text:
\section{Preliminaries}
In 1914,  H. Bohr \cite{Bohr-14} proved that if the power series $f(z)=\sum_{k=0}^{\infty} a_kz^k$  converges in the open unit disk
$\ID :=\{z\in\IC:\, |z|<1\}$ and $|f(z)|<1$ for all $z\in \ID$, then the majorant series
$M_f(r):=\sum_{k=0}^{\infty}|a_k|r^k $ is less than or equal to $1$ for all
$|z|=r\leq 1/6$. The largest $r\leq 1$ such that the above inequality holds is referred to as the Bohr radius for the unit disk case.
The fact that the constant $1/3$ is best possible was established independently by F. Wiener, M. Riesz and I. Schur.
  Other proofs of this result were later
obtained by Sidon and Tomic.   Bohr's idea naturally extends to  functions of several complex variables and thus, a variety of results
on Bohr's theorem in higher dimension appeared recently. In this contexts and in other respects, we suggest the reader to refer
\cite{Abu,Abu2, Abu4, AAD2,AizenTark-01-2,Aizen-00-1,Aizen-05-3,BaluCQ-2006,BenDahKha,BoasKhavin-97-4,PaulSingh-04-11,Pop} and
the references there. For a detailed account of the development, we refer to the recent survey article
on this topic \cite{AAPon1} and the references therein. More recently, the present authors obtained the following result
as a corollary to a general result for symmetric functions and thereby settling the recent conjecture of
Ali et al. \cite{AliBarSoly}.

\begin{Thm}{\rm (\cite[Corollary 1]{KayPon1})}\label{KP2-odd1}
If $f(z)=\sum_{k=0}^{\infty} a_{2k+1}z^{2k+1}$ is odd analytic function in $\ID$ and $|f(z)| \leq 1$ in $\ID$, then
$M_f(r) \leq 1$  for $r \leq r_2=0.789991\ldots .$
The extremal function has the form $z(z^2-a)/(1-az^2)$.
\end{Thm}

Motivated by the work of Ali et al.\cite{AliBarSoly} and Theorem \Ref{KP2-odd1}, we raise the following

\bprob {\rm{(\cite{AliBarSoly})}\label{KP2-prob1}}
Given $p \in \mathbb{N}$ and $0\leq m \leq p$, determine the Bohr radius for the class of functions
$f(z)=\sum_{k=0}^{\infty} a_{pk+m}z^{pk+m}$ analytic in $\ID$
and $|f(z)| \leq 1$ in $\ID$.
\eprob

The case $m=1$ has been handled by the authors in \cite{KayPon1}.

One of the aims of this article is to solve this problem  completely  and present several of its consequences.
As remarked in \cite{KayPon1}, it brings serious difficulties because if we use sharp the inequalities
$|a_n| \leq 1-|a_0|^2$ $(n\geq 1)$ simultaneously (as in the classical case) then we will not be able to obtain
sharp result due to the fact that in the extremal case  $|a_0|<1$.
Also it is important that in the classical case there is no extremal function while in our case there is.

The paper is organized as follows. The solution to Problem \ref{KP2-prob1} is presented in Section \ref{KP2-sec2a} (see Theorem \ref{KP2-th3}) and its proof
is given in Section \ref{KP2-sec2}. In Section \ref{KP3-sec4}, we introduce the concept of $p$-Bohr radius for the class of bounded harmonic functions defined on the
unit disk and discuss the  Bohr inequality for planar harmonic functions as a special case. We expect that our approach will lead to several new investigations on the
general notion of the so-called Bohr's phenomenon.

