text:
\begin{document}

\title{On Fr\'echet-Hilbert Algebras}

\date{\today}

\author{M. M\u antoiu and R. Purice \footnote{
\textbf{Primary  46L65, Secundary 35S05, 46K15.}
\newline
\textbf{Key Words:}  Quantization, Hilbert algebra, Fr\'echet space, Moyal algebra.}

}
\date{\small}
\maketitle \vspace{-1cm}

\begin{abstract}
We consider Hilbert algebras with a supplementary Fr\'echet topology and get various extensions of the algebraic structure by using duality techniques.
In particular we obtain  optimal multiplier-type involutive algebras, which in applications are large enough to be of significant practical use. The setting covers many situations arising from quantization rules, as those involving square-integrable families of bounded operators
\end{abstract}

\bigskip

\bigskip
{\bf Address}

\medskip
Departamento de Matem\'aticas, Universidad de Chile,

Las Palmeras 3425, Casilla 653, Santiago, Chile

\emph{E-mail:} mantoiu@uchile.cl

\emph{E-mail:} Radu.Purice@imar.ro

\bigskip
\medskip
{\bf Acknowledgements:}
The authors have been supported by the Chilean Science Foundation {\it Fondecyt} under the Grant 1120300.
RP acknowledges the partial support of a grant of the Romanian National Authority for Scientific Research, CNCS-UEFISCDI, project number
PN-II-ID-PCE-2011-3-0131 and the hospitality of the Universidad de Chile where part of this work has been done.

\maketitle

\section*{Introduction}\label{intro}

Hilbert algebras \cite{Di1} (see also Definition \ref{hilbalg}) play an important role in the theory of von Neumann algebras, group representations and Tomita-Takesaki theory. Each Hilbert algebra defines canonically two semifinite von Neumann algebras, each one being the commutant of the other. Reciprocally, a von Neumann algebra endowed with a faithful normal semi-finite trace defines a Hilbert algebra. In physics they often arise in the context of quantization theory, being non-commutative deformations of some classical commutative structure. Having such physical applications in view, we deal with the problem of restricting and (especially) extending the algebraical information contained in a Hilbert algebra to different spaces. Our aim is to perform this in a model-independent way, with a minimal extra structure involved.

In many concrete situations, besides the Hilbert norm, there is a second stronger Fr\'echet topology compatible with the involutive algebra structure.
We codify the entire structure in the first section of the paper under the name {\it Fr\'echet-Hilbert algebra}. In the second section we show
that in such a setting the algebraic structure extends considerably by duality techniques. In particular, one gets naturally
an optimal multiplier-type algebra that we call {\it the Moyal algebra}. Such an object has been studied in connection with the Weyl pseudodifferential calculus \cite{An1,GBV1,GBV2}, starting with the natural algebraico-topological structure of the Schwartz space $\S(\mathbb R^n)$. An adaptation for the Gelfand-Shilov spaces is contained in \cite{So}\,. Moyal algebras for the magnetic pseudodifferential theory \cite{IMP} were introduced and used in \cite{MP1}.

As said before, the abstractization we propose here works under somewhat minimal assumptions and does not need the setting of spaces of functions or distributions. It also opens the way to some developments and applications that will be the subject of a forthcoming article, which will also contain complements on the relevant topologies on the Moyal algebras, a representation theory by operators in locally convex spaces and applications to the matrix-valued pseudodifferential theory on compact Lie groups initiated in \cite{RT1,RT2,RTW}.

In \cite{BBM} a very general form of a symbolic calculus has been introduced and studied. It is defined by a family of bounded Hilbert space operators
indexed by a space $\Si$ and having a property of square-integrability with respect to a measure $\mu$ on $\Si$. The main novelty is that no topology or group properties are involved in the development of this calculus. Besides the usual pseudodifferential theory on $\mathbb R^n$
\cite{Fo,GBV1,GBV2}, many topics are particular cases of this approach, as is shown in \cite{BBM}. This includes twisted convolution algebras associated to projective group representations, the magnetic pseudodifferential calculus \cite{IMP,KO,MP1,MP4}, Weyl operators on nilpotent groups \cite{BB1,BB4,BB5,Pe}, pseudodifferential operators on Abelian locally compact groups \cite{GS,Ha,Se,We} and others. In a final section we are going to review briefly the constructions of \cite{BBM}, putting them in the perspective of Fr\'echet-Hilbert algebras. In this way the formalism of Moyal algebras will become available to the examples covered in \cite{BBM}, which is a novelty for some of them.

\section{Fr\'echet-Hilbert algebras}\label{mapsymb}

\begin{defn}\label{hilbalg}
A Hilbert algebra {\rm is a $^*$-algebra $(\A,\#,^\#)$ endowed with a scalar product $\<\cdot,\cdot\>:\A\times\A\rightarrow\mathbb C$ such that
\begin{enumerate}
\item
one has $\,\<g^\#,f^\#\>=\<f,g\>,\ \forall\,f,g\in\A$,
\item
one has $\,\<f\#g,h\>=\<g,f^\#\# h\>,\ \forall\,f,g,h\in\A$,
\item
for all $f\in\A$, the map ${\sf L}_f:\A\rightarrow\A,\,{\sf L}_f(g):=f\#g$ is continuous,
\item
$\A\#\A$ is total in $\A$.
\end{enumerate}
A complete Hilbert algebra is called an} $H^*$-algebra.
\end{defn}

Clearly one also has
\begin{equation*}\label{treica}
\<f\#g,h\>=\<f,h\# g^\#\>,\ \quad\forall\,f,g,h\in\A
\end{equation*}
and the map ${\sf R}_f:\mathscr A\rightarrow\A,\,{\sf R}_f(g):=g\#f$
is also continuous; therefore $\A\times\A\overset{\#}{\rightarrow}\A$ is separately continuous.

The completion $\mathscr B$ of $\A$ is a Hilbert space but in general it is no longer an algebra.
But the mappings ${\sf L}_f$ and ${\sf R}_f$ do extend to elements of $\mathbb B(\B)$, defining nondegenerate commuting representations ${\sf L,R}:\A\rightarrow\mathbb B(\B)$.
We can regard ${\sf L}$ and ${\sf R}$ as giving separately continuous bilinear extensions
\begin{equation*}\label{alta}
\A\times\B\overset{\#}{\longrightarrow}\B,\quad{\rm and}\quad\B\times\A\overset{\#}{\longrightarrow}\B.
\end{equation*}

By taking weak closures in $\mathbb B(\mathscr B)$ one gets von Neumann algebras $\mathcal L(\A)$ and $\mathcal R(\A)$
which are the commutant of each other.

An element $f$ of $\mathscr B$ is called {\it bounded} if the mapping $\A\ni g\rightarrow f\# g\in\mathscr B$ is continuous
(or, equivalently, if the mapping $\A\ni g\rightarrow g\# f\in\mathscr B$ is continuous).
We denote by $\A^\flat\subset\mathscr B$ the space of all bounded elements;
it becomes naturally a Hilbert algebra containing $\A$ densely.
One defines \cite{Di1} on $\mathcal L(\A)$ (resp. $\mathcal R(\A)$)
{\it a normal faithful semifinite trace} $\tau_L$ (resp. $\tau_R$)
for whom the finite-trace operators correspond to the elements of ${\mathsf L}_{\A^\flat}$ (resp. ${\mathsf R}_{\A^\flat}$).

\begin{defn}\label{stricat}
A Fr\'echet $^*$-algebra {\rm is a $^*$-algebra $(\A,\#,^\#)$ with a Fr\'echet locally convex space topology
$\mathscr T$ such that the involution
$$
\A\ni f\rightarrow f^\#\in \A
$$
is continuous and the product
$$
\A\times\A\ni(f,g)\rightarrow f\# g\in\A
$$
is separately continuous.}
\end{defn}

\begin{rem}\label{mishto}
{\rm By Theorem 41.2 in \cite{Tr}, separate continuity of the map $\#$ implies hypocontinuity. This means that for any bounded subset $A$
of $\A$ the families of linear maps
$$
\{\A\ni g\mapsto f\# g\in\A\mid f\in A\}\quad{\rm and}\quad\{\A\ni g\mapsto g\# f\in\A\mid f\in A\}
$$
are equicontinuous.
}
\end{rem}

On the dual $\A^\dagger$ one can consider various topologies
$\mathcal T_\nu$ which are stronger than the weak$^*$-topology $\mathcal T_\sigma$ but weaker than the strong topology $\mathcal T_\beta$.
Such a topology $\mathcal T_\nu$ will be called {\it admissible} if it has as a basis of neighborhoods of the origin the polars
of a family $\mathcal B_\nu$ of bounded subsets of $\A$ satisfying the following:
\begin{enumerate}
\item
if $A,B\in\mathcal B_{\nu}$, there exists $C\in\mathcal B_\nu$ such that $A\cup B\subset C$,
\item
if $A\in\mathcal B_\nu$ and $\alpha\in\mathbb C$, then $\alpha A\subset B$ for some $B\in\mathcal B_\nu$,
\item
if $A\in\mathcal B_\nu$ and $f\in \A$ then $f\# A\in\mathcal B_\nu$ and $A\# f\in\mathcal B_\nu$.
\end{enumerate}

Let us write $\A^\dag_\nu$ for the dual $\A^\dag$ when considered with the
topology $\mathcal T_\nu$. A net $\{F_\lambda\}_{\lambda\in\Lambda}$ converges to $0$ in $\A_\nu^\dag$ if and only if $F_\lambda(g)$ converges to $0$ uniformly in $g\in A$ for every $A\in\mathcal B_\nu$. Both the weak$^*$-topology $\mathcal T_\sigma$ and the strong topology
$\mathcal T_\beta$ are admissible topologies; for this one takes $\mathcal B_{\sigma}$ the family of all finite subsets of $\A$
and $\mathcal B_\beta$ the family of all bounded subsets of $\A$. Another interesting example is the family $\mathcal B_\gamma$
of all compact subsets of $\A$, leading to the topology $\mathcal T_\gamma$ of convergence which is uniform on compact sets.
By \cite[Th. 33.1]{Tr}, the bounded subsets of $\A^\dag_\nu$ are the same for all admissible topologies $\mathcal T_\nu$ (and are exactly the equicontinuous subsets).

\begin{rem}\label{pripri}
{\rm Weak duals of infinite-dimensional Fr\'echet spaces are definitely not complete. But it is known that they are quasi-complete.
On the other hand, the strong dual $\A^\dag_\beta$ (the dual $\A^\dag$ endowed with the strong topology $\mathcal T_\beta$ described above) of any Fr\'echet space is complete.
}
\end{rem}

We introduce now our main mathematical object.

\begin{defn}\label{tot}
A Fr\'echet-Hilbert algebra $(\A,\#,^\#,\mathscr T,\<\cdot,\cdot\>)$ {\rm is both a Fr\'echet $^*$-algebra and a Hilbert algebra,
the topology $\mathscr T$ being assumed finer that the norm topology associated to the scalar product.}
\end{defn}

As before, we denote by $\B$ the Hilbert space completion of $\A$; by the Riesz Lemma it is identified with its strong dual $\B^\dagger$.
We also denote by $\iota$ the canonical inclusion of $(\A,\mathscr T)$ into $(\B,\parallel\cdot\parallel)$; it is continuous and has dense range.
For each admissible topology $\mathcal T_\nu$ on the dual, using Riesz' identification, one gets a continuous linear injection
$\iota^\dagger:(\mathscr B,\parallel\cdot\parallel)\rightarrow(\A^\dagger,\mathcal T_\nu)$;
thus any Fr\'echet-Hilbert algebra $\A$ generates {\it a Gelfand triple} $(\A,\B,\A^\dagger_\nu)$.
Actually we are going to treat $(\A,\B,\A^\dagger)$ as the Gelfand triple and specify the topology $\mathcal T_\nu$ when needed. The duality between $\A$ and $\A^\dag$ will be denoted by $\<\cdot,\cdot\>$ because it is consistent with the scalar product of $\B$.

\begin{rem}\label{pripri}
{\rm Note that $\B$ (and hence $\A$) is dense in $\A^\dag_\sigma$ \cite[Prop. 35.4]{Tr}.

}
\end{rem}

\section{Extensions of the product and Moyal algebras}\label{injurac}

Let us fix a Fr\'echet-Hilbert algebra $(\A,\#,^\#,\mathscr T,\<\cdot,\cdot\>)$ and an admissible topology $\mathcal T_\nu$ on the topological dual $\mathscr A^\dagger$, given by a family $\mathcal B_\nu$ of bounded subsets of $\A$ as above.

\begin{prop}\label{fosforos}
\begin{enumerate}
\item
The composition law $\#$ extends to bilinear separately continuous mappings
$\#:\A\times\A^\dagger_\nu\rightarrow\A^\dagger_\nu$ and $\#:\A^\dagger_\nu\times\A\rightarrow\A^\dagger_\nu$.
\item
For any $f\in\A^\dagger$ and $g,h\in\A$ one has
\begin{equation*}\label{urce}
f\#(g\# h)=(f\# g)\#h,
\end{equation*}
\begin{equation*}\label{urcu}
h\#(g\# f)=(h\# g)\#f,
\end{equation*}
\begin{equation*}\label{urca}
(g\# f)\# h=g\#(f\# h).
\end{equation*}
\item
The involution $^\#$ extends to a topological anti-linear isomorphism $^\#:\A_\nu^\dag\rightarrow\A_\nu^\dag$ which is an involution,
such that for every $f\in\A^\dag$ and $g\in\A$ one has
\begin{equation*}\label{stuchid}
(f\# g)^\#=g^\#\# f^\#,\quad\ (g\# f)^\#=f^\#\# g^\#.
\end{equation*}
\end{enumerate}
\end{prop}

\begin{proof}
1. We are going to justify only the second extension; the first one follows in the same way. For $f\in\A^\dag$ and $g,h\in\A$ one sets
\begin{equation*}\label{diul}
\<f\# g,h\>:=\<f,h\# g^\#\>.
\end{equation*}
Clearly this defines an element $f\# g$ of the topological dual of $\A$, which coincides with the one given by the Hilbert algebra structure of $\A$ if $f\in\A$.

