text:
\section{Introduction}

Graph Languages and Graph Grammars were introduced in theoretical
computer science as an extension of the theory of formal
languages (linear languages), in order to model various types of parallelism in computation,
\cite{Nagl}, \cite{HEPT}, \cite{EKR}, \cite{Roz}. Instead of replacing nonterminal symbols with combinations of nonterminals and terminals in a linear string of characters,
the production rules of graph grammars replace a subgraph of a graph
with a new graph. The latter is obtained either by gluing along a common subgraph, or by
first performing an excision of a subgraph and then replacing it with a new graph.
An analog of the Chomsky hierarchy of grammars exists for Graph Languages, see \cite{Nagl}.
In particular, the context-free Graph Grammars are those where the left-hand-side of the
production rules is always a single vertex. Namely, no ``context" in the graph is taken into consideration
in deciding when a production rule can be applied: it just applies to any vertex. In this context-free
case the production rules then consist of inserting a new graph at a vertex of another graph.
This operation is reminiscent of the insertion operation that defines the Lie algebras of
Feynman graphs in the algebraic approach to renormalization in quantum field
theory pioneered in \cite{CoKr} (see also \cite{CoKr2}, \cite{CoMa}, \cite{EFBP}). In this paper we
show that, indeed, to certain classes of Graph Grammars (both context-free and context-sensitive)
it is possible to associate a Lie algebra, obtained by constructing a pre-Lie insertion operator
using the production rules of the grammar. We also show that the Feynman graphs of a given
quantum field theory are a graph language in the sense of the theory of formal languages.
This provides a new class of examples of graph languages, in addition to those arising in the
context of computer science (such as FFT networks, Petri nets, distributed parallelism), see
the articles in \cite{EKR} for several examples. Relations between the formalism
of algebraic renormalization in quantum field theory and aspects of the theory of
computation in theoretical computer science have already been investigated in \cite{Man},
see also the formulation of Dyson--Schwinger equations in the Hopf algebra of
flow charts in \cite{DelMar}. it would be interesting to see if a
theory of Dyson--Schwinger equations can be formulated
for Graph Languages, using the Lie theoretic approach of \cite{Foissy}.

\subsection{The Insertion Lie algebra of Quantum Field Theory}

In perturbative quantum field theory, one computes expectation values
as a formal series of Feynman amplitudes labeled by Feynman graphs.
These graphs are finite and the allowed valencies are constrained to match
the exponents in the interaction monomial in the Lagrangian of the field theory.
Graphs have a number of internal edges (connecting pairs of vertices)
and external edges (half edges).
The corresponding Feynman amplitude is a finite dimensional integral over a space of
momenta flowing through the graph, with assigned external momenta carried
by the external edges, and with conservation laws at the vertices.
These Feynman integrals are typically divergent, which leads to
the crucial problem of renormalization. The goal of a renormalization procedure
is a consistent extraction of finite values from all these integrals that takes into account the
combinatorics of how divergent subgraphs are nested inside larger
graphs. Since the work of Kreimer \cite{Kr} and Connes--Kreimer \cite{CoKr},
it has become clear that the renormalization procedure can be formulated
algebraically in terms of a Hopf algebra of Feynman graphs. The
algebraic Feynman rules are seen as algebra homomorphisms to a target
commutative algebra determined by a choice of regularization procedure, and endowed
with a ``pole-subtraction" operation (Rota--Baxter algebra). See \S 1 of \cite{CoMa}
for an overview.

\smallskip

The Hopf algebra ${\mathcal H}$ of Feynman graphs is a graded connected commutative
Hopf algebra generated by the 1PI Feynman graphs of the given quantum
field theory. The 1PI (one-particle irreducible) condition means that the graphs
are connected and cannot be disconnected by removal of a single edge.
A standard argument in quantum field theory reduces the combinatorics of
Feynman graphs to the connected case, and further to the 1PI case, see \cite{ItZu}.
The coproduct in the Hopf algebra is not co-commutative. It is given by
\begin{equation}\label{CKcoprod}
\Delta(G)=G \otimes 1 + 1 \otimes G + \sum \gamma \otimes  G/\gamma,
\end{equation}
where the sum is over all the (not necessarily connected) subgraphs $\gamma \subset G$,
such that the quotient graph $G /\gamma$ (obtained by shrinking each component of $\gamma$
to a vertex) is a 1PI Feynman graph of the theory. The Hopf algebra is
dual to a pro-unipotent affine group scheme that is entirely determined by its Lie algebra.
Connes and Kreimer gave a very explicit geometric description of this {\em insertion Lie
algebra} \cite{CoKr2} (see also \S 1 of \cite{CoMa}).
On the vector space spanned by all 1PI Feynman graphs of the theory, one can define a Lie bracket
by setting
\begin{equation}\label{QFTLiebracket}
[G_{1},G_{2}]=\sum_{v}G_{1}\circ_{v} G_{2}-\sum_{v'} G_{2}\circ_{v'} G_{1},
\end{equation}
where the sums are over all vertices in $G_{1}$ and $G_{2}$.
The expression $G_{1}\circ_{v}G_{2}$ denotes the graph resulting from
the insertion of $G_{2}$ at the vertex $v$ of $G_{1}$.
One can insert one graph into another by matching external edges to
the edges incident at the vertex. In the sum one counts all the possible
inequivalent ways in which the graph can be inserted at the given
vertex. This bracket indeed satisfies the Jacobi
identity and defines a Lie algebra, which can be related to the
primitive elements in the dual Hopf algebra of Feynman graphs \cite{CoKr2}
(see also \S 1 of \cite{CoMa}). A detailed survey of the use of Lie algebra methods
in Quantum Field Theory can be seen in \cite{EFBP}. The language
of Lie algebras in Quantum Field Theory provides an elegant formulation
of the Dyson--Schwinger equations (the quantum equations of motion
of the theory), and a general method for solving them in the Lie algebra
of Feynman graphs \cite{Foissy}.

\section{Graph Grammars and Lie algebras}

\subsection{Two descriptions of graphs}

It is convenient to consider two slightly different ways of assigning the data of a
finite graph. The first is the one most commonly used in Combinatorics, while
the second is more frequently used in Physics.

\medskip

\subsubsection{Version 1:} A graph $G$ consists of a set of vertices $V(G)$ and a set of edges $E(G)$
together with a boundary map $\partial: E(G)\to V(G)\times V(G)$ assigning
to an edge $e\in E(G)$ its (unordered) pair of boundary
vertices $\partial(e)=\{ v_1, v_2 \}$. The graph can have looping edges if we
allow $v_1=v_2$ and it can have multiple parallel edges if $\partial^{-1}(v_1,v_2)$
can consist of more than one element. If the graph $G$ is oriented (directed)
then the boundary map consists of two maps (source and target)
$s,t: E(G)\to V(G)$. A system of vertex and edge labeling consists of two
sets $\Sigma_V$, $\Sigma_E$ of vertex and edge labels, respectively, and
functions $L_{V,G}: V(G) \to \Sigma_V$ and $L_{E,G}: E(G) \to \Sigma_E$.

\medskip

\subsubsection{Version 2:}
A graph $G$ consists of a set $C(G)$ of {\em corollas} (vertices $v$ with valence
${\rm val}(v)$ with ${\rm val}(v)$ half-edges attached to it) and an involution
${\mathcal I}: {\mathcal F}(G)\to {\mathcal F}(G)$ on the set ${\mathcal F}(G)$ of all half-edges (flags) attached to
all the corollas. The set $E_{int}(G)$ of internal edges of $G$ corresponds to all the pairs
$(f,f')$ with $f\neq f'$ in ${\mathcal F}(G)$ and with $f'={\mathcal I}(f)$. The set $E_{ext}(G)$ of external
(half)edges of $G$ consists of all the $f\in {\mathcal F}(G)$ such that ${\mathcal I}(f)=f$. A labeling system
is given by a set $\Sigma_{\mathcal F}$ of flag labels and a set $\Sigma_V$ of vertex
labels together with maps $L_{{\mathcal F},G}: {\mathcal F}(G)\to \Sigma_{\mathcal F}$ and $L_{V,G}: C(G)\to \Sigma_V$
with $L_{{\mathcal F},G}\circ {\mathcal I}= L_{{\mathcal F},G}$.

\medskip
\subsection{Insertion Graph Grammars}

Using the first description of graphs, we define an Insertion Graph Grammar  as follows.

\begin{defn}\label{IGG1}
An Insertion Graph Grammar consists of data
$$ (N_E, N_V, T_E, T_V, P, G_S) $$
where the set of edge labels of graphs is $\Sigma_E=N_E \cup T_E$,
with $N_E$ the nonterminal symbols and $T_E$ the terminal symbols,
and the set of vertex labels is given by $\Sigma_V = N_V\cup T_V$,
with non-terminal and terminal symbols given respectively by
$N_V$ and $T_V$. The start graph is $G_S$ and $P$ is a finite set of
production rules of the form $P =(G_L, H, G_R)$, with $G_L$ and
$G_R$ labelled graphs (respectively, the left-hand-side and the right-hand-side
of the production) and with
$H$ a labelled graph with isomorphisms
$$ \phi_L: H \stackrel{\simeq}{\to} \phi_L(H)\subset G_L, \ \ \
\phi_R: H \stackrel{\simeq}{\to} \phi_R(H)\subset G_R. $$
The isomorphism $\phi_L$ should be label preserving.
The production rule $P =(G_L, H, G_R)$ searches for a copy of $G_L$
inside a given graph $G$ and glues in a copy of $G_R$ by
identifying them along the common subgraph $H$, with new
labels matching those of $\phi_R(H)$.
\end{defn}

\medskip
\subsubsection{Context-free Graph Grammars}

We recall the notion of context-freeness for graph grammars from \cite{Nagl}.

\begin{defn}\label{contextfree}
An Insertion Graph Grammar as in Definition \ref{IGG1}
is {\em context-free} if $G_L =\{ v \}$ (hence $H=\{ v \}$ also).
It is {\em context-sensitive} if $G_L \neq \{ v \}$. In the context-sensitive case $G_L$
is called the {\em context} of the production rule.
\end{defn}

A Chomsky hierarchy for graph grammars is described in \cite{Nagl}.

\medskip
\subsubsection{Insertion Graph Grammars and Flags} If we consider the second
version of the definition of graphs given above, we can formulate a slightly different
notion of Insertion Graph Grammars. For a subgraph $G'\subset G$ the set of
external edges $E_{ext}(G';G)$ is defined as the union of the set ${\mathcal F}_{G'}\cap E_{ext}(G)$
and the set of pairs $(f.f')\in E(G)$ such that only one half-edge in the pair belongs to ${\mathcal F}_{G'}$
while the other belongs to ${\mathcal F}_G\smallsetminus {\mathcal F}_{G'}$.