\section{Bohr inequality for analytic functions of the form $\sum_k b_k z^{kp+m}$}\label{KP2-sec2a}

\begin{thm}\label{KP2-th3}
Let $p \in \mathbb{N}$ and $0\leq m \leq p$, $f(z)=\sum_{k=0}^{\infty} a_{pk+m}z^{pk+m}$ be analytic in $\ID$
and $|f(z)| \leq 1$ in $\ID$. Then
$$M_f(r) \leq 1 \mbox{ for } r \leq r_{p,m},
$$
where $ r_{p,m}$ is the maximal positive root of the equation
\be\label{KP3-eq4}
-6 r^{p-m} + r^{2(p-m)} + 8 r^{2p} +1= 0.
\ee
 The extremal function has the form $z^m(z^p-a)/(1-az^p)$, where
$$a=\left (1-\frac{\sqrt{1-{r_{p,m}}^{2p}}}{\sqrt{2}}\right )\frac{1}{{r_{p,m}}^p }.
$$
\end{thm}

The case $p=2$ and $m=1$ has a special interest which is indeed Theorem \Ref{KP2-odd1} and it provides a solution to the conjecture of
Ali, Barnard and Solynin \cite{AliBarSoly}. More generally, it is a simple exercise to see that for the case $p=m,2m,3m$, the Bohr radii give
$$r_{m,m}=1/\sqrt[2m]{2}, ~r_{2m,m}= \sqrt[m]{r_2}, ~\mbox{ and }~r_{3m,m}=\sqrt[2m]{\frac{7 + \sqrt{17}}{16}},
$$
respectively, where $r_2$ is given in Theorem \Ref{KP2-odd1}. It is worth pointing out from the last case that $r_{3,1}$ gives the value $(\sqrt{7 + \sqrt{17}})/4$. The result for $m=0$ is well known \cite{AliBarSoly} which we now recall because of its independent interest.

\begin{cor} \label{Lemma-1}
Let $p\geq 1$. If $f(z)=\sum_{k=0}^\infty a_{pk} z^{pk}$ is analytic in $\mathbb{D}$, and $\left|f(z)\right|\le 1$ in $\mathbb{D}$, then $M_f(r)\le 1$
for $0\leq r\leq r_{p,0}=1/\sqrt[p]{3}$.  The radius $r_{p,0}=1/\sqrt[p]{3}$ is best possible.
\end{cor}

If $a_0=0$, it follows from the proof of Theorem \ref{KP2-th3} that the number $r_{p,0}=1/\sqrt[p]{3}$ in Corollary \ref{Lemma-1} can be evidently replaced
by $r_{p,0}=1/\sqrt[2p]{2}$ which is the Bohr radius in this case.  Moreover, the radius $r=1/\sqrt[2p]{2}$ in this case is best possible as
demonstrated by the function
$$\varphi_\alpha (z)=z^p\left (\frac{\alpha-z^p}{1-\alpha z^p}\right )
$$
with $\alpha =1/\sqrt[2p]{2}.$ The case $p=1$ of this result coincides with $p=2$ of a general result in
Corollary \ref{KP3-cor3} (especially when $g(z)\equiv 0$).

\section{Proof of Theorem \ref{KP2-th3}}\label{KP2-sec2}

For the proof of Theorem \ref{KP2-th3}, we need the following lemmas.

\begin{lem}\label{KP2-lem1}
Let $0\leq m \leq p$ and $r_{p,m}$ be as in Theorem \ref{KP2-th3}. Then $2{r_{p,m}}^{p+m} \leq 1$.
\end{lem}
\begin{proof} Let $y=r_{p,m}^{p+m}$. Then \eqref{KP3-eq4} becomes a quadratic equation
$$(8 +1/r_{p,m}^{2m})y^2-6y+r_{p,m}^{2m}=0
$$
which has two solutions
$$
y=\frac{3 \pm 2\sqrt{2}\sqrt{1-r_{p,m}^2}}{8 +1/r_{p,m}^2} \leq \frac{3 +2\sqrt{2}\sqrt{1-r_{p,m}^2}}{8 +1/r_{p,m}^2}.
$$
Consequently,
$$2r_{p,m}^{p+1}=2y \leq \frac{6 +4\sqrt{2}\sqrt{1-r_{p,m}^2}}{8 +1/r_{p,m}^2} \leq \sup_{r \in (0,1]}\frac{6 +4\sqrt{2}\sqrt{1-r^2}}{8 +1/r^2}=1,
$$
which completes the proof of Lemma \ref{KP2-lem1}.
\end{proof}

