text:
\section*{Introduction}

A linear space $V$ with two bilinear operations $ \circ $ and $[\cdot,\cdot]$ is called
a {\em Gelfand--Dorfman algebra} (or simply $\GD$-algebra) if $(V,\circ)$ is a
Novikov algebra, $(V,[\cdot,\cdot])$ is a Lie algebra, and the following
additional identity holds:
\begin{gather}\label{eq:GD1}
b\circ [a,c]=[a,b\circ c]-[c,b\circ a]+[b,a]\circ c-[b,c]\circ a.
\end{gather}
Recall that the variety of Novikov algebras is defined by the following identities:
\begin{gather}
(a\circ b)\circ c-a\circ (b\circ c)=(b\circ a)\circ c-b\circ (a\circ c), \label{eq:LSymm} \\
(a\circ b)\circ c=(a\circ c)\circ b. \label{eq:RComm}
\end{gather}

The axioms above emerged in the paper \cite{GD83} as a tool for constructing Hamiltonian operators
in formal calculus of variations. Later it was shown \cite{Xu2000} that $\GD$-algebras are in one-to-one
correspondence with quadratic Lie conformal algebras playing an important role in the theory
of vertex operators.
The class of $\GD$-algebras is governed by a binary quadratic operad (\cite{GinzburgKapranov},
see also \cite{bremner-dotsenko}
for the definition) denoted $\GD $. As it was shown in \cite{KSO2019},
the Koszul dual operad $\GD^!$ corresponds to the class
of differential Novikov--Poisson algebras introduced in \cite{BCZ2018-b}. The latter algebras
play an important role in the combinatorics of derived operations on non-associative
algebras \cite{KSO2019}.

In the present paper, we study special $\GD$-algebras, i.e., those embeddable
into Poisson algebras with a derivation. We prove that the class of special $\GD$-algebras
is closed under homomorphic images and thus forms a variety.

Non-special GD-algebras exist: the examples were found implicitly in \cite{KSO2019}
and explicitly  in \cite{KolPan2019}. Note that all these examples
are of dimension~three. In this paper we apply the Gr\"obner--Shirshov bases
technique for Poisson algebras to prove that all 2-dimensional GD-algebras are special.

In section \ref{identities}, we give a technical method to find all special identities of $\GD$-algebras and explicitly find all special identities of GD-algebras up to degree $5$. In section \ref{programm}, we prove that all special identities up to degree $5$ are consequences of two independent special identities of degree $4$
found in \cite{KSO2019}. For that purpose, we convert the symmetric operad GD with special identities of degree $4$ into a shuffle operad \cite{bremner-dotsenko}
and compute the first five components of its Gr\"obner basis \cite{DotKhor2010} by means of the  computer algebra package \cite{DotsHij}.
In the same way, we calculate the dimension of GD operad up to degree 6.

\section{Special GD-algerbas}

Suppose $(P,\cdot,\{\cdot,\cdot\})$ is a Poisson algebra with a derivation~$d$.
Hereinafter, we will denote $d(x)$ by $x'$ for simplicity.
Hence, $(P,\cdot )$ is an associative and commutative algebra, $(P,\{\cdot,\cdot\})$
is a Lie algebra and the {\em Leibniz identity} holds:
\[
 \{x,yz\} = y\{x,z\}+z\{x,y\},\quad x,y,z\in P.
\]
The linear map $d:P\to P$ acts as a derivation relative to both operations:
\[
 (xy)'=xy'+x'y,\quad \{x,y\}' = \{x',y\}+\{x,y'\}, \quad x,y\in P.
\]
Then the same space $P$ equipped with operations
\[
x\circ y=x y', \quad [x,y]=\{x,y\},
\]
is a GD-algebra denoted $P^{(d)}$ \cite{Xu2002}.
A GD-algebra $V$ is said to be {\em special} if it can be embedded into a
GD-algebra $P^{(d)}$ for an appropriate differential Poisson algebra~$P$.

It is clear that the class of special GD-algebras is closed with respect to
subalgebras and Cartesian products, so the class of all homomorphic images
of special GD-algebras is a variety denoted $\SGD$.

The relation between differential Poisson algebras and GD-algebras described above
is similar to the well-known relation between
associative and Jordan algebras \cite{Cohn} (see also \cite[Chapter~3]{ZSSS}).
However, in contrast to the Jordan algebra case, the class of special
GD-algebras turns to be closed with respect to homomorphisms.

\begin{theorem}\label{thm:SpecialImage}
Let $V$ be a special GD-algebra.
Then every homomorphic image of $V$ is special.
\end{theorem}

\begin{proof}
As $V$ is a special GD-algebra, it is a homomorphic image of
the free special GD-algebra $\SGD\<X \>$ generated by an appropriate set~$X$.
Therefore, it is enough to show that for every set $X$ all homomorphic images of
$\SGD\<X\>$ are special.

Let us recall the structure of $\SGD\<X\>$ \cite{KSO2019}. Consider
the free differential Poisson algebra
$\Pois\Der\<X,d\>$ generated by the set $X$.
The operation set of this system
consists of a commutative multiplication, Poisson bracket
$\{\cdot,\cdot \}$, and a unary operation $d$ which acts as a derivation with
respect to both binary operations.
As above, we will write $u'$ for $d(u)$.
Define the {\em weight} of a monomial from $\Pois\Der\<X,d\>$ by induction as
follows:
\[
\begin{gathered}
\wt(x)=-1,\quad x\in X; \quad \wt(u') = \wt(u)+1; \\
\wt(\{u,v\})=\wt(u)+\wt(v)+1; \quad \wt(uv)=\wt(u)+\wt(v).
\end{gathered}
\]
For a weight-homogeneous polynomial  $f\in \Pois\Der\<X,d\>$
we denote by $\wt(f)$ the weight of its monomials.

As shown in \cite{KSO2019},
$\SGD\<X\>$ is isomorphic to the subspace of
$\Pois\Der\<X,d\>^{(d)}$ span\-ned by all monomials of
weight~$-1$.

Suppose $I$ is an ideal of the GD-algebra $\SGD\<X\>$, the latter is immersed into $ \Pois\Der\<X,d\>$.
Consider the ideal $\hat I$ of
$\Pois\Der\<X,d\>$ generated by~$I$.
It is enough to show
$\hat I \cap \SGD\<X\> = I$. If the latter holds, then
$\Pois\Der\<X,d\>/\hat I$ is a differential Poisson envelope
of the GD-algebra $V =\SGD\<X\>/I$.

An arbitrary element in $f\in \hat I$ may be obtained as
\[
f= \sum_i F_i(X,t)|_{t=u_i},
\]
where $F_i(X,t)\in \Pois\Der\<X\cup\{t\}, d\>$,
$u_i\in I\subset \SGD\<X\>\subset \Pois\Der\<X,d\>$.
Note that $\wt(f)=\wt(u_i)=-1$,
so we may assume
$\wt (F_i) = -1$ for all~$i$ (otherwise, it is enough to choose
the homogeneous components of weight $-1$ for each $F_i$).
Therefore,
\[
F_i(X,t) \in \SGD \<X\cup \{t\} \>,
\]
and
$F_i(X,u_i) \in I$. Hence, $f\in I$.
\end{proof}

\begin{corollary}\label{cor:SGD-Var}
 The class of special Gelfand--Dorfman algebras forms a variety.
\end{corollary}

\begin{remark}
Novikov algebras are particular cases of GD-algebras (with $[x,y]=0$).
It was shown in \cite{dzhuma} that the free Novikov algebra embeds into
the free differential commutative algebra. Hence, restricting the proof
of Theorem \ref{thm:SpecialImage} to Novikov algebras we obtain another
proof of speciality of all Novikov algebras \cite{BCZ2017}.
\end{remark}

The examples of non-special GD-algebras  were constructed in \cite{KSO2019}, \cite{KolPan2019}.
It is worth mentioning that all these examples are of
dimension~3.
In the next statement we show that dimension 3 is a minimal one: all 2-dimensional
GD-algebras are special.

\begin{theorem}\label{thm:2dimSpecial}
Let $V$ be a 2-dimensional GD-algebra. Then $V$ is special.
\end{theorem}

\begin{proof}
Let $X=\{u,v\}$ be a basis of a 2-dimensional GD-algebra $V$.
If $[u,v]=0$ then $V$ is a pure Novikov algebra and by \cite{BCZ2017} $V$ embeds
into the
commutative differential algebra $\Com\Der\<X\>/(xy'-x\circ y)$
which may be considered as a Poisson one with respect to the
trivial bracket.

If $V$ is not abelian as a Lie algebra then we may assume $[u,v]=v$.
It is straightforward do deduce from \eqref{eq:GD1}, \eqref{eq:LSymm},
and \eqref{eq:RComm} that the multiplication table for $\circ $ on $V$ has the
following form:
\[
\begin{gathered}
u\circ u=\alpha u+\delta v, \\
v\circ v=0, \\
u\circ v=\gamma v, \\
v\circ u=\alpha v,
\end{gathered}
\]
for $\alpha,\gamma,\delta \in \Bbbk $.
It is enough to show that $V$ can be embedded into a differential Poisson
algebra. We consider three cases:
(1) $\alpha\neq\gamma$,
(2) $\alpha=\gamma \neq 0$,
(3) $\alpha=\gamma=0$.

Case 1: $\alpha\neq\gamma$.
Note that
$u\circ v-v\circ u=(\gamma-\alpha)v=(\gamma-\alpha)[u,v]$,
i.e.,
\[
[u,v]={\frac{1}{\gamma-\alpha}}(u\circ v-v\circ u).
\]
Therefore, the structure of a GD-algebra on $V$ is almost completely similar to
the commutator GD-algebra on a Novikov algebra considered in \cite{KolPan2019}.
The only difference is the multiplicative constant $\frac{1}{\gamma-\alpha}$
which does not affect the construction of a differential Poisson envelope.
Namely, consider the free differential commutative algebra
$F=\Com\Der\<\{u,v\},d\>=\Com\<u,v,u',v',u'',v'',\dots  \>$,
and define a bracket $\{\cdot,\cdot\}$ on $F$ as follows.
On the generators, let
\begin{equation}\label{eq:Bracket1}
\{u^{(m)},v^{(n)}\}={\frac{1}{\gamma-\alpha}}
 ((n-1)u^{(m+1)}v^{(n)}-(m-1)u^{(m)}v^{(n+1)}), \quad
 n,m\ge 0.
\end{equation}
Then apply the Leibniz rule to expand the bracket on the entire~$F$.
The bracket obtained is proportional to the one considered in
\cite{KolPan2019}. Therefore, the operation
\eqref{eq:Bracket1}
satisfies the Jacobi identity, and
the operation $d$ is a derivation with respect to this bracket.
Consider the ideal $I_V$ of the free differential commutative algebra $F$
generated by $xy'-x\circ y$, $x,y\in \{u,v\}$.
The ideal $I_V$
is invariant under all operations $\{f,\cdot\}$, $f\in F$.
Hence, the universal differential commutative envelope
$F/I_V$ of the Novikov algebra $V$ is also a differential
Poisson envelope of $V$ as of a GD-algebra.

Case 2: $\alpha=\gamma\neq 0$. An obvious change of variables
$\frac{1}{\alpha}u -\frac{\delta}{\alpha^2}v \to u$ ($v\to v$) leads to the
following multiplication table in $V$:
\begin{equation}\label{eq:Mult_Table-2}
{}
[u,v]=\frac{1}{\alpha} v, \quad
u\circ u=u, \quad u\circ v=v, \quad v\circ u=v, \quad v\circ v=0.
\end{equation}

Consider the polynomial algebra $\Bbbk [x,e]$ equipped with two commuting
derivations
\[
d_1 = \dfrac{\partial}{\partial x},\quad
d_2 = \dfrac{e}{\alpha }\dfrac{\partial }{\partial e}.
\]
Then $\{f,g\} = d_1(f)d_2(g)-d_2(f)d_1(g)$,
$f,g\in \Bbbk [x,e]$,
is a Poisson bracket on $\Bbbk[x,e]$, and $d_1$ is a derivation on the Poisson
algebra obtained.
In particular,
\begin{equation}\label{eq:Bracket2}
\{x,e\} = \frac{1}{\alpha } e.
\end{equation}
Denote by $A$ the quotient of $\Bbbk [x,e]$ modulo the ideal $I$ generated by
$e^2$. As $d_1(I)=0$ and $d_2(I)\subseteq I$, the commutative algebra $A$
is a differential Poisson algebra generated by $x$ and $e$ with a bracket
defined via \eqref{eq:Bracket2} and with a derivation
\[
d(x)=1,\quad d(e)=\frac{1}{\alpha } e.
\]
Now, the initial GD-algebra $V$ embeds into $A^{(d)}$ by
\[
u \mapsto x,\quad v\mapsto ex.
\]
Indeed, it is straightforward to check that this
map preserves the multiplication table \eqref{eq:Mult_Table-2}.

