text:
\section{Introduction}

Let $I$ be a homogeneous ideal given in a standard graded polynomial ring $R$ in $n$ variables over a field $K$, and let $I_d$ denote the degree $d$ graded component of $I$.
Assuming the $K$-dimension of $I_d$ is known, it sounds a quite simple question to ask what one can say about the dimension of the graded component of $I$ in degree $d+1$, and yet it attracts a lot of attention in commutative algebra and algebraic geometry.  An answer to this question was given by Macaulay's breakthrough work \cite{Ma} by providing a numerical bound for the growth of Hilbert function depending on the value  at the preceding degree. He showed that Hilbert functions of  special monomial ideals, called \textit{lexicographic} ideals,  describe all possible Hilbert functions of homogeneous ideals in $R$.  Macaulay's result led to other classical results on Hilbert functions such as Gotzmann's Persistence Theorem and Green's Hyperplane Restriction Theorem (see \cite{BH} for nice treatments of all these theorems).

Generalizations of Macaulay's result on the extremal behavior of lexicographic ideals for Hilbert functions allows to relate a homogeneous ideal containing the powers of variables with a monomial ideal containing the same powers of variables (see \cite{Kr,Ka,CL}).

In their \emph{Higher Castelnuovo Theory} paper, Eisenbud, Green and Harris conjectured a further generalization of Macaulay's theorem for homogeneous ideals containing a regular sequence in certain degrees (see Conjecture \ref{EGH}).  Eisenbud-Green-Harris (EGH) conjecture, motivated by Cayley-Bacharach theorems, suggests a refinement of Macaulay's bound on the growth of the Hilbert function by involving the information of degrees of the regular sequence contained in the ideal minimally. Although there are notable works done on Eisenbud-Green-Harris (EGH) conjecture, it is still widely open after more than  25 years.

A survey on lex-plus-powers ideals by Francisco and Richert \cite{FR} provides a very good source to understand these special monomial ideals thoroughly, and it also discusses the EGH conjecture and its equivalent variations in details. Since there have been significant progress on the EGH conjecture since \cite{FR}, the main intent of our survey paper is to contribute the literature by providing the current state of the EGH conjecture and to assemble the results that have been obtained so far.

As plan of this paper, in \textsection 2 we state some preliminaries and review Macaulay's results on Hilbert function. Section 3 lays out the Eisenbud-Green-Harris (EGH) conjecture and its variations.  In \textsection 4, we present the results obtained on the EGH conjecture by grouping them in terms of their approaches. Finally, in \textsection 5 we point out the open cases of the EGH conjecture, and we recall a closely related conjecture known as the Lex-Plus-Powers conjecture. We conclude the final section with some applications of EGH.

 \section{Preliminaries and Macaulay's theorem on Hilbert functions}

We let $R$ be  the polynomial ring $K[x_1,\ldots, x_n]$ over a field $K$ with standard grading $ R = \oplus_{i\geq 0} R_i$ where $R_i$ is the $i$-th graded component.

We fix the lexicographic order as $x_1 >_{\text{lex}} x_2 >_{\text{lex}} \ldots >_{\text{lex}} x_n$.  Then we define the monomial order between two monomials of the same degree as  $x_1^{a_1}x_2^{a_2}\cdots x_n^{a_n} >_{\text{lex}} x_1^{b_1}x_2^{b_2}\cdots x_n^{b_n}$  if $a_i > b_i$ where $i$ is the smallest index such that $a_i \neq b_i$.  For the sake of simplicity, we use $>$ for $>_{\text{lex}}$.

\begin{definition} Let $I$ be a monomial ideal in $R$ minimally generated by monomials $m_1,\ldots, m_k$. We call $I$ a \textit{lexicographic ideal} or simply a \textit{lex  ideal} if it satisfies the following property: for any monomial  $m'$ in $R$ with $\deg m' = \deg m_i$ and  $m'>m_i$ for some $i=1,\ldots, k$, then $m' \in I$ as well.
\end{definition}
We next define another special type of monomial ideal in our context.

\begin{definition} For given $2\leq a_1 \leq a_2 \leq \ldots \leq a_n$, we call a monomial ideal $L$ a \textit{lex-plus-powers ideal associated with degree $(a_1,\ldots, a_n)$} if it can be written as
\[ L = (x_1^{a_1}, \ldots, x_n^{a_n}) + J\]
where $J$ is a lex ideal in $R$.
\end{definition}

For any homogeneous ideal $I$ we define the \textit{Hilbert function} of $R/I$ as
\[\Hf_{R/I}(t) = \dim_K (R/I)_t  = \dim_K R_t - \dim_K I_t\]
where $R_t$ is the degree $t$ graded component of $R$ with $\dim_K R_t = \binom{n-1+t}{t}$, and $I_t$ is the degree $t$ graded component of the ideal $I$.

\begin{example} In $K[x,y,z]$, consider the monomial ideal $I = (x^3, xy, y^4, yz, z^2)$. Then simple computations give us the graded components of $R/I$;
$(R/I)_1 = \text{$K$-}\Span\{ x,y,z\}$, $(R/I)_2 = \text{$K$-}\Span\{ x^2, xz, y^2 \}$, $(R/I)_3 = \text{$K$-}\Span\{ x^2z, y^3\}$ and $dim_K(R/I)_i = 0$ for $i\geq 4$. Therefore, one expresses the Hilbert function of $R/I$ as $(h_0, h_1, h_2, h_3) = (1, 3, 3, 2)$ where $h_i =\Hf_{R/I}(i)$ and $h_i=0$ for $i\geq 4$.
\end{example}

As we see in the above example, it is possible to compute the Hilbert function of monomial ideals even by hand, however for arbitrary homogeneous ideals it becomes challenging to calculate without using a software such as Macaulay 2 (see \cite{M2} package \textit{LexIdeals}, command \verb|hilbertFunct|).

We call a sequence $f_1,\ldots, f_r$ of forms in $R = K[x_1,\ldots, x_n]$  \emph{a regular sequence} of length $r$ if, for each $i=1,\ldots,r$, $f_i$ is a nonzero divisor for the ring $R/(f_1,\ldots, f_{i-1})$.  If $r = n$, then the regular sequence has full length, in this case, it is referred as a maximal regular sequence.

\begin{remark}\label{HF_of_ci}  Let $\mathfrak{c}$ be a homogeneous ideal in $R$ generated by a regular sequence $f_1,\ldots, f_n$ with $\deg f_i = a_i$ for $i=1,\ldots, n$. We call $\mathfrak{c}$ a \textit{complete intersection ideal of type $(a_1,\ldots, a_n)$} and the ring $R/\mathfrak{c}$ is called a complete intersection ring. Then the Hilbert function of $R/\mathfrak{c}$ is $\Hf_{R/\mathfrak{c}}(i) = \Hf_{R/(x_1^{a_1},\ldots, x_n^{a_n})}(i)$ for all $i\geq 0$.

Furthermore,  $\Hf_{R/(x_1^{a_1},\ldots, x_n^{a_n})}(i) =  \Hf_{R/(x_1^{a_1},\ldots, x_n^{a_n})}(s-i)$
where $s = \sum\limits_{i}^{n}(a_i-1)$.
\end{remark}

The following proposition provides a very useful relation between the Hilbert function of an ideal $I$ and the Hilbert function of another ideal generated by a regular sequence contained in $I$ under the liaison (see \cite[Theorem 3]{DGO}).
\begin{proposition}\label{dual} Let $I$ be a homogeneous ideal and $\mathfrak{c} \subseteq I$ a complete intersection ideal, and $s= \sum\limits_{i=1}^n(a_i-1)$. Then, for all $j\geq 0$,
\[  \Hf_{R/I}(j) = \Hf_{R/\mathfrak{c}}(j) - \Hf_{R/(\mathfrak{c}:I)}(s-j) \]
\end{proposition}

\medskip

For a given two positive integers $d$ and $i$,
the  \textit{$i$-th Macaulay representation of  $d$ } (also known as the $i$-th binomial expansion of $d$), denoted $d^{(a)}$, is given by $$d = \binom{m_i}{i}+\binom{m_{i-1}}{i-1} + \ldots + \binom{m_2}{2}+\binom{m_1}{1} $$ where $m_i > m_{i-1} > \ldots >  m_1 \geq 0$ are uniquely determined and are called the $i$-th Macaulay coefficients of $d$.  In this case, we let
 \[d^{\langle i \rangle} := \binom{m_i+1}{i+1}+\binom{m_{i-1}+1}{i} + \ldots + \binom{m_2+1}{3}+\binom{m_1+1}{2}.\]

To give a simple example, let $d=21$, $i=4$. Then $21^{(4)} = \binom{6}{4} + \binom{4}{3} +\binom{2}{2}+\binom{1}{1}$, therefore $21^{\langle 4 \rangle }  = \binom{7}{5} + \binom{5}{4} +\binom{3}{3} +\binom{2}{2}= 28$.

