text:
\subsection*{Structure and style of the paper}
In Section~\ref{s:prelim} we give a rapid definition of $\operatorname{\rm AM}_{\rm Z}(G)$ and establish some basic properties that will be needed. These properties were already established in \cite{AzSaSp}, using a Banach-algebraic perspective; here we shall give complete proofs that only need Schur orthogonality for finite groups, so that the reader does not need to take results from \cite{AzSaSp} on trust.

Also, following an idea used by Azimifard, Samei and Spronk in their proof of Theorem~\ref{t:ASSgap}, we introduce an auxiliary invariant $\operatorname{ass}(G)$ which is a lower bound for $\operatorname{\rm AM}_{\rm Z}(G)$ and is easier to estimate from below.
In Section~\ref{s:trivial-centre} we obtain a new lower bound on $\operatorname{ass}(G)$ for all groups with trivial centre (excluding the degenerate case of the $1$-element group). This is perhaps the main conceptually new idea in this paper that was missing from \cite{ACS_AMZL-2cd} and~\cite{AzSaSp}. Nevertheless, to get from here to the optimal constant in Theorem~\ref{t:mainthm} requires another ingredient, and that is to reduce the problem to the case of finite groups that are ``just non-abelian''. The precise statement is Theorem~\ref{t:bound for JNA cases}: the necessary definitions, and the proof of that theorem, take up all of Section~\ref{s:JNA cases}. We also need two technical results on just non-abelian groups (Lemma \ref{l:baby JN2} and Theorem~\ref{t:is-dihedral}): both can be extracted from the existing literature, but for the reader's convenience we provide full proofs in the appendix.

These sections, together with the Appendix, provide a complete proof of Theorem~\ref{t:mainthm}. Figure~\ref{fig:leitfaden} shows how the key results are assembled in this proof.
\begin{figure}[htpb]
\[ \left. \begin{array}{lcc}
 & &  \fbox{Corollary \ref{c:AMZL-of-prod-and-quot}\ref{li:quot}}  \\
 & & \\
 \left.   \begin{tabular}{c}
 \fbox{Theorem~\ref{t:bound for trivial centre}} \\
 \fbox{Theorem~\ref{t:is-dihedral}} \\
  \fbox{Lemma~\ref{l:baby JN2}}
\end{tabular} \right\}  & \Longrightarrow
& \fbox{Theorem~\ref{t:bound for JNA cases}}
\end{array} \right\} \Longrightarrow
\fbox{Theorem~\ref{t:mainthm}}
\]
\caption{Results needed to prove the main result}
\label{fig:leitfaden}
\end{figure}
For sake of completeness and interest: in Section \ref{s:ASS(perfect)} we use similar but easier ideas to those of Section~\ref{s:trivial-centre} to obtain a lower bound on $\operatorname{ass}(G)$ for all perfect groups $G$; and in Section~\ref{s:closing} we list several natural questions that we have not yet resolved, concerning the set of possible values of $\operatorname{\rm AM}_{\rm Z}(\cdot)$.

The style of this paper is intended to be somewhat expository. The aim, both here and in the earlier paper~\cite{ACS_AMZL-2cd}, has been to make the proofs accessible to researchers in abstract harmonic analysis who are interested in amenability constants of various Banach algebras, but who may not have much exposure to structure theory for finite groups. Therefore, the proofs are spelled out in some detail, and we have avoided more specialized results or techniques from finite group theory. On the other hand, this paper does not presume any knowledge of (or interest in) abstract harmonic analysis; so finite group theorists may find Theorem~\ref{t:mainthm} and its proof of some interest, and perhaps take the results obtained en route as a challenge to find sharper versions or better estimates.

\end{section}

\begin{section}{Preliminaries}\label{s:prelim}

\begin{subsection}{Notation and terminology}
We assume familiarity with basic character theory of finite groups over the complex field~$\mathbb C$, up to the Schur orthogonality relations.
 All of this can be found in Isaacs's book \cite{Isaacs_CTbook}; alternatively, see \cite[Chapter 7]{Cohn_ed2vol2} or \cite{JamLie}.

 Given a finite group $G$, we denote by $\operatorname{Conj}(G)$ the set of conjugacy classes of $G$, and we write $\operatorname{Irr}(G)$ for the set of irreducible characters of~$G$ (working over $\mathbb C$). The \dt{degree} of a character $\phi$ is just $\phi(e)$ (which equals the dimension of any representation affording that character) and we say that $\phi$ is \dt{linear} if $\phi(e)=1$.
If $C\in\operatorname{Conj}(G)$ and $\phi$ is a character on $G$, we write $\phi(C)$ for the value of $\phi$ on any element of $C$; thus $|\phi(C)|\leq\phi(e)$.

The element $\Delta_{\rm Z}(G)$ mentioned in the introduction is the \dt{diagonal element} or \dt{separability idempotent} for the commutative $\mathbb C$-algebra $Z(\mathbb C G)$. For the definition of these terms for general algebras over a field, see e.g.~\cite[Section 6.7]{Cohn_ed2vol3}; further explanation in the special case of the centre of a group algebra, and discussion of the connection with amenability for Banach algebras, can be found in \cite[Section~2.1]{ACS_AMZL-2cd}.

To keep our presentation self-contained, we will not use general properties of separability idempotents, and instead adopt a ``hands-on'' definition. First some terminology: if ${\mathbb I}$ is a non-empty finite indexing set, then by the ``natural'' $\ell^1$-norm on the vector space $\mathbb C^{\mathbb I}$ we simply mean $\norm{a}_1 \equiv \norm{a}_{\ell^1({\mathbb I})} := \sum_{i\in{\mathbb I}} |a_i|$.

\begin{dfn}\label{d:define diagonal}
Let $G$ be a finite group. If $\phi\in\operatorname{Irr}(G)$, we may regard $\phi\mathbin{\otimes}\phi$ as an element of $\mathbb C (G\times G)$ (since the underlying vector space of $\mathbb C (G\times G)$ is $\mathbb C^{G\times G}$).
Let
\[
\Delta_{\rm Z}(G) :=  \sum_{\phi\in\operatorname{Irr}(G)} \frac{\phi(e)}{|G|}\phi \mathbin{\otimes}  \frac{\phi(e)}{|G|}\phi \in \mathbb C(G\times G)\,.
\]
Then, equipping $\mathbb C^{G\times G}$ with its natural $\ell^1$-norm, define the \dt{ZL-amenability constant of~$G$} to be the $\ell^1$-norm of $\Delta_{\rm Z}(G)$.
\end{dfn}

We could of course define $\operatorname{\rm AM}_{\rm Z}(G)$ without explicitly naming $\Delta_{\rm Z}(G)$, but  some hereditary properties of $\operatorname{\rm AM}_{\rm Z}$ with respect to taking products and quotients of groups are best understood in terms of corresponding properties of $\Delta_{\rm Z}$. This will be done in the next section.
\end{subsection}

\begin{subsection}{Hereditary properties of the ZL-amenability constant}
The next two lemmas are known results, if one uses the definition of $\Delta_{\rm Z}(G)$ that is given in \cite{AzSaSp} and~\cite{ACS_AMZL-2cd}. We shall give character-theoretic proofs that just use Definition~\ref{d:define diagonal}.

\begin{lem}\label{l:diagonal of group product}
Let $G$ and $H$ be finite groups. If we identify $\mathbb C(G\times H\times G\times H)$ with $\mathbb C(G\times G) \mathbin{\otimes} \mathbb C(H\times H)$ in the natural way, then $\Delta_{\rm Z}(G\times H)$ is identified with $\Delta_{\rm Z}(G)\mathbin{\otimes} \Delta_{\rm Z}(H)$.
\end{lem}

\begin{proof}
This is a straightforward calculation using $\operatorname{Irr}(G\times H)=\{ \phi\mathbin{\otimes}\psi \mathbin{\colon} \phi\in\operatorname{Irr}(G), \psi\in\operatorname{Irr}(H)\}$ and $(\phi\mathbin{\otimes}\psi)(e_{G\times H}) = \phi(e_G)\psi(e_H)$.
\end{proof}

\begin{lem}\label{l:diagonal of quotient group}
Let $G$ and $H$ be finite groups, and suppose $q:G\to H$ is a surjective group homomorphism. Extend $q$ to an algebra homomorphism $Q:\mathbb C G \to \mathbb C H$ in the natural way. Then $(Q\mathbin{\otimes} Q)(\Delta_{\rm Z}(G))=\Delta_{\rm Z}(H)$.
\end{lem}

\begin{proof}
The map $\phi\mapsto \phi\circ q$ defines an injection $q^*: \operatorname{Irr}(H)\hookrightarrow \operatorname{Irr}(G)$.
Now
\[ Q\left(\sum_{y\in H} \sum_{x\in q^{-1}(y)} a_x\delta_x\right) = \sum_{y\in H} \left(\sum_{x\in q^{-1}(y)} a_x\right) \delta_y \qquad(a\in\mathbb C G).\]
In particular, for each $\phi\in\operatorname{Irr}(H)$,
\[ Q\left( \frac{q^*(\phi)(e_G)}{|G|} q^*(\phi)\right) = \frac{\phi(e_H)}{|G|} |\ker(q)|  \phi = \frac{\phi(e_H)}{|H|} \phi\;.
 \]

Let ${\mathcal R}$ denote the set $\operatorname{Irr}(G)\setminus q^*(\operatorname{Irr}(H))$. Then
\[ Q\left( \sum_{\psi \in\operatorname{Irr}(G)\setminus {\mathcal R}} \frac{\psi(e_G)}{|G|}\psi \mathbin{\otimes}   \frac{\psi(e_G)}{|G|}\psi \right) = \Delta_{\rm Z}(H)\,; \]
so to complete the proof, it suffices to show that ${\mathcal R}\subseteq \ker(Q)$.

Our argument is indirect. Given $f\in{\rm Z}\Cplx G$, for each $y\in H$ we pick $x\in q^{-1}(y)$ and note that
\[ \delta_y * Q(f) = Q(\delta_x* f) = Q(f*\delta_x) = Q(f)* \delta_y\,.\]
Hence $Q({\rm Z}\Cplx G)\subseteq {\rm Z}\Cplx H$. Since $\operatorname{Irr}(H)$ spans ${\rm Z}\Cplx H$, it now suffices to prove that
\[ \sum_{y\in H} Q(\theta)(y)\ \overline{\phi(y)} = 0 \qquad\text{for all $\theta\in {\mathcal R}$ and all $\phi\in\operatorname{Irr}(H)$.} \]
Equivalently, it suffices to prove that for each $\theta\in {\mathcal R}$ and each $\phi\in \operatorname{Irr}(H)$,
\[ 0 = \sum_{y\in H} \left( \sum_{x\in q^{-1}(y)} \theta(x) \right) \overline{\phi(y)} = \sum_{x\in G} \theta(x)\overline{(\phi\circ q)(x)}\,. \]
But this is immediate from Schur orthogonality for $\operatorname{Irr}(G)$.
\end{proof}

\begin{cor}[${\rm ZL}$-amenability constants of products and quotients]
\label{c:AMZL-of-prod-and-quot}\
\begin{YCnum}
\item\label{li:prod}
 Let $G_1$ and $G_2$ be finite groups. Then $\operatorname{\rm AM}_{\rm Z}(G_1\times G_2)=\operatorname{\rm AM}_{\rm Z}(G_1)\operatorname{\rm AM}_{\rm Z}(G_2)$.
\item\label{li:quot}
 Let $G$ be a finite group and let $N \unlhd G$. Then $\operatorname{\rm AM}_{\rm Z}(G/N)\leq \operatorname{\rm AM}_{\rm Z}(G)$.
\end{YCnum}
\end{cor}

\begin{proof}
For any finite indexing sets ${\mathbb I}$ and ${\mathbb J}$, the $\ell^1$-norm has the following property: for each $a\in \mathbb C^{\mathbb I}$ and $b\in \mathbb C^{\mathbb J}$\/,
we have $\norm{a \mathbin{\otimes} b}_{\ell^1({\mathbb I}\times{\mathbb J})} = \norm{a}_{\ell^1({\mathbb I})}\norm{b}_{\ell^1({\mathbb J})}$\/.
Now, using Lemma \ref{l:diagonal of group product}, part~\ref{li:prod} follows.

