text:
\section{Introduction}

Let $V=\K^\ell$, $S=\mbox{Sym}(V^*) =\K[x_1,\ldots,x_\ell]=\oplus_{d=0}^\infty S_d$ with the standard polynomial degree grading, and $\Der S=\oplus_{i=1}^\ell S \partial_{x_i}$. An \textbf{arrangement of hyperplanes} $\A$ is a finite set of linear hyperplanes in
$V$.
In the study of hyperplane arrangements, the most interesting and difficult problem is
to determine whether some property of $\A$ depends only on combinatorics. For example, when $\K=\CC$, the cohomology ring
$H^*(V \setminus \cup_{H \in \A} H;\Z)$ is shown to be combinatorial in \cite{OS}. However in general, to determine whether some property of $\A$ is combinatorial or not
is a hard problem. In this atricle, we will show that an important property of
algebra of $\A$ is combinatorial, which completes the recent sequental works by the author
in \cite{A}, \cite{A2}, \cite{A4} and \cite{A5}. For that, let us introduce a notation for algebra of $\A$.

For each $H \in \A$, let $\alpha_H$ denote a fixed defining linear form of $H$. Now the \textbf{logarithmic derivation module} $D(\A)$ of $\A$ is defined by
$$
D(\A):=\{\theta \in \Der S \mid \theta(\alpha_H) \in S \alpha_H\ (\forall H \in \A)\}.
$$
$D(\A)$ is an $S$-graded reflexive module of rank $\ell$, but not a free $S$-module in general.
We say that $\A$ is \textbf{free} with \textbf{exponents} $\exp(\A)=(d_1,\ldots,d_\ell)$ if there is a homogeneous $S$-basis $\theta_1,\ldots,\theta_\ell$ for $D(\A)$ such that $\deg \theta_i=d_i$ for
all $i$. Here we say that a non-zero derivation $\theta \in \Der S$ is \textbf{homogeneous of degree $d$} if
$\theta(\alpha) \in S_d$ for all $\alpha \in V^*$. For example, the \textbf{Euler derivation} $\theta_E:=\sum_{i=1}^\ell x_i \partial_{x_i}$, which is always contained in $D(\A)$, is homogeneous of degree $1$.

Because of several important results which relate the algebra $D(\A)$, topology and
combinatorics of $\A$, the logarithmic derivation module and free arrangements have been being studied. For example, see Theorem \ref{factorization}. However, it is not easy to check the freeness. The most useful result to check the freeness is the following addition-deletion theorem due to Terao in 1980.

\begin{theorem}[Addition-deletion theorem, \cite{T1}]
Let $H \in \A$, $\A':=\A \setminus\{H\}$ and $\A^H:=\{L \cap H \mid L \in \A'\}$. Then two of the following three imply the third:

\begin{itemize}
\item[(1)]
$\A$ is free with $\exp(\A)=(d_1,\ldots,d_{\ell-1},d_\ell)$.

\item[(2)]
$\A'$ is free with $\exp(\A')=(d_1,\ldots,d_{\ell-1},d_\ell-1)$.

\item[(3)]
$\A^H$ is free with $\exp(\A^H)=(d_1,\ldots,d_{\ell-1})$.
\end{itemize}

In particular, all the three above are true if both $\A$ and $\A'$ are free.
\label{additiondeletion}
\end{theorem}

Though Theorem \ref{additiondeletion} is proved almost 40 years before, it is the most useful to construct free arrangements, and to check the freeness and non-freeness even now. Related to this the relation between the freeness and combinatorics of $\A$ is well-studied too. Namely, the \textbf{intersection lattice} $L(\A)$ is defined by
$$L(\A):=\{
\cap_{H\in \B} H \mid
\B
\subset
\A\}.
$$
This is considered to be combinatorial information of $\A$. Then \textbf{Terao's conjecture} asks whether
the freeness depends only on $L(\A)$. Recently, the relation between Theorem \ref{additiondeletion} and combinatorics is intensively studied. See Theorems \ref{division} and  \ref{deletioncomb}
for example. In particular, the deletion theorem ((1)+(3) $\Rightarrow$ (2) in
Theorem \ref{additiondeletion}) is shown to be combinatorial in Theorem \ref{deletioncomb}.
Actually, in these results, not the total structure of $L(\A)$ but the following three
combinatorial objects play the key roles:

(i)\,\, The \textbf{localization}
$$\A_X:=\{H \in \A \mid X \subset H\}$$
of $\A$ at $X \in L(\A)$.

(ii)\,\,
The \textbf{restriction} $$
\A^H:=\{L \cap H \mid L \in \A \setminus \{H\}\}
$$
of $\A$ onto $H$,
which is an arrangement in $H =\K^{\ell-1}$. Also, for $X \in L(\A^H)$, define
$$
\A_X^H:=(\A_X)^H=(\A^H)_X.
$$
It is easy to show the second equality above.

(iii)\,\,
The \textbf{characteristic polynomial} $\chi(\A;t)$ of $\A$, see Definition \ref{cha}.

To state the main result in this article, let us introduce the following new
combinatorial property:

\begin{define}
Let $H \in \A$. We say that $\A$ is \textbf{divisional along $H$} if
$\chi(\A_X^H;t) \mid \chi(\A_X;t)$ for all $X \in L(\A^H)$.
We say that $\A$ is \textbf{locally (resp. globally) divisional along $H$} if
$\chi(\A_X^H;t) \mid \chi(\A_X;t)$ for all $X \in L(\A^H) \setminus \{\cap_{Y \in
\A^H} Y\}$ (resp. $\chi(\A^H;t) \mid \chi(\A;t))$.
\label{divisional}
\end{define}

By the divisibility along $H$, we can prove a combinatorial version of Theorem \ref{additiondeletion} as follows:

\begin{theorem}[Combinatorial addition-deletion theorem]
Let $H \in \A$, $\A':=\A \setminus\{H\}$ and $\A^H:=\{L \cap H \mid L \in \A'\}$. Then two of the following four imply the third and fourth:

\begin{itemize}
\item[(i)]
$\A$ is free.

\item[(ii)]
$\A'$ is free.

\item[(iii)]
$\A^H$ is free, and $\A$ is globally divisional along $H$.

\item[(iv)]
$\A$ is divisional along $H$.
\end{itemize}

In particular, all the four above are true if (iii) is true.
\label{additiondeletion2}
\end{theorem}

In Theorem \ref{additiondeletion2}, $(iii) \Rightarrow (i), (ii), (iv)$ is
the division theorem in \cite{A2} (Theorem \ref{division}). $(i)+(ii) \Rightarrow
(iii)+(iv)$ follows from Theorem \ref{additiondeletion} and Lemma \ref{LD}.
$(i)+(iv) \Rightarrow
(ii)+(iii)$ follows from \cite{A4} (Theorem \ref{deletioncomb}). So the rest part is
$(ii)+(iv) \Rightarrow
(i)+(iii)$, i.e., whether the addition theorem ((2)+(3) $\Rightarrow$ (1) in
Theorem \ref{additiondeletion})
is combinatorial or not.
This is the main part of
this article, as follows.

\begin{theorem}
For $H \in \A$, let $\A':=\A \setminus \{H\}$.
Assume that $\A'$ is free.
Then $\A$ is free
if and only if $\A$ is divisional along $H$. Hence whether $\A$ is free or not
depends only on $L(\A)$ when $\A'$ is free.
\label{main}
\end{theorem}

From the viewpoint of Theorems \ref{additiondeletion2} and \ref{main}, the addition-deletion theorem may be regarded as the result that implies the freeness of all the
triple $\A,\A \setminus \{H\},\A^H$ by (A) the freeness of one of them, and (B) one
combinatorial condition. So the following is the new interpretation of Theorem \ref{additiondeletion} from this viewpoint.

\begin{theorem}[Free triple theorem]
For $H \in \A$, assume that one of the following holds:
\begin{itemize}
\item[(i)]
$\A$ is free and $\A$ is divisional along $H$.
\item[(ii)]
$\A \setminus \{H\}$ is free and $\A$ is divisional along $H$.
\item[(iii)]
$\A^H$ is free and $\A$ is
globally divisional along $H$.
\end{itemize}
Then all of $\A,\A \setminus \{H\},\A^H$ are free.
\label{three}
\end{theorem}

Also, we can define the following class of free arrangements, which is very natural to consider, and
by which we have constructed a lot of free arrangements.

\begin{define}[Additionally free arrangements]
The set $\mathcal{AF}$ consists of
arrangements $\A$ such that there is a filtration $$
\emptyset=\A_0 \subset \A_1 \subset \cdots \subset \A_n=\A
$$
such that
$\A_i$ is divisional along $H_i$, where $\A_i \setminus \A_{i-1}=\{H_i\}$ for
$=1,\ldots,n$. Such a filtration is called the \textbf{additional filtration} of $\A$, and an arrangement $\A \in \mathcal{AF}$ is called an
\textbf{additionally free arrangement}.
\label{AF}
\end{define}

\begin{rem}
The additional filtration is the same as the \textbf{free filtration} introduced in \cite{AT0}. Because we call $\mathcal{AF}$ the set of additionally free arrangements, we use the terminology ``additional filtration'' here.
\end{rem}

In other words, an arrangement is additionally free if and only if that can be constructed from the empty arrangement by using the addition theorem, which is a really natural class of free arrangements. In fact, we can show the following contribution to Terao's conjecture in terms of additionally free arrangements.

\begin{theorem}
$\A$ is free if $\A \in \mathcal{AF}$. In particular, Terao's
conjecture is true in the class of additionally free arrangements.
\label{AFTC}
\end{theorem}

It was known that Terao's conjecture is true
in the class of inductively free arrangements (Definition \ref{IF}) which can be
constructed by using the addition and restriction theorem. In \cite{A2}, the new class so called the
divisionally free arrangement is defined (Definition \ref{DF}) in which Terao's conjecture is
true, and can be constructed by only using the division theorem (Theorem \ref{division}). Additionally free arrangements are the other direction of this generalization. Actually, we can show that the freeness of ideal-Shi arrangements depends only on
combinatorics  by using the additional freeness (Theorem \ref{idealShi}). Moreover, we define the largest class of free arrangements $\mathcal{SF}$ in which Terao's conjecture is true.
The class $\mathcal{SF}$ is called the \textbf{stair-free arrangements}, constructed in \S4 by combining both additionally and
divisionally free arrangements.

The organization of this article is as follows. In \S2 we recall several results on arrangements.
In \S3 we prove Theorem \ref{main}. In \S4 we study additionally free arrangements and its extension joining with the divisionally free arrangements. In \S5 we apply our main result to
ideal-Shi arrangements, proving that their freeness depends only on combinatorics.
\medskip

\noindent
\textbf{Acknowledgements}.
The author is
partially supported by KAKENHI, JSPS Grant-in-Aid for Scientific Research (B) 16H03924, and
Grant-in-Aid for Exploratory Research 16K13744.