We next state Macaulay's well-known theorem on Hilbert functions.

\begin{theorem}[\cite{Ma, BH}] Let $I$ be a homogeneous ideal in the polynomial ring $R$.
\begin{itemize}
\item[(a)]  There is a lex ideal in $R$ with the same Hilbert function, and this lex ideal is uniquely determined.
\item[(b)] [\textit{Macaulay's bound}]  If $\Hf_{R/I}(i) = d$ then
\[ \Hf_{R/I}(i+1) \leq d^{\langle i \rangle}.\]
\end{itemize}
\end{theorem}

\begin{example} \label{MacBound-R}
Notice that $\Hf_R(i) = \binom{n+i-1}{i} = d$ and so the $i$-th Macaulay representation of $d$ is $d^{( i )}= \binom{n-1+i}{i}$. Computing $\Hf_R(i+1) = \binom{n+i}{i+1}$ shows that $R$ attains exactly Macaulay's bound $d^{\langle i \rangle}$.
\end{example}

The following important theorem shows when a homogeneous ideal carries a similar behavior of attaining Macaulay's bound as in Example \ref{MacBound-R}.

\begin{theorem}[Gotzmann's Persistence Theorem \cite{Gotzmann,BH}
Let $I$ be a homogeneous ideal in $R$ generated by forms of degree $\leq d$. If the Hilbert function of $R/I$ achieves Macaulay's bound in the next degree $d+1$, that is $\Hf_{R/I}(d+1) = \Hf_{R/I}(d)^{\langle d\rangle},$
then \[ \Hf_{R/I}(j) = \Hf_{R/I}(j-1)^{\langle d\rangle} \quad \text{for all} \quad j \geq d.\]
\end{theorem}

Another classical result on the growth of Hilbert functions worth to mention is given by Green.  This result was also used to give an elegant proof of Macaulay's theorem.

\begin{theorem}[Green's Hyperplane Restriction Theorem \cite{Gr, BH}]
Let $I$ be a homogeneous ideal in $R$, and let $t\geq 1$ be a given degree. Then
\[ \Hf_{R/I+(\ell)}(t) \leq \binom{m_t-1}{t}+\binom{m_{t-1}-1}{t-1} + \ldots + \binom{m_2-1}{2}+\binom{m_1-1}{1}\]
where \[\Hf_{R/I}(t)^{(t)} = \binom{m_t}{t}+\binom{m_{t-1}}{t-1} + \ldots + \binom{m_2}{2}+\binom{m_1}{1}\] is the $t$-th Macaulay  representation of $\Hf_{R/I}(t)$.
\end{theorem}

\section{The EGH conjecture}

A generalization of Macaulay's results was considered by studying  homogeneous ideals in  $S=R/(x_1^{a_1},\ldots, x_n^{a_n})$ instead of homogeneous ideals in the polynomial ring $R=K[x_1,\ldots, x_n]$.  The existence of the lex ideal in $S$ with the same Hilbert function was shown by Kruskal \cite{Kr} and Katona \cite{Ka} when $a_1=a_2=\ldots=a_n = 2$ and more generally when $2\leq a_1\leq \ldots \leq a_n$ was done by Clements and Lindstr\"om \cite{CL} and also by Greene-Kleitman \cite{GrKl}. These results were obtained in set theoretical and combinatorial settings.

A question can be raised for a similar behavior for the homogeneous ideals in $R/(f_1,\ldots, f_n)$ where $f_1,\ldots, f_n$ is a regular sequence in $R$.  In \cite{EGH}, Eisenbud, Green and Harris initially stated the following conjecture for the case of  regular sequence of quadrics.

\begin{conjecture}\label{orig-EGH} Given homogeneous ideal $I$ in $R$ containing a full length regular sequence of quadratic forms.  Let $\Hf_{R/I}(i) = h$ and the $i$-th Macaulay representation of $h$ be $$h^{(i)} = \binom{m_i}{i}+\binom{m_{i-1}}{i-1} + \ldots + \binom{m_2}{2}+\binom{m_1}{1}$$ where $m_i > m_{i-1} > \ldots \geq m_1 \geq 0$.

Then \[\Hf_{R/I}(i+1) \leq \binom{m_i}{i+1}+\binom{m_{i-1}}{i} + \ldots +\binom{m_1}{2}.\]
\end{conjecture}

The new bound proposed by Conjecture \ref{orig-EGH} is finer than Macaulay's bound. We can see this
using the binomial identity $\binom{m+1}{k} = \binom{m}{k}+\binom{m}{k-1}$,
\begin{align*}
h^{\langle i \rangle } &= \binom{m_i+1}{i+1}+\binom{m_{i-1}+1}{i} + \ldots +\binom{m_1+1}{2}  \\
&=  \binom{m_i}{i+1} + \binom{m_i}{i} + \binom{m_{i-1}}{i}+\binom{m_{i-1}}{i-1} + \ldots + \binom{m_1}{2}+\binom{m_1}{1} \\
&= h^{(i)} + \binom{m_i}{i+1}+\binom{m_{i-1}}{i} + \ldots +\binom{m_1}{2}\\
&> \binom{m_i}{i+1}+\binom{m_{i-1}}{i} + \ldots +\binom{m_1}{2}.
\end{align*}

\begin{example} Suppose $I\subseteq K[x_1,\ldots,x_7]$ is a homogeneous ideal containing a regular sequence of quadratic forms $f_1,\ldots, f_7$ and $\Hf_{R/I}(2) = 17$. The possible growth for the Hilbert function in degree $3$ by Macaulay's bound is $\Hf_{R/I}(3)\leq 38$, but Conjecture \ref{orig-EGH} claims that $\Hf_{R/I}(3)\leq 21$.
\end{example}

If the homogeneous ideal $I$ is generated by generic quadrics, it is already known that Conjecture \ref{orig-EGH} is true by Herzog and Popescu \cite{HP} when the characteristic is zero. When $K$ has arbitrary characteristic, this was shown by Gasharov \cite{Gasharov}.

The main motivation behind Conjecture \ref{orig-EGH} about homogeneous ideals containing quadratic regular sequence was another conjecture, known as the Generalized Cayley-Bacharach conjecture, stated in \cite{EGH} in more geometric perspective.

\begin{conjecture}\label{GCB}[Generalized Cayley-Bacharach Conjecture for quadrics]
Let $\Gamma$ be a complete intersection of $n$ quadrics in $\mathbb{P}^n$.  Any hypersurface $X \subset \mathbb{P}^n$ of degree $d$ containing a subscheme $\Omega\subset \Gamma$ of degree strictly greater than $2^n - 2^{n-d}$ must contain $\Gamma$.
\end{conjecture}
Conjecture \ref{orig-EGH} implies the Generalized Cayley-Bacharach conjecture for quadrics.

In the same article \cite{EGH}, Eisenbud, Green and Harris dropped the quadratic condition on the regular sequence, and further conjectured the same statement for homogeneous ideals containing regular sequences with any degrees $2\leq a_1 \leq a_2 \ldots \leq a_n$.

\begin{conjecture}[Eisenbud-Green-Harris (EGH) Conjecture, \cite{EGH}] \label{EGH} Let $I$ be a homogeneous ideal in $R = K[x_1,\ldots, x_n]$ containing a regular sequence $f_1,\ldots, f_n$ with degrees $a_1,\ldots, a_n$ such that $2\leq a_1\leq \ldots \leq a_n$. Then there is a lex-plus-powers ideal $L = (x_1^{a_1}, \ldots, x_n^{a_n}) + J$ with a lex ideal  $J$ in $R$ such that
\[\Hf_{R/I}(i) = \Hf_{R/L}(i) \quad \text{for all} \quad i\geq 0.\]
\end{conjecture}

From now on, we will refer to this conjecture as the EGH conjecture. We will also use $\EGH_{(a_1,\ldots, a_n), n}$ to emphasize the degrees of the regular sequence and also that it is a full length-$n$ regular sequence in $R=K[x_1,\ldots, x_n]$.

Notice that in Remark \ref{HF_of_ci}, we observed a very trivial version of this statement when $I =(f_1,\ldots, f_n)$ and $L=(x_1^{a_1},\ldots, x_n^{a_n})$.

Another statement of the Generalized Cayley-Bacharach conjecture that does not require quadrics was given in \cite[Conjecture CB12]{EGH2}.  In 2013, Geramita and Kreuzer reformulated this version of Generalized Cayley-Bacharach conjecture for arbitrary degrees by dividing it into intervals. They also strengthened the Conjecture CB12 in \cite[Conjecture 3.5]{GK}. In $\mathbb{P}^3$, they provided a proof for it. In $\mathbb{P}^n$, they confirmed \cite[Conjecture 3.5]{GK} for some intervals. The EGH conjecture which is the concern of this paper is stronger than \cite[Conjecture CB12]{EGH2} as well.

One of the variations of the EGH conjecture in the literature is when one allows to have a regular sequence that is not of full length.

\begin{conjecture}[$\EGH_{n, (a_1,\ldots, a_r), r}$] \label{non-max_regseq} Let $I$ be a homogeneous ideal in $R=K[x_1,\ldots, x_n]$ containing a regular sequence of length $r<n$ with degrees $a_1,\ldots, a_r$ such that $2\leq a_1\leq \ldots \leq a_r$. Then there is a lex-plus-powers ideal $L = (x_1^{a_1}, \ldots, x_r^{a_r}) + J^{\text{lex}}$ in $R$ with the same Hilbert function as $I$.
\end{conjecture}

\begin{remark}\label{Rmk3.5} The equivalence between Conjectures \ref{EGH} and \ref{non-max_regseq} was discussed by Caviglia and Maclagan \cite{CM}.  For $2\leq a_1 \leq \ldots \leq a_r$ fixed, they showed that  if $\EGH_{(a_1',\ldots, a_n'), n}$ holds for all $2\leq a_1'\leq \ldots \leq a_n'$ where $a_i' = a_i$ for $i=1,\ldots, r$ then $\EGH_{n, (a_1,\ldots, a_r), r}$ holds (see \cite[Propositions 9]{CM}). They also showed that if $\EGH_{(a_1,\ldots, a_r), r}$ holds then $\EGH_{n, (a_1,\ldots, a_r), r}$ holds for all $n\geq r$ (see \cite[Propositions 10]{CM}).
\end{remark}
\section{Results on the EGH conjecture}

Richert  \cite{Ri} proved that the EGH conjecture is true for $R=K[x, y]$.  Thus, for any homogeneous ideal $I$ in two variables containing a regular sequence $f_1, f_2$ with degrees $2\leq a_1\leq a_2$,  we have  a lex plus powers ideal $L=(x^{a_1}, y^{a_2})+J$ such that
\[\dim_K I_i = \dim_K L_i \quad \text{ for all } \quad i\geq 0.\]

For $K[x_1,\ldots, x_n]$ with  $n>2$, we put together the known results on EGH  depending on the approaches were used.