To prove part~\ref{li:quot}, note that $Q:\mathbb C G \to \mathbb C H$ does not increase the $\ell^1$-norm of elements. The same is true for $Q\mathbin{\otimes} Q:\mathbb C(G\times G) \to \mathbb C(H\times H)$. Therefore $\norm {(Q\mathbin{\otimes} Q)(\Delta_{\rm Z}(G))}_1\leq \norm{\Delta_{\rm Z}(G)}_1$, and applying Lemma~\ref{l:diagonal of quotient group} does the rest.
\end{proof}

\begin{rem}\label{r:retread}
Both parts of Corollary~\ref{c:AMZL-of-prod-and-quot}
 were already proved in \cite[\S1]{AzSaSp}, using the abstract characterization of $\Delta_{\rm Z}(G)$ as the unique ``diagonal element'' or ``separability idempotent'' for the algebra ${\rm Z}\Cplx G$. Our approach has the benefit of just using Definition~\ref{d:define diagonal} and Schur orthogonality for finite groups, but the approach in \cite{AzSaSp} works in a somewhat broader context.
\end{rem}

\end{subsection}

\begin{subsection}{An explicit formula and some comments}
In \cite[Theorem 1.8]{AzSaSp} the authors give the following concrete formula for the ZL-amenability constant of a finite group:
\begin{equation}\label{eq:define AMZL}
\operatorname{\rm AM}_{\rm Z}(G) = \frac{1}{|G|^2} \sum_{C\in\operatorname{Conj}(G)}\sum_{D\in\operatorname{Conj}(G)} |C| |D| \left\vert \sum_{\phi\in\operatorname{Irr}(G)} \phi(e)^2 \phi(C)\overline{\phi(D)}\right\vert\;.
\end{equation}
 We could have taken this as the \emph{definition} of $\operatorname{\rm AM}_{\rm Z}(G)$, if we wished to avoid discussing $\Delta_{\rm Z}(G)$ and $\ell^1$-norms, but then the proof of Corollary~\ref{c:AMZL-of-prod-and-quot} would become even more opaque.

\begin{rem}
The presence of complex conjugates in Equation~\eqref{eq:define AMZL} may need some explanation. First, note that the same expression without any complex conjugates follows immediately from Definition~\ref{d:define diagonal} and the definition of the $\ell^1$-norm on $\mathbb C(G\times G)$. Then observe that there is a bijection on $\operatorname{Conj}(G)$ which exchanges the conjugacy class of $x$ with that of $x^{-1}$; and if we denote this bijection by $D\longleftrightarrow D^{-1}$, then $|D^{-1}|\phi(D^{-1}) = |D|\overline{\phi(D)}$ for each $D\in\operatorname{Conj}(G)$ and each character $\phi$.
\end{rem}

\begin{eg}[ZL-amenability constants of finite abelian groups]
\label{eg:AMZL(abelian)}
Let $G$ be a finite abelian group. It was noted in \cite{AzSaSp} that in this case
$\Delta_{\rm Z}(G) = |G|^{-1} \sum_{x\in G} \delta_x \mathbin{\otimes} \delta_{x^{-1}}$, which clearly has $\ell^1$-norm exactly~$1$. (See the remarks in \cite[Section 2.1]{ACS_AMZL-2cd} for further explanation of this identity.) Here we give a more direct proof.
By Equation~\eqref{eq:define AMZL}
\[ \operatorname{\rm AM}_{\rm Z}(G)
 = \frac{1}{|G|^2}\sum_{x,y\in G} \left\vert
\sum_{\phi\in \widehat{G}} \phi(x)\overline{\phi(y)} \right\vert \,.
\]
By Schur (column) orthogonality, most terms in this sum vanish and summing the remaining ones gives
\[ \operatorname{\rm AM}_{\rm Z}(G) = |G|^{-2} \sum_{x\in G} \sum_{\phi\in\widehat{G}} 1 = 1, \]
as required.
\end{eg}

In general it seems hard to get good \emph{lower} bounds on the expression in \eqref{eq:define AMZL}, since there could be considerable cancellation when we sum over~$\phi$, as in Example~\ref{eg:AMZL(abelian)}. A key idea in \cite{AzSaSp} is to introduce a minorant for $\operatorname{\rm AM}_{\rm Z}(G)$ that is easier to estimate well from below. The authors of \cite{AzSaSp} do not give this invariant a name, but since it is the main object of study in Section~\ref{s:trivial-centre} it deserves some {\it ad hoc}\/ notation.

\begin{dfn}[An auxiliary minorant]\label{d:ASS}
For $G$ a finite group, we define
\begin{equation}
\operatorname{ass}(G) = \frac{1}{|G|^2} \sum_{C\in\operatorname{Conj}(G)}\sum_{\phi\in\operatorname{Irr}(G)} |C|^2 |\phi(C)|^2 \phi(e)^2\ .
\end{equation}
\end{dfn}

\begin{prop}[Azimifard--Samei--Spronk, \cite{AzSaSp}]
\label{p:ASS(non-abelian)}
$\operatorname{\rm AM}_{\rm Z}(G)\geq \operatorname{ass}(G)\geq 1$ for every finite group~$G$. Moreover, if $G$ is non-abelian then $\operatorname{ass}(G)>1$.
\end{prop}

\begin{proof}
This is demonstrated in the proof of Theorem 1.8 and Corollary 1.9 in \cite{AzSaSp}, but for sake of completeness we include the argument here.
Note that $\operatorname{ass}(G)$ is obtained by considering the right-hand side of \eqref{eq:define AMZL} and only summing over the terms indexed by $\{(C,C) \mathbin{\colon} C\in\operatorname{Conj}(G)\} \subset \operatorname{Conj}(G)\times\operatorname{Conj}(G)$. Thus $\operatorname{\rm AM}_{\rm Z}(G)\geq\operatorname{ass}(G)$.
Moreover, since $|C|^2\geq |C|$, we have
\[ \begin{aligned}
\operatorname{ass}(G) & \geq
 |G|^{-2} \sum_{C\in\operatorname{Conj}(G)}\sum_{\phi\in\operatorname{Irr}(G)} |C| |\phi(C)|^2 \phi(e)^2  \\
& = \frac{1}{|G|}\sum_{\phi\in\operatorname{Irr}(G)}  \phi(e)^2 \sum_{C\in\operatorname{Conj}(G)} \frac{|C|}{|G|} |\phi(C)|^2 & = 1\,,
\end{aligned} \]
where the final equality follows from the Schur orthogonality relations (row and column).
Finally, if $G$ is non-abelian, then there is at least one $C\in\operatorname{Conj}(G)$ and $\phi\in\operatorname{Irr}(G)$ such that $|C|^2|\phi(C)|^2 > |C| |\phi(C)|^2$, so that running the argument above we see that the inequality is strict.
\end{proof}

\begin{rem}
Combining Proposition~\ref{p:ASS(non-abelian)} with Rider's hard result (Theorem~\ref{t:rider-thm}) one sees that if $G$ is non-abelian then $\operatorname{\rm AM}_{\rm Z}(G)\geq 1+ 1/300$, i.e.~we get a proof of Theorem~\ref{t:ASSgap}.
However, it is important to note that this does not tell us anything about
\[ \inf\{ \operatorname{ass}(G) \mathbin{\colon} \text{$G$ is finite and non-abelian} \}. \]
Indeed, the present paper is unable to determine if this infimum is strictly greater than~$1$.
\end{rem}

A closer inspection of the proof of Proposition~\ref{p:ASS(non-abelian)} yields the inequality
\[ \operatorname{ass}(G) - 1 \geq \frac{s(G)^2-s(G)}{|G|} \]
where $s(G) = \min \{ |C| \mathbin{\colon} C\in \operatorname{Conj}(G), |C| > 1 \}$.
However, in cases where $Z(G)=\{e\}$, we can do much better. This will be the topic of the next section.

\end{subsection}
\end{section}

\begin{section}{Bounds on $\operatorname{ass}$ for groups with trivial centre}
\label{s:trivial-centre}
In this section we obtain a new lower bound for $\operatorname{ass}(G)$ when $G$ has trivial centre. This is the key idea needed for a new proof of Theorem~\ref{t:ASSgap} with a much better constant than that of~\cite{AzSaSp}.

\begin{thm}[Lower bound on $\operatorname{ass}(G)$ when $Z(G)=\{e\}$]
\label{t:bound for trivial centre}
Let $G$ be a finite group with trivial centre, and let $s$ be the smallest size of its non-trivial conjugacy classes. Then
\[ \operatorname{ass}(G) -1 \geq \frac{s-1}{s+1} \cdot \frac{2s}{s+1} \geq \frac{4}{9}\ .\]
\end{thm}

\begin{rem}
The author has not found any groups with trivial centre for which $\operatorname{ass}(G)=13/9$. Probably one can improve on this lower bound by investigating the examples with $s=2$ more closely.
\end{rem}

The proof of Theorem~\ref{t:bound for trivial centre} will occupy the rest of this section.
To streamline some of the formulas which follow, we
introduce the matrix $A:\operatorname{Irr}(G)\times \operatorname{Conj}(G) \to [0,1]$ defined by
\begin{equation}\label{eq:ignore phase}
A_{\phi,C} = |G|^{-1} |C| \ |\phi(C)|^2 \qquad(\phi\in\operatorname{Irr}(G),C\in\operatorname{Conj}(G)).
\end{equation}
Note that rows and columns of $A$ add up to~$1$.
We define
\[ \mu:= \max_{\phi\in\operatorname{Irr}(G)} A_{\phi,e} = \max_{\phi\in\operatorname{Irr}(G)} |G|^{-1} \phi(e)^2 \in (0,1).\]

\begin{prop}\label{p:BS-1st-step}
Let $G$ be a finite group with trivial centre, and let $s=s(G)$ be as defined in the statement of Theorem~\ref{t:bound for trivial centre}.
Then
\begin{equation}\label{eq:BS-1st-step}
\frac{\operatorname{ass}(G)-1}{s-1} \geq \sum_{\phi\in\operatorname{Irr}(G)} A_{\phi,e}( 1- A_{\phi,e})
\geq \max(2\mu,1)(1-\mu).
\end{equation}
\end{prop}

\begin{proof}
We have
\[ \begin{aligned}
\operatorname{ass}(G)
 &  =  \sum_{\phi\in\operatorname{Irr}(G)} \sum_{C\in \operatorname{Conj}(G)} |C| A_{\phi,C} A_{\phi,e} \\
 & \geq \sum_{\phi\in\operatorname{Irr}(G)} \left(A_{\phi,e}^2+ \sum_{C\in \operatorname{Conj}(G)\setminus\{e\}} sA_{\phi,C} A_{\phi,e} \right)
    & \text{(definition of $s$)} \\
 & = (1-s)\sum_{\phi\in\operatorname{Irr}(G)} A_{\phi,e}^2 + s\sum_{\phi\in\operatorname{Irr}(G)}\sum_{C\in \operatorname{Conj}(G)} A_{\phi,C} A_{\phi,e}  \\
 & = (1-s)\sum_{\phi\in\operatorname{Irr}(G)} A_{\phi,e}^2 + s\sum_{\phi\in\operatorname{Irr}(G)}  A_{\phi,e}     & \text{(rows of $A$ sum to $1$)} \\
 & = (s-1)\sum_{\phi\in\operatorname{Irr}(G)} A_{\phi,e}\left( 1- A_{\phi,e}\right) + \sum_{\phi\in\operatorname{Irr}(G)} A_{\phi,e} \,.
\end{aligned} \]
Since columns of $A$ sum to $1$, we obtain
the first inequality in \eqref{eq:BS-1st-step}.