\section{Preliminaries}

In this section let us collect several definitions and results on arrangements which will be used in this article. See \cite{OT} for general reference. First let us recall several combinatorics of arrangements.

\begin{define}
For the intersection lattice $$
L(\A):=\{\cap_{H \in \B} H \mid \B \subset \A\}
$$
of an arrangement $\A$,
define
$$
L_k(\A):=\{X \in L(\A) \mid \codim_V X=k\}.
$$
The \textbf{rank $r(\A)$} of an arrangement $\A$ is defined by
$$
r(\A):=\codim \cap_{H \in \A}H.
$$
For $X \in L(\A)$, the \textbf{localization}
$\A_X$ of $\A$ at $X$ is defined as
$$
\A_X:=\{H \in \A \mid X \subset H\}.$$
For $H \in \A$, the \textbf{restriction}
$\A_X$ of $\A$ onto $H$ is defined as
$$
\A^H:=\{H \cap L\mid L\in \A \setminus \{H\}\}.$$
Let $$
0_X:=\cap_{H \in \A_X}H \in L(\A_X)$$
for $X \in L(\A)$. So
$\rank (\A_X)=\codim_V 0_X$.
\end{define}

Next recall the most important combinatorial invariant of $\A$.

\begin{define}
(1)\,\,
The \textbf{M\"obius function} $\mu:L(\A) \rightarrow \Z$ is defined by
$\mu(V)=1$, and by
$$
\mu(X)=-
\sum_{Y \in L(\A),\ X \subsetneq
Y \subset V} \mu(Y)
$$
for $Y \in L(\A) \setminus \{V\}$.

(2)\,\,
The \textbf{characteristic polynomial} $\chi(\A;t)$ is defined by
$$
\chi(\A;t):=\sum_{X \in L(\A)} \mu(X)t^{\dim X}.
$$
\label{cha}
\end{define}

\begin{rem}
By definition $\chi(\A;t)$ is a combinatorial invariant of $\A$. Moreover, when $\K=\mathbb{C}$,
$(-t)^\ell \chi(\A;-1/t)$ coincides with the topological Poincar\'e polynomial of
$\mathbb{C}^\ell \setminus \cup_{H \in \A} H$. So it is a topological invariant too.
\end{rem}

To compute $\chi(\A;t)$, the following \textbf{deletion-restriction formula} is the most useful. For the proof,
see Corollary 2.57 in \cite{OT} for example.

\begin{theorem}
For $H \in \A$, it holds that
$$
\chi(\A;t)=\chi(\A \setminus \{H\};t)-\chi(\A^H;t).
$$
\label{DR}
\end{theorem}

We also use the following property, which directly follows from the definition of M\"obius
function in Definition \ref{cha} (1).

\begin{prop}
When $\A$ is not empty,
$\chi(\A;t)$ is divisible by $t-1$. Define the
\textbf{reduced characteristic polynomial}
$\chi_0(\A;t):=\chi(\A;t)/(t-1)$.
\label{reducedcha}
\end{prop}

The localization behaves nicely in the sense of freeness as follows, see
Theorem 4.37 in \cite{OT} for the proof.

\begin{theorem}
If $\A$ is free, then $\A_X$ is free
for all $X \in L(\A)$.
\label{localization}
\end{theorem}

We will see the relation between the localization and freeness for details in the following.

\begin{define}
Let $X \in L_k(\A)$, and let
$\emptyset_\ell$ denote the \textbf{empty arrangement} in $\K^\ell$. Then
$\A_X$ decomposes into
$$
\A_X=\B_X \times \emptyset_{\ell-k}$$
for some arrangement $\B_X$ in $V/X$ with
$\mbox{rank}(\B_X)=k$, and
the empty arrangement $\emptyset_{\ell-k}$ in $X$.
We say that $\B_X$ is the \textbf{essential part} of $\A_X$.
\label{essentialpart}
\end{define}

For the proof of the following, see Proposition 4.14 in \cite{OT} for
example.

\begin{prop}
Let $X \in L_k(\A)$ and $\B_X$ be the essential part of $\A_X$. Then
$\A_X$ is free if and only if $\B_X$ is free.
\label{directfree}
\end{prop}

Let us introduce the local version of freeness.

\begin{define}
We say that $\A$ is \textbf{locally free} if $\A_X$ is free for all $X \in L(\A) \setminus \{
0_V\}$.
\label{locallyfree}
\end{define}

It is known that $\A$ is locally free if and only if $\widetilde{D(\A)}$ is a vector bundle on
$\P^{\ell-1}=\mbox{Proj}(S)$.
We use the following famous criterion for freeness. For the proof of the arrangement version, see Theorem 4.19 in \cite{OT}.

\begin{theorem}[Saito's criterion, \cite{Sa}]
$\A$ is free if and only if there are homogeneous derivations $\theta_1,\ldots,\theta_\ell \in D(\A)$ such that they are independent over $S$, and that
$|\A|=\sum_{i=1}^\ell \deg \theta_i$.
\label{Saito}
\end{theorem}

The freeness is related to combinatorics and topology of arrangements by the following
famous result.

\begin{theorem}[Teraro's factorization, Main Theorem in \cite{T2}]
Let $\A$ be free with $\exp(\A)=(d_1,\ldots,d_\ell)$. Then
$$
\chi(\A;t)=\prod_{i=1}^\ell (t-d_i).
$$
\label{factorization}
\end{theorem}

Based on the division property in Definition \ref{divisional} with the theory of multiarrangements, following two generalizations of
Theorem \ref{additiondeletion} have been
obtained.

\begin{theorem}[Division theorem, Theorem 1.1, \cite{A2}]
Let $H \in \A$. If $\A^H$ is free, and $\chi(\A^H;t) \mid
\chi(\A;t)$, then $\A$ is free.
\label{division}
\end{theorem}

\begin{theorem}[Theorem 1.2, \cite{A4}]
Let $\A$ be free, $H \in \A$ and let
$\A':=\A \setminus \{H\}$. Then $\A'$ is free if and only if $\A$ is divisional along $H$.
\label{deletioncomb}
\end{theorem}

There are several class of free arrangements in which Terao's conjecture is true. Let us
recall two of them.

\begin{define}[\cite{OT}]
Let $\mathcal{IF}_\ell$ denote the set of arrangements in $\K^\ell$ defined in the following manner.

\begin{itemize}
\item[(1)] $\emptyset_\ell \in \mathcal{IF}_\ell$ for all $\ell \ge 0$.

\item[(2)]
$\A \in \mathcal{IF}_\ell$ if there is $H \in \A$ such that $\A \setminus \{H\}
\in \mathcal{IF}_\ell$, and $\A^H \in \mathcal{IF}_{\ell-1}$.
\end{itemize}
The arrangement $\A$ belonging to
$$
\mathcal{IF}:=\cup_{\ell \ge 0} \mathcal{IF}_\ell
$$
is called the \textbf{inductively free arrangement}.
\label{IF}
\end{define}

\begin{define}[\cite{A2}]
The class $\mathcal{DF}_\ell$ of arrangements in $\K^\ell$ consists of arrangements
$\A$ such that, there is $X_i \in L_i(\A)\ (i=0,\ldots,\ell-2)$ such that
$$
X_0=V \supset X_1 \supset \cdots \supset X_{\ell-2},
$$
and $\chi(\A^{X_i};t) \mid \chi(\A^{X_{i-1}};t)$ for $i=1,\ldots,\ell-2$. The set
$$
\mathcal{DF}:=\cup_{\ell \ge 0} \mathcal{DF}_\ell
$$
is called the set of \textbf{divisionally free arrangements}, and the above flag $\{X_i\}_{i=0}^{\ell-2}$ of
$\A$ is called a \textbf{divisional flag} of $\A$.
\label{DF}
\end{define}

\begin{theorem}[Theorems 1.3 and 1.6, \cite{A2}]
$$
\mathcal{IF} \subsetneq \mathcal{DF},
$$
and $\A \in \mathcal{DF}$ is free. Moreover, Terao's conjecture is true for divisionally
free arrangements.
\label{IFDF}
\end{theorem}

For the proof of our main result, let us recall the following key results. For the proof, see Proposition 4.41 in \cite{OT} for example.

\begin{prop}[\cite{T1}]
Let $H \in \A$. Then there is a polynomial $B$ of degree $|\A|-|\A^H|-1$ such that
$\alpha_H \nmid B$, and
$$
\theta(\alpha_H) \in (\alpha_H,B)
$$
for all $\theta \in D(\A \setminus \{H\})$.
\label{B}
\end{prop}

The following submodule of $D(\A)$ plays a key rote too in the proof of Theorem \ref{main}.

\begin{prop}[e.g., Lemma 1.33, \cite{Y3}]
For $H \in \A$, define
$$
D_H(\A):=\{
\theta \in D(\A) \mid \theta(\alpha_H)=0\}.
$$
Then
$$
D(\A)=S\theta_E \oplus D_H(\A).
$$
In other words,
$$
D(\A)/S\theta_E \simeq D_H(\A)$$
as $S$-graded modules.
Thus $\theta_E$ is a part of basis when $\A \neq \emptyset$ is free. So $\exp(\A)=(1,d_2,\ldots,d_\ell)$ if $\A$ is not empty.
\label{DH}
\end{prop}

Then we have the following algebro-geometric
interpretation of $\chi(\A;t)$ by using $D_H(\A)$.

\begin{theorem}[Theorem 4.1, \cite{MS}]
Assume that $\A$ is locally free,
$H \in \A$ and
let
$\widetilde{D_H(\A)}$ denote the sheafification of $D_H(\A)$ onto
$\P^{\ell-1} \simeq \mbox{Proj}(S)$. If $c_t(\widetilde{D_H(\A)})$ denotes the
Chern polynomial of $\widetilde{D_H(\A)}$, then
$$
t^{\ell-1} c_{1/t}(\widetilde{D_H(\A)})=\chi_0(\A;t).
$$
\label{MS}
\end{theorem}

\section{Proof of main results}

In this section we prove Theorem \ref{main}. First let us show the following
easy results.

\begin{lemma}
Let  $H \in \A$ and $\A':=\A \setminus \{H\}$. Assume that $\A'$ is free, and
$X \in L(\A^H)$. Then $\A_X':=\A_X \setminus \{H\}$ is free.
\label{LD0}
\end{lemma}

\noindent
\textbf{Proof}.
First assume that
$X \in L(\A')$ too. Then $(\A')_X=\A_X \setminus \{H\}$. Since $\A'$ is free, Theorem \ref{localization} shows that $\A_X'$ is free.
Next assume that $X \not \in L(\A')$. Then there is $Y \in L(\A')$ such that
$\codim Y+1=\codim X$ and that $X =Y
\cap H$. Moreover, there are no $L \in \A'$ such that $L \cap Y=
X$. Then $\A_X \setminus \{H\}=(\A')_Y$, which is free by Theorem \ref{localization}.\owari
\medskip

\begin{lemma}
If $\A$ and $\A':=\A \setminus \{H\}$ are both free, then $\A$ is divisional along $H$.
\label{LD}
\end{lemma}

\noindent
\textbf{Proof}.
Apply Theorems \ref{additiondeletion}, \ref{factorization}, and Lemma \ref{LD0}.
\owari
\medskip

\begin{lemma}
Assume that $\A$ is divisional along $H \in \A$. Then for any $X \in L(\A^H)$,
$\A_X$ is divisional along $H$. Moreover, if $\B_X$ denotes
the essential part of $\A_X$, then $\B_X$ is divisional along $H$ too.
\label{localdivisional}
\end{lemma}

\noindent
\textbf{Proof}.
Immediate by Definitions \ref{divisional}, \ref{cha} and \ref{essentialpart}. \owari
\medskip