\smallskip

In this setting, we describe an Insertion Graph Grammar as follows.

\begin{defn}\label{IGG2}
An Insertion Graph Grammar consists of data
$(N_{\mathcal F}, N_V, T_{\mathcal F}, T_V, P, G_S)$, as in Definition \ref{IGG1}, with
$\Sigma_{\mathcal F} = N_{\mathcal F} \cup T_{\mathcal F}$ the non-terminal and terminal
labels for flags. The production rules $P =(G_L, H, G_R)$ are as in Definition \ref{IGG1},
with the additional requirement that $\phi_L(E_{ext}(H,G_R))\subset E_{ext}(G_L,G)$
and $\phi_R(E_{ext}(H,G_L))\subset E_{ext}(G_R)$, where $G$ is any graph
the production rule is applied to, with $G_L\subset G$.
\end{defn}

\smallskip

The reason for this modified definition is that the notion of gluing of two
graphs $G_L \cup_H G_L$ along a common subgraph $H$ is formulated
by taking as set of corollas $$C_{G_L \cup_H G_L} =C_{G_L} \cup_{C_H} C_{G_R},$$
identifying the corollas around each vertex of $H$ in $G_L$ and $G_R$ and
then matching half-edges by the involution  ${\mathcal I}(f)=f'$ with $f'={\mathcal I}_L(f)$ when
both $f,f'\in {\mathcal F}_{G_L}$, with $f\neq f'$, and $f'={\mathcal I}_R(f)$ when $f,f'\in {\mathcal F}_{G_R}$, with $f\neq f'$.
If ${\mathcal I}_1(f)=f$ and $f\in \phi_L(E_{ext}(H,G_R))\subset E_{ext}(G_L,G)$ with
$f=\phi_L(f')$, when ${\mathcal I}(f)=f'$ and similarly for $\phi_R(E_{ext}(H,G_L))\subset E_{ext}(G_R)$.

\smallskip

In this setting, because vertices are always endowed with a corolla of half-edges,
we cannot state the context-free condition by requiring that $G_L=H=\{ v \}$. An
appropriate replacement of the context free condition is given by the following.

\begin{defn}\label{contextfree2}
An Insertion Graph Grammar as in Definition \ref{IGG2} is {\em context-free}
if $G_L=H=C(v)$, the corolla $C(v)$ of a vertex $v$, and all the vertices of graphs in
the graph language have the same valence.
\end{defn}

In the case where graphs contain vertices of different
valences, these would still be context-sensitive graph grammars, with the
context specified by the valence of $C(v)$.

\medskip
\subsection{Insertion-elimination Graph Grammars}

We consider another variant of the definition of graph grammars,
where the production rules consist of {\em replacing} a subgraph by another one,
instead of gluing them along a subgraph. While the version discussed above
reflects the notion of graph grammars considered for instance in \cite{Nagl},
the version we discuss here reflects the use in other references (see for instance
\cite{Roz2}).

\smallskip

In order to formulate this version of graph grammars with the first notion of
graphs, we need to define the operation of removal of a subgraph from a graph.
Let $G'\subset G$ be a subgraph. Let
\begin{equation}\label{EGGprime}
E_G(G')=\{ e\in E(G)\smallsetminus E(G')\,|\, \partial(e)\cap V(G')\neq \emptyset \}.
\end{equation}
We define
$G\smallsetminus G'$ as the subgraph of $G$ with $V(G\smallsetminus G')=
V(G)\smallsetminus V(G')$ and with edges $E(G)\smallsetminus (E(G')\cup E_G(G'))$.
Thus, for example, removing a vertex $G'=\{ v \}$ means removing the vertex $v$ along
with its star of edges. We then define Insertion-elimination Graph Grammars as follows.

\begin{defn}\label{IEGG1}
An Insertion-elimination Graph Grammar consists of data $${\mathcal G}=(N_E, N_V, T_E, T_V, P, G_S)$$
as in Definition \ref{IGG1}, where the production rule $P=(G_L,H,G_R)$ acts by
searching for a copy of $G_L$ in $G$, removing $G_L\smallsetminus H$
and replacing it with the graph $G_R$ glued along $H$.
\end{defn}

\smallskip

Using the second description of graphs, the removal of a subgraph $G'\subset G$ is defined by
cutting all edges in $E_{ext}(G',G)$ into pairs of half-edges, one attached to $G'$ and one to
$G\smallsetminus G'$. Thus, the set of corollas $C(G\smallsetminus G')$ is given by
the difference $C(G)\smallsetminus C(G')$ and the set of flags is given by
${\mathcal F}(G\smallsetminus G') = {\mathcal F}(G)\smallsetminus {\mathcal F}(G')$, with involution
${\mathcal I}_{G\smallsetminus G'}(f)={\mathcal I}_G(f)$ if both $f$ and ${\mathcal I}_G(f)$ are in ${\mathcal F}(G)\smallsetminus {\mathcal F}(G')$
and ${\mathcal I}_{G\smallsetminus G'}(f)=f$ for $f\in {\mathcal F}(G)\smallsetminus {\mathcal F}(G')$ with ${\mathcal I}_G(f) \in {\mathcal F}(G')$.
Notice that the two notions of removal of subgraphs differ in the way the edges connecting
a vertex of the subgraph to a vertex of the complement are treated: in the first case they are
removed, while in the second case a half-edge remains as an external edge of the
complement graph. We then have the following formulation.

\begin{defn}\label{IEGG2}
An Insertion-elimination Graph Grammar consists of data $${\mathcal G}=(N_{\mathcal F}, N_V, T_{\mathcal F}, T_V, P, G_S)$$
as in Definition \ref{IGG2}, with the requirement that
$$  E_{ext}(\phi_L(H),G)=E_{ext}(\phi_L(H),G_L)= E_{ext}(\phi_R(H),G_R)=E_{ext}(G_R) . $$
The production rule $P(G_L,H,G_R)$ acts by searching for a copy of $G_L$ inside $G$,
removing $G_L\smallsetminus \phi_L(H)$ and replacing it with a copy of
$G_R\smallsetminus \phi_R(H)$, by matching the half-edges of $E_{ext}(G_R)$ to the
half-edges of $E_{ext}(G_L,G)$.
\end{defn}

\medskip
\subsection{Pre-Lie structures}

A (right) pre-Lie structure on a vector space $V$ is a bilinear map $$\triangleleft : V\otimes V \to V$$
satisfying the identity of associators under the exchange $y \leftrightarrow z$,
\begin{equation}\label{prelie}
(x \triangleleft y)\triangleleft z - x \triangleleft (y \triangleleft z) = (x \triangleleft z)\triangleleft y
 - x \triangleleft (z \triangleleft y), \ \ \ \forall x,y,z\in V.
\end{equation}

\medskip

A Lie algebra is a vector space $V$ endowed with a bilinear bracket $[\cdot ,\cdot]$
satisfying antisymmetry $[x,y]=-[y,x]$ and the Jacobi identity
\begin{equation}\label{jacobi}
 [x,[y,z]]+[z,[x,y]]+[y,[z,x]]=0, \ \ \  \forall x,y,z \in V.
\end{equation}

\medskip

A pre-Lie structure determines a Lie algebra by setting
\begin{equation}\label{prelietolie}
[x,y] := x \triangleleft y - y \triangleleft x .
\end{equation}
The pre-Lie identity ensures that the Jacobi identity is satisfied.
A detailed survey of occurrences of pre-Lie algebras in geometry, physics, and
the theory of formal languages can be found in \cite{Burde}.

\medskip

One can obtain a group structure from a pre-Lie algebra structure
(see \cite{AgraGam} and \cite{Manchon}) by considering formal series
$$ W(x)= x + \frac{1}{2} x \triangleleft x + \frac{1}{6} (x \triangleleft x) \triangleleft x + \cdots $$
with the multiplication operation
$$ W(x)\star W(y) =W(C(x,y)), $$
where $C(x,y)$ is the Baker--Campbell--Hausdorff formula
$$ C(x,y)=x+y+\frac{1}{2}[x,y]+\frac{1}{12} ( [x,[x,y]]+[y,[y,x]]) + \cdots $$

\medskip
\subsection{Lie algebras of context-free grammars of directed acyclic graphs}

Consider the case of a context-free Insertion Graph Grammar ${\mathcal G}$ as in
Definitions \ref{IGG1} and \ref{contextfree}, where the start graph $G_S$
is a single vertex and all the graphs $G$ are directed acyclic with a marked (root)
source vertex. The production rules are of the form $P(v,v_2,G_2)$, where
$v_2$ is the root vertex of $G_2$ and $v$ is a vertex of the graph $G_1$, to which
the rule is applied. The resulting graph
$$ G_1\triangleleft_{v} G_2= P(v,v_2,G_2)(G_1) = G_1\cup_{v\equiv v_2} G_2 $$
obtained by applying the production rule to $G_1$ is also a directed acyclic graph
with root vertex the root $v_1$ of $G_1$.

Let ${\mathcal V}$ be the vector space spanned by the set ${\mathcal W}_{\mathcal G}$ all the graphs
obtained by repeated application of production rules, starting with $G_S$.
The set ${\mathcal W}_{\mathcal G}$ is different from the graph language ${\mathcal L}_{\mathcal G}$, as it
also contains graphs whose vertices and edges are labelled by non-terminal
symbols.

We then define the insertion operator $\triangleleft : {\mathcal V}\otimes {\mathcal V} \to {\mathcal V}$ as
\begin{equation}\label{insopCF}
G_1 \triangleleft G_2 = \sum_{v\in V(G_1)} P(v,v_2,G_2)(G_1)
=\sum_{v\in V(G_1)} G_1\triangleleft_v G_2 .
\end{equation}

\begin{prop}\label{preLie1ins}
Given a context-free Insertion Graph Grammar ${\mathcal G}$ as above,
the insertion operator \eqref{insopCF} defines a pre-Lie
structure on the vector space ${\mathcal V}$.
\end{prop}

\proof We need to check that \eqref{prelie} is satisfied. We have
$$ (G_1\triangleleft G_2)\triangleleft G_3 =  \sum_{v\in V(G_1)}\,\,
\sum_{v'\in V(G_1\triangleleft_v G_2)} \,\, (G_1\triangleleft_v G_2) \triangleleft_{v'} G_3 $$
where $v$ and $v'$ are glued, respectively, to the root source vertices
$v_2$ and $v_3$ of $G_2$ and $G_3$. The choice of
$v'$ can be subdivided into the two cases where $v'$ is a vertex of $G_2$
or $v'$ is a vertex of $G_1$, including the case $v' = v$. Thus, we have
$$ (G_1\triangleleft G_2)\triangleleft  G_3 =\sum_{v\in V(G_1)}\,\, \sum_{v'\in V(G_2)} \,\,  (G_1 \triangleleft_v G_2) \triangleleft_{v'} G_3 +
\sum_{v, v'\in V(G_1)} \,\, (G_1\triangleleft_v G_2) \triangleleft_{v'} G_3. $$
Similarly, we have
$$ G_1\triangleleft (G_2 \triangleleft G_3) = \sum_{v\in V(G_1)} \sum_{v'\in V(G_2)} \,\,
G_1\triangleleft_{v} (G_2 \triangleleft_{v'} G_3), $$
where $v$ is glued to is the base vertex $v_2$ of $G_2 \triangleleft_{v'} G_3$, which is the same as
the base vertex of $G_2$.
Thus, we obtain
$$ (G_1\triangleleft G_2) \triangleleft G_3 - G_1\triangleleft (G_2 \triangleleft G_3)
= \sum_{v, v' \in V(G_1)} G_1\cup_{v\equiv v_2} G_2 \cup_{v' \equiv v_3} G_3 $$
and similarly
$$ (G_1\triangleleft G_3)\triangleleft G_2 - G_1\triangleleft (G_3 \triangleleft G_2)
= \sum_{v, v' \in V(G_1)} G_1\cup_{v'\equiv v_3} G_3 \cup_{v\equiv v_2} G_2, $$
which proves \eqref{prelie}.
\endproof