To get the second inequality: fix $\psi\in\operatorname{Irr}(G)$ with $A_{\psi,e}=\mu = \max_{\phi\in\operatorname{Irr}(G)} A_{\phi,e}$. For each $\phi\in\operatorname{Irr}(G)\setminus\{\psi\}$ we have $1-A_{\phi,e}\geq 1-\mu$; also, since columns of $A$ sum to $1$,
\[   1-A_{\phi,e} = \sum_{\gamma\in\operatorname{Irr}(G)\setminus\{\phi\}} A_{\gamma,e} \geq A_{\psi,e}=\mu\,. \]
Hence
\[ \begin{aligned}
\sum_{\phi\in\operatorname{Irr}(G)} A_{\phi,e}\left( 1- A_{\phi,e}\right)
& = \mu(1-\mu) + \sum_{\phi\in\operatorname{Irr}(G)\setminus\{\psi\}} A_{\phi,e}\left(1-A_{\phi,e} \right) \\
& \geq \mu(1-\mu) + \sum_{\phi\in\operatorname{Irr}(G)\setminus\{\psi\}} A_{\phi,e}\max(\mu,1-\mu) \\
& = \mu(1-\mu)+\max(\mu,1-\mu)(1-\mu)
    & \text{(rows of $A$ sum to $1$)} \\
& =(1-\mu)\max(2\mu,1),
\end{aligned} \]
as required.
\end{proof}

\begin{eg}\label{eg:affine}
Let $q$ be a prime power $\geq 3$ and consider $G=\operatorname{Aff}({\mathbf F}_q)$, the affine group of the finite field with $q$ elements. The character theory of this group is well known, and gives
$\mu = |G|^{-1}(q-1)^2 = 1- q^{-1}$.
For this group $s=q-1$, so that
\begin{equation}\label{eq:crude bound for Aff}
\operatorname{ass}(\operatorname{Aff}({\mathbf F}_q)) -1
  \geq (s-1)(1-\mu)\max(2\mu,1)
  = (q-2) \cdot \frac{1}{q}\cdot 2\left(1-\frac{1}{q}\right);
\end{equation}
and the right-hand side of \eqref{eq:crude bound for Aff}, being a monotone increasing function of $q$, is minimized when $q=3$ (with minimum value $4/3$). In fact an explicit but tedious calculation shows that $\operatorname{ass}(\operatorname{Aff}({\mathbf F}_q))=3-4q^{-1}$, so that our lower bound is a slight underestimate.
\end{eg}

\medskip
Example~\ref{eg:affine} shows that even within the class of groups with trivial centre, $1-\mu$ can be arbitrarily small. However, note that in this example, when $1-\mu$ is small $s$ is large. The following result shows that this happens more generally.

\begin{lem}\label{l:BS-2nd-step}
Let $G$ be a finite group. Then
$1\leq (1+s)(1-\mu)$.
\end{lem}
\begin{proof}
Choose $\psi$ with $A_{\psi,e}=\mu$ and choose $C$ with $|C|=s$.
Then $A_{\psi,C} \leq 1-\mu$ (since rows of $A$ sum to $1$), while
\[ A_{\phi,C} = \frac{s}{|G|} |\phi(C)|^2 \leq  \frac{s}{|G|} \phi(e)^2 = s A_{\phi,e} \quad\text{for all $\phi\in\operatorname{Irr}(G)$.} \]
Since the columns of $A$ sum to $1$, this gives
\[ 1 = A_{\psi,C} + \sum_{\phi\neq\psi}A_{\phi,C} \leq (1-\mu) + s\sum_{\phi\neq \psi} A_{\phi,e} = (1-\mu)+s(1-\mu) \]
as required.
\end{proof}

\begin{rem}\

\begin{YCnum}
\item
The idea behind the proof of Lemma~\ref{l:BS-2nd-step} comes from considering cases such as Example~\ref{eg:affine}, where the ratio $\mu$ is large in the sense of being close to~$1$. Picking $\phi\in\operatorname{Irr}(G)$ with $A_{\phi,e}=\mu$ and $C\in\operatorname{Conj}(G)$ with $|C|=s$, one reasons as follows:
\begin{itemize}
\item since the column labelled by $e$ adds up to $1$,
 $A_{\phi,e}$ must be small for every $\phi\in \operatorname{Irr}(G)\setminus\{\psi\}$;
\item since the row labelled by $\psi$ adds up to~$1$, $A_{\psi,C}$ has to be small;
\item
in turn, this forces $A_{\phi,C}$ to be large for every $\phi\in\operatorname{Irr}(G)\setminus\{\psi\}$.
\end{itemize}

\item
There are groups for which equality is achieved in Lemma~\ref{l:BS-2nd-step}. For instance, if $G$ is the dihedral group of order~$6$, we have $s=2$, $\mu= 2^2/6 = 2/3$, $(s+1)(1-\mu) = 1$.

\item
Lemma~\ref{l:BS-2nd-step} implies that if a finite group has an irreducible character with ``relatively high degree'', not only must the group have trivial centre, but all conjugacy classes other than the identity have to be ``quite large''.
Such observations are probably not new, but we are unaware of any references in the literature which provide such inequalities.
\end{YCnum}
\end{rem}

\begin{proof}[Proof of Theorem~\ref{t:bound for trivial centre}]
Let $h:[0,1] \to [0,\infty)$ be the function $h(t) = (1-t)\max(2t,1)$. This is a continuous, decreasing function. By Proposition~\ref{p:BS-1st-step} and Lemma~\ref{l:BS-2nd-step},
\[ \operatorname{ass}(A)-1 \geq (s-1)h(\mu) \geq (s-1)h\left(\frac{s}{s+1}\right) \ . \]
Now since $s/(s+1) \geq 2/3 > 1/2$\/,
\[
(s-1)h\left(\frac{s}{s+1}\right) = \frac{s-1}{s+1} \cdot \frac{2s}{s+1} \, ;
\]
and the right hand side is minimized when we take $s-2$, with value $4/9$.
\end{proof}

\begin{rem}
We could have tried the same approach for a non-abelian finite group $G$ with non-trivial centre.  The proof of Proposition~\ref{p:BS-1st-step} goes through much as before\footnotemark, but this time we merely obtain
\footnotetext{We need to know that $|\phi(x)|=\phi(e)$ for all $x\in Z(G)$ and all $\phi\in\operatorname{Irr}(G)$. This is an easy consequence of Schur's lemma, see e.g.~\cite[Lemma 2.25]{Isaacs_CTbook}.}
\[ \operatorname{ass}(G)-1 \geq (s-1) \sum_{\phi\in\operatorname{Irr}(G)} \frac{\phi(e)^2}{|G|} \left( 1- \frac{|Z(G)| \phi(e)^2}{|G|}\right). \]
Now if we define $\mu=\max_{\phi\in\operatorname{Irr}(G)}|G|^{-1}\phi(e)^2$ as before, then $|Z(G)| \mu\leq 1$, but this time we can have equality, which means that some of the terms in the sum above will be zero. It is not clear how to obtain good lower bounds on the remaining contributions to the sum.
\end{rem}

\end{section}

\begin{section}{Bounds on $\operatorname{\rm AM}_{\rm Z}$ for just non-abelian groups}
\label{s:JNA cases}

\begin{subsection}{JNA groups: definitions and two results}\label{ss:JNA background}

Throughout, we denote the \dt{derived subgroup} (or \dt{commutator subgroup}) of $G$ by $[G,G]$ rather than $G'$.

\begin{dfn}
We say that a quotient of a group $G$ is \dt{proper} if it is not equal to $G$ itself. A group $G$ is \dt{just non-abelian} (JNA for short) if it is non-abelian yet all its proper quotients are abelian.
\end{dfn}

\begin{rem}
Since every homomorphism from $G$ to an abelian group factors through the quotient homomorphism $G\to G/[G,G]$, a group $G$ is JNA if and only if all its non-trivial normal subgroups contain $[G,G]$. In alternative terminology: if $G$ is JNA then $[G,G]$ is the minimal non-trivial normal subgroup of~$G$.
\end{rem}

Recall that by Corollary~\ref{c:AMZL-of-prod-and-quot}\ref{li:quot}, if $H$ is a quotient of $G$ then $\operatorname{\rm AM}_{\rm Z}(G)\geq \operatorname{\rm AM}_{\rm Z}(H)$. Now if $G$ is a finite, non-abelian group, there exists a quotient of $G$ which is JNA. (The proof is a routine induction which we omit.) Hence
\[ \inf\{\operatorname{\rm AM}_{\rm Z}(G)\mathbin{\colon} \text{$G$ is finite and non-abelian}\} =
  \inf\{\operatorname{\rm AM}_{\rm Z}(G)\mathbin{\colon} \text{$G$ is finite and JNA}\}. \]
Therefore Theorem~\ref{t:mainthm} will follow from the following special case.

\begin{thm}\label{t:bound for JNA cases}
Let $G$ be a finite JNA group. Then $\operatorname{\rm AM}_{\rm Z}(G)\geq 7/4$.
\end{thm}

Our proof of Theorem~\ref{t:bound for JNA cases} works by looking at various cases.
The main point is that if $G$ is finite JNA and has non-trivial centre, then it can be handled by the main result of \cite{ACS_AMZL-2cd}. For this we use the following lemma, which is a simpler (but weaker) version of \cite[Lemma 12.3]{Isaacs_CTbook}.

\begin{lem}\label{l:baby JN2}
Let $G$ be a finite JNA group with non-trivial centre.
Then every non-linear $\phi\in\operatorname{Irr}(G)$ either has degree $1$ (i.e.~is linear) or has degree $|G:Z(G)|^{1/2}$.
Moreover, there is a prime $p$ such that $|G|$ is a power of $p$; $|[G,G]|=p$; and every conjugacy class has size $1$ or size~$p$.
\end{lem}

We will also need the following result, which the author was unable to find stated explicitly in the literature.

\begin{thm}[Folklore?]\label{t:is-dihedral}
Let $G$ be a finite JNA group which has trivial centre and has a conjugacy class of size $2$. Then $G$ is isomorphic to the dihedral group of order $2p$, for some odd prime~$p$.
\end{thm}

To keep this article as self-contained as possible, the Appendix includes
elementary proofs of Lemma~\ref{l:baby JN2} and Theorem~\ref{t:is-dihedral}.
In the rest of this subsection we offer some remarks on just non-abelian groups to provide extra context for the reader, and to acknowledge the sources for the ideas behind Lemma~\ref{l:baby JN2} and Theorem~\ref{t:is-dihedral}. The reader who just wants to get on with the proof of Theorem~\ref{t:bound for JNA cases} can skip to Section~\ref{ss:AMZL-of-JNA}.

\begin{rem}[JN2 groups and JM groups]\label{r:JM-JN2-history}
If $G$ is a JNA group with non-trivial centre, then $\{e\}\subsetneq [G,G]\subseteq Z(G)$, so that $G$ is $2$-step nilpotent.
Finite nilpotent JNA groups were classified by M. F. Newman \cite{Newman_JN2gp}, who christened them finite \dt{JN2 groups}; notable examples are the extra-special $p$-groups for any given prime~$p$.
His earlier paper \cite{Newman_JMgp} gives a classification of finite \emph{solvable} JNA groups with trivial centre, calling them finite \dt{JM groups}; examples include the affine groups of finite fields (cf.\ Example~\ref{eg:affine}). In general, every finite JM group is necessarily of the form
${\mathbf F}\rtimes {\bf D}$ for some finite field ${\mathbf F}$ and some subgroup ${\bf D} \leq {\mathbf F}^\times$. This is a necessary condition and not a sufficient one: see \cite[Theorem 6.4]{Newman_JMgp} for the full classification.
\end{rem}

\begin{rem}[Looking for a proof of Theorem~\ref{t:is-dihedral}]
If we knew the hypotheses of Theorem~\ref{t:is-dihedral} force $G$ to be solvable, we could extract this result from Newman's classification of JM groups. However, doing so is somewhat cumbersome: one has to analyze the conjugacy classes in ${\bf F}\rtimes {\bf D}$ to show that $|{\bf D}|=2$, explain why $|{\bf F}|$ is an odd prime, and so on. Alternatively, once we know $G$ is solvable, we could appeal to the results of Isaacs and Passman that were mentioned earlier; but this seems to require some form of Frobenius's theorem on malnormal subgroups (\cite[Theorem 7.2]{Isaacs_CTbook} or \cite[Chapter~7, Theorem 8.6]{Cohn_ed2vol2}). In both cases, of course, we would still need a separate argument to explain why the conditions in Theorem~\ref{t:is-dihedral} imply that the group is solvable! This is one reason why we chose to give a self-contained proof in the Appendix.
\end{rem}

\begin{rem}[Groups with two character degrees]
The present paper also owes an indirect debt to work of Isaacs and Passman
 on ``groups with two character degrees''. Rather than try to pick out those parts of the original papers which are relevant to the present article, the reader can find a good exposition in Chapter 12 of \cite{Isaacs_CTbook}. In particular, the dichotomy theorem contained in \cite[Lemma 12.3]{Isaacs_CTbook} implies that every finite JM group is Frobenius with abelian complement and kernel, so that its ZL-amenability constant  can be calculated explicitly using the methods of \cite[\S3.2]{ACS_AMZL-2cd}.
In~fact, the idea of looking at ZL-amenability constants of JNA groups arose from reading about this work in \cite[Chapter 12]{Isaacs_CTbook}, while seeking to extend the results of \cite{ACS_AMZL-2cd}.
\end{rem}