Now let us show the main result in this
section, which is the key to prove Theorem \ref{main}.

\begin{theorem}
Assume that $\A$ is locally free, and $\A'$ is free. Then $\A$ is
free if and only if $\chi(\A^H;t) \mid \chi(\A;t)$.
\label{maincor}
\end{theorem}

\noindent
\textbf{Proof}.
The statement is true if $\A'=\emptyset$. Assume not in the following.
If $\A$ and $\A'$ are free, then Lemma \ref{LD} implies that
$\A$ is divisional along $H$. So it suffices to show that
$\A$ is free if $\A'$ is free, $\A$ is locally free and
$\chi(\A^H;t) \mid \chi(\A;t)$.
If $\exp(\A')=(1,d_2,\ldots,d_\ell)$ with
$d_2 \le \cdots \le d_\ell$, then Theorem \ref{factorization} implies that
$$
\chi_0(\A';t)=\prod_{i=2}^\ell (t-d_i).
$$
Since $\chi(\A^H;t) \mid
\chi(\A;t)$, Theorem \ref{DR} shows that $\chi(\A^H;t) \mid
\chi(\A';t)$ too, and thus
$|\A'|-|\A^H|=d_j$ for some $j$. We may assume that $d_j<d_{j+1}$, or $j=\ell$. Let
$I:=\{2,\ldots,\ell\}
\setminus \{j\}$.
Then by Theorems \ref{additiondeletion} and \ref{factorization},
\begin{eqnarray*}
\chi_0(\A;t)&=&(t-d_j-1)\prod_{i \in I} (t-d_i),\\
\chi_0(\A^H;t)&=&\prod_{i \in I} (t-d_i).
\end{eqnarray*}
Let $\theta_1=\theta_E,\theta_2,\ldots,\theta_\ell$ be a homogeneous
basis for $D(\A')$ with $\deg \theta_i=d_i$. If $\theta_k
\not \in D(\A)$ for some $k$ with $d_k=d_j$, then we can show that
$\A$ is free. Namely, by Proposition \ref{B}, there is a homogeneous polynomial $B$ of
degree $|\A'|-|\A^H|=d_j$ such that
$$
\theta(\alpha_H) \in (\alpha_H,B)
$$
for all $\theta \in D(\A')$. So $\theta \in D(\A')$ is in $D(\A)$ if
$\deg \theta <d_j$. Now assume that $\theta_k \not \in D(\A)$ for some $k$ with
$d_k=d_j$. We may assume that $k=j$. Then $\theta_j
(\alpha_H)=B$ modulo $\alpha_H$. Let $\theta_i(\alpha_H)=a_i \alpha_H+b_i B$. Then replacing
$\theta_i$ by $\theta_i-b_i \theta_j$, we may assume that
$\theta_i \in D(\A)$ if $i \neq j$. Thus Theorem \ref{Saito} shows that
$\theta_1,\theta_2,\ldots,\theta_{j-1},\alpha_H\theta_j,\theta_{j+1},\ldots,\theta_\ell$ form a
basis for $D(\A)$.

Assume not, i.e.,
$$
D':=\oplus_{i=2}^j S\theta_i \subset D(\A).
$$
By replacing $\theta_i \ (i=2,\ldots,j)$ by
$\theta_i-(\theta_i(\alpha_H)/\alpha_H)\theta_E$, we may assume that
$\theta_i \in D_H(\A)$ for $i=2,\ldots,j$.
By composing the inclusion $D(\A) \subset D(\A')$ and
the canonical projection $D(\A') \rightarrow D'$, we have a
surjection
$$
p:D(\A) \rightarrow D'
$$
which has a canonical section by the above. Hence
$$
D(\A)=S\theta_E \oplus D' \oplus M
$$
for some $S$-subumodule $M$ of $D(\A)$.
By Proposition \ref{DH},
$$
D_H(\A) \simeq D(\A)/S\theta_E=D' \oplus M.
$$
Since $\A$ is locally free, Theorem \ref{MS} implies that
$$\chi_0(\A;t)=t^{\ell-1} c_{1/t}(\widetilde{D_H(\A)}).
$$
Since $D_H(\A) \simeq D' \oplus M$, it holds that
$$
c_{1/t}(\widetilde{D_H(\A)})=
c_{1/t}(\widetilde{D'})c_{1/t}(\widetilde{M}).
$$
Noting that $D'=\oplus_{i=2}^j S[-d_i]$,
$$
t^{\ell-1}c_{1/t}(\widetilde{D_H(\A)})=t^{\ell-j}c_{1/t}(\widetilde{M})\prod_{i=2}^j (t-d_i).
$$
Here
$$
\rank(\widetilde{M})=
\rank(\widetilde{D_H(\A)})-
\rank(\widetilde{D'})=\ell-j.
$$
Since $\widetilde{D_H(\A)}$ is locally free, so is
$\widetilde{M}$. Hence
$\deg c_t(\widetilde{M})=\ell-j$. So
$c(t):=c_{1/t}(\widetilde{M})t^{\ell-j} \in \Z[t]$.
Therefore
$$
\chi_0(\A;t)=(t-d_j-1)
\prod_{i \in I}^\ell (t-d_i)=
c(t)\prod_{i=2}^j(t-d_i).
$$
Assume that the multiplicity of $d_j$ in $\chi_0(\A';t)$ is $a \ge 1$. Then
$d_j<d_{j+1}$ or $j=\ell$ implies that
$C(t):=c(t)\prod_{i=2}^j(t-d_i)$ is divisible by
$(t-d_j)^a$, but $\chi_0(\A;t)$ is not, a contradiction. \owari
\medskip

\noindent
\textbf{Proof of Theorem \ref{main}}.
The statement is true if $\A'=\emptyset$. Assume not in the following. If $\A$ is free, then
$\A$ is divisional along $H$ by Lemma \ref{LD}. So it suffices to show that $\A$ is free if $\A'$ is free, and
$\A$ is divisional along $H$.

We prove by induction on $\rank(\A) \ge 0$. We may assume that $\rank(\A)=\ell$ by
Definition \ref{essentialpart}. The statement is true if $\ell \le 3$ by \cite{A}, or
Theorems \ref{factorization} and \ref{division}. Assume that $\ell \ge 4$. By Theorem \ref{maincor}, it suffices to show that $\A$ is locally free. Since
$\A'$ is free, $\A_X=(\A')_X$ is free too if $X \not \subset H$.
So let
$X \in L_i(\A) \setminus \{0_V\}$ with $
X \subset H$.
By Lemma \ref{LD0}, $\A_X \setminus \{H\}$ is free.
Let $\B_X$ be the essential part of
$\A_X$ as in Definition \ref{essentialpart}. In other words, $\A_X =\B_X \times \emptyset_{\ell-i}$, and $i =\rank(\B_X)< \rank (\A)=\ell$. Then by
Proposition \ref{directfree}, $\B_X \setminus \{H\}$ is free too.
Since $\B_X$ is of rank $i<\ell$, divisional along $H$ by Lemma \ref{localdivisional}, and
$\B_X \setminus \{H\}$ is free, the induction hypothesis on $\ell$ shows that $\B_X$ is free.
Thus $\A$ is locally free, and Theorem \ref{maincor} completes the proof. \owari
\medskip

\noindent
\textbf{Proof of Theorems \ref{additiondeletion2} and \ref{three}}. Immediate by
Theorems \ref{additiondeletion}, \ref{main}, \ref{factorization}, \ref{division}, and \ref{deletioncomb}. \owari
\medskip

Finally we pose several conjectures related to Theorem \ref{main}.

\begin{conj}
For $H \in \A$, let $\A':=\A \setminus \{H\}$. Then
\begin{itemize}
\item[(1)]
$\A$ is free if $\A'$ is free, and $\A$ is globally divisional along $H$.
\item[(2)]
Assume that
\begin{eqnarray*}
\chi_0(\A;t)&=&(t-d_2-1)\prod_{i=3}^\ell (t-d_i),\\
\chi_0(\A';t)&=&(t-d_2)\prod_{i=3}^\ell (t-d_i)
\end{eqnarray*}
for some integers $d_2,\ldots,d_\ell$.
Then $\A$ and $\A'$ are both free.
\end{itemize}
\label{conj}
\end{conj}

Note that Conjecture \ref{conj} (1) and (2) are true
if $\ell \le 3$ by Theorem \ref{main}, or Theorem \ref{division}.

\section{Additionally and stair-free arrangements of hyperplanes}

Theorem \ref{AFTC} is clear by Theorem \ref{main}.
Let us show the relation among
inductively, divisionally and additionally free arrangements.

\begin{prop}
It holds that
$$
\mathcal{AF} \supset \mathcal{IF} \subsetneq \mathcal{DF}.
$$
Moreover, $\mathcal{DF}  \not \subset \mathcal{AF}$.
\end{prop}

\noindent
\textbf{Proof}.
$\mathcal{IF} \subset \mathcal{AF}$ is clear by Definitions \ref{AF} and \ref{IF}.
$\mathcal{IF} \subsetneq \mathcal{DF}$ follows by Theorem \ref{IFDF}. Moreover,
by Lemma 3.13 in \cite{M},
we know that
$\A(G_{31}) \in \mathcal{DF}  \setminus \mathcal{AF}$. Here
$G_{31}$ is one of the finite unitary reflection groups
classified and labeled by Shephard and Todd in \cite{ST}. So
$\A(G_{31})$ is the corresponding unitary reflection arrangement in
$\mathbb{C}^4$.
\owari
\medskip

Now we have two class $\mathcal{DF}$ and $\mathcal{AF}$ of free arrangements which are larger than $\mathcal{IF}$. Since their definition is easy to deal with, we can join these two to
obtain the new largest class of free arrangements in which
Terao's conjecture is true.

\begin{define}[Stair-free arrangements]
The set $\mathcal{SF}_\ell$ consists of hyperplane arrangements $\A$
in $\K^\ell$ which satisfy the following; there are $H_1^i,\ldots,H_{j_i}^i, H_i\
(i=3,\ldots,\ell-1)$ in $L_{\ell-i}(\A)$ such that
\begin{itemize}
\item[(1)]
$\A_\ell:=\A,\
\A_{i}:=(\A_{i+1} \setminus \{H_j^i\}_{j=1}^{j_i})^{H_i}$. So
$H_j^i ,\ H_i\in \A_{i+1}$.
\item[(2)]
$\A_i^k:=\A_i \setminus \{ H_j^{i-1}\}_{j=1}^k$ is disivional
along $H_{k+1}^{i-1}$ for all $i,k$.
\item[(3)]
$\chi(\A_i;t)$ divides $
\chi(\A_{i+1}^{j_i};t)$.
\end{itemize}
$\A \in \mathcal{SF}$ is called a \textbf{stair-free arrangement of hyperplanes}.
\label{ADF}
\end{define}

By Theorems \ref{main} and \ref{division}, the following is clear.

\begin{theorem}
\begin{itemize}
\item[(1)]
$\A$ is free if $\A \in \mathcal{SF}$.