We still have to show separate continuity.

First fix $g\in\A$ and pick a net $\{f_\lambda\}_{\lambda\in\Lambda}$ converging to $0$ in $\A^\dag_\nu$.
This means that for every $B\in\mathcal B_\nu$ one has $\<f_\lambda,k\>\rightarrow 0$ uniformly in $k\in B$.
For any $A\in\mathcal B_\nu$, since $B:=A\#g^\#\in\mathcal B_\nu$, one has
$$
\<f_\lambda\# g,h\>=\<f_\lambda,h\# g^\#\>\rightarrow 0\ \,{\rm uniformly\ in}\ h\in A,
$$
thus $f_\lambda\# g\rightarrow 0$ in $\A^\dag_\nu$.

Now fix $f\in\A^\dag$ and assume that $g_\lambda\rightarrow 0$ in $\A$.
For any $A\in\mathcal B_\nu$ it follows from Remark \ref{mishto} that $h\# g_\lambda^\#\rightarrow 0$ uniformly in $h\in A$. Then
$$
\< f\# g_\lambda,h\>=\<f,h\# g^\#_\lambda\>\rightarrow 0\ \,{\rm uniformly\ in}\ h\in A,
$$
thus $f\# g_\lambda\rightarrow 0$ in $\A^\dag_\nu$ and we are done.

\medskip
2. Recall that $\A$ is dense in $\A^\dag_\sigma$. Then the associativity properties  follow easily
 by approximation from the associativity of the composition law $\#$ in $\A$ and from the continuity property we have just proved, with $\nu=\sigma$.

\medskip
3. The extension of $^\#$ is defined by transposition: if $f\in\A_\nu^\dag$ one sets
$$
\<f^\#,h\>:=\overline{\<f,h^\#\>},\quad\ \forall\,h\in\A.
$$
It follows immediately that it is an involution. It is a topological isomorphism for any $\nu$ because the family $\mathcal B_\nu$ is stable under the involution of $\mathscr A$. For $f\in\A^\dag$ and $g,h\in\A$ we compute using the definition of the involution and the axioms of a Hilbert algebra
$$
\begin{aligned}
\<(f\# g)^\#,h\>&=\overline{\<f\# g,h^\#\>}=\overline{\<f,h^\#\# g^\#\>}=\overline{\<(h^\#\# g^\#)^\#,f^\#\>}\\
&=\overline{\<g\# h,f^\#\>}=\<f^\#,g\#h\>=\<g^\#\# f^\#,h\>,
\end{aligned}
$$
so the first equality is proven. The second follows similarly.
\end{proof}

\begin{defn}
{\rm Let $\mathscr A$ be a Fr\'echet-Hilbert algebra with topological dual $\A^\dagger$\,. One introduces}
\begin{enumerate}
\item {\it the right Moyal algebra}
$\,\M_R:=\left\{f\in\A^\dagger\,\mid\, \A\# f\subset \A\right\}$,

\item {\it The left Moyal algebra} $\,\M_L:=\left\{f\in\A^\dagger\,\mid\, f\#\,\A\subset \A\right\}$,
\item {\it The (bi-sided) Moyal algebra}

$\,\M:=\M_R\cap \M_L=\left\{f\in\A^\dagger\,\mid\, f\#\,\A\subset \A\supset\A\# f\right\}$.
\end{enumerate}
\end{defn}

\noindent
Thus we have bilinear maps
\begin{equation*}\label{oaial}
\A\times\M_R\overset{\#}{\longrightarrow}\A,\quad\ \M_L\times\A\overset{\#}{\longrightarrow}\A.
\end{equation*}
If $f\in\A,\,g\in\M_L$ and $h\in\M_R$, by using Proposition \ref{fosforos} and the definitions we check immediatly that the identity $(g\# f)\# h=g\#(f\# h)$ holds in $\A$. One also shows easily that $\M_R^\#=\M_L$ and $\M_L^\#=\M_R$, implying that
$\M^\#=\M$.

\begin{prop}\label{ptak}
One gets bilinear extensions
\begin{equation*}\label{bil}
\M_R\times\A^\dag\overset{\#}{\longrightarrow}\A^\dag,\ \quad\A^\dag\times\M_L\overset{\#}{\longrightarrow}\A^\dag.
\end{equation*}
If $f\in\A^\dag,\,g\in\M_R$ and $h\in\M_L$ we have
$$
(g\# f)\# h=g\#(f\# h).
$$
\end{prop}

\begin{proof}
We treat the first one. For $f\in\M_R$ and $g\in\A^\dag$ define $f\# g\in\A^\dag$ by
\begin{equation*}\label{incerc}
\<f\# g,h\>:=\<g,f^\#\# h\>,\quad\ \forall\,h\in\A.
\end{equation*}
The right-hand side is clearly linear in $h$, but we also need to justify continuity.
By a simple application of the Closed Graph Theorem, if $k\in\M_L$ then $\A\ni h\mapsto k\# h\in\A$ is continuous.
Taking $k=f^\#$ and recalling that $g$ is a continuous functional, we conclude that $f\# g\in\A^\dag$ indeed.

Associativity follows if we apply the definitions and the associativity properties already obtained.
\end{proof}

We gather the basic properties of $\M_L,\M_R$ and $\M$ in the following statement.

\begin{prop}\label{complecar}
\begin{enumerate}
\item
Both $\M_L$ and $\M_R$ are algebras.
\item
The involution $^\#$ on $\A^\dagger$ restricts to reciprocal anti-linear isomorphisms $^\#:\M_L\rightarrow\M_R$ and $\,^\#:\M_R\rightarrow\M_L$ satisfying
$$
(f\# g)^\#=g^\#\# f^\#,\ \ \ \ \ \forall f,g\in \M_L \ (or\ \forall f,g\in \M_R).
$$
\item
$\M$ is an involutive algebra.
\item
One has $\A\#\M_R\subset\A$ and $\M_L\#\A\subset\A$\,, thus $\A$ a self-adjoint bi-sided ideal in the $^*$-algebra $\M$.
\end{enumerate}
\end{prop}

\begin{proof}
1. We treat $\M_L$. Let $f,g\in\M_L$ and $h\in\A$; by Proposition \ref{ptak} we need to show that $(f\# g)\# h\in\A$. This follows from the definition of
$\M_L$ if the equality $(f\# g)\# h=f\#(g\# h)$ is established. We are going to get it working weakly on any $k\in\A$:
$$
\begin{aligned}
\<(f\# g)\# h,k\>&=\<f\# g,k\#h^\#\>=\<f,(k\# h^\#)\# g^\#\>\\
&=\<f,k\#(h^\#\# g^\#)\>=\<f,k\#(g\# h)^\#\>\\
&=\<f\#(g\# h),k\>.
\end{aligned}
$$
During the computation we used an associativity relation that is already known.

There is still some associativity to prove, but we leave it to the reader.

\medskip
2. This has been mentioned above and is easy.

\medskip
3 follows from 1 and 2, since $\M$ is the intersection of $\M_L$ and $\M_R$.

\medskip
4 is obvious.
\end{proof}

\begin{rem}\label{prip}
{\rm By inspection it can be shown that the locally convex space $(\A,\mathscr T)$ only needs to be barrelled and $B_r$-convex \cite{Sch}, the main issue being the validity of the Closed Graph Theorem. This extra generality, needed to cover Gelfand-Shilov spaces in connection with the Weyl calculus \cite{So}, will not be considered here
}
\end{rem}

\begin{rem}\label{pripi}
{\rm In \cite{GBV1,MP1,So}, studying the particular case of the (magnetic) Weyl composition on Schwartz or Gelfand Shilov spaces, it is shown that the Moyal algebras are much larger than the initial $\A$. In the Weyl quantization, many symbols from the Moyal algebra are turned into unbounded operators on $L^2$.
}
\end{rem}

\begin{rem}\label{pipri}
{\rm However the Hilbert space $\B$ and the Moyal algebras are not comparable in general. For the Weyl pseudodifferential calculus with $\A=\S(\mathbb R^{2n})$ one has $\B=L^2(\mathbb R^{2n})$ and $\A^\dagger$ is the space of tempered distributions. In the Schr\"odinger representation, the Weyl correspondence maps isomorphically $\B$ to the ideal of Hilbert-Schmidt operators in $L^2(\mathbb R^n)$, while $\M_L$
corresponds to linear continuous operators in $\S(\mathbb R^n)$ and $\M_R$ corresponds to linear continuous operators in $\S'(\mathbb R^n)$.
Then clearly no inclusion is available (think of rank-one operators $|u\rangle\langle v|$ with various types of vectors $u,v$).
}
\end{rem}

\begin{ex}\label{rice}
{\rm Many examples of Fr\'echet-Hilbert algebras are constructed from the Gelfand triple
$\left(\A=\mathcal S(\mathbb R^m),\,\mathscr B=L^2(\mathbb R^m),\,\A^\dag=\mathcal S'(\mathbb R^m)\right)$,
where $\mathcal S(\mathbb R^m)$ is the Schwartz space endowed both with the $L^2$ scalar product and with its standard Fr\'echet topology
and $\mathcal S'(\mathbb R^m)$ is the space of all tempered distributions on $\mathbb R^m$. The simplest non-trivial situation
is $f\# g=fg$ (pointwise product) and $f^\#=\overline{f}$ (complex conjugation). In this case one has
$\A^\flat=L^2(\mathbb R^m)\cap L^\infty(\mathbb R^m)$, $\mathcal L(\A)=\mathcal R(\A)\cong L^\infty(\mathbb R^m)$
and $\mathscr M_L=\mathscr M_R=\mathscr M=C^\infty_{{\rm pol}}(\mathbb R^m)$
($C^\infty$ functions with polynomially bounded derivatives).
One sees from this example that although $\mathscr M$ can be quite big (containing all the H\"ormander classes of symbols \cite{Fo}), it does not contain the Hilbert space $\mathscr B$ or at least
$\A^\flat$. It is also clear that $\A\#\A^\dag$ is not contained in any of the Moyal algebras.}
\end{ex}

\section{Fr\'echet-Hilbert algebras associated to square-integrable families of bounded operators}\label{endix}

We are given a measure space $(\Si,\mu)$ (as a locally compact topological space endowed with a Radon measure, for instance). Let $\{\pi(s)\!\mid\! s\in\Si\}\subset\mathbb B(\H)$ be a family of bounded operators in the separable complex Hilbert space $\H$. We assume that $s\to\pi(s)$ is weakly measurable and satisfies the condition
$$
\int_\Si\! d\mu(s)\,|\<\pi(s)u,v\>_\H|^2=\,\parallel\!u\!\parallel^2\,\parallel\!v\!\parallel^2,\quad\forall\,u,v\in\H.
$$
The space $\Si$ is not a group and we don't know anything on the products $\pi(s)\pi(t)$. We are going to describe some constructions from \cite{BBM}, to which we send for proofs, technical details and further information.

Let us set $\Phi_{u\otimes v}(\cdot):=\<\pi(\cdot)u,v\>$; this defines an isomorphism from $\H\otimes\overline\H$ to a closed subspace $\mathscr B^2(\Si)$ of $L^2(\Si)$ (we set $\overline\H$ for the opposite of the Hilbert space $\H$ and $\otimes$ for the Hilbert space tensor product). In many (but not all) particular cases one has $\mathscr B^2(\Si)=L^2(\Si)$. Then a correspondence $\Pi:\mathscr B^2(\Si)\rightarrow \mathbb B(\H)$ is defined essentially by
$$
\Pi(f):=\int_\Si\!d\mu(s)f(s)\pi(s)^*.
$$
This correspondence sends isomorphically $\mathscr B^2(\Si)$ in $\mathbb B_2(\H)$ (the ideal of all Hilbert-Schmidt operators on $\H$).
Actually one has
$$
\<\Pi(f),\Pi(g)\>_{\mathbb B^2(\H)}:={\rm Tr}[\Pi(f)\Pi(g)^*]=\int_\Si\!d\mu(s)f(s)\overline{g(s)}=:\<f,g\>_{L^2(\Si)}.
$$
The mapping $\Pi$ is uniquely defined by the relation $\Pi(\Phi_{u\otimes v})=\<\cdot,v\>u$, specifying the way we obtain the rank one operators.
Being a closed subspace of $L^2(\Si)$, the set of "symbols" $\mathscr B^2(\Si)$ is a Hilbert space. We complete this structure to a $H^*$-algebra (Def. \ref{hilbalg}) $\(\mathscr B^2(\Si),\#,^\#,\<\cdot,\cdot\>_{L^2(\Si)}\)$
by transporting the $^*$-algebra structure of $\mathbb B(\H)$. So the product is defined by
$f\#g:=\Pi^{-1}[\Pi(f)\Pi(g)]$ and the involution by $f^\#:=\Pi^{-1}[\Pi(f)^*]$.
More explicit expressions are deduced in \cite{BBM} but they are not needed here.

