text:
\section{Introduction}

Theories of gravity with the Lagrangian of the form
\be
{ L}={ L}(g_{ab},R_{abcd},\nabla_{a_1}R_{bcde},\dots,\nabla_{a_1\dots a_p}R_{bcde})\label{Lagr}
\ee
are natural geometric generalizations of  Einstein gravity.
Many theories of this form, such as Einstein-Weyl gravity, quadratic gravity, cubic gravity, L(Riemann) gravity and their solutions, have been studied in recent years, often motivated by attempts to understand a quantum description
of the gravitational field (see e.g. \cite{Lu:2015cqa,Hennigar2016,Bueno2017,Gullu2011,Gurses2013,Pravda2017PRD} and references therein).

The complexity of the field equations is in general increasing considerably with each term added to the  Einstein-Hilbert action. Thus, very few exact solutions to generalized theories of gravity are known and naturally,  { to examine various mathematical and physical aspects of  these  theories,  } authors often resort to perturbative or  numerical methods.

However, there exists  a special class of spacetimes, {\it universal spacetimes},
that simultaneously solve vacuum field equations of  all theories of gravity
with the Lagrangian of the form
\eqref{Lagr}. {Particular examples of such spacetimes were first discussed in the context of string theory \cite{AmaKli89,HorSte90}  and in the context of spacetimes with vanishing quantum corrections \cite{Coleyetal08}. }
The formal definition of universal metrics  reads \cite{Coleyetal08}
\begin{defn}
	\label{univ}
	A metric is {\it universal} if all conserved symmetric rank-2 tensors constructed
	from  the  metric, the Riemann tensor and its covariant  derivatives
	of arbitrary order are multiples of the metric.
\end{defn}
{Note that from the conservation of the Einstein tensor, it immediately follows that
universal spacetimes are necessarily Einstein spaces.}

In previous works \cite{HerPraPra14,univII}, we studied  necessary and sufficient conditions for universal
spacetimes in an arbitrary dimension. For example, we have proved  \cite{HerPraPra14}

\begin{prop}
	\label{prop_univCSI}
	{ A universal  spacetime is necessarily a CSI spacetime\footnote{CSI {  (constant scalar curvature invariant)} spacetimes  are spacetimes, for which  all curvature invariants constructed from the metric, the Riemann tensor and its covariant derivatives of arbitrary order are constant, see e.g. \cite{ColHerPel06}.
		}.}
\end{prop}
Note that CSI is a necessary but not a sufficient condition for universality.

For type  N {(employing the higher-dimensional algebraic  classification of tensors \cite{classletter})}, we have  found necessary and sufficient conditions for universality \cite{HerPraPra14}:

\begin{prop}
	\label{prop_typeN}
	{ A type N spacetime is universal if and only if it is an Einstein Kundt spacetime\footnote{Kundt spacetimes are spacetimes admitting null geodetic conguence with vanishing shear, expansion and twist (see e.g. \cite{Stephanibook,OrtPraPra12rev}).}.}
\end{prop}

For type III \cite{classletter}, we have found sufficient conditions for universality
\cite{HerPraPra14}:
\begin{prop}
	\label{prop_typeIII}
	Type III, $\tau_i=0$ Einstein Kundt spacetimes obeying
	\be
	C_{acde} C_{b}^{\ cde}=0 \label{QGterm}
	\ee
	are universal.
\end{prop}
Note that the $\tau_i=0$ condition implies that the {null} Kundt direction $\bl$ is recurrent\footnote{{Recurrent null vector $\bl $  obeys $ \ell_{a;b} \propto \ell_a \ell_b$.}}
 and that the cosmological constant $\Lambda$ vanishes. Thus, these spacetimes are Ricci-flat.

In \cite{univII}, we have studied type II  and D universal spacetimes. It has turned out that this problem is dimension dependent. For instance, we have proved the non-existence of such spacetimes in five dimensions, while we have provided examples of type D universal spacetimes in any composite  number  dimension as well as  examples of type II universal
 spacetimes in  various dimensions.

Note that while all known universal spacetimes in dimension $d \geq4 $ \cite{Coleyetal08,HerPraPra14,univII} are algebraically special\footnote{Algebraically special spacetimes are spacetimes of Weyl/Petrov types II, D, III, N, and O.} and Kundt, the existence {algebraically general (type I or  G)} or non-Kundt universal spacetimes  has not been  excluded.

{Although the results  stated above valid in all dimensions considerably  constrain the space of universal spacetimes by giving various necessary conditions, so far the full
set of neccessary and sufficient conditions for universality has been known only
for Weyl type  N spacetimes (proposition \ref{prop_typeN}). }

In this work, we focus on the case of four dimensions.  {This leads to a simplification of the problem and in fact it allows us to find necessary and sufficient conditions for universality for all algebraic types except of the type II.}

 In section \ref{sec-typeI}, we prove the {\em non-existence of Petrov type I  universal
spacetimes in four dimensions}. In fact, in combination with further results presented here and in \cite{HerPraPra14}, we find that
\begin{prop}\label{propalgspec}
Four-dimensional  universal spacetimes are necessarily algebraically special and Kundt.
\end{prop}

Section \ref{secD} is devoted to Petrov type D spacetimes.
The main result of this section are necessary and sufficient conditions for universality for type D.
\begin{prop}
	\label{propD}
	A four-dimensional type D spacetime is universal if and only if it is a direct product of two 2-spaces of  constant curvature with  the Ricci scalars of the both 2-spaces being equal.
\end{prop}

{Section \ref{secII} focuses on type II universal spacetimes. This is the only  case
for which we do not arrive at a  full set of necessary and sufficient conditions for universality. Nevertheless, we obtain certain necessary conditions. In  particular,}
 we find that these spacetimes
{\em necessarily admit a recurrent Kundt null direction and that they are Kundt extensions of type D universal backgrounds} discussed above. Thus, they represent gravitational waves propagating on these backgrounds.  Furthermore, we prove that to examine necessary conditions for universality, it is sufficient to consider only rank-2
tensors linear or quadratic  in $\nabla^{(k)} C$, $k \geq 1$, or rank-2 tensors not containing derivatives of the Weyl tensor.

In section \ref{secIII}, we study type III universal spacetimes and we arrive at necessary and sufficient conditions.
\begin{prop}
	\label{propIII}
	A four-dimensional type III spacetime is universal if and only if it is an Einstein Kundt spacetime  obeying $F_2\equiv C^{pqrs}_{\phantom{pqrs};a} C_{pqrs;b}=0$.
\end{prop}
We also present an explicit type III Kundt Ricci-flat metric
with $\tau_i \not=0$ and vanishing $F_2$, providing thus an  example of type III non-recurrent universal metric.

Finally, {in section \ref{concl} we briefly summarize the main results and}  { in table \ref{table1} we compare known necessary/sufficient conditions for  universality for various algebraic types in four and  higher  dimensions. }  We also point out that VSI spacetimes (spacetimes with all scalar curvature invariants vanishing \cite{Pravdaetal02}) are not necessarily universal.

 Note that all results in the following sections apply to  four dimensions and often this will  not be  stated
	explicitly.
	We will employ the standard four-dimensional Newman-Penrose formalism  summarized e.g. in \cite{Stephanibook}. Occasionally, to connect with previous higher-dimensional results, we will also refer to  the four-dimensional version of the higher-dimensional real null frame formalism (see e.g. \cite{OrtPraPra12rev} and references therein).

\section{Type I universal spacetimes do not exist}
\label{sec-typeI}

In  this section, we prove the non-existence of type I universal spacetimes. By proposition
\ref{prop_univCSI}, we can restrict ourselves to CSI spacetimes.

It has been shown in \cite{ColHerPel09b} that CSI spacetimes
in four dimensions are either (locally) homogeneous or CSI degenerate Kundt metrics\footnote{Degenerate Kundt spacetimes  \cite{ColHerPel09a}  are Kundt spacetimes  with the Riemann tensor and  its covariant derivatives of arbitrary order  aligned and of type II or more special. For example, all Einstein Kundt spacetimes are degenerate Kundt.}. Degenerate Kundt metrics are algebraically special. Thus, it remains to study type I
locally homogeneous spacetimes.

Theorem 12.5  of \cite{Stephanibook} and the results given below this theorem imply that ``there are no homogeneous Einstein spaces with $\Lambda \not= 0$ of types I or II''.
Thus for type I universal spacetimes, we have to restrict ourselves to the Ricci-flat case.

Theorem 12.1  of \cite{Stephanibook} states that all non-flat Ricci-flat homogeneous solutions with a multiply transitive group are certain plane waves (of type N).
Theorem 12.2 of \cite{Stephanibook} states that the only vacuum solution admitting a simply transitive $G_4$ as its maximal group of motions is given by
\be
k^2 \d s^2 = \d x^2 + \mathrm{e}^{-2x} \d y^2 + \mathrm{e}^x \left[\cos {\sqrt{3}x} (\d z^2 - \d t^2) - 2 \sin {\sqrt{3}x} \d z \d t        \right] \label{typeICSI}
\ee
with $k$ being an arbitrary constant.
Thus, this metric
 is the only type I CSI Einstein metric and  the only type I candidate for a universal metric.

However, it can be shown by a direct calculation that for metric \eqref{typeICSI},
a rank-2 tensor $F_2 \equiv C^{pqrs}_{\phantom{pqrs};a} C_{pqrs;b}$ is conserved
and not proportional to the metric $(F_2)^a_{\ b}=$diag$(0,-48k^2\delta_{ij})$. Thus metric \eqref{typeICSI} is not universal. We conclude with

\begin{lem}
Universal spacetimes in four dimensions are necessarily algebraically special.
\end{lem}

\section{Type D universal spacetimes}
\label{secD}

Let us proceed with examining type D universal spacetimes.

Without loss of generality, we choose a frame aligned with both multiple principal null directions (PNDs), for which the following frame components of the Weyl tensor vanish
\be
\Psi_0=\Psi_1=\Psi_3=\Psi_4=0.\label{Dkomp}
\ee

The standard complex curvature invariant $I$ (see e.g. \cite{Stephanibook}) can be expressed in terms of the Weyl components as
\be
I=\Psi_0 \Psi_4 -4 \Psi_1 \Psi_3 + 3 {\Psi_2}^2=3 {\Psi_2}^2.
\label{I}
\ee
Thus the CSI condition implies
\be \Psi_2={\rm const}.
\ee
Then, the Bianchi equation (7.32e)  of \cite{Stephanibook} for
Einstein spacetimes reduces to
\be
D \Psi_2 + \frac{1}{12} D R = 3 \rho \Psi_2=0.\label{rho0}
\ee
Since $\Psi_2\not= 0$, $\rho$ vanishes and therefore a type D  Einstein CSI
spacetime is necessarily Kundt.

Further Bianchi equations (7.32a)-(7.32h)  of \cite{Stephanibook} imply
\beq
&& \kappa=0,\ \ \nu=0, \nonumber  \\
 && \sigma=0,\ \   \lambda=0, \nonumber \\
&&\rho=0,\ \ \mu=0, \nonumber \\
&& \tau=0,\ \ \pi=0. \label{typeDspincoeffs}
\eeq

Thus
\begin{lem}
	Type D  Einstein CSI spacetimes are doubly Kundt and both Kundt directions are recurrent.
\end{lem}

Furthermore, taking into account \eqref{typeDspincoeffs}, Ricci identity (7.21h) gives
\be
\Psi_2+\frac{R}{12}=0
\ee
and therefore,
\begin{lem}
	\label{lem_typeDRFCSI}
	Type D Ricci-flat CSI  spacetimes do not exist.
\end{lem}

Let us prove the following lemma:
\begin{lem}
	\label{}
	Type D  Einstein CSI spacetimes are symmetric (i.e. $R_{abcd;e}=0$).
\end{lem}

\begin{proof}
	This can be more easily shown using spinors, see e.g. \cite{Stephanibook}. The type D Weyl spinor in an adapted frame reads
	\be
	\Psi_{ABCD}= 6 \Psi_2 \sod{(A} \sod{B} \sid{C}\sid{D)}  .
	\ee
	Since due to \eqref{typeDspincoeffs}, the derivatives of the basis spinors satisfy
	\beq
	\dsu{A}{A} \sou{B}&=&\ T\smallsup{A\!\dot{A}}\sou{B},\\
	\dsu{A}{A} \siu{B}&=&- T\smallsup{A\!\dot{A}}\siu{B},
	\eeq
	where
	\be
	T\smallsup{A\!\dot{A}}=\gamma \sou{A}\csou{A}-\alpha \sou{A}\csiu{A}
	-\beta\siu{A}\csou{A}+\epsilon\siu{A}\csiu{A},
	\label{TAA}
	\ee
	and
	\be
	\dsu{A}{A}\Psi_2=0,
	\ee
	we get
	\be
	\dsu{A}{A}\Psi_{ABCD}=0.
	\ee

	Thus,
	\be
	\nabla_e C_{abcd}=\nabla_e R_{abcd}=0
		\ee
	and these spacetimes are symmetric.
\end{proof}

In four dimensions, type D symmetric spaces are necessarily direct products of two 2-spaces of constant curvature (see chapter 35.2 of \cite{Stephanibook}).
Such a product space is Einstein if and only if the Ricci scalars of both spaces are equal.
It has been shown in \cite{univII} that such direct product spaces are universal.
{This concludes the proof of proposition \ref{propD}.}

\section{Type II universal spacetimes}
\label{secII}

In this section, let us study type II universal spacetimes.