We then obtain the associated Lie algebra.

\begin{cor}\label{Lialg1cor}
Let ${\mathcal G}$ be a context-free Insertion Graph Grammar of rooted directed acyclic graphs,
with start graph $G_S=\{ v\}$ and production rules $P(v,v_2,G_2)$, with $v_2$ the
source of $G_2$. Then there is an associated Lie algebra ${\rm Lie}_{\mathcal G}$ given by the vector space
${\mathcal V}$ spanned by the graphs of ${\mathcal W}_{\mathcal G}$ with the Lie bracket $[G_1,G_2]=G_1\triangleleft G_2-
G_2 \triangleleft G_1$.
\end{cor}

\medskip

\begin{rem}\label{remsink}{\rm
A variant of the above construction that makes it (very mildly) context sensitive is obtained by
requiring that the marked source vertex of the graph $G_2$ in a production rule
$P(v,v_2,G_2)$ is glued to a {\em sink} vertex of the graph $G_1$, to which the rule is applied.
The argument is exactly as before, and one obtains a pre-Lie insertion operator and
a Lie algebra. We will generalize this context-sensitive version to more general gluing data
in Propositions \ref{preLie2ins} and \ref{preLie3ins} below.}
\end{rem}

\medskip
\subsection{Some Lie algebras of context-sensitive grammars of directed graphs}

We now consider a variant of the case of Proposition \ref{preLie1ins} where
we consider an example of context-sensitive graph grammars.
We still assume, as above,
that ${\mathcal G}$ is an Insertion Graph Grammar ${\mathcal G}$ as in Definition \ref{IGG1},
with start graph $G_S$ a single vertex, and where all the graphs $G\in {\mathcal W}_{\mathcal G}$
are directed. We no longer require that they are acyclic, hence graphs will
generally have oriented loops. An oriented loop $\gamma$ in a graph $G$
is an attractor if all the edges in $E_{G}(\gamma)$ (defined as in \eqref{EGGprime}) are incoming,
that is, $\partial(e) \cap V(\gamma)=t(e)$. It is a repeller if all edges in $E_{G}(\gamma)$
are outgoing, $\partial(e) \cap V(\gamma)=s(e)$. In general, there will be also oriented loops
that are neither attractors not repellerts. We modify the previous context-free construction
by considering, in addition to the production rules that glue a vertex of one graph
to a source vertex of another, also context-sensitive production rules that glue an attractor
loop of the first graph to a repeller loop of the second,
\begin{equation}\label{Ploop}
G_1\triangleleft_\gamma G_2 := P(\gamma, \gamma_2, G_2)(G_1)=G_1 \cup_{\gamma\equiv \gamma_2} G_2,
\end{equation}
where the two graphs are glued by identifying the two oriented loops $\gamma$ and $\gamma_2$
(which necessarily have to have the same number of edges). The insertion operator is then
defined as
\begin{equation}\label{insopH}
G_1 \triangleleft G_2 = \sum_{\substack{\gamma \subset G_1 \\ \gamma \text{ attractor loop}
\\ \gamma \simeq \gamma_2}} G_1 \triangleleft_\gamma G_2 =\sum_{\substack{\gamma \subset G_1 \\ \gamma \text{ attractor loop}\\ \gamma \simeq \gamma_2}}
P(\gamma, \gamma_2, G_2)(G_1).
\end{equation}

\begin{prop}\label{preLie2ins}
Given a context-sensitive Insertion Graph Grammar ${\mathcal G}$ as above,
the insertion operator \eqref{insopCF} defines a pre-Lie
structure on the vector space ${\mathcal V}$ spanned by the graphs in ${\mathcal W}_{\mathcal G}$.
\end{prop}

\proof The composition $G_1 \triangleleft (G_2 \triangleleft G_3)$ is given by
$$ G_1 \triangleleft (G_2 \triangleleft G_3) =
\sum_{\gamma \subset G_1} \,\, \sum_{\gamma' \subset G_2}
G_1 \triangleleft_\gamma (G_2 \triangleleft_{\gamma'} G_3) $$
while the composition $(G_1 \triangleleft G_2) \triangleleft G_3$ is
$$ (G_1 \triangleleft G_2) \triangleleft G_3 =
\sum_{\gamma \subset \Gamma_1} \,\, \sum_{\gamma'\subset  G_1 \triangleleft_\gamma G_2}
(G_1 \triangleleft_\gamma G_2) \triangleleft_{\gamma'} G_3. $$
In the last sum, the choice of $\gamma' \subset G_1 \triangleleft_\gamma G_2$ can be
broken down into the case where $\gamma' \subset G_1$,
the case where $\gamma' \subset G_2\smallsetminus \gamma$, and the case where it intersects
both, $\gamma' \cap G_1\neq \emptyset$ and $\gamma' \cap G_2\smallsetminus \gamma\neq \emptyset$.
In fact, because of our assumptions on the production rules, only the first two
possibilities can occur, and the first one can occur only with $\gamma'\cap \gamma =\emptyset$
To see this, suppose $\gamma'$ intersects both sets. Then it must intersect $\gamma$, since
$\gamma'$ is connected and $\gamma$ is the frontier between $G_1$ and $G_2$.
Under our assumptions, $\gamma'$ is an attractor loop for $G_1\triangleleft_\gamma G_2$,
hence all edges in $E_{G_1\triangleleft_\gamma G_2}(\gamma')$ must be incoming to $\gamma'$.
On the other hand, $\gamma$ is an attractor loop for $G_1$ and a repeller loop for $G_2$,
so inside $G_1\triangleleft_\gamma G_2$, there are vertices of $\gamma$ that have both incoming
and outgoing edges in $E_{G_1\triangleleft_\gamma G_2}(\gamma)$.
Consider a vertex $v$ in the intersection $\gamma'\cap \gamma$. Either $\gamma'$ and $\gamma$
have an adjacent edge in common, or they cross each other transversely at $v$. If they are
transverse, then the incoming and outgoing edges of $\gamma$ at $v$ show that $\gamma'$
cannot be an attractor loop for $G_1\triangleleft_\gamma G_2$. If $\gamma$ and $\gamma'$ have
at least one edge adjacent to $v$ in common, then that edge is either incoming or outgoing
at $v$. If it is incoming, then the next edge of $\gamma$ is outgoing and that suffices
to show $\gamma'$ is not an attractive loop. If it is outgoing, then one can argue the same
way with the next vertex. Thus, we can rewrite the sum above as
$$ (G_1 \triangleleft G_2) \triangleleft G_3 =
\sum_{\gamma \subset \Gamma_1} \,\, \sum_{\gamma'\subset  G_1 \smallsetminus \gamma}
(G_1 \triangleleft_\gamma G_2) \triangleleft_{\gamma'} G_3 +
\sum_{\gamma \subset \Gamma_1} \,\, \sum_{\gamma'\subset G_2\smallsetminus \gamma}
(G_1 \triangleleft_\gamma G_2) \triangleleft_{\gamma'} G_3. $$
Notice that, in the sum describing $G_1 \triangleleft (G_2 \triangleleft G_3)$
we also have $\gamma' \cap \gamma =\emptyset$ because $\gamma' \subset G_2\triangleleft_{\gamma'} G_3$
is not a repelling loop so it cannot intersect the repelling loop $\gamma_2$ that is glued to $\gamma$.
We then obtain
$$ (G_1\triangleleft G_2) \triangleleft G_3 - G_1\triangleleft (G_2 \triangleleft G_3)
= \sum_{\substack{\gamma, \gamma' \subset G_1 \\ \gamma,\gamma' \text{ attractor loops}}}
G_1\cup_{\gamma\equiv \gamma_2} G_2 \cup_{\gamma' \equiv \gamma_3} G_3 $$
and similarly
$$ (G_1\triangleleft G_3)\triangleleft G_2 - G_1\triangleleft (G_3 \triangleleft G_2)
= \sum_{\substack{\gamma, \gamma' \subset G_1 \\ \gamma,\gamma' \text{ attractor loops}}}
G_1\cup_{\gamma'\equiv \gamma_3} G_3 \cup_{\gamma \equiv \gamma_2} G_2, $$
which proves \eqref{prelie}.
\endproof

We obtain an associated Lie algebra ${\rm Lie}_{\mathcal G}$, as in Corollary \ref{Lialg1cor}. This
construction can be further generalized to other context-sensitive grammars, in the following way.
Assume again that ${\mathcal G}$ is an Insertion Graph Grammar ${\mathcal G}$ as in Definition \ref{IGG1},
with start graph $G_S$ a single vertex, and where all the graphs are oriented.
In addition to the production rules that
glue a vertex of one graph to a source vertex of another,
as in Proposition \ref{preLie1ins}, we also allow for
context-sensitive production rules of the form $P(G_L,H,G_R)$ where $H$ is
a connected, oriented graph with no sources or sinks. Moreover, we require that all the
edges in $E_{G_L}(H)$ are incoming to $H$ and all the edges in $E_{G_R}(H)$
are outgoing from $H$. We define the insertion operator by setting
\begin{equation}\label{GLHins}
 G_1\triangleleft_{G_L,H} G_2:= P(G_L,H,G_2)(G_1) = G_1\cup_H G_2
\end{equation}
to be the gluing of $G_1$ and $G_2$ along $H$, whenever $G_1$ contains a pair of subgraphs
isomorphic to $H\subset G_L$, with the orientation requirements as specified above.
We then have
\begin{equation}\label{GLHinsop}
 G_1\triangleleft G_2:=\sum_{G_L \subset G_1} P(G_L,H,G_2)(G_1),
\end{equation}
where the sum is over all the production rules and over all the possible
ways of identifying $G_L$ with a subgraph of $G_1$.
The result is zero if $G_1$ does not contain any subgraph isomorphic to $G_L$.
We then have the following straightforward generalization of Proposition \ref{preLie2ins}.

\begin{prop}\label{preLie3ins}
Given a context-sensitive Insertion Graph Grammar ${\mathcal G}$ as above,
the insertion operator \eqref{GLHinsop} defines a pre-Lie
structure on the vector space ${\mathcal V}$ spanned by the graphs in ${\mathcal W}_{\mathcal G}$.
\end{prop}

\proof The argument follows along the same lines as the previous cases.
Observe that, since $H$ is a directed graphs with neither sinks nor sources,
at every vertex $v\in V(H)$ there are at least one incoming and one outgoing
edge in $E(H)$. We then argue exactly as in Proposition \ref{preLie2ins}.
In the composition
$$ (G_1 \triangleleft G_2) \triangleleft G_3 =
\sum_{H\subset G_L \subset \Gamma_1} \,\, \sum_{H'\subset G_L' \subset  G_1 \triangleleft_{G_L,H} G_2}
(G_1 \triangleleft_{G_L,H} G_2) \triangleleft_{G_L',H'} G_3, $$
if the subgraph $H' \subset G_1 \triangleleft_{G_L,H} G_2$ has nontrivial intersection
with both $G_1$ and $G_2\smallsetminus H$, then it must intersect $H$. Then the
conditions on the orientations imply that $H'$ cannot have only incoming edges in
$E_{G_1 \triangleleft_{G_L,H} G_2}(H')$, which contradicts the orientation
requirements for $H'$. So the only compositions that give non-trivial terms are
the ones where $H'$ is fully contained in either $G_1$ (in fact $G_1\smallsetminus H$)
or in $G_2\smallsetminus H$. We then write the composition above by separating out
the sums for these two cases, and the rest of the argument follows exactly as in the
previous proposition.
\endproof