Suppose now given a Fr\'echet space $\G$ continuously and densely embedded in the Hilbert space $\H$. This is very common in applications, where $\H$ might be a $L^2$-space, $\G$ could be a space of more regular functions, with a natural Fr\'echet topology, and the topological dual $\G^\dagger$ is some space of distributions. No compatibility between $\G$ and the basic family $\{\pi(s)\!\mid\! s\in\Si\}$ is needed. Let us denote by $\G\widehat\otimes\overline\G$ the completed projective tensor product and define $\mathscr G(\Si):=\Phi(\G\widehat\otimes\overline\G)$ with the  topology transported by $\Phi$ from $\G\widehat\otimes\overline\G$. It is shown in \cite{BBM} that $\mathscr G(\Si)$ is a Fr\'echet-Hilbert algebra; thus we can take $\A=\mathscr G(\Si)$ and $\mathscr B=\mathscr B^2(\Si)$ in the preceding sections.

\begin{rem}\label{puzau}
{\rm Actually the Fr\'echet-Hilbert algebra described above admits a representation in the Gelfand triple $(\mathcal G,\mathcal H,\mathcal G^\dag)$. By definition, such a representation is an isomorphism $\Pi:\A^\dag\rightarrow\mathbb B(\mathcal G,\mathcal G^\dag)$ which restricts to an isomorphism $\Pi:\A\rightarrow\mathbb B(\mathcal G^\dag,\mathcal G)$ and satisfies
$\Pi(f\# g)=\Pi(f)\Pi(g)$ and $\Pi(f)^*=\Pi(f^\#)$ for all $f,g\in\A$.
In such a setting one shows easily that $\A\#\A^\dag\#\A\subset\A$, which implies immediately that $\A\#\A^\dag\subset\mathscr M_L$ and $\A^\dag\#\A\subset\mathscr M_R$. As noticed in Example \ref{rice}, this does not hold for all Fr\'echet-Hilbert algebras.
}
\end{rem}

Concrete examples are given in \cite{BBM}, along the lines described in the Introduction; it can be seen that, although $\mathscr G(\Si)$ is defined indirectly, in applications it coincides with some known useful Fr\'echet space. However its $^*$-algebra structure can be highly non-trivial. Our approach makes available the Moyal algebra formalism for all these situations.


TITLE: Question about a function in the argument of a Delta function (distribution)
QUESTION [0 upvotes]: referring to
https://physics.stackexchange.com/questions/160503/what-is-the-wave-propagated-away-from-an-impulsively-excited-spherical-shell
my question concerns the last part of the answer which I quote in the following:
"
$$\frac{R}{2z_o} \int_{-R}^{R}  \delta\left(z_s-\frac{R^2+z_o^2 -c^2t^2}{2z_o}\right) d z_s$$
Now $\int_{-R}^{R}  \delta\left(z_s-\frac{R^2+z_o^2 -c^2t^2}{2z_o}\right) d z_s$ is either one or zero. We know from the way we picked the delta function that the integral is one when $|z_o−R|<ct<z_o+R$ and the integral is zero otherwise. Therefore
$$f(z_o,t)=\begin{cases}
\frac{R}{2 z_o}, & |z_o-R|<ct<z_o+R \\
0, & \textrm{otherwise.}
\end{cases}$$
"
( where $f(z_0,t)$ is the result of the integral. )
My question is:
Is there a way to 'analytically' deduce this from the argument of the delta distribution?
The existing answer's method does not seem totally 'mathematical' otherwise I think the answer may be correct.   Refer to the link for the complete question and answer.  The following figure also may help.

REPLY [1 votes]: Yes. All of the mass is concentrated on $z_s = \frac{R^2+z_0^2-c^2 t^2}{2z_0}$. Since this quantity is fixed, there is only one such point $z_s$ in the integration range.
Such a value $z_s$ must lie in $[-R,R]$, hence $-2Rz_0 < R^2 +z_0^2 -c^2 t^2 < 2R z_0$ thus you can see that the inequality follows from this.


TITLE: Negative friction force, positive normal force
QUESTION [1 upvotes]: I realize that formulas don't always tell the whole story, but there seems to be some information missing from the equation: $F_f=\mu F_N$
Imagine we say up and right is positive. Then if an object is moving across a surface to the right, the friction force should be negative (acting against the positive applied force), and the normal force should be positive (acting against the gravitational force).
Let's say: $F_N=882\ \text{N}$ and $\mu=0.600$, then: $F_f=0.600\times822=529\ \text{N}$, which is positive, when it should be negative.
I realize that this formula converts between axes, so there can't be a reliable way to relate the signs, but then what is the convention? Does this formula just provide an absolute value for friction, and you can set it positive or negative, depending on the situation?

REPLY [0 votes]: No, the minus sign is not used to show direction.
A vector is just a direction and magnitude and cannot be negative. But it may be subtracted from another vector. This minus sign does not show direction but just a mathematical procedure.
But it turns out that subtracting a vector $\vec A$ form another vector $\vec B$: $\vec A-\vec B$ is the same as adding the vector $\vec A$ to the reversed vector $\vec B$: $\vec A+(-1)*\vec B=\vec A+\vec B_{reverse}$
Thus is fundamental (and pretty obvious), but this is where the issue starts. The minus sign is not defining a direction of the vector, but if it appears in our math it cab be interpreted to flip a vector around.
In general in physics, especially with forces: If you ever happen to get a minus sign in a result of the magnitude of a force, it just means that the vector is flipped around compared to whatever direction you defined to begin with (if that direction somehow is included in the calculation). If you do not get a minus sign, it doesn't mean anything at all about direction.


TITLE: Almost disjoint set (finite case)
QUESTION [3 upvotes]: I'm interested in the following:
Given a set $S_{n,k}$ of binary sequences of length $n$ with $k$ many 1-entries, what is the maximal size of a subset $S'_{k'}\subset S_{n,k}$ such that for every subset $T\subset S'_{k'}$ of size $k'$ there is no set of $k'$ many indices on which all elements of $T$ are $1$?
It seems to be quite a natural question to me, so i suppose there are some results. Unfortunately, I didn't find much on finite almost disjoint sets.
Thanks in advance!

REPLY [1 votes]: So recast the problem as:
$Q(n,k,m,t):$  Given the family  of all $k$ element subsets of $\{{1,\cdots,n\}}$, how large can a sub-family $\mathcal{F}$ be if we require that no  $m$ of them have an intersection of size $t$?
Then your question is $Q(n,k,k',k')$. A very rough bound is that at the absolute largest we could imagine that every subset of of size $t$ is in $m-1$ members of $F$. If $N$ is the size of $F$ then counting in two ways all ways to select  a member of $\mathcal{F}$ along with a further subset of size $t$ we have that $$N\binom{k}{t} \le (m-1)\binom{n}{t}$$ so $$N \le \frac{(m-1)\binom{n}{t}}{\binom{k}{t}} \tag{*}$$
It might be the case that that is not as crude as one might think in the sense that one can prove that there is for fixed $k,m,s$ always a family of size $N$ (rounded down to an integer) provided that $n$ is larger than some $n(k,m,s)$. However that could be a result of the form "there is at least one way to pick the family $\mathcal{F}$ which works" without any real guidance on how to get an actual specimen (nor an assurance that there are any examples with a compact description).
To get equality in $(*)$ requires the right hand side to be an integer (of course). Then $\mathcal{F}$ is called a $t$-design (with $\lambda=m-1$) and a Steiner system when $m=2$. A very exciting recent paper (link and description here) claims that this integrality condition is sufficient for existence, provided that $n$ is large enough.


TITLE: Smooth homotopy of proper maps
QUESTION [4 upvotes]: Suppose $M$ is a manifold. We call a map $F:M \rightarrow \mathbb{R}^m$ proper if for every compact subset $K \subset \mathbb{R}^m$ we have that $F^{-1}(K) \subset M$  is compact.
Suppose now that $F: M \times [0,T) \rightarrow \mathbb{R}^m$ is a smooth family of immersions such that $F(\cdot ,0)$ is proper. Is it true that $\forall t \in [0,T): F(\cdot, t)$ is proper?

REPLY [3 votes]: This is false as stated. For an (analytic) counterexample, take $M = \mathbb{R}$ and define $F:\mathbb{R} \times [0,\pi] \to \mathbb{R}$ via
$$F(x,t):= (1-\sin(t))x + \sin(t) \arctan(x).$$
For any fixed $t\in [0,\pi]$, $F(\cdot,t)$ is an immersion because the maximum slope of $\arctan$ is $1$. Clearly $F(\cdot,0)$ is proper. But $F(\cdot,\pi/2) = \arctan$, which is bounded and therefore not proper.
The conjecture is still false if $m > \text{dim}(M)$. As a counterexample, take $M= \mathbb{R}$ again, let $F$ be as above and define the map $G: \mathbb{R}\times [0,\pi]\to\mathbb{R}^m$ via
$$ G(x,t):= (F(x,t),\ldots,F(x,t)).$$
$G(\cdot,t)$ is an immersion for any $t \in [0,\pi]$ because $F(\cdot,t)$ is, and $G(\cdot,0)$ is proper, but $G(\cdot,\pi/2)$ is not proper because it is given by the product of $\arctan$ functions.


TITLE: If you take 200 rods of plutonium what would happen due to it’s half-life?
QUESTION [1 upvotes]: If you take 200 rods of plutonium then stick them together would 100 full rods worth of plutonium be remaining? And after half-life #2 here would be 50 rods worth of plutonium left and then after half-life #3 would there be on 25 rods of plutonium rods left and etc.?

REPLY [2 votes]: The half life doesn't mean that if you start with two rods after the half life one rod has disappeared. After the half life half the plutonium atoms originally present have disappeared, and which atoms disappear is random.
So let's say each rod has $N$ plutonium atoms in it, where we can calculate the number $N$ by dividing the mass of the rod by the mass of a plutonium atom. Then after one half life we still have two rods but now each rod now contains only $N/2$ plutonium atoms and a whole lot of fission products.
In practice the decay of so many of the atoms in the rods would probably make the rods crumble into dust. So after one half life you end up with two piles of dust - each pile containing $N/2$ plutonium atoms. After two half lives each pile would contain $N/4$ plutonium atoms, and so on.

REPLY [2 votes]: Plutonium has several different isotopes, with a wide range of half-lives. So if your rods aren't composed of a single isotope they don't have a well-defined half-life. But if the rods are all composed of a single isotope then they will decay as you describe.
However, if the rods are a fissile isotope (Pu-239 or Pu-241) sticking them together will increase the rate that fission occurs, since the neutrons released by spontaneous fission will have a higher probability of colliding with a Pu nucleus and inducing it to fission. If the bundle of rods is large enough, this chain reaction will rapidly become explosive.

REPLY [1 votes]: If it is plutonium 239, you most probably will have a melt down  the minute you stick them together, because  of the critical mass of plutoniumPu-239

Of all the common nuclear fuels, Pu-239 has the smallest critical mass. A spherical untamped critical mass is about 11 kg (24.2 lbs),3 10.2 cm (4") in diameter.

11kg/200 means each rod should be smaller than 55 grams, so one should make sure of this before "sticking them together".

Plutonium weighs 19.86 gram per cubic centimeter,

the critical mass limit says the rods  should be of pencil dimensions.
For other isotopes  or for Pu239   below the above limit, John's answer applies.


TITLE: Compactness of Set of Probability Measures
QUESTION [2 upvotes]: I'm currently studying Information Theory from Csiszar and Korner, "Information Theory: Coding theorems for Discrete Memoryless Sources". There are several questions pertaining to the set of all probability measures on the underlying space. I figured if I prove the following lemma, it will make things easier:
Let $\mathcal{X}$ be a finite set. Let $\mathcal{P}=\mathcal{P}(\mathcal{X})$ be the set of all probability measures that can be defined on $\mathcal{X}$. Under the total variation metric i.e. for any two measures $P,Q \in \mathcal{P}$
$$d_{TV}(P,Q) := \sum_{x \in \mathcal{X}}|P(x)-Q(x)|$$
Prove that $(\mathcal{P},d_{TV})$ is compact.
I'm actually looking for a simple proof of this. For instance, I wanted to use the result that a metric space is compact iff it is complete and totally bounded. Completeness was easy as I could just use the completeness of $\mathbb{R}$ in a pointwise fashion and show the pointwise limit is a measure. Finiteness of $\mathcal{X}$ helps greatly here.
So all that is left is Totally Boundedness. Unfortunately, while I did start by assuming an $\epsilon$-cover of $\mathcal{P}$, I could not find a way to pick finitely many that would cover the whole space. I'm sure that finiteness of $\mathcal{X}$ would help here as well.
The other idea, in fact the property I wanted to use, was to show sequential compactness since this is equivalent to compactness. But even here I'm not sure how to go about things.
Kindly help me in this regard in the form of subtle hints and tips. If I get it, I'll post my solution and credit the most helpful hint.
Update: It turns out I could use the fact that $\mathcal{P}$ is convex. Since any measure is a convex combination of degenerate distributions... Could I use this I wonder?

REPLY [2 votes]: When $X$ is finite, you can treat $\mathcal P$ as a subset of $\Bbb R^X$. Total variation is the norm there, and all the norms over finite-dimensional spaces are equivalent, so the induced metric is equivalent to the Eucledian one. Embedded $\mathcal P$ is bounded and closed in Eucledian metric, hence compact.


text:
\chapter{Microlocal analysis of generalized pullbacks of Colombeau functions}

This chapter is based on the article \cite{HalSim:07}.

The pullback of a general distribution by a $C^{\infty}$-function in classical distribution theory,
as defined in \cite[Theorem 8.2.4]{Hoermander:V1}, exists if the normal bundle of the $C^{\infty}$-function
intersected with the wave front set of the distribution is empty.
These microlocal restrictions reflect also the well-known fact that in general distribution theory one cannot carry out multiplications unrestrictedly,
since the product of two distributions can formally be written as the pullback of a tensor product of the two factors by the diagonal map $\delta:x \mapsto (x,x)$.

Generalized functions in the sense of Colombeau extend distribution theory in a way that it becomes a differential algebra with a
product that preserves the classical product $\cdot: C^{\infty} \times C^{\infty} \rightarrow C^{\infty}$.
In addition \cite[Proposition 1.2.8]{GKOS:01} states that the Colombeau algebra of generalized functions allows the definition of
a pullback by any c-bounded generalized function.
The classical concept of a wave front set has been extended to generalized functions of Colombeau type in \cite{DPS:98,Hoermann:99,NPS:98}.

In this Chapter we investigate how the generalized wave front set of a Colombeau function $u$ transforms under the
pullback by a c-bounded map $f$. Our main theorem is a result corresponding to the classical Theorem \cite[Theorem 8.2.4]{Hoermander:V1}.

