text:
\section*{Introduction}

Roughly speaking, the term {\it natural operation} makes reference to those operations in differential
geometry whose definition does not depend on choices of coordinates. As an example, if $M$ is a smooth manifold, the exterior differential
$$ \mathrm{d} \colon \Lambda^p T^*M \, \rightsquigarrow \, \Lambda^{p+1} T^*M $$ is an $\mathbb{R}$-linear differential operator whose definition
can be made on a chart, but it is independent on the chosen coordinates.

The study of these constructions is already present in the very beginnings of
differential geometry, although its modern and systematic development was established
in the 1970s, due to the works of Nijenhuis (\cite{Nijenhuis_72}), Atiyah-Bott-Patodi
(\cite{Atiyah_73}), Epstein-Thurston (\cite{Thurston_76}) and Terng
(\cite{Terng_78}), among many others. In a sense, this development culminated in the
monograph by Kol\'{a}\v{r}-Michor-Slov\'{a}k (\cite{KMS}), which has become the standard
reference in the subject since then.

In this paper, we are interested in the following characterization of the exterior
differential, due to Palais:

\medskip
\noindent {\bf Theorem (\cite{Palais_59}):} {\it Let $\,M\,$ be a smooth manifold. If
$$\,P \colon \Lambda^p T^*M
\rightsquigarrow \Lambda^{p+1} T^*M \,$$ is a natural and $\mathbb{R}$-linear differential
operator, then there exists $\,\lambda \in \mathbb{R}\,$ such that $\,P \, =
\, \lambda \, \mathrm{d}$.}
\medskip

This result was later improved by Krupka-Mikol\'{a}\v{s}ov\'{a} (\cite{Krupka_84}), who
dropped the hypothesis of linearity, and considered (non-linear) differential operators of
arbitrary order $\leq r$:

\medskip
\noindent {\bf Theorem (\cite{Krupka_84}):} {\it Let $\,M\,$ be a smooth manifold and let $p> 0$. If
$$\,P \colon \Lambda^p T^*M
\rightsquigarrow \Lambda^{p+1} T^*M \,$$ is a natural differential operator of order
$\leq r$, then there exists $\,\lambda \in \mathbb{R}\,$ such that $\,P \, = \, \lambda\, \mathrm{d}$.}
\medskip

The case $p=0$ lies outside this characterization; as an example, the differential
operator $\, \mathcal{C}^\infty (M) \rightsquigarrow T^*M \, , \, f \mapsto f \, \mathrm{d} f
\,$ is natural and it is not a constant multiple of the exterior differential.

In their monograph, Kol\'{a}\v{r}-Michor-Slov\'{a}k (\cite{KMS}, Prop. 25.4) refined this statement
even more, by considering regular (and natural) local operators. Theirs is a two-step proof: firstly,
they proved that regular local operators are indeed differential operators--the Peetre-Slov\'{a}k's
theorem--and then, a version of the result above applies.

Quite recently, there have been a couple of reformulations of this result
(\cite{Katsylo_08}, \cite{Navarro_2015}) that, essentially, confirm that
the only natural operation between differential forms is the exterior differential.
These theorems have been used for several different applications (\cite{FreedHopkins},
\cite{Katsylo_08}, \cite{Mason_2014}, \cite{Navarro_2015}, \cite{Timashev_15}). As an example, these ideas have been succesfully extended to study differential operations on forms on contact manifolds, allowing a deeper understanding of the Rumin operator (\cite{Bernig}).

In this paper, we aim to study up to what extent this statement remains true for holomorphic forms over a complex analytic manifold.

Katsylo and Timashev (\cite{Katsylo_08}, Theorem 3.1) have already considered natural, $\mathbb{C}$-linear differential operators, obtaining--among other things--a result analogous to the aforementioned theorem of Palais.

Our main result goes one step further and removes the hypothesis of linearity (see also Theorem \ref{General}):

\medskip
\noindent {\bf Theorem \ref{Main}:} {\it Let $X$ be a complex analytic manifold of
dimension $n$.

Let $\,\mathbb{C}\{ u,v\}\,$ be the anti-commutative algebra of polynomials on the
variables $\,u,v$, of degree $\,\mathrm{deg}\, u=p$, $\,\mathrm{deg}\, v=p+1$ (Definition \ref{AntiConmutativos}).

Any natural differential operator of order $\leq r$
$$ P\colon \Lambda^{p} T^*X \  \rightsquigarrow\ \Lambda^q T^*X \ ,  $$
where $0\leq q\leq n= \dim X$, can be written as
$$P\left(j^r_x \omega \right)\,=\, \mathbf{P}(\omega,\mathrm{d}\omega )_x \quad , $$
for a unique homogeneous polynomial $\,\mathbf{P}(u,v)\in\mathbb{C}\{ u ,v \}\,$ of
degree $\,q$.}
\medskip

The proof closely follows the one given in \cite{Navarro_2015} for the case of differentiable manifolds, which, in turn, applies
techniques already used by the Czech school (\cite{KMS}, \cite{Krupka_84}).

Nevertheless, in order to accomplish our task it is necessary to deal with natural bundles in the category of complex manifolds and holomorphic maps. Among experts, it is well-known that a great part of the theory in the smooth case  is purely algebraic, so that it extends without difficulties to this setting. This is the approach followed by Katsylo and Timashev (\cite{Katsylo_08}), although their definition of $k$-th order natural bundles over a manifold as associated bundles to the $k$-th order coframe bundle is not conceptually the most fundamental one. We will see that it assumes, as a starting point, a key result of the theory: the theorem of equivalence (\ref{TeoremaEquivalencia}). On the other hand, the usual approach to natural bundles (\cite{KMS}) as functors on the category of manifolds and local diffeomorphisms is the most general possible. However in this way one is not able to capture an important internal property, the Galoisian structure, enjoyed by natural bundles over a fixed manifold.

Thus,  we devote the first section of this paper to developing the basics of  the theory on natural holomorphic bundles over a fixed manifold from the most fundamental definitions and at the same time to make transparent its natural Galoisian structure. This will also serve to fix the notation used in the rest of the paper. We follow an approach that clearly resembles the modern expositions of Galois theory, where the role of the Galois group is played here by the group of finite order jets of biholomorphisms leaving a point fixed.
Our exposition is for the most part rather schematic and we just prove several key points, since most of the ideas carry along from
the usual presentations in the differentiable case.

However, we have identified several important results in the smooth theory
whose holomorphic counterparts do not follow straightforwardly and therefore we do not know yet wether they do hold or not in the holomorphic category. In our opinion these problems deserve further attention in order to have a complete understanding of the theory of holomorphic natural bundles and its relation with its smooth counterpart. We plan to address these questions in the future but now let us
just point out three of them:

\begin{itemize}
\item Is every holomorphic natural bundle a finite order one?

\item Can the regularity condition in the definition of natural bundle be deduced from the other two?

\item Does Peetre-Slov\'{a}k's theorem on the finiteness of the order of regular local operators hold in the holomorphic category?
\end{itemize}

Probably, the solution of these open problems would pave the road to put this holomorphic theory on an equal footing
to the smooth, classical one.

\section{Holomorphic natural bundles}

Let us present in this section a non-standard definition of natural bundle, first proposed in the smooth case by Juan B. Sancho Guimer\'{a}. We use it due to its simplicity and its benefits in order to state a Galois-type theorem. Later, in Section \ref{AlternativeDefi}, we will explain the equivalence with a functorial exposition that is analogous to the one commonly used in the smooth theory.

\medskip

Let us fix a complex analytic manifold $X$ of dimension $n$.

If $\,\pi\colon F \to X\,$ is a holomorphic fibre bundle (i.e., a holomorphic map isomorphic, locally, on $X$,  to the trivial fibration $U\times S \to U$), then a {\it lift to $F$} of an analytic isomorphism $\,\tau \colon U \to V\,$ between open sets in $X$ is any
analytic isomorphism $\,\ell^F(\tau) \colon F_{U} := \pi^{-1}(U) \to F_{V} := \pi^{-1}(U)\,$ such that the following square is commutative:
$$
\xymatrix{
F_{U} \ar[r]^-{\ell^F(\tau)} \ar[d]_-{\pi}  & F_{V} \ar[d]^-{\pi} \\
U \ar[r]^-{\tau} &  V}
$$

If $Z$ is a complex manifold we denote by  $\,\mathrm{Bihol}(Z) \, $ the set of analytic isomorphisms
$\tau \colon U \to V$ between open sets in $\,Z$.

\begin{definition}\label{DefinitionJS}
A natural bundle over $X$ is a pair formed by a holomorphic fibre bundle $F \to X$, together with a lifting of biholomorphisms:
$$\ell^F\colon \mathrm{Bihol}(X)  \to \textrm{Bihol}(F) \quad , \quad \tau \ \mapsto \ \ell^F(\tau) $$ that
satisfies:
\begin{itemize}
\item {\it Functoriality:} $\ell^F(\mathrm{Id}) = \mathrm{Id}$ and $\ell^F(\tau \circ \tau')=\ell^F( \tau)\circ \ell^F(\tau')$.
\item {\it Locality:} For any analytic isomorphism $\tau \colon U \to V$ between open sets on $X$, and for any open set $U' \subset U$:
$$
\ell^F(\tau_{|U'}) = \ell^F(\tau)_{|F_{U'}} \ .
$$
\item {\it Regularity:}  If $\{ \tau_t \colon U_t \to V_t \}_{t\in T}$ is an analytic family of biholomorphisms between open sets in $X$, parametrized by an analytic manifold $\,T$, then the collection of lifts $\,\{ \ell^F(\tau_{t} )\colon F_{U_t} \to F_{V_t} \}_{t \in T}\,$ is an analytic family of biholomorphisms between open sets in $\,F$.
\end{itemize}

Let $F, \bar{F} \to X$ be natural bundles over $X$. A morphism of natural bundles (or natural morphism) on an open set $W$
is a morphism of bundles  $\varphi \colon F_W \to \bar{F}_W\, $ that commutes with the action of $\,\mathrm{Bihol}(X) \,$; that is, such that
for any  analytic isomorphism $\tau \colon U \to V$ between open sets on $W$, the following square is commutative:
$$
\xymatrix{
F_{U} \ar[r]^-{\varphi} \ar[d]_-{\ell^F(\tau) }  & \bar{F}_{U} \ar[d]^-{\ell^{\bar F}(\tau)} \\
F_{V} \ar[r]^-{\varphi}  & \bar{F}_{V}
}
$$

A natural bundle $F \to X$ has order $k$ if for any biholomorphisms $\tau,\tau'\colon U\to V$ defined between open sets of $X$, and any $x\in U$, it holds
$$j^k_x\tau=j^k_x\tau'\quad\Rightarrow\quad \ell^F(\tau)=\ell^F(\tau')\quad\text{when restricted to the fibre }F_x \ .$$

The category of natural bundles of order $ k$ over $X$ will be denoted $\textbf{Nat}_X^k$.
\end{definition}

\begin{remark} From now on, if ($F\to X,\ell^F)$ is a natural bundle and $\tau \colon U \to V$ is a biholomorphism between open sets of $X$, whenever there is no risk of confusion we will denote the lift $\ell^F(\tau)\colon F_U\to F_V$ simply by $\tau_*\colon F_U\to F_V$.
\end{remark}

\begin{examples}
The trivial bundle $F = Y \times X \to X$, with the lifting $\tau_* (y, x) = \left( y , \tau (x) \right) $, is a natural bundle of order 0.

The tangent bundle $TX\to X$ is a natural bundle: the lifting of an analytic isomorphism $\tau\colon U\to V$ is its tangent linear map $\tau_*\colon TU\to TV$. More generally, any tensor bundle $\otimes^pT^*X\otimes^qTX$ is natural, of order $1$.

If $F\to X$ is a natural bundle of order $k$, then its bundle of $r$-jets, $\,J^rF\to X$, is also a natural bundle, of order $k+r$.
\end{examples}

\begin{example}[Universal natural bundle of order $ k$]\label{FibradoUniversal} Let us fix a po\-int $\, x_0 \in X$.

It is straightforward to check that the set $\mathcal{U}^k_{x_0}(X)$ formed by the jets  $j^k_{x_0}\sigma$ of germs of analytic isomorphisms $\,\sigma\colon U\rightarrow V\,$ between open sets of $X$ such that $x_0\in U$, has a natural structure of complex manifold that makes the projection:
$$\begin{CD}
\pi \colon \mathcal{U}^k_{x_0} (X)@>>> X\qquad ,\qquad j^k_{x_0}\sigma\mapsto \sigma(x_0)\ .
\end{CD}$$
into a holomorphic locally trivial bundle that is called the universal bundle of order $k$.