Case 3: $\alpha=\gamma=0$.
If $\delta=0$ then $V$ is a pure Lie algebra and thus embeds
into the associated graded Poisson algebra $P(V)=\mathrm{gr}\,U(V)$ of the
universal associative envelope of~$V$ with zero derivation.

If $\delta\neq 0$ then, up to a change of variables, we may assume
\[
[u,v]=v,\quad
u\circ u=v,\quad v\circ v=u\circ v=v\circ u=0.
\]

Consider the polynomial algebra $F=\Bbbk [u,v,u',v']$
in four formal variables. Define a skew-symmetric
bracket $\{\cdot ,\cdot \}$ on $F$ by the Leibniz rule
starting from
\begin{equation}\label{eq:GSB-2}
\begin{gathered}
\{u,v\}=v, \quad \{u,u'\}=u',\quad \{u,v'\}=2v', \\
\{v,u'\}=v',\quad \{v,v'\}=\{u',v'\}=0.
\end{gathered}
\end{equation}
This is a Poisson bracket since it is straightforward
to check the Jacobi identity on the generators.

Let $I$ be the ideal of $F$ generated by
\[
uu'-v,\ uv',\ vu',\ vv',\ vv,\ u'u'-v',\ u'v',\ v'v'.
\]
These polynomials form a Gr\"obner basis in $F=\Bbbk [u,v,u',v']$,
so the quotient $A=F/I$ contains $V$ as a subspace.
The ideal $I$ is closed under applying the bracket $\{x,\cdot \}$ for
$x=u,v,u',v'$. Hence, $A=F/I$ is a Poisson algebra.

The ideal $I$ is closed under the derivation $d$ of $F$ (as of commutative algebra) defined by
\[
d(u)=u',\quad d(v)=v',\quad d(u')=0,\quad d(v')=0.
\]
Therefore, $d$ induces a derivation on $A$.

It remains to check that $d$ is a derivation
relative to the Poisson bracket on $A$,
i.e., $d\{f,g\} =  \{d(f),g\} + \{f,d(g)\}$ for  $f,g\in A$.
 Note that the linear basis of $A$ consists of reduced monomials relative
 to the Gr\"obner basis \eqref{eq:GSB-2}: $u^n$, $u^mv$, for $n,m\ge 0$.
By definition,
\[
 d(u^n) = \begin{cases}
           0,& n=0,\\
           u',& n=1, \\
           nu^{n-2}v, & n\ge 2,
          \end{cases}
\quad
 d(u^mv) = \begin{cases}
            v', & m=0, \\
            0, & m\ge 1.
           \end{cases}
\]
If $f=u$, $g=v$ then
\[
 \{d(f),g\}+\{f,d(g)\} = \{u',v\}+\{u,v'\}=-v'+2v'=v' = d\{f,g\}.
\]
If $f=u^n$, $g=v$ then
\begin{multline}\nonumber
 \{d(f),g\}+\{f,d(g)\} = \{nu^{n-2}v,v\}+\{u^nv',v'\}\\
  =n(n-2)u^{n-3} vv + 2nu^{n-1}v'v'= 0= d\{f,g\}.
\end{multline}
For the remaining pairs of basic elements $f$, $g$, the derivation property
for $d$ may be checked in a similar way.

Hence, $A$ is a differential Poisson algebra.
It follows from the definition of $d$ and $A$ that
$V$ as a GD-algebra embeds into $A^{(d)}$.
\end{proof}

\section{Special identities of GD-algebras}\label{identities}

Since both classes $\GD $ and $\SGD$ are varieties and thus defined
by some set of identities, the interesting problem is
to determine a list of those identities that hold in
$\SGD$ but do not hold in $\GD $. As in the case of Jordan algebras, let us call such identities {\em special}.

In \cite{KSO2019}, the following description
of $\SGD $ was proposed: the operad of special
$\GD$-algebras is a sub-operad in the Manin white product $\GD^!\circ \Pois$.
Here $\Pois $ is the operad of Poisson algebras
and $\GD^! $ is Koszul dual to $\GD $.
However, finding special identities
in this way is technically hard since the
entire operad $\GD^!\circ \Pois$ is pretty large.

In this section, we develop another approach which
allows us to show that the two identities found in
\cite{KSO2019} exhaust all special identities of degree~4
and find the complete list of special identities of degree~5.
In the next section, we apply Gr\"obner basis technique
for operads to show that the identities of degree 5 found in this section follow from the identities of degree~4.

Let $X=\{x_1,x_2,\dots \}$ be a countable set,
and let
$\GD\<X\>$ be the free GD-algebra generated by~$X$.
Suppose $B$ is a linear basis of $\GD\<X\>$, $X\subset B$, equipped with a linear order $\le $.
Note that finding the explicit form
of $B$, or at least of the multilinear
part of $B$, is a separate interesting problem.
Let us construct $U=\Pois\Der\<B,d\>/I$, where
$I$ is the differential ideal generated by
\[
ad(b)-a\circ b,\quad a,b\in B,
\]
\[
\{ a,b\} - [a,b],\quad a,b\in B, \ a>b.
\]
The algebra $U$ constructed is the universal Poisson differential envelope of $\GD\<X\>$.
The kernel of the natural homomorphism
$\tau: \GD\<X\> \to U$, $\tau(b)= b+I$,
is exactly the set of special identities of $\GD$-algebras.

Denote $B^{(\omega )} = \{b^{(n)} \mid b\in B, n\ge 0 \}$
and expand the order $\le $ to $B^{(\omega )}$ by the following rule:
\begin{equation}\label{eq:Order_diff}
b^{(n)}<a^{(m)} \iff (b,-n)<(a,-m)\ \text{lexicographically}.
\end{equation}
The ordering is motivated by the Shirshov's argument \cite{Shir62}
concerning the standard bracketing on Lyndon--Shirshov words.

Let us identify $\Pois\Der\<B,d\>$ with $\Pois\<B^{(\omega )}\>$ assuming $d^n(b)=b^{(n)}$,
then $I$ coincides with the ideal in
$\Pois\<B^{(\omega )}\>$
generated by
\begin{gather}
ab^{(n)}-(a\circ b)^{(n-1)}+\sum_{i=1}^{n-1} \binom{n-1}{i} a^{(i)} b^{(n-i)},  \ n\ge1,
                  \label{eq:IdealGenerators1}
\\
\{a,b^{(n)}\} -[a,b]^{(n)} + \sum\limits_{i=1}^{n} \binom{n}{i} \{a^{(i)},b^{(n-i)} \},\ a>b,\ n\ge 0.
                        \label{eq:IdealGenerators2}
\end{gather}
In order to find the kernel of $\tau $
it is enough to calculate the intersection
of $I$ with $\Bbbk B^{(0)}$.
The latter can be done if we know
the Gr\"obner--Shirshov basis (GSB) of $I$ in the free
Poisson algebra $\Pois\<B^{(\omega )}\>$ \cite{BCZ2019}.

Let us proceed as follows. Consider the Lie algebra
$L= \Lie \<B^{(\omega )} \>/J$, where $J$ is the ideal
generated by \eqref{eq:IdealGenerators2}.
Then $\Pois\<B^{(\omega )}\>/I$
is  the same as
the quotient of the symmetric algebra
$S(L)$ modulo the ideal
generated by all elements of the form
\begin{equation}\label{eq:IdealGen-Pois}
\{a_1^{(n_1)},\{a_2^{(n_2)}, \dots , \{a_k^{(n_k)}, f\}\dots \}\},
\end{equation}
for all  $f$ in \eqref{eq:IdealGenerators1},
$a_i\in B$, $n_i\ge 0$.
Thus it is enough to calculate the intersection
of $L$ with the Gr\"obner basis of an ideal
in $S(L)$ generated by relations \eqref{eq:IdealGen-Pois}.
Let us first find the Gr\"obner--Shirshov basis
of the Lie algebra~$L$ in a slightly more general
context.

\begin{lemma}\label{lem:DiffGSB}
Let $\mathfrak g$ be a Lie algebra with a linearly ordered
basis $B$. Then \eqref{eq:IdealGenerators2} is
a Gr\"obner--Shirshov basis in
$\Lie \langle B^{(\omega )}\rangle $ relative to the deg-lex ordering based on \eqref{eq:Order_diff}.
\end{lemma}

\begin{remark}
Given a Lie algebra $\mathfrak g$ with a linear
basis $B$ as above,
the Lie algebra generated by $B^{(\omega )}$ with
defining relations \eqref{eq:IdealGenerators2} is
the universal differential envelope of~$\mathfrak g$.
\end{remark}

\begin{proof}
One may simply check that all compositions of intersection
of the relations \eqref{eq:IdealGenerators2} are trivial
in the sense of \cite{Shir62} (see also \cite{BokChen14}).
Note that a specific technique for calculating
Gr\"obner--Shirshov bases in differential Lie algebras
was proposed in \cite{CCLiu2009}, but we have a different ordering.

A more conceptual way is based on the following observation.
Denote by $L$ the quotient of
$\Lie \langle B^{(\omega )}\rangle $
by the Lie ideal generated by \eqref{eq:IdealGenerators2}.
The multiplication table on $\mathfrak g$ corresponds to $n=0$.
Add a new letter $t$ to the alphabet $B$ (assuming $t<B$) and construct
\[
U = \Lie \<B,t \mid \{a,b\} - [a,b], \ a>b \>.
\]
Since the multiplication table is always a Gr\"obner--Shirshov basis, the linear basis of $U$ consists of all Lyndon--Shirshov
words $[u]$ in $B\cup \{t\}$ such that their associative images
$u$ do not contain subwords $ab$ for $a>b$.
Such a word $u$ is either equal to $t$ or may be written as
\[
u = u_1a_1t^{k_1}\dots u_2a_2t^{k_2} \dots u_ma_mt^{k_m},
\]
for $u_i\in B^*$, $a_i\in B$, $k_i\ge 1$.
As shown in \cite{Shir62}, the standard bracketing $[u]$ on $u$
is constructed in the same way as on
\[
u_1a_1^{(k_1)}\dots u_2a_2^{(k_2)}\dots u_ma_m^{(k_m)},
\]
where $a^{(k)} = \{\{\dots \{a,t\},\dots , t\}, t \}$
and the order is defined by \eqref{eq:Order_diff}.
Hence, the linear basis of $U$ consists of $t$ and of all those
Lyndon--Shirshov words that are reduced modulo \eqref{eq:IdealGenerators2}.
Therefore, the latter relations form a G\"obner--Shirshov basis
in $\Lie \<B^{(\omega )} \>$, and, in particular,
$U \simeq  \Bbbk t \ltimes L$,
where $\{f,t\} = f'$ for $f\in L$.
\end{proof}

\begin{corollary}\label{cor:LBasis}
The linear basis of $L$ consists of all nonassociative Lyndon--Shirshov words of the form
\[
[x_{11}\dots x_{1l_1}a_1^{(k_1)} \dots x_{m1}\dots x_{ml_m} a_m^{(k_m)}],
\quad k_i\ge 1,\ x_{i1}\le \dots \le x_{il_i}\le a_{i}.
\]
\end{corollary}

In order to find the intersection of $L$ with the ideal in $S(L)$
generated by \eqref{eq:IdealGen-Pois} let us define a rewriting system in $S(\Lie \<B^{(\omega )}\>)$ based on the relations \eqref{eq:IdealGenerators2}
with principal parts $\{a,b^{(n)}\}$, $a>b$,
and on the relations
\eqref{eq:IdealGen-Pois}
by choosing the principal parts
as
\[
a LS(\{a_1^{(n_1)},\{a_2^{(n_2)}, \dots , \{a_k^{(n_k)}, b^{(n)}\}\dots \}\}), \quad n_i\ge 0, \ n\ge 1,
\]
where $LS(u)$ stands for the principal Lyndon--Shirshov word
in the expansion of $u\in \Lie \<B^{(\omega )}\>$.

