguments
underlying \eqref{eq:2.17}, one obtains
\[
w^{(i)} X^{(i)} \mathrel{\stackrel{\mathrm{d}}{=}} w^{(j)} Y^{(j)}
,\quad
i,j\in\cubr{1,\ldots,d}
\ldotp
\]
Hence the random vectors $wX$ and $wY$ satisfy the balanced tails condition \eqref{eq:3.15}, whereas their components are mutually ordered with respect to $\mathrel{\preceq_{\mathrm{apl}}}$.
Finally, Theorem~\ref{theo:3.4}(\ref{item:t4.2}) and invariance of canonical spectral measures under componentwise rescalings yield
\begin{equation*}
\Psi^\ast(\alpha,C) = \Psi^\ast_{wX}
\mathrel{\preceq_{\Gcalalpha}}
\Psi^\ast_{wY} = \Psi^\ast(\alpha,D)
\ldotp
\qed
\end{equation*}
\end{myproof}
\par
The subsequent result extends Theorem~\ref{theo:3.4} to random vectors that do not have balanced tails.
\par
\begin{theorem}
\label{theo:3.8}
Let $X$ and $Y$ be multivariate regularly varying random vectors on $\R^{d}$
with tail index $\alpha\in(0,\infty)$
and canonical spectral measures $\Psiast_X$ and $\Psiast_Y$.
Further, assume that $\abs{X^{(i)}} \mathrel{\preceq_{\mathrm{apl}}} \abs{Y^{(i)}}$ with
\begin{equation}
\label{eq:3.27}
\lambda_i
:=
\lim_{t\to\infty}
\frac{\mathrm{P}\cubrfl{\absfl{X^{(i)}}>t}}{\mathrm{P}\cubrfl{\absfl{Y^{(i)}}>t}}
\in(0,1]
\end{equation}
for $i=1,\ldots,d$ and that the vector $v=\robr{v^{(1)},\ldots,v^{(d)}}$ defined by
\begin{equation}
\label{eq:3.28}
v^{(i)}:=\lambda_i^{-1/\alpha}
\end{equation}
satisfies
\begin{equation}
\label{eq:3.29}
X \mathrel{\preceq_{\mathrm{apl}}} v X
\quad\text{or}\quad
v^{-1} Y \mathrel{\preceq_{\mathrm{apl}}} Y
\ldotp
\end{equation}
Then $\Psiast_X \mathrel{\preceq_{\Gcalalpha}} \Psiast_Y$ implies $X \mathrel{\preceq_{\mathrm{apl}}} Y$.
\end{theorem}
\par
\begin{myproof}
According to Proposition~\ref{prop:3.3}(\ref{item:L39.1}), there exists $w\in\Rplus^{d}$
such that $wY$ satisfies the balanced tails condition~\eqref{eq:3.15}.
Furthermore, the tails of the random vector
\[
vwX:=\robrfl{v^{(1)} w^{(1)} X^{(1)} ,\ldots, v^{(d)} w^{(d)} X^{(d)}}
\]
with $v$ defined in~\eqref{eq:3.27} are also balanced. Indeed, it is easy to see that
\[
\lim_{t\to\infty}
\frac{\mathrm{P}\cubrfl{\absfl{w^{(i)} Y^{(i)}} >t}}{\mathrm{P}\cubrfl{\absfl{Y^{(i)}} >t}}
=
\lim_{t\to\infty}
\frac{\mathrm{P}\cubrfl{\absfl{v^{(i)} w^{(i)} X^{(i)}} >t}}{\mathrm{P}\cubrfl{\absfl{v^{(i)} X^{(i)}} >t}}
=
\robrfl{w^{(i)}}^{\alpha}
\]
for $i=1,\ldots,d$. Analogously one obtains
\[
\lim_{t\to\infty}
\frac{\mathrm{P}\cubrfl{\absfl{v^{(i)} X^{(i)}} >t}}{\mathrm{P}\cubrfl{\absfl{X^{(i)}} >t}}
=
\robrfl{v^{(i)}}^{\alpha} = \lambda_i^{-1}
\]
and, as a result,
\begin{align*}
\hspace{2em}&\hspace{-2em}
\lim_{t\to\infty}
\frac{\mathrm{P}\cubrfl{\absfl{v^{(i)} w^{(i)} X^{(i)}} >t}}{\mathrm{P}\cubrfl{\absfl{w^{(i)} Y^{(i)}} > t}}\\
&=
\lim_{t\to\infty}
\frac{\mathrm{P}\cubrfl{\absfl{v^{(i)} X^{(i)}} >t}}{\mathrm{P}\cubrfl{\absfl{Y^{(i)}} > t}}\\
&=
\lim_{t\to\infty}\robrfl{
\frac{\mathrm{P}\cubrfl{\absfl{v^{(i)} X^{(i)}} >t}}{\mathrm{P}\cubrfl{\absfl{X^{(i)}} > t}}
\cdot
\frac{\mathrm{P}\cubrfl{\absfl{X^{(i)}} > t}}{\mathrm{P}\cubrfl{\absfl{Y^{(i)}} > t}}
}\\
&=
\lambda_i^{-1}
\cdot
\lim_{t\to\infty}\frac{\mathrm{P}\cubrfl{\absfl{X^{(i)}} > t}}{\mathrm{P}\cubrfl{\absfl{Y^{(i)}} > t}}\\
&=
1
\end{align*}
for $i=1,\ldots,d$. Hence the balanced tails condition for $wY$ implies that the tails of $vwX$ are also balanced.
\par
Furthermore, invariance of canonical spectral measures under
componentwise rescaling yields
\[
\Psi^\ast_{vwX} = \Psiast_X \mathrel{\preceq_{\Gcalalpha}} \Psiast_Y = \Psi^\ast_{wY}
\ldotp
\]
Thus, applying Theorem~\ref{theo:3.4}(\ref{item:t4.1}), one obtains
\begin{equation}
\label{eq:3.30}
vwX \mathrel{\preceq_{\mathrm{apl}}} wY
\ldotp
\end{equation}
Since $v^{(i)}=\lambda_i^{-1/\alpha} > 0$ for $i=1,\ldots,d$,
condition~\eqref{eq:3.30} is equivalent to
\begin{equation}
\label{eq:3.31}
wX \mathrel{\preceq_{\mathrm{apl}}} v^{-1} wY
\ldotp
\end{equation}
Moreover, assumption~\eqref{eq:3.29} implies
\begin{equation}
\label{eq:3.32}
wX \mathrel{\preceq_{\mathrm{apl}}} vwX
\quad\text{or}\quad
v^{-1}wY\mathrel{\preceq_{\mathrm{apl}}} wY.
\end{equation}
Combining this ordering statement
with \eqref{eq:3.30} and~\eqref{eq:3.31},
one obtains
\[
wX \mathrel{\preceq_{\mathrm{apl}}} wY
\ldotp
\]
Finally,
invariance of $\mathrel{\preceq_{\mathrm{apl}}}$ with respect to componentwise rescaling yields
$X \mathrel{\preceq_{\mathrm{apl}}} Y$.
\end{myproof}
\par
In the special case of random vectors in $\Rplus^{d}$ Theorem~\ref{theo:3.8}
can be simplified to the following result.
\par
\begin{corollary}
\label{cor:3.9}
Let $X$ and $Y$ be multivariate regularly varying random vectors on $\Rplus^{d}$
with tail index $\alpha\in(0,\infty)$ and canonical
spectral measures $\Psiast_X$ and $\Psiast_Y$.
Further, suppose that
\begin{equation}
\label{eq:3.33}
\lambda_i
:=
\limsup_{t\to\infty}
\frac{\mathrm{P}\cubrfl{\absfl{X^{(i)}}>t}}{\mathrm{P}\cubrfl{\absfl{Y^{(i)}}>t}} \in (0,1],
\quad
i = 1,\ldots,d
\ldotp
\end{equation}
Then $\Psiast_X \mathrel{\preceq_{\Gcalalpha}} \Psiast_Y$ implies $X \mathrel{\preceq_{\mathrm{apl}}} Y$.
\end{corollary}
\par
\begin{myproof}
Assumption~\eqref{eq:3.33} yields that the rescaling vector $v$
defined in \eqref{eq:3.28} is an element of $[1,\infty)^d$.
Thus $v-(1,\ldots,1)\in\Rplus^{d}$ and, since $X$ takes values in $\Rplus^{d}$,
we have
\[
X \mathrel{\preceq_{\mathrm{apl}}} X + \robr{v - (1,\ldots,1)}X = vX
\ldotp
\]
Similar arguments yield $v^{-1} Y\mathrel{\preceq_{\mathrm{apl}}} Y$.
Hence condition~\eqref{eq:3.29} of Theorem~\ref{theo:3.8} is satisfied.
\end{myproof}
\par
The final result of this section is due to the indifference of
$\mathrel{\preceq_{\Gcalalpha}}$ for $\alpha=1$ mentioned in
Remark~\ref{rem:3.1}(\ref{item:r14.1}).
This special property of spectral measures on $\Simp^d$ allows to reduce
$\mathrel{\preceq_{\mathrm{apl}}}$ to the ordering of components.
It should be noted that this result cannot be extended to the
general case of spectral measures on $\Sbb^d_1$.
\par
\begin{lemma}
\label{lem:3.10}
Let $X$ and $Y$ be multivariate regularly varying on $\Rplus^{d}$ with tail
index $\alpha=1$.
Further, suppose that $Y$ satisfies the non-degeneracy
condition~\eqref{eq:4} and that
$X^{(i)} \mathrel{\preceq_{\mathrm{apl}}} Y^{(i)}$ for $i=1,\ldots,d$. Then $X \mathrel{\preceq_{\mathrm{apl}}} Y$.
\end{lemma}
\par
\begin{myproof}
According to Proposition \ref{prop:3.3}(\ref{item:L39.1}),
there exists $w\in(0,\infty)^d$ such that $wY$ satisfies the balanced tails
condition~\eqref{eq:3.15}.
Furthermore, due to the invariance of $\mathrel{\preceq_{\mathrm{apl}}}$ under componentwise
rescaling, $X \mathrel{\preceq_{\mathrm{apl}}} Y$ is equivalent to $wX \mathrel{\preceq_{\mathrm{apl}}} wY$.
\par
Thus it can be assumed without loss of generality that $Y$ has balanced tails.
This yields
\[
\lambda_i
:=
\limsup_{t\to\infty}\frac{\mathrm{P}\cubrfl{X^{(i)} >t}}{\mathrm{P}\cubrfl{Y^{(i)} >t}}
=
\limsup_{t\to\infty}\frac{\mathrm{P}\cubrfl{X^{(i)} >t}}{\mathrm{P}\cubrfl{Y^{(1)} >t}}
,\quad
i=1,\ldots,d
\ldotp
\]
Hence the assumption $X^{(i)} \mathrel{\preceq_{\mathrm{apl}}} Y^{(i)}$ for $i=1,\ldots,d$
implies $\lambda_i\in[0,1]$ for all $i$.
Moreover, the balanced tails condition for $Y$ yields
\begin{equation}
\label{eq:3.34}
\gamma_{e_1}(Y)=\ldots=\gamma_{e_d}(Y)
\ldotp
\end{equation}
\par
Now consider the random vector $X$ and denote
\[
j:=\mathop{\mathrm{arg\,max}}\displaylimits_{i\in\cubr{1,\ldots,d}} \gamma_{\ei}(X)
\ldotp
\]
Recall that $\gamma_{\ei}(X)=\nu_X\robr{\cubr{x\in\Rplus^{d}: x^{(i)}>1}}$
with $\nu_X$
denoting the exponent
measure of $X$ and that $\nu_X$ is non-zero. This yields $\gamma_{\ej}(X)>0$
even if $X$ does not satisfy the non-degeneracy condition~\eqref{eq:4}.
Moreover, for $\alpha=1$, the mapping
$\xi\mapsto\gamma_\xi(X)$ is linear. This implies
\begin{equation}
\label{eq:3.35}
\gamma_\xi(X)
=
\sum_{i=1}^{d} \xi^{(i)} \cdot \gamma_{\ei}(X)
\le
\gamma_{\ej}(X)
,\quad
\xi\in\Simp^d
\end{equation}
and \eqref{eq:3.34} yields
\begin{equation}
\label{eq:3.36}
\gamma_\xi(Y) = \sum_{i=1}^{d} \xi^{(i)} \cdot \gamma_{\ei}(Y) = \gamma_{e_1}(Y)
,\quad
\xi\in\Simp^d
\ldotp
\end{equation}
Hence
\begin{align*}
\hspace{2em}&\hspace{-2em}
\limsup_{t\to\infty}
\frac{\mathrm{P}\cubrfl{\xi^{\top} X > t}}{\mathrm{P}\cubrfl{\xi^{\top} Y >t}}\\
&=
\limsup_{t\to\infty}\robrfl{
\frac{\mathrm{P}\cubrfl{\xi^{\top} X > t}}{\mathrm{P}\cubrfl{X^{(j)} > t}}
\cdot
\frac{\mathrm{P}\cubrfl{X^{(j)} > t}}{\mathrm{P}\cubrfl{Y^{(1)} > t}}
\cdot
\frac{\mathrm{P}{\cubrfl{Y^{(1)} > t}}}{\mathrm{P}\cubrfl{\xi^{\top} Y >t}}
}\\
&=
\frac{\gamma_\xi(X)}{\gamma_{\ej}(X)}
\cdot
\lambda_j
\cdot
\frac{\gamma_{e_1}(Y)}{\gamma_\xi(Y)} \\
&\le
1
\end{align*}
due to $\lambda_j\le 1$, \eqref{eq:3.35}, and \eqref{eq:3.36}.
\end{myproof}
\section{Relations to convex and supermodular orders}\label{sec:4}
As mentioned in Remark~\ref{rem:apl.1}(\ref{item:apl.2}),
dependence orders $\mathrel{\preceq_{\mathrm{sm}}}$, $\mathrel{\preceq_{\mathrm{dcx}}}$
and convexity orders $\mathrel{\preceq_{\mathrm{cx}}}$, $\mathrel{\preceq_{\mathrm{icx}}}$, $\mathrel{\preceq_{\mathrm{plcx}}}$
do not imply $\mathrel{\preceq_{\mathrm{apl}}}$ in general.
However, it turns out that the relationship between $\mathrel{\preceq_{\mathrm{apl}}}$ and
the ordering of canonical spectral measures by $\mathrel{\preceq_{\Gcalalpha}}$ allows
to draw conclusions of this type
in the special case of multivariate regularly varying models.
The core result of this section is stated in Theorem~\ref{thm:5}.
It entails
a collection of sufficient
criteria for $\mathrel{\preceq_{\mathrm{apl}}}$ in terms of convex and supermodular order relations,
with particular interest paid to the
inversion of diversification effects for $\alpha<1$.
An application to copula based models is given in Proposition~\ref{prop:4.4}.
\par
This approach was applied by \citet{Embrechts/Neslehova/Wuethrich:2009}
to the ordering of risks for the portfolio vector
$\xi=(1,\ldots,1)$ and for a specific family of multivariate
regularly varying models with identically distributed, non-negative margins
$X^{(i)}$ (cf. Example~\ref{ex:2} in Section~\ref{sec:5}).
\par
The next theorem is the core element of this section.
It generalizes the arguments of \citet{Embrechts/Neslehova/Wuethrich:2009}
to multivariate regularly varying random vectors in $\R^{d}$ with balanced
tails and tail index $\alpha\ne 1$.
The case $\alpha=1$ is not included for two reasons.
First, this case is partly trivial due to the indifference of $\mathrel{\preceq_{\Gcalalpha}}$
for spectral measures on $\Simp^d$ (cf.\ Remark~\ref{rem:3.1}(\ref{item:r14.1})).
Second, Karamata's theorem used in the proof of the integrable case $\alpha>1$ does not yield the desired result for random variables with tail index $\alpha=1$.
\par
\begin{theorem}
\label{thm:5}
Let $X$ and $Y$ be multivariate regularly varying on $\R^{d}$ with identical
tail index $\alpha\ne 1$. Further, assume that $X$ and $Y$ satisfy
the balanced tails condition~\eqref{eq:3.15}.
\begin{enumerate}[(a)]
\item
\label{item:t5.1}
For $\alpha>1$ let
\begin{equation}
\label{eq:309}
\limsup_{t\to\infty}
\frac{\mathrm{P}\cubrfl{\absfl{X^{(1)}}>t}}{\mathrm{P}\cubrfl{\absfl{Y^{(1)}}>t}}
= 1
\end{equation}
and let there exist $u_0>0$ such that with $h_u(t):=\robr{t-u}_{+}$
\begin{equation}
\label{eq:292a}
\forall u \ge u_0 \ \forall \xi\in\Simp^d
\quad
\mathrm{E} h_u\robrfl{\xi^{\top} X}
\le
\mathrm{E} h_u\robrfl{\xi^{\top} Y}
\ldotp
\end{equation}
Then $\Psiast_X \mathrel{\preceq_{\Gcalalpha}} \Psiast_Y$.
\vspace{0.5em}
\item
\label{item:t5.2}
For $\alpha<1$ suppose that $\abs{X^{(1)}}$ and $\abs{Y^{(1)}}$ are
equivalent with respect to $\mathrel{\preceq_{\mathrm{apl}}}$, i.e.,
\begin{equation}
\label{eq:310a}
\absfl{X^{(1)}} \mathrel{\preceq_{\mathrm{apl}}} \absfl{Y^{(1)}}
\quad\text{and}\quad
\absfl{Y^{(1)}} \mathrel{\preceq_{\mathrm{apl}}} \absfl{X^{(1)}},
\end{equation}
and let there exist $u_0 >0$ such that with $f_u(t):=-(t \wedge u)$,
\begin{equation}
\label{eq:292b}
\forall u \ge u_0 \ \forall \xi\in\Simp^d
\quad
\mathrm{E} f_u\robrfl{\robrfl{\xi^{\top} X}_{+}}
\le
\mathrm{E} f_u \robrfl{\robrfl{\xi^{\top} Y}_{+}}
\ldotp
\end{equation}
Then $\Psiast_Y \mathrel{\preceq_{\Gcalalpha}} \Psiast_X$.
\end{enumerate}
\end{theorem}
\par
The proof will be given after some conclusions and remarks.
In particular, it should be noted that the relation between
$\mathrel{\preceq_{\Gcalalpha}}$ and $\mathrel{\preceq_{\mathrm{apl}}}$ established in Theorem~\ref{theo:3.4}
immediately yields the following result.
\begin{corollary}
\label{cor:8}
\begin{enumerate}[(a)]
\item
\label{item:c8.1}%
If random vectors $X$ and $Y$ satisfy conditions of Theorem~\ref{thm:5}(\ref{item:t5.1}),
then $X \mathrel{\preceq_{\mathrm{apl}}} Y$;
\item
\label{item:c8.2}%
If $X$ and $Y$ satisfy conditions of Theorem~\ref{thm:5}(\ref{item:t5.2}),
then $Y \mathrel{\preceq_{\mathrm{apl}}} X$.
\end{enumerate}
\end{corollary}
\par
It should also be noted that
conditions \eqref{eq:292a} and \eqref{eq:292b} are asymptotic forms of
the increasing convex ordering $\xi^{\top} X \mathrel{\preceq_{\mathrm{icx}}} \xi^{\top} Y$
and the decreasing convex ordering $\xi^{\top} X \mathrel{\preceq_{\mathrm{decx}}} \xi^{\top} Y$,
respectively.
The consequences can be outlined as follows.
\begin{remark}
\label{rem:12}
\begin{enumerate}[(a)]
\item
The following criteria are sufficient for~\eqref{eq:292a}
and \eqref{eq:292b} to hold:
\begin{enumerate}[(i)]
\vspace{0.5em
\item
$\robr{\xi^{\top} X}_{+} \mathrel{\preceq_{\mathrm{cx}}} \robr{\xi^{\top} Y}_{+}$
for all $\xi\in\Simp^d$,
\item
$X$ and $Y$ are restricted to $\Rplus^{d}$ and
$X \mathrel{\preceq} Y$ with $\mathrel{\preceq}$ denoting either
$\mathrel{\preceq_{\mathrm{plcx}}}$, $\mathrel{\preceq_{\mathrm{lcx}}}$, $\mathrel{\preceq_{\mathrm{cx}}}$, $\mathrel{\preceq_{\mathrm{dcx}}}$, or $\mathrel{\preceq_{\mathrm{sm}}}$.
\end{enumerate}
\vspace{0.5em}
\item
Additionally, condition~\eqref{eq:292a} follows from
$X \mathrel{\preceq} Y$ with $\mathrel{\preceq}$ denoting either
$\mathrel{\preceq_{\mathrm{plcx}}}$, $\mathrel{\preceq_{\mathrm{lcx}}}$, $\mathrel{\preceq_{\mathrm{cx}}}$, $\mathrel{\preceq_{\mathrm{dcx}}}$, or $\mathrel{\preceq_{\mathrm{sm}}}$.
\end{enumerate}
\end{remark}
Finally, a comment should be made upon
convex ordering of non-integrable random variables and
diversification for $\alpha<1$.
The so-called \emph{phase change} at $\alpha=1$,
i.e., the inversion of diversification effects
taking place when the tail index $\alpha$ crosses this critical value,
demonstrates that the implications of convex ordering are essentially different
for integrable and non-integrable random variables.
Indeed, it is easy to see that if a random variable $Z$ on $\R$
satisfies $\mathrm{E} \sqbr{Z_{+}} = \mathrm{E} \sqbr{Z_{-}}=\infty$,
then the only integrable convex functions of $Z$ are the constant ones.
Moreover, if $Z$ is restricted to $\R_{+}$ and $\mathrm{E} Z =\infty$,
then any integrable convex function of $Z$ is necessarily non-increasing.
\vspace{0.5em}
\par
\begin{myproofx}{of Theorem~\ref{thm:5}}(\ref{item:t5.1})
Consider the expectations in~\eqref{eq:292a}.
It is easy to see that for $u>0$
\begin{align*}
\oneby{u}\mathrm{E} h_u\robrfl{\xi^{\top} X}
&=
\oneby{u}\int_{(u,\infty)}\mathrm{P}\cubrfl{\xi^{\top} X > t} \mathrm{d} t\\
&=
\int_{(1,\infty)} \mathrm{P}\cubrfl{\xi^{\top} X > tu} \mathrm{d} t
\end{align*}
and, as a consequence,
\[
\frac{u^{-1}\mathrm{E} h_u\robrfl{\xi^{\top} X}}{\mathrm{P}\cubrfl{\absfl{X^{(1)}}>u}}
=
\frac{\mathrm{P}\cubrfl{\xi^{\top} X > u}}{\mathrm{P}\cubrfl{\absfl{X^{(1)}}>u}}
\int_{(1,\infty)}
\frac{\mathrm{P}\cubrfl{\xi^{\top} X > tu}}{\mathrm{P}\cubrfl{\xi^{\top} X > u}}
\, \mathrm{d} t
\ldotp
\]
Moreover, Proposition~\ref{prop:3.3}(\ref{item:L39.3}) implies
\begin{equation}
\label{eq:313}
\lim_{u\to\infty}
\frac{\mathrm{P}\cubrfl{\xi^{\top} X > u}}{\mathrm{P}\cubrfl{\absfl{X^{(1)}}>u}}
=
\frac{\gamma_\xi(X)}{\gamma_{e_1}(X)+\gamma_{-e_1}(X)}
=
\PsiastXg_{\xi,\alpha}
\end{equation}
and Karamata's theorem
\citep[cf.][Theorem B.1.5]{de_Haan/Ferreira:2006}
yields
\[
\lim_{u\to\infty}
\int_{(1,\infty)}
\frac{\mathrm{P}\cubrfl{\xi^{\top} X > tu}}{\mathrm{P}\cubrfl{\xi^{\top} X > u}}
\, \mathrm{d} t
=
\int_{(1,\infty)}
t^{-\alpha} \mathrm{d} t
=
\oneby{\alpha-1}
\ldotp
\]
As a result one obtains
\begin{equation*}
\lim_{u\to\infty}
\frac{u^{-1} \mathrm{E} h_u\robrfl{\xi^{\top} X}}{\mathrm{P}\cubrfl{\absfl{X^{(1)}}>u}}
=
\oneby{\alpha-1}\Psiast_X g_{\xi,\alpha}
\end{equation*}
and, analogously,
\begin{equation*}
\lim_{u\to\infty}
\frac{u^{-1}\mathrm{E} h_u\robrfl{\xi^{\top} Y}}{\mathrm{P}\cubrfl{\absfl{Y^{(1)}}>u}}
=
\oneby{\alpha-1}\Psiast_Y g_{\xi,\alpha}
\ldotp
\end{equation*}
Hence \eqref{eq:292a} and \eqref{eq:309} yield
\begin{align*}
1
&\ge
\limsup_{u\to\infty}\frac{u^{-1}\mathrm{E}h_u\robrfl{\xi^{\top} X}}{u^{-1}\mathrm{E} h_u\robrfl{\xi^{\top} Y}}\\
&=
\limsup_{u\to\infty}\robrfl{
\frac{u^{-1} \mathrm{E}h_u\robrfl{\xi^{\top} X}}{\mathrm{P}\cubrfl{\absfl{X^{(1)}}>u}}
\cdot
\frac{\mathrm{P}\cubrfl{\absfl{Y^{(1)}}>u}}{u^{-1} \mathrm{E} h_u\robrfl{\xi^{\top} Y}}
\cdot
\frac{\mathrm{P}\cubrfl{\absfl{X^{(1)}}>u}}{\mathrm{P}\cubrfl{\absfl{Y^{(1)}}>u}}
}\\
&=
\frac{\PsiastXg_{\xi,\alpha}}{\PsiastYg_{\xi,\alpha}}
\end{align*}
for all $\xi\in\Simp^d$, which exactly means
$\Psiast_X \mathrel{\preceq_{\Gcalalpha}} \Psiast_Y$.
\par
\medskip
(\ref{item:t5.2})
Note that~\eqref{eq:310a} implies
\begin{equation}
\label{eq:310}
\lim_{t\to\infty}
\frac{\mathrm{P}\cubrfl{\absfl{X^{(1)}}>t}}{\mathrm{P}\cubrfl{\absfl{Y^{(1)}}>t}}
= 1
\end{equation}
and that~\eqref{eq:292b} yields
\begin{equation}
\label{eq:311}
\forall u>u_0 \ \forall v\ge 0
\quad
\mathrm{E} f_{u+v}\robrfl{\xi^{\top} X} - \mathrm{E} f_{u+v}\robrfl{\xi^{\top} Y} \le 0
\ldotp
\end{equation}
Furthermore, it is easy to see that any random variable $Z$ in $\R_{+}$
satisfies
\begin{align*}
\mathrm{E}\sqbrfl{Z\wedge u}
&=
\int_{(0,\infty)} \robrfl{t \wedge u} \mathrm{d} \mathrm{P}^Z(t)\\
&=
\int_{(0,\infty)}\int_{(0,\infty)} 1\cubr{s<t} \cdot 1\cubr{s<u} \, \mathrm{d} s \, \mathrm{d} \mathrm{P}^Z(t)\\
&=
\int_{(0,\infty)}1\cubr{s<u} \int_{(0,\infty)} 1\cubr{s<t} \, \mathrm{d} \mathrm{P}^Z(t) \, \mathrm{d} s\\
&=
\int_{(0,u)}\mathrm{P}\cubrfl{Z>s} \mathrm{d} s
\ldotp
\end{align*}
This implies
\[
\mathrm{E} f_{u+v}(Z) = \mathrm{E} f_u(Z) - \int_{(u,u+v)} \mathrm{P}\cubrfl{Z>t} \mathrm{d} t
\ldotp
\]
Consequently, \eqref{eq:311} yields
\begin{equation}
\label{eq:312}
\forall u\ge u_0\ \forall v>0
\quad
\mathrm{E} f_u\robrfl{\xi^{\top} X} - \mathrm{E}f_u\robrfl{\xi^{\top} Y}
\le
I(u,v)
\end{equation}
where
\begin{align*}
I(u,v)
&:=
\int_{(u, u+v)}
\robrfl{\mathrm{P}\cubrfl{\xi^{\top} X >t } - \mathrm{P}\cubrfl{\xi^{\top} Y > t}}
\, \mathrm{d} t\\
&\phantom{:}=
\int_{(u,u+v)}
\phi(t) \cdot \mathrm{P}\cubrfl{\absfl{X^{(1)}}>t}
\,\mathrm{d} t
\end{align*}
with
\[
\phi(t)
:=
\frac{\mathrm{P}\cubrfl{\xi^{\top} X > t} - \mathrm{P}\cubrfl{\xi^{\top} Y > t}}
{\mathrm{P}\cubrfl{\absfl{X^{(1)}}>t}}
\ldotp
\]
Moreover, \eqref{eq:310}, \eqref{eq:313}, and an
analogue of~\eqref{eq:313} for $Y$ yield
\begin{align}
\phi(t)
&=\nonumber
\frac{\mathrm{P}\cubrfl{\xi^{\top} X > t}}{\mathrm{P}\cubrfl{\absfl{X^{(1)}}>t}}
-
\frac{\mathrm{P}\cubrfl{\xi^{\top} Y > t}}{\mathrm{P}\cubrfl{\absfl{Y^{(1)}}>t}}
\cdot
\frac{\mathrm{P}\cubrfl{\absfl{Y^{(1)}}>t}}{\mathrm{P}\cubrfl{\absfl{X^{(1)}}>t}}\\
&\to\label{eq:314}
\PsiastXg_{\xi,\alpha} - \PsiastYg_{\xi,\alpha}
,\quad
t\to\infty
\ldotp
\end{align}
\par
Now suppose that $\Psiast_Y \mathrel{\preceq_{\Gcalalpha}} \Psiast_X$ is not satisfied,
i.e., there exists $\xi\in\Simp^d$ such that
$\PsiastYg_{\xi,\alpha} > \PsiastXg_{\xi,\alpha}$.
Then~\eqref{eq:314} yields $\phi(t) \le - \varepsilon$
for some $\varepsilon>0$ and sufficiently large $t$.
This implies
\begin{equation}
\label{eq:315}
I(u,v)
\le
-\varepsilon
\int_{(u, u+v)}
\mathrm{P}\cubrfl{\absfl{X^{(1)}}> t} \, \mathrm{d} t
\end{equation}
for sufficiently large $u$ and all $v\ge0$.
Moreover, regular variation of $\absfl{X^{(1)}}$ with tail index
$\alpha<1$ implies $\mathrm{E}\absfl{X^{(1)}}=\infty$.
Consequently, the integral on the right side of~\eqref{eq:315} tends to
infinity for $v\to\infty$:
\[
\forall u>0
\quad
\lim_{v\to\infty}
\int_{(u, u+v)}
\mathrm{P}\cubrfl{\absfl{X^{(1)}}> t} \, \mathrm{d} t
=
\infty
\ldotp
\]
Hence, choosing $u$ and $v$ sufficiently large, one can achieve
$I(u,v)<c$ for any $c\in\R$. In particular, $u$ and $v$ can be
chosen such that
\[
I(u,v) < \mathrm{E} f_u\robrfl{\xi^{\top} X} - \mathrm{E}f_u\robrfl{\xi^{\top} Y},
\]
which contradicts~\eqref{eq:312}. Thus
$\PsiastYg_{\xi,\alpha} > \PsiastXg_{\xi,\alpha}$ cannot be true
and therefore it necessarily holds that $\Psiast_Y \mathrel{\preceq_{\Gcalalpha}} \Psiast_X$.
\end{myproofx}
\par
Now let us return to the ordering criterion in terms of the supermodular
order $\mathrel{\preceq_{\mathrm{sm}}}$ stated in Remark~\ref{rem:12}. The invariance of $\mathrel{\preceq_{\mathrm{sm}}}$
under non-decreasing component transformations
allows to transfer these criteria to copula models.
Furthermore, since we are interested in the ordering of the asymptotic
dependence structures represented by the canonical spectral measures,
$\Psiast_1$ and $\Psiast_2$, we can take
any copulas that yield $\Psiast_1$ and $\Psiast_2$ as asymptotic
dependence structures.
\par
A natural choice is given by the \emph{extreme value copulas}, defined
as the copulas of \emph{simple max-stable distributions} corresponding to
$\Psi^\ast_i$, i.e., the distributions
\begin{equation}\label{eq:apl.4}
G^\ast_i(x):=\exp\robrfl{-\nu^{\ast}_i\robrfl{-[\infty,x]^\mathrm{c}}}
,\quad
x\in\Rplus^{d}
\end{equation}
where $\nu^{\ast}_i$ is the canonical exponent associated with $\Psi^\ast_i$
via~\eqref{eq:apl.3}. For further details on max-stable and simple max-stable
distributions we refer to \citet{Resnick:1987}.
Since extreme value copulas and canonical spectral measures can be
considered as
alternative parametrizations of the same asymptotic dependence structures,
we obtain the following result.
\par
\begin{proposition}
\label{prop:4.4}
Let $\Psiast_1$ and $\Psiast_2$ be canonical spectral measures on $\Simp^d$.
Further, for $i=1,2$, let $C_i$ denote the copula of the
simple max-stable distribution $G^\ast_i$ induced by $\Psi^\ast_i$
according to~\eqref{eq:apl.4} and~\eqref{eq:apl.3}.
Then $C_1 \mathrel{\preceq_{\mathrm{sm}}} C_2$ implies
\begin{enumerate}[(a)]
\item
$\Psiast_1 \mathrel{\preceq_{\Gcalalpha}} \Psiast_2$ for $\alpha\in(1,\infty)$;
\vspace{0.5em}
\item
$\Psiast_2 \mathrel{\preceq_{\Gcalalpha}} \Psiast_1$ for $\alpha\in(0,1)$.
\end{enumerate}
\end{proposition}
\par
\begin{myproof}
Let $\nu^{\ast}_i$ denote the canonical exponent measures corresponding
to $\Psi^\ast_i$ and $G^\ast_i$.
It is easy to see that the transformed measures
\[
\nu_{\alpha,i}:=\nu^{\ast}_i\circ T^{-1}
,\quad i=1,2,
\]
with $\alpha>0$ and the transformation $T$ defined as
\[
T : x \mapsto
\robrfl{\robrfl{x^{(i)}}^{1/\alpha},\ldots,\robrfl{x^{(d)}}^{1/\alpha}}
,
\quad
x\in\Rplus^{d},
\]
exhibit the scaling property with index $-\alpha$:
\begin{align*}
\nu_{\alpha,i}\robr{tA}
&=
t^{-\alpha} \nu_{\alpha,i}(A)
,\quad
A\in\mathcal{B}\robr{\Rplus^{d}\setminus\cubr{0}}
\ldotp
\end{align*}
Hence the transformed distributions
\begin{equation}
\label{eq:320}
G_{\alpha,i}(x)
:=
G^\ast_i\circ T^{-1}(x)
=
\exp\robrfl{-\nu_{\alpha,i}\robrfl{[0,x]^c}}
\end{equation}
are max-stable with exponent measures $\nu_{\alpha,i}$.
\par
It is well known that max-stable distributions with identical heavy-tailed margins are multivariate regularly varying \citep[cf.][]{Resnick:1987}.
Moreover, the limit measure $\nu$ in the multivariate regular variation condition can be chosen equal to the exponential measure associated with the property of max-stability.
Consequently, the probability distributions $G_{\alpha,i}$ for $i=1,2$ and $\alpha>0$ are multivariate regularly varying with tail index $\alpha$ and canonical spectral measures $\Psi^\ast_i$.
\par
Furthermore, it is easy to see that $X\simG_{\alpha,1}$ and $Y\simG_{\alpha,2}$
have identical margins:
\[
X^{(i)} \mathrel{\stackrel{\mathrm{d}}{=}} Y^{(j)}
,\quad
i,j\in\cubr{1,\ldots,d}
\ldotp
\]
Moreover, due to the invariance of $\mathrel{\preceq_{\mathrm{sm}}}$ under non-decreasing marginal
transformations, $C_1 \mathrel{\preceq_{\mathrm{sm}}} C_2$ implies
\[
G_{\alpha,1} \mathrel{\preceq_{\mathrm{sm}}} G_{\alpha,2}
\]
for all $\alpha>0$.
Thus an application of the ordering criteria from Remark~\ref{rem:12}
to $X\simG_{\alpha,1}$ and $Y\simG_{\alpha,2}$ completes the proof.
\end{myproof}
\section{Examples}
\label{sec:5}
This section concludes the paper by a series of examples with parametric models illustrating the results from the foregoing sections.
Examples~\ref{ex:6} and \ref{ex:2} demonstrate application of Proposition~\ref{prop:4.4} to copula based models and the phenomenon of phase change for random vectors in $\Rplus^{d}$.
The fact that the phase change does not necessarily occur in the general case is demonstrated by multivariate Student-t distributions in Example~\ref{ex:3}.
\par
\begin{example}
\label{ex:6}
Recall the family of Gumbel copulas given by
\begin{equation}
C_\theta(u)
:=
\exp\robrfl{- \robrfl{\sum_{i=1}^{d}\robrfl{-\log u^{(i)}}^\theta}^{1/\theta}}
,\quad
\theta\in[1,\infty)
\ldotp
\end{equation}
Gumbel copulas are extreme value copulas, i.e., they are copulas of simple max-stable distributions.
According to \citet{Hu/Wei:2002}, Gumbel copulas with dependence parameter $\theta\in[1,\infty)$ are ordered by $\mathrel{\preceq_{\mathrm{sm}}}$:
\begin{equation}
\label{eq:322}
\forall \theta_1,\theta_2\in[1,\infty)
\quad
\theta_1 \le \theta_2
\Rightarrow
C_{\theta_1} \mathrel{\preceq_{\mathrm{sm}}} C_{\theta_2}
\ldotp
\end{equation}
Consequently, Proposition \ref{prop:4.4} applies to the family of
canonical spectral measures $\Psi^\ast_\theta$ corresponding to
the Gumbel copulas $C_\theta$.
Thus $1\le\theta_1\le\theta_2<\infty$ implies
$\Psi^\ast_{\theta_1} \mathrel{\preceq_{\Gcalalpha}} \Psi^\ast_{\theta_2}$ for $\alpha>1$
and there is a phase change when $\alpha$ crosses the value $1$, i.e.,
for $\alpha\in(0,1)$ there holds
$\Psi^\ast_{\theta_2} \mathrel{\preceq_{\Gcalalpha}} \Psi^\ast_{\theta_1}$.
\par
Applying Theorem~\ref{theo:3.4}, one obtains ordering with respect to $\mathrel{\preceq_{\mathrm{apl}}}$ for random vectors $X$ and $Y$ on $\Rplus^{d}$ that are multivariate regularly varying with canonical spectral measures
of Gumbel type and have balanced tails ordered by $\mathrel{\preceq_{\mathrm{apl}}}$.
In particular,
this is the case if $X$ and $Y$ have identical
regularly varying marginal distributions and
Archimedean copulas that satisfy appropriate
regularity conditions
\citep[cf.][]{Genest/Rivest:1989, Barbe/Fougeres/Genest:2006}.
\par
Moreover, it is also worth a remark that
multivariate regularly varying random vectors with Archimedean copulas
can only induce extreme value copulas of Gumbel type
\citep[cf.][]{Genest/Rivest:1989}.
\par
Figure~\ref{figure:32} illustrates the resulting diversification effects
in the bivariate case,
including indifference to portfolio diversification for $\alpha=1$ and
the phase change occurring when $\alpha$ crosses this critical value.
The graphics show the function
$\xi^{(1)}\mapsto\Psi^\ast_\theta\,g_{\xi,\alpha}$ for selected values of
$\theta$ and $\alpha$.
Due to $X\in\Rplus^{d}$, representation
$\Psi^\ast_\theta\,g_{\xi,\alpha}=\gamma_\xi/(\gamma_{e_1} + \gamma_{-e_1})$ simplifies to
$\Psi^\ast_\theta\,g_{\xi,\alpha}=\gamma_\xi/\gamma_{e_1}$
and therefore
\[
\Psi^\ast_\theta \, g_{e_1,\alpha} = \Psi^\ast_\theta \, g_{e_2,\alpha} = 1
\ldotp
\]
\par
\begin{figure
\centering
\subfigure[Varying $\alpha$ for $\theta=1.4$]
{\includegraphics[width=.45\textwidth]{Graphics/Fig1a-gumbel-eri-1}}
\subfigure[Varying $\alpha$ for $\theta=2$]
{\includegraphics[width=.45\textwidth]{Graphics/Fig1b-gumbel-eri-2}}
\\
\subfigure[Varying $\theta$ for $\alpha>1$]
{\includegraphics[width=.45\textwidth]{Graphics/Fig1c-gumbel-eri-3}}
\subfigure[Varying $\theta$ for $\alpha<1$]
{\includegraphics[width=.45\textwidth]{Graphics/Fig1d-gumbel-eri-4}}
\caption{Bivariate Gumbel copulas: Diversification effects represented by functions $\xi^{(1)}\mapsto\Psi^\ast_\theta \,g_{\xi,\alpha}$ for selected values of $\theta$ and $\alpha$.}
\label{figure:32
\end{figure}
\end{example}
As already mentioned above, Theorem~\ref{thm:5} generalizes some
arguments from \citet{Embrechts/Neslehova/Wuethrich:2009}.
The next example concerns Galambos copulas as addressed in
that original publication.
\par
\begin{example}
\label{ex:2}
Another family of extreme value copulas that are ordered by $\mathrel{\preceq_{\mathrm{sm}}}$
is the family of $d$-dimensional
\emph{Galambos copulas}
with parameter $\theta\in(0,\infty)$:
\begin{equation}
C_\theta(u)
:=
\exp\robrfl{\sum_{I\subset\cubr{1,\ldots,d}}
(-1)^{\abs{I}}\robrfl{\sum_{i\in I} \robrfl{-\log u^{(i)}}^{-\theta}}^{-1/\theta}
}
\ldotp
\end{equation}
According to \citet{Hu/Wei:2002},
$\theta_1 \le \theta_2$ implies $C_{\theta_1} \mathrel{\preceq_{\mathrm{sm}}} C_{\theta_2}$.
Thus Proposition~\ref{prop:4.4} yields ordering of the corresponding canonical spectral measures $\Psi^\ast_\theta$ with respect to $\mathrel{\preceq_{\Gcalalpha}}$.
Similarly to the case of Gumbel copulas, $\theta_1\le\theta_2$ implies $\Psi^\ast_{\theta_1} \mathrel{\preceq_{\Gcalalpha}} \Psi^\ast_{\theta_2}$ for $\alpha>1$
and $\Psi^\ast_{\theta_2} \mathrel{\preceq_{\Gcalalpha}} \Psi^\ast_{\theta_1}$ for $\alpha\in(0,1)$.
\par
Finally, it should be noted that Galambos copulas correspond to the
canonical exponent measures of random vectors $X$ in $\Rplus^{d}$ with
identically distributed regularly varying margins $X^{(i)}$ and
dependence structure of $-X$ given by an Archimedean copula with a
regularly varying generator $\phi(1-1/t)$. Models of this type were
discussed in recent studies of aggregation effects for extreme risks %
\citep[cf.][]{Alink/Loewe/Wuethrich:2004, %
Alink/Loewe/Wuethrich:2005, %
Embrechts/Neslehova/Chavez-Demoulin:2006, %
Barbe/Fougeres/Genest:2006, %
Embrechts/Lambrigger/Wuethrich:2008, %
Embrechts/Neslehova/Wuethrich:2009}.
\end{example}
\par
The final example illustrates results established in Proposition~\ref{prop:3.7} and Theorem~\ref{theo:2.4}. In particular, it shows that elliptical distributions do not exhibit a phase change at $\alpha=1$.
\par
\begin{example}
\label{ex:3}
Recall multivariate Student-t distributions and
consider the case with equal degrees of freedom, i.e.,
\begin{equation}
X\mathrel{\stackrel{\mathrm{d}}{=}} \mu_X + R A_X U,
\quad
Y\mathrel{\stackrel{\mathrm{d}}{=}} \mu_Y + R A_Y U,
\end{equation}
where $R\mathrel{\stackrel{\mathrm{d}}{=}}\abs{Z}$ for a Student-t distributed random variable $Z$ with degrees of freedom equal to $\alpha\in(0,\infty)$.
Further, let the generalized covariance matrices $C_X=C(\rho_X)$ and $C_Y=C(\rho_Y)$ be defined as
\begin{equation}
\label{eq:146}
C(\rho):=
\left(
\begin{array}{cc}
1 &\rho\\
\rho & 1
\end{array}
\right)
\end{equation}
and assume that $\rho_X \le \rho_Y$.
\par
As already mentioned in Remark~\ref{rem:2.6}(\ref{item:rem:2.6.a}),
$C_X$ and $C_Y$ satisfy condition~\eqref{eq:3.26} and
Proposition~\ref{prop:3.7} yields $X \mathrel{\preceq_{\mathrm{apl}}} Y$. Moreover,
Proposition~\ref{prop:3.7} implies a uniform ordering of diversification
effects in the sense that
\[
\Psiast_X=\Psi^\ast_{\alpha,\rho_X} \mathrel{\preceq_{\Gcalalpha}} \Psi^\ast_{\alpha,\rho_Y}=\Psiast_Y
\]
for all $\alpha\in(0,\infty)$.
\par
Figure~\ref{figure:7} shows functions $\xi^{(1)} \mapsto \Psi^\ast_{\alpha,\rho}\,g_{\xi,\alpha}$ for selected parameter values $\rho$ and $\alpha$ that illustrate the ordering of asymptotic portfolio losses by $\rho$ and the missing phase change at $\alpha=1$. The indifference to portfolio diversification for $\alpha=1$ is also absent.
Moreover, symmetry of elliptical distributions implies $\gamma_{-e_1} = \gamma_{e_1}$ and, as a result,
\[
\Psi^\ast_{\alpha,\rho}\, g_{e_1,\alpha}
=
\Psi^\ast_{\alpha,\rho}\, g_{e_2,\alpha}
=
1/2
\ldotp
\]
Thus the standardization of the plots in Figure~\ref{figure:7} is different from that in Figure~\ref{figure:32}.
\par
\begin{figure
\centering
\subfigure[Varying $\alpha$ for $\rho>0$]
{\includegraphics[width=.45\textwidth]{Graphics/Fig2a-ellipt-1}}
\subfigure[Varying $\alpha$ for $\rho<0$]
{\includegraphics[width=.45\textwidth]{Graphics/Fig2b-ellipt-2}}
\\
\subfigure[Varying $\rho$ for $\alpha>1$]
{\includegraphics[width=.45\textwidth]{Graphics/Fig2c-ellipt-3}}
\subfigure[Varying $\rho$ for $\alpha<1$]
{\includegraphics[width=.45\textwidth]{Graphics/Fig2d-ellipt-4}}
\caption{Bivariate elliptical distributions with generalized covariance matrices defined in~\eqref{eq:146}: Diversification effects represented by functions $\xi^{(1)}\mapsto\Psi^\ast_{\alpha,\rho}\, g_{\xi,\alpha}$ for selected values of $\rho$ and $\alpha$.}
\label{figure:7
\end{figure}
\end{example}
\par
\begin{remark}
\label{rem:15}%
All examples the authors are aware of suggest that the diversification
coefficient $\Psiastg_{\xi,\alpha}$ is decreasing in $\alpha$.
This means that risk diversification is stronger for lighter component tails
than for heavier ones.
\par
However, it should be noted that the influence of the tail index $\alpha$
on risk aggregation is different from that. The asymptotic risk aggregation coefficient
\[
q_d := \lim_{t\to\infty}\frac{\mathrm{P}\cubrfl{X^{(1)}+\ldots+X^{(d)} > t}}{\mathrm{P}\cubrfl{X^{(1)}>t}}
\]
introduced by %
\citet{Wuethrich:2003} is known to be increasing in $\alpha$ when the loss components
$X^{(i)}$ are non-negative %
\citep[cf.][]{Barbe/Fougeres/Genest:2006}.
It is easy to see that the restriction to non-negative $X^{(i)}$ implies
\[
q_d
=
\lim_{t\to\infty}\frac{\mathrm{P}\cubrfl{\norm{X}_1 > t}}{\mathrm{P}\cubrfl{X^{(1)}>t}}
=
\oneby{\gamma_{e_1}}
\ldotp
\]
Moreover, denoting the uniformly diversified portfolio by $\eta$,
\[
\eta:=d^{-1}\robr{1,\ldots,1}
,
\]
one obtains
\[
q_d
=
\lim_{t\to\infty}\frac{\mathrm{P}\cubrfl{\eta^{\top} X > d^{-1} t}}{\mathrm{P}\cubrfl{X^{(1)}>t}}
=
d^{\alpha} \frac{\gamma_\eta}{\gamma_{e_1}}
\ldotp
\]
Thus $q_d$ is a product of the factor $d^{\alpha}$, which is
increasing in $\alpha$, and the ratio $\gamma_\eta/\gamma_{e_1}$,
which is closely related to to the diversification coefficient
$\Psiastg_{\xi,\alpha}$.
\par
In particular, given equal marginal weights, i.e.,
\[
\gamma_{e_1}=\ldots=\gamma_{e_d},
\]
Proposition~\ref{prop:3.3}(\ref{item:L39.3}) yields
\[
\frac{\gamma_\eta}{\gamma_{e_1}} = \Psi^\ast g_{\eta,\alpha}
\ldotp
\]
As already mentioned above, the coefficients $\Psiastg_{\xi,\alpha}$
with $\xi\in\Simp^d$ are decreasing in all examples considered here.
This means that the aggregation and the diversification of risks are
influenced by the tail index $\alpha$ in different, maybe even always
contrary ways.
\par
The question for the generality of this contrary influence is currently open.
One can easily prove that the extreme risk index $\gamma_\xi= \Psif_{\xi,\alpha}$ is decreasing in $\alpha$ for $\xi\in\Simp^d$. However, this result cannot be extended to $\Psiastg_{\xi,\alpha}$ directly since $\Psiastg_{\xi,\alpha}$ is related to $\Psif_{\xi,\alpha}$ by the normalizations~\eqref{eq:3.17} and~\eqref{eq:3.18}.
The question
whether $\Psiastg_{\xi,\alpha}$ with arbitrary $\xi\in\Simp^d$ or at least
$\Psi^\ast g_{\eta,\alpha}$ is generally decreasing in $\alpha$
is an interesting subject for further research.
\end{remark}
\section{Acknowledgements}
The research underlying this paper was done at the University of Freiburg.
Georg Mainik would also like to thank RiskLab for financial support.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 8,336 |
Q: Add sequence number to split comma separated values in new rows I have the following set of data:
Col A Col B
Prod1 SKU-A, SKU-B
Prod2 SKU-C, SKU-D, SKU-E
I need this outputted as:
Col A Col B Col C
Prod1 SKU-A 1
Prod1 SKU-B 2
Prod2 SKU-C 1
Prod2 SKU-D 2
Prod2 SKU-E 3
Basically i need to split the comma separated values in column B into separate rows, which i think i can do via various macros available on this site, however i need to add (another) column C that assigns a numeric sequence to each split row for every Product given in column A.
I hope this makes sense!
EDIT to add in VBA i am using:
Sub SliceNDice()
Dim objRegex As Object
Dim X
Dim Y
Dim lngRow As Long
Dim lngCnt As Long
Dim tempArr() As String
Dim strArr
Set objRegex = CreateObject("vbscript.regexp")
objRegex.Pattern = "^\s+(.+?)$"
'Define the range to be analysed
X = Range([a1], Cells(Rows.Count, "b").End(xlUp)).Value2
Redim Y(1 To 2, 1 To 1000)
For lngRow = 1 To UBound(X, 1)
'Split each string by ","
tempArr = Split(X(lngRow, 2), ",")
For Each strArr In tempArr
lngCnt = lngCnt + 1
'Add another 1000 records to resorted array every 1000 records
If lngCnt Mod 1000 = 0 Then Redim Preserve Y(1 To 2, 1 To lngCnt + 1000)
Y(1, lngCnt) = X(lngRow, 1)
Y(2, lngCnt) = objRegex.Replace(strArr, "$1")
Next
Next lngRow
'Dump the re-ordered range to columns C:D
[c1].Resize(lngCnt, 2).Value2 = Application.Transpose(Y)
End Sub
Do let me know if you need any other details, thank you in advance :-)
A: With data in columns A and B, running this short macro will produce the desired output in columns C through E
Sub reorg()
Dim i As Long, N As Long
Dim K As Long, KK As Long
N = Cells(Rows.Count, 1).End(xlUp).Row
K = 1
For i = 1 To N
prod = Cells(i, 1).Value
ary = Split(Cells(i, 2).Value, ", ")
For KK = LBound(ary) To UBound(ary)
Cells(K, 3).Value = prod
Cells(K, 4).Value = ary(KK)
Cells(K, 5).Value = KK + 1
K = K + 1
Next KK
Next i
End Sub
A: Yours is a two part question.
This is the answer for the second part of the question.
How to count and update the number of times PRODxhas appeared.
I have implemented your requirement using formulas (but feel free to try your hand with VBA)
The formula is =COUNTIF($A$2:A2,A2). Enter the formula for your first row and just drag along the length of your data and your will be able to get the count of PRODx
| {
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} | 1,527 |
How to get rid of homeless. Volume one and two.
by Matteo Bittanti
Features volume one: Soft cover, 440 pages, illustrations, full color.
Features volume two: Soft cover, 160 pages, illustrations, full color.
Format: 7 x 7 inches = 18 x 18 cm
ISBN volume 1: 9781320335096
Price volume one: $149.99
Price volume two: $69.99
LIMITED EDITION: 99 copies
PURCHASE HERE: VOLUME 1 & VOLUME 2
How to get rid of homeless is a monumental project.
A 600-page epic split in two volumes documenting the so-called "homeless scandal" that affected the newly released game SimCity (Maxis/Electronic Arts, 2013), How to get rid of homeless reproduces dozens of threads concerning "homelessness" that appeared in Electronic Arts' online forum between 2012 and 2013.
Matteo Bittanti collected, selected, and transcribed thousands of messages exchanged by the forum members who first experienced and then tried to "eradicate" the phenomenon of homelessness that "plagued" SimCity. From surprise to despair, from shock to resignation, these posts highlight the pitfalls of simulation, the not-so-subtle effects of ideology on game design, and the interplay between play and society, politics and entertainment. Decontextualized from their original source and reproduced on paper sans the majority of online communication hallmarks (e.g. author's signatures, side banners, avatar pictures etc.), these textual exchanges create a peculiar narrative. Some of the dialogues' absurdist tones evoke Ionesco's plays. Others reveal racist and classist biases, and forcefully introduce - or, rather, reintroduce - a highly political vision that the alleged "neutral" algorithms were supposed to overcome.
Matteo Bittanti Artist, writer, curator, publisher, translator, and scholar, Matteo Bittanti works with toys, games, and technology. His academic research focuses on the cultural, social, and theoretical aspects of emerging technologies, with an emphasis on their effects on communication, visual culture, and the arts. His approach is interdisciplinary in nature and his practice is situated at the intersection of media studies, game studies, visual studies, and contemporary art. Born in Milan, Italy Bittanti resides in Northern California. Like most Americans, he lives a life of quiet desperation.
READ THE CRITICAL TEXT
CRITICAL TEXT
MATTEO BITTANTI READS FROM HOW TO GET RID OF HOMELESS #1 | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 2,210 |
Q: Create 2 separate exes from one .Net Core project? I have two simple, standalone, command line programs. Each is a single source file (say Program1.cs & Program2.cs), each containing a static void Main(..), and each should compile to a single, standalone binary. They're related so I want to keep in a single directory for simplicity.
With .Net pre-Core, it's easy to build with csc Program<x>.cs & out comes the exe.
Is there a way to replicate that in .Net Core? Or does it mean 2 separate projects? So far, I can't find a way to generate 2 binaries, each with a separate entry point, in one project.
Thanks.
--
*
*These are simple demos. I know it's perhaps not representative of typical projects, where there would be multiple source files with a single entry point. However ability to scale down to simple cases is important too.
A: What you can do is to use separate configuration for each program and in that configuration set which files to compile.
In your case, the relevant section of your project.json might look like this:
"configurations": {
"Program1": {
"buildOptions": {
"compile": {
"exclude": "Program2.cs"
}
}
},
"Program2": {
"buildOptions": {
"compile": {
"exclude": "Program1.cs"
}
}
}
}
You can then build Program1 with dotnet build -c Program1 and run it with dotnet run -c Program1 and similarly for Program2.
| {
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} | 3,522 |
{"url":"https:\/\/projecteuclid.org\/euclid.die\/1356060473","text":"## Differential and Integral Equations\n\n### On the domains of elliptic operators in $L^1$\n\n#### Abstract\n\nWe prove optimal embedding estimates for the domains of second-order elliptic operators in $L^1$ spaces. Our procedure relies on general semigroup theory and interpolation arguments, and on estimates for $\\nabla T(t)f$ in $L^1$, in $L^\\infty$, and possibly in fractional Sobolev spaces, for $f\\in L^1$. It is applied to a number of examples, including some degenerate hypoelliptic operators, and operators with unbounded coefficients.\n\n#### Article information\n\nSource\nDifferential Integral Equations Volume 17, Number 1-2 (2004), 73-97.\n\nDates\nFirst available in Project Euclid: 21 December 2012\n\nLunardi, Alessandra; Metafune, Giorgio. On the domains of elliptic operators in $L^1$. Differential Integral Equations 17 (2004), no. 1-2, 73--97.https:\/\/projecteuclid.org\/euclid.die\/1356060473","date":"2018-01-22 06:16:50","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.5162421464920044, \"perplexity\": 1206.2549179546331}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2018-05\/segments\/1516084891105.83\/warc\/CC-MAIN-20180122054202-20180122074202-00438.warc.gz\"}"} | null | null |
Q: Delete all character after particular character in string I have a string and I want to delete some character after particulate character, I tried using with substring but nothings working
A: "word" is a String Variable here where you can apply your needs and "T" is a character after which you want to delete all other characters
if let index = word.range(of: "T")?.lowerBound {
let substring = word[..<index]
let string = String(substring)
self.birthdayField.text = string
}
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 376 |
Hung Liu, American artist of Chinese descent with an exhibition at the National Portrait Gallery, recalls
Hung Liu in his studio with Rat Year 2020, 2020; oil on linen canvas and mixed media on wood panel, each: 64 x 100 in. (162.6 x 254 cm), Diptych. Photo by John Janca. Artwork © Hung Liu
Artist Hung Liu, born in Changchun, China, in 1948, died at the age of 73 in California. (Find out more in ARTnews)
Kim Sajet, director of the Smithsonian's National Portrait Gallery, made the following statement on the passing of Hung Liu, just weeks before the museum's upcoming exhibition "Hung Liu: Portraits of Promised Lands", the first major presentation of the art. from Liu on the east coast:
"The National Portrait Gallery mourns the death of Hung Liu, whose extraordinary artistic vision reminds us that even in the midst of despair, there is hope, and when people help each other, there is joy. She believed in the power of art and portraiture to change the world, "said Kim Sajet, director of the Smithsonian's National Portrait Gallery.
"We are grateful that during Hung Liu's last week, Dorothy Moss, the curator of the exhibition, was able to travel to Oakland, Calif., To meet the artist," Sajet noted. Liu, who was instrumental in the design of the installation and its accompanying catalog, approved the final layout of the exhibition and admired the completed book. She shared her belief in the potential from the exhibition to conveying his hopes for the future, a future based on a foundation of empathy for others. "
"The museum's upcoming portrait retrospective of Liu will celebrate a life and career dedicated to honoring the stories of those at risk of being forgotten. Devoid of platitude or cliché, Liu's artistic practice has always been anchored in history by transforming marginalized subjects into monumental, heroic and contemporary figures. Her generous and vibrant spirit is alive in everything she has done and is powerfully present throughout the installation of what will be the final presentation of her curated art during her lifetime.
About his work, Liu said, "We don't need a language, but we can communicate across time and space; there is something beyond that connects us. The exhibition "Hung Liu: Portraits of Promised Lands" opens on August 27.
Indra Nooyi inducted into the National Portrait Gallery
London's National Portrait Gallery to close for three years for major renovation
Baby Yoda gets official portrait at National Portrait Gallery
The Smithsonian's National Portrait Gallery presents "Hung Liu: Portraits of Promised Lands"
Parkes' National Portrait Gallery reopened after being evacuated | Canberra weather
Exhibition review: Australian Love Stories, National Portrait Gallery, ACT | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 6,181 |
package ec2
import (
"context"
"github.com/aws/aws-sdk-go-v2/aws"
"github.com/aws/aws-sdk-go-v2/internal/awsutil"
)
type CancelImportTaskInput struct {
_ struct{} `type:"structure"`
// The reason for canceling the task.
CancelReason *string `type:"string"`
// Checks whether you have the required permissions for the action, without
// actually making the request, and provides an error response. If you have
// the required permissions, the error response is DryRunOperation. Otherwise,
// it is UnauthorizedOperation.
DryRun *bool `type:"boolean"`
// The ID of the import image or import snapshot task to be canceled.
ImportTaskId *string `type:"string"`
}
// String returns the string representation
func (s CancelImportTaskInput) String() string {
return awsutil.Prettify(s)
}
type CancelImportTaskOutput struct {
_ struct{} `type:"structure"`
// The ID of the task being canceled.
ImportTaskId *string `locationName:"importTaskId" type:"string"`
// The current state of the task being canceled.
PreviousState *string `locationName:"previousState" type:"string"`
// The current state of the task being canceled.
State *string `locationName:"state" type:"string"`
}
// String returns the string representation
func (s CancelImportTaskOutput) String() string {
return awsutil.Prettify(s)
}
const opCancelImportTask = "CancelImportTask"
// CancelImportTaskRequest returns a request value for making API operation for
// Amazon Elastic Compute Cloud.
//
// Cancels an in-process import virtual machine or import snapshot task.
//
// // Example sending a request using CancelImportTaskRequest.
// req := client.CancelImportTaskRequest(params)
// resp, err := req.Send(context.TODO())
// if err == nil {
// fmt.Println(resp)
// }
//
// Please also see https://docs.aws.amazon.com/goto/WebAPI/ec2-2016-11-15/CancelImportTask
func (c *Client) CancelImportTaskRequest(input *CancelImportTaskInput) CancelImportTaskRequest {
op := &aws.Operation{
Name: opCancelImportTask,
HTTPMethod: "POST",
HTTPPath: "/",
}
if input == nil {
input = &CancelImportTaskInput{}
}
req := c.newRequest(op, input, &CancelImportTaskOutput{})
return CancelImportTaskRequest{Request: req, Input: input, Copy: c.CancelImportTaskRequest}
}
// CancelImportTaskRequest is the request type for the
// CancelImportTask API operation.
type CancelImportTaskRequest struct {
*aws.Request
Input *CancelImportTaskInput
Copy func(*CancelImportTaskInput) CancelImportTaskRequest
}
// Send marshals and sends the CancelImportTask API request.
func (r CancelImportTaskRequest) Send(ctx context.Context) (*CancelImportTaskResponse, error) {
r.Request.SetContext(ctx)
err := r.Request.Send()
if err != nil {
return nil, err
}
resp := &CancelImportTaskResponse{
CancelImportTaskOutput: r.Request.Data.(*CancelImportTaskOutput),
response: &aws.Response{Request: r.Request},
}
return resp, nil
}
// CancelImportTaskResponse is the response type for the
// CancelImportTask API operation.
type CancelImportTaskResponse struct {
*CancelImportTaskOutput
response *aws.Response
}
// SDKResponseMetdata returns the response metadata for the
// CancelImportTask request.
func (r *CancelImportTaskResponse) SDKResponseMetdata() *aws.Response {
return r.response
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 6,101 |
Black & White - ScotlandFramed.co.uk - Scenic photography of Scotland.
"Forth Railway Bridge BW (1)"
"St Leonards In The Fields"
"Tay Stree in Winter "
Some Black & White images can also be found in "Special Effects" and "Panoramic". Some the places featured here can also be found in other galleries such as The Hermitage at Dunkeld, Balthayock near Perth, Rua Reidh near Melvaig and Loch Maree go on have a look round enjoy the other photographs of Scotland and see if you can spot where they are. | {
"redpajama_set_name": "RedPajamaC4"
} | 7,173 |
Pentacros margaritatus, unique représentant du genre Pentacros, est une espèce d'opilions laniatores de la famille des Podoctidae.
Distribution
Cette espèce est endémique de Waigeo en Nouvelle-Guinée occidentale en Indonésie.
Description
Le mâle syntype mesure .
Publication originale
Roewer, 1949 : « Über Phalangodidae II. Weitere Weberknechte XIV. » Senckenbergiana, , .
Liens externes
genre Pentacros :
espèce Pentacros margaritatus :
Notes et références
Podoctidae
Espèce d'opilions (nom scientifique)
Faune endémique de Nouvelle-Guinée occidentale | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 3,071 |
DISPOSAL OF PUBLIC LAND | BREHENY PLACE, WEST ULVERSTONE
Notice is given that at its meeting on Monday 13 December 2021, the Council resolved to dispose (transfer) a parcel of land located adjacent 12 Breheny Place, West Ulverstone (Certificate of Title 24214 Folio 245). (Parcel highlighted in red on the plan below.)
The parcel to be transferred comprises an area of approx. 93m2 and is to be transferred to the Crown, with costs being met either by the Crown or Housing Tasmania. The parcel is to be combined with adjoining parcels to allow the Director of Housing to develop land at 12 and 27 Breheny Place, West Ulverstone for residential purposes.
The disposal (transfer) will be in accordance with the provisions of s.178 of the Local Government Act 1993 (the Act) – Sale, exchange and disposal of public land.
In accordance with the Act, any person may, by writing to the General Manager, Central Coast Council, PO Box 220, Ulverstone TAS 7315, within 21 days, object to the proposed disposal (transfer) of said public land. Any objections received will be considered by the Council and, in accordance with s.178A of the Act, further appeal provisions apply.
The expected timeframe for the proposed disposal of land is 60 days.
For further information, please contact the Council's Director Corporate Services, Ian Stoneman via email at [email protected] or tel. 6429 8900.
Notices of objection will be received until 12 noon on Monday 24 January 2022.
Our Facebook Feed | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 8,013 |
Q: How can I redirect a user if they are not authenticated using router In Angular 2 how can I redirect a user if they are not authenticated using router version 2.0.0-rc.1
I'm looking to do this in my app.component where my routes are store.
I want to detect if the user is authenticated before it hits a pages component.
So I can then redirect them to the login.
I've tried canActivate which doesnt seem to work for my router version.
I'm looking for a solution for this version of the router: "@angular/router": "2.0.0-rc.1"
OR
A solution for the latest version of the router. If one is provided can you show me how to update my router version using git bash with this solution.