\begin{lem}\label{New} Let $0\leq m \leq p$ and $r_{p,m}$ be as in Theorem \ref{KP2-th3}. Then
$$\frac{1}{r^{p-m}}\left(3-2 \sqrt{2}\sqrt{1-r^{2p}} \right)=1.
$$
\end{lem}
\begin{proof} At first, suppose that $m<p$ and let $y=r^{p-m}$. Then \eqref{KP3-eq4} reduces to a quadratic equation
$$-6 y + y^2 + 8 r^{2p} +1= 0,
$$ which has two solutions $y_1=3 +2 \sqrt{2}\sqrt{1-r^{2p}}$ and $y_2=3 -2 \sqrt{2}\sqrt{1-r^{2p}}$.
The solution $y=y_1$ is impossible because all positive roots of the initial equation are less than $1$. Consequently
$y=y_2$.

Now, consider the case $m=p$. In this case $r_{m,m}=1/\sqrt[2m]{2}$ so that
$$ \frac{1}{r^{p-m}}\left(3-2 \sqrt{2}\sqrt{1-r^{2p}} \right)=3-2 \sqrt{2}\sqrt{1-r^{2m}}=3-2 \sqrt{2}\sqrt{1-1/2}=1
$$
and the proof is complete.
\end{proof}

We now recall the following lemma from \cite{KayPon1}.

\begin{Lem}\label{KP2-lem2}
 Let $|a|<1$ and $0 < R \leq 1$. If $g(z)=\sum_{k=0}^{\infty} b_kz^k$ is analytic and satisfies the inequality $|g(z)| \leq 1$ in $\ID$, then
the following sharp inequality holds:
\begin{equation}\label{KP2-eq3}
\sum_{k=1}^\infty |b_k|^2R^{pk} \leq R^{p}\frac{(1-|b_0|^2)^2}{1-|b_0|^2R^{p}}.
\end{equation}
\end{Lem}

\begin{proof}[Proof of Theorem {\rm \ref{KP2-th3}}]
The case $m=p$ follows from known result (\cite{KayPon1}) and thus, we may assume that $0\leq m < p$.
Following the idea from \cite{KayPon1}, we consider $f(z)=\sum_{k=0}^{\infty} a_{pk+m} z^{pk+m}$, where $|f(z)|\le 1$ for $z\in\ID$.
Also, let $r=r_{p,m}$. At first, we remark that the function $f$ can be represented as $f(z)=z^m g(z^p)$, where $|g(z)| \leq 1$ in $\ID$ and
$g(z)=\sum_{k=0}^{\infty} b_kz^k$ is analytic in $\ID$  with $b_k=a_{pk+m}$.  Let $|b_0|=a$. Choose any $\rho >1$ such that $\rho r \leq 1$. Then it follows that
\begin{eqnarray*}
\sum_{k=1}^{\infty} |a_{pk+m}|r^{pk} &=& \sum_{k=1}^{\infty}|b_k| r^{pk}  \\
&\leq & \sqrt{\sum_{k=1}^{\infty}|b_k|^2 \rho^{pk}r^{pk}} \sqrt{\sum_{k=1}^{\infty} \rho^{-pk}r^{pk}} \\
&\leq & \sqrt{r^p\rho^p\frac{(1-a^2)^2}{1-a^2r^p\rho^p}}\, \sqrt{\frac{\rho^{-p}r^p}{1-\rho^{-p}r^{p}}}\\
&= &  \frac{r^p(1-a^2)}{\sqrt{1-a^2r^p\rho^p}}\,  \frac{1}{\sqrt{1-\rho^{-p}r^{p}}}.
\end{eqnarray*}

In the second and the third steps above we have used the classical Cauchy-Schwarz inequality and \eqref{KP2-eq3} with $R=\rho r$, respectively.
 Hence
\begin{equation} \label{Additional}
\sum_{k=1}^{\infty} |a_{pk+m}|r^{pk} \leq \frac{r^p(1-a^2)}{\sqrt{1-a^2r^p\rho^p}}\,  \frac{1}{\sqrt{1-\rho^{-p}r^{p}}}.
\end{equation}

We need to consider the cases $a \ge r^p$ and $a<r^p$ separately.