For example, if
$f = ac'-a\circ c$,
$b> c$,
then the relation
$\{b,f\} = \{b,ac'\}-\{b,a\circ c \}$ from
\eqref{eq:IdealGen-Pois}
gives rise to the following rewriting rule:
\[
a\{b,c'\} \to [a,b]\circ c + [b,a\circ c].
\]
Similarly, $\{c,ab'-a\circ b\}$ gives us
\[
a\{c,b'\} \to [a,c]\circ b + [c,a\circ b].
\]
On the other hand, we have a rule
\[
\{b,c'\} \to \{c,b'\} + [b,c]'.
\]
from \eqref{eq:IdealGenerators2}.

Here we have an ambiguity in $a\{b,c'\}$: there are
two ways how to rewrite it.
This particular critical pair
is convergent by \eqref{eq:GD1}:
\[
a\{b,c'\} \to a\{c,b'\} + a[b,c]'\to
[a,c]\circ b + [c,a\circ b] + a\circ [b,c] =
[a,b]\circ c + [b,a\circ c].
\]

Denote by $\mathcal G(B)$ the oriented graph
with verices $S(\Lie \<B^{(\omega )}\>)$ and
edges defined by the rewriting rules based
on \eqref{eq:IdealGenerators2} and \eqref{eq:IdealGen-Pois}.
These rules preserve the weight ($\wt$) of differential
polynomials in $B^{(\omega )}$ as well as the degree in~$X$.
The set $\Bbbk B$ is homogeneous of weight $-1$.
Hence, $\mathcal G(B)$ splits into connected components $\mathcal G_{n,w}$ that contain vertices of degree $n$ in $X$ of weight~$w$.
Denote $\mathcal G_{n,-1}$ by $\mathcal G_n(B)$.
This graph has no infinite chains (i.e., it is a rewriting system):
as shown in \cite{KSO2019},
these rules applied to a differential
monomial of weight $-1$ rewrite it to an element of
$\Bbbk B$ in a finite number of steps.
This observation allows us
not to define any ordering on commutative monomials in
$S(\Lie \<B^{(\omega )}\>)$.

In order to find the special identities
of degree $n$ (with respect to $X$) it is enough to
find the relations in $\Bbbk B$ that make the rewriting
system $\mathcal G_{n}(B)$ confluent.

Given a fixed integer $n\ge 3$,
it is enough to consider all multilinear differential
 monomials of weight $-1$ and of degree $n$
in $X$, and expand all critical pairs
in the rewriting system $\mathcal G_n(B)$.
Note that all such pairs without brackets (i.e., those from $\Com \<B^{(\omega )}\rangle $) are confluent
by \cite{BCZ2017} and we do not need to consider them. Similarly,
Lemma~\ref{lem:DiffGSB} shows that all ``pure Lie'' critical pairs
$f \leftarrow u\to g$, $u\in \Lie \<B^{(\omega )} \>$,
are trivial.

For $n=3$, the remaining critical pairs
correspond to the ambiguities of the form
$a\{b,c'\}$, $b>c$.
As shown above, these critical pairs are convergent.

For $n=4$, there are five potential ambiguities:
\begin{itemize}
    \item[(A1)] $a\{b,\{c,d'\}\}$, $c>d$;
    \item[(A2)] $a\{\{b,c'\},d\}$, $c\le d$, $b>c$;
    \item[(A3)] $ab'\{c,d'\}$, $c\ge d$;
    \item[(A4)] $ab\{c',d'\}$, $c>d$;
    \item[(A5)] $ab \{c,d''\}$, $c\ge d$.
\end{itemize}

In the case (A1), one may apply either
$\{c,d'\} \to \{c',d\} + [c,d]'$
or the rule coming from
$\{b,\{c, ad'-a\circ d \}\}$ in \eqref{eq:IdealGen-Pois}:
\[
a\{b,\{c,d'\}\} \to f_1= a\{b,\{c',d\}\} + a\{b,[c,d]'\}
\]
or
\[
a\{b,\{c,d'\}\} \to f_2 = \{a,b\}\{c,d'\} + \{a,c\}\{b,d'\} - [b,[c,a]]d' +\{b,\{c,a\circ d\}\}.
\]
Then apply the rewriting rules
coming from \eqref{eq:IdealGen-Pois} to get
\[
f_2\to [c, [a,b]\circ d] - [c,[a,b]]\circ d
 + [b,[a,c]\circ d] +[b,[c,a\circ d]]
\]
and, similarly,
\[
f_1\to -[d, [a,b]\circ c] + [d,[a,b]]\circ c
 - [b,[a,d]\circ c] - [b,[d,a\circ c]] +[b, a\circ [c,d]]
 - [b,a]\circ [c,d]
\]
One may see that
$f_1=f_2$ due to \eqref{eq:GD1}, namely,
$f_2-f_1$ is zero modulo the Gel'fand--Dorfman relations
at $(a,c,d)$ and $(c,[a,b], d)$.

The ambiguity (A2) also gives rise to a
convergent critical pair modulo
\eqref{eq:GD1}, as one may
check in a similar way.

In the case (A3), we have three possible rewriting rules:
\[
\begin{aligned}
\{c,d'\} &{}\to \{c',d\} + [c,d]',\quad \text{if } c>d, \\
a\{c,d'\} &{}\to [c, a\circ d] + [a,c]\circ d, \\
ab' &{} \to a\circ b.
\end{aligned}
\]
The first one, combined with either of other rules, leads
to a convergent critical pair due to \eqref{eq:GD1}.
Consider the pair coming from the last two rules:
\[
\begin{gathered}
ab'\{c,d'\} \to [c,a\circ d]\circ b + ([a,c]\circ d)\circ b, \\
ab'\{c,d'\} \to (a\circ b)\{c,d'\} \to [c,(a\circ b)\circ d] - [c,a\circ b]\circ d.
\end{gathered}
\]
The relation obtained
\begin{equation}\label{eq:Spec1}
[c,a\circ d]\circ b + ([a,c]\circ d)\circ b =
 [c,(a\circ b)\circ d] - [c,a\circ b]\circ d
\end{equation}
is one of the special identities found in \cite{KSO2019}.

For (A4), it is enough to consider the following rules:
\[
\begin{aligned}
a\{c',d'\} &{} \to \{a,c'\}d' + \{c',a\circ d\}, \\
b\{c',d'\} &{}\to \{b,c'\}d' + \{c',b\circ d\}, \\
b\{c',d'\} &{}\to -\{b,d'\}c' - \{d',b\circ c\}.
\end{aligned}
\]
For example, the second rule allows us to rewrite
$ab\{c',d'\}$ as
\[
\{b, (a\circ d)\circ c \} - \{b, a\circ d\}\circ c + \{a\circ c, b\circ d\} -\{a,b\circ d\}\circ c.
\]
Now use \eqref{eq:Spec1} to replace the first term with
$[b, (a\circ c)]\circ d - ([b,a]\circ c)\circ d + [b, a\circ d]\circ c$. As a result, we obtain
\begin{equation}\label{eq:A4case}
ab\{c',d'\} \to
[b,a\circ c]\circ d - ([b,a]\circ c)\circ d + [a\circ c, b\circ d]
- [a,b\circ d]\circ c
\end{equation}

The convergence of the corresponding three critical pairs starting
at $ab\{c',d'\}$ is equivalent to the conditions that
the right-hand side of \eqref{eq:A4case}
is symmetric with respect to the permutation $(a,b)$ and skew-symmetric by $(c,d)$.

Either of these two conditions (due to \eqref{eq:RComm} and anti-commutativity of $[\cdot,\cdot ]$)
gives rise to the following relation:
\begin{multline}\label{eq:Spec2}
2([a,b]\circ c)\circ d
=
[b\circ c, a\circ d] - [a\circ c, b\circ d] \\
+ ([a,b\circ c] -[b,a\circ c])\circ d
+ ([a,b\circ d] - [b,a\circ d])\circ c.
\end{multline}
This is another special identity found in \cite{KSO2019}.

In the case (A5), we have two types of critical pairs:
the first one coming from the rules based on
$\{c,bd''-(b\circ d)' + b'd'\}$ and $\{c,ad''-(a\circ d)' + a'd'\}$,
the second one (for $c>d$) coming from either of the rules above
and $\{c,d''\} + 2\{c',d'\} + \{c'',d\} - [c,d]''$ in
\eqref{eq:IdealGenerators2}.

For example,
\[
ab\{c,d''\} \to f_1 = a\{c,(b\circ d)'\} - a\{c,b'\}d' -ab'\{c,d'\} - a[c,b]d''.
\]
The polynomial $f_1$ rewrites in $\mathcal G_4(B)$ as follows:
\[
f_1 \to ([c,a], b,d) - [c, (a,b,d)] + (a,[c,b], d).
\]
Here $(x,y,z)$ stands for $(x\circ y)\circ z - x\circ (y\circ z)$.
The expression obtained is symmetric with respect to $(a,b)$
by \eqref{eq:LSymm}
which means the convergence of the first type critical pair.

The second type critical pairs based on (A5) appear
when we rewrite
\[
ab[c,d'']\to f_2 = ab\{d,c''\} - 2ab\{c',d'\} + ab[c,d]''.
\]
We already know how to rewrite all terms in the right-hand side to get an element in $\Bbbk B$.
Comparing the result with what is obtained from $f_1$
we obtain a relation which is a corollary of
\eqref{eq:GD1}, \eqref{eq:Spec1}, and \eqref{eq:Spec2}.
(Note that \eqref{eq:GD1} and \eqref{eq:Spec1}
allow us to rewrite both summands in $[c,(a,b,d)]$
via shorter terms, like those in \eqref{eq:Spec2}.)

In \cite{KSO2019} it was proved that
\eqref{eq:Spec1} and \eqref{eq:Spec2} are independent identities
on a GD-algebra.  Now we may state the following

\begin{proposition}
All special identities of $\GD$-algebras of degree  $\le 4$ are
consequences of (\ref{eq:Spec1}) and (\ref{eq:Spec2}).
\end{proposition}

In a similar way, we may find the complete list of special identities
of degree 5.
The list of ambiguities in $\mathcal G_5(B)$ is relatively long: we have to consider all Poisson differential monomials
of degree 5 and weight~$-1$:
\[
\begin{gathered}{}
[a,[b,[c,d']]]e,\ [a,[b,c'']]de,\
[a,[b',c']]de,\ [a',[b,c']]de,\
[a,[b,c']]de',\ [a,b'][c,d']e, \\
[a,b^{(3)}]cde,\ [a',b'']cde,\ [a,b'']cde',\ [a',b']cde',\ [a,b']cde'',\ [a,b']cd'e'.
\end{gathered}
\]
As a result of the same computations as for $n=4$, we obtain
three special identities of degree~5:
\begin{multline}\label{eq:Spec4}
[d\circ a,[b,e\circ c]]=[e\circ a,[b,d\circ c]]+[d,[b,e\circ c]\circ
a]-[d,[b,e]\circ a]\circ c+([d,[b,e]]\circ c)\circ a  \\
-[d,[b,e\circ c]]\circ a-[e\circ a,[b,d]\circ c]-[e,[b,d\circ c]]\circ
a+[e,[b,d]\circ c]\circ a+[d\circ a,[b,e]\circ c] \\
+[d,[b,e\circ c]]\circ a-[d,[b,e]\circ c]\circ a-[e,[b,d\circ c]\circ
a]+[e,[b,d]\circ a]\circ c-([e,[b,d]]\circ c)\circ a \\
+[e,[b,d\circ c]]\circ a,
\end{multline}
\begin{multline}\label{eq:Spec3}
[c,[a,e\circ b]\circ d]=[a,[c,e\circ d]\circ b]-[a,[c,e\circ d]]\circ
b-[a,[c,e]\circ b]\circ d+([a,[c,e]]\circ d)\circ b \\
+[c,[a,e]\circ d]\circ b+[c,[a,e\circ b]]\circ d-([c,[a,e]]\circ d)\circ b,
\end{multline}
and
\begin{multline}\label{eq:Spec5}
[a,d\circ b]\circ(c\circ e)=[a,c\circ b]\circ(d\circ e)+([a,d\circ b]\circ
c)\circ e+([a,d]\circ(c\circ e))\circ b-(([a,d]\circ c)\circ e)\circ b \\
-([a,c\circ b]\circ d)\circ e-([a,c]\circ(d\circ e))\circ b+(([a,c]\circ d)\circ
e)\circ b.
\end{multline}
Other critical pairs are convergent modulo  \eqref{eq:Spec1}, \eqref{eq:Spec2}, \eqref{eq:Spec4}, \eqref{eq:Spec3}, and \eqref{eq:Spec5}.
We do not state the details here since in the next section
we show that in fact all special identities of degree $\le 5$
are corollaries of \eqref{eq:Spec1} and \eqref{eq:Spec2}.