This is my current code:
export class AppComponent implements CanActivate {
authService: AuthService;
userService: UserService;
constructor(_authService: AuthService, _userService: UserService, private location: Location, private router: Router) {
this.authService = _authService;
this.userService = _userService;
}
canActivate(route: ActivatedRouteSnapshot, state: RouterStateSnapshot): Observable < boolean > | boolean {
//This doesnt get hit
console.log("here");
return true;
}
}
A: Use an Interceptor:
import {bootstrap} from '@angular/platform-browser-dynamic';
import {provide} from '@angular/core';
import {HTTP_PROVIDERS, Http, Request, RequestOptionsArgs, Response, XHRBackend, RequestOptions, ConnectionBackend, Headers} from '@angular/http';
import {ROUTER_PROVIDERS, Router} from '@angular/router';
import {LocationStrategy, HashLocationStrategy} from '@angular/common';
import { Observable } from 'rxjs/Observable';
import * as _ from 'lodash'
import {MyApp} from './app/my-app';
class HttpInterceptor extends Http {
constructor(backend: ConnectionBackend, defaultOptions: RequestOptions, private _router: Router) {
super(backend, defaultOptions);
}
request(url: string | Request, options?: RequestOptionsArgs): Observable<Response> {
return this.intercept(super.request(url, options));
}
get(url: string, options?: RequestOptionsArgs): Observable<Response> {
return this.intercept(super.get(url,options));
}
post(url: string, body: string, options?: RequestOptionsArgs): Observable<Response> {
return this.intercept(super.post(url, body, this.getRequestOptionArgs(options)));
}
put(url: string, body: string, options?: RequestOptionsArgs): Observable<Response> {
return this.intercept(super.put(url, body, this.getRequestOptionArgs(options)));
}
delete(url: string, options?: RequestOptionsArgs): Observable<Response> {
return this.intercept(super.delete(url, options));
}
getRequestOptionArgs(options?: RequestOptionsArgs) : RequestOptionsArgs {
if (options == null) {
options = new RequestOptions();
}
if (options.headers == null) {
options.headers = new Headers();
}
options.headers.append('Content-Type', 'application/json');
return options;
}
intercept(observable: Observable<Response>): Observable<Response> {
return observable.catch((err, source) => {
if (err.status == 401 && !_.endsWith(err.url, 'api/auth/login')) {
this._router.navigate(['/login']);
return Observable.empty();
} else {
return Observable.throw(err);
}
});
}
}
bootstrap(MyApp, [
HTTP_PROVIDERS,
ROUTER_PROVIDERS,
provide(LocationStrategy, { useClass: HashLocationStrategy }),
provide(Http, {
useFactory: (xhrBackend: XHRBackend, requestOptions: RequestOptions, router: Router) => new HttpInterceptor(xhrBackend, requestOptions, router),
deps: [XHRBackend, RequestOptions, Router]
})
])
.catch(err => console.error(err));
A: You can use this
import { Injectable } from '@angular/core';
import { HttpInterceptor, HttpRequest, HttpHandler, HttpEvent, HttpErrorResponse } from '@angular/common/http';
import { Observable, throwError } from 'rxjs';
import { catchError } from 'rxjs/operators';
import { Router } from '@angular/router';
@Injectable()
export class ErrorInterceptor implements HttpInterceptor {
constructor(private router: Router) {}
intercept(req: HttpRequest<any>, next: HttpHandler): Observable<HttpEvent<any>> {
return next.handle(req)
.pipe(
catchError(
(err: HttpErrorResponse) => {
if (this.router.url !== '/login' && err.status === 401) {
this.router.navigate(['/login']);
}
return throwError(err);
}
)
);
}
}
And in app.module.ts
{
provide: HTTP_INTERCEPTORS,
useClass: ErrorInterceptor,
multi: true,
},
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 623 |
Coventry Utd set to be saved from liquidation after takeover agreed
Peter Parker January 4, 2022
Women's Championship side Coventry United are set to be saved from liquidation after a Midlands-based energy company reached a verbal agreement to take over the club.
Last month, Coventry United announced they had entered voluntary liquidation, having only gone fully professional this season.
But Energy Angels chief Lewis Taylor, who is willing to invest between £200,000 to £250,000 to ensure Coventry United can finish this season, is set to purchase the club after an agreement was reached with existing shareholders.
It comes after Taylor held constructive talks on Tuesday with the existing shareholders, who were willing to negotiate as they did not want to place the club into liquidation.
As a result of the takeover talks, Coventry's next match at Bristol City on Sunday has been postponed.
Taylor told Sky Sports News on Monday: "The deal is solely between us and the existing shareholders, so we're not reliant on any third parties to say we can or can't do this.
"I envisage the existing shareholders to just be happy someone else is willing to take it all on.
"We're not doing this just to keep the team together this season. That's the immediate concern, but I wouldn't be here doing this if it was only for this season.
"There's a potential to play WSL here, which should always be the goal."
Coventry United manager Jay Bradford has met prospective new buyer Lewis Taylor
Taylor revealed he held virtual meetings on New Year's Day with manager Jay Bradford and the club's players to explain his takeover bid.
He said: "We've had a really positive response from the squad.
"We will shore the situation up very quickly and get the players paid for a start and make sure we're training on Wednesday morning.
"I want to reverse everything to December 22 when everything was okay and give reassurances that the players have the confidence to go and train, play and get results."
Coventry, who are playing in their third season in the second tier of the women's game, are 11th in the 12-team Championship, with six points from 11 games.
Previous Previous post: Celtic target Jota & Carter-Vickers deals
Next Next post: Rangers set to sign USA international Sands on loan | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 7,974 |
https://blogs.wsj.com/cfo/2017/10/04/treasurers-make-strides-on-analytics-shared-services/
CFO Journal.
Treasurers Make Strides on Analytics, Shared Services
Nina Trentmann
BiographyNina Trentmann
@Nina_Trentmann
nina.trentmann
nina.trentmann@wsj.com
Business 'card' for Lego A/S group treasurer Jesper Broskov. Photo: Nina Trentmann
CFO Journal's Nina Trentmann is in Barcelona this week for EuroFinance 2017, a global event focused on the corporate treasury function. This year's discussions are centered around using technology and data to boost treasury's value to a company. Below are highlights from Day One:
Lego to build a model -- for FX cash flow risks
Danish toymaker Lego A/S plans develop its own way to assess cash flow risks stemming from foreign exchange moves, in the next three to six months. "This is currently not part of our modeling," said group treasurer Jesper Broskov.
The plan is a response to Lego's exposure to recent weakness in the Russian ruble and the Brazilian real, Mr. Broskov said. "At the time, we did not know whether we should stop selling [Lego products in those countries] or what the impact would be," Mr. Broskov said.
About a year ago, Lego started making changes to its foreign exchange hedging strategy, towards a portfolio-based approached. "Before, most of our hedging was done in silos, we spent a lot of time on very simple hedging," Mr. Broskov said. "We are trying to become more scientific about managing our risk," he added.
The company has started building clauses into its contracts allowing it to make changes with a shorter notice period than before. This also is a consequence of recent foreign exchange moves, Mr. Broskov said. He has a mandate to hedge 24 months in advance -- with spots, forwards and options -- but tends to hedge for 12 months or less, he said.
Statoil lets 'Roberta' handle some invoices
Norwegian oil and gas firm Statoil ASA has started utilizing a robot for some of its treasury tasks, Tor Stian Kjøllesdal, the company's head of internal treasury said. The robot, called "Roberta", is searching for due payments and sends payment reminders, Mr. Kjøllesdal said. "This is the first deployment of a robot in our treasury function," he added.
Statoil is trying to boost the efficiency of its treasury function with the help of artificial intelligence and robotics, Mr. Kjøllesdal said. "We should be very open-minded and look at all the opportunities," he added. The move to digitize the treasury finance however will result in job cuts. "We will run the treasury with less people in a couple of years," Mr. Kjøllesdal said. The company currently employs around 50 people in its treasury function, including those working in its shared service center.
Bose goes deep on money-market management
U.S. audio equipment company Bose Corp. is introducing a global portal to manage its money-market fund investments, said Steven Gomes, treasury operations manager. The tool, to be operational in November, allows regional treasury managers to manage day-to-day, short term money-market fund investments and provides its Framingham, Mass. headquarters with a central overview over investments. Longer term investments will be managed centrally, Mr. Gomes said.
The move is part of a wider centralization drive at the company. Bose is trying to forecast its liquidity on a central level and also harmonizes the number of bank accounts it operates. Despite these efforts, the process remains challenging, Mr. Gomes said. "Our bank relations are very split up. It is difficult to have one bank only," Mr. Gomes said. He did not disclose how many banks Bose works with or how many bank accounts the company has.
More Roche affiliates relying on pharma's bank
Swiss pharmaceutical company Roche Holding AG is serving a growing number of affiliates by its own, in-house bank, Martin Schlageter, head of treasury operations. "Fifteen percent of our businesses don't have a bank account anymore," Mr. Schlageter said. Out of around 300 entities around the globe, 40 are served by the firm's in-house bank in the Netherlands. "That number will increase over time," Mr. Schlageter said. "We will have more affiliates without their own bank account."
This sometimes involves "tough discussions" with the management teams of local business units, he said. The move is part of a drive towards a more centralized treasury function. "The more centralized you are [as a company], the more standardized your processes are, the better the position is that you are in," Mr. Schlageter said. Roche currently operates around 700 bank accounts globally.
Royal Dutch Shell moving more functions to shared service centers
Oil and gas giant Royal Dutch Shell PLC plans to move more treasury functions into its shared service centers, said Frances Hinden, vice president of treasury operations. "We continue to migrate tasks, mainly for cost reasons but also to do better," Ms. Hinden said. The company operates five shared service centers in addition its treasury centers in London and Singapore. "There is very little you could not move [to a shared service center]", Ms. Hinden said.
Senior and strategic positions -- including Shell's debt and capital markets team and the front and back office of the treasury centers -- will not be transferred, she added. Other tasks, such as daily cash forecasting have already been relocated. At the moment, Shell is moving the support teams for its pension plans from London to Glasgow. Ms. Hinden did not provide additional examples for tasks that will be moved out of London. Headcount in her department stands at around 80 people, down from 125 people five years ago.
Spain's NH Group cut costs through centralization
Centralizing treasury functions has helped Spanish hotel group NH Hotel Group SA become more efficient, said Luis Martinez Jurado, SVP of treasury and financing. Five years ago, most of the roughly 30 countries NH operates in had their own treasury function, Mr. Jurado said. "We had a fully decentralized treasury function," Mr. Jurado said.
The company streamlined payments, merged business units and cut the number of bank accounts from around 1000 to 800 as part of this process, Mr. Jurado said.
The head count in the company's central treasury function went down to around 15 to 20 people, he said. He declined to state how many treasury jobs were lost as part of the efficiency drive.
Robotics and artificial intelligence could help to further increase efficiency of the treasury function, Mr. Jurado said. All back office activities have been outsourced to a third-party shared service center that is already using these technologies, he added.
In the long run, robotics could also be utilized for liquidity planning, Mr. Jurado said. "Once we see that these robots don't go crazy, we can envisage many more activities for them," he said. At the moment however, robotics are only used for low-risk accounting tasks by NH's service center operator, Mr. Jurado said.
NH holds around €140 million ($164.7 million) in cash, mainly with banks in Spain and Luxembourg. The firm's deposits currently don't incur negative interest rates, despite the negative interest rate policy by the European Central Bank. "This however does not mean we are not under pressure from banks," Mr. Jurado said. But, so far he managed to escape negative rates. NH does not invest in money-market funds.
Previous Tesco to Review Contracts and Centralize Orders, CFO Says Next The Morning Ledger: Companies That Perform Best Don't Pay CEOs the Most
SHOW CONVERSATION HIDE CONVERSATION | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 2,199 |
This document describes the algorithm implemented in the `generate_cask_token` script, and covers detailed rules and exceptions which are not needed in most cases.
* [Purpose](#purpose)
* [Finding the Simplified Name of the Vendor's Distribution](#finding-the-simplified-name-of-the-vendors-distribution)
* [Converting the Simplified Name To a Token](#converting-the-simplified-name-to-a-token)
* [Cask Filenames](#cask-filenames)
* [Cask Headers](#cask-headers)
* [Cask Token Examples](#cask-token-examples)
* [Token Overlap](#token-overlap)
## Purpose
The purpose of these stringent conventions is to:
* Unambiguously boil down the name of the software into a unique identifier
* Minimize renaming events
* Prevent duplicate submissions
The token itself should be:
* Suitable for use as a filename
* Mnemonic
Details of software names and brands will inevitably be lost in the conversion to a minimal token. To capture the vendor's full name for a distribution, use the [`name`](CASK_LANGUAGE_REFERENCE.md#name-stanza-details) within a Cask. `name` accepts an unrestricted UTF-8 string.
## Finding the Simplified Name of the Vendor's Distribution
### Simplified Names of Apps
* Start with the exact name of the Application bundle as it appears on disk, such as `Google Chrome.app`.
* If the name uses letters outside A-Z, convert it to ASCII as described in [Converting to ASCII](#converting-to-ascii).
* Remove `.app` from the end.
* Remove from the end: the string "app", if the vendor styles the name like "Software App.app". Exception: when "app" is an inseparable part of the name, without which the name would be inherently nonsensical, as in [rcdefaultapp.rb](../Casks/rcdefaultapp.rb).
* Remove from the end: version numbers or incremental release designations such as "alpha", "beta", or "release candidate". Strings which distinguish different capabilities or codebases such as "Community Edition" are currently accepted. Exception: when a number is not an incremental release counter, but a differentiator for a different product from a different vendor, as in [pgadmin3.rb](../Casks/pgadmin3.rb).
* If the version number is arranged to occur in the middle of the App name, it should also be removed. Example: [IntelliJ IDEA 13 CE.app](../Casks/intellij-idea-ce.rb).
* Remove from the end: "Launcher", "Quick Launcher".
* Remove from the end: strings such as "Mac", "for Mac", "for OS X". These terms are generally added to ported software such as "MAME OS X.app". Exception: when the software is not a port, and "Mac" is an inseparable part of the name, without which the name would be inherently nonsensical, as in [PlayOnMac.app](../Casks/playonmac.rb).
* Remove from the end: hardware designations such as "for x86", "32-bit", "ppc".
* Remove from the end: software framework names such as "Cocoa", "Qt", "Gtk", "Wx", "Java", "Oracle JVM", etc. Exception: the framework is the product being Casked: [java.rb](../Casks/java.rb).
* Remove from the end: localization strings such as "en-US"
* If the result of that process is a generic term, such as "Macintosh Installer", try prepending the name of the vendor or developer, followed by a hyphen. If that doesn't work, then just create the best name you can, based on the vendor's web page.
* If the result conflicts with the name of an existing Cask, make yours unique by prepending the name of the vendor or developer, followed by a hyphen. Example: [unison.rb](../Casks/unison.rb) and [panic-unison.rb](../Casks/panic-unison.rb).
* Inevitably, there are a small number of exceptions not covered by the rules. Don't hesitate to [contact the maintainers](../../../issues) if you have a problem.
### Converting to ASCII
* If the vendor provides an English localization string, that is preferred. Here are the places it may be found, in order of preference:
- `CFBundleDisplayName` in the main `Info.plist` file of the app bundle
- `CFBundleName` in the main `Info.plist` file of the app bundle
- `CFBundleDisplayName` in `InfoPlist.strings` of an `en.lproj` localization directory
- `CFBundleName` in `InfoPlist.strings` of an `en.lproj` localization directory
- `CFBundleDisplayName` in `InfoPlist.strings` of an `English.lproj` localization directory
- `CFBundleName` in `InfoPlist.strings` of an `English.lproj` localization directory
* When there is no vendor localization string, romanize the name by transliteration or decomposition.
* As a last resort, translate the name of the app bundle into English.
### Simplified Names of `pkg`-based Installers
* The Simplified Name of a `pkg` may be more tricky to determine than that of an App. If a `pkg` installs an App, then use that App name with the rules above. If not, just create the best name you can, based on the vendor's web page.
### Simplified Names of non-App Software
* Currently, rules for generating a token are not well-defined for Preference Panes, QuickLook plugins, and several other types of software installable by homebrew-cask. Just create the best name you can, based on the filename on disk or the vendor's web page. Watch out for duplicates.
Non-app tokens should become more standardized in the future.
## Converting the Simplified Name To a Token
The token is the primary identifier for a package in our project. It's the unique string users refer to when operating on the Cask.
To convert the App's Simplified Name (above) to a token:
* Convert all letters to lower case.
* Expand the `+` symbol into a separated English word: `-plus-`.
* Expand the `@` symbol into a separated English word: `-at-`.
* Spaces become hyphens.
* Hyphens stay hyphens.
* Digits stay digits.
* Delete any character which is not alphanumeric or hyphen.
* Collapse a series of multiple hyphens into one hyphen.
* Delete a leading or trailing hyphen.
We avoid defining Cask tokens in the repository which differ only by the placement of hyphens. Prepend the vendor name if needed to disambiguate the token.
## Cask Filenames
Casks are stored in a Ruby file named after the token, with the file extension `.rb`.
## Cask Headers
The token is also given in the header line for each Cask.
## Cask Token Examples
These illustrate most of the rules for generating a token:
App Name on Disk | Simplified App Name | Cask Token | Filename
-----------------------|---------------------|------------------|----------------------
`Audio Hijack Pro.app` | Audio Hijack Pro | audio-hijack-pro | `audio-hijack-pro.rb`
`VLC.app` | VLC | vlc | `vlc.rb`
`BetterTouchTool.app` | BetterTouchTool | bettertouchtool | `bettertouchtool.rb`
`LPK25 Editor.app` | LPK25 Editor | lpk25-editor | `lpk25-editor.rb`
`Sublime Text 2.app` | Sublime Text | sublime-text | `sublime-text.rb`
# Token Overlap
When the token for a new Cask would otherwise conflict with the token of an already existing Cask, the nature of that overlap dictates the token (for possibly both Casks). See [Finding a Home For Your Cask](https://github.com/caskroom/homebrew-cask/blob/master/CONTRIBUTING.md#finding-a-home-for-your-cask) for information on how to proceed.
# <3 THANK YOU TO ALL CONTRIBUTORS! <3
| {
"redpajama_set_name": "RedPajamaGithub"
} | 6,872 |
Crkva sv. Filipa u Makarskoj, na Rivi, zaštićeno kulturno dobro
Opis dobra
Crkva sv. Filipa jednobrodna je građevina s polukružnom apsidom i zvonikom s rastvorenom ložom na prvom katu. Pročelje crkve naglašeno je portalom. U unutrašnjosti crkve nalazi se pet baroknih mramornih oltara te grob biskupa Blaškovića ukrašen mramornom intarzijom. Na ogradi kora s ostacima Nakićevih orgulja, nalaze se barokne slike. Uz crkvu je danas pregrađeni samostan filipinaca, koji se sastojao od nekoliko kuća oko unutrašnjeg dvorišta, a sagradio ga je biskup Stjepan Blašković u 18. st. Na ulazu u samostan nalazi se natpis koji govori o gradnji samostana i njegovom donatoru.
Zaštita
Pod oznakom Z-5063 zavedena je kao nepokretno kulturno dobro - pojedinačno, pravna statusa zaštićena kulturnog dobra, klasificirano kao "sakralna graditeljska baština".
Izvori
Zaštićene sakralne građevine u Splitsko-dalmatinskoj županiji
Katoličke crkve u Makarskoj | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 7,798 |
La serie Radeon 500 es una serie de procesadores gráficos desarrollados por AMD. Estas tarjetas se basan en la cuarta iteración de la arquitectura Graphics Core Next, con GPU basadas en chips Polaris 30, Polaris 20, Polaris 11 y Polaris 12. Por lo tanto, la serie RX 500 utiliza la misma microarquitectura y conjunto de instrucciones que su predecesor, al tiempo que utiliza mejoras en el proceso de fabricación para permitir velocidades de reloj más altas.
Los chips GCN de tercera generación se producen en un proceso CMOS de 28 nm. Los chips Polaris (GCN de cuarta generación) (excepto Polaris 30) se producen en un proceso FinFET de 14 nm, desarrollado por Samsung Electronics y con licencia para GlobalFoundries. Los chips Polaris 30 se producen en un proceso FinFET de 12 nm, desarrollado por Samsung y GlobalFoundries.
Modelos
Los estándares de visualización admitidos son: DisplayPort 1.4 HBR, HDMI 2.0b, color HDR10.
También se admiten Dual-Link DVI-D y DVI-I con resoluciones de hasta 4096 × 2304, a pesar de que los puertos no están presentes en las tarjetas de referencia.
También se admiten puertos VGA con resoluciones de hasta 2048x1536, a pesar de que los puertos no están presentes en las tarjetas de referencia, aunque los puertos VGA se encuentran principalmente en tarjetas vendidas exclusivamente en el este de Asia.
Véase también
AMD Radeon Pro
AMD FireStream
Unidades de procesamiento de gráficos de AMD
Referencias
Tarjetas gráficas
Unidades de procesamiento gráfico
Introducciones relacionadas a la ciencia de la computación de 2017
Advanced Micro Devices | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 9,665 |
\section{Introduction}
\vspace{1cm}
In this article, we are interested in functors from the category of free $R$-modules to that of $R$-modules (for $R$ a commutative ring) and particularly in the functor from such a module $M$ to the module of $d$-fold invariant tensors $ (M^{\otimes d})^{\Sigma_d}$. This module of invariant tensors is canonically isomorphic to the degree $d$ components $\Gamma^d_R(M)$ of the divided power algebra $\Gamma_R(M)$. Though these are non-additive functors of $M$ whenever $d>1$, they may be derived by the Dold-Puppe theory \cite{d-p}. This associates to $M$ a family of $i$th-derived functors $L_i\Gamma^d(M,n)$ which depends on an additional positive integer $n$. In a wider perspective, $L_i\Gamma^d_R(M,n)$ is the $i$th homotopy group of Quillen's left-derived object $L\Gamma^d_R(M[n])$, where $M[n]$ is the module $M$, viewed as a chain complex concentrated in degree $n$ \cite{quillen:homot}.
When $R$ is a field, the functors $L_i\Gamma^d_R(M,n)$ are known. Our aim is to compute certain of its values when $R = {\mathbb{Z}}$. Such computations are motivated by applications in both algebraic topology and in representation theory.
\bigskip
In algebraic topology, the functors $L_i\Gamma^d_{\mathbb{Z}}(A,n)$ where $A$ is an arbitrary abelian group are closely related to the integral homology of the Eilenberg-Mac Lane spaces $K(A,n+2)$, i.e those spaces whose only non-trivial homotopy group is $A$ in degree $n+2$. Indeed, even though a truly functorial description of the homology groups $H_\ast(K(A,n+2);{\mathbb{Z}})$ is complicated, they are endowed with a natural filtration inducing functorial isomorphism
\begin{equation}
\bigoplus_{d\ge 0} L_{*-2d}\Gamma^d_{\mathbb{Z}}(A,n) \simeq \mathrm{gr}\, (H_*(K(A,n+2);{\mathbb{Z}}))\label{h-ast}
\end{equation}
on the associated graded components. This filtration splits functorially when we restrict ourselves to the subcategory of free abelian groups, so that there are then functorial isomorphisms
\begin{equation}
\label{h-ast2} \bigoplus_{d\ge 0} L_{*-2d}\Gamma^d_{\mathbb{Z}}(A,n) \simeq \,H_\ast(K(A,n+2);{\mathbb{Z}}) \end{equation}
for all $n \geq 0$. When $A$ non free, such an isomorphism still exists, but is no longer functorial in $A$ (we refer to our appendix \ref{han} for a further discussion of this issue). The abelian groups $L_i\Gamma^d_{\mathbb{Z}}(A,n)$ are thus quite fundamental for topology, and the description of their dependence on the group $A$ carries much useful information which remains hidden when such a (finitely generated) group is decomposed into a direct sum of cyclic groups.
\bigskip
In representation theory, the representations of integral Schur algebras can be described in terms of the strict polynomial functors of Friedlander and Suslin. Such strict polynomial functors can be thought of as functors from finitely generated free abelian groups to abelian groups, equipped with an additional strict polynomial structure. The derived category of weight $d$ homogeneous strict polynomial functors $\mathbf{D}(\mathcal{P}_{d,{\mathbb{Z}}})$ is equipped with a Ringel duality operator $\Theta$, which is a self-equivalence of $\mathbf{D}(\mathcal{P}_{d,{\mathbb{Z}}})$.
For $A$ free finitely generated there is an isomorphism
$$ H^\ast(\Theta^n\Gamma^d_{\mathbb{Z}}(A))\simeq L_{nd - \ast}\Gamma^d_{\mathbb{Z}}(A,n) $$
Such an isomorphism holds for all strict polynomial functors, but the case of divided powers is fundamental since these functors constitute a family of projective generators of the category. In this context, the description of the $L_*\Gamma^d_{\mathbb{Z}}(A,n)$ as functors of $A$ is essential, since the functoriality is necessary in order to determine the action of the Schur algebra, hence to obtain the expressions $ H_*(\Theta^n\Gamma^d_{\mathbb{Z}}(A))$ as representations, and not simply as abelian groups. Actually we need more than the functoriality in order to understand the action of the Schur algebra, we really need to describe the strict polynomial structure of these functors.
\bigskip
Let us briefly review what was known previously on the derived functors $L_*\Gamma^d_{\mathbb{Z}}(A,n)$. The integral homology of Eilenberg-Mac Lane spaces was computed by H. Cartan: a non-functorial description of these homology groups is given in \cite{cartan} Exp. 11, Thm 1, and a functorial one in \cite[Exp. 11, Thm 5]{cartan}. Work in this direction was pursued by two students of Mac Lane, Hamsher \cite{hamsher} and Decker \cite{decker}, who studied the cases $n=1$ and $n \geq 1$ respectively. By the isomorphism \eqref{h-ast}, it is possible to retrieve from these results a description of the derived functors $L_*\Gamma^d_{\mathbb{Z}}(A,n)$. The answers however are very complicated. For example, the homology groups $H_i(K(A,n);{\mathbb{Z}})$ for a finitely generated abelian group $A$ are described functorially by an infinite list of generators and relations, even though they are of finite type. Some additional progress in computing the functors
$L_\ast\Gamma^d_{\mathbb{Z}}(A,n)$
was made by a direct study of its properties by Bousfield in \cite{bousfield-hom}, \cite{bousfield-pens}, and a complete description of these functors for any $A$ was obtained for $d=2$ by Baues and Pirashvili \cite{baupira} and in \cite{BM} for $d=3$. On the other hand, no description of the strict polynomial structure of the functors $L_\ast\Gamma^d_{\mathbb{Z}}(A,n)$ has been given so far.
\bigskip
We now list the results which we obtain in this paper. We give full description (as strict polynomial functors) :
\begin{enumerate}
\item[(1)] of the $L_*\Gamma^d_{\mathbb{Z}}(A,1)$ for all $d$ and $A$ free.
\end{enumerate}
By \eqref{h-ast2}, this determines a new and functorial description of the groups $H_\ast(K(A,3);{\mathbb{Z}})$ for $A$ a free abelian group.
\begin{enumerate}
\item[(2)] of the $L_*\Gamma^d_{\mathbb{Z}}(A,n)$ for $d \leq 4$ and $A$ free.
\end{enumerate}
Our treatment is elementary and self-contained, and does not use computations from the literature. The strict polynomial structures come in as a help in the computations. Our computations have a number of interesting byproducts, in particular:
\begin{enumerate}
\item[(3)] short proofs of some computations first obtained in \cite{antoine} including that of $L_*\Gamma^d_\mathbb{k}(V,n)$ where $\mathbb{k}$ is a field of characteristic $2$,
\item[(4)] a new family of exact complexes `of Koszul type' involving divided powers over a field of characteristic 2.
\end{enumerate}
The kernels of these new complexes yield new families of functors, related to the $2$-primary component of $L_*\Gamma^d_{\mathbb{Z}}(A,1)$. We describe these functors in a variety of ways. The simplest is the 2-primary component of $L_3\Gamma^4(A,1)$, which we denote by $\Phi^4(A)$, and which can be described as the cokernel of the natural transformation $\Lambda^4_{\mathbb{F}_2}(A/2) \to \Gamma^4_{\mathbb{F}_2}(A/2)$ determined by the algebra structure of $\Gamma^\ast_{\mathbb{F}_2}(A/2)$ (such a map only exists in characteristic 2).
\begin{enumerate} \item[(5)] We give a description of the groups $H_{n+i}(K(A,n);{\mathbb{Z}})$ for $A$ free in the range $0 \leq i \leq 10$ which is both more natural and more precise than Cartan's in \cite{cartan}.
\end{enumerate}
Finally in section \ref{conj}, we return to the derived functors $L_*\Gamma^d_{\mathbb{Z}}(A,n)$ for all $n$, $d$ and all abelian groups $A$. We present a conjectural description of these functors, up to a filtration, in terms of simpler derived functors of divided and exterior powers. We then make use here of our results and of those from \cite{BM} in order to provide some evidence in support of this conjecture.
\bigskip
We will now discuss the content of this article in more detail. In sections \ref{sec-classical} and \ref{sec-der} we collect the main properties of divided power algebras, strict polynomial functors and derived functors which will be required.
In section \ref{qt-fil}, we introduce the useful notion of a quasi-trivial filtration which will allow us to prove that certain spectral sequences degenerate. For $V$ a vector space over a characteristic $p$ field $\mathbb{k}$, the divided power algebra $\Gamma^\ast_\mathbb{k}(V)$ possesses such a filtration, which we call the principal filtration. We use this filtration to compute in
section \ref{descr-gvn} those values of the derived functors $L_i\Gamma^d_\mathbb{k}(V,n)$ which will be needed in the sequel.
\bigskip
In sections \ref{der1-gamma-z} and \ref{sec-der1-gamma-z-bis} we compute the strict polynomial functors $L_i\Gamma^d_{\mathbb{Z}}(A,1)$ for any free abelian group $A$ with the help of the Bockstein maps. These may be viewed as a family of differentials on the mod $p$ graded groups $L_\ast\Gamma^d_{\mathbb{F}_p}(A/p,1)$ and can be explicitly described. While in odd characteristic these mod $p$ derived functors together the Bockstein differentials simply provide us with a standard dual Koszul complex, the result is slightly different in characteristic 2. In that case the Bockstein differentials determine on $L_\ast\Gamma^d_{\mathbb{F}_2}(A/2,1)$ a sort of dual Koszul complex structure, in which the differentials are twisted by a Frobenius map. We call this characteristic 2 complex the skew Koszul complex. In both odd and even characteristic the integral homology groups $L_\ast\Gamma^d_{\mathbb{Z}}(A/p,1)$ are simply the groups of cycles in these
Koszul and twisted Koszul complexes, and we are able to analyze these more precisely in a number of situations.
\bigskip
In section \ref{sec-max} to \ref{der-gamma4-sec}, we present another method for computing the derived functors of
$\Gamma^d_{\mathbb{Z}}(A)$. Here we use the adic filtration associated to the
augmentation ideal of the algebra $\Gamma^\ast_{\mathbb{Z}}(A)$. We call this filtration the maximal filtration on $\Gamma_{\mathbb{Z}}(A)$. It was the
filtration chosen in \cite{BM} in order to study the derived functors
of $\Gamma^3_{\mathbb{Z}}(A)$. With the help of the associated spectral sequence, we determine the values of
$L_\ast\Gamma^4_{\mathbb{Z}}(A,n)$.
The easiest cases $n= 1,2$ are studied in section \ref{sec-derG12} where we pay particular attention to the first new functor occurring
among the derived functors of $\Gamma^4_{\mathbb{Z}}(A)$. This is the functor
$\Phi^4(A)$ mentioned above, the 2-primary component of $ L_{3}\Gamma^4_{\mathbb{Z}}(A,1)$, which we describe in three distinct ways.
As $n$ increases the situation becomes more involved, and a detailed study of the differentials in the maximal filtration spectral sequence for $\Gamma^4_{\mathbb{Z}}(A,n)$ for a general $n$ is carried out in section \ref{der-gamma4-sec}. The computation of certain differentials is delicate, and involves functorial considerations and careful dimension counts.
This yields an inductive formula for $L_\ast\Gamma^4_{\mathbb{Z}}(A,n)$ (theorem \ref{deriveddescr}), from which the complete description
of the derived functors of $\Gamma^4_{\mathbb{Z}}(A,n)$ for all $n$ follows directly (theorem \ref{thm-G4Z}).
\bigskip
In section \ref{conj}, we return to the derived functors $L_*\Gamma^d_{\mathbb{Z}}(A,n)$ for all $n$, $d$ and all abelian groups $A$.
We propose a conjectural description, up to a filtration, of the functors $L_r\Gamma^d_{\mathbb{Z}}(A,n)$. To state this requires that we first discuss the stable homology groups $H_r(K(A,n);{\mathbb{Z}})$ , i.e. those for which $n \leq r < 2n$. These additive groups are discussed in a number of sources, and we first review here for the reader's convenience certain aspects of the admissible words formalism of Cartan. We then reformulate in theorem \ref{thm-new-formula} his admissible words in terms of a simpler labelling which only involves those words which do not involve the transpotence operation. We refer in proposition \ref{prop-new} to Cartan's computation of these stable values, and this is the only place in the text where we make use, for simplicity, of his results. We then state our conjecture \ref{conject}, and we verify that it is compatible with theorems \ref{thm-calcul-LG-un-Z} and \ref{thm-G4Z} as well as with the results in \cite{BM}.
\bigskip
We view the appendices as an integral part of our text.
In appendix \ref{app-comput}, we review some classical methods of
computations of $\rm Hom$'s and $ {\rm Ext}^1$'s in functor
categories which are used many times in our arguments. Appendix \ref{han} begins with a
discussion of the relation between the derived functors of the functor
$\Gamma^d_{\mathbb{Z}}(A)$ and the integral homology of $K(A,n)$. We then
display a complete table of the functorial values of the groups
$H_{n+i}(K(A,n);{\mathbb{Z}})$ for $A$ free in the range $0 \leq i \leq
10$. While the constraints in choosing this range of values of $i$ were
to some extent typographical, the fact that we only know the complete set of
derived functors of $\Gamma^d_{\mathbb{Z}}(A)$ for $d \leq 4$ would have precluded
the display of a complete table for much higher values of $i$. This
table already features most of the unexpected phenomena which we
encounter in our computations, as will be seen from the
discussion which precedes the table. A final appendix provides for
convenience a complete list of non-trival derived functors
$L_{*}\Gamma^4_{\mathbb{Z}}(A,n)$ for $n \leq 4$.
\bigskip
The following diagram summarizes the relations between the various parts of our text:
\[\xymatrix@R=30pt@C=15pt{
& \fbox{Sections 2-5}\ar[dl] \ar[dr]&& \ar[ll]\ar[llld]\ar[ld] \fbox{Apppendix A}\\
\fbox{Sections 6,7} \ar[dr]&& \fbox{Sections 8-10} \ar[dl]&\\
&\fbox{Section 11 and Appendix B} &&
}
\]
\thanks{The authors wish to thank A.K. Bousfield for correspondence concerning topics related to the matter discussed here. }
\bigskip
\thanks{The research of the second author is supported by the Chebyshev Laboratory (Department of Mathematics and Mechanics, St.
Petersburg State University) under RF Government grant 11.G34.31.0026 and by JSC "Gazprom Neft", as well as by the RF Presidential grant MD-381.2014.1.}
\bigskip
\noindent {\it Notation}. Throughout the text the notation $A/p$, where $A$ is an abelian group and $p$ prime number, stands for $A \otimes_{\mathbb{Z}} {\mathbb{Z}}/p$. For any abelian group $A$, the notation $\Gamma (A)$ will stand for $\Gamma_{\mathbb{Z}}(A)$, the divided power algebra associated to the ${\mathbb{Z}}$-module $A$ unless otherwise specified. On the other hand for any $\mathbb{F}_p$-vector space $V$, the notation $\Gamma (V)$ stands for the divided power algebra $\Gamma_{\mathbb{F}_p}(V)$ in the category of $\mathbb{F}_p$-vector spaces.
\section{Classical functorial algebras}\label{sec-classical}
\subsection{The divided power algebra} Let $M$ be an $R$-module, where $R$ be a commutative ring with unit. The symmetric algebra $S_R(M)$ and
the exterior algebra $\Lambda_R(M)$ are well-known. The
divided power algebra $\Gamma_R(M)$ deserves equal attention. We recall here its basic properties, refering to \cite{roby} for the proofs. The algebra $\Gamma_R(M)$ is defined as the commutative $R$-algebra generated, for all $x \in M$ and all non-negative integers $i$, by elements $ \gamma_i(x)$ which satisfy the following relations for all $x,y \in M$ and $\lambda \in R$:
\begin{align} & 1)\ \gamma_0(x) =
1 \hspace{1cm} x \neq 0\label{rel1} \\
\label{rel3a} & 2)\
\gamma_s(x)\gamma_t(x)=\binom{s+t}{s}\gamma_{s+t}(x) \\\label{rel4}
& 3)\ \gamma_n(x+y)=\sum_{s+t=n}\gamma_s(x)\gamma_t(y),\ n\geq 1\\
& 4)\
\ \gamma_n(\lambda x)=\lambda^n\gamma_n(x), \ n\geq 1 \label{rel5}
\end{align}
Setting $s=t=1$ in \eqref{rel3a} one finds by induction on $n$ that $x^n = n!\gamma_n(x)$, which justifies the name of divided power algebra for $\Gamma_R(M)$.
The definition $\Gamma_R(M)$ is functorial with respect to $M$: an $R$-linear map $f:M\to N$ induces a morphism of $R$-algebras $\Gamma_R(M)\to \Gamma_R(N)$ which sends $\gamma_i(x)$ to $\gamma_i(f(x))$.
The relations \eqref{rel1}-\eqref{rel5} are homogeneous, so that $\Gamma_R(M)$ can be given a graded algebra structure by setting $\text{deg}\,\gamma_i(x) = i$. We denote by $\Gamma^d_R(M)$ the homogeneous component of degree $d$, so that we have functorial $R$-linear isomorphisms $\Gamma^0_R(M)\simeq R$ and $\Gamma^1_R(M)\simeq M$ and a functorial decomposition
$$\Gamma_R(M)=\bigoplus_{d\ge 0}\Gamma^d_R(M)\;.$$
Consider the graded commutative algebra $TM=\bigoplus_{d\ge 0} V^{\otimes d}$, equipped with the shuffle product, and let us denote by $TS(M)$ the graded subalgebra of invariant tensors $TS(M)=\bigoplus_{d\ge 0} (V^{\otimes d})^{\Sigma_d}$.
There is a well defined functorial homomorphism of graded algebras
\begin{align}\Gamma_R(M)\to TS(M)\;,\label{eq-identification-dp}\end{align}
which sends $\gamma_n(x)$ to $x^{\otimes n}$. The morphism \eqref{eq-identification-dp} is an isomorphism if $M$ is projective \cite[Prop. IV.5]{roby}. In particular, for a projective $R$-module $M$ we have functorial isomorphisms
\[
\label{inv} \Gamma^d_R(M) \simeq (M^{\otimes \,
d})^{\Sigma_d}. \]
This description of $\Gamma^d(M)$ by invariant tensors is no longer valid for arbitrary modules $M$. For example, it follows by inspection from the relations \eqref{rel1}-\eqref{rel5} above that \begin{equation}
\label{gammators}\Gamma^2_{\mathbb{Z}}({\mathbb{Z}}/2) \simeq {\mathbb{Z}}/4,\end{equation} so that this group cannot live in ${\mathbb{Z}}/2\, \otimes \,{\mathbb{Z}}/2$. This example also shows that divided power algebras behave differently from symmetric and exterior algebras with respect to torsion. Indeed, $S_{\mathbb{Z}}(M)$ and $S_{\mathbb{F}_p}(M)$ coincide for any $\mathbb{F}_p$-module $M$, as do $\Lambda_{\mathbb{Z}}(M)$ and $\Lambda_{\mathbb{F}_p}(M)$, but the similar assertion is not true for $\Gamma_{\mathbb{Z}}(M)$ and $\Gamma_{\mathbb{F}_p}(M)$. In particular, for any $\mathbb{F}_p$-module $M$, we will always carefully distinguish between the functors $\Gamma_{\mathbb{Z}}(M)$ and $\Gamma_{\mathbb{F}_p}(M)$.
\subsection{Exponential functors}
A functor $F(M)$ from $R$-modules to $R$-algebras is said to be an
exponential functor if for all pair of $R$-modules $M$ and $N$, the composite
\begin{align}F(M)\otimes F(N)\xrightarrow[]{F(i_1)\otimes F(i_2)} F(M\oplus N)\otimes F(M\oplus N)\xrightarrow[]{\mathrm{mult}} F(M\oplus N)\;, \label{eq-exp-prop}\end{align}
induced by the canonical inclusions $i_1$, $i_2$ of $M$ and $N$ into $M\oplus N$, is an isomorphism.
We refer to the map \eqref{eq-exp-prop} as the \emph{exponential isomorphism} for
$F(M)$.
For example, the algebras $S_R(M)$, $\Lambda_R(M)$, and $\Gamma_R(M)$ \cite[Thm III.4]{roby} determine exponential functors.
Exponential functors are endowed
with a canonical bialgebra structure. Indeed, there is a coproduct induced by the diagonal
inclusion $\Delta$ of $M$ into $M\oplus M$:
$$F(M)\xrightarrow[]{F(\Delta)} F(M\oplus M)\simeq F(M)\otimes F(M)\;. $$
It follows from \eqref{rel4}
that the coalgebra structure obtained in this way on $\Gamma_R(M)$ is the one determined by the comultiplication map $\gamma_i(x)\mapsto \sum_{i= j+k}\gamma_j(x) \otimes \gamma_k(x)$.
\subsection{Duality}
\label{duality}
Given a $R$-module $M$, we let $M^\vee:=\mathrm{Hom}_R(M,R)$.
The dual $\Gamma_R (M)^\sharp$ of the divided power algebra is the $R$-module
$$\Gamma_R (M)^\sharp :=\bigoplus_{d\ge 0} (\Gamma^{d}_R(M^\vee))^\vee\;.$$
The bialgebra structure on $\Gamma_R (M^\vee)$ defines a bialgebra structure on $\Gamma_R (M)^\sharp$. The dual of symmetric and exterior algebras are defined similarly.
An explicit computation shows that for all projective $R$-modules $M$, there are a natural isomorphisms of $R$-bialgebras
$$\Gamma_R(M)^\sharp \simeq S_R(M)\;,\qquad S_R(M)^\sharp\simeq \Gamma_R(M)\;,\qquad\Lambda_R(M)^\sharp \simeq
\Lambda_R(M)\;.$$
\subsection{Base change}
For any $R$-module $M$ and any commutative $R$-algebra $A$, there is a base change isomorphism of $A$-algebras \cite[Thm III.3]{roby}, which sends $\gamma_n(x)\otimes 1$ to $\gamma_n(x\otimes 1)$:
\begin{equation}
\label{basechange}
\Gamma_R(M) \otimes_R A \simeq \Gamma_{A}(M\otimes_R A).
\end{equation}
There are similar base change isomorphisms for symmetric and exterior algebras:
$$S_A(M) \otimes_R A \simeq S_{A}(M\otimes_R A)\;,\qquad \Lambda_R(M) \otimes_R A \simeq \Lambda_{A}(M\otimes_R A)\;.$$
\section{Derived functors and strict polynomial functors}\label{sec-der}
\subsection{Derived functors}
Let $R$ be a commutative ring. The Dold-Kan correspondence states that the
normalized chain complex functor $\mathcal{N}$ is an equivalence of categories preserving
homotopy equivalences, with inverse $K$ (also preserving homotopy equivalences):
$$\mathcal{N}: \mathrm{simpl}(R\text{-Mod})\rightleftarrows \mathrm{Ch}_{\ge
0}(R\text{-Mod}): K\;. $$
If $M$ is a
$R$-module, and $F(M)$ denotes an endofunctor of the category of $R$-modules,
Dold and Puppe defined \cite{d-p} its derived functor $L_iF(M,n)$ by the formula:
$$L_iF(M,n) = \pi_iF K(P^M[n])\;,$$
where $P^M$ is a projective resolution of $M$, and $[n]$ is the degree $n$ shift
of complexes (i.e. $C[n]_i=C_{i+n}$). More generally, if $C$ is a complex of $R$-modules, we denote by
$LF(C)$ the simplicial $R$-module $F(K(P))$, where $P$ is a complex of projective $R$-modules quasi-isomorphic to $C$,
and by $L_iF(C)$ its homotopy groups.
\begin{remark}
The definition of the derived functors of $F$ only depends on the restriction of $F$ to the category of free $R$-modules. Furthermore, if $F$ commutes with directed colimits of free $R$-modules (as the divided power functors do), then $F$ is completely determined by its restriction to the category of free finitely generated $R$-modules. For example, for a free $R$-module $M$, we have an isomorphism $L_*\Gamma^d(M,n)=\lim_iL_*\Gamma^d(M_i,n)$, where the limit is taken over the directed system of free finitely generated submodules $M_i$ of $M$.
\end{remark}
\subsection{Strict polynomial functors}\label{subsec-str_der}
We are mainly interested in the divided powers functors $\Gamma^{d}_R(M)$ and functors related to these.
All these functors actually belong to a class of very rigid functors called
`strict polynomial functors', introduced by Bousfield \cite{bousfield-hom} in the context of derived functors (Bousfield called them `homogeneous functors') and independently by Friedlander and Suslin \cite{FS} in the context of the cohomology of affine algebraic group schemes. We also recommend \cite{krause} for a presentation of strict polynomial functors.
We will only recall here the basic facts required for our computations.
\subsubsection{The category of strict polynomial functors}
Strict polynomial functors can be thought of as functors from the category of finitely generated projective $R$-modules to the category of $R$-modules, equipped with an additional `scheme-theoretic' structure. Morphisms of strict polynomial functors are natural transformations which preserve this additional structure.
The category $\mathcal{P}_R$ of strict polynomial functors is abelian.
Let $\mathcal{F}_R$ denote the category of functors from finitely generated projective $R$-modules to $R$-modules.
There is an exact forgetful functor:
$$\mathcal{P}_R\to \mathcal{F}_R \;.$$
The tensor product $F(M)\otimes_R G(M)$ of two strict polynomial functors has a canonical structure of a strict polynomial functor, as well as the composition $F(G(M))$ provided that $G(M)$ has values in finitely generated projective $R$-modules.
The additional scheme-theoretic structure of a strict polynomial functor $F(M)$ determines the weight of $F(M)$ (this weight is called `degree' in \cite{FS}. In the present article, we prefer to reserve the term degree for the homological degrees). A strict polynomial functor $F(M)$ of weight $d$ is \emph{homogeneous of weight $d$} if for all $G(M)$ of weight less than $d$, $\mathrm{Hom}_{\mathcal{P}_R}(G(M),F(M))$ is zero. The full subcategory of $\mathcal{P}_R$ whose objects are homogeneous strict polynomial functors of weight $d$ is abelian. It is usually denoted by $\mathcal{P}_{d,R}$, and there is an isomorphism of abelian categories:
$$\mathcal{P}_{R}\simeq \bigoplus_{d\ge 0}\mathcal{P}_{d,R}\;.$$
In practice, the weights can be determined by the following rules. The functors $S^d_R(M)$, $\Lambda^d_R(M)$, $\Gamma^d_R(M)$ and $M^{\otimes d}$ are homogeneous of weight $d$. If $F(M)$ and $G(M)$ are homogeneous of degree $d$, resp. $e$, then $F(M)\otimes G(M)$ is homogeneous of degree $d+e$ and $F(G(M))$ is homogeneous of degree $de$.
A useful structure on the category of strict polynomial functors is the (weight preserving) duality functor
\begin{equation}
\label{duality1}
^\sharp: \mathcal{P}_{R}^{\mathrm{op}}\to \mathcal{P}_{R}\;,\end{equation}
which sends a strict polynomial functor $F(M)$ to the functor $F(M)^\sharp:=F(M^\vee)^\vee$, where `$^\vee$' denotes $R$-linear duality, i.e. $M^\vee=\mathrm{Hom}_R(M,R)$. For example we have $S_R^d(M)^\sharp=\Gamma^d_R(M)$ and $\Lambda^d_R(M)^\sharp=\Lambda^d_R(M)$. When $R$ is a field, the duality functor is exact.
The advantage of working with strict polynomial functors rather than with ordinary functors from finitely generated projective $R$-modules to $R$-modules is twofold.
\begin{enumerate}
\item The category of strict polynomial functors is graded by the weight. In all computations involving strict polynomial functors this information is automatically and transparently carried by the `scheme theoretic' structure of the functors, see remark \ref{rk-poids}. The price to pay is that one has be a take care that a given ordinary functor may sometimes be given several non-isomorphic scheme-theoretic structures, as the example of the Frobenius twists in section \ref{subsubsec-Frob} shows.
\item It is easier to compute extensions in $\mathcal{P}_R$ than in $\mathcal{F}_R$, and there are often less extensions possible in $\mathcal{P}_R$. This fact will be of great help in solving extension problems coming from spectral sequences. We have gathered in appendix \ref{app-comput} some standard methods and results regarding the computation of extensions in these categories. An illustration of the difference between extensions in $\mathcal{P}_R$ and in $\mathcal{F}_R$ is provided by comparing the results from lemma \ref{lm-calc2} with those of lemma \ref{lm-calc3}.
\end{enumerate}
\subsubsection{Frobenius twists functors}\label{subsubsec-Frob}
We now take a field $\mathbb{k}$ of positive characteristic $p$ as our ground ring $R$, and we let $V$ be a generic finite dimensional $\mathbb{k}$-vector space.
We denote by $V^{(r)}$ the strict polynomial subfunctor of $S_\mathbb{k}^{p^r}(V)$ generated by
the $p^r$-th powers of elements of $V$. Equivalently, $V^{(r)}$ is the the kernel of the map $S_\mathbb{k}^{p^r}(V)\to \bigoplus_{k=1}^{p^r-1}S_\mathbb{k}^{k}(V)\otimes S_\mathbb{k}^{p^r-k}(V)$ induced by the comultiplication of the symmetric power bialgebra. The functor $V^{(r)}$ is called
the \emph{$r$-th
Frobenius twist functor}. It has an important role in the theory of representations of affine algebraic group schemes in positive characteristic \cite{FS}, and it will also appear in our computations. It enjoys the following basic properties.
\begin{enumerate}
\item The functor $V^{(r)}$ is a homogeneous strict polynomial functor of weight $p^r$.
\item The functor $V^{(r)}$ is additive.
\item The dimension of $V^{(r)}$ is equal to the dimension of $V$.
\item The Frobenius twist functors can be composed together $(V^{(r)})^{(s)}=V^{(r+s)}$, and the functor $V^{(0)}$ is the identity
functor.
\item The Frobenius twist functors are self dual: $(V^{(r)})^\sharp\simeq V^{(r)}$, so that $V^{(r)}$ is the cokernel of the map $ \bigoplus_{k=1}^{p^r-1}\Gamma_\mathbb{k}^{k}(V)\otimes \Gamma_\mathbb{k}^{p^r-k}(V)\to \Gamma_\mathbb{k}^{p^r}(V)$ induced by the multiplication of the algebra of divided powers.
\end{enumerate}
The inclusion $V^{(r)}\hookrightarrow S_\mathbb{k}^{p^r}(V)$ is called the Frobenius map and the dual
epimorphism $\Gamma^{p^r}_\mathbb{k}(V)\twoheadrightarrow V^{(r)}$ therefore deserves to be called the
Verschiebung map following the usage in arithmetic. As explained in appendix \ref{app-comput}, these two morphisms provide bases of the vector spaces $\mathrm{Hom}_{\mathcal{P}_\mathbb{k}}(V^{(r)},S_\mathbb{k}^{p^r}(V))$ and $\mathrm{Hom}_{\mathcal{P}_\mathbb{k}}(\Gamma_\mathbb{k}^{p^r}(V), V^{(r)})$ respectively.
Observe that if $\mathbb{k}=\mathbb{F}_q$ with $q$ dividing $p^r$, the strict polynomial functors $V^{(nr)}$, $n\ge 0$, are not isomorphic to each other since they do not have the same weight. However, if we forget their scheme-theoretic structure and view them as ordinary functors, that is as objects of $\mathcal{F}_R$, they all become isomorphic to the identity functor.
\subsection{Derived functors and differential graded $\mathcal{P}_R$-algebras}
Derivation in the sense of Dold and Puppe yield functors for all nonnegative integers $i$ and $n$:
$$
\begin{array}{ccc}
\mathcal{P}_{d,R} &\to &\mathcal{P}_{d,R}\\
F(M) & \mapsto & L_iF(M,n)
\end{array}
$$
We shall denote by $\mathcal{N} F(M,n)$ the normalized chains of the simplicial object $F(K(M[n]))=F(K(R[n])\otimes M)$. Thus, $\mathcal{N} F(M,n)$ is a complex of homogeneous strict polynomial functors of weight $d$.
\begin{remark}\label{rk-derfrobtw}
If $\mathbb{k}$ is a field of positive characteristic and $G(V)=F(V^{(r)})$, the additivity of the Frobenius twist yields an isomorphism
$\mathcal{N} G(V,n)\simeq \mathcal{N} F(V^{(r)},n)$. Thus precomposition by Frobenius twist is harmless in computations, i.e. there are isomorphisms:
\begin{align*}L_iG(V,n)\simeq L_iF(V^{(r)},n)\;.\end{align*}
\end{remark}
By composing the shuffle map in the Eilenberg-Zilber theorem and the product of the divided power algebra:
\begin{align*}\mathcal{N}\Gamma^d_R(M,n)\otimes \mathcal{N}\Gamma^e_R(M,n)\to \mathcal{N}(\Gamma^d_R\otimes \Gamma^e_R)(M,n)\to \mathcal{N}\Gamma^{d+e}_R(M,n)\;, \end{align*}
we obtain an algebra structure on the direct sum $\mathcal{N}\Gamma_R(M,n)=\bigoplus_{d\ge 0}\mathcal{N}\Gamma^d_R(M,n)$.
The following definition axiomatizes the properties of this direct sum, as well as similar objects which will appear in our computations.
\begin{definition}\label{def-PR-alg}
Let $M$ be a projective finitely generated module over a ring $R$. A
\emph{differential graded strict polynomial algebra} (dg-$\mathcal{P}_R$-algebra, for short) is a
bigraded $R$-algebra
$A(M)=\bigoplus_{d\ge 0, i\ge 0} A^d_i(M)$, endowed with a differential
$\partial:A^d_i(M)\to A^d_{i-1}(M)$, and
satisfying the following
properties.
\begin{enumerate}
\item The $A^d_i(M)$ are homogeneous strict polynomial functors
of weight $d$ with respect to $M$. We call these the homogeneous components of degree
$i$ and weight
$d$ of $A(M)$.
\item The multiplication $A^d_i(M)\otimes A^e_j(M)\to
A^{d+e}_{i+j}(M)$ and the differential $\partial$ are morphisms
of strict polynomial functors.
\end{enumerate}
A morphism of differential graded $\mathcal{P}_R$-algebras
$f:A(M)\to B(M)$ is a morphism of differential graded
$R$-algebras provided by a family of morphisms of strict polynomial
functors $f^d_i:A^d_i(M)\to B^d_i(M)$.
\end{definition}
Definition \ref{def-PR-alg} admits obvious variants for dg-$\mathcal{P}_R$-coalgebras, dg-$\mathcal{P}_R$-bialgebras, etc. whose formulation is left to the reader.
A dg-$\mathcal{P}_R$-algebra with zero differential is simply called a graded
$\mathcal{P}_R$-algebra, and a graded $\mathcal{P}_R$-algebra concentrated in degree zero is simply called a $\mathcal{P}_R$-algebra. Here are some basic examples of graded $\mathcal{P}_R$-algebras.
\begin{itemize}
\item We denote by $\Gamma_R(M[i])$ the divided power algebra generated by a
finitely generated projective $R$-module $M$ placed in degree $i$. It is a
graded $\mathcal{P}_R$-algebra whose homogeneous component of degree $di$ and weight $d$ is $\Gamma^d_R(M)$.
\item If $R=\mathbb{k}$ is a field of positive characteristic $p$, and $V$ is a finite dimensional $\mathbb{k}$-vector space, we similarly denote by
$\Gamma_\mathbb{k}(V^{(r)}[i])$ the divided power algebra generated by a copy of the
Frobenius twist functor $V^{(r)}$ placed in degree $i$. It is a graded
$\mathcal{P}_R$-algebra with $\Gamma^d_\mathbb{k}(V^{(r)})$ as homogeneous part of degree $di$
and weight $dp^r$.
\item We will use the similar notations $\Lambda_R(M[i])$,
$\Lambda_\mathbb{k}(V^{(r)}[i])$, $S_R(M[i])$ and $S_\mathbb{k}(V^{(r)}[i])$ for the corresponding
exterior and symmetric algebras.
\end{itemize}
Derivation of functors (at the level of chain complexes) can be restated as a functor:
\begin{align}\begin{array}{ccc}
\left\{\text{$\mathcal{P}_R$-algebras}\right\} & \to &
\left\{\text{dg-$\mathcal{P}_R$-algebras}\right\} \\
A(M) & \mapsto & \mathcal{N} A(M,n)
\end{array}\;.\label{deriv-dgp}
\end{align}
More generally, if $A(M)$ is a dg-$\mathcal{P}_R$-algebra, we define $\mathcal{N} A(M,n)$ by placing the degree $j$ object of the complex $\mathcal{N} A^d_i(M)$ in weight $d$ and degree $i+j$, with product defined by the shuffle map and the product of $A(M)$, and with differential defined as the sum of the differential of $A(M)$ and the differential coming from the simplicial structure. We thus obtain a functor:
\begin{align}\begin{array}{ccc}
\left\{\text{dg-$\mathcal{P}_R$-algebras}\right\} & \to &
\left\{\text{dg-$\mathcal{P}_R$-algebras}\right\} \\
A(M) & \mapsto & \mathcal{N} A(M,n)
\end{array}\;.\label{deriv-dgp2}
\end{align}
\begin{remark}
When $A(M)$ is graded commutative and exponential, the dg-$\mathcal{P}_R$-algebra $\mathcal{N} A(M,n)$ coincides (up to homotopy equivalence) with the $n$-fold bar construction of $A(M)$ \cite[Chap X]{ML}. For arbitrary dg-$\mathcal{P}_R$-algebras, these two constructions are different.
\end{remark}
The homology of a dg-$\mathcal{P}_R$-algebra is a graded $\mathcal{P}_R$-algebra. For example
we have the graded $\mathcal{P}_R$-algebra
\begin{align}L_*\Gamma_R(M,n)=\bigoplus_{d\ge
0}L_*\Gamma^d_R(M,n)\;.\label{eq-gP-alg}\end{align}
\begin{remark}\label{rk-poids}
The left hand side of \eqref{eq-gP-alg} does not display the integer
$d$. However, if we know $L_*\Gamma_R(M,n)$ as a graded $\mathcal{P}_R$-algebra, it is always
easy to retrieve $L_*\Gamma^d_R(M,n)$ as its homogeneous component of weight $d$. In
this paper, we will usually not explicitly write down the weight of functors,
because this information is already implicitly encoded in the strict polynomial
structure of the objects. For example, if $R$ is a field of characteristic $2$
and $M$ a finite dimensional $R$-vector space, then the weight
$4$ component of $\Gamma_R(M[1])\otimes \Gamma_R(M^{(1)}[3])$ is equal to
$\Gamma_R^4(M)\oplus \Gamma^2_R(M)\otimes M^{(1)}\oplus \Gamma^2_R(M^{(1)})$, where these three summands live in degrees $4$, $5$ and $6$ respectively.
\end{remark}
\subsection{Some basic facts regarding derived functors of the divided power algebra}
Before starting our computations, we recall in this section some basic facts on derived functors of the
symmetric algebras, the exterior algebras and the divided powers algebras, which
give a clearer picture of the situation. The following formula is due to Bousfield \cite{bousfield-hom} and Quillen \cite{quillen-rings} (see also \cite{illusie} vol. I, chapter I,\: \S 4.3.2).
\begin{proposition}[D\'ecalage]
Let $R$ be a commutative ring, and let $M$ be a finitely generated projective $R$-module.
There are isomorphisms of graded
$\mathcal{P}_R$-algebras:
\begin{align}&\bigoplus_{i,d\ge 0}L_{i}\Gamma_R^d(M, n)\simeq \bigoplus_{i,d\ge 0}L_{i+d}\Lambda_R^d(M,n+1)
\simeq \bigoplus_{i,d\ge 0} L_{i+2d}S_R^d(M,n+2)\;.\label{dec} \end{align}
\end{proposition}
If $R$ is a $\mathbb{Q}$-algebra, there is an isomorphism of graded
$\mathcal{P}_R$-algebras $S_R(M[i])\simeq \Gamma_R(M[i])$. Thus the d\'ecalage formula
implies the following statement.
\begin{corollary}
If $R$ is a $\mathbb{Q}$-algebra, and $M$ is a finitely generated
$R$-module, then the graded $\mathcal{P}_R$-algebra $L_*\Gamma_R(M, n)$ is isomorphic to $\Gamma_R(M[n])$ if $n$ is even and to $\Lambda_R(M[n])$ if $n$ is odd.
\end{corollary}
There is thus no issue in computing derived functors of divided power algebras
over $\mathbb{Q}$-algebras $R$. We now give some elementary properties of divided powers
when $R$ is noetherian.
\begin{proposition}\label{lm-free-tors}
Let $R$ be noetherian and let $M$ be a finitely generated projective
$R$-module. Then $L_j\Gamma_R^d(M, n)$ is
\begin{itemize}
\item zero if $j<n$ or $j>nd$,
\item a finitely generated $R$-module if $n\le j\le nd$. If $R={\mathbb{Z}}$,
then for $n\le j<nd$, $L_j\Gamma_R^d(M, n)$ is a finite abelian group.
\end{itemize}
Finally, the graded $\mathcal{P}_R$-subalgebra
$$\bigoplus_{d\ge 0} L_{nd}\Gamma_R^d(M, n) \subset \bigoplus_{d,j\ge 0}
L_{j}\Gamma_R^d(M, n)=L_*\Gamma_R(M, n)$$
is equal to
$\Lambda_R(M)$ if $n$ is odd and $2\ne 0$ in $R$, and to $\Gamma_R(M)$ otherwise.
\end{proposition}
\section{A quasi-trivial filtration of the divided power algebra}
\label{qt-fil}
In this section, we fix a field $\mathbb{k}$ of positive characteristic $p>0$. All the
functors considered are strict polynomial functors defined over $\mathbb{k}$. We write $\Gamma^d$,
$\Lambda^d$ and $S^d$ for $\Gamma^d_\mathbb{k}$, $S^d_\mathbb{k}$ and $\Lambda^d_\mathbb{k}$.
A generic finite dimensional $\mathbb{k}$-vector space will be denoted by the
letter `$V$'. The goal of this section is to introduce some particularly nice filtrations of
dg-$\mathcal{P}_\mathbb{k}$-algebras, which we call `quasi-trivial', and to exhibit a
quasi-trivial filtration of the divided power algebra.
\subsection{Quasi-trivial filtrations}
A nonnegative decreasing filtration on a dg-$\mathcal{P}_\mathbb{k}$ algebra $A(V)$ is a family
of subfunctors of $A(V)$
$$\textstyle\bigcap_{i\ge 0}F^iA(V)=0\subset\dots\subset F^{i+1}A(V)\subset F^i A(V)\subset\dots\subset F^0A(V)=A(V)\;,$$
satisfying the following conditions
\begin{enumerate}
\item $F^iA(V)=\bigoplus_{d\ge 0,j\ge 0} (F^iA(V))^d_j$ with
$(F^iA(V))^d_j\subset A_j^d(V)$.
\item The differential and the multiplication in $A(V)$ restrict to morphisms
$d:F^iA(V)\to F^iA(V)$ and $F^iA(V)\otimes F^jA(V)\to F^{i+j}A(V)$.
\end{enumerate}
The associated graded object is then a dg-$\mathcal{P}_\mathbb{k}$ algebra. The grading which we will consider is the one coming from $A(V)$, not the filtration grading.
We will simply write
$$\mathrm{gr}\, A(V)=\bigoplus_{i,j,d} \mathrm{gr}\,^i A^d_j(V)\;,\quad\text{ and } (\mathrm{gr}\, A)^d_j(V)=\bigoplus_{i} \mathrm{gr}\,^i A^d_j(V)\;.$$
\begin{definition}\label{def-quasi-trivial}
Let $A(V)$ be a dg-$\mathcal{P}_\mathbb{k}$-algebra equipped with a nonnegative decreasing filtration
$(F^iA(V))_{i\ge 0}$. We will say that this filtration is quasi-trivial if
\begin{enumerate}
\item the algebra $\mathrm{gr}\, A(V)$ is exponential and the components $\mathrm{gr}\, A_j^d(V)$ have
finite dimensional values for all $j$ and $d$, and
\item there is a weight preserving isomorphism of differential graded
$\mathbb{k}$-algebras \linebreak $\varphi:A(\mathbb{k})\xrightarrow[]{\simeq}\mathrm{gr}\, A(\mathbb{k})$.
\end{enumerate}
\end{definition}
The next lemma gives some straightforward consequences of definition
\ref{def-quasi-trivial}. The property (c) says that the graded $\mathcal{P}_\mathbb{k}$-algebras $A(V)$ and $\mathrm{gr}\, A(V)$ are `as
close as possible': the filtration modifies the functoriality but not the algebra structure of $A(V)$.
\begin{lemma}\label{lm-triv}
Let $A(V)$ be a dg-$\mathcal{P}_\mathbb{k}$-algebra equipped with a quasi-trivial filtration.
Then:
\begin{enumerate}
\item[(a)] The filtration of each summand $A^d_j(V)$ is bounded.
\item[(b)] The algebra $A(V)$ is exponential.
\item[(c)] The choice of a basis of $V$ determines a
\emph{non-functorial} weight-preserving isomorphism of differential graded $\mathbb{k}$-algebras $A(V)\simeq \mathrm{gr}\, A(V)$.
\end{enumerate}
\end{lemma}
\begin{proof}
(a) Since each $A_j^d(V)$ is a strict polynomial functor with finite dimensional values, it is a finite functor, in particular all filtrations are bounded \cite[lemma 14.1]{antoine}.
(b) The map $\psi:A(V)\otimes A(W)\to A(V\oplus W)$ induced by the
multiplication preserves filtrations, and the associated graded map
$\mathrm{gr}\,\psi :\mathrm{gr}\, A(V)\otimes \mathrm{gr}\, A(W)\xrightarrow[]{\simeq}\mathrm{gr}\, A(V\oplus W)$ is an
isomorphism by the first condition of definition \ref{def-quasi-trivial}.
If we restrict ourselves to homogeneous components of a given weight $d$, the filtrations on
$A(V)\otimes A(W)$ and $A(V\oplus W)$ are finite, so that $\psi$ is an isomorphism.
(c) A basis of $V$ determines an isomorphism $V\simeq \mathbb{k}^s$. We obtain the
required isomorphism of differential graded algebras as the composite
$$A(\mathbb{k}^s)\simeq A(\mathbb{k})^{\otimes s}\xrightarrow[\simeq]{\varphi^{\otimes s}} (\mathrm{gr}\,
A(\mathbb{k}))^{\otimes s}\simeq \mathrm{gr}\, A(\mathbb{k}^s)\;, $$
where the first and third isomorphisms are induced by the multiplications
(and are isomorphisms since $A(V)$ and $\mathrm{gr}\, A(V)$ are exponential).
\end{proof}
A crucial property of quasi-trivial filtrations is that they commute
with derivation.
\begin{proposition}\label{prop-quasi-trivial-collapse}
Let $A(V)$ be a graded $\mathcal{P}_\mathbb{k}$-algebra, endowed with a quasi-trivial
filtration. For all $n\ge 0$, there exists a filtration of the graded
$\mathcal{P}_\mathbb{k}$-algebra $L_*A(V,n)$ and a
functorial isomorphism of graded $\mathcal{P}_\mathbb{k}$-algebras:
$$ \mathrm{gr}\, (L_*A(V,n))\simeq L_*(\mathrm{gr}\, A)(V,n)\;.$$
\end{proposition}
\begin{proof}
The filtration of $A(V)$ induces a filtration of the dg-$\mathcal{P}_\mathbb{k}$-algebra
$\mathcal{N} A(V,n)$. The associated spectral sequence of graded $\mathcal{P}_\mathbb{k}$-algebras has
the form:
\begin{align}E^1_{i,j}=H_{i+j}(\mathrm{gr}\,^{-i}\mathcal{N} A(V,n))\Longrightarrow
H_{i+j}(\mathcal{N} A(V,n))\;.\label{eq-ss-filtr}\end{align}
While this is a second quadrant spectral sequence, there
is no problem with convergence since it splits as a direct sum of spectral
sequences of homogeneous strict polynomial functors of given weight $d$, and
the lemma \ref{lm-triv}(a) ensures that each summand bounded converges \cite[Thm 5.5.1]{weibel}.