We choose a frame with
\be
\Psi_0 = 0 = \Psi_1.
\ee
Then the curvature invariant $I$ is  given by
\eqref{I} as in type D
and thus the CSI condition again implies
\be
\Psi_2={\rm const}. \label{Phi2const}
\ee
For type II Einstein spacetimes, the Goldberg-Sachs theorem implies
\be
\kappa=0=\sigma
\ee
and eq. (7.32e)  of \cite{Stephanibook} again reduces to
\eqref{rho0}
and thus a type II Einstein CSI spacetime is Kundt.
Then, eq. (7.32h)  of \cite{Stephanibook}
reduces  to
\be
\tau \Psi_2=0
\ee
and therefore
\begin{prop}\label{lem_K}
Genuine\footnote{Meaning $\Psi_2 \not=0$.} type II Einstein CSI  spacetimes are degenerate  Kundt with a recurrent principal null direction.
\end{prop}

Now, let us study behaviour of the covariant derivatives of the Weyl
tensor.

\begin{lem}
	For a type II Einstein CSI Kundt spacetime with a recurrent principal null direction, boost
	order of the first covariant derivative of the Weyl tensor is at most $-1$.
\end{lem}

\begin{proof}
	This can be more easily shown using spinors. The type II Weyl spinor in an adapted frame reads
	\be
	\Psi_{ABCD}= 6 \Psi_2 \sod{(A} \sod{B} \sid{C}\sid{D)}
	- 4 \Psi_3 \sod{(A} \sod{B} \sod{C}\sid{D)}
	+  \Psi_4 \sod{A} \sod{B} \sod{C}\sod{D}.\label{Weylspinor}
	\ee

	We choose an affinely parametrized Kundt congruence $\bk$ and a
	frame parallelly propagated along $\bk$
	\be
	0=\kappa=\sigma=\rho=\tau=\epsilon=\pi.
	\ee
	Then,  the derivatives of the basis spinors read
	\beq
	\dsu{A}{A} \sou{B}&=&\ T\smallsup{A\!\dot{A}}\sou{B},	\label{der_spin_o}\\
	\dsu{A}{A} \siu{B}&=&- T\smallsup{A\!\dot{A}}\siu{B}
	+(-\lambda \sou{A}\csiu{A}-\mu \siu{A}\csou{A}+\nu\sou{A}\csou{A})\sou{B},
	\label{der_spin_i}
	\eeq
	where
	\be
	T\smallsup{A\!\dot{A}}=\gamma \sou{A}\csou{A}-\alpha \sou{A}\csiu{A}
	-\beta\siu{A}\csou{A}.
	\ee
	Note that the covariant derivative $\dsu{A}{A}$ does not increase the boost order
	of the frame spinors $\sou{A}$ and $\siu{A}$.
	 The Bianchi identity (7.32g) from \cite{Stephanibook} reduces to
	 \be
	 D\Psi_3=0.\label{Bianchi_DPsi3}
	 \ee
	 Thus, taking into account
	\be
	\dsu{A}{A} =\siu{A}\csiu{A}D
	+\sou{A}\csou{A}\Delta
	-\siu{A}\csou{A}\delta
	-\sou{A}\csiu{A}{\bar{\delta}},\label{spinorder}
	\ee
	it follows that
		\be
	\dsu{A}{A}(- 4 \Psi_3 \sod{(A} \sod{B} \sod{C}\sid{D)}
	+  \Psi_4 \sod{A} \sod{B} \sod{C}\sod{D})
	\ee
	contains only b.w. negative terms.

	However,
	since $\dsu{A}{A}\Psi_2=0$,
	it follows from  \eqref{der_spin_o} and \eqref{der_spin_i}	that
	\be
	\dsu{A}{A} (6 \Psi_2 \sod{(A} \sod{B} \sid{C}\sid{D)})
	\ee
	also contains only b.w. negative terms, cf. also \eqref{derWeylspinor}.
\end{proof}

\begin{lem}\label{lem_balanII}
	For a type II Einstein CSI Kundt spacetime with a recurrent
	principal null direction,
	 boost
	order of an arbitrary covariant derivative of the Weyl tensor is at most $-1$.
\end{lem}

\begin{proof}
	The Ricci equations (7.21d,e,f,g,h,i,q) from \cite{Stephanibook}, using also
	 \eqref{Bianchi_DPsi3} and \eqref{Phi2const},  imply
	\beq
	D\alpha&=&0,\label{Da}\\
	D\beta&=&0,\\
	D\gamma&=&\Psi_2-\frac{R}{24}\ \ \rightarrow\ \ D^2\gamma=0,\\
	D\lambda&=&0,\\
	D\mu&=&\Psi_2+\frac{R}{12},\label{Dmu}\\
	D\nu&=&\Psi_3\ \ \rightarrow\ \ D^2\nu=0,\\
	0&=&\Psi_2+\frac{R}{12},\label{Psi2R}
	\eeq
	respectively. Eq. \eqref{Psi2R} implies that  eq. \eqref{Dmu}
	reduces to
	\be
	D\mu=0
	\ee
	and that $\Psi_2$ is real
	\be
	\Psi_2=-\frac{R}{12}.
	\ee

	From the Bianchi equation
	(7.32c) in \cite{Stephanibook}, it follows
	\be
	D\Psi_4={\bar{\delta}}\Psi_3+2\alpha\Psi_3-3\lambda\Psi_2.
	\label{Bianchi_DPsi4}
	\ee
	Applying the operator $D$ on \eqref{Bianchi_DPsi4} and using the commutator
	\be
	D\delta-\delta D=-({\bar{\alpha}}+\beta)D,\label{comut_Ddelta}
	\ee
	we arrive at
	\be
	D^2\Psi_4=0.\label{D2Psi4}
	\ee

	Applying the covariant derivative \eqref{spinorder} on the Weyl spinor \eqref{Weylspinor}, we obtain
		\beq
	\dsu{E}{E}\Psi_{ABCD}\!\!\!&=&\!\!\!
	4\sod{(A} \sod{B} \sod{C}\sid{D)}
	[\sou{E}\csiu{E} ({\bar\delta}{\Psi_3}-3\lambda\Psi_2+2\alpha\Psi_3)
	\nonumber\\
	&&\!\!\!
		+\siu{E}\csou{E}(\delta\Psi_3-3\mu\Psi_2+2\beta\Psi_3)
	+\sou{E}\csou{E}(-\Delta\Psi_3+3\nu\Psi_2-2\gamma\Psi_3)]
	\nonumber\\
	&&\!\!\!
	+\sod{A} \sod{B} \sod{C}\sod{D}
	[\siu{E}\csiu{E}D\Psi_4+\sou{E}\csiu{E}(-{\bar{\delta}}\Psi_4+4\lambda\Psi_3-4\alpha\Psi_4)
	\nonumber\\
	&&\!\!\!
		+\siu{E}\csou{E}(-\delta\Psi_4+4\mu\Psi_3-4\beta\Psi_4)
		\nonumber\\
		&&\!\!\!
	+\sou{E}\csou{E}(\Delta\Psi_4-4\nu\Psi_3+4\gamma\Psi_4)]
	.\label{derWeylspinor}
	\eeq

	Let us employ the balanced scalar/tensor approach in a parallelly propagated frame introduced in \cite{Pravdaetal02}. A scalar $\eta$ with a b.w. b under a constant boost is a balanced
	scalar if $D^{-{\rm b}}\eta=0$ for b$<0$ and $\eta=0$
	for b$\geq 0$. A tensor, whose components are all balanced scalars,
	is a balanced tensor. Obviously, balanced tensors have only b.w.
	negative components.

	While in our case, the Weyl tensor itself is not balanced, we will show that its first derivative \eqref{derWeylspinor} is balanced.

	For the first derivative of \eqref{derWeylspinor} to be balanced, we have to show that $D^{\rm b}$
	on a component of b.w. b vanishes. B.w. $-1$,  $-2$,
	and  $-3$   components of \eqref{derWeylspinor}
	read
	\beq
	&& \delta\Psi_3,\ {\bar\delta}{\Psi_3},\ 	D\Psi_4,\ \lambda\Psi_2, \
	 \ \mu\Psi_2,\  \alpha\Psi_3 ,\
	\beta\Psi_3,\ \\
 &&
\Delta\Psi_3, \ \delta\Psi_4,\ 	{\bar{\delta}}\Psi_4,\ \nu\Psi_2,\ \gamma\Psi_3,\
\lambda\Psi_3,\ \mu\Psi_3, \ \alpha\Psi_4,\
	\beta\Psi_4,
	\\ &&
	\Delta\Psi_4, \ \nu\Psi_3,\ \gamma\Psi_4,
	\eeq
	respectively.

	Using the Bianchi and Ricci equations and commutators \eqref{comut_Ddelta}
	and
	\be
	\Delta D-D\Delta =(\gamma +\bar{\gamma})D,\label{comut_DDelta}
	\ee
	we arrive at
	\beq
	0&=& 	D({\bar\delta}{\Psi_3})= D(\lambda\Psi_2)
    =D(	\alpha\Psi_3)
	=D(\delta\Psi_3) =D( \mu\Psi_2) =D( \beta\Psi_3)
	=D^2 \Psi_4,\nonumber\\
	0&=&D(\Delta\Psi_3) =D^2( \nu\Psi_2)=D^2(\gamma\Psi_3)
	=D^2({\bar{\delta}}\Psi_4) =D (\lambda\Psi_3)\nonumber\\
	 &=&D^2 (\alpha\Psi_4)=D^2
	(\delta\Psi_4) =D( \mu\Psi_3) =D^2( \beta\Psi_4),\nonumber\\
	0&=&D^3(\Delta\Psi_4)=D^2( \nu\Psi_3) =D^3 (\gamma\Psi_4).
	\eeq
	This implies that the first derivative of the Weyl tensor is balanced\footnote{Note that the term $\gamma\Psi_2$ that is not balanced
	does not appear in \eqref{derWeylspinor}.}.

	In fact, a covariant derivative of a balanced tensor in a degenerate Kundt spacetime is again
	a balanced tensor (see Lemma B.3 of  \cite{VSIpforms}) and thus   all derivatives of the Weyl tensor are balanced. This concludes the proof.

	\end{proof}

As a consequence of lemma \ref{lem_balanII},
all tensors of the form $\underbrace{\nabla^{(k_1)}C\otimes\dots\otimes \nabla^{(k_p)}C}_{p\ times}$, $k_i>0$, have boost order $\leq -p$. Since
a rank-2 tensor has in general boost order $\geq   -2$, all rank-2 tensors
constructed from the Riemann tensor and its covariant derivatives
containing more than two terms of the form $\nabla^{(k)}C$, $k> 0$,  vanish.

{Therefore,  } further necessary conditions for universality may
follow only from rank-2 tensors linear or quadratic
in $\nabla^{(k)}C$, $k> 0$, or from terms not containing
derivatives of the Weyl tensor.
Now, let us study some of these rank-2 tensors.