\noindent
\vspace{6pt}
{\bf Case 1: $a \ge r^p$.} In this case set $\rho=1/\sqrt[p]{a}$ and
obtain
\begin{equation}\label{KP2-eq4}
\sum_{k=0}^{\infty}|a_{pk+m}|r^{pk+m} \leq r^m \left(a+r^p\frac{(1-a^2)}{1-r^p a}\right).
\end{equation}
For convenience, we may let $\alpha =r^p$ and consider
$$\psi (x)=x+\alpha  \frac{(1-x^2)}{1- \alpha x}, \quad x\in [0,1].
$$
As in \cite{KayPon1}, we find that $\psi (x) $ attains its maximum at $x=x_1$, where
$$x_1=\left (1-\frac{\sqrt{1-\alpha ^2}}{\sqrt{2}}\right )\frac{1}{\alpha }, \quad \alpha \ge \frac{1}{3},
$$
and thus, $\psi (x)\leq \psi (x_1)$. Consequently, by \eqref{KP2-eq4},  we find for the $r=r_{p,m}$ defined in Theorem \ref{KP2-th3} that
\begin{equation} \label{Finish1}
\sum_{k=0}^{\infty}|a_{pk+m}|r^{pk+m} \leq \frac{1}{r^{p-m}}\left(3-2 \sqrt{2}\sqrt{1-r^{2p}} \right)=1,
\end{equation}
where we have used Lemma \ref{New}.

\noindent
\vspace{6pt}
{\bf Case 2: $a < r^p$.} In this case we set $\rho=1/r$ and apply the inequality \eqref{Additional}. As a result we get
\begin{equation} \label{Finish2}
\sum_{k=0}^{\infty} |a_{pk+m}|r^{pk+m} \leq r^m(a+r^p \sqrt{1-a^2}/\sqrt{1-r^{2p}}) \leq 2r^{p+m} \leq 1.
\end{equation}
Here we omitted the critical point $a=\sqrt{1-r^{2p}}$ because it is less than or equal to $r^p$ only in the case $r^{2p}>1/2$ which
contradicts Lemma \ref{KP2-lem1}.

The last inequality in \eqref{Finish2} is derived in Lemma \ref{KP2-lem1}.

Inequalities (\ref{Finish1}) and (\ref{Finish2}) finish the proof of the first part of Theorem \ref{KP2-th3}. Now we have to say a few words about extremal.
We set $f(z)=z^m(z^p-a)/(1-az^p)$ with
$a=r^{-p}\left (1-\frac{\sqrt{1-{r}^{2p}}}{\sqrt{2}}\right )$ and then calculate the Bohr radius for it. It coincides with $r$.

Certainly, an extremal function is unique up to a rotation of $a$. To see this we just trace our inequalities and
see that the equality holds only when $|b_0|=a$.
\end{proof}