\item[(2)]
It holds that
$$\mathcal{IF}
 \subsetneq \mathcal{DF}\cup
\mathcal{AF} \subset \mathcal{SF}.
$$

\item[(3)]
Terao's conjecture is true in the class of stair-free arrangements.
\end{itemize}
\label{ADFTC}
\end{theorem}

When you apply the division theorem, the arrangement moves vertically, i.e., the direction
of dimensions, and the addition theorem horizontally, i.e., the direction of the cardinality of hyperplanes. If you connect these paths, then it looks like a stair. That is the reason of the name in
Definition \ref{ADF}. A typical example of a stair-free arrangement is the Catalan and Shi  arrangements, which will be explained in \S5. In \cite{AT}, it is shown that the freeness of Catalan arrangement is combinatorial. Seeing the proof, \cite{AT} shows that Catalan arrangement is stair-free.

Related to these new classes, we have the following conjecture.

\begin{conj}
\begin{itemize}
\item[(1)]
$$
\mathcal{IF} \subsetneq \mathcal{AF}.
$$
\item[(2)]
$$
\mathcal{DF} \not \supset \mathcal{AF}.
$$
\item[(3)]
$$
\mathcal{DF} \cup \mathcal{AF} \subsetneq \mathcal{SF}.
$$
\end{itemize}
\end{conj}

\section{Applications to root systems}

In this section we apply Theorem \ref{main} to explicit examples, showing the combinatorial dependency of their freeness.

In this section let $\K=\R$, thus $V=\R^\ell$. Let $W$ an irreducible crystallographic Weyl group of rank $\ell$ acting on $V$.
Let $\Phi$ be the corrsponding root system to $W$, $\Phi^+$ the set of positive roots, and
$\Delta=\{\alpha_1,\ldots,\alpha_\ell\}$ the set of simple roots.

A lot of arrangements related to $W$ are check to be free, and their freeness are often
proved to be combinatorial.
For example, if $\A_W$ denotes the set of all reflecting hyperplanes of reflections in $W$, then $\A_W \in \mathcal{IF}$, thus its freeness depends only on combinatorics.
To review some of them, let us introduce some notation.

For $\alpha \in \Phi^+$, let $H_\alpha$ denote the reflecting hyperplane of $\alpha$, and
$$
H_\alpha^k:=\{\alpha=k\} \subset V
$$
be an affine hyperplane for $k \in \Z$. Let $cH_\alpha^k$ be the cone $\alpha=kz$ of $H_\alpha^k$ by the new coordinate in $z$, i.e., $cH_\alpha^k$ is a hyperplane in $\R^{\ell+1}=
\mbox{Spec}(S[z])$.
Then for $m \in \Z_{>0}$, the \textbf{Catalan arrangement} $\Cat^m$ is defined by
$$
\Cat^m:=\{z=0\} \cup \{cH_\alpha^k\}_{\alpha \in \Phi^+,-m \le k \le m},
$$
and the \textbf{Shi arrangement} $\Shi^m$ is defined by
$$
\Shi^m:=\{z=0\} \cup \{cH_\alpha^k\}_{\alpha \in \Phi^+,-m+1 \le k \le m}.
$$
Both are free by \cite{Y1}, and their freeness depends only on the lattice by
\cite{A2} for Shi arrangements, and \cite{AT} for Catalan arrangements. Also, we say that
$I \subset \Phi^+$ is a \textbf{lower ideal} if
$\alpha \in I,\ \beta \in \Phi^+,\
\alpha-\beta \in \sum_{i=1}^\ell \Z_{\ge 0} \alpha_i$ implies $\beta \in I$. Then it was shown that
$\A_I:=\{H_\alpha \mid \alpha \in I\}$ is free in \cite{ABCHT}. Moreover, if we define the
\textbf{ideal Shi arrangements}
\begin{eqnarray*}
\Shi_{+I}^m:&=&\Shi^m \cup \{cH_\alpha^{-m}\mid
\alpha \in I\},\\
\Shi_{-I}^m:&=&\Shi^m \setminus \{cH_\alpha^{m}\mid
\alpha \in I\},
\end{eqnarray*}
then it is shown in Theorem 1.6 in \cite{AT0} that they are both free, and
the freeness of $\Shi_{+m}^I$ depends only on combinatorics. So the left problems is that of $\Shi_{-I}^m$. Since ideal-Shi arrangements have a free filtration by
Theorem 1.5 in \cite{AT0}, which is nothing but the additional filtration,
Theorem \ref{AF} implies the following.

\begin{theorem}
It holds that
$$
\Shi_{\pm I}^m \in \mathcal{AF}.
$$
Thus, the freeness of $\Shi_{\pm I}^m$ depends only on
$L(\Shi_{-I}^m)$ for all lower ideal $I \subset \Phi^+$ and $m \in \Z_{\ge 0}$.
\label{idealShi}
\end{theorem}

Since $\Shi_{+\Phi^+}^m=\Cat^m$ and $\Shi_{\emptyset}^m=\Shi^m$, Theorem \ref{idealShi} gives a uniform proof of the combinatorial dependency of their freeness.

redpajama_set_name: RedPajamaArXiv


Game 9 of the 2018 International WFTDA Championships New Orleans between Angel City Derby of Los Angeles, Calif., U.S. and Montréal Roller Derby of Montréal, Québec, Canada, the final consolation game of the weekend, was as intense as the road was for both teams this weekend. Despite a close first period, Angel City came out with the solid win 199-118.
Both teams won their first game of the weekend. Montréal won in a nailbiter against Crime City Rollers (184-180) and Angel City won big over Jacksonville Roller Derby (248-85). The teams proceeded to play the two top-seeded teams — Montréal took on Rose City Rollers Friday night and lost in a respectable game (222-107). Angel City who lost to the reigning Hydra Cup champions, Victorian Roller Derby League (170-95).
This game had all of the ingredients of a close match up, and up until 19 minutes into the game when Montréal led by 8 points, it was. At that point Angel City earned multiple lead jams in a row, and slowed down Montréal's scoring. With much help from a 19-point jam by Rachel Rotten, Angel City headed into halftime with a healthy 115-66 lead.
There isn't enough space here to describe the agility demonstrations and strength and determination by both team's jammers, or the grueling defense of both teams' blockers, not to mention the swarming mixed-up packs that had fans wondering who was doing what.
Angel City was able to lock down and force drawbacks of Montréal's jammers more often than Montréal was able to do to Angel City's jammers and the precious time it took to recycle impacted Montréal's ability to score and put away the win.


Perched above Newcastle Harbour, this sensational apartment offers extraordinary living. Located within the iconic Breakwater Apartments, a waterfront complex offering just 43 tightly held apartments, this is truly something special! With famous views up the Harbour towards Nobby's and across to Dyke Point, Stockton and beyond, it is a thrilling sun-lit gem.
Offering 3 bedrooms, all with great space, the master suite being the piece de resistance - a generous room with extensive northerly views over Newcastle Harbour. The open plan living/ dining area also has a sunny opulence, opening to a wonderful wrap around balcony, while the kitchen features stone benches and top-line integrated appliances. With the huge bonus of parking for 2 cars and 2 lockable storage areas, this is a magnificent apartment offering absolute Newcastle Harbour living.


text:
Arts & Culture, news 1 Comment
Photos: This Year's Bushwig Was Simply Divine
September 25, 2017 By Scott Lynch
A tribute to John Waters legend Divine. (Photo: Scott Lynch)
The fifth annual Bushwig festival turned the Knockdown Center into a rowdy two-day celebration of "Drag, Music & Love" this weekend, as nearly 150 queens and kings performed for thousands of fans of all things cheeky, flirty, pretty, raw, and/or raunchy. This was the biggest Bushwig yet, with two stages–one in the venue's cavernous main hall, and another out in the rambling backyard–plus several first-rate food vendors, plenty of access to booze, and a whole separate room devoted to "Bushswag!"
If you've never been, the basic Bushwig entertainment runs like this: a performer– usually in drag but sometimes just rocking a wild, sexy outfit, gets up on stage and lip-syncs her/his/their way through a single song. These are all spectacularly good, with tight choreography, elaborate makeup and props, and an abundance of exuberance and charm.
Audience members shower the best performers with dollar bills, but Bushwig isn't a competition. In fact, the overall atmosphere is enormously supportive, inclusive, and loving. As the official website states, "We like to think our attendees do not just go to a festival, they join a family. Our mission is to change the face of queer event culture."
Click through our slideshow to witness this "wondrous world of discovery and adventure."
« Bob Dylan Is the Village Voice's Final Cover Boy, But He Wasn't the First to Record 'Blowin' in the Wind'	» At Home With Amy Sedaris Is a Treat For Strangers With Candy Fans, Colbert and All
Jolene Oldham
Sep 29, 2017 @ 06:30
I think this is wonderful! I love the non-competitive spirit of fun! Everyone looks wonderful!
I especially love the shot of the Divines in the header…that's my girl – Regina Stargazer – in the center! I'm a very proud mommy!

redpajama_set_name: RedPajamaCommonCrawl


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Watch Live: Senate Impeachment Trial of President Trump Begins
NBA Star Derrick Rose Takes the Stand in Rape Trial: 'I Didn't Do Anything Wrong'
Posted 10:08 PM, October 7, 2016, by Los Angeles Times, Updated at 10:10PM, October 7, 2016
NBA star Derrick Rose took the stand Friday, denying allegations that he and two friends raped a former girlfriend.
Derrick Rose waits for a reporter's question during a press conference at the New York Knicks Media Day on Sept. 26, 2016, in White Plains, New York. (Credit: EDUARDO MUNOZ ALVAREZ/AFP/Getty Images)
Rose, a player for the New York Knicks, and the two other men are being sued in federal civil court by a woman who claims that they drugged her while she was drunk and later sneaked into her apartment to have sex with her while she was incapacitated. The plaintiff has asked to be awarded $21 million.
The men have not been charged criminally in the case. Los Angeles police officials have said they are continuing to investigate the 2013 incident, which the woman reported in 2015.
Rose, 28, was called to testify by his accuser's attorney in the afternoon and answered questions for several hours until U.S. District Judge Michael W. Fitzgerald ended testimony for the day. Rose will continue his testimony Tuesday when the trial resumes. The other two defendants, Randall Hampton and Ryan Allen, childhood friends of Rose's whom he now employs, are expected to testify later.
Click here to read the full story on LATimes.com.
Civil Rape Trial of NBA Star Derrick Rose Underway in L.A.
Derrick Rose's Accuser in Rape Allegation Lawsuit Must Reveal Her Identity: Federal Judge in L.A.
NBA Star Derrick Rose Under Investigation for Allegedly Raping Intoxicated Woman With 2 Friends: LAPD
Topics: Derrick Rose, Los Angeles County Superior Court, NBA
5th Body Found at Tijuana Home Where Garden Grove Couple Was Discovered Buried: Authorities
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Robert Durst Told Police a Body Was in Susan Berman's Benedict Canyon Home: Lawyers
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Surge in Anti-Semitic Attacks Weighs Heavily on U.S. Jewish Youth