{All rank-2 order-4 tensors constructed from the Riemann tensor and its derivatives can be expanded on the FKWC basis \cite{FKWC,Decanini2008}  of rank-2 order-4.}
For Einstein spacetimes, the FKWC basis  of rank-2 order-4 tensors  without derivatives
of the Weyl tensor
 reduces to the four-dimensional identity
\be
C_{aefg}{C_b}^{efg}=\frac{1}{4}g_{ab} C_{efgh}C^{efgh}, \label{4did}\\
\ee
while the FKWC basis of rank-2 order-6 tensors  without derivatives
reduces to
\be
R^{pqrs}R_{pqta}{{R_{rs}}^{t}}_{b}, \ \
R^{prqs}{R^t}_{pqa}R_{trsb},\ \
{R^{pqr}}_s R_{pqrt}{{{{R^s}}_a}^t}_b.
\ee
It turns out that in our case, all these tensors  are either zero or proportional to the metric and thus they do not yield any further
necessary conditions for universality.

A lengthy but straightforward computation
of the FKWC basis of rank-2, order-6 Weyl polynomials
containing derivatives of the Weyl tensor
 \cite{FKWC}
\be
F_1 \equiv C^{pqrs} C_{pqrs;ab}, \ \ \ F_2 \equiv C^{pqrs}_{\ \ \ \ \ ;a} C_{pqrs;b},
\ \ \ F_3 \equiv C^{pqr}_{\ \ \ \ a;s} C_{pqrb}^{\ \ \ \ \ ;s} \label{eqF}
\ee
in the Newman-Penrose formalism gives
\be
F_2=0=F_3.
\ee
For CSI spacetimes, by differentiating  the identity \eqref{4did}
twice,  we  obtain
\be
F_1+F_2=0\label{F1F2}
\ee
and thus vanishing of $F_2$ implies vanishing of $F_1$.
Thus, all rank-2 order-6 tensors in the FKWC basis either vanish or
are proportional to the metric and give no further necessary conditions for universality.

Explicit examples of type II spacetimes in the context of universality
were studied in \cite{Coleyetal08} and \cite{univII}. It has been found that
further necessary conditions follow from rank-2 tensors involving higher derivatives of the Weyl tensor, for instance, e.g., from the rank-2 tensor
\be
{R^{cg}}_{eh}{R^{dh}}_{fg}\nabla^{(e}\nabla^{f)}C_{acbd} . \label{GT-RRnablaC}
\ee
Thus, the necessary conditions for universality of type II spacetimes,
Einstein, CSI, Kundt, and recurrent, clearly are not  sufficient.
To find the full set of necessary conditions for type II
at the general level is beyond the scope of this paper.

\subsection{Case $D\Psi_4=0$}

Note that using the Bianchi equations \eqref{Bianchi_DPsi4}, and (7.32d)
and (7.32f) in\cite{Stephanibook}
\beq
\Delta\Psi_3-\delta\Psi_4&=&4\beta\Psi_4 -2(2\mu+\gamma)\Psi_3
+3\nu\Psi_2,\\
-\delta\Psi_3&=&2\beta\Psi_3-3\mu\Psi_2,
\eeq
respectively,
the first derivative of the Weyl spinor simplifies to
\beq
\dsu{E}{E}\Psi_{ABCD}\!\!\!&=&\!\!\!
\underbrace{D\Psi_4 [4\sod{(A} \sod{B} \sod{C}\sid{D)}
\sou{E}\csiu{E} +\sod{A} \sod{B} \sod{C}\sod{D}
\siu{E}\csiu{E}]}_{b.w. -1}\nonumber\\ &&
+\underbrace{(-\delta\Psi_4+4\mu\Psi_3-4\beta\Psi_4)
[\sod{A} \sod{B} \sod{C}\sod{D}
\siu{E}\csou{E} +4\sod{(A} \sod{B} \sod{C}\sid{D)} \sou{E}\csou{E}]}_{b.w. -2}\nonumber\\ &&
+\underbrace{(-{\bar{\delta}}\Psi_4+4\lambda\Psi_3-4\alpha\Psi_4) \sod{A} \sod{B} \sod{C}\sod{D}\sou{E}\csiu{E} }_{b.w. -2}
\nonumber\\
&&\!\!\!
+\underbrace{(\Delta\Psi_4-4\nu\Psi_3+4\gamma\Psi_4) \sod{A} \sod{B} \sod{C}\sod{D}\sou{E}\csou{E}}_{b.w. -3}
,\label{derWeylspinorsimpl}
\eeq
where
\be
D(-\delta\Psi_4+4\mu\Psi_3-4\beta\Psi_4)=0.\label{d1}
\ee
Thus, there is a special subcase of  type II CSI Einstein Kundt spacetimes characterized by $D\Psi_4=0$, for which the first derivative of the Weyl tensor \eqref{derWeylspinorsimpl} contains only b.w.$\leq -2$ terms. Furthermore,
\beq
D(-{\bar{\delta}}\Psi_4+4\lambda\Psi_3-4\alpha\Psi_4)&=&0,\label{d2}\\
D^2(\Delta\Psi_4-4\nu\Psi_3+4\gamma\Psi_4)&=&D(-4\Psi_3^2+6\Psi_2\Psi_4)=0.\label{d3}
\eeq
Thus, in this case, the first derivative of the Weyl tensor is
1-balanced\footnote{A scalar $\eta$ with a b.w. b under a constant boost is $1$-balanced
	 if $D^{-{\rm b}-1}\eta=0$ for b$<-1$ and $\eta=0$
	for b$\geq -1$. A tensor, whose components are all $1$-balanced scalars,
	is a $1$-balanced tensor. Obviously, $1$-balanced tensors have only  components of b.w. $\leq -2$ .}.
Using \eqref{Bianchi_DPsi3},  \eqref{Da}--\eqref{D2Psi4}, \eqref{comut_DDelta}, and \eqref{d1}--\eqref{d3}
the same proof as in sec. 4 of \cite{HerPraPra14} or in sec. 7.1.
of \cite{univII} applies to our case and thus a {\it covariant derivative of
a 1-balanced tensor
 is 1-balanced}. Therefore, all covariant derivatives of the Weyl tensor are 1-balanced and hence they contain only b.w.$\leq -2$
components. This implies that while studying universality within this class, it is sufficient to  study only  rank-2 tensors linear in derivatives of the Weyl tensor.

Note that the Khlebnikov-Ghanam-Thompson metric discussed
in the context of universality in \cite{Coleyetal08} and \cite{univII} in four and higher dimensions, respectively,  are explicit examples of spacetimes belonging to the
$D\Psi_4=0$ class.

\subsection{Seed metric for type II universal spacetimes}
\label{sec_seedII}

All type II Einstein recurrent Kundt spacetimes have the metric of the form
\be
\d s^2=2\d u (\d v+H\d u+W_x\d x+W_y\d y)+h^2(u,x,y)(\d x^2+\d y^2),\label{dsKundt}
\ee
where $H=v^2 \Lambda/2
+vH^{(1)}(u,x,y)+H^{(0)}(u,x,y)$, and $W_i=
W^{(0)}_i(u,x,y)$.
The CSI condition implies further that
$h$ does not depend on $u$.

Consider the one-parameter group of diffeomorphisms of the metric \eqref{dsKundt} defined by $\phi_\lambda: (u,v)\mapsto (ue^{-\lambda},ve^{\lambda})$. This map gives a rescaling of the functions as follows:
\beq
&&\left(H^{(1)}(u,x,y), H^{(0)}(u,x,y)\right)\longmapsto

  \left(e^{-\lambda}H^{(1)}(ue^{-\lambda},x,y), e^{-2\lambda}H^{(0)}(ue^{-\lambda},x,y)\right), \\
&& W^{(0)}_i(u,x,y)\longmapsto e^{-\lambda}W^{(0)}(ue^{-\lambda},x,y).
\eeq
This map is a diffeomorphism and leaves the invariants invariant and is the Lorentizan version of the limiting map in \cite{HHY}. Let $p$ be the fixed point of $\phi_\lambda$ given by $(u,v,x^i)=(0,0,x^i)$. Then note that the map $d\phi_\lambda$ induces a boost on the tangent space $T_pM$ which aligns with the natural null-frame of (\ref{dsKundt}). Hence, given an arbitrary curvature tensor $R$ of  (\ref{dsKundt}) with boost weight decomposition $R=\sum_{b\leq 0}(R)_b$, then at $p$
\[ \phi_\lambda^*R=\sum_{b\leq 0}e^{b\lambda}(R)_b=(R)_0+e^{-\lambda}(R)_{-1}+e^{-2\lambda}(R)_{-2}+...\]
Consequently,
\[ \lim_{\lambda\rightarrow\infty}\phi_\lambda^*R=(R)_0.\]
However, in the limit $\lambda \rightarrow \infty$, the metric is a type D metric with the same invariants as the type II metric. We also note that the universality requirement is invariant under this diffeomorphism\footnote{This follows from the fact that $T_{ab}=k g_{ab}$ and $\phi^*_{\lambda}g_{ab}=g_{ab}$ at $p$.}, as well as in its limit, and hence, in this limit,  the metric turns into a universal type D metric having identical invariants. This implies that the ``background'' metric for universal type II metrics are universal type D metrics.

Example \cite{PodOrt03}
\be
\d s^2 = \d s^2_b
+ [ f(\zeta,u) + {\bar f}({\bar\zeta}, u)] \d u^2 ,
\ee
where $\d s^2_b$
\be
\d s^2_b=\frac{2\d\zeta\d {\bar\zeta}}{\left(1+\frac{1}{2}\Lambda\zeta{\bar\zeta}\right)^2}+2\d u\d v+\Lambda v^2\d u^2,
\ee
is the metric of the (anti-)Nariai vacuum universe with
$\Lambda > 0$ ($\Lambda  < 0$), and $f(\zeta,u)$
is an arbitrary holomorphic (in $\zeta$)  function
characterizing the profile of the gravitational
wave. This metric is a special case of metrics considered in \cite{univII} with
\be
H= [ f(\zeta,u) + {\bar f}({\bar\zeta}, u)]
\ee
and it was conjectured there that such a metric is universal if
\be
 (\Box^{(1)})^P  H=0, \label{GT-sumBoxP}
\ee
where $P=1, 2$ (note that $\Box^{(0)} H=0$ identically and that
 the vacuum Einstein equations with the cosmological constant $\Box^{(1)} H=0$ imply  $(\Box^{(1)})^2 H=0$).

\section{Type III universal spacetimes}
\label{secIII}

It follows from the results of Sec. 5.2. of \cite{HerPraPra14} that type III universal spacetimes in four dimensions are Kundt.

In four dimensions for type III, the following identity holds
\be
C_{acde} C_{b}^{\ cde}=0 .\label{QGterm}
\ee
As a consequence of \eqref{QGterm}, theorem 1.4. of \cite{HerPraPra14} reduces to
\begin{prop}
\label{prop_typeIII}
Type III, recurrent ($\tau_i=0$) Einstein Kundt spacetimes     are universal.
\end{prop}

Note that it follows directly from the Ricci identity (7.21q) of \cite{Stephanibook} that $\tau=0$ implies that Ricci scalar vanishes and thus these spacetimes are in fact Ricci-flat,  as observed in \cite{HerPraPra14}.
An explicit example of such a metric is given in \cite{HerPraPra14}.

 Thus, in this section we focus on the non-recurrent ($\tau_i\not=0$) case
 which also allows for $\Lambda\not=0$.