\medskip

\subsection{EGH depending on the degrees $(a_1,\ldots, a_n)$.}
Let $n\geq 2$. For a fixed degree $d\geq 1$, when the Hilbert function of $R/I$ at degree $d$ is known,
the EGH conjecture proposes a maximal growth for degree $d+1$.  One of the adopted approaches in the literature focuses on the growth at certain degree.  Hence, the following definition states a partial version of EGH conjecture for consecutive degrees.

\begin{definition}[$\EGH_{(a_1,\ldots, a_n), n}(d)$] \label{weakerEGH} For any homogeneous ideal $I$ in $R$ containing a regular sequence $f_1,\ldots, f_n$ of degrees $2\leq a_1\leq \ldots \leq a_n$ respectively, if there exists a lex-plus-powers ideal $L$ associated with degrees $a_1,\ldots, a_n$ such that
\[\Hf_{R/I}(d) = \Hf_{R/L}(d) \quad \text{ and } \quad \Hf_{R/I}(d+1) = \Hf_{R/L}(d+1),\]
 then we say that $\EGH_{(a_1,\ldots, a_n), n}(d)$ holds.
\end{definition}

The following proposition is given by Francisco \cite{Fr} for the almost complete intersection ideals.

\begin{proposition}\label{Fr-prop} Let $I = (f_1,\ldots, f_n, g)$ be a homogeneous ideal where $f_1, \ldots, f_n$  is a regular sequence with degrees $a_1,\ldots, a_n$ and $\deg g =d\geq a_1$. Then $\EGH_{(a_1,\ldots, a_n), n}(d)$ is true for $I$.
\end{proposition}

In \cite{CDS2}, for an almost complete intersection $I = (f_1,f_2, f_3, g) \subset K[x_1,\ldots, x_n]$ where $f_1, f_2, f_3$ is a regular sequence of length three with $\deg f_i = a_i$, $i=1,2,3$ and $\deg g = d \leq a_1+a_2+a_3-3$,  Caviglia-De Stefani showed $\Hf_{R/I}(i) \leq \Hf_{R/L}(i)$ for all $i\geq 0$ where $L = (x_1^{a_1}, x_2^{a_2}, x_3^{a_3}, \mathrm{m})$ with $\mathrm{m}$ is the largest monomial of degree $d$ with respect to lexicographic order that is not in $(x_1^{a_1}, x_2^{a_2}, x_3^{a_3})$. Their work on such almost complete intersections also recovered the result of \cite{GK} for $\mathbb{P}^3$.

To show that $\EGH_{(d,\ldots, d), n}(d)$ holds  it suffices to show that the statement in Definition \ref{weakerEGH} holds for the homogeneous ideals $I = (f_1,\ldots, f_n, g_1,\ldots, g_m)\subseteq K[x_1,\ldots, x_n]$ generated by degree $d$ forms $f_1,\ldots, f_n, g_1,\ldots, g_m$ where $f_1,\ldots, f_n$ form a regular sequence (see \cite[Lemma 2.6]{GH}.)  In other words, it is enough to show the statement for the ideals where not only the regular sequence $f_1,\ldots, f_n$ in the generators have degree $d$, but also rest of the generators $g_1,\ldots, g_m$ have degree $d$ too.
Thus, focusing on the case $d=2$ we have the following remark.

\begin{remark}\label{rmk-quadrics} In order to show that $\EGH_{(2,\ldots, 2), n}(2)$ is true, it suffices to study the ideals generated by only quadrics containing a maximal regular sequence.
\end{remark}

In \cite{CM}, Caviglia-Maclagan provided the following lemma about this weaker version of  the EGH. Due to its importance as a tool for studying the EGH conjecture, we would like to present its proof given in \cite[Lemma 12]{CM}.

\begin{lemma}\label{lemma} Given $2\leq a_1 \leq \ldots \leq a_n$, set $s= \sum\limits_{i=1}^n (a_i-1)$. Let $d\geq 1$. Then
$$ \mbox{
$\EGH_{(a_1, \ldots, a_n), n}(d)$ holds if and only if $\EGH_{(a_1, \ldots, a_n), n}(s-d-1)$ holds.}$$
Furthermore,
$$\mbox{
if $\EGH_{(a_1, \ldots, a_n), n}(d)$ holds for all $0\leq d \leq \floor{\frac{s-1}{2}}$ then  $\EGH_{(a_1, \ldots, a_n), n}$ holds.}$$
\end{lemma}

\begin{proof} Suppose that $\EGH_{(a_1, \ldots, a_n), n}(d)$ holds. Then given any homogeneous ideal containing a regular sequence with degrees $a_1,\ldots, a_n$, there is lex-plus-powers ideal associated with degrees $a_1,\ldots, a_n$ such that Hilbert functions of both ideals agree at degrees $d$ and $d+1$.
Let $I$ be a homogeneous ideal in $R$ containing a regular sequence $f_1,\ldots, f_n$ with $\deg f_i=a_i$, for $i=1,\ldots, n$.  Then by Proposition \ref{dual} we get
\begin{align*}
\Hf_{R/I}(j) = \Hf_{R/(f_1,\ldots,f_n)}(j) - \Hf_{R/((f_1,\ldots,f_n) : I)}(s-j).
\end{align*}

Since the colon ideal $((f_1,\ldots,f_n) : I)$ contains the regular sequence $f_1,\ldots, f_n$, then by assumption there is a lex-plus-powers ideal $L' = (x_1^{a_1},\ldots, x_n^{a_n}) + J'$ such that
\begin{align}\label{eqn2}
\Hf_{R/((f_1,\ldots,f_n): I)}(d) =\Hf_{R/L'}(d)      \quad \text{ and } \quad \Hf_{R/((f_1,\ldots,f_n) : I)}(d+1) = \Hf_{R/L'}(d+1).
\end{align}
On the other hand, we also have
\begin{align}\label{eqn3}
\Hf_{R/L'}(j) = \Hf_{R/(x_1^{a_1},\ldots,x_n^{a_n})}(j) - \Hf_{R/((x_1^{a_1},\ldots,x_n^{a_n}) : L')}(s-j).
\end{align}

Thus, for $j=s-d-1$ and $j=s-d$,
\begin{align*}
\Hf_{R/I}(j) &= \Hf_{R/(f_1,\ldots,f_n)}(j) - \Hf_{R/((f_1,\ldots,f_n): I)}(s-j)  \\     &=  \Hf_{R/(f_1,\ldots,f_n)}(j) - \Hf_{R/(x_1^{a_1},\ldots,x_n^{a_n})}(s-j) + \Hf_{R/((x_1^{a_1},\ldots,x_n^{a_n}) : L')}(j) \\
    &=  \Hf_{R/((x_1^{a_1},\ldots,x_n^{a_n}) : L')}(j).
\end{align*}
The second equality follows from \eqref{eqn2} and \eqref{eqn3}. Then the last equality is by Remark \ref{HF_of_ci} and Proposition \ref{dual}.

Finally, since the ideal $((x_1^{a_1},\ldots,x_n^{a_n}) : L')$ contains the regular sequence $x_1^{a_1},\ldots,x_n^{a_n}$, by Clement-Lindstr\"om result mentioned previously, there exists a lex-plus-powers ideal $L$ associated with degrees $a_1,\ldots, a_n$ such that $\Hf_{R/((x_1^{a_1},\ldots,x_n^{a_n}) : L')}(i) = \Hf_{R/L}(i)$ for all $ i\geq 0$.

Hence,
$$\Hf_{R/I}(s-d-1) = \Hf_{R/L}(s-d-1) \quad \text{ and } \quad \Hf_{R/I}(s-d) = \Hf_{R/L}(s-d).$$

\end{proof}

\begin{remark}\label{EGH(0)}  We know $\Hf_{R/I}(0)=1$.  If $\Hf_{R/I}(1)=n$, then any lex-plus-powers ideal $L = (x_1^{a_1},\ldots, x_n^{a_n}) +J$ where the lex ideal $J$ does not contain any linear form  has $\Hf_{R/L}(0)=1$ and $\Hf_{R/L}(1)=n$. If $\Hf_{R/I}(1)=r<n$, that is $I$ has $n-r$ linear generators, then it is enough to pick the lex ideal $J$ containing $x_1,\ldots, x_{n-r}$, then $\Hf_{R/L}(1)=r$ as well. Therefore, we see that $\EGH_{(a_1,a_2,\ldots, a_n), n}(0)$ is always true.
\end{remark}

\begin{theorem}[\cite{CM,CDS}] For  $2\leq a_1\leq \ldots \leq a_n$  such that $a_j \geq \sum\limits_{i=1}^{j-1}(a_i-1)$, the $\EGH_{(a_1,\ldots,a_n), n}$ is true.
\end{theorem}

The strict inequality was shown by Caviglia-Maclagan in \cite{CM}. Their proof used an inductive argument on $n$, and followed from Lemma \ref{lemma} and Remark \ref{Rmk3.5}. Very recently, Caviglia-De Stefani \cite{CDS} extended this degree growth condition by including the equality. Their work actually provided a stronger case.  They showed that if a homogeneous ideal $I \subseteq R$ contains a regular sequence $f_1,\ldots, f_{n-1}$ with degrees $2\leq a_1 \leq \ldots \leq a_{n-1}$ satisfies the $\EGH_{n, (a_1,\ldots, a_{n-1}), n-1}$, then $I+(f_n)$ satisfies the EGH conjecture for any $f_n$ where $f_1,\ldots, f_n$ form a regular sequence and $\deg f_n \geq \sum\limits_{i=1}^{n-1}(a_i-1)$. (see \cite[Theorem 3.6]{CDS}).

The result of Caviglia-Maclagan with the recent improvement by Caviglia-De Stefani provides an affirmative answer for the EGH conjecture for a large case in terms of the degrees $a_1,\ldots, a_n$.  One of the interesting case that is not covered by this result is when $a_1=a_2=\ldots = a_n$, more specifically as in Conjecture \ref{orig-EGH} when $n\geq 4$.  We will focus on the quadratic case $a_i=2$ for all $i=1,\ldots, n$ separately (see Subsection \ref{subsec-quadrics}).

In \cite{Cooper}, Cooper approached the EGH conjecture for $n=3$ in a geometric setting by investigating the Hilbert functions of the subsets of complete intersections in $\mathbb{P}^2$ and $\mathbb{P}^3$. She showed that the $\EGH_{(a_1,a_2, a_3), 3}$ is true for the degrees $(2, a, a)$ for $a\geq 2$ and $(3, a, a)$ for $a\geq 3$.

Another result for $n=3$ for the Gorenstein ideals containing a regular sequence with degrees $2\leq a_1 \leq a_2 \leq a_3$ was proven by Chong \cite{Chong}.

\medskip

\subsection{EGH via liaison.}
Chong's work covers EGH beyond Gorenstein ideals in $K[x,y,z]$. It uses the linkage theory and studies a special subclass of licci ideals.  First, we would like to review some definitions and concepts related to linkage theory for ideals in $R=K[x_1,\ldots, x_n]$ to present Chong's result.