text:
Families Separated at US Border Now Fear Extortion Attempts
WASHINGTON - For the 30-year-old Honduran woman, the worst seemed to be over. She's been reunited with her son who, as a 6-year-old, was separated from her under the Trump administration. She's working construction in North Carolina. And attorneys were negotiating a payment for families like hers that endured separations.
But reports about those negotiations have created a new worry: extortion attempts stemming from the mistaken belief that she received a huge payout. Her family has already received demands for $5,000 a month.
'Apparently, I am a millionaire now,' said the woman, who, like others interviewed by The Associated Press, spoke on condition of anonymity due to fears for her family's safety. 'I don't have the money to pay for something like that and I don't know what to do. I am desperate, really.'
While specific reports are isolated, widespread extortion in Central America explains why many seek asylum in the United States in the first place. Some advocates fear prospects of large payments will fuel many more threats. An attorney for the woman and other families has asked U.S. officials to consider admitting more relatives because of the threats.
It is far from clear whether families will receive any money at all from the U.S. government. Negotiations to settle claims for damages ended amid political outrage over payments erupted following a report in the The Wall Street Journal that the Justice Department was considering $450,000 a person to compensate for suffering - or $900,000 for a parent and child. A person familiar with the talks who spoke to the AP on condition of anonymity because discussions were private confirmed that figure had been floated.
'People here think that I have lots of money,' said a 47-year-old business owner in northern Guatemala whose wife was separated from their son. He has become more nervous because of news reports on the settlement talks and now changes his cellphone number every two weeks.
The man lives in Guatemala with his 14-year-old daughter, while his wife and now 18-year-old son live in Atlanta after being separated at the border for more than a month in 2018. The man said he was getting text messages at the time threatening to kidnap his son if he didn't pay money.
'My neighbor told me the other day, 'So you have money, because money was given to people who were separated in the United States.' And I told him that I did not know anything about that,' he said.
FILE - Unaccompanied migrant minors, aged 3 to 9, watch television inside a playpen at the U.S. Customs and Border Protection facility, the main detention center for unaccompanied children in the Rio Grande Valley, in Donna, Texas, March 30, 2021.
The man said he and his daughter tried going to the U.S. in 2019. They were kidnapped in Mexico for two weeks, released to Mexican authorities after paying more than $3,000 and deported to Guatemala.
'I don't live in peace,' he said. 'I am always looking over my shoulder.'
Ricardo de Anda, an attorney for the Honduran woman and Guatemalan man, said five of the 72 families he represents have told him they were threatened after news coverage of the possible payments. One in Guatemala was targeted in an attempted kidnapping.
'These families have told us that they are now the subject of rumors in their communities as to the apparent wealth of family members in the U.S., that they have been subjected to surveillance by apparent criminal elements, and that they have been warned to be vigilant as criminal gangs are considering them the subjects of extortion,' he wrote to Michelle Brane, executive director of the Homeland Security Department's Family Reunification Task Force. 'As a result of the (news) leaks, family members in the U.S., and those trapped in home country, now live in constant fear.'
The task force, which is aiming to reunite nearly 2,000 children with their parents in the United States, had planned for the possibility of extortion, realizing that such threats are common in Central America, and set up a system to channel reports through the U.N. refugee agency, Brane said in an interview last month.
Brane said she had not yet received any specific reports but the potential danger underscores the need for the task force to complete its work.
'If families are in unsafe situations and need reunification, we are here to work and get that done as soon as possible,' she said.
The task force has reunited about 112 children with their parents in the U.S. as of last week. They are being granted permission to stay in the country for at least three years while they pursue asylum or seek permanent status through another program.
FILE - David Xol-Cholom, of Guatemala, hugs his son Byron Xol-Cholom at Los Angeles International Airport as they reunite after being separated during the Trump administration's wide-scale separation of immigrant families, in Los Angeles, Jan. 22, 2020.
Other attorneys for the families said they had no direct knowledge of threats tied to possible payments but said they were inevitable, if they haven't happened already. The attorneys suspect some attempts have gone unreported or word hasn't reached them.
'I have no doubt that it's happening in more cases than we know about,' said Trina Realmuto, executive director of the National Immigration Litigation Alliance, which was involved in settlement talks over financial compensation.
The talks are delicate for the administration, which has been criticized for considering large payouts. President Joe Biden himself said, 'That's not going to happen,' when asked in November about the $450,000 figure, and later clarified that he backed some compensation.
Last month, the Justice Department withdrew from talks over financial compensation after eight months but didn't rule out an agreement.
'While the parties have been unable to reach a global settlement agreement at this time, we remain committed to engaging with the plaintiffs and to bringing justice to the victims of this abhorrent policy,' the department said in a statement.
This month, attorneys for families renewed a request for the administration to turn over troves of records on how the policy was conceived and executed, signaling a potentially long court battle.
Talks have continued over non-monetary issues, including reunifying families in the United States and other services, such as mental health, attorneys said.
De Anda asked the administration to consider admitting family members who were threatened since the news reports. The administration has focused on parents and children who were separated but says it will consider additional families case by case.
The Honduran woman said her 56-year-old mother has received notes asking for $5,000 a month. The mother cares for the woman's other children, an 11-year-old son and 7-year-old daughter. The woman wants all to join her in North Carolina.
Acquaintances have warned that the children may be unsafe in Honduras.
'I am scared,' said the woman, who takes pills for anxiety and went to the emergency room with chest pains after the threats against her mother. 'I don't know what can happen to my kids.'


RM Auctions announces the sale of a private, single-owner collection of iconic American hot rods, vintage racers, and one-off custom show cars. This one-day auction event, billed as "Icons of Speed & Style" is set for September 26, 2009, at the Petersen Automotive Museum in Los Angeles, California.
Approximately 80 vehicles and several lots of period hot rod and "kustom kulture" memorabilia will be offered "without reserve". The eclectic collection represents the owner's enthusiasm for and lifelong love of cars, and its offering will be a once-in-a-lifetime buying opportunity for collectors.
Legendary cars such as the famous Dodge "Little Red Wagon" Drag Truck, the dominating Ray Allen LS6 Chevelle Convertible, "Big John" Mazmanian's Corvette gasser, the original "Road Agent" show car by Ed "Big Daddy" Roth, as well as famous customs from the Ayala Brothers, Sonny Mazza, the Alexander Brothers, George Barris, Daryl Starbird, and Boyd Coddington will be on offer.
As specialists for estate and private collection sales, RM has attained considerable recognition in the collector car market by providing an unsurpassed level of marketing and cataloging services for such diverse clients as Formula One racing CEO Bernie Ecclestone, Orange County motoring legend Joe McPherson, broadcasting entrepreneur Art Astor, Michigan businessman John McMullen, actor Nicolas Cage and kustom kulture collectors, the Brucker Brothers. The Icons of Speed & Style collection has been extensively photographed in detail for the full-color collector's catalog and is available for $80 U.S. (plus shipping) by calling 519-352-4575 and admits two to the auction. The list and descriptions of the offered cars, in addition to event times, bidding options, and directions can be found www.rmauctions.com.


No electricity or running water meant no showers for a week. Though I'm not averse to a little improvising, I was kind of worried about my hair getting greasy and looking disgusting.
To my surprise, there was something about the mountain air that kept it completely clean. Still, after a week of wearing hunting gear and napping in the dirt, I was in for the best shower of my life.
The first elk that is caught in the hunt, is celebrated by eating the heart. This heart was sliced into strips and sauteed in a cast-iron skillet. It was so large that it fed 8 grown men and 2 women.
Wild elk meat is incredibly tender and tastes fairly similar to beef, as elk feed on grasses too. The elk that is sold in restaurants however, isn't wild meat, but farmed, and the taste does not quite compare.