Let us start with proposition 5.1. of \cite{HerPraPra14}
\begin{prop}
\label{prop-balanced}
For type III Einstein  Kundt spacetimes, the boost order of  $\nabla^{(k)} C$  (a covariant derivative  of an arbitrary order of the Weyl tensor)
{  {with respect to the multiple WAND}}  is at most $-1$.
\end{prop}

A straightforward consequence of the above proposition is
a generalization of Lemma 5.2. of \cite{HerPraPra14} to Einstein spacetimes:
\begin{lem}
\label{lemma_quadraticC}
For  type III Einstein Kundt spacetimes, a non-vanishing rank-2
tensor constructed from the metric, the Weyl tensor and its covariant
derivatives of arbitrary order is at most quadratic in the Weyl tensor
 and its covariant derivatives.
\end{lem}
It has been shown  in the proof of proposition 5.1. of \cite{HerPraPra14}  that
for type III Einstein  Kundt spacetimes, the Weyl tensor and its covariant derivatives of arbitrary order are  balanced.  Thus it follows:
\begin{cor}
\label{typeIIIcons}
For type III Einstein  Kundt spacetimes, all rank-2 tensors constructed
from the Weyl tensor and its covariant derivatives of arbitrary order
quadratic in the Weyl tensor and its covariant derivatives are conserved.
\end{cor}

In the following, we will employ the formula for
the commutator for an arbitrary tensor:
\be
[\nabla_a,\nabla_b]T_{c_1.\dots c_k}=T_{d\dots c_k}{R^d}_{c_1 ab}+\dots
+T_{c_1\dots d}{R^d}_{c_kab} \label{commut_der}.
\ee

\subsection{The Ricci-flat case}
\label{sec_III_Rflat}

In the Ricci-flat case,  covariant derivatives in a rank-2 tensor quadratic in the Weyl tensor and its derivatives
effectively commute thanks to lemma \ref{lemma_quadraticC} and \eqref{commut_der}.
Thus, using the Bianchi identities, one can generalize lemmas 5.3. and 5.4. of \cite{HerPraPra14} to the $\tau_i\not=0$ case
\begin{lem}
\label{lemma_insum}
For  type III Ricci-flat Kundt spacetimes, a rank-2
tensor constructed from the metric, the Weyl tensor and its
 covariant derivatives of arbitrary order quadratic in $\nabla^{(k)} C$,  $k\geq 0$,
vanishes if it contains a summation within $\nabla^{(k)} C$.
\end{lem}

\begin{lem}
\label{lemma_der}
For  type III, Ricci-flat Kundt spacetimes, let us assume that
a certain rank-2 polynomial quadratic in $\nabla^{(k)} C$ vanishes.
Symbolically we will write $C^{(1)} C^{(2)} =0$. Then also
$C^{(1)}_{\ ;f} C^{(2)\ \! ;f}
=0$.
\end{lem}

First, let us examine conserved rank-2 tensors quadratic in the Weyl tensor from
 the FKWC basis \cite{FKWC} of rank-2, order-6 Weyl polynomials
 \eqref{eqF}.
In our case, $F_3$  vanishes identically as a consequence of \eqref{QGterm}
and lemma \ref{lemma_der}.

On the other hand, $F_2$ is in general non-vanishing (see sec. \ref{sec_F2van}), however, in this case, $F_2=0$ is a necessary condition for universality and  will be assumed in the rest of this section.
From \eqref{F1F2}, vanishing of $F_2$ implies vanishing of $F_1$.

For spacetimes satisfying $F_2=0$,
the FKWC basis of rank-2, order-6 tensors  vanishes and thus also all rank-2, order-6 Weyl polynomials.

Now, let us  prove universality in  the Ricci-flat case.

\begin{prop}
Type III, Ricci-flat Kundt spacetimes, obeying $F_2=0$ are universal.
\end{prop}

\begin{proof}

By lemma \ref{lemma_quadraticC}, we can limit ourselves to the discussion
of rank-2 tensors which are linear or quadratic in $\nabla^{(k)} C$, where $k=0,1,\dots$. We start with the quadratic case.

The key tools in the proof are Lemmas \ref{lemma_insum} and \ref{lemma_der}
and the observation that covariant derivatives in a rank-2 tensor quadratic in the Weyl tensor and its derivatives effectively commute.

First, consider rank-2 tensors quadratic in the Weyl tensor and its derivatives with both free indices appearing in the first term $\nabla^{(k)} C$.
Symbolically, such tensors will be written as
$$
C_{ab..;\dots  }C^{....;\dots}, \ C_{a.b.;\dots  }C^{....;\dots}, \
C_{a...;b\dots  }C^{....;\dots}, \ C_{....;ab\dots  }C^{....;\dots},
$$
etc., where $a,b$ are free indices and the dots represent various combinations of dummy indices. We understand that covariant derivatives are of arbitrary high order.

Using symmetries of the Weyl tensor, the Bianchi identities, by lemma \ref{lemma_insum}, and the fact that here covariant derivatives commute, all above rank-2 tensors can be reduced to
\be
C_{a.b.;\dots  }C^{....;\dots} = \nabla^{(n)} C_{a.b.} \nabla^{(n-2)} C^{....}\,.
\label{typeIIIproof}
\ee
All indices in  $\nabla^{(n)}$ are dummy indices and by lemma \ref{lemma_insum},
to obtain a non-zero result, they should be contracted with the dummy indices in the second term $\nabla^{(n-2)} C_{....}$.  Due to the symmetries of the Weyl tensor, only two of them can be contracted with  $C_{....}$, while remaining indices are contracted with those of $\nabla^{(n-2)}$. Now by lemma \ref{lemma_der}, the tensor \eqref{typeIIIproof} vanishes since
$$
\nabla^{(2)} C_{a.b.}  C^{....} =0,
$$
as a consequence of vanishing of the  rank-2, order-6 Weyl FKWC basis.

Next, consider rank-2 tensors quadratic in the Weyl tensor and its derivatives with the free indices appearing in both terms. Such tensors reduce
to
\be
C_{a...;\dots  }C^{b...;\dots} = \nabla^{(n)} C_{a...} \nabla^{(n)} C^{b...}.
\ee
In order to get a non-zero result, at most two dummy indices in  $\nabla^{(n)}$
in the first term can be contracted with $C^{b...}$ in the second term. Thus
$n-2$ indices will appear in both $ \nabla^{(n)}$ terms. By lemma \ref{lemma_der}, the problem thus reduces to determining whether
\be
C_{a...;..  }C^{b...;..} = \nabla^{(k)} C_{a...} \nabla^{(k)} C^{b...},\ \ \ \ k\leq 2,
\ee
vanishes. Cases $k=0,1$ are trivial.
For $k=2$, to obtain a non-trivial result, the indices in the first $ \nabla^{(2)}$ have to be contracted with $ C^{b...}$ and similarly with the second $\nabla^{(2)}$. Taking into account the symmetries of the Weyl tensor,
we arrive at the form
\be
C_{acde;fg}{C_{b\ }}^{fge;cd}=-C_{acde;fg}{C_{b\ }}^{fcg;ed}
-C_{acde;fg}{C_{b\ }}^{fec;gd}=0,\label{CC22}
\ee
where the first term vanishes due to the symmetries of  the Weyl tensor and its derivatives and the second term due to lemma \ref{lemma_der}  and vanishing of the  rank-2, order-6 Weyl FKWC basis.

Above, we have proven vanishing of all rank-2 tensors quadratic in  $\nabla^{(k)} C$.  Due to this result and \eqref{commut_der}, covariant derivatives in a rank-2 tensor linear in $\nabla^{(k)} C$ commute. Vanishing of these linear terms is then a trivial consequence of the Bianchi identities and
tracelessness of the Weyl tensor.

\end{proof}

\subsection{The Einstein case}

In the Einstein case, all rank-2 tensors constructed from the Weyl tensor without derivatives vanish due to \eqref{QGterm}.

Let us proceed with conserved rank-2 tensors quadratic in the Weyl tensor containing derivatives.
The FKWC basis \cite{FKWC} of rank-2, order-6 Weyl polynomials
reduces again to \eqref{eqF}.

Differentiating \eqref{QGterm} twice, we obtain
\be
{C^{pqr}_{\ \ \ \ a;s}}^s C_{pqrb}+2C^{pqr}_{\ \ \ \ a;s} C_{pqrb}^{\ \ \ \ \ ;s}
+C^{pqr}_{\ \ \ \ a} {C_{pqrb;s}^{\ \ \ \ \ \ s}}=0. \label{derQGterm}
\ee
Using the Bianchi identities, \eqref{commut_der}, and the fact that all rank-2 tensors quadratic in the Weyl tensor vanish,
we
find that the first and the last terms in \eqref{derQGterm} vanish.
Consequently, from \eqref{derQGterm}
\be
F_3=0.
\ee

As in the Ricci flat case \ref{sec_III_Rflat}, we demand
\be
F_2=0.\label{vanF2}
\ee
Then from \eqref{F1F2}, $F_1=0$.

As in the Ricci flat case, for spacetimes satisfying $F_2=0$,
the FKWC basis of rank-2, order-6 vanishes and thus do also all rank-2,
order-6 Weyl polynomials.

Using \eqref{commut_der} and vanishing of the FKWC basis, it follows that covariant derivatives in a rank-2 tensor of the form
$\nabla^{(2)}C\nabla^{(2)}C$, $\nabla^{(3)}C\nabla^{(1)}C$,
and   $C\nabla^{(4)}C $  commute. If there is a summation within
one term $\nabla^{(2)}C$ in $\nabla^{(2)}C\nabla^{(2)}C$
or in one term in  $\nabla^{(3)}C\nabla^{(1)}C$, or in  $C\nabla^{(4)}C $ then the resulting rank-2 tensor vanishes due
to the Bianchi identities, tracelessness of the Weyl tensor, and commuting of  covariant derivatives.
Then vanishing of all rank-2 order-6 tensors that we write symbolically as
$C^{(1)}C^{(2)}=0$ implies
\be
C^{(1)}_{;f}C^{(2);f}=0.\label{C1fC2f}
\ee
Hence, $\nabla^{(3)}C\nabla^{(1)}C$
and   $C\nabla^{(4)}C $  vanish and
the only rank-2 possibly non-vanishing tensor of the form $\nabla^{(2)}C\nabla^{(2)}C$ is
\eqref{CC22} that still vanishes using the same arguments as given for \eqref{CC22}.

Thus, we have proven

\begin{lem}
For type III, Einstein Kundt spacetimes, obeying $F_2=0$, all rank-2
tensors of the form $\nabla^{(k)}C\nabla^{(l)}C$, $k+l\leq 4$ vanish.
\end{lem}

Let us prove using mathematical induction

\begin{prop}\label{prop_kvadrat}
	For type III, Einstein Kundt spacetimes, obeying $F_2=0$, all rank-2
	tensors of the form $\nabla^{(k)}C\nabla^{(l)}C$  vanish.
\end{prop}

 We start by assuming that all rank-2
tensors of the form $\nabla^{(k)}C\nabla^{(l)}C$, $k+l\leq p$  vanish.
Then
\begin{lem}\label{lem_commute}
 If all rank-2
 tensors of the form $\nabla^{(k)}C\nabla^{(l)}C$, $k+l\leq p$,  vanish then the covariant derivatives in rank-2 tensors of the form
 $\nabla^{(r)}C\nabla^{(s)}C$,
$r+s\leq p+2$,
 commute.
\end{lem}
\begin{proof}
 When commuting derivatives using \eqref{commut_der}, the additional terms  are
 rank-2 tensors of the form $\nabla^{(r)}C\nabla^{(s)}C$, $r+s\leq p$
 that vanish by our assumption.
 \end{proof}
Then obviously,
\begin{lem}
	\label{lem_sumLambda}
	If all rank-2
	tensors of the form $\nabla^{(k)}C\nabla^{(l)}C$, $k+l\leq p$,  vanish then
	rank-2 tensors of the form
	$\nabla^{(r)}C\nabla^{(s)}C$, $r+s\leq p+2$,
vanish if there is a summation within one term.
\end{lem}
\begin{proof}
	We commute the repeated dummy indices to the first position and then employ the Bianchi
	identities and the tracelessness of the Weyl tensor.
\end{proof}

This further implies,
\begin{lem}\label{lem_C1C2lambda}
	If all rank-2 tensors of the form $C^{(1)}C^{(2)}=\nabla^{(k)}C\nabla^{(l)}C$, $k+l\leq p$ vanish then also
	 $C^{(1)}_{;e}C^{(2);e}=0$.
\end{lem}
\begin{proof}
	This can be shown by differentiating $C^{(1)}C^{(2)}=0$ twice and using
	lemma \ref{lem_sumLambda}.
\end{proof}

\begin{proof}
	Now let us prove proposition \ref{prop_kvadrat}.

	We have assumed that all rank-2
	tensors of the form $\nabla^{(k)}C\nabla^{(l)}C$, $k+l\leq p$,
	vanish. We want to show that then also all rank-2
	tensors of the form
	$\nabla^{(r)}C\nabla^{(s)}C$, $r+s\leq p+2$,
	vanish.