\section{The $p$-Bohr radius of bounded harmonic functions}\label{KP3-sec4}

Suppose that $f=h+\overline{g}$ is a harmonic mapping of $\mathbb{D}$ such that  $|f(z)|<1$ with
$h(z)=\sum_{k=0}^{\infty}a_{k}z^{k}$ and $g(z)=\sum_{k=1}^{\infty}b_{k}z^{k}$. Then $|a_{0}|<1$. Also, following
the proof from \cite{CPW2-jmaa} (see also \cite{CPW2-fil}), for $\theta\in [0, 2\pi )$, let
$$v_{\theta}(z)={\rm Im\,}(e^{i\theta}f(z))
$$
and observe that
$$v_{\theta}(z)  = {\rm Im\,}( e^{i\theta}h(z)+\overline{e^{-i\theta}g(z)})
={\rm Im\,}(e^{i\theta}h(z)-e^{-i\theta}g(z)).
$$
Because $|v_{\theta}(z)| <1$, it follows that
\be\label{KP3-eq1}
e^{i\theta}h(z)-e^{-i\theta}g(z)\prec K(z),
\ee
where $\prec$ denotes the usual subordination (cf. \cite{Abu,AAPon1,KayPon1}),
$$K(z)=\lambda+\frac{2}{\pi}\log \left (\frac{1+\xi z}{1-z} \right )=\lambda +\frac{2}{\pi}\sum_{k=1}^\infty \frac{(1-(-\xi)^k)}{k} z^n,
$$
$\xi=e^{-i\pi\mbox{Im}(\lambda)}$ and $\lambda=e^{i\theta}h(0)=e^{i\theta}a_0$. Note that
$K(z)$ maps $\ID$ onto a convex domain with $K(0)=\lambda$
and $K'(0)=\frac{2}{\pi}(1+\xi)$, and therefore, by the definition of subordination, it follows that
$$e^{i\theta}h(z)-e^{-i\theta}g(z) = K(w(z)) =\lambda+\frac{2}{\pi}\log \left (\frac{1+ \xi w(z) }{1-w(z)} \right ),
$$
where $w$ is analytic in $\ID$  satisfying $w(0)=0$ and $|w(z)|<1$ for $z\in\ID$.

Now, we may set $a_0=0$. Then $\lambda =0$ and hence, $\xi=1$. In this case, \eqref{KP3-eq1} reduces to
\be\label{KP3-eq2}
e^{i\theta}h(z)-e^{-i\theta}g(z)\prec \frac{2}{\pi}\log \left (\frac{1+ z}{1-z} \right )=\frac{4}{\pi}\sum_{k=1}^\infty \frac{1}{2k-1} z^{2k-1}.
\ee
Thus,
$$|h(z)-e^{-2i\theta}g(z) | \leq \frac{2}{\pi}\log\left (\frac{1+|z|}{1-|z|} \right )
$$
and hence,
$$|h(z)|+|g(z)| \leq  \frac{2}{\pi}\log \left ( \frac{1+|z|}{1-|z|}\right )
$$
as a consequence of the arbitrariness of $\theta$ in $[0, 2\pi )$. Thus, we have

\begin{thm}\label{KP3-th1}
Let $f=h+\overline{g}$ be a harmonic mapping in $\ID$ such that $f(0)=h(0)=0$ and $|f(z)| < 1$ in $\ID$.
Then $|h(z)|+|g(z)| \leq 1$
for $|z| \leq r_0=\tanh(\pi/4)$. The number $r_0$ is sharp as the functions
$$f_0(z)=\frac{2e^{i\alpha}}{\pi}\mbox{\rm Im}\left(\log\frac{1+e^{i\beta} z}{1-e^{i\beta} z}\right)
$$
show, whose values are confined to a diametral segment of the disk $\ID$. Here $\alpha$ and $\beta$ are real numbers.
\end{thm}

In view of Theorem \ref{KP3-th1} it is natural to ask question about the Bohr radius for a pair $(h,g)$ of  analytic functions in the disk $\ID$
that satisfies the inequality $|h(z)|+|g(z)| \leq 1$ in $\ID$. This provides several new information on the Bohr radius.

\bprop\label{prop1}
Assume that $h(z)=\sum_{k=0}^\infty a_k z^k$ and $g(z)=\sum_{k=0}^\infty b_k z^k$ are  two analytic functions in the disk $\ID$
satisfying the condition $|h(z)|+|g(z)| \leq 1$ in $\ID$. If $g(0)=0$, then we have
\be\label{KP3-eq8}
|a_0|+\sum_{k=1}^\infty (|a_k|+|b_k|)r^k \leq 1 ~\mbox{ for $|z|=r \leq 1/3$}.
\ee
\eprop
\bpf
The proof is almost identical to the classical proof of Bohr's theorem. Let us consider the function $\Phi(z)=h(z)+e^{i\theta}g(z)$,
where $\theta$ is arbitrary. This function satisfies the inequality $|\Phi(z)| \leq 1$ in the unit disk $\ID$ and
consequently, $|a_n+e^{i\theta}b_n| \leq 1-|a_0|^2$. Hence, $|a_n|+|b_n| \leq 1-|a_0|^2$ for $n\geq 1$.
It means that
$$|a_0|+\sum_{k=1}^\infty (|a_k|+|b_k|)r^k \leq |a_0|+(1-|a_0|^2)\frac{r}{1-r} \leq 1 ~\mbox{ for }~ r \leq 1/3,
$$
and thus, \eqref{KP3-eq8} follows.