text:
Laura Pole, RN, MSN, OCNS, BCCT Senior Researcher
Ms. Pole is an oncology clinical nurse specialist who has been providing integrative oncology clinical care, navigation, consultation and education services for more than 30 years. View profile.
Nancy Hepp, MS, BCCT Project Manager
Ms. Hepp is a science researcher and communicator who has been writing and editing educational content on varied health topics for more than 20 years. View profile.
Also known by these names
Polyphenolic flavonoid
3,3',4',5,7-pentapentahydroxyflavone
Before using this therapy, consult your oncology team about interactions with other treatments and therapies. Also make sure this therapy is safe for use with any other medical conditions you may have.
Quercetin is a dietary flavonoid found in certain fruits and vegetables and some medicinal plants.
Quercetin is used widely as an antioxidant.
Quercetin may have anti-inflammatory and cancer prevention effects, although it's also been shown to block apoptosis and to worsen estrogen-induced breast tumors.
Quercetin may increase the anti-tumor effects of some chemotherapy drugs, but it may also interfere with the actions of some chemotherapy drugs.
Quercetin may help manage some symptoms related to treatment and/or cancer.
Quercetin may reduce side effects of some conventional cancer therapies.
No published clinical trials to date establish quercetin as an anti-cancer therapy in humans, although preclinical studies show promising effects.
Quercetin is of interest to BCCT because of its potential anticancer, conventional treatment enhancement and side-effect management properties.
Quercetin is included in a number of integrative care cancer protocols.
Although oral quercetin has low toxicity and is usually well-tolerated, it can cause side effects as well as react with some prescription drugs. Caution and medical supervision are advised.
The flavonoid quercetin is a natural antioxidant that is found in several foods:
Black, green and buckwheat tea
Quercetin is also found in some medicinal plants such as ginkgo biloba and St. John's Wort. It is also available as a supplement.
Treating the Cancer
Working against cancer growth or spread, improving survival, or working with other treatments or therapies to improve their anticancer action
Studies have found both benefits and concerns regarding effects of quercetin on cancer. Concerns are listed in the Cautions section below.
Lab and Animal Evidence
Killed cancer cells or prevented cancer cells from growing1
May increase cancer cell sensitivity to topotecan and gemcitabine.2
Either promoted or blocked cell death (apoptosis) or promoted or reduced tumor growth in preclinical studies, depending on dose3
Breast cancer:
Induced cell death (apoptosis) in cell studies and reduced tumor volume in mice bearing estrogen receptor‑positive breast cancer cells.4
Enhanced antitumor activities of doxorubicin, paclitaxel, and vincristine and reversed multidrug resistance in breast cancer cells5
Rutin, a quercetin glycoside, reversed multidrug resistance and restored chemosensitivity to cyclophosphamide and cyclophosphamide of chemoresistant, triple-negative breast cancer cells6
Induced cell death (apoptosis ) in cell studies and reduced tumor volume in mice bearing estrogen receptor‑positive breast cancer cells.7
Sensitized resistant breast cancer cells to doxorubicin8
Reversed multidrug resistance and restored chemosensitivity to cyclophosphamide and cyclophosphamide of human chemoresistant, triple-negative breast cancer cells with rutin, a quercetin glycoside9
Colorectal cancer: Induced cell death (apoptosis) in colon cancer cells10
Liver cancer: Sensitized resistant liver cancer cells to doxorubicin11
Ovarian cancer: Increased sensitivity of ovarian cancer cells to cisplatin when used in combination with curcumin12
Prostate cancer:
Inhibited prostate cancer growth in animal studies.13
Resensitized enzalutamide-resistant prostate cancer cells to enzalutamide treatment in mouse xenografts.14
Stomach cancer: promoted activity to fight Helicobacter pylori (H. pylori) infection, which is associated with an increased risk of stomach (gastric cancer)15
Managing Side Effects and Promoting Wellness
Reduced fatigue and other common side effects of sunitib when isoquercetin was used with vitamin C and vitamin B3 among people with kidney cancer16
Protected cells from radiation damage, mainly by reducing cellular stress17
Reducing the risk of developing cancer or the risk of recurrence
Reduced risk of all cancers combined, breast and ovarian cancers with high dietary intake of quercetin or of flavanols including quercetin18
No reduced risk of ovarian cancer was found at at levels commonly consumed in a typical diet.19
Reduced risk of colorectal cancer in epidemiological and case-control studies with increasing intake of quercetin or of flavanols including quercetin20
Reduced risk of smoking-related cancers (lung cancer and aerodigestive tract cancer such as oral and esophageal cancers) in smokers, but not nonsmokers21
Optimizing Your Terrain
Quercetin either increased or decreased aromatase enzyme activity in adrenocortical carcinoma cells, depending on dose.22
uppressed tumor-promoting responses in breast cancer cells.23
No impact on markers of inflammation24
May have either a protective or damaging effect of reactive oxygen species on DNA, depending dose or on the concentration of chelating metal ions25
Quercetin is relatively nontoxic and is considered "possibly safe" orally in doses of 500 mg or less.26 However, quercetin can cause side effects and drug interactions.27
Quercetin may interfere with the actions of some chemotherapy drugs.28 Consult with your pharmacist for interactions and discuss using quercetin with your doctor. See The Definitive Guide to Cancer, 3rd Edition: An Integrative Approach to Prevention, Treatment, and Healing, page 170, for a list of chemotherapy drugs that interact with quercetin and with which quercetin should not be taken.
In lab and animal studies, quercetin has shown worrisome effects, blocking cell death (apoptosis) in one lab study and worsening estrogen-induced breast tumors in one animal study.29
Quercetin is widely available in food sources and in supplements.
BCCT does not recommend therapies or doses, but only provides information for patients and providers to consider as part of a complete treatment plan. Patients should discuss therapies with their physicians, as contraindications, interactions and side effects must be evaluated. Levels of active ingredients of natural products can vary widely between and even within products. See Quality and Sources of Herbs, Supplements and Other Natural Products.
Clinical trials have not established optimal quercetin dose during or after cancer treatment. Dosage recommendations are available from these sources:
Solutions 4 Health:
Bladder Cancer Wellness Plan
Breast Cancer Wellness Plan
Colorectal Cancer Wellness Plan
Lung Cancer Wellness Plan
Lymphoma Wellness Plan
Prostate Cancer Wellness Plan
Natural Medicines Database (requires purchase)
Also see the protocols below.
Integrative Programs, Protocols and Medical Systems
For more information about programs and protocols, see our Integrative Programs and Protocols page.
Integrative oncology clinicians such as Keith Block, Dwight McKee and Lise Alschuler incorporate quercetin supplementation into care during active cancer treatment and/or in post-treatment care to prevent recurrence or secondary cancers.
Programs and protocols
Alschuler & Gazella complementary approaches30
Bastyr University Integrative Oncology Research Center protocol for stage IV breast cancer31
Block program32
Anti-inflammatory terrain modification
Chemotherapy enhancement
Natural molecular target modification
Reduction of risk of secondary cancers
Lemole, Mehta & McKee protocols33
MacDonald breast cancer program34 advises avoiding quercetin
McKinney protocols35
Brain/nerve cancer
Integrative oncologist and BCCT advisor Dr. Keith Block uses quercetin as a natural molecular target modifier (IGF-I). Targeted therapies home in on particular growth pathways that cancer cells use to proliferate while sparing normal cells.36 Dr. Block also uses quercetin in multiple vitamin combinations, including with vitamin C to tackle inflammation alterations of the terrain of the human body.37 Quercetin may also help manage cancer-related and/or treatment-related symptoms such as diarrhea and depression.38 Please see Dr. Block's book for details: Life over Cancer: The Block Center Program for Integrative Cancer Treatment.
Non-cancer Uses of Quercetin
A 2017 review listed these effects and uses of quercetin:39
Treating allergies
Preventing cardiovascular disease
Treating chronic obstructive pulmonary disease (COPD)
Treating neurodegenerative diseases
Reducing systolic blood pressure and plasma oxidized LDL concentrations
Treating ulcers and gastritis
Treating bacterial and viral infections
BCCT has not reviewed the effectiveness of this therapy for non-cancer uses.
Note: BCCT has not conducted an independent review of research of quercetin. This summary draws primarily from Memorial Sloan Kettering Cancer Center's About Herbs and other sources as noted.
Haghi A, Azimi H, Rahimi R. A comprehensive review on pharmacotherapeutics of three phytochemicals, curcumin, quercetin, and allicin, in the treatment of gastric cancer. Journal of Gastrointestinal Cancer. 2017 Dec;48(4):314-320.
Block KI. Life Over Cancer. New York: Bantam Dell. 2009. p. 486
Aiyer HS, Warri AM, Woode DR, Hilakivi-Clarke L, Clarke R. Influence of berry polyphenols on receptor signaling and cell-death pathways: implications for breast cancer prevention. Journal of Agricultural and Food Chemistry. 2012 Jun 13;60(23):5693-708.
Hashemzaei M, Delarami Far A et al. Anticancer and apoptosis‑inducing effects of quercetin in vitro and in vivo. Oncology Reports. 2017 Aug;38(2):819-828.
Li S, Zhao Q et al. Quercetin reversed MDR in breast cancer cells through down-regulating P-gp expression and eliminating cancer stem cells mediated by YB-1 nuclear translocation. Phytotherapy Research. 2018 Aug;32(8):1530-1536; Li S, Yuan S et al. Quercetin enhances chemotherapeutic effect of doxorubicin against human breast cancer cells while reducing toxic side effects of it. Biomed Pharmacother. 2018 Apr;100:441-447.
Iriti M, Kubina R et al. Rutin, a quercetin glycoside, restores chemosensitivity in human breast cancer cells. Phytotherapy Research. 2017 Oct;31(10):1529-1538.
Scambia G, Ranelletti FO et al. Quercetin potentiates the effect of adriamycin in a multidrug-resistant MCF-7 human breast-cancer cell line: P-glycoprotein as a possible target. Cancer Chemotherapy and Pharmacology. 1994;34(6):459-64; Wang G, Zhang J, Liu L, Sharma S, Dong Q. Quercetin potentiates doxorubicin mediated antitumor effects against liver cancer through p53/Bcl-xl. PLoS One. 2012;7(12):e51764.
Kuppusamy P, Yusoff MM et al. Nutraceuticals as potential therapeutic agents for colon cancer: a review. Acta Pharm Sin B. 2014 Jun;4(3):173-81.
Chan MM, Fong D, Soprano KJ, Holmes WF, Heverling H. Inhibition of growth and sensitization to cisplatin-mediated killing of ovarian cancer cells by polyphenolic chemopreventive agents. Journal of Cellular Physiology. 2003 Jan;194(1):63-70.
Yang F, Song L et al. Quercetin in prostate cancer: chemotherapeutic and chemopreventive effects, mechanisms and clinical application potential (Review). Oncology Reports. 2015 Jun;33(6):2659-68.
Tummala R, Lou W, Gao AC, Nadiminty N. Quercetin targets hnRNPA1 to overcome enzalutamide resistance in prostate cancer cells. Molecular Cancer Therapeutics. 2017 Dec;16(12):2770-2779.
Buonerba C, De Placido P et al. Isoquercetin as an adjunct therapy in patients with kidney cancer receiving first-line sunitinib (QUASAR): results of a phase I trial. Frontiers in Pharmacology. 2018 Mar 16;9:189.
Fischer N, Seo EJ, Efferth T. Prevention from radiation damage by natural products. Phytomedicine. 2018 Aug 1;47:192-200.
Mokbel K, Mokbel K. Chemoprevention of breast cancer with vitamins and micronutrients: a concise review. In Vivo. 2019 Jul-Aug;33(4):983-997.
Parvaresh A, Razavi R et al. Quercetin and ovarian cancer: an evaluation based on a systematic review. Journal of Research in Medical Sciences. 2016 May 9;21:34.
Chang H, Lei L et al. Dietary flavonoids and the risk of colorectal cancer: an updated meta-analysis of epidemiological studies. Nutrients. 2018 Jul;10(7):950; Kocic B, Kitic D, Brankovic S. Dietary flavonoid intake and colorectal cancer risk: evidence from human population studies. Journal of B.U.ON.: Official Journal of the Balkan Union of Oncology. Jan-Mar 2013;18(1):34-43.
Woo HD, Kim J. Dietary flavonoid intake and smoking-related cancer risk: a meta-analysis. PLoS One. 2013 Sep 19;8(9):e75604.
Khan SI, Zhao J, Khan IA, Walker LA, Dasmahapatra AK. Potential utility of natural products as regulators of breast cancer-associated aromatase promoters. Reproductive Biology and Endocrinology. 2011 Jun 21;9:91.
Yasuda MT, Sakakibara H, Shimoi K. Estrogen- and stress-induced DNA damage in breast cancer and chemoprevention with dietary flavonoid. Genes and Environment. 2017 Feb 1;39:10.
Peluso I, Raguzzini A, Serafini M. Effect of flavonoids on circulating levels of TNF-α and IL-6 in humans: a systematic review and meta-analysis. Molecular Nutrition & Food Research. 2013 May;57(5):784-801.
Yasuda MT, Sakakibara H, Shimoi K. Estrogen- and stress-induced DNA damage in breast cancer and chemoprevention with dietary flavonoid. Genes and Environment. 2017 Feb 1;39:10; Griffiths K, Aggarwal BB et al. Food antioxidants and their anti-inflammatory properties: a potential role in cardiovascular diseases and cancer prevention. Diseases. 2016 Aug 1;4(3). pii: E28.
TRC Natural Medicines Database. Quercetin: Monograph. Viewed December 28, 2017.
Andres S, Pevny S et al. Safety aspects of the use of quercetin as a dietary supplement. Molecular Nutrition & Food Research. 2017 Nov 11.
Alschuler LN, Gazella KA. The Definitive Guide to Cancer, 3rd Edition: An Integrative Approach to Prevention, Treatment, and Healing. Berkeley, California: Celestial Arts. 2010. p. 170.
Alschuler LN, Gazella KA. The Definitive Guide to Cancer, 3rd Edition: An Integrative Approach to Prevention, Treatment, and Healing. Berkeley, California: Celestial Arts. 2010; Alschuler LN, Gazella KA. The Definitive Guide to Thriving after Cancer: A Five-Step Integrative Plan to Reduce the Risk of Recurrence and Build Lifelong Health. Berkeley, California: Ten Speed Press. 2013.
McKinney N. Naturopathic Oncology, 3rd Edition. Victoria, BC, Canada: Liaison Press. 2016. p. 316.
Block KI. Life over Cancer: The Block Center Program for Integrative Cancer Treatment. New York: Bantam Dell. 2009.
Lemole G, Mehta P, McKee D. After Cancer Care: The Definitive Self-Care Guide to Getting and Staying Well for Patients with Cancer. New York, New York: Rodale, Inc. 2015.
MacDonald B. The Breast Cancer Companion: A Complementary Care Manual: Third Edition. (self-published, Amazon, 2016).
McKinney N. Naturopathic Oncology, 3rd Edition. Victoria, BC, Canada: Liaison Press. 2016.
Block KI. Life over Cancer: The Block Center Program for Integrative Cancer Treatment. New York: Bantam Dell. 2009. p. 525.
Block KI. Life Over Cancer. New York: Bantam Dell. 2009. pp. 337 and 341.
Block KI. Life over Cancer: The Block Center Program for Integrative Cancer Treatment. New York: Bantam Dell. 2009. p. 143; Memorial Sloan Kettering. About Herbs: Quercetin. December 6, 2016. viewed December 28, 2017.
Anand David AV, Arulmoli R, Parasuraman S. Overviews of biological importance of quercetin: a bioactive flavonoid. Pharmacognosy Reviews. 2016 Jul-Dec;10(20):84-89.
Memorial Sloan Kettering Cancer Center's About Herbs: Quercetin
TRC Natural Medicines website (subscription is required): in-depth information, ratings of effectiveness and safety and evaluation of specific quercetin products
Consumer Labs: (subscription required): Quercetin
Moss Reports (purchase required): Select from the list of cancers down the left side of the page for a report describing uses of conventional, complementary, alternative and integrative therapies related to that cancer. Ralph Moss is among the most knowledgeable and balanced researchers of integrative cancer therapies. The cost of his Moss Reports is not negligible, but many patients find them of considerable value. Moss is also available for consultations.
Dwight McKee, MD, editor: Clinical Pearls
Donald I. Abrams, MD, and Andrew T. Weil, MD: Integrative Oncology, 2nd Edition
Neil McKinney, BSc, ND: Naturopathic Oncology, 3rd Edition
Lise Alschuler, ND, FABNO, and Karolyn Gazella: The Definitive Guide to Cancer, 3rd Edition
Keith I. Block, MD: Life over Cancer: The Block Center Program for Integrative Cancer Treatment
Lone Star Medical Group: Natural Alternative Treatments


