text:
\section{Introduction}

The second of the six famous Painlev\'e equations reads
\begin{equation}\label{P2}
 y''= 2 y^3 +z y + a,
\end{equation}
where $'$ denotes the derivative with respect to $z$ and $a$ is an arbitrary complex constant. A matrix version of the Painlev\'e--Kovalevskaya test was proposed in \cite{Balandin_Sokolov_1998}, where it was proved that it holds for \eqref{P2} in the case when $y(z)$ is a matrix of arbitrary size $n\times n$ and $a$ is a scalar matrix.

One of possible generalizations of the matrix equation \eqref{P2} is as follows. It is clear that in the non-commutative case one can change the principal differential-homogeneous part $y''= 2 y^3$ of this equation by adding the term of the same weight $\kappa [y, y']$, where $\kappa\in{\mathbb C}$. In particular, from the results of Section 2 it follows that the equation
$$
 y''= \kappa [y,y'] +2y^3 +z y +a,\qquad a\in \mathbb C
$$
satisfies the matrix Painlev\'e--Kovalevskaya test if and only if $\kappa = 0, \pm 1, \pm 2.$ Note that equations with opposite signs of $\kappa$ are related by the changes $y\to -y$, $a\to -a$ or $y\to y^T$.

Another direction for generalizations was suggested by the results of \cite{Retakh_Rubtsov_2010}, where the equation P$_2$ with matrix coefficients appeared. More precisely, in this paper the term linear in $y$ was written as $\tfrac{1}{2}(zy+yz)$ where $z$ denoted a non-commutative dependent variable such that $z'= 1$. For our purposes it is more convenient to replace this variable with $z+2b$, where $z$ is a commutative independent variable and $b$ is a matrix constant.

In this paper, we study matrix generalizations of the P$_2$ equation of the general form
\begin{equation}\label{gP2}
 y''= \kappa[y,y'] +2y^3 +z y +b_1 y +y b_2 +a,
\end{equation}
where $a$, $b_1$ and $b_2$ are matrix constants and $\kappa$ is a scalar constant.

\begin{remark}
Equation (\ref{gP2}) is invariant under the change
\begin{equation}\label{bbb}
b_1\to b_1+\beta_1I,\qquad  b_2\to b_2+\beta_2I,\qquad z\to z-\beta_1-\beta_2,
\end{equation}
where $\beta_i\in{\mathbb C}$ and $I$ is the identity matrix.
\end{remark}

In the second section, by use of the matrix version of Painlev\'e--Kovalevskaya test, we examine the sets of $\kappa$, $a$, $b_1$ and $b_2$ which are candidates for integrable cases and prove the following statement.

\begin{theorem}
The equation \eqref{gP2} satisfies the Painlev\'e--Kovalevskaya test if and only if it is reduced by the transformation \eqref{bbb} to one of the following cases:
\begin{alignat}{2}
\label{P20}\tag*{P$_2^0$}
 & y''= 2y^3+zy+by+yb+\alpha I, &\qquad& \alpha\in{\mathbb C},~~ b\in\Mat_n,\\
\label{P21}\tag*{P$_2^1$}
 & y''= \pm[y,y']+2y^3+zy+a, && a\in\Mat_n,\\
\label{P22}\tag*{P$_2^2$}
 & y''= \pm2[y,y']+2y^3+zy+by+yb+a, && a,b\in\Mat_n,~~ [b,a]=\pm2b.
\end{alignat}
\end{theorem}

Note that \ref*{P20} is exactly the equation from \cite{Retakh_Rubtsov_2010}. The equations \ref*{P21} and \ref*{P22} appear to be new.

In Section 3, the integrability of the found cases is substantiated by construction of the isomonodromic Lax pairs
\begin{equation}\label{AB}
 A'=B_\zeta+[B,A],
\end{equation}
where $\zeta$ is a spectral parameter, $A(z,\zeta)$ and $B(z,\zeta)$ are $2\times2$ matrices with non-commutative entries. For equation \ref*{P20}, such a representation was found in \cite{Irfan_2012}.

One of the methods for obtaining representations \eqref{AB} is based on the observation that group-invariant solutions of evolution equations admitting zero curvature representations satisfy ordinary differential equations of the Painlev\'e type. In this case, the isomonodromic representation is obtained from the zero curvature representation by a standard procedure. In particular, it is well known that the P$_2$ equation corresponds to Galilean-invariant solutions of the nonlinear Schr\"odinger equation, as well as to scale-invariant solutions of the mKdV equation. Here we generalize these reductions to the matrix case.

One source of non-commutative coefficients in \eqref{gP2} is the arbitrary matrices contained in the symmetry groups of non-Abelian \cite{Marchenko_1986, OS_1998a} evolutionary equations. For example, the matrix nonlinear Schr\"odinger equation
\begin{equation}\label{NLS}
     u_t=u_{xx}+2uvu ,\qquad v_t= -v_{xx}-2vuv
\end{equation}
admits transformations of the form $u\to b_1 u b_2$, $v\to b_2^{-1} v b_1^{-1}$, where $b_i\in\Mat_n.$ Two well-known matrix generalizations of the mKdV equation \cite{Marchenko_1986, Khalilov_Khruslov_1990}
\begin{gather*}
 u_t= u_{xxx}-3u^2u_x-3u_xu^2,\\[1.5mm]
 u_t= u_{xxx}+3[u,u_{xx}]-6uu_xu
\end{gather*}
obviously admit the transformation group $u \to b u b^{-1}.$

Another source of matrix coefficients in \eqref{gP2} is related with matrix coefficients in integrable evolution equations themselves. In this paper, we use the following integrable version of the matrix mKdV equation:
$$
 u_t= u_{xxx}+3[u, u_{xx}]-6u u_x u + (u_x+u^2) b + b (u_x-u^2), \qquad b\in\Mat_n.
$$
Its zero curvature representation is given in Section 3. We do not know whether this generalization appeared in the literature.

In Section 3, we show that equations P$_2^i$ from Theorem 1 are obtained from the above matrix evolution equations by some group-invariant reductions. This made possible to find the isomonodromic Lax pairs \eqref{AB} for the matrix Painlev\'e II equations.

\section{Proof of Theorem 1}\label{s:P2-test}

\subsection{The Painlev\'e--Kovalevskaya matrix test}

The Painlev\'e--Kovalevskaya matrix test \cite{Balandin_Sokolov_1998} for equation (\ref{gP2}) is based on counting of arbitrary scalar constants in a formal solution of the form
\begin{equation}\label{y}
 y=\frac{p}{z-z_0}+c_0+c_1(z-z_0)+\cdots\,, \qquad
 p,c_j \in\Mat_n,\quad z_0\in{\mathbb C}.
\end{equation}
Here $z_0$ is one of these arbitrary constants. In order for the series $y$ to represent a generic solution, it is necessary that the matrices $p$ and $c_j$ contain additionally $2n^2-1$ arbitrary constants. We assume that the Painlev\'e--Kovalevskaya test is fulfilled if such matrices exist. Note that there may also exist other formal solutions of the form \eqref{y} that contain fewer arbitrary constants.

\begin{remark}
For any nondegenerate matrix $T$, the series $T y T^{-1}$ satisfies equation (\ref{gP2}), where $b_i\to \bar b_i \stackrel{def}{=} T b_i T^{-1}$ and $a \to \bar a\stackrel{def}{=}T a T^{-1}$. Hence, equation (\ref{gP2}) satisfies the Painlev\'e--Kovalevskaya test simultaneously with the equation corresponding to the coefficients $\bar b_i$ and $\bar a$.
\end{remark}

Substituting the series into the equation and collecting the coefficients at powers of $z-z_0$, we obtain relations of the form
\begin{gather}
\label{ep3}
 p^3=p,\\
\label{eLc}
 L_{\frac{j+1}{2}\kappa}(c_j) -\frac{j(j-1)}{2}c_j =f_j(z_0,p,c_0,\dots,c_{j-1}),\quad j\ge 0,
\end{gather}
where
\[
 L_\sigma(c) \stackrel{def}{=} p^2c + pcp+ cp^2 +\sigma(pc-cp).
\]
The first few functions $f_i$ are easy to compute explicitly, for instance,
\[
 f_0=0,\qquad f_1=-pc^2_0-c_0pc_0-c^2_0p-\frac{1}{2}(z_0p+b_1p+pb_2).
\]
Since $f_j$ in the right-hand sides of (\ref{eLc}) do not contain $c_j$, the matrices $c_j$ can be calculated from these equations recursively. If the linear operator in the left-hand side is invertible, then $c_j$ is uniquely determined. If, for some $j$, the corresponding operator is degenerate then, firstly, the answer contains arbitrary constants in the amount equal to the dimension of the kernel; secondly, the solvability conditions lead to some constraints on $f_j$, from which the conditions on the parameters $b_1$, $b_2$ and $a$ should be extracted.

We start from calculating the possible total number of arbitrary constants. The number of arbitrary constants in the matrix $p$ is equal to the dimension of the orbit of its Jordan form. It is easy to see that the Jordan form of any matrix $p$ satisfying \eqref{ep3} is
\[
 p=\diag(E_k,\, -E_m,\, 0_{n-k-m}).
\]
When a group acts on a manifold, the dimension of the orbit of a point is equal to the difference of the manifold dimension and the dimension of the stabilizer of this point, that is, the subgroup that leaves the point fixed. In our case, the dimension of the manifold is $n^2$, the stabilizer consists of non-degenerate matrices which commute with $p$. It is easy to see that its dimension is $k^2+m^2+(n-k-m)^2$; whence it follows that the dimension of the orbit of $p$ is equal to $2m(n-m)+2k(n-k)-2km$.