This bundle is natural of order $ k$. The lift of a biholomorphism  $\tau\colon U\to V$ between open sets of $X$ is defined as:
$$\begin{CD}
\mathcal{U}^k_{x_0}(X)_{|U} @>{\tau_*}>> \mathcal{U}^k_{x_0}(X)_{|V}\quad ,\qquad \tau_*(j^k_{x_0}\sigma):=j^k_{x_0}(\tau\circ\sigma)\ .
\end{CD}$$

Moreover, let $\,G^k_{x_0}\,$ be the complex Lie group of $k$-jets $\,j^k_{x_0}\xi\,$ of germs of biholomorphisms $\xi\colon U\rightarrow U'$  leaving the point $x_0$ fixed. This group acts on the right on $\mathcal{U}^k_{x_0}(X)$:
$$ (j^k_{x_0}\sigma) \cdot(j^k_{x_0}\xi):=j^k_{x_0}(\sigma\circ  \xi). $$ Via this action, the universal bundle becomes a principal $G^k_{x_0}$-bundle.
\end{example}

\subsection{Theorem of equivalence}

Let $X$ be a complex analytic manifold, and let $x_0 \in X$ be any point. Recall that $\,G^k_{x_0}\,$ denotes the complex Lie group of $k$-jets $\,j^k_{x_0}\xi\,$ of germs of analytic isomorphisms $\xi \colon U \rightarrow  U'\,$  leaving the point $x_0$ fixed.

\subsubsection*{Fibre functor}

The condition of a natural bundle $F \to X$ being of order $\leq k$ amounts to saying that $G^k_{x_0}$ acts on the left on the fibre $F_{x_0}$ over the point $x_0 \in X$: for any point $\,e \in F_{x_0}\,$ and any $\,g= j^k_{x_0} \xi \in G^k_{x_0}$, define
\begin{equation}\label{Action}
g \cdot e := \xi_* (e) \ .
\end{equation}

This action is holomorphic: in a chart around $\,x_0$, it is easy to check the existence of an analytic family of isomorphisms $\,\tau_g \colon U_g \to V_g$, parametrized by $\, g\in G^k_{x_0}$, such that $j^k_{x_0} \tau_g = g $. By the regularity hypothesis, $(\tau_g)_*$ is an analytic family of isomorphisms of $F_{x_0}$; hence the action defined in (\ref{Action}) is holomorphic in $g$.

Moreover, if $F$, $\bar{F}$ are natural bundles of order $ k$, then any morphism of natural bundles $\varphi \colon F \to \bar{F}$ obviously defines a $G^k_{x_0}$-equivariant analytic map $\varphi_{x_0} \colon F_{x_0} \to \bar{F}_{x_0}$.

Thus, if $G^k_{x_0}\textbf{-Man}$ denotes the category of analytic $G^k_{x_0}$-manifolds, we have constructed a ``fibre functor'':
$$
\mathcal V_{x_0}\colon \textbf{Nat}_X^k  \xrightarrow{\ \  \ \ } G^k_{x_0}\textbf{-Man}    \quad , \quad F \ \mapsto \ F_{x_0}  \ .
$$

\subsubsection*{Associated bundle functor}

Let $F_0$ be a $G^k_{x_0}$-manifold; that is, $F_0$ is a complex manifold together with an analytic left action:
$$\begin{CD}
 G^k_{x_0}\times F_0 @>{\cdot}>> F_0  \ .
 \end{CD}$$

Let $\,\mathcal{U}^k_{x_0}(X)\, $ be the universal natural bundle of order $k$ (Example \ref{FibradoUniversal}). The group $G^k_{x_0}$ acts on the right on $\,\mathcal{U}^k_{x_0}(X)\times F_0\,$: $$ (j^k_{x_0}\sigma,e) \cdot(j^k_{x_0}\xi):=((j^k_{x_0}\sigma) \cdot(j^k_{x_0} \xi), (j^k_{x_0}\xi)^{-1}\cdot(e)). $$

\begin{definition}
For any $G^k_{x_0}$-manifold  $F_0$, its associated bundle is defined as:
$$\begin{CD}
F := (\mathcal{U}^k_{x_0}(X) \times F_0)/G^k_{x_0} @>>> X\quad ,\qquad [(j^k_{x_0}\sigma, e)]\mapsto \sigma(x_0) \ .
\end{CD}$$
\end{definition}

The associated bundle $F\to X$ is natural of order $\leq k$: the lifting of a biholomorphism $\tau\colon U\to V$ between open sets of  $X$ is defined by the formula:
 $$\begin{CD}
 F_U=(\mathcal{U}(X)^k_{\, U}\times F_0)/G^k_{x_0} @>{\tau_*}>> (\mathcal{U}(X)^k_{\, V}\times F_0)/G^k_{x_0}=F_V
  \end{CD}$$
  $$\tau_*\left([j^k_x\sigma,e]\right):=[\tau_*(j^k_x\sigma),e] \ . $$

This lifting is well defined because the actions of $\tau_*$ and $G^k_{x_0}$ on $\mathcal{U}^k_{x_0}(X)$ commute:
\begin{align*}
\tau_*\left((j^k_{x_0}\sigma)\cdot(j^k_{x_0}\xi)\right) \, &= \, \tau_*\left(j^k_{x_0}(\sigma\circ\xi)\right)
\, = \, j^k_{x_0}\left(\tau\circ\sigma\circ\xi\right) \\
&= \,\left(j^k_{x_0}(\tau\circ\sigma)\right)\cdot (j^k_{x_0}\xi) \, = \, \tau_*(j^k_{x_0}\sigma)\cdot(j^k_{x_0}\xi)\ .
\end{align*}

This construction is functorial: it transforms a morphism of $G^k_{x_0}$-manifolds $f \colon F_0 \to \bar{F_0}$ into the following morphism of natural bundles:
$$F=(\mathcal{U}^k_{x_0}(X)\times F_0)/ G^k_{x_0} \xrightarrow{\ \ \text{Id} \times f \ \ } (\mathcal{U}^k_{x_0}(X)\times \bar{F_0}) / G^k_{x_0}=\bar{F} ,$$ that maps  $[j^k_x\sigma , e] $ to $\ [j^k_x\sigma , f(e) ] \ . $

Thus, we have constructed a functor:
$$
\mathcal A\colon G^k_{x_0}\textbf{-Man}  \xrightarrow{\ \  \ \ } \textbf{Nat}_X^k   \quad , \quad F_0 \ \mapsto \mathcal A(F_0) :=\ (\mathcal{U}^k_{x_0}(X) \times F_0) / G^k_{x_0}  \ .
$$

\subsubsection*{Theorem of equivalence}

\begin{theorem}\label{TeoremaEquivalencia}
Let $X$ be a complex analytic manifold and $x_0\in X$ be any point. Consider the complex Lie group $G^k_{x_0}$ of $k$-jets  of germs of analytic isomorphisms leaving the point $x_0$ fixed.

The ``associated bundle" functor $\mathcal A$ establishes an equivalence of categories:
$$\begin{CD}
 \mathcal A\colon  G^k_{x_0}\mathrm{\bf \text{-}Man} @= \mathrm{\bf Nat}_X^k  \ ,
\end{CD}$$
whose inverse functor is the fibre functor $\mathcal V_{x_0}\colon {\bf Nat}_X^k  \xrightarrow{\ \  \ \ } G^k_{x_0}{\bf \text{-}Man}$ .
\end{theorem}

\noindent \textit{Proof: }  Let us check that both functors are inverse to each other: if $F_0$ is a $G^k_{x_0}$-manifold and $F\to X$ is the associated bundle, then there is an isomorphism
$$\begin{CD}
F_0 @= \left[(\mathcal{U}^k_{x_0}(X)\times F_0)/G^k_{x_0}\right]_{x_0}=F_{x_0}\\
e & \mapsto & [(j^k_{x_0}\text{Id},e)]
\end{CD}$$
which is $G^k_{x_0}$-equivariant:
\begin{align*}
 j^k_{x_0}\xi\cdot [j^k_{x_0}\text{Id},e] &= \, \xi_*[j^k_{x_0}\text{Id},e]=
[\xi_*(j^k_{x_0}\text{Id}),e] \, = \, [j^k_{x_0}(\xi\circ\text{Id}),e] \\
& = \, [j^k_{x_0}\xi,e]=[j^k_{x_0}(\text{Id}\circ\xi),e)]=[j^k_{x_0}\text{Id},j^k_{x_0}\xi\cdot e] \ .
\end{align*}

Conversely, if $F \to X$ is a natural bundle of order $\leq k$, then the following map is a bundle isomorphism between $F$ and the bundle associated to $F_{x_0}$:
$$\begin{CD}
(\mathcal{U}^k_{x_0}(X)\times F_{x_0} ) / G^k_{x_0} @= F \\
[j^k_{x_0}\sigma , e_{x_0} ]  & \mapsto &  \sigma_* e_{x_0} \ .
\end{CD}$$

It is well defined: if we change the pair $(j^k_x\sigma , e_{x_0})$ by the equivalent element $(j^k_{x_0}\sigma , e_{x_0})\cdot \,j^k_{x_0}\xi=(j^k_{x_0}(\sigma\circ\xi),(j^k_{x_0}\xi)^{-1}\cdot e_{x_0})=(j^k_{x_0}(\sigma\circ\xi),\xi^{-1}_*e_{x_0})$, then:
$$[j^k_{x_0}(\sigma\circ\xi),\xi^{-1}_*e_{x_0}]\,\mapsto\, (\sigma\circ\xi)_*\xi^{-1}_* e_{x_0}=\sigma_* e_{x_0} \ .
$$

\hfill $\square$

\begin{examples}
Natural vector bundles of order $ 1$ correspond to holomorphic linear representations of the complex general linear group $$G^1_{x_0}=\mathrm{GL}_{\mathbb{C}}(T_{x_0}X).$$ As an example, the natural bundle associated to the standard action of this group on $\otimes^pT_{x_0}^*X\otimes^qT_{x_0}X$ is the vector bundle of $(p,q)$-tensors.

Natural coverings of order $ k$ correspond to discrete $G^k_{x_0}$-manifolds. The complex group $G^k_{x_0}$ is connected, so that any (connected) holomorphic natural covering is trivial, in contrast with the smooth theory, where there exists the orientation covering (observe that any complex manifold is orientable).

If $\mathcal{O}_{x_0}$ denotes the ring of germs of holomorphic functions around $x_0 \in X$, then the group $G^k_{x_0}$ acts on $P^k_{x_0}:=\{ j^k_{x_0}f:f\in\mathcal{O}_{x_0}\}$ as follows: $\,(j^k_{x_0}g)\cdot(j^k_{x_0}f)=j^k_{x_0}(f\circ g^{-1})$. The corresponding natural bundle is the bundle $\, J^k(X,\mathbb{C})\to X\,$ of $k$-jets of holmorphic functions on $X$.
\end{examples}

\begin{corollary}\label{MorfismosNaturalesK} Let $F,\bar F\to X$ be natural bundles of order $k$.
Given any point $x_0 \,$ in an open set $U$, the assignment $\varphi\mapsto\varphi_{x_0}$ defines a bijection:
$$\begin{CD}
\left\{\begin{aligned}
&\mathrm{Morphisms\ of\ natural\ bundles \, } \\
& \hskip .7cm \ \, \ \varphi \colon F_U\to\bar F_U
\end{aligned}\right\} @=
\left\{\begin{aligned}
& G^k_{x_0}\text{--}\,\mathrm{equivariant\ analytic\ maps\, }\\
& \hskip 1cm \varphi_{x_0} \colon F_{x_0} \to\bar F_{x_0}
\end{aligned}\right\}
\end{CD} \ . $$

In particular, any morphism $\varphi$ of natural bundles is globally defined and covers the identity map on $X$ (i.e. $\varphi (F_y) \subseteq \bar{F}_y$, for any point $y \in X$).
\end{corollary}

\begin{example} The only morphisms of  vector bundles $\varphi\colon TX\to TX$ that are natural are the homotheties, $\varphi(D)=\lambda D$, because:
$$\mathrm{Hom}_{\mathrm{nat}}(TX,TX)\, =\, \mathrm{Hom}_{G^1_{x_0}}(T_{x_0}X,T_{x_0}X)\, =\, \mathrm{Hom}_{\mathrm{GL}_{\mathbb{C}} (T_{x_0}X)}(T_{x_0}X,T_{x_0}X) \, = \,  \mathbb{C}\ .
$$
\end{example}

\subsection{Functorial definition}\label{AlternativeDefi}

Let $X, Y$ be complex manifolds of the same dimension $n$, and let $x_0 \in X$, $y_0 \in Y$ be arbitrary points. Any local isomorphism $\tau$ such that $\tau (x_0) = y_0$ induces an isomorphism $G^k_{x_0} \simeq G^k_{y_0}$. Hence, Theorem \ref{TeoremaEquivalencia} implies that a natural bundle over a complex manifold produces a natural bundle over any other complex manifold of the same dimension. This motivates a functorial approach to natural bundles, that we proceed to scketch in analogy with the standard references for smooth bundles (\cite{KMS}, \cite{Nijenhuis_72}).