Let $I, I' \subseteq R$ be homogeneous ideals of height $r\leq n$.  If there exists a regular sequence $g_1,\ldots, g_r$ such that the complete intersection $\mathfrak{c} = (g_1,\ldots, g_r) \subseteq I\cap I'$, and $I = \mathfrak{c} : I'$ and $I' = \mathfrak{c} : I$, then we say that  $I$ and $I'$ are \textit{linked (algebraically) via $\mathfrak{c}$}. We express this linkage as $I\linkby[\mathfrak{c}] I'$.   If $I$ minimally contains a regular sequence $f_1,\ldots, f_r$ with degrees $2\leq a_1\leq \ldots \leq a_r$ and if the link $\mathfrak{c}$ is a complete intersection of type $(a_1,\ldots, a_r)$ then we say $\mathfrak{c}$ is a \textit{minimal link}

Suppose that there is a finite sequence of links $I \linkby[\mathfrak{c}_1] I_1 \linkby[\mathfrak{c}_2] \cdots \linkby[\mathfrak{c}_t]I_t$ where $I_t$ is a complete intersection, we say that $I$ is in the linkage class (a.k.a. liaison class) of the complete intersection $I_t$.  An ideal in the linkage class of a complete intersection is called \textit{licci}.  We next state the work of Chong on this.

\begin{theorem}\cite{Chong} Let $2\leq a_1\leq \ldots \leq a_n$, and $I\subseteq R$ be a a homogeneous ideal containing a regular sequence of degrees $a_1,\ldots, a_n$.
If $I$ is licci such that $I \linkby[\mathfrak{c}_1] I_1 \linkby[\mathfrak{c}_2] \cdots \linkby[\mathfrak{c}_t]I_t$ where each link $\mathfrak{c}_i$ has type $\bar{a}_i$ for $i=1,\ldots, n$ with $\bar{a}_1\geq \bar{a}_2 \geq \ldots \geq \bar{a}_n$, and $\mathfrak{c}_1$ is a minimal link, that is $\bar{a}_1 = (a_1,\ldots, a_n)$, then there is a lex-plus-powers ideal associated with degrees $a_1,\ldots, a_n$ with the same Hilbert function as $I$.
\end{theorem}

In the same paper \cite{Chong}, Chong also proved that $\EGH_{n, (a_1,\ldots, a_r), r}$ holds for the licci ideals where the types of the links satisfy the ascending condition and the first link is minimal. His result on Gorenstein ideals when $n=3$ we mentioned in previous subsection is a consequence of this result.
\medskip

\subsection{EGH via the structure of the regular sequence.}

Let $I \subseteq R=K[x_1,\ldots, x_n]$ be a homogeneous ideal containing a regular sequence $f_1,\ldots, f_n$ with degree $2\leq a_1 \leq \ldots \leq a_n$, respectively. By Clements-Lindstr\"om's result, we already know that $\EGH_{(a_1,\ldots,a_n), n}$ is true when $f_i = x_i^{a_i}$ for all $i=1,\ldots, n$.

Mermin and Murai \cite{MM} proved a special case of $\EGH_{n, (a_1,\ldots, a_r), r}$, $r<n$, when $\Char K =0$. For the homogeneous ideals containing a regular sequence $f_1,\ldots, f_r$ formed by monomials with degrees $2\leq a_1 \leq \ldots \leq a_r$, they showed the existence of lex-plus-powers ideal associated with $(a_1,\ldots, a_r)$ with the same Hilbert function.

Another notable work regarding the structure of the regular sequence contained in the ideal is done by Abedelfatah in \cite{Abed}. He showed that if a homogeneous ideal $I$ containing a regular sequence $f_1,\ldots, f_n$ such that $\deg f_i =a_i$, $i=1,\ldots,n$ and each $f_i$ splits into linear factors completely, then $I$ has the same Hilbert function of a lex-plus-powers ideal $(x_1^{a_1},\ldots, x_n^{a_n})+J$  in $R$.

Shortly after, Abedelfatah extended this result in \cite{Abed2}.

\begin{theorem} Let $\mathfrak{a}$ be an ideal generated by the product of linear forms and contain a regular sequence $f_1,\ldots, f_r$ with degrees $a_1,\ldots, a_r$. Let $I$ be a homogeneous ideal in $R$ such that $(f_1\ldots, f_r) \subset \mathfrak{a} \subset I$ then the Hilbert function of $I$ is the same as the lex-plus-powers ideal containing powers $x_1^{a_1},\ldots,x_r^{a_r}$.
\end{theorem}

The previous result in \cite{Abed} is simply the case when $\mathfrak{a} = (f_1,\ldots, f_r)$.  Let $I \subset R$ be a height $r$ monomial ideal containing a regular sequence of degrees $a_1\leq \ldots \leq a_r$, then this theorem of Abedelfatah confirms that $I$ has the same Hilbert functions as a lex-plus-powers containing $x_1^{a_1},\ldots, x_r^{a_r}$. This also improves another related result given by Caviglia-Constantinescu-Varbaro in \cite{CCV} for height $r$ monomial ideals generated by quadrics.
\medskip

\subsection{When $\mathbf{a_1=\ldots = a_n=2}$.}\label{subsec-quadrics}

Finally we focus on the case when the regular sequence is formed by quadrics as originally conjectured by the Eisendbud-Green-Harris as in Conjecture \ref{orig-EGH}. For simplicity, we refer it as
$\EGH_{(2,2,\ldots, 2), n} = \EGH_{{\bf \bar 2}, n}$.

We have already mentioned the cases when $n=2$ by Richert \cite{Ri} as his result shows EGH for any degree when $n=2$. Moreover, in $K[x,y,z]$, we have seen that the EGH conjecture for the degrees $(2,2,2)$ was covered by Cooper \cite{Cooper} and Caviglia-De Stefani \cite{CDS} separately, and quadratic monomial ideal case by \cite{CCV}.

In terms of the weaker version of the EGH conjecture for consecutive degrees given in Definition \ref{weakerEGH}, using Proposition \ref{Fr-prop} given by Francisco,  $\EGH_{{\bf \bar 2}, n}(2)$ is true for almost complete intersections $I = (f_1,\ldots, f_n, g)$ where $\deg f_i =2$ for all $i=1,\ldots,n$ and $\deg g=2$.  More precisely,  $\Hf_{R/I}(3) \leq \Hf_{R/L}(3)$ where $L $ is the lex-plus-powers ideal containing the squares of the variables.

An analogous result on  $\EGH_{{\bf \bar 2}, n}(2)$ for the ideals generated by a quadratic regular sequence plus two more generators is given in the following theorem.

\begin{theorem}\cite{GH}\label{GH}  For $n\geq 5$, $\EGH_{{\bf \bar 2}, n}(2)$ holds for homogeneous ideals minimally generated by a regular sequence of quadrics and two more generators whose degrees are at least $2$.
\end{theorem}
Thanks to \cite[Lemma 2.6]{GH}, which is mentioned in Remark \ref{rmk-quadrics}, to prove Theorem \ref{GH} it was enough to show the statement for an ideal $I$ generated by $n+2$ quadrics containing a maximal regular sequence.  More precisely, it sufficed to show the Hilbert function of the lex-plus-powers ideal $(x_1^2,\ldots, x_n^2, x_1x_2, x_1x_3)$ in degree $3$ is greater than or equal to $\Hf_{R/I}(3)$.  This was shown by analyzing the linear relations among the generators of the ideal $I$.

\vspace*{1ex}

For a homogeneous ideal $I = (f_1,\ldots, f_n, g_1,\ldots, g_m)$ containing quadratic regular sequence $f_1,\ldots, f_n$, it is easy to see that if each $g_i$ has degree $>2$ then they don't contribute the dimension in degree $2$ and $\dim_K I_2= n$.  Therefore,  any lex plus power ideal $L=(x_1^2,\ldots, x_n^2)+J$ where $J$ is generated by monomials of degree $> 2$ gives $\dim_KL_2=n$ as well.

We finish this section by presenting the known cases of the original EGH conjecture for $n\geq 4$.

\begin{theorem} $\EGH_{{\bf \bar 2}, n}$ is true when
\begin{itemize}
\item[(a)] $n=4$ by Chen \cite{Chen},
\item[(b)] $n=5$ by the author and Hochster \cite{GH}.
\end{itemize}
\end{theorem}

\begin{proof}
Notice that when $a_1=\ldots = a_n=2$ we get $s= \sum\limits_{i=1}^n (a_i-1) =  n$. Then by Lemma \ref{lemma}, we get $\EGH_{{\bf \bar 2}, n}(d)$ holds if and only if $\EGH_{{\bf \bar 2}, n}(n-1-d)$ holds.

Using this symmetry, when $n=4$, by Remark \ref{EGH(0)} we trivially have $\EGH_{{\bf \bar 2}, 4}(0)$, therefore we have $\EGH_{{\bf \bar 2}, 4}(3)$. It is enough to show $\EGH_{{\bf \bar 2}, 4}(1)$ and therefore we also get $\EGH_{{\bf \bar 2}, 4}(2)$.

Similarly, when $n=5$, we know  $\EGH_{{\bf \bar 2}, 5}(0)$ holds, so does $\EGH_{{\bf \bar 2}, 5}(4)$. Then we need to show only  $\EGH_{{\bf \bar 2}, 5}(1)$ and $\EGH_{{\bf \bar 2}, 5}(2)$ because $\EGH_{{\bf \bar 2}, 5}(1)$ implies $\EGH_{{\bf \bar 2}, 5}(3)$.

By \cite[Proposition 2.1]{Chen},  we know that $\EGH_{{\bf \bar 2}, n}(1)$ holds for any $n\geq 2$.
Thus this completes the proof of (a).

On the other hand, the proof of (b) is done as well since $\EGH_{{\bf \bar 2}, 5}(2)$ is true as a result of Theorem \ref{GH} and Remark \ref{rmk-quadrics}.
\end{proof}

\section{Open cases of EGH and more connections.}

Although there has been a significant progress on the EGH conjecture, it is fair to say that the conjecture is still broadly open.  In this section, we discuss the open cases, and also state another well known conjecture related to the EGH conjecture.