Next, we calculate the dimensions of the eigenspaces of the linear operator $L_\sigma:\Mat_n\to\Mat_n$.

\begin{lemma}\label{l:dim}
The eigenvalues of the operator $L_\sigma$ belong to the set
\[
 \lambda_0=0,\quad \lambda_{\pm 1}=1\pm \sigma, \quad \lambda_{\pm 2}=1\pm 2 \sigma, \quad \lambda_3=3.
\]
The space $\Mat_n$ decomposes into the direct sum of eigenspaces
\[
 \Mat_n=V_0\oplus V_{-1}\oplus V_1\oplus V_{-2}\oplus V_2\oplus V_3,\qquad L_\sigma V_i= \lambda_iV_i,
\]
with dimensions
\begin{gather*}
\dim V_0 = (n-k-m)^2, \qquad  \dim V_{\pm1} = k(n-k)+m(n-m) - 2 k m,\\
\dim V_{\pm2} = k m, \qquad  \dim V_3 = k^2+m^2.
\end{gather*}
In the case when the eigenvalues coincide, the dimensions of the corresponding eigenspaces add up.
\end{lemma}

\begin{proof}
We represent $c$ as a $3\times3$ block matrix with the block sizes determined by the Jordan form of $p$:
\[
 c=\bordermatrix{
    & k      & m      & l \cr
  k & c_{11} & c_{12} & c_{13} \cr
  m & c_{21} & c_{22} & c_{23} \cr
  l & c_{31} & c_{32} & c_{33}},
 \qquad l=n-k-m.
\]
Then it is easy to check that
\[
 L_\sigma(c)=\begin{pmatrix}
  3c_{11} & (1+2\sigma)c_{12} & (1+\sigma)c_{13}\\
  (1-2\sigma)c_{21} & 3c_{22} & (1-\sigma)c_{23}\\
  (1-\sigma)c_{31}  & (1+\sigma)c_{32} & 0
 \end{pmatrix},
\]
that is, each eigenspace consists of one or two blocks of the matrix $c$. The dimensions of the eigenspaces are calculated by summing the block sizes.
\end{proof}

The operator in the left hand side of (\ref{eLc}) degenerates if one of the eigenvalues of the operator $L_{\frac{j+1}{2}\kappa}$ coincides with $\frac{j(j-1)}{2}$ (we call this the resonance condition). In particular, the eigenvalue $\lambda_0=0$ appears twice: for $j=0$ and for $j=1$, and the eigenvalue $\lambda_3=3$ appears once for $j=2$. The conditions that the eigenvalues $\lambda_{\pm1}$ and $\lambda_{\pm2}$ are resonant for some index $j$, are given, respectively, by equations
\[
 1\pm \frac{\kappa (j+1)}{2}=\frac{j(j-1)}{2} \quad\text{and}\quad
 1\pm \kappa (j+1)=\frac{j(j-1)}{2}.
\]
Cancelling this by $j+1$ (recall that $j\ge0$), we obtain the resonance conditions
\[
 \lambda_{\pm1}:~ j=2 \pm \kappa,\qquad
 \lambda_{\pm2}:~ j=2 \pm 2\kappa.
\]
We see that for a fixed $\kappa$ any of eigenspaces $V_{\pm 1}$, $V_{\pm 2}$ can occur at its resonant $j$ no more than once. This makes possible to estimate from above their total dimension, that is, the number of possible arbitrary constants. Namely, if both $\lambda_{\pm1}$ and $\lambda_{\pm2}$ appear in (\ref{eLc}), then the total sum of dimensions is
\begin{multline*}
 \qquad 2\dim V_0+\dim V_{-1}+\dim V_1+\dim V_{-2}+\dim V_2+\dim V_3\\
 = \dim V_0 +n^2 = (n-k-m)^2+n^2.\qquad
\end{multline*}
By adding the dimension of the orbit of $p$ we obtain that the total number of arbitrary constants is equal to $2n^2-k^2-m^2$. This is equal to $2n^2-1$ only for $k=1$, $m=0$ or for $k=0$, $m=1$. The second case is reduced to the first one by the change $y\to -y$. In the first case the Jordan form of $p$ is $\diag(1,0,\dots,0)$ and its orbit ${\cal O}$ consists of the matrices of the form $u \, v^T$, where $u$ and $v$ are column vector such that $u^T v=1.$

Thus, we assume that $p\in {\cal O}$. Then the eigenvalues $\lambda_{\pm 2}$ from the Lemma \ref{l:dim} disappear. To provide the required number of arbitrary constants, the series \eqref{y} must exist for the general element of the orbit. More precisely, we require that:
\begin{itemize}
    \item for any column vectors $u$ and $v$ such that $u^T v=1$, there exists a formal solution of the form
    $$
 y=\frac{u\,v^T}{z-z_0}+c_0+c_1(z-z_0)+\cdots\,, \qquad
 p,c_j \in\Mat_n,\quad z_0\in{\mathbb C},
    $$
such that its coefficients $c_i$ contain in total $2n^2-2n+1$ arbitrary constants.
\end{itemize}

{\bf Proof of Theorem 1.} The above counting of arbitrary constants shows that a necessary condition for passing the Painlev\'e--Kovalevskaya test is that {\em both} eigenvalues $\lambda_{\pm 1}$ must be resonant. The resonance conditions for $j=2\pm\kappa$ imply that the parameter $\kappa$ must be such that both numbers $2\pm\kappa$ are non-negative integers. This leaves the admissible values
\[
 \kappa=0,\,\pm 1,\,\pm 2.
\]
The case $\kappa<0$ is reduced to $\kappa>0$ by the change $y\to y^T$. Therefore, further it is sufficient to analyze equations (\ref{eLc}) for each of the three cases $\kappa=0,1$ and 2.