\end{subsection}

\begin{subsection}{The ZL-amenability constant in the JNA cases}
\label{ss:AMZL-of-JNA}
To fully exploit Lemma~\ref{l:baby JN2} and Theorem~\ref{t:is-dihedral}, we will apply the main theorem of \cite{ACS_AMZL-2cd}, stated below for clarity.

\begin{thm}[{\cite[Theorem 2.4]{ACS_AMZL-2cd}}]\label{t:ACSformula}
Let $m$ be an integer greater than or equal to $2$, and suppose
 $G$ is a finite non-abelian group such that $\phi(e)\in\{1,m\}$ for all $\phi\in\operatorname{Irr}(G)$. Then
\[ \operatorname{\rm AM}_{\rm Z}(G)-1 = 2(m^2-1)\sum_{C\in\operatorname{Conj}(G)} \frac{|C|}{|G|} \left( 1- \frac{|C|}{|[G,G]|} \right). \]
\end{thm}

The following calculation was done in \cite[Example 3.13]{ACS_AMZL-2cd} for extraspecial $p$-groups, but works in greater generality. For convenience we give the proof.

\begin{prop}\label{p:AMZL of JN2}
Let $G$ be a finite JNA group with non-trivial centre. Then there is a prime $p$ and an integer $k\geq 1$ such that
\begin{equation}\label{eq:AMZL-of-JN}
 \operatorname{\rm AM}_{\rm Z}(G) -1 = 2\left(1-\frac{1}{p^{2k}}\right)\left(1- \frac{1}{p}\right).
\end{equation}
In particular, $\operatorname{\rm AM}_{\rm Z}(G)\geq 7/4$.
\end{prop}

\begin{proof}
Let $p$ be the prime from Lemma~\ref{l:baby JN2} and let $k$ satisfy $|G:Z(G)|^{1/2}=p^k$.  By the lemma every $\phi\in\operatorname{Irr}(G)$ has degree $1$ or $p^k$, so $k$ is a positive integer (strictly positive since $G$ is non-abelian). Moreover, Lemma~\ref{l:baby JN2} also implies
 \[ \frac{|C|}{|[G,G]|}  =
 \begin{cases} p^{-1} & \quad\text{if $C\in Z(G)$} \\ 1 & \quad\text{if $C\in\operatorname{Conj}(G)\setminus Z(G)$.}
\end{cases} \]
Applying Theorem~\ref{t:ACSformula} yields
\[ \begin{aligned}
\operatorname{\rm AM}_{\rm Z}(G)-1
 & = 2(p^{2k}-1) \sum_{C\in\operatorname{Conj}(G)} \frac{|C|}{|G|} \left( 1- \frac{|C|}{|[G,G]|}\right) \\
 & = 2(p^{2k}-1) |Z(G)| \cdot \frac{1}{|G|} \left( 1- \frac{1}{p}\right) \\
 & = 2\frac{(p^{2k}-1)}{|G:Z(G)|} \left(1-\frac{1}{p}\right) \\
 & = 2\left( 1- \frac{1}{p^{2k}}\right)\left(1-\frac{1}{p}\right).
\end{aligned} \]
To finish: note that the right-hand side of Equation~\eqref{eq:AMZL-of-JN} is an increasing function of $k$ and of~$p$. Taking $k=1$ and $p=2$ gives the minimum value $3/4$, as required.
\end{proof}

\begin{eg}[JNA groups achieving the minimal ZL-amenability constant]\label{eg:when AMZL=7/4}
The calculations in  \cite[Example 3.13]{ACS_AMZL-2cd} show that the only extraspecial $p$-groups with ZL-amenability constant equal to $7/4$ are the dihedral and quaternion groups of order~$8$.
However, for any $n\in\mathbb N$ the group denoted in \cite{Newman_JN2gp} by $M(2^n)$ is a JNA group of order $2^{n+2}$ whose centre has order $2^n$. The proof of  Proposition~\ref{p:AMZL of JN2} then yields $\operatorname{\rm AM}_{\rm Z}(M(2^n)) = 7/4$.
\end{eg}

We now turn to JNA groups with trivial centre. If $G$ is such a group and it has a conjugacy class of size~$2$, then (Theorem~\ref{t:is-dihedral}) $G$ is isomorphic to the dihedral group $C_p\rtimes C_2$ for some odd prime~$p$.
As shown in \cite[Section 3.1]{ACS_AMZL-2cd}, Theorem~\ref{t:ACSformula} applies to $C_p\rtimes C_2$ and straightforward calculations yield
\begin{equation}\label{eq:dihedral-cases}
\operatorname{\rm AM}_{\rm Z}(C_p\rtimes C_2)  = 1+3\left(1-\frac{1}{p}\right)^2 \geq 1+ 3\left(1-\frac{1}{3}\right)^2 = \frac{7}{3}.
\end{equation}

It remains to deal with JNA groups where every conjugacy class, except that of $e$, has size $\geq 3$. (Recall that this class includes all finite simple non-abelian groups!) Here, it is no longer clear that we can exploit the results of \cite{ACS_AMZL-2cd}, but instead we can turn to the results of Section~\ref{s:trivial-centre}.

\begin{prop}\label{p:generic}
Let $G$ be a finite group with at least two elements and trivial centre. Suppose that $G$ has no conjugacy classes of size $2$. Then $\operatorname{\rm AM}_{\rm Z}(G)\geq \operatorname{ass}(G) \geq 7/4$.
\end{prop}

\begin{proof}
Let $s= \min \{ |C| \mathbin{\colon} C\in \operatorname{Conj}(G)\setminus\{e\} \}$. The conditions put on our group ensure that $s\geq 3$. Then, by Theorem~\ref{t:bound for trivial centre},
\[ \begin{aligned}
\operatorname{ass}(G)-1
 & \geq \frac{s-1}{s+1} \frac{2s}{s+1} \\
 & = 2\left(1- \frac{2}{s+1}\right)  \left(1-\frac{1}{s+1}\right) \\
 & \geq  2\left(1- \frac{2}{4}\right)  \left(1-\frac{1}{4}\right) & = \frac{3}{4}\,,
\end{aligned} \]
as claimed.
\end{proof}

\begin{proof}[Proof of Theorem~\ref{t:bound for JNA cases}]
Combine Proposition~\ref{p:AMZL of JN2}, Proposition~\ref{p:generic} and the inequality~\eqref{eq:dihedral-cases}.
\end{proof}

\begin{rem}
When $G$ has trivial centre and has a conjugacy class of size $2$, the direct application of Theorem~\ref{t:bound for trivial centre} merely gives $\operatorname{\rm AM}_{\rm Z}(G)\geq \operatorname{ass}(G)\geq 13/9$, which is not good enough. This is why we had to deal with such cases separately (and also use the JNA condition, which is not used in Proposition~\ref{p:generic}).
\end{rem}

\end{subsection}

\end{section}

\begin{section}{A lower bound for perfect groups}\label{s:ASS(perfect)}
Since every non-abelian simple group is JNA, and since we proved Theorem~\ref{t:mainthm} by passing from the given non-abelian finite group to some JNA quotient, it is natural to wonder what can be said about the ZL-amenability constant of a finite simple group. In this section we obtain a lower bound that is far better than what we obtain from Theorem~\ref{t:bound for trivial centre}, although it is probably far short of the true infimum.
 In fact, our bound holds for all finite perfect groups, and our approach does not require any structure theory.

\begin{prop}[An alternative bound for $\operatorname{ass}$]\label{p:dual bound for ASS}
Let $G$ be a finite group, and let $m=\min\{ \phi(e) \mathbin{\colon} \phi\in\operatorname{Irr}(G)\text{ and } \phi(e)>1\}$. Then
\[ \operatorname{ass}(G)-1 \geq (m^2-1) \sum_{C\in\operatorname{Conj}(G)} \frac{|C|}{|G|}\left( 1 - \frac{|C|}{|[G,G]|} \right). \]
Equality holds if every non-linear irreducible character of $G$ has degree~$m$.
\end{prop}

\begin{proof}
This is like the proof of \cite[Theorem 2.4]{ACS_AMZL-2cd}, only simpler; in some sense it is dual to the argument used in the proof of Proposition~\ref{p:BS-1st-step}.

As in Equation~\ref{eq:ignore phase}, let $A_{\phi,C}:= |G|^{-1}|C| |\phi(C)|^2$. Let $\varepsilon$ denote the trivial character of $G$, sometimes called the \dt{augmentation character}, which takes the value $1$ on every conjugacy class. Then
\[
\operatorname{ass}(G) =\sum_{C\in\operatorname{Conj}(G)} A_{\varepsilon,C}\sum_{\phi\in \operatorname{Irr}(G)} \phi(e)^2A_{\phi,C}  \,.
\]

Let ${\mathcal L}=\{ \phi \in\operatorname{Irr}(G)\mathbin{\colon} \phi(e)=1\}$.
Then for each $C\in\operatorname{Conj}(G)$,
\[ \begin{aligned}
\sum_{\phi\in\operatorname{Irr}(G)} \phi(e)^2 A_{\phi,C}
& \geq \sum_{\phi\in{\mathcal L}} A_{\phi,C}  + m^2 \sum_{\phi\in\operatorname{Irr}(G)\setminus{\mathcal L}} A_{\phi,C}
  & \text{(definition of $m$)} \\
 & = (1-m^2) \sum_{\phi\in{\mathcal L}} A_{\phi,C} +  m^2 \sum_{\phi\in\operatorname{Irr}(G)} A_{\phi,C} \\
 & = (1-m^2)\sum_{\phi\in{\mathcal L}} A_{\phi,C}  + m^2  &\text{(columns of $A$ sum to $1$)} \\
 & = (1-m^2) |{\mathcal L}| A_{\varepsilon,C}  + m^2  &\text{(definition of ${\mathcal L}$).} \\
\end{aligned} \]

Multiplying this inequality by $A_{\varepsilon,C}$ and summing over all $C$,
\[ \operatorname{ass}(G) \geq \sum_{C\in\operatorname{Conj}(G)} A_{\varepsilon,C} \;+\; (m^2-1) \sum_{C\in\operatorname{Conj}(G)} A_{\varepsilon,C}( 1- |{\mathcal L}|A_{\varepsilon,C}).\]
Since rows of $A$ sum to $1$,
 and $|{\mathcal L}|$ equals the order of the quotient group $G/ [G,G]$, the result follows.
\end{proof}

The following result is surely folklore, but we include a proof since it is straightforward.

\begin{lem}\label{l:big conjugacy class}
Let $G$ be a finite group. Suppose it has a conjugacy class $C$ of size $|G|/2$. Then $K=G\setminus C$ is an index $2$ subgroup of $G$.
In particular, $G$ is not perfect.
\end{lem}

\begin{proof}
We divide the proof into several steps.

\para{Step 1}
Let $t\in C$. Then its centralizer $Z_G(t)$ has order $|G|/|C|=2$. Since $\gen{t}\subseteq Z_G(T)$, $t$ is an involution and $Z_G(t)=\{1,t\}$.

\para{Step 2} If $x\in G$ is an involution, and $y\in C$ and $xy$ is an involution, then $x=e$ or $x=y$.

\noindent[{\it Proof}\/:
we have $xy=(xy)^{-1}=y^{-1}x^{-1}=yx$, so $x\in Z_G(y)$; now use Step~1.]