If Oregon voters are ever going to approve a sales tax, two conditions have to be met. It has to be simple, and it has to replace one of the other big taxes we have, either the income tax or the property tax. The proposal discussed by the state Senate tax committee on April 15 — a fitting date — meets neither of those conditions.
It would take effect only if voters approve it, and it might be placed on the ballot in 2014 or '16. But take a look at the proposal as it stands so far. Its complexities are spelled out in the 75 pages of Senate Bill 824.
The income tax would be lowered but not repealed. Filing the state return would become more complicated so that lower-income people could claim a credit to partially reimburse them for paying sales tax.
The property tax would exempt the first $50,000 of assessed value of homes occupied by their owners. The burden of annual bills remains, and establishing that a place is owner-occupied makes the system even more cumbersome for tax collectors.
always asking for more when in fact Oregon seems to have done satisfactory during the recession with less revenue. They seem to do better if required to come up for a special freeway lane for Burdick or flushing toilets….no wait. Perhaps if they were to pull their collective heads from their behinds more flushing will be needed. Taxes and death…. said before.


Theresa Mayberry, Ph.D. has more than 40 years of experience in leading and transforming organizations, successful programs and policy development concentrating on improved service delivery systems for children, families, and communities. For decades, she has invested in public policy solutions to improve the opportunities for all segments of society, particularly the least served.
– Bureau Deputy Director for the Illinois Department of Children & Family Services.
She is an Annie E. Casey Children and Family Fellows alumnus, where she was involved with the design and implementation of the national policy group, Corporate Voices for Working Families. Her volunteer work includes service as a Trustee for the Missouri Foundation for Health, a Trustee for the St. Louis City Department of Health, and a Board member for The Soulard School.
Theresa works on strategy and leadership projects to transform organizations and improve conditions for the disadvantaged and those institutions that serve them.


John & Jodi Brinich 07/06/2018 $125.00 Always proud of the person you are. Good luck & have fun!
Margaret Desiderio 07/06/2018 $25.00 Cancer research saves lives!!
Michele Spoto 06/12/2018 $25.00 Thank you, Mike. You are awesome to take part in this!
Charles Sears 04/30/2018 $100.00 God bless your dedication to this game and this community! You are an outstanding coach, father and friend. We are proud to support such a great guy for a great cause. Keep up the great work! Have fun and stay hydrated!!!
Jeff Kadlub 04/22/2018 $50.00 Make it count!
Damian Cirincione 04/20/2018 $40.00 Sorry for your loss and thankful your doing this. Good lunch bro and God Bless!!!
Jill DiJames 04/19/2018 $25.00 It's awesome you're doing this....Good luck.
Chris Pieri 04/15/2018 $25.00 Good luck Mike!
Anonymous Friend 04/14/2018 $10.00 Way to Go Mike!!!!!
Kendra Vankos 04/05/2018 $25.00 go mike!!
Gary Miller 04/05/2018 $25.00 Go Leafs!
Maggi McDonald 04/04/2018 $20.00 Good luck!
Jeffrey DeKing 04/04/2018 $20.00 Good luck!


text:
Triepeolus rugosus är en biart som beskrevs av Mitchell 1962. Triepeolus rugosus ingår i släktet Triepeolus och familjen långtungebin. Inga underarter finns listade i Catalogue of Life.

Källor

Långtungebin
rugosus

redpajama_set_name: RedPajamaWikipedia


text:
Q: SQL - Filtering within first_value (or window function in general) I have some log data tracking invoice processing, like the example below:
Invoice  Activity              Date
--------------------------------------
A        Creation              12-Mar
A        Quantity change       13-Mar
A        Quantity change       14-Mar
A        Payment               17-Mar
B        Creation              20-Apr
B        Payment               24-Apr
B        Payment               29-Apr

I need to show, for every invoice, when the first and last of each activity occurred. For example, for invoice A there were two quantity changes, and I am interested in that date. I need to display everything in an aggregated table with 1 row per invoice as shown below:
Invoice    Creation date    First quantity change     Last payment
---------------------------------------------------------------------
A          12-Mar           13-Mar                    17-Mar
B          20-Apr           NULL                      29-Apr

I have explored a couple of different options but nothing works so far.
The most obvious one is to join the table on itself, using the invoice id as the join key. However, this is not possible because of performance issues, as the tables are very large and this would require too many joins.
Another option is to use the first_value and last_value functions, but I am not able to set them up in a way that gives me the results I need, because I can't find a way to somehow put a filter in it.
I have tried this, which doesn't work, but kind of shows what I'm trying to do:
SELECT
Invoice
, first_value(CASE WHEN  Activity = 'Quantity Change' THEN Activity ELSE NULL END)
  OVER (PARTITION BY Invoice ORDER BY Date)

FROM
Data

Does anyone have any suggestion on how to do this?
I am running these transformations in google big query.
Many thanks,
Alessandro

A: Using PIVOT query,
SELECT * FROM (
  SELECT Invoice, Activity,
         FORMAT_DATE('%d-%b', MIN(date0) OVER (PARTITION BY Invoice, Activity)) first,
         FORMAT_DATE('%d-%b', MAX(date0) OVER (PARTITION BY Invoice, Activity)) last,
    FROM sample_table, UNNEST([PARSE_DATE('%d-%b', Date)]) date0
)  PIVOT (ANY_VALUE(first) first, ANY_VALUE(last) last FOR REPLACE(Activity, ' ','_') IN ('Creation', 'Payment', 'Quantity_change'));

You can get following results:

And you can make above query more general using a dynamic sql, but I don't think you want to have a table with 100,000 columns.
So, I think below query and a table schema is more practical than a pivoted table.
SELECT DISTINCT Invoice, Activity,
       FORMAT_DATE('%d-%b', MIN(date0) OVER (PARTITION BY Invoice, Activity)) first,
       FORMAT_DATE('%d-%b', MAX(date0) OVER (PARTITION BY Invoice, Activity)) last,
  FROM sample_table, UNNEST([PARSE_DATE('%d-%b', Date)]) date0;

A sameple table used in above queries:
create temp table sample_table as
select 'A' Invoice, 'Creation' Activity, '12-Mar' Date union all
select 'A', 'Quantity change', '13-Mar' union all
select 'A', 'Quantity change', '14-Mar' union all
select 'A', 'Payment', '17-Mar' union all
select 'B', 'Creation', '20-Apr' union all
select 'B', 'Payment', '24-Apr' union all
select 'B', 'Payment', '29-Apr';

A: You can achieve this by using the MIN and MAX aggregate functions.
WITH inv AS
(
  SELECT "A" AS Invoice, 'Creation' as Activity, DATE '2022-03-12' as Date UNION ALL
  SELECT "A" AS Invoice, 'Quantity change' as Activity, DATE '2022-03-13' as Date UNION ALL
  SELECT "A" AS Invoice, 'Quantity change' as Activity, DATE '2022-03-14' as Date UNION ALL
  SELECT "A" AS Invoice, 'Payment' as Activity, DATE '2022-03-17' as Date UNION ALL
  SELECT "B" AS Invoice, 'Creation' as Activity, DATE '2022-04-20' as Date UNION ALL
  SELECT "B" AS Invoice, 'Payment' as Activity, DATE '2022-04-24' as Date UNION ALL
  SELECT "B" AS Invoice, 'Payment' as Activity, DATE '2022-04-29' as Date
)
SELECT
  Invoice,
  MIN(IF(inv.Activity = 'Creation', Date, NULL)) as CreationDate,
  MIN(IF(inv.Activity = 'Quantity change', Date, NULL)) as FirtsQuantityChange,
  MAX(IF(inv.Activity = 'Payment', Date, NULL)) as LastPayment
FROM inv
GROUP BY Invoice

A: This is the solution with the dynamic columns.
BEGIN
DECLARE columns STRING;

CREATE TEMP TABLE inv
AS
SELECT "A" AS Invoice, 'Creation' as Activity, DATE '2022-03-12' as Date UNION ALL
SELECT "A" AS Invoice, 'Quantity change' as Activity, DATE '2022-03-13' as Date UNION ALL
SELECT "A" AS Invoice, 'Quantity change' as Activity, DATE '2022-03-14' as Date UNION ALL
SELECT "A" AS Invoice, 'Payment' as Activity, DATE '2022-03-17' as Date UNION ALL
SELECT "B" AS Invoice, 'Creation' as Activity, DATE '2022-04-20' as Date UNION ALL
SELECT "B" AS Invoice, 'Payment' as Activity, DATE '2022-04-24' as Date UNION ALL
SELECT "B" AS Invoice, 'Payment' as Activity, DATE '2022-04-29' as Date
;

SET columns = (
  SELECT STRING_AGG(
    CASE Activity WHEN 'Payment' THEN
    CONCAT("MAX(IF(inv.Activity = '", Activity ,"', Date, NULL)) as Last",REPLACE(Activity,' ','') , " ")
    ELSE
    CONCAT("MIN(IF(inv.Activity = '", Activity ,"', Date, NULL)) as First",REPLACE(Activity,' ','') , " ")
    END
  )
  FROM (SELECT DISTINCT Activity FROM inv)
);

SELECT columns;

EXECUTE IMMEDIATE format("""SELECT
  Invoice,%s
  FROM inv
  GROUP BY Invoice
  """,columns);

END;

You must use the case statement in order to decide for which column you want the MAX and for which one you want the MIN as the aggregation function and also the EXECUTE IMMEDIATE in order to form the final statement.
Anyway, if you really have 100.000 distinct values for Activity then you should use another table schema for your results like @Jaytiger suggested.