Besides Richert's result on EGH when $n=2$ in \cite{Ri}, we still do not know if the EGH conjecture is true when $n\geq 3$ without assuming any conditions on the degrees or on the homogeneous ideal.

\begin{question}
 Is $\EGH_{(a_1,a_2,a_3), 3}$ true for any given degrees $2\leq a_1 \leq a_2 \leq a_3$?
\end{question}

 We have seen that the works by Cooper, Caviglia-Maclagan and Caviglia-De Stefani cover many cases of  $(a_1,a_2,a_3)$ already. On the other hand, Chong's and Abedelfatah's results require certain conditions on the homogeneous ideals. As a result,  we can conclude that it is not known if $\EGH_{(a_1,a_2,a_3), 3}$ is true for the ideals in the following scenario: Let $I$ be a homogeneous ideal containing a regular sequence  $f_1, f_2, f_3$ such that
 \begin{itemize}
 \item the degrees $\deg f_i =a_i$, $i=1,2,3$ satisfy $ 4 \leq a_1\leq a_2 \leq a_3$ and  $a_2 \leq a_3 < a_1+a_2-2$,
 \item $I$ is not a Gorenstein ideal, and
  \item $f_1, f_2, f_3$ cannot be split into linear factors.
 \end{itemize}

For example, $\EGH_{(4,4,5), 3}$ and $\EGH_{(4,5,5), 3}$ are two open cases with small degrees.

\begin{remark}
If we focus on the original EGH conjecture, that is, when $a_1=\ldots = a_n = 2$, $ \EGH_{{\bf \bar 2}, n}$ is still open when $n\geq6$.
\end{remark}

For a given homogeneous ideal $I$ in $R=K[x_1,\ldots, x_n]$, the graded Betti number of $I$ is $\beta_{i,j}(R/I) = \dim_K (\Tor^R_i(R/I,K))_j$.
Another well-known conjecture motivated by the EGH conjecture is given by Evans and Charalambous if these graded Betti numbers are also concerned (see lex-plus-powers ideals survey \cite{FR}).  This conjecture can be also considered analogous to Bigatti-Hulett-Pardue Theorem \cite{Bi,Hu,Pa} which is a generalization of the Macaulay's theorem for the graded Betti numbers, more precisely, it shows the extremal behavior of lex ideals for the graded Betti numbers among the homogeneous ideals with the same Hilbert function.

\begin{conjecture}[Lex-Plus-Powers (LPP) conjecture]
Let  $I$ be a homogeneous ideal containing a regular
sequence of degrees $2\leq a_1\leq \ldots \leq a_n$ in $R=K[x_1,\ldots, x_n]$.  Suppose that there exists a lex-plus-powers ideal $L$ with the same Hilbert function as $I$. Then the graded Betti numbers of $R/I$ cannot exceed those of $R/L$.
That is,
$$\mbox{$\beta_{i,j}(R/I) \leq \beta_{i,j}(R/L)$ for all $i$ and $j$.}$$
\end{conjecture}

Just like the EGH conjecture, the LPP conjecture remains widely open.  Nevertheless, there have been remarkable results obtained.  In \cite{Ri}, Richert showed the equivalence of the EGH conjecture and the LPP conjecture when $n=2,3$. Therefore, the LPP conjectures holds when $n=2$.

Similar to Conjecture \ref{non-max_regseq} where ideal contains a regular sequence of length $\leq n$, one can restate the LPP conjecture by allowing non-maximal regular sequences $f_1,\ldots, f_r$ with $r<n$.  Caviglia and Kummini \cite{CK} showed that this case can be also reduced to Artinian case like the EGH conjecture.  In \cite{MM}, the LPP conjecture is shown to be true when the homogeneous ideal containing monomial regular sequence. Thus, when the regular sequence has full length then the LPP conjecture is true for the homogeneous ideals containing $x_1^{a_1},\ldots, x_n^{a_n}$.

 In \cite{CS}, when characteristic of $K$ is 0, it was shown that the LPP conjecture holds for the homogeneous ideals containing a regular sequence with degrees $2\leq a_1 \leq \ldots \leq a_n$ if $a_i \geq\sum\limits_{ j=1}^{i-1} (a_j - 1) + 1$ for $i\geq 3$.

When $i=1$, the numbers $\beta_{1,j}(R/I)$ tells us how many generators the homogeneous ideal $I$ has in degree $j$.
For a given homogeneous $I$ containing a regular sequence with degrees $0\leq a_1\leq \ldots \leq a_n$, the EGH conjecture claims the existence of the lex-plus-powers ideal $L$ associated with degrees $0\leq a_1\leq \ldots \leq a_n$ such that $\dim_K I_j = \dim_K L_j$ for all $j\geq 0$. Then it is well-known that this implies $\beta_{1,j}(R/L) \geq \beta_{1,j}(R/I)$. Thus, the EGH conjecture covers this particular case of the LPP conjecture when $i=1$.

\begin{remark}
Let  $I=(f_1,\ldots, f_n, g_1, g_2)\subseteq K[x_1,\ldots, x_n]$ be a homogeneous quadratic ideal where $f_1,\ldots, f_n$ is a regular sequence. Then, by Theorem \ref{GH}, we see that $\EGH_{{\bf \bar 2}, n}(2)$ holds for such quadratic ideals. Therefore, we get  $\dim_K I_3 \geq n^2+2n -5 = \dim_KL_3,$ where $L$ is the lex-plus-powers ideal $(x_1^2,\ldots, x_n^2, x_1x_2, x_1x_3)$.  This shows us that the number of the independent linear relations among the generators $f_1, \ldots, f_n$, $g_1$, $g_2$ is always at most $5$.
In other words, we obtain $\beta_{2,3}(R/I) \leq 5 = \beta_{2,3}(R/L)$ as well.
\end{remark}

Richert and Sabourin, in \cite{RS}, showed that the following conjecture, a special case of the LPP conjecture when $i=n$, is equivalent the EGH conjecture.

\begin{conjecture} Let $I$ be a homogeneous ideal in $R$ containing a regular sequence of degrees $2\leq a_1 \leq \ldots \leq a_n$ and let $L$ be a lex-plus-powers ideal associated with degrees $a_i$ such that $\Hf_{R/I}(i) = \Hf_{R/L}(i)$ for all $i\geq 0$. Then the dimension of the socle of $I$ is at most the dimension of socle of $L$ in every degree. In other words,
$\beta_{n,j}(R/L) \geq \beta_{n,j}(R/I)$ for all $j\geq 0$.
\end{conjecture}

The LPP conjecture seems much harder than the EGH conjecture due to its strong claim on every graded Betti numbers, yet focusing on certain Betti numbers as its special cases opens up many new directions to work on.

Finally, there are some interesting applications of the EGH conjecture worth to mention.
In \cite{HWW}, Harima-Wachi-Watanabe show that, assuming the EGH conjecture is true, every graded complete intersection has the Sperner property, which simply says for a graded complete intersection $A = R/(f_1,\ldots, f_n)$,
$\max\{ \mu(I)\ | \ I \text{ is an ideal in } A\} = \max\{\dim_K A_i  \ | \ i=0,1,2,\ldots \}$, where $\mu(I)$ is the number of minimal generators of $I$.  It is also known that the Sperner property is related to the so-called Weak Lefschetz property.

Due to geometric background of the EGH conjecture as a result of its connection to Cayley-Bacharach theory, EGH has applications in more geometric settings as well.  For instance, a recent work by Jorgenson \cite{Jor} points out that an affirmative answer for EGH has an implication on the sequence of secant indices of Veronese varieties of $\mathbb{P}^n$ (see Question 3.2 in \cite{Jor}.)

Another application of the EGH conjecture coincides with a very famous problem on decomposing real polynomials in $n$ variables as a sum of squares of real polynomials.  The cone of real polynomials that can be decomposed as a sum of squares of real polynomials is simply referred as SOS cone. In a recent work by Laplagne and Valdettaro\cite{LV}, they show that, when EGH holds, for a strictly positive polynomial on the boundary of the SOS cone, they provide bounds for the maximum number of polynomials that can appear in a SOS decomposition and the maximum rank of the matrices in the Gram spectrahedron.

\section*{Acknowledgements} The author thanks Mel Hochster for introducing and proposing to work on the EGH conjecture during her postdoctoral research. She is deeply grateful for all of their conversations.  The author thanks the referee for their valuable feedback and comments.  She also thanks Martin Kreuzer for pointing out their work.
\bigskip

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Home / Sports / Ducks
Published Nov. 9, 2018 at 11:31PM
Even if unspoken, No. 3 Ducks have destination in mind
By James Crepea
No. 18 Syracuse at No. 3 Oregon
When: 2 p.m. Saturday
EUGENE — When preseason practice began, there was an addition to the Oregon women's basketball locker room courtesy of the coaching staff: A picture of the Tampa skyline.
For as little as the future or long-term goals are discussed by the Ducks, the image of the site of this year's NCAA Women's Final Four was all that needed to be said to strike a chord with a team that's been to back-to-back Elite Eights.
"I love the idea because we won't talk about it, but it'll be there," said Sabrina Ionescu, Oregon's leading scorer and a national player of the year candidate. "I think everyone has that in the back of their mind, that the ultimate goal would be to achieve more than we did these last few years, which is the Elite Eight. Having that picture in there, everyone understands that's where we want to get but no one talks about it and it's not really something that we talk about too much."
The Ducks have bought in to coach Kelly Graves' message of focusing on the daily task at hand and not even setting team goals for the season.
"We don't have a goal for going to the Final Four, winning a national championship, winning the Pac-12. That's not what we've ever been about," Graves said. "Our goal is to get better each and every day. If you only think March and April then I think you do a disservice to your basketball team. I think it's about the process, it's about getting better each and every day. … I think if we achieve our full potential, then who knows what the limit is? We aren't thinking that far ahead."
Oregon opened the season with a 115-36 win over Alaska-Fairbanks on Tuesday in a homecoming for power forward Ruthy Hebard that will count in the record books, though it was an exhibition for the Division II Nanooks and will not count toward Oregon's RPI or NET rankings.
But the real tests begin on Saturday afternoon when No. 3 Oregon hosts No. 18 Syracuse.
When the Ducks take the floor at Matthew Knight Arena for the first time this season, the players will pass the picture of the destination where they hope their 2018-19 journey will end.
"We know what we can do and where we are at. I think seeing it every day is going to be a big reminder that we can make it there," Hebard said. "We were talking about how we should slap it before every game or something just for a little more motivation."
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How I Met Your Mother Series Finale: Funeral Montage Cut!
Many fans were left in shock at the speedy death of the titular character, but it has been revealed that there was in fact a funeral scene filmed for Ted to grieve.
The deleted scene in question was described by Hannigan as a "one-second" montage of title character Tracy's funeral.
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Would you like to see the scene or prefer it be kept far away from your still grieving heart?


text:
Q: Changing $scope property and watching for changes OK, I'm 1-day new to AngularJS, so this has to be ultra-basic (I hope not an ignorant one about how AngularJS works - I admit MVC is not really my thing, but anyway...).
My Code :
var PP = angular.module('PP', []);

PP.controller('documentController', function ($scope) {
    $scope.documents = [
        {
            'filename': 'Untitled-1',
            'content': 'Content-1'
        },
        {
            'filename': 'Untitled-2',
            'content': 'Content-2'
        },
        {
            'filename': 'Untitled-3',
            'content': 'Content-3'
        },
    ];
});

So, I have declared a module (PP), with a controller (documentController) and a property(?) in it (documents).