\medskip
\subsection{Lie algebras and gluing along half-edges}
We now consider Insertion Graph Grammars ${\mathcal G}$ as in Definition \ref{IGG2}
and we consider the case where the left-hand-side of the production rules are
corollas $C(v)$, though we do not require that they are all of the same valence, so that
we include grammars that are (mildly) context-sensitive. Graphs are not
necessarily oriented, but we assume that they have a base vertex.
For a graph $G$ and a vertex $v\in V(G)$ we write $E_{ext}(G,v) \subset E_{ext}(G)$
for the subset of external edges of $G$ that are attached to the vertex $v$.
The condition on external
edges in Definition \ref{IGG2} corresponds in this case to the requirement that
the number of external edges $E_{ext}(G_2,v_2)$ attached to the base vertex
$v_2$ is at least equal to the valence of $C(v)$.
These external edges of $G_2$ are identified by the production rule with
the half-edges of the corolla $C(v) \subset G_1$, where $G_1$ is the
graph the production is applied to. The argument is then exactly as in
Proposition \ref{preLie1ins}.  The base vertex of a production $P(C(v), C(v_2), G_2)(G_1)$
is the base vertex $v_1$ of $G_1$

\begin{prop}\label{preLie4ins}
Let ${\mathcal G}$ be an Insertion Graph Grammars ${\mathcal G}$ as in Definition \ref{IGG2}, where
all the production rules are of the form $P(C(v),C(v_2),G_2)$ with ${\rm val}(C(v))={\rm val}(C(v_2))$
and $E_{ext}(C(v))\subset E_{ext}(G_2,v_2)$. Assume the start graph $G_S$ of ${\mathcal G}$ is also a
corolla $C(v)$. The insertion operator
\begin{equation}\label{insopCF2}
G_1\triangleleft G_2 = \sum_{\substack{v \in V(G_1) \\ {\rm val}(v)={\rm val}(C(v_2))}}
P(C(v),C(v_2),G_2)(G_1) =\sum_{\substack{v \in V(G_1) \\ {\rm val}(v)={\rm val}(C(v_2))}}
G_1\triangleleft_{C(v)} G_2
\end{equation}
defines a pre-Lie structure on the vector space ${\mathcal V}$ generated by the graphs in ${\mathcal W}_{\mathcal G}$.
\end{prop}

\proof We have
$$ G_1 \triangleleft (G_2 \triangleleft G_3) =
\sum_{\substack{v \subset V(G_1) \\ {\rm val}(v)\leq \# E_{ext}(G_2\triangleleft_{C(v')} G_3,v_2)}} \,\,
\sum_{\substack{v' \subset V(G_2)\\ {\rm val}(v') \leq \# E_{ext}(G_3,v_3)}}
G_1 \triangleleft_{C(v)} (G_2 \triangleleft_{C(v')} G_3) $$
$$ (G_1 \triangleleft G_2) \triangleleft G_3 =
\sum_{\substack{v \subset V(G_1) \\ {\rm val}(v)\leq \# E_{ext}(G_2,v_2)}} \,\, \sum_{\substack{v'\subset  V(G_1 \triangleleft_\gamma G_2)\\ {\rm val}(v')\leq \# E_{ext}(G_3,v_3) }}
(G_1 \triangleleft_{C(v)} G_2) \triangleleft_{C(v')} G_3. $$
In the first sum $E_{ext}(G_2\triangleleft_{C(v')} G_3,v_2)$ is $E_{ext}(G_2,v_2)$ since the
base vertex $v_2$ of $G_2\triangleleft_{C(v')} G_3$ is the base vertex of $G_2$. We then
separate out the last sum of the second expression into the cases where $v' \in V(G_1)$
or $v'\in V(G_2)$. We obtain, as in the previous cases,
$$ (G_1\triangleleft G_2) \triangleleft G_3 - G_1\triangleleft (G_2 \triangleleft G_3)
= \sum_{v, v' \in V(G_1)} G_1\cup_{C(v)} G_2 \cup_{C(v')} G_3 =
 (G_1\triangleleft G_3)\triangleleft G_2 - G_1\triangleleft (G_3 \triangleleft G_2) . $$
\endproof

Again we obtain an associated Lie algebra ${\rm Lie}_{\mathcal G}$.
One can also similarly extend the case of gluing along oriented loops, by assigning orientations to the
attached half-edges, with incoming/outgoing requirements as in Proposition \ref{preLie2ins}, or in
the case of gluing along more general graphs $H$ with orientation requirements as in
Proposition \ref{preLie3ins}, reformulated in terms of half-edges. These analogs of Proposition \ref{preLie2ins}
and Proposition \ref{preLie3ins} are completely straightforward and are proved by essentially the same
argument, so we will not state them explicitly here.
Within this setting, however, one cannot further extend the construction to more general context-sensitive
cases, beyond what we have seen in Proposition \ref{preLie3ins},
because the cases where the gluing data graph $H'$ intersects both $G_1$
and $G_2\smallsetminus H$
in $G_1\triangleleft_H G_2$ creates terms that do not cancel in the difference
$(G_1\triangleleft G_2) \triangleleft G_3 - G_1\triangleleft (G_2 \triangleleft G_3)$ and that
are not symmetric with respect to exchanging $G_2$ and $G_3$. To see more
precisely where the difficulty lies, we can write out the expression above, as before, in
the form
$$ (G_1\triangleleft G_2) \triangleleft G_3 - G_1\triangleleft (G_2 \triangleleft G_3)=
\sum_{\substack{H\subset G_1 \\ H'\subset G_1\triangleleft G_2}} (G_1\cup_H G_2)\cup_{H'} G_3
- \sum_{\substack{H\subset G_1 \\ H'\subset G_2}} G_1\cup_H (G_2\cup_{H'} G_3). $$
We can separate out, in the first sum the cases where $H'$ is completely contained in $G_1$
or completely contained in $G_2$, and where it intersects both graphs. The latter case,
in general, cannot be decomposed further, because it is not necessarily true that, if a
certain graph $H'$ is the gluing data of a production rule, subgraphs $G_1\cap H'$
and $G_2\cap H'$ would also occur in production rules. Thus, the term involving
subgraphs $H'$ intersecting both $G_1$ and $G_2$ does not cancel in the difference
between $(G_1\triangleleft G_2) \triangleleft G_3$ and $G_1\triangleleft (G_2 \triangleleft G_3)$
and at the same time is not symmetric with respect to interchanging $G_2$ and $G_3$,
hence one would not obtain a pre-Lie insertion operator.

\smallskip

We conclude this section by discussing a special example, which we will return to
in our application to Feynman graphs. We denote by $G_e$ the graph consisting of
a single edge $e$, identified with the union of two half-edges $e=(f,f')$ glued together by an
involution ${\mathcal I}(f)=f'$.
Let ${\mathcal G}$ be an Insertion Graph Grammars ${\mathcal G}$ as in Definition \ref{IGG2},
with start graph $G_S$ and with production rules:
\begin{enumerate}
\item $P(G_S,\{ f,f '\}\subset {\mathcal F}_{G_S},G_e)$, where the single edge graph $G_e$
is glued onto a pair $(f,f')$ of external half-edges of $G_S$,
\item $P(G_S,\{ f \}\subset {\mathcal F}_{G_S},G_S\cup_{f'} G_e)$, where the copy of $G_S$ in the right-hand-side
of the production is glued to the one on the left-hand-side by matching the remaining external half-edge
of $G_e$ to the half-edge $f$ to form a graph $G_S\cup_{e=(f,f')} G_S$ consisting of two copies of $G_S$
glued together along an edge.
\end{enumerate}

\begin{prop}\label{glueextedges}
Let ${\mathcal G}$ be an Insertion Graph Grammar as above.
Then the insertion operator
\begin{equation}\label{insopext}
G_1 \triangleleft G_2 =\sum P(G_S,{\mathcal F}_{G_S},G_2)(G_1)
\end{equation}
defines a pre-Lie structure on the vector space ${\mathcal V}$ spanned by the graphs in ${\mathcal W}_{\mathcal G}$.
\end{prop}

\proof It suffices to notice that, by the form of the production rules,
in the composition $(G_1 \triangleleft G_2) \triangleleft G_3$
the gluing of $G_3$ to $G_1 \triangleleft G_2$ happens along some of the external
half-edges of $G_1 \triangleleft G_2$. The previous gluing of $G_2$ to $G_1$, in turn,
glues some external half-edges of $G_2$ to some external half-edges of $G_1$.
Thus, the remaining half-edges of $G_1 \triangleleft G_2$ are either in $G_1\smallsetminus H$
or in $G_2\smallsetminus H$, where $H$ is the set of half-edges along which the gluing
of $G_2$ to $G_1$ happened (which are no longer external edges in $G_1 \triangleleft G_2$).
This suffices then to get the pre-Lie condition, exactly as in the cases discussed previously.
\endproof

We will see in \S \ref{FeynSec} below that this provides a different way of constructing
of Lie algebras of Feynman graphs, which is not equivalent to the insertion
Lie algebra of \cite{CoKr2}, \cite{EFBP}. More notably, it shows that the set of Feynman
graphs of a given quantum field theory is a graph language in the sense
of the theory of formal languages.

\medskip
\subsection{Lie algebras of insertion-elimination graph grammars}

We now consider the case of insertion-elimination graph grammars, as in Definitions
\ref{IEGG1} and \ref{IEGG2}.