\subsection{Case $\kappa = 0$}\label{kap0}

Let $p=\diag(1,0,\dots,0)$. In this case the operator $L_\sigma$ is of the form
\[
 L_\sigma:\quad
 \bordermatrix{ & 1 & n-1 \cr
  ~~~1 & c_{11} & c_{12}\cr
  n-1  & c_{21} & c_{22}}
 \mapsto
 \begin{pmatrix}
  3c_{11} &  (1+\sigma)c_{12}\\
  (1-\sigma)c_{21} & 0
 \end{pmatrix}.
\]
In order to analyze equations (\ref{eLc}), we write them block-wise by dividing the involved matrices into blocks of appropriate sizes:
\[
 c_j=\begin{pmatrix}
   c_{j,11} & c_{j,12} \\
   c_{j,21} & c_{j,22}
 \end{pmatrix},\quad
 b_1=\begin{pmatrix}
   b_{1,11} & b_{1,12} \\
   b_{1,21} & b_{1,22}
 \end{pmatrix},\quad
 b_2=\begin{pmatrix}
   b_{2,11} & b_{2,12} \\
   b_{2,21} & b_{2,22}
 \end{pmatrix},\quad
 a=\begin{pmatrix}
   a_{11} & a_{12} \\
   a_{21} & a_{22}
 \end{pmatrix}.
\]
Conditions on the matrices $b_1$, $b_2$ and $a$ arise from the requirement of existence of solutions for inhomogeneous linear systems \eqref{eLc} for the resonance values of $j$. In this case, the homogeneous system has a nontrivial solution and the conditions of the Kronecker--Capelli theorem must be satisfied.

Let us consider equations (\ref{eLc}) for $j=0,1,2$ and $3$. For $j=0$ we find $c_{0,11}=c_{0,12}=c_{0,21}=0$ and the block $c_{0,22}$ is arbitrary.

Next, for $j=1$ we obtain
\[
 c_{1,11}=-\frac{1}{6}(z_0+b_{1,11}+b_{2,11}),\qquad
 c_{1,12}=-\frac{1}{2}b_{2,12},\qquad
 c_{1,21}=-\frac{1}{2}b_{1,21}
\]
and $c_{1,22}$ is arbitrary. So far, no conditions for the coefficients $b_1,b_2$ and $a$ appear.

For $j=2$ we find, by substituting the obtained values, that
\[
 c_{2,11}=-\frac{1}{4}(1+a_{11}),\qquad
 c_{2,22}=\frac{1}{2}(a_{22}+b_{1,22}c_{0,22}+c_{0,22}b_{2,22}+z_0c_{0,22})+c^3_{0,22},
\]
and the blocks $c_{2,12}$ and $c_{2,21}$ are arbitrary. In addition, we obtain the restrictions on the coefficients in the form of the relations
$$
 (b_{2,12}-b_{1,12})c_{0,22}=a_{12},\qquad c_{0,22}(b_{1,21}-b_{2,21})=a_{21}.
$$
Since the block $c_{0,22}$ is arbitrary, this implies that
\begin{equation}\label{cond1}
 b_{1,12}=b_{2,12},\qquad b_{1,21}=b_{2,21},\qquad a_{12}=a_{21}=0.
\end{equation}

Finally, for $j=3$ it is sufficient to write down the condition for the $1\times1$ block
\begin{equation}\label{cond2}
 b_{1,12}b_{1,21}-2b_{2,12}b_{1,21}+b_{2,12}b_{2,21}=0,
\end{equation}
because all other equations are solved uniquely with respect to $c_{3,12}$, $c_{3,21}$ and $c_{3,22}$. The condition \eqref{cond2} follows from \eqref{cond1}. For $j>3$, there are no resonances and therefore all equations for $c_j$ are solved uniquely.

Thus, we proved that equations (\ref{eLc}) with $\kappa=0$ are solved with arbitrary blocks $c_{0,22}$, $c_{1,22}$, $c_{2,12}$, $c_{2,21}$ and $c_{3,11}$, which gives, as one can easily see, $2(n-1)^2+2(n-1)+1$ arbitrary constants. Moreover, we obtained the solvability conditions in the form of relations (\ref{cond1}) which mean that the matrices $a$ and $b_1-b_2$ commute with $p=\diag(1,0,\dots,0)$.

By choosing as $p$ the matrices of the form $\diag(0,0,\dots, 1, \dots,0)$ which belong to $\cal O$, we obtain that $a$ and $b_1-b_2$ must commute with any such matrix, which means that they are diagonal. According to Remark 1, the matrices $T a T^{-1}$ and $T (b_1-b_2) T^{-1}$ must be diagonal for any nondegenerate $T$. This means that $a$ and $b_1-b_2$ are scalar matrices. Obviously, the condition $b_1=b_2+2\beta I$ is reduced to $b_1=b_2$ by the transform \eqref{bbb}. As the result, we arrive at equation \ref{P20}.

\subsection{Case $\kappa = 1$}

Like in the case $\kappa=0$, it is sufficient to analyze equations (\ref{eLc}) for $j=0,1,2$ and $3$.

Let $p=\diag(1,0,\dots,0)$. Then we find for $j=0$ that $c_{0,11}=c_{0,12}=c_{0,21}=0$ and the block $c_{0,22}$ remains arbitrary.

For $j=1$ we have
\[
 c_{1,11}=-\frac{1}{6}(b_{1,11}+b_{2,11}+z_0),\qquad c_{1,12}=-\frac{1}{4}b_{2,12},
\]
both blocks $c_{1,21}$ and $c_{1,22}$ are arbitrary and $b_1$ should satisfy the restriction
\[
 b_{1,21}=0.
\]
Now we repeat the reasoning from section \ref{kap0}. By choosing the matrices $\diag(0,\dots,1,\dots,0)$ as $p$, we obtain that $b_1$ must be diagonal. Then we prove that it is scalar. Finally, we make use of the transformation \eqref{bbb} and set $b_1=0$.

Next, for $j=2$ all blocks $c_2$ are uniquely determined:
\begin{gather*}
 c_{2,11}=-\frac{1}{4}(1+a_{11}),\qquad
 c_{2,12}=\frac{1}{12}(b_{2,12}c_{0,22}-4a_{12}),\\
 c_{2,21}=\frac{1}{3}(a_{21}+c_{0,22}b_{2,21}+3c_{0,22}c_{1,21}),\\
 c_{2,22}=\frac{1}{2}(a_{22}+z_0c_{0,22}+c_{0,22}b_{2,22}
  +[c_{0,22},c_{1,22}])+c^3_{0,22}.
\end{gather*}

For $j=3$, the blocks $c_{3,11}$ and $c_{3,12}$ remain arbitrary, and the blocks $c_{3,21}$ are $c_{3,22}$ uniquely determined. Moreover, the following relations appear as the solvability conditions for the linear system:
\[
 b_{2,12}(b_{2,21}+4c_{1,21})=0,\qquad b_{2,12}(z_0+b_{2,22}+2c_{1,22}+2c^2_{0,22})=0.
\]
Taking the arbitrariness of $c_{1,22}$ and $c_{0,22}$ into account, this amounts just to the condition $b_{2,12}=0$ which implies, as before, that the matrix $b_2$ is also scalar and then it also can be set to zero. There are no resonance values for $j>3$ and all other coefficients of the series \eqref{y} are uniquely found.  Thus, equations (\ref{eLc}) with $\kappa=1$ are solvable for vanishing $b_1$ and $b_2$, and with the arbitrariness in the blocks $c_{0,22}$, $c_{1,21}$, $c_{1,22}$, $c_{3,11}$ and $c_{3,12}$. Like in the previous case, this gives the required number of arbitrary constants. There are no restrictions for the matrix $a$, since it is not involved in the solvability conditions at all, and we obtain equation \ref{P21}.

\subsection{Case $\kappa = 2$}

In this case we have to analyze equations for $j=0,1,2,3$ and $4$.

For $j=0$ we find that $c_{0,11}=c_{0,12}=0$. In contrast to the previous cases, two blocks $c_{0,21}$ and $c_{0,22}$ remain arbitrary.

For $j=1$ we have
\[
 c_{1,11}=-\frac{1}{6}(b_{1,11}+b_{2,11}+z_0),\qquad
 c_{1,12}=-\frac{1}{6}b_{2,12},\qquad
 c_{1,21}=\frac{1}{2}b_{1,21}+c_{0,22}c_{0,21},
\]
the block $c_{1,22}$ is arbitrary, no solvability condition appears.

For $j=2$ all blocks of the matrix $c_2$ are found uniquely:
\begin{gather*}
 c_{2,11}=-\frac{1}{4}(a_{11}+1)+\frac{1}{12}(b_{2,12}-3b_{1,12})c_{0,21},\qquad
 c_{2,12}=-\frac{1}{6}(a_{12}+b_{1,12}c_{0,22}),\\
 c_{2,21}=\frac{1}{6}\big(a_{21}+b_{1,22}c_{0,21}-c_{0,21}b_{1,11}+c_{0,22}(2b_{1,21}+b_{2,21})\big)
    +c^2_{0,22}c_{0,21},\\
 c_{2,22}=\frac{1}{2}(a_{22}+z_0c_{0,22}+b_{1,22}c_{0,22}+c_{0,22}b_{2,22})
  +[c_{0,22},c_{1,22}]+\frac{1}{6}c_{0,21}b_{2,12}+c^3_{0,22}.
\end{gather*}

For $j=3$ the block $c_{3,11}$ is arbitrary, the other blocks are uniquely determined and the first solvability condition appears:
\[
 3b_{1,12}b_{1,21}-2b_{2,12}b_{1,21}-b_{2,12}b_{2,21}
  +6(b_{1,12}-b_{2,12})c_{0,22}c_{0,21}=0.
\]
Due to arbitrariness of $c_{0,22}$ and $c_{0,21}$ we conclude from here that $b_{1,12}-b_{2,12}=0$. By repeating the same reasoning as in the previous cases we prove that the matrices $b_1$ and $b_2$ coincide up to a scalar matrix; then we can set $b_1=b_2=b$ by transformation \eqref{bbb}. It is easy to see that after this the remaining terms in the above relation cancel out.

Finally, for $j=4$ the block $c_{4,12}$ can be choosen arbitrarily if the following condition is fulfilled:
\[
 b_{11}a_{12}+b_{12}a_{22}=a_{11}b_{12}+a_{12}b_{22}+2b_{12}.
\]
This is nothing but $([b,a]-2b)_{12}=0$, and again, by the reasoning based on the arbitrariness of $p$, we conclude that the matrices $a$ and $b$ must satisfy the relation\footnote{Under the transposition, $\kappa=2$ is replaced with $\kappa=-2$ and the sign of the commutator is changed as well.}
\[
 [b,a]=2b.
\]
The rest blocks of $c_4$ and all subsequent $c_j$ are found uniquely. Thus, under the above condition the coefficients of the series are determined with the arbitrariness in the blocks $c_{0,21}$, $c_{0,22}$, $c_{1,22}$, $c_{3,11}$ and $c_{4,12}$ which have the required total dimension. As the result, we obtain equation \ref{P22}.

The sufficiency of the obtained conditions on the coefficients $\kappa$, $a$ and $b$ for the existence of the series \eqref{y} with arbitrary matrix $p\in{\cal O}$ and the required number of arbitrary constants is proved by the same reasoning in each of three cases. Since the series \eqref{y} with $p=\diag(1,0,\dots,0)$ exists for any  $\kappa$, $a$ and $b$ satisfying the conditions of the theorem, we can replace them with $\kappa$, $\bar a = T a T^{-1}$ and $\bar b=T b T^{-1}$ where $T$ is any non-degenerate matrix. Then the series $T^{-1} y T$ satisfies the equation with parameters $\kappa$, $a$ and $b$ and it has the arbitrary element of the orbit ${\cal O}$ as $p$. \qquad The theorem is proved.

\section{Reductions of partial differential equations}

\subsection{Matrix nonlinear Scr\"odinger equation}

The matrix NLS equation \eqref{NLS} admits the following reduction (which defines solutions that are invariant with respect to some one-parametric subgroup in the group generated by the Galilean transformation, the translation of $t$ and the conjugation by the matrix exponent of a constant matrix):
\begin{equation}\label{NLS.Gal}
 u=e^rp(z),\qquad v=q(z)e^{-r},\qquad r=\frac{1}{6}(t^3-3xt)-tb,\qquad z=x-\frac{1}{2}t^2,\qquad b\in\Mat_n.
\end{equation}
One can easily check that this substitution reduces equations (\ref{NLS}) to the following ODE system with respect to $p$ and $q$ (cf. with equation (3.1) in \cite{Retakh_Rubtsov_2010}):
\begin{equation}\label{NLS.red}