\medskip

Let ${\bf Man _{\mathbb{C}}}[n]$ be the category of complex analityc manifolds of dimension $n$ and local analytic isomorphisms between them.

On the other hand, let ${\bf Bund}$ be the category whose objects are fibre bundles $F \to X$, and whose morphisms $$\,(f, \bar{f}) \colon \{ F\xrightarrow{\pi} X\} \longrightarrow \{\bar{F}\xrightarrow{\bar{\pi}} \bar{X}\}\,$$ are pairs of analytic maps $\,(f, \bar{f})\,$ making the following square commutative:
$$
\xymatrix{
F \ar[r]^-{f} \ar[d]_-{\pi }  & \bar{F} \ar[d]^-{\bar{\pi}} \\
X \ar[r]^-{\bar{f}}  & \bar{X}
}
$$

\begin{definition}\label{StandardDefi}
A natural bundle in dimension $n$ is a covariant functor:
$$ \mathfrak{F} \colon {\bf Man_{\mathbb{C}}}[n] \to {\bf Bund} $$ satisfying the following properties:
\begin{itemize}
\item If $\mathfrak{B} \colon {\bf Bund} \to {\bf Man_{\mathbb{C}}}[n]$ denotes the base functor, $\mathfrak{B} (F \to X) := X$, then the composition $\mathfrak{B} \circ \mathfrak{F} \colon {\bf Man_{\mathbb{C}}}[n] \to {\bf Man_{\mathbb{C}}}[n]$ is the identity functor $\text{Id} _{{\bf Man_{\mathbb{C}}}[n]} $.

For any complex $n$-manifold $X$, let us write $\,\mathfrak{F}(X)=\{\pi_X \colon \mathfrak{F}_X \to X \}$.

\item {\it Locality:} For any open inclusion $i\colon U \hookrightarrow  X$, the bundle $\mathfrak{F}_U$ is identified with $\pi_X^{-1} (U)$ via the map $\mathfrak{F}(i)$.

\item {\it Regularity:} If $\,\{ f_t\colon X_t\to Y_t\}_{t\in T}\,$ is an analytic family of local isomorphisms between $n$-manifolds, parametrized by a complex manifold $T$, then  $\,\{\mathfrak{F} (f_t) \colon\mathfrak{F}(X_t)\to\mathfrak{F}(Y_t)\}_{t\in T}\,$ is also an analytic family.
\end{itemize}

A morphism of natural bundles is a morphism of functors $\Phi \colon \mathfrak{F} \to \bar{\mathfrak{F}}$.

A natural bundle $\mathcal{F}$ has order $ k$ if for any local isomorphisms between $n$-manifolds $f,g \colon X \to Y$ and any point $x \in X$, the following holds:
$$ j^k_x f = j^k_x g \quad \Rightarrow \quad \mathfrak{F} (f)  = \mathfrak{F} (g) \  \mbox{ when restricted to the fibre } (\mathfrak{F}_X)_x \ .$$

Let us write $\mathrm{\bf Nat}^k_n$ fot the category of natural bundles in dimension $n$ of order $k$ just defined.
\end{definition}

\begin{proposition}\label{EquivalenciaDefiniciones} Let $\mathrm{\bf Nat}^k_n$ be the category of natural bundles in dimension $n$, of order $ k$ (Definition \ref{StandardDefi}).

Let $X$ be a complex analytic manifold of dimension $n$, and let $\mathrm{\bf Nat}^k_X$ denote the category of natural bundles over $X$, of order $\leq k$ (Definition \ref{DefinitionJS}).

The functor $\mathfrak{F}  \mapsto \mathfrak{F}(X)$ establishes an equivalence of categories:
$$\begin{CD}
\mathrm{\bf Nat}^k_n @= \mathrm{\bf Nat}^k_X
\end{CD}$$
\end{proposition}

\noindent \textit{Proof: } Fix any point $x_0\in X$.

The universal bundle can be simultaneously defined  for all $n$-manifolds: given an $n$-manifold $Z$, let $\mathcal{U}(Z)^k$ be the manifold formed by jets  $j^k_z\sigma$ of analytic
isomorphisms $\, \sigma\colon X \supset U \rightarrow \, V \subset Z\,$ defined between open sets, such that $x_0 \in U$. The universal bundle for $Z$ is the projection $\,\mathcal{U}(Z)^k\to Z$, $j^k_{x_0}\sigma\mapsto \sigma (x_0)$.

As a consequence, the ``associated bundle'' functor is defined for any $n$-manifold. This allows to extend, {\it mutatis mutandis}, the arguments in the proof of Theorem \ref{TeoremaEquivalencia} to obtain an equivalence of categories:
$$\begin{CD}
  G^k_{x_0}\mathrm{\bf \text{-}Man} @= \mathrm{\bf Nat}_n^k\\
 F_0 & \longmapsto & \mathfrak{F}
\end{CD}
\qquad ,\qquad \mathfrak{F}(Z):=(\mathcal{U}^k_{x_0} (Z) \times F_0)/G^k_{x_0}\ .$$

Combining it with Theorem \ref{TeoremaEquivalencia}, it readily follows the equivalence $\,\mathrm{\bf Nat}_n^k=\mathrm{\bf Nat}_X^k$.

\hfill $\square$

\subsection{Natural differential operators}

Let $F , \bar{F} \to X$ be natural bundles.

\begin{definition}\label{DiffOperatNatu}
A natural holomorphic differential operator $P \colon F_U \rightsquigarrow \bar F_U$ of finite order $\leq r$ on an open set $U \subset X$ is a morphism of natural bundles:
$$ P \colon J^r F_U \to \bar{F}_U \ . $$
\end{definition}

If $F\to X$ is a natural bundle of order $ k$, then $J^rF\to X$ is a natural bundle of order $k+r$. Therefore, Corollary  \ref{MorfismosNaturalesK} implies:

\begin{proposition}\label{Terng}
Let $F,\bar F\longrightarrow X$ be natural bundles of order $k$.
Given any point $x_0\,$ in an open set $U$, the assignment $P \mapsto P_{x_0}$ defines a bijection:
$$\begin{CD}
\left\{\begin{aligned}
&\mathrm{Natural\ holomorphic\ operators \ } \\
& \hskip .4cm P \colon F_U \rightsquigarrow \bar F_U \ \ \mathrm{\ of\ order\ } \leq r \
\end{aligned}
\right\}
@=
\left\{\begin{aligned}
&G^{k+r}_{x_0}\text{--}\,\mathrm{equivariant\ analytic\ maps\ }\\
& \hskip 1.2cm P_{x_0} \colon J^r_{x_0} F\to\bar F_{x_0}
\end{aligned}
\right\}
\end{CD} \ . $$

In particular, any natural holomorphic differential operator is globally defined.
\end{proposition}

\medskip

\section{Natural differential operators on holomorphic forms}

Let $\,X\,$ be a compelx analytic manifold of dimension $\,n$ and let $\,T^*X$ denote the holomorphic cotangent bundle.

At any point $\,x_0 \in X$, the space of jets of holomorphic $p$-forms is isomorphic to the space of jets of $p$-forms at the origin of $\,\mathbb{C}^n$:
$$ J^r_{x_0} \left( \Lambda^p T^* X \right) \, \simeq \, J^r_0 \left( \Lambda^p T^* \mathbb{C}^n \right) \ . $$

Let us also introduce the following notations for the exterior and symmetric powers of the holomorphic cotangent space of $\,\mathbb{C}^n\,$ at the origin:
$$\,\Lambda^p:=\Lambda^p(T^*_0\mathbb{C}^n)\,\quad \mbox{ and } \quad \, S^p:=S^p(T^*_0\mathbb{C}^n) \ .$$

There exists an analytic isomorphism:
$$\begin{CD}
J^r_0(\Lambda^pT^*\mathbb{C}^n) @= (\Lambda^p)\oplus(S^1\otimes \Lambda^p)\oplus\cdots\oplus(S^r\otimes \Lambda^p)\\
j^r_0\omega & \longmapsto & (\omega^0,\omega^1,\dots  ,\omega^r)\qquad\qquad
\end{CD}$$
where $\,\omega^s\,$ is the homogeneous component of degree $\,s\,$ in the Taylor expansion (on cartesian coordinates) at the origin of the holomorphic $p$-form $\,\omega$.

If we write $\, \nabla\,$ for the standard (flat) covariant derivative of $\,\mathbb{C}^n$, then
$$\omega^s \, = \, (\nabla^s \omega)_{z=0} \ . $$

This isomorphism is not invariant under arbitrary changes of coordinates, but it is so under linear changes of coordinates. In other words, this isomorphism is equivariant with respect to the action of the linear group $\,\mathrm{GL}_n(\mathbb{C})\simeq G^1_0 \subseteq G^{r+1}_0$, although it is not equivariant with respect to the whole group $\,G^{r+1}_0$.

Proposition \ref{Terng}, combined with the previous isomorphism, imply the following:

\medskip

\begin{lemma}\label{2.1} Let $\,X\,$ be a complex analytic manifold of dimension $\,n$. There exists an injective map:
$$\begin{CD}
\left\{\begin{aligned}\mathrm{Natural\ differential\ operators\ of \ order } \leq r\\
P\colon\Lambda^{p} T^* X \,\rightsquigarrow\,\Lambda^q T^*X \qquad\qquad\end{aligned}\right\}\\
|\bigcap^{\phantom{\bigcap}}
_{\phantom{\bigcap}} \\
 \left\{\begin{aligned} &\mathrm{Analytic\ }\mathrm{GL}_n(\mathbb{C})\text{--}\,\mathrm{equivariant \ maps \ }\quad\\ \tilde P\colon  &� (S^0\otimes \Lambda^{p})\oplus\cdots\oplus(S^r\otimes \Lambda^{p})\,\longrightarrow\, \Lambda^q \, \end{aligned}\right\}
\end{CD}$$
where the relation between $\,P\,$ and $\,\tilde P\,$ is determined by the equality
\begin{align*}
P(j^r_x \omega) \, &=\,\tilde P(\omega^0,\dots,\omega^r ) \ .
\end{align*}
\end{lemma}

Later on, we will prove that this inclusion is in fact a linear isomorphism.

\subsection{Homogeneity}

Equivariance with respect to homotheties produces the following homogeneity condition:

\begin{lemma}\label{2.2} Any $\,\mathrm{GL}_n(\mathbb{C})$-equivariant map
$$\begin{CD} ( S^0\otimes \Lambda^{p})\oplus\cdots\oplus(S^r\otimes \Lambda^{p}) @>{\tilde P}>> \Lambda^q
\end{CD}$$
satisfies, for any complex number $\, \lambda\neq 0$, the following homogeneity condition
\begin{equation}\label{CondicionHomog}
\tilde P\left(\lambda^{p+0}\omega^0,\dots,\lambda^{p+r}\omega^r \right)\,=\,\lambda^q\tilde P\left(\omega^0,\dots,\omega^r \right) \ .
\end{equation}
\end{lemma}

\noindent \textit{Proof: } Let $\,\tau_\lambda\colon\mathbb{C}^n\to\mathbb{C}^n$, $\tau_\lambda(z)=\lambda z$, be the homothety of ratio $\,\lambda\neq 0$.

On the one hand, for any tensor $\,\omega^j\in S^j\otimes\Lambda^{p}$, covariant of order $\,p+j$, it holds $\,\tau_\lambda^*\omega^j=\lambda^{p+j}\omega^j$.