	Using lemma \ref{lem_commute} and the Bianchi identities, without loss of generality, all case reduce to the following two cases
	\beq
	& &
	C_{a. b.;\underbrace{....}_{r}}C_{....;\underbrace{....}_{s}}\,,\\
	& &	C_{a. ..;\underbrace{....}_{r}}C_{b...;\underbrace{....}_{s}}\,.
	\eeq
		If there is a summation within one term then by lemma
	\ref{lem_sumLambda}, the rank-2 tensor vanishes. Otherwise,
	$r=s+2$ or $r=s$, respectively. Then by lemma \ref{lem_C1C2lambda}, it reduces to (non-)vanishing of
	$C_{a.b.;..}C^{....}$ and 	\eqref{CC22}, respectively,
	which was discussed earlier.

\end{proof}

The discussion of rank-2 tensors linear in $\nabla^{(k)}C$
is straightforward. The derivatives in  $\nabla^{(2)}C$
commute due to eqs. \eqref{QGterm} and \eqref{commut_der}.
Then all such rank-2 tensors vanish due to Bianchi identities
and tracelessness of the Weyl tensor. If all rank-2
tensors linear in $\nabla^{(k)}C$ vanish then using
\eqref{commut_der} and proposition \ref{prop_kvadrat},
 all rank-2
tensors linear in $\nabla^{(k+2)}C$   vanish as well.
Thus, by mathematical induction all rank-2
tensors linear in $\nabla^{(p)}C$ for arbitrary $p$ vanish.

{ This  concludes the proof of proposition \ref{propIII}.}

\subsection{A type III  non-recurrent universal metric }
\label{sec_F2van}

Let us present an explicit example of a type III universal spacetime
with $\tau\not= 0$. In this section we use the real null basis
and corresponding formalism (see e.g. \cite{OrtPraPra12rev}).

In this case, the necessary condition for universality $F_2=0$ (see proposition \ref{propIII})
 reads
\beq
F_2\!\!\!&=&\!\!\!48\ell_a\ell_b
\Psi'_i\tau_j (2\Psi'_j\tau_i-\Psi'_i\tau_j)\\
\!\!\!&=&\!\!\!
48\ell_a\ell_b[\tau_2 (\Psi'_3+ \Psi'_2)+\tau_3 (\Psi'_3- \Psi'_2)]
[\tau_2 (\Psi'_2- \Psi'_3)+\tau_3 (\Psi'_2+ \Psi'_3)]=0,
\nonumber
\eeq
hence
\be
\tau_2 (\Psi'_3\pm \Psi'_2)=\tau_3 (\Psi'_2\mp \Psi'_3).\label{condF2}
\ee

Type III
 Ricci-flat Kundt spacetimes  with $\tau\not=0$ admit a metric
 \cite{Stephanibook}
\be
\d s^2= -2\d u (\d r+ W_2\d x - W_3\d y +H\d u)+\d x^2 + \d y^2,
\ee
where
\beq
W_2&=&-\frac{2r}{x}+W^0_2 (u,x,y),\\
W_3&=&W^0_3 (u,x,y),\\
H&=&-\frac{r^2}{2x^2}+r \left(\frac{W^0_2}{x}+h_1(u)\right)+ H^0(u,x,y),
\eeq
where
\beq
W_2^0,_x&=&W_3^0,_y,\nonumber\\
W_2^0,_y&=&-W_3^0,_x \label{holom}
\eeq
(in the complex notation, the function $W_2^0+{\rm i} W_3^0$ is holomorphic)
and $H^0$ is subject to an additional b.w. $-2$ Einstein equation
\cite{Stephanibook}.

In the adapted null frame
\beq
\bl&=&\d u,\\
\bn&=& - (\d r+ W_2\d x - W_3\d y +H\d u),\\
\bm^{(2)}&=&\d x,\\
\bm^{(3)}&=&\d y,
\eeq
we obtain
\beq
\tau_2\!\!\!&=&\!\!\!-1/x, \ \ \ \ \ \ \ \ \ \
\tau_3=0,\\
\Psi_2'\!\!\!&=&\!\!\!-\frac{1}{2x}W^0_2,_x,\ \ \
\Psi_3'=-\frac{1}{2x}W^0_2,_y.\label{III_Psi3}
\eeq

The condition $F_2=0$
 \eqref{condF2} implies
\be
\Psi'_3=\mp\Psi'_2,\label{condF2j}
\ee
which gives
\beq
W_2^0(u,x,y)=g(x\pm y)+f_2(u),\\
W_3^0(u,x,y)=g(y\mp x)+f_3(u).
\eeq
By \eqref{holom}, this reduces to
\beq
W_2^0(u,x,y)&=&F(u)(x\pm y)+c_2(u),\\
W_3^0(u,x,y)&=&F(u) (y\mp x)+c_3(u).
\eeq

\section{Conclusions}
\label{concl}

In four dimensions, we have obtained stronger results on universal spacetimes than in  previous works in arbitrary  dimensions \cite{HerPraPra14,univII}.

In four dimensions, we have proved that universal spacetimes are necessarily algebraically special and Kundt.  Furthermore, in addition to the necessary and sufficient conditions for universality for type  N already known in arbitrary dimension, we have found necessary  and sufficient conditions for type III.  We have pointed out that apart from type III spacetimes with a  recurrent null vector,
the non-recurrent case is also universal provided $F_2$ (as defined in proposition \ref{propIII}) vanishes.

For type D, the universality condition is very restrictive, allowing only for direct products of two 2-spaces of  constant and equal curvatures. Type II universal spacetimes then reduce to these type D backgrounds in an appropriate limit. In contrast to types III and N, type II and D universal spacetimes necessarily  admit  recurrent null vector.

{{In table \ref{table1}, known necessary/sufficient conditions for universality
for various algebraic types in four and higher dimensions are summarized. }}

\begin{table}[h]

	\begin{center}
		\begin{tabular}{|c||l|l|}
			\hline
			type & 4D & HD

			\\[0.5mm]
			\hline\hline

			I/G  & $\not\exists$ (prop. \ref{propalgspec})& \\[1mm]\hline
			II  & N: & \\[1mm]
		    &	$\bullet$  E+K+ $(\tau=0)\ (\Rightarrow\Lambda\not= 0)$ (prop. \ref{lem_K})

			& $\bullet$ $\not\exists $ 5D (theorem 1.2 \cite{univII})\\[1mm]
			&$\bullet$ additional conditions (e.g. from eq. \eqref{GT-RRnablaC})  & $\bullet$ $\tau=0$ is not necessary \\[1mm]
			& $ \bullet $
			  extensions of univ. type D
			(sec. \ref{sec_seedII})
			& $ \bullet $  $\tau=0\ \Rightarrow\ \Lambda\not=0$ (prop. 5.1 \cite{univII})\\[1mm]
			 & &
			$ \bullet $ S: universal Kundt extensions  \\[1mm]
			 & & of type D univ. spacetimes (prop. 6.2 \cite{univII})\\[1mm]\hline
			 	D  & NS: direct product of 2 2-spaces
			 & S: direct product of $N$ max. sym. $n$-spaces \\[1mm]
			 & with the same Ricci scalar
			   (prop. \ref{propD}) & with the  same  Ricci scalar
			   (prop. 6.1 \cite{univII})\\[1mm] \hline
			III    & NS: E+K + $(F_2 =0)$
			 (prop. \ref{propIII}) &
			S: E+K + $({C_a}^{cde}C_{bcde}=0)

			+ $(\tau=0)$ \\ & & (theorem 1.4. \cite{HerPraPra14})

			\\[1mm]\hline
			N  & NS: E+K (theorem 1.3. \cite{HerPraPra14}) & NS: E+K (theorem 1.3. \cite{HerPraPra14})  \\[1mm]
			\hline
		\end{tabular} \\[2mm]
		\caption{Universal spacetimes  in four and higher dimensions (HD), known necessary (N)/sufficient (S) conditions for various algebraic types. All universal spacetimes are Einstein (E) and CSI (theorem 1.2 \cite{HerPraPra14}) and in four dimensions, they are all necessarily Kundt (K) (proposition \ref{propalgspec}). }
		\label{table1}
		\label{tbl:01}
	\end{center}
\end{table}

Let us conclude with a discussion of universality for VSI spacetimes (spacetimes with all scalar curvature invariants vanishing \cite{Pravdaetal02}).
Although all curvature invariants
in VSI spacetimes vanish, conserved
rank-2 tensors may be non-vanishing
(in contrast to what seems to be suggested in \cite{ColeyLett}).
For example, as noted
in \cite{HerPraPra14}, in higher dimensions $C_{acde} C_{b}^{\ cde}$ is in general non-vanishing  for type III VSI spacetimes
and $F_2$ is in general non-vanishing for type III VSI spacetimes with $\tau_i \not=0$ even in four dimensions.
Thus, although many VSI spacetimes are universal and thus represent  an interesting class of spacetimes in this context,  VSI is neither a sufficient, nor a necessary condition for universality.

\section*{Acknowledgements}
AP and VP would like to thank University of Stavanger for its hospitality while part of this work was carried out.
This work was supported from the research plan RVO: 67985840, the research grant GA\v CR 13-10042S (VP, AP) and  through the Research Council of Norway, Toppforsk
grant no. 250367: \emph{Pseudo-Riemannian Geometry and Polynomial Curvature Invariants:
Classification, Characterisation and Applications} (SH).

\providecommand{\href}[2]{#2}

redpajama_set_name: RedPajamaArXiv


Larry Coleman was born in 1950s in the small farming community of Vidalia, Georgia. When he was young his family moved to Massachusetts and then later to Alabama, but eventually returned to Vidalia when Larry was in middle school. This is where Larry later graduated from high school.
Larry's love of photography dates back to the late sixties when he helped his father develop B&W negative film and print those to photographic paper in a makeshift darkroom his father had set up in an upstairs bathroom of their home. To this day Larry does not remember if his assistance in this darkroom was a willing participation or a requirement, but acknowledges that this was the spark that ignited his interest in photography. In the early seventies Larry's father gave him his old hand-me-down Honeywell Pentax 35mm camera that he had purchased years earlier from a news reporter who had used it for many years. The old camera had no light meter or any of the other conveniences that even at that time could be found on new cameras. Once again, Larry praises his father's move as one that taught him the fundamentals of photography.
Larry graduated from college with a B.S. degree in Criminal Justice. At first he worked in his family's beverage distribution business but soon decided to pursue a career in law enforcement. For the next ten years Larry rose through the ranks from patrolmen to detective and finally worked as an undercover narcotics officer in a larger city in Georgia. During that time he also worked for over a year as a task force officer with the Drug Enforcement Administration. Eventually he moved to Tucson, Arizona where he worked for the Aviation Branch of U.S. Customs pursuing major air borne drug smugglers attempting to fly drugs into the United States through Mexico.
During all his early years, through high school, college, and eventually law enforcement, Larry continued to devote every minute that he could to his love of photography. He even gave a couple of courses to other law enforcement officers in the use of a camera in crime scene photography. Fellow officers and friends remember him as the guy who always had a camera within arms' reach, both on and off duty.
By 1990 Larry had made the decision to leave law enforcement and return to Vidalia, Georgia to work in the family wholesale beverage distribution business that he had grown up around. He continued in this business until his retirement in 2007.
Somewhere around 2001, Larry, like many other photographers, moved to the new world of digital photography. He had religiously stored and cataloged his collection of negatives and positive film (35mm Slides) that he had accumulated for thirty years and when he finally completed the slow process of scanning and digitizing this collection of film even he was a little surprised that he had accumulated over thirty thousand images. Today Larry's collection totals over one hundred thousand images that span the last forty-four years. The amazing story is that this collection was taken by an amateur photographer who has never received any formal training or sold any of his work to the general public. Larry's collection is a unique look at one man's travel through life with a camera at his side. His work reflects how he sees the world around him and those things that impress him enough to make him push the shutter button.
In 2013, after six years of retirement and several trips back to Arizona, Larry and his wife made the big decision to move to the northern part of that state. Larry looks at the move as putting him right in the middle of a photographer's dream land where some of the greatest natural wonders are within a day's drive. 2017 will mark the year Larry Coleman enters the demanding and competitive world of professional photography by presenting his works to the general public for sale.