redpajama_set_name: RedPajamaStackExchange


text:
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Marlon Brando's Rolex GMT Watch Sells for $1.95 Million At Auction
Roberta Naas
I write about the worlds of time, luxury watches and jewelry.
The bidding in New York City for this rare Rolex watch — once owned by Marlon Brando and worn in the movie Apocalypse Now — was intense. It seems a great many people around the world were interested in acquiring the watch that Brando carved his name on the back of nearly five decades ago. In the end, after about 15 minutes of bidding, Lot 30 sold for $1.61 million, translating to $1.952 million with fees.
Rolex GMT Master watch owned by Marlon Brando sells for $1.6 million at auction tonight by Phillips ... [+] in Association with Bacs & Russo.
Phillips in Association with Bacs & Russo
Quipping that the watch could make records — for being the longest bid-on piece of the night — Aurel Bacs of Phillips in Association with Bacs & Russo called this one of the greatest Rolex GMT Master watches, worn by one of the greater actors.
Tonight's Game Changers auction had a lot of hype around it. The watches being sold — 75 lots — are all part of game-changing events or belonged to game-changing people. The watches expected to fetch the highest amounts of money were this watch and the Jack Nicklaus Rolex, lagoon with a few others.
Marlon Brando in Apocalypse Now wearing his Rolex GMT Master watch.
This watch, which we wrote about in detail here this past summer, was actually purchased by Brando in the 1970's. He opted to wear it in his role as Colonel Walter E. Kurtz, former Green Beret who went rogue during the Vietnam War. Because the filmmakers wanted him to remove the watch in the movie, which Brando did not want to do, he removed just the bezel, giving it an unusual look. This year marks the 40th anniversary of the legendary movie.
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Marlon Brando engraved his name on the back of the watch in crude letters.
Brando gifted the watch to his daughter in 1995, and this was teh first time it has gone up for auction. The watch, sold as it was worn in the movie (without the bezel) will also benefit charity, as Brandon's daughter Petra and her husband are donating a portion of the proceeds to assist children living through hardship.
The 39 mm Rolex GMT-Master Superlative Chronometer with date (with cyclops magnifying window) is created in steel with luminous markers. Circa 1972, the Ref. 1675 is in well-preserved state. It was sold with two signed provenance letters from Petra Brando Fischer and Caroline Barrett, and two additional, custom "M B" straps designed by Attila Aszodi.
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One of the things that makes bowhunting so appealing is that it's so difficult. That's why I love the challenge of hiking into the middle of the wilderness with nothing but my pack and my bow.
As a bowhunter, you need complete faith in your physical ability to hike in to your hunting ground, stalk your prey, pull your bow stealthily and pack out your animal. That means it's essential to train for your hunt.
Whether you are a whitetail hunter trying to get more speed from your bow by pulling more weight, or an elk hunter trekking through the top of the Rocky Mountains, physical strength and endurance is vitally important.
If you are an average hunter like me, you save up what few vacation days you have each year to spend time in the woods. You spend all year thinking about those one or two big hunts. Don't you owe it to yourself to make sure you have the best opportunity for success?
You would never go hunting without tuning your bow to precision, yet most bowhunters don't think about training their body prior to a hunt. Proper training will lead to more consistent success, a more enjoyable hunt and the ability to hunt hard long after your hair turns grey.
The exercises I've developed below are designed specifically for the needs of bowhunters. They'll help you gain more leg and core strength, which is grounded by strong glutes and hamstrings, you'll be able to pull a 5-year-old buck out of the weeds to field dress it, or haul yourself up the tallest peak to scope for game.
Increasing your lower body endurance will enable you to hike farther and faster, covering more ground to reach the trophy you're after. You'll also shoot your bow straighter and with more consistency when you train your core, which increases your shoulder strength and stability. You will attain the upper body strength to smoothly pull back a heavy enough bow to bring down a brown bear.
All it takes is two, 40-minute workouts per week to master these 10 essential workout routines for bowhunters and to improve your chances for success this year.
Adam Holt lives in Denver, Colo., and has been a personal trainer and strength & conditioning coach for 11 years. He's worked with many athletes, from the high school to professional level, in a variety of sports. He was introduced to bowhunting four years ago by his uncle; now it's an obsession. He's passionate about helping other bowhunters enjoy the sport through better fitness.


The grounds maintenance and landscaping work that we undertake ensures that your outdoor spaces to look great. We're also able to undertake a range of land cultivation work, and this enables you to get the most out of your land.
For sports turf management that really works, choose our team of experts today. We offer excellent services to all, and the drainage systems that we install ensure that pitches and outdoor spaces are always usable.
Our tree work services allow us to transform your outdoor spaces into something truly amazing. These services, combined with our lawn care services, come together to give you the garden of your dreams.
As well as work in gardens and grounds, we also provide high-quality gritting and jet washing services to clients in the local area. For a great range of services, undertaken by professionals, get in touch with our team of experts.
Call us, in Macclesfield, Cheshire, for more information on our landscaping and grounds maintenance services.
We've always covered much more than lawn care — you're as likely to find us undertaking landscaping, installing drainage for a sports pitch, or offering grounds maintenance at a business park as you are sitting on a mower. Based in Macclesfield, Cheshire, we decided we needed a name that more accurately represented what we do. The result was LCS Grounds and Landscapes. We have more than 30 years of experience, and our free quotations provide you with all of the information up front.
Company owner Huw Jones is an experienced grounds manager, holding both sports ground management and horticultural qualifications. Before launching the business, Huw had a career in golf course management. During this time, he managed teams of gardeners and was responsible for both course construction and maintenance.
We work for customers based within a 25-mile radius of Macclesfield. That includes South Manchester, Wilmslow, Knutsford, Sale, and Hale, plus, of course, local villages such as Prestbury and Mottram St Andrew.


Through The Simple Bus Life Project we hope to inspire people to go "tiny" and leave a smaller footprint on Mother Earth. We truly feel that changes in our current living / building practices need to occur. There are so many perks to going tiny and saving our beautiful planet!
We have made it our mission to make the lives of future generations (and our own children) better and view ourselves as "tiny house crusaders." We believe that more attention needs to be paid to mother nature and her limited natural resources. Living sustainably, eco-friendly and downsizing will help to solve a lot of the current issues we are facing this day and age when it comes to renewable energy, water, and forest depletion.
Tiny house living is our Valhalla! Our paradise and heaven on earth. You too can live the simple yet complicated bus life. While the work is never truly done, we feel that we have learned so much over the past two years that we are eager to share with all of you! If you have the same beliefs as we do, please "like" our page on Facebook, follow us on Instagram (latest IG pics below), and subscribe to our blog as we will be posting updates on our journey.
OUR STORY ON THE RECORD?
The One Love Bus is our pride and joy. We are so excited to share with you the story of creating our home in the Virgin Islands! Some of you may have heard the rumors, and yes they are true, we have followed our dreams and are living sustainably in our eco-friendly school bus by the bay. Kinda sounds like a song right? We feel so blessed to be living on such an enchanted property that we lovingly refer to as Valhalla.
Tiny house living, while it's incredibly rewarding, can have it's uphill struggles at times. When it comes to a new way of thinking there are a lot of details to be worked out with the government and "old school thinkers," aka "cranky stuck up neighbors". This is a battle that we gladly face and feel fortunate to be a part of a movement that we are so passionate about. With the proper education on all of the various ways of living creatively and sustainably, we are confident that before we know it "tiny house living" will be the norm.
We want to take this opportunity to share our journey, and with good intentions, hope to educate others on how to live self-sufficiently, have financial freedom and live a life full of experiences. We are pretty sure that our beautiful planet needs it ;) and you may even find yourselves less strapped for cash at times when you want to take that extra night out without having to worry about money.
Follow us on our social media channels below as we make updates involving the early planning stages, the buildout, and the final living space of the One Love Bus.
Through our website and blog we will be offering up the full scope of our project from our buses journey from Chicago to St. Thomas, where we live, to the construction and daily life of living in an off the grid home. There have surely been lots of laughs and high fives, sometimes some tears and cursing too.


text:
Q: flutter run cannot access files of my secondary hard disk I have a SSD and a HDD in my system.I am using ssd for store ubuntu and windows and saving my project files in HDD. But i flutter cant access files in the HDD.
It is throwing this error message
Oops; flutter has exited unexpectedly: "FileSystemException: Cannot delete file,
path = '/path to project/.flutter-plugins' (OS Error:
Read-only file system, errno = 30)".
Unable to generate crash report due to secondary error: FileSystemException: Cannot create file, path = '/path to project/flutter_01.log' (OS Error: Read-only file system, errno = 30)
Please report a bug at https://github.com/flutter/flutter/issues.
Oops; flutter has exited unexpectedly: "FileSystemException: Cannot delete file,
path = '/path to project/.flutter-plugins' (OS Error:
Read-only file system, errno = 30)".
Unable to generate crash report due to secondary error: FileSystemException: Cannot create file, path = '/path to project/flutter_01.log' (OS Error: Read-only file system, errno = 30)
Please report a bug at https://github.com/flutter/flutter/issues.
Unhandled exception:
ProcessExit: 1
#0      _handleToolError (package:flutter_tools/runner.dart:172:7)

#1      run.. (package:flutter_tools/runner.dart:86:7)

A: My secondary harddisk was mounting as read only file system.
I have remounted it to solve the issue.


When we discussed kitchen remodeling mistakes to avoid earlier this year, we mentioned inadequate track lighting as a major fault with many kitchen designs. Don't make this mistake in your home! After all, the kitchen is one of the most important areas of your home and you need lights so you can cook, entertain, and enhance the other components in your kitchen design.
Kitchen cabinet lighting is one way to improve the overall appearance of your room, whether you choose over cabinet lighting or under cabinet lighting—and we can help! At Bath, Kitchen & Tile Center, we'll incorporate custom cabinetry lighting into your kitchen design to help you make the most out of your space.
Contact us today if you live in the Delaware area!
If you spend a lot of time in the kitchen, you know how important it is to have cabinetry lighting. Under cabinet lighting is one form of kitchen cabinet lighting that makes your space fully functional and adds an element of style to the room.
Task lighting is under cabinet lighting that is installed beneath cabinets found directly over the kitchen sink and countertops. This type of kitchen cabinet lighting is used mainly to improve the functionality of your kitchen, but it can also enhance its beauty.
Accent lighting, on the other hand, is under cabinet lighting that serves as a finishing touch to your kitchen design. Used mainly for aesthetics, this type of cabinetry lighting is a great way to highlight your granite countertops or kitchen backsplash, among other things.
Over cabinet lighting is another form of kitchen cabinet lighting that adds more light to your kitchen. Used mainly to add ambience to the room, over cabinet lighting can make your kitchen feel more inviting by making your ceiling glow—and it can add the illusion of height, which is ideal for small, cramped kitchens. This type of cabinetry lighting can also serve as a night light to increase safety, which is especially useful if you have young children.
Over cabinet lighting usually consists of small, uniform lights placed subtly over cabinets or entertainment centers so they can't be seen. LED rope lights tend to be popular in over cabinet lighting because they preserve energy but can be left on for as long as you'd like.
As a full-service design-build company, our in-house staff provides everything you need for a kitchen remodel or addition, including installation, electrical, dry wall, painting—and cabinetry lighting! This means our design and remodeling professionals can help you completely upgrade your cabinets, installing lighting to enhance your overall room design.
Looking to update your cabinets with kitchen cabinet lighting? Bath, Kitchen & Tile Center provides cabinetry installation throughout Delaware, New Jersey, Pennsylvania, and Maryland's Eastern Shore, and we pay attention to all the details—including kitchen cabinet lighting! Call us today!