The first page of the spectral sequence may be rewritten as
$E^1_{i,j}=L_{i+j}(\mathrm{gr}\,^{-i}A)(V,n)$. To prove proposition
\ref{prop-quasi-trivial-collapse} it therefore suffices to prove that the spectral
sequence \eqref{eq-ss-filtr} degenerates at $E_1$. This will be the case if we are
able to prove that for all $i$ and $d$, the homogeneous components of degree $i$ and weight
$d$ of the graded $\mathcal{P}_\mathbb{k}$-algebras $L_*A(V,n)$ and $L_*(\mathrm{gr}\, A)(V,n)$ have the
same dimension (note that we already know that they both are finite-dimensional by proposition \ref{lm-free-tors}).
This equality of dimensions follows directly from the observation that $A(K(V,n))$ and $\mathrm{gr}\, A(K(V,n))$
coincide as semi-simplicial $\mathbb{k}$-vector spaces.
Indeed the graded $\mathbb{k}$-algebras
with weights $A(\mathbb{k})$ and $\mathrm{gr}\, A(\mathbb{k})$ are isomorphic, and we claim that for exponential graded $\mathcal{P}_\mathbb{k}$-algebras
$E$, the graded $\mathbb{k}$-algebra with weights $E(\mathbb{k})$ determines completely
the semi-simplicial $\mathbb{k}$-vector space $E(K(V,n))$.
The latter claim follows from the explicit construction
of the Dold-Kan functor $K$
\cite[Sec. 8.4]{weibel}. The simplicial $\mathbb{k}$-vector
space $K(V[n])$ is degreewise finite-dimensional, say of some dimension $d_k$ in
degree $k$. If we choose a basis of $V$, each $K(V[n])_k$ has a canonical basis determined by the
basis of $V$, and if we use coordinates relative to these bases, then for each
$i$ the face operators $\partial_i:K(V[n])_k\to K(V[n])_{k-1}$ is given by a formula
$\partial_i(x_1,\dots,x_{d_k})=(y_1,\dots,y_{d_{k-1}})$, where $y_j=\sum_{i\in
I_j} x_i$ and $I_1,\dots,I_{d_{k-1}}$ is some partition of the set
$\{1,\dots,d_k\}$. We therefore have commutative diagrams
\begin{align*}\xymatrix{
E(\mathbb{k})^{\otimes d_k}\ar[rr]^-{\simeq}_-{\mu}\ar[d]^-{\overline{\partial_i}} &&
E(K(V[n])_k)\ar[d]^-{{\partial_i}} \\
E(\mathbb{k})^{\otimes d_{k-1}}\ar[rr]^-{\simeq}_-{\mu}&& E(K(V[n])_{k-1})
}
\end{align*}
where the horizontal isomorphisms $\mu$ are induced by the multiplication of
$E(V)$, and where $\overline{\partial_i}$ sends $x_1\otimes\dots\otimes x_{d_k}$
to $y_1\otimes\dots\otimes y_{d_{k-1}}$, with $y_j=\prod_{i\in I_j} x_i$. This proves our claim, and finishes the proof of proposition
\ref{prop-quasi-trivial-collapse}.
\end{proof}
Another useful property of quasi-trivial filtrations is their compatibility with
kernels. To be more specific,
let $A(V)$ be a filtered dg-$\mathcal{P}_\mathbb{k}$-algebra, and denote by $Z(V)$ the
subalgebra of cycles of $A(V)$. The filtration of $A(V)$ induces a
filtration on $Z(V)$ by setting $F^iZ(V)=F^iA(V)\cap Z(V)$. Let $Z'(V)$ be the
subalgebra of cycles of $\mathrm{gr}\, A(V)$. Then we have a canonical injective morphism
of algebras:
\begin{align}\mathrm{gr}\, Z(V)\hookrightarrow Z'(V)\;.\label{eq-Z}\end{align}
In general, this morphism is not surjective, but it turns out to be the case if
the filtration of $A(V)$ is quasi-trivial.
\begin{proposition}\label{compat-Z}
Let $A(V)$ be a dg-$\mathcal{P}_\mathbb{k}$-algebra, endowed with a quasi-trivial filtration.
Let us denote by $Z(V)$ the cycles of $A(V)$ and by $Z'(V)$ the cycles of $\mathrm{gr}\,
A(V)$.
The canonical morphism \eqref{eq-Z} is an isomorphism of graded $\mathcal{P}_\mathbb{k}$-algebras.
\end{proposition}
\begin{proof}
It suffices to prove that the homogeneous components $(\mathrm{gr}\, Z)^d_i(V)$ and $Z^d_i(V)$
have the same dimension for all degrees $i$ and all weights $d$, or equivalently that the maps $\partial:A^d_i(V)\to A^d_{i-1}(V)$ and
$\mathrm{gr}\, \partial: (\mathrm{gr}\, A)^d_i(V)\to (\mathrm{gr}\, A)^d_{i-1}(V)$ have the same rank. This follows
from lemma \ref{lm-triv}(c).
\end{proof}
\subsection{Truncated polynomial algebras}\label{subsec-trunc}
The truncated polynomial algebra $Q(V)$ is the $\mathcal{P}_\mathbb{k}$-algebra obtained as
the quotient of $S(V)$ by the ideal generated by $V^{(1)}$.
For all $i\ge 0$ we denote by $Q(V[i])$ the truncated polynomial algebra
on a generator $V$ placed in degree $i$. This is defined in a similar way, as the
quotient of $S(V[i])$ by the ideal generated by $V^{(1)}[i]$.
Truncated polynomial algebras enjoy the following properties.
\begin{enumerate}
\item If $p=2$, $Q(V)=\Lambda(V)$ (but this
is no longer true in odd characteristics).
\item The $\mathcal{P}_\mathbb{k}$-algebra $Q(V)$ is an exponential functor (in particular a $\mathcal{P}_\mathbb{k}$-bialgebra).
\item\label{it-3} Let us denote by $\varphi:S(V)\to \Gamma(V)$ the unique morphism of $\mathcal{P}_\mathbb{k}$-algebras whose restriction $V=S^1(V)\to\Gamma^1(V)=V$ to the summand of weight one is equal to the identity. It follows that $Q(V)$ is equal to the image of $\varphi$.
\item\label{it-4} $Q(V)$ is self-dual, namely, there is an isomorphism of $\mathcal{P}_\mathbb{k}$-bialgebras $Q(V)\simeq Q^\sharp(V)$.
\end{enumerate}
All these properties are well-known and easy to check, we just indicate how to retrieve \eqref{it-4} from \eqref{it-3}. Since $\varphi$ is a morphism between exponential $\mathcal{P}_R$-algebras, it is actually a morphism of $\mathcal{P}_R$-bialgebras. Hence its dual yields a morphism of $\mathcal{P}_R$-bialgebras $\varphi^\sharp:S(V)\simeq\Gamma^\sharp(V)\to S^\sharp(V)\simeq\Gamma(V)$. Since $\varphi$ and $\varphi^\sharp$ coincide when restricted to the homogeneous summand of weight $1$, they must be equal. So we obtain: $Q(V)=\mathrm{Im}\,\varphi\simeq \mathrm{Im}\,\varphi^\sharp = Q^\sharp(V)$.
We finish this paragraph with a slightly less known result on trunctated polynomials, namely the construction of functorial resolution of $Q(V)$.
We equip the graded $\mathcal{P}_{\mathbb{k}}$-algebra $S(V)\otimes
\Lambda(V^{(1)}[1])$ with a differential $\partial $ defined as the composite:
$$S^d(V)\otimes \Lambda^e(V^{(1)})\to S^d(V)\otimes V^{(1)}\otimes
\Lambda^{e-1}(V^{(1)})\to S^{d+p}(V)\otimes \Lambda^{e-1}(V^{(1)}) $$
where the first map is induced by the comultiplication in
$\Lambda(V^{(1)})$ and the second by composition of the inclusion
$V^{(1)}\hookrightarrow S^p(V)$ and the multiplication in $S(V)$.
The composite morphism of graded $\mathcal{P}_\mathbb{k}$-algebras $S(V)\otimes
\Lambda(V^{(1)}[1])\twoheadrightarrow S(V)\twoheadrightarrow Q(V)$
induces a morphism of differential graded
algebras
\begin{equation}
\label{resq}
f: (S(V)\otimes \Lambda(V^{(1)}[1]),\partial)\to (Q(V),0)\;. \end{equation}
\begin{proposition}\label{prop-qis-trunc}
The morphism $f$ is a quasi-isomorphism.
\end{proposition}
\begin{proof} The graded $\mathcal{P}_\mathbb{k}$-algebras $S(V)\otimes \Lambda(V^{(1)}[1])$ and
$Q(V)$ are exponential functors. Hence for $V=\mathbb{k}^d$ there is a commutative diagram of differential graded $\mathbb{k}$-algebras, whose
vertical isomorphisms are induced by the multiplication:
$$\xymatrix{
\left(S(\mathbb{k})\otimes \Lambda(\mathbb{k}^{(1)}[1])\right)^{\otimes d}\ar[rr]^-{f^{\otimes
d}}\ar[d]^-{\simeq}&& Q(\mathbb{k})^{\otimes d}\ar[d]^-{\simeq}\\
S(\mathbb{k}^d)\otimes \Lambda(\mathbb{k}^{d\,(1)}[1])\ar[rr]^-{f}&& Q(\mathbb{k}^d)
}\;.$$
Thus, by the K\"unneth formula, the proof reduces to the easy case $V=\mathbb{k}$.
\end{proof}
\begin{example}
In characteristic 2, the weight 4 component of the morphism of differential
graded algebras \eqref{resq} determines the following resolution of $
\Lambda^4(V)$, where $\partial_1(x \wedge y) = x^2 \otimes y - y^2 \otimes x$ and $\partial_0(xy \otimes z)
= xyz^2$:
\begin{equation}
\label{s12cx}
\xymatrix{
0 \ar[r] & \Lambda^2(V^{(1)}) \ar[r]^(.47){\partial_1} & S^2(V) \otimes V^{(1)}
\ar[r]^(.6){\partial_0}&
S^4(V) \ar[r]^{f_4} & \Lambda^4(V) \ar[r] & 0
}
\end{equation}
\end{example}
\subsection{The principal filtration on the divided power algebra}
\label{filt-1}
We denote by $\mathcal{I}(V)$ the ideal of $\Gamma(V)$ generated by $V=\Gamma^1(V)$. We
call this ideal the principal ideal of $\Gamma(V)$ (although it is not strictly
speaking a principal ideal). The adic filtration relative to $\mathcal{I}(V)$ will be called the
principal filtration of $\Gamma(V)$. The associated graded object is the
$\mathcal{P}_\mathbb{k}$-algebra:
\[
\mathrm{gr}\, \Gamma(V):= \bigoplus_{n\ge 0} \mathrm{gr}\,^n (\Gamma(V)) = \bigoplus_{n\ge
0}\mathcal{I}(V)^n/\mathcal{I}(V)^{n+1}\;.
\]
In this section, we will compute in proposition
\ref{prop-filtration-Gamma} the graded object associated to this principal
filtration. The result which will be obtained in proposition
\ref{prop-filtration-Gamma} below deserves to be compared to
the following well-known assertion \cite[Expos\'e 9, p.9-07]{cartan}:
\begin{proposition}\label{prop-iso-Nonnat}
The choice of a basis of the finite dimensional vector space $V$ determines a
(non-natural) weight-preserving algebra isomorphism
$$\Gamma(V)\xrightarrow[]{\simeq}
Q(V)\otimes\Gamma(V^{(1)})\;.$$
\end{proposition}
\begin{proof}
By the exponential properties of $\Gamma(V)$ and $Q(V)\otimes\Gamma(V^{(1)})$
(as in the proof of lemma \ref{lm-triv}), the proof reduces to
the case $V=\mathbb{k}$ which is a straightforward
computation.
\end{proof}
To describe the $\mathcal{P}_\mathbb{k}$-algebra $\mathrm{gr}\, \Gamma(V)$, we first need to interpret $\mathcal{I}(V)$ as a kernel.
By the universal property of the symmetric algebra, the inclusion
$V^{(1)}\hookrightarrow S^{p^r}_\mathbb{k}(V)$, induces an injective morphism of $\mathcal{P}_\mathbb{k}$-algebras:
$ S_\mathbb{k}(V^{(1)})\hookrightarrow S_\mathbb{k}(V)$.
Since $S_\mathbb{k}(V)$ is an exponential functor, and
since Frobenius twists are additive functors, $S_\mathbb{k}(V^{(1)})$ is also an
exponential functor. Thus the natural inclusion above is also a
morphism of $\mathcal{P}_\mathbb{k}$-bialgebras and it induces by duality an
epimorphism of $\mathcal{P}_\mathbb{k}$-(bi)algebras
$\Gamma(V)\twoheadrightarrow \Gamma(V^{(1)})$.
\begin{lemma}\label{prop-cokernel} The principal ideal $\mathcal{I}(V)$ is the kernel of the morphism $\Gamma(V)\twoheadrightarrow \Gamma(V^{(1)})$. In other words,
the multiplication in $\Gamma(V)$ yields a short exact sequence:
\begin{align}V\otimes \Gamma(V) \xrightarrow[]{\mathrm{mult}} \Gamma(V)\twoheadrightarrow
\Gamma(V^{(1)})\to 0\;. \label{ex-sq}\end{align}
\end{lemma}
\begin{proof}
If $V=V_1\oplus V_2$, by using the exponential properties of
$\Gamma(V)$ and $\Gamma(V^{(1)})$, we obtain that \eqref{ex-sq} is isomorphic to the
short exact sequence:
$$\begin{array}{c}
V_1\otimes\Gamma(V_1)\otimes\Gamma(V_2)\oplus\\
\Gamma(V_1)\otimes V_2\otimes\Gamma(V_2)
\end{array}\to \Gamma(V_1)\otimes \Gamma(V_2)\twoheadrightarrow
\Gamma(V_1^{(1)})\otimes \Gamma(V_2^{(1)})\to 0\;.$$
Hence it suffices to check exactness for $V=\mathbb{k}$, which is easy.
\end{proof}
\begin{proposition}\label{prop-filtration-Gamma}
There is an
isomorphism of $\mathcal{P}_\mathbb{k}$-algebras, which maps $\mathrm{gr}\,^n \Gamma(V)$ isomorphically onto $Q^n(V)\otimes
\Gamma(V^{(1)})$:
$$
\mathrm{gr}\, \Gamma(V)\simeq
Q(V)\otimes \Gamma(V^{(1)})\;.$$
\end{proposition}
\begin{proof}
Let $\mathcal{I}(V)_d^n$ be the direct summand of $\mathcal{I}(V)^n$
contained in $\Gamma^d(V)$, so that $$\mathrm{gr}\,^n (\Gamma^d(V))=
\mathcal{I}(V)_d^n/\mathcal{I}(V)_d^{n+1}\;.$$
Then $\mathcal{I}(V)^n_d=0$ if $d< n$, and for $d\ge n$ the multiplication
of $\Gamma(V)$
induces an epimorphism:
\begin{equation}\label{eqn-local-1}
V^{\otimes n}\otimes \Gamma^{d-n}(V)\twoheadrightarrow \mathcal{I}(V)^n_d\;.
\end{equation}
Since the multiplication $V^{\otimes n}\to \Gamma^n(V)$ factors
through the canonical inclusion of $Q^n(V)$ in $\Gamma^n(V)$, the maps
(\ref{eqn-local-1}) induce commutative diagrams:
$$\xymatrix{
Q^n(V)\otimes
V\otimes \Gamma^{d-n-1}(V)\ar[d]^-{Q^n(V)\otimes\mathrm{mult}} \ar@{->>}[r]&
\mathcal{I}(V)^{n+1}_{d}\ar@{^{(}->}[d]\\
Q^n(V)\otimes \Gamma^{d-n}(V)\ar@{->>}[r]&\mathcal{I}(V)^n_{d}
}\;.$$
By lemma \ref{prop-cokernel}, the cokernel of the multiplication
$V\otimes \Gamma^{d-n-1}(V)\to \Gamma^{d-n}(V)$ is equal to
$\Gamma^{(d-n)/p}(V^{(1)})$ if $p$ divides $d-n$, and to zero
otherwise. Hence, if $p$ divides $d-n$, the map $Q^n(V)\otimes
\Gamma^{d-n}(V)\twoheadrightarrow \mathcal{I}(V)^n_{d}$ induces an
epimorphism
$$ Q^n(V)\otimes \Gamma^{(d-n)/p}(V^{(1)})\twoheadrightarrow
\mathcal{I}(V)^n_d/\mathcal{I}(V)^{n+1}_d\;,
$$
and the quotient $\mathcal{I}(V)^n_d/\mathcal{I}(V)^{n+1}_d$ equals zero if $p$ does
not divide $d-n$. We thus have a surjective morphism of
$\mathcal{P}_\mathbb{k}$-algebras:
\begin{equation}\label{eqn-local-2}Q(V)\otimes
\Gamma(V^{(1)})\twoheadrightarrow
\bigoplus_{n\ge 0} \mathrm{gr}\,^n \Gamma(V)\;,
\end{equation}
which sends $Q^n(V)\otimes
\Gamma(V^{(1)})$ onto $\mathrm{gr}\,^n \Gamma(V)$.
To finish the proof, we observe that epimorphism \eqref{eqn-local-2} is
actually an isomorphism for dimension reasons: it follows from
proposition
\ref{prop-iso-Nonnat} that the
direct summands of a given weight $d$ of the source and the target of the epimorphism
\eqref{eqn-local-2} have the same finite dimension.
\end{proof}
Propositions \ref{prop-iso-Nonnat} and
\ref{prop-filtration-Gamma} have the following consequence.
\begin{corollary}\label{cor-qtf}
There exists a quasi-trivial filtration of $\Gamma(V)$, and an isomorphism of
$\mathcal{P}_\mathbb{k}$-algebras $\mathrm{gr}\,\Gamma(V)\simeq Q(V)\otimes \Gamma(V^{(1)})$.
\end{corollary}
In the previous statements, we considered the divided power algebra
$\Gamma(V)$ as a nongraded algebra (or equivalently as a graded algebra
concentrated in degree zero). But we can define an extra degree on the
divided power algebra by placing the generator $V$ in degree $i$. In that
case the statements of propositions \ref{prop-iso-Nonnat},
\ref{prop-filtration-Gamma} and corollary \ref{cor-qtf} remain valid with
`$V$' replaced by $V[i]$ and `$V^{(1)}$' replaced by $V^{(1)}[pi]$, since all
the morphisms in those propositions preserve the weights, and the extra degree
is equal to $i$ times the weight. By iterating corollary \ref{cor-qtf} we then
obtain the following result.
\begin{corollary}\label{cor-iterated-princ}For any non-negative integer $i$,
there exists a quasi-trivial filtration on $\Gamma(V[i])$, and an isomorphism of
graded $\mathcal{P}_\mathbb{k}$-algebras
$$\mathrm{gr}\, \Gamma(V[i])\simeq \bigotimes_{r\ge 0} Q(V^{(r)}[ip^r])\;.$$
\end{corollary}
\section{The derived functors of $\Gamma^d_\mathbb{k}(V)$ in positive characteristic}
\label{descr-gvn}
In this section, $\mathbb{k}$ is a field of positive characteristic $p$. All the
functors considered are strict polynomial functors defined over $\mathbb{k}$. In
particular, we write $\Gamma^d$,
$\Lambda^d$ and $S^d$ for $\Gamma^d_\mathbb{k}$, $S^d_\mathbb{k}$ and $\Lambda^d_\mathbb{k}$.
A generic finite dimensional $\mathbb{k}$ vector space will be denoted by the
letter `$V$'.
The main results of this section are theorems \ref{thm-derived-p2} and \ref{thm-derived-podd}, which
describe the derived functors of $\Gamma(V)$. These results were already proved by one of us
in \cite{antoine}, where the proof was rather technical and relied heavily on the computations of
Cartan \cite{cartan}. The proofs which we will give here are more elementary and independent
of \cite{antoine, cartan}.
We will require theorems \ref{thm-derived-p2} and \ref{thm-derived-podd} as an input for the computations
of sections \ref{der1-gamma-z}, \ref{sec-derG12}, and \ref{der-gamma4-sec}.
\subsection{The description of $L_*\Gamma^d(V,n)$ in characteristic $2$}
\begin{theorem}\label{thm-derived-p2}
Let $\mathbb{k}$ be a field of characteristic $2$, and let $V$ be a finite
dimensional $\mathbb{k}$-vector space. For all $n\ge 1$, there is an isomorphism of
graded
$\mathcal{P}_\mathbb{k}$-algebras
\begin{align}
\label{lgammavn}
&L_*\Gamma(V,n)\simeq \bigotimes_{r_1,\dots, r_n\ge 0}
\Gamma \left(V^{(r_1+\dots
+r_n)}[2^{r_2+\dots+r_n}+2^{r_3+\dots+r_n}+\dots+2^{r_n}
+1]\right)\;.
\end{align}
\end{theorem}
For example, there is an isomorphism
$L_*\Gamma(V,1)\simeq \bigotimes_{r\ge 0} \Gamma(V^{(r)}[1])$.
The homogeneous component of weight $d$ of the graded
$\mathcal{P}_\mathbb{k}$-algebra $L_*\Gamma(V,n)$ provides us the derived functors of $\Gamma^d(V)$.
Let us spell this out in the $d=4$ case.
\begin{example}\label{ex-G4}
For $n\ge 1$, the derived functors $L_*\Gamma^4(V,n)$ are given by the following formula (where $F(V)\,[k]$ means a copy of the strict polynomial functor $F(V)$ placed in degree $k$).
\begin{align*}
L_*\Gamma^4(V,n)\simeq &\qquad \Gamma^4(V)\;[4n]\\
&\oplus\;\bigoplus_{i=1\dots n} \Gamma^2(V)\otimes V^{(1)}\;[3n+i-1]\\
&\oplus\;\bigoplus_{1\le i<j\le n} V^{(1)}\otimes V^{(1)}\;[2n+i+j-2]\quad \oplus \;\bigoplus_{i=1\dots n} \Gamma^{2}(V^{(1)})\;[2n+2i-2]\\
&\oplus\;\bigoplus_{1\le i\le j\le n} V^{(2)}\;[n+2i+j-3]\;.
\end{align*}
\end{example}
\begin{proof}[Explanation of example \ref{ex-G4}]
In order to unpackage the compact formula \eqref{lgammavn}, we list those generators of the graded $\mathcal{P}_\mathbb{k}$-algebra $L_*\Gamma(V,n)$ which can contribute (after applying a divided power functor or after taking tensor products) to a summand of weight $4$ of $L_*\Gamma(V,n)$ .
These generators are of the following four distinct types:
\begin{enumerate}
\item[(i)] one generator $V[n]$, corresponding to the $n$-tuple $(0,\dots,0)$,
\item[(ii)] $n$ generators of the form $V^{(1)}[n+i-1]$, corresponding to the $n$-tuples $(r_1,\dots,r_n)$ with $r_i=1$, $r_{k}=0$ if $k\ne i$,
\item[(iii)] $n$ generators of the form $V^{(2)}[n+3i-3]$, corresponding to the $n$-tuples $(r_1,\dots,r_n)$ with $r_i=2$, $r_{k}=0$ if $k\ne i$,
\item[(iv)] $n(n-1)/2$ generators of the form $V^{(2)}[n+j+2i-3]$, corresponding to the $n$-tuples $(r_1,\dots,r_n)$ with $r_i=r_j=1$ for a given pair $\{i,j\}$ with $i<j$, and $r_{k}=0$ if $k\ne i,j$.
\end{enumerate}
Then we determine all possible manners in which these generators can contribute to a direct summand of weight $4$ of $L_*\Gamma(V,n)$:
\begin{itemize}
\item The generator (i) can contribute to a summand of weight $4$ in two ways namely (a) via a summand $\Gamma^4(V)$, and (b) via a summand $\Gamma^2(V)\otimes V^{(1)}$, where $V^{(1)}$ is a generator of type (ii).
\item The generators of type (ii) can contribute to a summand of weight $4$ in three ways. First of all by the method (b) listed before, secondly via a summand $V^{(1)}\otimes V^{(1)}$ where two generators of type (ii) are involved, or thirdly via a summand $\Gamma^2(V^{(1)})$.
\item The generators of type (iii) and (iv) are already of weight $4$, hence they can only contribute to the part of weight $4$ as summands of the form $V^{(2)}$.
\end{itemize}
Finally, we compute the degree of each of these summands of weight $4$ and thereby obtain the sought-after expression for $L_\ast\Gamma^4(V,n)$.
\end{proof}
More generally, we may extract from theorem \ref{thm-derived-p2}
the homogeneous component of an arbitrary given weight $d$. This
yields the following result.
\begin{corollary}\label{cor-derived-p2-Gd}
Let $\mathbb{k}$ be a field of characteristic $2$, $d$ be a positive integer and
$V$ be a finite dimensional $\mathbb{k}$-vector space. There exists an isomorphism of
strict polynomial functors:
$$L_i\Gamma^d(V,n)\simeq \bigoplus_{\delta}\:
\bigotimes_{r_1,\dots,r_n\ge 0} \Gamma^{\delta(r_1,\dots,r_n)}(V^{(r_1
+\dots+r_n)}[2^{r_2+\dots+r_n}+2^{r_3+\dots+r_n}+\dots+2^{r_n}
+1]) $$
where the sum is taken over all the maps $\delta:\mathbb{N}^n\to \mathbb{N}$ satisfying the
following two summability
conditions:
\begin{align*}
& \sum_{r_1,\dots,r_n\ge 0} \delta(r_1,\dots,r_n)2^{r_1+\dots +r_n}=d\;,& (1)\\
& \sum_{r_1,\dots,r_n\ge 0}
\delta(r_1,\dots,r_n)\left(2^{r_2+\dots+r_n}+2^{r_3+\dots+r_n}+\dots+2^{r_n}
+1\right)=i\;. &(2)
\end{align*}
\end{corollary}
\subsection{Proof of theorem \ref{thm-derived-p2}}
In this proof, we will constantly use the graded $\mathcal{P}_\mathbb{k}$-algebras $\Gamma(V^{(r)}[i])$ and $\Lambda(V^{(r)}[i])$. To keep formulas in a compact form and to handle the degrees and the twists in a confortable way, we denote these graded $\mathcal{P}_\mathbb{k}$-algebras respectively by $\Gamma^{(r,i]}(V)$ and $\Lambda^{(r,i]}(V)$. For example, the homogeneous summand of weight $dp^r$ and degree $di$ of $\Gamma^{(r,i]}(V)$ is $\Gamma^d(V^{(r)})$. And the homogeneous summand of weight $dp^r$ and degree $di+j$ of $L_*\Gamma^{(r,i]}(V,n)$ is equal to $L_{j+di}\Gamma^d(V^{(r)},n)$. With these notations, theorem
\ref{thm-derived-p2} appears the special case $r=i=0$ of the following theorem,
which is the statement that we actually prove.
\begin{theorem}\label{thm-derived-p2-prime}
Let $\mathbb{k}$ be a field of characteristic $2$, and let $V$ be a finite
dimensional $\mathbb{k}$-vector space. For all $n\ge 1$ and all $r,i\ge 0$, the graded
$\mathcal{P}_\mathbb{k}$-algebra $L_*\Gamma^{(r,i]}(V,n)$ is isomorphic to the tensor
product
\begin{align*}
\bigotimes_{r_1,\dots, r_n\ge 0}
\Gamma \left(V^{(r+r_1+\dots
+r_n)}[i2^{r_1+r_2+\dots+r_n}+2^{r_2+r_3+\dots+r_n}+\dots+2^{r_n}
+1]\right)\;.\end{align*}
\end{theorem}
\begin{proof}
{\bf Step 1: The quasi-trivial filtration of the divided power
algebra.}
Corollary \ref{cor-iterated-princ} yields a quasi-trivial filtration of
$\Gamma^{(r,i]}(V)$ with associated graded $\mathcal{P}_\mathbb{k}$-algebra
\begin{align}\mathrm{gr}\, \Gamma^{(r,i]}(V)\simeq \bigotimes_{s\ge 0}\Lambda^{(r+s,i2^s]}(V)\;.\label{eq-iso2-1}\end{align}
By proposition
\ref{prop-quasi-trivial-collapse}, derivation commutes with quasi-trivial filtrations and by the
Moreover, by the Eilenberg-Zilber theorem and the K\"unneth formula, derivation commutes with tensor products. We therefore have isomorphisms of graded $\mathcal{P}_\mathbb{k}$-algebras
\begin{align}\mathrm{gr}\, (L_* \Gamma^{(r,i]}(V,n))\simeq L_*(\mathrm{gr}\, \Gamma^{(r,i]})(V,n)\simeq \bigotimes_{s\ge 0}
L_*\Lambda^{(r+s,i2^s]}(V,n) \;.\label{eq-iso2-2}\end{align}
{\bf Step 2: D\'ecalage.}
Now we use the d\'ecalage formula of proposition \ref{dec}:
\begin{align*} L_*\Lambda^{(r+s,i2^s]}(V,n)\simeq
L_*\Gamma^{(r+s,i2^s+1]}(V,n-1)\; \end{align*}
to rewrite the right-hand side of isomorphism \eqref{eq-iso2-2}. In this way we
obtain for all
$n\ge 1$ an isomorphism:
\begin{align}\mathrm{gr}\, (L_*\Gamma^{(r,i]}(V,n))\simeq \bigotimes_{s\ge 0}
L_*\Gamma^{(r+s,i2^s+1]}(V,n-1)\label{eq-iso2-4} \end{align}
{\bf Step 3: Induction.}
We now prove theorem \ref{thm-derived-p2-prime} by induction on $n$. For $n=1$,
$L_*\Gamma^{(r+s,i2^s+1]}(V,0)$ is isomorphic to
$\Gamma^{(r+s,i2^s+1]}(V)$ so that theorem \ref{thm-derived-p2-prime} holds.
Let us assume that we have computed $L_*\Gamma^{(r+s,i2^s+1]}(V,n-1)$.
By inserting the formula giving $L_*\Gamma^{(r+s,i2^s+1]}(V,n-1)$ in the
right hand side of isomorphism \eqref{eq-iso2-4}, we obtain that the isomorphism
of theorem \ref{thm-derived-p2-prime} holds, up to a filtration.
To prove theorem \ref{thm-derived-p2-prime}, it remains to verify that the
filtration on the left-hand side of
of isomorphism \eqref{eq-iso2-4} is trivial. This is a direct consequence of the
following proposition.
\end{proof}
\begin{proposition}[{\cite[Prop 14.5]{antoine}}]\label{prop-split}
Let $\mathbb{k}$ be a field of characteristic $2$, and let $A(V)$ be a filtered
graded commutative $\mathcal{P}_\mathbb{k}$-algebra, whose summands $A^d_i(V)$ are finite
dimensional.
Assume that $\mathrm{gr}\, A(V)$ is isomorphic to a tensor product of graded
$\mathcal{P}_\mathbb{k}$-algebras of the form $\Gamma(V^{(r)}[i])$. Then there exists an
isomorphism of graded $\mathcal{P}_\mathbb{k}$-algebras $A(V)\simeq \mathrm{gr}\, A(V)$.
\end{proposition}
\begin{proof}
The proof is already given in \cite{antoine}, we sketch it here for the sake of completeness.
The starting point is the vanishing lemma \ref{prop-vanish}, which yields an isomorphism of graded strict polynomial functors $f:A(V)\xrightarrow[]{\simeq} \mathrm{gr}\, A(V)$. The isomorphism $f$ is not an isomorphism of algebras, but we can use it to build one in the following way.
Let us first recall that $\mathrm{gr}\, A(V)$ is exponential, hence $A(V)$ also is by the same reasoning as in lemma \ref{lm-triv}.
In particular, both algebras also have a coalgebra structure determined by the multiplication, as explained in section \ref{sec-classical}. One can check that the primitives of an exponential functor form an additive functor. So the isomorphism $f$ shows that the primitives of $A(V)$ form a direct summand of the primitives of $\mathrm{gr}\, A(V)$. Let us denote by $F(V)$ the primitive part of $\mathrm{gr}\, A(V)$, that is the direct sum of all the generators $V^{(r)}[i]$.
Then $f$ induces a monomorphism $f:A(V)\hookrightarrow F(V)$. But $\mathrm{gr}\, A(V)$ is the universal cofree coalgebra on $F(V)$, hence $f$ extends uniquely to a morphism of graded $\mathcal{P}_\mathbb{k}$-coalgebras $\overline{f}:A(V)\to \mathrm{gr}\, A(V)$. Since $\overline{f}$ induces an injection between the primitives of $A(V)$ and those of $\mathrm{gr}\, A(V)$, it is injective, and it is an isomorphism for dimension reasons. Finally, the coalgebra structure of an exponential functor uniquely determines its algebra structure and vice versa, so the isomorphism $\overline{f}$ is also a morphism of algebras.
\end{proof}
\subsection{The computation of $L_*\Gamma^d(V,1)$ over a field of odd
characteristic}
\begin{theorem}\label{thm-derived-podd}
Let $\mathbb{k}$ be a field of odd characteristic $p$, and let $V$ be a finite
dimensional $\mathbb{k}$-vector space. There is an isomorphism of
graded
$\mathcal{P}_\mathbb{k}$-algebras
\begin{align}
\label{lgammav1odd}
&L_*\Gamma(V,1)\simeq \bigotimes_{r\ge 1}
\Gamma \left(V^{(r)}[2]\right) \otimes \bigotimes_{r\ge 0}\Lambda\left(V^{(r)}[1]\right)\;.
\end{align}
\end{theorem}
\begin{remark}
The reader might find it surprising that derived
functors of $\Gamma$ are so different in characteristic $2$ from
what they are in odd characteristic. However, proposition
\ref{prop-iso-Nonnat} shows that theorem \ref{thm-derived-podd} is
valid in characteristic $2$ in a nonnatural way. And proposition
\ref{prop-filtration-Gamma} shows that theorem
\ref{thm-derived-podd} remains valid in characteristic $2$, up to a
filtration.
\end{remark}
\begin{proof}
The proof is similar to the characteristic $2$ case, i.e. to the proof of theorem \ref{thm-derived-p2-prime}.
{\bf Step 1: The quasi-trivial filtration of the divided power
algebra.}
Corollary \ref{cor-iterated-princ} yields a quasi-trivial filtration of
$\Gamma(V)$ with associated graded $\mathcal{P}_\mathbb{k}$-algebra
$\bigotimes_{s\ge 0}Q(V^{(s)})$.
Thus, by proposition
\ref{prop-quasi-trivial-collapse} and the Eilenberg-Zilber theorem, the graded $\mathcal{P}_\mathbb{k}$-algebra
$L_*\Gamma(V,1)$ is filtered and we have isomorphisms
\begin{align}\mathrm{gr}\, (L_* \Gamma(V,1))\simeq
L_*(\mathrm{gr}\, \Gamma)(V^{(s)},1)\simeq \bigotimes_{s\ge 0} L_* Q(V^{(s)},1)\;.\label{eq-isoodd-1bis}\end{align}
{\bf Step 2: Derived functors of truncated polynomials.}
In characteristic $2$, the derived functors of truncated polynomials (i.e. of exterior powers) can be computed by the d\'ecalage formula of proposition \ref{dec}.
This is not the case anymore in odd characteristic. The aim of step 2 is to prove an analogue of the d\'ecalage formula (for $n=1$), namely an isomorphism of
graded $\mathcal{P}_\mathbb{k}$-algebras:
\begin{align} L_* Q(V^{(s)},1)\simeq \Lambda(V^{(s)}[1])\otimes \Gamma(V^{(s+1)}[2])\;. \label{eq-decpodd}\end{align}
Since derivation commutes with precomposition by the Frobenius twist (cf. remark \ref{rk-derfrobtw}), it suffices to prove \eqref{eq-decpodd} for $s=0$. Let us denote by $A(V)$ the differential graded $\mathcal{P}_\mathbb{k}$-algebra $(S(V)\otimes\Lambda(V^{(1)}[1]),\partial)$ introduced in \eqref{resq}. Thus there is a quasi-isomorphism: $f:A(V)\to Q(V)$, where the target has zero differential. By deriving this quasi-isomorphism, we obtain a morphism of differential graded $\mathcal{P}_\mathbb{k}$-algebras:
\begin{align}\mathcal{N} A(V,1)\to \mathcal{N} Q(V,1)\;.\label{eq-qis-f}\end{align}
By definition, $\mathcal{N} A(V,1)$ is the total object of a bigraded $\mathcal{P}_\mathbb{k}$-algebra equipped with two differentials: the differential $\partial$ of the dg-$\mathcal{P}_\mathbb{k}$-algebra $A(V)$ and the differential $d$ coming from the simplicial structure. Thus we have two spectral sequences of graded $\mathcal{P}_\mathbb{k}$-algebras converging to the homology of $\mathcal{N} A(V,1)$. The first one is obtained by computing first the homology along the differential $\partial$, and secondly the homology along the differential $d$. Thus, its second page is given by
$ E^2_{0,t}= L_t Q(V,1)$ and $E^2_{s,t}=0$ if $s\ne 0$, and this proves that \eqref{eq-qis-f} is a quasi-isomorphism. The second spectral sequence is given by computing first the homology along the simplicial differential $d$. By the d\'ecalage formula of proposition \ref{dec}, its first page is given by
$$ 'E^1_{s,t} =
\Lambda^{t-s}(V)\otimes \Gamma^{s}(V^{(1)})\;,
$$
with the convention that $\Lambda^{t-s}(V)$ is zero if $t-s<0$. We observe that $ 'E^1_{s,t}$ is a strict polynomial functor of weight $t+(p-1)s$. Since the differentials of the spectral sequence are weight preserving maps: $d_i:\,'E^i_{s,t}\to \,'E^i_{s+i,t-i+1}$, they must be zero for lacunary reasons. Hence $'E^1_{s,t}=\,'E^\infty_{s,t}$. Thus we can conclude that there exists a filtration on the graded-$\mathcal{P}_\mathbb{k}$-algebra $L_*Q(V,1)$ such that the quasi-isomorphism \eqref{eq-qis-f} induces an isomorphism of graded $\mathcal{P}_\mathbb{k}$-algebras:
$$ \mathrm{gr}\, (L_* Q(V^{(s)},1))\simeq \Lambda(V^{(s)}[1])\otimes \Gamma(V^{(s+1)}[2])\;.$$
This is almost the formula \eqref{eq-decpodd} that we want to prove. To finish the proof, we have to get rid of the filtration on the left hand side. This follows from proposition \ref{prop-split-odd} below.
{\bf Step 3: Conclusion.}
The isomorphisms \eqref{eq-isoodd-1bis} and \eqref{eq-decpodd} together yield the formula of theorem \ref{thm-derived-podd} up to a filtration. To finish the proof of theorem \ref{thm-derived-podd}, we prove that this filtration splits by applying one more time proposition \ref{prop-split-odd}.
\end{proof}
In the course of the proof of theorem \ref{thm-derived-podd}, we have made use of the following statement whose proof is similar to the one of proposition \ref{prop-split}.
\begin{proposition}[\cite{antoine}]\label{prop-split-odd}
Let $\mathbb{k}$ be a field of odd characteristic, and let $A(V)$ be a filtered
graded commutative $\mathcal{P}_\mathbb{k}$-algebra, whose summands $A^d_i(V)$ are finite
dimensional.
If $\mathrm{gr}\, A(V)$ is isomorphic to a tensor product of graded
$\mathcal{P}_\mathbb{k}$-algebras of the form $\Gamma(V^{(r)}[2i])$ or $\Lambda(V^{(r)}[2i+1])$, then there exists an
isomorphism of graded $\mathcal{P}_\mathbb{k}$-algebras $A(V)\simeq \mathrm{gr}\, A(V)$.
\end{proposition}
\section{The first derived functors of $\Gamma$ over the integers}
\label{der1-gamma-z}
In this section, we work over the ground ring ${\mathbb{Z}}$.
In particular, we write $\Gamma^d$,
$\Lambda^d$ and $S^d$ for $\Gamma^d_{\mathbb{Z}}$, $S^d_{\mathbb{Z}}$ and $\Lambda^d_{\mathbb{Z}}$.
A generic free finitely generated abelian group will be denoted by the
letter `$A$', and we will denote by `$A/p$' the quotient $A/pA$.
Strict polynomial functors defined over prime fields will enter the picture under the following
form: if $F\in\mathcal{P}_{d,{\mathbb{F}_p}}$, the functor $A\mapsto F(A/p)$ lives in $\mathcal{P}_{d,{\mathbb{Z}}}$.
In particular, Frobenius twist functors yield strict polynomial functors $(A/p)^{(r)}$.
We most often drop the parentheses and simply denote those functors by $A/p^{\,(r)}$.
We denote by $\Gamma^d_{\mathbb{F}_p}$, $\Lambda^d_{\mathbb{F}_p}$ and $S^d_{\mathbb{F}_p}$ the symmetric,
exterior and divided powers functors considered as objects of $\mathcal{P}_{d,{\mathbb{F}_p}}$.
The goal of this section is to compute the derived functors $L_*\Gamma(A,1)$.
The main result in this section is theorem \ref{thm-calcul-LG-un-Z},
which gives a first description of these derived functors. In section \ref{descr-LGA1} we present and illustrate this result. The proof will be given in section \ref{sec-calcul-LG-un-Z}, while sections \ref{subsec-Koszuldiff} and \ref{subsec-SkewKoszuldiff} contain preliminary results which will be needed for this proof.
We will further elaborate on this theorem in section \ref{sec-der1-gamma-z-bis} to obtain other forms of the result, in the hope that one or another of these descriptions will be of direct use to the reader.
\subsection{The description of $L_*\Gamma(A,1)$}\label{descr-LGA1}
Recall from proposition \ref{lm-free-tors} that the graded $\mathcal{P}_{\mathbb{Z}}$-algebra $L_*\Gamma(A,1)$ decomposes as
$$L_*\Gamma(A,1)= \underbrace{\bigoplus_{d\ge 0} L_d\Gamma^d(A,1)}_{D(A)}\oplus\underbrace{\bigoplus_{0<i< d} L_i\Gamma^d(A,1)}_{I(A)}$$
where the diagonal subalgebra $D(A)$ is isomorphic to $\Lambda(A[1])$, and where the ideal $I(A)$ has values in torsion abelian groups.
Thus, if $_{(p)} L_*\Gamma(A,1)$ denotes the $p$-primary part of the abelian group $L_*\Gamma(A,1)$, there is an equality:
$$I(A)=\bigoplus_{\text{$p$ prime}} {_{(p)}} L_*\Gamma(A,1)\;.$$
To describe the graded $\mathcal{P}_{\mathbb{Z}}$-algebra $L_*\Gamma(A,1)$, it therefore suffices to compute the $p$-primary summands ${_{(p)}}L_*\Gamma(A,1)$ as graded $\mathcal{P}_{\mathbb{Z}}$-algebras (without unit), and to describe their $D(A)$-module structure
$D(A)\otimes {_{(p)} L_*\Gamma(A,1)}\to {_{(p)} L_*\Gamma(A,1)}$.
We shall describe the $p$-primary summands ${_{(p)}} L_*\Gamma(A,1)$ by the means of `Koszul kernel algebras' (for odd $p$) and `skew Koszul kernel algebras' (for $p=2$), which we now introduce.
\begin{definition}\label{def-dKos}
Let $p$ be a prime. (1) We denote by
$\partial_{\mathrm{Kos}}$ the unique differential of graded $\mathcal{P}_{\mathbb{Z}}$-algebras
on the connected algebra
$$\Lambda_{\mathbb{F}_p}(A/p[1])\otimes \bigotimes_{r\ge 1}
\left(\;\Gamma_{\mathbb{F}_p}\left(A/p^{(r)}[2]\right)\otimes
\Lambda_{\mathbb{F}_p}\left(A/p^{(r)}[1] \right)
\;\right) $$
sending the
generators $A/p^{(r)}[2]=\Gamma^1(A/p^{(r)}[2])$ identically to
$A/p^{(r)}[1]=\Lambda^1(A/p^{(r)}[1])$.
(2) The Koszul kernel algebra $K_{\mathbb{F}_p}(A/p)$ is the graded $\mathcal{P}_{\mathbb{Z}}$-subalgebra
of $L_*\Gamma_{\mathbb{F}_p}(A/p,1)$ consisting of the cycles
relative to the differential $\partial_{\mathrm{Kos}}$.
\end{definition}
The notation $\partial_{\mathrm{Kos}}$ and the name `Koszul kernel algebra' are justified by the fact (which will be proved in section \ref{subsec-Koszuldiff}) that the differential graded $\mathcal{P}_{\mathbb{Z}}$-algebra $(L_*\Gamma_{\mathbb{F}_p}(A/p,1),\partial_{\mathrm{Kos}})$ is the tensor product of all the algebras $\Gamma_{\mathbb{F}_p}\left(A/p^{(r)}[2]\right)\otimes \Lambda_{\mathbb{F}_p}\left(A/p^{(r)}[1] \right)$ with a Koszul differential and of the algebra $\Lambda_{\mathbb{F}_p}(A/p[1])$ with the zero differential.
We will prove in corollary \ref{cor-filtr-SK} that the Koszul kernel algebra $K_{\mathbb{F}_2}(A/2)$ is very closely related to the following skew Koszul kernel algebra $SK_{\mathbb{F}_2}(A/2)$.
\begin{definition}\label{def-dSKos} (1)
We let
$\partial_{\mathrm{SKos}}$ be the unique differential of graded
$\mathcal{P}_{\mathbb{Z}}$-algebras
on
$$\bigotimes_{r\ge 0}\Gamma_{\mathbb{F}_2}(A/2^{(r)}[1])$$
whose restriction to the each of the summands $\Gamma^2_{\mathbb{F}_2}(A/2^{(r)}[1])$, $r\ge 0$ is
equal to the Verschiebung map $\Gamma^2_{\mathbb{F}_2}(A/2^{(r)}[1])\stackrel{\pi}{\to} A/2^{(r+1)}[1]$.
(2) The skew Koszul kernel algebra $SK_{\mathbb{F}_2}(A/2)$ is the graded
$\mathcal{P}_{\mathbb{Z}}$-subalgebra
of $L_*\Gamma_{\mathbb{F}_2}(A/2,1)$ consisting of the
cycles relative to the differential $\partial_{\mathrm{SKos}}$.
\end{definition}
The following theorem provides our first description of the graded $\mathcal{P}_{\mathbb{Z}}$-algebra $L_*\Gamma(A,1)$. It will be proved in section \ref{sec-calcul-LG-un-Z}.
\begin{theorem}\label{thm-calcul-LG-un-Z}
Let $A$ be a finitely generated free abelian group.
\begin{enumerate}
\item[(i)] The diagonal algebra $D(A)$ is isomorphic to $\Lambda(A[1])$.
\item[(ii)] For any prime number $p$, the $p$-primary component $_{(p)} L_*\Gamma(A,1)$ is entirely $p$-torsion. In particular, there is an isomorphism of graded $\mathcal{P}_{\mathbb{Z}}$-algebras
$$L_*\Gamma(A,1)\otimes{\mathbb{F}_p}\simeq D(A)\otimes{\mathbb{F}_p}\;\oplus \; {_{(p)}} L_*\Gamma(A,1)\;.$$
\item[(iii)] There are isomorphisms of
graded $\mathcal{P}_{\mathbb{Z}}$-algebras:
\begin{align}
&L_*\Gamma(A,1)\otimes{\mathbb{F}_p}\simeq K_{\mathbb{F}_p}(A/p)\quad \text{ if $p$ is an odd prime,}\\
&L_*\Gamma(A,1)\otimes{\mathbb{F}_2}\simeq SK_{\mathbb{F}_2}(A/2)\quad\text{ if $p=2$.}
\end{align}
\end{enumerate}
\end{theorem}
\begin{remark}
The description of the $D(A)$-module structure on $_{(p)} L_*\Gamma(A,1)$ is contained in part (iii) of theorem \ref{thm-calcul-LG-un-Z}.
Indeed, part (ii) yields an isomorphism
$$D(A)\otimes {_{(p)}} L_*\Gamma(A,1)\simeq (D(A)\otimes{\mathbb{F}_p})\otimes {_{(p)}} L_*\Gamma(A,1)$$
so that the $D(A)$-module structure is obtained by restriction to $(D(A)\otimes{\mathbb{F}_p})\otimes {_{(p)}} L_*\Gamma(A,1)$ of the multiplication of $L_*\Gamma(A,1)\otimes{\mathbb{F}_p}$.
\end{remark}
The differentials $\partial_{\mathrm{Kos}}$ and $\partial_{\mathrm{SKos}}$ will be
very precisely described in sections \ref{subsec-Koszuldiff} and
\ref{subsec-SkewKoszuldiff} so that one can easily write down explicitly the
homogeneous component of weight $d$ of each of the differential graded algebras
$(L_*\Gamma_{\mathbb{F}_p}(A/p,1),\partial_{\mathrm{Kos}})$ and
$(L_*\Gamma_{\mathbb{F}_2}(A/2,1),\partial_{\mathrm{SKos}})$, and thereby compute
explicitly $L_*\Gamma^d(A,1)$ for a given $d$. More details regarding
the systematic description of $L_*\Gamma^d(A,1)$ will be given in section \ref{sec-der1-gamma-z-bis}.
For the moment, we simply provide the flavour of the explicit description of $L_*\Gamma^d(A,1)$ by writing down in detail the homogeneous summands of weight $d$ of $(L_*\Gamma_{\mathbb{F}_2}(A/2,1),\partial_{\mathrm{SKos}})$, for low $d$. The family of complexes of functors obtained here does not seem to have appeared elsewhere in the literature.
In weight $1$, the complex consists simply of a copy of $A/2$, placed in degree $1$. The complexes corresponding to the homogeneous summands of weight $d$ ranging from $2$ to $6$ are the following ones, where each differential can be characterized as the unique non-zero morphism having the specified strict polynomial functors as source and target:
\begin{align}\underbrace{\Gamma_{\mathbb{F}_2}^2(A/2)}_{\text{deg $2$}}\xrightarrow[]{f_2}
& \underbrace{A/2^{(1)}}_{\text{deg $1$}}\;,\\
\underbrace{\Gamma_{\mathbb{F}_2}^3(A/2)}_{\text{deg $3$}}\xrightarrow[]{f_3}
&\underbrace{A/2\otimes A/2^{(1)}}_{\text{deg $2$}}\;,\\
\underbrace{\Gamma_{\mathbb{F}_2}^4(A/2)}_{\text{deg $4$}}\xrightarrow[]{f_4}
&\underbrace{\Gamma_{\mathbb{F}_2}^2(A/2)\otimes A/2^{(1)}}_{\text{deg $3$}}\xrightarrow[]{g_4} \underbrace{\Gamma_{\mathbb{F}_2}^{2}(A/2^{(1)})}_{\text{deg $2$}}\xrightarrow[]{h_4} \underbrace{A/2^{(2)}}_{\text{deg $1$}}\;,\\
\intertext{}
\underbrace{\Gamma_{\mathbb{F}_2}^5(A/2)}_{\text{deg $5$}}\xrightarrow[]{f_5}
&\underbrace{\Gamma_{\mathbb{F}_2}^3(A/2)\otimes A/2^{(1)}}_{\text{deg $4$}}\xrightarrow[]{g_5} \underbrace{A/2\otimes\Gamma_{\mathbb{F}_2}^2(A/2^{(1)})}_{\text{deg $3$}}\xrightarrow[]{h_5} \underbrace{A/2\otimes A/2^{(2)}}_{\text{deg $2$}}\;,\\
\underbrace{\Gamma_{\mathbb{F}_2}^6(A/2)}_{\text{deg $6$}}\xrightarrow[]{f_6}
&\underbrace{\Gamma_{\mathbb{F}_2}^4(A/2)\otimes A/2^{(1)}}_{\text{deg $5$}}
\xrightarrow[]{g_6}
\underbrace{\Gamma_{\mathbb{F}_2}^2(A/2)\otimes\Gamma_{\mathbb{F}_2}^2(A/2^{(1)})}_{\text{deg $4$}}\nonumber\\
& \qquad \xrightarrow[]{\text{\footnotesize $\left[\begin{array}{c}h_6 \\h_6'\end{array}\right]$}}
\underbrace{\begin{array}{c}
\Gamma_{\mathbb{F}_2}^2(A/2)\otimes A/2^{(2)}\\
\oplus \Gamma_{\mathbb{F}_2}^3(A/2^{(1)})
\end{array}
}_{\text{degree $3$}}
\xrightarrow[]{k_6+f_3^{(1)}}\underbrace{A/2^{(1)}\otimes A/2^{(2)}}_{\text{degree $2$}}\;.
\end{align}
More explicitly, the morphisms $f_n$ above are defined as the composites
$$ \Gamma_{\mathbb{F}_2}^n(A/2)\to \Gamma_{\mathbb{F}_2}^{n-2}(A/2)\otimes \Gamma^2_{\mathbb{F}_2}(A/2)\to \Gamma_{\mathbb{F}_2}^{n-2}(A/2)\otimes A/2^{(1)}\;, $$
where the first map is induced by the comultiplication of $\Gamma_{\mathbb{F}_2}(A/2)$, and the second one by the Verschiebung morphism. The morphisms $g_n$ are defined as the composites
$$ \Gamma_{\mathbb{F}_2}^{n-2}(A/2)\otimes A/2^{(1)}\to \Gamma_{\mathbb{F}_2}^{n-4}(A/2)\otimes \Gamma^2_{\mathbb{F}_2}(A/2)\otimes A/2^{(1)}\to \Gamma_{\mathbb{F}_2}^{n-4}(A/2)\otimes \Gamma_{\mathbb{F}_2}^2(A/2^{(1)})\;, $$
where the first map is induced by the comultiplication of $\Gamma_{\mathbb{F}_2}(A/2)$, and the second one by the Verschiebung morphism and the multiplication $A/2^{(1)}\otimes A/2^{(1)}\to \Gamma^2_{\mathbb{F}_2}(A/2^{(1)})$. The maps $h_n$ and $k_n$ are induced by the Verschiebung morphism, and the map $h'_6$ is induced by the Verschiebung morphism and the multiplication of the algebra $\Gamma_{\mathbb{F}_2}(A/2)$.
\begin{example}\label{ex-2tG4}
The $2$-primary component of $L_*\Gamma^4(A,1)$ is given by
$$_{(2)}L_*\Gamma^4_{\mathbb{Z}}(A,1)= A/2^{(2)}[1]\;\oplus\;
\Lambda^2_{\mathbb{F}_2}(A/2^{(1)})[2]\;\oplus\;
\Phi^4(A)[3]\;,$$
where $\Phi^4(A)$ is the kernel of the morphism $g_4:\Gamma_{\mathbb{F}_2}^2(A/2)\otimes A/2^{(1)}\to\Lambda^2_{\mathbb{F}_2}(A/2^{(1)})$, and a term `$F(A)[i]$' means a copy of the functor $F(A)$ placed in degree $i$.
\end{example}
\subsection{The Koszul kernel algebra}\label{subsec-Koszuldiff}
The purpose of this section is to justify definition \ref{def-dKos}, that is to define the differential $\partial_{\mathrm{Kos}}$ and to study some of its properties.
\subsubsection{Koszul algebras}
Let $M$ be a projective finitely generated module over a commutative ring $R$.
We equip the graded $\mathcal{P}_R$-algebra
$\Gamma_R(M[2])\otimes\Lambda_R(M[1])$ with the
differential $d_{\mathrm{Kos}}$ defined as the composite
$$\Gamma^d_R(M)\otimes \Lambda^e_R(M)\to \Gamma^{d-1}_R(M)\otimes M\otimes
\Lambda^{e}_R(M)\to \Gamma^{d-1}_R(M)\otimes \Lambda^{e+1}_R(M) $$
where the first map is induced by the comultiplication in
$\Gamma_R(M)$ and the second one by the multiplication in
$\Lambda_R(M)$ (if $d=0$, the differential with source
$\Lambda^e_R(M)$ is zero). The resulting commutative
differential graded $\mathcal{P}_R$-algebra $(\Gamma_R(M[2])\otimes \Lambda_R(M[1]),
d_{\mathrm{Kos}})$ is called the Koszul algebra (on $M$).
\begin{proposition}\label{prop-homology-Koszul}
The homology of the Koszul algebra is equal to $R$ in degree zero
and is zero in all other degrees.
\end{proposition}
\begin{proof}
Using the fact that $\Gamma_R(M[2])\otimes \Lambda_R(M[1])$ is exponential (proceed as in the proof of proposition \ref{prop-qis-trunc}), the proof reduces to the case elementary $M=R$.
\end{proof}
\begin{remark}
The Koszul algebra is a particular case of more general constructions
\cite{illusie, FFSS}.
Its name is illustrated by the fact that its summand of weight
$d$ is the dual (via the duality $^\sharp$) of the more familiar Koszul
complex:
\begin{equation*}
\label{kos-v}
\Lambda^d_R(M)\to S^1_R(M)\otimes \Lambda^{d-1}_R(M)\to \dots\to S^{d-1}_R(M)\otimes
\Lambda^1_R(M)\to S^{d}_R(M)\;.
\end{equation*}
\end{remark}
\subsubsection{The Koszul differential on $L_*\Gamma_{{\mathbb{F}_p}}(A/p,1)$}
Let $p$ be a prime integer. To be concise, we denote by $\mathcal{L}_{\mathbb{F}_p}(A/p)$ the graded commutative $\mathcal{P}_{\mathbb{Z}}$-algebra
\begin{align}\mathcal{L}_{\mathbb{F}_p}(A/p):= \Lambda_{\mathbb{F}_p}(A/p [1])\,\otimes
\,\bigotimes_{r\ge
1}\left(\;\Gamma_{\mathbb{F}_p}\left(A/p^{(r)}[2]\right)\otimes
\Lambda_{\mathbb{F}_p}\left(A/p^{(r)}[1] \right)
\;\right)\;.\label{notation}\end{align}
If $p$ is odd,
$\mathcal{L}_{\mathbb{F}_p}(A/p)$ is isomorphic to the derived functors $L_*\Gamma_{{\mathbb{F}_p}}(A/p,1)$ by theorem \ref{thm-derived-podd},
but this is not the case for $p=2$. However, the algebra $\mathcal{L}_{\mathbb{F}_2}(A/2)$ will be considered later on.
We can endow $\mathcal{L}_{\mathbb{F}_p}(A/p)$ with the structure of a commutative
dg-$\mathcal{P}_{\mathbb{Z}}$-algebra in the following way.
Let us consider the factor $\Lambda_{\mathbb{F}_p}(A/p [1])$ as a differential graded
algebra with zero differential, and the other factors of \eqref{notation} as
Koszul algebras on the vector spaces $A/p^{(r)}$. We define
$(\mathcal{L}_{{\mathbb{F}_p}}(A/p),\partial_{\mathrm{Kos}})$ to be the tensor product of these differential graded
$\mathcal{P}_{\mathbb{Z}}$-algebras. By the K\"unneth formula and proposition \ref{prop-homology-Koszul} we have:
\begin{proposition}\label{prop-HKos}
The homology of $(\mathcal{L}_*\Gamma_{\mathbb{F}_p}(A/p,1),\partial_{\mathrm{Kos}})$ is isomorphic to the graded $\mathcal{P}_{\mathbb{Z}}$-algebra $\Lambda_{\mathbb{F}_p}(A/p[1])$.
\end{proposition}
We will now justify definition \ref{def-dKos}, that is we will characterize the differential $\partial_{\mathrm{Kos}}$ by its values on the generators $\Gamma^1(A/p^{(r)}[2])$ of $\mathcal{L}_{{\mathbb{F}_p}}(A/p)$. For this, we use the following tool.
\begin{lemma}[Uniqueness principle]\label{lm-unique}
Let $\mathbb{k}$ be a field of positive characteristic $p$, and let $V$ be a finite dimensional $\mathbb{k}$-vector space.
Let $A(V)$ be a graded commutative $\mathcal{P}_\mathbb{k}$-algebra of the form
$$ A(V)=\left(\bigotimes_{k}\Gamma_\mathbb{k}(V^{(r_k)}[i_k])\right)\otimes \left(\bigotimes_{\ell}\Lambda_\mathbb{k}(V^{(r_\ell)}[j_\ell])\right)\;.$$
We denote by $G(V)$ the graded functor
$$G(V)= \left(\bigoplus_{k}\Gamma^1_\mathbb{k}(V^{(r_k)}[i_k])\right)\oplus\left(\bigoplus_{\ell}\Lambda^1_\mathbb{k}(V^{(r_\ell)}[j_\ell])\right)\;,$$
and by $\pi:A(V)\twoheadrightarrow G(V)$ the surjection induced by the projections of $\Gamma_\mathbb{k}(V^{(r_k)}[i_k])$, resp. $\Lambda_\mathbb{k}(V^{(r_\ell)}[j_\ell])$ onto $V^{(r_k)}[i_k]$, resp. $V^{(r_\ell)}[j_\ell]$. Then the map $\partial\mapsto \pi\circ\partial$ induces a injection
between the set of differentials on the graded $\mathcal{P}_\mathbb{k}$-algebra $A(V)$ and the set of degree $-1$ morphisms of graded functors $A(V)\to G(V)$:
$$\begin{array}{ccc}
\left\{\text{
\begin{tabular}{c}
differentials\\
on $A(V)$
\end{tabular}
}\right\}
& \hookrightarrow & \mathrm{Hom}_{-1}(A(V),G(V))\\
\partial & \mapsto & \pi\circ\partial
\end{array}\;.$$
\end{lemma}
\begin{proof}[Proof of lemma \ref{lm-unique}]
Since $A(V)$ is exponential and graded commutative, the multiplication in $A(V)$ induces an isomorphism of bialgebras $\varphi:A(V)\otimes A(W)\simeq A(V\oplus W)$. A morphism $\partial:A(V)\to A(V)$ of degree $-1$ is a derivation if and only if $\partial$ commutes with $\varphi$, which holds if and only if $\partial^\sharp$ commutes with $\varphi^\sharp$, which holds if and only if $\partial^\sharp: A(V)^\sharp\to A(V)^\sharp$ is a derivation.
By duality, lemma \ref{lm-unique} is therefore equivalent to the statement that differentials on $A^\sharp(V)$ are completely determined by their restriction $G^{\sharp}(V)\to A^\sharp(V)$. The latter statement is true since $A^\sharp(V)$ is the free graded commutative algebra on $G^{\sharp}(V)$.
\end{proof}
\begin{proposition}\label{prop-charact}
Let $\partial$ be a differential on the graded $\mathcal{P}_{\mathbb{Z}}$-algebra $\mathcal{L}_{{\mathbb{F}_p}}(A/p)$.
\begin{enumerate}
\item[(i)] The differential $\partial$ is determined by its restriction to the summands $\Gamma^1_{\mathbb{F}_p}(A/p^{(r)}[2])$, for all $r\ge 1$.
\item[(ii)] The restriction of $\partial$ sends the summand $\Gamma^1_{\mathbb{F}_p}(A/p^{(r)}[2])$ into $\Lambda^1_{\mathbb{F}_p}(A/p^{(1)}[1])$.
\end{enumerate}
\end{proposition}
\begin{proof}
To prove (ii), we observe that by definition, $\partial$ must send the summand $\Gamma^1(A/p^{(r)}[2])$ into the homogeneous summand of degree $1$ and weight $p^r$, which is equal to $\Lambda^1(A/p^{(1)}[1])$.
We now prove (i). By lemma \ref{lm-unique}, $\partial$ is uniquely determined by the morphism
\begin{align}\pi\circ\partial_{\mathrm{Kos}}:\mathcal{L}_{{\mathbb{F}_p}}(A/p)\to A/p[1]\oplus \bigoplus_{r\ge 0}(A/p^{(r)}[2]\oplus A/p^{(r)}[1])\;. \label{eq-pid}\end{align}
If we denote by $\pi_r$, resp $\pi'_r$, resp. $\pi'_0$, the canonical projection of the right-hand side of isomorphism \eqref{eq-pid} onto the summands $A/p^{(r)}[2]$, $A/p^{(r)}[1]$, and $A/p[1]$ respectively, then
$$\pi\circ \partial=\pi_0'\circ(\pi\circ \partial)+\sum \pi'_r\circ(\pi\circ \partial)+\sum\pi_r\circ(\pi\circ \partial)\;.$$
The source of the morphism $\pi_0'\circ (\pi\circ \partial)$ is the direct summand of weight $1$ and degree $2$ of $\mathcal{L}_{{\mathbb{F}_p}}(A/p)$, which is equal to zero, so that $\pi_0'\circ(\pi\circ \partial)=0$.
The source of $\pi'_r\circ(\pi\circ \partial)$ is the summand of weight $p^r$ and degree $2$ of $\mathcal{L}_{{\mathbb{F}_p}}(A/p)$, which equals
$\Gamma^1_{\mathbb{F}_p}(A/p^{(r)}[2])$ if $p$ is odd, and $\Gamma^1_{\mathbb{F}_2}(A/2^{(r)}[2])\oplus \Lambda^2_{\mathbb{F}_2}(A/2^{(r-1)}[1])$ if $p=2$. Now an easy computation shows that $\mathrm{Hom}_{\mathcal{P}_{{\mathbb{F}_2}}}(\Lambda^2(V^{(r-1)}), V^{(r)})$ is zero so that for any prime $p$, $\pi'_r\circ(\pi\circ \partial)$ is determined by the restriction of $\partial$ to the direct summand $\Gamma^1_{\mathbb{F}_p}(A/p^{(r)}[2])$.
Similarly, the source of $\pi_r\circ(\pi\circ \partial)$ is the summand of weight $p^r$ and degree $3$ of $\mathcal{L}_{{\mathbb{F}_p}}(A/p)$. The latter is equal to $0$ is $p\ge 5$, to $\Lambda^3_{\mathbb{F}_3}(A/3A^{(r-1)}[1])$ is $p=3$, and to
$$ \Gamma^1_{\mathbb{F}_2}(A/2^{(r-1)}[2])\otimes \Lambda^1_{\mathbb{F}_2}(A/2^{(r-1)}[1]) \,\oplus\, \Lambda^2_{\mathbb{F}_2}(A/2^{(r-2)}[1])\otimes \Lambda^1_{\mathbb{F}_2}(A/2^{(r-1)}[1]) $$
if $p=2$.
An easy computation shows that there are no nonzero morphisms from such functors to the functor $A/p^{(r)}$ so that $\pi_r\circ(\pi\circ \partial)=0$.
It follows that $\pi\circ \partial$ (hence $\partial$) is completely determined by the restriction of $\partial$ to the summands $\Gamma^1_{\mathbb{F}_p}(A/p^{(r)}[2])$.
\end{proof}
\begin{corollary}\label{cor-partialKos}
The morphism $\partial_{\mathrm{Kos}}$ is the unique differential on the graded
$\mathcal{P}_{\mathbb{Z}}$-algebra $\mathcal{L}_{{\mathbb{F}_p}}(A/p)$ which sends the generators
$\Gamma^1_{\mathbb{F}_p}(A/p^{(r)}[2])$ identically to
$\Lambda^1_{\mathbb{F}_p}(A/p[1])$.