So, I have 2 questions :

*

*How (any possible way is welcome) could I programmatically have access to and change the documents array? (insert/remove/etc)

*How can I set up an event which fires up when the array has been changed, e.g. an element has been inserted?

A: Accessing the array
In your controller, you just use:
$scope.documents.push({filename: 'Another', content: 'Document'})

You can do that in any controller function because the array is a variable shared in the scope. Since it's in the scope, you can also access it in your view like so:
<button ng-click="documents.push({filename: 'Another', content: 'Document'})">
  Add document
</button>

Usually, I'd define convenience functions for this behavior in the controller to be used in views, but in this simple example, this should do.
Detecting changes in the array
To watch changes in a collection, you can use $scope.$watchCollection:
$scope.$watchCollection('documents', function(newDocuments, oldDocuments) {
  // do something when the documents array changes
});

For a comparison between $watch and $watchCollection, see Scope $watch() vs. $watchCollection() In AngularJS.
Note that watches can be tricky: It's not always trivial to detect changes correctly, in my experience those functions are called way more often than they should.
Depending on your use-case, you might get by without any watches (Angular provides many ways to react on changes without using a single watch), and you probably shouldn't use $watch in your controllers anyway.

A: You access documents the same way you'd access any other javascript array. Meaning the usual operations are available.
This is how you listen for changes:
$scope.$watchCollection('documents', function(val){
    // documents have changed!
});

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Home Society Climate Change
Iman, the Solar-Powered Heroine Inspiring Change in Rural Yemen
byUNDP
in Climate Change, Environment, Equal Rights, SDG Series, Society
"Being featured in the BBC list of 100 Most Influential Women for 2020 gives great pride to me, my business partners, and Yemen as a whole," says Iman, a solar power plant manager from Abss, Yemen.
Iman Hadi, an entrepreneur from Yemen's rural north, is no stranger to the great poverty and misery that plagues much of the country's remote villages where basic services are almost non-existent. But she felt a sense of responsibility and decided to do something about it. "I realized to what extent people in my area, women in particular, are suffering so I decided to go beyond social and cultural barriers and take action," she says.
Iman leads a group of 10 women who installed a solar power plant to provide clean, low-impact energy for residents in Abss, Hajjah Governorate. "The high cost of diesel-generated power has deprived many poor families of electricity access. But now, the solar power plant provides affordable energy for 43 households…so far," says Iman. "I can say with confidence, we could alleviate the suffering of vulnerable people and ease the economic burden from their shoulders," she adds.
This was the beginning of a journey of real change in society's convictions regarding women, their role, their rights, and the importance of giving them an opportunity to prove themselves,
— Iman Hadi
Recently, Iman was announced as one of the world's most influential women for leading positive change and making a difference during Yemen's turbulent times.
When Iman first heard the news, she was taken aback. "I was in Sana'a accompanying my sister to the hospital for cancer treatment. I got a call from one of the solar energy engineers who told me the great news. I was so surprised! It took me time to realize the great work I have done," she says. "Four days later, I returned home with my sister and my family welcomed me warmly, despite their great sadness for my sister. They prepared a cake and congratulated me. Even my brothers – who were vehement opponents to the idea of me running a solar power plant – hugged me tightly," explains Iman.
Her father was also opposed to Iman's work in the beginning, but through discussion and increasing their knowledge about the importance of her work, she was able to convince her family to give her the chance to run her own business. "This was the beginning of a journey of real change in society's convictions regarding women, their role, their rights, and the importance of giving them an opportunity to prove themselves," she says proudly.
Since she started her business two years ago, with the support of the United Nations Development Programme (UNDP) and the European Union, the number of customers has increased from 25 to 43 and the net profit has reached 1,200,000 Yemeni Riyals (approximately US $2,070) – a significant sum in rural Yemen
In the Photo: Iman standing in front of solar panels. Photo Credit: UNDP Yemen/2021
"We have purchased five additional batteries to meet increasing demand and are currently working on expanding the plant to employ more women and extend electricity to more families," describes Iman. "In the near future, I plan to establish another solar power plant in one of the neighboring areas where 47 families need power. In the long run, I dream to launch a large solar power plant to extend solar services for all 3060 households in the Al-Ghadi area," she announces confidently
Iman is also aware that the current situation for many Yemenis is not easy. One-in-four Yemenis have lost their jobs and more are underemployed – barely able to make ends meet. To help her community and to pay her good fortune forward, Iman has an innovative idea for Yemen. "I plan to give easy-to-pay loans from the net profits of the solar plant, to allow my business partners to open small projects such as grocery stores, bakeries and clothing stores. This will create employment opportunities and increase the number of services available to the community."
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Iman's work has pervaded through her whole community, inspiring and providing hope to her neighbors and friends. Faleha Mohammad, a resident of the Al Ghadi area, lives with her elderly parents in a hut. The family is mainly dependent upon their son who sells cold water for income. But the reliable power supply provided by Iman's solar plant encouraged her to buy a sewing machine and sell women's clothes. As a result, as the family's income increased, they were able to construct a new home made of brick and a large room.
Ibraheem Ali, a father of seven and a daily-wage laborer also wanted to improve living conditions for his family. He approached Iman to ask if it would be possible to buy a welding gun and run it on her solar power. Iman saw the merit in his idea and quickly encouraged him to bring it to reality. Now, after taking professional welding classes, he is welding two to three water tanks a day. He has also increased his daily income to YER 8000 from YER 3000 (US$ 14 from US $5).
In the Photo: At the solar plant, the local women of Abs are asking for electricity supply services. Photo Credit: UNDP Yemen/2021
"Women and men from my village came to me and asked for advice on how to improve their living conditions. This reflects how their views on working women have changed," explains Iman. "Zahra, for example, a mother of four who lost her husband at the beginning of the war recently visited me."
Since losing her husband, Zahra has endured great hardship – she is alone in a country suffering from the ravages of war. The difficult circumstances forced her eldest son – a mere 15 years old – to work as day laborer for very little money. But sadly, business slowed and has now come to a complete stop due to the COVID-19 outbreak. This has resulted in increased suffering for the family, who struggle to afford food and medicine.
"One evening, Zahra started complaining about the difficult life she is experiencing with her kids. I suggested that she might consider bringing electricity to her home, buy an electric mixer, and make incense that she could sell for a profit," recounts Iman. Zahra liked the idea and quickly took action. "She has now started her own business and is a productive woman with a sustainable source of income for her and her family."
On 8 March, International Women's Day, we asked Iman what she would like to say to Yemen's women. She had three messages to convey.
1. "The first message is to all great Yemeni women: do not stand where you fail. Strive with determination and confidence and your dreams will come true."
2. "The second message is for the community: women are not ONLY housewives, we are half of society; we have the right to equal opportunities in all life's aspects."
3. "Thirdly and finally, I would like to thank all the men in my village who were once against women working outside the home, but today offer support and encouragement."
In the Photo: Iman and her business partners. Photo Credit: UNDP Yemen/2021
In total, UNDP and its partner the Sustainable Development Foundation (SDF) have installed two solar power systems in two communities. 20 individuals were trained in solar power system installation, maintenance and business management to help establish local solar power businesses and ensure the sustainability of renewable energy systems in Yemen.
These activities were implemented as a part of the Supporting Resilient Livelihoods and Food Security in Yemen Joint Programme (ERRYJP II) in partnership with Sustainable Development Foundation (SDF) and generous funding and support from the European Union.
Editor's Note: The opinions expressed here by Impakter.com columnists are their own, not those of Impakter.com — In the Featured Photo: Iman standing in front of two solar panels. — Featured Photo Credit: UNDP Yemen/2021
Tags: entrepreneurshipIman Hadisolar powerUNDPUNDP YemenwomenYemen
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UNDP works in some 170 countries and territories, helping to achieve the eradication of poverty, and the reduction of inequalities and exclusion. We help countries to develop policies, leadership skills, partnering abilities, institutional capabilities and build resilience in order to sustain development results.