 p''= -\frac{1}{2}zp-bp-2pqp,\qquad q''= -\frac{1}{2}zq-qb-2qpq,\qquad b\in\Mat_n.
\end{equation}
The order of this system can be reduced by one due to the first integral
\begin{equation}\label{NLS.int}
 qp'-q'p=c,\qquad c\in\Mat_n.
\end{equation}
Another way to reduce the order is by passing to the logarithmic derivatives of $p$ and $q$. It turns out that to use both methods we can retain only one of the matrix constants $b$ or $c$. The second one should be chosen scalar. Let us consider these two possibilities.
\medskip

1) Let $b\in\Mat_n$ and $2c=\gamma\in\mathbb C$. Then we introduce the variables
\begin{equation}\label{NLS.sub1}
 f=p'p^{-1},\qquad g=q^{-1}q',\qquad h=2pq,
\end{equation}
and the system (\ref{NLS.red}) takes the form
\[
 f'=-f^2-\frac{1}{2}z-b-h,\qquad g'=-g^2-\frac{1}{2}z-b-h,\qquad h'=fh+hg.
\]
Since $\gamma$ is scalar, the first integral (\ref{NLS.int}) can be expressed in terms of $f,g$ and $h$, too:
\[
 f-g=\gamma h^{-1}.
\]
By its use, we eliminate $g$ and arrive at equations
\begin{equation}\label{fh0}
 f'=-f^2-\frac{1}{2}z-b-h,\qquad h'=fh+hf-\gamma.
\end{equation}
Finally, we find $h$ from the first equation and substitute into the second one; this brings to equation \ref{P20} for the variable $y=f$ and with $a=\gamma-\frac{1}{2}$.

\begin{remark}
The symmetry of equations with respect to $f$ and $g$ immediately implies that $g$ also satisfies the same equation with the free term $-\gamma-\frac{1}{2}$. By combining this with the change of sign $g\to -g$ we obtain the B\"acklund transformation for \ref{P20}. One can prove that iterations of this transformation in terms of $p$ or $q$ are governed by the non-Abelian Toda lattice. Vice versa, it is possible to derive \ref{P20}, starting from the Toda lattice and imposing the corresponding reduction (see \cite{Retakh_Rubtsov_2010}).
\end{remark}

2) Now let $c\in\Mat_n$ and $b$ be a scalar. In this case, according to \eqref{bbb}, we can set $b=0$. Now we apply another substitution
\begin{equation}\label{NLS.sub2}
 f=p^{-1}p',\qquad g=q'q^{-1},\qquad h=2qp
\end{equation}
(alternatively, we can use (\ref{NLS.sub1}) as before, but replace (\ref{NLS.int}) with another first integral $p'q-pq'=\tilde c$ which appears for $b=0$). As a result, the system (\ref{NLS.red}) with $b=0$ turns into
\[
 f'=-f^2-\frac{1}{2}z-h,\qquad g'=-g^2-\frac{1}{2}z-h,\qquad h'=hf+gh,
\]
and the first integral (\ref{NLS.int}) takes the form
\[
 hf-gh=2c.
\]
By eliminating $g$, we obtain the system
\begin{equation}\label{fh1}
 f'=-f^2-\frac{1}{2}z-h,\qquad h'=2hf-2c.
\end{equation}
In this case, $f$ satisfies equation \ref{P21} with $\kappa=-1$ and $a=2c-\frac{1}{2}$.

An isomonodromic Lax pair for the system (\ref{NLS.red}) can be easily obtained by extending the substitutions (\ref{NLS.Gal}) with the change $\zeta=\lambda-\tfrac{1}{4}t$ and applying this to the well-known representation $U_t=V_x+[V,U]$ of the system (\ref{NLS}), with the matrices
\[
 U=\begin{pmatrix}
 \lambda & -v \\
  u & -\lambda
 \end{pmatrix},\quad
 V=-2\lambda U+\begin{pmatrix}
  -vu & v_x \\
  u_x & uv
 \end{pmatrix}.
\]
After some simple algebraic manipulations, this leads to the representation \eqref{AB} for equation (\ref{NLS.red}) with the matrices
\[
 B=\begin{pmatrix}
  \zeta & -q\\ p & -\zeta
 \end{pmatrix},\qquad
 A=\begin{pmatrix}
  8\zeta^2+4qp+z & -8\zeta q-4q'\\[1mm]
  8\zeta p-4p' & -8\zeta^2-4pq-z-4b
 \end{pmatrix}.
\]
From here, it is possible to obtain also the Lax pairs for the above systems in variables $f,h$, by applying gauge transformations. For the system (\ref{fh0}) we have
\[
B=\begin{pmatrix}
   \zeta+f & -\tfrac{1}{2}h\\ 1 & -\zeta
 \end{pmatrix},\qquad
  A=\begin{pmatrix}
   8\zeta^2+2h+z & -4\zeta h-2hf+2\gamma\\[1mm]
   8\zeta -4f & -8\zeta^2-2h-z-4b
  \end{pmatrix}.
 \]
Note that equation \ref{P20} admits also another representation (\ref{AB}) with
\[
 B=\begin{pmatrix}
  \zeta & y\\
  y & -\zeta
 \end{pmatrix},
 \qquad
 A=-4\zeta B+\begin{pmatrix}
  2y^2+z+2b & -2y'-\frac{a}{\zeta}\\[1mm]
  2y'-\frac{a}{\zeta} & -2y^2-z-2b
 \end{pmatrix},
\]
which is equivalent to representation from \cite{Irfan_2012}. We were not able to establish a gauge equivalence between these two Lax pairs.

In the case of the system (\ref{fh1}) we obtain \eqref{AB} with
\[
  B=\begin{pmatrix}
   \zeta & -\tfrac{1}{2}h\\ 1 & -\zeta-f
  \end{pmatrix},\qquad
  A=\begin{pmatrix}
   8\zeta^2+2h+z & -4\zeta h-2hf+4c\\[1mm]
   8\zeta -4f & -8\zeta^2-2h-z
  \end{pmatrix}.
\]
This representation for equation \ref{P21} is equivalent to the Lax pair obtained in the next section starting from the mKdV equation.

\subsection{Matrix mKdV equations}

The scalar P$_2$ equation can be obtained also from the mKdV equation
\[
 u_t=u_{xxx}-6u^2u_x
\]
by the self-similar reduction related with the scaling group. This brings to equation $y'''=6y^2y'+y+zy'$ which reduces to P$_2$ by integration. As usual, the zero curvature representation for mKdV turns into an isomonodromic Lax pair. We describe this procedure in the matrix setting.

The first matrix analog of the mKdV \cite{Marchenko_1986} reads
\begin{equation}
\label{mKdV1}
 u_t= u_{xxx}-3u^2u_x-3u_xu^2
 \end{equation}
and admits the representation $U_t=V_x+[V,U]$ with
\[
 U=\begin{pmatrix}
  \lambda & u\\
  u & -\lambda
 \end{pmatrix},\qquad
 V=4\lambda^2U +
 \begin{pmatrix}
  -2\lambda u^2+[u,u_x] & 2\lambda u_x+u_{xx}-2u^3\\[1mm]
  -2\lambda u_x+u_{xx}-2u^3 & 2\lambda u^2+[u,u_x]
 \end{pmatrix}.
\]
In the matrix case, we combine the scaling group with the group of conjugations and apply the self-similar Ansatz
\begin{equation}\label{mKdV.self}
 u=\varepsilon\tau e^{\log(\tau)d}y(z)e^{-\log(\tau)d},\qquad
 \tau=t^{-1/3},\qquad z=\varepsilon\tau x,\qquad 3\varepsilon^3=-1,\qquad d\in\Mat_n,
\end{equation}
which turns equation (\ref{mKdV1}) into
\begin{equation}\label{mKdV1.red}
 y'''=3y^2y'+3y'y^2+y+zy'+[d,y].
\end{equation}
In contrast to the scalar case, this equation does not have the first integral even for $d=0$. Nevertheless, the reduction of order is still possible due to a {\em partial} first integral. It turns out that (\ref{mKdV1.red}) admits special solutions described by a second-order equation. This can be easily proved by direct elimination of the third and second derivatives in virtue of equation (\ref{gP2}) of the general form. The coefficients are determined by equating the remaining terms and we obtain that (\ref{mKdV1.red}) is satisfied by the equation \ref{P21} with $a=-\kappa d$. To obtain the corresponding Lax pair, we supplement the substitution (\ref{mKdV.self}) with the change $\lambda=\varepsilon\tau\zeta$. Then the dependence of $U$ and $V$ on $\tau$ is separated out:
\[
 U=\varepsilon\tau e^{\log(\tau)d}Be^{-\log(\tau)d},\qquad V=(\varepsilon\tau)^3e^{\log(\tau)d}Ke^{-\log(\tau)d},
\]
where $B$ and $K$ depend on $\zeta$, $z$, $y$ and $y'$ (the second derivative in $K$ is replaced according to \ref{P21}).  For the derivations, this change yields
\[
 \partial_t=(\varepsilon\tau)^3(\tau\partial_\tau+z\partial_z-\zeta\partial_\zeta),\qquad
 \partial_x=\varepsilon\tau\partial_z.
\]
As a result, the equation for $U$ and $V$ takes the form
\[
 -\zeta B_\zeta+zB'+B+[dI,B]=K'+[K,B]
\]
and further change $A=-\zeta^{-1}(K-zB-dI)$ brings to the standard form of the Lax pair \eqref{AB}. The calculations by this scheme give, for equation \ref{P21},
\[
 B=\begin{pmatrix}
  \zeta & y\\
  y & -\zeta
 \end{pmatrix},\qquad
 A= -4\zeta B +\begin{pmatrix}
  2y^2+z & -2y'\\
  2y' & -2y^2-z
 \end{pmatrix}
 -\zeta^{-1}(\kappa[y,y']+a)
 \begin{pmatrix} \kappa & 1\\ 1 &\kappa \end{pmatrix}.
\]

The second matrix analog of mKdV was introduced in \cite{Khalilov_Khruslov_1990}. It turns out that, in contrast to (\ref{mKdV1}), an arbitrary matrix constant can be introduced directly into this equation:
\begin{equation}\label{mKdV2}
 u_t= u_{xxx}+3[u,u_{xx}]-6uu_xu-3(u_x+u^2)c-3c(u_x-u^2),\qquad c\in\Mat_n.
\end{equation}
The origin of this constant is related with the Miura map for the matrix KdV equation which is constructed by solution of the linear Schr\"odinger equation
$$
\psi''+ v \psi + \psi c = 0, \qquad c\in\Mat_n.
$$
The zero curvature representation for (\ref{mKdV2}) is given by the matrices
\begin{gather*}
U=\begin{pmatrix}
  0 & 1\\
  c-\lambda & -2u
 \end{pmatrix},\\[1.5mm]
V=2\begin{pmatrix}
  2u(c-\lambda) &
  -u_x-u^2-c-2\lambda\\[1mm]
  (u_x-u^2-c-2\lambda)(c-\lambda) &
  2\lambda u-u_{xx}-[u,u_x]+2u^3+2cu+2uc
 \end{pmatrix}.
\end{gather*}
At first glance, introducing the terms with $c$ into (\ref{mKdV2}) makes the self-similar substitution (\ref{mKdV.self}) impossible, because the homogeneity of the equation is violated. However, we can complement this substitution as follows:
\[
 c=(\varepsilon\tau)^2e^{\log(\tau)d}c_0e^{-\log(\tau)d},
\]
where $c_0$ is an arbitrary constant matrix such that $2c_0+[d,c_0]=0$. Indeed, the differentiation with respect to $\tau$ shows that this relation implies that the matrix $c$ is constant as well (notice, that for scalars this gives $c=c_0=0$, that is, this trick is only possible in the non-abelian case). As the result, (\ref{mKdV2}) is reduced to the equation
\begin{equation}\label{mKdV2.red}
 y'''=3[y'',y]+6yy'y+y+zy'+3(y'+y^2)c_0+3c_0(y'-y^2)+[d,y].
\end{equation}