\end{corollary}
The following variant of corollary \ref{cor-partialKos} will be useful in the proof of theorem \ref{thm-calcul-LG-un-Z}.
\begin{corollary}\label{cor-versal-prop}
Let $\partial$ be a differential of the graded $\mathcal{P}_{\mathbb{Z}}$-algebra $\mathcal{L}_{{\mathbb{F}_p}}(A/p)$. Assume that all the summands $\Lambda^1(A/p^{(r)}[1])$, $r\ge 1$, lie in the image of $\partial$. Then there exists an isomorphism of dg-$\mathcal{P}_{\mathbb{Z}}$-algebras:
$$(\mathcal{L}_{{\mathbb{F}_p}}(A/p),\partial)\simeq (\mathcal{L}_{{\mathbb{F}_p}}(A/p),\partial_{\mathrm{Kos}})\;. $$
\end{corollary}
\begin{proof}
The only morphisms of strict polynomial functors $f:A/p^{(r)}\to A/p^{(r)}$ are the scalar multiples of the identity. By proposition \ref{prop-charact}, $\delta$ is completely determined by its restrictions to $\Gamma^1(A/p^{(r)}[2])$. These restrictions must be nonzero in order to ensure that the expressions $\Lambda^1(A/p^{(r)}[1])$ lie in the image of $\delta$, so they are of the form $\lambda_r{\mathrm{Id}}$ with $\lambda_r\in \mathbb{F}_p^{\ast}$. Now the required isomorphism is induced by the automorphism of the graded $\mathcal{P}_{\mathbb{Z}}$-algebra $\mathcal{L}_{{\mathbb{F}_p}}(A/p)$, which sends the generators $A/p[1]$ and $A/p^{(r)}[1]$, $r\ge 1$ identically to themselves and whose restrictions to the generators $A/p^{(r)}[2]$, $r\ge 1$ are equal to $\lambda_r{\mathrm{Id}}$.
\end{proof}
\subsection{The skew Koszul kernel algebra}\label{subsec-SkewKoszuldiff}
The purpose of this section is to justify definition \ref{def-dSKos}, that is to define the differential $\partial_{\mathrm{SKos}}$ and to study some of its properties. We also prove that the skew Koszul kernel algebra is, up to a filtration, isomorphic to the Koszul kernel algebra introduced in definition \ref{def-dKos}.
\subsubsection{The skew Koszul algebras in characteristic $2$}
Let $\mathbb{k}$ be a field of characteristic $2$, and let $V$ be a finite dimensional $\mathbb{k}$ vector space.
Consider the graded $\mathcal{P}_\mathbb{k}$-algebra
$\Gamma_\mathbb{k}(V[1])\otimes\Gamma_\mathbb{k}(V^{(1)}[1])$, equipped
with the differential $d_{\mathrm{SKos}}$ defined as a composite:
$$\Gamma^d_\mathbb{k}(V)\otimes \Gamma^e_\mathbb{k}(V^{(1)})\to \Gamma^{d-2}_\mathbb{k}(V)\otimes
\Gamma^2_\mathbb{k}(V)\otimes
\Gamma^{e}_\mathbb{k}(V^{(1)})\to \Gamma^{d-2}_\mathbb{k}(V)\otimes \Gamma^{e+1}_\mathbb{k}(V^{(1)})
$$
where the first map is induced by the comultiplication of
$\Gamma_\mathbb{k}(V)$ and the second one is induced by the Verschiebung map
$\Gamma^{2}_\mathbb{k}(V)\twoheadrightarrow V^{(1)}$ and the multiplication
of $\Gamma_\mathbb{k}(V^{(1)})$ (if $d\le 1$, the differential with source
$\Gamma^0_\mathbb{k}(V)\otimes \Gamma^e_\mathbb{k}(V^{(1)})$ is zero). The resulting commutative
differential graded $\mathcal{P}_\mathbb{k}$-algebra $(\Gamma_\mathbb{k}(V[1])\otimes
\Gamma_\mathbb{k}(V^{(1)}[1]),
d_{\mathrm{SKos}})$ will be called the \emph{skew Koszul algebra} (on $V$). This name
is justified by the following result, which is a differential graded algebra version of corollary \ref{cor-qtf}.
\begin{proposition}\label{prop-qtfiltr-SK}
Let $V$ be a finite dimensional vector space over a field $\mathbb{k}$ of characteristic $2$.
The tensor product of the principal filtrations of $\Gamma_\mathbb{k}(V[1])$ and of $\Gamma_\mathbb{k}(V^{(1)}[1])$ yields a quasi-trivial filtration of the skew Koszul algebra, and the associated graded object is isomorphic to the dg-$\mathcal{P}_\mathbb{k}$-algebra:
$$\Lambda_\mathbb{k}(V[1])\otimes \left(\Gamma_\mathbb{k}(V^{(1)}[2])\otimes
\Lambda_\mathbb{k}(V^{(1)}[1])\right)\otimes \Gamma_\mathbb{k}(V^{(2)}[2])\quad,\quad{\mathrm{Id}}_{\Lambda_\mathbb{k}(V[1])}\otimes
d_{\mathrm{Kos}}\otimes {\mathrm{Id}}_{\Gamma_\mathbb{k}(V^{(2)}[2])}\;.$$
\end{proposition}
\begin{proof}
Let us denote by $(A(V),d_{\mathrm{SKos}})$ the skew Koszul algebra and by $(B(V),d)$ the tensor product of $(\Lambda_\mathbb{k}(V[1]),0)$, of the Koszul algebra on a generator $V^{(1)}[1]$, and of $(\Gamma_\mathbb{k}(V^{(2)}[2],0)$.
The tensor products of the principal filtration of $\Gamma_\mathbb{k}(V[1])$ and $\Gamma_\mathbb{k}(V^{(1)}[1])$ coincides with the adic filtration of $A(V)$ relative to the ideal $J(V)$ generated by $\Gamma^1_\mathbb{k}(V[1])\oplus \Gamma^{1}_\mathbb{k}(V^{(1)}[1])$. By definition, the image of the differential $d_{\mathrm{SKos}}$ is contained in the image of the multiplication $A(V)\otimes V^{(1)}[2]\to A(V)$. In particular, $d_{\mathrm{SKos}}$ sends $J(V)$ to $J(V)$ so the $J(V)$-adic filtration yields a filtration on the dg-$\mathcal{P}_\mathbb{k}$-algebra
$(A(V),d_{\mathrm{SKos}})$.
By Proposition \ref{prop-filtration-Gamma}, we have an isomorphism of graded $\mathcal{P}_\mathbb{k}$-algebras $\mathrm{gr}\, A(V)\simeq B(V)$. We have to prove that $\mathrm{gr}\,(d_{\mathrm{SKos}})=d$. By proposition \ref{prop-charact} it suffices to show that the restriction of $\mathrm{gr}\,(d_{\mathrm{SKos}})$ to the direct summand $\Gamma^{1}_\mathbb{k}(V^{(1)}[2])=V^{(1)}[2]$ sends this generator identically to $V^{(1)}[1]=\Lambda^1_\mathbb{k}(V^{(1)}[1])$. To prove this, we write down explicitly the homogeneous weight $2$ component of the $J(V)$-adic filtration. We have:
$$A(V)_2=\Gamma^2_\mathbb{k}(V[1])\oplus \Gamma^1_\mathbb{k}(V^{(1)}[1])\;,\quad J(V)_2= \Lambda^2_\mathbb{k}(V[1])\;,$$
and the component of weight $2$ of the power $J(V)^n$ is zero for all $n\ge 2$. The restriction of $d_{\mathrm{SKos}}$ to $\Gamma^2_\mathbb{k}(V[1])$ is the Verschiebung map so that $\mathrm{gr}\,(d_{\mathrm{SKos}}): \Lambda^2_\mathbb{k}(V[1])\oplus V^{(1)}[2]\to V^{(1)}[1]$ is zero on the summand $\Lambda^2_\mathbb{k}(V[1])$ and maps $ V^{(1)}[2]$ identically to $V^{(1)}[1]$.
The
fact that $(A(\mathbb{k}),d_{\mathrm{SKos}})$ is isomorphic to $(B(\mathbb{k}),d)$ is easily proved by direct inspection.
\end{proof}
\subsubsection{The skew Koszul differential on $L_*\Gamma_{{\mathbb{F}_2}}(A/2,1)$}
We are going to define a differential $\partial_{\mathrm{SKos}}$ on the graded
$\mathcal{P}_{\mathbb{Z}}$-algebra
$$L_*\Gamma_{\mathbb{F}_2}(A/2,1)\simeq \bigotimes_{r\ge
0}\Gamma_{\mathbb{F}_2}(A/2^{(r)}[1])\;.$$
To do this, we consider for all $r\ge 0$ the factor
$\Gamma_{\mathbb{F}_2}(A/2^{(r)}[1])\otimes \Gamma_{\mathbb{F}_2}(A/2^{(r+1)}[1])$ as the skew Koszul
algebra (on the vector space $A/2^{(r)}$). Tensoring by identities on the left
and the right, this defines a differential $\partial_r$ on
$L_*\Gamma_{\mathbb{F}_2}(A/2,1)$. In other words, each element in
$L_*\Gamma_{\mathbb{F}_2}(A/2,1)$ can be written as a finite tensor product of
elements $x_i\in \Gamma_{{\mathbb{F}_2}}(A/2^{(i)}[1])$ and $\partial_r$ is given by:
$$\partial_r(x_0\otimes \dots \otimes x_r\otimes x_{r+1}\otimes \dots \otimes
x_n)=x_0\otimes \dots \otimes d_{\mathrm{SKos}}(x_r\otimes x_{r+1})\otimes
\dots \otimes x_n\;.$$
\begin{lemma}The differentials $\partial_r$ commute with each other.
\end{lemma}
\begin{proof}
We have to check that $\partial_i\circ\partial_j=\partial_j\circ\partial_i$.
Since the $\partial_i$ are differentials of algebras, we can use the
exponential property to reduce the proof to the trivial case in which $A/2$ is one-dimensional.
\end{proof}
We define the skew Koszul differential $\partial_{\mathrm{SKos}}$ as the sum:
$\partial_{\mathrm{SKos}}=\sum_{r\ge 0}\partial_r $
(this infinite sum reduces to a finite one on each summand of given degree
and weight). We will now justify definition \ref{def-dSKos}, that is characterize
$\partial_{\mathrm{SKos}}$ is by its value on the summands $\Gamma^2_{\mathbb{F}_2}(A/2^{(r)})$.
The following proposition is proved in the same way as proposition \ref{prop-charact}.
\begin{proposition}\label{prop-charact-p2}
Let $\partial$ be a differential of the graded $\mathcal{P}_{\mathbb{Z}}$-algebra $L_*\Gamma_{{\mathbb{F}_2}}(A/2,1)$.
\begin{enumerate}
\item[(i)] The differential $\partial$ is determined by its restriction to the summands $\Gamma^2_{{\mathbb{F}_2}}(A/2^{(r)}[1])$, for $r\ge 0$.
\item[(ii)] The restriction of $\partial$ sends the summand $\Gamma^2_{{\mathbb{F}_2}}(A/2^{(r)}[1])$ into $\Gamma^1_{{\mathbb{F}_2}}(A/2^{(r+1)}[1])$.
\end{enumerate}
\end{proposition}
\begin{corollary}
The morphism $\partial_{\mathrm{SKos}}$ is the unique differential on the graded
$\mathcal{P}_{\mathbb{Z}}$-algebra
$$L_*\Gamma_{\mathbb{F}_2}(A/2,1)=\bigotimes_{r\ge 0}\Gamma_{\mathbb{F}_2}(A/2^{(r)}[1])$$
whose restriction to the summands $\Gamma^2_{\mathbb{F}_2}(A/2^{(r)}[1])$, $r\ge 0$
equals the Verschiebung map $\Gamma^2_{\mathbb{F}_2}(A/2^{(r)}[1])\to A/2^{(r+1)}[1]$.
\end{corollary}
We also have the analogue of corollary \ref{cor-versal-prop}. Since the Verschiebung is the only nonzero morphism $\Gamma^2_{\mathbb{F}_2}(A/2^{(r)})\to A/2^{(r+1)}$, this characteristic $2$ analogue yields a slightly stronger statement.
\begin{corollary}\label{cor-versal-prop-p2}
The differential $\partial_{\mathrm{SKos}}$ is the unique differential on the graded
$\mathcal{P}_{\mathbb{Z}}$-algebra $L_*\Gamma_{\mathbb{F}_2}(A/2,1)$ whose image contains all the generators $\Gamma^1_{{\mathbb{F}_2}}(A/2^{(r)}[1])$ for $r\ge 1$.
\end{corollary}
\subsubsection{Koszul versus skew Koszul kernels} The definition of the Koszul differential on $\mathcal{L}_{\mathbb{F}_2}(A/2)$ and of the Skew Koszul differential on $L_*\Gamma_{\mathbb{F}_2}(A/2,1)$ are completely parallel. We now compare explicitly these two constructions.
The following proposition follows directly from proposition \ref{prop-qtfiltr-SK}.
\begin{proposition}\label{prop-qt-SK}
The tensor product of the principal filtrations on the graded $\mathcal{P}_{\mathbb{Z}}$-algebras $\Gamma_{\mathbb{F}_2}(A/2^{(r)}[1])$, $r\ge 1$, yields a quasi-trivial filtration on $(L_*\Gamma_{{\mathbb{F}_2}}(A/2,1),\partial_{\mathrm{SKos}})$, whose associated graded object is the differential graded $\mathcal{P}_{\mathbb{Z}}$-algebra $(\mathcal{L}_{{\mathbb{F}_2}}(A/2,1),\partial_{\mathrm{Kos}})$.
\end{proposition}
\begin{corollary}
\label{cor-HSKos}
The homology of $(L_*\Gamma_{\mathbb{F}_2}(A/2,1),\partial_{\mathrm{SKos}})$ is isomorphic to the graded $\mathcal{P}_{\mathbb{Z}}$-algebra $\Lambda_{\mathbb{F}_2}(A/2[1])$.
\end{corollary}
\begin{proof}[Proof of corollary \ref{cor-HSKos}]
The morphism of algebras $\Lambda_{\mathbb{F}_2}(A/2)\hookrightarrow \Gamma_{\mathbb{F}_2}(A/2)$ induces a morphism of algebras
$\Lambda_{\mathbb{F}_2}(A/2[1])\hookrightarrow L_*\Gamma_{\mathbb{F}_2}(A/2,1)\;.$
The image of this morphism consists of cycles, and since $(L_*\Gamma_{\mathbb{F}_2}(A/2,1))^d_i$ is zero for $i>d$ there is an injective morphism
\begin{align}\Lambda_{\mathbb{F}_2}(A/2[1])\hookrightarrow H_*(L_*\Gamma_{\mathbb{F}_2}(A/2,1),\partial_{\mathrm{SKos}})\;.
\label{eq-inj}\end{align}
We want to prove that the morphism \eqref{eq-inj} is an isomorphism. For this purpose, it suffices to check that its source and its target have the same dimensions in each summand of a given weight and degree. But proposition \ref{prop-qt-SK} and lemma \ref{lm-triv}(c) yield a \emph{non functorial} isomorphism of differential graded algebras
which preserves the weights:
$$(L_*\Gamma_{\mathbb{F}_2}(A/2,1),\partial_{\mathrm{SKos}})\simeq
(\mathcal{L}_{\mathbb{F}_2}(A/2),\partial_{\mathrm{Kos}}) \;.$$
By proposition \ref{prop-descr-Kos}, the homology of $(\mathcal{L}_{\mathbb{F}_2}(A/2),\partial_{\mathrm{Kos}})$ is isomorphic to $\Lambda_{\mathbb{F}_2}(A/2[1])$. Hence the dimensions of the source and the target of morphism \eqref{eq-inj} agree.
\end{proof}
We also mention another consequence of proposition \ref{prop-qt-SK}, which shows that the skew Koszul kernel algebra is very close to the Koszul kernel algebra. It follows directly from the properties of quasi-trivial filtrations (lemma \ref{lm-triv} and proposition \ref{compat-Z}).
\begin{corollary}\label{cor-filtr-SK}
Let $A$ be a finitely generated free abelian group. The choice of a basis of $A$ determines a \emph{non functorial} isomorphism of algebras which preserves the weights:
$$SK_{\mathbb{F}_2}(A/2)\simeq K_{\mathbb{F}_2}(A/2)\;.$$
Moreover, there is a filtration of the graded $\mathcal{P}_{\mathbb{Z}}$ algebra $SK_{\mathbb{F}_2}(A/2)$ and a functorial isomorphism of graded $\mathcal{P}_{\mathbb{Z}}$-algebras
$$\mathrm{gr}\, SK_{\mathbb{F}_2}(A/2)\simeq K_{\mathbb{F}_2}(A/2)\;.$$
\end{corollary}
\subsection{Proof of theorem \ref{thm-calcul-LG-un-Z}}
\label{sec-calcul-LG-un-Z} Theorem
\ref{thm-calcul-LG-un-Z}(i) is already known by proposition \ref{lm-free-tors}.
In this section, we will prove theorem \ref{thm-calcul-LG-un-Z}(ii) and (iii)
in corollary \ref{cor-proof-11}. The proof proceeds by running the Bockstein spectral sequence.
We first need the following result.
\begin{lemma}\label{prop-calc-G1}
Let $r$ be a positive integer, and let $p$ be a prime integer.
Then the $p$-primary part of the functor $L_1\Gamma^{p^r}(A,1)$ is equal to
$A/p^{(r)}$.
Moreover, the morphism induced by mod $p$ reduction yields an
isomorphism:
$${_{(p)}}L_1\Gamma^d(A,1)=L_1\Gamma^d(A,1)\otimes{\mathbb{F}_p} \xrightarrow[]{\simeq}
L_1\Gamma^d_{\mathbb{F}_p}(A/p,1)\;.$$
\end{lemma}
\begin{proof}
The complex $\mathcal{N}\Gamma^{p^r}(A,1)\otimes {\mathbb{F}_p}$ is isomorphic to the
complex $\mathcal{N}\Gamma^{p^r}_{\mathbb{F}_p}(A/p,1)$, and $\mathcal{N}\Gamma^{p^r}(A,1)$
is zero in degree zero. Thus, from the long exact sequence associated to the
short exact sequence of complexes
$$0\to \mathcal{N}\Gamma^{p^r}(A,1)\xrightarrow[]{\times p} \mathcal{N}\Gamma^{p^r}(A,1)\to
\mathcal{N}\Gamma_{\mathbb{F}_p}^{p^r}(A/p,1)\to 0$$
we obtain an isomorphism
$L_1\Gamma^{p^r}(A,1)\otimes{\mathbb{F}_p} \simeq
L_1\Gamma^{p^r}_{\mathbb{F}_p}(A/p,1)$.
But we know by \cite[Korollar 10.2]{d-p} that the $p$-primary part of
$L_1\Gamma^d(A,1)$
only contains $p$-torsion. The result follows.
\end{proof}
Let $p$ be a prime number. The Bockstein spectral sequence \cite[5.9.9]{weibel}
is a device which enables one to recover the $p$-primary part of homology of a
complex $C$ of free abelian groups from the homology of the complex
$C\otimes{\mathbb{F}_p}$ (provided one is able to compute the
differentials in the spectral sequence).
We consider the Bockstein spectral sequence for the complex $C=\mathcal{N}\Gamma(A,1)$, which computes the derived functors of $\Gamma(A)$. Since $\mathcal{N}\Gamma(A,1)\otimes {\mathbb{F}_p}$ is isomorphic to $\mathcal{N}\Gamma_{\mathbb{F}_p}(A/p,1)$, the Bockstein spectral sequence is a spectral sequence of graded $\mathcal{P}_{\mathbb{Z}}$-algebras, starting at page $0$
$$ E^0(A)_i=L_i\Gamma_{\mathbb{F}_p}(A/p,1)\Longrightarrow \left(L_i\Gamma(A,1)/\mathrm{Torsion}\right)\otimes {\mathbb{F}_p} = \Lambda_{\mathbb{F}_p}(A/p[1])\, ,$$
with differentials $d^r: E^r(A)_i\to E^r(A)_{i-1}$.
The differentials in this spectral sequence are related to the $p$-primary torsion of $L_*\Gamma(A,1)$ in the following way.
\begin{enumerate}
\item[(i)] Given an integer $r\ge 0$, the $p$-torsion of $L_*\Gamma(A,1)$ is
killed by multiplication by $p^r$ if and only if $E^{r}(A)=E^\infty(A)$.
\item[(ii)] The injective map induced by the mod $p$ reduction of the complex
$L\Gamma(A,1)$
\begin{align}L_*\Gamma(A,1)\otimes{\mathbb{F}_p}\to L_*\Gamma_{\mathbb{F}_p}(A/p,1)
=E^0(A)_*\;,\label{eq-psi}\end{align}
has an image contained in the permanent cycles of the spectral sequence, and the
image of the
$p$-torsion part of $L_*\Gamma(A,1)$ lies in the image of $d^0$.
\end{enumerate}
We will now completely compute this Bockstein spectral sequence.
\begin{proposition}\label{prop-Bockstein}
The Bockstein spectral sequence degenerates at the first page: $E^1(A)=E^\infty(A)$, and the kernel $d^0$ equals $K_{\mathbb{F}_p}(A/p)$ if $p$ is odd, and $SK_{\mathbb{F}_2}(A/2)$ if $p=2$.
\end{proposition}
\begin{proof}
By condition (ii) and proposition \ref{prop-calc-G1}, we know that the
image of $d^0$ must contain the generators $A/p^{(1)}[1]$. Thus by corollary
\ref{cor-versal-prop} in odd characteristic $p$ or by corollary
\ref{cor-versal-prop-p2} in characteristic $p=2$, we know that the
dg-$\mathcal{P}_{\mathbb{Z}}$-algebra is isomorphic to
$(L_*\Gamma_{\mathbb{F}_p}(A/p,1),\partial_{\mathrm{Kos}})$ if $p$ is odd and to
$(L_*\Gamma_{\mathbb{F}_2}(A/2,1),\partial_{\mathrm{SKos}})$ if $p=2$. This proves the
statement about the kernel of $d^0$.
Both dg-$\mathcal{P}_{\mathbb{Z}}$ algebras have homology equal to $\Lambda_{\mathbb{F}_p}(A/p[1])$,
whence the degeneration of the spectral sequence at the $E^1$ page.
\end{proof}
\begin{corollary}[Theorem \ref{thm-calcul-LG-un-Z}(ii)-(iii)]
\label{cor-proof-11}
For all primes $p$, the $p$-primary component of $L_*\Gamma(A,1)$ consists only of $p$-torsion, and there are isomorphisms of
graded $\mathcal{P}_{\mathbb{Z}}$-algebras:
\begin{align}
&L_*\Gamma(A,1)\otimes{\mathbb{F}_p}\simeq K_{\mathbb{F}_p}(A/p)\quad \text{ if $p$ is an odd prime,}\\
&L_*\Gamma(A,1)\otimes{\mathbb{F}_2}\simeq SK_{\mathbb{F}_2}(A/2)\quad\text{ if $p=2$.}
\end{align}
\end{corollary}
\begin{proof}
Since the Bockstein spectral sequence degenerates at the page $E^1$, the $p$-primary component of $L_\ast\Gamma (A,1)$ is entirely $p$-torsion.
In particular, if $0<i<d$ the $p$-primary component $_{{(p)}} L_i\Gamma^d(A,1)$ is isomorphic to $L_i\Gamma^d(A,1)\otimes{\mathbb{F}_p}$. Hence the map \eqref{eq-psi} sends $_{(p)} L_i\Gamma^d(A,1)$ to the weight $d$ and degree $i$ boundaries of $d^0$. The homogeneous part of $E^1(A)$ of degree $i$ and weight $d$ is zero (since this is the case at the $E^\infty(A)$ level). It follows that these boundaries are equal to the weight $d$ and degree $i$ cycles of $d^0$.
\end{proof}
\section{Further descriptions of the derived functors of $\Gamma$ over the integers}\label{sec-der1-gamma-z-bis}
We keep the notations of section \ref{der1-gamma-z}, in particular the notation $\mathcal{L}_{\mathbb{F}_p}(A/p)$ from \eqref{notation}. The purpose of this section is to give a more detailed description of the derived functors $L_i\Gamma^d(A,1)$ than the one given in theorem \ref{thm-calcul-LG-un-Z}.
By theorem \ref{thm-calcul-LG-un-Z}, we know that for all $d>0$:
$$L_i\Gamma^d(A,1)\simeq \begin{cases}
0 & \text{ if $i>d$ or $i=0$,}\\
\Lambda^d(A) & \text{ if $i=d$,}\\
\displaystyle SK_{\mathbb{F}_2}(A/2)^d_i\,\oplus \,\bigoplus_{\text{$p$ odd prime}} K_{\mathbb{F}_p}(A/p)^d_i & \text{ if $0<i<d$,}
\end{cases}$$
where the functors $K_{\mathbb{F}_p}(A/p)^d_i$ and $SK_{\mathbb{F}_2}(A/2)^d_i$ denote the homogeneous components of weight $d$ and degree $i$ of the (skew) Koszul kernel algebra.
Thus, to give a more detailed description of $L_i\Gamma^d(A,1)$, we need to describe these functors more precisely.
The following description of $K_{\mathbb{F}_p}(A/p)^d_i$ follows directly from the definition of the Koszul kernel algebra and the computation of its homology in proposition \ref{prop-HKos}.
\begin{proposition}\label{prop-descr-Kos}
Let $p$ be a prime. Then
\begin{enumerate}
\item[(0)] $K_{\mathbb{F}_p}(A/p)^d_d = \Lambda_{\mathbb{F}_p}^d(A/p)$
\item[(1)] $K_{\mathbb{F}_p}(A/p)^d_i = 0$ if $i<d<p$ or if $d-p+1<i<d$.
\item[(2)] The nontrivial component $K_{\mathbb{F}_p}(A/p)^d_i$ of highest degree $i<d$ is given by
$$K_{\mathbb{F}_p}(A/p)^d_{d-p+1}=\begin{cases} \Lambda^{d-p}_{\mathbb{F}_p}(A/p)\otimes A/p^{(1)} & \text{ if $p\ne 2$,}\\
\bigoplus_{1\le k\le d/p} \Lambda^{d-kp}_{\mathbb{F}_2}(A/2)\otimes \Gamma_{{\mathbb{F}_2}}^k(A/2^{(1)}) & \text{ if $p= 2$.}
\end{cases}
$$
\item[(3)] For any positive integer $i<d-p+1$, $K_{\mathbb{F}_p}(A/p)^d_i$ can be equally described as:
\begin{enumerate}
\item the kernel of the map $\partial_{\mathrm{Kos}}:(\mathcal{L}_{\mathbb{F}_p}(A/p))^d_i\to (\mathcal{L}_{\mathbb{F}_p}(A/p))^d_{i-1}$,
\item the image of the map $\partial_{\mathrm{Kos}}:(\mathcal{L}_{\mathbb{F}_p}(A/p))^d_{i+1}\to (\mathcal{L}_{\mathbb{F}_p}(A/p))^d_i$,
\item the cokernel of the map $\partial_{\mathrm{Kos}}:(\mathcal{L}_{\mathbb{F}_p}(A/p))^d_{i+2}\to (\mathcal{L}_{\mathbb{F}_p}(A/p))^d_{i+1}$.
\end{enumerate}
\end{enumerate}
\end{proposition}
\begin{proof}
Let us describe explicitly the terms of highest degrees in the homogeneous component of weight $d$ of $\mathcal{L}_{\mathbb{F}_p}(A/p)$ (an expression such as $F(A)\,[k]$ means a copy of the functor $F(A)$ placed in degree $k$ and the exterior powers with a negative weight are by convention equal to zero, so that the following direct sums are actually finite):
\begin{align*}\mathcal{L}_{\mathbb{F}_p}(A/p) = &\;\Lambda^d_{\mathbb{F}_p}(A/p)\,[d]\;\;
\oplus \bigoplus_{k\ge 1} \Lambda^{d-kp}_{\mathbb{F}_p}(A/p)\otimes \Gamma_{\mathbb{F}_p}^{k}(A/p^{(1)})\,[d-kp+2k]\;\\
&\oplus \;\bigoplus_{k\ge 1} \Lambda^{d-kp}_{\mathbb{F}_p}(A/p)\otimes \Gamma^{k-1}_{\mathbb{F}_p}(A/p^{(1)})\otimes \Lambda^1_{\mathbb{F}_p}(A/p^{(1)})\,[d-kp+2k-1]\;\\
&\oplus \;\text{terms of degree $< d-kp+2k-1$.}\end{align*}
The differential $\partial_{\mathrm{Kos}}$ vanishes on the term $\Lambda^d_{\mathbb{F}_p}(A/p)[d]$, and maps the rest of the first line injectively into the terms of the second line. This proves (0) and (1). Moreover, by proposition \ref{prop-HKos} the homogeneous component of weight $d$ of the complex $(\mathcal{L}_{\mathbb{F}_p}(A/p),\partial_{\mathrm{Kos}})$ is exact in degrees less than $d$. Thus, the degree $d-p+1$ component of the kernel of $\partial_{\mathrm{Kos}}$ is given by some terms from the first line. If $p=2$ all the terms of the second line have degree $d$, so all of them but $\Lambda^d_{\mathbb{F}_p}(A/p)[d]$ contribute to the degree $d-1$ of the kernel of $\partial_{\mathrm{Kos}}$, whereas the only contribution is the one of $\Lambda^{d-p}_{\mathbb{F}_p}(A/p)\otimes A/p^{(1)}$ if $p\geq 3$.
Finally, (3) follows from the decomposition of the homogeneous component of weight $d$ of the complex $(\mathcal{L}_{\mathbb{F}_p}(A/p),\partial_{\mathrm{Kos}})$ into short exact sequences.
\end{proof}
The following description of the expressions $SK_{\mathbb{F}_p}(A/2)^d_i$ is proved exactly in the same fashion as proposition \ref{prop-descr-Kos}.
\begin{proposition}\label{prop-descr-SKos} The following holds.
\begin{enumerate}
\item[(0)] $SK_{\mathbb{F}_2}(A/2)^d_d = \Lambda_{\mathbb{F}_2}^d(A/2)$
\item[(1)] $SK_{\mathbb{F}_2}(A/2)^d_{d-1}=\Phi^d(A)$, where $\Phi^1(A)=0$, $\Phi^2(A)=A/2^{(1)}$, $\Phi^3(A)=A/2\otimes A/2^{(1)}$ and for $d\ge 4$, $\Phi^d(A)$ can be equivalently described as:
\begin{enumerate}
\item the kernel of the unique nonzero morphism (induced by the comultiplication of $\Gamma_{\mathbb{F}_2}(A/2)$, the Verschiebung morphism $\Gamma^2_{\mathbb{F}_2}(A/2)\twoheadrightarrow A/2^{(1)}$, and the multiplication $(A/2^{(1)})^{\otimes 2}\to \Gamma_{\mathbb{F}_2}^2(A/2^{(1)})$)
$$\Gamma^{d-2}_{\mathbb{F}_2}(A/2)\otimes A/2^{(1)}\to\Gamma^{d-4}_{\mathbb{F}_2}(A/2)\otimes\Gamma^2_{\mathbb{F}_2}(A/2^{(1)})\;.$$
\item the image of the unique nonzero morphism (induced by the comultiplication of $\Gamma_{\mathbb{F}_2}(A/2)$ and the Verschiebung morphism $\Gamma^2_{\mathbb{F}_2}(A/2)\twoheadrightarrow A/2^{(1)}$)
$$\Gamma^d_{\mathbb{F}_2}(A/2)\to\Gamma^{d-2}_{\mathbb{F}_2}(A/2)\otimes(A/2)^{(1)}\;.$$
\item the cokernel of the canonical inclusion
$$\Lambda^d_{\mathbb{F}_2}(A/2)\hookrightarrow \Gamma^d_{\mathbb{F}_2}(A/2)\;.$$
\end{enumerate}
\item[(2)] More generally, for $i<d-1$, $SK_{\mathbb{F}_2}(A/2)^d_i$ can be equivalently described as:
\begin{enumerate}
\item the kernel of the map $\partial_{\mathrm{SKos}}:(L_*\Gamma_{\mathbb{F}_2}(A/2,1))^d_i\to (L_*\Gamma_{\mathbb{F}_2}(A/2,1))^d_{i-1}$,
\item the image of the map $\partial_{\mathrm{SKos}}:(L_*\Gamma_{\mathbb{F}_2}(A/2,1))^d_{i+1}\to (L_*\Gamma_{\mathbb{F}_2}(A/2,1))^d_i$,
\item the cokernel of the map $\partial_{\mathrm{SKos}}:(L_*\Gamma_{\mathbb{F}_2}(A/2,1))^d_{i+2}\to (L_*\Gamma_{\mathbb{F}_2}(A/2,1))^d_{i+1}$.
\end{enumerate}
\end{enumerate}
\end{proposition}
Propositions \ref{prop-descr-Kos} and \ref{prop-descr-SKos} yield descriptions of the $L_i\Gamma^d(A,1)$ as kernels, images or cokernels of some very explicit complexes. However, most of these kernels, cokernels or images yield `new' functors which are not direct sums of some familiar functors.
For example, by corollary \ref{cor-filtr-SK}, the functors $\Phi^d(A)$ are filtered, with associated graded object equal to the functor $K_{\mathbb{F}_2}(A/2)^d_{d-1}$ described in proposition \ref{prop-descr-Kos}. In particular for $d=4$ there is a short exact sequence, which is non split (as we prove it in proposition \ref{prop-nonsplit1}).
\begin{equation}
\label{seq-phi4}
0\to \Lambda^2_{\mathbb{F}_2}(A/2)\otimes A/2^{(1)}\to \Phi^4(A)\to \Gamma^2_{\mathbb{F}_2}(A/2^{(1)})\to 0\;.\end{equation}
To describe the derived functors $L_i\Gamma^d(A,1)$ in more familiar terms (in terms of classical Weyl functors), we will describe them up to a filtration in theorem \ref{thm-expr} below.
By corollary \ref{cor-filtr-SK}, the description of the derived functors $L_i\Gamma^d(A,1)$ up to a filtration reduces to describe the Koszul kernel algebras $K_{\mathbb{F}_p}(A/p)$ up to a filtration for all primes $p$. These $\mathcal{P}_{\mathbb{Z}}$-graded algebras are the cycles of the tensor product of the algebra $\Lambda_{\mathbb{F}_p}(A/p[1])$ with trivial differential, and of the Koszul algebras $(\Lambda_{\mathbb{F}_p}(A/p^{(r)}[1])\otimes\Gamma_{\mathbb{F}_p}(A/p^{(r)}[2]),d_{\mathrm{Kos}})$ for $r\ge 1$. To obtain a description of the cycles of such a tensor product of complexes, we will use the following result from elementary algebra.
\begin{lemma}\label{lm-filtr-cplx}
Let $C_\bullet(A)$ and $D_\bullet(A)$ be finite exact complexes of functors with values in finite dimensional ${\mathbb{F}_p}$-vector spaces, and denote by $\delta$ the differential on the tensor product $C_\bullet(A)\otimes D_\bullet(A)$. There is a natural filtration of $\ker\delta$, whose associated graded object is given by
$$\mathrm{gr}\,((\ker\delta)_k)\simeq \bigoplus_{i+j=k-1}\mathrm{Ker}\,d^C_{i}\otimes \mathrm{Ker}\,d^D_{j}\,\oplus\, \bigoplus_{i+j=k}\mathrm{Ker}\,d^C_{i}\otimes \mathrm{Ker}\,d^D_{j}$$
\end{lemma}
\begin{proof}
The proof is very much in the spirit of the quasi-trivial filtrations of section \ref{qt-fil}.
Let us denote by $X_k(A)$ the kernel of the differential $d^C_k:C_k(A)\to C_{k-1}(A)$.
We define a two-step decreasing filtration of each functor $C_k(A)$ by
\begin{align*}
F^{k+1}C_k(A) =0\;, \quad F^kC_k(A) = X_k(A)\;,\quad F^{k-1}C_k(A) = C_k(A)\;.
\end{align*}
Then $C_\bullet(A)$ becomes a filtered complex, and the associated graded complex is isomorphic to the split exact complex:
\begin{align} \dots\to\underbrace{X_k(A)\oplus X_{k-1}(A)}_{\text{degree $k$}}\xrightarrow[]{(0,{\mathrm{Id}})}\underbrace{X_{k-1}(A)\oplus X_{k-2}(A)}_{\text{degree $k-1$}}\to \dots \label{eq-splitcplx}\end{align}
Moreover there is a \emph{non functorial} isomorphism between $C_\bullet(A)$ and the split complex \eqref{eq-splitcplx}. The complex $D_\bullet(A)$ is filtered similarly. The tensor product of these two filtrations yields a filtration of the complex $C_\bullet(A)\otimes D_\bullet(A)$, whose associated graded complex $(\mathrm{gr}\,(C_\bullet(A)\otimes D_\bullet(A)),\mathrm{gr}\, \delta)$ is the tensor product of two split complexes. Moreover, the filtration of the complex $C_\bullet(A)\otimes D_\bullet(A)$ induces a filtration of its cycles $\mathrm{Ker}\, \delta$, defined by the rule $F^i(\mathrm{Ker}\, \delta):= \left(\mathrm{Ker}\, \delta\right)\cap F^i\left(C_\bullet(A)\otimes D_\bullet(A)\right)$, and there is a canonical injection:
\begin{align}\mathrm{gr}\, (\mathrm{Ker}\, \delta)\hookrightarrow \mathrm{Ker}\,(\mathrm{gr}\, \delta)\label{eq-can-gr}.\end{align}
Now the complex $(\mathrm{gr}\,(C_\bullet(A)\otimes D_\bullet(A)),\mathrm{gr}\, \delta)$ is non-functorially isomorphic to the complex $(C_\bullet(A)\otimes D_\bullet(A),\delta)$, hence the ranks of the differentials $\delta$ and $\mathrm{gr}\, \delta$ are equal. Thus the source and the target of the canonical morphism \eqref{eq-can-gr} have the same dimension, and the morphism \eqref{eq-can-gr} is an isomorphism.
It follows that, up to a filtration, $\mathrm{Ker}\, \delta$ is the kernel of the differential of the tensor product of the split complex \eqref{eq-splitcplx} and the similar complex for $D_\bullet(A)$. The formula of lemma \ref{lm-filtr-cplx} follows.
\end{proof}
The following result follows from lemma \ref{lm-filtr-cplx} by induction on $n$:
\begin{lemma}\label{lm-filtr-n-cplx}
Let $C^i_\bullet(A)$, $1\le i\le n$, be a family of finite exact complexes of functors with values in finite dimensional ${\mathbb{F}_p}$-vector spaces, and let us denote by $\delta$ the induced differential on the $n$-fold tensor product $\bigotimes_{i=1}^n C^i_\bullet(A)$. There is a natural filtration of $\ker\delta$, whose associated graded object is given by
$$\mathrm{gr}\,((\ker\delta)_k)\simeq \bigoplus_{j=0}^{n-1}\;\bigoplus_{i_1+\dots+i_n=k-j}\; \left(\;\mathrm{Ker}\,d^{C^1}_{i_1}\otimes\dots \otimes \mathrm{Ker}\,d^{C^n}_{i_n}\;\right)^{\oplus \binom{n-1}{j}}\;.$$
\end{lemma}
To state our description of the derived functors $L_i\Gamma^d(A,1)$, we need to introduce the following combinatorial device. For a fixed prime number $p$, we can consider all possible decompositions of a positive integer $k$ as a sum $k=\sum_i k_i p^{r_i}$ where the $k_i$ are positive integers, and the $r_i$ are distinct positive integers. Since the $r_i$ are positive, the existence of such a decomposition implies that $p$ divides $k$. There might however be many such decompositions. Each of these may be identified with a finite sequence of pairs of integers $((r_1,k_1),\dots, (r_i,k_i),\dots)$ satisfying the following conditions:
\begin{enumerate}
\item[(i)] The integers $r_i$ and $k_i$ are positive.
\item[(ii)] For all $i$, $r_i<r_{i+1}$.
\item[(iii)] $\sum k_ip^{r_i}=k$.
\end{enumerate}
We denote by $\mathrm{Decomp}(p,k)$ the set of all such finite sequences.
\begin{theorem}\label{thm-expr}
Let $V$ be a finite dimensional ${\mathbb{F}_p}$-vector space and let $W_{k,{\mathbb{F}_p}}^d(V)\in\mathcal{P}_{d,{\mathbb{F}_p}}$ denote the kernel of the Koszul differential
$$d_{\mathrm{Kos}}: \Gamma^k_{\mathbb{F}_p}(V)\otimes \Lambda^{d-k}_{\mathbb{F}_p}(V)\to \Gamma^{k-1}_{\mathbb{F}_p}(V)\otimes \Lambda_{\mathbb{F}_p}^{d-k+1}(V)\;.$$
By convention, $W_{k,{\mathbb{F}_p}}^d(V)$ is zero if $k>d$ or if $k<0$.
For $0<i<d$, the $p$-primary part of $L_i\Gamma^d(A,1)$ is \emph{up to a filtration} isomorphic to the direct sum:
$$\bigoplus_{k=0}^{d} \bigoplus_{
\text{\footnotesize
$
\begin{array}{c}
((r_1,k_1)\dots (r_n,k_n))\\
\in\mathrm{Decomp}(p,k)
\end{array}
$
}
}
\bigoplus_{j=0}^{n-1}\;
\bigoplus_{
\text{
\footnotesize
$
\begin{array}{c}
i_1+\dots+i_n\\
=i+k-d-j
\end{array}
$
}
}\;\left(\Lambda^{d-k}_{\mathbb{F}_p}(A/p)\otimes\bigotimes_{\ell=1}^n W^{k_\ell}_{i_\ell-k_\ell,{\mathbb{F}_p}}(A/p^{(r_\ell)})\right)^{\,\oplus\binom{n-1}{j}}. $$
\end{theorem}
\begin{proof}
Up to a filtration, the $p$-primary part of $L_i\Gamma^d(A,1)$ is given by the functor $K_{\mathbb{F}_p}(A/p)^d_i$. Let us denote by $\kappa^d_\bullet(V)$ the complex given by the homogeneous component of weight $d$ of the Koszul algebra $(\Lambda_{\mathbb{F}_p}(V[1])\otimes\Gamma_{\mathbb{F}_p}(V[2]),d_{\mathrm{Kos}})$. By definition, $K_{\mathbb{F}_p}(A/p)^d_i$ is equal to the cycles of degree $i$ in the complex
$$ \bigoplus_{k=0}^{d} \Lambda^{d-k}_{\mathbb{F}_p}(A/p[1])\otimes \left(\bigoplus_{
(r_i,k_i)\in\mathrm{Decomp}(p,k)
} \kappa^{k_1}_\bullet(A/p^{(r_1)})\otimes \dots \otimes \kappa^{k_n}_\bullet(A/p^{(r_n)})\right)\;, $$
with the convention that direct sums over empty sets are equal to zero.
The left-hand factors $\Lambda^{d-k}_{\mathbb{F}_p}(A/p[1])$ are concentrated in degree $d-k$. We therefore need to compute the cycles of degree $i+k-d$ of the right-hand factors. The homogeneous summand of degree $i_\ell$ of the cycles of the complex $\kappa^{k_\ell}_\bullet(A/p^{(r_\ell)})$ is $W^{k_\ell}_{i_\ell-k_\ell,{\mathbb{F}_p}}(A/p^{(r_\ell)})$.
Given a sequence $((r_1,k_1),\dots, (r_n,k_n))$, we use lemma \ref{lm-filtr-n-cplx} to verify that the cycles of degree $i+k-d$ in the tensor product $\kappa^{k_1}_\bullet(A/p^{(r_1)})\otimes \dots \otimes \kappa^{k_n}_\bullet(A/p^{(r_n)})$ are (up to a filtration) isomorphic to
$$\bigoplus_{j=0}^{n-1}\;\bigoplus_{i_1+\dots+i_n=(i+k-d)-j}\;
W^{k_1}_{i_1-k_1,{\mathbb{F}_p}}(A/p^{(r_1)})\otimes \cdots\otimes W^{k_n}_{i_n-k_n,{\mathbb{F}_p}}(A/p^{(r_n)})^{\;\oplus\binom{n-1}{j}}.$$
This proves the formula of theorem \ref{thm-expr}.
\end{proof}
The notation $W_{k,{\mathbb{F}_p}}^d(V)$ has been chosen in order to remind the reader that these kernels of the Koszul complex are well-known to representation theorists as Weyl functors \cite{ABW,BB}. To be more specific, recall that given a partition $\lambda=(\lambda_1,\dots,\lambda_n)$ of $d$ and a commutative ring $R$, the Weyl functor $W_\lambda(M)\in \mathcal{P}_{d,R}$ is the dual of the Schur functor associated to the partition $\lambda$. The functor $W_\lambda(M)$ may be defined as the image of a certain composite map
$$\Gamma^{\lambda_1}(M)\otimes\dots\otimes\Gamma^{\lambda_n}(M)\hookrightarrow M^{\otimes d}\xrightarrow[]{\sigma_{\lambda'}}M^{\otimes d}\twoheadrightarrow \Lambda^{\lambda'_1}(M)\otimes\dots\otimes\Lambda^{\lambda'_k}(M)\;,$$
where $\sigma_{\lambda'}$ is a specific combinatorial isomorphism and $\lambda'=(\lambda'_1,\dots,\lambda'_k)$ is the partition dual to $\lambda$. The Weyl functor $W_\lambda(M)$ is denoted by $K_{\lambda}(M)$ in \cite{ABW,BB}. For example $K_{(d)}(M)=\Gamma^d_R(M)$ and $K_{(1^d)}(M)=\Lambda^d_R(M)$. In particular, our functors $W^d_{k,{\mathbb{F}_p}}(V)$ are the Weyl functors associated to hook partitions, i.e. there is an isomorphism: $W^d_{k,{\mathbb{F}_p}}(V)\simeq W_{(k+1,1^{d-k-1})}(V)$ (
see e.g. \cite[Chap III.1]{BB} for more details).
\section{The maximal filtration on $\Gamma(A)$}\label{sec-max}
In this section, we work over the ground ring ${\mathbb{Z}}$. We use the same notations as in section \ref{der1-gamma-z}
In particular, we write $\Gamma^d$,
$\Lambda^d$ and $S^d$ for $\Gamma^d_{\mathbb{Z}}$, $S^d_{\mathbb{Z}}$ and $\Lambda^d_{\mathbb{Z}}$.
A generic free finitely generated abelian group will be denoted by the
letter `$A$'. We will denote by `$A/p$' the quotient $A/pA$
and we will abuse notations and write `$A/p^{(r)}$' instead of $(A/pA)^{(r)}$ for shortening.
The purpose in this section is to introduce the maximal filtration of $\Gamma(A)$, and the associated spectral sequence, which will be our main tool for the computation of the derived functors $L_*\Gamma^d(A,n)$ for low $d$ and all $n$. As a warm-up, we finish the section by running the spectral sequence in the baby cases $d=2$ and $d=3$.
\subsection{The maximal filtration}
We denote by $\mathcal{J}(A)$ the augmentation ideal of the divided power algebra $\Gamma(A)$:
$$\mathcal{J}(A): = \Gamma^{>0}(A) = \ker (\Gamma(A) \rightarrow \Gamma^0(A)={\mathbb{Z}})\;.$$
The adic filtration relative to $\mathcal{J}(A)$ will be called the maximal filtration on $\Gamma(A)$ (even though $\mathcal{J}(A)$ is not strictly speaking a maximal ideal in this algebra).
The associated graded object is the
$\mathcal{P}_{\mathbb{Z}}$-algebra:
\[
\mathrm{gr}\, \Gamma(A):= \bigoplus_{i\ge 0} \mathrm{gr}\,_{-i} (\Gamma(A)) = \bigoplus_{i\ge
0}\mathcal{J}(A)^i/\mathcal{J}(A)^{i+1}\;.
\]
\begin{remark}
The maximal filtration is different from the principal filtration on $\Gamma(A)$ defined in section \ref{filt-1} (compare the definition of $J(A)$ with lemma \ref{prop-cokernel}).
\end{remark}
Restricting ourselves to the weight $d$ component of $\Gamma(A)$, the principal filtration yields a filtration
of $\Gamma^d(A)$, with
\[
F_{-i}\,\Gamma^d(A) := \mathcal{J}(A)^i \cap \Gamma^d(A).
\]
and associated graded components
\[gr_{-i}\Gamma^d(A):= F_{-i}\Gamma^d(A)/F_{-i-1}\Gamma^d(A).\]
By definition, the terms of the filtration can be concretely described as follows \begin{equation}
\label{deffi1}
F_{-i}\Gamma^d(A) =
\mathrm{Im}( \bigoplus \Gamma^{k_1}(A) \otimes \dots \otimes \Gamma^{k_i}(A) \longrightarrow \Gamma^m(A)) \end{equation}
where the sum is taken over all $i$-tuples of positive integers $(k_1,\dots,k_i)$ whose sum equals $d$. In particular $F_{-i}\Gamma^d(A)=0$ for $i>d$, so that the filtration is bounded and
\begin{equation}
\label{deffi3}gr_{-d}\,\Gamma^d(A) = F_{-d}\Gamma^d(A) = \mathrm{Im}(A^{\otimes \,d} \longrightarrow \Gamma^d(A)) =
S^d(A),\end{equation}
where the inclusion of $S^d(A)$ into $\Gamma^d(A)$ is determined by the commutative algebra structure on $\Gamma(A)$.
It is also easy to identify the graded component $gr_{-1}\Gamma^d(A)$.
\begin{lemma}\label{lm-gr-1}
For any free abelian group $A$, $\mathrm{gr}\,_{-1}\Gamma^d(A)=0$ if $d$ is not a power of a prime $p$, and $\mathrm{gr}\,_{-1}\Gamma^d(A)=A/p^{(r)}$ if $d=p^r$.
\end{lemma}
\begin{proof}
The composite $\Gamma^d(A)\xrightarrow[]{comult} \Gamma^{k}(A)\otimes \Gamma^{\ell}(A)\xrightarrow[]{mult} \Gamma^d(A)$ equals the multiplication by $\binom{d}{k}$, so by \eqref{deffi1} the integral torsion of $gr_{-1}\Gamma^d(A)$ is bounded by the g.c.d. of the binomials $\binom{d}{k}$, $0<k<d$. This g.c.d. equals $p$ if $d$ is a power of a prime $p$, and $1$ otherwise. Hence
$gr_{-1}\Gamma^d(A) = 0$ if $d$ is not a power of a prime, and is a ${\mathbb{F}_p}$-vector space if $d=p^r$. In the latter case, by base change $gr_{-1}\Gamma^d(A)$ identifies with the cokernel of the map $\bigoplus \Gamma^k_{\mathbb{F}_p}(A/pA)\otimes \Gamma^\ell_{\mathbb{F}_p}(A/pA)\xrightarrow[]{mult} \Gamma^d_{\mathbb{F}_p}(A/pA)$, where the sum is taken over all pairs $(k,\ell)$ of positive integers with $k+\ell=d$. Whence the result.
\end{proof}
For $1<i<d$, the graded components $gr_{-i}\Gamma^d(A)$ are more complicated and their description involves
new classes of functors. We will use the strict polynomial functors $\sigma_{(1,n)}(V)$, defined for $n\ge 2$ on the category of ${\mathbb{F}_2}$-vector spaces by
\begin{align}
&\label{sig12} \sigma_{(1,n)}(V)=\mathrm{Coker}\;\left(\Lambda^2_{\mathbb{F}_2}(V^{(1)})\otimes S^{n-2}_{\mathbb{F}_2}(V)\xrightarrow[]{u} V^{(1)}\otimes S^{n}_{\mathbb{F}_2}(V)\right)
\end{align}
where $u: (x\wedge y)\otimes z \mapsto x \otimes (y^2z) - y \otimes (x^2z)$. The map $u$ appears in the resolution of truncated polynomials introduced in section \ref{subsec-trunc}. In particular, proposition \ref{prop-qis-trunc} implies that
$\sigma_{(1,n)}(V)$ lives in a characteristic $2$ exact sequence
\begin{align}\label{sig12bis}0\to \sigma_{(1,n)}(V)\to S^{n+2}_{\mathbb{F}_2}(V)\to \Lambda_{\mathbb{F}_2}^{n+2}(V)\to 0\;.\end{align}
\begin{remark}
The functors $\sigma_{(1,n)}(V)$ belong to a family of functors $^p\sigma^e_{(\alpha, \beta)}(V)$ introduced by F. Jean in \cite[Appendix A]{jean}.
\end{remark}
\begin{lemma}\label{lm-gr-d+1}
Let $d\ge 4$. For any free abelian group $A$,
$\mathrm{gr}\,_{-d+1}\Gamma^d(A)\simeq \sigma_{(1,d-2)}(A/2)$.
\end{lemma}
\begin{proof}
By definition of the maximal filtration, there is a commutative diagram with exact rows (the exactness of the upper row follows from lemma \ref{lm-gr-1} for $d=2$):
$$\xymatrix{
S^{d-2}(A)\otimes S^2(A)\ar[r]^-{mult}\ar@{->>}[d]^-{mult} &
S^{d-2}(A)\otimes\Gamma^2(A)\ar[r]\ar@{->>}[d]^-{mult} &
S^{d-2}(A)\otimes A/2^{(1)}\ar[r]&0\\
F_{-d}\Gamma^d(A)\ar[r]&F_{-d+1}\Gamma^d(A)\ar[r]&gr_{-d+1}\Gamma^d(A)\ar[r]&0
}.$$
Hence the surjective morphism $S^{d-2}(A)\otimes\Gamma^2(A)\twoheadrightarrow gr_{-d+1}\Gamma^d(A)$ factors into a surjective morphism
\begin{align} S^{d-2}(A)\otimes A/2^{(1)}\twoheadrightarrow gr_{-d+1}\Gamma^d(A).\label{eq-surj}\end{align}
Now we check that the composite of the morphism $u$ appearing in \eqref{sig12} and the morphism \eqref{eq-surj} is zero. For this, let us take $x\in S^{d-4}(A)$, $y,z\in A$ and let us denote by $\overline{y}$, resp. $\overline{z}$ the image of $\gamma_2(y)$, resp $\gamma_2(z)$ in $A/2^{(1)}$. Then it suffices to check that the surjection \eqref{eq-surj} sends $xy^2\otimes \overline{z}-xz^2\otimes\overline{y}$ to zero.
This follows from the fact that the elements $xy^2\otimes \gamma_2(z)$ and $xz^2\otimes \gamma_2(y)$ are both sent to the same element $2x\gamma^2(y)\gamma_2(z)$ in $F_{-d+1}\Gamma^d(A)$.
Hence the morphism \eqref{eq-surj} factors to a surjective morphism
\begin{align} \sigma_{(1,d-2)}(A/2)\twoheadrightarrow gr_{-d+1}\Gamma^d(A).\label{eq-surj2}\end{align}
Finally we check that \eqref{eq-surj2} is an isomorphism by dimension counting (the source and the target are ${\mathbb{F}_2}$-vector spaces). Let $(a_1,\dots,a_n)$ be a basis of the free abelian group $A$. Then the products $a_{i_1}\dots a_{i_d}$ with $i_1\le\dots\le i_d$ form a basis of $F_{-d}\Gamma^d(A)$. A basis of $F_{-d+1}\Gamma^d(A)$ is given by the products $a_{i_1}\dots a_{i_d}$ for $i_1<\dots<i_d$, and the products $a_{i_1}\dots a_{i_{d-2}}\gamma_2(a_{i_{d-1}})$ for $i_1\le \dots \le i_{d-2}$. The latter elements can be written as the elements $\frac{1}{2}a_{i_1}\dots a_{i_d}$ for $i_1\le \dots \le i_{d}$ where at least two of the
$i_k$ are equal. Thus the dimension of $gr_{-d+1}\Gamma^d(A)$ is equal to the number of $d$-tuples $(i_1,\dots,i_d)$ with $i_1\le \dots\le i_d$ and at least two $i_k$ are equal. By the short exact sequence \eqref{sig12bis}, this is exactly the dimension of $\sigma_{(1,n)}(A/2)$.
\end{proof}
We can identify $\mathrm{gr}\,_{-2}\Gamma^d(A)$ in a similar (and slightly simpler) fashion.
\begin{lemma}
Let $d\ge 3$. For any free abelian group $A$, $\mathrm{gr}\,_{-2}\Gamma^d(A)$ is a torsion abelian group, whose $p$-primary part $_{(p)}\mathrm{gr}\,_{-2}\Gamma^d(A)$ is given by:
$$_{(p)}\mathrm{gr}\,_{-2}\Gamma^d(A)= \begin{cases} 0 & \text{ if $d$ is not a sum of two powers of $p$,}\\ A/p^{(k)}\otimes A/p^{(\ell)} & \text{ if $d=p^k+p^\ell$ with $k\ne \ell$,}\\
S^2_{\mathbb{F}_p}(A/p^{(k)})& \text{ if $d=2p^k$.}
\end{cases}$$
\end{lemma}
\begin{proof}
Let $T^2(A)=\bigoplus_{k+\ell=d}\Gamma^k(A)\otimes \Gamma^\ell(A)$ and let $T^3(A)$ denote the direct sum
$$T^3(A):=\bigoplus_{k+\ell=d}\left(\bigoplus_{k_1+k_2=k} \Gamma^{k_1}(A)\otimes\Gamma^{k_2}(A)\otimes \Gamma^\ell(A)\, \oplus\, \bigoplus_{\ell_1+\ell_2=\ell} \Gamma^{k}(A)\otimes\Gamma^{\ell_1}(A)\otimes \Gamma^{\ell_2}(A)\right).$$
The multiplication of the divided power algebra defines a map $T^3(A)\to T^2(A)$. Let us denote by $C(A)$ its cokernel. By lemma \ref{lm-gr-1}, $C(A)$ is a torsion abelian group, and its $p$-primary part equals $A/p^{(k)}\otimes A/p^{(\ell)}$ if $d=p^k+p^\ell$ for a prime $p$ and zero otherwise.
By definition of the maximal filtration, there is a commutative diagram
$$\xymatrix{
T^3(A)\ar[r]\ar[d]^-{mult}& T^2(A)\ar@{->>}[d]^-{mult}\\
F_{-3}\Gamma^d(A)\ar[r]& F_{-2}(A)
}
$$
so the canonical surjection $T^2(A)\twoheadrightarrow gr_{-2}\Gamma^d(A)$ factors into a surjective morphism
$C(A)\twoheadrightarrow gr_{-2}\Gamma^d(A)$. In particular, $gr_{-2}\Gamma^d(A)$ is a torsion abelian group, and given prime $p$, we have $_{(p)} gr_{-2}\Gamma^d(A)=0$ if $d$ is not of the form $p^k+p^\ell$, and we have a surjective morphism
\begin{align}
A/p^{(k)}\otimes A/p^{(\ell)}\twoheadrightarrow _{(p)} gr_{-2}\Gamma^d(A)\;. \label{eq-surj3}
\end{align}
If $k=\ell$, the commutativity of the product in $\Gamma(A)$ implies that the morphism \eqref{eq-surj3} factors further as a surjective morphism
\begin{align}
S^2_{\mathbb{F}_p}(A/p^{(k)})\twoheadrightarrow _{(p)} gr_{-2}\Gamma^d(A)\;. \label{eq-surj4}
\end{align}
To finish the proof, it suffices to check that the morphism \eqref{eq-surj3}, resp. \eqref{eq-surj4}, are isomorphisms for $d=p^k+p^\ell$ with $k\ne \ell$, resp. $d=2p^k$. This follows from a dimension counting argument similar to the one in the proof of lemma \ref{lm-gr-d+1}.
\end{proof}
We collect the descriptions of the graded components $gr_{-i}\Gamma^d(A)$ for low $d$ in the following example. The cases $d\le 3$ already appear in \cite{BM}.
\begin{example}
\label{ga23}
For any free abelian group $A$, $gr_{-1}\,\Gamma^1_{\mathbb{Z}}(A)$ and
\begin{align}
&gr_{-i}\,\Gamma^2(A)&& = \begin{cases}A/2^{(1)} & i=1
\label{s2}
\\ S^2 (A) \ \ \ \ \ & i= 2
\end{cases}
\\
&gr_{-i}\,\Gamma^3(A) &&= \begin{cases}A/3^{(1)} & i=1\\ A\otimes A/2^{(1)}
&i=2\hspace{1cm}\\ S^3(A) & i=3 \label{s3}\end{cases}\\
&\label{filg4}
gr_{-i} \Gamma^4(A) &&= \begin{cases}\ A/2^{(2)} & i = 1\\
(A \otimes A/3^{(1)}) \oplus S^2_{\mathbb{F}_2}(A/2^{(1)}) & i= 2\\
\;\sigma_{(1,2)}(A/2) & i = 3\\
\;S^4(A) & i=4
\end{cases}
\end{align}
\end{example}
\subsection{The spectral sequence associated to the maximal filtration}
In order to study derived functors of $\Gamma^d(A)$ for $d >2$, we consider the spectral sequence associated to the maximal filtration of $\Gamma^d(A)$. This is obtained by filtering the simplicial abelian group $\Gamma^dK(A[n])$ componentwise according to the maximal filtration. This yields the following homological spectral sequence:
\begin{equation}
\label{filss}
E^1_{p,q} = L_{p+q}(gr_p\Gamma^d)(A,n) \Longrightarrow L_{p+q}\Gamma^d(A,n)
\end{equation}
with $d^r_{p,q}: E^r_{p,q} \to E^r_{p-r,q+r-1}$ as usual.
The maximal filtration on $\Gamma^d(A)$ is bounded for
any fixed integer $d$, so that the associated spectral sequence \eqref{filss} converges to $ L_{\ast}\Gamma^d(A,n)$ in the strong sense.
We now use this spectral sequence in order to compute the most elementary cases, that is the derived functors $L_i\Gamma^d(A,n)$ for $A$ free and $d=2$ or $3$.
\subsection{The derived functors of $\Gamma^2(A)$ for $A$ free}
For $d=2$, there are only two non-trivial terms in the graded object associated to the maximal filtration of $\Gamma^2(A)$ by \eqref{s2}, so that the spectral sequence \eqref{filss} has only two non-zero columns, namely:
\begin{align}
&E^1_{-1,q} = A/2^{(1)}\; \text{ if $q=n+1$, and zero otherwise.}
\\
&E^2_{-2,q}=L_{q-2}S^2(A,n)\;.
\end{align}
If $n=1$, then $L_iS^2(A,1)=\Lambda^2(A)$ if $i=4$ and zero otherwise. The spectral sequence degenerates at $E^1$ for lacunary reasons and we obtain:
\begin{equation}
\label{quad1}
L_*\Gamma^2(A,1) = A/2^{(1)}[1]\;\oplus\;\Lambda^2(A)[2]\;.
\end{equation}
If $n\ge 2$, $L_{i}S^2(A,n)=L_{i-4}\Gamma^2(A,n-2)$ by double d\'ecalage.
So the spectral sequence gives a relation between $L_*\Gamma^2(A,n)$ and $L_*\Gamma^2(A,n-2)$. This gives us the following result.
\begin{proposition}\label{prop-calcul-d2Z}
For any $n \geq 0$ and all free abelian group $A$, the only nontrivial values of $ L_i\Gamma^2(A,n)$ are:
\begin{equation}
\label{quadratic}
L_i\Gamma^2(A,n) = \begin{cases}A/2^{(1)} & i = n, n+2, n+4, \ldots, 2n-2 \qquad n \ \text{even}\\A/2^{(1)} & i = n, n+2, n+4, \ldots, 2n-1 \ \qquad n\ \text{odd}
\\ \Gamma^2(A) & i= 2n \qquad n \ \text{even}\\
\Lambda^2(A)& i= 2n \ \qquad n\ \text{odd.}\end{cases} \end{equation}
\end{proposition}
\begin{proof}
The assertion is trivial for $n=0$, and it is known for $n=1$ by \eqref{quad1}. We prove by induction that if it is
true for $n-2$, then it is also valid for $n$.
Indeed, the spectral sequence degenerates at $E^1$ for lacunary reasons. Thus the result for $n$ is valid up to a filtration. And since there is at most one nonzero term in each total degree in the spectral sequence, the filtration on the abutment is trivial.
\end{proof}
\subsection{The derived functors of $\Gamma^3(A)$ for $A$ free}
\label{subsec:ga3}
By \eqref{s3}, there are only three nontrivial terms in the graded object associated to the maximal filtration of $\Gamma^3(A)$. So for $d=3$, the spectral sequence \eqref{filss}
has only three nonzero columns, namely:
\begin{align}
E^1_{-1,q} &=
A/3^{(1)} \quad\text{ if $q =n+1$ and zero if $q\neq n+1$}
\\
E^1_{-2,q} &=
A\otimes (A/2)^{(1)} \quad\text{ if $q= 4$ and zero if $ q \neq 4$}
\\
E^3_{-3,q}&=L_{q-3}S^3(A,n)
\end{align}
If $n=1$, then by d\'ecalage $L_{*}S^3(A,1)=\Lambda^3(A)[3]$, so the spectral sequence degenerates at $E^1$ for lacunary reasons, and we obtain
\begin{equation}
\label{lig3a1} L_*\Gamma^3(A,1) = A/3^{(1)}[1]\;\oplus\;A \otimes A/2^{(1)}[2]\;\oplus\;
\Lambda^3(A)[3]\;.
\end{equation}
If $n\ge 2$, then $L_{i}S^3(A,n)=L_{i-6}\Gamma^3(A,n-2)$ by double d\'ecalage, so the spectral sequence essentially gives a relation between $L_*\Gamma^3(A,n)$ and $L_*\Gamma^3(A,n-2)$. The following result is proved exactly in the same way as proposition \ref{prop-calcul-d2Z}.
\begin{proposition}
\label{genliga3} Let $A$ be a free abelian group.
{\it i}) For any odd positive integer $n$, the only non trivial values of $L_i\Gamma^3(A,n)$ are:
\begin{equation}
\label{liga3odd}
L_{i}\Gamma^3(A,n) = \begin{cases}
A/3^{(1)} & i = n, n+4,n+8, \dots,3n-2\\
A \otimes A/2^{(1)}& i = 2n, 2n + 2, 2n+4, \dots, 3n-1\\
\Lambda^3(A) & i=3n.
\end{cases}
\end{equation}
{\it ii}) For any even nonnegative integer $n$, the only non trivial values of $L_i\Gamma^3(A,n)$ are:
\begin{equation}
\label{liga3even}
L_{i}\Gamma^3(A,n) = \begin{cases}
A/3^{(1)} & i= n, n+4, n+8, \dots, 3n -4\\
A \otimes A/2^{(1)} & i = 2n, 2n+2, 2n+4, \dots,3n -2\\
\Gamma^3(A) & i=3n.
\end{cases}
\end{equation}
\end{proposition}
\begin{remark}
For a general abelian group $A$ the values of the derived functors $L_{*}\Gamma^2(A,n)$ and $L_{*}\Gamma^3(A,n)$ are more complicated and we refer to \cite[Prop 4.1, Thm. 5.2]{BM} for a complete discussion.
The present discussion of the derived functors $L_*\Gamma^3(A,n)$ is
related to that in \cite{BM} since the
since the functor $W_3(A)$ emphasized there is simply the quotient group $F_{-1}\Gamma^3(A)/F_{-3}\Gamma^3(A)$ for the maximal filtration.
\end{remark}
\section{ The derived functors $L_*\Gamma^4(A,1)$ and $L_*\Gamma^4(A,2)$ for $A$ free}\label{sec-derG12}
In this section, we keep the notations of section \ref{sec-max}, in particular $A$ denotes a generic free finitely generated abelian group and $\Gamma(A)$ stands for $\Gamma_{\mathbb{Z}}(A)$. We compute the derived functors $L_*\Gamma^4(A,1)$ and $L_*\Gamma^4(A,2)$, by the same methods as in propositions \ref{prop-calcul-d2Z} and \ref{genliga3}. The situation is slightly more complicated here. Indeed, although the spectral sequence \eqref{filss} degenerates at $E^1$ for lacunary reasons as in the computations of the derived functors of $\Gamma^2(A)$ and $\Gamma^3(A)$, we will have to solve nontrivial extension problems both to compute the $E^1$ page of the spectral sequence and to recover the the derived functors of $\Gamma^4(A)$ from the $E^\infty$-term of the spectral sequence.