Let ${\mathcal G}$ be an insertion-elimination graph grammar as in Definition \ref{IEGG2}.
We assume the following hypotheses:
\begin{enumerate}
\item In all the production rules $P(G_L,H,G_R)$ the graph $G_L$ is connected and
we have $H\subset {\mathcal F}_{G_L}$, a set of flags (half-edges),
with $H=E_{ext}(G_R)$
\item All the graphs in ${\mathcal W}_{\mathcal G}$ are oriented and in all production rules the half-edges in $H$
are incoming to both $G_L$ and $G_R$.
\end{enumerate}

\begin{prop}\label{IEGGLie}
Let ${\mathcal G}$ be an insertion-elimination graph grammar satisfying the two conditions above.
Then the insertion operator
\begin{equation}\label{IEGGins}
G_1 \triangleleft G_2 = \sum_{G\subset G_1} P(G, H, G_2)(G_1)
\end{equation}
is a pre-Lie operator.
\end{prop}

\proof With the notation $G_1\triangleleft_{G,H} G_2=P(G, H, G_2)(G_1)$, we have
$$ G_1 \triangleleft (G_2 \triangleleft G_3) = \sum_{G,G'} G_1\triangleleft_{G,H} (G_2 \triangleleft_{G',H'} G_3), $$
with $G\subset G_1$ and $H\subset {\mathcal F}_{G_1}$, $H=E_{ext}(G_2\triangleleft_{G',H'} G_3)$,
and with $G'\subset G_2$, $H'\subset {\mathcal F}_{G_2}$ and $H'=E_{ext}(G_3)$. Since all the external edges
of $G_3$ are glued to flags of $G_2$, in the identification along $H'=E_{ext}(G_3)$, we have
$H=E_{ext}(G_2\triangleleft_{G',H'} G_3)= E_{ext}(G_2)$. When composing in the opposite order, we
have
$$ (G_1 \triangleleft G_2) \triangleleft G_3 = \sum_{G,G'} (G_1\triangleleft_{G,H} G_2) \triangleleft_{G',H'} G_3, $$
with $G\subset G_1$, $H\subset {\mathcal F}_{G_1}$, $H=E_{ext}(G_2)$, and with
$G'\subset G_1\triangleleft_{G,H} G_2$, $H'\subset {\mathcal F}_{G_1\triangleleft_{G,H} G_2}$,
$H'=E_{ext}(G_3)$. If the graph $G'$ intersects nontrivially both $G_1\smallsetminus (G\smallsetminus H)$
and $G_2$, then by connectedness $H'\cap H\neq \emptyset$, but the orientation conditions on the
edges of $H$ and of $H'$ are incompatible, so $G'$ must be contained in either
$G_1\smallsetminus G$ or in $G_2$. We then obtain
$$ (G_1 \triangleleft G_2) \triangleleft G_3 - G_1 \triangleleft (G_2 \triangleleft G_3) = \sum_{\substack{G,G'\subset G_1 \\ G\cap G' =\emptyset}} (G_1\smallsetminus ((G\smallsetminus H)\cup (G'\smallsetminus H'))) \cup_H G_2 \cup_{H'} G_3 $$
which is symmetric in exchanging $G_2$ and $G_3$.
\endproof

We obtain an associated Lie algebra ${\rm Lie}_{\mathcal G}$.

\bigskip

\medskip
\subsection{The insertion Lie algebra of Quantum Field Theory and primitive graphs}

In the previous sections we have shown that one can associate to certain
classes of graph grammars an insertion Lie algebra, that behaves very
similarly to the insertion Lie algebra of Quantum Field Theory of \cite{CoKr2}, \cite{EFBP}.
It is then natural to ask whether the insertion Lie algebra of Quantum Field Theory is
itself obtained from a Graph Grammar via the same procedure discussed above.
This is {\em not} the case, because it would violate the
property that graph grammars have a {\em finite} number of production rules.
In fact,  the Lie algebra of Feynman graphs is
generated by the primitive elements of the Hopf algebra. We would like to
obtain all 1PI Feynman graphs of the theory from a graph grammar
that has a single start graph $G_S$ and a finite number of production
rules $P(G_L,H,G_R)$, in such a way that the Lie bracket $[G_1,G_2]$
of the Lie algebra of quantum field theory would agree with the
Lie bracket defined by the graph grammar,
$$ [G_1,G_2] = \sum_{G_L\subset G_1} P(G_L,H,G_2)(G_1) - \sum_{G_L\subset G_2} P(G_L,H,G_1)(G_2). $$
For this to be the case, we see that we would need a production rule for each insertion
of a primitive graph $G$ of the Hopf algebra of the theory into a vertex with valence equal
to the number of external edges of $G$. Since there are infinitely many primitive graphs,
this would violate the requirement that the graph grammar has only finitely many production rules.
We will see in \S \ref{FeynSec} that, despite this negative result, the
Feynman graphs of a given quantum field theory are a graph language,
obtained from a graph grammar with finitely many production rules.
These graph grammars in turn define Lie algebras, by the procedure
discussed in the previous sections, which are in general not
equivalent to the insertion Lie algebra of \cite{CoKr2}, \cite{EFBP}.

\bigskip

\section{Feynman diagrams as graph languages}\label{FeynSec}

Motivated by the insertion Lie algebra of quantum field theory, \cite{CoKr2}, \cite{EFBP},
we have shown in the previous section that, under certain conditions on the production
rules, one can associate Lie algebras to Graph Grammars.
In this section, we return to the motivating example of Feynman graphs and we show that
the Feynman graphs of certain quantum field theories are examples of graph languages.
Our generative description of Feynman graphs in terms of graph
grammars can be seen as ``reading in reverse" the procedure described in \cite{BKP}, \cite{KPKB},
that generates all Feynman graphs starting with the vacuum bubbles (no external edges) and
progressively cutting internal edges into pairs of external half-edges.

\medskip
\subsection{The $\phi^4$ Graph Language}

We first analyze the example of the $\phi^4$-theory.
This is the scalar quantum field theory with (Euclidean) Lagrangian
density
\begin{equation}\label{Lagphi4}
{\mathcal L}(\phi)=\frac{1}{2}(\partial\phi)^{2}+\frac{1}{2}m^{2}\, \phi^{2}+\frac{1}{4!}\lambda\,\phi^{4}.
\end{equation}
The Feynman graphs of this theory have all vertices of valence four.
More precisely, we should also include valence two vertices that correspond to
the mass and kinetic terms, but we will not mark them explicitly in the diagrams.

\smallskip

\begin{prop}\label{phi4GG}
The Feynman graphs of the $\phi^4$-theory are the elements of the graph language ${\mathcal L}_{\mathcal G}$
generated by a graph grammar ${\mathcal G}$ as in Proposition \ref{glueextedges}, with start graph
$G_S$ given by a $4$-valent corolla, and two production rules: one of the form
$P(G_S,\{ f,f '\}\subset {\mathcal F}_{G_S},G_e)$, which glues together two external edges of $G_S$
and one of the form $P(G_S,\{ f \}\subset {\mathcal F}_{G_S},G_S\cup_{f'} G_e)$, which glues together
two copies of $G_S$ along an edge.
At each stage in the application of one of the production rules, the external edges of the
resulting graph are marked either with a terminal or with a non-terminal symbol.
\end{prop}

\proof Whenever the external edges of a graph in ${\mathcal W}_{\mathcal G}$ are marked by non-terminal symbols
one can continue to apply production rules to them, while if all the external edges are marked by
terminals the resulting graph is in ${\mathcal L}_{\mathcal G}$. Thus, a graph is in ${\mathcal L}_{\mathcal G}$ if either it has no more
external edges, in which case it is a vacuum bubble of the $\phi^4$-theory, or if all the external
edges are marked by terminals, in which case, it can be identified with the result of cutting a number
of edges of a vacuum bubble into half edges. This produces all Feynman graphs of the $\phi^4$
theory, \cite{KPKB}.
\endproof

\smallskip

The production procedure is illustrated in Figure \ref{phi4GG}.
The graph in the upper left-hand corner is $G_S$ for this
theory. Single arrows indicate the first type of production gluing
external edges to make a loop. Double arrows indicate the second type
of production joining base graphs along an edge to make a new
graph with no loops.
the graphs with no external edges are represented on the right, while
the graphs with as many external edges as possible are on the left.
Note that the latter type of graphs are trees.
t is clear that for this theory, any allowed Feynman graph
can be transformed into a tree such as the ones above. Consider the
graphs on one internal vertex. They all can be constructed by gluing
the edges of the base graph $G_S$, via repeated application of the
production rule represented by the single arrows. A similar statement
can be made of the graphs on two internal vertices, except that, after cutting
all edges into pairs of half edges, one is now left with a disjoint union of
two copies of the base graph $G_S$. In general, any (connected) $\phi^4$ graph with
$k$ valence $4$ vertices can be constructed by gluing together pairs of half-edges,
starting from $k$ copies of $G_S$, hence by repeated applications of the two
types of production rules.

\begin{figure}
\includegraphics[scale=0.6]{phi4GG.jpg}
\caption{Generative grammar for Feynman graphs: $\phi^4$-theory. \label{phi4GG}}
\end{figure}

\medskip
\subsection{Graph grammar for $\phi^k$-theories}

The case discussed above of the $\phi^4$-theory can easily be generalized to
the case of (Euclidean) Lagrangian densities
\begin{equation}\label{LagPphi}
{\mathcal L}(\phi)=\frac{1}{2}(\partial\phi)^{2}+\frac{1}{2}m^{2}\, \phi^{2}+P(\phi)
\end{equation}
where the interaction term is a single monomial $P(\phi) =\frac{1}{k!} \lambda \, \phi^k$.

\begin{prop}\label{phikGG}
The Feynman graphs of the $\phi^k$-theory are the elements of the graph language ${\mathcal L}_{\mathcal G}$
generated by a graph grammar ${\mathcal G}$ as in Proposition \ref{glueextedges}, with start graph
$G_S$ given by a $k$-valent corolla, and two production rules: one of the form
$P(G_S,\{ f,f '\}\subset {\mathcal F}_{G_S},G_e)$, which glues together two external edges of $G_S$
and one of the form $P(G_S,\{ f \}\subset {\mathcal F}_{G_S},G_S\cup_{f'} G_e)$, which glues together
two copies of $G_S$ along an edge.
At each stage in the application of one of the production rules, the external edges of the
resulting graph are marked either with a terminal or with a non-terminal symbol.
\end{prop}

\proof
Let $G$ be a connected graph of the $\phi^k$ theory. Thus, all
vertices $v\in V(G)$ have valence ${\rm val}(v)=k$. We neglect for the moment
the possible presence of valence $2$ vertices associated
to the kinetic and mass terms in ${\mathcal L}(\phi)$. Consider all possible ways
of cutting internal edges, so that they are replaced by a pair of external half-edges,
that leave the graph connected. The number of possible such cuts is the
degree of edge-connectedness of the graph. Stop when no further such cuts
remain. If we denote the resulting graph by $G'$, then it is clear that $G$
is obtained from $G'$ by repeatedly applying the first production rule.
Every connected graph has a decomposition into a tree with insertions at the vertices
of 1PI graphs (one particle irreducible, also known as 2-edge-connected)
that have a number of external edges equal to the valence of the tree vertex.
By repeatedly cutting non-disconnecting edges, and using this decomposition,
it is clear that the resulting graph $G'$ is a tree. Since all the vertices of $G'$
have valence ${\rm val}(v)=k$, the tree can be constructed by repeated application of
the second production rule. One can consider also the presence
of valence two vertices, with each gluing of a copy of $G_e$ in the production
rules involving a valence two vertex inserted in the middle of an edge
connecting two valence $k$ vertices, and the argument remains essentially the same.
The difference between valence two vertices coming from the kinetic and the
mass terms can be taken care of by using two different terminal symbols
labeling the vertices.
\endproof

\medskip
\subsection{Graph grammars for arbitrary scalar field theories}

We then consider the case of a scalar field theory with Lagrangian density \eqref{LagPphi}
where the interaction term is a polynomial
\begin{equation}\label{Pphi}
P(\phi)=\sum_{k\geq 3}\frac{\lambda_{k}}{k!}\phi^{k}.
\end{equation}