Like in the case of equation (\ref{mKdV1.red}), its order can be reduced by a partial first integral. By eliminating $y'''$ and $y''$ in virtue of equation of the form (\ref{gP2}) we find that if $\kappa=-2$, $3c_0=b$ and $d=-a$ then (\ref{mKdV2.red}) is a consequence of equation \ref{P22}. Further manipulations with the matrices $U$ and $V$ differ from the previous example only by insignificant details (one has to set $\lambda=(\varepsilon\tau)^2\zeta$ and to apply the conjugation by a suitable constant matrix), and this leads to the Lax representation \eqref{AB} with
\begin{gather*}
 B=\begin{pmatrix}
  0 & 1\\
  \frac{1}{3}b-\zeta & -2y
 \end{pmatrix},\\[1.5mm]
 A= \frac{1}{\zeta}\begin{pmatrix}
  2\zeta y-\frac{2}{3}yb-\frac{1}{2}(a+1) & 2\zeta+y'+y^2+\frac{1}{3}b+\frac{z}{2}\\[1mm]
  (2\zeta-y'+y^2+\frac{1}{3}b+\frac{z}{2})(\frac{1}{3}b-\zeta) &
  -2\zeta y -[y,y']+\frac{1}{3}by+\frac{1}{3}yb+\frac{1}{2}a
 \end{pmatrix},
\end{gather*}
for equation \ref{P22} with $\kappa=-2$.

\section{Conclusion}

We have demonstrated that there are at least three matrix generalizations of the second Painlev\'e equation that satisfy the Painlev\'e--Kovalevskaya test and admit isomonodromic Lax pairs. Of course, a similar diversity should be expected for other Painlev\'e equations, too. Although the literature on their non-Abelian generalizations is quite rich, the question on the number of different versions remains open, in particular, the question of how the non-Abelian constants can enter into the equations.

As one of examples, we present the following matrix version of the P$_4$ equation
\[
 y''= \frac{y'^2}{2\, y}  +\frac{3}{2}y^3+4zy^2 +2(z^2-\alpha)y+\frac{\beta}{y},
\]
which contains the matrix constant $c\in\Mat_n$ and the scalar constant $\alpha\in{\mathbb C}$:
\[
 y''= \frac{1}{2}(y'+2c)y^{-1}(y'-2c) +\frac{1}{2}[y,y'] +\frac{3}{2}y^3+4zy^2 +2(z^2-\alpha)y+cy+yc.
\]
This equation can be obtained by eliminating $h$ from the equivalent system
\[
 -y'=y^2+2hy+2zy+2c,\qquad h'=h^2+yh+hy+2zh+\alpha-1
\]
which admits the representation \eqref{AB} with the matrices
\[
 B=\begin{pmatrix}
  -2\zeta^2 & 2\zeta \\
  \zeta h & y-h
 \end{pmatrix},\qquad
 A=\begin{pmatrix}
  4\zeta^3-2\zeta(h+2z)+2\zeta^{-1}c & -4\zeta^2+2y+2h+4z \\
  -2\zeta^2h-hy-\alpha+1 & 2\zeta h+\alpha\zeta^{-1}
 \end{pmatrix}.
\]
This example, as well as a set of other versions of P$_4$, can be obtained by self-similar reduction from equations of nonlinear Schr\"odinger type (in this case we have used equation S$'_5(0,1)$ from \cite{Adler_Sokolov_2020}). For such equations, the reduction procedure brings to systems of two second order equations and  further reduction of order is usually more complicated than in the scalar setting.

\subsubsection*{Acknowledgements}

The authors are grateful to R.Conte and V.Rubtsov for useful discussions. This work was carried out under the State Assignment 0029-2021-0004 (Quantum field theory) of the Ministry of Science and Higher Education of the Russian Federation.

redpajama_set_name: RedPajamaArXiv


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Children's Spine Surgery - What You Need to Know!
Children's spine surgery is used to correct the problems with the spinal cord that exist from time of birth in children and adolescents. The problems in the spine are known as the "congenital anomalies: and include scoliosis (curving of the spine), spondylolysis (stress fracture of the spine), kyphosis (increasing round back of the spine) and spondylolisthesis (movement of one part of the spine to another). These spinal disorders along with other multiple syndromes and conditions are linked with the spinal deformity.
In this article, we'll discuss about the types of children's spine surgery, causes, symptoms, diagnosis and treatments.
There are a huge number of medical treatments and surgeries addressing the spine disorders that offer the next generation care to the infants, children and adolescents. The children's spine surgery is designed to cure the pediatric spine problems and the deformities. This will ensure that the children are well-treated for their spine condition.
The spine curves very quickly during the skeletal growth in the early onset scoliosis in children. If the fusion of the spine is performed at a very young age, then the children's lung will continue its growth. This will increase the likelihood for a deformity in front of the chest called "crank-shaft deformity" to continue its growth and rotate while the back is still fused. In such a case, a limit fusion with instrumentation is placed along the spine that will direct the growth of the spine.
Spondylosis refers to the fracture in the spine, which is usually seen among the high level athletes. This condition may progress to the spondylolisthesis where a vertebrae begins to slip forward on the other.
This condition is also known as a hunchback, in which the spine in the upper back has an excessive curvature. The upper back is thought to have a slight natural curve, if you have this condition, then you'll have a visible hump on the upper back. This kind of the spinal disorder will lead to the excessive pressure on the spine and will cause you pain.
It refers to an abnormal curvature of the spine that is usually seen in late childhood or the adolescence. The spine expands a side-to-side curvature along in elongated the "S" or the "C" shape, instead of growing straight.
This condition usually occurs during the adolescent growth spurt that is a time when the children are growing rapidly. In most cases, the abnormal spinal curve is stable while in some children the curve is progressive. For some unknown reasons, the severe and progressive curves are frequently seen in girls than the boys.
Before surgery, the child will be given general anesthesia so that he/she becomes unconscious and do not feel pain during the surgery. The surgeon will make at least one cut (either posterior or anterior) to get to the child's spine. After making the cut, the surgeon will move the muscle aside so that the joints between the different vertebrae can be removed. Often bone grafts are used as a replacement along with the metal instruments like hooks, rods, wires or screws to hold the spine together. The surgeon will observe the nerves that come from the spine using the special equipments to ensure no damage occurs to any of the nerve roots. Usually, this surgery will take about 4-6 hours and it may be longer or shorter for some children.
Surgery can dramatically improve the rigid or progressive kyphosis greater than 80 degrees. It is recommended to discuss about the procedure with your child's doctor.
Spinal Fusion: This procedure commonly includes the placement of rods, hooks and screws in the spine so as to prevent the worsening of the spinal deformity and improve the appearance. India pediatric orthopedic surgeons have a great deal of experience in treating all the types of spinal conditions in children, including the mild and most severe ones.
Growing Rod: The growing rods do not grow actually, but they can be lengthened with the minor surgery that gets repeated every 6 to 8 months. The goal of this procedure is to allow the continuous growth while correcting the curve. Every six months, the child returns to the doctor to lengthen the rod, which will help the child's growth. When the child achieves full growth, then the rods are replaced and the spinal fusion is performed.
Vertical Expandable Prosthetic Titanium Rib (VEPTR): This procedure is used for managing severe scoliosis in young patients. This technique will support the continuous growth of the spine in younger children, thereby allows the normal development and functioning.
This surgery will stabilize the levels of the spine that are slipping by placing the bone graft to fuse the spine bone together. Small crews or metal rods will be used for older children and adolescents with significant slippage. After this fusion surgery, it will take about 4-8 months for the bones to solidly mend together.
Children's spinal surgery: Is it effective?
The children's spinal surgery is effective as it is performed by the best pediatric spine surgeon having vast experience in treating the pediatric spinal surgery patients. The spine surgeons typically treat children from the newborn stages through the teen years and treat children in a unique way by making them feel cooperative and comfortable.
Recovery after the children's spinal surgery is often faster. Your child will require to stay in hospital for about 3-4 days post surgery. A tube may be inserted into the chest of your child to drain the fluid build-up and it will be removed after 24-72 hours. Your child may be given pain medication.
The repaired spine should be kept in proper position to keep it aligned. Doctor will give you instructions on how to care for your child at home. Your child's spine will look much straighter after surgery. Children having this surgery will possibly gain height from having a straighter spine.
In India you will get about 60-85% savings for children's spinal surgery as compared to the rates in the US and the UK. Our highly skilled pediatric spine surgeons who have access the latest technological advancement in children's spine treatments and surgery. We understand your anxiety and concern and works with the child and family to provide complete care throughout your medical treatment.
If you are really seeking for the Children's Spine Surgery, kindly fill up the form for a free consultation with our surgeons. You will be provided with thorough analysis and suggestions regarding the Children's Spine Surgery you are seeking for.


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redpajama_set_name: RedPajamaCommonCrawl


text:
package com.drillster.api2.general;