\para{Step 3}
Let $x,y\in C$. Then $xy\in K$.

\noindent[{\it Proof}\/:
if $x=y$ then $xy=e$; while if $x$ and $y$ are distinct elements of $C$, then by Step~2, $xy$ cannot be an involution, so in particular $xy\notin C$.]

\para{Step 4}
Let $t\in C$. Then the functions $L_t, R_t:G\to G$ defined by $L_t(x)=tx$ and $R_t(x)=xt$ are involutive bijections $C\leftrightarrow K$.

\noindent[{\it Proof}\/:
by Step 3, $L_t(C)\subseteq K$; and since $t$ is an involution, $L_t$ is injective. So $|L_t(C)|\leq K$. But $|K| = |G\setminus C| = |C|$, hence $L_t:C\to K$ is surjective. The argument for $R_t$ is similar.]

\para{Step 5} $K$ is closed under multiplication.

\noindent[{\it Proof}\/:
let $a,b\in K$. Pick $t\in C$; by Step 4, $at,tb\in C$. Since $t$ is an involution, $ab=(at)(tb)\in C\cdot C \subseteq K$ by Step~3.]

\para{Final step} Since $C$ is the conjugacy class of an involution, it is closed under inversion. Hence $K=G\setminus C$ is also closed under inversion. Combined with Step 5, this shows $K$ is a subgroup of $G$; and since $K$ is the union of conjugacy classes, it is normal. Finally, since $G/K$ is abelian of order $2$, $G$ is not perfect.
\end{proof}

\begin{rem}
In Step 1 of the proof we showed, in effect, that $C$ consists of self-centralizing involutions. So if $t\in C$ then $\gen{t}$ is \emph{malnormal} in $G$, hence by Frobenius's theorem (\cite[Theorem 7.2]{Isaacs_CTbook} or \cite[Chapter~7, Theorem 8.6]{Cohn_ed2vol2}) it admits a complement, and Lemma~\ref{l:big conjugacy class} follows.
However, Frobenius's theorem remains quite a deep result and seems excessive in this instance.
\end{rem}

\begin{thm}\label{t:ASS(perfect)}
Let $G$ be a finite perfect group. Then $\operatorname{ass}(G)\geq 3$.
\end{thm}

\begin{proof}
If $C\in\operatorname{Conj}(G)$ then $|C|\leq |G|/3$ (since $|C|$ is a proper factor of $|G|$, and it cannot be $|G|/2$ by Lemma~\ref{l:big conjugacy class}).
Applying Proposition~\ref{p:dual bound for ASS} and using $[G,G]=G$ yields
\[ \operatorname{ass}(G) -1 \geq (m^2-1) \sum_{C\in\operatorname{Conj}(G)} \frac{|C|}{|G|} \left(1 - \frac{1}{3}\right) = \frac{2(m^2-1)}{3}\,.\]
Since $m\geq 2$ the result follows.
\end{proof}

\begin{rem}\label{r:simple-no-2dim-irrep}
If $G$ is simple one can improve this lower bound. A finite simple group has no irreducible $2$-dimensional representations over $\mathbb C$ (see e.g.~\cite[Corollary 22.13]{JamLie});
hence $m\geq 3$ when $m$ is as defined in Proposition~\ref{p:dual bound for ASS}. Running the argument in the proof of Theorem~\ref{t:ASS(perfect)} then yields $\operatorname{ass}(G)\geq 19/3$.

Moreover: since quotients of perfect groups are perfect, every finite perfect group $G$ has a quotient group $K$ which is non-abelian and simple. It follows that if $G$ is a finite perfect group,
\[ \operatorname{\rm AM}_{\rm Z}(G)\geq \operatorname{\rm AM}_{\rm Z}(K) \geq \operatorname{ass}(K)\geq \frac{19}{3}\,.\]
While this argument does not imply $\operatorname{ass}(G)\geq 19/3$ for all finite perfect groups, we know of no finite perfect group whose associated minorant is less than $19/3$.
\end{rem}

\end{section}

\begin{section}{Questions for future investigation}
\label{s:closing}

Our first two questions concern the associated minorant $\operatorname{ass}(G)$. Although it is technically easier to deal with than $\operatorname{\rm AM}_{\rm Z}(G)$ for particular groups, we do not know if it has the same good hereditary properties.

\begin{qn}
Is there a finite group $G$ and $N\lhd G$ such that $\operatorname{ass}(G) < \operatorname{ass}(G/N)$?
\end{qn}

If the answer is negative, then we can obtain lower bounds on $\operatorname{ass}(G)$ by passing to JNA quotients, and hence the next question would have a \emph{positive} answer.

\begin{qn}
Is $\inf\{ \operatorname{ass}(G) \colon \text{$G$ non-abelian and finite} \}$ strictly greater than $1$?
\end{qn}

To motivate the next two questions, we present a small consequence of our earlier work on the JN2 cases.

\begin{thm}\label{t:nilpotent}
Let $G$ be a finite nilpotent group. If $\operatorname{\rm AM}_{\rm Z}(G) < 59/27$ then $G$ is the product of a $2$-group with an abelian group.
\end{thm}

\begin{proof}
Since $G$ is nilpotent it factorizes as a direct product of finite $p$-groups, say $G=H_1\times\dots\times H_m$.
(More precisely, it is the direct product of its Sylow $p$-subgroups; see~e.g.\
\cite[Section 9.8, Theorem 5]{Cohn_ed2vol1}.)

Suppose at least one of these groups, say $H_1$, is a non-abelian $p$-group where $p\geq 3$. Then $H_k$ admits a JNA quotient $K$, and $K$ must also be a $p$-group; the proof of Proposition~\ref{p:AMZL of JN2} then implies that
\[ \operatorname{\rm AM}_{\rm Z}(K)-1 \geq 2( 1- p^{-2})(1-p^{-1}) \geq 2\left(1-\frac{1}{9}\right)\left(1-\frac{1}{3}\right) = \frac{32}{27}\,. \]
But since $\operatorname{\rm AM}_{\rm Z}(G)\geq\operatorname{\rm AM}_{\rm Z}(H_1)\geq\operatorname{\rm AM}_{\rm Z}(K)$, this contradicts the assumption on $\operatorname{\rm AM}_{\rm Z}(G)$.
Thus each of the groups $H_1,\dots, H_m$ is either abelian or a $2$-group, and the result follows.
\end{proof}

\begin{qn}\label{q:small AMZL forces nilpotent}
Does Theorem~\ref{t:nilpotent} remain true if one drops the word ``nilpotent''?
\end{qn}

Note that a positive answer to Question~\ref{q:small AMZL forces nilpotent} would yield a positive answer to the next question.

\begin{qn}
Is $\inf\{\operatorname{\rm AM}_{\rm Z}(G) \colon \text{$G$ a finite, non-nilpotent group}\}$ strictly greater than $7/4$? Can we at least show that a group with ZL-amenability constant $7/4$ has to be nilpotent?
\end{qn}

\begin{rem}
An explicit calculation shows that $\operatorname{ass}(C_3\rtimes C_2)=5/3$. Therefore, with our current methods we have no hope of proving results of the form ``if $G$ has trivial centre then $\operatorname{\rm AM}_{\rm Z}(G) \geq 2$'', because without the JNA condition the best we can do for groups with trivial centre is to use the lower bound provided by $\operatorname{ass}(G)$.
\end{rem}

Moving away from questions of lower bounds, we could look at the structure of the set
\[ \mathcal{ZL}=\{ \operatorname{\rm AM}_{\rm Z}(G) \colon \text{$G$ a finite group}\}. \]
  of possible values for ZL-amenability constants.

\begin{qn}
Is $\mathcal{ZL}$ closed? If not, does it contain any non-isolated points?
\end{qn}

The results summarized in \cite[Section 3.4]{ACS_AMZL-2cd} show that there exist strictly increasing sequences in $\mathcal{ZL}$ which converge. The same results suggest that these limiting values are limit points of $\mathcal{ZL}$, and hence that $\mathcal{ZL}$ is not closed, but they are not conclusive.

\begin{qn}
Does $\mathcal{ZL}$ contain any strictly decreasing sequences? That is, does there exist a sequence $(G_n)$ of finite groups such that $\operatorname{\rm AM}_{\rm Z}(G_{n+1})< \operatorname{\rm AM}_{\rm Z}(G_n)$ for all $n$?
\end{qn}

Our final question connects the study of $\operatorname{\rm AM}_{\rm Z}(G)$ back to the world of infinite-dimensional Banach algebras.
\begin{qn}
 Consider an inverse system of finite groups
\[ \dots G_3 \to G_2 \to G_1 \to G_0 \]
in which all connecting maps are surjective group homomorphisms.
 If $\sup_n\operatorname{\rm AM}_{\rm Z}(G_n) < \infty$, is the inverse limit $\varprojlim_n G_n$ always virtually abelian?
\end{qn}

If the answer is positive, then as noted in \cite{AzSaSp}, we could prove the following: if $G$ is a profinite group which is not virtually abelian, then the centre of $L^1(G)$ is non-amenable.
\end{section}

\subsection*{Acknowledgements}
The main ideas needed for the proof of Theorem~\ref{t:mainthm} were originally worked out in 2013, while the author worked at the University of Saskatchewan.
 He thanks M. Alaghmandan and E. Samei for their interest and encouragement.

Other important parts, such as an early version of Theorem~\ref{t:is-dihedral}, were obtained while attending the {\it Conference in Abstract Harmonic Analysis}\/ held in Granada, May 2013. The author thanks the conference organizers for an engaging conference and a stimulating atmosphere.  During this time the author was supported by NSERC Discovery Grant 402153-2011.

The results of Section~\ref{s:ASS(perfect)} were obtained during the author's present position at Lancaster University, which is also where this paper took shape as various proofs were streamlined, rearranged and improved.
Thanks are due to the referee for an attentive reading and constructive feedback, in particular for drawing his attention to the reference cited in Remark~\ref{r:simple-no-2dim-irrep}.