The derived functors $L_*\Gamma^4(A,1)$ were already computed in section \ref{der1-gamma-z} and \ref{sec-der1-gamma-z-bis}. The proof given here is more elementary, and independent from the techniques developed there.
\subsection{The derived functor $L_*\Gamma^4(A,1)$ for $A$ free}
\label{derga4-12}
The description of $\mathrm{gr}\,\Gamma^4(A)$ is given in example \ref{ga23}. Most of the graded terms are elementary, so that the corresponding initial terms of the spectral sequence \eqref{filss} for $m=4$ and $n=1$ are easy to compute:
\begin{align*}
E^1_{-1,q} & = A/2^{(2)} \qquad \text{ if $q=2$, and zero if $q\neq 2$,}\\
E^1_{-2,q}& = A \otimes A/3^{(1)}\; \oplus \;\Lambda^2_{\mathbb{F}_2}(A/2^{(1)})\qquad\text{if $q=4$, and zero if $q \neq 4$,}\\
E^1_{-4,q}&= \Lambda^4(A) \qquad \text{if $q=8$, and zero if $q\neq 8$.}
\end{align*}
To complete the description of the first page, we have to describe the column $E^1_{-3,*}$, that is the derived functors of $\mathrm{gr}\,_{-3}\Gamma^4(A)=\sigma_{(1,2)}(A/2)$.
The presentation \eqref{sig12} of $\sigma_{(1,2)}(A/2)$
determines for each $n$ an exact sequence of simplicial $\mathbb{F}_2$-vector spaces (since for $\sigma_{(1,2)}(A/2)$, the map $u$ is injective):
\begin{equation}\label{swws}
0\to \Lambda^2_{\mathbb{F}_2} K(A/2^{(1)}[n])\to S^2_{\mathbb{F}_2} K(A/2[n])\otimes K(A/2^{(1)}[n])
\to \sigma_{(1,2)}K(A/2[n])\to 0 \end{equation}
For $n=1$ this induces by d\'ecalage a short exact sequence
\begin{equation}
\label{gr3ga4a1}
0 \to \Lambda^2_{\mathbb{F}_2}( A/2)\otimes (A/2)^{(1)} \to \pi_3(\sigma_{1,2}K(A/2[1])) \to \Gamma^2_{\mathbb{F}_2}(A/2^{(1)}) \to 0
\end{equation}
in the category of $\mathbb{F}_2$-vector spaces which describes $E^1_{-3,6} $ as the middle term in \eqref{gr3ga4a1}. It also follows from \eqref{swws} that the terms $E^1_{-3,q}$ are trivial for $q \neq 3$
and this also shows that $E^1_{-3,q}= 0 $ for $q\neq 6$.
The spectral sequence degenerates for lacunary reasons, and since it has only has one non-trivial term in each total degree, there is no filtration issue on the abutment. So we immediately obtain the following result.
\begin{proposition}\label{prop-GA41}
\begin{equation}
\label{l4ga4a21}
L_i\Gamma^4(A,1) = \left\{ \begin{array}{ll} A/2^{(2)} & i=1\\
(A \otimes A/3^{(1)}) \oplus \Lambda^2_{\mathbb{F}_2}(A/2^{(1)}) & i=2\\
\Lambda^4(A) & i=4\end{array} \right.
\end{equation}
and there is a short exact sequence
\begin{equation}
\label{gr3ga4a11}
0 \to \Lambda^2_{\mathbb{F}_2} (A/2) \otimes A/2^{(1)} \to L_3\,\Gamma^4(A,1) \to \Gamma^2_{\mathbb{F}_2} (A/2^{(1)}) \to 0
\end{equation}
\end{proposition}
To solve the extension issue involved in this description of $L_3\Gamma^4(A,1)$, we prove the following statement.
\begin{proposition}\label{prop-nonsplit1}
The extension \eqref{gr3ga4a11} is not split. In fact
\begin{equation}\label{extgroup} {\mathrm{Ext}}^1(\Gamma^2_{\mathbb{F}_2}(A/2^{(1)}), \Lambda^2_{\mathbb{F}_2}(A/2)\otimes A/2^{(1)})\simeq \mathbb Z/2\;,
\end{equation}
where the extension group is computed in the category of strict polynomial functors or in the category of ordinary functors. Thus $L_3\,\Gamma^4(A,1)$ can be characterized as the unique non-trivial extension of $\Gamma^2_{\mathbb{F}_2} (A/2^{(1)})$ by $\Lambda^2_{\mathbb{F}_2} (A/2) \otimes A/2^{(1)}$.
\end{proposition}
\begin{proof}
Suppose that the exact sequence \eqref{gr3ga4a11} is functorially split, so that the $2$-primary component $_{(2)}L_3\Gamma^4(A,2)$ is isomorphic to $ \Lambda^2_{\mathbb{F}_2}(A/2)\otimes A/2^{(1)} \,\oplus \,\Gamma^2_{\mathbb{F}_2}(A/2^{(1)})$. In that case the universal coefficient theorem yields a short exact sequence:
\begin{align*}
0 \to & L_3\Gamma^4_{\mathbb{Z}}(A,1) \otimes {\mathbb{Z}}/2 \to L_3\Gamma^4_{\mathbb{F}_2}(A/2,1)
\to \mathrm{Tor}(L_2\Gamma^4_{\mathbb{Z}}(A,1), {\mathbb{Z}}/2) \to 0\;.
\end{align*}
The term in the middle is the value for $V:=A/2$ of $L_3\Gamma^4_{\mathbb{F}_2}(V,1)$. By
example \ref{ex-G4}, this is equal to $\Gamma^2_{\mathbb{F}_2}(A/2)\otimes A/2^{(1)}$. We would therefore have an injective map
$$ \Lambda^2_{\mathbb{F}_2}(A/2)\otimes A/2^{(1)} \,\oplus \,\Gamma^2_{\mathbb{F}_2}(A/2^{(1)})\hookrightarrow \Gamma^2_{\mathbb{F}_2}(A/2)\otimes A/2^{(1)}$$
By the methods of paragraph \ref{comput-hom}, we see that there is no non-zero morphism from $\Gamma^2_{\mathbb{F}_2}(A/2^{(1)})$ to $\Gamma^2_{\mathbb{F}_2}(A/2)\otimes A/2^{(1)}$, hence
the exact sequence \eqref{gr3ga4a11} cannot be functorially split. We refer to lemma \ref{lm-calc1} for the proof of formula \eqref{extgroup}.
\end{proof}
\subsection{The derived functor $L_*\Gamma^4(A,2)$ for $A$ free}\label{filga4}
We compute the derived functors of $\mathrm{gr}\,_{-3}\Gamma^4(A)=\sigma_{(1,2)}(A/2)$ as in section \ref{derga4-12}, that is by the exact sequence (\ref{swws}). If $n\ge 2$, there is no extension problem arising from the induced long exact sequence and we obtain
\begin{equation}
\label{lisig12}
L_i\,\sigma_{(1,2)}(A/2,n)=\begin{cases}
\Gamma^2_{\mathbb{F}_2}(A/2)\otimes A/2^{(1)},\ i=3n
\\A/2^{(1)}\otimes A/2^{(1)},\ 2n+2 \leq i \leq 3n-1
\\\Gamma^2_{\mathbb{F}_2}(A/2^{(1)}),\ i=2n+1
\\A/2^{(2)}, n+2 \leq i \leq 2n
\end{cases}
\end{equation}
So we can compute the $E^1$ page of the spectral sequence \eqref{filss} for $\Gamma^4$ for $n=2$. The result is displayed in table \ref{table-ga4a2} below.
\begin{table}[ht]
{\small
\renewcommand{\arraystretch}{1.8}
\begin{tabular}{|r||c|c|c|c|c|}
\hline $E^1_{p,q}$ & $p=-4$&-3&-2&-1
\\\hline \hline
$q=12$ & $\Gamma^4(A)$ & 0 & 0 & 0
\\
\hline 11 & 0 & $0$ & $0$ & 0
\\ \hline 10 & 0 & 0 & 0 & 0
\\ \hline 9 & 0 & $\Gamma^2_{\mathbb{F}_2}(A/2)\otimes A/2^{(1)}$ &
0 & 0
\\ \hline 8 & 0 & $\Gamma^2_{\mathbb{F}_2}(A/2^{(1)})$ &
0 & 0
\\ \hline 7 & 0 & $A/2^{(2)}$ &
0 & 0
\\ \hline 6 & 0 & 0 & $\Gamma^2_{\mathbb{F}_2}(A/2^{(1)})\oplus (A\otimes A/3^{(1)})$ & 0
\\ \hline 5 & 0 & 0 & 0 & 0
\\ \hline 4 & 0 & 0 & 0 & 0
\\ \hline 3 & 0 & 0 &
0 & $A/2^{(2)}$
\\ \hline
\end{tabular}
}
\vspace{.5cm} \caption{The initial terms of the maximal filtration spectral sequence for $L\Gamma^4(A,2)$}
\label{table-ga4a2}
\end{table}
In particular, the spectral sequence degenerates at $E^1$ for lacunary reasons.
So we obtain the following result.
\begin{proposition}\label{prop-GA42}
For $A$ free we have:
\begin{equation}
\label{liga4table}
L_i\Gamma^4(A,2)=\begin{cases} \Gamma^4(A),\ i=8\\ 0,\ i=3,7\\
\Gamma^2_{\mathbb{F}_2}(A/2)\otimes A/2^{(1)},\ i=6\\
\Gamma^2_{\mathbb{F}_2}(A/2^{(1)}),\ i=5\\ A/2^{(2)},\
i=2,\end{cases}
\end{equation}
as well as a short exact sequence \begin{equation} \label{l4ga4a2} 0\to
A/2^{(2)}\to L_4\Gamma^4(A,2)\to \Gamma^2_{\mathbb{F}_2}(A/2^{(1)})\oplus (A\otimes A/3^{(1)})\to 0. \end{equation}
\end{proposition}
To complete the description of $L_*\Gamma^4(A,2)$, we have to solve the extension issue involved in the description of the functor $L_4\Gamma^4(A,2)$. This is the purpose of the following result.
\begin{proposition}\label{prop-nonsplit2}
The extension \eqref{l4ga4a2} is not split. Actually we have
\begin{equation}\label{extgroup2} {\mathrm{Ext}}^1_{\mathcal{P}_{\mathbb{Z}}}(\Gamma^2_{\mathbb{F}_2}(A/2^{(1)}), A/2^{(2)}\,\oplus\, A\otimes A/3^{(1)})=\mathbb Z/2\;,
\end{equation}
so that $L_4\,\Gamma^4(A,2)$ can be characterized as the unique nontrivial extension of $\Gamma^2_{\mathbb{F}_2} (A/2^{(1)})$ by $A/2^{(2)}\oplus A\otimes A/3^{(1)}$. In particular, we obtain
$$L_4\Gamma^4(A,2)\simeq \Gamma^2_{\mathbb{Z}}(A/2^{(1)})\oplus A\otimes A/3^{(1)}\;.$$
\end{proposition}
\begin{proof}The universal coefficient theorem yields a short exact sequence:
\begin{align}
0 \to L_5\Gamma^4_{\mathbb{Z}}(A,2) \otimes {\mathbb{Z}}/2 \to L_5\Gamma^4_{\mathbb{F}_2}(A/2,2)
\to \mathrm{Tor}(L_4\Gamma^4_{\mathbb{Z}}(A,2), {\mathbb{Z}}/2) \to 0\;.
\label{kunl5}
\end{align}
The term in the middle was already computed in
example \ref{ex-G4}. If the extension \eqref{l4ga4a2} was functorially split, then $_{(2)}L_4\Gamma^4(A,2)$ would be isomorphic to $ A/2^{(2)} \,\oplus \,\Gamma^2_{\mathbb{F}_2}(A/2^{(1)})$, so that the short exact sequence \eqref{kunl5} could be restated as
\[
\xymatrix{
0 \ar[r] &\Gamma^2_{\mathbb{F}_2}(A/2^{(1)}) \ar[r]^(.35)i & \left(A/2^{(1)} \otimes A/2^{(1)} \right)\oplus A/2^{(2)}
\ar[r]^(.55)j & A/2^{(2)} \,\oplus \,\Gamma^2_{\mathbb{F}_2}(A/2^{(1)}) \ar[r] & 0}
\]
A dimension count in the category of $\mathbb{F}_2$-vector spaces makes it clear that such a short exact sequence cannot exist. It follows that \eqref{l4ga4a2} is not split.
The formula \eqref{extgroup2} and the identification of $L_4\Gamma^4(A,2)$ follow from lemma \ref{lm-calc2}.
\end{proof}
\section{The derived functors of $\Gamma^4(A)$ for $A$ free}
\label{der-gamma4-sec}
In this section we assume that $A$ is a free abelian group. We will
give a complete description of the derived functors
$L_i\Gamma^4(A,n)$ for all $i \geq 0 $ and $n\geq 1$.
\subsection{The description of $L_*\Gamma^4(A,n)$ for $A$ free}
The main result of section \ref{der-gamma4-sec} is the following computation.
\begin{theorem}\label{thm-G4Z}
Let $n$ be a positive integer. If $n=2m+1$, then we have an isomorphism of graded strict polynomial functors (where $F(A)[k]$ denotes a copy of $F(A)$ placed in degree $k$, and sums over empty sets mean zero):
\begin{align*}
L_*\Gamma^4(A,n) \simeq & \Lambda^4(A)[4n] \oplus \Phi^4(A)[4n-1]\oplus \bigoplus_{i=0}^{m-1}\Gamma^2_{\mathbb{F}_2}(A/2)\otimes A/2^{(1)}[3n+2i]\\
& \oplus \bigoplus_{i=0}^{m} \Lambda^2_{\mathbb{F}_2}(A/2^{(1)})[2n+4i] \oplus \bigoplus_{i=0}^{m-1}\Gamma^2_{\mathbb{F}_2}(A/2^{(1)})[2n+4i+1]\\
& \oplus \bigoplus_{i=0}^{m-1}\bigoplus_{j=2i}^{n-3}A/2^{(1)}\otimes A/2^{(1)}[2n+2i+j+2]
\oplus \bigoplus_{i=0}^{m}A/3\otimes A/3^{(1)}[2n+4i]\\
& \oplus \bigoplus_{i=0}^{m} A/2^{(2)}[n+6i]\oplus \bigoplus_{i=0}^{m-1}\bigoplus_{j=2i}^{n-2}A/2^{(2)}[n+4i+j+2]\;,
\end{align*}
Here $\Phi^4(A):=L_3\Gamma^4(A,1)$ is, as shown in propositions \ref{prop-GA41} and \ref{prop-nonsplit1}, the unique nontrivial extension of $\Gamma^2_{\mathbb{F}_2}(A/2^{(1)})$ by $\Lambda^2_{\mathbb{F}_2}(A/2)\otimes A/2^{(1)}$.
Similarly, if $n=2m$ there is an isomorphism of graded strict polynomial functors:
\begin{align*}
L_*\Gamma^4(A,n) \simeq & \Gamma^4(A)[4n] \oplus \bigoplus_{i=0}^{m-1}\Gamma^2_{\mathbb{F}_2}(A/2)\otimes A/2^{(1)}[3n+2i]\\
& \oplus \bigoplus_{i=0}^{m-1} \Gamma^2_{\mathbb{Z}}(A/2^{(1)})[2n+4i] \oplus \bigoplus_{i=0}^{m-1}\Gamma^2_{\mathbb{F}_2}(A/2^{(1)})[2n+4i+1]\\
& \oplus \bigoplus_{i=0}^{m-2}\bigoplus_{j=2i}^{n-3}A/2^{(1)}\otimes A/2^{(1)}[2n+2i+j+2]
\oplus \bigoplus_{i=0}^{m-1}A/3\otimes A/3^{(1)}[2n+4i]\\
& \oplus \bigoplus_{i=0}^{m-1} A/2^{(2)}[n+6i]\oplus \bigoplus_{i=0}^{m-2}\bigoplus_{j=2i}^{n-3}A/2^{(2)}[n+4i+j+2]\;.
\end{align*}
\end{theorem}
Although the formulas describing the derived functors $L_*\Gamma^4(A,n)$ look very similar for $n$ even and odd, there are at least two major differences for the
$2$-primary part. First of all, if $n$ is even there is some $4$-torsion in $L_*\Gamma^4(A,n)$, provided by the summands $\Gamma_{\mathbb{Z}}^2(A/2^{(1)})$. On the contrary, when $n$ is odd, $L_*\Gamma^4(A,n)$ has only $2$-torsion. Secondly, the functor $\Phi^4(A)$ appears as a direct summand (with multiplicity one) in $L_*\Gamma^4(A,n)$ when $n$ is odd and does not appear in the formula when $n$ is even. We observe that the formula for $L_*\Gamma^4_{\mathbb{F}_2}(V,n)$ in example \ref{ex-G4} did not depend on the parity of $n$.
For $n=1$, theorem \ref{thm-G4Z} is equivalent to proposition \ref{prop-GA41}, and for $n=2$, it is equivalent to proposition \ref{prop-GA42}. Now theorem \ref{thm-G4Z} easily follows, by induction on $n$, from the following statement.
\begin{theorem}\label{deriveddescr}
Let $n\ge 3$. If $n$ is odd, there is an isomorphism of graded strict polynomial functors
\begin{multline}
\label{desg4}
L_\ast \Gamma^4(A,n)=L_\ast\Gamma^4(A,n-2)[8]\oplus\bigoplus_{i=n,\ i\neq
n+1}^{2n}A/2^{(2)}[i]\oplus
\bigoplus_{i=2n+2}^{3n-1}A/2^{(1)}\otimes A/2^{(1)}[i]\oplus\\
\Gamma^2_{\mathbb{F}_2}(A/2)\otimes A/2^{(1)}[3n]\oplus \Lambda^2_{\mathbb{F}_2}(A/2^{(1)})[2n]\oplus \Gamma^2_{\mathbb{F}_2}(A/2^{(1)})[2n+1]\oplus A\otimes
A/3[2n].
\end{multline}
Similarly, if $n$ is even, there is an isomorphism
\begin{multline}
\label{desg41}
L_\ast\Gamma^4(A,n)=L_\ast\Gamma^4(A,n-2)[8]\oplus\bigoplus_{i=n,\ i\neq
n+1}^{2n-1}A/2^{(2)}[i]\oplus
\bigoplus_{i=2n+2}^{3n-1}A/2^{(1)}\otimes A/2^{(1)}[i]\oplus\\
\Gamma^2_{\mathbb{F}_2}(A/2)\otimes A/2^{(1)}[3n]\oplus \Gamma^2_{\mathbb{Z}}(A/2^{(1)})[2n]\oplus
\Gamma^2_{\mathbb{F}_2}(A/2^{(1)})[2n+1]\oplus A\otimes A/3[2n]
\end{multline}
\end{theorem}
The remainder of section \ref{der-gamma4-sec} is devoted to the proof of theorem \ref{deriveddescr}. The proof goes along the same lines as the computation of $L_*\Gamma^4(A,1)$ and $L_*\Gamma^4(A,2)$ in section \ref{sec-derG12}, that is the relation between $L_\ast\Gamma^4(A,n)$ and $L_\ast\Gamma^4(A,n-2)$ is provided by the analysis of the spectral sequence \eqref{filss} induced by the maximal filtration of $\Gamma^4(A)$. However, for $n\ge 3$, the analysis of the spectral sequence is more delicate than in section \ref{sec-derG12} as there now are nontrivial differentials in the spectral sequence. Also, we will have to solve extension problems in order to recover $L_\ast\Gamma^4(A,n)$ from the $E^\infty$-page of the spectral sequence.
\subsection{Proof of theorem \ref{deriveddescr}}
From now on $n\ge 3$ and we assume that theorem \ref{thm-G4Z} has been proved for $n-2$. We are going to prove that theorem \ref{deriveddescr} holds for $n$. The first page of the maximal filtration spectral sequence
(\ref{filss}) for $d=4$ and $n \ge 3$ can be computed by the methods
introduced in paragraphs \ref{derga4-12} and \ref{filga4} for $n= 1, 2$. In particular, the terms in the $p= -3$ column follow from \eqref{lisig12} and those in the $p= -4$ column from the double d\'ecalage formula \eqref{dec}. The $E^1$ page therefore
has the form depicted in table 2.
\begin{table}[H
{\small
\renewcommand{\arraystretch}{1.8}
\begin{tabular}{|r||c|c|c|c|c|}
\hline $E^1_{p,q}$ & $p=-4$&-3&-2&-1
\\\hline \hline
$q=4n+4$ & $L_{4n-8}\Gamma^4(A,n-2)$ & 0 & 0 & 0
\\
\hline $4n+3$ & $L_{4n-9}\Gamma^4(A,n-2)$ & $0$ & $0$ & 0
\\ \hline \dots & \dots & \dots & \dots & \dots
\\ \hline $3n+4$ & $L_{3n-8}\Gamma^4(A,n-2)$ & 0 &
0 & 0
\\ \hline 3n+3 & $L_{3n-9}\Gamma^4(A,n-2)$ & $\Gamma^2_{\mathbb{F}_2}(A/2)\otimes A/2^{(1)}$ &
0 & 0
\\ \hline 3n+2 & $L_{3n-10}\Gamma^4(A,n-2)$ & $A/2^{(1)}\otimes A/2^{(1)} $ &
0 & 0
\\ \hline \dots & \dots & \dots & \dots & \dots
\\ \hline 2n+5 & $L_{2n-7}\Gamma^4(A,n-2)$ & $ A/2^{(1)}\otimes A/2^{(1)} $ & 0 & 0
\\ \hline 2n+4 & $\mathbf{L_{2n-8}\Gamma^4(A,n-2)}$ & $\Gamma^2_{\mathbb{F}_2}(A/2^{(1)})$ &
0 & 0
\\ \hline 2n+3 & $L_{2n-9}\Gamma^4(A,n-2)$ & $\mathbf{A/2^{(2)}}$ & 0 &
0
\\ \hline 2n+2 & $L_{2n-10}\Gamma^4(A,n-2)$ & $A/2^{(2)}$ & $\begin{array}{c}\mathbf{\Gamma^2_{\mathbb{F}_2}(A/2^{(1)})} \\\oplus \mathbf{A\otimes A/3^{(1)}}\end{array}$ & 0
\\ \hline 2n+1 & $L_{2n-11}\Gamma^4(A,n-2)$ & $A/2^{(2)}$ & $A/2^{(2)}$ & $\mathbf{0}$
\\ \hline \dots & \dots & \dots & \dots & \dots
\\ \hline n+5 & $L_{n-7}\Gamma^4(A,n-2)=0$ & $A/2^{(2)}$ & $A/2^{(2)}$ & 0
\\ \hline n+4 & 0 & 0 & $A/2^{(2)}$ & 0
\\ \hline n+3 & 0 & 0 & 0 & 0
\\ \hline n+2 & 0 & 0 & 0 & 0
\\ \hline n+1 & 0 & 0 & 0 & $A/2^{(2)}$
\\ \hline
\end{tabular}
}
\vspace{.5cm} \caption{The initial terms of the maximal filtration spectral sequence for $\Gamma^4(A,n)$ with $n >2$}
\label{table-ga4an}\end{table}
The boldface expressions in table 2 are the terms of total degree $2n$. They will play a special role in the proof. We will indeed prove that all the differentials of the spectral sequence are zero, except some of the differentials with terms of total degree $2n$ as source or target.
In order to obtain some information regarding the differentials of the spectral sequence, we are going to use mod $2$ reduction, in the spirit of the proof of propositions \ref{prop-nonsplit1} and \ref{prop-nonsplit2}. The universal coefficient theorem yields short exact sequences of strict polynomial functors (where $_2G$ denotes the $2$-torsion subgroup of an abelian group $G$):
\begin{align}
&0\to L_i\Gamma^4(A,n)\otimes{\mathbb{F}_2}\to L_i\Gamma^4_{\mathbb{F}_2}(A/2,n)\to {_2L_{i-1}\Gamma^4(A,n)}\to 0\;.\label{uct}\\
&0\to L_i\Gamma^4(A,n-2)\otimes{\mathbb{F}_2}\to L_i\Gamma^4_{\mathbb{F}_2}(A/2,n-2)\to {_2L_{i-1}\Gamma^4(A,n-2)}\to 0\;.\label{uctbis}
\end{align}
Moreover, we have already computed $L_i\Gamma^4_{\mathbb{F}_2}(A/2,n)$ in example \ref{ex-G4}. The following mod $2$ analogue of theorem \ref{thm-G4Z} is a straightforward consequence of example \ref{ex-G4}.
\begin{lemma}\label{lm-m2descr}
There are isomorphisms of strict polynomial functors
$$L_i\Gamma^4_{\mathbb{F}_2}(A/2,n)\simeq L_{i-8}\Gamma^4_{\mathbb{F}_2}(A/2,n-2)\oplus C_i(A,n)$$
where
$$
C_i(A,n)\simeq
\begin{cases}
0 & i>3n+1\\
\Gamma^2_{\mathbb{F}_2}(A/2)\otimes A/2^{(1)} & i=3n+1\\
\Gamma^2_{\mathbb{F}_2}(A/2)\otimes A/2^{(1)}\,\,\oplus \,\,\, A/2^{(1)}\otimes A/2^{(1)}\,\,\, & i=3n\\
A/2^{(1)}\otimes A/2^{(1)}\,\, \oplus \,\,\, A/2^{(1)}\otimes A/2^{(1)} & 2n+2<i<3n \\
A/2^{(1)}\otimes A/2^{(1)}\,\, \oplus \,\,\, \Gamma^2_{\mathbb{F}_2}(A/2^{(1)}) & i=2n+2 \\
A/2^{(1)}\otimes A/2^{(1)}\,\, \oplus \,\,\, A/2^{(2)} & i=2n+1\\
\Gamma^2_{\mathbb{F}_2}(A/2^{(1)}) \oplus A/2^{(2)} & i=2n\\
A/2^{(2)}\oplus A/2^{(2)} & n+2<i<2n\\
A/2^{(2)} & n\le i\le n+2
\end{cases}$$
\end{lemma}
Let $E_i^r$ denote the part of total degree $i$ of the $r$-th page of the spectral sequence, $E_i^r:=\bigoplus_{j+k=i}E^r_{j,k}$, and let $d^r_i:E^r_i\to E^r_{i-1}$ denote the total differential.
We distinguish three steps in the analysis of the spectral sequence.
\begin{enumerate}
\item[$\bullet$] We first analyze the spectral sequence in total degrees $i<2n$. Formulas \eqref{eq-lown}, \eqref{eq-lown+1} and \eqref{eq-prop-basdeg} show that theorem \ref{deriveddescr} holds in degrees $i<2n$.
\item[$\bullet$] Then we analyze the spectral sequence in total degrees degrees $i>2n$. Formulas \eqref{eq-triviale} and \eqref{eq-suiv} show that theorem \ref{deriveddescr} holds in degrees $i>2n$.
\item[$\bullet$] Finally, we analyze the spectral sequence in total degree $2n$. Formulas \eqref{eq-2neven} and \eqref{eq-2nodd} show that theorem \ref{deriveddescr} holds in degree $2n$.
\end{enumerate}
\subsubsection{The spectral sequence in total degree $i<2n$}
For $i=n$ or $n+1$, we have $E^1_i=E^\infty_i$ for lacunary reasons. Since there is only one nontrivial term in total degree $i$ there is no extension issue to recover the abutment. Hence we obtain:
\begin{align}&L_n\Gamma^4(A,n)= A/2^{(2)}= A/2^{(2)}\oplus L_{n-8}\Gamma^4(A,n-2)\;,\label{eq-lown}
\\
&L_{n+1}\Gamma^4(A,n)=0= L_{n+1-8}\Gamma^4(A,n-2)\;.\label{eq-lown+1}
\end{align}
The case $i\ge n+2$ is slightly more involved.
\begin{proposition}\label{prop-lowdeg}
For $n+2\le i\le 2n-1$ we have
\begin{align}L_i\Gamma^4(A,n)\simeq L_i\Gamma^4(A,n-2)\oplus A/2^{(2)}\;.\label{eq-prop-basdeg}
\end{align}
Moreover, if $n$ is even, the differentials $d^1_{2n}:E^1_{2n}\to E^1_{2n-1}$ and $d^2_{2n}:E^2_{2n}\to E^2_{2n-1}$ are zero. If $n$ is odd, one of the two differentials $d^1_{2n}$, $d^2_{2n}$ has $A/2^{(2)}$ as its image and the other one is the zero map.
\end{proposition}
\begin{proof}
First of all, by theorem \ref{thm-G4Z} (which we assume to be proved for $n-2$) we know the terms in the column $E^1_{-4,*}$. In particular all the expressions $E^1_i$ for $i<2n$ are direct sums of terms $A/2^{(2)}$. A subquotient of a direct sum of copies of $A/2^{(2)}$ is once again a direct sum of copies of $A/2^{(2)}$, so that $E_i^\infty$ is also a direct sum of copies of $A/2^{(2)}$ if $i<2n$. Finally the strict polynomial functor $A/2^{(2)}$ has no self-extensions of degree $1$, so that:
$$E^\infty_i\simeq L_i\Gamma^4(A,n)\quad\text{for $i<2n$}\;.$$
Since only functors of the form $A/2^{(2)}$ appear in total degrees $i<2n$ in the spectral sequence, there is no functorial issues involved in these degrees and analyzing this part of the spectral sequence amounts to analyze a spectral sequence of ${\mathbb{F}_2}$-vector spaces.
To be more specific, if $d_i(n)$ denotes the dimension of the ${\mathbb{F}_2}$-vector space $L_i\Gamma^4({\mathbb{Z}},n)$ for $i<2n$, then the formula \eqref{eq-prop-basdeg} is equivalent to the following equality for $n+2\le i\le 2n-1$:
\begin{align} d_i(n)=d_i(n-2)+1\;.\label{eq-interm}\end{align}
We now prove \eqref{eq-interm} by induction on $i$. We have $d_{n+1}(n)=d_{n+1}(n-2)=0$ by \eqref{eq-lown+1}, and if we denote by
$\delta_i(n)$ the dimension of the ${\mathbb{F}_2}$-vector space $L_i\Gamma^4_{\mathbb{F}_2}({\mathbb{Z}}/2,n)$, the exact sequences \eqref{uct} and \eqref{uctbis} and lemma \ref{lm-m2descr} yield equalities:
$$d_{n+2}(n)=\delta_{n+2}(n),\quad d_{n+2}(n-2)=\delta_{n+2}(n-2),\quad \delta_{n+2}(n)=\delta_{n+2}(n-2)+1\;.$$
This shows that \eqref{eq-interm} holds for $i=n+2$. Now assume that $n+2<i<2n$. Then
the exact sequences \eqref{uct} and \eqref{uctbis} and lemma \ref{lm-m2descr} yield equalities:
$$d_{i}(n)=\delta_{i}(n)-d_{i-1}(n),\quad d_{i}(n-2)=\delta_{i}(n-2)-d_{i-1}(n-2),\quad \delta_{n+2}(n)=\delta_{n+2}(n-2)+2\;.$$
Thus, assuming that \eqref{eq-interm} holds for $i-1$, we obtain that \eqref{eq-interm} holds for $i$.
It remains to prove the assertion on the differentials of the spectral sequence. By the formula \eqref{eq-prop-basdeg}, for $n+2\le i<2n$, $E^1_i$ has one more copy of $A/2^{(2)}$ than $E^\infty_i$. Hence there are only two possibilities for each $i$.
\begin{enumerate}
\item[$(a_i)$] The maps $d^1_i$ and $d^2_i$ are zero, one of the maps $d^1_{i+1}$, $d^2_{i+1}$ is zero, and the other one has image $A/2^{(2)}$.
\item[$(b_i)$] One of the maps $d^1_{i}$, $d^2_{i}$ is zero and the other one has image $A/2^{(2)}$, and the maps $d^1_{i+1}$ and $d^2_{i+1}$ are zero.
\end{enumerate}
We observe that $(a_i)\Longrightarrow (b_{i+1})\Longrightarrow (a_{i+2})$.
Since $E_{n+1}^1=0$, $(a_{n+2})$ holds. We can therefore deduce the result by induction on $i$.
\end{proof}
\subsubsection{The spectral sequence in total degree $i>2n$}
If $i>3n$ then $E^1_i=E^\infty_i$ for lacunary reasons, and since there is only one nontrivial term in total degree $i$ we have:
\begin{align}
L_i\Gamma(A,n)\simeq L_{i-8}\Gamma(A,n-2)\;. \label{eq-triviale}
\end{align}
To analyze the spectral sequence in total degree $i$ with $2n<i\le 3n$, we will use mod $2$ reduction. So we first recall basic facts regarding mod $p$ reduction. First of all, for any finite abelian group $G$, the $p$-torsion subgroup $_pG$ has the same dimension (as an ${\mathbb{F}_p}$-vector space) as the mod $p$ reduction $G\otimes {\mathbb{F}_p}$. Using this basic fact, one easily proves the following very rough estimation, which will be useful to compare the $E^1$ and the $E^\infty$ pages of the spectral sequence modulo $p$.
\begin{lemma}\label{lm-modp2}
Let $(G_*,\partial_*)$ be a degreewise finite differential graded abelian group with $\partial_i:G_i\to G_{i-1}$. Given a prime $p$, we denote by $(_{p}G_*,{_{p}\partial})$ the subcomplex of $p$-torsion elements of $G_*$. Then we have
\begin{align*}
\dim_{{\mathbb{F}_p}}(H_i(G)\otimes{\mathbb{F}_p})\le \dim_{\mathbb{F}_p}( G_i\otimes{\mathbb{F}_p}) - \mathrm{rk}(_p\partial_i)\;.
\end{align*}
\end{lemma}
The next elementary lemma is useful for a modulo $p$ comparison of $L_*\Gamma^4(A,n)$ and the $E^\infty$ page of the spectral sequence.
\begin{lemma}\label{lm-modp1}
Let $G$ be a filtered finite abelian group. For all primes $p$ we have
\begin{align*}\dim_{\mathbb{F}_p}\left( G\otimes{\mathbb{F}_p}\right)\le \dim_{\mathbb{F}_p} \left(\mathrm{gr}\,(G)\otimes{\mathbb{F}_p}\right)\;.
\end{align*}
Moreover, the equality holds if and only if there exists an isomorphism of groups between the $p$-primary parts $_{(p)} G\simeq {_{(p)} \mathrm{gr}\,(G)}$.
\end{lemma}
\begin{proposition}\label{prop-high-deg}Let $2n<i\le 3n$. There is an isomorphism:
\begin{align}L_i\Gamma^4(A,n)\simeq L_{i-8}\Gamma^4(A,n-2)\oplus \begin{cases}
\Gamma^2_{\mathbb{F}_2}(A/2)\otimes A/2^{(1)} & i=3n\\
A/2^{(1)}\otimes A/2^{(1)} & 2n+1<i<3n\\
\Gamma^2_{\mathbb{F}_2}(A/2^{(1)}) & i=2n+1\\
\end{cases}\label{eq-suiv}\end{align}
Moreover, the differential $d^1_{-3,2n+4}:\Gamma^2_{\mathbb{F}_2}(A/2^{(1)})\to L_{2n-8}\Gamma^4(A,n-2)$ is zero.
\end{proposition}
\begin{proof}Lemmas \ref{lm-modp1} and \ref{lm-modp2} yield inequalities:
\begin{align}&\dim \left(E^1_i\otimes {\mathbb{F}_2}\right) - \mathrm{rk} ( d^1_{i}) \ge \dim \left(E^\infty_i\otimes {\mathbb{F}_2}\right)\ge \dim \left(L_i\Gamma^4(A,n)\otimes {\mathbb{F}_2}\right),\label{eq-ineqbis}\\
&\dim \left(_2E^1_{i-1}\right) - \mathrm{rk} ( d^1_{i-1}) \ge \dim \left(_2E^\infty_{i-1}\right)\ge \dim \left(_2L_{i-1}\Gamma^4(A,n)\right).\label{eq-ineqter}
\end{align}
We will now verify that the expressions \eqref{eq-ineqbis}, \eqref{eq-ineqter} are actually equalities for $2n+1< i\le 3n$, thereby proving that the total differential $d^1_i$ is zero in degrees $2n<i\le 3n$. The
universal coefficient exact sequence \eqref{uct} yields an equality
\begin{align}\dim \left(L_i\Gamma^4(A,n)\otimes{\mathbb{F}_2}\right) + \dim \left({_2L_{i-1}\Gamma^4(A,n)}\right)= \dim L_i\Gamma^4_{\mathbb{F}_2}(A/2,n)\;.\label{eq-dep1}\end{align}
The short exact sequence \eqref{uctbis} and lemma \ref{lm-m2descr} imply that
\begin{align*}
\dim L_i\Gamma^4_{\mathbb{F}_2}(A/2,n)=\dim \left(E^1_{-4,i+4}\otimes {\mathbb{F}_2}\right)+\dim \left(_2E^1_{-4,i+3}\right)+\dim C_i(A/2,n)\;.
\end{align*}
We observe that $E^1_{-3,i+3}\oplus E^1_{-3,i+2}\simeq C_i(A/2,n)$ for $2n+1<i\le 3n$. It follows that
\begin{align}
\dim L_i\Gamma^4_{\mathbb{F}_2}(A/2,n)=\dim \left(E^1_i\otimes {\mathbb{F}_2}\right)+\dim \left(_2E^1_{i-1}\right).\label{eq-dep2}
\end{align}
By comparing the sum of the inequalities \eqref{eq-ineqbis} and \eqref{eq-ineqter} with the equality provided by \eqref{eq-dep1} and \eqref{eq-dep2}, we can now conclude that the expression \eqref{eq-ineqbis} and \eqref{eq-ineqter} are actually equalities for $2n+1< i\le 3n$.
Since total differential $d^1_i$ is zero in degrees $2n<i\le 3n$ (and $d^1_{3n+1}$ is zero by lacunarity), we have $d^1_{-3,2n+4}=0$ and $E^1_i=E^\infty_i$ for $2n<i\le 3n$. Furthermore, since \eqref{eq-ineqbis} and \eqref{eq-ineqter} are equalities, lemma \ref{lm-modp1} yields a \emph{non functorial} isomorphism
\begin{align}E^\infty_{i-4,4}\oplus E^\infty_{i-3,3}= E^\infty_i\simeq L_i\Gamma^4(A,n)\label{eq-nonfunct}\;.\end{align}
To finish the proof, we have to prove a \emph{functorial} isomorphism
$E^\infty_{i-4,4}\oplus E^\infty_{i-4,3}\simeq L_i\Gamma^4(A,n)$. But the $2$-primary part of $E^\infty_{i-4,4}$ is a direct sum of functors of the following types:
$$A/2^{(2)}\;,\; \Gamma^2_{\mathbb{F}_2}(A/2^{(1)})\;,\; A/2^{(1)}\otimes A/2^{(1)}\;,\;\Gamma^2_{\mathbb{Z}}(A/2^{(1)})\;,\; \Lambda^2_{\mathbb{F}_2}(A/2^{(1)}) $$
(as no term of the form $\Gamma^4(A)$, $\Lambda^4(A)$, $\Gamma^2_{\mathbb{F}_2}(A/2)\otimes A/2^{(1)}$ or $\Phi^4(A)$ occurs in the degrees which we are considering here).
In addition, $E^\infty_{i-4,3}$ is one of the following functors:
$$\Gamma^2_{\mathbb{F}_2}(A/2^{(1)})\;,\; A/2^{(1)}\otimes A/2^{(1)}\;,\; \Gamma^2_{\mathbb{F}_2}(A/2)\otimes A/2^{(1)}\;.$$
It follows from the ${\mathrm{Ext}}^1$ computations of appendix \ref{app-comput} that there can be no nonsplit extension of $E^\infty_{i-4,3}$ by $E^\infty_{i-4,4}$, except in the case $E^\infty_{i-4,3}=\Gamma^2_{\mathbb{F}_2}(A/2^{(1)})$. In the latter case, the only possible nontrivial extension is an extension of $\Gamma^2_{\mathbb{F}_2}(A/2^{(1)})$ by a functor of the form $(A/2^{(2)})^{\oplus \,k}$. The middle term of such a nontrivial extension is a functor $\Gamma_{\mathbb{Z}}^2(A/2^{(1)})\oplus (A/2^{(2)})^{\oplus \, k-1}$, which has $4$-torsion. Such a nontrivial extension is therefore excluded, as this would contradict the isomorphism \eqref{eq-nonfunct}. Since all possible extensions are split, we obtain an isomorphism of functors $E^\infty_i\simeq L_i\Gamma^4(A,n)$. This finishes the proof of proposition \ref{prop-high-deg}.
\end{proof}
\subsubsection{The spectral sequence in total degree $i= 2n$}
The study of the spectral sequence in total degrees $i>2n$ and $i<2n$ has already provided us with some partial information regarding the situation for $i=2n$.
Let us sum up what we know so far.
\begin{itemize}
\item The differential $d^1_{2n+1}$ is zero, hence $E^\infty_{2n}$ is a subfunctor of $E^1_{2n}$.
\item If $n$ is even, then $d^1_{2n}$ and $d^2_{2n}$ are zero, hence $E^\infty_{2n}=E^1_{2n}$.
\item If $n$ is odd, then one of the differentials $d^1_{2n}$ and $d^2_{2n}$ is zero, and the other has image equal to $A/2^{(2)}$. Hence we have only two possibilities:
\begin{enumerate}
\item[(a)] $E^\infty_{2n} = L_{2n-8}\Gamma^4(A,n-2)\oplus \Gamma^2_{\mathbb{F}_2}(A/2^{(1)})\oplus A\otimes A/3^{(1)}$
\item[(b)] $E^\infty_{2n} = L_{2n-8}\Gamma^4(A,n-2)\oplus A/2^{(1)}\oplus \Lambda_{\mathbb{F}_2}^2(A/2^{(1)})\oplus A\otimes A/3^{(1)}$
\end{enumerate}
\end{itemize}
\begin{proposition}\label{prop-middegneven}
If $n$ is even, then
\begin{align}L_{2n}\Gamma^4(A,n)= L_{2n-8}\Gamma^4(A,n-2)\oplus \Gamma^2_{\mathbb{Z}}(A/2^{(1)})\oplus A\otimes A/3^{(1)}\;.\label{eq-2neven}\end{align}
\end{proposition}
\begin{proof}
We already know that $E_{2n}^1=E_{2n}^\infty$, so we just have to retrieve $L_{2n}\Gamma^4(A,n)$ from $E_{2n}^\infty$.
By theorem \ref{thm-G4Z} (which we assume to be proved for $n-2$), $L_{2n-8}\Gamma^4(A,n-2)$ is a sum of copies of $A/2^{(2)}$. Since $A/2^{(2)}$ has no self-extension of degree one, there is no extension problem between the columns $p=-4$ and $p=-3$. Hence the extension problem on the abutment can be restated as a short exact sequence:
\begin{align*}0\to L_{2n-8}\Gamma^4(A,n-2)\oplus A/2^{(2)}\to L_{2n}\Gamma^4(A,n) \to \Gamma^2_{\mathbb{F}_2}(A/2^{(1)})\oplus A\otimes A/3^{(1)}\to 0\;. \end{align*}
But the only nontrivial extension of $\Gamma^{2}_{\mathbb{F}_2}(A/2^{(1)})$ by a functor of the form $(A/2^{(2)})^{\oplus \, k}$ is given by a functor of the form $\Gamma^{2}_{\mathbb{Z}}(A/2^{(1)})\oplus (A/2^{(2)})^{\oplus\, k-1}$. Thus we have only two possibilities for $L_{2n}\Gamma^4(A,n)$, namely:
\begin{align*}
&(i)\quad L_{2n-8}\Gamma^4(A,n-2)\oplus \Gamma_{\mathbb{Z}}(A/2^{(1)})\oplus A\otimes A/3^{(1)}\;,\\
&(ii)\quad L_{2n-8}\Gamma^4(A,n-2)\oplus A/2^{(2)}\oplus \Gamma^2_{\mathbb{F}_2}(A/2^{(1)})\oplus A\otimes A/3^{(1)}\;.
\end{align*}
The difference between $(i)$ and $(ii)$ can be seen in the dimension of the ${\mathbb{F}_2}$-vector space $L_{2n}\Gamma^4(A,n)\otimes{\mathbb{F}_2}$. So we can use mod $2$ reduction to determine which of the two possibilities is correct.
Let $d_i(n)$, resp. $\delta_i(n)$, be the dimension of $L_{i}\Gamma^4(A,n)\otimes{\mathbb{F}_2}$, resp. $L_{i}\Gamma^4_{\mathbb{F}_2}(A/2,n)\otimes{\mathbb{F}_2}$. By proposition \ref{prop-lowdeg} $L_{2n-1}\Gamma^4(A,n)$ is an ${\mathbb{F}_2}$-vector space, hence $d_{2n-1}(n)=\dim ({_2}L_{2n-1}\Gamma^4(A,n))$. Similarly, $d_{2n-9}(n-2)=\dim ({_2}L_{2n-1}\Gamma^4(A,n))$. We have
\begin{align*}
d_{2n}(n)&= \delta_{2n}(n) - d_{2n-1}(n) && \text{ by \eqref{uct}}\\
&= \delta_{2n-8}(n-2)+\dim\Gamma^2_{\mathbb{F}_2}(A/2^{(1)})+\dim A/2^{(2)} - d_{2n-1}(n) && \text{ by lemma \ref{lm-m2descr}}\\
&= \delta_{2n-8}(n-2)+\dim\Gamma^2_{\mathbb{F}_2}(A/2^{(1)})+d_{2n-9}(n-2) && \text{ by prop. \ref{prop-lowdeg}}\\
&= d_{2n-8}(n-2)+\dim\Gamma^2_{\mathbb{F}_2}(A/2) && \text{ by \eqref{uctbis}}
\end{align*}
Thus the possibility $(ii)$ is excluded for dimension reasons so that $(i)$ holds.
\end{proof}
\begin{proposition}\label{prop-middegnodd}
If $n$ is odd, then
\begin{align}L_{2n}\Gamma^4(A,n)= L_{2n-8}\Gamma^4(A,n-2)\oplus \Lambda^2_{\mathbb{F}_2}(A/2^{(1)})\oplus A/2^{(2)}\oplus A\otimes A/3^{(1)}\;.\label{eq-2nodd}\end{align}
\end{proposition}
\begin{proof}
Recall that we have only two possibilities for $E_{2n}^\infty$, namely
\begin{enumerate}
\item[(a)] $E^\infty_{2n} = L_{2n-8}\Gamma^4(A,n-2)\oplus \Gamma^2_{\mathbb{F}_2}(A/2^{(1)})\oplus A\otimes A/3^{(1)}$
\item[(b)] $E^\infty_{2n} = L_{2n-8}\Gamma^4(A,n-2)\oplus A/2^{(1)}\oplus \Lambda_{\mathbb{F}_2}^2(A/2^{(1)})\oplus A\otimes A/3^{(1)}$
\end{enumerate}
We now list the possibilities for $L_{2n}\Gamma^4(A,n)$. By the same reasoning as in proposition \ref{prop-middegneven}, one finds the following three possibilities, where $L_{2n-8}\Gamma^4(A,n-2)'$ denotes the expression $L_{2n-8}\Gamma^4(A,n-2)$ with one copy of $A/2^{(2)}$ deleted:
\begin{align*}
&(i)\quad L_{2n-8}\Gamma^4(A,n-2)'\oplus \Gamma_{\mathbb{Z}}(A/2^{(1)})\oplus A\otimes A/3^{(1)}\;,\\
&(ii)\quad L_{2n-8}\Gamma^4(A,n-2)\oplus \Gamma^2_{\mathbb{F}_2}(A/2^{(1)})\oplus A\otimes A/3^{(1)}\;,\\
&(iii)\quad L_{2n-8}\Gamma^4(A,n-2)\oplus A/2^{(2)}\oplus \Lambda^2_{\mathbb{F}_2}(A/2^{(1)})\oplus A\otimes A/3^{(1)}\;.
\end{align*}
To be more specific, $(i)$ and $(ii)$ correspond to the possible reconstructions of $L_{2n}\Gamma^4(A,n)$ if (a) holds, and $(ii)$ and $(iii)$ correspond to the possible reconstructions if (b) holds.
We can exclude $(i)$ for dimension reasons. Indeed, we can compute the dimension of $L_{2n}\Gamma^4_{{\mathbb{F}_2}}(A/2,n)\otimes{\mathbb{F}_2}$ as in the proof of proposition \ref{prop-middegneven}. We find that $\dim L_{2n}\Gamma^4(A,n)\otimes{\mathbb{F}_2} = \dim E^\infty_{2n}\otimes{\mathbb{F}_2}$. By lemma \ref{lm-modp1} this implies that $L_{2n}\Gamma^4(A,n)$ is (non functorially) isomorphic to $E^\infty_{2n}$, hence it is a ${\mathbb{F}_2}$-vector space. Thus $(i)$ does not hold.
On the other hand, we can exclude $(ii)$ for functoriality reasons. Indeed, the universal coefficient theorem yields a surjective morphism of strict polynomial functors
\begin{align*}
L_{2n+1}\Gamma^4(A/2,n)\twoheadrightarrow {_2}L_{2n}\Gamma^4(A,n)\;.
\end{align*}
If $(ii)$ holds, then we can compose this surjective morphism with the projection onto the summand $\Gamma^2_{\mathbb{F}_2}(A/2^{(1)})$ of $L_{2n}\Gamma^4(A,n)$ to get a surjective morphism
\begin{align}
L_{2n+1}\Gamma^4(A/2,n)\twoheadrightarrow \Gamma^2_{\mathbb{F}_2}(A/2^{(1)})\;.\label{eq-surjection}
\end{align}
But the source of this map was computed in example \ref{ex-G4}. It is of the form
$$L_{2n+1}\Gamma^4(A/2,n) \simeq (A/2^{(1)}\otimes A/2^{(1)})\, \oplus\, (A/2^{(2)})^{\oplus k} $$
for some value of $k$. By the $\mathrm{Hom}$ computations of appendix \ref{app-comput}, there is no nonzero map from $A/2^{(2)}$ to $\Gamma^2_{\mathbb{F}_2}(A/2^{(1)})$, and the only nonzero morphism from $A/2^{(1)}\otimes A/2^{(1)}$ to $\Gamma^2_{\mathbb{F}_2}(A/2^{(1)})$ is not surjective. This contradicts the existence of the surjective morphism \eqref{eq-surjection}. Thus $(ii)$ does not hold.
It follows that $(iii)$ holds.
\end{proof}
\section{A conjectural description of the functors $L_i\Gamma^d_{\mathbb{Z}}(A,n)$}
\label{conj}
In this section, we return to the study of the derived functors $L_i\Gamma^d(A,n)$ for arbitrary abelian groups $A$. We therefore drop strict polynomial structures, and consider the derived functors as genuine functors of the abelian group $A$. The combinatorics of weights however, will still be present in the picture.
We begin with a new description of the stable homology of Eilenberg-Mac Lane spaces, equivalent to the Cartan's classical one. We then use this
new parametrization of the stable homology groups in order to
formulate a conjectural functorial description of the derived functors $L_*\Gamma^d(A,n)$ for all abelian groups $A$ and positive integers $n,d$.
We finally show that the computations of the present article as well as some computations of \cite{BM} agree with the conjecture.
\subsection{A new description of the stable homology} \label{new-descr}
We begin this section by quoting Cartan's computation of the stable homology:
$$H^{\mathrm{st}}_i(A):=\lim_n H_{n+i}(K(A,n),{\mathbb{Z}})\simeq H_{n+i}(K(A,n),{\mathbb{Z}})\text{ if $i<n$}\;.$$
To this purpose, we first recall Cartan's admissible words for the reader's convenience. Fix a prime number $p$. A $p$-admissible word is a non empty word $\alpha$ of finite length, formed with the letters $\varphi_p$, $\gamma_p$ and $\sigma$, satisfying the following two conditions:
\begin{enumerate}
\item[(1)] the word $\alpha$ starts with the letter $\sigma$ or $\varphi_p$,
\item[(2)] the number of letters $\sigma$ on the right of each letter $\gamma_p$ or $\varphi_p$ in $\alpha$ is even.
\end{enumerate}
A $p$-admissible word is of first type (`de premi\`ere esp\`ece') if it ends with a $\sigma$, and of second type (`de deuxi\`eme esp\`ece') if it ends with a $\varphi_p$ (words finishing on the right with the letter $\gamma_p$ will not be considered here).
There are two basic integers associated to a $p$-admissible word $\alpha$.
\begin{enumerate}
\item[$\bullet$] The degree of $\alpha$ is the integer $\deg(\alpha)$ defined recursively as follows. The degree of the empty word is zero, and
$$\qquad\deg(\varphi_p\beta)=2+p\deg(\beta)\;,\quad \deg(\sigma\beta)=1+\deg(\beta)\;,\quad \deg(\varphi_p\beta)=p\deg(\beta)\;.$$
\item[$\bullet$] The height of $\alpha$ is the integer $h(\alpha)$ corresponding to the number of letters $\sigma$ and $\varphi_p$ in $\alpha$.
\end{enumerate}
\begin{theorem}[{\cite[Exp. 11, Thm. 2]{cartan}}]\label{thm-cartan-stable}
There is a graded isomorphism, functorial with respect to the abelian group $A$:
$$H^{\mathrm{st}}_*(A)\simeq A[0]\,\oplus\, \bigoplus_{p\text{ prime}}\left(\bigoplus_{\alpha\in X_1(p)} A/p\,[\deg(\alpha)-h(\alpha)]\;\oplus\;\bigoplus_{\alpha\in X_2(p)} {_p A}[\deg(\alpha)-h(\alpha)]\right)\;, $$
where $X_i(p)$ stands for the set of $p$-admissible words of $i$-th type, starting on the left by the letters $\sigma\gamma_p$.
\end{theorem}
\begin{remark}
Cartan's proof of theorem \ref{thm-cartan-stable} is based on the integral computation of \cite[Exp. 11, Thm. 1]{cartan}. However, it can also be deduced from the mod $p$ computations of \cite[Exp. 9 and exp. 10]{cartan} (see also \cite{betley}) and the universal coefficient theorem, if one knows in advance that the stable homology only contains $p$-torsion for primes $p$. A simple proof of the latter fact is contained in \cite[Korollar 10.2]{d-p}.
\end{remark}
We now propose a compact reformulation of theorem \ref{thm-cartan-stable}.
Denote by $\EuScript A$ the set of sequences
$\alpha=(t_1,\dots, t_m)$ (for some $m\geq 1$) such that $t_1\geq
\dots \geq t_m>0.$ For every $\alpha\in {\EuScript A}$ let $o(\alpha)$ denote the
number of distinct strictly positive integers $t_j$ in the sequence $\alpha=(t_1,\dots, t_m)$.
For any abelian group $A$, we define an object $\EuScript B(A)$ of the derived category of abelian groups by the following formula:
\begin{equation}
\label{def-ba}
{\EuScript S}\mathrm{t}(A):=\bigoplus_{\alpha=(t_1,\dots, t_m)\in \EuScript A}\,\,
\bigoplus_{p\ \text{prime}}A\buildrel{L}\over\otimes \mathbb Z/p^{\buildrel{L}\over\otimes
o(\alpha)}[2(p^{t_1}+\dots+p^{t_m}-m)]
\end{equation}
Cartan's description of the integeral stable homology $H^{\rm st}_\ast(A)$ of an Eilenberg-Mac Lane space $K(A,n)$ may then be rephrased as follows:
\begin{theorem}\label{thm-new-formula}
There exists, functorially in the abelian group $A$, a graded isomorphism:
\begin{equation}
\label{hista}
H_*^{\rm st}(A)\simeq H_*(A\oplus {\EuScript
S}\mathrm{t}(A)).
\end{equation}
\end{theorem}
\begin{proof}
We observe that there is a bijection $\xi:X_1(p)\xrightarrow[]{\simeq} X_2(p)$ where $\xi(\alpha)$ is obtained from $\alpha$ by replacing the last two letters $\sigma^2$ of $\alpha$ by the single letter $\varphi_p$. This bijection preserves the degree, and decreases the height by one, so that the $p$-primary part in the description of $H_*^{\mathrm{st}}(A)$ theorem \ref{thm-cartan-stable} can be rewritten as:
\begin{align}\bigoplus_{\alpha\in X_1(p)} \left(\; A/p\,[\deg(\alpha)-h(\alpha)]\;\oplus\;{_p A}[\deg(\alpha)-h(\alpha)+1]\;\right)\;.\label{eq-pr-1}
\end{align}
We define an equivalence relation $\mathcal{R}$ on $X_1(p)$ in the following way. For a word $\alpha\in X_1(p)$ its \emph{$\sigma^2\gamma_p$-substitution} is the word of $X_1(p)$ obtained by replacing each occurence of the letter $\varphi_p$ by the group of letters $\sigma^2\gamma_p$. For example, the $\sigma^2\gamma_p$-substitution of $\sigma\gamma_p\varphi_p\sigma^4\varphi_p\sigma^2$ is the word $\sigma\gamma_p\sigma^2\gamma_p\sigma^6\gamma_p\sigma^2$. We say that two words of $X_1(p)$ are equivalent if they have the same $\sigma^2\gamma_p$-substitution.
Let $\alpha$ be a $p$-admissible word beginning with the letter $\sigma\gamma_p$. We say that a word $\alpha$ is \emph{restricted} if it is composed only of the letters $\sigma$ and $\gamma_p$ (i.e. no $\varphi_p$ occurs in the word $\alpha$).
The set $R(p)$ of restricted $p$-admissible words form a subset of $X_1(p)$. Each equivalent class of $X_1(p)$ contains exactly one restricted admissible word so that we can rewrite the direct sum \eqref{eq-pr-1} as:
\begin{align}\bigoplus_{\alpha\in R(p)}\, \bigoplus_{\beta\mathcal{R}\alpha} \left(\; A/p\,[\deg(\beta)-h(\beta)]\;\oplus\;{_p A}[\deg(\beta)-h(\beta)+1]\;\right)\;.\label{eq-pr-2}
\end{align}
We can replace in \eqref{eq-pr-2} the indexing set $R(p)$ by the set $\EuScript A$. Indeed there is a bijection $\chi:R(p)\xrightarrow[]{\simeq} \EuScript A$ defined as follows. Each restricted $p$-admissible word has the form:
\begin{align}\sigma\gamma_p\underbrace{(\sigma^2)\dots(\sigma^2)}_{\text{$k_1$ terms}}\gamma_p\underbrace{(\sigma^2)\dots(\sigma^2)}_{\text{$k_2$ terms}}\gamma_p\dots \gamma_p\underbrace{(\sigma^2)\dots(\sigma^2)}_{\text{$k_s$ terms}}\;, \label{eq-pr-3}\end{align}
where $s$ is the number of occurences of $\gamma_p$ and the $k_i$ are nonnegative. Our bijection $\chi$ is defined by sending such a restricted $p$-admissible word to the sequence of integers:
\begin{align}(\underbrace{s,\dots,s}_{\text{$k_s$ terms}}, \underbrace{s-1,\dots,s-1}_{\text{$k_{s-1}$ terms}},\dots,\underbrace{1,\dots,1}_{\text{$k_1$ terms}})\;. \label{eq-pr-4}\end{align}
If we define the degree of a sequence $(t_1,\dots,t_n)\in\EuScript A$ as the sum $2(p^{t_1}+\dots+p^{t_n})$, and its height as $2n$, then the bijection $\chi$ preserves the degree and the height.
Now we describe the $\mathcal{R}$-equivalent classes in $X_1(p)$. Let $C$ be one such equivalent class, containing a restricted admissible word $\alpha$ of the form \eqref{eq-pr-3}. The element in $C$ are all the words which can be obtained from $\alpha$ by substituting some groups of letters $(\sigma^2)\gamma_p$ by the letter $\varphi_p$. Observe that the number of groups of letters $(\sigma^2)\gamma_p$ available for substitution is equal to the number of positive $k_i$, which equals $o(\alpha)-1$ (where $o(\alpha)$ is the number of distinct positive integers in the associated sequence \eqref{eq-pr-4}). Thus $C$ contains $2^{o(\alpha)-1}$ elements. Moreover there are $\binom{o(\alpha)-1}{i}$ distinct words $\beta$ obtained from $\alpha$ by exactly $i$ substitutions, and they all satisfy the conditions $\deg(\beta)=\deg(\alpha)$ and $h(\beta)=h(\alpha)-i$. As a consequence \eqref{eq-pr-2} can be rewritten as a direct sum:
\begin{align}\bigoplus_{\alpha\in \EuScript A}\, \bigoplus_{i=0}^{o(\alpha)-1} \left(\; A/p\,[\deg(\alpha)-h(\alpha)-i]\;\oplus\;{_p A}[\deg(\alpha)-h(\alpha)-i+1]\;\right)^{\oplus\binom{o(\alpha)-1}{i}}\;.\label{eq-pr-5}
\end{align}
Finally the graded abelian group $_pA[1]\oplus A/p[0]$ is the homology of the complex $A\buildrel{L}\over\otimes {\mathbb{Z}}/p$, and the object ${\mathbb{Z}}/p\buildrel{L}\over\otimes {\mathbb{Z}}/p$ is isomorphic to ${\mathbb{Z}}/p[0]\oplus {\mathbb{Z}}/p[1]$ in the derived category. This determines a functorial isomorphism of graded abelian groups by induction on $n$:
\begin{align}
H_*(A\buildrel{L}\over\otimes {\mathbb{Z}}/p^{\buildrel{L}\over\otimes n})\simeq \bigoplus_{i=0}^{n-1} ({_pA}[1+i]\oplus A/p[i])^{\oplus\binom{n-1}{i}}\label{eq-pr-6}
\end{align}
The statement of theorem \ref{thm-new-formula} follows
by combining \eqref{eq-pr-5} and \eqref{eq-pr-6}.
\end{proof}
Just as one speaks of stable homology, one defines the stable derived functors \cite[(8.3)]{d-p}:
$$ L^{\rm st}_iF(A)=\lim_n L_{n+i}F(A,n)\simeq L_{n+i}F(A,n) \text{ if $i<n$}\;. $$
As explained in appendix \ref{han}, there exists a graded isomorphism, natural with respect to the abelian group $A$:
\begin{align}
H_*^{\rm st}(A)\simeq \bigoplus_{d\ge 0} L_*^{\rm st}S^d(A)\;.\label{eq-isost}
\end{align}
By d\'ecalage the (stable) derived functors of symmetric powers are isomorphic to the (stable) derived functors of the divided power functors. Hence \eqref{eq-isost} can be used to obtain a description of $L_*^{\rm st}\Gamma^d(A)$. However, to obtain such a description, we have to separate the various summands. In other words, we have to determine which summands $_pA$ and $A/p$ contribute to which stable derived functors $L_*^{\rm st}\Gamma^d(A)$. To this purpose, we define yet another integer associated to $p$-admissible words.
\begin{enumerate}
\item[$\bullet$] Let $r(\alpha)$ be the number of occurences of the letters $\varphi_p$ and $\gamma_p$ in a $p$-admissible word $\alpha$. Then the weight $w(\alpha)$ is defined by $w(\alpha)=p^{r(\alpha)}$ if $\alpha$ is of first type, and $w(\alpha)=p^{r(\alpha)-1}$ if $\alpha$ is of second type.
\end{enumerate}
Then as explained e.g. in \cite[section 10.2]{antoine} the direct summand of $H_*^{\rm st}(A)$ corresponding to $L_*^{\rm st}S^d(A)$ is given by the admissible words of weight $d$. We have to translate this in the new indexation provided by theorem \ref{thm-new-formula}. We obtain the following description
of the stable derived functors of the functor $\Gamma^d(A)$.
\begin{proposition}\label{prop-new} If the integer $d$ is not a prime power, then
\label{lprop-new}
\[L_i^{\mathrm{st}}\Gamma^d(A)=0\ \ {\it for \ all} \ i\ge 0 .\]
For any prime number $p$ and any integer $r>0$ the following is true:
$$L_i^{\mathrm{st}}\Gamma^{p^r}(A)=H_i\left(\bigoplus_{\alpha=(t_1,\dots, t_m)\in \EuScript A,\ t_1=r} A\buildrel{L}\over\otimes \mathbb Z/p^{\buildrel{L}\over\otimes
o(\alpha)}[2(p^{t_2}+\dots+p^{t_m}-m)]\right).
$$
\end{proposition}
\begin{proof}
Returning to the proof of theorem \ref{thm-new-formula}, we see that the bijection $\xi$ preserves the weights. Moreover, all the words in the same equivalent class in $X_1(p)$ also have the same weight. Thus all the terms corresponding to the homology of the summand $A\buildrel{L}\over\otimes{\mathbb{Z}}/p^{\otimes o(\alpha)}$ have the same weight as $\alpha$. Now if $\alpha$ is a restricted $p$-admissible word corresponding to the sequence $\chi(\alpha)=(t_1,\dots,t_n)\in \EuScript A$ then $w(\alpha)=p^{t_1}$. Proposition \ref{lprop-new} follows.
\end{proof}
\subsection{The conjecture}
We observed in proposition \ref{lprop-new} that the stable derived functors of $\Gamma^d(A)$ may conveniently be repackaged by appropriately deriving those summands $A/p$ which most obviously occur in Cartan's computation, these being the summands which correspond to {\it restricted} admissible words, that is the words which do not involve the transpotence operation $\varphi_p$.
Our contention in what follows is that a similar mechanism also works unstably.
To be more specific, let
$\EuScript A_{\le m}\subset \EuScript A$ denote the subset containing all the sequences of length less or equal to $m$. We can reformulate the definition of ${\EuScript A}_{\le m}$ as:
\begin{align}{\EuScript A}_{\le m}=\{ (t_1,\dots,t_m)\;:\; t_1\ge t_2\dots\ge t_m\ge 0 \text{ and } t_1\ne 0\}\;.\end{align}
For every $\alpha\in {\EuScript A}_{\le m}$ we still denote by $o(\alpha)$ the
number of distinct strictly positive integers $t_j$ in the sequence $\alpha=(t_1,\dots, t_m)$. The $\EuScript A_{\le m}$ form an increasing family of subsets which exhaust $\EuScript A$. Moreover $\EuScript A_{\le m}$ can be interpreted as an indexing set for those stable summands $A/p$ corresponding to restricted admissible words which already appear in the unstable homology of $K(A,2m+1)$, or equivalently in the derived functors $L_*\Gamma(A,2m-1)$.
Our conjecture asserts that the derived functors $L_*\Gamma(A,n)$ are filtered, and that the associated graded pieces have the following description. If $n=2m+1$ is odd there is an isomorphism (functorial with respect to an arbitrary abelian group $A$):
\begin{align}
\mathrm{gr}\, ( L_*\Gamma(A,n))\simeq \pi_*\left(L\Lambda(A)\otimes \bigotimes_{\text{$p$ prime}}\;\bigotimes_{\alpha\in {\EuScript A}_{\le m+1}} \, L\Lambda(A\buildrel{L}\over\otimes {\mathbb{Z}}/p^{\buildrel{L}\over\otimes o(\alpha)})\right)\;,\label{conj-u1}
\end{align}
and if $n=2m$ there is an isomorphism (functorial with respect to an arbitrary abelian group $A$):
\begin{align}
\mathrm{gr}\, ( L_*\Gamma(A,n))\simeq \pi_*\left(L\Gamma(A)\otimes \bigotimes_{\text{$p$ prime}}\;\bigotimes_{\alpha\in {\EuScript A}_{\le m}} \, L\Gamma(A\buildrel{L}\over\otimes {\mathbb{Z}}/p^{\buildrel{L}\over\otimes o(\alpha)})\right)\;.\label{conj-u2}
\end{align}
The two isomorphisms above do not preserve the homological grading. We will explain below how to introduce appropriate shifts of degrees on the right hand side so as to obtain graded isomorphisms. For the moment, we keep things simple by discussing the ungraded version of the conjecture.