On the other, as the map $\, \tilde P\,$ is $\,\mathrm{GL}_n(\mathbb{C})$-equivariant, it holds $\,\tilde P\circ\tau_\lambda^*=\tau_\lambda^*\circ\tilde P$. Consequently:
\begin{align*}
\tilde P \left(\lambda^{p+0}\omega^0,\dots,\lambda^{p+r}\omega^r \right)\, &=\,
\tilde P\left( \tau_\lambda^*\omega^0,\dots,\tau_\lambda^*\omega^r \right) \\
& =\,\tau_\lambda^*\tilde P\left( \omega^0,\dots,\omega^r \right)  \,=\,\lambda^q\tilde P\left( \omega^0,\dots,\omega^r \right)\ .
\end{align*}

\hfill $\square$

Any holomorphic map between vector spaces satisfying such a homogeneity condition has necessarily to be a polynomial, in virtue of the following elementary result  (e. g. \cite{KMS}, Thm. 24.1):

\medskip
\noindent {\bf Homogeneous Function Theorem:} {\it Let $E_1,\dots, E_k$ be finite dimensional $\mathbb{C}$-vector spaces.

Let $\, f \, \colon  \prod E_i \to \mathbb{C}$ be a holomorphic function such that there exist natural numbers $\, a_i ,  w \in \mathbb{N}\,$ satisfying:
\begin{equation}\label{CondicionHomogeneidadLemma}
 f ( \lambda^{a_1} e_1 , \ldots , \lambda^{a_k} e_k) = \lambda^w \, f(e_1 , \ldots , e_k)
\end{equation}
 for any complex number $\lambda \neq 0$ and any vector
$(e_1 , \ldots , e_k) \in \prod E_i$.

Then $f$ is a sum of monomials of
degree $(d_1,\dots,d_k)$ in the variables $e_1,\dots,e_k$ satisfying the relation
\begin{equation}\label{CondicionMonomios}
 a_1 d_1 + \cdots + a_k d_k = w \ .
\end{equation}

If there are no natural numbers  $d_1,\dots,d_k \in \mathbb{N} \cup \{ 0 \}$ satisfying this equation, then $f$ is the zero map. }
\medskip

In other words, for any finite dimensional vector space $W$, there exists a $\mathbb{C}$-linear isomorphism:
$$\begin{CD}
\left[ \text{Holomorphic maps }\, f \colon  \prod E_i \to W\,\text{ satisfying }  (\ref{CondicionHomogeneidadLemma})\right]  \\
@| \\
\bigoplus \limits _{(d_1 , \ldots , d_k)}\text{Hom}_{\mathbb{C}}(S^{d_1} E_1 \otimes \ldots
\otimes S^{d_k} E_k,\, W)
\end{CD}$$
where $(d_1, \ldots , d_k)$ runs over the non-negative integers solutions of equation (\ref{CondicionMonomios}).

A holomorphic map $\,f\colon\prod E_i\to W$, satisfying (\ref{CondicionHomogeneidadLemma}), and the corresponding linear map $\,\oplus f^{d_1\dots d_k}\in \bigoplus_{(d_1 , \ldots , d_k)}\text{Hom}_{\mathbb{C}}(S^{d_1} E_1 \otimes \ldots
\otimes S^{d_k} E_k,\, W)$, are related by the equality
$$f(e_1,\dots,e_k)\,=\,\sum_{(d_1 , \ldots , d_k)}f^{d_1\dots d_k}\left((e_1\otimes\overset{d_1}{\dots}\otimes e_1)  \otimes\cdots\cdots\otimes (e_k\otimes\overset{d_k}{\dots}\otimes e_k) \right)\ .$$

\medskip

\subsection{Invariant theory of $\mathrm{GL}_n(\mathbb{C})$}

Let $\,V\,$ be a $\mathbb{C}$-vector space of finite dimension $\,n$, and let $\,\mathrm{GL}_n(\mathbb{C})\,$ be the complex Lie group of its $\mathbb{C}$-linear automorphisms.

As any linear representation of $\,\mathrm{GL}_n(\mathbb{C})\,$ decomposes as a direct sum of irreducible representations, the following holds:

\begin{proposition}\label{Semisimple}
Let $\,E\,$ and $\,F\,$ be linear representations of $\,\mathrm{GL}_n(\mathbb{C})$, and let $\,E'\subset E\,$ be a sub-representation.
Any equivariant linear map $\,E' \to F\,$ is the restriction of an equivariant linear map $\,E \to F$.
\end{proposition}

On the other hand, the Main Theorem of the invariant theory for the general linear group states that the only $\,\mathrm{GL}_n(\mathbb{C})$-equivariant linear maps $\,\otimes^p V \longrightarrow \otimes^p V\,$ are the linear combinations of permutations of indices (\cite{LieGroups}).

In this paper, we will only use the following consequence:

\begin{proposition}\label{CorolarioMain}  The only $\,\mathrm{GL}_n(\mathbb{C})$-equivariant linear maps $\, \otimes^p V \longrightarrow \Lambda^p V\,$ are the multiples of the skew-symmetrisation operator.
\end{proposition}

Combining the previous two propositions, it follows:

\begin{corollary}\label{0.3}  Let $\,E\subseteq\,\otimes^pV\,$ be a $\,\mathrm{GL}_n(\mathbb{C})$-sub-representation. The only $\,\mathrm{GL}_n(\mathbb{C})$-equi\-va\-riant linear maps $\,E \longrightarrow \Lambda^p V\,$ are the multiples of the skew-symme\-trisation operator.
\end{corollary}

\begin{remark}\label{0.4} We will make use of the following properties of the skew-sy\-mme\-trisa\-tion operator $\,h$:\smallskip

-- $h(T\otimes T')=h(T)\wedge h(T')\,$ for any covariant tensors $\,T,T'$,\smallskip

-- $h(\omega)=q!\,\omega$ for any $q$-form $\,\omega$,\smallskip

-- $h(\nabla \omega)=q!\,\mathrm{d} \omega$ for any holomorphic $q$-form $\omega$ on $\,\mathbb{C}^n$, where $\,\nabla\,$ denotes the standard flat connection on the affine space.
\end{remark}

\subsection{Computation}

\begin{definition}\label{AntiConmutativos} Let us fix a finite sequence of positive integers $\,(p_1,\dots,p_m)$. Let us denote $\,\mathbb{C}\{ u_1,\dots,u_m\}\,$ the {\it anti-commutative} algebra of polynomials with complex coefficients in the variables $\,u_1,\dots, u_m$, where each variable $\,u_i\,$ is assigned degree $\,p_i$. The anti-commutative character of this algebra is expressed by the relations
$$u_iu_j\,=\, (-1)^{p_ip_j}u_ju_i\ .$$

The degree of a monomial $\,u_1^{a_1}\dots u_m^{a_m}\,$ is defined as $\,\sum a_ip_i$.

A polynomial $\,\mathbf{P}(u_1,\dots, u_m)\in \mathbb{C}\{ u_1,\dots,u_m\}\,$  is said homogeneous of degree $\,q\,$ if it is a linear combination of monomials of degree $\,q$.
\end{definition}

Let $\,\mathbf{P}(x_1,\dots, x_m)\in \mathbb{C}\{ u_1,\dots,u_m\}\,$ be a homogeneous polynomial of degree $\,q$, and let $\,\omega_1,\dots,\omega_m\,$ differential forms of degree $\,p_1,\dots,p_m\,$ on a complex manifold $\,X$. Then $\,\mathbf{P}(\omega_1,\dots,\omega_m)$, where the product of variables is replaced by the exterior product of forms, is a differential form of degree $\,q\,$ on $\,X$.

\begin{theorem}\label{Main} Let $\,p\,$ be a positive integer,  and let $\,\mathbb{C}\{ u,v\}\,$ be the anti-commutative algebra of polynomials on the variables $\,u,v$, of degree $\,\mathrm{deg}\, u=p$, $\,\mathrm{deg}\, v=p+1$.

Any natural differential operator of order $\leq r$
$$ P\colon \Lambda^{p} T^*X \  \rightsquigarrow\ \Lambda^q T^*X \ ,  $$
where $0\leq q\leq n= \dim X$, can be written as
$$P\left(j^r_x \omega \right)\,=\, \mathbf{P}(\omega,\mathrm{d}\omega )_x \quad , $$
for a unique homogeneous polynomial $\,\mathbf{P}(u,v)\in\mathbb{C}\{ u ,v \}\,$ of degree $\,q$.
\end{theorem}

\noindent \textit{Proof: } Due to Lemma \ref{2.1}, any such a differential operator $\,P\,$ is determined by a $\mathrm{GL}_n(\mathbb{C})$-equivariant analytic map
$$\tilde P\colon (S^0\otimes \Lambda^{p})\oplus\cdots\oplus(S^r\otimes \Lambda^{p}) \,\longrightarrow\, \Lambda^q\ ,$$
satisfying the homogeneity condition (\ref{CondicionHomog}).

The Homogeneous Function Theorem then assures that $\,\tilde P\,$ has to be a polynomial; that is to say, it is a sum $\,\oplus_{\{d_{j}\}}\, \tilde P^{\{d_{j}\}}\,$ of $\mathrm{GL}_n(\mathbb{C})$-equivariant linear maps:
$$ \tilde P^{\{ d_{j}\}}:  S^{d_{0}} (S^0\otimes\Lambda^{p}) \otimes\cdots\otimes S^{d_{r}} (S^r\otimes\Lambda^{p}) \, \longrightarrow \, \Lambda^{q} $$ where each sequence $\,\{ d_{j}\}\,$ of non-negative integers is a solution to the equation:
\begin{equation*}
p d_{0}+\cdots+(p+r)d_{r} \, = \, q \ .
\end{equation*}

Observe that this condition implies that $\,\tilde P^{\{ d_{j}\}}\,$ is in fact defined on a vector subspace of $\,\otimes^qT_0^*\mathbb{C}^n$.
Then, by Proposition \ref{0.3}, the linear map $\,\tilde P^{\{ d_{j}\}}\,$ is a multiple of the skew-symmetrisation operator $\,h$,
$$\tilde P^{\{ d_{j}\}}\,=\, \lambda^{\{ d_{j}\}}\cdot h\ .$$

The skew-symmetrisation of two symmetric indices vanishes, so that we may assume, from now on, that
$$
d_{2} =d_{3} =\ldots =d_{r}= 0 \quad  \ .$$

That is to say, we only consider solutions $\,\{ d_{0},d_{1} \}\,$ to the equation:
\begin{equation*}
p_1d_{0}+(p+1)d_{1} \, = \, q \ .
\end{equation*}

Bringing all this together,
$$P(j^r_0 \omega )\,=\,\tilde P\left((\omega)_0,(\nabla \omega )_0 \right)
$$
$$=\,\sum_{\{ d_{0}, d_1\}}\tilde P^{\{ d_{j}\}}\left(\left[(\omega)_0\otimes\overset{d_{0}}{\dots}\otimes(\omega)_0\right]\otimes\left[(\nabla \omega)_0\otimes\overset{d_{1}}{\dots}\otimes(\nabla \omega)_0\right] \right) $$
$$=\,\sum_{\{ d_{0} , d_1\}}\lambda^{\{ d_{0}, d_1\}}h\left(\left[(\omega)_0\otimes\overset{d_{0}}{\dots}\otimes(\omega)_0\right]\otimes\left[(\nabla \omega)_0\otimes\overset{d_{1}}{\dots}\otimes(\nabla \omega)_0\right]\right) $$
(using Properties \ref{0.4} of the skew-symmetrisation operator)
$$=\,\sum_{\{ d_{0} , d_1\}}\mu^{\{ d_{0} , d_1\}} \left[(\omega)_0\wedge\overset{d_{0}}{\dots}\wedge(\omega)_0\right]\wedge\left[(\mathrm{d}\omega)_0\wedge\overset{d_{1}}{\dots}\wedge(\mathrm{d}\omega)_0\right] $$
where $\mu^{\{ d_{0} , d_1\}}:=\lambda^{\{ d_{0} , d_1\}}(p!)^{d_{0}} ((p+1)!)^{d_{1}}$.

Therefore, taking
$$\mathbf{P}(u,v ):=\,\sum_{\{ d_{0}, d_1\}}\mu^{\{ d_{0}, d_1\}}u^{d_{0}}v^{d_{1}}$$
it follows
$$P(j^r_0 \omega ) \,=\, \mathbf{P}(\omega,\mathrm{d}\omega )_{x=0}\ .$$

By naturalness, we conclude
\begin{equation}\label{CondUnicidad}
P(j^r_x\omega)\,=\, \mathbf{P}(\omega,\mathrm{d}\omega )_x
\end{equation}
for any jet $\,j^r_x \omega\,$ at any point $\,x \in X$.