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IMTEX Forming 2022 to feature over 350 exhibitors
IMTEX FORMING 2022 IMTEX Forming 2022 to feature over 350 exhibitors
Indian Machine Tool Manufacturers' Association (IMTMA) will organise the seventh edition of IMTEX Forming, IMTEX Forming 2022 - Southeast Asia's largest exhibition on metal forming technologies, at Bangalore International Exhibition Centre (BIEC) in Bengaluru from June 16 - 21, 2022. In this show, over 350 exhibitors will be showcasing their metal forming technologies.
Dies & Moulds
metal tooling
IMTMA
Indian Machine Tool Manufacturers' Association (IMTMA) recently announced that it will be organising the seventh edition of IMTEX Forming, IMTEX Forming 2022 at Bangalore International Exhibition Centre (BIEC) in Bengaluru from June 16-21, 2022. The concurrent shows, Tooltech 2022, which focuses on machine tool accessories, forming tools, die & moulds, metrology, CAD/CAM, etc and Digital Manufacturing, which features latest innovations in Additive Manufacturing and Industry 4.0, etc will be held along with IMTEX Forming 2022.
Over 350 exhibitors from 16 countries would be participating in the exhibition which is being held after a gap of more than two years. IMTEX Forming 2022 along with Tooltech and Digital Manufacturing will span around 33,000 square metres of exhibition space with companies from Austria, China, Czech Republic, France, Germany, India, Italy, Japan, Korea, Malaysia, Singapore, Switzerland, Taiwan, Turkey, UK and USA, exhibiting there.
IMTEX Forming 2022 and Tooltech 2022 will bring diverse groups and stakeholders on a common platform for knowledge sharing and transacting business. Technologies in metal forming, such as presses, bending, welding & joining, high speed laser machines, robotics & automation in sheet metal working, Additive Manufacturing, metrology, and CAD/CAM essential for manufacturing will be displayed 'live', enabling visitors to take informed decisions.
Expressing his thoughts, Ravi Raghavan, President, IMTMA, cited, "IMTEX Forming is Southeast Asia's leading show on metal forming and manufacturing technologies. With the country having just recovered from the global pandemic, the exhibition will be crucial in advancing India's manufacturing capabilities."
Further, P J Mohanram, Principal Advisor, IMTMA, added, "Technology enthusiasts will have a lot to unearth on various developments in forming machinery, near net manufacturing, Additive Manufacturing, digital manufacturing, shop floor automation and evolutionary Industry 4.0."
The six-day exhibition will have visitors from various industry sectors such as auto, auto components, aerospace, defence, railways, power, medical equipment, white and brown goods, oil & gas equipments, ship building, and many more industries.
The highlights of the show are –
More than 350 exhibitors
Exhibitors from 16 countries
700+ live machines
Focus on digital manufacturing
Around 33,000 square metres of exhibition space
i2 Academia Pavilion (an event for academia and industry to interact and explore possible tie-ups)
Aatmanirbhar Bharat Pavilion - showcasing select indigenous technological capabilities and solutions developed within India
For more information about the event, log on to www.imtex.in
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If you are reading this exhibition epilogue, that means you had survived 2012 safe and sound.
People of Hong Kong that live in this era would still be writing something about world ends thing, no matter what is the reason behind, is a laughable matter. After all, we are being intimidated by this world ends thing from the day when we were born. Not so long ago, in the eighties, our relatives, friends flee for foreign land, fight for British nationality selection scheme. Those who did not plan to leave or did not have the ability to leave, left themselves a hole in their hearts. The return of sovereign right to China in 1997, Mr. Tung's, mother tongue tutoring, Asian financial crisis. 911 New York, principal officials accountability system, SARS pandemic, 1st July rally, Lehman brothers, financial tsunami, HSBC share subscribe, bird flu contagion. We are so used to this 'centennial level' of incidents, so called crisis, for what history taught us is to let those fatal disease, bankruptcy and catastrophe be awaited, for tomorrow is another day !
Hong Kong people are already immune to this world ends syndrome, we have it inherited in our genes, despite that, we still get numb, get annoyed. We pardon those who do not have this kind of experience as we did, it's just not the kind of experience that an average earthling should have. '2012' is a bland subject for us, way beyond compared with Valentine's Day or Christmas commercially. Stella and me are typical Hong Kong people. We committed ourselves monthly to discuss tedious matters of our exhibition, there's no other kind of impractical behavior to make us feel more like : we have been there before.
When I employed myself in this series of photography, while imagining in every corner, every constant, the drained people of Hong Kong leave marks in their city, traces of their civilization. Of course, who will stay until the end ? Who's the audience ? If you are alive, that would be you !
When I took the photographs of the new government buildings, I imagine the chief executive looking at the Victoria Harbour coastline, photographing the last glimpse of his vision of Hong Kong. Is he looking at the West Kowloon across the harbour ? Or the new government buildings that are haunted by the soul of the veterans ? Imagine the bus drivers when they drive pass Tai Po road, encounter the high voltage electric wire that symbolises civilization, or the professionals in the water treatment plant, the football fans in grass sports ground, the housewives in public housing or luxurious mansions, the lonely salesman at Star ferry pier, the tour guide outside Ocean Park, the wedding photographer underneath the Tsing Ma bridge, the school kids on the beach waiting the sun rays to flow. I set my tripod up, open the lens, seeing the inverted image through the ground glass, the glass surface review another reflection from behind. With naked eyes, I use the magnify glass head on with ground glass for viewing and focus; in relation to drawing, when you set up the easel, pen holding far reach for an arm length, narrow your eyes to view the perspective of world, the proportion, the structure and form in front of us. This narrowing of our eyes when we draw, the white and blur scenery coincide with the action when the world ends, the great explosion force our eyes to narrow down again so as not to pierce by its radiant light.
Layers and layers of images could easily arouse sensuality, on the other hand, metaphysical sense overlaps with reality. The creative dialogues repetitively bring abstract thinking back to our living self, or the spaceship for universe exploration never ever had a successful launch, she took off and then to be found fallen on the ground again. The reason behind the story never falls from the polluted city, education and culture. To the people of Hong Kong, we are forced into indifference, take it for granted already.
The time has come. Adieu, traditional Chinese characters and Cantonese, Hong Kong movies, television shows, the last two Cantonese lyricists, beds in hospitals, infant milk powder, food safety, Dongjiang water supply, the last China White dolphin. The replacement would be the political poll survey, politician and their propaganda, Home Affair Bureau thinks that recreation is top cultural priority. Academic freedom and university campus security were brought to an unprecedented level, the core value of publishing only rests on consumerism and marketing.
Live and be prosperous. The process is more important than the result. As people of Hong Kong, what we have is an undated goodbye, doomsday without timeline. Make every day the last day of your life, tomorrow is another no better day. I do not know, whenever there is a doomsday, how many people could survive. At this very moment, I sincerely thank the brave Stella for sharing with me this creative experience that has no past.
The dust from the scans of film negative shown on the monitor is the larger and brighter than the star Sirius hanging outside my window. Imagine to travel sixteen light years back from Sirius to earth, the conscience priority of the dust that rest on the monitor would be so minute as if it ceases to exist. When the thyself of 2013 look back to this very moment of your present self, you might as well forget it completely already.