Both sides of \eqref{conj-u1} and \eqref{conj-u2} are equipped with weights and our conjectural isomorphisms preserve the weights. To be more specific, the weight is defined on the left hand sides of \eqref{conj-u1} and \eqref{conj-u2} by viewing $\mathrm{gr}\,(L_*\Gamma^d(A,n))$ as the homogeneous summand of weight $d$. The weights on the right hand sides of \eqref{conj-u1} and \eqref{conj-u2} are defined as follows. We view $L_*\Lambda^d(A)$ and $L_*\Gamma^d(A)$ as functors a weight $d$, we wiew the factors $L\Lambda^d(A\buildrel{L}\over\otimes {\mathbb{Z}}/p^{\buildrel{L}\over\otimes o(\alpha)})$ and $L\Gamma^d(A\buildrel{L}\over\otimes {\mathbb{Z}}/p^{\buildrel{L}\over\otimes o(\alpha)})$ corresponding to a sequence $\alpha=(t_1,\dots)$ as functors of weight $p^{t_1}$ (compare proposition \ref{prop-new}) and weights are additive with respect to tensor products.
To describe more concretely the homogeneous component of weight $d$ of the right hand sides of isomorphisms \eqref{conj-u1} and \eqref{conj-u2}, we introduce the following notation.
Given a prime integer $p$, a nonnegative integer $m$, a nonnegative integer $d_0$, and a family of nonnegative integers $(d_\alpha)_{\alpha\in {\EuScript A}_{\le m}}$ which is not identically zero, we denote by $d(d_0,(d_\alpha),m;p)$ the positive integer defined by
\begin{align} d(d_0,(d_\alpha),m;p):= d_0+\sum_{\alpha\in {\EuScript A}_{\le m}} d_\alpha p^{t_1(\alpha)}\;,\end{align}
where $t_1(\alpha)$ denotes the first integer in the sequence $\alpha$, that is $\alpha=(t_1(\alpha),\dots)$. The integer $d(d_0,(d_\alpha),m;p)$ is the weight of the following objects of the derived category
\begin{align}&\mathcal{E}(d_0,(d_\alpha),m;p) := L\Lambda^{d_0}(A)\,\otimes \,\bigotimes_{\alpha\in{\EuScript A}_{\le m}}\, L\Lambda^{d_\alpha}(A\buildrel{L}\over\otimes {\mathbb{Z}}/p^{\buildrel{L}\over\otimes o(\alpha)})\;,
\label{conj-termE}
\\&\mathcal{D}(d_0,(d_\alpha),m;p) := L\Gamma^{d_0}(A)\,\otimes \,\bigotimes_{\alpha\in{\EuScript A}_{\le m}}\, L\Gamma^{d_\alpha}(A\buildrel{L}\over\otimes {\mathbb{Z}}/p^{\buildrel{L}\over\otimes o(\alpha)})\;.\label{conj-termD}
\end{align}
With these notations, the homogeneous component of weight $d$ of the right hand side of the isomorphism \eqref{conj-u1} is given by the homotopy groups of the direct sum of $L\Lambda^d(A)$ and all the terms $\mathcal{E}(d_0,(d_\alpha),m+1;p)$ such that $d(d_0,(d_\alpha),m+1;p)=d$, for all nonnegative integers $d_0$, all families of integers $(d_\alpha)$ which are not identically zero and all prime integers $p$. The right hand side of the isomorphism \eqref{conj-u2} has an obvious similar description.
Finally, we introduce suitable shifts in order to transform the isomorphisms \eqref{conj-u1} and \eqref{conj-u2} into graded isomorphisms.
Given a prime integer $p$, a nonnegative integer $m$ and a nonnegative integer $d_0$, and a family of nonnegative integers $(d_\alpha)_{\alpha\in {\EuScript A}_{\le m}}$ which is not identically zero, we set:
\begin{align}
& \ell(d_0,(d_\alpha),m;p):=(2m+1)d_0+\sum_{\alpha\in {\EuScript A}_{\le m}} \ell(\alpha;p)\,d_\alpha
\end{align}
where the integer $\ell(\alpha;p)$ associated to a sequence $\alpha=(t_1,\dots,t_m)$ is given by
\begin{align}
& \ell(\alpha;p):=\begin{cases}2p^{t_2}+...+2p^{t_m}+1 & \text{if}\ m>1\;,\\
1 & \text{if} \ m=1.
\end{cases}
\end{align}
We also set:
\begin{align}
e(d_0,(d_\alpha),m;p):=\left(\sum_{\alpha\in {\EuScript A}_{\le m}} d_\alpha\right)-d_0\;.
\end{align}
We may now state the graded version of our conjecture.
\begin{conjecture}
\label{conject}
Let $s$ be a positive integer, an let $n$ be a positive integer.
\begin{enumerate
\item Assume that $n=2m+1$ is odd. Then there exists a filtration on $ L_s\Gamma^d(A, n)$ such that
the associated graded functor is isomorphic to
the direct sum of the term
\begin{align*}
L_{s-nd}\Lambda^d(A)
\end{align*}
together with the terms
\begin{align*}
\pi_s\left(\;\mathcal{E}(d_0,(d_\alpha),m+1;p)\; [\ell(d_0,(d_\alpha),m+1;p)] \;\right)
\end{align*}
for all primes $p$, all nonnegative integers $d_0$ and all famillies of integers $(d_\alpha)_{\alpha\in {\EuScript A}_{\le m+1}}$ which are not identically zero, and satisfying
$d(d_0,(d_\alpha),m;p)=d$.
\item Similarly, if $n=2m$, there exists a filtration on
$L_s\Gamma^d(A,n)$ such that the associated graded functor is isomorphic to
the direct sum of the term
$$L_{s-nd}\Gamma^d(A)$$
together with the terms
\begin{align*}
\pi_s\left(\;\mathcal{D}(d_0,(d_\alpha),m;p)\; [\ell(d_0,(d_\alpha),m;p)+ e(d_0,(d_\alpha),m;p)] \;\right)
\end{align*}
for all primes $p$, all nonnegative integers $d_0$ and all famillies of integers $(d_\alpha)_{\alpha\in {\EuScript A}_{\le m}}$ which are not identically zero, and satisfying
$d(d_0,(d_\alpha),m;p)=d$.
\end{enumerate}
\end{conjecture}
The remainder of the present section is devoted to proving that the conjecture holds in a certain number of cases.
\subsection{The cases $d=2$, $d=3$, for all $A$ and all $n$.}
For $s,m\geq 1,$ and $d= 2,3$, there is no filtration to consider, and conjecture \ref{conject} reduces
to the natural
isomorphisms
\begin{equation}\label{gamma2description}
L_s\Gamma^2(A,2m+1)\simeq
\pi_s\left(\bigoplus_{i=0}^{m}\left(A\buildrel{L}\over\otimes
{\mathbb{Z}}/2[2m+2i+1]\right)\oplus L\Lambda^2(A)[4m+2]\right)
\end{equation}
\begin{equation}
L_s\Gamma^2(A,2m)\simeq
\pi_s\left(\bigoplus_{i=0}^{m-1}\left(A\buildrel{L}\over\otimes
{\mathbb{Z}}/2[2m+2i]\right)\oplus L\Gamma^2(A)[4m]\right)
\end{equation}
and
\begin{multline}
L_s\Gamma^3(A,2m+1)\simeq\\
\pi_s\left(\bigoplus_{i=0}^{m}\left(A\buildrel{L}\over\otimes {\mathbb{Z}}/3[2m+4i+1]\oplus
A\buildrel{L}\over\otimes A\buildrel{L}\over\otimes {\mathbb{Z}}/2[4m+2i+2]\right)\oplus
L\Lambda^3(A)[6m+3]\right),\label{gamma3description}
\end{multline}
\begin{multline}
L_s\Gamma^3(A,2m)\simeq\\
\pi_s\left(\bigoplus_{i=0}^{m-1}\left(A\buildrel{L}\over\otimes {\mathbb{Z}}/3[2m+4i]\oplus
A\buildrel{L}\over\otimes A\buildrel{L}\over\otimes {\mathbb{Z}}/2[4m+2i]\right)\oplus
L\Gamma^3(A)[6m]\right).\label{gamma3description-a}
\end{multline}
consistently with the results in paragraphs 4 and 5 of \cite{BM}.
\subsection{The case $d=4$, for $A$ free and $n$ odd.}
We now proceed to prove that the conjecture agrees with our previous computations for $d=4$, $A$ free and $n=2m+1$ with $m\ge 0$. In that case the conjecture says that (up to a filtration), $L_*\Gamma^4(A,n)$ is isomorphic to the homology groups of the following complexes \eqref{eq:1A}-\eqref{eq:1G}.
At the prime $p=3$, there is a single sort of complex:
\begin{subequations}\label{eq:1}
\renewcommand\theequation{\roman{equation}}
\begin{align} A\buildrel{L}\over\otimes A\buildrel{L}\over\otimes \mathbb Z/3\ [4m+4k+2] &\hspace{1cm} 0\leq k \leq m. \label{eq:1A}
\intertext{At the prime $p=2$, we have the five following types of complexes:}
L\Lambda^2(A)\buildrel{L}\over\otimes A\buildrel{L}\over\otimes \mathbb
Z/2\ [6m+2k+3] & \hspace{1cm} 0 \leq k \leq m \label{eq:1B} \\
L\Lambda^2(A\buildrel{L}\over\otimes \mathbb Z/2)[4m+4k+2]&\hspace{1cm} 0 \leq k \leq m
\label{eq:1C}
\\
A\buildrel{L}\over\otimes \mathbb Z/2\buildrel{L}\over\otimes \mathbb
Z/2\ [2m+6k+2l+1]& \hspace{1cm}0 \leq k+l\leq m, \ \ l
\neq 0
\label{eq:1D}
\\
A\buildrel{L}\over\otimes \mathbb Z/2\ [2m+6k+1] &\hspace{1cm}
0 \leq k \leq m\label{eq:1E} \\
A\buildrel{L}\over\otimes A\buildrel{L}\over\otimes \mathbb Z/2\buildrel{L}\over\otimes
\mathbb Z/2\ [4m+2k+2l+2] & \hspace{1cm} 0\leq l < k \leq m \label{eq:1F}
\intertext{together with the final term:}
L\Lambda^4(A)[8m+4]. \label{eq:1G} &
\end{align}
\end{subequations}
To verify that these expressions coincide with our computations from section \ref{der-gamma4-sec}, we proceed by induction on $m$.
For $m=0$, the formula for $L_*\Gamma^4(A,1)$ was described in proposition \ref{prop-GA41}. We filter the term $L_3\Gamma^4(A,1) = \Phi^4(A)$ in \eqref{gr3ga4a11}
and replace it here by the direct sum $\Lambda^2(A)\otimes A/2)\oplus\Gamma^2_{{\mathbb{F}_2}}(A/2)$. The result of proposition \ref{prop-GA41} then coincides with the present $m=0$ case, once we observe that $L_\ast\Lambda^2(A/2)\simeq \Lambda^2(A/2)[0]\oplus\Gamma^2_{{\mathbb{F}_2}}(A/2)[1]$ (\cite{BM} \S 2.2,
\cite{baupira}).
\bigskip
To prove that the formulas provided by the conjecture agree for a general $n=2m+1$ with the computations of theorem \ref{thm-G4Z}, it suffices to show that the additional summands predicted by the conjecture when passing from $L_*\Gamma^4(A,n-2)[8]$ to $L_*\Gamma^4(A,n)$ agree with those obtained in \eqref{desg4}. Let us denote by $G(m)$ the sum of all the terms (i)-(vii). One verifies that $ G(m)=G(m-1)[8] \oplus \Delta(m) $, where $\Delta(m)$ is given by:
\begin{align*}\Delta(m)=\, &\, A\otimes A/3[2n]\,\oplus\, L\Lambda^2(A)\otimes A/2[3n]\,\oplus\, L\Lambda^2(A/2)[2n]\,\\
&\oplus \left(\bigoplus_{\ell=1}^m A\buildrel{L}\over\otimes{\mathbb{Z}}/2^{\buildrel{L}\over\otimes 2}\right)\,\oplus\, A/2[n]\oplus \left(\bigoplus_{k=1}^m A\buildrel{L}\over\otimes A\buildrel{L}\over\otimes {\mathbb{Z}}/2^{\buildrel{L}\over\otimes 2}[2n+2k]\right)\;.\end{align*}
In $\Delta(m)$ we replace the expression ${\mathbb{Z}}/2^{\buildrel{L}\over\otimes 2}$ by ${\mathbb{Z}}/2[0]\oplus{\mathbb{Z}}/2[1]$, $L\Lambda^2(A/2)$ by $\Lambda^2(A/2)[0]\oplus\Gamma^2_{{\mathbb{F}_2}}(A/2)[1]$. Then $\Delta(m)$ coincides with the additional summands occuring when one passes for $n$ odd from $L_*\Gamma^4(A,n-2)[8]$ to $L_*\Gamma^4(A,n-2)$ in theorem \ref{deriveddescr}, provided we replace the summand $\Gamma^2_{\mathbb{F}_2}(A/2)\otimes A/2^{(1)}[3n]$ in \eqref{desg4} by the direct sum $\Lambda^2(A/2)\otimes A/2[3n]\oplus A/2\otimes A/2[3n]$. This proves the conjecture for $d=4$, $A$ free and $n=2m+1$ odd.
\subsection{The case $d=4$, for $A$ free and $n$ even.}
We now proceed to prove that the conjecture agrees with our previous computations for $d=4$ and $A$ free and $n=2m$ with $m\ge 1$.
It is straightforward to verify that the conjecture agrees for $m=1$ with the computation of proposition \ref{prop-GA42}, since we know that $L_\ast\Gamma^2(A/2)\simeq \Gamma^2_{\mathbb{Z}}(A/2)[0]\oplus \Gamma^2_{\mathbb{F}_2}(A/2)[1]$ (\cite{BM}, \cite{baupira} \S 4 ) (there is no filtration to consider in this situation).
The conjecture says that (up to a filtration) $L_*\Gamma^4(A,n)$ is isomorphic to the homology groups of the following complexes \eqref{eq:2A}-\eqref{eq:2G}.
At the prime $p=3$, there is a single sort of complex:
\begin{subequations}\label{eq:2}
\renewcommand\theequation{\alph{equation}}
\begin{align}
\renewcommand\theequation{\roman{equation}}
A\buildrel{L}\over\otimes A\buildrel{L}\over\otimes \mathbb Z/3\ [4m+4k] & \hspace{1cm} 0 \leq k \leq m-1 \label{eq:2A}\\
\intertext{For the prime $p=2$, we have the following complexes:}
L\Gamma^2(A)\buildrel{L}\over\otimes A\buildrel{L}\over\otimes \mathbb
Z/2\ [6m+2k]& \hspace{1cm} 0 \leq k \leq m-1 \label{eq:2B}\\
L\Gamma^2(A\buildrel{L}\over\otimes \mathbb Z/2)\ [4m+4k]& \hspace{1cm} 0 \leq k \leq m-1 \label{eq:2C}
\\A\buildrel{L}\over\otimes \mathbb Z/2\buildrel{L}\over\otimes \mathbb
Z/2\ [2m+6k+2l] & \hspace{1cm} 0 \leq k+l \leq m-1,
\ l\neq 0 \label{eq:2D}\\
A\buildrel{L}\over\otimes \mathbb Z/2\ [2m+6k] &\hspace{1cm}
0 \leq k \leq m-1 \label{eq:2E}\\
A\buildrel{L}\over\otimes A\buildrel{L}\over\otimes \mathbb Z/2\buildrel{L}\over\otimes
\mathbb Z/2\ [4m+2k+2l] & \hspace{1cm} 0\leq l < k \leq m-1 \label{eq:2F} \\
\intertext{together with the final term:}
L\Gamma^4(A)[8m] & \hspace{1cm}\label{eq:2G}
\end{align}
\label{lm-calc3}
\end{subequations}
To prove that the formulas provided by the conjecture agree for $n = 2m$ with the computations of theorem \ref{thm-G4Z}, it suffices to show as above that the additional summands predicted by the conjecture when passing from $L_*\Gamma^4(A,n-2)[8]$ to $L_*\Gamma^4(A,n)$ agree with those obtained in \eqref{desg41}. Let us denote by $J(m)$ the sum of all the terms (a)-(g). One verifies that $ J(m)=J(m-1)[8] \oplus \Delta'(m)$ where
\begin{align*}
\Delta'(m) =\, &\, A \otimes A/3[2n] \, \oplus \, \Gamma^2 A \otimes A/2 [3n] \, \oplus \, L\Gamma^2(A/2)[2n] \\
& \oplus \, \left( \bigoplus_{l=1}^{m-1} \, A \stackrel{L}{\ot} {\mathbb{Z}}/2 \stackrel{L}{\ot} {\mathbb{Z}}/2 [2n+2l] \right) \, \oplus \, A/2 \, \oplus \, \left( \bigoplus_{k=1}^{m-1} A/2 \stackrel{L}{\ot} A/2 [4m+2k]\right) .
\end{align*}
We replace in $\Delta'(m)$ the summand $L\Gamma^2(A/2)$ by its value $\Gamma^2_{\mathbb{Z}}(A/2)[0] \oplus \Gamma^2_{\mathbb{F}_2}(A/2)[1]$ (as in \cite{BM} \S 2.2,
\cite{baupira} \S 4), and once more replace
$ {\mathbb{Z}}/2 \stackrel{L}{\ot} {\mathbb{Z}}/2$ by ${\mathbb{Z}}/2[0]\oplus{\mathbb{Z}}/2[1]$. Then $\Delta'(m)$ coincides with the additional summands occuring when one passes for $n$ even from $L_*\Gamma^4(A,n-2)[8]$ to $L_*\Gamma^4(A,n-2)$ in theorem \ref{deriveddescr}. Note that in this $n$ even case, no summand in $\Delta'(m) $ requires any filtering.
\subsection{The case $n=1$, $A$ free and all $d$}
In the case $n=1$ and $A$ free, the conjecture asserts that, up to a filtration, the $p$-primary part of $L_*\Gamma^d(A,1)$ is isomorphic to
\begin{align}\pi_*\left(\bigoplus_{(k_0,,k_1,\dots,k_d)}\Lambda^{k_0}(A)\buildrel{L}\over\otimes L\Lambda^{k_1}(A/p)\buildrel{L}\over\otimes\dots\buildrel{L}\over\otimes L\Lambda^{k_d}(A/p)\;[k_0+\dots+k_d]\right)\;,\label{eq-conjn1}\end{align}
where the sum runs over all sequences of nonnegative integers $(k_0,k_1,\dots, k_d)$ of length exactly $d+1$, satisfying $\sum k_ip^{i}=d$.
On the other hand, we have explicitly computed the derived functors $L_*\Gamma(A,1)$ in section \ref{der1-gamma-z}. By theorem \ref{thm-calcul-LG-un-Z} and proposition \ref{prop-qt-SK}, the $p$-primary part of $L_i\Gamma^d(A,1)$ is concentrated in degrees $i<d$ and it is isomorphic (up to a filtration) in these degrees to the homogeneous component of degree $i$ and weight $d$ of the cycles of the tensor product of an exterior algebra with trivial differential and a family of Koszul algebras:
\begin{align}\left(\Lambda_{\mathbb{F}_p}(A/p[1]),0\right)\otimes \bigotimes_{r\ge 1} \left(\Lambda_{\mathbb{F}_p}(A/p^{(r)}[1])\otimes\Gamma_{\mathbb{F}_p}(A/p^{(r)}[2]),d_\mathrm{Kos}\right)\;. \label{eq-Kos-conj}\end{align}
We will now reformulate this result in a different form, closer to \eqref{eq-conjn1}. For this, we consider the following modification of the Koszul algebra over ${\mathbb{Z}}$, namely the dg-$\mathcal{P}_{\mathbb{Z}}$-algebra $(\Gamma(A[2])\otimes \Lambda(A[1]),pd_{\mathrm{Kos}})$ with the same underlying graded $\mathcal{P}_{\mathbb{Z}}$-algebra, but whose differential is the Koszul differential multiplied by $p$.
We denote by $C^k(A)$ the complex of functors given by its homogeneous component of weight $k$.
\begin{lemma}\label{lm-transfo-1}
For $i<d$, the $p$-primary part of the functor $L_i\Gamma^d(A,1)$ is isomorphic to the homology of the complex
$$\bigoplus_{(k_0,\dots,k_d)}\left(\Lambda^{k_0}(A)[k_0],0\right)\otimes C^{k_1}(A)\otimes \dots \otimes C^{k_d}(A) $$
where the sum runs over all sequences of nonnegative integers $(k_0,k_1,\dots, k_d)$ satisfying $\sum k_ip^{i}=d$.
\end{lemma}
\begin{remark}
The isomorphism of lemma \ref{lm-transfo-1} is really an isomorphism of functors, not an isomorphism of strict polynomial functors. For example, this isomorphism does not preserve the weights.
\end{remark}
\begin{proof}[Proof of lemma \ref{lm-transfo-1}]
{\bf Step 1.} Since we are interested in the homogeneous component of weight $d$ of \eqref{eq-Kos-conj}, we can limit ourselves to the differential graded subalgebra of \eqref{eq-Kos-conj}:
\begin{align}\left(\Lambda_{\mathbb{F}_p}(A/p[1]),0\right)\otimes \bigotimes_{1\le r\le d} \left(\Lambda_{\mathbb{F}_p}(A/p^{(r)}[1])\otimes\Gamma_{\mathbb{F}_p}(A/p^{(r)}[2]),d_\mathrm{Kos}\right)\;. \label{eq-Kos-sub}\end{align}
By forgeting the strict polynomial structure, the strict polynomial functors $A/p^{(r)}$ become isomorphic to $A/p$, and the differential graded algebra \eqref{eq-Kos-sub} is functorially isomorphic to the differential graded algebra:
\begin{align}\left(\Lambda_{\mathbb{F}_p}(A/p[1]),0\right)\otimes \bigotimes_{1\le r\le d} \left(\Lambda_{\mathbb{F}_p}(A/p[1])\otimes\Gamma_{\mathbb{F}_p}(A/p[2]),d_\mathrm{Kos}\right)\;. \label{eq-Kos-sub2}\end{align}
Moreover, under this isomorphism, the homogeneous summand of weight $d$ of \eqref{eq-Kos-sub} corresponds to the homogeneous summand of \eqref{eq-Kos-sub2} supported by the subfunctors
\begin{align}
\bigoplus \Lambda_{\mathbb{F}_p}^{k_0}(A/p[1])\otimes \Lambda_{\mathbb{F}_p}^{a_1}(A/p)\otimes \Gamma^{b_1}_{\mathbb{F}_p}(A/p)\otimes \dots \otimes \Lambda_{\mathbb{F}_p}^{a_\ell}(A/p)\otimes \Gamma^{b_\ell}_{\mathbb{F}_p}(A/p)\;.
\label{eq-summand1}
\end{align}
where the sum runs over all sequences $(k_0,a_1,b_1,\dots,a_{\ell},b_{\ell})$ satisfying $k_0+\sum (a_i+b_i)p^i=d$.
{\bf Step 2.} We claim that the subalgebra of cycles of positive degree of functorial graded algebra \eqref{eq-Kos-sub2} is isomorphic to the homology algebra of:
\begin{align}\left(\Lambda_{\mathbb{Z}}(A[1]),0\right)\otimes \bigotimes_{1\le r\le d} \left(\Lambda_{\mathbb{Z}}(A[1])\otimes\Gamma_{\mathbb{Z}}(A[2]),pd_\mathrm{Kos}\right)\;, \label{eq-Kos-sub3}\end{align}
and we claim that the isomorphism sends the kernels of the summand \eqref{eq-summand1} of the differential graded algebra \eqref{eq-Kos-sub2} isomorphically to the homology of the following summand of the differential graded algebra \eqref{eq-Kos-sub3}
$$\bigoplus_{(k_0,\dots,k_d)}\left(\Lambda^{k_0}(A[1]),0\right)\otimes C^{k_1}(A)\otimes \dots \otimes C^{k_d}(A), $$
where the sum runs over all sequences of nonnegative integers $(k_0,k_1,\dots, k_d)$ satisfying $\sum k_ip^{i}=d$.
The statement of lemma \ref{lm-transfo-1} follows from this claim, so to finish the proof of lemma \ref{lm-transfo-1}, we only have to justify our claim.
Koszul algebras are acyclic in positive degrees by proposition \ref{prop-homology-Koszul}. Hence, the positive degree homology of the differential graded algebra \eqref{eq-Kos-sub3} is equal to the mod $p$ reduction of the algebra formed by the cycles of positive degree of
$$\left(\Lambda_{\mathbb{Z}}(A[1]),0\right)\otimes \bigotimes_{1\le r\le d} \left(\Lambda_{\mathbb{Z}}(A[1])\otimes\Gamma_{\mathbb{Z}}(A[2]),d_\mathrm{Kos}\right)\;.$$
The latter is isomorphic to the algebra formed by the cycles of positive degree of the algebra \eqref{eq-Kos-sub2}. This justifies our claim.
\end{proof}
We are now going to check that the graded functor \eqref{eq-conjn1} can also be rewritten as the homology of the complex of lemma \ref{lm-transfo-1}. This will follow from the following lemma. Recall that $C^k(A)$ is the homogeneous part of weight $k$ of $(\Gamma(A[2])\otimes \Lambda(A[1]),pd_{\mathrm{Kos}})$, hence its desuspension $C^k(A)[-k]$ is the homogeneous part of weight $k$ of $(\Gamma(A[1])\otimes \Lambda(A[0]),pd_{\mathrm{Kos}})$.
\begin{lemma}\label{lm-smallmodel}
Let $A$ be a free abelian group, and let $k$ be a positive integer. The normalized chains of $L\Lambda^k(A/p)$ are naturally isomorphic to the complex $C^k(A)[-k]$.
\end{lemma}
\begin{proof}
We have $L\Lambda(A/p)=\Lambda(K(A\xrightarrow[]{\times p}A))$. The functor $K$ is explicit, and we readily check that for all complexes $C_1\xrightarrow[]{\partial} C_0$, the normalized chains of the simplicial object $\Lambda(K(C_1\xrightarrow[]{\partial} C_0))$ is the complex whose degree $n$ component is
$$\Lambda^{>0}(C_1)^{\otimes n}\otimes \Lambda(C_0)\;,$$
where $\Lambda^{>0}(C_i)$ stands for the augmentation ideal of the exterior algebra,
and whose differential maps an element $x_1\otimes \dots \otimes x_n\otimes y$ to the sum:
$$\sum_{i=1}^{n-1}(-1)^i x_1\otimes \dots\otimes x_i x_{i+1}\otimes\dots\otimes x_n\otimes y+(-1)^n x_1\otimes\dots\otimes x_{n-1}\otimes \Lambda(\partial)(x_n)y\;. $$
Consider the normalized chains $\mathcal{N}\Lambda(A/p)$ of $L\Lambda(A/p)$ as a differential graded $\mathcal{P}_{\mathbb{Z}}$-algebra. Its homogeneous component of weight $k$ yields the normalized chains $\mathcal{N}\Lambda^k(A/p)$ of $L\Lambda^k(A/p)$. These normalized chains contain $C^k(A)[-k]$ as a subcomplex. Indeed the inclusion is simply given by seeing the object of degree $i$ of $C^k(A)[-k]$, that is the functor $\Gamma^i(A)\otimes \Lambda^{k-i}(A)$, as a subfunctor of $(\Lambda^1(A))^{\otimes i}\otimes \Lambda^{k-i}(A)$ by the canonical inclusion of invariants into the tensor product.
One readily checks that the differential of $C^k(A)[-k]$ coincides with the restriction of the differential of $\mathcal{N}\Lambda^k(A/p)$.
To finish the proof, it remains to show that the inclusion of complexes
\begin{align}C^k(A)[-k]\hookrightarrow \mathcal{N}\Lambda^k(A/p)\label{eq-qios}\end{align}
is a quasi-isomorphism. For this, we filter both complexes by the weight of the exterior power on the right, that is the term
$F_s(C^k(A)[-k])$ of the filtration is the subcomplex of $C_k(A)$ supported by the $\Gamma^i(A)\otimes \Lambda^{k-i}(A)$ for $k-i\ge s$, and the term $F_s(\mathcal{N}\Lambda^k(A/p))$ is the subcomplex of $\mathcal{N}\Lambda^k(A/p)$ supported by the $\Lambda^{i_1}(A)\otimes\dots\otimes \Lambda^{i_n}(A)\otimes\Lambda^{k-\sum i_j}(A)$ for $k-\sum i_j\ge s$. Both filtrations have finite length, and the inclusion of complexes preserves the filtrations, whence a morphism:
\begin{align}\mathrm{gr}\, (C^k(A)[-k])\to \mathrm{gr}\, (\mathcal{N}\Lambda^k(A/p))\;.\label{eq-qiosgr}\end{align}
Since the filtrations are finite, the fact that \eqref{eq-qios} is a quasi-isomorphism will follow from the fact that \eqref{eq-qiosgr} is. But we readily check that $\mathrm{gr}\, (C^k(A)[-k])$ is equal to the complex $(\bigoplus_{i=0}^k\Gamma^i(A)\otimes\Lambda^k(A),0)$ (with zero differential) that $ \mathrm{gr}\, (\mathcal{N}\Lambda^k(A/p))$ is equal to the complex
$\bigoplus_{i=0}^k \mathcal{N}\Lambda^i(A)\otimes (\Lambda^k(A),0)$, and that the morphism \eqref{eq-qiosgr} is simply the morphism constructed from the quasi-isomorphisms $\Gamma^i(A)[i]\hookrightarrow \mathcal{N}\Lambda^i(A)$. This proves that \eqref{eq-qiosgr} is a quasi-isomorphism, hence that \eqref{eq-qios} is a quasi-isomorphism.
\end{proof}
The results of lemma \ref{lm-transfo-1} and \ref{lm-smallmodel} together prove the following proposition.
\begin{proposition}
The conjecture holds for $n=1$ and $A$ a free abelian group.
\end{proposition}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 6,978 |
Q: In AWS Lambda Docker Container, LibreOffice `soffice` binary executes from BASH but fails from Python (subprocess) Following this guide I have an (Alpine) Docker Image running on AWS Lambda.
The image contains an app.py which is a simple .docx -> .pdf document converter.
At the core is the following code, which works in the Docker Container on my local dev box, but raises subprocess.CalledProcessError on an actual Lambda deployment:
def handler(event, context):
src_filename = event['filename']
filename_body, _ = os.path.splitext(src_filename)
src_filepath = '/tmp/test-template.docx'
shutil.copyfile('/home/app/test-template.docx', src_filepath) # for testing
print( subprocess.check_output(['ls', '-l', '/tmp'] ) )
# ^ -rw-rw-r-- 20974 bytes test-template.docx
LIBRE_BINARY = '/usr/bin/soffice'
print( subprocess.check_output(['ls', '-l', LIBRE_BINARY] ) )
# ^ lrwxrwxrwx /usr/bin/soffice -> /usr/lib/libreoffice/program/soffice
MAX_TRIES = 3
success = False
print(f'Processing file: {src_filepath} with LibreOffice')
for kTry in range(MAX_TRIES):
print(f'Conversion Attempt #{kTry}')
try:
# https://stackoverflow.com/questions/4256107/running-bash-commands-in-python
result = subprocess.run(
[
LIBRE_BINARY,
'--headless',
'--invisible',
'--nodefault',
'--nofirststartwizard',
'--nolockcheck',
'--nologo',
'--norestore',
'--convert-to', 'pdf:writer_pdf_Export',
'--outdir', TMP_FOLDER,
src_filepath
],
stdout=subprocess.PIPE,
stderr=subprocess.PIPE,
shell=False,
check=True,
text=True
)
except subprocess.CalledProcessError as e:
raise RuntimeError(f"\tGot exit code {e.returncode}. Msg: {e.output}") from e
continue
Response string is:
[ERROR] RuntimeError: Got exit code 77. Msg:
Traceback (most recent call last):
File "/home/app/app.py", line 82, in handler
raise RuntimeError(f"\tGot exit code {e.returncode}. Msg: {e.output}") from e
How is it possible that this succeeds on my local machine but fails on AWS?
It is the same container image executing.
It is entirely self-contained.
The problem is definitely coming from this subprocess.run command.
Here is my aws lambda create-function:
aws lambda create-function \
--function-name $AWS_LAMBDAFUNC_NAME \
--role $role_arn \
--code ImageUri=$full_url \
--package-type Image \
--memory-size 8192 \
--timeout 300 \
--publish
I've used a large memory size and a large timeout.
I have read that writing to the file system outside of my /home/app folder and outside of /tmp might be problematic. So I am careful to use no such writes.
So what could be the problem?
It works from BASH
If I perform this processing in my entry.sh it works:
#!/bin/sh
/usr/bin/soffice \
--headless \
--invisible \
--nodefault \
--nofirststartwizard \
--nolockcheck \
--nologo \
--norestore \
--convert-to pdf:writer_pdf_Export \
--outdir /tmp \
/home/app/test-template.docx \
&> /home/app/output_and_error_file
ls /tmp >> /home/app/output_and_error_file
exec python -m awslambdaric $1
output_and_error_file:
{"response": "convert /home/app/test-template.docx -> /tmp/test-template.pdf using filter : writer_pdf_Export
hsperfdata_root
test-template.pdf"}
So it must be that something about subprocess is grating against the Lambda runtime.
Test: Using os.system
os.system(
f'export HOME=/home/app && {LIBRE_BINARY}' \
f' --headless --invisible --nodefault --nofirststartwizard' \
f' --nolockcheck --nologo --norestore' \
f' --convert-to pdf:writer_pdf_Export' \
f' --outdir {TMP_FOLDER}' \
f' {src_filepath}'
)
This produces a more descriptive error:
START RequestId: f2c18863-977e-46e4-a138-c1db80759406 Version: $LATEST
Executing 'app.handler' in function directory '/home/app'
b'total 24\n-rw-rw-r-- 1 sbx_user 990 20974 Jan 8 11:01 test-template.docx\n'
/usr/bin/soffice
b'lrwxrwxrwx 1 root root 36 Jan 8 03:56 /usr/bin/soffice -> /usr/lib/libreoffice/program/soffice\n'
Processing file: /tmp/test-template.docx with LibreOffice
Conversion Attempt #0
javaldx failed!
Warning: failed to read path from javaldx
LibreOffice 6.4 - Fatal Error: The application cannot be started.
User installation could not be completed.
Unknown error with saving to S3: <class 'FileNotFoundError'>
END RequestId: f2c18863-977e-46e4-a138-c1db80759406
REPORT RequestId: f2c18863-977e-46e4-a138-c1db80759406 Duration: 2698.32 ms Billed Duration: 5585 ms Memory Size: 8192 MB Max Memory Used: 175 MB Init Duration: 2885.69 ms
soffice --version works
result = subprocess.run(
[ LIBRE_BINARY, '--version' ],
stdout=subprocess.PIPE,
stderr=subprocess.PIPE,
shell=False,
check=True,
text=True
)
This works fine!
A: It's happening because libreoffice needs to create a dir called .cache/dconf in your user's home directory.
But on AWS lambdas, the user's home dir is read-only, so libreoffice fails with Fatal Error: The application cannot be started.
On lambdas, you can only write to temporary directories.
So, the solution is to set a temporary dir as your home directory while you call subprocess.run(...).
import subprocess
import tempfile
temp_dir = tempfile.TemporaryDirectory()
temp_dir_path = temp_dir.name
subprocess.run(
f"soffice --headless --convert-to pdf {temp_dir_path}/input.xlsx --outdir {temp_dir_path}",
shell=True,
check=True,
# libreoffice needs to create a dir called .cache/dconf in the HOME dir.
# So HOME must be writable. But on aws lambda, the default HOME is read-only.
env={"HOME": temp_dir_path},
)
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 7,372 |
Q: Set attr selectively using jQuery? Setting all a elements on a page to have target="_blank" is a simple line of jQuery: $("a").attr("target", "_blank");
But what I can't figure out is how to do this selectively. I'd like to set all element a targets to _blank except for those that have id="self".
So far, I have the below non-functional code. Am I on the right path?
$("a").attr("target", function(val){if ($(this).attr("id") == "self") {return "_self";} else {return "_blank";}});
A: There are many ways to do this, one would be:
$("a").not("#self").attr("target", "_blank");
You can always take a look at the jQuery docs if you're trying to figure out how to do something new.
Side note: It's invalid for more than one element on a page to have the same id, so make sure you do not have two elements with id="self".
A: There is a .not method specifically for the purpose of excluding an element (or collection of elements) from the selection.
$("a").not("#self").attr("target", "_blank");
A: Use each method:
$('a').each(function(){
if($(this).attr('id')=='self'){
return $(this).attr('target','self');
} else{
return $(this).attr('target','_blank');
}
});
But I would recommend to use class instead of checking id as simple as like this:
$('a').each(function(){
if(!$(this).hasClass('self')){
return $(this).attr('target','_blank');
}
});
A: One of the simplest ways to do this would be in the selector itself like this:
$('a[id!="self"]').attr("target", "_blank");
a more flexible approach would be to use jQuery's filter method like this:
$('a').filter(function(){
return $(this).attr('id') != 'self'; // Returning false means the element will not be used.
}).attr("target", "_blank");
A: You can select elements based on their attribute values. See this
$("a[id!='self']").attr("target", "_blank");
Refer this JSFiddle
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 3,706 |
Australia's Poor Wage Growth is Destroying its Prime Ministers
Australia has swapped Prime Ministers again–this time the Liberal Party replaced Malcolm Turnbull with Scott Morrison. Morrison will be Australia's sixth Prime Minister in the last then years. This level of turnover at the top is remarkable. The UK has only had three Prime Ministers during the same period. Canada has only had two. Why are Australian politics so volatile? I couldn't find any explanation online which satisfied me, so I'm writing my own. I think it has to do with a combination of wages and the way Australia's political parties choose their leaders.
The Liberal Party has a very old fashioned way of choosing its leaders. Unlike Britain, where the Labour and Conservative parties both permit rank and file party members to vote in leadership contests, the Australian Liberal Party–and until recently, its Labour Party–permit members of parliament to change the party leader (and consequently, the Prime Minister) on their own, without going to the members. This means leadership contests–or "spills", as they call them in Australia–can happen quickly and easily, facilitating turnover. The Labour Party eliminated this in 2013, but by that point the party had already been shattered by two mid-term leadership changes in a six year period. The Liberal Party won the 2013 election, and it has now replicated Labour's instability, swapping Prime Ministers twice in just five years.
That said, Australia's party leader selection process cannot be the primary cause of the turnover, because Australia has often gone through periods of tremendous stability under these rules. While the most recent six Prime Ministers have ruled over a ten year period, the six Prime Ministers before that ruled for 36 year combined. The instability which the party rules encourages comes and goes. Australia again had six Prime Ministers in the decade between 1965 and 1975. This period also coincided with economic unrest and stagflation. Immediately before that, a single Prime Minister ruled the roost for seventeen years. But yet again, prior to that, the country endured seven Prime Ministers between 1939 and 1949–during the war years and their aftermath. When things are running smoothly in Australia, politics stabilises, but as soon as things come off the rails the trademark instability reasserts.
So what's gone wrong in Australian politics these last ten years? At first glance, we might think not much–Australia is less dependent on the European and North American markets and was hit much less hard by the economic crisis of 2008 than most of the other rich countries:
Of course, you might notice that Australia's growth rate hasn't been altogether high–it only topped 3% once during this period, in 2012. Australia avoided the misery, but it hasn't produced consistently superior results in the years that followed, especially recently. But still, looking at the growth rate, you wouldn't think Australia were in some sort of malaise. This only becomes clear when we look at wage data. While Australian GDP shrugged off 2008, wage growth has never been the same:
For quite some time, Australian wages have grown at well below the rate they grew even during the worst part of the global economic crisis. To make matters worse, in the last few years Australia's inflation rate has frequently managed to exceed the wage growth rate, which means in real terms the entire annual wage increase is now being torched:
In the United States, we're all too familiar with wage stagnation. Real median household incomes in America have hardly budged in 20 years. But Australia was accustomed to rapid wage growth, especially in the decade before the crisis:
Australians are reluctant to accept wage stagnation as a new normal, and political parties are likely to pay a price for it at the ballot box. Australia is just now beginning to develop its very own productivity/wage gap:
The Americans have one just like this, but it got going back in the 70s:
Australia has long been more equal than many rich countries–its top 1% receive only 9% of income, compared to 14% in the UK and over 20% in America. In the early 80s, its top 1% share was a stunningly low 4.4% of income, almost a full point lower than Denmark's lowest recorded top 1% share. Australia was so equal that even the Soviet Union occasionally recorded higher top 1% income shares.
That's been changing–the 9% it runs today is almost double Denmark's current rate. The appearance of stagnating wages and a productivity/wage gap indicates that Australia is likely to get considerably less equal going forward. Its voters are accustomed to a remarkable level of egalitarianism–they won't like this. Indeed, they already don't. Australian Prime Ministers' parties increasingly blow their good will with the Australian people quickly, falling behind in the polls. Under Turnbull, the Liberals once led Labour by nearly 8 points in polling. Today the Liberals trail by 10. In a desperate bid to prop up poll numbers and keep their seats, Australian MPs are dumping their leaders and hoping that Australian voters will mistake changes in leadership for changes of policy. For quite a while, this worked pretty well, but it appears Australians have started to catch on. New Prime Ministers usually get the benefit of the doubt, but this time Morrison is getting shredded:
Australia's next election likely won't be until next year, but Morrison doesn't seem to have any bold new ideas to turn the wage situation around. It's likely the Labour Party returns to power, and once that happens, its new leadership selection rules will probably enable the new Labour Prime Minister to successfully finish a term without falling victim to another spill. But a second term may not be in the offing if that Labour PM allows Australian workers to get continuously kicked in the face.
Tags: Australia : Economics : Household Income : Julia Gillard : Kevin Rudd : Malcolm Turnbull : Party Dynamics : Productivity : Scott Morrison : Tony Abbott : Wage Growth : Wages | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 4,902 |
Nghe An (vietnamita: Nghệ An) è una provincia del Vietnam, della regione di Bắc Trung Bộ. La provincia di Nghe An è quella che consta il maggior numero di comuni di tutto il Vietnam: in tutto, sono presenti 437 comuni. Occupa una superficie di 16.499 km² e ha una popolazione di 3.327.791 abitanti.
La capitale provinciale è Vinh.
Distretti
La provincia è amministrativamente suddivisa in una città (Vinh), tre città minori (Cửa Lò, Thái Hoà e Hoàng Mai), e 17 distretti.
Di questa provincia fanno parte i distretti:
Anh Sơn
Con Cuông
Diễn Châu
Đô Lương
Hưng Nguyên
Kỳ Sơn
Nam Đàn
Nghi Lộc
Nghĩa Đàn
Quế Phong
Quỳ Châu
Quỳ Hợp
Quỳnh Lưu
Tân Kỳ
Thanh Chương
Tương Dương
Yên Thành
Note
Voci correlate
Suddivisioni del Vietnam
Altri progetti
Collegamenti esterni | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 6,226 |
\section{Introduction}
Deep Neural Networks (DNNs) have demonstrated outstanding performance on a wide range of tasks, including image classification ~\cite{image-classification}, speech recognition ~\cite{speech} etc. These networks typically consists of multiple convolution layers with a large number of parameters. The models are trained on high performance servers typically with GPUs and are deployed on lower-end machines, i.e. mobile or IoT devices, for inference tasks. Improved inference accuracy usually comes with millions of model parameters and high computation cost. For example, the largest Mobilenet v1 model ~\cite{mobilenet} has 4.2 million parameters and 569 million floating point MAC per inference ~\cite{TF-mb-int8:}. For applications that demand high inference accuracy, low latency and low power consumption, the large memory requirements and computation costs are a significant challenge for constrained platforms.
\parskip = \baselineskip
To achieve efficient inference, one approach is to design compact network architectures from scratch ~\cite{compact-network-1}~\cite{compact-network-2}~\cite{compact-network-3}~\cite{compact-network-4}. Alternatively, existing models can be optimized for efficiency. There are several optimization techniques that boost efficiency when applied to pretrained models: weight pruning ~\cite{pruning-cluster}~\cite{pruning-svd-cluster}, weight clustering ~\cite{pruning-cluster}~\cite{pruning-svd-cluster}, singular value decomposition (SVD) ~\cite{svd} and quantization ~\cite{50}~\cite{quant-2}~\cite{quant-3}~\cite{quant-4:}. The basic principle is to reduce the number of parameters and/or lower the computation cost of inference. Weight pruning techniques remove parameters while minimizing the impact on inference accuracy. Weight clustering clusters similar weights to shrink the overall size of a model. The $\textit{SVD}$ method potentially reduces both model size and computation cost through discarding small singular values. Quantization techniques convert normal floating-point values to narrow and cheaper integer or fixed point i.e. 8-bits, 4-bits or binary multiplication operations without incurring significant loss in the accuracy. There are three major benefits to quantization: reduced memory bandwidth, reduced memory storage, and higher throughput computation. The predominant numerical format used for training neural networks is IEEE fp32 format. There is a potential 4x reduction in overall bandwidth and storage if one can quantize fp32 floating point to 8-bits for both weight and activation. The corresponding energy and area saving are 18x and 27x ~\cite{dally} respectively. The efficient computation kernel libraries for fast inference, i.e. Arm CMSIS ~\cite{arm-cmsis}, Gemmlowp ~\cite{gemmlowp}, Intel MKL-DNN ~\cite{mkl}, Nvidia TensorRT ~\cite{nvidia} and custom ASIC hardware, are built upon the reduced precision numerical forms.
\parskip = \baselineskip
The Straight-Through Estimator (STE) \cite{ste:}\cite{Bengio2013STE} is widely implemented in discrete optimization using SGD due to its effectiveness and simplicity. STE is an empirical workaround to the gradient vanishing issue in Backprop; however it lacks complete mathematical justification especially for large-scale optimization problems \cite{Yin2018STE}. In this paper, we propose a novel optimization technique, termed alpha-blending ($\textbf{AB}$), for quantizing full precision networks to lower precision representations(8-bits, 4-bits or 1-bit). AB does not rely on the concept of STE to back-propagate the gradient update to weights; AB instead replaces the weight vector $\textbf{ w }$ in the loss function by the expression $\textbf{w}_{ab} = (1 - \alpha) \textbf{w} + \alpha \textbf{w}_q $, which is the affine combination of the $\textbf{w}$ and its quantization $\textbf{w}_q $. During training, we gradually increase the non-trainable parameter $\alpha$ from 0.0 to 1.0. This formulation isolates the quantized weights $\textbf{w}_q$ from the full-precision trainable weights $\textbf{w}$ and therefore avoids the challenges arising from the use of Straight-Through Estimation (STE).
\parskip = \baselineskip
To evaluate the performance of the proposed method, we trained single-bit BinaryNet \cite{binary} on CIFAR10 and 4-bits, 8-bits MobileNet v1, ResNet v1 and v2 models on the ImageNet dataset. $\textbf{AB}$ outperforms previous state-of-art STE based quantization 0.9\% for 1-bit BinaryNet and 2.9\% for 4-bits weight and 8-bits activation (4-8) ~\cite{TF-int4-8:} in top-1 accuracy. Moreover, we have applied our AB approach to quantize MobileNet v1, ResNet v1,2 networks with both 4-bit weight as well as 4-bit activation (4b/4b). In this configuration, our 4b/4b quantization delivers similar accuracy level as the best known 4b/8b quantization approach \cite{TF-int4-8:}.
\section{Related works}
There is a significant body of research on neural network quantization techniques from the deep learning community. BinaryConnect ~\cite{50} binarizes the weights of neural networks using the sign function. Binary Weight Network ~\cite{42} has the same binarization while introducing a scaling factor. BinaryNet ~\cite{52}~\cite{binary} quantizes both weights and activations to binary values. TWN ~\cite{compact-network-4} constructs networks with ternary values 0, +/-1 to balance the accuracy and model compression compared to the binary quantization.
STE \cite{ste:} is used to approximate the gradient of quantization function during the learning process. Once they are quantized, these models eliminate majority of the floating-point multiplications, and therefore exhibit improved power efficiency by using SIMD instructions on commodity micro-processor or via special hardware. On the downside, the single bit quantization schemes often lead to substantial accuracy drop on large scale dataset while achieving good results on simple dataset such as MNIST, CIFAR10.
\parskip = \baselineskip
Another approach is to train the network in full floating-point domain, then statically quantize the model parameter into reduced numerical forms and keep the activation in floating-point. Google's Tensorflow provides a post-training quantization flow ~\cite{TF-quant:} to convert float-point weights into 8-bits of precision from – INT8. Its uniform affine quantization maps a set of floating-point values to 8-bits unsigned integers by shifting and scaling ~\cite{TF-int4-8:}. The minimum and maximum values correspond to quantized value 0 and 255 respectively. Another mapping scheme is uniform symmetric quantizer, which scales the maximum magnitude of floating-point values to maximum 8-bit integer e.g. 127 and the floating-point zero always mapped to quantized zero. The conversion is done once, and reduction of model size is up to 4X. A further improvement dynamically quantizes activations into 8-bits as well at inference. With 8-bits weight and activation, one can switch the most compute-intensive operations e.g. convolution, matrix multiply (GEMM) from original floating-point format to the cheaper operation, and reduces the latency as well.
\parskip = \baselineskip
The main drawback of such post-processing approach is the degradation in model accuracy. To overcome this accuracy drop, "quantization aware training" ~\cite{TF-quant:} techniques have been developed to ensure that the forward pass uses the reduced precision for both training and inference. To achieve this, full precision weights and activations values flow through "fake quantization" nodes, then quantized values feed through convolution or matrix multiply. Applying the Straight-Through Estimator (STE) approximation ~\cite{ste:}~\cite{binary}~\cite{arXiv180805240Y}, the operations in the back propagation phase are still at full precision as this is required to provide sufficient precision in accumulating small adjustment to the parameters.
\section{Alpha-blending, the proposed method ($\textbf{AB}$)}
We introduce an optimization methodology, alpha-blending ($\textbf{AB}$), for quantizing neural networks. Section \ref{sect:3.1} describes the scheme of $\textbf{AB}$ and weights quantization; section \ref{sect:3.2} sketches the quantization of activation using $\textbf{AB}$.
\subsection{Alpha-blending \textbf{AB} and quantization of weights} \label{sect:3.1}
During quantization-aware training, the full precision weights are quantized to low precision values ${ { \textbf{w}_q } }$. Mathematically, we want to minimize a convex function $L(\textbf{w})$ as equation \ref{eq:1} with the additional constraint that $\textbf{w}$ must be n-bit signed integers i.e. $\textbf{w} \in \textbf{Q} = [-(2^{n-1}-1), 2^{n-1}-1]$.
\begin{equation} \label{eq:1}
\min\limits_{s.t.\ w \in \textbf{Q}} L(\textbf{w})
\end{equation}
Previous approaches i.e. ~\cite{TF-quant:},~\cite{binary} insert quantizer nodes in the computation graph. These nodes receive full precision input $\textbf{w}$ and generate quantized output $\textbf{w}_q = \textit{q}(\textbf{w})$, between the full precision weights $\textbf{w}$ and computation nodes as in Figure \ref{fig:bp_ste}. The quantized weights $ \textbf{w}_q = \textit{q}(\textbf{w}) $ are used in the forward and backward pass while the gradient update to the full precision weight uses full precision to ensure smooth updates to the weights. But the quantization function has zero gradient almost everywhere $ {\partial{\textbf{w}_q}}/{\partial{\textbf{w}}} \underset{a.e.}{=} 0 $, which prevents further backpropagation of gradients and halts learning. The $\textit{Straight-Through Estimator}$ (STE) ~\cite{ste:}~\cite{binary}~\cite{TF-int4-8:} was developed to avoid the vanishing gradient problem illustrated in Figure \ref{fig:bp_ste}. \textbf{STE} approximates quantization with the identity function $ \textit{I}(\textbf{w}) = \textbf{w}$ in $\textbf{Backprop}$ as eq. \ref{eq:2'}. Therefore with STE, the gradient of the quantization function with respect to the full precision weight is approximated using the quantized weight as in equation \ref{eq:2}. We hypothesize that the error introduced by this approximation may impact the accuracy of the gradient computation, thereby degrading overall network accuracy, especially for very low precision (1-bit or 4-bit) networks.
\begin{figure}[h!]
\centering
\includegraphics[scale=0.6]{bp-ste-0.png}
\caption{ Gradient update to the full precision weight in backprop using STE approximation as eq. \ref{eq:2'} and \ref{eq:2}.}
\label{fig:bp_ste}
\end{figure}
\begin{equation} \label{eq:2'}
\frac{\partial{\textbf{w}_q}}{\partial{\textbf{w}}} = \frac{\partial{\textit{q}(\textbf{w})}}{\partial{\textbf{w}}} \underset{\scriptsize{STE}}{ \approx } \frac{\partial{\textit{I}(\textbf{w})}}{\partial{\textbf{w}}} = 1
\end{equation}
\begin{equation} \label{eq:2}
\frac{\partial{L(\textbf{w})}}{\partial{\textbf{w}}} =
\frac{\partial{L(\textbf{w}_q)}}{\partial{\textbf{w}_q}} \cdot \frac{\partial{\textbf{w}_q}}{\partial{\textbf{w}}} \underset{STE}{\approx} \frac{\partial{L(\textbf{w}_q)}}{\partial{\textbf{w}_q}}
\end{equation}
Our proposed method, alpha-blending (\textbf{AB}), does not rely on the $\textit{Straight-Through Estimator}$ (STE) to overcome the quantizer's vanishing gradient problem in $\textbf{Backprop}$, therefore it eliminates the quantization error due to equation \ref{eq:2}. \textbf{AB} replaces the weight term in the loss function by $ (1-\alpha)\textbf{w} + \alpha \textbf{w}_q $, an affine combination of the original full precision weight term and its quantized version with coefficient $ \alpha $. The new loss function $L_{ab}(\textbf{w},\alpha)$ for a neural network is shown in equation \ref{eq:10}. The gradient of $L_{ab}(\textbf{w}, \alpha)$ with respect to the weights is in equation \ref{eq:12}, accepting the $\textbf{zero}$ gradient of quantization function $ {\partial{\textbf{w}_q}}/{\partial{\textbf{w}}} \underset{a.e.}{=} 0 $ without STE approximation. Its Backprop flow is illustrated in figure \ref{fig:bp_no_ste}.
\begin{equation} \label{eq:10}
\begin{aligned}
L_{ab}(\textbf{w}, \alpha) = L((1-\alpha)\textbf{w} + \alpha \textbf{w}_q)
\end{aligned}
\end{equation}
\begin{equation} \label{eq:12}
\begin{aligned}
\frac{\partial{L_{ab}}}{\partial{\textbf{w}}} = (1-\alpha+ \underset{=0\ a.e.}{\xcancel{\alpha \frac{\partial{\textbf{w}_q}}{\partial{\textbf{w}}}}})
\left.\frac{\partial{L(\textbf{w}^{'})}}{\partial{\textbf{w}^{'}}} \right|_{\textbf{w}^{'}= (1-\alpha)\textbf{w} +\alpha\textbf{w}_q}
\end{aligned}
\end{equation}
\begin{figure}[h!]
\centering
\includegraphics[scale=0.6]{bp-ste-1.png}
\caption{\textbf{AB} quantization performs the convolution using an affine combination of the full precision weights and the quantized weights. The coefficient $\alpha$ is gradually increased from 0 to 1 during training. This approach avoids back-propagation through the Quantizer, eliminating the gradient vanishing path (from the quantized weight node in light green to the weight node in blue). There is no need to apply $\textit{Straight Trough Estimator}$ (STE) during the $\textit{Backprop}$. The actual weight gradient update goes through the $(1-\alpha)$ path, where the gradient, eq. \ref{eq:12}, is well-defined.}
\label{fig:bp_no_ste}
\end{figure}
The $\textbf{AB}$ flow gradually increases the non-trainable parameter $ \alpha $ from 0 to 1 using a function of the form shown in equation \ref{eq:13} for training steps in the optimization window $[T_0, T_1]$. An example is shown in Figure \ref{fig:3_curves}. The function in equation \ref{eq:13} is not unique, for example, an alternative choice is $ A(step, \lambda) = 1 - e^{- \lambda \cdot step} $. The optimization window $[T_0, T_1]$, during which $\alpha$ is increased, is a user-defined hyper parameter.
We use algorithm \ref{alg:IPQ} described in section \ref{sect:exp-ppq} to convert $\textbf{w}$ to $\textbf{w}_q = {\gamma}_w \cdot \textbf{q}_w $, where $ {\gamma}_w$ is a scaling factor and $\textbf{q}_w \in \textbf{Q}$, at certain frequency, $\textit{quantizing\_frequency}$, in training steps.
\begin{equation} \label{eq:13}
\begin{aligned}
A(step) =
\begin{cases}
0 & step \le T_0 \\
1 - (\frac{T_1 - step}{T_1-T_0})^3 & T_0 < step \le T_1 \\
1 & T_1 < step
\end{cases}
\end{aligned}
\end{equation}
\begin{algorithm}[tb]
\caption{Alpha-blending optimization ($\textit{ABO}$)}
\label{alg:ABO}
\begin{algorithmic}
\STATE {\bfseries Input:} $\textit{derivative loss function} \ L(\textbf{w})$
\STATE {\bfseries Def.} function: $L_{ab}(\textbf{w}, \textbf{w}_q, \alpha)$ = $ L((1-\alpha)\textbf{w} + \alpha \textbf{w}_q) $
\STATE {\bfseries Initialize:} {$ \textbf{w} \gets w_0, \alpha \gets 0, \varepsilon \gets {learning\_rate}, \textit{f} \gets {optimization\_frequency}, \textit{T}_0,\textit{T}_1 \gets {traing\_window} $}
\FOR{$step=0$ {\bfseries to} $\textit{T}\ \ $}
\STATE $ w_q \gets $ Algorithm \ref{alg:IPQ} ${\textit{PPQ}}(\textbf{w})$ or other optimization function $(\textbf{w})$
\STATE $\textbf{w} \gets \textbf{w} - {\left. \varepsilon \cdot (1-\alpha)\ \frac{\partial{L(\textbf{w}^{'}) }}{\partial{\textbf{w}^{'}}}\right|_{\textbf{w}^{'} = (1-\alpha)\textbf{w} + \alpha \textbf{w}_q}} $
\IF {$step \ \% \ \textit{f} = 0 $ and $ \alpha < 1 $}
\STATE $\alpha \gets A(step) \ \ \ \ \ $
\COMMENT{Raising $\alpha$ toward to 1.0; eq \ref{eq:13}}
\ENDIF
\ENDFOR
\STATE {\bfseries Output:} $ \textbf{w}_q $
\end{algorithmic}
\end{algorithm}
Algorithm \ref{alg:ABO} summarizes the $\textbf{AB}$ optimization procedure, in which the original learning rate $\varepsilon$ is scaled by the factor $(1-\alpha)$ to act as an effective learning rate $\varepsilon \cdot (1-\alpha)$.
To visualize the process, figure \ref{fig:AB_3d} demonstrates how to solve the trivial example, $\underset{s.t. \ w \in \textbf{Q}}{arg \ min} {(w - 5.7)^2} = 6 $ using AB.
To compare the $\textbf{AB}$ optimization concept with $\textbf{STE}$, we trained the single bit 8-layer BinaryNet defined in \cite{binary} on the CIFAR10 dataset in section \ref{sect:ab_ste}, figure \ref{fig:alpha-ste}. The top-1 accuracy score achieved with $\textbf{AB}$ is 0.9\% higher compared to the accuracy achieved with $\textbf{STE}$.
Figure \ref{fig:3_curves} shows a more practical example of $\textbf{AB}$ quantization using MobileNet\_1.0\_0.25/128 v1 on the ImageNet dataset. The $\textbf{AB}$ quantization flow gradually transforms the full precision model at $\alpha=0$ to a model with quantized weights $\textbf{w}_q$ at $\alpha=1.0$ with an accuracy loss of 0.6\% versus the full precision model.
\begin{figure}[h!]
\centering
\includegraphics[scale=0.45]{AB_3d_3.png}
\caption{Apply \textbf{AB} to minimize a trivial example $ Loss(w) = (w - 5.7)^2 $, equivalently to find the minimal of 2D surface
$ Loss(w, \alpha) = ((1 - \alpha)w + \alpha w_q)^2 $ using \textbf{SGD} while alpha $(\alpha)$ has changed from 0 to 1 using function ${\textit{A}}(\cdot)$ in eq. \ref{eq:13}, and $w_q=round(w)$. ${w}$ started at the initial value $(w,\alpha)$=$(2.0,0)$ and moved along the ${\color{red}\small{X}}$ trace, its corresponding quantized weights are marked by $\color{green}+$. In 20 steps, the iteration converged to ($\textbf{w}$,$\alpha$)=(5.506555, 1). ${w}_q$ = 6 is the final quantized solution.}
\label{fig:AB_3d}
\end{figure}
\begin{figure}
\resizebox {\columnwidth} {!} {
\begin{tikzpicture}
\begin{axis}[
axis y line*=left,
title={Mobilenet\_v1\_0.25/128, top-1 accuracy with weight $\textbf{w}$ and quantized weight $\textbf{q}$ },
xlabel={training step},
ylabel={Accuracy},
xmin=0, xmax=100000,
ymin=0.2, ymax=0.5,
xtick={0,20000,40000,60000,80000,100000},
ytick={0.20,0.30,0.40,0.5},
legend pos=north west,
ymajorgrids=true,
grid style=dashed,
]
\addplot[
color=blue,
mark=o,
]
coordinates {
(0,0.415)(234,0.41)(3770,0.412)(7306,0.41)(10995,0.407)(17445,0.398)(20907,0.39)(24375,0.386)(31320,0.371)(34818,0.363)(41830,0.349)(48828,0.34)(55746,0.329)(66537,0.318)(77303,0.313)(94916,0.308)(98354,0.307)
}; \label{fig:3_curves_weight}
\addlegendentry{weight $ \textbf{ w } $}
\addplot[
color=green,
mark=triangle,
]
coordinates {
(0,0)(234,0.27)(3770,0.289)(7306,0.305)(10995,0.319)(17445,0.346)(20907,0.357)(24375,0.368)(34818,0.389)(41830,0.398)(48828,0.404)(55746,0.406)(66537,0.408)(77303,0.407)(94916,0.41)(98354,0.41)
};\label{fig:3_curves_quantized}
\addlegendentry{quantized weight $\textbf{w}_q $}
\end{axis}
\begin{axis}[
axis y line*=right,
hide x axis,
ylabel={ $\alpha \ (\textbf{alpha}$)},
ylabel near ticks,
xmin=0, xmax=100000,
ymin=0.0, ymax=1.1,
legend pos=south east,
]
\addplot[
color=red,
mark=none,
]
coordinates {
(0,0)(234,0.007)(3770,0.109)(7306,0.20355)(10995,0.2949)(17445,0.4374)(20907,0.505)(24375,0.567)(31320,0.676)(34818,0.723)(41830,0.803)(48828,0.866)(55746,0.913)(66537,0.9625)(77303,0.988)(94916,0.9998)(98354,1.0)(100000,1.0)
};\label{fig:3_curves_alpha}
\addlegendentry{$ \alpha=\textit{A}(step)$}
\end{axis}
\end{tikzpicture}
}
\captionof{figure}{Two accuracy curves, evaluated with the full precision weights $\textbf{w}$ and 8-bits quantized weights $\textbf{w}_q $ during $\textbf{AB}$ quantization training with Mobilenet 0.25/128 V1 for 2.5 epochs. The $ \alpha $ curve \ref{fig:3_curves_alpha} is the $ \textbf{A} $ function in eq. \ref{eq:13}. The accuracy \ref{fig:3_curves_weight} corresponding to full precision weights has dropped 10\% during the training while the \ref{fig:3_curves_quantized}, with the quantized weights, has gradually increased to approach its maximum accuracy 40.9\% when $\alpha$ = 1.0. The final quantized model has 0.6\% accuracy loss compared to full precision one.
}
\label{fig:3_curves}
\end{figure}
\subsection{Quantization of activation} \label{sect:3.2}
$\textbf{AB}$ uses the $\textit{PPQ}$, algorithm \ref{alg:IPQ} in section 4.2, to quantize the input feature maps or activation ${\textbf{a}}$ to ${\textbf{a}}_q$ as well, and accumulates the scaling factor $ {\gamma}_a $ via exponential moving average with the smoothing parameter being close to 1, e.g. 0.99. Thus $\textbf{a}$ can be approximated as $
\textbf{a} \approx {\gamma}_{a} \cdot {\textbf{q}}_a $
For inference ($\alpha$=1), the floating point computation of the $k^{th}$ layer in forward pass is $\textbf{a}^{(k+1)}$ = $\delta({{\textbf{w}}}^{(k)}{\textbf{a}}^{(k)} + {\textbf{b}}^{(k)})$. With the quantizaton of both weight and activation, the same calculation becomes eq. \ref{eq:51}.
\begin{equation} \label{eq:51}
\begin{split}
\delta(\textbf{w}\cdot{\textbf{a}}+{\textbf{b}}) \approx
\delta({\gamma}_{w} \textbf{q}_w \cdot {{\gamma}_a} {\textbf{q}}_a + {\textbf{b}}) \\ =
\delta((\gamma_w {{\gamma}_a})(\textbf{q}_w \cdot {\textbf{q}}_a) + {\textbf{b}})
\end{split}
\end{equation}
$(\textbf{q}_w \cdot {\textbf{q}}_a) $ in \ref{eq:51} is the compute-intensive operation of matrix multiply or convolution $(\textbf{GEMM})$ in low precision quantized values, which will gain significant power efficiency compared to the original floating-point version. Other relatively unimportant terms in \ref{eq:51} e.g. $(\gamma_w\gamma_a)$ and $\textbf{b}$ can be represented by higher precision fixed points.
\section{Experiments}
To evaluate the AB quantization methodology, we performed several experiments. The first one, in section \ref{sect:ab_ste}, is a single bit (1-bit) control test between $\textbf{STE}$ and $\textbf{AB}$ on CIFAR10. Section \ref{sect:exp4-8} presents results for Mobilenet v1 and ResNet v1,2 with the ImageNet ILSVRC 2012 dataset. All evaluations were performed on a x86\_64 ubuntu Linux based Xeon server, Lenovo P710, with a TitanV GPU.
\subsection{BinaryNet with alpha-blending \textbf{AB} and Straight-Through Estimator ($\textbf{STE}$)} \label{sect:ab_ste}
To evaluate AB's function directly, 1-bit BinaryNet (BNN) \footnote{https://github.com/itayhubara/BinaryNet.tf} \cite{binary} on CIFAR-10 was trained on Tensorflow using AB and STE respectively. Both weight and activation are quantized into +1 or -1 (single bit) by the same binarization function, $ binarize(x) = Sign(x) $. Figure \ref{fig:alpha-ste} shows the results of these experiments. The AB method achieves a top-1 accuracy of 88.1\%. Using STE, we achieve 87.2\%. The FP32 baseline accuracy is 89.6\%.
\begin{figure}[h!]