Finally, the uniqueness of the polynomial $\,\mathbf{P}\,$ is proved applying Lemma \ref{2.4} below to the difference of two polynomials satisfying (\ref{CondUnicidad}).

\hfill $\square$

\begin{lemma}\label{2.4}
Let $\,\mathbb{C}\{ u,v\}\,$ be as in Theorem \ref{Main}.

Let $\,\mathbf{P}\in\mathbb{C}\{ u,v \}\,$ be a non-zero homogeneous polynomial of degree $\,q\leq n$. Then, the natural differential operator of order $\leq r$
$$ P\colon \Lambda^{p} T^*\mathbb{C}^n \  \rightsquigarrow \ \Lambda^q T^*\mathbb{C}^n \quad ,\quad P(j^r_x\omega )=\mathbf{P}(\omega ,\mathrm{d}\omega )_x  $$
is not identically zero.
\end{lemma}

\noindent \textit{Proof: } Up to a scalar factor, $\,\mathbf{P}\,$ is a monomial of one of the following four types, depending on the parities of $\,p\,$ and $\,q$,
$$\mathbf{P}(u,v)\, = \left\{\begin{aligned}
&\ u^0v^s\qquad (p\text{ odd, }q \text{ even}) \\
&\ u^1v^s\qquad (p\text{ odd, }q\text{ odd}) \\
&\ u^sv^0\qquad (p\text{ even, }q \text{ even}) \\
&\ u^sv^1\qquad (p\text{ even, }q\text{ odd})\end{aligned}\right.$$
where $\,s\,$ has to be taken in each case, for the monomial to have degree $\,q$.

Depending on the case, let us consider the following $\,p$-forms, which involve $q$ variables each:
$$\omega\,=\left\{\begin{aligned}
\sum_{j=1}^s z_{j , 0}\mathrm{d} z_{j , 1}\wedge\dots\wedge\mathrm{d} z_{j, {p}} \quad , \quad
\text{case }u^0v^s\\
\mathrm{d} z_1\wedge\dots\wedge\mathrm{d} z_{p }+\sum_{j=1}^s z_{j , 0}\mathrm{d} z_{j , 1}\wedge\dots\wedge\mathrm{d} z_{j , {p}}\quad , \quad \text{case }u^1v^s\\
\sum_{j=1}^s\mathrm{d} z_{j , 1}\wedge\dots\wedge\mathrm{d} z_{j ,{p}} \quad , \quad
\text{case }u^sv^0\\
z_0\mathrm{d} z_1\wedge\dots\wedge\mathrm{d} z_{p}+\sum_{j=1}^s \mathrm{d} z_{j , 1}\wedge\dots\wedge\mathrm{d} z_{j, {p}} \quad , \quad \text{case }u^sv^1
\end{aligned}\right.$$

In each of the four cases, it holds
$$P(j^r\omega)\,=\,\mathbf{P}(\omega,\mathrm{d}\omega)\,=\, s! \, \mathrm{d} z_1 \wedge \ldots \wedge \mathrm{d} z_q \ . $$

Hence, the differential operator $\,P\colon \Lambda^{p} T^*\mathbb{C}^n\rightsquigarrow \Lambda^q T^*\mathbb{C}^n\,$ is not identically zero.

\hfill $\square$

\medskip
For the sake of completeness, let us remark that the following statement also holds, with similar reasonings in its proof:

\begin{theorem}\label{General} Let $\,p_1,\dots,p_m\,$ be positive integers,  and let $\,\mathbb{C}\{ u_1,v_1,\dots,u_m,v_m\}\,$ be the anti-commutative algebra of polynomials on the variables $\,u_1,v_1,\dots,u_m,v_m$, of degree $\,\mathrm{deg}\, u_i=p_i$, $\,\mathrm{deg}\, v_i=p_i+1$.

Any natural differential operator of order $\leq r$
$$ P\colon \Lambda^{p_1} T^*X \oplus \ldots \oplus \Lambda^{p_k} T^*X \  \rightsquigarrow\ \Lambda^q T^*X  $$
where $0\leq q\leq n = \dim X$, can be written as
$$P\left(j^r_x \omega_1,\dots,j^r_x \omega_r \right)\,=\, \mathbf{P}(\omega_1,\mathrm{d}\omega_1,\dots,\omega_k,\mathrm{d}\omega_k)_x \quad , $$
for a unique homogeneous polynomial $\,\mathbf{P}(u_1,v_1,\dots,u_k,v_k)\in\mathbb{C}\{ u_1,v_1,\dots,u_k,v_k\}\,$ of degree $\,q$.
\end{theorem}

\medskip
\noindent {\bf Acknowledgements:} We acknowledge J. A. Navarro and J. B. Sancho, who explained the theory of smooth natural operations to the first two authors, and A. Chirvasitu for raising interesting questions.

The first author has been partially supported by ICMAT Severo Ochoa project
SEV-2015-0554 (MINECO), MTM2013-42135-P (MINECO) and ANR-12-BS01-0002 (ANR).
The second author has been partially supported by Junta de Extremadura and FEDER funds. The third author has been supported by the research grant MTM2013-45935-P, Ministerio de Econom\'{\i}a y Competitividad, Spain.

redpajama_set_name: RedPajamaArXiv


AMTA-MA Chapter Wins 2016 LLEAD Grant!
Your AMTA-MA Chapter was the honored recipient of The LLEAD (Legislative and Legal Education, Advocacy and Defense) Grant Application. The purpose of the LLEAD fund is to provide assistance and funding to aid the efforts of chapters in their legislative efforts that seek to safeguard the interests of the AMTA membership at large by promoting the most favorable legal climate for the profession of massage therapy. We are the only professional organization with a lobbyist on hand to help with state licensing in MA. If you have questions or want to get involved, just email info@massamta.org.
In the meantime, check out all that we've been a part of at the state and local level in the past few years by reading our updates on Massachusetts Laws and Legislation. These are the issues that affect massage therapists. If you are not currently receiving our chapter emails and newsletters, please email communications@massamta.org and we'll add you to our distribution list.
Oh, and if you are wondering, we'll definitely be applying for the grant again in 2017!


text:
L'Ordine al merito bavarese (in tedesco Bayerischer Verdienstorden), è un ordine cavalleresco del Libero Stato di Baviera, all'interno della Repubblica Tedesca.

Storia
L'Ordine al merito bavarese viene concesso dal Primo Ministro di Baviera "in riconoscimento di contributi particolari apportati al Libero Stato di Baviera con servizi all'estero".

Insegne
 La medaglia dell'Ordine consiste in una croce maltese smaltata di bianco e bordata d'azzurro avente al proprio interno un medaglione circolare bordato d'oro che riporta sul diritto, a sfondo nero, un leone rampante dorato, mentre sul retro si trova a tutto campo la bandiera scaccata d'azzurro e di bianco della Baviera.
 Il nastro dell'Ordine è azzurro in centro bordato di blu, avente su ciascun lato una striscia bianca e una azzurra.

Insigniti notabili
Heinrich von Brentano di Tremezzo
Benedetto XVI
Johannes Friedrich
Ludwig Schick
Jutta Speidel

Fonti
  (PDF-Datei; 13 kB)

 Datenbank

Altri progetti

Onorificenze tedesche

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This is the allocutio I will deliver this evening at our Legion meeting. It comes from Ch. 24, part 4: St. Michael. The allocutio is a spiritual talk given by the spiritual director or, when he is not there, the president, during the Legion of Mary meeting.
Why should we make friends with the angels, and what role do they play in our lives? Personally, I know I neglect my thought and prayer toward them, aside from maybe asking for their protection ahead of going on a trip. With all the other figures we can pray to, the angels are simply not at the top of my list – I am sad to say I forget about them.
We should make friends with the angels because they are powerful figures in our lives and the lives of all those we interact with, and their intercession is a powerful tool. If you are arguing with a friend and family member, you can ask their guardian angel to help or intercede, along with your own. If you are worried for your safety, you can ask your guardian angel to be at your side. Angels have been credited with stopping traffic incidents, shielding people from harm and even bringing couples together for marriage, as was the case with Tobia, who was led by the angel Raphael to his future wife, Sarah, as we learned in the recent readings from the Book of Tobit.
Just the idea of angels is really astonishing – beings whose whole reason for existence is to look after us. They are our best friends and allies, and they share in our joy and sadness equally with us. We must make friends of them and lean on them, because that is what God designed them and us for.
We are fortunate in the Legion to have St. Michael the Archangel watching over us. It is particularly fitting given the military lexicon in the Legion – we are Mary's army, and he leads the army of angels. We know him as the angel who cast Satan from Heaven. He is the guardian of the church, patron of police officers, paramedics and the military. He also has a special role related to death – he carries souls as they depart, giving them a chance to come back to God and redeem themselves. He is depicted with a sword and, often, scales, where he is measuring souls with mercy. The sword is his weapon against evil, and he wields it on our behalf.
St. Michael will always come to our aid, and it is a good idea to invoke him before performing our works each week. He is, as Frank Duff noted, zealous in his willingness to render service to Mary's servants (that's us).
Be our defense against the wickedness and the snares of the Devil.


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We Tried Zappos' New Personal Shopping Service
Zappos sort of found me a Mansur Gavriel bag. It just took a while.
Dhani Mau
Have you ever seen something worn in a photograph or on the street that you instantly needed to have, but had no idea how to find it? Now, there are a slew of new services willing to do that research for you, so long as you have a photo of the item.
The latest is Ask Zappos: The online retailer promises to find any item you're looking for, even if it's not actually sold by Zappos. All you need to do is send in a photo through the Ask Zappos site (on desktop or mobile), or by email, text message or Instagram (by tagging a photo #askzappos).
A few apps offer a similar service: Kate Bosworth's StyleThief uses image recognition technology to find items, ASAP54 uses a combination of image recognition and an in-house team of searchers, while Wheretoget and The Hunt use crowdsourcing to find items.
So does Ask Zappos work? Um, sort of.
I "asked Zappos" through a few different channels: I uploaded a photo of a Mansur Gavriel bucket bag to its website, text messaged a photo of a pair of (my own) Superga sneakers, and had a friend email in a photo of Miranda Kerr, calling attention to the leopard print mules she was wearing.
To avoid misleading my Instagram followers/shamelessly promoting Zappos to them, I did not personally try it on Instagram, but I searched the image platform to find people who had. It appeared that everyone who had posted an image of a legitimate accessory or clothing item with the appropriate hashtag received a response (see left).
As for my own accessory inquiries, I did get responses to all of them ... though not within the 24-hour timeframe Zappos had promised. I sent my requests in at 1 p.m. Wednesday afternoon, and didn't hear back until 6:30 p.m. Thursday.
Each of the emails included a link to the Zappos site with a nicely presented photo and links to the items I was looking for and/or an assortment of similar items, including a few from sites other than Zappos. Interestingly, Zappos is getting a commission on at least some of the external items it suggests. A link to Warby Parker went through affiliate revenue company Shareasale before landing on the eyewear brand's site.
My results were mixed: All of my inquiries were answered by a girl named Nicole, who had a nice, friendly tone despite being implausibly slow. She impressively found a place where I could pre-order a Mansur Gavriel bag (Barney's) and found the bag available from resale site Avenue K. She also found Miranda Kerr's Miu Miu heels. She could not, however, find my Superga sneakers, or any Superga sneakers, even though they, um, sell the exact same ones at Zappos.
So, there are some bugs and delays – but as a human-powered service, we have to allow for human errors. By comparison, StyleThief gets you results immediately through its image recognition technology, but it, too, is not always accurate and at the moment you can't search accessories.
Ask Zappos promises it will soon be better, though. The company said it's planning to add image recognition technology, in addition to building a larger team of "stylists," if the service takes off.
ask zapposzappos
Zappos Is Piloting a Personal Shopping Service on Instagram With #nextootd
Zappos is the latest in a string of companies trying to monetize Instagram.
By Rachel Arthur
Zappos Tests the Brick and Mortar Waters With Vegas Holiday Shop
The retailer also wants to remind shoppers that it sells more than just shoes.
The "Holacracy" of Zappos and the Future of Zappos Couture
Zappos's executive team discusses the company's new management structure and the future of their higher-end fashion division, Zappos Couture.
By Ann Binlot
What to Expect From Andre Leon Talley's New Zappos Gig
The man who brokered the deal tells us why Andre Leon Talley has been named artistic director of Zappos Couture.