The normal garage doors evaluate around a hundred pounds and also normally use an expansion spring for the heaviness of the Pro Garage Door Minneapolis. This becomes unsafe if the springs break, creating the door to malfunction. Undoubtedly, the garage door is a fantastic possibility of harm and also crash at home. To assist you stay clear of garage door accidents, attempt adhering to these advices. If you are still planning to have your door mounted, select a trustworthy as well as professional solution installer to do it. Be meticulous about safety, they must have the essential skill in this kind of work, particularly in the setting of the spring, which is the extremely important yet important part of the work.
An incorrectly installed door will certainly offer you a frustration and also dissatisfaction regardless of the number of repairs you make. Nonetheless, if you are installing it by yourself, make certain to check out the guidelines as well as follow it carefully. There have to be no area for errors. After the installation, you need to acquaint and look after the push-button control, if ever before there is a garage door opener. Do not leave it existing around. Ask any garage door installer and also they will provide you one principle: Keep your children secure. As well as by this, it suggests that you should driven your kids (as well as any other family members) about garage door safety. Tell them that playing near it is prohibited, no playing with the remote and also describe to them the consequence of being trapped under as well as behind the doors.
Regular monthly exams and also inspections ought to be done. Try to find imbalanced doors, hidden things, loosened bolts, damaged cords, rollers, cables and the button wall must be out of reach of the youngsters. Contact local technicians if problems are observed. Your garage door is extremely supported by the springtime under high tension. Ensure to replace old springs. Broken or loosened springtime may appeal your vehicle or even worse, on an individual.
Normally, doors have two springtimes, if the other is damaged, make sure and transform the two. Door openers ought to have a reversing function, or else, replace it. The lack of the said function will trigger damage, the fact that it could caught something while it is shutting, due to the fact that it will make contact to the door prior to closing. All garage doors are required by the authority to install turning around attribute. For additional knowledge, garage door openers are mechanized tool made use of to open as well as close garage doors. One of the most usual design of driver is the trolley operator (usually in the house).
The electric motor hangs near the facility of the ceiling with rail guides and a trolley across the ceiling to the top of the entrance. The trolley is connected to the door arm that is after that affixed to the top of the door. When the cart goes across the rail the door opens as well as shuts Extra functions that have actually been added for many years, including and automatic lights that turn on if the door opens up and also immediately turn off with a preset time, cordless keypads and also keychain remotes, which are commonly made use of today.
One of the leading errors people make is leaving the Choice Garage Door Denver remote in their lorry. Thieves get into your automobile and take the remote providing direct access to your residence. The most effective means to stop this from happening and still have easy access is to simply buy a keychain design remote.
Keep GPS gadgets stashed when car park at shopping center, movie theaters or sporting events. Offenders will certainly get into your auto steal and your opener remote then hit the "home" and it leads them to your vacant home where they will certainly have gain access to for several hrs while you are appreciating your entertainment. If your transmitter does obtain stolen make certain to instantly remove all the codes in your opener and reprogram all your remotes and keypads. This avoids that stolen device from opening your door. If you do unknown ways to clear your codes please call the door opener manufacturer for help.
Protect your garage door emergency launch mechanism using a zip connection or product developed to avoid the "6 second" break-in. Products like the no-entry latch guard are simple to install and are commonly under $20 and make this break-in method nearly impossible. Lawbreakers will certainly make use of a wedge and layer hanger to grab the detach mechanism on your garage door opener and can be in your garage (and home if your door to your house is unlocked) in seconds.
Maintain the door secured in between your garage and residence. We recommend making use of a deadbolt. Also make sure the door from the garage to your house is as durable as your front door. A steel door is the very best and anti kick gadgets are additionally recommended. It truly is not that large of a trouble to unlock the door when you consider the piece of mind you will have knowing your home is safe. Another practical hint is to install a great wide-angle peephole to check into your garage from securely inside your house incase you do hear odd sounds coming from your garage.
Do not make it easy on the crooks and fail to remember to close your door when you go to sleep or leave for job. This gives them free access to anything in your garage and if they shut your door behind them plenty of time to additionally get into your residence without being seen. There are a couple products on the market like a garage door monitor that can tell you if your garage door is open or closed from your bedside or throughout your residence. Genie currently has a garage door opener that can be fitted with a device called a shut verify remote that will tell you if your garage door closed appropriately behind you.
If you do have windows in garage door or side home windows in your garage make certain to obtain odd glass or frozen glass to avoid burglars from looking in to see just what excellent things you have to use or more notably if somebody might be home or otherwise. Maintain pricey things like golf clubs, bikes and power devices out of site. These are items easy to offer and pawn.
Constantly padlock the slide lock on your door when taking a trip or far from your house for extended periods of time. If you do not have a slide lock on your garage door they can be bought for under $10 from an equipment store or on-line garage door parts seller. This makes certain no one will be able to open your garage door by hand or with a remote. This will offer you a piece of mind that your garage is safeguarded while taking a trip. A little tip though make sure to either disconnect your opener or transform the vacation activate your wall station to stop you from mistakenly pressing the switch when you return residence and damage your door, opener or both because of it being locked down.
It seems basic sufficient: Your garage door isn't really functioning, so you make a decision to take on the problem. You're unsure just what the issue is. Possibly the opener has actually gone bad, or among the tracks needs a bit of Tender Loving Care. However, it's a garage door repair Milwaukee WI. Just how hard could it be to deal with?
Garage door repairs are among one of the most difficult to do, thanks to the auto mechanics and thickness of the door. In more situations than not, it's commonly best to go with an expert garage door repair company to repair troubles and issues, instead of attempting to tackle it by your lonely. Why? There are several excellent needs to offer a few dollars to someone to look after the issue rather than going it alone.
These kinds of doors are heavy. It is just one of the reasons they're run by openers with tracks, springs and so forth. They are so heavy that some have been wounded – and worse – when they obtain entraped below a closing or falling door. These doors are a charming convenience to have, but can be dangerous in the wrong circumstances.
Choosing to repair your garage door on your own is an unsafe decision. What occurs if something falls on you, and there is nobody around in order to help? Most expert garage door companies send out a minimum of two to three individuals whenever a job requires done. They know that it takes more than one person to appropriately repair a garage door, so no injuries or hazards occur throughout the process.
Exactly how familiar are you with just how this sort of door jobs? Do you understand what a torsion spring is? What troubleshooting procedures should be done if the opener isn't really functioning? If you're not able to address these concerns, after that it's best to put down the tools and pick up a phone to obtain in touch with your neighborhood door repair carrier.
The persons that help business in the industry recognize the technicians – inside and outside – of just how your door must work. The majority of have handled comparable concerns numerous times in the past, so they will not have to finagle or mess around with equipment or items in a vain initiative to obtain your door functioning once again.
If you handle your own garage door repairs, exactly how are they protected versus feasible future failings? Are you able to offer on your own a warranty for components and labor? The answer is most likely no, but repair business can. Lots of deal substantial guarantees on components and labor, so you won't have to stress over exactly what to do ought to repairs be necessary again.
When it involves garage doors, many property owners with basic expertise on technical stuffs have the ability to address the straightforward issues that ail a garage door. However anything complicated should be delegated the experts alone. Fixing garage door requires high abilities and might prove dangerous for property owner. There are many companies who will have the ability to successfully resolve your facility with their most recent devices in nearly no time at all.
Nonetheless, if there are just simple troubles that bother your garage door and also you believe you could handle them; you have to understand the basics of repairing a garage door. Well, below they are!
Hammers, pliers, cable cutters and testers are needed when fundamental repairs and also maintenance can be found in to concern. One of the fundamental problems is when dirt and particles obstruct the tracks that are important for the smooth performance of the garage door. Prior to going on with the cleaning as well as fixing, owners must guarantee to obtain the supply switched off to the opener. If the proprietor is negligent here, he/she might encounter a crash.
After a lengthen use, the opener's batteries head out. With a tester, one will be able to gauge the life of the batteries and replace them. This straightforward task could conserve you on a significant quantity of cash. Also, garage doors create dents on several areas years after an excellent very long time. Utilizing a hammer, a property owner will be able to solve the issue. One need to be careful not to hammer it too hard as that might generate even more damages. If you are feeling unclear, call the service provider who mounted it or a service provider better to your house.
When you believe your garage door's tracks need lubrication, you need to get the lubricants that work well with the door. In instance you are not knowledgeable about the lubricating substances, get in touch with the specialists instead. These are the factors which will certainly assist you to avoid any straightforward trouble. Nevertheless, you need to recognize that garage door elements weigh and also may prove hazardous to someone with much less experience. Ensure to phone the specialist for repairing any type of significant issue on your garage door.
Ask your friends and family for recommending you the names of the business which they have actually employed in the past. When it comes to finding specialists for garage door fixing, homeowners can contact today!
The majority of A1 Garage Door Repair – Sierra Vista treatments can be less complicated compared to home owners believe. Although, this is not to state that experts for these types of doors are unneeded because there are some sort of fixer-uppers that actually require the skill and knowledge of those who are educated to take care of these points up. These experts can be hired to repair just what the house owner has no concept ways to take care of yet there are some remedies that house owners can do as well as try prior to actually calling the big guns.
A number of these entranceways work via making use of a collection of cable televisions or tracks on which they travel. These tracks could quickly collect dirt, dust and debris specifically if your house is found in a dusty road or in an environment where there is little rainfall and cool. Garage door repair service for filthy tracks is to actually try and also cleanse them up as well as try to examine which component has the haze build-up of dust, dirt or particles and also aim to reach it to clean it. Some of these contraptions have tracks that are extremely simple to accessibility, thus making it equally as easy for the property owner to clean. Oil or lube may have likewise hardened therefore developing a barrier for the panels to travel. These can be conveniently cleansed but the house owner needs to exercise treatment when relocating when the blockage is gotten rid of. The rollers that take a trip on the tracks could additionally have actually built up particles, which can be detrimental to their correct feature. Often, it is likewise beneficial to inspect the rollers as well as the tracks. In connection with debris on the rollers and tracks, movement of the entranceway panels could be restricted due to absence of lubrication.
Other than debris blocking the tracks on the walls of the carport, another reason that there could be a trouble opening or closing them could be because one or more of the mounting braces which hold up the tracks may be loose. Mounting braces are screwed to the wall so they can line up and keep the tracks directly, as soon as one or more hangs, positioning might remain in risk and panels could not take a trip on the tracks efficiently or in any way. To examine if the tracks are properly straightened, the house owner can use a degree to see if everything remains in their correct area. Touching the tracks back right into the correct placement might benefit small imbalances yet larger ones may need to be check by the experts since there is a risk of the panels injuring a person.
Shielded garage doors with weather stripping can make your entire home much more energy efficient. This post will certainly assist you uncover just how your garage door can influence your energy expense. Click here sierra vista garage door repair for more information.
In general, property owners strive to reduce their power prices. Despite having protected home windows and also home air conditioning as well as heating unit upgrades, energy costs will certainly still reflect one of the biggest areas of your home: the garage. This space has a really high impact on the thermal comfort of your house and also the garage itself.
Garage doors are made from various materials such as timber, steel, fiberglass, vinyl, and light weight aluminum. If you do not know it currently, you ought to figure out the insulation value of your garage door. A greater R-value implies much better thermal effectiveness. Insulated doors have a metal skin over an inflexible core, which is made from either polystyrene or polyurethane. It is essential to keep in mind that the latter, polyurethane, supplies two times the R-value of polystyrene. In addition to a far better insulation core, higher-quality doors may have a distinctive, tinted, or contoured surface.
Rates differ, so thoroughly compare both efficiency and also style. If buying a new door is not within your current spending plan, take into consideration purchasing a foam setup set, or calling a garage door fixing service for recommendations on furnishing your current door.
Energy cost savings are one of the most apparent advantage. As energy expenses remain to increase, a brand-new door will begin to pay for itself in cost savings soon after installment. There are, nonetheless, numerous other benefits to installing a shielded door.
With boosted temperature level control, the garage area as well as adjacent areas in your house will be kept cooler in the summertime and also warmer in the wintertime. This additionally makes the garage a lot more favorable for use as a workshop or entertainment location. In addition, insulation will help soundproof your garage doors.
A more thermal reliable garage also implies a decrease in temperature level extremes inside your vehicle. This will certainly lower the amount of time had to cool or heat up the air inside your cars and truck prior to you enter, while lowering your level of individual discomfort as a result of temperature modifications as you exit your cars and truck.
Including a weather condition seal or boundary climate seal system around the garage door is exceptionally vital for temperature control, in addition to prevent rain and snow from permeating right into the garage. In addition to real garage door repairs, the other entryways, such as a back entrance or side windows, ought to be considered. If your residence has an affixed garage with a door leading to the interior, this door should likewise have insulation and also climate stripping.
How do you understand when it's time to explore Sterling Garage Door Repair work? It's pretty simple to inform – You hit the remote. Absolutely nothing takes place. Or, something does occur but it's not just what you desired. The thing squeaks open shateringly, hanging halfway off your house, and threatens to diminish anytime!
Garage door repair service belongs of your upkeep routine. Like every other part of the house, it sometimes does not function, and that's when it's time to obtain it taken care of.
Before you surrender hope, dabble with things a little and see if it isn't just your remote. Buying a new remote is much better on your finances than obtaining new doors. You could attempt opening it manually with the switch typically situated inside. This button is generally found by the door that introduces your home.
You may be surprised to discover that this is one of the most common garage door fixing trouble. Prior to you head to your neighborhood supplier to purchase a new remote, attempt cleaning off the sensors. Your control box (the little thing that remains on the ceiling) needs to have "eyes" that find the remote's transmissions. Clean them tidy with a dustcloth. While you're up there, see to it all the cables are securely secured.
If the wall button won't open it, that suggests you've obtained no juice. Make certain everything's connected in as well as check the breaker as well as fuses.
Your doors have a terrific safety function that maintains you or your loved ones from obtaining squashed under their remarkable weight. If your doors are shutting, yet stopping somewhere along the road and then heading back up, this implies that you've got this security attribute set incorrect.
You could want to break out the user's manual and also look up the "restriction switch." There ought to be a switch someplace on package that permits you to manage where the doors stop. Set this closer to the ground and also it must work appropriately.
You must also look for blockages. If there's something on the tracks, even something little, it could be striking that and also stopping.
If your door does not open (as well as it's not electrical), or if it opens slowly or unevenly, you may have a torsion spring problem. Over the top of your door you've obtained a torsion spring system. The torsion turn lift drums and also wind cable televisions that lift the heavy doors off the ground.
You can fix your torsion spring system yourself however it is really dangerous. I do not indicate to terrify you, yet it could be lethal. The reason is that these springs, in order to lift such a hefty piece of residence, are under an incredible amount of tension. If one ought to break, it can trigger severe injury or even death. For a job such as this, you could wish to leave it to a professional.
If you're in over your head Sterling Garage Door Repair and also the stupid thing simply won't open no matter what, it's far better to contact a specialist. They've got years of experience with repair, as well as the tools to get the job done right. It costs a little modification, however it finishes the job and also you can be sure it'll work the method it's expected to.
Any home improvement job is close to a property owner's heart. However, these tasks are very essential too due to the fact that you depend on them for the structural and practical stability of your home. Garage doors are an essential element of your house. Typically these doors take quite a lot of beating with frequent use. You also have to take into consideration the periodic bumps and swellings from beginning drivers in your home. Taking all this into stride, it is anticipated to tackle regular garage door maintenance jobs. Below is all you need to understand before you begin with your following repair on your garage doors. Visit this site http://elginilgaragedoorrepair.com/ for more information.
The extent of the job that has to be done and the final cost of the project depend upon exactly how thoroughly you investigate the condition of your garage door before you begin the task. Here are a couple of tips to guide you in the direction.
· Degree of the damage: The sort of the door and the level of damage that it has actually sustained need to be the beginning facet of your examination. The problems can be as small as small imperfections and breaking to major troubles with the technical receptors of the door.
· Door Designs: The difference in the design of the door will majorly impact the last job to be done on the door. Every different style has various devices to deal with the design.
· Products used for the door: The market includes a vast array of choice of materials to be used for doors. Subsequently the type of materials and the components used to construct the door are very crucial for the repair approach.
Below are a few tips that you could utilize to actively decrease your overall project costs for your garage door repairs and general garage transformation. These are also upkeep standards, which could help you boost the long life of your garage, doors, therefore, requiring less repairs down the line.
· Maintain the door tracks tidy: The garage tracks often gather particles and dust that may make it arduous for the door to work appropriately. Maintaining the tracks cleanse will apply lesser stress on the doors over time.
· Realignment of the sensing units: Garage doors usually leave alignment because of regular usage. Relying on the type of sensing units that you are using, there need to be a caution light blinking to advise you of this scenario. Adjustment can be managed manually. You ought to additionally check for blockages or dust and dampness damages that could be creating the concern.
· Modification of loose components: Nuts, screws, and even springs in the mechanism could come loose. This puts in unnecessary instability over the entire mechanism of the door. Leaving this concern neglected might create numerous problems in the future. Rather check for these abnormalities and just provide a great conditioning and tightening to prevent heavier repair expenses in the future.


Justice wraps your profile perfectly in a unique carbon fiber pattern acetate and debuts our newest logo plaque, the cross that is found within the Eyes of Faith logo, in a sporty, faithful call-out on each temple. You'll appreciate wearing a little inspiration with the Isaiah 1:17 passage, ̱Learn to do right; seek justice. Defend the oppressed,̨ printed inside Justice̍s temple, and outdoor and sports enthusiasts will love that Justice is finished with polarized lenses.