\section{Transformation of wave front sets}\label{sectionwfset}

In order to obtain a microlocal inclusion (corresponding to \cite[eq. 8.2.4]{Hoermander:V1}) for generalized pullbacks,
we need the notion of a generalized normal bundle and find a method to transform wave front set under
c-bounded generalized maps.

In this section we consider a c-bounded generalized map $f\in \colmap{\Omega_1}{\Omega_2}$,
where $\Omega_1 \subseteq \mathbb{R}^n,\ \Omega_2 \subseteq \mathbb{R}^m$ are open sets and assume that
$\Gamma \subseteq \Omega_2 \times S^{m-1}$ is a closed set.

Although the pullback of a Colombeau function $u$ by any c-bounded generalized map $f$ is well-defined,
we cannot derive a general microlocal inclusion
for the pullback without requiring further properties for the generalized map $f$.
We define an open subdomain $\domainset{f}$ of $\Omega_1 \times S^{m-1}$,
where the generalized map $(x,\eta) \mapsto {}^T\! df_{\varepsilon}(x) \eta$ has certain properties
which are needed to obtain a microlocal inclusion relation. This leads to the notion of a generalized normal bundle.

\begin{definition} \label{unfavdef}
Let $f \in\colmap{\Omega_1}{\Omega_2}$ be a c-bounded generalized map, then
we define the open set $\domainset{f}$ by
\begin{equation*} \begin{split}
\domainset{f} := &\{ (x,\eta) \in \Omega_1 \times S^{m-1} \mid \exists\  \rm{\ neighborhood\ }X \times V \subseteq \Omega_1 \times S^{m-1} \rm{\ of\ } (x,\eta) \\
&\rm{\ and\ a\ positive\ net\ of\ slow\ scale\ }
(\sigma_{\varepsilon})_{\varepsilon}, \exists \alpha, \beta\in ]0,\infty[,\exists \varepsilon'\in ]0,1]:\\
&\inf_{(x,\eta)\in X\times V}
   |\sigma_{\varepsilon}  {}^T\! df_{\varepsilon}(x) \eta | \ge \alpha  \ \rm{and}\\
&\sup_{(x,\eta) \in X\times V^{\perp}} |\sigma_{\varepsilon}  {}^T\! df_{\varepsilon}(x) \eta | \le \beta \rm{\ for\ all\ } \varepsilon < \varepsilon'  \},
\end{split}
\end{equation*}
where $V^{\perp}:= \{\eta \in S^{m-1} \mid \exists \eta_0 \in V: \langle \eta, \eta_0 \rangle = 0 \}$.
Then the \emph{generalized normal bundle} \index{normal bundle!generalized} of $f$ is defined by
\begin{equation*}
\gnormalb{f} := \{(y,\eta) \in \Omega_2 \times S^{m-1} \mid (x,y) \in \ggraph{f}, (x,\eta) \not \in \domainset{f} \}.
\end{equation*}
This is in correspondence to the classical normal bundle.

Furthermore, we define the \emph{wave front unfavorable support} \index{wave front unfavorable support} of
$f$ with respect to a closed set $\Gamma\subseteq \Omega_2 \times S^{m-1}$ by
\begin{eqnarray*}
\unfavsupp{f}{\Gamma}:= \{ x\in \mathbb{R}^n \mid  (x,y) \in \ggraph{f}, (x,\eta) \not \in \domainset{f},
(y,\eta) \in \Gamma \}.
\end{eqnarray*}
\end{definition}

\begin{example}\label{mgexample}
Consider the c-bounded generalized map defined by
\begin{eqnarray*}
f_{\varepsilon}(x,y)=(x+ \gamma_{\varepsilon} y, x-\gamma_{\varepsilon} y),
\end{eqnarray*}
where $(\gamma_{\varepsilon})_{\varepsilon}$ is a positive net converging to zero.
\end{example}
Then the transposed Jacobian is
\begin{eqnarray*}
{}^T\!df_{\varepsilon}(x,y) := \left( \begin{array}{cc}
1 &  1 \\
\gamma_{\varepsilon}  & - \gamma_{\varepsilon}
\end{array}\right),
\end{eqnarray*}
which is constant with respect to the $(x,y)$ variable, so we put $M_{\varepsilon} := {}^T\!df_{\varepsilon}(x,y)$.

We observe that $M_{\varepsilon} \eta= (\eta_1 +\eta_2, \gamma_{\varepsilon}(\eta_1-\eta_2))$.
Setting $\nu_1:=( 1/\sqrt{2}, - 1/\sqrt{2})$ and $\nu_2:=( 1/\sqrt{2}, 1/\sqrt{2})$ we observe
$$
|\sigma_{\varepsilon} M_{\varepsilon} \nu_1| = \sqrt{2} \sigma_{\varepsilon}  \gamma_{\varepsilon}, \quad
|\sigma_{\varepsilon} M_{\varepsilon} \nu_2| = \sqrt{2} \sigma_{\varepsilon}
$$
and $\langle M_{\varepsilon} \nu_1, M_{\varepsilon} \nu_2 \rangle = 0$.

We can write any $\eta \in S^1$  in terms of the orthonormal basis $(\nu_k)_{k=1,2}$ by
$\eta= \langle \eta, \nu_1 \rangle  \nu_1 + \langle \eta, \nu_2 \rangle \nu_2$.
Note that $\langle \eta, \nu_1 \rangle^2 +\langle \eta, \nu_2 \rangle^2=1$ since $|\eta|=1$.

For any $\eta \in S^1$ we have
\begin{multline*}
|\sigma_{\varepsilon} M_{\varepsilon} \eta|^2 =
|\sigma_{\varepsilon} M_{\varepsilon} ( \langle \eta, \nu_1 \rangle \nu_1 + \langle \eta, \nu_2 \rangle \nu_2) |^2 \\
=  \langle \eta, \nu_1 \rangle^2 |\sigma_{\varepsilon} M_{\varepsilon} \nu_1|^2  +
\langle \eta, \nu_2 \rangle^2 |\sigma_{\varepsilon} M_{\varepsilon} \nu_2|^2 \\
= 2 \sigma_{\varepsilon}^2 (\langle \eta, \nu_1 \rangle^2 \gamma_{\varepsilon}^2 +
 \langle \eta, \nu_2 \rangle^2 ) = 2 \sigma_{\varepsilon}^2 (1 -  \langle \eta, \nu_1 \rangle^2  (1-\gamma_{\varepsilon}^2 )).
\end{multline*}

Now for any $\eta_0 \in S^1$ with $\eta_0 \ne \pm \nu_1$ we can find a neighborhood $V_0$ such that there exists a
$\delta \in ]0,1]$ with the property that $|\langle \eta, \nu_1 \rangle| \le 1-\delta$ holds for all $\eta \in V_0$.
Choosing $\sigma_{\varepsilon} := 1$
we obtain
\begin{multline*}
\inf_{(x,\eta)\in \mathbb{R}^n  \times V_0} |\sigma_{\varepsilon}  {}^T\! df_{\varepsilon}(x) \eta | =
\sqrt{2}\sigma_{\varepsilon} \inf_{\eta \in V_0}  \sqrt{1 -  \langle \eta, \nu_1 \rangle^2  (1-\gamma_{\varepsilon}^2 )} \\
= \sqrt{2} \sigma_{\varepsilon} \sqrt{1- (1-\delta)^2  (1-\gamma_{\varepsilon}^2 )} \rightarrow \sqrt{2}  \sqrt{2 \delta-\delta^2}
\end{multline*}
and
\begin{multline*}
\sup_{ (x,\eta)\in \mathbb{R}^n  \times V_0^{\perp}}    |\sigma_{\varepsilon}  {}^T\! df_{\varepsilon}(x) \eta | =
\sqrt{2} \sigma_{\varepsilon} \sup_{\eta \in S^1}  \sqrt{1 -  \langle \eta, \nu_1 \rangle^2  (1-\gamma_{\varepsilon}^2 )} \\
\le  \sqrt{2}\sigma_{\varepsilon}   =  \sqrt{2},
\end{multline*}
so it follows immediately that $\domainset{f} \supseteq \mathbb{R}^2 \times (S^1 /\{\pm \nu_1\})$.

It remains to check whether $\mathbb{R}^2 \times \{\pm \nu_1\}$ belongs to $\domainset{f}$:
Let $V \subseteq S^1$ be a neighborhood of $\nu_1$, then it follows that
$$
\inf_{(x,\eta)\in \mathbb{R}^n  \times V} |\sigma_{\varepsilon}  {}^T\! df_{\varepsilon}(x) \eta | =
\sqrt{2} \sigma_{\varepsilon} \inf_{\eta \in V}  \sqrt{1 -  \langle \eta, \nu_1 \rangle^2  (1-\gamma_{\varepsilon}^2 )} =
\sqrt{2} \sigma_{\varepsilon}   \gamma_{\varepsilon} ,
$$
and
$$
\sup_{ (x,\eta)\in \mathbb{R}^n  \times V^{\perp}}    |\sigma_{\varepsilon}  {}^T\! df_{\varepsilon}(x) \eta | =
\sqrt{2} \sigma_{\varepsilon} \sup_{\eta \in V^{\perp}}  \sqrt{1 - \langle \eta, \nu_1 \rangle^2  (1-\gamma_{\varepsilon}^2 )}
= \sqrt{2} \sigma_{\varepsilon},
$$
since the set $V^{\perp}$ by definition contains a vector $\nu'$ with $\langle \nu_1, \nu^{\prime}\rangle =0$.
Any slow scaled net $(\sigma_{\varepsilon})_{\varepsilon}$ with the property that $\sqrt{2} \sigma_{\varepsilon} \gamma_{\varepsilon}$ is bounded
away from zero, satisfies $\sigma_{\varepsilon} \rightarrow \infty$ (since $\gamma_{\varepsilon}$ tends to zero).
It immediately follows that $\domainset{f} \cap (\mathbb{R}^2 \times \{\pm \nu_1\}) = \emptyset$ ,
thus $\domainset{f} = \mathbb{R} \times (S^1 \backslash \{\pm \nu\})$.

\begin{remark} \label{closed_set_remark}
Note that any non-empty closed set $\Gamma \subseteq \Omega_2 \times S^{m-1}$ can be
considered as a set-valued map
$$
\supp{(\Gamma)}:= \pi_1(\Gamma) \rightarrow S^{m-1}, \quad y \mapsto \Gamma_y,
$$
where $\Gamma_y := \pi_2(\Gamma \cap(\{y\}\times S^{m-1}))$.
Then Theorem \ref{set_valued_map_main_theorem} yields that $y \mapsto \Gamma_y$ is upper semi-continuous, i.e.
for all $y\in \supp{(\Gamma)}$ and $W$ some open neighborhood of $\Gamma_y$, there exists some open neighborhood $Y \subseteq \Omega_2$
of $y$ such that $\Gamma_Y = \Gamma_{Y\cap\supp{(\Gamma)}} \subseteq W$, where  $\Gamma_Y = \bigcup_{y\in Y} \Gamma_y$
as in Definition \ref{set_valued_map_def}.
Due to Proposition \ref{usc_corr_proposition} the upper semi-continuity of $\Gamma$ yields that $\supp{(\Gamma)}$ is a closed subset of $\Omega_2$, so
we obtain for all $y\not\in \supp{(\Gamma)}$, that there exists some neighborhood $Y$ of $y$ such that $\Gamma_Y =\emptyset$.

Finally it is straight-forward to prove that for all $Y_0 \csub \Omega_2$ and $W$ some open neighborhood of
$\Gamma_{Y_0}$, there exists an open neighborhood $Y \subseteq \Omega_2$ of $Y_0$ such that
$\Gamma_Y \subseteq W$.
\end{remark}

\begin{lemma} \label{normal_bundle_lemma}
The generalized normal bundle $\gnormalb{f}$ and the wave front unfavorable support $\unfavsupp{f}{\Gamma}$ of $f$ (with respect to $\Gamma$) are closed sets.
If $\gnormalb{f} \cap \Gamma =\emptyset$, then $\unfavsupp{f}{\Gamma} = \emptyset$.
\end{lemma}
\begin{proof}

Let $x_0 \not \in \unfavsupp{f}{\Gamma}$ and put $Y_0 := \ggraph{f}_{x_0}$, then since $\domainset{f}$ is open there exist open
neighborhoods $X_0$  of $x_0$ and  $V_0$ of $\Gamma_{Y_0}$ such that $X_0 \times V_0 \subseteq \domainset{f}$.
As pointed out in Remark \ref{closed_set_remark} there exists a neighborhood $Y$ of $Y_0$, such that
$V_0$ is still a neighborhood of $\Gamma_Y$.  The upper semi-continuity of $\ggraph{f}$ provides a neighborhood $X'$ of
$x_0$ such that $\ggraph{f} \cap (X' \times \Omega_2) \subseteq X' \times Y$.
Now we can choose a smaller neighborhood $X_1 \subset X' \cap X_0$ (such that $X' \cap X_0$ is a neighborhood of $X_1$)
of $x_0$,
such that $X_1 \times V_0 \subseteq \domainset{f}$ and $\ggraph{f} \cap (X_1 \times \Omega_2) \subseteq X_1 \times Y$.
For all $x_1\in X_1$ and $y\in \Omega_2$ with $(x_1,y)\in \ggraph{f}$, it holds that $(x_1,y) \in X_1 \times Y$ and if
$(y,\eta) \in \Gamma$ it follows
that $\eta \in \Gamma_Y \subseteq V$ and thus $(x_1,\eta) \in \domainset{f}$.
It follows that $x_1\not\in \unfavsupp{f}{\Gamma}$ for all $x_1\in X_1$, $X_1 \cap \unfavsupp{f}{\Gamma} =\emptyset$
and thus $\unfavsupp{f}{\Gamma}^c$ is an open set.
\end{proof}