redpajama_set_name: RedPajamaCommonCrawl


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← I'm not stalking him, he is stalking me
Ed Marinero Talking Blues I mean Bruise →
Leonard Frey of 201 Streeter
Posted on December 11, 2014	by markweiss86
Leonard Frey, Dartmouth class of 1949, lived at 201 Streeter, one of the older dorms, in the so-called Gold Coast, west campus, closer to the river, by my sense. When Harry Hillman died, Frey contributed $5 to a memorial fund. In 2003, when Richard Thorner, my classmate, sold me Hillman's scrapbook, I researched other info on the famous coach and Olympian, and on a hunch cross-referenced his name with Blunt Alumni records, who reported then-current contact info, in San Diego. Ten minutes ago, for no reason whatsoever other than, as Shakespeare sometimes suggests, storms portend, I fished a 2003 notebook from my storage space, here at Oak Creek. This is the first item in the book.
here is a prologue, courtesy of a recent Dartmouth Alumni Magazine class notes, for 1949s Dr. Frey's class:
Gene Miller, with his health (no pills or walkers) and memory intact, found an old reminiscence of the 49 '49ers (including me) who entered in March 1945. Some tidbits: "[We] were housed mostly in Richardson, Wheeler or a barely habitable Crosby. The campus was awash with 'Swabbies' and 'Gyrenes.' War raged in Europe and Asia. The immediate future looked grim [for us] mostly 16- and 17-year olds. For [us smokers] begging inferior brands at Allen's Drug or the Indian Bowl was a major pastime. The pool table in the back of a basement barbershop was the center of a pitiable recreation effort (which included water fights. dodging Wormwood (campus security) and carrying the Dartmouth tradition to Smith, Colby, et. al., usually by hitchhiking on gasless, deserted roads)." Gene currently resides in Greensburg, Pennsylvania.
"Interview with Dr. Leonard Frey, PhD '49"
(619) 295-XXXX – I tried the number but it is disconnected; the search-injun offers little immediate clue to my main question: How is Dr. Frey?
"In the package sent by Barbara Krieger at Rauner/Baker, I noticed that one of the correspondences was a donation to the Hillman fund from Dr. Frey. Written at the bottom of the letter was "$5 – 201 Streeter" — a dorm address — and I inferred first that this donation was from a student and second that the student may still be living. Sure enough, a check of my alumni directory (2001) showed that Dr. Frey lived in San Diego. I called directory assistance and then reached him.
First, he hung up on me when I said I was calling from Dartmouth. Then I called back and said I was calling about Hillman and he did indeed want to talk to me. "
Ok, I am interrupting the flow here, of transcribing my notes from 2003 to Plastic Alto, to paste in a woman, Meisue Francis, who in 2006, i.e. during the time I was more avidly a Hillmanite, won the 2006 championship 4×400 with some LSU teammates and she went to the same high school Brooklyn Boys and Girls, as did Hillman 100 years prior. This was in my notes, or the version of such I posted here in June, 2014. Not sure how I got there but worth noting.
Brooklyn Boys and Girls, since 1886, is the oldest public school in Brooklyn, in Bed-Stuy and has 3,600 students. I recall taking subways and buses to get there, hoping to flip thru the yearbook archives to find more info on Hillman, and when I got there the guards had no record of my appointment so I was denied access. So meeting a Dartmouth student, in front of Hanover Inn, waiting to get home for holidays, from the same school was kind of a thrill for me. For a second here, 11 years later I thought the LSU athlete and the Dartmouth co-ed were the same person. I wonder if the 2006 female athlete was as fast as the 1904 male Olympian. Will check that. 1700 Fulton Street, Brooklyn, NY 11213
Well, no, the 2006 NCAA women's featuring Meisue on leg 2 and Brooklyn Morris on leg 1 they each ran about 51 seconds to Hillman's 49 and one-fifth.
And not to detract from our subject, dear Dr. Frey, but here is more of Meisue so to speak in 2010 tweeting under the name "tracshortie01" which I think translates into "attractive person who runs fast, the first":
I was noting, at Phil Hanlon's lecture, that times have changed a wee bit since my day but I am fairly certain that this former track champion is someone "Men of Dartmouth" and alumni would give a rouse for.
If we could catch her, I mean.
"Frey entered Dartmouth in the summer of 1945. Hillman died shortly thereafter, in August of 1945. (August 9, 1945) (Frey had gone up early)
He was from Northern NJ. Near Morriston. He went to public school (a good one, that offered six languages). His family summered in New Hampshire so he knew the area. He was an English major, but switched to linguistics. He went on to teach German and linguistics at San Diego State but is now — as of 2003 — retired. He's 76. (Ok, he is 86 or 87 by now, lwatcdr.).
He also did some work as a journalist. He worked for the New York Herald Tribune, as a sportswriter. He worked with Red Smith. He called Smith "a really nice guy…from Notre Dame".
He contributed to a Sports Encyclopedia published in Iowa, by a David Porter. He said he wrote 40 to 50 stories. He said there is an article on Hillman which he is sending along. He is also sending an article on Guy Broberg the basketball player, father of Pete Broberg. Ok I gotta paste in a Topps Broberg or I may have one in my apartment to Swede in, Rangers, 1972.
Meanwhile and this is egregious Plastic Alto bull roar here is a book on Palm trees recommended by Pete Broberg, the former Dartmouth and Rangers hurler, because Dr. Frey told me what a hero Broberg's father was, the War hero pilot who lost his arm in a crash but went on to be a lawyer and judge and father of another Dartmouth hero. Gus Broberg had just passed away when I researched all this.
http://www.amazon.com/Betrocks-Essential-Guide-Palms-Leaser/dp/0962976199
And I did spend 90 minutes today bending the ears of Steve Staiger of PAHA warming up for being the moderator of a panel on jazz history; I said worst case scenario I read my 20,000 word treatise, or repeat half of what I said just now. And I pulled a copy of Errol Morris "Fast Cheap and out of Control" out of my pack: Rod Brooks of MIT robotics, the lion tamer, the topiary gardener and expert on mole rats, 1996, re-wired me brain.
the cap is airbrushed because Washington had just gone to Texas, worth $3
I am sparing you the link to the Errol Morris film but here is info on Streeter, which I presume still has a 201 although my old dorm Richardson went from about 50 rooms to about 30, because they added fire escapes and took out all the end singles:
Heat Control in Streeter Hall
For Streeter, each room has a small thermostat that is usually tan or gray and has numbers on it. It is located on one end of the radiator. This thermostat is really a valve that allows differing amounts of steam to enter your particular radiator; the more steam that enters the radiator, the warmer you will be. As noted above, the times steam is delivered to your building is controlled by a computerized program, and that steam is either all the way on or all the way off. Your room thermostat will only affect the heat output of your radiator when the building is receiving steam. If you have trouble setting this thermostat, call the ORL Operations Office at 6-1203 during normal business hours, 8 AM until 4:30 PM, M-F. Report the problem and we will send someone to your room to check it with a digital thermometer. If it is found that the heat is below 66°F or above 70°F, we will call Facilities, Operations and Management and ask them to send someone to correct the problem. After business hours and on weekends, you may call 6-2344 to report heating problems.
If you have any questions, call or blitz: Woody Eckels, Director of Residential Operations, 646-1203.
Streeter has 67 beds to 61 for Richardson. The second floor is now all female, I think Streeter was still all-male in my day. For comparison, Dartmouth was all male, 1769 to 1972, for instance for Dr. Frey. Here is a floor plan, 124 square feet.
Here is a picture of the glorious Richardson, where presumably Frey sometimes called on friends:
richardson, not streeter, and I'm sure we beat them in football
I lived in 204 Richardson, with Teddy Conway of New Orleans and David Williams of Greenwich for freshman fall and winter, 1982-83, then switched up to a double with Brian Gaul of Chevy Chase or DC area, for the spring. Our 204 is probably now part of 202, which they call a three room quad to our two room triple.
Frey lived at 201 Streeter for three of his four years meanwhile, back at the ranch, and then Butterfield. His father had seen Hillman compete so Leonard arranged to introduce him to Hillman "And he was thrilled.
"But I (Frey) only knew him (Hillman) briefly. He was an awfully nice man. Very kindly. A fatherly sort."
He was succeeded at the helm of the Dartmouth track and field program by Elliot Noyes.
"I have a picture right here next to me of myself standing next to Elliot Noyes. It's the 1945 Dartmouth cross country team picture " he told me in 2003.
And then at the bottom of the page I write: I should ask him about Moe Berg!
And then at the top of the page I jot: "Earl Brown" in quotes and "Goering" not in quotes. I think I pulled this file when I was introduced or made aware of Goering a former basketball player of that era. As in a couple years later, ten years ago.
Dartmouth archive only lists 3 of 5 players, with Coach Brown
And continuing I realize that although I claim to find the new public radio Ira Glass stuff annoying maybe it is an influence Frey was a fan of basketball and said that Dartmouth had a dynasty won seven Ivy titles and played for the national championship twice, against Stanford in 1942 and Utah in 1944. He attended the games at Madison Square Garden in 1944 (i.e. before matriculation)
Well if you are still with me you are in for a real treat in that here is 27 minutes of highlights from that 1944 NCAA game, I was researching just the other day, Dartmouth versus Utah, so you can see what you Leonard Frey saw, and maybe you can actually see him in the crowd shots.
The other hoop athletes he mentioned were Jim "Chip" Coleman* and "Gut". (I struck thru the second guy)
Broberg was a pilot during WWII got shot down, and lost his arm in the crash. He went on to law school and became a municipal judge in Florida. Frey interviewed him.
Frey was a classmate of Slade Gorton "of the Gorton codfish family". He was a Senator (and I mean in Congress, unlike young Broberg the baseball variety Ranger and maybe Senator0 from Washington (I think we mean West Coast, above Oregon) for 24 years, "an awfully nice fellow".
Dartmouth was national champs in football in 1925, undefeated (Swede Oberlander era, I write in later, should be a ?)
The track at Alumni Gym was "a big track". It was 6 laps to the mile, compared to 11 laps at Madison Square Garden. He remembers Glen Cunningham's record performance in Hanover as "4 minutes 4 seconds and 4 tenths" (4:04:04, I had not previously noted it, although there is actually a clip that Hillman had put in his book, loosely marking that event).
"It was so unusual to have a track that big and a real pleasure to run on it" Frey recalled, in my 2003 interview by phone.
He recalled that he followed Jessie Owens' career and that Hitler snubbed him, but suggested that Roosevelt snubbed him, too.
He recalled that Marty Glickman later an announcer, was replaced from the mile relay in Berlin '36 Olympics because he was Jewish, "two of them were Jewish".
Frey remembers a Dartmouth Professor of German named "Schlossbacher" who was very strict — "he had a scar we surmised was from dueling".
"Schlossbacher once called on a another '49 Van Hamburg first and Van got the answer wrong and the prof said 'You are a discredit to your name' and called on me next."
Frey reported that despite these fond memories he had not been back, as of 2003 to Hanover, New Hampshire campus of Dartmouth since the nineteen-fifties. He said he remembered, nonetheless, how remarkably beautiful is the campus. He was a Kappa Kappa Kappa "a luca" — no idea — but "went inactive". It was a track team fraternity.
"I was in tears when I head that Harry Hillman died."
He said Hillman seemed shorter than the 5'foot 11 I suggested.
Tesreau (I was quizzing him, or cross-referencing my various notes): "I knew him well"
Eddie Jeremiah – hockey
Ozzie Cowles — hoops
"Gus" "McGloughery" I guess this means that I need to proof the names – football
and yeah so that I had Ozzie Cowles here, I must have pulled this when I got a chance to talk by phone to a former Dartmouth 1944 collegiate championship game player named Vince Goering, toward the end of his life, a year or so later. I had them ask him "did he play for Ozzie Cowles?" and the response was no, Earl Brown. Which I had to check later. Cowles had just retired.
Blaik — went to Army to coach, Blanchard and Davis slash West Point
Tesreau he noted was a teammate of Christy Mathewson. Big Six (and what was Griffin Boninis statement about an all-time great nickname of a pitcher of that era("Meal Ticket" Carl Hubbell?)
45 minutes: 5:45 stop (and here I am about 7:30 or about two hours into this trip down a couple simultaneous memory lanes.
The next page in my book has NYAC New York Athletic Club, a 212 number, then below that, a separate entry, dated 11/5/03 four days later "Brooklyn Boys and Girls" some basic facts like A Train C Train a Ms. Bell librarian for 14 years x550 or x215 which again was a wild goose chase or dead end when I actually tried to visit. I remember being someone out of place on the bus.
The next page has two numbers in 603 for C. Everett Koop and I know in the file is a short note back that he barely knew Hillman. And then I have a cite about a new book in 2003 about the 1903 Word Serious "Autumn Glory"
http://www.amazon.com/Autumn-Glory-Baseballs-First-World-ebook/dp/B008BJ3MNE/ref=la_B001IOH5O6_1_7?s=books&ie=UTF8&qid=1418355571&sr=1-7
and there is a listing for a current Class of '49 columnist I may ring him or offer to fax this over, although people might be put off by the rambling style or lack of editorial discipline: John Adler, 1623 Pelican Cove Road, BA123, Sarasota, FL 34231; (203) 622-9069; note that he does not list an email address.
edit to add: Tesreau note the proper spelling, and he deserves his own post, even in Plastic Alto, died in September 1946 so if asked Frey would or may have to someone expressed sadness at being in close proximity to that occurrence, in Hanover. Here is a pic.
And Judge and former Titan hoops star Griffin Bonini is absolutely correct that the pitcher and HOF of the next generation of Giants Carl Hubbell was known as King Carl or "Meal Ticket" check out the plaque:
edit to add, next morning; I am tempted to call this number and tell Pete Broberg's secretary that there is a Dartmouth alum writing an article about basketball and see if he picks up. (561) 655-5166, in Florida. I asked Rick Kimball a Dartmouth trustee if he thinks Murray Bowden the star of the 1970 football team would take a call from a writer asking him about Ed Marinero: "Of course!". We shall see. But first, let's get some update on Dr. Frey. I will have to go search for his clips he sent me. And I think not far from those would be a 1972 Topps Broberg, so I can say "I have your card, right here — I bought it when I was 8". And Kimball ad libbed the other night after being introduced as a Yaz fan by Martha Beattie that he could also recite some stats on her man Jim Beattie.
later that afternoon: I did reach Mr. Adler the '49 newsletter editor or class scribe who says that his records show that same phone number for Dr. Frey as of August of this year, as current, living and breathing and we hope supporting the Old Mother. He gave me some leads on other athletic memoirs of that era and, on another thread, set me straight about the V-12 program. Enjoying his winter in Florida when not in Greenwich himself.
*from Feb. '47 Crimson: A revamped tribe of Dartmouth Indians will do its inhospitable best tonight to spoil the Varsity basketball team's annual hegira to the hills of Hanover, where the teams meet in an Ivy League contest.
Since the Green suffered a 58 to 45 defeat at the hands of the Varsity earlier this month, Coach Elmer Lampe has lost the services of two of his players. Center Aud Brindley was graduated, and forward Paul Campbell was declared scholastically ineligible.
But the Indians' new lineup, which includes Andy Carstensen at center and Emil Hudak at guard–Captain Chip Coleman was moved to forward–has shown surprising strength. Dartmouth gave Holy Cross a dogged battle before bowing by 11 points last week, and held league-leading Columbia to a five-point victory at Hanover Saturday night. Always tough on its home floor, the Big Green has lost only six games in ten years on native grounds.
edit to add: Ozzie Cowles lured away by Michigan, then featured on Wheaties box:
This entry was posted in media, sports and tagged barbara krieger, betrocks, brooklyn, brooklyn boys and girls, dartmouth, harry hillman, jeff tesreau, leonard frey, madison square garden, meisue francis, rauner library, slade gorton, woody eckels. Bookmark the permalink.