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2019 Serbia 1-oz Silver Nikola Tesla Remote Control NGC MS69 First Releases - Tesla Label
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Pay Tribute to the 120th Anniversary of Remote Control With Sold-Out Silver Coin!
Thanks to modern technology, we don't have to direct our kids to change the channel for us like our parents did. We simply pick up a little hand-held device and change the channel by remote control. The "remote" may seem like a relatively new innovation, but in fact it was first debuted by renowned Serbian-American inventor Nikola Tesla at the Electrical Exhibition of 1898 held in New York City's Madison Square Garden.
In honor of the 120th anniversary of this groundbreaking invention, Serbia has issued the second release in its series of coins commemorating the genius of Nikola Tesla. These legal-tender coins are struck in a full ounce of highly-pure 99.99% silver and feature not only the image of the scientific pioneer, but one of his greatest discoveries – the radio-controlled boat he used to demonstrate his invention. The low-lying steal craft was about four feet in length and easily skimmed across a small pool of water in a tour de force that cemented Tesla's patent for his work. Tesla had hoped that the United States Navy would buy the idea of his remote-controlled boats but was rejected by the Navy Brass on the grounds that they thought the technology was too flimsy for war. Ironically, less than two decades later the German navy used remotely controlled boats loaded with explosives to attack Allied ships during World War I.
Your 2019 Serbia Nikola Tesla Remote Control One-Ounce Silver Coin has been graded by the Numismatic Guaranty Corporation (NGC) as being in near-perfect Mint State-69 (MS69) condition, a distinction that means your almost-flawless coin will showcase an impressive full strike and indistinguishable imperfections – well above and beyond the quality of the vast majority of coins in the series. In addition, NGC has graded your coin to be a First Releases (FR) coin, a designation that means it was received by NGC or an NGC-approved depository within the first 30 days of the its release. It will come sonically sealed in an acrylic holder with an exclusive Nikola Tesla label.
You'd think with a résumé such as his that Tesla's name would be a household word. Unfortunately, Tesla sold most of his patents, and the fame generated by his discoveries was bestowed onto others. Discarded and forgotten, he died in 1943 alone and penniless in a New York City hotel. It wasn't until a 2017 episode of the hit CBS comedy "Big Bang Theory" that Tesla's name started to get its due among the general public. During the episode, the gang debates whether they are "Teslas," who make their own discoveries, or "Edisons," who build upon existing research. The spike in Google searches about Tesla was felt immediately.
Tesla and his remarkable invention take center stage on this beautiful coin. History buffs, science aficionados and silver stackers alike are sure to appreciate the design and high silver content. Just 50,000 of these coins have been struck and they are completely sold out at the mint! Don't wait – get one for yourself and several others for the budding scientists in your life. Add them to your cart today!
COMMEMORATIVE DESIGN OF ONE OF THE WORLD'S GREATEST INVENTORS – Though he made most of his discoveries after immigrating to the United States in 1884, Nikola Tesla was born in a region of Serbia that is now part of Croatia. The former Serb is commemorated on the obverse of the stunning silver coin released by the prestigious National Bank of Serbia. The coin's reverse features one of his most famous inventions: the teleautomaton – a wireless remote-controlled boat. The design also features the patent number and date of patent US 613809A NOV 8. 1898. Inscribed above the design is 120th ANNIVERSARY REMOTE CONTROL.
STRUCK IN HIGHLY PURE SILVER – These 2019 Serbia Nikola Tesla Remote Control Silver Coins were struck in an ounce of highly pure 99.99% silver.
LEGAL-TENDER COINAGE – Each Nikola Tesla One-Ounce Silver Coin is 100 Dinara legal tender in Serbia.
GRADED IN NEAR-PERFECT UNCIRCULATED CONDITION – Your 2019 Serbia Nikola Tesla Remote Control One-Ounce Silver Coin has been graded by the Numismatic Guaranty Corporation (NGC) as being in near-perfect Mint State-69 (MS69) condition, a distinction that means your almost-flawless coin will showcase an impressive full strike and indistinguishable imperfections – well above and beyond the quality of the vast majority of coins in the series.
KEY FIRST RELEASES CERTIFICATION – NGC has graded your coin to be a First Releases (FR) coin, a designation that means it was received by NGC or an NGC-approved depository within the first 30 days of the its release. It will come sonically sealed in an acrylic holder with an exclusive Nikola Tesla label.
SOLD-OUT LIMITED WORLDWIDE MINTAGE – The Serbian Mint has set a maximum worldwide mintage for this coin at just 50,000 – and they are completely sold out at the mint!
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First Releases
Currency Type
Mint Name
Serbian Mint
Coin Weight
31.1 Grams - g
1 Ounce - oz
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Turnips are flowering, but not producing much of the turnip itself. Is this because they are transplanted plants?
Turnips will bolt to flower when conditions are not right. They might have got too dry, or too much of a setback when transplanted. Root crops like turnips are best sown straight into the ground.
The most likely cause of your turnips flowering before developing good bulbs is that the temperature dropped below 5ºC (40ºF) over a series of nights fairly soon after they were transplanted followed by warmer weather.
This can be a signal to turnips that vernalization (winter then spring) is happening and triggers the development of flowering stems.
Normally turnips are biennial and grow leaves and swollen roots the first year, and the next year after a cold winter they reproduce by flowering in spring. But if adverse weather comes with near freezing temperatures which mimic winter vernalization, turnips, especially vulnerable transplanted seedlings, are likely to flower in their first year.
Nothing to lose by eating them as long as they are big enough to get a meal from. They might be a bit woody, but they won't grow any bigger, so get them out and put them in your next soup or stew.


I have had this recipe from Favorite Family Recipes saved for awhile now and having company last weekend finally gave me a good reason to make them! There is just something about stuffed chicken that instantly seems fancy and how can you go wrong with anything wrapped in bacon?! Even if it is turkey bacon :-). These were so delicious and definitely should be tried! I served them with a mushroom wild rice mix and green salad and it hit the spot!
◦**Optional Cooking Tip: When I wrap chicken breasts with bacon, I like the bacon a little crisper. If you don't like it crisp, feel free to wrap raw bacon around chicken breasts. Either way works, it just looks better and I think tastes better when you partially cook the bacon first! You can partially fry the bacon or cook in the oven.
◦Oven Method: Preheat oven to 400 degrees. Place bacon slices on cookie sheet lined with aluminum foil. Bake for 12 minutes. Remove partially cooked bacon and pat off excess fat.
◦Chop up 6 pieces of bacon to use in the cheese filling and reserve remaining 6 slices to wrap around chicken.
◦In bowl, combine bacon, green onion, sour cream, gorgonzola cheese and garlic.
◦Cut a 1 1/2 inch slit (about 2 inches deep) into the thickest side of each chicken breast. Using a spoon make a larger pocket inside of chicken breast.
◦Spoon in equal portions of the cheese mixture into each chicken breast. Lay a slice of bacon on top or wrap around. Sprinkle with Montreal steak seasoning.
◦Combine gorgonzola cheese and sour cream. Microwave about 30 seconds and stir until smooth. Drizzle over cooked chicken.
Combine mayonnaise and chipotle, set aside.
Dry the beans well after washing, extra moisture keeps the burgers from sticking. In a medium bowl, mash beans with a fork until thick and pasty.
Freeze at least 2 hours before cooking or keep frozen until ready to cook.
Heat a lightly sprayed skillet to medium heat and cook frozen burgers about 7 minutes on each side.
If grilling, preheat grill over medium heat, and lightly oil a sheet of aluminum foil; grill 7-8 minutes on each side or you can bake in the oven at 375° on a lightly oiled baking sheet.


text:
25.10.2021 - 25.10.2021 | Greifswald
How biotrophic fungi infect plants: unexpected insights from essential effectors
Öffentlicher Abendvortrag von Professorin Dr. Regine Kahmann (Max-Planck-Institut für terrestrische Mikrobiologie, Marburg)
Plant pathogenic fungi colonizing living plant tissue secrete a cocktail of novel effector proteins to suppress plant immunity and reprogram host cells. Although many of these effectors function inside host cells, delivery systems used by pathogenic bacteria to translocate effectors into host cells have not been detected in fungi. During the course of studying such novel effectors in Ustilago maydis, a biotrophic fungus causing smut disease in corn, we have identified five unrelated effectors and two membrane proteins which form a stable protein complex. The corresponding genes appear co-regulated and are only expressed during colonization. Single mutants in any of these seven genes arrest in the epidermal layer, fail to suppress host defense responses and fail to induce non-host resistance, two reactions likely depending on translocated effectors. The complex is anchored in the fungal membrane, protrudes into host cells and contacts channel forming plant plasma membrane proteins, suggesting that we identified a translocation system. The system could be reconstituted in a surface-exposed form in cultured U. maydis cells. As orthologs of the complex-forming proteins are conserved in smut fungi, the complex may become an interesting fungicide target.
Regine Kahmann studied biology at the Georg August University in Göttingen. She did her doctorate from 1972 to 1974 at the Max Planck Institute for Molecular Genetics and the Free University of Berlin. From 1980 to 1982 she worked as a research assistant at the Max Planck Institute for Biochemistry in Martinsried. In 1982 she started an independent research group at the Max Planck Institute for Molecular Genetics in Berlin. Between 1992 and 2001 she was Professor for Genetics at the Ludwig Maximilians University in Munich. Regine Kahmann was Director and Head of the Department of Organismic Interactions at the Max Planck Institute for Terrestrial Microbiology in Marburg from 2000-2019.
Moderation: Professorin Dr. Ulla Bonas
https://www.wiko-greifswald.de/programm/allgemeines/veranstaltungskalender/
Stiftung Alfried Krupp Kolleg Greifswald
info@wiko-greifswald.de
Vortrag / Kolloquium / Vorlesung
Priscilla Primpke
Alfried Krupp Wissenschaftskolleg Greifswald


School Services of California, a well-respected school finance and advocacy firm, has published comparison reports for total employee compensation and health and welfare benefits. Each School Services of California report contains local district averages, comparative group averages, the statewide average, and regional average. These reports show how Evergreen School District compares to these averages. Because 2012-13 school year is incomplete, the 2011-12 data is the most current comparison data available.


text:
Arguments for Draconian Program Cuts Don't Survive Scrutiny
Home / blog / Federal Budget / Arguments for...
Newly revised CBPP reports show that two main Republican arguments for deep cuts to programs for low-income Americans — that program growth across the federal budget is driving the nation's fiscal problems and that programs for people of low or modest incomes are mushrooming in cost — don't survive scrutiny.
President Trump and House Republicans have both proposed 2019 budgets that would deeply cut key domestic investments, including programs that help millions of families pay the rent, put food on the table, and get health care. Those budgets, by the way, came in the aftermath of December's tax law, whose tax cuts were disproportionately tilted to the wealthy. Programs cuts that disproportionately target programs on which low-income families rely have been a consistent feature of Republican budgets in recent years.
But consider the findings of our revised reports:
The first report finds that federal program spending outside Social Security and Medicare is below its 40-year average as a percent of the economy and is projected to keep falling. Total federal program spending (not counting interest payments on the debt) outside Social Security and Medicare will be 11.3 percent of gross domestic product (GDP) in 2018, below the 40-year average of 11.9 percent. And it's projected to fall to 10.0 percent in 2028. Combined Social Security and Medicare spending is projected to keep rising as a percent of GDP, from 7.9 percent in 2018 to 10.0 percent in 2028, well above the 40-year average of 6.4 percent. But this reflects the aging of the population and rising costs across the U.S. health care system (both public programs and private insurance), not more generous benefits.
The second report finds that federal spending for low-income programs outside health care has fallen to its historical average as a percent of GDP and is projected to decline further.  After rising during the Great Recession, spending on these programs has returned to its 40-year average of 2.0 percent of GDP and is projected to fall to 1.5 percent by 2028 — the lowest level since 1970. (See chart.) Spending on low-income health programs is projected to rise as a percent of GDP over the decade, largely due to the aging of the population and rising costs throughout the U.S. health care system. But the projected decline in spending on low-income programs outside health will fully offset this increase. Total spending on low-income programs — health and non-health — is expected to fall slightly between 2018 and 2028, from 4.4 percent of GDP to 4.3 percent.
To be sure, the nation faces long-term fiscal challenges, as we've documented. Federal spending and revenues will need to rise as a share of the economy due to an aging population and rising health and interest costs. Unfortunately, the 2017 tax law moved in precisely the wrong direction, weakening revenues and directing its benefits disproportionately toward those who least need them. Rather than doubling down on an agenda that combines large tax cuts with damaging program cuts, policymakers should put the budget on a more sustainable path that protects key domestic investments and raises sufficient revenue to keep pace with the nation's needs.
K-12 Funding Cuts Include Capital Spending to Build and Renovate Schools
Trump Budget Deeply Cuts Health, Housing, Other Assistance for Low- and Moderate-Income Families
Contrary to Rhetoric, Trump Budget Would Make It Harder for Many to Work, Gain Skills to Get Ahead
Federal Spending and Revenues Will Need to Grow in Coming Years, Not Shrink
New CBO Budget Projections Are Cause for Concern — Not Alarm


The story began when a student in IIT-KGP and his mates bought an I-phone for a price that they realised had a much lowered price on another portal just two days later. The student, Gaurav Dahake, shocked and disgruntled that there was no way to get a comparison of prices and other information, took the problem to his head. Then came the Hatke idea. Gaurav partnered with Prashant Singh, and Srikanth Sethumadhavan, and together the IITians opened BuyHatke.com that solves a customer's pain-point. Today BuyHatke is a major compare site that takes on biggies in the online product discovery space.
We Indians love a good bargain, and the BuyHatke team gives us the best deal across the online e-commerce sites, be it shopping or travel. And in turn the company's growth seems to be booming like the Indian population. Gaurav Dahake's entrepreneurial journey proves there are many Indians who can take a problem and solve it remarkably well. Let's prove to the world that India can be the best in what she puts her mind to.


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