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Earthquake Warning California
Utilities Toolkit & Resources
The California Governor's Office of Emergency Services (Cal OES) launched the "Don't Get Caught Off Guard" campaign in October 2020 to help inform communities throughout California about the Earthquake Warning California system. Utility providers can find the latest materials, resources, and news relating to the nation's first statewide, publicly available earthquake warning system on the newly revamped website: earthquake.ca.gov
For utility providers in California, earthquakes are not a matter of if, but when. Utility locations often span a wide region and service thousands of customers that might live along a fault line or in an area that is prone to earthquakes. Utility providers should have a plan in place to help mitigate risk and recover quickly following an earthquake. They should also take measures to establish a communication strategy for customers experiencing service disruptions as a result of an earthquake. They should also consider including earthquake early warning technology in their disaster planning and measures that could be taken to automatically adjust processes in the event of an earthquake to reduce damage and speed recovery.
It is important to inform and educate customers on the necessary steps to check for hazards after an earthquake, such as gas leaks, damaged electrical wiring, and the potential for flooding. By helping share resources with employees and customers, utility providers can play a key role in making Californians more resilient and keeping people safe when the next major earthquake occurs.
Share the following information via newsletter or social media! For questions relating to Earthquake Warning California, email earthquakeinfo@caloes.ca.gov
Newsletter Language
Use the language below to copy/paste and share with target audiences through email blasts, newsletters, or other channels. If you belong to a housing association, club, or social group, we encourage you to share this material. Please feel free to tailor or personalize it.
April is Earthquake Preparedness Month in California! We know that there are many things to be worried about, but when it comes to earthquakes, it is not a matter of if one will occur, but when. The California Governor's Office of Emergency Services (Cal OES) is encouraging residents to take the time to update their emergency plan, download the MyShake app, and adjust their phone settings. Many things in life may catch us off guard; but now, earthquakes don't have to!
GAME CHANGER: Californians can now get warnings for earthquakes from Earthquake Warning California. By downloading the FREE MyShake app and adjusting phone settings, people can get a jump start, in some cases before shaking can be felt, to take protective actions such as to DROP, COVER, and HOLD ON (or LOCK, COVER, and HOLD ON if using a wheelchair).
If you receive a warning or feel shaking, there are moments to act. There may be circumstances where a warning is issued, but no shaking occurs. It is always better to be on the side of caution when it comes to earthquakes. We encourage you to share this information with friends and family in California to help spread the word about the technology available to help keep them safe.
Here are some of the ways to receive earthquake warnings:
MyShake App. An app that can be downloaded for mobile devices at no-cost from Google Play and the Apple App Store. App location services must be set to "always-on" so that ground sensors can alert those in the nearby area when shaking is detected.
Android Earthquake Alerts. Included in new or updated Android devices, the system uses the same technology as the MyShake App.
Wireless Emergency Alerts (WEAs). Text-like messages from the government during emergency situations. This includes: Presidential, Imminent Threats (fire, earthquake, floods, etc.), and AMBER alerts.
When an earthquake warning is issued from any of these sources, individuals should quickly take protective actions to stay safe, such as dropping to the ground, covering their head with their arms, and holding onto their neck with both hands until shaking stops. Do not stand in doorways or near glass windows.
[Insert organization name] encourages our [employees/customers] to have a plan for how to protect themselves in the event of an earthquake, be it onsite, on the road, or at home. If you receive a warning, be sure to react with the assumption that shaking will occur soon after. There may be circumstances where a warning is issued, but no shaking occurs. It is always better to be on the side of caution when it comes to earthquakes. We encourage you to share this information with friends and family to help spread the word about the technology available to help keep them safe..
Please visit earthquake.ca.gov to learn more about the latest tools and resources, and how to use them. You can also sign up for a regular partner e-newsletter on the website from Cal OES that includes the latest news and information about Earthquake Warning California.
Sector-specific social media images are available at https://earthquake.ca.gov/get-prepared.
Please consider using social media to help spread awareness about Earthquake Warning California resources. Hashtags (#) help others looking for information to see trending posts while using the @ symbol tags other organizations (use @Cal_OES for Twitter and @CaliforniaOES for Facebook).
[Insert Organization]wants to remind our customers about no-cost resources from #EarthquakeWarningCA that are designed to provide a few moments warning before an #earthquake. Find out more at earthquake.ca.gov
The #MyShake App, #AndroidEarthquakeAlerts, and #WEAs are tools that can give California residents earthquake warning. Learn how to be prepared at earthquake.ca.gov.
[Insert Organization] is helping our employees learn about #EarthquakeWarningCA that can provide a few moments notice before shaking so they can take protective actions. Learn more at earthquake.ca.gov.
Earthquakes can cause other disasters such as fires, floods, and power outages. Be one step ahead with #EarthquakeWarningCA and visit earthquake.ca.gov.
Social Media (Graphics)
The following can be tailored and used on personal social media pages or by organizations to help spread awareness about Earthquake Warning California resources. Hashtags (#) help others looking for information to see trending posts, while using the @ symbol tags other organizations (use @Cal_OES for Twitter and @CaliforniaOES for Facebook).
Right click an image above to save it, or click it to enlarge it.
Click the top right icon to reveal the video playlist


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What is the rarest poison dart frog?
Are dart frogs poisonous to humans?
Is it illegal to own a poison dart frog?
Are pink frogs real?
Can you keep a poison dart frog as a pet?
How much do dart frogs cost?
What is the species of a poison dart frog?
What are the characteristics of a poison dart frog?
Most collectors consider the Lehmann's poison frog to be the holy grail among poison dart frogs for its rarity.
Are all poison dart frogs poisonous? Most poison dart frogs aren't dangerous to humans, though some are lethal to the touch. For example, the golden poison dart frog, at just 2 inches long, has enough poison to kill 10 grown men. Dart frogs raised by humans, such as those at Berkshire Museum, are not poisonous at all.
Is it poisonous to touch a poison dart frog?
The frogs' poison is found in their skin, making them too toxic to touch. While most frogs are considered toxic but not deadly, they are distasteful to a predator and can even be fatal. The poison can cause serious swelling, nausea, and muscular paralysis.
What is the prettiest poison dart frog?
blue poison dart frog
The blue poison dart frog (Dendrobates tinctorius "azureus") is unquestionably beautiful—like sapphire.
Just because a species is available, does not mean it is legal to own. Unfortunately, dart frogs are still readily smuggled and available wild-caught. You should always search for healthy, captive bred specimens that have been long established in the hobby.
Pink Frogs (Lipstick Frogs) Do Not Exist.
Are dart frogs good pets?
In captivity, poison dart frogs do not eat the toxic plants that cause them to be dangerous. As such, these tiny little animals can make excellent pets. With minimal maintenance, a beautiful habitat, and a remarkably long lifespan, a poison dart frog can be a perfect choice for your next pet.
Can you survive a poison dart frog?
Most poison frog species are considered toxic but not deadly. The poison in their skin can cause swelling, nausea, and paralysis if touched or eaten without necessarily being fatal.
between $30 and $70
Expect to pay between $30 and $70 for a single poison dart frog. More sought-out colors or species will be more expensive. It is always best to buy captive-bred frogs for your hobby. Not only are these frogs healthier and hardier, but they do not deplete natural frogs in the wild.
Are dart frogs good for beginners?
Anthony's poison dart frog is typically considered an easy species for experienced frog keepers, and beginners may also have success with these frogs. They should not be kept in mixed-species tanks, and they typically fare best when kept in a tank with only one, opposite-sex companion.
Is there a blue frog?
tinctorius azureus is also known by its indigenous Tirio name, okopipi. Its scientific name comes from its azure (blue) color….
Species: D. tinctorius
Subspecies: D. t. "azureus"
Trinomial name
Dendrobates tinctorius "azureus" (Hoogmoed, 1969)
Ranitomeya. Silverstoneia. However, frogs from most of these genera are not available in the pet trade – at least, not in significant numbers. Instead, the vast majority of poison dart frogs in captivity are members of a handful of genera, including Dendrobates, Phyllobates, Ranitomeya and Epipedobates.
Fascinating Facts. Poison dart frogs get their toxicity from some of the insects they eat.
Physical Characteristics.
Habitat/Diet.
Social Behavior.
Status In The Wild.
Location in the Zoo.
What type of skin does a poisonous dart frog have?
The blue poison dart frog is recognized by its distinctive, blue color. When blue poison dart frogs are bred in captivity and fed a varied diet, their skin does not become poisonous. What is the most poisonous thing in the world?
What is the classification of a poison dart frog?
Scientists currently recognize approximately 200 species of poison dart frog. It's difficult to identify the exact number of described species, as authorities often differ on some of the minor points of poison dart frog taxonomy. Nevertheless, these species are all part of a single group, called the family Dendrobatidae.


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Hackney Gazette > News > Education
Noam Chomsky attacks government over £1.75m Hackney Community College cuts
By Emma Bartholomew
Published: 4:47 PM May 20, 2014 Updated: 10:35 AM October 14, 2020
Noam Chomsky - Credit: Archant
Famed American academic and political activist Noam Chomsky has praised campaigners at Hackney Community College who are fighting what he has dubbed "the war which is being waged against our most valuable institutions".
Hackney Community College. - Credit: Archant
The college in Falkirk Street is currently consulting about £1.75m proposed cuts as part of the government's ongoing deficit reduction strategy, to be implemented in the forthcoming academic year.
The cuts come on top of a £1m reduction in the funding for adult education last year and £2m the previous year.
Chomsky, who believes those in power do all they can to maintain an unequal social status quo, said: "The attack on education in England and particularly in the US is a particularly malicious and destructive element on the neoliberal assault of the general population in the past generation.
The US-based author and lecturer, who is sometimes described as the "father of modern linguistics" continued: "Hackney Community College (HCC) is an outstanding example of what should be done if policy were devoted to the needs of people, rather than enhancing privilege power and authority.
"It's not surprising then that the college should be targeted in the general assault."
Chomsky's video message was played at a public meeting at the college last Tuesday night organised by the University and College Union branch union, at which managers, governors, and trade unions agreed the cuts will unfairly impact the further education sector nationwide.
The college is consulting with trades unions and staff over the proposals, which will effectively see a £1.5m budget cut because the college has sourced some funding through business streams.
The college has not yet ruled out redundancies.
Ian Ashman, HCC principal said: "We are consulting about how the college's structure can be re-shaped to work within this reduced funding, and how we continue to provide high quality training and courses that meet the needs of local communities and businesses."
On Tuesday Mr Ashman, HCC's chairman of governors, staff and students met with Hackney MPs Diane Abbott and Meg Hiller at Westminster to explain the implications the cuts will have.
The pair have promised to organise a parliamentary debate on adult further education funding.
A spokesman for the Department for Business, Innovation and Skills spokesperson said the adult further education and skills budget of £4.1 billion would be prioritised where it will make the most impact.
"Our priorities remain focused on apprenticeships, traineeships, supporting unemployed people's work skills and increasing standards in maths and English," he said.
"We are working closely with colleges so that they understand the need to generate a greater range of income streams from local partners and employers."


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La Pézize baie (Peziza badia) est un champignon ascomycète de la famille des pézizacées.

Description
Cette pézize a une hauteur de 1 à . Son pied minuscule est caché par l'apothécie érigée de 2 à  de diamètre formant un réceptacle en forme de coupe qui s'étale avec l'âge et devient irrégulière, lobée. La face interne correspond à l'hyménium brun châtaigne virant au brun-olive, produisant une sporée blanche. La face externe vésiculeuse est brun ocre puis brun rougeâtre et foncée, d'où son nom vernaculaire de Pézize noirâtre. La chair mince et brun rouge renferme un suc aqueux.