A substantial step in the proof of the main theorem is the application of a generalized stationary phase theorem (cf. Appendix B).
In order to obtain a lower bound for the gradient of the occurring phase function we have to consider the map
\begin{eqnarray*}
\Omega_1 \times S^{m-1} \times S^{n-1} & \rightarrow& S^{n-1} \\
(x,\eta,\xi)	&	\mapsto & \left|\frac{{}^T df_{\varepsilon}(x) \eta}{|{}^T df_{\varepsilon}(x) \eta|} -\xi \right|.
\end{eqnarray*}

\begin{lemma}
Let $W \subseteq S^{n-1}$ be a neighborhood of some $\eta_0 \in S^{n-1}$, then we have for any
$\eta_1,\eta_2 \in W$ that
$$
\frac{\eta_1 -\eta_2}{|\eta_1 - \eta_2|} \in W^{\perp} := \{\eta \in S^{n-1} \mid \exists \eta' \in W : \langle \eta', \eta\rangle = 0 \}.
$$
\end{lemma}

\begin{proof}
Assume $W \subseteq C_{\delta}(\eta_0) := \{\eta \in S^{n-1} \mid \langle \eta_0, \eta \rangle \ge 1- \delta\}$ for some
$\delta \in ]0,1[$. If no such $\delta \in ]0,1[$ exists $C_{0}(\eta_0) \subset W$ implies $W^{\perp} = S^{n-1}$, so the statement
is trivial.
Observe that
$$
\langle \frac{\eta_1 +\eta_2}{|\eta_1 + \eta_2|} , \eta_0 \rangle =
\frac{1}{|\eta_1 + \eta_2|} \left( \langle \eta_1, \eta_0 \rangle+
\langle \eta_2 , \eta_0 \rangle \right) \ge \frac{2 (1-\delta)}{\sqrt{2(1+ \langle \eta_1 ,\eta_2 \rangle})} \ge 1-\delta
$$
and since
$$
\langle \frac{\eta_1 +\eta_2}{|\eta_1 + \eta_2|} , \frac{\eta_1 -\eta_2}{|\eta_1 - \eta_2|} \rangle
= \frac{|\eta_1|^2 - |\eta_2|^2}{|\eta_1 + \eta_2||\eta_1 - \eta_2|}=0
$$
which implies $\frac{\eta_1 -\eta_2}{|\eta_1 - \eta_2|} \in W^{\perp}$.
\end{proof}

So we introduce the following notation:
\begin{lemma}\label{notionlemma}
We define the Colombeau map $M \in \colmap{\domainset{f}}{S^{n-1}}$ by
\begin{eqnarray*}
M_{\varepsilon}(x,\eta):=\frac{{}^T\!df_{\varepsilon}(x) \eta }{|{}^T\!df_{\varepsilon}(x) \eta|}
\end{eqnarray*}
on the domain $\domainset{f}$, then $\eta \mapsto M_{\varepsilon}(x,\eta)$ defines an equi-continuous Colombeau function for fixed $x$.
\end{lemma}
\begin{proof}
The map $g:\xi \rightarrow \frac{\xi}{|\xi|}$
is equi-continuous on $\mathbb{R}^n / B_{\delta}(0)$ for any fixed $\delta>0$ since
\begin{eqnarray*}
\left| \frac{\xi}{|\xi|} - \frac{\eta}{|\eta|} \right| = \frac{1}{|\xi|} \left| \xi - \frac{\eta|\xi|}{|\eta|} \right| =\frac{1}{|\xi|}
\left| \xi -\eta + \frac{\eta}{|\eta|} \left( |\eta|-|\xi| \right) \right| \le \frac{2}{|\xi|} |\xi-\eta|.
\end{eqnarray*}
Let $(x_0,\eta_0) \in D_{f}$, then we can find some neighborhood $X\times V \subseteq \Omega_1 \times S^{m-1}$ of $(x_0,\eta_0)$ such that
\begin{eqnarray*}
\inf_{(x,\eta)\in X\times V}  | \sigma_{\varepsilon}  {}^T\! df_{\varepsilon}(x) \eta|  &\ge& \alpha \quad \text{and } \sup_{(x,\eta) \in X\times V^{\perp}} |  \sigma_{\varepsilon}  {}^T\! df_{\varepsilon}(x) \eta | \le \beta
\end{eqnarray*}
for some slow scaled net $(\sigma_{\varepsilon})_{\varepsilon}$ and $\alpha, \beta \in ]0, \infty[$.
Then we conclude that
\begin{equation*}\begin{split}
&|M_{\varepsilon}(x,\eta)- M_{\varepsilon}(x,\xi)| \le \frac{2}{  | {}^T df_{\varepsilon}(x) \eta |} \left|{}^T\! df_{\varepsilon}(x) \eta-{}^T\! df_{\varepsilon}(x) \xi  \right| \\ &=  \frac{2}{ |  \sigma_{\varepsilon} {}^T\! df_{\varepsilon}(x) \eta| }
\left| \sigma_{\varepsilon} {}^T\! df_{\varepsilon}(x) (\eta-\xi)  \right| \le
2 \frac{ \sup_{(x,\zeta) \in X\times V^{\perp}} |\sigma_{\varepsilon} {}^T\! df_{\varepsilon}(x) \zeta |}{
 \inf_{(x,\eta)\in X\times V}   |\sigma_{\varepsilon} {}^T\! df_{\varepsilon}(x) \eta| } \\
&\cdot |\xi -\eta| \le  2 |\xi- \eta| \frac{\beta}{\alpha}
\end{split}
\end{equation*}
uniformly for all $x\in X_0$ and $\xi,\eta \in V$.
\end{proof}

\begin{remark}
By the preceding Lemma we have obtained that $\left[(\eta \mapsto M_{\varepsilon}(x,\eta)\mid_{\domainset{f}})_{\varepsilon}\right]$ is an equi-continuous Colombeau map, thus it follows by Proposition \ref{generalized_graph_equicontinuous_proposition} that
\begin{eqnarray*}
\ggraph{M}_{x,\eta} &=&
 \left\{ \xi \in S^{n-1} \mid \exists \text{\ a \ net \ }(x_{\varepsilon})_{\varepsilon}
\text{\ in\ } \Omega_1: \lim_{\varepsilon\rightarrow 0} x_{\varepsilon}=x \in \Omega_1 \right.
\\ &&\left. \text{ and } y\in\cp{(M_{\tau(\varepsilon)}(x_{\varepsilon},\eta))_{\varepsilon}}
 \text{for\ some\  map}\ \tau \in \mathcal{T} \right\},
\end{eqnarray*}
which simplifies the determination of the generalized graph $\ggraph{M}$ considerably.
\end{remark}

\begin{definition} \label{transformed_wf_set}
Let $f\in\colmap{\Omega_1}{\Omega_2}$ be a c-bounded generalized map and assume that $\domainset{f} \ne \emptyset$.
We define the pullback of some closed set $\Gamma \subseteq  \Omega_1 \times S^{n-1}$ by
$$
f^{\ast} \Gamma := \ggraph{M} \circ ( \{\bullet\} \times (\Gamma \circ \ggraph{f})_{\bullet} ),
$$
where $\{\bullet\} \times (\Gamma \circ \ggraph{f})_{\bullet}$ denotes the set-valued map defined by
$x \mapsto \{x\} \times (\Gamma \circ \ggraph{f})_{x}$,
using the notation from Definition \ref{set_valued_map_def} and \ref {set_valued_map_inverse}.

If $\domainset{f} = \emptyset$ we put $(f^{\ast} \Gamma)_x =  S^{n-1}$ for all $x\in\Omega_1$.
\end{definition}

\begin{remark}
Obviously the set-valued map $f^{\ast} \Gamma$ is defined as the composition of upper semi-continuous and
locally bounded set-valued maps, thus $f^{\ast} \Gamma$ is again an upper semi-continuous and locally bounded set-valued map
$\Omega_1 \mapsto \mathcal{F}_0(S^{n-1})$. According to Theorem \ref{set_valued_composition_theorem} the pullback is
defined by
$$
(f^{\ast} \Gamma)_x = \bigcup_{\eta \in \widetilde{\Gamma}_x}\ggraph{M}_{x,\eta}
$$
where $\widetilde{\Gamma}_x:= \bigcup_{y\in \ggraph{f}_x} \Gamma_y$.
The upper semi-continuity of $f^{\ast} \Gamma$ yields that for any neighborhood $W \subseteq S^{n-1}$ of $(f^{\ast} \Gamma)_x$ there exists
some neighborhood $X$ of $x$ such that $(f^{\ast} \Gamma)_X \subseteq W$.
\end{remark}

\section{Generalized pullbacks of Colombeau functions}

In this section we prove the main result which gives a microlocal inclusion relation for the generalized pullback of a Colombeau function.
The proof of the theorem relies on a generalized stationary phase theorem, the details of which are discussed in the Appendix B.

\begin{definition} \label{specialsets}
Let $f \in \colmap{\Omega_1}{\Omega_2}$ be a c-bounded generalized map. Then we call $f$ slow scaled in all
derivatives on the open set $X_0\subseteq \Omega_1$, if for all $\alpha\in\mathbb{N}^n$ there exists slow scaled nets $(r_{\alpha,\varepsilon})_{\varepsilon}$
such that
\begin{eqnarray}\label{slowscale}
\sup_{x \in X_0} |\partial^{\alpha} f_{\varepsilon}(x)| \le C_{\alpha} r_{\alpha,\varepsilon} \ \rm{as} \ \varepsilon \rightarrow 0
\end{eqnarray}
holds, where $C_{\alpha}$ are constants. Furthermore we call
$f$ \emph{slow scaled in all derivatives} \index{slow scaled!in all derivatives} at $x_0\in\Omega_1$, if there exists a neighborhood $X_0$ of $x_0$ such that (\ref{slowscale}) holds for all $\alpha \in \mathbb{N}^n$.

Define the sets
\begin{eqnarray*}
S^f := \{ x \in \Omega_1 \mid f \rm{\ is\ slow scaled\ in\ all\ derivatives\ at\ } x\}
\end{eqnarray*}
and
\begin{eqnarray*}
K^f(u) := \bigcap_{k \in  \widetilde{\Omega_2}  } \pi_1(\ggraph{f} \cap \rm{supp}(u-k) \times \Omega_2) .
\end{eqnarray*}
\end{definition}

\begin{remark}
The set $\pi_1(\ggraph{f} \cap \Omega_1 \times \rm{supp}(u-k))$ is closed for all $k \in \widetilde{\Omega_2}$ due to Proposition \ref{usc_corr_proposition} since $\ggraph{f}$ is upper semi-continuous and $\rm{supp}(u-k)$ is closed. This implies that $\ksupp{f}(u)$ is closed in the relative topology of $\Omega_1$.
\end{remark}

\begin{lemma}\label{lemmacool}
Let $f \in \colmap{\Omega_1}{\Omega_2}$ be a c-bounded generalized map. Furthermore let $x_0 \not \in \unfavsupp{f}{\Gamma} \cup (\slsupp{f})^c$ and $Y_0 := \pi_2(\ggraph{f} \cap \{x_0\}\times \Omega_2)$. If $W$ is some open neighborhood of $(f^{\ast} \Gamma)_{x_0}$ ,
then there exist neighborhoods $X$ of $x_0$, $V$ of $\Gamma_{Y_0}$
and $Y$ of $Y_0$
with the following properties:
\begin{equation*}\begin{split}
&X \times V \subseteq \domainset{f}(\Gamma),\ (f^{\ast} \Gamma)_X \subseteq W,\
f_{\varepsilon}(X) \subseteq Y \ \rm{for\ all\ }\ \varepsilon < \varepsilon', \\
&(f_{\varepsilon}) \rm{\ is\ slow scaling\ in\ all\ derivatives\ on\ } X \rm{\ and\ }
\Gamma_{Y} \subseteq V \end{split}
\end{equation*}
Furthermore there exists a positive constant $c>0$, such that
\begin{equation*}
\inf_{(x,\eta,\xi)\in X \times V \times W^c} \left|M_{\varepsilon}(x,\eta) - \xi\right|>c \ \rm{for\ all\ }\varepsilon< \varepsilon'
\end{equation*}
holds (we are using the notation from Lemma \ref{notionlemma}).
\end{lemma}

\begin{proof}
From $x_0 \not \in \unfavsupp{f}{\Gamma}$ it follows that $\{x_0 \}\times \Gamma_{Y_0}$
is a compact subset of the open set $\domainset{f}$. Thus we can find some
neighborhood $X' \times V' \subseteq \domainset{f}$ of $\{x_0\} \times \Gamma_{Y_0}$.
Since $x_0 \in \slsupp{f}$ we can assume without loss of generality that $f$ is slow scale in all derivatives on
the compact set $X'$.