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All posts tagged "NCAA Basketball"
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Weiss' Weekly Watch
Kentucky's convincing 85-69 victory over Tennessee at Rupp Arena on Saturday could have created a sea change in our bracketology. The...
Team USA Roster Announced for NIKE Hoop Summit
The roster for the U.S. team at the annual NIKE Hoop Summit in Portland, Oregon was announced last week. Most of...
Kentucky Looks like Final Four contender after win over Tennessee
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The NCAA tournament selection committee gave us a sneak peek of March Madness with the release of its initial rankings of...
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The New York Knicks are tired of being a punch line for so many bad jokes about the fact their team...
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Full-Court Pressure: A Year in Kentucky Basketball


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Investment in Indian education: size, sources and effectiveness
Pandit, Hirday Nath
IIEP Occasional Papers, 43
Methods and tools for educational planning
This study discusses investment in the Indian educational system with respect to measuring methods of its size, sources and effectiveness. The multi-source system of financing of education shows this sector has become a very important enterprise in the socio-economic life of the country, generating economic and non economic effects. The growth of aggregate expenditure on education one the one hand changes in the participation rates of the central and State governments in its financing on the other hand, have been analysed in a historical prospective. Subsequently, an estimation has been made of the ressources cost of the educational system during the three five-year plans at current and constant prices. An attempt has been made to identify social, institutional and private unit costs of education. And finally, the major trends emerging out of the studies on investement effectiveness have been used to evaluate the allocation of resources during the fourth and fifth plans.
Publications Homepage
Faire des partenariats publics-privés dans la formation professionnelle un levier fort de l'émergence économique en Afrique: synthèse des travaux de l'atelier régional de partage et de renforcement mutuel sur les partenariats publics-privés dans la
L'Enseignement privé et spontané dans le système éducatif tchadien
Péano, Serge; Esquieu, Paul
Budgetary processes in the Ministry of Education and Sports, Slovenia
Bitenc, Mira
School grants and the right to education: the case of the Honduran Community-Based Education
Sourrouille, Florencia; Souto Simao, Marcelo


Who doesn't like tasty doughnuts? This past May, we visited Portland, OR for a long weekend with the goal of sampling all the tasty vegan food we could. Voodoo Doughnuts was near the top of the list due to all the positive things we'd read online about them.
While we were walking around downtown, we kept seeing people carrying pink Voodoo boxes. We must be getting close. A short walk from the train, and we were there. The line was out the door and around the corner. They must be popular.
It was quite the scene inside. We waited at the door to be called in to one of the many employees helping customers. There were carts of doughnuts in the back.
Our faces were plastered to the vegan doughnut case. The bearclaw was calling my name. I could almost taste it through the glass. It looked really good.
We exercised a little restraint while ordering and managed to make it out of the store without buying a dozen doughnuts. It was hard!
We quickly sat down outside and ripped open our pink box to eat the tastiness inside. It started to mist, but we weren't deterred.
The bearclaw was delicious, and it was totally worth the wait. My wife ate her doughnut as well. We saved one for a snack after the movie. The box was a bit unwieldy to deal with, so we put the last two doughnuts into her purse and continued to explore the downtown area.
I am ashamed to say I still have not ben to VooDoo Donuts. I must rectify that if hubby ever has a day off. He just started 11 days straight between work and Team Oregon.
And damn I am craving a doughnut now.


Alice-Anne graduated with a Cert. IV & Dip. Remedial Massage in 1999. During this time she had a part time home massage business working for Hunter Area Health.
For 10 years, Alice-Anne served as a volunteer massage therapist and treasurer for Westlakes Palliative Cafe for Cancer Patients. Unfortunately the organisation was closed down in 2016 due to a lack of volunteers.
In 2016, after 25 years as Supervisor Sterilizing Dept. for Hunter Area Health, she was invited by ATUNE Health Centres to be part of a new off-site service for aged care/palliative care clients. She found this new venture very exciting and rewarding, as she got to be part of a group that offered services to help clients who were unable to attend a clinic for specialised treatment.
Alice-Anne loves her family, friends, scrap booking, travel, cooking and staying healthy.


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Is Freezing Your Eggs Worth the Cost?
By Karen Rowan published 17 April 15
(Image credit: Image Point Fr/Shutterstock.com)
For women who may want to have children in their 40s, freezing their eggs in their 20s or 30s costs less and has a higher success rate than waiting until age 40 to try to get pregnant, a new study finds.
Scientists ran all the numbers — the cost of egg freezing, the odds of having a baby at age 40 without in vitro fertilization and the cost of IVF for women who will need it in order to have a baby — and found that it costs about $15,000 less, on average, for women to freeze their eggs at age 35 and use them at age 40, rather than wait until age 40 and try to become pregnant.
Egg freezing remained the more cost-effective option for women up until age 38, the researchers said.
The study shows that "if a woman invested in having a genetically related child at age 40, egg banking at least once at age 35 is a cost-effective approach," said Dr. Wendy Vitek, an assistant professor of obstetrics and gynecology at the University of Rochester Medical Center in Rochester, New York.
"This is very valuable information for women to consider, as the financial burden of egg banking and storage is often a limiting factor for interested women," said Vitek, who was not involved in the study.
The researchers also found that a woman's chances of eventually having a baby are better if she freezes her eggs, according to the study published online March 24 in the journal Fertility and Sterility. [Egg Freezing: 5 Things You Need to Know]
Why freeze your eggs?
Some women are beginning to embrace egg freezing in order to keep open the possibility of having kids in their 40s, Dr. Kate Devine of the National Institutes of Health and her colleagues wrote in their study.
In egg freezing, women inject themselves with hormones for about two weeks, and then a doctor places a needle into the vagina to retrieve mature egg cells from the ovary. The egg cells are then frozen for later use in IVF.
The procedure to retrieve the eggs costs between $10,000 and $17,000 per cycle (and some women may undergo more than one cycle to get enough eggs). Other fees may include those for storage (up to $3,000 for five years) and thawing ($3,400 to $6,800), according to the study.
In comparison, IVF with fresh eggs costs about $13,000 to $17,000 per cycle, according to the study.
The researchers decided to calculate how these costs might play out in real life. In their calculations, they made the following assumptions based on previous studies: 16 percent of women get pregnant at age 40 within six months of trying, without the need for any interventions; one cycle of IVF at age 40 has a 17 percent chance of success; and women will undergo a second cycle of IVF if the first one doesn't work.
They found that 62 percent of women who freeze their eggs at age 35 and try to get pregnant at age 40 would successfully have a baby, with the average total cost of the procedures leading to the birth coming to $39,946.
Just 42 percent of women who tried to get pregnant at age 40 using IVF with newly retrieved eggs would have a baby, with costs totaling $55,060, on average.
Under a third scenario, women freeze their eggs at age 35, and then at age 40, they try conventional IVF. Only if those newly retrieved eggs don't work do they proceed to use frozen eggs. Women in this situation would spend an average of $61,887 — making it the most costly option in the study. But this scenario also had the highest success rate, with 74 percent eventually giving birth, the researchers said.
Vitek said she wasn't surprised that egg freezing remains cost effective up until age 38. "We know that while fertility is individual and most [women] will experience a progressive decline, there does seem to be a sharper decline at age 38 in population data." [Conception Misconceptions: 7 Fertility Myths Debunked]
Who pays for it?
Egg freezing is not usually covered by insurance, so women must typically pay for it themselves, Vitek said. For IVF (with either frozen or fresh eggs), insurance coverage varies, with some companies paying for almost all costs for procedures and medications, and others not covering any costs.
Though Vitek thinks the study results are plausible, she said the assumptions made for the calculations could be problematic. For example, the authors assumed that live birth rates from frozen eggs would be similar to birth rates using eggs that had just been retrieved, but researchers do not know if this would hold true for eggs that were retrieved from women who are nearing the end of their reproductive years, she said.
The researches also didn't account for the fact that 40-year-old women who don't get pregnant after two cycles of IVF may turn to adoption, or IVF using donor eggs, which are both quite costly, Vitek noted. If those costs were considered, egg freezing would be even more cost-effective compared with other options, she said.
Follow Live Science @livescience, Facebook & Google+. Originally published on Live Science.
Karen Rowan
Health Editor
Karen came to LiveScience in 2010, after writing for Discover and Popular Mechanics magazines, and working as a correspondent for the Journal of the National Cancer Institute. She holds an M.S. degree in science and medical journalism from Boston University, as well as an M.S. in cellular biology from Northeastern Illinois University. Prior to becoming a journalist, Karen taught science at Adlai E. Stevenson High School, in Lincolnshire, Ill. for eight years.
Striking satellite photo captures Mount Vesuvius peering through a hole in the clouds
New study provides first evidence of non-random mutations in DNA


If your business is in Lincoln or North Hykeham, you need to join this free Onlincolnshire event to help you make the most of the online revolution.
The event will inform businesses about the support being offered by Onlincolnshire, and will include taster sessions on how businesses can deal with change, developing a digital strategy and social media.
Masterclasses are also available, looking in depth at e-commerce and internet marketing, as well as social media surgeries on Twitter, LinkedIn and Facebook. One-to-one advice and support is available through the project.
You can join in on Wednesday, May 21, at the Bentley Hotel in South Hykeham, just south of Lincoln. Networking starts from 8.30am and includes a free breakfast. The event then runs until 11.15am.
This free event is open to all businesses across Lincoln and North Kesteven and is part of a wider project of support across Lincolnshire and Rutland being run by Onlincolnshire. It is part-funded by the European Regional Development Fund (ERDF).
To book your place on this free awareness session please call 0300 80 80 120 or email onlincolnshire@lincs-chamber.co.uk.