public class Coordinates {

	private double latitude;
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	public Coordinates setLatitude(double latitude) { this.latitude = latitude; return this; }
	public Coordinates setLongitude(double longitude) { this.longitude = longitude; return this; }

	public double getLatitude() { return this.latitude; }
	public double getLongitude() { return this.longitude; }

}

redpajama_set_name: RedPajamaGithub


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PPG's Power Generation team also provides preventative maintenance for the critical information hub. By proactively managing the maintenance needs of Infomart's backup power system, Pacific Power Group is able to help Infomart increase its effectiveness and their peace of mind in having power at all times.
In 2015 our expert sales consultant team worked with a Hillsboro, OR data center to supply a complete backup power generation solution. Seven 2250 kWe and two 2000 kWe MTU Onsite Energy generators were supplied, installed and commissioned by our team, along with enclosures and fuel tanks. With its new power system in place the data center has peace of mind knowing that if a power outage or other emergency strikes, its customers are protected with 10 second or less load acceptance, ISO 8528 rated load factor and long service intervals.


text:
http://www.beecrowd.com.br/judge/problems/view/1020

# Age in Days

Read an integer value corresponding to a person's age
in days and print it in years, months and days,
followed by its respective message "ano(s)", "mes(es)", "dia(s)".

Note: only to facilitate the calculation, consider
the whole year with 365 days and 30 days every month.

## Input

The input file contain 1 integer number.

## Output

Print the output like the following example.


text:
Cumminsiella wootoniana är en svampart som först beskrevs av Joseph Charles Arthur, och fick sitt nu gällande namn av Joseph Charles Arthur 1933. Cumminsiella wootoniana ingår i släktet Cumminsiella och familjen Pucciniaceae.   Inga underarter finns listade i Catalogue of Life.

Källor

Basidiesvampar
wootoniana

redpajama_set_name: RedPajamaWikipedia


text:
package org.broadinstitute.hellbender.utils.dragstr;

import org.testng.Assert;
import org.testng.annotations.DataProvider;
import org.testng.annotations.Test;

import java.util.Arrays;
import java.util.Iterator;
import java.util.stream.IntStream;

/**
 * Unit tests for {@link DoubleSequence}
 */
public class DoubleSequenceTest {

    @Test(dataProvider = "correctSequenceData")
    public void testCorrectSequence(final String spec, final double[] expectedValues, final double epsilon) {
        final DoubleSequence subject = new DoubleSequence(spec);
        Assert.assertEquals(subject.size(), expectedValues.length);
        // check single value access:
        for (int i = 0; i < expectedValues.length; i++) {
            Assert.assertEquals(subject.get(i), expectedValues[i], epsilon);
        }
        // check array access:
        final double[] actualValues = subject.toDoubleArray();
        Assert.assertNotNull(actualValues);
        Assert.assertEquals(actualValues, expectedValues, epsilon);
    }

    @Test(dataProvider = "badSpecsData", expectedExceptions = RuntimeException.class)
    public void testBadSpecs(final String spec) {
        new DoubleSequence(spec);
    }

    @DataProvider(name = "badSpecsData")
    public Iterator<Object[]> badSpecsData() {
        return Arrays.stream(new String[] {null, "", "::", "1:2", "2:2:2:2:2", "2,2", "4;5;100", "10:-1:100"})
                .map(s -> new Object[] { s }).iterator();
    }

    @DataProvider(name = "correctSequenceData")
    public Iterator<Object[]> correctSequenceData() {
        final String STD_EPSILON = "0.001";
        final String[] testCases = {
                "0:1:10",      "0,1,2,3,4,5,6,7,8,9,10", STD_EPSILON,
                "-1:2.5:5",    "-1,1.5,4", STD_EPSILON,
                "1:1e-1:2.3",  "1,1.1,1.2,1.3,1.4,1.5,1.6,1.7,1.8,1.9,2.0,2.1,2.2,2.3", STD_EPSILON,
                "5:-1:1",      "5,4,3,2,1", STD_EPSILON,
                "1:0.00001:1", "1", STD_EPSILON,
                "5:-123:5",    "5", STD_EPSILON,
                "8:1:10.00001", "8,9,10.00001", "0.0000000001",
        };
        return IntStream.range(0, testCases.length / 3)
                 .map(i -> 3 * i)
                 .mapToObj(i -> {
                     final String spec = testCases[i++];
                     final String strValues = testCases[i++];
                     final double epsilon = Double.parseDouble(testCases[i]);
                     final double[] dblValues = Arrays.stream(strValues.split("\\s*,\\s*")).mapToDouble(Double::parseDouble).toArray();
                     return new Object[]{spec, dblValues, epsilon};
                 }).iterator();
    }
}


text:
from collections import namedtuple
import contextlib
import logging
import sys
import threading
import time

import six

# Django
from django.conf import settings, UserSettingsHolder
from django.core.cache import cache as django_cache
from django.core.exceptions import ImproperlyConfigured
from django.db import ProgrammingError, OperationalError

# Django REST Framework
from rest_framework.fields import empty, SkipField

# Tower
from awx.main.utils import encrypt_field, decrypt_field
from awx.main.utils.db import get_tower_migration_version
from awx.conf import settings_registry
from awx.conf.models import Setting
from awx.conf.migrations._reencrypt import decrypt_field as old_decrypt_field

# FIXME: Gracefully handle when settings are accessed before the database is
# ready (or during migrations).

logger = logging.getLogger('awx.conf.settings')

# Store a special value to indicate when a setting is not set in the database.
SETTING_CACHE_NOTSET = '___notset___'

# Cannot store None in memcached; use a special value instead to indicate None.
# If the special value for None is the same as the "not set" value, then a value
# of None will be equivalent to the setting not being set (and will raise an
# AttributeError if there is no other default defined).
# SETTING_CACHE_NONE = '___none___'
SETTING_CACHE_NONE = SETTING_CACHE_NOTSET

# Cannot store empty list/tuple in memcached; use a special value instead to
# indicate an empty list.
SETTING_CACHE_EMPTY_LIST = '___[]___'

# Cannot store empty dict in memcached; use a special value instead to indicate
# an empty dict.
SETTING_CACHE_EMPTY_DICT = '___{}___'

# Expire settings from cache after this many seconds.
SETTING_CACHE_TIMEOUT = 60

# Flag indicating whether to store field default values in the cache.
SETTING_CACHE_DEFAULTS = True

__all__ = ['SettingsWrapper', 'get_settings_to_cache', 'SETTING_CACHE_NOTSET']

@contextlib.contextmanager
def _log_database_error():
    try:
        yield
    except (ProgrammingError, OperationalError) as e:
        if get_tower_migration_version() < '310':
            logger.info('Using default settings until version 3.1 migration.')
        else:
            logger.warning('Database settings are not available, using defaults (%s)', e, exc_info=True)
    finally:
        pass

def filter_sensitive(registry, key, value):
    if registry.is_setting_encrypted(key):
        return '$encrypted$'
    return value

class EncryptedCacheProxy(object):

    def __init__(self, cache, registry, encrypter=None, decrypter=None):
        """
        This proxy wraps a Django cache backend and overwrites the
        `get`/`set`/`set_many` methods to handle field encryption/decryption
        for sensitive values.

        :param cache: the Django cache backend to proxy to
        :param registry: the settings registry instance used to determine if
                         a field is encrypted or not.
        :param encrypter: a callable used to encrypt field values; defaults to
                          ``awx.main.utils.encrypt_field``
        :param decrypter: a callable used to decrypt field values; defaults to
                          ``awx.main.utils.decrypt_field``
        """

        # These values have to be stored via self.__dict__ in this way to get
        # around the magic __setattr__ method on this class.
        self.__dict__['cache'] = cache
        self.__dict__['registry'] = registry
        self.__dict__['encrypter'] = encrypter or encrypt_field
        self.__dict__['decrypter'] = decrypter or decrypt_field

    def get(self, key, **kwargs):
        value = self.cache.get(key, **kwargs)
        value = self._handle_encryption(self.decrypter, key, value)

        # python-memcached auto-encodes unicode on cache set in python2
        # https://github.com/linsomniac/python-memcached/issues/79
        # https://github.com/linsomniac/python-memcached/blob/288c159720eebcdf667727a859ef341f1e908308/memcache.py#L961
        if six.PY2 and isinstance(value, six.binary_type):
            try:
                six.text_type(value)
            except UnicodeDecodeError:
                value = value.decode('utf-8')
        logger.debug('cache get(%r, %r) -> %r', key, empty, filter_sensitive(self.registry, key, value))
        return value

    def set(self, key, value, log=True, **kwargs):
        if log is True:
            logger.debug('cache set(%r, %r, %r)', key, filter_sensitive(self.registry, key, value),
                         SETTING_CACHE_TIMEOUT)
        self.cache.set(
            key,
            self._handle_encryption(self.encrypter, key, value),
            **kwargs
        )

    def set_many(self, data, **kwargs):