Since $W$ is a neighborhood of $(f^{\ast}\Gamma)_{x_0}$ we have that
\begin{eqnarray*}
(f^{\ast}\Gamma)_{x_0}  = \pi_2( \ggraph{M} \cap \{x_0\} \times \Gamma_{Y_0} \times S^{n-1}) \csub W
\end{eqnarray*}
where $\ggraph{M}$ denotes the generalized graph of the generalized map defined by $(x,\eta) \mapsto M_{\varepsilon}(x,\eta)$ on the open domain $\domainset{f}$.
By Lemma \ref{generalized_graph_approximation_property}
there exist neighborhoods $X'', V''$ of $x_0$ resp. $\Gamma_{Y_0}$ such that
\begin{eqnarray} \label{propcool}
M_{\varepsilon}(X'' \times V'') \subseteq W
\end{eqnarray}
for all $\varepsilon < \varepsilon'$. Let $V:= V' \cap V''$, then by Remark \ref{closed_set_remark} we can find some neighborhood $Y$ of $Y_0$ such that $\Gamma_Y \subseteq V$. By Lemma \ref{generalized_graph_approximation_property} there exists some neighborhood $X'''$ of $x_0$ such
that $f_{\varepsilon}(X''') \subseteq Y$ for small $\varepsilon$. Let $X:=X' \cap X'' \cap X'''$ and
$Z:= X \times V \times W^c$. Then there exists $(x_{\varepsilon},y_{\varepsilon},\xi_{\varepsilon})\in Z$ (note that $(x,\eta,\xi) \mapsto |M_{\varepsilon}(x,\eta) - \xi |$ is a continuous function for each $\varepsilon \in ]0,1]$ and $Z$ is a compact set) such that
\begin{equation*}
c_{\varepsilon} := \inf_{(x,\eta,\xi) \in Z} |M_{\varepsilon}(x,\eta) - \xi | = |M_{\varepsilon}(x_{\varepsilon},\eta_{\varepsilon}) - \xi_{\varepsilon} |
\end{equation*}
holds for some net $(x_{\varepsilon}, \eta_{\varepsilon}, \xi_{\varepsilon})_{\varepsilon}$.
By (\ref{propcool}) we have that $c_{\varepsilon} > c >0$ holds for all $\varepsilon < \varepsilon'$, where $c$ is a constant.
\end{proof}

\begin{lemma} \label{minilemma}
Let $v,w \in \mathbb{R}^n$ , then the inequality
$$
|\alpha v -(1-\alpha)w|^2 \ge \frac{|v|^2 |w|^2}{|v+w|^2} \left(1- (\langle v/|v|, w/|w|\rangle)^2\right)
$$
holds for all $\alpha \in [0,1]$.
\end{lemma}
\begin{proof}
Observe that
\begin{multline*}
|\alpha v -(1-\alpha)w|^2 = \alpha^2 |v|^2 - 2\alpha(1-\alpha) \langle v, w \rangle + (1-\alpha)^2 |w|^2 \\
 = \alpha^2 |v+w|^2  -2 \alpha (|w|^2 + \langle v,w \rangle) + |w|^2
\end{multline*}
and differentiation in $\alpha$ shows that the expression has a local extremum at $\alpha_0:=(\langle v,w \rangle + |w|^2)/|v+w|^2$.
Since the second derivative with respect to $\alpha$ is $|v-w|^2 > 0$ for $v\ne -w$ (the case $v=-w$ is trivial), we have a local minimum at $\alpha_0$. Finally we obtain
$$
|\alpha v -(1-\alpha)w|^2  \ge |\alpha_0 v -(1-\alpha_0)w|^2 = \frac{|v|^2 |w|^2}{|v+w|^2} \left(1- \langle v/|v|, w/|w|\rangle^2\right),
$$
by checking the boundary case $\alpha\in \{0,1\}$ we verify that it is a global lower bound.
\end{proof}

\begin{theorem} \label{maintheorem}
Let $f \in \colmap{\Omega_1}{\Omega_2}$ be a c-bounded generalized map with representative $(f_{\varepsilon})_{\varepsilon}$.
For $u \in \col{ \Omega_2}$ with representative $(u_{\varepsilon})_{\varepsilon}$, we define the pullback  $f^{\ast} u$  by
\begin{eqnarray*}
f^{\ast} u := (u_{\varepsilon}(f_{\varepsilon}(x)))_{\varepsilon} +\mathcal{N}(\Omega_1).
\end{eqnarray*}
It satisfies the microlocal inclusion relation
\begin{eqnarray*}
\wf{f^{\ast} u} \subseteq f^{\ast} \wf{u}\ \bigcup\ \left(\unfavsupp{f}{\wf{u}} \times S^{n-1}\right)\ \bigcup\
\left(\left(\ksupp{f}(u) \cap (\slsupp{f})^c \right) \times S^{n-1}\right).
\end{eqnarray*}
\end{theorem}

\begin{remark}
It is apparent that the microlocal inclusion relation for $\wf{f^{\ast} u}$ is splitted into three different parts:
\begin{itemize}
\item $f^{\ast} \wf{u}$: This part corresponds to its classical counterpart (cf. \cite[eq. 8.2.4.]{Hoermander:V1}).
\item $\unfavsupp{f}{\wf{u}} \times S^{n-1}$: According to Lemma \ref{normal_bundle_lemma} this set is non-empty whenever
 $\gnormalb{f} \cap \wf{u}$ is non-empty.
 In the classical case the pullback of a distribution $u$ by a smooth map $f$ is not defined if $\gnormalb{f} \cap \wf{u}$
 is non-empty, so there is no classical counterpart to this set.
\item $\left(\ksupp{f}(u) \cap (\slsupp{f})^c \right) \times S^{n-1}$:
$\left(\ksupp{f}(u) \cap (\slsupp{f})^c \right)$ is the set of all points $x\in \Omega_1$, where the generalized map
$f$ is not of slow-scale in all derivatives, whenever the Colombeau map $u$ is not constant in a neighborhood of $\ggraph{f}_x$.
This set is required since we allow non-regular Colombeau maps $f$, which are only c-bounded. So the singularities of the map $f$ may cause
singularities of $f^{\ast} u$. There is no classical correspondence, since in \cite[Theorem 8.2.4]{Hoermander:V1}
the map $f$ is assumed to be smooth.
\end{itemize}
\end{remark}

\begin{proof}[Proof of Theorem \ref{maintheorem}]
According to \cite[Proposition 1.2.8]{GKOS:01} the pullback $f^{\ast} u$ is a well-defined Colombeau function in $\mathcal{G}(\Omega_1)$. Let $(x_0,\xi_0) \in \Omega_1 \times S^{n-1}$ and $Y_0 =\{ y \in \Omega_2 \mid (x_0, y) \in \ggraph{f}(\Omega_1) \}$.
We set $\Gamma:=\wf{ u }$ and $\Gamma_y:=\{\eta \mid (y,\eta) \in \Gamma\}$.
In order to prove the statement we show that if $(x_0,\xi_0) \not \in f^{\ast} \Gamma  \bigcup \unfavsupp{f}{\Gamma} \times S^{n-1} \bigcup \left(\ksupp{f}(u) \cap (\slsupp{f})^c \right) \times S^{n-1}$ it follows that $(x_0, \xi_0)\not \in \wf{f^{\ast} u} $.

So let us assume that $(x_0,\xi_0) \not \in f^{\ast} \Gamma  \bigcup \unfavsupp{f}{\Gamma} \times S^{n-1} \bigcup \left(\ksupp{f}(u) \cap (\slsupp{f})^c \right) \times S^{n-1}$. Then
we have $\xi_0 \not \in (f^{\ast} \Gamma)_{x_0}$ and $x_0 \not \in \unfavsupp{f}{\Gamma} \bigcup \ksupp{f}(u) \cap (\slsupp{f})^c$. This allows splitting the proof in two parts for the cases $x_0 \not\in \ksupp{f}(u) \cup
\unfavsupp{f}{\Gamma}$ and
$x_0 \not\in (\slsupp{f})^c \cup \unfavsupp{f}{\Gamma}$.\\

In the first case ($x_0 \not\in \ksupp{f}(u) \cup \unfavsupp{f}{\Gamma}$): Since $x_0 \not\in \ksupp{f}(u)$
there exists a generalized constant $k\in \widetilde{\mathbb{R}}^n$ with representative $(k_{\varepsilon})_{\varepsilon}$, such that $ \ggraph{f} \cap \{x_0\}\times \rm{supp}(u-k) = \emptyset$. Hence
for all $y \in Y_0$ it holds that $y \not \in \rm{supp}(u-k)$.
Since $Y_0$ and $\rm{supp}(u-k)$ are two disjoint closed sets,
we can find a closed neighborhood $Y$ of $Y_0$ such that $Y \cap \rm{supp}(u)=\emptyset$.
Let $\chi \in C_{c}^{\infty}(\Omega_2)$ with the property that $\chi \equiv 1$ on some compact neighborhood $Y' \subseteq Y^{\circ}$ of $Y_0$
and $\rm{supp}(\chi) \subseteq Y$. By Lemma \ref{generalized_graph_approximation_property}
there exists a neighborhood $X'$ of $x_0$ and $\varepsilon'\in]0,1]$ such that $f_{\varepsilon}(X') \subseteq Y'$ for $\varepsilon<\varepsilon'$.

In order to show that $\xi_0 \not \in (\wf{f^{\ast}u})_{x_0}$ for $x_0 \not\in \ksupp{f}(u)$ we have to find a smooth function $\varphi$ with support on a neighborhood of $x_0$, such that $\mathcal{F}(f^{\ast}u \varphi)$ is rapidly decreasing on some neighborhood of $\xi_0$.
We choose $\varphi$ to be a smooth function with $\rm{supp}(\varphi) \subseteq X'$. For all $\varepsilon < \varepsilon'$ the
identity $f_{\varepsilon}^{\ast} u \cdot \varphi = f_{\varepsilon}^{\ast}(\chi u) \cdot \varphi$ holds, since the functions $\chi$ and $\varphi$ where chosen
such that $ (\chi \circ f_{\varepsilon})\cdot \varphi \equiv 1$ for $\varepsilon < \varepsilon'$.
We have that $\chi \cdot (u_{\varepsilon}-k_{\varepsilon}) \in \colneg{\Omega_2}$ and it follows that
$f^{\ast}u\cdot \varphi$ is $\mathcal{G}^{\infty}$, so $\xi_0 \not \in (\wf{f^{\ast}u})_{x_0}$. \\

The second case ($x_0 \not \in \unfavsupp{f}{\Gamma} \cup (\slsupp{f})^c$ and $\xi_0 \not \in (f^{\ast} \Gamma)_{x_0}$) is more difficult to prove:
Again we have to find a smooth cutoff function $\varphi$ with compact support containing $x_0$, such that
$\mathcal{F}(f^{\ast}u \varphi)$ is rapidly decreasing on some neighborhood of $\xi_0$. The main step will be the application of the generalized
stationary phase theorem (cf. Appendix B), which requires a suitable lower bound on the norm of the gradient
of the occurring phase function (it will turn out that $x_0 \not \in \unfavsupp{f}{\Gamma}$ is essential) and slow-scaledness of the phase function in all derivatives
in a neighborhood of $x_0$.

First of all we note that $(f_{\varepsilon})$ is slow scaling in all derivatives in some neighborhood of $x_0$, since $x_0\in \slsupp{f}$.

Now let $W \subseteq S^{n-1}$ be an open neighborhood of $(f^{\ast} \Gamma)_{x_0}$  such that $W^c$ is a neighborhood of $\xi_0$ (this is possible since $\xi_0 \not \in (f^{\ast} \Gamma)_{x_0}$), then Lemma \ref{lemmacool} implies that there exist neighborhoods $X$ of $x_0$, $Y$ of $Y_0$,
and $V$ of $\Gamma_{Y_0}$ such that $(f^{\ast} \Gamma)_X \subseteq W$, $f_{\varepsilon}(X) \subseteq Y$ for all $\varepsilon < \varepsilon'$,
$\Gamma_{Y} \subseteq V$, and
\begin{equation}\label{eq1}
\inf_{(x,\eta_1,\xi_1) \in X\times V \times W^c}  \left|M_{\varepsilon}(x,\eta_1) -  \xi_1 \right| > d >0
\end{equation}
holds for all $\varepsilon < \varepsilon'$, where $d$ is some positive constant. Without loss
of generality $(f_{\varepsilon})$ is slow scaling in all derivatives on $X$.

Let $\chi \in C_{c}^{\infty}(\Omega_2)$ with the property that $\chi \equiv 1$ on some compact neighborhood $Y' \subseteq Y^{\circ}$ of $Y_0$
and $\rm{supp}(\chi) \subseteq Y$. By Lemma \ref{generalized_graph_approximation_property} there exists a neighborhood $X'$ of $x_0$ and $\varepsilon'\in]0,1]$ such that
$f_{\varepsilon}(X') \subseteq Y'$ for $\varepsilon<\varepsilon'$. Without loss of generality we can assume that $X' \subseteq X$.

In order to show that $\xi_0 \not \in \wf{f^{\ast} u}_{x_0}$ we are going to prove that
there exists a smooth function $\varphi$ with support on a neighborhood of $x_0$, such that $\mathcal{F}(f^{\ast}u \varphi)$ is rapidly decreasing on $W^c$ (which is a neighborhood of $\xi_0$), which implies $(x_0, \xi_0) \not \in \wf{f^{\ast} u}$.

We choose $\varphi$ to be a smooth function with $\rm{supp}(\varphi) \subseteq X'$. For all $\varepsilon < \varepsilon'$ the
identity $(f_{\varepsilon}^{\ast}) u_{\varepsilon}\ \varphi = f_{\varepsilon}^{\ast}(\chi u_{\varepsilon})\ \varphi$ holds, since the functions $\chi$ and $\varphi$ where chosen such that $ (\chi \circ f_{\varepsilon})\cdot \varphi \equiv 1$ for $\varepsilon < \varepsilon_1$.