\smallskip

\begin{prop}\label{phikGG}
The Feynman graphs of a scalar field theory with interaction polynomian $P(\phi)$ as in \eqref{Pphi} of degree $N$
are the elements of the graph language ${\mathcal L}_{\mathcal G}$
generated by a graph grammar ${\mathcal G}$ as in Proposition \ref{glueextedges}, with start graph
$G_S$ given by a $k$-valent corolla, where $k$ is the smallest term in \eqref{Pphi}
with $\lambda_k\neq 0$ and three production rules:
\begin{enumerate}
\item The first kind of production rules is of the form
$P(G_S,\{ f,f '\}\subset {\mathcal F}_{G_S},G_e)$, which glues together two external edges of $G_S$.
\item The second kind of production rule
$P(G_S, G_e, G_{S,f})$ glues a copy of $G_e$ to the start graph $G_S$
by identifying one half edge of $G_e$ with one of the half-edges of $G_S$ and leaving
the other half edge $f$ as a new external edge, thus creating a corolla of valence $k+1$.
If $\lambda_{k+1}\neq 0$ the vertex of the resulting graph $G_{S,f}$ can be labeled by either a terminal
or a nonterminal symbol, if $\lambda_{k+1}=0$ it is labeled by a nonterminal symbol.
Production rules $P(G_{S,f_1,\ldots,f_r}, G_e, G_{S,1,\ldots,f_r,f'})$ can be further applied to previously produced
corollas $G_{S,f_1,\ldots,f_r}$ with vertex labeled by nonterminals, until ${\rm val}(G_{S,f_1,\ldots,f_r})=N-1$:
in this case the vertex in the resulting $G_{S,1,\ldots,f_r,f'}$ can only be labelled by a terminal.
\item The third kind of production rules $P(G_{S,f_1,\ldots, f_r}, \{ f_i \}\subset {\mathcal F}_{G_S},G_{S,f_1,\ldots, f_r}\cup_{f_i=f_j'} G_{S,f'_1,\ldots, f'_s})$, which glues together along an edge two corollas $G_{S,f_1,\ldots, f_r}$
and $G_{S,f'_1,\ldots, f'_s}$ produced by the previous type of production rules.
\end{enumerate}
At each stage in the application of one of the production rules, the external edges of the
resulting graph are marked either with a terminal or with a non-terminal symbol.
\end{prop}

\proof The argument is similar to the previous case: one starts from an arbitrary connected
Feynman graph $G$ of the theory and performs the maximal number of cuts of internal
edges into pairs of external half-edges that leaves the graph connected. The only
difference in the argument is that the resulting graph $G'$ is now a tree with vertices of
valences ranging among the values $3\leq k \leq N$ for which $\lambda_k\neq 0$ in \eqref{Pphi}.
These are then obtained by repeated application of the production rules of the second and
third type that produce corollas of the right valences and glue them together along edges to form the
tree $G'$.
\endproof

\smallskip

Figure \ref{phi3phi4GG} illustrates the generative grammar of Proposition \ref{phikGG}
for the scalar field theory with
$$ {\mathcal L}(\phi) = \frac{1}{2}(\partial\phi)^{2}+\frac{1}{2}m^{2}\, \phi^{2}+\frac{1}{6} \lambda_3\, \phi^3 +\frac{1}{24} \lambda_4\, \phi^4. $$

\begin{figure}
\includegraphics[scale=0.6]{phi3phi4GG.jpg}
\caption{Generative grammar for $P(\phi)=\frac{1}{6} \lambda_3\, \phi^3 +\frac{1}{24} \lambda_4\, \phi^4$. \label{phi3phi4GG}}
\end{figure}

\medskip
\subsection{Graph grammar for the $\phi^2 A$-theory}

This theory has two different propagators for the fields $\phi$ and $A$, which one
represents by drawing straight edges for the $\phi$-propagator and wavy edges
for the $A$-propagator. The cubic interaction terms implies that the Feynman graphs
have vertices of valence $3$ with two straight and one wavy external half-edges.
The Feynman graphs of this theory were analyzed in \cite{KPKB}, with $\phi$
representing fermions and $A$ the photon. Note that in this theory
edges representing photons are always internal, unlike what happens in
quantum electrodynamics, where photons can be external, \cite{BKP}.

\begin{prop}\label{phi2AGG}
The Feynman graphs of the $\phi^2 A$-theory are the elements of the graph language ${\mathcal L}_{\mathcal G}$
generated by a graph grammar ${\mathcal G}$ as in Proposition \ref{glueextedges}. The start graph
$G_S$ has two trivalent vertices, one internal wavy edge connecting them, and four external
straight half-edges. There are two kinds of production rules: one of the form
$P(G_S,\{ f,f '\}\subset {\mathcal F}_{G_S},G_e)$, which glues together two external edges of $G_S$
and one of the form $P(G_S,\{ f \}\subset {\mathcal F}_{G_S},G_S\cup_{f'} G_e)$, which glues together
two copies of $G_S$ along an external edge.
At each stage in the application of one of the production rules, the external edges of the
resulting graph are marked either with a terminal or with a non-terminal symbol.
\end{prop}

\proof The graph grammar ${\mathcal G}$ is illustrated in Figure \ref{phi2AGG}.
Notice that, if photon edges were allowed to be external, then the argument would
be the same as in the $\phi^{3}$-theory, except that the labeling of the edges
as bosonic or fermionic must be taken into account when inserting the base graph
in the tree. The fact that we require bosonic edges to be internal means that these
edges cannot be cut in the process that leads from $G$ to $G'$.
SInce all vertices in $G$ have valence $3$
with two fermion and one boson line, after all the internal fermion lines
are cut, one still obtains a tree $G'$, which we now view as
being formed out of repeated application of the second production
rule applied to the start graph.
\endproof

The properties of external and internal edges of the $\phi^2 A$-theory
discussed in \cite{KPKB} are reflected here in the fact that, in the
production rules, it is only possible to join
base graphs along fermion edges. The fact that photon
edges are only internal is taken into account by the choice
of the start graph having two vertices instead of one, with one
internal bosonic edge.

\begin{figure}
\includegraphics[scale=0.5]{phi2AGG.jpg}
\caption{Generative grammar for the $\phi^2A$-theory. \label{phi2AGG}}
\end{figure}

\medskip
\subsection{A general procedure}

We can summarize all the cases discussed above, for the different theories, in
a common general procedure, as follows.

\begin{itemize}

\item Fix $n\in\mathbb{Z}^{+}$. This is the number of distinct conditions.
\item Fix $t\in\mathbb{Z}^{+}$. This is the number of edge types. Let $v\in\{0,1\}^{t}$
where $v(i)$ is 0 if type $i$ edges cannot be external and 1 otherwise.
\item For each $k\in\{1,...,n\}$:
\begin{itemize}
\item Let $C_{k}\in M_{t,2}(\mathbb{Z})$ where $C_{k}(i,1)=i$ for all
$i\in\{1,...,t\}$ (the edge type) and with $C_{k}(i,2)$ the number
of edges of type $i$ allowed by this condition at each vertex.
\item Let $G_{k}$ be the star graph with edges determined by $C_{k}$
\end{itemize}
\item Let $G$ be a connected graph that satisfies the conditions of $v$
and each of the $C_{k}$.
\item There is a sequence of production rules that glue
edges of finitely many copies of the graphs $G_{k}$ to make $G$.
\item Any $G$ that satisfies these conditions can be constructed
from these initial graphs using the production rules.
\item In order to have a single start graph one needs to add further
production rules that derive higher valence star graphs $G_{k}$
from lower valence ones, marking the vertex with a terminal label
when the process should stop.
\end{itemize}

\begin{ex}\label{ex1}{\rm
In a $\phi^{2}A$ theory, $n=1$, $t=2$, $v=(1,0)$
and
$C_{1}=\left(\begin{array}{cc}
1 & 2\\
2 & 1
\end{array}\right)$.}
\end{ex}

\begin{ex}\label{ex2}{\rm
In a theory where $P(\phi)=\frac{\lambda_{3}}{3!}\phi^{3}+\frac{\lambda_{4}}{4!}\phi^{4}$,
$n=2$, $t=1$, $v=(1)$, $C_{1}=(\begin{array}[t]{cc}
1 & 3\end{array})$ and $C_{2}=(\begin{array}[t]{cc}
1 & 4\end{array})$.}
\end{ex}

\bigskip
\bigskip

\noindent {\bf Acknowledgment} The first author is supported by NSF grants DMS-1007207,
DMS-1201512, PHY-1205440. The second author was supported by a Summer Undergraduate
Research Fellowship at Caltech.

redpajama_set_name: RedPajamaArXiv


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Newswise — ALBUQUERQUE, N.M. — Sandia National Laboratories is transforming how it assesses nuclear weapons in a stockpile made up of weapons at different stages in their lifecycles — some systems that have existed for decades alongside those that have undergone life extension programs.
Back when the United States was developing new nuclear weapon systems, weapons typically were either in production or were retired before they aged much more than about 10 years. The U.S. today is no longer designing new systems, so scientists and engineers refurbish weapons to ensure the stockpile will function as intended and that weapons are safe, secure and reliable.
Sandia is responsible for developing as much as 97 percent of a weapon system's non-nuclear components. It has created ever-more sophisticated tests and computer models to qualify those systems under its stockpile stewardship role — certifying they always will work as designed when authorized by the president but will never work in any other circumstance, said Scott Holswade, deputy chief engineer.
The stockpile surveillance program assesses each nuclear weapon system to detect or anticipate potential problems.
Components of nuclear weapons age, and scientists and engineers address that through life extension programs or less comprehensive alterations. A life extension program refurbishes components nearing the end of their life, remanufacturing or redesigning them. Some components are reused by being requalified to go back into a weapon without change. Remanufacturing means using the original specifications to remake components that have aged. However, sometimes the original technology is no longer available, so Sandia redesigns parts using modern technology — think switching out vacuum tubes for solid state technology.
The U.S. last conducted underground nuclear testing in 1992 and has been in a moratorium ever since. Since then, Sandia has used non-nuclear tests, experiments and computer simulations to study environments weapons might face, such as vibration, radiation or extreme cold or heat. Sandia must design and engineer systems to handle those conditions and do extensive testing to make sure designs meet requirements established by the departments of Defense and Energy, said Toby Townsend, stockpile systems engineering senior manager. Once refurbished weapons enter the stockpile, Sandia conducts tests to assure the systems continue to meet requirements as they age.
"We're often asked, 'How much longer could the weapon potentially be good for?' That's a major part of the stewardship mission," Townsend said.
When nuclear weapons were first developed, they were expected to be useful for a so-called protected period, the lifespan of the design. To mimic that lifespan, engineers use environmental test chambers to speed up conditions the weapon could face over time, a process called accelerated aging. They also do a multitude of tests on refurbished weapons coming into the stockpile, such as the W76-1, to catch possible "birth defects" due to design or manufacturing problems.
However, the question for a weapon in the stockpile for decades is different: What happens as it ages?
"That's one thing we're trying to get to: What are the risks and knowledge gaps, and focus on those rather than continuing to do things that aren't returning a lot of differentiating information," Holswade said.
Life extension programs and alterations test everything from individual components to nearly complete systems without nuclear material.
Jay Vinson, senior manager for integrated stockpile evaluation, said assessing non-nuclear systems includes laboratory testing, system-level flight testing of gravity bombs, cruise missile flights, ballistic missile flights and submarine-launched missile flights, all with the nuclear explosive packages removed. In addition, numerous sites, including Sandia, the Kansas City National Security Campus and the Weapon Evaluation Test Laboratory (WETL), test non-nuclear components.
Once a weapon is in production, the program must assure it conforms to the design, an example of an early lifecycle concern. "You can have production errors, or you might have missed something in the design process that only reveals itself as you get to higher volumes," Holswade said.
As production ramps up, Sandia pulls units before they go into the stockpile to catch possible production errors or early-onset failures, Vinson said.
Weapons also are pulled from the field for tests later in their lifecycle, where they've been handled by military personnel and might have been on a submarine or a missile. Engineers want to study whether field conditions exposed a problem.
Some weapons from the field are built into joint test assemblies, mock weapons without nuclear materials but fitted with sensors and instrumentation to assess performance. Flight testing is part of the qualification process for refurbished weapons, and its main objective is to obtain reliability, accuracy and performance data under operational conditions. Scientists and engineers use the test data in computer simulations developed by Sandia to evaluate systems' reliability and to verify they functioned as designed.
Last June, the W80-1 Air Launched Cruise Missile Surveillance Flight Test Program conducted a long flight of a joint test assembly to a target and performed arming maneuvers.
Weapons in some flight tests contain high explosives along with instruments that send out data during the flight until the explosives detonate. In other tests without explosives, additional sensors check that the detonation chain would have functioned. In all cases, the nuclear material is removed before the test.
Sensors in non-explosives tests send data during flight, but engineers also recover the system to gather more diagnostics. For B61-12 flight tests at Tonopah Test Range, telemetry data gathered during flight are verified at the range's test operations center and main telemetry ground station, and transmitted to Sandia in Albuquerque.
Engineers do additional tests on components and subsystems for so-called "corner cases" — like the hottest day with the biggest shock condition or the coldest day with more vibration.
"We can't fly every potential trajectory because you're limited by where you can launch from and where you can land. We can't fly every weather condition because you're stuck with the weather you have when the flight's scheduled," Holswade said.
One problem with testing a non-nuclear system as a whole is that the system works or it doesn't, and the test can't always pinpoint something specific that might have worked, but only marginally. Weapons programs supplement system testing with component testing, allowing engineers to understand whether there were changes in the components.
Sandia's large environmental test facilities can evaluate subsystems and components under all kinds of conditions. That doesn't mean testing is easy — combining environments is especially challenging. Engineers might test how well a component works in a radiation environment and separately in a shock environment, but it's difficult to combine radiation and shock testing, for example.
So researchers use simulation. First they perform experiments to gather information to anchor their models, then use those models to simulate thousands of different environments.