A natural question is to whether the inequality \eqref{KP3-eq8} is true if we replace $|a_0|$ in \eqref{KP3-eq8} by $|a_0|+|b_0|$.
It turns out that this proposition is false without an additional information on $g$ or $h$. To prove this, we let $a>0$ and consider the functions
$$h(z)=\frac{z+a}{2\sqrt{1+a^2}}, \quad g(z)=\frac{z-a}{2\sqrt{1+a^2}}
$$
so that $h(0)=a/(2\sqrt{1+a^2})$ and $h(0)=-g(0)$. Also, we have
\beqq
(|h(z)|+|g(z)|)^2&=&|h(z)|^2+|g(z)|^2+2|h(z)||g(z)|\\
&=&\frac{2(|z|^2+a^2)+2|z^2-a^2|}{4(1+a^2)} \\
&\leq & \frac{2(|z|^2+a^2)+2(|z|^2+a^2)}{4(1+a^2)} < 1
\eeqq
for $z \in \ID$. On the other hand, for this function, we obtain that
$$\sum_{k=0}^\infty (|a_k|+|b_k|)r^k = \frac{r+a}{\sqrt{1+a^2}} > 1 \mbox{ for } r>r_0(a)=\sqrt{1+a^2}-a.
$$
But,
$$r_0(a)=\frac{1}{\sqrt{1+a^2}+a} \to 0 \mbox{ as } a \to +\infty.
$$
This example shows that Proposition \ref{prop1} has a precise form and that the condition $g(0)=0$ cannot be ignored.
Otherwise, there will be no positive Bohr radius. Thus, $|a_0|$ in \eqref{KP3-eq8} cannot be replaced by $|a_0|+|b_0|$.
\epf

However, there is a generalization of the classical Bohr theorem in which the initial condition $g(0)=0$ in Proposition \ref{prop1}
can be omitted.

\bprop\label{prop2}
Assume that $h(z)=\sum_{k=0}^\infty a_k z^k$ and $g(z)=\sum_{k=0}^\infty b_k z^k$ are  two analytic functions in the disk $\ID$
satisfying the condition $|h(z)|+|g(z)| \leq 1$ in $\ID$. Then, we have
\be\label{KP3-eq6}
\sqrt{|a_0|^2+|b_0|^2}+ \sum_{k=1}^\infty \left (\sqrt{|a_k|^2+|b_k|^2}\right )r^k \leq 1 \mbox{ for } r \leq 1/3.
\ee
\eprop\bpf
We again consider the function $\Phi(z)=h(z)+e^{i\theta}g(z)$ so that $|\Phi(z)| \leq 1$ for $z\in\ID$ and consequently,
the classical Wiener inequality gives
$$|a_n+e^{i\theta}b_n| \leq 1-|a_0+e^{i\theta}b_0|^2  ~\mbox{ for $n\ge 1$}.
$$
This inequality together with the inequality $|a_n-e^{i\theta}b_n| \leq 1-|a_0-e^{i\theta}b_0|^2$ gives
$$|a_n+e^{i\theta}b_n| +|a_n-e^{i\theta}b_n| \leq 2-|a_0+e^{i\theta}b_0|^2 -|a_0-e^{i\theta}b_0|^2=2(1-|a_0|^2-|b_0|^2).
$$
From here, we derive the following inequality
$$(|a_n+e^{i\theta}b_n| +|a_n-e^{i\theta}b_n|)^2 \leq 4(1-|a_0|^2-|b_0|^2)^2
$$
which is equivalent to the inequality
$$|a_n|^2+|b_n|^2+|a_n^2-e^{i 2\theta}b_n^2| \leq 2(1-|a_0|^2-|b_0|^2)^2.
$$
Now, we choose $\theta=\pi/2+\arg a_n-\arg b_n$ and obtain
$$|a_n|^2+|b_n|^2 \leq (1-|a_0|^2-|b_0|^2)^2
$$
which gives  $\sqrt{|a_n|^2+|b_n|^2} \leq 1-|a_0|^2-|b_0|^2$ for $n\geq 1$. Therefore,
\beqq
\sum_{k=0}^\infty \left (\sqrt{|a_k|^2+|b_k|^2}\right )r^k &\leq &  \sqrt{|a_0|^2+|b_0|^2} + (1-|a_0|^2-|b_0|^2)\frac{r}{1-r}\\
& \leq & \sqrt{|a_0|^2+|b_0|^2} + (1-\sqrt{|a_0|^2+|b_0|^2})\frac{2r}{1-r} \\
&\leq 1& ~\mbox{ for $r \leq 1/3$}
\eeqq
and the proof is completed.
\epf