Set $\widetilde{V}:=\{\eta \in S^{m-1}\mid  \exists \eta_1 \in V, \lambda \in \mathbb{R}_+: \eta=\lambda \cdot \eta_1 \}$ and
$\widetilde{W}:=\{\xi \in S^{m-1}\mid  \exists \xi \in W, \lambda \in \mathbb{R}_+: \xi=\lambda \cdot \xi_1 \}$.
Obviously $\four{u \chi}$ is rapidly decreasing on $V^c$. We have that
\begin{eqnarray*}
&&
|\four{( f^{\ast} u)_{\varepsilon} \varphi}(\xi)| = |\four{(f^{\ast} \chi u)_{\varepsilon} \varphi } (\xi)| \\
&=&\left| \int_{\mathbb{R}^m} \widehat{\chi u_{\varepsilon}}(\eta)
\left(\int_{\mathbb{R}^n} \exp{(i \langle f_{\varepsilon} (x), \eta \rangle - i\langle x, \xi \rangle )} \varphi(x) \m{x} \right) \m{\eta}\right|\\
&=&\left|\int_{\mathbb{R}^m} \widehat{\chi u_{\varepsilon}}(\eta) I_{\varepsilon}(\xi, \eta) \m{\eta} \right|
= \left|\int_{\widetilde{V}} \widehat{\chi u_{\varepsilon}}(\eta) I_{\varepsilon}(\xi, \eta) \m{\eta} \right| +
\left|\int_{\widetilde{V}^c} \widehat{\chi u_{\varepsilon}}(\eta) I_{\varepsilon}(\xi, \eta) \m{\eta} \right|,
\end{eqnarray*}
where we have set
\begin{eqnarray*}
I_{\varepsilon}(\xi, \eta)  := \int \exp{(i \langle f_{\varepsilon} (x), \eta \rangle - i\langle x, \xi \rangle )} \varphi(x) \m{x}.
\end{eqnarray*}
We intend to apply the stationary phase theorem \ref{statphas} (cf. the Appendix B) with
\begin{eqnarray*}
\omega&:=&|\xi|+|\eta| \\
\phi_{\varepsilon} &:=& \langle f_{\varepsilon} (x), \frac{\eta}{|\eta|+|\xi|} \rangle - \langle x, \frac{\xi}{|\eta|+|\xi|} \rangle.
\end{eqnarray*}
Thus we have to find a bound for the gradient of the phase function $|\phi_{\varepsilon}'(x)| = \left|  {}^T\!df_{\varepsilon}(x) \alpha \eta_1  -  (1-\alpha)\xi_1 \right|$ from below for all $\alpha\in[0,1]$, $\eta_1 = \eta/|\eta|$, $\xi_1=\xi/|\xi|$ and $x \in X$.
By optimization in the parameter $\alpha$ (according to Lemma \ref{minilemma}) and using the notation of Lemma \ref{notionlemma}  we can bound the expression
\begin{eqnarray*}
\inf_{x \in X,\eta_1 \in V, \xi_1 \in W,\alpha\in[0,1]}  \left| {}^T\!df_{\varepsilon}(x) \alpha \eta_1  -  (1-\alpha) \xi_1 \right|
\end{eqnarray*}
from below by
\begin{eqnarray*}
&&
\inf_{(x,\eta_1,\xi_1) \in X\times V \times W^c} \frac{|
{}^{T}\! df_{\varepsilon}(x) \eta_1| }{   |{}^{T}\! df_{\varepsilon}(x) \eta_1 + \xi_1|}
\sqrt{\left|
1- \Big \langle M_{\varepsilon}(x,\eta_1) , \xi_1
\Big \rangle^2 \right|} \\
&&=
\inf_{(x,\eta_1,\xi_1) \in X\times V \times W^c} \frac{1}{2} \frac{ |  {}^{T}\! df_{\varepsilon}(x) \eta_1| }
{| {}^{T}\! df_{\varepsilon}(x) \eta_1 + \xi_1|}
\left| M_{\varepsilon}(x,\eta_1) -  \xi_1 \right|  \\
&&\ge   \inf_{(x,\eta_1,\xi_1) \in X\times V \times W^c} \frac{1}{4}
\frac{  |{}^{T}\! df_{\varepsilon}(x) \eta_1| }{
    |{}^T\!df_{\varepsilon}(x) \eta_1| + 1}
\inf_{(x,\eta_1,\xi_1) \in X\times V \times W^c}
  \left|M_{\varepsilon}(x,\eta_1)
 -  \xi_1 \right| \\
 &&\ge   C \sigma_{\varepsilon}^{-1} \inf_{(x,\eta_1,\xi_1) \in X\times V \times W^c}  \left|
M_{\varepsilon}(x,\eta_1)
 -  \xi_1 \right|,
\end{eqnarray*}
where $\sigma_{\varepsilon}$ is the slow scaling net from the Definition \ref{unfavdef}.
By (\ref{eq1}) we have
\begin{equation*} \label{supernet}
\inf_{(x,\eta_1,\xi_1) \in X\times V \times W^c}  \left|
M_{\varepsilon}(x,\eta_1)
 -  \xi_1 \right| > d >0
\end{equation*}
for $\varepsilon< \varepsilon'$, where $d$ is a positive constant.
Thus the gradient of the phase function is uniformly bounded from below by
$|\phi_{\varepsilon}'(x)| \ge  C d \cdot \sigma_{\varepsilon}^{-1}$
for all  $(x,\xi ,\eta) \in X \times \widetilde{V} \times \widetilde{W}^c$. The stationary phase theorem \ref{statphas} yields
\begin{eqnarray}\label{superest1}
\left|I_{\varepsilon}(\xi, \eta) \right| &\le& C_q \varepsilon^{-1} (1+|\xi|+|\eta|)^{-q}
\end{eqnarray}
for all $q \in \mathbb{N}_0$. Note that we use $C, C_p, C_q$ and $C_k,l$ as generic constants.
In the case where $(x, \eta, \xi) \in X \times \widetilde{V}^c \times \widetilde{W}^c$ the stationary phase theorem (now with the phase function $\exp{(-i\langle x, \xi \rangle)}$)
gives
\begin{equation} \begin{split} \label{superexpr}
&|\xi|^{k} |I_{\varepsilon}(\xi, \eta)|=| \xi|^{k}  \left|\int  \exp{(- i\langle x, \xi \rangle )}  \exp{(i \langle f_{\varepsilon} (x), \eta \rangle)
 } \varphi(x)  \m{x} \right| \\
&\sum_{|\alpha| \le k} \left|D^{\alpha}_x \left( \exp{(i |\eta| \langle f_{\varepsilon} (x), \eta/|\eta| \rangle )} \varphi \right)\right|  \\
\end{split}
\end{equation}
and by repeated use of the chain rule we obtain the estimate
\begin{equation*}
\begin{split}
&\le\sum_{|\alpha| \le k} \sum_{\beta \le \alpha} c_1(\beta) \sup_{x\in X_0} \left|\partial^{\alpha-\beta} \varphi(x)\right| \sum_{l=1}^{|\beta|} |\eta|^l \cdot \\
&\sum_{\gamma_1+...+\gamma_l = \beta} d(\gamma_1, ... ,\gamma_l)  \prod_{1\le i \le l}
\sup_{x\in X_0} \left|\partial^{\gamma_i} \langle  f_{\varepsilon}(x), \eta/|\eta| \rangle\right|
\end{split}
\end{equation*}
where $\gamma_1 + \gamma_2 + \cdots + \gamma_l$ denotes a partition of the multi-index $\beta$ in exactly
$l$ multi-indices, that add up componentwise to $\beta$. Using the
notation $|g|_{k} := \max_{|\alpha|=k} |\partial^{\beta} f(x)|$, we can bound the expression by
\begin{equation*}
\begin{split}
&\le
C_{k,2} \sum_{|\alpha| \le k} \sum_{\beta \le \alpha}  \sum_{l=0}^{|\beta|} |\eta|^l \max_{\gamma_1+...+\gamma_l = \beta}{\left( \prod_{1\le i \le l}
\sup_{x\in X, \eta \in \widetilde{V}^c} \left|\langle  f_{\varepsilon}(x), \eta/|\eta| \rangle\right|_{|\gamma_i|} \right)} \\
&\le
C_{k,2} \sum_{|\alpha| \le k} \sum_{\beta \le \alpha}  \sum_{l=0}^{|\beta|} |\eta|^l \max_{0\le j \le |\beta| - l + 1}{
\left( \sup_{x\in X, \eta \in \widetilde{V}^c} \left|\langle  f_{\varepsilon}(x), \eta/|\eta| \rangle\right|_{j}  \right)^l } \\
&\le
C_{k,3}  (1+|\eta|)^k \max_{l=1}^{k} \max_{0\le j \le k - l + 1}{
\left( \sup_{x\in X, \eta \in \widetilde{V}^c} \left|\langle  f_{\varepsilon}(x), \eta/|\eta| \rangle\right|_{j}  \right)^l }
.\end{split}
\end{equation*}
Since $f_{\varepsilon}(x)$ is of slow scale in all derivatives at $x_0$, it holds that for all $j \in \mathbb{N}_0$ there exists
constants $C_j$ and slow scaled nets $r_{\varepsilon,j}$ such that
\begin{equation*}
\sup_{(x,\eta_1) \in X\times V^c}{\left|\langle  f_{\varepsilon}(x), \eta_1  \rangle\right|_{j}} \le C_j r_{\varepsilon,j}
\end{equation*}
holds for small $\varepsilon$.
Summing up the estimates of the form (\ref{superexpr}) give that there exists constants $C_p$ such that
\begin{eqnarray}\label{superest2}
|I_{\varepsilon}(\xi, \eta)| \le C_p (1+|\eta|)^{-p} \varepsilon^{-1} (1+|\xi|)^{p}
\end{eqnarray}
holds for all $p\in \mathbb{N}_0$ and $(\eta,\xi) \in \widetilde{V}^c \times \widetilde{W}$.
We observe that
\begin{equation*}\label{last}
\begin{split}
|\four{( f^{\ast} u)_{\varepsilon} \varphi}(\xi)| =& \left|\int_{\mathbb{R}^m} \widehat{\chi u_{\varepsilon}}(\eta) I_{\varepsilon}(\xi, \eta) \m{\eta} \right|\\
\le&
C_q  \varepsilon^{-1} \int_{\widetilde{V}} |\widehat{\chi u_{\varepsilon}}(\eta)|  (1+|\xi|+|\eta|)^{-q} \m{\eta}
\\ &+
C_p (1+|\xi|)^{-p} \varepsilon^{-1}  \int_{\widetilde{V}^c} |\widehat{\chi u_{\varepsilon}}(\eta)|  (1+|\eta|)^{p} \m{\eta}
\end{split}
\end{equation*}
holds and using (\ref{superest1}) and (\ref{superest2}) leads to the upper bound
\begin{equation*}
\begin{split}
& C_q \varepsilon^{-1} \sup_{\eta \in \tilde{V} }\left|(1+|\eta|)^{-k} \widehat{\chi u_{\varepsilon}}(\eta) \right|  \int_{\tilde{V}}    (1+|\xi|+|\eta|)^{k-q} \m{\eta}  \\
& + C_p (1+|\xi|)^{-p} \varepsilon^{-1-n} \int_{\widetilde{V}^c} (1+|\eta|)^{p-l}  \m{\eta}.
\end{split}
\end{equation*}
Finally we set $k:=q-p-m$ and $l:=p+n-1$ and obtain
\begin{equation*}
\begin{split}
& C_q \varepsilon^{-1} \sup_{\eta \in \tilde{V} }\left|(1+|\eta|)^{-k} \widehat{\chi u_{\varepsilon}}(\eta) \right|  \int_{\tilde{V}}    (1+|\xi|+|\eta|)^{k-q} \m{\eta}  \\
& + C_p (1+|\xi|)^{-p} \varepsilon^{-1-n} \int_{\widetilde{V}^c} (1+|\eta|)^{p-l}  \m{\eta} \\
& \le C_{p,q,m,n} \varepsilon^{-1-n} (1+|\xi|)^{-p}
\end{split}
\end{equation*}
for all $\xi \in \widetilde{W}^c$ and $C_{p,q,m,n}$ some constant depending on $p,q,m$ and $n$.
It follows that $(x_0,\xi_0)  \not \in  \wf{f^{\ast}u}$.
\end{proof}

\section{Examples}
\begin{example}[Multiplication of Colombeau functions]
This example was presented in \cite[Example 4.2]{HK:01}  in order to show that
an inclusion relation like in \cite[Theorem 8.2.10]{Hoermander:V1} for the wave front set of a product of distributions,
cannot be extended to Colombeau function with wave front sets in unfavorable position.

Consider the Colombeau functions $u$ and $v$ defined by
\begin{eqnarray*}
u_{\varepsilon}&:=& {\varepsilon}^{-1} \rho({\varepsilon}^{-1} (x + \gamma_{\varepsilon} y))\\
v_{\varepsilon}&:=& {\varepsilon}^{-1} \rho({\varepsilon}^{-1} (x - \gamma_{\varepsilon} y)),
\end{eqnarray*}
where $\gamma_{\varepsilon}$ is some net with $\lim_{\varepsilon \rightarrow 0} \gamma_{\varepsilon} = 0$. Note that
in \cite[Example 4.2]{HK:01} $\gamma_{\varepsilon} := \varepsilon^{1/2}$.
These Colombeau functions are both associated to $\delta(x) \otimes 1(y)$ and the wave front sets $\wf{u}=\wf{v} =
\{0\}\times\mathbb{R} \times \{(\pm 1,0)\}$ are in an unfavorable position.
\end{example}
We are going to apply Theorem (\ref{maintheorem}) in order to calculate $WF(u\cdot v)$. First we rewrite $u \cdot v = f^{\ast} \iota(\delta)$,
where $f_{\varepsilon}(x,y)=(x+ \gamma_{\varepsilon} y, x-\gamma_{\varepsilon} y)$
with $\gamma_{\varepsilon}$ some net tending to zero. In Example (\ref{mgexample}) we already showed that $\domainset{f}= \mathbb{R}^2 \times S^1 / (\pm 1, \mp1)$.
From \cite[Theorem 15]{HdH:01} it follows that $WF(\iota(\delta)) = \{(0,0)\} \times S^{1}$.
The wave front unfavorable support of $f$ with respect to $WF(\iota(\delta))$ according to Definition \ref{unfavdef} is
\begin{eqnarray*}
\unfavsupp{f}{(0,0)\times S^1} = \{(0,0)\}.
\end{eqnarray*}
Since $f_{\varepsilon}(x,y)$ is of slow scale in all derivatives at all $x\in\mathbb{R}^2$, it follows that $(\slsupp{f})^c=\emptyset$.
Now Theorem \ref{maintheorem} gives that
\begin{eqnarray*}
\wf{u \cdot v} \subseteq (0,0) \times S^{1},
\end{eqnarray*}
which is consistent with the result in \cite[Example 4.2]{HK:01}.

\begin{example}[Hurd-Sattinger]
Let us consider the initial value problem
\begin{equation} \label{ivp} \begin{split}
&\partial_t u + \Theta \partial_x u + \Theta' u  = 0   \\
&u(0,x)  = u_0 \in \col{\mathbb{R}},
\end{split}
\end{equation}
where $\Theta \in \col{\mathbb{R}^2}$  is defined by $\Theta_{\varepsilon}(x) =  \rho_{\gamma_{\varepsilon}} \ast H(-\cdot)$
 with $\rho_{\gamma_{\varepsilon}} = \frac{1}{\gamma_{\varepsilon}}\rho (\cdot /  \gamma_{\varepsilon})$
where $\rho \in \mathcal{S}(\mathbb{R})$ and $\int \rho(x) \m{x}=1$ and $\gamma_{\varepsilon}= 1/\log{(1/\varepsilon)}$
is a net of slow scale. For the initial value we choose $u_0 := \iota(\delta_{-s_0})$ a delta like singularity at $-s_0$
(for a positive $s_0>0$).
The Hurd-Sattinger example was first given in \cite{HS:68} (it was shown that it is not solvable in $L^1_{\rm{loc}}$,
 when distributional products are employed).
It was further investigated in \cite{HdH:01} with methods from Colombeau theory. In \cite{GH:05b}
the wave front set $\rm{WF}_{\gamma}$  (with respect to the slow scale net $\gamma$) of the Colombeau solution was calculated.
For sake of simplicity we do only consider the standard generalized wave front set $\rm{WF}$, which is smaller since
it neglects the singularities coming from the coefficient $\Theta$.
\end{example}
We can write
\begin{eqnarray*}
\Theta_{\varepsilon}(x) = g(x/\gamma_{\varepsilon}),
\end{eqnarray*}
where  $g(x):=\int_{x}^{\infty} \rho(z) \m{z}$ with $\lim_{x \rightarrow -\infty} g(x)=0$ and $\lim_{x \rightarrow +\infty} g(x)=1$
(this implies that $\overline{\ran{g}}=[\alpha_{+},\alpha_{-}]$ for some $\alpha_{-}\le 0, \alpha_{+}\ge 1$).