\section{On the Gr\"obner basis of the Gelfand--Dorfman operad with special identities}\label{programm}

A (symmetric) operad $\mathcal P$ in the category $\Vecc_\Bbbk $ of linear
spaces over a field $\Bbbk $ is
a collection of spaces $\mathcal P(n)$, $n\geq 1$,
equipped with linear composition maps
\[
\gamma^m _{n_1,\dots,n_m} :\mathcal P(m)\otimes \mathcal P(n_1)\otimes\ldots
\otimes \mathcal P(n_m)\rightarrow \mathcal P(n_1+\ldots +n_m)
\]
for all integers $m,n_1,\ldots,n_m\geq 1$,
each $\mathcal P(n)$ is a module over the symmetric group $S_n$.
These data have to satisfy the following conditions:
the composition is associative and equivariant relative to the
action of $S_n$; the space $\mathcal P(1)$ contains an element $1$
which acts as an identity relative to the composition.

Every operad may be considered as an image of an
appropriate free operad, i.e., a quotient
modulo an operad ideal.

Namely, for every graded space
$V = \bigoplus_{n\ge 1} V(n)$ there exists a uniquely
defined free operad $\mathcal F(V)$ generated by $V$.
An operad ideal of $\mathcal F(V)$
may be presented as a minimal one that contains a given series of elements from
$\bigcup\limits_{n\geq 1} \mathcal F(V)(n)$.
Therefore, an operad may be defined by generators and relations.

For example, the operad $\Lie $
governing the variety of Lie algebras
is generated by $V_1=V_1(2)$, where $\dim V_1(2)=1$,
this is a skew-symmetric $S_2$-module.
The free operad $\mathcal F(V_1)$ is exactly the operad
of anti-commutative algebras. The set of defining relations
of $\Lie $ consists of the Jacobi identity:
\[
\gamma^2_{2,1} (\mu,\mu,1) + \gamma^2_{2,1} (\mu,\mu,1)^{(123)} + \gamma^2_{2,1}
(\mu,\mu,1)^{(132)}.
\]

The operad $\Nov $ of the variety of Novikov algebras
is generated by $V_2 = V_2(2)$, $\dim V_2=2$, this is the
regular $S_2$-module. Namely, a basis of
$V_2(2)$ consists of $\nu$, $\nu^{(12)}$.
The free operad $\mathcal F(V_2)$ is exactly the magmatic one.
The defining relations of $\Nov $ include left symmetry and right
commutativity:
\[
\begin{gathered}
\gamma^2_{2,1} (\nu,\nu,1) - \gamma^2_{2,1} (\nu,\nu,1)^{(12)} -
\gamma^{2}_{1,2} (\nu,1,\nu)
+  \gamma^{2}_{1,2} (\nu,1,\nu)^{(12)}, \\
\gamma ^2_{1,2}(\nu,1,\nu) -  \gamma^2_{1,2} (\nu,1,\nu)^{(23)}.
\end{gathered}
\]

There is an intermediate notion between nonsymmetric and symmetric operads, known as a {\em shuffle operad}.
By definition, a shuffle operad $\mathcal V$ is a collection of spaces $\mathcal V(n)$, $n\ge 1$, equipped with
a collection of compositions
\[
 \gamma_\pi : \mathcal P(m)\otimes \mathcal P(n_1)\otimes\ldots
\otimes \mathcal P(n_m)\rightarrow \mathcal P(n_1+\ldots +n_m)
\]
where
$\pi $ is a {\em shuffle partition} of the set $\{1,\dots , n\}$,
$n=m_1+\dots + m_n$, into $m$ disjoint subsets $I_j$, $j=1,\dots, m$,
such that $\min I_1<\min I_2<\dots <\min I_m$.
Thus, a shuffle operad has no symmetric module structure, but its composition
structure still carries some information about the order of arguments.

Shuffle operads provide a convenient framework for the computation of
Gr\"obner bases and normal forms
in an operad defined by generators and relations \cite{DotKhor2010}.
Every symmetric operad may be considered as a shuffle one \cite{bremner-dotsenko} provided that the operation set
is extended.

Following \cite{bremner-dotsenko}, we  may convert the symmetric operad $\GD$ into
a shuffle operad $\GD^f$, a homomorphic image of the shuffle tree operad $\mathcal{T}_\Sha(\mathcal{X})$
for an appropriate language $\mathcal X$.

Let us replace the operations $\nu $ and $\mu $ with the set of three operations
$\mathcal X=\{x,y,z\}$, where $x$ and $y$ represent $\nu $ and $\nu^{(12)}$,
$z$ represents~$\mu $. We will use the natural notation $x(1\;2) = \nu (x_1,x_2)$,
$y(1\;2)=\nu(x_2,x_1)$, etc.
To convert the defining relations of the operad $\GD$
into elements of
$\mathcal{T}_\Sha(\mathcal{X})$
we may use the equivariance property of the composition (see \cite{bremner-dotsenko} for details).
As a result, \eqref{eq:LSymm} and \eqref{eq:RComm} turn into the following relations for
$x$ and $y$:
\[
\begin{gathered}
x(x(1\;2)\;3) - x(1\; x(2\; 3)) - x(y(1\; 2)\; 3) + y(x(1\; 3)\; 2), \\
x(x(1\; 3)\; 2) - x(1\; y(2\; 3)) - x(y(1\; 3)\; 2) + y(x(1\; 2)\; 3),\\
y(1\; x(2\; 3)) - y(y(1\; 3)\; 2) - y(1\; y(2\; 3)) + y(y(1\; 2)\; 3),\\
x(x(1\; 2)\; 3) - x(x(1\; 3)\; 2),\\
x(y(1\; 2)\; 3) - y(1\; x(2\; 3)),\\
x(y(1\; 3)\; 2) - y(1\; y(2\; 3)).
\end{gathered}
\]
The Jacobi identity may be expressed in terms of $z$
as
\[
z(z(1\; 2)\; 3) - z(1\; z(2\; 3)) - z(z(1\; 3)\; 2).
\]
Similarly, converting \eqref{eq:GD1} (namely, all relations in the symmetric group orbit of the
defining relation), we obtain
\[
\begin{gathered}
z(1\; x(2\; 3)) + z(y(1\; 2)\; 3) - x(z(1\; 2)\; 3) - y(1\; z(2\; 3)) -
y(z(1\; 3)\; 2), \\
-z(x(1\; 3)\; 2) + z(x(1\; 2)\; 3) + x(z(1\; 2)\; 3) - x(z(1\; 3)\; 2) - x(1\;
z(2\; 3)), \\
-y(z(1\; 2)\; 3) + z(1\; y(2\; 3)) + z(y(1\; 3)\; 2) - x(z(1\; 3)\; 2) + y(1\;
z(2\; 3)).
\end{gathered}
\]
The elements ({\em shuffle tree polynomials} \cite{bremner-dotsenko}) obtained generate the ideal of defining relations for the operad
$\GD^f$, a quotient of $\mathcal{T}_{\Sha}(\mathcal{X})$.

Calculating the Gr\"obner base of $\GD^f$ by means of the package \cite{DotsHij},
we get the following result for $\dim \GD(n)=\dim \GD^f(n)$.
\begin{center}
\begin{tabular}{c|cccccc}
 $n$ & 1 & 2 & 3 & 4 & 5 & 6 \\
 \hline
 $\dim(\GD(n)) $ & 1 & 3 & 17 & 140 & 1524 & 20699
\end{tabular}
\end{center}

The first five terms of the sequence coincide with the number of certain planar graphs (see OEIS A322137, A291842). However, the sixth one is different. Finding a linear basis of the free $\GD$-algebra, or at least the sequence $\GD(n)$
is an interesting open problem. Note that the Poincar\'e series of this operad is not a rational function (the operad $\GD $ is not Koszul since so is $\Nov$ \cite{dzhuma1}).

By Corollary \ref{cor:SGD-Var}, the class $\SGD$ is defined by identities.
There should exist identities separating $\SGD$ from $\GD$, i.e., independent special identities of $\GD$-algebras.
Let us consider the class of $\GD$-algebras with additional identities (\ref{eq:Spec1}) and
(\ref{eq:Spec2}). Denote this class by $\wSGD$ (weak special Gelfand-Dorfman algebra).

As above, we may convert the defining relations of $\wSGD$ into shuffle tree polynomials
 adding the orbits of relations (\ref{eq:Spec1}) and (\ref{eq:Spec2}) to $\GD^f$.
Namely, (\ref{eq:Spec1}) and (\ref{eq:Spec2}) give rise to the following elements of
$\mathcal{T}_{\Sha}(\mathcal{X})$:
\[
\begin{gathered}
z(1\; x(x(2\; 3)\; 4)) - x(z(1\; x(2\; 3))\; 4) - x(z(1\; x(2\; 4))\; 3) +
x(x(z(1\; 2)\; 3)\; 4), \\
z(1\; x(y(2\; 3)\; 4)) - x(z(1\; y(2\; 3))\; 4) - x(z(1\; x(3\; 4))\; 2) +
x(x(z(1\; 3)\; 2)\; 4), \\
z(1\; y(2\; y(3\; 4))) - x(z(1\; y(3\; 4))\; 2) - x(z(1\; y(2\; 4))\; 3) +
x(x(z(1\; 4)\; 2)\; 3),\\
-z(x(x(1\; 3)\; 4)\; 2) + x(z(x(1\; 3)\; 2)\; 4) + x(z(x(1\; 4)\; 2)\; 3) - x(x(z(1\; 2)\; 3)\; 4), \\
-z(x(y(1\; 3)\; 4)\; 2) + x(z(y(1\; 3)\; 2)\; 4) - y(1\; z(2\; x(3\; 4))) + x(y(1\; z(2\; 3))\; 4), \\
-z(x(y(1\; 4)\; 3)\; 2) + x(z(y(1\; 4)\; 2)\; 3) - y(1\; z(2\; y(3\; 4))) + x(y(1\; z(2\; 4))\; 3), \\
-z(x(x(1\; 2)\; 4)\; 3) + x(z(x(1\; 2)\; 3)\; 4) + x(z(x(1\; 4)\; 3)\; 2) - x(x(z(1\; 3)\; 2)\; 4), \\
-z(x(y(1\; 2)\; 4)\; 3) + x(z(y(1\; 2)\; 3)\; 4) + y(1\; z(x(2\; 4)\; 3)) - y(1\; x(z(2\; 3)\; 4)), \\
-z(x(y(1\; 4)\; 2)\; 3) + x(z(y(1\; 4)\; 3)\; 2) + y(1\; z(y(2\; 4)\; 3)) + y(1\; y(2\; z(3\; 4))), \\
-z(x(x(1\; 2)\; 3)\; 4) + x(z(x(1\; 2)\; 4)\; 3) + x(z(x(1\; 3)\; 4)\; 2) - x(x(z(1\; 4)\; 2)\; 3), \\
-z(x(y(1\; 2)\; 3)\; 4) + x(z(y(1\; 2)\; 4)\; 3) + y(1\; z(x(2\; 3)\; 4)) - x(y(1\; z(2\; 4))\; 3), \\
-z(x(y(1\; 3)\; 2)\; 4) + x(z(y(1\; 3)\; 4)\; 2) + y(1\; z(y(2\; 3)\; 4)) - x(y(1\; z(3\; 4))\; 2)
\end{gathered}
\]
and
\begin{multline*}
z(x(1\; 2)\; x(3\; 4)) - x(z(x(1\; 2)\; 3)\; 4) - x(z(1\; x(3\; 4))\; 2) + 2 x(x(z(1\; 3)\; 2)\; 4) \\
+ z(x(1\; 4)\; y(2\; 3)) - x(z(1\; y(2\; 3))\; 4) - x(z(x(1\; 4)\; 3)\; 2),
\end{multline*}
\begin{multline*}
z(x(1\; 3)\; x(2\; 4)) - x(z(1\; x(2\; 4))\; 3) - x(z(x(1\; 3)\; 2)\; 4) + 2 x(x(z(1\; 2)\; 3)\; 4) \\
+ z(x(1\; 4)\; x(2\; 3))- x(z(1\; x(2\; 3))\; 4) - x(z(x(1\; 4)\; 2)\; 3),
\end{multline*}
\begin{multline*}
z(y(1\; 2)\; y(3\; 4)) - y(1\; z(2\; y(3\; 4))) - x(z(y(1\; 2)\; 4)\; 3) + 2 y(1\; x(z(2\; 4)\; 3))  \\
- z(y(1\; 4)\; x(2\; 3))+ x(z(y(1\; 4)\; 2)\; 3) - y(1\; z(x(2\; 3)\; 4)),
\end{multline*}
\begin{multline*}
z(y(1\; 2)\; x(3\; 4)) - y(1\; z(2\; x(3\; 4))) - x(z(y(1\; 2)\; 3)\; 4) + 2 y(1\; x(z(2\; 3)\; 4)) \\
- z(y(1\; 3)\; x(2\; 4)) + x(z(y(1\; 3)\; 2)\; 4) - y(1\; z(x(2\; 4)\; 3)),
\end{multline*}
\begin{multline*}
z(y(1\; 3)\; y(2\; 4)) + y(1\; z(y(2\; 4)\; 3)) - x(z(y(1\; 3)\; 4)\; 2) + 2 y(1\; y(2\; z(3\; 4))) \\
- z(y(1\; 4)\; y(2\; 3)) - y(1\; z(y(2\; 3)\; 4)) + x(z(y(1\; 4)\; 3)\; 2),
\end{multline*}
\begin{multline*}
z(x(1\; 2)\; y(3\; 4)) - x(z(1\; y(3\; 4))\; 2) - x(z(x(1\; 2)\; 4)\; 3) + 2 x(x(z(1\; 4)\; 2)\; 3) \\
+ z(x(1\; 3)\; y(2\; 4)) - x(z(x(1\; 3)\; 4)\; 2) - x(z(1\; y(2\; 4))\; 3).
\end{multline*}