Ride waves – surf life – save the ocean! A film that will empower your personal growth by connecting you with the power of the ocean.
88 backers pledged €10,525 to help bring this project to life.
Ride Waves – Surf Life!
Passion can overcome limits – even when things are really tough. There was a time, when I lost my dreams. But giving up was not an option. Now, I have the chance of a lifetime to participate in the Stand Up Paddling (SUP) World Cup.
Living "landlocked" in Hamburg, being tiny, an amateur and over 50 – impossible? No – it is never too late to live your dream. I dreamt big, felt deep, climbed up and used the challenging energy to propel it into something better.
With my story and the story of other people who overcame hardship, I will give everybody a great inspiration on how we can use nature as a teacher to discover and empower our inner strength to live life more fully.
And how valuable the ocean is – worth to protect as a powerful resource. A beautiful place, where you can feel like you matter…. and overcome everything.
Make your dream come true – no matter how many obstacles there are – getting stronger – that is what I experience and want to share in this documentary.
JUST GO FOR IT – cause Betty would definitely go!
Let me take you on my journey! I want to encourage everybody to live their dreams with a documentary which shows all ups and downs. And furthermore, include other stories of inspirational people.
The goal of the Kickstarter campaign is to raise minimum 10.000 EUR to produce this documentary. The money will be 100% to cover costs for e.g. travelling, team members (project-management, creation, production) and film-equipment. So far we plan to travel to China, Denmark and some German spots.
Next step will be my participation at the Standup Paddling World Cup in China in Nov 2018, which will be one of the key points of the documentary.
So far the documentary is planned to be released by May 2019.
I'm just a 151 cm tall pocket-sized powerhouse, a surfer, a mother and an entrepreneur who despite a few rounds in the ring with depression have defied all odds and at age 50 got a ticket for her first World Cup.
Not only have I done this but that was the reason to make this documentary now – about my journey, through both the ups and downs – to share how surfing has been my life coaching tool both socially and psychologically.
With this attitude of FLOW LIVING I always LOVE to empower others. So, when I spoke to my filmmaker-friend Andrea, she was also hooked on the idea to share the story and the idea of making a movie came up. An authentic, life inspiring documentary for everyone.
In this connection, we also want to highly empower the message of SAVE THE OCEAN – the most wonderful & important resource for our planets life !
We'll make this documentary with people who have worked on world-class documentaries and filming projects. So as you can see in our campaign video, we have the highest expectation in our production quality and storytelling. We want you to come out of watching this documentary not only uplifted by the story, but also visually pleased of the beautiful journey we took you through.
Bettina is a professional Feng Shui consultant, marketing-expert and ocean-lover living with her family in Hamburg, Germany. She loves stand up paddling and ... she has a dream. Her big goal is to compete at the SUP World in China and to produce this documentary.
Last year she failed, due to the tragic suicide of her lovely mom. After an emotional-roller coaster she qualified for this year event by winning 2nd place at German SUP Wave Championships.
Back on track she goes for her dreams even stronger. Even, going through the ups and downs of life she was always the kind of person, that doesn't give up. Who tries – with positive and powerful energy – to follow the ways life is unfolding.
Her mantra is "Do whatever is good for you".
Andrea is a Hamburg based filmmaker, director, writer and producer who's passionate about good stories and visions in films. Her love and ambition for the ocean shows on some international documentaries called "100years of California Beach Culture" or "Hawaii – Magic & Sublime".
She also has been the creative director on several TV shows, commercial and web videos. For over a decade, she produced not only feature documentaries but also films that made a change for each one of us, like her award winning fundraiser film for severely handicapped children or violence against children.
Interviews TV Station "Hamburg 1"
Contributions: Support our Kickstarter campaign – every little cent helps! As a contributor we keep you informed about the creative process so you are part of the whole experience over the next 8 month.
Spread the word: Email your friends! Post to Facebook! Tweet it!
Share our campaign with bloggers, journalists and businesses that share our values. Does your group have an email-newsletter, a Facebook page, or a blog? It would mean a lot if you could share our link with your subscribers!
Tell us and the world that you have contributed to the documentary by using the hashtag #igoformydreamnow. Once you have made your contribution please take a photo of yourself with the hashtag #igoformydreamnow and use social media to spread the message.
Follow Bettina on Facebook, Instagram and Twitter!
German SUP Wave Championships 2017 - pic by "FotobyKLOTZI"
Kickstarter is a funding platform dedicated to making creative projects possible with direct support. When you back our project, you're saying that you want this documentary to be made. We have chosen Kickstarter because it is designed to make a pledge as convenient as possible.
We set a funding goal. Our supporters make pledges. If we reach our goal, your pledge will be processed by Kickstarter, and we can start filming. If we don't reach our goal, our supporters will not be charged for their pledge. It's all or nothing. For example our target is € 10.000 if we reach € 8.000 we get nothing. We must reach the € 10.000 target to make this documentary dream a reality.
So whether your contribution is small or big, it doesn't matter, you will play a great part in making the world a better place !
We don't want you to feel like you're giving us money and not getting value in return. We're keeping the reward levels simple and few. We hope you value the rewards like you would any other purchase.
As we are producing this film more or less on our own, WE NEED YOU! The whole project depends mostly on the money we can collect via this Kickstarter campaign. We are highly encouraged to get this valuable heart-project flying!
Bettina recently ran a private GoFundMe campaign to cover her travel costs already to participate at the SUP World Cup in China, which will be one of the key points of the documentary. We are very thankful for her great contributors so far.
Of course the success of this campaign will also be important while looking for further potential sponsors.
Finally, the more budget we get the more quality we can deliver!
We have fantastic ideas we want to get alive to empower others. And we have great (potential) team members & film crew who are convinced of the concept.
One of the further bigger challenges is getting the distribution – "Betty would go" does not have a distributor attached yet. We are truly doing this all ourselves so far. But we are confident that we will inspire interest in the right people to partner with us. If for any reason we are forced to go this alone, then that's exactly what we will try to do.
In terms of partnering with a sponsor or distributor, there will be no compromise of our integrity or values, we will choose to only work with "good" people with the right intentions, the right focus and as much positive energy as we do!
If you fund the project, we intend to start pre-production right away, shooting from October until Spring 2019. Completing of all the post production is scheduled by May 2019.
We want to keep focused on making an awesome documentary!
We don't want to rush making this documentary and we want to be able to allow ourselves time to travel and share stories from inspiring people. But of course it also depends on risks like broken bones & material, bad weather conditions etc.
Please note we anticipate the rewards will be fulfilled by the dates mentioned in their description and we will do our best to keep the dates. Some rewards are depending on the finishing date of the movie.
You get "GOOD KARMA POINTS" & I ride a PERSONAL WAVE for you.
I draw a HEART IN THE BEACH with your name & email it to you with a PERSONAL NOTE.
You get a link for a DIGITAL MOVIE DOWNLOAD as soon as it gets released.
You get a link for a DIGITAL MOVIE DOWNLOAD as soon as it gets released & EXCLUSIVE access to our closed group-video-call "BEHIND THE SCENES" about the creation process of the film incl. open question section & Beach heart drawing.
Your are NAMED IN THE FILM-CREDITS & You get EXCLUSIVE access to our closed group-video-call "BEHIND THE SCENES" & Link for a DIGITAL MOVIE DOWNLOAD as soon as it gets released.
You get 1 HOUR SUP LESSON with me in Hamburg in summer 2019 ! Enjoy paddling in this amazing city and get great tipps to improve your paddling & EXCLUSIVE access to our closed-group video-call "BEHIND THE SCENES" & Link for a DIGITAL MOVIE DOWNLOAD as soon as it gets released.
You get 1 HOUR ONLINE FENG SHUI COACHING SESSION with me to get great impulses for more FLOW in your lifestyle & EXCLUSIVE access to our closed group-video-call "BEHIND THE SCENES" & Link for a DIGITAL MOVIE DOWNLOAD as soon as it gets released.
Your name or logo in the VIP FILM -CREDITS & EXCLUSIVE access to our closed group-video-call "BEHIND THE SCENES" & Link for a DIGITAL MOVIE DOWNLOAD as soon as it gets released.
You get 2 HOUR INTENSIVE COACHING SESSION for your BREAKTROUGH in your Business and/or Lifestyle with me and International Strategy Consultant Doreen Hegemann / Globalleading & EXCLUSIVE access to our closed group-video-call "BEHIND THE SCENES" & Link for a DIGITAL MOVIE DOWNLOAD as soon as it gets released.
VIP SPECIAL – Get it all & more !
2) EXCLUSIVE access to our closed group-video-call "BEHIND THE SCENES".
3) 3 HOUR PERSONAL SUP TRAINING with me in Hamburg in summer 2019 ! Enjoy paddling in this amazing city and get great tipps to improve your paddling.
4) 1 HOUR ONLINE FENG SHUI COACHING SESSION with me to get great impulses for more FLOW in your rooms & lifestyle.
5) Link for DIGITAL MOVIE DOWNLOAD as soon as it gets released.


South Africa is set to explore more opportunities in Africa's oil-rich countries, with government planning to use President Jacob Zuma's visit to Egypt later this month to discuss possible future oil deals.
International Relations and Cooperation Minister Maite Nkoana-Mashabane confirmed this week that further discussions for oil exploration by PetroSA in Egypt will be on the agenda when Zuma visits that country on 19 October.
PetroSA, the largest national oil company, recently revealed its plans to pursue investment opportunities internationally, with a special focus on Africa.
A pioneer in gas-to-liquids (GTL) technology, the company is currently at an advanced stage of a project that could see the establishment of a world-class crude refinery in the Eastern Cape. The project, known as Project Mthombo, will create the biggest refinery on the continent.
Earlier this year, a delegation led by Zuma for a state visit to Uganda sought to explore investment opportunities in Uganda's newly found oil wealth. While the country does not yet produce oil, its oil fields are believed to have large potential.
While Nkoana-Mashabane could not divulge the details of the possible oil deal with Egypt, she did say that South Africa had outstanding bilateral agreements with that country in the areas of ICT, environment, health, oil and gas and Zuma's visit aims to conclude those agreements.
Despite the energy sector making up only about 15 percent of Egypt's GDP, South Africa still sees that country as one of the continent's important producers of oil and gas, even though its oil reserves are said to be on the decline.
As the second largest economy in Africa ($450 billion), Zuma's visit to Egypt is expected to be closely followed by local investors.
Politically, Egypt is an influential player in the Middle East peace process and currently chairs the Non-Aligned Movement, a body representing the interests and priorities of 115 developing countries.