We have already considered (cf. Example \ref{hs_flow_example}) the ordinary differential equation for the characteristic curves
$$
\partial_s \xi_{\varepsilon}(s) = \Theta_{\varepsilon}(\xi_{\varepsilon}(s)),\quad\xi_{\varepsilon}(t)=x,
$$
and we obtained
$$
\ggraph{\xi(0;\bullet)}_{t,x} \subseteq \left\{
\begin{array}{ll}
\{\min{(x-t,0)}\} & x < 0 \\
\left[-t,0\right] & x = 0, 0 \le  t \\
\{x\} & x > 0.
\end{array}\right.
$$

If we set $f_{\varepsilon}(t,x):=(t, \xi_{\varepsilon}(0;t,x))$ and $f := [(f_{\varepsilon})_{\varepsilon}]$
then
\begin{equation*}
u = f^{\ast} (1 \otimes u_{0})  \cdot \partial_x f(t,x)
\end{equation*}
is a solution of the initial value problem (\ref{ivp}).

We observe that the derivatives of the characteristic flow satisfies
\begin{eqnarray*}
\partial_t \xi_{\varepsilon}(0;t,x) &=& - \Theta_{\varepsilon}(\xi_\varepsilon(0;t,x)) \ \text{and} \
\partial_x \xi_{\varepsilon}(0;t,x) = \frac{\Theta_{\varepsilon}(\xi_\varepsilon(0;t,x))}{\Theta_\varepsilon(x) },
\end{eqnarray*}
which follows from Remark \ref{ode_derivatives}.
So $\partial_x f_{\varepsilon}(t,x)$ is slow scaled in all derivatives,
due to the fact that $\Theta_{\varepsilon}(x) = g(x/\gamma_{\varepsilon})$ where
$\gamma_{\varepsilon}$ is a net of slow scale. We may conclude that
$\wf{u} \subseteq \wf{f^{\ast} (1 \otimes u_{0})}$, which enable us the apply Theorem \ref{maintheorem}
in order to estimate the wave front set of $u$.

We put $M=[(M_{\varepsilon})_{\varepsilon}] \in \colmap{\domainset{f}}{S^1}$ defined by
$$
M_{\varepsilon}(t,x,\eta) = \frac{{}^T df_{\varepsilon}(t,x)\eta}{|{}^T df_{\varepsilon}(t,x)\eta|}
$$
as in Lemma \ref{notionlemma}. Note that
$$
 {}^T df_{\varepsilon}(t,x)
= \left( \begin{array}{rr}
1 &  - \Theta_{\varepsilon}(\xi_\varepsilon(0;t,x))\\
0 & \frac{\Theta_{\varepsilon}(\xi_\varepsilon(0;t,x))}{\Theta_\varepsilon(x) }
\end{array} \right).
$$
For $\eta_{\pm} := (0, \pm 1) \in S^{1}$
(which are the only irregular directions coming from the wave front set of $1\otimes u_0$),
we obtain
\begin{eqnarray*}
M_{\varepsilon}(t,x,\eta_{\pm}) &=&\frac{df_{\varepsilon}(t,x)^T \eta_{\pm}}{|df_{\varepsilon}(t,x)^T \eta_{\pm}|}\\
&=&\pm \left( \frac{-1}{\sqrt{1+ \Theta_\varepsilon(x)^{-2}}}, \frac{1}{\sqrt{1+\Theta_\varepsilon(x)^2}} \right).
\end{eqnarray*}
Remarkably the result does only depend on the coefficient $\Theta_{\varepsilon}(x)$. In Example
\ref{heaviside_ggraph_example} we obtained
$$
\ggraph{(\Theta_{\varepsilon})_{\varepsilon}}
=
\left(\{x\in\mathbb{R}\mid x<0\} \times \{1\}\right) \cup \left(\{0\} \times \overline{\ran{g}} \right) \cup \left(\{x\in\mathbb{R}\mid x>0\} \times \{0\}\right),
$$
so Theorem \ref{generalized_graph_composition} yields
\begin{multline*}
\ggraph{M(\cdot,\eta_{\pm})}_{t,x} \subseteq
 \left( \left( \frac{\mp 1}{\sqrt{1+ \bullet^{-2}}}, \frac{\pm 1}{\sqrt{1+\bullet^2}} \right) \circ \ggraph{\Theta} \right)_{t,x} =
\\  \left\{
\begin{array}{ll}
\{\frac{1}{\sqrt{2}}(\mp 1, \pm 1)\}   &  \text{for}\ x<0\\
\{(0,\pm 1)\} &  \text{for}\ x>0\\
\bigcup_{\alpha\in[\alpha_{-},\alpha_{+}]} \{\left(\mp \frac{\alpha}{\sqrt{1+\alpha^2}} ,\pm  \frac{1}{\sqrt{1+\alpha^2}} \right)\}
 ,\ &\text{for}\ x=0.
\end{array} \right.
\end{multline*}
Furthermore we observe that
\begin{multline*}
(\wf{1\otimes u_0} \circ \ggraph{f})_{t,x} \\ \subseteq
 \left\{
\begin{array}{ll}
\wf{1\otimes u_0}_{t,\min{(x-t,0)}}  & x < 0 \\
\bigcup_{z\in \left[-t,0\right]}  \wf{1\otimes u_0}_{t,z}   & x = 0 \\
\wf{1\otimes u_0}_{t,x} = \emptyset & x > 0
\end{array}\right\}
=\left\{
\begin{array}{ll}
\{(0, \pm 1)\}  & x < 0, t=x-s_0 \\
\{(0, \pm 1)\}  &x = 0, t \ge  -s_0 \\
\emptyset & x > 0
\end{array}\right. \\
\end{multline*}
and according to Definition \ref{transformed_wf_set} we obtain
\begin{multline*}
(f^{\ast}(\wf{1 \otimes u_0}))_{t,x}= \ggraph{M} \circ ( \{(t,x)\} \times (\wf{1\otimes u_0} \circ \ggraph{f})_{t,x})
\\
=
\left\{ \begin{array}{ll}
 \{ (\mp 1, \pm 1) \}  & t<s_0, x=t-s_0 \\
\bigcup_{\alpha \in [\alpha_{-}, \alpha_{+}]}  \{ (\frac{\mp \alpha}{\sqrt{1+\alpha^2}}, \frac{\pm 1}{\sqrt{1+\alpha^2}} ) \} & t=s_0,x=0  \\
\{ (0, \pm 1 )\}  & t>s_0, x=0
\end{array}
\right.
\end{multline*}
and Theorem \ref{maintheorem} yields $\wf{u} \subseteq f^{\ast}(\wf{1 \otimes u_0})$.

config: arxiv


TITLE: Is every seminorm induced by a linear operator into a normed space?
QUESTION [6 upvotes]: I'm reading an analysis textbook chapter on convex topological vector spaces, and there is this statement that (one of) the most common way(s) to define a topology on a vector space $X$ is by requiring the continuity of certain linear maps, (i.e via the semi-norms induced by given linear maps $T\colon X\to Y$ where $Y$ is a normed space).
This makes sense, but then there's this that bugs me: "Can any semi-norm on $X$ be shown to be induced by some linear map T, to some normed linear space?" I can't find a counter-example to this, or show that it is true. Any insight would be appreciated.

REPLY [3 votes]: Let $p$ be a seminorm on a space $X$. Then $X/ \{x\in X\colon p(x)=0\}$ is a normed space under the norm $\|\pi(x)\|=p(x)$. Then the seminorm $p$ is implemented by the quotient map $\pi$.


TITLE: Geometry triangle question
QUESTION [6 upvotes]: In the figure below, AB=BC=CD. If the area of triangle CDE is 42, what is the area of triangle ADG?

I think triangles are similar. Are there any properties of similar triangles regarding their area. Help me get the answer with explanation please
ans :378

REPLY [0 votes]: Here is the Easiest Way :
since Theorem says- In two similar triangles, the ratio of their areas is the square of the ratio of their sides.
area of CDE/area of ADG=(CD/DA)^2
Here in figure suppose CD=x,then AD=3x(Already mentioned in question AB=BC=CD)
so
area of CDE/area of ADG=(x/3x)^2
42/area of ADG=1/9
area of ADG=378  ANSWER:378


TITLE: strongly convex normed space
QUESTION [0 upvotes]: A space is said to be convex if the unit ball is a convex set.
This definition gives the intuition about the convex normed space. Now I am wondering what would be the similar definition for a strongly convex space?
P.S. This is more related to Randers-Finsler spaces.

REPLY [0 votes]: I don't think the term "strongly convex" is commonly used for normed spaces. But if it was, then by analogy with strongly convex functions, which satisfy
$$
f\left(\frac{x+y}2\right) \le \frac{f(x)+f(y)}{2} - c\|x-y\|^2
$$
they would be "spaces with quadratic modulus of convexity".
The modulus of convexity is defined as the function
$${\displaystyle \delta (\varepsilon )=\inf \left\{1-\left\|{\frac {x+y}{2}}\right\|\,:\,\|x\|=\| y\| = 1,\ \|x-y\|\geq \varepsilon \right\},}$$
The space is uniformly convex if $\delta(\varepsilon)>0$ for every $\varepsilon \in (0, 2]$. This is already stronger than "strictly convex".
A stronger property yet is having quadratic modulus of convexity, which means there exists $c>0$ such that
$$
\delta(\varepsilon) \ge c\varepsilon^2, \quad \forall \ \varepsilon \in (0, 2]
$$
This is the strongest kind of convexity that makes sense for a normed space. Examples of such spaces are $L^p$ and $\ell^p$ with $1<p\le 2$.


TITLE: Is there any function such that $f(x)\to0$ as $x\to \infty$ but $x\log(x) f'(x)\to +\infty$?
QUESTION [3 upvotes]: My question es exactly the one in the title. I am trying to find any smooth function $f\in C^\infty((1,\infty))$ such that $$
f(x)\xrightarrow{x\to+\infty}0 \qquad \hbox{but} \qquad x\log(x)f'(x)\xrightarrow{x\to+\infty}+\infty.
$$
So far I've tried with $1/\log(x)$ and $1/\log(\log(x))$, etc, with no success. Now I am wondering if that kind of function can actually exists. A priory looks like a quite reasonable requirement, but surprisingly I cannot find any such function.

REPLY [7 votes]: If $x\log(x)f'(x)\to+\infty$ as $x\to+\infty$, then there exists $M$ such that
$f'(x)\ge\frac{1}{x\log(x)}$ whenever $x>M$. Integrating gives
$$f(x)\ge f(M)+\int_M^x\frac{dt}{t\log(t)}$$
whenever $x>M$. And this integral can be evaluated as
$$\log\log (x)-\log\log(M)$$
which tends to $+\infty$ as $x\to+\infty$, contradicting $\lim_{x\to+\infty}f(x)=0$.


TITLE: Exponential function negative: $\left(\frac{81}{4}\right)^{1/4}\left(\frac{1}4\right)^{-3/4}$
QUESTION [1 upvotes]: This is another example. $\left(\dfrac{81}{4}\right)^{1/4}\left(\dfrac{1}4\right)^{-3/4}$
Multiply on both sides equals  $\dfrac{81^{1/4}}{4^{1/4}}\cdot \dfrac{1^{-3/4}}{4^{-3/4}}$
This should be $\dfrac{3}{4^{1/4}}\cdot \dfrac{1^{-3/4}}{4^{-3/4}}$ which I get to -3/$\sqrt 4^{-1/2}$
Is that correct? I should get the answer 6.

REPLY [0 votes]: An alternative way (bring everything under a single exponent):
$$(\frac{81}{4})^\frac 14 \cdot (\frac{1}{4})^\frac {-3}{4} = (\frac{81}{4})^\frac 14 \cdot [(\frac 14)^{-3}]^{\frac 14} = (\frac{81}{4})^\frac 14 \cdot(4^{3})^{\frac 14}  = (\frac{81}{4}\cdot 64)^{\frac 14} = (81 \cdot 16)^{\frac 14} = 3 \cdot 2 = 6$$