Calculating the Gr\"obner base of the operad $\wSGD^f$ by means of the package \cite{DotsHij},
we get the following result:
\begin{center}
\begin{tabular}{c|ccccc}
 $n$ & 1 & 2 & 3 & 4 & 5 \\
 \hline
 $\dim(\wSGD (n)) $ & 1 & 3 & 17 & 130 & 1219
\end{tabular}
\end{center}
In particular,  we have $\dim \SGD(n)=\dim \wSGD(n)$ for $n\le 5$.
Hence, all special identities of degree $\le 5$ are corollaries of (\ref{eq:Spec1}) and (\ref{eq:Spec2}). We obtain the following result:

\begin{corollary}
The relations
(\ref{eq:Spec4}), (\ref{eq:Spec3}) and (\ref{eq:Spec5}) are consequences
of (\ref{eq:Spec1}) and (\ref{eq:Spec2}).
\end{corollary}

It remains an open problem whether (\ref{eq:Spec1}) and (\ref{eq:Spec2}) exhaust all
independent special identities of GD-algebras.

redpajama_set_name: RedPajamaArXiv


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text:
What To Do About Herod
Sermon for January 3, 2016 (Christmas 2, Year C)
Offered by Nathan Ferrell at The Episcopal Church of Saint Mary
Texts: Jeremiah 31:7-14; Psalm 84:1-8; Matthew 2:13-23
Title: What to do about Herod
Still rage the fires of hate today, and innocents the price must pay,
while aching hearts in every land, cry out, "We cannot understand."
[My friends]: What are we to do about Herod?
This King Herod crashes down right in the middle of our Christmas celebration, and he brings with him his terrifying reign of violence and bloodshed.
Why? Why must there be a Herod to interrupt every one of our celebrations?
And what are we to do about Herod?
On December 28 of each year, the Church remembers the slaugher of the Holy Innocents. This is one of our forgotten and overlooked commemorations, since it comes right after Christmas and in the middle of a vacation week.
I think it is ironic that our current calendars do not permit a robust remembrance of this tragedy.
I say ironic, because our society has undertaken such great pains to remove death and pain and suffering from our everyday lives.
Just think about it:
We have flooded our population with pain-killing medications.
We do not want the needs of our elders to interrupt our lives, so we put them away in homes to be cared for by professionals.
When people die, we no longer take care of the bodies and clean and dress them in our living rooms, in our parlors. Now we call the professionals and have them take the dead away as soon as possible.
Usually, when that happens, all we get back nowadays is a small box of ashes – cremains they are properly called – in a clean, small container.
Let's face it: the only place where we like to see death and pain and suffering in our time is on TV or in the movies. But never, ever in real life.
We do not actually observe the commemoration of the Holy Innocents, so I changed the Gospel reading for today.
You see, on this second Sunday in Christmas, verses 16, 17, 18 of Matthew's second chapter are always left out – skipped over! – so that all we get is the Holy Family's flight to, and return from, Egypt.
The reason, I suppose, is that we already did this story on December 28.
But do you know what I think? I think they are left out, because we do not want the reality of death and pain and suffering to interrupt our Christmas frivolity.
Keep those uncomfortable things safely and cleanly contained on December 28 – even though no one pays attention that day!
We are obligated to remember the Holy Innocents of Bethlehem. We will do it, but let's not bring them out into public view. If only life were that simple, right?
But this story from Matthew's Gospel reminds us of the truth:
Life on this planet earth is incredibly complex, and life and death are always intertwined in an inseparable bond.
Let's go back to the story for a bit and look at it in more detail.
There are at least FIVE different layers at work when we read this story of the Holy Innocents.
The story of Rachel, the beloved wife of the patriarch Jacob, who died in labor while giving birth to Benjamin, and who is referenced by the prophet Jeremiah.
The story of Pharaoh, the archenemy of the Israelites, who ordered the death of all male Hebrew children in Egypt.
The story of the exiles who were oppressed by the King of Babylon and dispossessed from their ancestral lands, which is the cause of Rachel's weeping in Jeremiah's text.
The story of the Holy Family fleeing in search of safety due to Herod's desire to crush any potential competitor for his throne.
The story of our own day in which unprecedented numbers of human beings all across the earth are fleeing in search of safety due to violence, tyranny and oppression.
What are you and I to do about Herod?
We need to know at least a bit about the original Herod.
This is important, because the shadow of Herod looms over all of the Gospel stories and it lingers in the background of everything that we can say about the life of Jesus.
Herod was a Jew. He was born around the year 74 Before Christ and his father was a high-ranking official in Idumea, just south of Judea.
Ambitious from birth, Herod earned an appointment as governor of Galilee when he was a young man. By supporting the various leaders of Rome, he was eventually awarded by the Roman Senate with the title of "King of the Jews".
Through bloody battles and sinister deal-making, he finally won control over all of Judea in the year 37 BC and so began the Herodian dynasty.
The House of Herod was established.
Because he licked the boots of every Roman emperor, Rome allowed Herod to rule his kingdom with an iron fist.
He was King of the Jews, and he made sure that everyone knew it!
He had coins minted for his realm which bore the title: "the kingdom of Herod".
Every potential rival for his throne was eliminated, including three of his very own sons! Because of this, Caesar Augustus famously remarked that it was safer to be one of Herod's pigs than one of his sons! As a Jew, of course, Herod did not eat pork! But he did kill his own family, if he suspected them of possible treason.
When Matthew's Gospel reports that the Magi came to Herod and asked,
"Where is the child who has been born king of the Jews?" (Matthew 2:2) you can be certain that Herod must have erupted in rage!
Who would DARE to usurp that title which he had won for himself by hard battle and effort and sacrifice? Anyone who tried to take it would be killed.
Truth be told, we don't actually know if this horrible incident in Bethlehem took place. Matthew's Gospel is the only record of it.
We DO know that Herod would not have thought twice about it!
As his own death approached, the historian Josephus stated that Herod was so concerned that no one would mourn his passing, that he commanded a large group of distinguished men to come to Jericho, and he gave an order that they should be slain by the sword on the day of his death so that the whole region would be filled with grieving mourners. Thankfully, Herod died and his children refused to carry out this command.
I know that this is a dark and sad story, and that we all want to sing a few more happy Christmas carols before we have to let them go until next December, but this is part of the Christmas story as well – and it needs to be remembered.
Life and death are always intertwined in a complex story.
And it makes me wonder:
Do you think that Jesus knew about these deaths in Bethlehem? Of course, the Gospels tell us that he was an infant when the slaughter took place. And the Holy Family was not present when it happened.
But did Mary and Joseph hear about it? And did they tell Jesus about it when he became older? And if so, what did they say? And how did he feel about this?
About dozens of young boys being killed because of him?
Did he struggle with survivor's guilt?
And how did he not grow up to be entirely bitter and angry toward Herod?
Not seeking revenge against Herod for his evil brutality?
When we consider what to do about the Herod's of our day, how can we learn from our Lord's teaching and example?
What are we to do about Herod?
The House of Herod is long since gone, but the bitter effects of tyranny and oppression will always be with us.
Herod thinks that, by killing every threat to his rule, his reign will continue and he will gain glory and honor for himself and his family.
But he is sadly mistaken.
Pope Leo the Great, Bishop of Rome in the 5th century, once famously preached about this story and directed his words straight at every single Herod:
"The fear that racks you, Herod, is wasted; in vain do you try to vent your rage on the infant you suspect. He, whom you do not wish to reign in Judæa, [already] reigns everywhere!"
My friends: what are we to do about Herod?
Resist him, we must. And strive always an ceaselessly to protect the innocents.
But Christ already reigns everywhere, even in this complex world of life and death.
Thanks be to God that WE do not have to fix this world, or even to save it.
With trust in that reality, we are free to sing – and rejoice – and to pray!
Please bow your heads as we pray this collect for the commemoration of the Holy Innocents:
"We remember today, O God, the slaughter of the holy innocents of Bethlehem by King Herod. Receive, we pray, into the arms of your mercy all innocent victims; and by your great might frustrate the designs of evil tyrants and establish your rule of justice, love, and peace;" In the name of Jesus Christ, the great Prince of Peace, we pray. May it be so. Amen.
TOPIC: Violence
SCRIPTURE: New Testament
Sermon Details

redpajama_set_name: RedPajamaCommonCrawl


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April 06, 2021 at 7:00pm - 8:15pm
Countering prejudice and building cohesion: The More in Common way
The Common Room Series #4
Decades of research, supported by our own experiences, show that regular positive interactions with people different to ourselves are a powerful and effective way to reduce prejudice, counter harmful stereotypes and build community cohesion. Yet we know that many people have little contact with people who are different to them, an issue the pandemic has made even harder to overcome.
How can we - as people who care about our communities - create opportunities for people to meet and build meaningful relationships with those different to themselves? What are the creative ways we can counter hatred and prejudice through community-led initiatives?
Join us in the Common Room for a collaborative workshop, exploring how we can use social connection to counter prejudice within our communities. The session will be co-hosted by Co-Executive Director of Protection Approaches, Andy Fearn and our Communities Manager Emily Danby.
The Common Room brings together people from across the UK who share a passion for building stronger, more connected communities.
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'I told him not to let go, and he did not let go': Family mourns devoted father killed in Whatcom flood
Mayowa Aina
Jose Garcia, 50, was one of the few fatalities from the flood. Garcia was married with four children: three daughters and a son. His children describe him as a loving and hardworking man who did whatever he could to put a smile on their faces. "I just got to stick by my mom's side," said Garcia's son, Luis Garcia, 24. "She's really hurting. She lost her other half."
Single Dads By Choice: More Men Going It Alone
Jennifer Ludden
A small but growing number of men — gay and straight — are deciding to have children on their own, just as more women began doing two decades ago. Single dads say they still face a social stigma but find parenting emotionally rewarding.