Habitat
Cette pézize vient en fin d'été et en automne sur les bords de chemins, de fossés, dans des endroits très humides (bois de bouleaux ou de conifères, tourbières).

Comestibilité
Toxique à l'état cru, elle est un comestible agréable une fois cuite.

Habitat
Sur sol sablonneux et terre nue.

Notes et références

Liens externes

Pezizomycetes
Espèce de champignons (nom vernaculaire)

redpajama_set_name: RedPajamaWikipedia


Unions help make possible what would be impossible for individuals acting alone. It's how we were able to lift teachers' salaries in New York City by double digits before the 2008 recession, so they were in line with surrounding suburbs, and how teachers in West Virginia are fighting for a living wage and to stop skyrocketing healthcare premiums. Unions advocate for good public schools for all our kids, affordable higher education and healthcare, and a voice at our jobs and in our democracy. After yet another school gun massacre, we are redoubling our efforts to make schools safe sanctuaries, not armed fortresses. And we are fighting with special urgency against arming teachers and for the school safety and mental health funds that President Trump's new budget eliminates.
We have daily reminders of the voice and agency that public employee unions afford their members so they can do their jobs well and support themselves, their families, the people they serve, and their communities. There are numerous examples within the pages of this magazine.
If the Houston Federation of Teachers (HFT) had not researched the deficiencies of using value-added methods in teacher evaluations, and built political strength to elect a school board that was responsive to the evidence, students in Houston would still be losing good teachers tarnished by this indecipherable test-based formula. Beyond teaching and learning conditions, after Hurricane Harvey, I saw firsthand how Zeph Capo and members of the HFT doubled down to care for union and community members who suffered terrible losses.
Also in this issue, Jon Shelton examines the powerful connections between teachers unions' interests and the needs of the broader community. As he writes, the Chicago Teachers Union bargained for "the schools Chicago's students deserve"—negotiating for smaller class sizes; wraparound services for students; relevant, high-quality professional development; and an end to institutional racism in Chicago schools. And Emily Gasoi and Deborah Meier write eloquently about how teachers uphold the vital importance of public education as a keystone of American democracy.
You don't hear any of that from the billionaires backing the Janus case. They simply want the unions that public employees belong to out of the picture.
I am not prone to hyperbole, but we face a clear and present threat to American democracy by those who want to further rig the system toward the already powerful. They're going after unions, public education, and the right to vote. Why? Because those are the vehicles for regular folks to secure a better life.
Janus is just one part of this. The conservative State Policy Network has pledged to spend $80 million to "defund and defang" unions. The Kochs, after receiving the Trump tax cut, upped the ante with $400 million to undermine public education and "break" the teachers unions. Why? Because unions fight for a better life for working people, and the right-wing sees that as a threat to their political and economic power.
"Never again" has been the cri de coeur for many—those opposing genocide, of course, and, more recently, those decrying mass gun violence. It is also fitting for those who insist that our country must not revert to a time when workers were systematically denied even the most fundamental rights—a voice and a better life.


text:
Leonhard "Leo" Georg Fischer (* 29. Juli 1981 in München) ist ein deutscher Journalist, Literaturwissenschaftler, Buchautor und Mitglied der Regierung der Stadt Frankfurt am Main. Von 2008 bis 2013 war er Chefredakteur des Satiremagazins Titanic.

Leben
Fischer, der aus einer Arztfamilie stammt, legte im Jahr 2001 am Albrecht-Altdorfer-Gymnasium Regensburg sein Abitur ab. An der Freien Universität Berlin sowie in Lausanne studierte er Allgemeine und Vergleichende Literaturwissenschaft, Philosophie und Publizistik.

Ab 2007 war er Redakteur der Satirezeitschrift Titanic, deren Chefredakteur er von Oktober 2008 bis Oktober 2013 war.

Seit Mai 2012 ist er Kolumnist der Wochenzeitung Jungle World. Von 2014 bis 2015 verfasste er "Briefe von der Heimatfront" für das Nachrichtenportal Watson. Er schrieb für das Musikmagazin Intro, die "Wahrheit"-Seite der Taz, die Tageszeitung Neues Deutschland (die Kolumne "Das kann weg") sowie für die Monatszeitschrift Konkret. Im August 2017 betreute Fischer, wie andere Prominente zuvor, den Twitteraccount des ZEITMagazins, bis er von dieser Aufgabe wegen in satirischer Absicht veröffentlichter Falschmeldungen entbunden wurde.

2008 wurde Fischer zum Mitglied des Bundesvorstands der Partei für Arbeit, Rechtsstaat, Tierschutz, Elitenförderung und basisdemokratische Initiative (Die PARTEI) ohne besonderen Geschäftsbereich gewählt. 2011 kandidierte er bei der Kommunalwahl in Frankfurt am Main auf der Liste der PARTEI. Wie schon bei der Bundestagswahl 2013 war Fischer auch bei der Bundestagswahl 2017 Direktkandidat im Wahlkreis Frankfurt II und stand zugleich auf der Landesliste der PARTEI. Im Juni 2019 trat er aus Protest gegen das zögerliche Handeln der Partei bei parteiinternem Sexismus gemeinsam mit Martina Werner aus dem Bundesvorstand aus.

Aktuell (2021) ist Fischer bei der Bildungsstätte Anne Frank in Frankfurt am Main für die Presse- und Öffentlichkeitsarbeit zuständig.

Bei den Kommunalwahlen in Hessen 2021 am 14. März kandidierte Fischer auf der Liste der Wählervereinigung ÖkoLinX-ARL für einen Sitz im Stadtverordnetenrat des Frankfurter Römer.
Am 8. September 2021 wurde er zum ehrenamtlichen Magistratsmitglied gewählt.

Bücher
 Generation "Gefällt mir". Lappan Verlag, 2012, ISBN 978-3-8303-3302-9.
 Die Titanic-Bibel: Segen, Sünden, Sauereien – der Weltbestseller erstmals unzensiert! Hrsg. mit Tim Wolff und Michael Ziegelwagner. Rowohlt Verlag, Berlin, 2013, ISBN 978-3-87134-766-5.
 Fröhliche Hundegeschichten, mit Illustrationen von Leonard Riegel. Eichborn Verlag, 2014, ISBN 978-3-8479-0557-8.
 Niemand sagt's Angela: Das supergeheime Abhörprojekt der NSA. Satirischer Roman. Bastei Entertainment, 2014, E-Book, ISBN 978-3-8387-5710-0.
 Warum ich links bin … : ...und immer noch meine Deutschlehrerin hasse (Essay). Murmann Verlag, 2015, ISBN 978-3-86774-458-4.
 Leo Fischer ohne Markus Söder – Gottes Werk und mein Beitrag: Die komplett erfundene Autobiografie. Riva Verlag, 2018, ISBN 978-3-7423-0812-2.
 Der Kaffee am Arsch der Welt: Eine Erzählung über den Unsinn pseudophilosophischer Weisheiten. riva Verlag, 2019, ISBN 978-3-7423-0946-4.
 Der Kaffee am Arsch der Welt. Hörbuch, eingelesen von Michael A. Grimm. riva Verlag, 2020, EAN 9783745314755.
 Die kürzeste Menschheit der Geschichte. riva Verlag, ISBN 978-3-742318893 (angekündigt für 2022).

Weblinks

 Leo Fischer bei Twitter
 Leo Fischer bei Facebook
 Leo Fischer bei abgeordnetenwatch.de

Einzelnachweise

Autor
Literatur (21. Jahrhundert)
Literatur (Deutsch)
Satire
Journalist (Deutschland)
Kultur (Frankfurt am Main)
Deutscher
Geboren 1981
Mann


I found a nice open source package to edit the videos into a nice compilation.
Pixel is a very special cat. Apart from being able to fly, he can make funny noises, and in general be very cuddly.
Next story (AVR-)Dude, Arduino crashes fixed!


text:
Q: What's the meaning of the bold sentence in this paragraph?
Feedback includes any information you get about yourself. In the broadest sense, it's how we learn about ourselves from our experiences and from other people—how we learn from life. It's your annual performance review, the firm's climate survey, the local critic's review of your restaurant. But feedback also includes the way your son's eyes light up when he spots you in the audience and the way your friend surreptitiously slips off the sweater you knitted her the minute she thinks you're out of view. It's the steady renewal of services by a longtime client and the lecture you get from the cop on the side of the road. It's what your bum knee is trying to tell you about your diminishing spryness, and the confusing mix of affection and disdain you get from your fifteen-year-old.
So feedback is not just what gets ranked; it's what gets thanked, commented on, and invited back or dropped. Feedback can be formal or informal, direct or implicit; it can be blunt or baroque, totally obvious or so subtle that you're not sure what it is.

(source)

A: In the boldfaced sentence, the authors are making the point that feedback need not always be explicit (what gets ranked). In fact, feedback is often implicit and open to interpretation (what gets thanked, commented on, and invited back or dropped).
Examples of explicit feedback, such as a performance review or a critic's review of your restaurant, are often such that they can be mutually compared and even ranked. For instance, based on the reviews of the food critic, one could conceivably rank the restaurants in the town, and your restaurant will earn a definite place, be it high or low, on that list. Similarly, performance reviews of different employees can often be directly compared, so that employee A can be said to have gotten a better review than employee B. When the authors talk about the kinds of feedback that can be ranked, that is what they mean.
However—and that's the point of the boldfaced sentence—not all feedback is like that. For example, suppose you notice your friend surreptitiously slipping off the sweater you knitted her the minute she thinks you're out of view. That is also valid feedback: your friend clearly doesn't like the sweater. And this is true even though she presumably said that she did. This kind of feedback is implicit in her actions, where the friend is even trying to hide the actions themselves. The sweater is an example of what gets dropped.
Let's take another example of implicit feedback, this time of something that gets invited back (or not). Suppose some coworker, whom you barely know, decides to have a big birthday party. Many people get invited, including yourself. Next year, she has a much smaller party, but you still get invited. That's very positive feedback: clearly, she liked you being there. But since she actually didn't say anything to you, this feedback is implicit.