\centering
\includegraphics[scale=0.40]{alpha-ste.png}
\caption{Training curves of BinaryNet on CIFAR-10 dataset. The dashed lines represent the validation Loss and the continuous lines are the corresponding validation accuracy. The blue curve of fp32 baseline has max top-1 accuracy 0.896. BNN which utilized $\textbf{STE}$ in training, blue line, converges to 0.872, while the red line of $\textbf{AB}$ yields a better top-1 accuracy 0.881.}
\label{fig:alpha-ste}
\end{figure}
\subsection{4-bits and 8-bits quantization with MobileNet and ResNet} \label{sect:exp4-8}
In this section, we describe the iterative quantization scheme we use to quantize FP32 values to low-precision, Progressive Projection Quantization ($\textbf{PPQ}$). We apply PPQ to convert floating point values into 4-bits or 8-bits integers, then utilize $\textbf{PPQ}$ and $\textbf{AB}$ to quantize MobileNet and ResNet into 4-bits or 8-bits and compare the result with existing results. All results are consolidated into Figure \ref{fig:cha_1} for easy comparison.
\subsubsection{Progressive projection quantization, $\textbf{\emph{PPQ}}$} \label{sect:exp-ppq}
To quantize a set of N floating-point values $ \textbf{x} = \big\{ x_i | i \in \big[0,N-1\big] \big\} $ to symmetric n-bits signed integer set $ \textbf{x}_q = \big\{ q_i\ |\ q_i\in{ \textbf{Q}=\big(0,\pm1,\pm2,...,\pm\big(2^{n-1}-1\big)} \big), i \in \big[0,N-1\big]\big\} $ with a positive scaling factor $\gamma$, we can approximate the initial quantization by rounding $ \left.{\frac{x_i}{\gamma}}\right|_{\gamma = \frac{max|x|}{2^{n-1}-1}} $ to the nearest-neighbor in $ \textbf{Q} $ as equation \ref{eq:40}. Then we can improve $ \gamma $ by equation \ref{eq:41}.
\begin{equation} \label{eq:40}
\begin{aligned}
\textbf{x}_q = round(\frac{\textbf{x}}{\gamma})
\end{aligned}
\end{equation}
\begin{equation} \label{eq:41}
\begin{aligned}
\gamma = \frac{\langle \textbf{x} ,\textbf{x}_q\rangle}{\langle \textbf{x}_q,\textbf{x}_q\rangle}
\end{aligned}
\end{equation}
\parskip = \baselineskip
$\textit{PPQ}$ is an iterative procedure: by repeatedly applying eq. \ref{eq:40} and \ref{eq:41}, as described in algorithm \ref{alg:IPQ}, projects vector $ \textbf{x} $ onto space $ \textbf{Q} $ to determine $ \gamma $ progressively \cite{admm}.
The procedure is guaranteed to converge to a local minimum. In practice, convergence is very fast and 3 iterations is enough. Thus, $\textbf{x}$ can be approximated by the product of the scalar $\gamma$ and $\textbf{x}_q $: $
\textbf{x} \approx \gamma \textbf{x}_q = \gamma \cdot round(\frac{\textbf{x}}{\gamma}) $
\begin{algorithm}[tb]
\caption{Progressive Project Quantization ($\textit{PPQ}$)}
\label{alg:IPQ}
\begin{algorithmic}
\STATE {\bfseries Input:} full precision vector $ \textbf{x} = \big\{ x_i | i \in \big[0,N-1\big] \big\}$, scaling factor $ \gamma $
\IF{$ \gamma \leq 0$}
\STATE Initialize $ \gamma \gets \frac{max\big(
|\textbf{x}| \big)}{2^{n - 1}-1} $
\ENDIF
\REPEAT
\STATE $ {\gamma}_0 \gets \gamma $
\FOR{$i=0$ {\bfseries to} $N-1$}
\STATE $ q_i \gets round(\frac{x_i}{{\gamma}_0}) $
\ENDFOR
\STATE $ \gamma \gets \frac{\langle \textbf{x} ,\textbf{q}\rangle}{\langle \textbf{q},\textbf{q}\rangle} $
\UNTIL{$\gamma$ = ${\gamma}_0$}
\STATE {\bfseries Output:} ${\textbf{q}, \gamma }$
\end{algorithmic}
\end{algorithm}
\subsubsection{Evaluation of 8-bits weight and activation (INT8-8)} \label{sect:4.2}
The top-1 accuracy for 8-bit weight and 8-bit activation quantization are listed in table 1. The $2^{nd}$ column gives the fp32 accuracy of the pre-trained models ~\cite{TF-mb-int8:}. The $3^{rd}$ column contains the quantization results \cite{TF-mb-int8:}~\cite{TF-rn-int8:}. The last column gives the best results that $\textbf{AB}$ generated.
Both quantization approaches delivered roughly the same top-1 accuracy, although AB has slightly (0.82\%) better accuracy on average.
\begin{table}[h]
\centering
\caption{top-1 accuracy of fp32 pre-trained models, Tensorflow's INT8-8 and $\textbf{AB} $ 8-8. $^*$ \protect\cite{TF-int4-8:}}
\label{table1}
\begin{tabular}{|l|c|c|c|}
\hline
Model name & \multicolumn{1}{l|}{
{\begin{tabular}[c]{@{}l@{}}
{fp32} \\ {\ \%} \end{tabular}}
} & \multicolumn{1}{l|}{
{\begin{tabular}[c]{@{}l@{}}
{TF8-8} \\ {\ \ \ \%} \end{tabular}}
} & \multicolumn{1}{l|}{
{\begin{tabular}[c]{@{}l@{}}
{AB8-8} \\ {\ \ \ \ \%} \end{tabular}}
} \\ \hline
MB\_1.0\_224v1 & 70.9 & 70.1 & 70.9 \\ \hline
MB\_1.0\_128v1 & 65.2 & 63.4 & 65.0 \\ \hline
\small{MB\_0.75\_224v1} & 68.4 & $67.9$ & 68.2 \\ \hline
\small{MB\_0.75\_128v1} & 62.1 & $59.8$ & 61.6 \\ \hline
MB\_0.5\_224v1 & 63.3 & $62.2$ & 63.0 \\ \hline
MB\_0.5\_128v1 & 56.3 & 54.5 & 55.8 \\ \hline
\small{MB\_0.25\_224v1} & 49.8 & 48 & 49.2 \\ \hline
\small{MB\_0.25\_128v1} & 41.5 & 39.5 & 40.9 \\ \hline
ResNet\_50v1 & 75.2 & $75^{\small{*}}$ & 75.1 \\ \hline
ResNet\_50v2 & 75.6 & $75^{\small{*}}$ & 75.4 \\ \hline
\end{tabular}
\end{table}
\makeatletter
\newdimen\legendxshift
\newdimen\legendyshift
\newcount\legendlines
\newcommand{1mm}{1mm}
\newcommand{\bclegend}[3][7mm]{%
\legendxshift=0pt\relax
\legendyshift=0pt\relax
\xdef\legendnodes{}%
\foreach \lcolor/\ltext [count=\ll from 1] in {#3}%
{\global\legendlines\ll\pgftext{\setbox0\hbox{\bcfontstyle\ltext}\ifdim\wd0>\legendxshift\global\legendxshift\wd0\fi}}%
\@tempdima#1\@tempdima0.5\@tempdima
\pgftext{\bcfontstyle\global\legendxshift\dimexpr\bcwidth-\legendxshift-1mm-\@tempdima-0.72em}
\legendyshift\dimexpr5mm+#2\relax
\legendyshift\legendlines\legendyshift
\global\legendyshift\dimexpr\bcpos-2.5mm+1mm+\legendyshift
\begin{scope}[shift={(\legendxshift,\legendyshift)}]
\coordinate (lp) at (0,0);
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\node[anchor=north, minimum width=#1, minimum height=5mm,fill=\lcolor] (lb\ll) at (lp) {};
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}
\node[draw, inner sep=1mm,fit=\legendnodes] (frame) {};
\end{scope}
}
\makeatother
\begin{figure}
\scalebox{0.20}{
\centering
\begin{bchart}[step=50,max=140,unit=\%,scale=1.5]
\bcbar[label=MB\_10\_224 v1, color=yellow]{70.9}
\bcbar[color=blue!60]{70.1}
\bcbar[color=green!60]{70.9}
\bcbar[color=blue!30]{65.0}
\bcbar[color=purple!60]{68.7}
\bcbar[color=purple!30]{69.6}
\bcbar[color=gray!90]{61.8}
\bcbar[color=gray!40]{64.3}
\smallskip
\bcbar[label=MB\_1.0\_128 v1, color=yellow]{65.2}
\bcbar[color=blue!60]{63.4}
\bcbar[color=green!70]{65.0}
\smallskip
\bcbar[label=MB\_0.75\_224 v1, color=yellow]{68.4}
\bcbar[color=blue!60]{67.9}
\bcbar[color=green!60]{68.2}
\smallskip
\bcbar[label=MB\_0.75\_128 v1, color=yellow]{62.1}
\bcbar[color=blue!60]{59.8}
\bcbar[color=green!60]{61.6}
\smallskip
\bcbar[label=MB\_0.5\_224 v1, color=yellow]{63.3}
\bcbar[color=blue!60]{62.2}
\bcbar[color=green!60]{63.0}
\smallskip
\bcbar[label=MB\_0.5\_128 v1, color=yellow]{56.3}
\bcbar[color=blue!60]{54.5}
\bcbar[color=green!60]{55.8}
\smallskip
\bcbar[label=MB\_0.25\_224 v1, color=yellow]{49.8}
\bcbar[color=blue!60]{48}
\bcbar[color=green!60]{49.2}
\smallskip
\bcbar[label=MB\_0.25\_128 v1, color=yellow]{41.5}
\bcbar[color=blue!60]{39.5}
\bcbar[color=green!60]{40.9}
\smallskip
\bcbar[label=ResNet\_50 v1, color=yellow]{75.2}
\bcbar[color=blue!60]{75}
\bcbar[color=green!60]{75.1}
\bcbar[color=blue!30]{73.2}
\bcbar[color=purple!60]{73.8}
\bcbar[color=purple!30]{74.3}
\bcbar[color=gray!90]{69.6}
\bcbar[color=gray!40]{71.2}
\smallskip
\bcbar[label=ResNet\_50 v2, color=yellow]{75.6}
\bcbar[color=blue!60]{75}
\bcbar[color=green!60]{75.4}
\bcbar[color=blue!30]{72.0}
\bcbar[color=purple!60]{74.6}
\bcbar[color=purple!30]{75.1}
\bcbar[color=gray!90]{71.6}
\bcbar[color=gray!40]{72.2}
\smallskip
\bclegend{6pt}{
yellow/$\underset{baseline}{fp32}$,
blue!60/${TF 8-8}$,
green!60/${AB 8-8}$,
blue!30/$\underset{per-channel}{TF 4-8}$,
purple!60/$\underset{per-layerl}{AB 4-8}$,
purple!30/$\underset{per-channel}{AB 4-8}$,
gray!90/$\underset{per-layer}{AB 4-4}$,
gray!40/$\underset{per-channel}{AB 4-4}$
}
\bcxlabel{ Top-1 accuracy }
\end{bchart}}
\caption{Top-1 accuracy of fp32, Tensorflow(TF)'s and Alpha-blending(AB) optimization with 8-bits or 4-bits numerical forms. 8-8: 8-bits weight and activation; 4-8: 4-bits weight and 8-bits activation; 4-4: 4-bits weight and activation.}
\label{fig:cha_1}
\end{figure}
\subsubsection{Evaluation of 4-bits weight and 8-bits activation (INT4-8)} \label{sect:4.3}
\cite{TF-int4-8:} reported that accuracy of 4-bits weight and 8-bits activation (INT4-8) is within 5\% of the fp32 baseline for Mobilenet v1 and ResNet networks. We ran the same models using AB quantization, and have listed the result in the $4^{th}$ and $5^{th} $columns in table \ref{table2}. The $4^{th}$ one is per-layer quantization, and $5^{th}$ is per-channel.
AB INT4-8 achieved a 1.53\% accuracy drop on average compared to the fp32 baseline for per-layer quantization, and a 0.9\% accuracy drop for per-channel quantization. Moreover, AB's INT4-8 per-channel performance outperforms the prior result ~\cite{TF-int4-8:} in $3^{rd}$ col. by 2.93\%.
\begin{table}[h]
\caption{ Top-1 accuracy: the pre-trained accuracies are in $2^{nd}$ col.; Tensorflow's INT4-8 - 4bits weight and 8-bits activation are in $3^{rd}$ col.; $\textbf{AB}$ INT4-8 - 4-bits weight and 8-bits activation in $4^{th}$ and $5^{th}$ cols. Note: for MobileNet in table \ref{table2}, the first layer and all depth-wise convolution layers, which are only 1.1\% of all the weights and consume 5.3\% of total MAC operations for inference are quantized into 8-bits. For ResNet v1 and v2, weight and activation of the first layer are quantized into 8-bits. $^+$\protect\cite{TF-int4-8:} }
\label{table2}
\begin{tabular}{|l|c|c|c|c|}
\hline
Model name & \multicolumn{1}{l|}{
{\begin{tabular}[c]{@{}l@{}}
{fp32} \\ {\ \ \%} \end{tabular}}
} & \multicolumn{1}{l|} {
{\begin{tabular}[c]{@{}l@{}}
\small{TF4-8} \\ {\ \ \ \%} \end{tabular}}
} & \multicolumn{1}{l|} {
{\begin{tabular}[c]{@{}l@{}}
\small{AB4-8} \\ {\scriptsize{per-layer\ \%}} \end{tabular}}
} & \multicolumn{1}{l|} {
{\begin{tabular}[c]{@{}l@{}}
\small{AB4-8} \\ {\scriptsize{per-channel\ \%}} \end{tabular}}} \\ \hline
{\small MB1.0\_224v1} & 70.9 & $65.0^{\small{+}}$ & 68.7 & 69.6 \\ \hline
{\small MB0.75\_224v1} & 68.4 & - & 65 & - \\ \hline
{\small MB0.50\_224v1} & 63.3 & - & 58.4 & - \\ \hline
{\small MB0.25\_224v1} & 49.8 & - & 43.8 & - \\ \hline
ResNet\_50v1 & 75.2 & $73.2^{\small{+}}$
& 73.8 & 74.3 \\ \hline
ResNet\_50v2 & 75.6 & $72^{\small{+}}$ & 74.6 & 75.1 \\ \hline
\end{tabular}
\end{table}
\subsubsection{Evaluation of 4-bits weight and 4-bits activation (INT4-4)} \label{sect:4.4}
Finally, we quantized the well-known neural networks, MobileNet\_1.0\_224 v1 and ResNet\_50 v1/v2, using 4-bit weights and 4-bit activations. The $4^{th}$ column in table \ref{table-3} is for per-layer quantization, whose accuracy is 5.5\% lower than fp32's in average. The per-channel quantization in the $5^{th}$ column has 4.66\% accuracy loss. AB's INT4-4 result, using per-channel quantization, achieves similar accuracy as the TF4-8 scheme \cite{TF-int4-8:}, which has 4-bits weight and 8-bits activation as shown in the $3^{rd}$ column.
\begin{table}[h]
\caption{top-1 accuracy of fp32, Tensorflow's INT4-8 and $\textbf{AB} $ INT4-4 quantization. The first layers, all depth-wise layers and the last layer are quantized in to 8-bits, and all other layers are in 4-bits both for weight and activation. $^+$\protect\cite{TF-int4-8:}}
\label{table-3}
\begin{tabular}{|l|c|c|c|l|}
\hline
Model name & \multicolumn{1}{l|}{
{\begin{tabular}[c]{@{}l@{}}
{fp32} \\ {\ \%} \end{tabular}}
} & \multicolumn{1}{l|}{
{\begin{tabular}[c]{@{}l@{}}
{\small{TF4-8}} \\ {\ \ \ \%}\end{tabular}}
} & \multicolumn{1}{l|}{\begin{tabular}[c]{@{}l@{}}
{\small{AB4-4}} \\ {\scriptsize{per-layer \%}} \end{tabular}} & \begin{tabular}[c]{@{}l@{}}{\small{AB4-4}}\\ {\scriptsize{per-channel \%}} \end{tabular} \\ \hline
{\small MB1.0\_224v1} & 70.9 & $65.0^{\small{+}}$ & 61.8 & 64.3 \\ \hline
\small{ResNet\_50v1} & 75.2 & $73.2^{\small{+}}$ & 69.6 & 71.2 \\ \hline
\small{ResNet\_50v2} & 75.6 & $72^{\small{+}}$ & 71.6 & 72.2 \\ \hline
\end{tabular}
\end{table}
\section{Conclusion and future work}
We have introduced alpha-blending ($\textbf{AB}$), an alternative method to the well-known $\textit{Straight-Through Estimator}$ ($\textbf{STE}$) for learning low precision neural networks using SGD. $\textbf{AB}$ accepts the almost everywhere zero gradient of quantization function during Backprop, and uses an affine combination of the original full-precision weights and corresponding quantized values as the actual weights in the loss function. This change allows the gradient update to the full-precision weights in backward propagation to be performed through the full-precision path incrementally, instead of applying STE to the quantization path.
To measure the impact on network accuracy using the AB methodology, we have trained a single-bit BinaryNet(BBN) \cite{binary} on CIFAR10 to show that $\textbf{AB}$ generates equivalent or better accuracy compared to training with $\textbf{STE}$. Moreover, we have applied the AB metholody to larger, more practical networks such as MobileNet and ResNet to compare with $\textbf{STE}$ based quantization. The top-1 accuracy of 8-bits weight and 8-bits activation is 0.82\% better than the existing state-of-art results \cite{TF-mb-int8:}\cite{TF-rn-int8:}. For 4-bits weight and 8-bits activation quantization, AB has 2.93\% higher top-1 accuracy on average compared to that reported in \cite{TF-int4-8:}.
$\textbf{AB}$ can also be applied to several other network optimization techniques besides quantization. We plan to investigate AB on clustering and pruning in a future work.
\bibliographystyle{named}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 9,601 |
Amorotto Trail
Il primo ultratrail dell'Appennino Reggiano
AUT – Amorotto Trail (80 Km)
MVT – Monte Valestra Trail (57 Km)
SVT – San Vitale Trail (20 Km)
PST – Poiago Short Trail (12 km)
CCV – Castello delle Carpinete Vertical (1,6 Km)
Nordic Walking dei Briganti
Nordic Walking – 10k and 20k
Eco commitment
You are here: Home / The races / Trail / SVT – San Vitale Trail (20 Km)
The Track:
Start from the race hospitality, then turn rigt around the park and after to get on Via Crispi, the main street through the village and leading up to the Lamola township where, turning left and then right, you take the first stretch of dirt road. The route continues slightly downhill to the hamlet of Montecchio where, follow the signs after a short stretch of asphalt, the two routes are divided, for SVT turn right and follow the path down to the SP98, cross (be careful, the security personnel for crossing) and continue uphill to the Petrella Cross. Cross the village, turn left and take the path marked by ups and downs that leads to the village of Croveglia where the two paths come together (be careful in case of single track in the rain can be very slippery), continuing downhill along the track which is divided in the midst of forests and carpini and roverelle you arrive in Cerpiano where, turning right, you will take the paved road. Arrived at the intersection with SP98 cross (be careful, follow the instructions of the security personnel to cross) and keep the left side of the road doing the utmost attention to open traffic. After 100 meters, turn left and take the steep path: after about two km take a steep path on the left where you will find the ropes tied to trees to facilitate the climb. The effort will be rewarded with the splendid view that you will find on the top, with also the refreshment. After being regenerated continue along the track for about 1 km: at this point the tracks are divided, for SVT turn right and follow the track, which in some places is characterized by single track with argyle ground in rain is very slippery. Follow the signs to the intersection with the SP7: cross (be careful, follow the instructions of the security personnel for crossing) and climb along the dirt road that after a few km rejoins the path of the AUT. Continue on the steep gravel road that will take you about 1 km to the beautiful Pieve di San Vitale where you will find a well-deserved rest (18km – 803 on the sea level). From there you will take the trail on the ridge from which you can admire on the right the valley of the river Tresinaro and left the valley of the river Secchia (note which also exposed stretches of single track). Once the ridge you will find yourself in front of the spectacular scenery of the Castle of Carpinete, perched on Antognano Mountain. At the foot of the climb that leads inside the walls you will have to cross the SP76 (attention, follow the instructions of the security staff to the crossing). Continuing along the cobblestone road you will enjoy the right vs the ancient castle and the Romanesque church. At this point the paths are divided again. For SVT, just out from the walls of the castle, turn right and take the downhill on Matildica path after 1.5 km takes you back in the village center of Carpineti. Pay attention to some road crossings because the traffic is open and follow the instructions of the security staff. Continue the path obligated to Matilde Park: congratulations, you are a SVT finisher!!
Amorotto asd – Via Farioli, 15 – 42033 – Carpineti – RE – PI 02695570354 | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 5,371 |
{"url":"http:\/\/www.physicsforums.com\/showthread.php?t=139358","text":"# Meaning of irrational exponent\n\nby murshid_islam\nTags: exponent, irrational, meaning\n P: 361 i know what the meaning of $$a^p$$ is when p is an integer or rational. e.g., $$a^3 = a.a.a$$ or $$a^{\\frac{1}{5}}$$ is such a number that when multiplied five times gives the number a. but what is the menaing of $$a^p$$ when p is an irrational number?\n P: 76 same thing just a irrational amount of times instant of a integer\/rational times. like a^3,14 (simple part of pi) its the same as a^3*a^0,14=a^3*a^(7\/50)=a^3*(a^(1\/50))^7 pi and such can be explained as a infinite amount of sums like this and by that a infintie amount of parts like this\nP: 4\n Quote by murshid_islam i know what the meaning of $$a^p$$ is when p is an integer or rational. e.g., $$a^3 = a.a.a$$ or $$a^{\\frac{1}{5}}$$ is such a number that when multiplied five times gives the number a. but what is the menaing of $$a^p$$ when p is an irrational number?\nThis shows how bad things are in some schools. If you are going to consider irrational exponents then a better approch is needed.\n\nThe modern way is:\n1. Define the Natural logarithm using an integral, viz, $$ln(x)$$. This is continuous and continuously differentiable for $$x>0$$\n2. Its inverse will be $$e^x$$ for all real $$x$$.\n3. This means that $$a^p = e^(pln(a))$$. This is then taken to be the definition of $$a^p$$.\n4. This is or was A-level in England till the year 2000 but you don't need to know that, these days, as things are dummed down.\n\n P: 1,520 Meaning of irrational exponent Well, it depends. For example, the value 2^pi does not mean anything, meaning that all it represent is another irrational number, untill you give pi a rational approximation. However, something like 10^log 2 has a meaning as it is another expression of 2. Here it is important to notice that ln 2 is a limit, as in the more digits you assign to log 2, the closer the expression 10^log 2 is to 2.\nMath\nEmeritus\nThanks\nPF Gold\nP: 39,323\n Quote by Werg22 Well, it depends. For example, the value 2^pi does not mean anything, meaning that all it represent is another irrational number, untill you give pi a rational approximation. However, something like 10^log 2 has a meaning as it is another expression of 2. Here it is important to notice that ln 2 is a limit, as in the more digits you assign to log 2, the closer the expression 10^log 2 is to 2.\nI am tempted to echo eds' statement, \"This shows how bad things are in some schools\"! Are you claiming that irrational numbers do not exist or \"do not mean anything\"? Your last statement \"ln 2 is a limit, as in the more digits you assign to log 2, the closer the expression 10^log 2 is to 2.\" is correct but that means that \"10^log 2\" does in fact exist and \"mean something\"!\n\nYou certainly can define, as eds said, 2^pi as e^(pi ln 2) but since pi ln 2 is an irrational number itself, you still haven't answered the original question:\n Quote by murshad_islam i know what the meaning of $a^p$ is when p is an integer or rational. e.g.,$a^3= a.a.a$ or $a^\\frac{1}{5}$ is such a number that when multiplied five times gives the number a. but what is the meaning of when p is an irrational number?\nIf p is an irrational number, then there exist a sequence of rational numbers {ri} converging to p. We define ap to make the function ax continuous: $a^p= \\lim_{i\\rightarrow \\infty} a^{r_i}$.\n P: 361 so, here ri is a closer and closer approximation to the irrational number p as i becomes larger and larger. am i right? and can we say that $r_{i} \\rightarrow p$ as $i \\rightarrow \\infty$?\n Sci Advisor HW Helper P: 9,398 Werg, what do you mean by 'meaning'. pi, is just as meaningful as 1\/2, mathematically, and if you want to discuss the philosophy of it there is another forum entirely devoted to that. Of course, mathematicians bandy about these tongue in cheek statements, but I wouldn't condone doing so here where the opportunity for misapprehension is so large.\nP: 361\n Quote by HallsofIvy pi ln 2 is an irrational number itself\nmaybe pi ln 2 is irrational, but how do we know that?\nP: 1,520\n Quote by murshid_islam maybe pi ln 2 is irrational, but how do we know that?\nThere is a way to prove that the product of the irrational number gives another irrational number provided that we are dealing with roots of powers (such as 2^1\/2 and 2^1\/2).\nP: 361\n Quote by Werg22 There is a way to prove that the product of the irrational number gives another irrational number provided that we are dealing with roots of powers (such as 2^1\/2 and 2^1\/2).\ni don't get it. $\\sqrt{2}.\\sqrt{2} = 2$(rational)\n P: 1,520 Sorry I forgot to add the \"not\". It should be provided that we are not dealing...\nEmeritus\nPF Gold\nP: 4,500\n Quote by Werg22 There is a way to prove that the product of the irrational number gives another irrational number provided that we are dealing with roots of powers (such as 2^1\/2 and 2^1\/2).\nP: 1,520\n Quote by Office_Shredder What about e and 1\/e?\nAlso inverse operations don't count. I admit that I am not sure about the validity of what I said, but I remember seeing a problem that asked to prove that the product of irrational numbers, provided certain conditions, is also irrational. I'll check it right now, I'll get back at you as soon as I find it.\nP: 17\n Quote by murshid_islam i know what the meaning of $$a^p$$ is when p is an integer or rational. e.g., $$a^3 = a.a.a$$ or $$a^{\\frac{1}{5}}$$ is such a number that when multiplied five times gives the number a. but what is the menaing of $$a^p$$ when p is an irrational number?\nHere is one definition: $$a^p$$ is the unique number y>0 such that $$\\int_{1}^{y} \\frac{dt}{t}=p\\int_{1}^{a} \\frac{dt}{t}$$\n HW Helper P: 3,352 I prefer the integral definitions. A good way to approximate them by hand if you ever lose your calculator :P Its also interesting when your a is not equal to 0 or 1, you get a nice transcendental number. Look up Gelfond-Schnieder Theorem\nEmeritus","date":"2014-07-26 01:12:26","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 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\": 0.8618587255477905, \"perplexity\": 610.748306956547}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2014-23\/segments\/1405997894931.59\/warc\/CC-MAIN-20140722025814-00163-ip-10-33-131-23.ec2.internal.warc.gz\"}"} | null | null |
{"url":"https:\/\/wiki.openelectrical.org\/index.php?title=Sequence_Networks","text":"# Sequence Networks\n\nReduced sequence networks: (a) Positive sequence, (b) Negative sequence, (c) Zero sequence\n\nIn symmetrical component analysis (e.g. for unbalanced faults), a balanced three-phase electrical network can be broken down into three sequence networks, which are independent, de-coupled sub-networks comprising only quantities in the same sequence, i.e. the positive sequence network contains only positive sequence quantities, the negative sequence network contains only negative sequence quantities and the zero sequence network contains only zero sequence quantities.\n\nFor any location in the system, the sequence networks can be reduced to Thevenin equivalent circuits (see the figure to the right). Notice that only the positive sequence network has a voltage source. This is because in a balanced system, there are no negative sequence or zero sequence voltages.\n\n## Constructing Sequence Networks\n\nSequence networks are constructed from two-port sequence networks of individual elements. This is best illustrated by example. Given the system below:\n\nThe individual positive, negative and zero sequence networks for each of the network elements are shown in the figure below:\n\nThese individual networks are then connected together to form the positive, negative and zero sequence networks for the overall system:\n\n## Reduction of Sequence Networks\n\nAt any location in the system, the overall sequence networks developed in the previous section can be reduced to a Thevenin equivalent circuit (positive sequence) or equivalent impedances (negative and zero sequence).\n\nFor example, consider a fault on the grid side of cable C1 as follows:\n\nFor this fault location, we can represent the sequence networks as reduced sequence networks:\n\nwhere\n\n$Z_{1} = \\left[ \\left( X^{''}_{d} + X_{1,T1} \\right) || Z_{1,L1} + Z_{1,C} \\right] || \\left( X_{1,T2} + X_{1,S} \\right) || Z_{1,L2} \\,$\n$Z_{2} = \\left[ \\left( X_{2,G1} + X_{2,T1} \\right) || Z_{2,L1} + Z_{2,C} \\right] || \\left( X_{2,T2} + X_{2,S} \\right) || Z_{2,L2} \\,$\n$Z_{0} = \\left( X_{0,T1} || Z_{0,L1} + Z_{0,C} \\right) || \\left( X_{0,T2} + X_{0,S} \\right) || Z_{0,L2} \\,$","date":"2019-08-23 13:31:20","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\": 0.8258398771286011, \"perplexity\": 1157.8822631558444}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2019-35\/segments\/1566027318421.65\/warc\/CC-MAIN-20190823130046-20190823152046-00370.warc.gz\"}"} | null | null |
{"url":"https:\/\/chemistry.stackexchange.com\/questions\/97340\/is-avogadros-number-an-integer?noredirect=1","text":"# Is Avogadro's number an integer? [duplicate]\n\nI have heard that Avogadro's number, $N_\\mathrm A=6.022 \\times 10^{23}$, is the number of atoms contained in $12$ grams of $\\ce{^{12}C}$. I think it should be an integer, but I couldn't find the exact number. Is it an integer?\n\u2022 The old (1971 I guess) definition of mole that you stated surely leads to floating point numbers (although only if you measure weights with more than 23 significant digits). The new definition (2018) fixes one mole to be exactly $6.02214076 \\times 10^{23}$, which is an integer. \u2013\u00a0Felipe S. S. Schneider May 22 '18 at 16:09","date":"2019-11-19 23:12:44","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\": 0.7817296981811523, \"perplexity\": 413.92417235559606}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2019-47\/segments\/1573496670268.0\/warc\/CC-MAIN-20191119222644-20191120010644-00013.warc.gz\"}"} | null | null |
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\section{Further results and problems}
\label{sec1}
\def\Dot#1{{\bf\dot#1}}
\noindent{\bf Question:} can the phenomenological notion of friction be
represented in alternative ways?
\noindent{\bf Related Question:} is it possible to set up a theory of
statistical ensembles, and their equivalence, extending to
stationary non-equilibria the ideas behind the canonical
and microcanonical ensembles, \cite{Ma867-b,SJ993}.
\*
A guide could be provided by the existence of a fundamental
symmetry like "time reversal" which cannot be "spontaneouly
broken".\footnote{\small In the subatomic world time reversal is
not a symmetry but another more fundamental symmetry, CPT, could
replace it in the following discussion}
Therefore even the stationary states of dissipative systems ought
to be describable via time reversible equations. Clearly the
question is not an easy one: hence it will be better to
specialize here on a paradigmatic example, namely the
Navier-Stokes (NS) fluid in a $2\pi$-periodic box, in $2$ or $3$
dimensions (2D or 3D), viscosity $\n$.
The $d$D equation, $d=2,3$, is:
\begin{equation}}\def\ee{\end{equation}\eqalign{
\dot{\V u}}\def\kk{{\V k}}\def\xx{{\V x}}\def\ff{{\V f}(\xx)_a=&-(({\V u}}\def\kk{{\V k}}\def\xx{{\V x}}\def\ff{{\V f}\cdot{\mbox{\boldmath$ \partial$}}){\V u}}\def\kk{{\V k}}\def\xx{{\V x}}\def\ff{{\V f})_a-
{\mbox{\boldmath$ \partial$}}_a p\cr
&+\n\D{\V u}}\def\kk{{\V k}}\def\xx{{\V x}}\def\ff{{\V f}_a+{\bf F}_a=0, \quad {\mbox{\boldmath$ \partial$}}\cdot{\V u}}\def\kk{{\V k}}\def\xx{{\V x}}\def\ff{{\V f}=0\cr}\Eq{e1.1}\ee
where ${\V u}}\def\kk{{\V k}}\def\xx{{\V x}}\def\ff{{\V f}(\xx)$ is the velocity field that in 2D can be represented
via Fourier's series as:
\begin{equation}}\def\ee{\end{equation} {\V u}}\def\kk{{\V k}}\def\xx{{\V x}}\def\ff{{\V f}(\xx)_a = \sum_{\kk\ne\bf0} i u_{\kk} \frac{\kk^\perp_a}{||\kk||}
e^{-i\kk\cdot\xx}\Eq{e1.2}\ee
and $\bf F$ is likewise represented via its Fourier's series.
In
terms of the complex scalars $u_\kk=\lis{\V u}}\def\kk{{\V k}}\def\xx{{\V x}}\def\ff{{\V f}_{-\kk}$ the 2D NS
equation is:
\begin{equation}}\def\ee{\end{equation} \eqalign{\dot u_\kk =& -\sum_{\kk_1+\kk_2=\kk}
\frac{(\kk_1^\perp\cdot\kk_2)(\kk_2\cdot\kk)}{||\kk_1 ||\,||\kk_2 ||\,||\kk ||}
u_{\kk_1} u_{\kk_2}\cr&-\n\kk^2u_\kk+F_\kk\cr}
\Eq{e1.3}\ee
Although the 2D-NS admit general smooth solution $t\to S_u{\V u}}\def\kk{{\V k}}\def\xx{{\V x}}\def\ff{{\V f}$
starting from smooth initial data ${\V u}}\def\kk{{\V k}}\def\xx{{\V x}}\def\ff{{\V f}$, it is convenient (aiming
to discuss also the 3D-NS) to imagine them as truncated at
$|\kk|=\max_i|k_i|\le N$. The ultraviolet (UV) cut-off $N$ will be
temporarily fixed. The 2D-NS become $M_N=(2N+1)^2-1$ dimensional
ODE's, on phase space $\MM_N$.
In the 3D case the 3D-NS
equations with UV cut-off $N$ can be likewise written on a
$M_N=2((2N+1)^3-1)$ dimensional phase space $\MM_N$.
The time reversal transformation $I{\V u}}\def\kk{{\V k}}\def\xx{{\V x}}\def\ff{{\V f}=-{\V u}}\def\kk{{\V k}}\def\xx{{\V x}}\def\ff{{\V f}$
does not imply $IS_t{\V u}}\def\kk{{\V k}}\def\xx{{\V x}}\def\ff{{\V f} = S_{-t} I{\V u}}\def\kk{{\V k}}\def\xx{{\V x}}\def\ff{{\V f}$ if $\n>0$: hence these
are irreversible equations.
Let ${\V u}}\def\kk{{\V k}}\def\xx{{\V x}}\def\ff{{\V f}$ be an initial state: then $t\to S_t{\V u}}\def\kk{{\V k}}\def\xx{{\V x}}\def\ff{{\V f}{\buildrel def\over=} {\V u}}\def\kk{{\V k}}\def\xx{{\V x}}\def\ff{{\V f}(t)$ evolves and
it is easly seen that $||S_t{\V u}}\def\kk{{\V k}}\def\xx{{\V x}}\def\ff{{\V f}||$ will verify an {\it a priori } estimate on
$\|{\V u}}\def\kk{{\V k}}\def\xx{{\V x}}\def\ff{{\V f}(t)\|^2_2=\sum_\kk |u_\kk(t)|^2 $:
\begin{equation}}\def\ee{\end{equation} \|{\V u}}\def\kk{{\V k}}\def\xx{{\V x}}\def\ff{{\V f}(t)\|^2_2=\le
\|{\V u}}\def\kk{{\V k}}\def\xx{{\V x}}\def\ff{{\V f}(0)\|^2_2+\frac{\|\bf F\|^2_2}{\n^2}< \Big(\frac{\|\bf
F\|^2_2}{\n^2}\Big)_+\Eq{e1.4} \ee
where the last inequality abridges stating that if
$\|{\V u}}\def\kk{{\V k}}\def\xx{{\V x}}\def\ff{{\V f}(0)\|^2_2$ is larger than $\frac{\|\bf F\|^2_2}{\n^2}$ then
it will decrease to become eventually, smaller than any target
$>\frac{\|\bf F\|^2_2}{\n^2}$ (in a time depending on
${\V u}}\def\kk{{\V k}}\def\xx{{\V x}}\def\ff{{\V f}(0)$ and on the prefixed target).
\*
\noindent{\bf Definition:} {\sl A smooth dynamical system $S_t$ on a manifold $\MM$
will be called {\it regular} if in $\MM$ there are a finte number
of open sets ${\mathcal A}}\def\CC{{\mathcal C}}\def\FF{{\mathcal F}_i, i=1,2,\ldots,n$ which:
\\
(1) are $S_t$-invariant, \\
(2) their union is $\MM$ up to a set of zero volume,
(3) the evolution of almost all data ${\V u}}\def\kk{{\V k}}\def\xx{{\V x}}\def\ff{{\V f}\in{\mathcal A}}\def\CC{{\mathcal C}}\def\FF{{\mathcal F}_i$ assigns an
average value to the observables $O$ ({\it i.e.\ } to functions on $\MM$)
$\m_i(O)$ where $\m_i$ is an invariant distribution.
\\
The $\m_i$ will be called ``physical distributions'' and if $n>1$
the system will be said to admit $n$ phases.}
\*
The distributions are called ``physical dustribution'' because
they allow to compute the time averages of functions $O({\V u}}\def\kk{{\V k}}\def\xx{{\V x}}\def\ff{{\V f})$ on
phase space: hence determine the statistical properties of almost
all data.
If the motion is chaotic then generates a ``stationary state'' on
$\MM_N$, {\it i.e.\ } a stationary probability distribution which, aside
exceptions on the initial data collected in a $0$-volume set in
$\MM_N$, will be supposed unique, for simplicity, and denoted
$\m_\n (d{\V u}}\def\kk{{\V k}}\def\xx{{\V x}}\def\ff{{\V f})$.
Stationary probability distributions generalize to the case of
much more general systems, the regular ones,
what in many classes of Hamiltonian systems are the equilibrium distributions
studied in equilibrium statistical mechanics
(SM).
Here we assume that the regularized NS equations, denoted
$INS^N$, on $\MM_N$ parameterized by the viscosity $\n$ are
regular in the above sense. And it is natural to define the
collection ${\mathcal E}}\def\DD{{\mathcal D}}\def\TT{{\mathcal T}^{N}_{viscosity}$ of all physical distributions
$\m^N_\n (d{\V u}}\def\kk{{\V k}}\def\xx{{\V x}}\def\ff{{\V f})$ on $\MM_N$. In the cases in which there are
several physical distributions corresponding to the same
viscosity a further label $\b=1,\ldots,n_\n^N$ will be attached as
$\m^N_{\n,j}$ to distinguish them.
Furthermore we fix once and for all the forcing $\F$, with $||\bf
F||_2=1$, and with $|F_\kk|=0$, if $|\kk|>k_{max}$ with
$k_max<\infty$: physically this is read that $\bf F$ is assumed
to be a large scale forcing.
\defFurther results and problems{Enstrophy ensemble}
\section{Further results and problems}
\label{sec2}
\setcounter{equation}{0}
Consider the new equation, \cite{Ga996b}, with UV-cut-off $N$:
\begin{equation}}\def\ee{\end{equation} \eqalign{\dot u_\kk =& -\sum_{\kk_1+\kk_2=\kk}
\frac{(\kk_1^\perp\cdot\kk_2)(\kk_2\cdot\kk)}{||\kk_1 ||\,||\kk_2 ||\,||\kk ||}
u_{\kk_1} u_{\kk_2}\cr&-\a({\V u}}\def\kk{{\V k}}\def\xx{{\V x}}\def\ff{{\V f})\kk^2u_\kk+F_\kk, \qquad
|\kk|,|\kk_1|,|\kk_2|\le N\cr}
\Eq{e2.1}\ee
differing from the Eq.\equ{e1.3} because the multiplier $\a({\V u}}\def\kk{{\V k}}\def\xx{{\V x}}\def\ff{{\V f})$
replaces the viscosity $\n$. The non linear term in Eq.\equ{e2.1}
will be denoted ${\bf n}({\V u}}\def\kk{{\V k}}\def\xx{{\V x}}\def\ff{{\V f},{\V u}}\def\kk{{\V k}}\def\xx{{\V x}}\def\ff{{\V f})_\kk$.
The multiplier $\a({\V u}}\def\kk{{\V k}}\def\xx{{\V x}}\def\ff{{\V f})$ will be so defined that the solutions of
the Eq.\equ{e2.1} will admidt
$\DD({\V u}}\def\kk{{\V k}}\def\xx{{\V x}}\def\ff{{\V f}){\buildrel def\over=}\sum_\kk\kk^2|u_\kk|^2$, usually called {\it enstrophy},
as an {\it exact constant of motion}. From Eq.\equ{e2.1} this means that
\begin{equation}}\def\ee{\end{equation}\a({\V u}}\def\kk{{\V k}}\def\xx{{\V x}}\def\ff{{\V f})=\frac{\sum_\kk \kk^2 \Big({\bf n}({\V u}}\def\kk{{\V k}}\def\xx{{\V x}}\def\ff{{\V f},{\V u}}\def\kk{{\V k}}\def\xx{{\V x}}\def\ff{{\V f})_\kk+F_\kk
\Big) \lis u_\kk}{\sum_\kk\kk^4|u_\kk|^2}\Eq{e2.2}\ee
(in 2D the term with $\bf n({\V u}}\def\kk{{\V k}}\def\xx{{\V x}}\def\ff{{\V f},{\V u}}\def\kk{{\V k}}\def\xx{{\V x}}\def\ff{{\V f})$ vanishes identically).
The new equation is reversible: $IS_t {\V u}}\def\kk{{\V k}}\def\xx{{\V x}}\def\ff{{\V f} = S_{-t} I{\V u}}\def\kk{{\V k}}\def\xx{{\V x}}\def\ff{{\V f}$ (as
$\a({\V u}}\def\kk{{\V k}}\def\xx{{\V x}}\def\ff{{\V f})$ is odd) and will be called $RNS^N$. So $\a$ can be
called a "reversible friction".\footnote{\small In the 3D case
the equations are very similar (for instance the $u_\kk$ are
replaced bu vectors orthogonal to $\kk$, ..): the main and
important difference is that the expression of $\a({\V u}}\def\kk{{\V k}}\def\xx{{\V x}}\def\ff{{\V f})$ receives
a contribution from the quadratic transport term which, although
present also in 2D, cancels from $\a$ essentially because in 2D
when $\n=0$ the $\DD({\V u}}\def\kk{{\V k}}\def\xx{{\V x}}\def\ff{{\V f})$ is conserved.}
The non Newtonian forces $\n\D{\V u}}\def\kk{{\V k}}\def\xx{{\V x}}\def\ff{{\V f}$, $\a({\V u}}\def\kk{{\V k}}\def\xx{{\V x}}\def\ff{{\V f})\D{\V u}}\def\kk{{\V k}}\def\xx{{\V x}}\def\ff{{\V f}$ can be imagined to play
the role of a thermostat:\cite{Ga021} the forcing performs work on the fluid
but the fluid density remains the same. This means that the
temperature must change so that the equation of state linking
pressure density and temperature remains fulfilled. Heat has to
be removed or inserted and both forces
can be viewed as external forces allowing to achieve respect of
the equation of state. Then it is natural to think that the two
equations should be equivalent.
This leads to an equivalence conjecture, \cite{Ga020b}, which we
formulate still making use of our knowledge of Statistical
Mechanics where examples of equivalences are well known and can
guide us.
The evolution with $RNS^N$ wil be assumed regular in the sense of
the above definition; it generates a family of stationary by the
distributions on phase space: $\m^N_D$ parameterized by the
constant value $D$ of the enstrophy $\DD({\V u}}\def\kk{{\V k}}\def\xx{{\V x}}\def\ff{{\V f})$. Again for a given
$D$ there may be $m^N_D>1$ physical states which will be
distinguished by extra label $j=1,\ldots,n_D^N$. The collection
of such distributions will form the ``enstrophy ensemble'',
${\mathcal E}}\def\DD{{\mathcal D}}\def\TT{{\mathcal T}^N_{enstrophy}$.
To formulate the connection between the two ensembles
${\mathcal E}}\def\DD{{\mathcal D}}\def\TT{{\mathcal T}^N_{viscosity},{\mathcal E}}\def\DD{{\mathcal D}}\def\TT{{\mathcal T}^N_{enstrophy}$ it is necessary do define
the {\it local observables} $O({\V u}}\def\kk{{\V k}}\def\xx{{\V x}}\def\ff{{\V f})$: these are functions on
phase space whose value depends on the harmonics $\kk$ contained in a
finite region $\D$ {\it independent} on $N$,\footnote{\small
Hence depend on finitely many Fourier's
harmonics of ${\V u}}\def\kk{{\V k}}\def\xx{{\V x}}\def\ff{{\V f}$, {\it i.e.\ } are ``large scale'' observables'', but
unlike the forcing, which only has the harmonics $|\kk|<k_{max}$
with $k_{max}$ fixed, no limit is set on the size scale}.
The local observables here are analogous to the SM observables on
phase space depending only on the positions and velocities of
particles located in finite region $\D$ {\it independent} on the
size $V$ of the container. So once more arises a similarity
between the SM of a system enclosed
in a volume $V$ and of a NS fluid with UV cut-off $N$: one can
say that locality in SM is in position space while in NS it is in
momentum space.
With this in mind the following conjecture has been proposed to
relate physical distributions $\m^N_{\n,\b}, \r^N_{D,\b'}$ in
${\mathcal E}}\def\DD{{\mathcal D}}\def\TT{{\mathcal T}^N_{viscosity},{\mathcal E}}\def\DD{{\mathcal D}}\def\TT{{\mathcal T}^N_{enstrophy}$ where $\b=1,\ldots,n^N_\n;
\b'=1,\ldots,m^N_D$ are labels distinguishing the physical
distributions, if more then one: note however that cases in which
$n^N_\n,m^N_D>1$ are expected to be rare. Then:\cite{Ga020b}
\*
{\bf Conjecture:} {\sl Let $\m^N_{\n,\b}\in{\mathcal E}}\def\DD{{\mathcal D}}\def\TT{{\mathcal T}^N_{viscosity}$ and
$\r^N_{D,\b'}\in{\mathcal E}}\def\DD{{\mathcal D}}\def\TT{{\mathcal T}^N_{enstrophy}$ be physical
distributions with the same enstrophy
\begin{equation}}\def\ee{\end{equation}\m^N_{\n,\b}(\DD)=D \Eq{e2.3}\ee
Then if $N$ is large enough it is
$n^N_\n=m^N_D$\ \footnote{\small
In many cases no ``intermittency'' is expected, {\it i.e.\ } $n^N_\n=m^N_D=1$.}
and for each
$\b$ there is $\b'$
\begin{equation}}\def\ee{\end{equation} \lim_{N\to\infty} \n^N_{\n,\b}(O)
=\lim_{N\to\infty} \r^N_{D,\b'}(O)\Eq{e2.4}\ee
for all local observables.}
\*
So the averages of large scale observables will show the same
statistical properties, as $N\to\infty$ in $INS^N$ and $RNS^N$
evolutions, under the correspondence condition of equal
enstrophy, Eq.\equ{e2.3}.
The $\a({\V u}}\def\kk{{\V k}}\def\xx{{\V x}}\def\ff{{\V f})$ in the $RNS^N$ evolution will fluctuate strongly in
turbulence regime and it will "self-average" to a constant $\n$
thus "homogenizing" the equation and turning it into the
$INS^N$ with friction $\n$.
The conjecture however does not mention a condition like for $\n$
small enough: it should hold also at high viscosity where often
$INS^N$ exhibits periodic attractors. The reason it is proposed
also in such cases is that in all cases the NS equations
should be regarded as yielding macroscopic descriptions of
microscopic, certainly chaotic, evolutions derived via scaling
limits without modifyig the microscopic equations, \cite{Le008}.
It is natural to think that there should be no condition for
strong chaos. The microscopic motion is always strongly chaotic
and the chaoticity condition should be always fulfilled even when
motion appears laminar.
The analogy with SM becomes even more clear: the $N\to\infty$
limit corresponds to the thermodynamic limit $V\to\infty$.
As a final remark other ensembles can be imagined: first is the
``energy ensemble'' formed by the physical distributions for the
equation Eq.\equ{e2.1} with $\a$ so defined that the energu
$\|{\V u}}\def\kk{{\V k}}\def\xx{{\V x}}\def\ff{{\V f}\|^2_2$ is an exact constant ({\it i.e.\ } $\a$ is given, in 2D and
also in 3D by Eq.\equ{e2.2} replacing $\kk^2$ with $1$ and
$\kk^4$ with $\kk^2$) and a similar analysis and conjecture be can be
set up: for results obtained with this approach see
\cite{SDNKT018}.
At this point it is convenient to pause and show a few
results from simulations which begin to test the equivalence
proposal.
\defFurther results and problems{Some 2D simulations}
\section{Further results and problems}
\label{sec3}
\setcounter{equation}{0}
Here are collected results the earlier (<2018) simulations that
might interest readers: skipping this part does not preclude
following the final comments in Sec.IV.
There is a first obvious test suggested by a rigorous consequence
of the conjecture based on the remark that the work per unit time
of the forcing is a local observable, being $W={\bf
F}\cdot{\V u}}\def\kk{{\V k}}\def\xx{{\V x}}\def\ff{{\V f}\equiv \sum_{|\kk|<k_max} F_\kk\lis u_\kk$.
Multiplying both sides of $RNS^N$ or $INS^N$ by $\lis u_{\kk}$
one finds the energy conservation identity:
\begin{equation}}\def\ee{\end{equation}\eqalign{\a({\V u}}\def\kk{{\V k}}\def\xx{{\V x}}\def\ff{{\V f})D=& {\bf F}\cdot{\V u}}\def\kk{{\V k}}\def\xx{{\V x}}\def\ff{{\V f},\qquad RNS^N\cr
\n \DD({\V u}}\def\kk{{\V k}}\def\xx{{\V x}}\def\ff{{\V f})=& {\bf F}\cdot{\V u}}\def\kk{{\V k}}\def\xx{{\V x}}\def\ff{{\V f},\qquad INS^N\cr}\Eq{e3.1}\ee
The conjecture implies that $\r^N_D(\a)D$ has to be equal in the
$\lim_{N\to\infty}$ to the average $W$ which in the $INS^N$ by the
second line of Eq.\equ{e3.1} is $\n\m^N_\n(\DD)$. Hence in
absence of intermittency the equivalence condition
$\m^N_\n(\DD)=D$ yields
\begin{equation}}\def\ee{\end{equation} \lim_{N\to\infty} \r^N_D(\a)=\n\Eq{e3.2}\ee
Hence a test is: i) fix $\n,N$ and run the $INS^N$ evolution from a
random initial ${\V u}}\def\kk{{\V k}}\def\xx{{\V x}}\def\ff{{\V f}$ until the average enstrophy value $D$ of the
enstrophy $\DD({\V u}}\def\kk{{\V k}}\def\xx{{\V x}}\def\ff{{\V f}(t))$ is numerically reached. The run the
$RNS^N$ with initial ${\V u}}\def\kk{{\V k}}\def\xx{{\V x}}\def\ff{{\V f}$ adjusted to have enstrophy $D$: the
result should be that if $N$ is large enough the running average
of $\a$,{\it i.e.\ } $\frac1t\int_0^t\frac{\a({\V u}}\def\kk{{\V k}}\def\xx{{\V x}}\def\ff{{\V f}(t'))}\n
dt'\tende{t\to\infty} 1$. An example is
\eqfig{185}{140}{}{FigA32-19-17-11.1-all}{
}
\noindent{\small Fig.1: The running average of the reversible friction
$\frac{\a({\V u}}\def\kk{{\V k}}\def\xx{{\V x}}\def\ff{{\V f}(t))}\n\equiv \frac1\n\frac{\sum_{\kk} \kk^2 2
f_{-\kk} u_{\kk}} {\sum_\kk \kk^4|u_\kk|^2}$, superposed to
the \alertb{conjectured value $1$} and to the fluctuating
values $\frac{\a({\V u}}\def\kk{{\V k}}\def\xx{{\V x}}\def\ff{{\V f}(t))}\n$: Evolution is by \alert{$RNS^N$},
\alertb{\bf R=2048}, 224 modes ($N=7$), Lyap.-max $\simeq 2$,
$x$-axis unit $2^{19}$, forcing only on modes
$\kk=\pm(2,-1)$. Data are obtained via a sequence of
integration steps of size $h=2^{-19}$ registered every
$4h$. The plot gives $4000$ successive registered results but
only every $10$ of them to avoid too dense a plot.}
\eqfig{185}{140}{}{FigA32-19-17-11.1-detail}{}
\noindent{\small Fig.1-detail: The running average of the reversible friction
$\frac{\a({\V u}}\def\kk{{\V k}}\def\xx{{\V x}}\def\ff{{\V f}(t))}\n$, superposed to the conjectured limit value
$1$. Same data as the Fig.1 but for the shorter time interval
$[0,300]$ to show the initial transient.}
The Fig.2 shows the preliminary evaluation of the average
enstrophy: it shows that in the case considered the average
avaleu of the enstrophy is reached quite rapidly in spite of the
strong fluctuations.
\eqfig{185}{140}{}{FigEN32-19-17-11.1}{}
\noindent{\small Fig.2: Running average of $\frac{W({\V u}}\def\kk{{\V k}}\def\xx{{\V x}}\def\ff{{\V f}(t))}\n$ (dark green) in $INS^N$;\\
the average $\frac1\n D$ of $\frac1\n \DD({\V u}}\def\kk{{\V k}}\def\xx{{\V x}}\def\ff{{\V f})$ in $INS^N$ (straight red line)\\
running average of $\frac1\n \DD({\V u}}\def\kk{{\V k}}\def\xx{{\V x}}\def\ff{{\V f}(t))$ in $INS^N$ 'converging'
to $\frac1\n D$ (very close to $\frac1\n D$)\\
large fluctuations are those of $\frac1\n \DD({\V u}}\def\kk{{\V k}}\def\xx{{\V x}}\def\ff{{\V f}(t))$. Data are
the same in the previous figures.}
It is natural to study the observable $\a({\V u}}\def\kk{{\V k}}\def\xx{{\V x}}\def\ff{{\V f})$ as it evolves
under the $INS^N$ equation. It is not covered by the conjecture,
that only implies that its average should be, under the equivalence condition,
close to $1$ in the $RNS^N$ evolution. An unexpected result is
that the running average of $\frac1\n\a({\V u}}\def\kk{{\V k}}\def\xx{{\V x}}\def\ff{{\V f}(t))$ also has running
average very close to $1$ as indicated by the following figure:
\eqfig{185}{140}{}{FigA32-19-17-11.0-all}{}
\noindent{\small Fig.7: Same as Fig.1 but for $INS^N$.
The running average of the reversible friction
$\frac{\a({\V u}}\def\kk{{\V k}}\def\xx{{\V x}}\def\ff{{\V f}(t))}\n$ in a $N=7$ regularized $INS^N$ evolution
forced on the mode $\kk=\pm(2,-1)$: the running average is in
the large fluctuations curve. }
This is one more example of a non local observable with equal
averages in corresponding physical distributions for the two
evolutions considered.
The equality to $\n$ under the equivalence condition
between the average value of
$\a({\V u}}\def\kk{{\V k}}\def\xx{{\V x}}\def\ff{{\V f})=\frac{\sum_\kk \kk^2 F_\kk\lis u_\kk}{\sum_\kk
\kk^4|u_\kk|^2}$ considered as an observable for both $RNS^N$
and $INS^N$ with $\n$ is perhaps surprising.
It is a theorem (consequence of the
conjecture) in the $RNS^N$ evolution but it might not be even expected
in the case of $INS^N$ because $\a({\V u}}\def\kk{{\V k}}\def\xx{{\V x}}\def\ff{{\V f})$ is not a local
observable.
Therefore it is tempting to test possible equality of the
averages of other nonlocal quantities, as such equalities are
well known to hold in the thermodynamic limit for several non
local observables.
One such observable is the spectrum of the symmetric part of the
Jacobian $J({\V u}}\def\kk{{\V k}}\def\xx{{\V x}}\def\ff{{\V f})$, the $M_N\times M_N$ matrix
$J_{\kk,\kk'}=\frac{{\partial}\cdot u_\kk}{{\partial} u_{\kk'}}$,
formally $\frac{{\partial} \cdot{\V u}}\def\kk{{\V k}}\def\xx{{\V x}}\def\ff{{\V f}}{{\partial} {\V u}}\def\kk{{\V k}}\def\xx{{\V x}}\def\ff{{\V f}}$. The spectrum can be
called the ``local Lyapunov spectrum''.
\eqfig{185}{140}{}{FigL16-30-15-13-11.01-15}{}
\noindent{\small Fig.3: The spectrum of the symmetrized Jacobian called the
``local Lyapunov exponents'' for $N=3$ ($48$ modes) truncation for
$INS^N$ and $RNS^N$ under the equivalence condition: they are
superposed but undistinguishable on the scale of the picture
which plots $(k,\l_k)$, same data as in the previous pictures
(but truncated at $N=3$).}
The above coincidence to some extent is due to the wide ordinate
scale used. It is expected the two spectra are subject to
computatinal errors which should be more visible near the $k$ to
which correspond $\l_k$'s close to $0$. This is clarified in
\eqfig{185}{140}{}{FigDiff16-30-15-13-11.01-15}{}
\noindent{\small Fig.4: Relative difference betweeen (local) Lyapunov exponents
in the previous Fig.3 ($\n=2048^{-1}$, $N=3$ ({\it i.e.\ } $48$ modes)).}
Certainly a cut-off at $N=3$ is much too small to be of any
significance. In fact agreement between the $INS^N$ and $RNS^N$
is expected also at fixed $N$ and small $\n$, \cite{Ga997}, as a
consequence of appearance of turbulence: which generates
apparently random fluctuations on $\a$ with a corresponding
homogenization phenomenon: and physically diferent phenomenon.
The following figure tests the equivalence in the case of a
higher UV cut-off
\eqfig{185}{140}{}{FigL32-19-17-11.01}{}
\noindent{\small Fig.5: Local Lyapunov spectra in a $15 \times 15$
($960$ modes) truncation for $INS^N$ and $RNS^N$ (keys ending
respectively in 0 or 1), with the $\l_k$ interpolated versus
$k$ by lines, $\n=\frac1{2048}$, forcing onthe
modes$\kk=(2,-1)$. Each of the $\l_k({\V u}}\def\kk{{\V k}}\def\xx{{\V x}}\def\ff{{\V f}(t))$ is evalueted every
$2^{19}$ integration steps and the graf reports the average of
of each $\l_k({\V u}}\def\kk{{\V k}}\def\xx{{\V x}}\def\ff{{\V f}(t))$ over $2200$ successive evaluations.}
The latter graph shows the spectra for both $INS^N$ and $RNS^N$
cases: again they are superposed. As in the previous case the
relative difference can be studied more clesely
\eqfig{185}{140}{}{FigDiff32-19-17-11.01}{}
\noindent{\small Fig.6: Relative difference betweeen (local) Lyapunov exponents
in the previous Fig.5}
\defFurther results and problems{Further results and problems}
\section{Further results and problems}
\label{sec4}
\setcounter{equation}{0}
{\bf(1)} The question of exhibiting examples of ``regular systems'' in the
sense of Sec.II has been essentially answered in the proposal by
Ruelle that I intepret as saying that ``generically'' all systems
exhibiting chaotic evolution should be ``regular''.
In more mathematically oriented works the idea emerges from the
theory of Anosov flows: they play the role, in chaotic dynamics,
of the harmonic oscillators in ordered dynamicsl they are the
paradigm of Chaos. This idea rests on fundamental works of Sinai
(on Anosov sys.), and Ruelle, Bowen (on Axioms A systems). A
strict, general, heuristic, interpretation of original ideas on
turbulence phenomena, \cite{Si968a,BR975,Bo975,Ru989,Ru978b}, led
to the, \cite{GC995b}:
\*
\noindent{\bf Chaotic hypothesis:} {\it A chaotic evolution takes place on a
smooth surface $A$, "attracting surface", contained in phase
space, and on $A$ the maps $S$ (or the flow $S_t$) is an Anosov
map (or flow).}
So a regular system is a system with $n\ge1 $ attractors $A_i$ whose
basins of attraction are open sets ${\mathcal A}}\def\CC{{\mathcal C}}\def\FF{{\mathcal F}_i$ whose union of the
entire phase space up to a set of zero volume.
\* {\bf(2)} The cahotic hypothesis is dismissed (by many) with
arguments like (1999) {\it More recently Gallavotti and Cohen
have emphasized the "nice" properties of Anosov systems. Rather
than finding realistic Anosov examples they have instead
promoted their "Chaotic Hypothesis": if a system behaved "like"
a [wildly unphysical but well-understood] time reversible
Anosov system there would be simple and appealing consequences,
of exactly the kind mentioned above. Whether or not
speculations concerning such hypothetical Anosov systems are an
aid or a hindrance to understanding seems to be an aesthetic
question.}\cite{HG012}
While giving up any evaluation of the statement I stress that
Statistical Mechanics, after Clausius, Boltzmann and
Maxwell was a simple and appealing consequence of the
"[wildly unphysical but well-understood]" periodicity
of motions of atoms in a gas.
\*
{\bf(3)} The same tests mentioned here (dated up to 2018) have been
made in 2D NS with up to $920$ modes and in 3D NS even up to
$\sim 5\cdot10^6$ modes.\cite{Ga021}
{\bf(4)} The 3D tests have suggested that the notion of local
observable should be made more strict defining $O$ a local observable
if $O({\V u}}\def\kk{{\V k}}\def\xx{{\V x}}\def\ff{{\V f})$ depends finitely many harmonics $\kk$, as above, further
verifying $|\kk|< c K_\n$
where $K_\n=(\frac\h{\n^3})^{\frac14})$ is Kolmogorov's scale and
$c$ is a constant and $\h=\n\media{W}$ is the average work per
unit time: it is believed widely the in the limit $\n\to0$ $\h$
should remain finite and positive.\cite{MBCGL022}
Although the 3D tests seem to suggest $c$ is of order $O(1)$ in
my view the possibility that $c=\infty$ should be studied with
further simulations.
{\bf(5)} one of the consequences of the chaotic hypothesis is that if
the system is time reversible then a general result is the
``fluctuation theorem'' (FT). The problem is that there is strong
evidence that in the NS the asymptotic motion is attracted on a
set of dimension lower than that of phase space: this is shown by
the fact that as soon as $N$ is large enough and $\n$ small
enough the attractor has dimension less that that of phase space:
this is shown by the $>0$ Lyapunov exponents seem to be less than
half the number of negative ones.\cite{Ga020b}
In 2D this seems to be the case with $\n=1/2048$ and $N=7$ and
$15$. There are a few examples in which even though the attractor
$A$ has dimension lower than that of phase space the motion on
$A$ admits a symmetry $I':A\to A$ which has the same properties
as time reversal ({\it i.e.\ } $I'S_t=S_{-t}I'$): but is is unclear that
the NS admit such a symmetry. Positive results in testing
validity of the FT have been found, \cite{Ga020}, in 2D in the
$RNS^N$ with $N=48, \n=1/2048$, way too small for being really
interesting, while already for the case $N=224, \n=1/2048$ it is
unclear if FT holds.