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Property / Conveyancing
Building your own home
Selling a residential investment property
Switch My Mortgage in 10 Steps
Purchasing from a receiver
Landlord and Tenancy Law
Managing a residential investment property
Making a Will
Extracting a Grant of Probate
Extracting a Grant where there is no Will
Spouse's rights to inherit
Civil Partner's right to inherit
Co-habitee's right to inherit
Cycling Personal Injuries
Recovery of debt below €15,000
Recovery of debt €15,000-€75,000
Recovery of debt over €75,000
Driving Without a Licence
Driving Without Insurance
Driving Without Reasonable Consideration
Drug Driving
Judicial Separation
Civil Partnerships
Personal Injury Claims *
MB Solicitors provide expert legal advice in relation to personal injuries including the following:
Slip and Trip
Injured while on a premises or property belonging to someone else
Children and personal injury
Frequently asked questions include:
What is Whiplash?
Do I need a solicitor?
What will my solicitor do for me?
What if the other party did not have insurance?
How much compensation am I entitled to?
Do I have to pay tax on compensation?
What does no win no fee mean?
Car Accidents often result in Personal Injury. At MB Solicitors you will be advised free of charge, as to whether you can bring a successful claim for compensation.
What should I do if I have been involved in a car accident?
Take details of the other driver involved e.g.
Registration number of their vehicle
Take details of any witnesses
Seek Medical assistance, preferably from your own doctor.Contact the Gardaí and request them to attend the scene of the accident. I f they refuse you should attend the nearest station and request that they record details of the accident.
Contact us before speaking to the insurance company. It may not be in your interest to settle your claim early. You may not know the extent of the injuries suffered. It is only when you have had the benefit of medical and legal advice will you be in a position to settle your claim.
What should I not do?
Do not admit liability at the scene.
Do not leave the scene without obtaining the other parties details.
Can I claim for damage to my car?
Yes, you are entitled to claim for any damage to your car. However, you should consult with us before accepting any offer. We will ensure that the payment does not prejudice your right to bring a claim for the personal injuries suffered.
If you have been injured in a shop or a shopping centre you could be entitled to compensation. The owner of a chop or shopping centre must take reasonable care to ensure that the shop is a safe pace for customers.
All supermarkets and shops should have a cleaning policy in place. If you slip on a wet floor resulting in personal injury, then you may be entitled to compensation.
I have slipped on a wet floor what should I do?
Take contact details for any witnesses
Take photographs of the where the accident occurred
Take a note of whether there were any wet floor signs
Report accident to a member of staff
Seek medical assistance for your injuries
Contact us so that we can advise you on the likelihood of success and ensure all of the evidence is preserved
How do I know whether I have a claim?
For a free assessment of your claim please leave brief details including the following in the contact form:
The date of the accident
The injury suffered
Brief description of the accident
Alternatively, please request a call back and a solicitor will contact you to discuss your case.
Employers have a duty of care to provide a safe place to work for all of their employees. If you are injured at work you may be entitled to financial compensation.
If you are involved in a accident at work you should do the following:
Report the accident
Take photographic evidence
Record details of the accident
Record contact details of any witnesses
Seek medical assistance
Contact MBSolicitors
For a free assessment of the strength of your case please contact us through the contact form or telephone us at 01 5677343
If you are injured on property/house or other premises that you are not the owner of you may be entitle to compensation. The Occupiers Liability Act 1995 is the relevant law governing such injuries. The law divides those who may be entitled to compensation for injuries into three types: visitors, recreational users and trespassers.
In relation to visitors, the owner/occupier of the property must take reasonable care to ensure that the visitor dopes not suffer injury or damage while on the property.
The responsibility towards a recreational owner or trespasser while it exists it is considerably less that that owed to a visitor.
In order to establish whether you are entitled to compensation while injured on property/land/house belonging to another person please contact us by phone or by using the contact form.
My child has suffered personal injury
If you child has suffered a personal injury the time within which proceedings can be take is two years from the date of the injury or where 2 years from the child's eighteenth birthday. However, it is always better to bring the case as soon as possible so that crucial evidence is not lost or witnesses have not moved house/forget the details of the accident.
The case can be brought by the child's "next friend" usually on of the child's parents. The "next friend" is responsible for initiating the legal proceedings and can accept an offer made by the defendant. However, any such offer must be considered or approved by the court.
Whiplash is a common term used to describe a strain to a personas neck which is caused by a high impact slip/trip or fall. It commonly occurs as a result of cash crashes or other road traffic accidents.
An Insurance company is a business. Business must make a profit. Therefore, it is in their interest to pay a reduced amount of compensation. They have dedicated claims handlers who are trained to negotiate settlement with the injured party. Unless you have the same training or have the benefit of legal advice you are at a disadvantage. Your solicitor is trained to negotiate on your behalf and it is our business to ensure that you obtain the maximum compensation which you are entitled to.
We will assess your claim and advise on the likelihood of success
We will write to the other party involved to ensure that all the evidence is preserved so that it can be assessed
We will liaise with your doctor to obtain a legal medical report
We will make the application for the Injuries Board on your behalf
We will negotiate with the insurance company on your behalf to ensure that you get the highest level of compensation which you are entitled to
If the claim cannot be settled, we will institute court proceedings
Motor insurance is compulsory in Ireland. However, there are cases where a driver involved in an accident does not have insurance. This does not mean that you cannot bring a claim as it may be possible for you to bring a successful claim to the Motor Insurers Bureau of Ireland commonly known as "MIBI".
For a free and quick assessment of your case please give brief details of the accident in the contact form and we will respond to you. Alternatively, please call us at 01 5380084
The amount of compensation that you are entitled to will depend on the injury sustained. The following is a guide as to the what compensation you may be entitled to for some of the more common injuries.
Head Injuries:
Minor up to €21,000
Moderate €19,000-€35,200
Severe €41,600 to €74,000
Eye Injuries:
Minor up to €9,800
Moderate €22,500 to €45,400
Severe up to €138,000
Facial Injuries:
Fractured Cheek minor €21,400 to €42,400
Moderate 37700 to €47,300
Nose Fractures
Minor €18,000 to €22,100
Moderate 22,100 to €32 200
Sever €44,500 to €63,900
Loss of milk tooth (baby tooth) €4000 to €7,000
Broken Tooth €7500 to €10,300
Loss of Tooth €10,300 to €12,700
Compensation is not payable on an award of compensation.
*In contentious business a solicitor may not calculate fees or other charges as a percentage or proportion of any award or settlement.
If you would like to find out if you are entitled to compensation, please contact our expert personal injury solicitors to by phone
on 01 567 7343 or use the contact form on the website.
Michael Burns Solicitors
97 Malahide Road
D03 X2P6
Office: 01 567 7343
Email: info@mbsolicitors.ie
Property & Conveyancing
Road Traffic – Drug Driving
Road Traffic – Drink Driving
*In contentious business, a solicitor may not calculate fees or other charges as a percentage or proportion of any award or settlement.


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Office 365 | Microsoft Office 365 | Office 365 Online
February 25, 2020 by quizzec
Office 365 is Microsoft product that has served millions of people for nearly three decades, the earlier versions of Microsoft office initially had a combination of Word Excel and PowerPoint which come in a discounted bundle.
It is now the new version of Microsoft office, differs from the older model in two aspects; first, its services come in the form of monthly subscription and no longer as a perpetual license. And secondly, it offers a combination of online services, which include traditional desktop apps, business-class mail, and cloud file storage.
Microsoft Office 365 is a subscription service that provides individuals with the necessary workforce they need for modern productivity; it enables users to get their work done quite easily. Office 365 makes it easy for you to access your documents and applications easily and get work done as long as you are online.
Office 365 refers to services that include cloud services like SkyDrive online storage, Skype minutes for home use, Lync web conferencing, or Exchange Online hosted email for businesses.
Office 365 as a cloud-based service, offers products to users based on their needs which helps them to get their work done faster and better. Now, customers can subscribe to enjoy office 365 services; they have access to emails, calendars, Outlook, OneNote, Excel, Word, PowerPoint, and other Office apps.
Essential Things You Need To Know About Microsoft Office 365
Office 365 users who have active subscriptions get notified of new updates and newer versions of office 365 applications.
Office 365 services include SharePoint Online, which enables the creation, management, and sharing of data, Users, Site pages, and other resources.
Furthermore, SkyDrive is the storage house for your documents, and you can easily access your documents through a web browser, PC, Mac, or other devices. You can easily upload files to your cloud storage and retrieve them whenever you want; you can access files that are important to you like notes, videos, photos, and documents.
SkyDrive makes it easy for you to access your documents and edit them anytime, anywhere, and on multiple devices. You can also access your documents when you are offline in the local SkyDrive folder on your computer.
Skype for business is another office 365 service which enables users to make international calls to people in the world. The Home Premium and University account both come with 60 minutes for international calls. Users can make these international calls on their mobile devices, TV, Windows 8 tablet, and other devices.
College students and staff members can opt for the University package, and they need to go through the academic verification process before they can start enjoying its services.
Then, customers who are not college members or staff members can go for the Home premium package.
Microsoft Team is a new hub where customers can chat, organize meetings, and share files as team members. This encourages teamwork and allows users to share ideas and collaborate as a group.
Finally, you can run your business using the Business apps available on the platform; this helps you manage customer scheduling, invoicing, referrals, and others from one place.
In conclusion, Office 365 has served its customers for years, you just have to subscribe based on your needs. Furthermore, you can gain access to an integration of services that suits your passion and grow your business. You can visit the Microsoft office to get to know more about Microsoft and its services.
Categories Tech Hub Tags microsoft office 365 home, Office 365, office 365 business, office 365 business essentials, office 365 home, office 365 login outlook, office 365 online, office 365 outlook, office 365 price, office 365 sign in, office 365 student, office 365 student login, office online
Yahoo Account Recovery | Yahoo Email Recovery | Recover Yahoo Mail
Marketplace Facebook | Facebook Marketplace App | FB Marketplace Near Me


"Retreat? That's completely impossible," Huai An said icily. His gaze was extremely cold, and there was heavy killing intent when he looked at Jian Chen in particular.
If it were not for the Primordial realm expert behind Jian Chen capturing Yaxi Lian, why would he, Huai An, be forced into negotiating?
This was all because of Jian Chen.
"I can never let this Jian Chen go. Even if I ignore how great of a problem he'll become once he matures, just what he has done to Yaxi Lian cannot be forgiven," killing intent filled Huai An inside.
At this moment, he was even tempted to ignore the divine king and directly try to kill Jian Chen. However, Yaxi Lian was in the divine king's hands right now, so he did not dare to act recklessly.
"Since your Empyrean Demon Cult won't retreat, I can only lower myself to personally kill the commander of the seventh army. With that, the strength of your cult will weaken," the divine king said icily. It was filled with firm killing intent.
He was also someone who could kill without hesitation. Since they could not come to terms, he had no need to keep Yaxi Lian alive!
Right when the divine king was about to kill Yaxi Lian, Huai An suddenly called out.
The divine king remained indifferent. Powerful laws had condensed around his hand on Yaxi Lian's head.
"Huai An, have you through it through?" The divine king asked as he glared at Huai An.
"Pingtian, I'll be honest. You can't kill Yaxi Lian, because if you do, your Divine Kingdom of Pingtian will definitely be annihilated. That's because Yaxi Lian is a direct disciple of one of the eight elders, the Fox Saint Elder. At the same time, she's the Fox Saint Elder's only disciple. You should be able to guess just how important she is to the Fox Saint Elder," Huai An growled.
"The Fox Saint Elder?" The divine king frowned. His understanding of the Empyrean Demon Cult was limited to only the branch on the Cloud Plane. Clearly, the Fox Saint Elder of the eight elders had already surpassed the limits of the Cloud Plane.
At this moment, an ancient voice suddenly boomed out in the divine king's head.
The divine king knew who the voice came from. He was a Primordial realm ancestor of the imperial family of the Heavenly Moon Empire. He remained with the imperial family. Thanks to his powerful soul, he was able to communicate with him from so far away.
Huai An was out of choices with the current situation. Yaxi Lian's identity was just too special. If something were to happen to her, he would not be able to bear the consequences.
"Seven days are too short. Give me a month," the divine king said coldly.
"Ten days!" Huai An also responded coldly.
In the end, they agreed to a ceasefire of half a month.
Although half a month was very short, the time would allow the side of the divine kingdom to greatly regroup, allowing them to deploy even more troops.
With the ceasefire, both the battle between the Overgods and the Godkings stopped at that moment. They all returned to their respective bases.
It was just that there were far fewer of them compared to the beginning of the battle.
A third of all the Overgods had passed away, while several Godkings had died as well.
However, when they saw Jian Chen standing beside the divine king in perfect condition, many of them became deeply surprised.
"Brother, this is fantastic. I knew you'd be fine." Sacredfeather arrived before Jian Chen as soon as possible. There was excitement and joy on his face that he struggled to hide.
Right now, Sacredfeather was covered in blood. There were many vicious and horrifying wounds on him.
Behind him, Houston, Rui Jin, Hong Lian, and Hei Yu all let out a breath in relief when they discovered that Jian Chen had returned safely. They all smiled from the bottom of their hearts.
They were also injured to varying degrees. Hei Yu was the most heavily injured out of all of them.
"Let's go back first!" Without talking too much, Jian Chen vanished into the fortress with them.
There was a military base within the fortress, and at the centre of it stood several divine halls. Each divine hall represented a major organisation.
The divine king had also placed a divine hall here for the Divine Kingdom of Pingtian as a place where his people could rest.
Jian Chen directly took Sacredfeather and the others into the divine hall. They directly made their way to a secret room. After that, he took out the best healing pills out of everything he had purchased from the Xuandao Empire.
"These are all high grade pills, so consume them and heal up," Jian Chen said to everyone.
They did not hold back, directly devouring the pills to heal their wounds and replenish their origin energy.
"Brother, sister Mu'er has been taken away," said Sacredfeather. He became rather depressed when he mentioned this.
"It's a sect called the Daoist Sect of Divine Sound…" Everyone explained everything that had happened in a few sentences.
"The Skyfire Plane, the Daoist Sect of Divine Sound," Jian Chen murmured. Instead, he calmed down after learning about this matter. Then, he raised his head and stared at the ceiling of the secret room quietly.
His gaze seemed to be able to penetrate all, reaching into outer space. He seemed to see the distant Skyfire Plane that he had never even heard about.
"Mu'er, I will come looking for you. That day won't be too far away," Jian Chen secretly made up his mind.