Helen trained as an actor at the Drama Studio, Sydney, 1982 – 84. Her first professional theatre work in New Zealand was with Mervyn Thompson and she performed in three of his plays over 1989 and 90. Since then her work in devising and physical theatre with Bert van Dijk was particularly inspiring and formative and she is drawing on this work in the direction of EARTHQUAKES AND BUTTERFLIES. Helen has devised, produced and toured 3 solo shows, the last of which, THE BONE KEEPER'S STORY, she co-wrote with Kathleen Gallagher. Helen's directorial work includes JESUS MY BOY and SCARAMOUCHE JONES with Martin Howells and many youth theatre productions. She sees EARTHQUAKES AND BUTTERFLIES as a community theatre event, a 7th anniversary ritual, which invites us to reflect on the transformation of our lives, our city and ourselves.
Fleur is an established New Zealand choreographer and has had works featured nationally in Arts Festivals throughout the last decade and is known for her diversity and connections with many different companies including Jolt Dance Theatre, Rebound, Southern Lights Dance Company ,Unitec, Footnote and Lyttelton Arts Factory. Fleur is currently artistic advisor and choreographer for Hagley Dance company.
Chris Reddington is a sculptor, award-winning set designer, composer, ensemble director and musician. He has a degree in Craft design from Christchurch Polytechnic and has worked for New Zealand sculptors Neil Dawson and Graham Bennett. He has completed sculptural commissions for the Christchurch City Council, CPIT and St Thomas of Canterbury College. Chris has designed and made many sets for the Free Theatre, IMAGINE Youth Theatre, One Man Banned Productions and Ben McDonald touring musical productions, most recently SOUND OF MUSIC and SATURDAY NIGHT FEVER. He is currently also designing the set for TITUS ANDRONICUS at the Court Theatre. In addition to designing the set and props for EARTHQUAKES AND BUTTERFLIES Chris has advised on sound for this production.
Soon after graduating from CPIT School of Fashion in 1996, Tina was appointed Wardrobe Supervisor for Southern Ballet. She joined The Court Theatre in 1998 and designed her first show The Big OE in 2002.
A year later Tina became the Costume Manager for Canterbury Opera before moving to Australia. Highlights from her five years in Sydney include styling and costuming for television, musicals and feature films, along with working on productions for Opera Australia; three years at the renowned Sydney Theatre Company; and touring through China and South Africa with The Phantom of the Opera.
Tina returned to The Court in 2013. Her costume design credits include Kings of the Gym; Snap!; When the Rain Stops Falling; Ache; Plum; Stag Weekend; Shepherd; Romeo and Juliet; Exit the King; Badjelly the Witch; Macbeth; and, most recently, The Streaker.
Kathleen Gallagher is a poet, playwright, filmmaker and novelist. She received the New Zealand Playwrights Award in 1993, and the Sonja Davies Peace Award in 2004 for the film Tau Te Mauri Breath Of Peace. She has authored three collections of poetry, 16 plays, six feature films, and one novel. Her films Earth Whisperers Papatuanuku, Water Whisperers Tangaroa and Sky Whisperers Ranginui, have played in cinema and film festivals throughout New Zealand and around the world.
Her most recent work is the novel Earthquakes & Butterflies - Otautahi Christchurch launched in Christchurch on September 2015, on the 5th anniversary of the beginning of the Christchurch earthquakes sequence (2010 - 2012).
Images used with permission or are in the public domain.
Website and social media management by Webfuture.


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GO Virginia Region 8 Completes Growth and Diversification Plan
A regional Economic Growth and Diversification Plan prepared by the GO Virginia Region 8 Council has received final approval from the GO Virginia State Board. The Region 8 Council's plan identifies five target industry sectors in the Shenandoah Valley where job growth is anticipated in the next five years. These jobs are expected to have an average wage of $41,000 or higher. The target industries include financial and business services, healthcare, light manufacturing, information technology/communications, and transportation and logistics.
The Region 8 Economic Growth and Diversification Plan identifies economic opportunities, needs, and challenges, establishes priorities, and outlines support necessary for multi-locality collaborative projects to potentially be funded under GO Virginia. The plan also recommends strategies for addressing workforce recruitment and training in target industry sectors. In addition to defining the five anticipated growth industries, Region 8 Council's plan also includes a set of broadly-defined framework initiatives intended to recruit/retain necessary talent, grow existing businesses, establish start-ups, and support joint economic development activities such as site development and workforce training.
George Pace, Chair of the Go Virginia Region 8 Council, said: "The Plan we have developed will provide the framework for prioritizing projects for GO Virginia funding. Our goal is to create more job opportunities that pay over $41,000 through incentivized collaboration between public and private organizations throughout Region 8."
GO Virginia Region 8 covers a large area from Winchester to Lexington, and includes ten counties and six cities in the Shenandoah Valley, with a combined population of 525,000. The 28-member council that oversees the GO Virginia program is made up of representatives from small and large business, higher and k-12 education, elected officials, and economic and workforce development organizations.
Approximately $1M in implementation funding will be made available to the region for projects that align with the approved Region 8 Council's Economic Growth and Diversification Plan. Projects must be collaborative, involving two or more localities and private sector partners, and be focused on growing and diversifying the region's economy to complement existing industry clusters as outlined in the Council's plan.
For more information on GO Virginia, visit online at Shenandoah Valley Partnership or Virginia Department of Housing and Community Development.
About GO Virginia: The Virginia Initiative for Growth and Opportunity (GO Virginia) is a voluntary, business-led, bipartisan initiative that was formed to foster private-sector growth and diversification across nine economic development regions in the Commonwealth of Virginia. State financial incentives designated for regional projects that encourage collaboration between private sector companies, workforce, education, and government are administered by the Virginia Growth and Opportunity Board.
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text:
Amazon Music might land in India in 2018
by Abhimanyu Ghoshal — in India
Amazon is reportedly gearing up to launch its streaming music service in India early next year, reports The Economic Times.
That's based on accounts from numerous production firm executives who say the company is in talks with local record labels to acquire streaming music rights in the country.
There's certainly good reason for Amazon to do so; According to a Deloitte report from 2015, India will see 273 million people subscribe to streaming music services by 2020. The market is estimated to grow to about $475 million by then.
Of course, there's stiff competition awaiting Amazon in India if it chooses to enter this space. Local incumbents like Ganna and Saavn are already claiming 28 million and 20 million monthly active users on their platforms respectively, while mobile carrier Airtel's Wynk Music crossed 25 million downloads last July.
Amazon India declined to comment. Should it launch a music service in India, it'll not only have to offer localized pricing (rival services charge between Rs. 99 ($1.5) and Rs. 120 ($1.85) per month) but also consider creating exclusive content to win subscribers over.
While Apple Music offers international programming like Beats 1 radio as an add-on, it's only managed to garner about 400,000 subscribers in India. Local players like Gaana and Saavn, which boast millions of paying customers, are working on unique pieces of content that includes exclusive releases, a platform for indie artists and reviews of new music from established performers.
Amazon previously launched its Prime Video service in India, as well as the rest of the world, last December. It bundled the service with its Prime delivery service and offered a mix of local and international movies and shows.
It'll also be interesting to see how Amazon intends to roll out tunes in India; it's Prime Music service in the US offers a limited catalog of about two million songs for free to Prime subscribers, while Amazon Music Unlimited serves up a much larger library and starts at $4 a month for Echo/Dot/Tap users, and goes up to $15 a month for a family plan that covers up to six people. It isn't immediately clear which of these Amazon will make available in the country if it goes ahead with the rumored plan.
Read next: Laptops and tablets are now banned in cabins on some US-bound flights [Updated]
AmazonWorld


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Q: Jsf 2.0 Custom Tag I get the following error on every page of my project when I am removing the MyFaces-2.0 entry from web.xml. My project is created using JSF 2.0.
Error:

One or more resources have the target of 'head', but no 'head'
  component has been defined within the view.

web.xml:
<context-param>
    <param-name>org.jboss.jbossfaces.JSF_CONFIG_NAME</param-name>
    <param-value>MyFaces-2.0</param-value>
</context-param>

After removing that entry my custom tag runs. If I put this in web.xml again, then the compiler doesn't go to the component class.

A: As the message you got says, you have no head component in your view (on your Facelet). Such a component is needed for other components that e.g. want to inject scripts and css resources into the head.
The remedy is to simply add this component on your Facelet, e.g.:
<!DOCTYPE html>
<html lang="en"
    xmlns="http://www.w3.org/1999/xhtml"
    xmlns:f="http://java.sun.com/jsf/core"
    xmlns:h="http://java.sun.com/jsf/html"
>
    <!-- The head component that was missing -->
    <h:head/>

    <h:body>
        <!-- Other components here -->
    </h:body>
</html>

redpajama_set_name: RedPajamaStackExchange


1. In this randomized controlled trial, the injection of mesenchymal precursor cells into the myocardium of patients undergoing a Left Ventricular Assist Device (LVAD) implant did not increase successful temporary weaning from device support.
2. Left ventricular ejection fractions (LVEFs) and quality of life scores were similar between groups.
Study Rundown: Implantation of Left Ventricular Assist Devices (LVADs) is a key surgical therapy for advanced heart failure patients unresponsive to medical therapy. Injection of mesenchymal precursor cells (MPCs) is a promising adjuvant therapy that may promote cardiac recovery through suppression of inflammatory cytokines that can incite infection, bleeding, and thrombosis. In this randomized controlled trial, injection of MPCs did not improve left ventricular recovery as determined by proportion of successful LVAD weans over 6 months. Similarly, left ventricular ejection fractions (LVEFs) and quality of life scores were similar between groups.
Injection of MPCs may not have been effective for this condition for a variety of reasons, with one such reason being that cell delivery through direct transepicardial injection may result in significant cell loss and low rates of retention. The study's conclusion is further limited by its enrollment of a wide spectrum of patients receiving 1 of 2 LVAD types, which may have increased variability in such a small trial.


Tuesday's REINSW Women in Real Estate conference featured one of the early voices of the #MeToo movement.
Nathan Vecchio, Mortgage Broker at Hunter Galloway's, goes through his six tips for increasing home equity today.
This Golden Mile residence is set on approx. 1.3 ha of park-like grounds with views reaching south across rural Victoria. For sale by Jock Langley and Emma Pierson of Abercromby's Armadale.
This brand new Mermaid Beach villa close to Nobbys Beach is for sale by Jordan Williams of JW Prestige Agents.
An approved development site on Brisbane's northside is for sale through Ray White Special Projects.
Behind the traditional period facade with showpiece full-width balcony, this Richmond terrace has had a warm and inviting modern transformation. For sale by Edward Hobbs and James Richer of Biggin & Scott Richmond.
With heritage accents and contemporary functionality this character-filled Carlton conversion is for sale by Rudy Van Der Berg and Sarah Case of RT Edgar Toorak.
Wishing to fill vacant space and increase customer traffic, many shopping center developers have been considering bringing in nonconventional tenants as an effective solution says Xuan Pham and Quynh Dang, from JLL Vietnam.
Serge Pun, NYDC CEO addresses topics raised about the New Yangon City a recent townhall meeting.
Queenstown Lakes District Council has granted Resource Consent for a hotel and related activities, including a wellness centre, restaurants, and conference and event facilities, at Waterfall Park.
Crown Group will open its third luxury serviced apartment hotel, SKYE Suites Green Square, mid-year as part of the iconic $575 million development Infinity by Crown Group.
The Fantauzzo Art Series Hotel at Howard Smith Wharves has appointed Shaun Fields as general manager of its brand new $100m Brisbane hotel, opening next Tuesday, 26 March 2019.