ShenZhen Focused Smartech Co.,Ltd, Established In 2012,Is An Industry-Leading Professional Manufacturer Of Smart Cards And RFID Label Products. With Total Assets Of 3 Million USD And a Land Area Of 3,000 M2, Which Is Always Committed To The Development, Production And Sales Of Smart Card Technology And Products, Has Formed Several Series Of Main Products Like Contactless IC Card, ,RFID Label, RFID Card Inlay, Key Chains, Epoxy Card, Rfid Wristband, RFID Reader/Writer Etc. It Has Already Provided Millions Of Smart Card Products For The Users At Home And Abroad. As At Present, The Company Has Two High-Precision Smart Card And RFID Label Production Lines And All Products Are Made Independently In Its Plant. Focused Will Keep Developing And Innovate To Better Prompt RFID Technology Application In Every Walk Of Life!
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text:
Microscopic theory of diffusive spin current with spin-orbit interaction
Kazuhiro Hosono, Akinobu Yamaguchi, Yukio Nozaki, Gen Tatara
We present the diagrammatic calculation of the spin current in the presence of the spin-orbit interaction due to impurity and electromagnetic field. By considering the vertex corrections for the normal impurity and the spin-orbit interaction, we derive both the field-driven and diffusion part of the spin current with side-jump and skew-scattering contributions. We found an alternative source of the spin-density generation, given by ×A•, where A is an electromagnetic vector potential. This generation mechanism is distinct from the conventional spin accumulation one, given by •A•.
「Microscopic theory of diffusive spin current with spin-orbit interaction」の研究トピックを掘り下げます。これらがまとまってユニークなフィンガープリントを構成します。
spin-orbit interactions Physics & Astronomy 100%
Orbits Engineering & Materials Science 81%
Electron Spin Chemical Compounds 77%
Electromagnetic fields Engineering & Materials Science 47%
Scattering Engineering & Materials Science 43%
Electromagnetic Field Chemical Compounds 40%
impurities Physics & Astronomy 30%
Hosono, K., Yamaguchi, A., Nozaki, Y., & Tatara, G. (2011). Microscopic theory of diffusive spin current with spin-orbit interaction. Physical Review B - Condensed Matter and Materials Physics, 83(14), [144428]. https://doi.org/10.1103/PhysRevB.83.144428
Microscopic theory of diffusive spin current with spin-orbit interaction. / Hosono, Kazuhiro; Yamaguchi, Akinobu; Nozaki, Yukio; Tatara, Gen.
In: Physical Review B - Condensed Matter and Materials Physics, Vol. 83, No. 14, 144428, 28.04.2011.
Hosono, K, Yamaguchi, A, Nozaki, Y & Tatara, G 2011, 'Microscopic theory of diffusive spin current with spin-orbit interaction', Physical Review B - Condensed Matter and Materials Physics, vol. 83, no. 14, 144428. https://doi.org/10.1103/PhysRevB.83.144428
Hosono K, Yamaguchi A, Nozaki Y, Tatara G. Microscopic theory of diffusive spin current with spin-orbit interaction. Physical Review B - Condensed Matter and Materials Physics. 2011 4 28;83(14). 144428. https://doi.org/10.1103/PhysRevB.83.144428
Hosono, Kazuhiro ; Yamaguchi, Akinobu ; Nozaki, Yukio ; Tatara, Gen. / Microscopic theory of diffusive spin current with spin-orbit interaction. In: Physical Review B - Condensed Matter and Materials Physics. 2011 ; Vol. 83, No. 14.
@article{1c77e5beafb74c3abb920992e3c10c2f,
title = "Microscopic theory of diffusive spin current with spin-orbit interaction",
abstract = "We present the diagrammatic calculation of the spin current in the presence of the spin-orbit interaction due to impurity and electromagnetic field. By considering the vertex corrections for the normal impurity and the spin-orbit interaction, we derive both the field-driven and diffusion part of the spin current with side-jump and skew-scattering contributions. We found an alternative source of the spin-density generation, given by ×A•, where A is an electromagnetic vector potential. This generation mechanism is distinct from the conventional spin accumulation one, given by •A•.",
author = "Kazuhiro Hosono and Akinobu Yamaguchi and Yukio Nozaki and Gen Tatara",
journal = "Physical Review B-Condensed Matter",
T1 - Microscopic theory of diffusive spin current with spin-orbit interaction
AU - Hosono, Kazuhiro
AU - Yamaguchi, Akinobu
AU - Nozaki, Yukio
AU - Tatara, Gen
N2 - We present the diagrammatic calculation of the spin current in the presence of the spin-orbit interaction due to impurity and electromagnetic field. By considering the vertex corrections for the normal impurity and the spin-orbit interaction, we derive both the field-driven and diffusion part of the spin current with side-jump and skew-scattering contributions. We found an alternative source of the spin-density generation, given by ×A•, where A is an electromagnetic vector potential. This generation mechanism is distinct from the conventional spin accumulation one, given by •A•.
AB - We present the diagrammatic calculation of the spin current in the presence of the spin-orbit interaction due to impurity and electromagnetic field. By considering the vertex corrections for the normal impurity and the spin-orbit interaction, we derive both the field-driven and diffusion part of the spin current with side-jump and skew-scattering contributions. We found an alternative source of the spin-density generation, given by ×A•, where A is an electromagnetic vector potential. This generation mechanism is distinct from the conventional spin accumulation one, given by •A•.
JO - Physical Review B-Condensed Matter
JF - Physical Review B-Condensed Matter

redpajama_set_name: RedPajamaCommonCrawl


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2018 is officially the year of paper flowers for me. I have recently launched a photo booth business and then realized my flower wall is actually the start of a flower making business. It's new, it's scary, but I'm incredibly happy at the end of the day.
And I am also very grateful. This new side of me, this happiness of making paper flowers drew in so many wonderful people who inspired and invited me to express myself.
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However, with new business opportunities and orders, I needed to find a way to make paper flowers easy. The Easter bunny was listening because I got a really nice bundle of Sizzix's new die cuts, all from the new collection, including the one in today's paper flower DIY tutorial. So this marks the beginning of a monthly series of tutorials, where I show you how to use different Sizzix dies to create paper flowers and more.
In today's tutorial, I'm going to show you how to make an easy paper flower using one of the newest flowers design framelits on the Sizzix website - a large rose die, created by wedding planner David Tutera. It contains4 dies that create a beautiful paper flower. Follow the step-by-step tutorial down the page to learn how to use this new large rose die.
Start making your paper flower by using the large and small petal die cuts plus the center. Bend an A4 piece of paper in half and place the die cuts with the narrow end at the folded end. Make sure you slightly move the narrow end on the folded end. That way when you press and cut, you will get a double petal, instead of a single one.
Do the same thing for the center of your flower.
Assembling all the paper petals is going to be the easiest step. Use the Crease and Curl tool from Sizzix or a crayon to give contour and volume to your petals. This will give your paper petals a realistic look.
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If you want to add leaves to your flower, you can use the die cut included in the package. Just place the die cut face down in your Sizzix sandwich and pass it through the machine. Repeat twice to get two large leaves and then glue them on the back of your paper flower.
Thanks for reading my DIY tutorial. I hope it will inspire you to start making your own beautiful paper flowers.
Visit my Pinterest board for more paper flower ideas, decorations and tutorials. PS: if you're in the photo booth business, make sure you check out my Flowers Walls board. It's chock full of beautiful ideas for your events.