text:
www.journalgazette.net
Toby Melville | Pool via Associated Press: In this Monday file photo, U.S. President Donald Trump inspects an honor guard during a welcome ceremony in the garden of Buckingham Palace in London. On Friday, the Associated Press has found that a video circulating on the internet showing a British army band playing Darth Vader's theme song during Trump's visit to Buckingham Palace was manipulated. The military band played U.S. Army, Navy and Air Force songs during the inspection of troops.
Friday, June 07, 2019 5:01 pm
NOT REAL NEWS: A look at what didn't happen this week
AMANDA SEITZ and BEATRICE DUPUY | Associated Press
A roundup of some of the most popular but completely untrue stories and visuals of the week. None of these is legit, even though they were shared widely on social media. The Associated Press checked these out. Here are the real facts:
CLAIM: Video shows British army band playing Darth Vader's theme song during Trump visit to Buckingham Palace.
THE FACTS: The sound on the video was manipulated to make it appear that the Grenadier Guards played "The Imperial March" from "Star Wars," as President Donald Trump inspected the guard during his state visit Monday. The march is the theme song for notorious villain Darth Vader in the films. Footage from the BBC's broadcast shows the military band played U.S. Army, Navy and Air Force songs during the inspection of troops, which Trump made with Prince Charles. The band, known for their tall bearskin hats and red uniforms, kicked off the pageantry by playing "Wild Blue Yonder," the official song of the United States Air Force. That was followed by "Anchors Aweigh," the official march of the U.S. Navy.
CLAIM: Photo shows women Red Cross workers arriving on a beach in Normandy, France, in 1944, to help injured troops.
THE FACTS: The photo, which circulated on social media around the 75th anniversary of the D-Day landing, is being misrepresented. The photo was taken January 15, 1945, seven months after D-Day, as Red Cross workers stepped from a U.S. Coast Guard-manned barge onto a beach in the Riviera region of France, far south of the Normandy beaches in northern France. The allied invasion in Normandy was June 6, 1944. The women seen in the photo were there to "carry out their duties and keep high the spirits" of fighting men, according to a 2016 American Red Cross story published on the Red Cross' website. The original caption with the photo in Getty's archive says it shows "Ladies Day" on the Riviera, where Red Cross women arrived on the beach with small packages to help the fighters. The photo can be found on the American Red Cross website and in Getty Images.
CLAIM: Photo shows large crowd in London protesting Tuesday against President Donald Trump's state visit, with three Trump baby blimps floating over demonstrators.
THE FACTS: Although there were protests during President Donald Trump's state visit to Britain, the photo in the post was taken during protests around a Trump visit in 2018. The photo was manipulated to add more blimps and larger posters. Facebook posts Tuesday paired the manipulated 2018 photo with a picture of Trump and a fabricated quote: "Where are the protesters? It's fake News! All I saw was love." The original photo, taken July 13, 2018, shows protesters in Trafalgar Square during the president's trip to London. It was taken by Getty/AFP photographer Niklas Halle'n. The image was altered to include larger anti-Trump signs and three Trump baby blimps, which were fashioned after a real 20-foot-tall blimp that depicts Trump as a screaming baby in diapers. That blimp made its debut during protests last year and flew over crowds at Tuesday's protest in London's government district.
CLAIM: Roe v. Wade was only passed because Roe lied about being raped. She admitted it on live TV.
THE FACTS: The U.S. Supreme Court did not consider how "Jane Roe" became pregnant when making its landmark decision in 1973. The caption on a video spreading on social media wrongly suggests that the Supreme Court ruled to legalize abortion because Norma McCorvey, who was known as "Jane Roe" in the lawsuit, claimed to have been raped. Whether McCorvey had been raped is not mentioned in the court's majority or dissenting opinions. And the Supreme Court has not weighed in on restricting or legalizing abortions based on why a woman seeks the procedure, said Mary Ziegler, a law professor at Florida State University who has written extensively about the ruling. The video, which has been shared on social media before, features edited clips of McCorvey talking about her life in a 1987 interview with WUSA-TV. During the interview, she admits to fabricating the rape as she sought an abortion in 1969 because she believed it would help her get access to the procedure. Abortion was illegal in Texas then, except to save a woman's life. The video recirculated in recent weeks as state legislatures across the country passed new restrictions on abortion procedures.
CLAIM: "If I were to run, I'd run as a Republican. They're the dumbest group of voters in the country. They believe anything on Fox News. I could lie and they'd still eat it up. I bet my numbers would be terrific." -- Donald Trump in 1998 People magazine interview
FALSE: The president did not make such a comment to People magazine. Singer and actress Bette Midler, who often speaks out against Trump, shared the false quote on her Twitter account Sunday. She apologized and deleted the quote after learning it was false. Trump responded with a tweet calling her "a washed up psycho." Julie Farin, a People magazine spokeswoman, told the Associated Press that the magazine looked into the claim exhaustively when it first surfaced years ago but did not find that the president said anything remotely like it. The quote first began circulating in 2015 and has been widely shared across social media platforms, including Facebook. It has been widely debunked since that time.
This is part of the Associated Press' ongoing effort to fact-check misinformation that is shared widely online, including work with Facebook to identify and reduce the circulation of false stories on the platform.

redpajama_set_name: RedPajamaCommonCrawl


CALGARY, ALBERTA--(Marketwire - Feb. 22, 2013) - The Fraser Institute's Report Card on Alberta's Elementary Schools 2013, the go-to source for measuring school improvement, will be released Saturday, February 23 at 6:30 a.m. (Mountain).
The report card provides parents with information they can't easily get anywhere else. In addition to five years of academic results, the report card shows which Alberta elementary schools have improved, or fallen behind, in terms of language arts mathematics, science, and social studies over the past five years.
A news release with additional information will be issued via Marketwire on Saturday, February 23 at 6:30 a.m. (Mountain).
School results will be published in the Saturday editions of the Calgary Herald and the Edmonton Sun. The complete results for 659 elementary schools will also be available at www.compareschoolrankings.org where visitors can compare schools on their results over the past five years.
Peter Cowley, co-author of the report card, will be in Calgary and available for media interviews on Saturday and Sunday.


text:
Q: Find all integers $a,b,c$ such that $\frac {abc}{a+b+c}$ is an integer. Find all integers $ (a,b,c) $ such that $ \frac{abc}{a+b+c} $ is an integer.
Could someone help me with this number theory problem?

A: Note that $$\frac{abt}{a+b+t}=ab-\frac{ab(a+b)}{a+b+t}.$$
So pick any $a,b$ and list the divisors (positive and negative} of $ab(a+b)$ and then solve $a+b+t=d$ for each divisor $d.$

A: Choose $a$ and $b$ randomly and denote $ab=x$ and $a+b=y$ .You want that :
$$c+y \mid cx$$ or equivalently $$c+y \mid xy$$ so just choose some random divisor $d$ of $xy$ and choose $c=d-y$ .So all solutions are the triplets :
$$(a,b,d-a-b)$$ where $d$ is a divisor of $$ab(a+b)$$

redpajama_set_name: RedPajamaStackExchange


?69.99, then with the voucher code taken off is: ?64.99.
Please note that the voucher code expires 22/02/2011.?
Hi there Geoff-thanks for the heads up on the weather-station offer.
My Oregon Scientific model is 6+years old is showing severe signs of senility, like the rest of us ;)and the rain and wind direction indications have both gone down.
One question- is the "house" display screen battery or battery and/or mains powered?.
I have found in the past that the battery only versions need frequent battery change-outs. The Oregon has batteries as back up but uses a mains supply via a transformer.
Just back from a few days away in Birmingham, so my apologies for the delay in responding.
I keep the indoor console plugged in the the USB port on my PC 24/7 so have never had to change the batteries as the power is picked up from the computer. The two batteries in the transmitter last(outdoors) up to a year.
Hi Geoff my unit arrived on Wednesday two days after ordering so please be on standby for technical back up . Very pleased so far as it sits in the box.
BTW there are cable ties on offer in Lidl's this week if you wished to organise your station's cables in a less "bird nest" way!!!!.
Well, Colin, I hope your getting ready for some REALLY dramatic weather to record!!
Good luck setting it up - I'm always ready to help!
Hi Geoffw-it looks as if the MHPA weather webpage is now up and running with Milford Docks, Pembroke Ferry, and Lawrenny Yacht Club met. data displayed.
I do not know if this is the final format but IMHO much prefer the old displays.
Anyway available at http://http://www.mhpa.co.uk/content.asp?article_id=1072&d=14&directory_id=14 if of interst.
Thank I'll bookmark th site excellent!
With my weather station we have the county covered!
Hi Geoffw -Just touching base since I have now installed and commissioned my new weather station and have loaded software and so far things seem to be talking to each other OK and am very pleased with display(s) etc.
One intriguing item, which I do not know if you can help with, is that I seem to lose outside temperature, outdoor humidity, wind speed and direction and rain measurement i.e. all outdoor inputs intermittently-say twice per day.. Loss is for approx 15 minutes or so and with no intervention from me just as quickly all return!!!. There does not "appear" to be a loss from the trend memory.
I fitted two Energiser Ultimate Lithium AA batteries in the outside unit thinking of longevity and reliability and am using 3 off 2700 ma rechargeable batteries in the base station. I am now connected via USB to my PC however.
Physical distance between outdoor mast and base station approx 60ft but through a few internal and external walls.
Worth me changing the batteries do you think? or any other suggestions would be gratefully received.
Glad you have got it set up and running. Are you using the EasyWeather software that came with it?
As for your queries about loss of contact - I think we all suffer them on occasions. There can be all sorts of sources of interference such as dodgy light switches and other wireless equipment. 60 feet should be OK, but stone of concrete walls can cause loss of signal, and make sure you do not have a fridge, cooker or washing appliance between the transmitter and receiver.
If they are new batteries I don't think changing them will help as they should be fine. Let us know how it settles down.


text:
Operating Department Practice
University of Gloucestershire - 'Always On' Open Day Content
Why Operating Department Practice? Our Operating Department Practice degree has been designed to give you the skills and knowledge to work successfully as a crucial member of an operating department team. The programme has been developed with input from our partners in the NHS and private healthcare to ensure you benefit from the very latest ideas, technologies and techniques and can meet the needs of health employers in modern society. Pending approval by the Health and Care Professions Council (HCPC), our course will enable you to apply to register and practice as an Operating Department Practitioner on graduation and begin a rewarding career delivering vital care to surgical patients. You'll study the fundamentals of anatomy, physiology and pathology and how to apply your knowledge to clinical situations. You'll work with cutting-edge specialist equipment and develop advanced technical skills, as you learn to manage patients through the 3 phases of perioperative care - anaesthesia, surgery and recovery. Work placements will form an essential element of your learning on the course and you'll have the chance to work with experienced healthcare practitioners in a variety of clinical settings. By the time you graduate therefore, you'll have built the experience and confidence necessary to undertake a key role working alongside other professionals, such as nurses, surgeons, anaesthetists and healthcare support workers. Why University of Gloucestershire? We've been using our expertise to support and inspire students since 1847. Join us and you'll benefit from a wide range of courses and professional-standard facilities in the beautiful, historic surroundings of Cheltenham and Gloucester. Our vibrant and tight-knit community of over 10,000 students and staff means you will enjoy a diverse range of sports clubs, societies and activities whilst making friends for life and securing valuable professional connections. Our specialised and experienced lecturers treat our students as more than just a number, and care about their futures. Alongside their support, guest speakers, networking events and work placements will help you prepare for life after graduation, whether that be in a successful career in your field or onto further study. Our students never tire of finding inspiration here in the dynamic county of Gloucestershire. Cheltenham and Gloucester offer the best of city and country living, with the rolling hills of the Cotswolds paired with vibrant nightclubs and stunning parks in the bustling town and city centres. After the course Your story with us doesn't end at graduation. We are proud of our record that 92% of University of Gloucestershire graduates are in work or further study within six months of completing their course*, and throughout your studies we are committed to working with you to develop your future plan. *Graduate Outcomes Survey published 2021 and based on 2018/19 leavers. Experience an open day Book yourself a place at a University of Gloucestershire open day at www.glos.ac.uk/BookAnOpenDay. Our open days have been designed to inform you, inspire you, and help you make the right decision about your next step. It's your chance to see the university for yourself, get a real insight into what we're about and meet your potential course tutors and lecturers. Our friendly student ambassadors will also be on hand to show you round your campus. To find out more information about this course, visit www.glos.ac.uk/OurCourses, email us on [email protected] or call 03330 14 14 14.
Oxstalls Campus - Gloucester
We welcome international students from around the world. Studying in the UK can be a great experience allowing you to meet new people, try new things and experience a new country. Come and join us. Please see the full details of entry requirements in your country: http://www.glos.ac.uk/study/international/in-your-region/pages/in-your-country.aspx
Grade C/4 in GCSE English Language and Mathematics (or equivalent)
International students will need IELTS, with a score of 7.0 in all areas.
http://www.glos.ac.uk/docs/download/International/english-Language-entry-requirements.pdf
There is no data available for this course. For further information visit the Discover Uni website.
EU £15000 Year 1
Additional costs and fee information can be found at: http://www.glos.ac.uk/life/finance/costs/pages/fees-and-additional-costs.aspx
GL50 2RH
Enquiries Office
Bachelor of Science (with Honours)
Full-time 3 years 23 January 2023 Available to apply, please continue to register for 2023-2024 to make an application
Course option 44de3f49-0e24-4e7f-9295-cbd320393c77
Operating Department Practice at University of Gloucestershire - UCAS


Synthetic biology is the chemical synthesis of biological components, or biosynthesis of chemicals, or putting engineering principles in designing novel biological systems. Hierarchy and Modular organization of biological systems allows synthetic biology to be possible. This is the newest field in bioscience and the already published papers thrust on DNA assembly, genetic regulation, chromosomal manipulation, cell signaling & coordination, cell state reporting, artificial cells and cellular structures, metabolic pathway optimization. It is very interesting; modeling synthetic libraries of intergenic sequence, synthetic protein scaffolding, markerless chromosomal deletion, modeling carboxysome- a bacterial micro-compartment. All these become possible with the new era of DNA assembly methodologies like- BioBrick, SLIC, Gibson, CPEC, MoClo, GoldenGate, GoldenBraid, USER, yeast approaches. There are broad areas open; microbial host engineering by genetic regulation without plasmid, real world therapeutics, crop engineering, bioremediation, bioprospecting.