        filtered_data = dict(
            (key, filter_sensitive(self.registry, key, value))
            for key, value in data.items()
        )
        logger.debug('cache set_many(%r, %r)', filtered_data, SETTING_CACHE_TIMEOUT)
        for key, value in data.items():
            self.set(key, value, log=False, **kwargs)

    def _handle_encryption(self, method, key, value):
        TransientSetting = namedtuple('TransientSetting', ['pk', 'value'])

        if value is not empty and self.registry.is_setting_encrypted(key):
            # If the setting exists in the database, we'll use its primary key
            # as part of the AES key when encrypting/decrypting
            obj_id = self.cache.get(Setting.get_cache_id_key(key), default=empty)
            if obj_id is empty:
                logger.info('Efficiency notice: Corresponding id not stored in cache %s',
                            Setting.get_cache_id_key(key))
                obj_id = getattr(self._get_setting_from_db(key), 'pk', None)
            elif obj_id == SETTING_CACHE_NONE:
                obj_id = None
            return method(
                TransientSetting(
                    pk=obj_id,
                    value=value
                ),
                'value'
            )

        # If the field in question isn't an "encrypted" field, this function is
        # a no-op; it just returns the provided value
        return value

    def _get_setting_from_db(self, key):
        field = self.registry.get_setting_field(key)
        if not field.read_only:
            return Setting.objects.filter(key=key, user__isnull=True).order_by('pk').first()

    def __getattr__(self, name):
        return getattr(self.cache, name)

    def __setattr__(self, name, value):
        setattr(self.cache, name, value)

def get_writeable_settings(registry):
    return registry.get_registered_settings(read_only=False)

def get_settings_to_cache(registry):
    return dict([(key, SETTING_CACHE_NOTSET) for key in get_writeable_settings(registry)])

def get_cache_value(value):
    '''Returns the proper special cache setting for a value
    based on instance type.
    '''
    if value is None:
        value = SETTING_CACHE_NONE
    elif isinstance(value, (list, tuple)) and len(value) == 0:
        value = SETTING_CACHE_EMPTY_LIST
    elif isinstance(value, (dict,)) and len(value) == 0:
        value = SETTING_CACHE_EMPTY_DICT
    return value

class SettingsWrapper(UserSettingsHolder):

    @classmethod
    def initialize(cls, cache=None, registry=None):
        """
        Used to initialize and wrap the Django settings context.

        :param cache: the Django cache backend to use for caching setting
        values.  ``django.core.cache`` is used by default.
        :param registry: the settings registry instance used.  The global
        ``awx.conf.settings_registry`` is used by default.
        """
        if not getattr(settings, '_awx_conf_settings', False):
            settings_wrapper = cls(
                settings._wrapped,
                cache=cache or django_cache,
                registry=registry or settings_registry
            )
            settings._wrapped = settings_wrapper

    def __init__(self, default_settings, cache, registry):
        """
        This constructor is generally not called directly, but by
        ``SettingsWrapper.initialize`` at app startup time when settings are
        parsed.
        """

        # These values have to be stored via self.__dict__ in this way to get
        # around the magic __setattr__ method on this class (which is used to
        # store API-assigned settings in the database).
        self.__dict__['default_settings'] = default_settings
        self.__dict__['_awx_conf_settings'] = self
        self.__dict__['_awx_conf_preload_expires'] = None
        self.__dict__['_awx_conf_preload_lock'] = threading.RLock()
        self.__dict__['_awx_conf_init_readonly'] = False
        self.__dict__['cache'] = EncryptedCacheProxy(cache, registry)
        self.__dict__['registry'] = registry

    def _get_supported_settings(self):
        return self.registry.get_registered_settings()

    def _preload_cache(self):
        # Ensure we're only modifying local preload timeout from one thread.
        with self._awx_conf_preload_lock:
            # If local preload timeout has not expired, skip preloading.
            if self._awx_conf_preload_expires and self._awx_conf_preload_expires > time.time():
                return
            # Otherwise update local preload timeout.
            self.__dict__['_awx_conf_preload_expires'] = time.time() + SETTING_CACHE_TIMEOUT
            # Check for any settings that have been defined in Python files and
            # make those read-only to avoid overriding in the database.
            if not self._awx_conf_init_readonly and 'migrate_to_database_settings' not in sys.argv:
                defaults_snapshot = self._get_default('DEFAULTS_SNAPSHOT')
                for key in get_writeable_settings(self.registry):
                    init_default = defaults_snapshot.get(key, None)
                    try:
                        file_default = self._get_default(key)
                    except AttributeError:
                        file_default = None
                    if file_default != init_default and file_default is not None:
                        logger.debug('Setting %s has been marked read-only!', key)
                        self.registry._registry[key]['read_only'] = True
                        self.registry._registry[key]['defined_in_file'] = True
                    self.__dict__['_awx_conf_init_readonly'] = True
        # If local preload timer has expired, check to see if another process
        # has already preloaded the cache and skip preloading if so.
        if self.cache.get('_awx_conf_preload_expires', default=empty) is not empty:
            return
        # Initialize all database-configurable settings with a marker value so
        # to indicate from the cache that the setting is not configured without
        # a database lookup.
        settings_to_cache = get_settings_to_cache(self.registry)
        setting_ids = {}
        # Load all settings defined in the database.
        for setting in Setting.objects.filter(key__in=settings_to_cache.keys(), user__isnull=True).order_by('pk'):
            if settings_to_cache[setting.key] != SETTING_CACHE_NOTSET:
                continue
            if self.registry.is_setting_encrypted(setting.key):
                setting_ids[setting.key] = setting.id
                try:
                    value = decrypt_field(setting, 'value')
                except ValueError, e:
                    #TODO: Remove in Tower 3.3
                    logger.debug('encountered error decrypting field: %s - attempting fallback to old', e)
                    value = old_decrypt_field(setting, 'value')

            else:
                value = setting.value
            settings_to_cache[setting.key] = get_cache_value(value)
        # Load field default value for any settings not found in the database.
        if SETTING_CACHE_DEFAULTS:
            for key, value in settings_to_cache.items():
                if value != SETTING_CACHE_NOTSET:
                    continue
                field = self.registry.get_setting_field(key)
                try:
                    settings_to_cache[key] = get_cache_value(field.get_default())
                    if self.registry.is_setting_encrypted(key):
                        # No database pk, so None will be passed to encryption algorithm
                        setting_ids[key] = SETTING_CACHE_NOTSET
                except SkipField:
                    pass
        # Generate a cache key for each setting and store them all at once.
        settings_to_cache = dict([(Setting.get_cache_key(k), v) for k, v in settings_to_cache.items()])
        for k, id_val in setting_ids.items():
            logger.debug('Saving id in cache for encrypted setting %s, %s',
                         Setting.get_cache_id_key(k), id_val)
            self.cache.cache.set(Setting.get_cache_id_key(k), id_val)
        settings_to_cache['_awx_conf_preload_expires'] = self._awx_conf_preload_expires
        self.cache.set_many(settings_to_cache, timeout=SETTING_CACHE_TIMEOUT)

    def _get_local(self, name):
        self._preload_cache()
        cache_key = Setting.get_cache_key(name)
        try:
            cache_value = self.cache.get(cache_key, default=empty)
        except ValueError:
            cache_value = empty
        if cache_value == SETTING_CACHE_NOTSET:
            value = empty
        elif cache_value == SETTING_CACHE_NONE:
            value = None
        elif cache_value == SETTING_CACHE_EMPTY_LIST:
            value = []
        elif cache_value == SETTING_CACHE_EMPTY_DICT:
            value = {}
        else:
            value = cache_value
        field = self.registry.get_setting_field(name)
        if value is empty:
            setting = None
            setting_id = None
            if not field.read_only or name in (
                # these two values are read-only - however - we *do* want
                # to fetch their value from the database
                'AWX_ISOLATED_PRIVATE_KEY',
                'AWX_ISOLATED_PUBLIC_KEY',
            ):
                setting = Setting.objects.filter(key=name, user__isnull=True).order_by('pk').first()
            if setting:
                if getattr(field, 'encrypted', False):