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robbie.cee wrote: Be ok as long as no anpr cams see ya, if your pulled over by an actual cop, least you can state 'on way to pre booked mot' but Cams and automatically issued fines wont account for that, as they are robots!
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text:
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text:
GM trumpets flexible EV architecture, starting with Cadillac EV to be unveiled in April
Bradley Berman | Feb 6 2020 - 12:16 pm PT
General Motors outlined its EV strategy yesterday at the company's Capital Markets Day. GM President Mark Reuss confirmed that the first vehicle to use GM's next-generation electric architecture will be badged as a Cadillac. It will be unveiled in April.
Reuss emphasized that GM's flexible architecture can be applied to a wide array of segments. "Nobody we know has this combination of these levels of flexibility, speed, and scale," he said.
Reuss took about 10 minutes to present GM's EV strategy — at an event featuring nearly three hours of presentations from executives. He promised more details at an "EV Day" next month.
Reuss prefaced the EV overview by speaking about the company's new digital system that can handle "4.5 terabytes per hour of data processing power." He said the new onboard system, which is five times faster than GM's current architecture, will pave the way to the company's electric and autonomous future.
We will unveil Cadillac's first EV in April. Cadillac will offer mostly electric vehicles by the end of this decade. And Cadillac's EVs will have names that are words — not alpha-numeric designations.
Reuss' strongest case for electrification was based on its flexible battery architecture. He compared it to an ice-cube tray. "You can put in as much water to make as many cubes as you need," he said. Reuss said that engineering was "pretty much done."
It allows us to use as many battery packs as the vehicle specifications call for. Six for a smaller EV, or we can go up to 8, 10, 12, or even 24 [modules] stacked on top of each other. It allows for significant improvements over the previous generations, including larger footprint [vehicles] with lower height, higher energy density, a flexible modular design, and DC fast charging.
One of the biggest advantages this approach gives us is how much possibility we can dial into any one program. The system allows us to be as agile as the market dictates, never locked into any one thing. We can meet the market head on, whatever it is. We can adjust on the fly if we need to. It also means we're not spending money validating way more designs than necessary. We just swap the configuration.
We can go to variant X, Y, and Z quickly. We will introduce multiple models a year.
Slide from GM capital markets day
Reuss referred to the EV battery design used by other automakers as "one-off snowflakes."
Instead, according to the GM president, GM will have more flexibility and speed than any other automaker. "The GMC Hummer EV battery-electric truck we launched last week is a great example of all of that," he said.
Referring to the Hummer EV, he said GM will offer one-, two-, and three-motor versions with different ranges, levels of performance, and price points.
If a customer wants a basic package, we will have that. If the customer wants true off-road capability, we will have that, too.
He believes that the GMC Hummer might be the first electric truck of its kind available in the marketplace.
Reuss also championed GM's vertical integration and intellectual property regarding electrification.
We'll be the first full-line OEM to vertically integrate and manufacture battery cells.
Reuss said that GM's battery chemistry, developed with its partner LG Chem, could become a "new revenue stream for us down the road." Reuss echoed statements from earlier in the day by Mary Barra, GM's CEO, about making the company's batteries completely recyclable.
The final question of the day in the Q&A session was about how the Hummer EV might stack up against Tesla's Cybertruck. Reuss declined to comment about the competition. Instead, he played up the GMC Hummer EV's exterior and interior design that will be revealed on May 20.
"We're not here to compete," said Reuss. "We're here to win."
Yesterday's remarks are becoming part of a steady drumbeat of EV messaging from General Motors. That's mostly a good thing. But we wish GM would tone down the rhetoric about being the "best" or the "only" company to have certain electric-vehicle technologies or capabilities. That's unnecessary, unhelpful, and slightly annoying at this stage.
We can be encouraged that the company continues to up the ante on its EV plans. Go, go, go. But until GM delivers zero-emission vehicles at scale, those claims are just words. (Reminder: The company still sides with the Trump administration against California clean-air laws.)
The company's brand managers might be better advised to tone down big EV claims and simply say that GM is working as hard as it can to bring about a zero-emissions future. And then make sure it delivers.
GM designs and manufactures a few electric vehic…
Bradley Berman
Bradley writes about electric cars, autonomous vehicles, smart homes, and other tech that's transforming society. He contributes to The New York Times, SAE International, Via magazine, Popular Mechanics, MIT Technology Review, and others.


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Nike has announced its new line of performance hijabs for female Muslim athletes around the world in a bold move which has reignited debate about women competing in sport in the conservative Islamic world.
"The Nike Pro Hijab may have been more than a year in the making, but its impetus can be traced much further back, to an ongoing cultural shift that has seen more women than ever embracing sport," Nike said in a statement, according to Al Arabiya English.
The sports hijab, which took 13 months to design with input from athletes across the world, including Emirati Olympic weightlifter Amna Al Haddad, will go on sale to the general public in Spring 2018. It will feature tiny holes which make the fabric more breathable than traditional hijabs, while remaining opaque in accordance with tradition.
Emirati figure skater Zahra Lari was among the first athletes to receive the new hijab and took to Instagram to express her joy at Nike's decision to produce sports performance attire that respects Muslim traditions.
A recent Nike ad released in the buildup to the announcement of the Pro Hijab, which showed Arab women fencing, boxing and ice skating, stirred debate online.
The controversial ad featured the rhetorical question, "What will they say about you?" apparently in reference to a common question asked of Arab women in the Muslim world when they attempt to challenge societal or cultural norms, according to CNN Money.
Reaction to the announcement online has been mixed so far.
The issue of Muslim women in sport has become a major issue in recent years, with Saudi Arabia's decision in February to license women-only gyms sparking debate both in the Kingdom itself and the wider region as a whole.
"It is not my role to convince the society, but my role is limited to opening the doors for our girls to live a healthy lifestyle," said Saudi Princess Reema bint Bandar, a senior official at the General Authority of Sports, as cited by Reuters.


In the BUDA (Belgaum Urban Development authority) meet held yesterday it was decided to allot a 3 acre plot for the police Commissionerate. For the time being once the same is established it would commence work from the SP's office.
90 acres land would be acquired on Sambra road near Poddar school would be developed into a new colony.
20 hitech bus shelters would be constructed soon.
Next articlePre Paid auto booths to be restarted in Belgaum ?
Sambra road has good fertile agricultural lands.. acquiring fertile agricultural lands for development is insane… example is vidhan souda .. how a beautiful/fertile agricultural village Halga turned into construction sites … Belgaum is already facing the fear of concrete jungle and with such plans i guess its going to be reality soon .. Such developments should be discouraged ..


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She Played 'Persephone' in The Matrix Movies. See Monica Bellucci Now at 58.
Monica Bellucci is an Italian actress and model, who is best known for her work in The Matrix movies, as well as being a bond girl in the movie Spectre.
She was born in Città di Castello, Umbria, Italy, on September 30th, 1964, in an Italian village to parents Brunella (née Briganti) and Pasquale Bellucci.
Bellucci is an only child and is fluent in Italian, English, French, Persian, and Spanish. She can also speak Portuguese.
Monica Bellucci's early career was as a model, and she began as early as 13. She studied law for a time, but by 1988 she left the University of Perugia and moved to Milan.
While in Milan, she signed with Elite Model Management with the aid of agent Piero Piazzi.
Within a year, she was one of the most prominent fashion models in Paris and New York City. In her career, she would pose for Dolce & Gabbana and the French Elle magazine.
She would begin acting in 1990 on the foundation of her looks and notoriety in the modeling world. In her modeling career, she has modeled for Esquire, Men's Health, Maxim, Dior, Vanity Fair Italy, GQ Italia and has walked the runway many times.
Her first acting jobs would be in two 1991 projects. One was a TV drama movie in Italy titled Via coi figli.
She would have second billing in the film. Her second project released that year was the comedy-drama La riffa (The Raffle), where she was the lead.
Monica Bellucci has been fortunate in her career to appear in many iconic films in Italy and the United States.
She made a splash among international audiences in a brief but iconic role in Bram Stoker's Dracula (1992) as one of the vampire's brides.
Francis Ford Coppola's landmark horror film was pivotal in the 1990s horror boom and would also serve as Monica Bellucci's US debut.
She would continue to make movies in Italy, and in 1996 would make L'appartement (The Apartment), where she appeared with Romane Bohringer and Vincent Cassel.
This thriller was a BAFTA winner and a critical hit. Bellucci would be nominated for a César Award for best-supporting actress.
She would make more frequent appearances in American cinema in the early 2000s but continued to make notable Italian films, such as Malena (2000), an influential coming-of-age film about a teenage boy in WWII who becomes infatuated with a beautiful woman played by Bellucci. The movie is often a favorite in modern Italian cinema.
Bellucci is no stranger to controversial films, either. She was notable for her role in the controversial drama Irréversible (2002), directed by Gaspar Noé, which features a traumatic sexual assault scene.
The film was famous for its frank depiction of such a traumatic moment and its reverse-chronological order structure.
Bellucci would end up making popular American films as well. In 2003 she appeared in Tears of the Sun, The Matrix Reloaded, and The Matrix Revolutions.
In the Matrix films, she would play the computer program Persephone. She would be one of the older programs from earlier incarnations of the digital world of The Matrix and an ally of sorts to the humans within the simulation.
In 2007 she would receive top billing alongside Clive Owen and Paul Giamatti in the action comedy Shoot 'Em Up. The film would become a cult hit, given the ridiculous action scenes and a strange sense of humor.
In 2015 landed a starring role in the James Bond film, Spectre, starring Daniel Craig. At the time of her appearance in that film, she was seen as the oldest Bond Girl at 51.
She is also no stranger to American television, having notable television appearances.
In 2016 she appeared in multiple episodes of the Amazon Prime series Mozart in the Jungle playing the role of Alessandra.
She would also play herself in David Lynch's Twin Peaks: The Return in 2017.
What is Monica Bellucci doing now?
Monica Bellucci is still acting today and in 2022 has appeared in three films, with the most recent being Diabolik – Ginko all'attacco!, an adaptation of a famous European crime and adventure comic book. She plays the role of Altea di Vallemberg.
Also in 2022, Monica starred in the movie Memory, alongside Liam Neeson.
She has two projects in post-production as well for 2023. The first is an American comedy titled Mafia Mamma, where Bellucci will star with Toni Collette.
She will also appear in six episodes of a TV mini-series about the life of Communist revolutionary Tina Modotti titled Radical Eye: The Life and Times of Tina Modotti.
(Monica Bellucci in Memory)
Bellucci, in an interview with Yahoo, discusses that she is embracing her age. This was especially after her role in Spectre, where the media made much about her age. In 2021 she appeared to be at ease given her career trajectory.
Monica Bellucci has been married twice. She was dating sculptor and former model Nicolas Lefebre, but she has kept quiet about her relationship.
They announced they were together during a Chanel show in 2019 but would end the relationship shortly afterward.
Bellucci's first marriage was to fashion photographer Claudio Carlos Basso from January 1990 to June 1990, but the marriage ended in a divorce. She was 20 at the time.
(Vincent Cassel)
Her second marriage was to actor Vincent Cassel. The two had met on the set of L'appartement in 1996 and dated for a few years.
The two would marry in August of 1999 in Monaco. They would also make nine films together from 1996 to 2006.
The couple announced their separation in August 2013 and divorced shortly after. The two remain close to this day, even living in the same city and co-parenting their two children.
(Deva Cassel)
Monica Bellucci had two daughters with her second husband, Vincent Cassel. Deva Cassel was born September 12, 2004. Léonie Cassel was born May 21, 2010.
Recently, Deva Cassel has begun a career of her own as a model, but the onset of the coronavirus pandemic disrupted the early stages of her career. She was discovered at a young age, much like her mother.
Monica Bellucci can be found on social media, but she has social media managers handle her accounts. She can be found on YouTube, Facebook, Instagram, and Twitter.
6 Recent Horror Movies That Deserve Cult Classic Status


In a few short days, I will hop on a plane and make the journey from Sierra Leone back home to Canada. The few weeks that I have been here have been jammed packed meeting with our team here, meeting with our beneficiaries, driving the very bumpy roads from one end of the country to what seems like the other end!!
They held a very lively and competitive game of football (soccer).
Last Thursday I had the pleasure to meet with our beneficiaries from POPDA – an organization focused on helping people who have been affected by polio and their children. I was blown away by their enthusiasm for supporting each other and their advocacy for people who have had polio. It was so awesome to see the dignity that has been restored to these men and women. Countless times they said to me that without the support of CAUSE Canada, they would be beggars on the street.


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GSI resources provide information and tools on subsidies globally.
GSI programs and research place a spotlight on subsidies and the corrosive effect they have on economic development, governance and environmental quality. We work with governments and partners to help remove subsidies that work against sustainable development.
The FFS Civil Society Organisation (CSO) Network is an international peer to peer email list for sharing information on fossil fuel subsidies and their reform. Launched at COP 21 in Peru upon recommendation from CSOs and research institutes, the FFS CSO network aims to inform a growing number of advocates for fair and sustainable fossil fuel subsidy reform within their countries. This list includes NGOs, research and policy institutes, academics, and some intergovernmental organisations.