The Conservatory at Whitworth Locke have begun their new French style after-work party complete with bottomless Prosecco and mezze every Thursday and Friday.
Thursday & Friday between 6-8pm you can sip on Bottomless Prosecco, wines or beers for just £27 at this globally-influenced French after work gathering.
Pair the drink of choice with The Conservatory's delicious mezze aperitif selection for the ultimate Parisian style after work vibes.
The spectacular atrium at Whitworth Locke, a former street on the corner of Whitworth Street and Princess Street, has been transformed into the stunning pop-up bar which will host a calendar of events over the next fourteen weeks.
The glass roof of the atrium reveals the changing colours of the Manchester sky, with its intricate detail complementing the dimly lit exposed brickwork. In the centre of the hotel atrium the bar dominates the space, inviting guests to sit and soak in the atmosphere throughout the afternoon and into the night.
The Conservatory team have spent a long time on their approach to mixology. Alcohol is taken seriously here with the quality of the ingredients taking centre-stage. Drinks are simple cocktails served properly, inspired by the relaxed afternoon drinking in France – expect Negronis and Aperol spritzes as well as wines and spirits.
As well as THECONSERVATORYBottomless, the bar will also offer AfterROCKteas, a disruptive afternoon tea, where drinks include kombucha and bubble tea, and sweet snacks are disguised as savoury ones, with a soundtrack of old school rock 'n' roll.
AfterROCKtea runs Wednesday and Thursday from 3pm till 5.30pm, with the last sitting 4pm. Booking is mandatory.
The Conservatory, Whitworth Locke, 74 Princess St, Manchester, M1 6JD.


In this lesson, you learn how to access your Basecamp data from a mobile device.
For professionals on the go, accessing your Basecamp system from a computer might not always be the most convenient option. The good news is that there are numerous add-on applications that enable you to view, manage, and update your Basecamp data from your mobile device, such as an iPhone, iPod Touch, BlackBerry, Treo, Nokia s60, and many others.


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NextUncharted 4: A Thief's End Review
How I Game
Yes. I took a picture of my corgi, Tali, to sucker people into clicking on this.
Shelby "Falcon509" Steiner on July 2, 2016
Want to know how I play? No? Well read on anyway.
I've been writing here for a little over five months now, usually trying to keep things interesting. I've had some hits (THE LONG DARK IMPRESSIONS: CANADA SIMULATOR was a high point), a few duds that I probably shouldn't have published (smh at MOBA-IZATION) because I'm honestly not even sure why I finished them, and a ton that I have thrown in the trash.
It all started with my About Me page really, and sort of expanded from there. I've touched on my gaming preferences a little, but didn't talk much more about it at all.
So how can I add anything to this, you may ask?
MORE BACKSTORY
First, I'm primarily an Xbox gamer but I am not going to drone on about that. I have my reasons why and that's something I elaborated on in another article. What I didn't talk about much was my gaming habits in general.
I've branched out a lot over the past decade of gaming really. Even just looking back about my teenage years where my gaming habits began to take hold, I played only shooters and real-time strategy games. I despised most racing games, horror titles, almost all Japanese games, and anything that didn't have guns in it. Those were the days of Blockbuster Video, the PlayStation 2, original Xbox, the underappreciated GameCube, and my family's PC.
Again, I branched out. The Elder Scrolls IV: Oblivion was my first purchase on the Xbox 360 alongside Ghost Recon Advanced Warfighter. Around that time I was enamored with World of Warcraft and studying to start a game development career. I had graduated from Blockbuster to Hastings for my game rentals and fell in love with role-playing games. Mass Effect became one of my favorite games of all time, melding my fascination with science fiction, firearms, and RPGs (not rocket propelled grenades).
But a tight budget constrained my love of gaming which forced me to miss out on many great things. I couldn't afford the ridiculously expensive PS3 and my PC was severely lacking. Not that I could afford to buy games on the PC anyway, as Steam had just started to take up roots. Remember the days before massive Steam sales? Yeah, me too.
I continued on until I took up a job at GameStop, where I took full advantage of the employee discount and game borrowing policies. I experimented with a ton of games as a result. That's more or less how I ended up the way I am today.
GAMERTYPE
I'm very much a marathon gamer. I like to plop down for hours at a time if I can, just whittling away at whatever game I'm currently stuck on. I go through bouts of gaming depression sometimes, with this last one being the longest. I haven't gotten sucked into a game in a long while, but it always passes.
I also exhibit the tendencies of an addict when it comes to games (but I'm not an addict, damn it!), spending weeks at a time stuck on a single game. I don't take breaks from anything I'm playing. For instance, I played The Elder Scrolls V: Skyrim for what I swear was a month straight, though I did stop to eat and see my girlfriend (she's my wife now so I didn't screw up my relationship with her, obviously).
As of right now at least the only things I don't play still are sports games and Japanese RPGs. I have nothing against them for the most part, even though a great many Japanese games have some fairly creepy tendencies when it comes to portraying women. Not that they're alone in that, Far Cry 3 and 4 being notable examples of gratuitous fan service.
CURRENT GEN PLATFORMS
I know I've said it before, but I do own both a PS4 and Xbox One personally. My wife has her own Xbox One and we share a desktop and laptop. I won't go so far as to list the games I own on any of those because it would take up most of my time to jot down everything. It was another perk of being a GameStop manager: getting free games from publishers. I'm kicking myself for giving away my code for Bloodborne, though I did buy it recently and am pleasantly surprised.
My multi-platform games are all on Xbox for the most part, with a few on my PCs. The PS4 won me over with the port of The Last of Us along with Infamous: Second Son, Bloodborne, and Uncharted 4. It was worth the cost of entry for those three games alone, though I'm still pretty stoked the No Man's Sky is coming out sometime this century. My PS4 is just a standard model with the 500GB HDD and I don't see the reason to go through the moderate hassle of increasing it.
My PCs are a different story altogether. Prepare for a bunch of gibberish.
One is a cheap Dell that I bought for college:
An Intel processor with an integrated graphics chip that I don't care to check for because it's a piece of garbage
8GB of RAM
A 5200RPM 1TB HDD
A Samsung 720p, 32″ TV for a monitor
It's nothing special but it gets the job done when I want to play an older game or write something on my blog when the wife is using the laptop.
Our laptop is a Sager:
An i7-4710MQ processor
An Nvidia GeForce GTX 960M with 2GB of GDDR5 VRAM
16GB of DDR3 RAM
A 13.3″ 1920 x 1080 LED screen with HDMI output so I can hook it up to my 46″ TV
It's also not top of the line but it gets me by with my Steam library. I won't exactly be playing Batman Arkham Knight but let's face it, who would play that on the PC anyway?
All in all, I can play pretty much anything I want to play. Granted, I don't have a Wii U, so Nintendo games are off the table. The thing is that I'm not really a Nintendo kinda guy anymore. I don't have a principled stance against Nintendo or an antipathy towards their games, I just don't want to play their games enough to plunk down $300+ to buy their system.
AND NOW WE COME TO A CLOSE
So this leaves me curious, dear readers. What do you game on? What are your preferences? Your hang-ups? Are you a Pony? An Xbot? Are you part of the PC master race? A… Nintendork?
Is that a thing? Nintendork?
Tagged in: Gaming, News, PC, Personal History, PS4, Video Game, Xbox One
Anidaan July 3, 2016 at 18:30
I played a lot of xbox and PC a number of years back with the focus on ESO, WoW, COD, and Halo.
Nowadays I just own a PS4 (switched to be able to play with friends) and spend a lot of time playing SMITE, Rocket League, Fallout 4, and COD: Black Ops 3. Not sure what game I am going to get next but I imagine I will stick with the PS4 at least for the next year or so!
Uncharted 4: A Thief's End Review
Elegy For A Dead World Review