\*
\def}\def\Alertb{{}\def\Alertb{}
\centerline{\bf Appendix: A path through the theme}
\*
\01) }\def\Alertb{{A first equivalence example:} \cite{SJ993}\\
2) \Alertb{Path to the conjecture:} \cite{Ga997b,Ga019c,Ga020b,MBCGL022}\\
3) }\def\Alertb{{3D enstrophy ensemble:} \cite{MBCGL022,JC021}\\
4) \Alertb{3D energy ensemble:} \cite{SDNKT018}\\
5) }\def\Alertb{{Shell model:} \cite{BCDGL018}\\
7) \Alertb{Stat-Mech:} \cite{Ru969,Ru977,Ru978b,Ru989,Ru012}\\
8) }\def\Alertb{{Turbulence physics:} \cite{Fr995,BF010,Ge013,BV019,Fe000}
\*
\noindent{Acknowledgements: This is a redacted and updated version of a
talk at the {\it DinAmici} meeting on 21/Dec/2018 at the
Accademia dei Lincei, Roma.}
\bibliographystyle{unsrt}
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{"url":"https:\/\/physics.stackexchange.com\/questions\/365789\/is-this-expression-for-radial-probability-flux-in-sakurais-modern-quantum-mecha","text":"# Is this expression for radial probability flux in Sakurai's Modern Quantum Mechanics wrong?\n\nThe section on Schrodinger's equation for central potentials in Sakurai's Modern Quantum Mechanics (p. 208, 2nd edition) contains the following expression for the radial probability flux, as part of his argument for ruling out the (asymptotic) $r^{-l}$ solution for the radial part of the wavefunction, which I believe is wrong: $$j_r = \\hat{\\textbf{r}} \\cdot \\textbf{j} = \\frac{\\hbar}{m} R_{El}(r) \\frac{dR_{El}(r)}{dr} \\tag{1}$$ Context:\n\nLet us assume that the potential-energy function is not so singular so that $\\lim_{r\\to0}r^2V(r) = 0$. Then, for small values of $r$, (3.7.9) [Radial part of Schrodinger equation as $r \\to 0$] becomes $$\\frac{d^2u_{El}}{dr^2} = \\frac{l(l+1)}{r^2}u_{El}(r)$$ [where $R_{El}(r)$ is the the radial part of the wavefunction and $u_{El}(r) = rR_{El}(r)$]\n\nwhich has the general solution $u(r) = Ar^{l+1} + Br^l$\n\n...\n\nConsider the probability flux. This is a vector quantity whose radial component is $$j_r = \\frac{\\hbar}{m}Im(\\psi^*\\frac{\\partial\\psi}{\\partial r}) = \\frac{\\hbar}{m}R_{El}(r)\\frac{d}{dr}R_{El}(r)$$\n\nThe textbook then goes on to substitute one by one, the asymptotic solutions $R = Ar^l$ and $R = Br^{-(l+1)}$, and shows that for the second case, $lim_{r\\to0}j_r \\neq 0$.\n\nNow, according to me, $$\\psi(r, \\theta, \\phi) = R_{El}(r)Y_l^m(\\theta, \\phi) \\\\ \\frac{\\partial \\psi}{\\partial r} = Y_l^m(\\theta, \\phi)\\frac{dR_{El}(r)}{dr} \\\\ Im\\left(\\psi^*\\frac{\\partial \\psi}{\\partial r}\\right) = \\left|Y_l^m\\right|^2Im\\left(R^*\\frac{dR}{dr}\\right)$$ For $R = Ar^{-(l+1)}$, $$R^*\\frac{dR}{dr} = \\left(A^* r^{-(l+1)} \\right) \\left(-(l+1)A r^{-(l+2)} \\right) = -|A|^2(l+1)r^{-(2l+3)} \\tag{2}$$ which is purely real, thus for this case\n\n$$j_r = \\frac{\\hbar}{m} \\left| Y_l^m \\right|^2 Im\\left(R^*\\frac{dR}{dr}\\right) = 0$$\n\nI looked up the errata (pdf) for Sakurai's book but the entry for page 208 only notes the absence of spherical harmonics from (1). Is there something wrong with my calculation (2)?\n\n(Edited to add details of calculation)\n\nConsider l=0, so you don't think this has something to do with polar coordinates: perfect spherical symmetry, $\\psi=R_E(r)\\equiv \\sqrt{\\rho (r)} e^{iS(r)}$, for real \u03c1 and S, in general.\n\nFor bound states like atoms, of course, S vanishes, R is real, as you noticed, as E < 0 and these systems are stationary: they stay put, $\\dot{\\rho}=0$, without leaking probability out, $j_r=0$. Atoms are stable.\n\nBut, for scattering states, of course, it does not: think of a spherical wave ($R\\sim e^{irk}\/r$, so $j_r=\\hbar k\/mr^2$), or a free spreading wave packet, below.\n\nThe probability density current is thus $$j_r= \\frac{\\hbar}{m}\\operatorname{Im} \\bar{R}\\partial_r R= \\frac{\\hbar}{m} \\rho \\partial_r S .$$\n\nThe continuity equation, $\\dot{\\rho}+ j_r(r)=0$ then implies that the probability P in a spherical volume V with surface area A, decreases as the current efflux from the surface shell, $$\\dot{P}=\\int dV ~\\dot{\\rho} = -\\oint dA ~j_r.$$\n\nSo, for example, for the freely (zero is a spherical potential!) diffusing Gaussian wave packet, unnormalized, starting out with squared-width a at the origin of time (t =0), $$R=\\left ( \\frac{\\sqrt{a}}{a+\\frac{i\\hbar t }{m}} \\right )^{3\/2} ~\\exp \\left (-r^2 \\frac{a-i\\hbar t\/m}{2(a^2+\\hbar^2t^2\/m^2)} \\right )~,$$ decidedly complex.\n\nEvaluating the above current density, $$j_r= \\frac{\\hbar}{m} \\rho \\frac{r\\hbar t }{m(a^2+\\hbar^2 t^2\/m^2 )}= \\frac{r t \\hbar^2}{m^2 a}\\left ( \\frac{1}{a+\\frac{\\hbar^2 t^2 }{m^2 a}} \\right )^{5\/2} ~\\exp \\left (-r^2\\frac{a}{a^2+\\hbar^2t^2\/m^2} \\right )~,$$ you check the current and efflux attenuate with r (probability is conserved at the shell at infinity). For $\\hbar t\/m \\gg a$, a is replaced by $\\hbar ^2t^2\/m^2 a$, a width-squared growing to infinity, as the Gaussian collapses to a constant and localization is lost completely.\n\nThe \"quantum flow velocity\", $$\\frac{j_r}{\\rho}=\\frac{r}{t}~\\left ( \\frac{1}{1+\\frac{m^2 a^2}{\\hbar^2 t^2}}\\right )$$ then collapses to r\/t for large times.\n\n\u2022 Edit on your amended question \"where's my error in (2)?\": Let us leave out the irrelevant spherical harmonics, etc, absorbing them as r-constants in the term A the authors wish to banish. So, in my language, $\\psi \\propto R(r)=\\sqrt{\\rho}e^{iS(r)}\\sim A\/r^{l+1}$ near the origin. From my expression, $j_r\\sim \\frac{\\hbar A A^*}{m r^{2l+2}}\\partial_r S ,$ so that the probability leakage from a shell near the origin is $$4\\pi r^2 j_r \\propto \\frac{\\hbar A A^*}{m r^{2l}}\\partial_r S \\propto r^{-(2l+1)},$$ for a reasonable S vanishing at the origin, as you encounter in scattering theory. (Of course, if S is a constant, as in bound states, it is useless and absorbable, and the answer vanishes.) As a result of this efflux explosion, you have to reject this singular solution as unphysical. By contrast, for l=0, $S=kr$ is safe, amounting to a spherical wave, $R\\sim \\exp(irk) \/r$.\n\n\u2022 A total aside, but while we are at it. For nonvanishing l and nonvanishing azimuthal q.n. m (not mass!), the bound state w.f. does have a complex behavior, $\\exp(im\\phi)$, leading to a nontrivial \u03c6-component to the probability current! So, even though the probability never flows out of a spherical shell, there is this azimuthal probability flow like the jet stream, going round and round and round!\n\n\u2022 I edited my question to add details of where I (or Sakurai) might be going wrong \u2013\u00a0Styg Nov 18 '17 at 15:14\n\u2022 I understood your answer earlier and realized I was wrong in assuming offhand that $R(r)$ will be real in general. My question was narrower in scope (hopefully clear after my edit): I am no longer asking for the probability flux in the general case, but the case with $l$ not necessarily $0$ and with $r$ near $0$ for a simultaneous eigenfunction of $L_z$, $L^2$ and $H$. In particular, the free spreading Gaussian is not an eigenfunction of $H$. \u2013\u00a0Styg Nov 18 '17 at 16:38","date":"2020-04-06 11:53:20","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\": 0.9341291189193726, \"perplexity\": 396.6007614844465}, \"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\/1585371624083.66\/warc\/CC-MAIN-20200406102322-20200406132822-00068.warc.gz\"}"} | null | null |
Green Camp Township is one of the fifteen townships of Marion County, Ohio, United States. The 2010 census found 1,179 people in the township, 374 of whom lived in the village of Green Camp.
Geography
Located in the southwestern part of the county, it borders the following townships:
Big Island Township - north
Marion Township - northeast
Pleasant Township - east
Prospect Township - southeast
Jackson Township, Union County - southwest
Bowling Green Township - west
The village of Green Camp is located in eastern Green Camp Township.
Name and history
It is the only Green Camp Township statewide.
Government
The township is governed by a three-member board of trustees, who are elected in November of odd-numbered years to a four-year term beginning on the following January 1. Two are elected in the year after the presidential election and one is elected in the year before it. There is also an elected township fiscal officer, who serves a four-year term beginning on April 1 of the year after the election, which is held in November of the year before the presidential election. Vacancies in the fiscal officership or on the board of trustees are filled by the remaining trustees.
References
External links
County website
Townships in Marion County, Ohio
Townships in Ohio | {
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If you're into classic rock, then who doesn't appreciate – if not love – The Doors and their contribution to the soundtrack of our youth? Their music is still burned into our minds and their live performances are the stuff of legend.
The DVD, Blu-Ray and digital video each feature a 16x9 high-definition digital transfer with both a stereo and 5.1 audio soundtrack as well as over an hour of bonus material. Included in the additional content are Echoes From The Bowl, The Doors' route to the Hollywood Bowl; You Had To Be There, memories of The Doors' performance at the Bowl; Reworking The Doors, an in-depth look at how the film was restored; and three bonus performances: Wild Child from The Smothers Brothers Show in 1968, Light My Fire from The Jonathan Winters Show in December 1967 and a version of Van Morrison's Gloria with specially created visuals.
Geoff Kempin, executive producer for Eagle Rock said "The Doors were one of THE most incredible live bands ever – we wanted to apply the top technology so that everyone can fully appreciate the phenomenon of The Doors captured at their height on 5 July 1968". | {
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NEW ORLEANS (AP) " New Orleans has become the 16th city to join a U.S. Drug Enforcement Administration initiative to fight heroin and prescription drug abuse.
Local, state and federal law enforcement officials announced the plan Tuesday at New Orleans City Hall.
The news conference also included officials from St. Bernard and Jefferson parishes. Those parishes are joining New Orleans in the DEA pilot program, which has been dubbed the "360 strategy."
The DEA says the program includes continuing law enforcement efforts aimed at drug traffickers and violent gangs. But it also seeks to engage drug manufacturers, marketers and medical professionals to increase awareness of opioid and prescription drug abuse. And it includes outreach efforts aimed at young people in classrooms, homes and after school organizations. | {
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Q: Wireless device Intel Corporation Device 9df0 (rev 30) DISABLED on 18.04.3 I'm new to Ubuntu. I will try to provide as much detail as possible in this question, but if I have left anything out, please let me know in the comments and I will provide it. If this question would be better in a different forum, please let me know which forum.
My wireless interface was working until 3 days ago when I believe an update I said OK to disabled it. When I do sudo lshw -C network, I get:
*-network DISABLED
description: Wireless interface
product: Intel Corporation
vendor: Intel Corporation
physical id: 14.3
bus info: pci@0000:00:14.3
logical name: wlo1
version: 30
serial: 00:bb:60:08:95:85
width: 64 bits
clock: 33MHz
capabilities: pm msi pciexpress msix bus_master cap_list ethernet physical wireless
configuration: broadcast=yes driver=iwlwifi driverversion=5.1.0-050100-generic firmware=46.6bf1df06.0 latency=0 link=no multicast=yes wireless=IEEE 802.11
resources: irq:16 memory:b431c000-b431ffff
When I do iwlist scan, I get:
wlo1 Failed to read scan data : Network is down
lo Interface doesn't support scanning
When I do sudo service network-manager restart and then sudo service network-manager status, I get this error in the output (abbreviated here):
<error> [1576682510.9833] sup-iface[0x5655383f5240,wlo1]: error adding interface: wpa_supplicant couldn't grab this interface.
Finally, when I do rfkill list all, I get:
0: asus-wlan: Wireless LAN
Soft blocked: no
Hard blocked: no
1: asus-bluetooth: Bluetooth
Soft blocked: no
Hard blocked: no
2: phy0: Wireless LAN
Soft blocked: no
Hard blocked: no
3: hci0: Bluetooth
Soft blocked: no
Hard blocked: no
Those are the main things I've looked at.
I've rebooted my computer a few times, restarted network-manager, and tried to do ip set wlo1 up and ifconfig set wlo1 up (this latter command is outdated for me and so doesn't work).
Now for my system info: I have an ASUS Zenbook and am dual-booting Ubuntu 18.04.3 LTS. I am using my phone in the meantime as an ethernet connection. When I run lsusb, I get:
Bus 002 Device 002: ID 2109:0813 VIA Labs, Inc.
Bus 002 Device 001: ID 1d6b:0003 Linux Foundation 3.0 root hub
Bus 001 Device 003: ID 13d3:56cb IMC Networks
Bus 001 Device 005: ID 8087:0aaa Intel Corp.
Bus 001 Device 007: ID 046d:c52b Logitech, Inc. Unifying Receiver
Bus 001 Device 006: ID 062a:4101 Creative Labs Wireless Keyboard/Mouse
Bus 001 Device 004: ID 0c76:161e JMTek, LLC.
Bus 001 Device 002: ID 2109:2813 VIA Labs, Inc.
Bus 001 Device 001: ID 1d6b:0002 Linux Foundation 2.0 root hub
When I run lspci, I get:
00:00.0 Host bridge: Intel Corporation Device 3e34 (rev 0b)
00:02.0 VGA compatible controller: Intel Corporation Device 3ea0
00:04.0 Signal processing controller: Intel Corporation Xeon E3-1200 v5/E3-1500 v5/6th Gen Core Processor Thermal Subsystem (rev 0b)
00:08.0 System peripheral: Intel Corporation Xeon E3-1200 v5/v6 / E3-1500 v5 / 6th/7th Gen Core Processor Gaussian Mixture Model
00:12.0 Signal processing controller: Intel Corporation Device 9df9 (rev 30)
00:14.0 USB controller: Intel Corporation Device 9ded (rev 30)
00:14.2 RAM memory: Intel Corporation Device 9def (rev 30)
00:14.3 Network controller: Intel Corporation Device 9df0 (rev 30)
00:14.5 SD Host controller: Intel Corporation Device 9df5 (rev 30)
00:15.0 Serial bus controller [0c80]: Intel Corporation Device 9de8 (rev 30)
00:15.1 Serial bus controller [0c80]: Intel Corporation Device 9de9 (rev 30)
00:15.3 Serial bus controller [0c80]: Intel Corporation Device 9deb (rev 30)
00:16.0 Communication controller: Intel Corporation Device 9de0 (rev 30)
00:19.0 Serial bus controller [0c80]: Intel Corporation Device 9dc5 (rev 30)
00:1c.0 PCI bridge: Intel Corporation Device 9db8 (rev f0)
00:1c.4 PCI bridge: Intel Corporation Device 9dbc (rev f0)
00:1d.0 PCI bridge: Intel Corporation Device 9db4 (rev f0)
00:1f.0 ISA bridge: Intel Corporation Device 9d84 (rev 30)
00:1f.3 Audio device: Intel Corporation Device 9dc8 (rev 30)
00:1f.4 SMBus: Intel Corporation Device 9da3 (rev 30)
00:1f.5 Serial bus controller [0c80]: Intel Corporation Device 9da4 (rev 30)
02:00.0 3D controller: NVIDIA Corporation GP107M [GeForce GTX 1050 Mobile] (rev a1)
03:00.0 Non-Volatile memory controller: Sandisk Corp Device 5003 (rev 01)
If I have left out any useful information, please let me know.
Thanks very much in advance.
A: You are using an unsupported mainline kernel.
I suggest to install the 5.3 HWE Ubuntu kernel.
Run in a terminal:
sudo apt install linux-generic-hwe-18.04-edge
and reboot.
| {
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import jQuery from 'jquery';
/*
The Company module exposes functionality needed in the company section
of the dashboard.
*/
const Inventory = (function PrivateInventory($) {
const postDeleteForm = (url) => {
$(`<form method="POST" action="${url}">` +
`<input type="hidden" name="csrfmiddlewaretoken" value="${
$('input[name=csrfmiddlewaretoken]').val()}"></form>`).submit();
};
return {
// Bind them buttons and other initial functionality here
init() {
$('#inventory-delete-item').on('click', function deleteItem(e) {
e.preventDefault();
$('.confirm-delete-item').data('id', $(this).data('id'));
});
$('.deletebatch').on('click', function deleteBatch(e) {
e.preventDefault();
$('.confirm-delete-batch').data('id', $(this).data('id'));
});
$('.confirm-delete-item').on('click', function confirmDeleteItem() {
const url = `/dashboard/inventory/item/${$(this).data('id')}/delete/`;
postDeleteForm(url);
});
$('.confirm-delete-batch').on('click', function confirmDeleteBatch() {
const itemId = $('#item_id').val();
const url = `/dashboard/inventory/item/${itemId}/batch/${$(this).data('id')}/delete/`;
postDeleteForm(url);
});
$('#inventory-add-batch').on('click', (e) => {
e.preventDefault();
$('#inventory-add-batch-form').slideToggle(200);
});
},
};
}(jQuery));
export default Inventory;
| {
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Asia Business Middle East
Japanese firms expected to spend $300 billion in GCC region…
Thu 14 Shaban 1435 12-6-2014
Japanese firms have their eyes set on the GCC region, which is expected to spend $300bn on water technology and desalination projects until 2022.
Tadashi Mitsugi, director, Japan Co-operation Centre for the Middle East (JCCME) said, "The Mena (Middle East and North Africa) region is one of the most exciting markets for clean technology development, water treatment and natural resources management in the world. Rising energy and potable water demands, combined with huge solar and wind potential and strong leadership in the move towards sustainability, make it an attractive market for Japanese environmental technology companies.
"Nowhere is this truer than in Abu Dhabi, and we look forward to exploring the emerging opportunities the region's flagship future energy event offers us."
Japan emerged as the only top 20 ranked Asian country in the 2013 Global Sustainable Competitiveness Index, released earlier this year. The East Asian nation was placed 12th among 176 countries, for its ability to maintain economic growth in a world of reduced natural resources. In the specific field of sustainable innovation, Japan ranked fourth.
"Japan is a global leader in the deployment of solar energy and a clean technology innovator," said Naji Haddad, show director of the World Future Energy Summit. "Their experience and knowledge can play a key role in enhancing the UAE's and wider region's renewable energy and clean technology industry as it enters a period of growth.
"The World Future Energy Summit and International Water Summit unite players from across the full value chain, offering innovators and technology players both large and small, an unrivalled platform to showcase and realise commercial opportunities."
The Middle East and North Africa is fast emerging as one of the world's most promising clean energy regions, particularly in the adoption of solar energy. The region has the potential to supply between 50 and 70% of the world's energy.
The Mena region has allocated $50bn to renewable energy deployment until 2020.
Responsible for nearly 60% of global desalination capacity, the region is also piloting desalination from renewable energy technology. Masdar recently awarded four companies with a contract to begin work on a pilot project.
Japan, an Asian leader in sustainability, has confirmed its participation at the 2015 edition of the World Future Energy Summit (WFES) and International Water Summit (IWS) in January, 2015.
The events, which take place during Abu Dhabi Sustainability Week – hosted by Masdar – will offer Japanese companies a window into the UAE and GCC region's growing clean energy and water industries.
The announcement was made following a visit to Tokyo this week by a UAE delegation that included Matthew Griffiths, Director of Water Strategy and Reuse at Abu Dhabi's Regulation and Supervision Bureau, and Naji El Haddad, WFES Show Director.
The delegation outlined the commercial opportunities for inward investment as the region diversifies its energy mix, water resources and steps up its sustainability ambitions. Source: Gulf Times
Stable credit outlook for Qatar on LNG lead…
Qatar is the country of millionaires with almost ¼ of the population… | {
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The Climate Odyssey (Odisseia pelo Clima) aims to engage communities in art-science-practice partnerships for place-based social change and climate action. It seeks to contribute to new approaches to climate change using the potential of artful and participatory elements to increase awareness and agency for the topic of climate change.
Climate Odyssey is a transdisciplinary approach involving artistic practices, social and natural sciences as well as local knowledges and collaborations. The project aims to co-create with the local communities a thematic trajectory ('odyssey'), which elicits local stories of change and transformation and make visible the various aspects (social, cultural, environmental) of climate change.
Between February and June 2019 the project promotes weekly interactive art-and science workshops engaging local participants in the co-creation of the community theatre Climate Odyssey to be presented to the public in a neighbourhood festival in the end of May 2019 (Festival de Telheiras). More updates soon! | {
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Constipation is a common complaint in older adults. Although constipation is not a physiologic consequence of normal aging, decreased mobility and other comorbid medical conditions may contribute to its increased prevalence in older adults. Functional constipation is diagnosed when no secondary causes can be identified, such as a medical condition or a medicine with a side effect profile that includes constipation. Empiric treatment may be tried initially for patients with functional constipation. Management of chronic constipation includes keeping a stool diary to record the nature of the bowel movements, counseling on bowel training, increasing fluid and dietary fiber intake, and increasing physical activity. | {
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Mega-rejections signal M&A's Taylor Swift moment
22 June 2015 By Jeffrey Goldfarb Follow @jgfarb
M&A is having its Taylor Swift moment. U.S. health insurer Cigna on Sunday brushed off a $54 billion merger proposal from rival Anthem just as pipeline operator Williams was doing the same to a $53 billion approach from Energy Transfer Equity. All the while, the 25-year-old pop idol renowned for her songs of relationship angst forced Apple to pay up for music. In a seller's market like this, love isn't usually unrequited for long.
Companies are chasing each other now like hormone-addled teenagers. Some $1.9 trillion of deals were announced worldwide through early June, a 36 percent rise from a year ago. The pursuits are getting bolder, too, with cross-border bids also on the rise. Along with the aggressive tactics of Anthem and Energy Transfer, Brazilian meat-packer JBS traversed nearly 6,000 miles of the Atlantic Ocean over the weekend to buy UK poultry producer Moy Park for $1.5 billion.
Suitors are putting on a nearly unprecedented show of bravado. There have been $250 billion of unsolicited or hostile deals unveiled so far this year, accounting for more than 12 percent of all M&A activity, according to Thomson Reuters data. Over the past two decades, the rate was higher only in exuberant 1999.
The high-school nature of current corporate mergers is further evidenced by a FOMO mentality, or a fear of missing out. As industries consolidate rapidly, failing to move fast enough could exclude one participant. In U.S. health insurance, for example, Cigna – along with Aetna – wants to hook up with Humana, even as Anthem pursues Cigna. A similar frenzy is occurring in pharmaceuticals, where Teva desires Mylan, while Mylan hearts Perrigo.
Forlorn dealmakers could fill a Swift box set. As the songstress herself showed this weekend, prey have the upper hand. Apple quickly reversed itself on a royalty payment plan after Swift threatened to yank her hit album "1989" from the company's new streaming service in protest. Corporate buyers with healthy stock prices, abundant cash and access to cheap debt are bound to give in to similar pressures. Many will find, however, that the objects of their desire at too high a price eventually bring nothing but heartache.
An earlier version of this item misstated the sequence of companies in paragraph four, and has been corrected.
Pipeline operator Energy Transfer Equity said on June 22 it had made a $53 billion offer to merge with Williams, a day after Williams disclosed that it had rejected a bid from an unnamed buyer that "significantly undervalues" the company.
U.S. health insurer Cigna on June 21 rebuffed a $54 billion merger proposal from rival Anthem, saying it was "deeply disappointed" with its suitor's actions.
Also on June 22, Apple reversed itself and said it would pay royalties to music labels and publishers during the three-month free trial of its Apple Music service, after pop star Taylor Swift protested the earlier decision and said she would hold back her hit album "1989."
Source: REUTERS/Lucas Jackson
Singer Taylor Swift performs on ABC's "Good Morning America" to promote her new album "1989" in New York, October 30, 2014. REUTERS/Lucas Jackson
Reuters: ETE confirms $48 billion bid for reluctant Williams Co
Reuters: Cigna rebuffs Anthem's "deeply disappointing" proposal
Reuters: Apple bows to Taylor Swift on paying for all music streaming
Acquirers can expect more M&A investor skeptics
An $18 bln deal straightens out twisted pipelines
M&A predators stalk Michael Milken's ball
Health insurer M&A is risky Obamacare side effect | {
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Blekinge var ett svenskt örlogsskepp som sjösattes 1682. Hon var det första krigsfartyget som man lät bygga på Vämö i Karlskrona. Skeppsbyggmästare var engelsmannen Robert Turner.
I tjänst
Året därpå seglade Blekinge på ett skär och sjönk, men bärgades och var därefter åter i bruk i ungefär tjugo år. Fartyget var ett linjeskepp uppemot 45 meter långt, som hade 68 kanoner och plats för en besättning på 450 man. Hon deltog bland annat i bombarderingen av Köpenhamn och i Karl XII:s landstigning vid Humlebæk 1700.
Blockskepp
Mycket talar för att skeppet medvetet sänktes 1713 när hon blivit för gammal och hon placerades på hamnbotten mitt i Karlskrona örlogshamn för att återanvändes i ett så kallat "blockhus", vilket betyder att skeppets nedre del stod förankrat på botten medan det övre batteridäcket fanns ovanför vattenytan. På så sätt kunde man använda kanonerna för att beskjuta och förhindra fientligt sinnade fartyg att ta sig in i hamnområdet. Antagligen saknades ekonomiska förutsättningar för att anlägga ordentliga försvarsanläggningar på grund av Karl XII:s kostsamma krigståg.
Vraket
Skeppet glömdes bort men på olika historiska kartor har en sådan vrakplats utpekats. Dykningar som gjordes i det aktuella området 2016 bekräftar att där ligger ett gammalt skeppsvrak, som med största sannolikhet är linjeskeppet Blekinge. Skeppsvraket är nu till stora delar nedsjunket i bottenslammet och täckt med sediment.
Utseendemässigt lär skeppet Blekinge ha liknat regalskeppet Solen.
Se även
Lista över svenska vrak
Lista över svenska seglande örlogsfartyg
Regalskeppet Solen (1669)
Källor och referenser
Här är Karlskronas första krigsskepp BLT den 1 februari 2017
Karlskronas första krigsskepp funnet på SVT Nyheter den 1 februari 2017
Värdefullt vrakfynd i Karlskrona – stadens första krigsskepp Blekinge i Hällekiskuriren den 1 februari 2017
Marinarkeologi
Blekinge (1682)
Svenska linjeskepp
Svenska segelfartyg
Fartyg sjösatta under 1680-talet
Skeppsvrak i Östersjön
Karlskronas historia
Fartyg byggda i Karlskrona
Skeppsvrak i Blekinge skärgård | {
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{"url":"https:\/\/tonyroyphilosophy.net\/textbook-blog\/","text":"## The Textbook Blog\n\nThis is a location for discussion of Symbolic Logic. Comments may range from general assessment to details of presentation. We can discuss substantive questions about logic too. You may submit a post of your own or add comments to another. If you require special symbols, $\\LaTeX$ code may be inserted between the markers $. . .$ \u2014 to start with this, see LaTeX Typing (you see the compiled result only when posted).\n\n## textbook commentary\n\nIt is surprising to me that the initiation of this textbook blog seems to have reduced, rather than expanded, commentary on SL (which I used to receive regularly by e-mail). It is not the case that fewer people are using the text. Perhaps the public nature the forum discourages participation? Or maybe nobody wants to be \u201cthe first\u201d? If you have an explanation, I would love to hear. For now, I can only encourage you that I welcome comments, and that the public nature of the forum lets your discussion benefit others (and email is still fine). T.R.\n\n## SLAPP: Symbolic Logic APP\n\nFor what it is worth, I am embarked on a (long-term) project to produce an open-source computer application that would be a context for creation, submission, and correction of exercises for Symbolic Logic. If you are contemplating study of Symbolic Logic, do not \u201cwait for it,\u201d it will be a long time coming. However over the next years I do hope to release the program bit by bit. Among goals are,\n\n\u2022 Exercises are cleaner and easier (more fun) in SLAPP than on paper. (This is not trivially true \u2013 see many existing web apps.)\n\u2022 Provides contextual feedback and checking, with goal that students always complete exercises correctly \u2013 or at least know that and where they have problems.\n\u2022 Runs on as many platforms as possible \u2013 but primarily on laptop \/ desktop.\n\nQuite generally, logic software is beset by a problem of resources: The market is not large enough to support full-scale commercial development (as for mathematics), and instructors may have neither the time nor training to develop full-fledged software projects on their own. I hope to overcome at least the time problem by the magic of \u201cretirement\u201d! T.R.\n\n## version updates\n\n3\/1\/20: The\u00a0SL version is updated from 8.2 to 8.3.\u00a0 Small changes, new cover by my wife!\u00a0 Adopted Creative Commons License and posted to the Merlot textbook repository.\n\n9\/10\/19: The\u00a0SL version number is updated from 8.1 to 8.2.\u00a0 Main structure remains the same.\u00a0 But there are enough changes to mark this as a substantive improvement over previous versions.\n\n4\/29\/19: The SL version number is incremented from 8.0 to 8.1. Changes to the main body are minor. The primary update is that Answers to Selected Exercises are moved online. This permits large pages and so improved formatting.\n\nT.R.\n\n## jolly good\n\nAs a retired engineer, I decided to get back to my passion of understanding ethics. Most of the works were clearly woolly and very soon I realized that I needed to enhance my skills in logic to make any headway in this direction. Most of the recent books in logic were loaded in formal mathematics and appeared way beyond my reach. It was then that Prof Roy\u2019s book on the Internet came as a pleasant surprise. Two years later, I managed to complete most of the exercises in his book and understand the principles. It is to Prof Roy\u2019s credit that someone like me with little training in formal mathematics can learn such a difficult subject and all along keep my interest in it without being daunted by the complexity of the subject. During this journey, Prof Roy gave his time unstintingly to respond to my queries. Nuggets of Russelian wit, veiled challenges to the daredevils, and his love for his family are sprinkled in every chapter in this book. What attracted me most were the problems at the end of each chapter; in every chapter you invariably find one which asks the student to draft a short note explaining the main ideas in the chapter in a language his teenage daughter can understand. It is well known that, if you understand something well, you ought to be able to communicate that to the non-initiated. And, equally if you teach someone what you learnt, your concepts become even clearer. As the Tamil poet says, knowledge is rare commodity that grows by giving. To recognize this and challenge the reader to try his hands at this, Prof Roy shows great intuition in the art of teaching.\n\nAs an old man living in the UK, I cannot say his book is cool, but perhaps I could say it is jolly good.\u00a0 R.V.\n\n## answers to exercises\n\nI have complete answers to nearly all the exercises in the text (and am working on the rest). It is easy to move answers in or out of the Selected Answers at the back of the book. Short of including them all, if you think some answer(s) should or should not be included among the Selected Answers, let me know. T.R.","date":"2020-10-01 15:08:01","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 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\": 1, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.49976012110710144, \"perplexity\": 1245.507376796561}, \"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-2020-40\/segments\/1600402131777.95\/warc\/CC-MAIN-20201001143636-20201001173636-00063.warc.gz\"}"} | null | null |
{"url":"https:\/\/docs.mantidproject.org\/nightly\/fitting\/fitfunctions\/Bessel.html","text":"$$\\renewcommand\\AA{\\unicode{x212B}}$$\n\n# Bessel\u00b6\n\n## Description\u00b6\n\n$A(t)=A_0J_0(2\\pi\\nu t+\\phi)$\n\nwhere,\n\n$$A_0$$ is the amplitude,\n\n$$\\nu$$ is the frequency of oscillation (MHz),\n\n$$\\phi$$ is the phase at $$t=0$$,\n\nand $$J_0(x)$$ is the Bessel Function of the first kind, its expression is given by:\n\n$J_0(x)=\\sum_{l=0}^{\\infty}\\frac{(-1)^l}{2^{2l}(l!)^2}x^{2l}.$\n\nName\n\nDefault\n\nDescription\n\nA0\n\n1.0\n\nAmplitude\n\nPhi\n\n0.1\n\nNu\n\n0.1\n\nFrequency(MHz)\n\n## References\u00b6\n\nCategories: FitFunctions | Muon\\MuonSpecific\n\n## Source\u00b6\n\nPython: Bessel.py","date":"2023-04-02 04:56:01","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 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\": 0.9493100047111511, \"perplexity\": 4038.9185377152694}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.3, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2023-14\/segments\/1679296950383.8\/warc\/CC-MAIN-20230402043600-20230402073600-00060.warc.gz\"}"} | null | null |
\section{Introduction }
Self-trapping is a ubiquitous phenomenon in nature,
for example, solitons in various systems \cite{Drazin1989Solitons,Guo2018Solitons,Kivshar2003,Kono2010Nonlinear,Kevrekidis2008Emergent,CarreteroGonzalez2008Nonlinear},
liquid Helium droplets \cite{Barranco2006Helium} are all some kind
of self-trapped states. The highly tunable atomic Bose-Einstein condensate
(BEC) provides an ideal platform for studying such phenomena. In the
BEC system, the attractive inter-atom interaction (s-wave collision)
results in a Kerr type self-trapping nonlinearity \citep{Dalfovo1999Theory},
and can support bright solitons \citep{Strecker2002Bright,Khaykovich2002Formatioin,Strecker2003Bright}.
If quantum fluctuation (Lee-Huang-Yang correction \cite{Lee1957Eigenvalues})
is included, self-trapped droplets can also be formed in the BEC system \cite{Kadau2016Oberving,FerrierBarbut2016Observation,Chomaz2016Quantum,Cabrera2018Quantum,Cheiney2018Bright}.
Interacting with electromagnetic fields can also lead to nonlinearity
in the BEC systems. When a BEC is illuminated by electromagnetic waves,
it feels a potential from the electromagnetic field. At the same time,
the BEC also serves as a medium, and will backwardly affect the propagation
of the electromagnetic waves. The affected electromagnetic field will
in turn further affect the dynamics of BEC. Due to this feedback mechanism,
nonlinear features arise in the system. Many interesting phenomena
resulting from this type of nonlinearity have been reported \citep{Ritsch2013Cold}.
For atomic gas in a cavity, because of this feedback effect, a dynamical rather than a static
optical lattice is produced. In such a dynamical lattice the atoms feel a friction force,
thus can be cavity cooled and self-organized
\citep{Domokos2001Semmiclassical,Niedenzu2011Kinetic,Ostermann2015Atomic}.
It also soften an optical lattice, and leads to asymmetric
matter wave diffraction \citep{Li2008Matter-Wave,Zhu2011Strong} and
polaritonic soliton \citep{Dong2013Polaritonic}. It also give
rise to phenomena such as spin-exchange \citep{Norcia2018Cavity,Davis2019Photon}
and long-range interactions \citep{Zhang2018Long,Guan2019Two}, self-structuring
\citep{Robb2015Quantum,Ostermann2016Spontaneous},
photon bubble \citep{Mendonca2012Photon,Rodrigues2016Photon}, bistability
\citep{Zhou2009Cavity,Zhou2010Spin,Zhou2011Cavity,Dalafi2017Instrinsic},
spin texture \citep{Landini2018Formation,Ostermann2019Cavity}, chaotic
dynamics \citep{Diver2014Nonlinear}, parametric resonance \citep{Li2019Nonlinear}
in the light-BEC interacting systems. And in the microwave-BEC interacting
systems, soliton \citep{Qin2015Hybrid,Qin2019Tial-free} and vortex
\citep{Qin2016Stable} phenomena have also been reported. Most recently,
it is found that due to such nonlinearity, supersolid can exist in
a driven-dissipative ring-cavity-BEC system \cite{Mivehvar2018Driven,Schuster2020Supersolid},
futhermore a precise gravimeter has been proposed based on the system \citep{Gietka2019Supersolid}.
And a type of novel crystalline droplet has also been predicted in an atom-cavity setup.
\citep{Karpov2019Crystalline}.
Motivated by these progresses, in this work we propose that self-trapped
matter wave can also be supported by the cavity-mediated nonlinearity
in a driven-dissipative cavity-BEC system, and study its stability
and dynamics
using the mean-field theory. In the considered system (see figure \ref{fig:Diagram}),
the cavity light field is built up by transversely illuminating the
BEC, then the built-up light field forms an optical lattice potential
for the BEC. We theoretically demonstrated that this induced optical
lattice can support a self-trapped wave packet. For a weak cavity
pumping, the induced optical lattice is shallow, the localized wave
packet can not be well trapped, therefore it spreads during the time
evolution. And for a strong pumping, the induced optical lattice can
be strong enough to support a long-time stable self-trapped wave packet.
The moving dynamics of these self-trapped waves show very different
features under different cavity decay rates. This is due to the adiabaticity
of the induced optical lattice. The induced optical lattice tends
to follow the movement of the self-trapped wave packet, however, can
not completely catch up. Thus, the self-trapped wave
packet feels a dragging force from the induced optical lattice falling
behind it, therefore it decelerates, and finally stops. When the
self-trapped wave packet stops, the optical lattice also catches up,
and the system reaches a steady state. If the cavity decay rate is
small, even the self-trapped wave packet has been decelerated to the
speed of zero, the optical lattice still can not catch up, so the
self-trapped wave packet will then be accelerated in the opposite
direction. The deceleration and acceleration alternately repeat several
times, and overall the self-trapped wave packet displays a damped
oscillation. In the bad cavity limit \cite{Cirac1995Laser,Horak2000Coherent}
(which means that the cavity decay rate is much larger
than the atoms-cavity coupling, the cavity light field quickly decays
to a steady state, and can instantaneously follow the dynamics of
BEC), the self-trapped wave packet feels no dragging force and will
constantly move with the initial given speed.
At last, we point out that in different atom-cavity setups the phenomena of
self-organization and friction force on atoms
are also predicted by the semiclassical theory in which
the atoms are treated as classical particles
\citep{Niedenzu2011Kinetic,Ostermann2015Atomic}. But here by using the
mean-field theory, the atoms are described by a Schr\"{o}dinger-like
equation, thus the effects of quantum pressure and tunneling of atoms
to neighboring lattice sites can also be included.
The paper is organized as follows: In section \ref{sec:Model}, the
physical model studied in this paper is presented. In section \ref{sec:SolitaryWave},
we show the existence of self-trapped wave packets in the system with
both the variational method and numerical simulation. Examples of
the self-trapped wave packets and their stability are also shown in
this section. In section \ref{sec:MovingDynamic}, the moving dynamics
of the self-trapped wave packets are studied in detail.
And, we briefly compare the main results obtained using mean-field theory
with their semiclassical correspondences in section \ref{sec:Semiclassical}.
At last, the paper is summarized in section \ref{sec:Summary}.
\section{Model\label{sec:Model}}
\begin{figure}
\begin{centering}
\includegraphics{fig1}
\par\end{centering}
\caption{Diagram of the considered system. A quasi-one-dimensional atomic BEC
is loaded into a ring-cavity with loss rate $\kappa$. The BEC atoms
interact with two degenerate counter-propagating modes ($\hat{a}_{+}$
and $\hat{a}_{-}$) of the ring-cavity. The system is pumped by transversely
shining a laser on the BEC, the pumping strength is $\eta$. \label{fig:Diagram}}
\end{figure}
We consider a ring cavity-BEC coupling system \citep{Mivehvar2018Driven}
which is schematically shown in figure \ref{fig:Diagram}. A two-level
atomic BEC is trapped along the cavity axis by a tight transverse
confining potential, thus can be reduced to one-dimensional. The atoms
are driven in the transverse direction by an off-resonant (with detuning
$\Delta_{a}$) pump laser, which induces a Rabi oscillation of frequency
$\Omega_{0}$ between the two internal atomic states. The transition
between the two atomic energy levels is also off-resonantly coupled
to the two counter-propagating cavity modes $\hat{a}_{\pm}e^{ik_{c}x}$
($k_{c}$ is the wave number of the cavity modes) with strength $\mathcal{G}_{0}$.
In the far-off-resonant regime $\left|\Delta_{a}\right|\gg\Omega_{0},\mathcal{G}_{0}$,
the excited atomic state can be adiabatically eliminated, and in the
rotating frame of the pump laser, the system can be described by the
following effective Hamiltonian
\begin{equation}
\mathcal{H}=-\hbar\Delta_{c}\left(\hat{a}_{+}^{\dagger}\hat{a}_{+}+\hat{a}_{-}^{\dagger}\hat{a}_{-}\right)+\int\hat{\psi}^{\dagger}H_{a}\hat{\psi}dx,\label{eq:Hamiltonian_eff}
\end{equation}
where the first term describes the two counter-propagating cavity
modes, and the second term accounts for the BEC and its interaction
with the light field. In this equation, $\hbar$ is the Planck constant,
$\Delta_{c}$ is the detuning between the cavity modes and pump laser,
$\hat{\psi}$ is the field operator of the BEC and $H_{a}$ is the
corresponding single-particle Hamiltonian
\begin{align}
H_{a} & =\frac{\hat{p}^{2}}{2m}+V_{ac}+V_{ap},\label{eq:Hamiltonian_atom}
\end{align}
with
\begin{equation}
V_{ac}=\hbar U_0\!\left[\hat{a}_{+}^{\dagger}\hat{a}_{+}+\hat{a}_{-}^{\dagger}\hat{a}_{-}+\left(\hat{a}_{+}^{\dagger}\hat{a}_{-}e^{-2ik_{c}x}+\mathrm{h.c.}\right)\right],\label{eq:V_ac}
\end{equation}
\begin{equation}
V_{ap}=\hbar\eta_0\left(\hat{a}_{+}e^{ik_{c}x}+\hat{a}_{-}e^{-ik_{c}x}+\mathrm{h.c.}\right).\label{eq:V_ap}
\end{equation}
Here, $\hat{p}^{2}/2m$ is the kinetic energy of the BEC atom, $V_{ac}$
is the optical potentials due to two-photon scattering between the
two cavity modes, and $V_{ap}$ is the optical potential due to two-photon
scattering between the pump and cavity modes. The meanings of the symbols
are as follows: $m$ is the mass of the BEC atom, $\hat{p}=-i\hbar\frac{\partial}{\partial x}$
is the momentum operator, $U_{0}=\hbar\mathcal{G}_{0}^{2}/\Delta_{a}$
describes the strength of optical potential $V_{ac}$, and $\eta_{0}=\hbar\mathcal{G}_{0}\Omega_{0}/\Delta_{a}$
is the effective cavity pump strength. In the following contents,
natural unit $m=\hbar=k_{c}=1$ will be applied for simplicity, i.e.,
the length, time, velocity, and energy will be measured in units of
$1/k_{c}$, $m/\left(\hbar k_{c}^{2}\right)$, $\hbar k_{c}/m$, and
$\hbar^{2}k_{c}^{2}/m$.
The BEC usually contains a large number of atoms. To support the
self-trapped wave which is the main subject of this paper, a strong
light field is also needed. Thus, the mean-field approximation
\citep{Zhang2008MeanField} can
be adopted (in reference \citep{Schuster2020Supersolid} which considers
a very similar system, the mean-field results fit the experimental
preservation well). The quantum mechanical operators
can be approximated by their corresponding
mean value c-numbers, $\hat{a}_{\pm}\rightarrow\alpha_{\pm}$ and
$\hat{\psi}\rightarrow\psi$. We further scale $\alpha_{\pm}$
and $\psi$ with the total atom number $N$, i.e., $\alpha_{\pm}\rightarrow\alpha_{\pm}/\sqrt{N}$,
$\psi\rightarrow\psi/\sqrt{N}$. Using such a scaling, the norm of
the wave function $\psi$ becomes
\[
\int\left|\psi\left(x\right)\right|^{2}dx=1.
\]
And we also introduce new parameters $\eta=\sqrt{N}\eta_{0}$ and
$U=U_{0}N$ to account for the many atoms. The equations governing
the dynamics of these mean-field variables can be obtained by taking
the mean values of the corresponding Heisenberg equations
\begin{equation}
i\frac{\partial}{\partial t}\alpha_{\pm}=\left(-\Delta_{c}+U-i\kappa\right)\alpha_{\pm}+UN_{\pm2}\alpha_{\mp}+\eta N_{\pm1},\label{eq:Meanfield_cavity}
\end{equation}
\begin{equation}
i\frac{\partial}{\partial t}\psi=\left[-\frac{1}{2}\frac{\partial^{2}}{\partial x^{2}}+\mathcal{V}_{\mathrm{eff}}\left(x\right)\right]\psi,\label{eq:Meanfield_atom}
\end{equation}
where the cavity loss with rate $\kappa$ has been introduced phenomenologically,
and for conciseness of the equations, here we also defined the following
quantities
\[
N_{\pm1}=\int\left|\psi\left(x\right)\right|^{2}e^{\mp ix}dx,
\]
\[
N_{\pm2}=\int\left|\psi\left(x\right)\right|^{2}e^{\mp2ix}dx,
\]
\[
\mathcal{V}_{\mathrm{eff}}\left(x\right)=\mathcal{V}_{ac}\left(x\right)+\mathcal{V}_{ap}\left(x\right),
\]
\[
\mathcal{V}_{ac}=U\left(\left|\alpha_{+}\right|^{2}+\left|\alpha_{-}\right|^{2}\right)+U\left(\alpha_{+}^{*}\alpha_{-}e^{-2ix}+\mathrm{c.c.}\right),
\]
\[
\mathcal{V}_{ap}=\eta\left(\alpha_{+}e^{ix}+\alpha_{-}e^{-ix}+\mathrm{c.c.}\right).
\]
Letting $\frac{\partial}{\partial t}\alpha_{\pm}=0$ and $\psi\left(x,t\right)=\psi\left(x\right)e^{-i\mu t}$
with $\mu$ being the BEC chemical potential, the steady state of
the system follows equations
\begin{equation}
\mu\psi\left(x\right)=\left[-\frac{1}{2}\frac{\partial^{2}}{\partial x^{2}}+\mathcal{V}_{\mathrm{eff}}\left(x\right)\right]\psi\left(x\right),\label{eq:steadypsi}
\end{equation}
\begin{equation}
\alpha_{+}=-\frac{\left(-\Delta_{c}+U-i\kappa\right)\eta N_{+1}-\eta UN_{+2}N_{-1}}{\left(-\Delta_{c}+U-i\kappa\right)^{2}-U^{2}N_{-2}N_{+2}},\label{eq:alpha_p}
\end{equation}
\begin{align}
\alpha_{-} & =-\frac{\left(-\Delta_{c}+U-i\kappa\right)\eta N_{-1}-\eta UN_{-2}N_{+1}}{\left(-\Delta_{c}+U-i\kappa\right)^{2}-U^{2}N_{-2}N_{+2}}.\label{eq:alpha_m}
\end{align}
Here we point out that equations (\ref{eq:alpha_p}) and (\ref{eq:alpha_m})
also describe the cavity field amplitudes of the system in the bad
cavity limit \cite{Cirac1995Laser,Horak2000Coherent}. In the bad
cavity limit, the cavity light field quickly decays to the steady
state, then $\partial_{t}\alpha_{\pm}\approx0$ can be approximately
applied, so equations (\ref{eq:alpha_p}) and (\ref{eq:alpha_m})
holds. Together with equation (\ref{eq:Meanfield_atom}), dynamics
of the cavity-BEC system in the bad cavity limit can be described.
At last, we see that the dynamics of BEC macroscopic wavefunction
are governed by a Schr{\"{o}}dinger-like equation (\ref{eq:Meanfield_atom}).
This equation is a nonlinear one, since the optical potentials $\mathcal{V}_{ac}\left(x\right)$
and $\mathcal{V}_{ap}\left(x\right)$ felt by the BEC recursively
depend on the wave function $\psi$ of the condensate. This nonlinearity
can support self-trapped waves in the system, which will be discussed
in the next section.
\section{Self-Trapped Matter Wave\label{sec:SolitaryWave}}
In the system, a super-radiation phase transition take place at the
critical pumping strength \citep{Mivehvar2018Driven}
\begin{equation}
\eta_{c}=\sqrt{\frac{\left(-\Delta_{c}+U\right)^{2}+\kappa^{2}}{8\left(-\Delta_{c}+U\right)}.}\label{eq:eta_c}
\end{equation}
Below the critical pumping strength ($\eta<\eta_{c}$), the cavity
light field is almost zero ($\alpha_{\pm}\approx0$), the atoms feel
a negligible optical potential, and will have a uniform distribution.
However, above the critical pumping strength ($\eta>\eta_{c}$), a
considerable intensity of cavity field can be built up, hence the
optical lattice potential acting on the atoms will play a crucial
role. And it will be natural to think that this induced optical lattice
potential can support a self-trapped matter wave packet. This will
be verified by both the variational method analysis and numerical
simulation in the following contents of this section.
\begin{figure}
\begin{centering}
\includegraphics{fig2}
\par\end{centering}
\caption{Variational Energy. The variational energy $E_{\mathrm{va}}$ is plotted
as a function of the variational parameter $\sigma$ (width of the
wavepacket) in the trial wave function (\ref{eq:psi_va}). The four
lines correspond to different pumping strength $\eta=6.0$, $10.0$,
$12.5$ and $15.0$. The black squares are the minimum points of the
lines. Other parameters used are $U=-0.5$, $\Delta_{c}=-1$, $\kappa=10$.
\label{fig:Variational}}
\end{figure}
To simplify the variational calculation, we further
neglect the terms related to $U,UN_{\pm2}$ (i.e., terms due to two-photon
scattering between the two cavity modes) in equations (\ref{eq:alpha_p})
and (\ref{eq:alpha_m}) under the assumption $\kappa\gg U,UN_{\pm2}$.
And the simplified optical field reads
\begin{equation}
\alpha_{\pm}\approx\frac{\eta N_{\pm1}}{\Delta_{c}+i\kappa}.\label{eq:BadCavity_alpha}
\end{equation}
We see the optical field is determined by the two-photon scattering
between the pump and cavity modes. Then, the ratio between amplitudes
of the two optical potentials $\mathcal{V}_{ac}$ and $\mathcal{V}_{ap}$
is calculated to be
\[
\frac{\mathcal{V}_{ac}}{\mathcal{V}_{ap}}\sim\frac{U\left|\alpha_{\pm}\right|^{2}}{\eta\left|\alpha_{\pm}\right|}=\frac{U\left|N_{\pm1}\right|}{\left|\Delta_{c}+i\kappa\right|}\ll1.
\]
So, compared to $\mathcal{V}_{ap}$, $\mathcal{V}_{ac}$ can be neglected.
Equation (\ref{eq:Meanfield_atom}) which governs the evolution of
atomic BEC can be simplified to the following nonlinear Schr{\"{o}}dinger
equation
\begin{equation}
i\frac{\partial}{\partial t}\psi=-\frac{1}{2}\frac{\partial^{2}}{\partial x^{2}}\psi+\left[\frac{\eta^{2}\left(N_{+1}e^{ix}+N_{-1}e^{-ix}\right)}{\Delta_{c}+i\kappa}+\mathrm{c.c.}\right]\psi.\label{eq:BadCavity_psi}
\end{equation}
The effective Hamiltonian corresponding to this equation can be written
as
\begin{align}
H_{\mathrm{eff}}= & \int\hat{\psi}^{\dagger}\left(x\right)\left(-\frac{1}{2}\frac{\partial^{2}}{\partial x^{2}}\right)\hat{\psi}\left(x\right)dx\nonumber \\
+ & \left[\frac{\eta^{2}}{\Delta_{c}+i\kappa}\left(\int\hat{\psi}^{\dagger}\left(x_{1}\right)e^{ix_{1}}\hat{\psi}\left(x_{1}\right)dx_{1}\right)\right.\nonumber \\
& \left.\cdot\left(\int\hat{\psi}^{\dagger}\left(x_{2}\right)e^{-ix_{2}}\hat{\psi}\left(x_{2}\right)dx_{2}\right)+\mathrm{h.c.}\right].\label{eq:Heff}
\end{align}
Taking a Gaussian wave packet localized at position $x=0$
\begin{equation}
\psi_{\mathrm{va}}\left(x\right)=\left(\frac{2}{\pi\sigma^{2}}\right)^{1/4}e^{-\left(\frac{x}{\sigma}\right)^{2}},\label{eq:psi_va}
\end{equation}
as the variational trial wave function where the wave packet width
$\sigma$ is the only variational parameter, the variational energy
is integrated to be
\begin{align}
E_{\mathrm{va}} & =\frac{1}{2\sigma^{2}}+\frac{2\Delta_{c}\eta^{2}}{\Delta_{c}^{2}+\kappa^{2}}\exp\left[-\frac{\sigma^{2}}{4}\right].\label{eq:E_va}
\end{align}
In figure \ref{fig:Variational}, the variational energy $E_{\mathrm{va}}$
is plotted as a function of variational parameter $\sigma$ for different
pumping strength $\eta$. We clearly see that there exists a minimal
point on the $E_{\mathrm{va}}\text{{-}}\sigma$ curve, which indicates
the existence of a self-trapped wave packet. And from the figure,
one also expects that the self-trapped wave packet will have a narrower
width under a stronger pumping strength (a larger value of $\eta$),
as the stronger pumping can produce a deeper optical lattice. These
conclusions will be further verified by the numerical simulations.
Numerically, the steady state of the system is found by propagating
equations (\ref{eq:Meanfield_atom}, \ref{eq:alpha_p}, \ref{eq:alpha_m})
with the imaginary time method from an initial trial narrow Gaussian
wave packet. Some examples of the numerically found self-trapped wave
packets and their variational counterparts for different pumping strength
$\eta=6$ (top), $10$ (middle), and $15$ (bottom) are shown in the
left panels of figure \ref{fig:SW}, where the induced optical lattice
potentials $\mathcal{V}_{\mathrm{eff}}$ are also plotted. Here the
optical lattices in fact have different bottom energies, but for the
convenience of comparison, we shift all of them to the value of zero.
We see that the variational and numerical results fit each other very
well. And as the pumping strength $\eta$ increases, the depth of
the induced optical lattice also increases, as a result, the width
of self-trapped wave packet deceases. This agrees with our variational
discussion in the previous paragraph.
\begin{figure}
\begin{centering}
\includegraphics{fig3}
\par\end{centering}
\caption{Some examples of the self-trapped wave packets and their stability.
Left panels: Density profiles $\left|\psi\right|^{2}$ of the self-trapped
waves and the corresponding induced optical lattice potential $\mathcal{V}_{\mathrm{eff}}$
for pumping strength $\eta=6$ (top panel), $10$ (middle panel),
and $15$ (bottom panel). The violet solid and green dashed lines
are numerical and variational results for $\left|\psi\right|^{2}$
respectively. The cyan dotted lines are numerical results for $\mathcal{V}_{\mathrm{eff}}$.
Right panels: Corresponding time evolution of the self-trapped waves
shown in the left panels. Other parameters are $U=-0.5$, $\Delta_{c}=-1$
and $\kappa=10$. \label{fig:SW}}
\end{figure}
We also examined the stability of these self-trapped waves by directly
simulating equations (\ref{eq:Meanfield_cavity}) and (\ref{eq:Meanfield_atom}).
The results are shown in the right panels of figure \ref{fig:SW}.
We found that for a weak pumping strength ($\eta=6$ in the top panel),
the induced optical lattice potential is not strong enough to retain
the atoms around a single lattice site, they can tunnel to the neighboring
sites, and the self-trapped wave packet spreads. When the pumping
strength is strong ($\eta=10$, $15$ in the middle and bottom panels),
the self-trapped wave packet can maintain its shape for quite a long
time.
\section{Dynamics\label{sec:MovingDynamic}}
Because of the dissipative nature of the system, the moving dynamics
of the self-trapped waves also show additional features. In the top panel
of figure \ref{fig:Moving}, we initially give the self-trapped wave
packet a velocity of $v_{0}=-3$ by imprinting a phase factor $\exp\left(-iv_{0}x\right)$
on the steady state wave packet \citep{Denschlag2000Generating},
and plot its density profile in the afterward evolution. Unlike the
constant-speed moving of the conventional atomic bright soliton supported
by inter-atom interaction (s-wave collision) \citep{Kevrekidis2008Emergent},
here we see that the self-trapped wave undergoes a decelerating motion.
This is because according to equations (\ref{eq:Meanfield_cavity})
and (\ref{eq:Meanfield_atom}), there is a scope of
timing delay between the change of light field and the moving of atomic
condensate, and the condensate will feel a dragging (friction) force
from the falling behind optical potential \citep{Gietka2019Supersolid}.
As shown in the bottom left panel, at $t=0.5$ the center of the wave
packet has traveled to $x=-1.23$ (black dashed line), but the bottom
of the optical lattice is still left behind at $x=-0.89$, and the
lattice will impede the moving of the condensate. After traveling
some distance, the self-trapped wave packet gets to stop, and the
light field also catches up, thus the system comes back to a steady
sate, see bottom right panel of the figure where the center of the
wave packet and the bottom of the lattice overlap again at $t=5.0$.
This friction force can be used to cool atomic gas
\citep{Domokos2001Semmiclassical,Niedenzu2011Kinetic}.
Moreover, it may also provide an opportunity to simplify
the engineering of the self-trapped waves. For a conventional BEC
bright soliton, if it is required to transfer from one place to another,
one needs to firstly accelerate it, and then one also needs to slow
down and stop it at the destination \citep{Kevrekidis2005Statics,Baines2018Soliton}.
But for the self-trapped waves considered here, the stopping process
can be omitted, one only needs to kick the self-trapped wave packet
with an appropriate initial velocity, then it will travel to and stop
at the destination automatically.
\begin{figure}
\begin{centering}
\includegraphics{fig4}
\par\end{centering}
\caption{Decelerating motion of the self-trapped wave packet. Initially, the
self-trapped wave packet locates at $x=0$ (white solid line), and
its velocity is set to $v_{0}=-3.$ Top panel: The afterward evolution
of the density profile. The limit traveling distance of the wave packet
is $x_{s}=2.88$ (white two-heads arrow). Bottom panels: The density
profiles (violet solid line) and corresponding induced optical lattice
potentials (green dashed line) at $t=0.5$ (left panel) and $t=5.0$
(right panel). The black dotted line is plotted to mark the center
of the wave packet. Other parameters used are $U=-0.5$, $\Delta_{c}=-1$,
$\kappa=10$ and $\eta=15$. \label{fig:Moving}}
\end{figure}
\begin{figure}
\begin{centering}
\includegraphics{fig5}
\par\end{centering}
\caption{Relation between mean friction force $\bar{f}$ and
cavity pumping strength $\eta$.
The squares and circles are data points collected
from mean-field and semiclassical numerical simulations respectively.
The lines are simple linear connections
of the data points to guide the eyes. Parameters used to plot this
line are $U=-0.5$, $\Delta_{c}=-1$, $\kappa=10$ and $v_{0}=-3$.}\label{fig:fVSeta}
\end{figure}
\begin{figure}
\begin{centering}
\includegraphics{fig6}
\par\end{centering}
\caption{Relation between the mean friction force $\bar{f}$ and the initial
wave packet moving speed $v_{0}$.
The squares and circles are data points collected
from mean-field and semiclassical numerical simulations respectively.
The lines are simple linear connections
of the data points to guide the eyes. Parameters used to plot this
line are $U=-0.5$, $\Delta_{c}=-1$, $\kappa=10$ and $\eta=15$.
\label{fig:fVSv0}}
\end{figure}
\begin{figure}
\begin{centering}
\includegraphics{fig7}
\par\end{centering}
\caption{Escaping of atoms from a fast-moving self-trapped wave packet. The
initial speed of the wave packet is $v_{0}=-8$. Left panel: Time
evolution of the atomic density $\left|\psi\right|^{2}$. The escaping
of atoms from the self-trapped wave packet is emphasized by the white
box. Right panel: The initial ($t=0$, violet solid line) and final
($t=10$, green dashed line) density profiles of the self-trapped
wave packet. Other Parameters are $U=-0.5$, $\Delta_{c}=-1$, $\kappa=10$
and $\eta=15$. \label{fig:Largev0}}
\end{figure}
Next, we denote the mean friction force felt by the self-trapped wave
as $\bar{f}$, and study its properties in detail. Its value can be
calculated from equation
\begin{equation}
\bar{f}x_{s}=\frac{Nmv_{0}^{2}}{2},\label{eq:meanFriction}
\end{equation}
where we equal the work done by the friction force and the initial
kinetic energy of the self-trapped wave. And here $x_{s}$ is the
limit traveling distance of the wave packet (as shown in figure \ref{fig:Moving}),
which is determined from numerical results.
We firstly examine the dependence of $\bar{f}$ on
pumping strength $\eta$, see figure \ref{fig:fVSeta}. As the cavity
pumping strength $\eta$ increases, the strength of the induced optical
potential also increases accordingly. As can be expected, the self-trapped
wave packet feels a stronger friction force at a larger pumping strength.
In figure \ref{fig:fVSv0}, we plot $\bar{f}$ as a function of the
initial speed $v_{0}$ of the self-trapped wave packet. The faster
the self-trapped wave packet moves, the severer the light field falls
behind, thus the friction force $\bar{f}$ is expected to be proportional
to $v_{0}$. This is numerically observed at small values of $v_{0}$
($v_{0}<3$). However, as $v_{0}$ further increases, the friction
force is saturated; and after $v_{0}>6$ the friction force decrease.
We found that this is caused by the escaping of atoms from the self-trapped
wave packet. When the wave packet moves with a fast speed, a considerable
fraction of atoms can escape from the self-trapped wave packet, thus
the atomic density, and therefore the depth of the optical lattice
is reduced. And a shallower optical lattice will have a weaker friction
effect. This is shown in figure \ref{fig:Largev0}, in the left panel
of which the evolution of atomic density for $v_{0}=8$ is plotted,
and in the right panel the initial and final density profile is compared.
In the figure, the escaping of atoms from the self-trapped wave packet
is characterized by the precursor in the white box. And integrating
the initial and final density profiles, we found that about $27\%$
of the atoms have been lost.
\begin{figure}
\begin{centering}
\includegraphics{fig8}
\par\end{centering}
\caption{Relation between the mean friction force $\bar{f}$ and decay rate
of the cavity $\kappa$.
The squares and circles are data points collected from
mean-field and semiclassical numerical simulations respectively.
The lines are simple linear connection
of the data points to guide the eyes. Parameters used to plot this
line are $v_{0}=-2$, $U=-0.5$, $\Delta_{c}=-1$, and $\eta=15$.\label{fig:fVSkappa}}
\end{figure}
\begin{figure}
\begin{centering}
\includegraphics{fig9}
\par\end{centering}
\caption{Damped oscillation of the self-trapped wave packet under a small cavity
decay rate ($\kappa=2$). Left panel: Time evolution of the atomic
density profile $\left|\psi\right|^{2}$. The black solid line is
the center of the self-trapped wave packet. Right panel: Atomic density
profile (violet solid line) and corresponding optical lattice potential
(green dashed line) at $t=0.96$ (the first time at which the speed
of the wave packet reaches 0, i.e., the first minimal point of the
black line in the left panel). The black dotted line is plotted to
mark the center of the wave packet. Other parameters used are $v_{0}=-2$,
$U=-0.5$, $\Delta_{c}=-1$ and $\eta=3.$\label{fig:DampedOscillatingMotion}}
\end{figure}
\begin{figure}
\begin{centering}
\includegraphics{fig10}
\par\end{centering}
\caption{Constant-speed moving of the self-trapped wave packet in the bad cavity
limit. To produce this figure, equations (\ref{eq:Meanfield_atom})
together with bad cavity limit optical field formulae (\ref{eq:alpha_p})
and (\ref{eq:alpha_m}) are solved numerically. Parameters used are
$v_{0}=-2$, $U=-0.5$, $\Delta_{c}=-1$, $\eta=75$ and $\kappa=50$.
\label{fig:BadCavityLimit}}
\end{figure}
Since the friction force results from the time delay of the cavity
light field relative to the moving of the atomic matter wave, one
can expect reducing the friction force by shortening the cavity relaxation
time, i.e., increasing the cavity decay rate $\kappa$. This is also
demonstrated by our numerical results, see figure \ref{fig:fVSkappa},
where the mean friction force $\bar{f}$ felt by the self-trapped
wave is plotted as a function of the cavity decay rate $\kappa$.
And we also found that for a small value of the cavity decay rate
$\kappa$, the moving dynamics of the self-trapped waves can show
new features. It undergoes a damped oscillation, as shown in figure
\ref{fig:DampedOscillatingMotion}. In the left panel, the moving
of the density profile (colormap) and center position (black solid
line) of the self-trapped wave is plotted. At the beginning stage,
the delayed light field seriously decelerates the self-trapped wave
packet. Because the light field falls too much behind in this case,
even the speed of the self-trapped wave packet has been decelerated
to zero at $t=0.96$ (the first minimal of the black solid line),
the induced optical lattice potential still can not catch up, see
the right panel where we plot the density profile and the induced
optical lattice potential at this time. As a result, in an afterward
time interval the still falling behind optical lattice accelerates
the self-trapped wave in the opposite direction. Then, the light field
catches up, and decelerates the condensate again. This deceleration-acceleration
process repeats several times, therefore the self-trapped wave undergoes
a damped oscillation.
At last, in the bad cavity limit, the light field can instantaneously
follow the moving self-trapped wave packet, thus it will have no friction
effect on the self-trapped wave. In such a case, we expect that the
self-trapped wave packet will move with its initial speed all the
afterward time. In figure \ref{fig:BadCavityLimit}, the moving of
a self-trapped wave packet is studied under the bad cavity approximation,
i.e., the simulation is done by numerically solving equations (\ref{eq:Meanfield_atom}),
(\ref{eq:alpha_p}), and (\ref{eq:alpha_m}). The expected constant
speed motion is demonstrated by the numerical result.
\section{Comparison to the Semiclassical Theory \label{sec:Semiclassical}}
Taking the atoms as classical polarizable particles, their motion can be approximately
described by a one-dimensional Vlasov equation \citep{Niedenzu2011Kinetic,Ostermann2015Atomic}
\begin{equation}
\frac{\partial f}{\partial t} + v \frac{\partial f}{\partial x}
- \frac{\partial \mathcal{V}_{\mathrm{eff}}}{\partial x} \frac{\partial f}{\partial v} = 0,
\label{eq:Vlasov}
\end{equation}
where $f(x,v,t)$ is the phase space distribution of the atoms with $v$ meaning the velocity.
And the light field is still governed by equation (\ref{eq:Meanfield_cavity}), except
that the variables $N_{\pm 1}$ and $N_{\pm 2}$ are now defined as
\[
N_{\pm 1} = \int \rho(x) e^{\mp i x} dx,
\]
and
\[
N_{\pm 2} = \int \rho(x) e^{\mp 2 i x} dx,
\]
with $\rho(x)$ being the spatial distribution of the atoms
\[
\rho(x,t) = \int f(x,v,t) dv.
\]
Such a semiclassical theory also predicts a self-trapped state of the
atoms, see the left panel of figure \ref{fig:semiClassical}.
In this figure, we also plot the
mean-field result for comparison. It is found that the semiclassical
theory gives a narrower spatial distribution of the atoms than the
mean-field theory. This is because the quantum pressure (the kinetic
term in Sch\"{o}dinger equation) which tends to spread
the atomic distribution is absent in the semiclassical theory
[while in the mean-field theory, the atoms are described by the
Schr\"{o}dinger-like equation (\ref{eq:Meanfield_atom}) which
can include the effect of quantum pressure].
Another difference between the mean-field and semiclassical theory results
is the instability of the self-trapped state under weak pumping strength.
Recalling that under weak pumping strength tunneling of atoms to the
neighboring lattice sites will lead to the spreading of self-trapped
state during its evolution (top right panel of figure \ref{fig:SW}).
However, the semiclassical theory treats the atoms as classical particles,
thus the tunneling phenomena can not be included, as a result, it gives
a non-spreading stable evolution of the self-trapped state, see the right
panel of figure \ref{fig:semiClassical} where the parameters
are chosen the same as in the top panels of figure \ref{fig:SW}.
We also compared the mean friction force predicted by
mean-field and semiclassical
theory, as shown in figures \ref{fig:fVSeta}, \ref{fig:fVSv0} and
\ref{fig:fVSkappa}. Because the semiclassical theory
predicts a narrower spatial distribution of the atoms, the
produced optical potential will also be tighter accordingly. And this will
make the semiclassical theory overestimate the friction force.
\begin{figure}
\includegraphics{fig11}
\caption{
An self-trapped state and its time evolution given by
the semiclassical theory. Left panel: Spatial distribution
(violet and green line)
of the self-trapped state atoms and the corresponding optical
lattice potential (cyan and brown line).
The solid lines are the mean-field result,
while the dotted lines are the semiclassical result. Right
panel: Semiclassical time evolution of the self-trapped state.
The parameters are $U=-0.5$, $\Delta_c=-1$, $\kappa=10$ and
$\eta=6$, which are the the same as in top panels of figure
\ref{fig:SW}. \label{fig:semiClassical}
}
\end{figure}
\section{Summary\label{sec:Summary}}
In summary, we have studied the self-trapped matter waves and their
moving dynamics in a driven-dissipative ring cavity-BEC system
within the mean-field theory. The
self-trapped wave packets have been found by both variational
and numerical methods, and the results fit with each other very well.
The stability of the self-trapped waves is verified by direct numerical
simulations. It is found that for a strong cavity pumping the self-trapped
wave packet can be stable for quite a long time, while for a weak
cavity pumping the self-trapped wave packet suffers a spatial spreading
during its evolution. We also found that the moving dynamics of these
self-trapped waves can be greatly affected by the cavity loss rate.
Three distinct types of motion of the self-trapped waves have been
identified in the system. For a cavity with a small decay rate, the
cavity light field can alternatively drag and push the self-trapped
wave packet, therefore the self-trapped wave packet endures a damped
oscillation. And for a cavity with a moderate decay rate, the self-trapped
wave packet always fells a dragging force from the cavity light field,
and undergoes a decelerating motion. In the bad cavity limit, the
friction force disappears, and the self-trapped wave packet constantly
moves with the initial speed.
The main results are also compared with a semiclassical
calculation where the atoms are treated as classical particles.
We found that the semiclassical theory predicts a narrower
spatial distribution of the atoms, and will overestimate the friction
force. It also misses the instability of the self-trapped state
under weak pumping.
These dynamical tunable self-trapped
waves may find potential applications in fields such as matter wave
interferometers \citep{Polo2013Soliton,McDonald2014Bright,Helm2015Sagnac,Wales2020Splitting}.
\begin{acknowledgments}
This work is supported by the National Natural Science
Foundation of China (Grants No. 11904063, No. 12074120,
No. 11847059, and No. 11374003), and the Science and Technology
Commission of Shanghai Municipality (Grant No.
20ZR1418500).
\end{acknowledgments}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 4,902 |
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