We observe that if the constant term $\sqrt{|a_0|^2+|b_0|^2}$ in \eqref{KP3-eq6} is replaced by $|a_0|^2+|b_0|^2$, then the number $1/3$ in
\eqref{KP3-eq6} can be replaced by $1/2$.

\br
{\rm
If $f=h+\overline{g}$ is a harmonic mapping of $\mathbb{D}$ such that  $|f(z)|<1$  and $h(0)=0=f(0)$, then the
subordination relation \eqref{KP3-eq2} gives the inequality
\be\label{KP3-eq3}
\sum_{k=1}^\infty |a_k -e^{-2i\theta}b_k|^2r^{2k} \leq \frac{16}{\pi ^2}\sum_{k=1}^\infty \frac{1}{(2k-1)^2}r^{2(2k-1)}
\ee
and the arbitrariness of $\theta$ in $[0, 2\pi )$ shows that
$$\sum_{k=1}^\infty (|a_k|^2 +|b_k|^2)r^{2k} \leq \frac{16}{\pi ^2}\sum_{k=1}^\infty \frac{1}{(2k-1)^2}r^{2(2k-1)}.
$$
From \eqref{KP3-eq3}, we also have
$$\sum_{k=1}^\infty |a_k -e^{-2i\theta}b_k|r^{k} \leq
\frac{4}{\pi }\sqrt{\sum_{k=1}^\infty \frac{1}{(2k-1)^2}(r\rho)^{2(2k-1)}} \left ( \frac{1}{\sqrt{\rho^{2}-1}}\right ),
$$
where $\rho >1$ such that $\rho r \leq 1$. The choice $\rho=1/r$ gives
\beqq
\sum_{k=1}^\infty |a_k -e^{-2i\theta}b_k|r^{k}  &\leq &
\frac{4}{\pi }\sqrt{\sum_{k=1}^\infty \frac{1}{(2k-1)^2}}  \frac{r}{\sqrt{1-r^2}}=\frac{4}{\pi }\sqrt{\frac{\pi^2}{8}}  \frac{r}{\sqrt{1-r^2}}=\frac{\sqrt{2}r}{\sqrt{1-r^2}}.
\eeqq
}\er

However, one can also obtain general results. To do this, let us recall some facts.
In the paper \cite{Abu} the Bohr radius for harmonic functions $f=h+\overline{g}$ was defined as the maximal $r$ for which
$$\sum_{k=1}^\infty {(|a_k| +|b_k|)}r^{k} \leq 1.
$$
However, one can give a more flexible definition of the Bohr radius in the case of harmonic functions.
For  $p \ge 1$, we say that $r_p$ is the $p$-Bohr radius for the harmonic function $f=h+\overline{g}$ of the unit disk $\ID$, where $g(0)=0$,
if $r_p$ is the largest value such that
$$\sum_{k=0}^\infty(|a_k|^p +|b_k|^p)^{1/p}r^{k} \leq 1  ~\mbox{ for $|z|=r \leq r_p$}.
$$
At first we remark that all these radii coincide in the analytic case. Moreover, essentially, they are equivalent in view of the estimates
$$\max(|a_k|,|b_k|) \leq (|a_k|^p +|b_k|^p)^{1/p}  \leq |a_k|+|b_k| \leq 2 \max(|a_k|,|b_k|).
$$
Note that the case $\max(|a_k|,|b_k|)$ corresponds to $p=\infty$. This approach can be used to prove the following general result.