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Home » Boldy's Late Goal Lifts Eagles Over UMass in Opener
Sports, Top Story, Hockey, Men's Hockey
Boldy's Late Goal Lifts Eagles Over UMass in Opener
November 28, 2020 Updated November 28, 2020 at 10:21 am
Boston College men's hockey saw its successful 2019-20 season come to a premature end before it could partake in their scheduled Hockey East quarterfinal against Providence. Over eight months later, a new squad of Eagles hit the ice to begin the 2020-21 season with a new mantra: accept the challenge.
Despite the key losses of former captain David Cotton, defenseman Ben Finkelstein, and Jesper and Julius Mattila, this Eagles squad remains extremely loaded with talent, which is reflected in BC's No. 2 spot on the USCHO rankings.
After their scheduled opening series against New Hampshire was postponed last weekend, the Eagles (1-0) were finally able to open up the new, highly anticipated campaign with a 4-3 victory over their instate rival UMass (1-1-1) on Friday night at home. Matt Boldy scored the game winner early in the third to break a 3-3 deadlock and open the weekend with a victory.
It is always interesting to see how Hall of Fame coach Jerry York will shuffle his lines in the opening games of the season, especially after losing so many key players like he did going into this season. He also found himself without reigning National Rookie of the Year sensation Alex Newhook, who will be competing in the World Juniors with Team Canada until January.
In the postgame press conference, York also mentioned that he expects a few more of his players to play in the tournament with Team USA. With World Juniors taking place during the college season, York will likely tinker with his lines all season long. With Newhook out of the lineup tonight, freshman winger Nikita Nesterenko and junior center Jack McBain suited up on the offensive first line alongside senior winger Logan Hutsko.
"We'll fit," York said in his postgame press conference. "We'll move people. Someone else will have to take quality minutes. I think it is a great reflection of the program and Boston College when we can have players of this caliber."
Both teams took penalties early in the first period, but neither team found the net on the power play. The contest remained scoreless behind strong play from both teams' netminders, Spencer Knight of BC and Matt Murray of UMass.
The Eagles finally broke the ice a little over halfway into the period on a goal scored by the Eagles' new top-line center McBain. Hustko deked by the UMass defense and found McBain in the slot, who wristed it right past Murray to give the Eagles the 1-0 lead, which held into the first intermission.
The Eagles found themselves on the power play yet again early in the second period. Despite having the man advantage, the Eagles let Minutemen defenseman Zac Jones streak down the ice and rip a shot past Knight's glove to tie up the game on the shorthanded goal.
On that same power play, however, the Eagles were able to take the lead right back within seconds. Mike Hardman received a pass from Boldy and beat Murray on the backhand to give the Eagles the 2-1 lead. With seven minutes to go in the period, the Eagles doubled their lead. McBain received a stretch pass from Nesterenko, took his time, and rocketed a wrist shot past Murray for his second goal of the day. With two goals in the opener, McBain surely proved he deserves his spot on the top line.
The lead seemed rather comfortable for the Eagles until the final five minutes of the second period. BC took another penalty, and this time the Minutemen managed a conversion. The Eagles' lead fell to only one goal after freshman forward Josh Lopina deflected the puck past Knight.
Only 13 seconds later, Lopina beat Knight yet again off of a feed from Oliver MacDonald to tie the score up at three. With only two periods of college hockey under his belt, freshman Lopina had already notched two goals. Despite a tie on the scoreboard, it seemed that the Eagles had lost the majority of their momentum heading into the second intermission.
"I thought the shorty was a key one that gave them a lot of momentum," York said. "They moved the puck pretty well on the power play. They moved pucks well and got a shot on goal. The third goal was a breakdown in our defensive zone coverage. All things that we can fix."
The Eagles quickly regained that momentum and regained the lead in the first minute of the final frame. Colby Ambrosio completed a stretch pass onto Boldy's stick, who fired it into the top corner past Murray for the goal to give BC the 4-3 lead, which held all the way through the end.
The Eagles had several chances to extend their lead, but they couldn't manage a score on two breakaway chances, one by Hutsko and the other by Ambrosio. Knight and the young BC defense killed off two penalties in the final half of the period and ultimately held on for the one-goal victory.
"It was a great battle for us," York said. "I thought that some of the keys were killing two penalties in the last ten minutes of the game and the six-on-five in the last 35 to 40 seconds. I thought we had standout performances from Jack McBain and Matt Boldy, and Spencer Knight was solid in goal again."
jerry york jack mcbain alex newhook spencer knight matt boldy mike hardman eamon powell colby ambrosio
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Warmest greeting from Mai Charming Hotel & Spa! Located in the center of Hanoi Old Quarter, well renovated Mai Charming Hotel & Spa is only few steps walking away from the famous centered Hoan Kiem Lake – This super location is perfected place for domestics and foreigners who like staying in the centered with all the restaurants and entertainments surrounded.
With 35 cozy rooms and suites each furnished with elegant interiors mixed with locals traditional styles, Mai Charming Hotel & Spa provides guests with the comforts of home and the sleek services of a five stars service in a three star hotel. We provide the home away from home for the families with teens or younger, couples, business men and groups of friends. Room amenities include comfort bedding and linens, flat screen TV's with multi-channel cable TV, water, coffee/tea facilities, refrigerator, complimentary wireless internet access in all hotel's corners and more. Included in the room tariff is a delicious buffet breakfast more than 30 dishes while other similar standard hotels offered poor breakfast.
Whether your travel to Hanoi involves business or pleasure, our friendly staffs can accommodate the most discerning guests. We are pleased to offer accommodations for short and long term stays. Check your email from the comfort of your room, catch up on the day's events with the complimentary Vietnam News, and receive calls in your room with your own phone. Allow our staff to book dinner reservations, pick up theater tickets, and stock your fridge. We are happy to offer suggestions on what you can do and see in and around our neighborhood and our fair city. Whether in town for a day, a weekend or an extended visit, the comfort and quality of our Mai Charming Hotel & Spa awaits you.


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Q: Variadic template call in subclass I'm struggeling with variadic templates in C++:
I want to call a variadic template function implemented in class A from class B which inherits from A:
a.hpp
class A{
protected:
    template<typename... Ts>
    void variadic(Ts... args);
};

a.cpp
#include "a.hpp"
template<typename... Ts>
void A::variadic(Ts... args) {/*something*/}

b.hpp
#include "a.hpp"
class B : public A {
public:
     void func() {
         variadic(4, 3.4, "hello");
     }
};

When I call the method (e.g. B testobj;testobj.func();) I always get the following linker error:
$ clang++ -std=c++11 main.cpp a.cpp -o main
/tmp/main-2e3fd9.o: In function `B::func()':
main.cpp:(.text._ZN1B4funcEv[_ZN1B4funcEv]+0x29): undefined reference to    `void A::variadic<int, double, char const*>(int, double, char const*)'
clang: error: linker command failed with exit code 1 (use -v to see invocation)

If I implement the variadic method directly in the header file I don't get any error. If I put func in class A it will also compile without error.
It seems to me that the variadic template isn't visible in class B.
Is there any way to solve my issue?


We would want our lovely guests to come in at least 15 minutes prior to your services. Bella Vita is a place to feel at ease the minute you step in! We ask our guests to fill out an intake form online on our website before you begin your Bella Vita experience. Enjoy a cup of our complimentary teas, water and fresh fruits/snacks.
We want all of our guests to be relaxed and not rush into your services.
Being on time for your appointment is greatly appreciated so that you may receive your full service and our other guests are not inconvenienced.
Prior to your service, you may be required to complete an Intake Form to safeguard your well-being and comfort, and to allow us to safely deliver our services by minimizing any risk of injury.
Bella Vita Boutique Spa assures and maintains confidentiality of the strictest standards.
The Spa is an adult oriented atmosphere; therefore we are not able to accept appointments for children under the age of 16 years, unless accompanied by a parent. Children under the age of 18 are not allowed in the spa unless they are receiving services. Children under the age of 18 may need a signed waiver and/or accompaniment by a parent or adult guardian for services. Online bookings cannot be made by or for persons under 18 years of age. Please contact the spa for further information.
We request that you disclose any medical conditions and/or prescription medications as some services may be contraindicated. Please advise our professional staff of any allergies or sensitivities or if you are pregnant.
For the comfort of all of our guests, Bella Vita Boutique Spa is a smoke-free environment.
So you may fully enjoy your spa experience, and as a courtesy to other guests, cell phone should be turned off.
Wear something comfortable to our spa, we will provide a basket to put your clothing and your valuables. Robes and slippers are provided for your comfort if you are booked for multiple services.
To secure your appointment, we will request a credit card or gift certificate number, however you can pay for your services on the day of your appointment with cash, or credit card (Visa, Master or Discover) or gift certificate.
We do not perform any massage treatments during the first trimester of pregnancy!
During your visit at Bella Vita Boutique Spa any sexual misconduct or any inappropriate language, actions will not be tolerated. The treatment will end immediately and the guest will be charged for the full price. The guest will then be asked to leave the property and NOT return if such incident takes place.
(Master Card, Visa, Discover…Sorry we do NOT take American Express Card or Checks).
Gratuities: Gratuities are highly appreciated and are solely at your discretion.


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RECREATIONAL SPORT
Trap Shooting is an excellent place to start if you are new to the shotgun sports!
Trap shooting is a sport for all levels of shooters, kids and adults. A round of trap is 25 targets. Five targets at each of five stations.
Trapshooting is a specific form of clay target shooting. Trapshooting is a game of movement, action and split-second timing. It requires the accuracy and skill to repeatedly aim, fire and break the clay targets which are hurled through the air at a speed of 42mph, simulating the flight path of a bird fleeing a hunter.
The shooter is required to shoot at a target after he/she calls "pull." It does not matter in scoring if the shooter hits only a small piece of the target or whether he/she shatters the target. The target is considered a "dead" or "lost" bird. If the target is hit it is "dead." It is the shooters responsibility to check his own score.
REGISTERED SPORT
Registered trap shooting is a competition that is regulated by the Amateur Trapshooting Association (ATA). Gun clubs hold shoots in accordance with ATA rules, but they must apply and register for each shoot. All participants of these shoots must be ATA members. The shooters scores are recorded in the ATA office where all records are kept and yearly averages computed. The records are used for handicapping and classifying shooters.
In registered trapshooting, the rules specify that targets must be thrown no less than 48 yards no more than 52 yards and should be between 8 and 12 feet high and 10 yards from the trap. Shooters stand a minimum of 16 yards from the trap houses.


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Another good example of why short poems are a mixed bag.
Short poems are great. They often pack a punch, have a high memorability rate, and can be easily recited or quoted. They are fun, light, universal and pristine, if done right.
Editing them is another issue. Because there isn's much there to work with, the poem might feel finished as is. Don't be fooled. Editing pertains to these poems MORE THAN EVER. This is a trap that even the best of us fall into (including those far better than me). The poem itself needs to be examined in all instances, from all angles before deemed finished by anyone.
For this poem, I use the same Sonnet Tactic from a previous posting. Remember, that a form can be a tool of editing as well as a structure for the piece. In this instance, the information was too long for a haiku, which was my first attempt, and it let me know that most of the poem was too important to remove. I did take a second look and took out the lines that seemed too overt.
What we are left with is a shorter, more informed piece. As a poet, everything feels important, especially in a poem of only 6 lines. As an editor, we know that there are always ways to change, and cutting might not be the right move, but it can still be done in a short poem.
Make it all count, don't let flop lines hide in your poems.
As long as you edit edit edit!


This resource has received more positive feedback from parents than anything else I've done in 15 years of ministry.
90 days before a student has their 13th birthday, send this full-color, step-by-step guide to their parents. That's it!
You can either email them the pdf or print it out and give it to them directly. Either way works.
There isn't any branding to remove.
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I've even included the email I send to parents so you can simply copy and paste it when you're ready!
What's in this 13-page guide?
Also included is an editable Word document so you can make any changes you want.
You've been talking about partnering with parents.
When you put this resource in the hands of a parent, you'll empower them to create a memorable moment in the life of their new teenager!
Buy it once. Give it to every parent in your ministry.