                    value = decrypt_field(setting, 'value')
                    setting_id = setting.id
                else:
                    value = setting.value
            else:
                value = SETTING_CACHE_NOTSET
                if SETTING_CACHE_DEFAULTS:
                    try:
                        value = field.get_default()
                        if getattr(field, 'encrypted', False):
                            setting_id = SETTING_CACHE_NONE
                    except SkipField:
                        pass
            # If None implies not set, convert when reading the value.
            if value is None and SETTING_CACHE_NOTSET == SETTING_CACHE_NONE:
                value = SETTING_CACHE_NOTSET
            if cache_value != value:
                if setting_id:
                    logger.debug('Saving id in cache for encrypted setting %s', cache_key)
                    self.cache.cache.set(Setting.get_cache_id_key(cache_key), setting_id)
                self.cache.set(cache_key, get_cache_value(value), timeout=SETTING_CACHE_TIMEOUT)
        if value == SETTING_CACHE_NOTSET and not SETTING_CACHE_DEFAULTS:
            try:
                value = field.get_default()
            except SkipField:
                pass
        if value not in (empty, SETTING_CACHE_NOTSET):
            try:
                if field.read_only:
                    internal_value = field.to_internal_value(value)
                    field.run_validators(internal_value)
                    return internal_value
                else:
                    return field.run_validation(value)
            except:
                logger.warning(
                    'The current value "%r" for setting "%s" is invalid.',
                    value, name, exc_info=True)
        return empty

    def _get_default(self, name):
        return getattr(self.default_settings, name)

    @property
    def SETTINGS_MODULE(self):
        return self._get_default('SETTINGS_MODULE')

    def __getattr__(self, name):
        value = empty
        if name in self._get_supported_settings():
            with _log_database_error():
                value = self._get_local(name)
        if value is not empty:
            return value
        return self._get_default(name)

    def _set_local(self, name, value):
        field = self.registry.get_setting_field(name)
        if field.read_only:
            logger.warning('Attempt to set read only setting "%s".', name)
            raise ImproperlyConfigured('Setting "%s" is read only.'.format(name))

        try:
            data = field.to_representation(value)
            setting_value = field.run_validation(data)
            db_value = field.to_representation(setting_value)
        except Exception as e:
            logger.exception('Unable to assign value "%r" to setting "%s".',
                             value, name, exc_info=True)
            raise e

        setting = Setting.objects.filter(key=name, user__isnull=True).order_by('pk').first()
        if not setting:
            setting = Setting.objects.create(key=name, user=None, value=db_value)
            # post_save handler will delete from cache when added.
        elif setting.value != db_value or type(setting.value) != type(db_value):
            setting.value = db_value
            setting.save(update_fields=['value'])
            # post_save handler will delete from cache when changed.

    def __setattr__(self, name, value):
        if name in self._get_supported_settings():
            with _log_database_error():
                self._set_local(name, value)
        else:
            setattr(self.default_settings, name, value)

    def _del_local(self, name):
        field = self.registry.get_setting_field(name)
        if field.read_only:
            logger.warning('Attempt to delete read only setting "%s".', name)
            raise ImproperlyConfigured('Setting "%s" is read only.'.format(name))
        for setting in Setting.objects.filter(key=name, user__isnull=True):
            setting.delete()
            # pre_delete handler will delete from cache.

    def __delattr__(self, name):
        if name in self._get_supported_settings():
            with _log_database_error():
                self._del_local(name)
        else:
            delattr(self.default_settings, name)

    def __dir__(self):
        keys = []
        with _log_database_error():
            for setting in Setting.objects.filter(
                    key__in=self._get_supported_settings(), user__isnull=True):
                # Skip returning settings that have been overridden but are
                # considered to be "not set".
                if setting.value is None and SETTING_CACHE_NOTSET == SETTING_CACHE_NONE:
                    continue
                if setting.key not in keys:
                    keys.append(str(setting.key))
        for key in dir(self.default_settings):
            if key not in keys:
                keys.append(key)
        return keys

    def is_overridden(self, setting):
        set_locally = False
        if setting in self._get_supported_settings():
            with _log_database_error():
                set_locally = Setting.objects.filter(key=setting, user__isnull=True).exists()
        set_on_default = getattr(self.default_settings, 'is_overridden', lambda s: False)(setting)
        return (set_locally or set_on_default)


text:
2019 Pole Vault Camp Registration
2019 Pole Vault Camp Medical Release Form
Matthews to Compete at U.S. Olympic Trials
EUGENE, Ore. – Georgia Tech track & field rising sophomore Bria Matthews will compete for a spot in the 2016 Olympic Games, joining four former Jackets at the U.S. Olympic Trials in Eugene, Ore. The competition takes place July 1-10 with Matthews competing in the preliminary rounds of the long jump (July 1) and the triple jump (July 4).
The final of the women's long jump is July 2 and the final of the women's triple jump is July 7.
The top three finishers at the Olympic Trials who have already met the Olympic Games qualifying standard will comprise the USA's entrants in each event.
Matthews is coming off of one of the most remarkable freshman campaigns in school history. In addition, she won both the long jump and triple jump at the U.S. Junior Nationals last weekend, earning a spot in the World Under-20 Championships.
In addition to Matthews, Tech alumni Chaunte Howard Lowe (2004, 2008 & 2012 Olympian), Angelo Taylor (2000, 2004, 2008 & 2012 Olympian and a three-time gold medalist), Alphonso Jordan and Jonathan Gardner will compete at the trials.
Current Georgia Tech Student-Athlete Participants:
Bria Matthews – Long Jump (Prelims 7/1 – 8:10 p.m. ET)
Bria Matthews – Triple Jump (Prelims 7/4 – 7:15 p.m. ET)
Former Yellow Jackets:
Jonathan Gardner – Triple Jump (Prelims 7/7 – 7:45 p.m. ET)
Chaunte Howard Lowe – High Jump (Prelims 7/1 – 7:15 p.m. ET)
Alphonso Jordan – Triple Jump (Prelims 7/7 – 7:45 p.m. ET)
Angelo Taylor – 400m Hurdles (Prelims 7/7 – 9:53 p.m. ET)
For the latest information on the Georgia Tech Yellow Jackets, follow us on Twitter (@GT_TrackNField), Facebook (Georgia Tech Track & Field/Cross Country) or visit us at www.ramblinwreck.com.
January 16, 2021 Jackets Close Out Clemson Invite
Tech concludes week two showdown at Clemson
Jackets Close Out Clemson Invite
January 15, 2021 Strong Day One at Clemson Invite for Tech
Yellow Jackets return to Clemson for Week Two bout
Strong Day One at Clemson Invite for Tech


text:
Fallout 76 Beta Begins in October
Xbox One / July 24, 2018
"Starting in October we'll be selecting people who have pre-ordered the game from a participating retailer to enter our B.E.T.A."
The Fallout 76 B.E.T.A. (Break-it Early Test Application) will begin in October and Bethesda will be "selecting people who have pre-ordered the game from a participating retailer."
Announced by community administrator Jurassica on Bethesda's Fallout 76 general discussion forum, she continued to say "we'll start small and grow over time as we prepare for launch."
Jurassica also confirmed that "During the Fallout 76 panel at QuakeCon, we'll be answering some of your questions."
The panel will take place on August 10 at 9am PT/12pm ET and will see Todd Howard and other Bethesda Game Studios developers dive deeper into the newest entry of the Fallout franchise.
Bethesda also detailed more of their plans for QuakeCon, which runs from August 9-12 in Dallas, TX, and it will feature more from Rage 2 and the gameplay reveal of Doom Eternal, which was announced at E3 2018.
Fallout 76 was revealed back in May with a teaser trailer, and was one of the centerpieces of Bethesda's E3 2018 press conference which also saw the announcement of Starfield and Elder Scrolls VI.
Fallout 76 is a prequel to all Fallout games, the first to be "entirely online," and will be "four times the size of Fallout 4." You, as the player, will explore West Virginia and will be "one of the first to emerge in an untamed and very different wasteland."
Since E3 2018, we've learned more about the game, including the consequences (and rewards) of Nukes and that Fallout 76 began as Fallout 4's multiplayer mode.
Fallout 76 will be released on November 14, 2018, on PlayStation 4, Xbox One, and PC.
Adam Bankhurst is a freelance writer for IGN and one of his favorite games of all time is Fallout 3. Needless to say, he's excited for Fallout 76 and hopes to see you in the wasteland. You can follow him on Twitter.
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