If you're interested with it, theres still no Metisse 0.4 packages for distros other that Mandriva thats has been out in the public yet. However, you can build it from source by following this Howto.
Sadly, It uses a modified Xserver (rather than available compositing from AIGLX/XGL) which immediately turned me off. I don't like polluting my system with from-source installations that might be hard to remove later.
As promised before in one of my previous post, a Howto on how to mirror and manage yum repositories using some of the utilities in yum-utils.
Reposync is a utility for mirroring and synchronizing local copy of a yum/rpmmetadata repository.
This utility is very useful if you wanted to make a yum repository mirror. Before this, I used "wget -R -np -N" but this method is a little bit tedious and it doesnt work with repos that didn't use directory listing. Plus, it also download together additional site stuff that I don't need/want and it doesn't verify checksum of the downloaded packages.
Yum-utils package have some useful tools for sysadmins who wanted to create and manage their own RPM metadata repository.
To UTP students, our UTPchat IRC network is now back functional in Village 1, 4 and 5. The servers have been linked up and connected with each other.
There are some problem with V2 and V3, and we are still finding a way to resolve it.
Another semester going to start, and yet another semester with only port 80 and 443 outbound to Internet *sigh*.
Struggling to re-setup the IRC servers because of room changes and new admins. Also, need to update my FC6 repository mirror of Extras, Updates and Livna. The UTP Open Source society also have lots of stuff to catch up.
Hopefully I can finish all of this by 2 weeks.
var feed_url = "/feeds/posts/default";in the code.
You can get the widget using a one-click interface here.
Students at the University of Bristol have recently been warned of the dangers of posting to social networking websites. They aren't the first to hear these warnings, and they won't be the last.
Prof. Nigel Smart of the Computer Science Department at the University of Bristol has expressed his concern at the worrying trend of people giving up their privacy on the internet via social networking websites. He told HEXUS: "I am concerned that from some of the posts I have seen, by colleagues, students and others, that there is a deep societal problem emerging of people giving up their privacy without realising it".
There's little point in worrying about ID cards, RFID tags and spyware when more and more people are throwing away their privacy anyway. And the potential consequences are dire.
OpenDesigns.org : Open Web Templates for your website!
I found something interesting from StumbleUpon yesterday. An OpenSource website template community!!
The Open Design Community (TODC) is a group of Open Source Website Designers providing free web design templates helping to make the global internet a prettier place.
Our goal is to create a community owned and managed site, where users can share their free website designs and templates, discuss web design, and promote their services.
I browsed through the templates hosted there, and they are all awesome!. Almost all of them are well polished and professional looking and a lot are as high quality as expensive commercial web templates.
While watching Death Note ep13 (thanks Animanda!) I realized that I can translate my name to Japanese name. Got the idea from Yagami Light's name. The word Light in his name is spelled as 月(Tsuki) but pronounced as "Light".
Feeling curious how my name will be in Japanese, I fired up Kiten and start hunting down words that can match with the meaning of my name.
Mohd Izhar Firdaus Ismail - I omitted my father's name and the first word of my name because both are names of ancient Phophets which doesn't have a direct literal meaning. That left me with Izhar Firdaus.
Sometimes, admins might want to extract some files from a RPM file for some purposes. If on a system with GUI, this can be easily done using file-roller or some other GUI based archive manager that supports RPM/CPIO archives. RPMs are basically modified CPIO archives, so we use rpm2cpio and cpio command to extract files from it in a CLI environment.
rpm2cpio <rpmpackage.rpm> |cpio -id <file> Note: extracting specific files requires you to put the full path of the file in the archive which you get from the cpio -t output.
Lets go straight to the point. First, to those you a newbie in editing userChrome.css, locate the file using this guide.
Then, just add these lines in userChrome.css.
Edit: For the Menu button, use the Compact Menu extension.
I was wondering why it took so long for ep 13 of Death Note, so I googled and found out the news. Dammit, the US licensed the anime but us in non-US countries who suffered.
(^-^) Anxiously waiting for the next episode.
I'm back from JB .. It was a great trip .. learned some new stuff and met some new and old friends .. Trying to find some ideas on what to write next ..
YUM is the default package manager provided with most RPM distros, especially Fedora and Redhat based distros. It is similar to APT of the Debian based distros. Admittedly, it is not the best package manager in term of performance, I personally feels APT is faster and easier to use. However, it do have some good points. One of it is the ease of creating a YUM RPM repositories.
Some users might want to store cached/downloaded RPM packages for backup or future offline use. Some of them might also want to easily resolve dependencies of downloaded RPMs from non-repository sources. By having their own repository, they can easily install the RPMs and avoid dependency hell.
To create our own repo, we use a tool called "createrepo". Its available in the Fedora DVD/CD and you just need to insert the DVD/CD and install it. You can also get it through yum.
Adding the widget into the template is easy. Open edit template html page, check expand widget, then copy the codes below, and paste it after another widget. Save the template and you're done. To rearrange the widget, just use the 'Page Elements' page.
After adding into your template, goto 'Page Elements' and click the edit button at the widget. Put your feed url into the widget content form. Be careful not to put any line breaks or other characters because it will go into your url.
Firefox 2 introduces tab scrolling feature similar to Epiphany and several other browsers. However, while this feature seems to help stop tabs from becoming too small and hide its title, its considered annoying to some users. Not be able to see all of the tabs in once glance might not be preferable to some.
Firefox 2 already have a session saver built-in with the browser. However, it only activates when firefox crashed. Theres no available menu in the firefox preferences to adjust this behaviour, and some of us might want Firefox to restore tabs from our previous session. Fear not, you can still adjust it by editing a key through the about:config interface and enable the built-in restore previous session functionality.
browser.startup.page = 3You can read here to know more about the key.
This is my second post about what I've learned from the new Blogger engine. This time, I'll explain the about the widgets, its tags, and how to create your first widget.
To those who didnt know it, Blogger widgets are those drag-and-drop able page elements in the new Blogger. By default, you can only choose to use the widgets provided by Google for your page. However, that is not the limit. The widget engine in the new Blogger offers good customization and allows developers to develop their own widgets to share with others!. Its uses HTML for layout and XML for content, therefore, its not hard to create your own widget if you know the basics.
Widgets are defined using <b:widget> tags. You can read Blogger's documentation about the tag here. The basic is, when you define a widget, you just need to give it an id and a type. For a list of available widget types and it data variables, visit here. Below is an example of a simple widget using the HTML type.
I stumbled upon a url to this site from an email in a mailing list. Being curious, I opened the site. At first glance, I thought it was yet-another-friend networking site like friendster and myspace, but I noticed theres something different with it. So, I click the site's about page and read it.
LinkedIn is an online network of more than 8.5 million experienced professionals from around the world, representing 130 industries.
I beleive lots of us is familiar with Yahoo!'s I'm Online images on websites, blogs, myspace, friendster, etc. Some of us might not prefer the over-fancy Yahoo Messenger service (I know I am), prefer Gtalk, and wanted an online indicator for their sites. Jon Burrows have a solution for you. He hosted a Gtalk Online Status bot that will check your online status and provide an image of your status.


The National Association of State Procurement Officials (NASPO) has published a new Guide to International Trade Agreements. It is designed to assist "state officials understand U.S. trade agreements and their obligations under the agreements" with respect to state government procurement. NASPO is a non-profit association made up of the directors of the central purchasing offices in the 50 states, the District of Columbia and U.S. territories.
International Agreements with State Procurement Obligations: The Guide provides an overview of international agreements, which includes a discussion of state authorization of covering procurement in agreements. The agreements with state procurement obligations are the WTO Government Procurement Agreement (GPA) — the first international agreement to cover state procurement, eight free trade agreements (FTAs) and a 1995 U.S.-European Union exchange of letters.
Scope and Coverage of State Procurement: This Guide sets out three tests for states to apply to determine whether a procurement is subject to an international agreement. In addition, it explains how to calculate the value of a procurement to determine whether it is covered under applicable thresholds.
State Obligations under Agreements: The Guide outlines the obligations of states that cover procurement under agreements. It describes what is meant by the fundamental principle of non-discriminatory treatment of goods, services and suppliers of GPA and FTA parties. It also outlines the procurement procedures that apply to each of eight steps in a procurement. In addition, other requirements such as disclosure of information and domestic review of state procurement are addressed.
Other Information: The Guide describes the role of the Office of the U.S. Trade Representative with respect to covering state procurement under trade agreements. It also responds to frequently asked questions such as whether USTR can automatically add states to international agreements. The Guide closes with a brief overview of current negotiations of agreements that may affect state procurement: the Trans-Pacific Partnership (TPP), Transatlantic Trade and Investment Partnership (TTIP) and the addition of countries to the GPA.
This entry was posted in GPA Market Access Commitments, U.S. Procurement Agreements, U.S. State Procurement, WTO Government Procurement Agreement (GPA) and tagged NASPO, states and international agreements, U.S. State Procurement by Jean Heilman Grier. Bookmark the permalink.


While I had yet more excellent adventures on Saturday and today, I want to return--before memories fade--to our day last Friday.
After Holy Mass at Calvary--albeit a little to the right--and after breakfast, we then visited the traditional site of our Lord's trial before Caiphas and the Sanhedrin, called Saint Peter in Gallicantu.
The front doors of Saint Peter in Gallicantu.
The main church on the upper level.
The guide gave reasons why he thought another site, closer to the temple, was more likely: namely, that this was--in our Lord's time--a poor area, and thus an unlikely place for the high priest to live. Yet the traditional site is such because of veneration by early Christians. There is a lesson here: in these matters, no matter what even well informed people tell you, these things are not as cut and dried as they think. Remember that when someone insists that there's no question the book of Isaiah had three authors, or Mark was written before Matthew, or that such-and-such couldn't have (or must have) happened here.
As it is, this site was compelling. The existing church sits on a steep hill. You enter the main church a bit below street level; then you go down, down, with chapels below, with some striking, Eastern-style icons depicting something else that happened here: Peter's denials.
This shows the moment our Lord turned and looked at Peter.
Note the use of colors: red for blood, gold for glory, and black for chaos.
This shows the moment--described in the last chapter of the Gospel of John--when our Lord restored Peter.
Then at the bottom is a pit that goes very deep. (Sorry, no more pictures.) Tradition has it that our Lord was thrown into it, and there spent the night before being taken, early in the morning of Good Friday, before Pontius Pilate.
We walked around the old city a lot, but I'll skip to the afternoon, when around 3 pm, we joined the Franciscans in the Via Crucis, along the Via Dolorosa. When we have the stations of the cross every Friday in Lent, it is this journey we are recreating. (The origin of stations in churches is this: when Christians were prevented from coming to the Holy Land to walk the Lord's walk, the Via Crucis was recreated in parish churches. Although they aren't a required element, it's rare to have a church without them (my Argentinian roommate confirms this is the same in his country).
This pilgrimage has included a lot of walking and climbing, and I was tempted not to join this part of the day. I'm glad I resisted the temptation. The walk along the Via included a good number of pilgrims; the Franciscans prayed in Spanish, English and Italian. (And you can easily understand why the traditional form includes an Our Father, Hail Mary and Glory Be--they cover more ground between stations on site than in our parish churches!) The way was along a tight street, with small shops on both sides, with Arab vendors attempting to make a living--although they were very respectful (no doubt quite familiar with this) as we came by. In fact most were; still, there were a good number of folks who simply wanted to go about their business and didn't take any interest in our ritual; and even a few who hurled insults.
The last several stations take place within the Church itself; and include an arduous climb up extremely steep steps to Calvary. We often refer to Mount Calvary, but the reality is a bit more complicated. Where our Lord died was believed to have been a quarry, situated even so on a hill outside the walls; but as it was a quarry, the crosses themselves might well have been out of view. However, the place where our Lord died was itself a mound of rock "shaped like a skull"--hence the name Calvary.
We completed our Way of the Cross at the sepulcher, of course; and at that point, we were permitted to enter. The tomb itself is a very small space; it's made of stone, and is situated maybe two hundred feet (I'm guessing) from Calvary. Our guide informed us that while in our Lord's time, this would have been in the side of the hill, all the stone around the tomb was taken away to make room for the church; but the tomb itself remains. On top of, and in front of, the tomb, the Greek Orthodox have built a square structure, before which stand gigantic candle stands, some of which now have electric lights instead of living flame. Nevertheless, lots of candles around, and also in the anteroom that stands before the tomb itself. When we were present that morning for Holy Mass, we observed our Greek brethren conducting the sacred liturgy around the tomb. Now they allowed the Catholic pilgrims to venerate.
Coming out of the tomb! What an exhilarating thing to do--and say! I might have explained already that nearby was a section of the church controlled by the Catholics, and there an organist added some beautiful music as we completed the journey.
But we weren't finished. The Franciscans have a procession that begins after this, that takes us around the interior of the church. They were on the third station before I realized my brother priests weren't at hand, and I went and found them. Everyone got a candle, and booklets were available in various languages. I grabbed a booklet, only to find out it was in Latin, German, Czech and (I think) Polish. At least I had Latin.
As we went along, the Franciscans led us with beautiful Latin chant. How sad that so many Catholics (at least in our country) loathe and despise this beautiful language and throw such fits over any attempt to include it!
I wanted to keep the booklet, but at the end of our procession--which came in a chapel very near Calvary, with exposition of the Blessed Sacrament--one of the Franciscans deftly retrieved it, along with the stub of the candle I still had. But the journey recalled many of the things that happened to our Lord at Calvary. While most of the chant was unfamiliar to me, we did sing the Stabat Mater and Ave Verum.
After this--feeling even more exhilarated!--we chanced upon yet another group of the faithful making a procession; this time, Armenian Christians. (I'm not clear if they were Catholic or Orthodox.) We found them at the tomb, with a priest at the door, and a deacon behind him, with about 12 young men in cassocks chanting beautifully. I guessed them to be seminarians. We watched and listened as they moved to an image nearby of the Lord's body being taken from the cross; here the priest chanted the Gospel; and afterward, everyone came up and was welcome to kiss the book--and the priest's hands. Some of our group did so, as did I.
I think we finished around 5 pm, after which we headed back to the guest house. As our Protestant brethren might say, "I think we've prayed today, church--can I get an amen?"
Amen! ;-) ~ Rosemary A.


NPT Fitting Probe - Princeton Electronics, Inc.
This is our NPT fitting style probe with separate electronics box. Available in 3 standard lengths of 12", 18" or 24" with an option of no bendable section, a 3" bendable section (used for a singular bend) or a 5" bendable section (used for when the probe needs to be bent in two directions). The NPT style probe is available in custom lengths for an additional charge. The electronics module or probe may be purchase separately for replacement purposes.
This is our NPT fitting style probe with separate electronics box. Available in 3 standard lengths of 12", 18" or 24" with no bendable section. The NPT style probe is available in custom lengths for an additional charge (additional charges apply to custom lengths over 24"). The electronics module or probe may be purchased separately for replacement purposes.
This is our NPT fitting style probe with separate electronics box. Available in 3 standard lengths of 12", 18" or 24" with a 3" bendable section (used for a singular bend). The NPT style probe is available in custom lengths for an additional charge (additional charges apply to custom lengths over 24"). The electronics module or probe may be purchased separately for replacement purposes.
This is our NPT fitting style probe with separate electronics box. Available in 3 standard lengths of 12", 18" or 24" with a 5" bendable section (used when a probe needs to be bent in 2 different directions). The NPT style probe is available in custom lengths for an additional charge (additional charges apply to custom lengths over 24"). The electronics module or probe may be purchased separately for replacement purposes.
This is our NPT fitting style probe with separate electronics box. This item is for custom length probes between 49" and 110". Please note the length required.