\begin{thm}\label{KP3-th3}
Let $f=h+\overline{g}$ be a harmonic mapping in $\ID$ such that $|f(z)| \leq 1$ in $\ID$, where
$h(z)=\sum_{k=0}^{\infty}a_{k}z^{k}$ and $g(z)=\sum_{k=1}^{\infty}b_{k}z^{k}$.
Then for any $p \ge 1$ and $r<1$, the following inequality holds:
\beqq
\sum_{k=1}^\infty(|a_k|^p +|b_k|^p)^{1/p}r^{k}  &\leq &
 \max \{2^{(1/p)-1/2},1\}\sqrt{1-|a_0|^2}\frac{r}{\sqrt{1-r^2}}.
\eeqq
\end{thm}
\bpf
We have $ 1 \ge |h+\overline{g}|^2=(h+\overline{g})(\overline{h}+g)=|h|^2+|g|^2+2 {\rm Re}\, (hg)$
and thus,
\beqq
1 &\geq & \frac{1}{2\pi}\int_0^{2\pi}  |f(re^{i\theta})|^2\, d\theta
=  \frac{1}{2\pi} \int_0^{2\pi} ( |h(re^{i\theta})|^2+|g(re^{i\theta})|^2)\, d\theta\\
 &=& |a_0|^2+\sum_{k=1}^\infty (|a_k|^2 +|b_k|^2)r^{2k}
\eeqq
so that letting $r$ approach $1$, we get
$$|a_0|^2+\sum_{k=1}^\infty (|a_k|^2 +|b_k|^2) \leq 1.
$$
As a result of it, we obtain
\beqq
\sum_{k=1}^\infty (|a_k|^p +|b_k|^p)^{1/p}r^{k} &\leq &\sqrt{\sum_{k=1}^\infty(|a_k|^p +|b_k|^p)^{2/p}}{\sqrt{\sum_{k=1}^\infty r^{2k}}}\\
&\leq& \sqrt{\max\{ 2^{(2/p)-1}, 1\}\sum_{k=1}^\infty(|a_k|^2 +|b_k|^2)}\, \frac{r}{\sqrt{1-r^2}}\\
& \leq & \max\{2^{(1/p)-1/2},1\}\sqrt{1-|a_0|^2}\frac{r}{\sqrt{1-r^2}}
\eeqq
and the proof is complete.
\epf

In particular, Theorem \ref{KP3-th3} for $p=1$ shows that for $r\leq 1/3$,
$$\sum_{k=1}^\infty (|a_k| +|b_k|) r^k \leq \frac{\sqrt{1-|a_0|^2}}{2}\leq \frac{1}{2}.
$$
It is worth pointing out that Abu Muhanna \cite[Theorem 1]{Abu} obtained a similar inequality with $2/\pi \approx 0.63662$
instead of $(\sqrt{1-|a_0|^2})/2$. This observation shows that the above inequality is a vast improvement over the result of \cite{Abu}.

Note that $\max \{2^{(1/p)-1/2},1\}$ equals $1$ for $p\geq 2$, and equals $2^{(1/p)-1/2}$ for $p\in [1,2]$. This observation for
$a_0=0$ in Theorem \ref{KP3-th3} gives the following.

\begin{cor}\label{KP3-cor3}
Suppose that $f=h+\overline{g}$ is a harmonic mapping of $\ID$ such that $|f(z)| \leq 1$ in $\ID$, where
$h(z)=\sum_{k=1}^{\infty}a_{k}z^{k}$ and $g(z)=\sum_{k=1}^{\infty}b_{k}z^{k}$. If $p \ge 2$. Then
$$
\sum_{k=1}^\infty (|a_k|^p +|b_k|^p)^{1/p}r^{k} \leq 1 \mbox{ for } r \leq \frac{1}{\sqrt{2}}.
$$
The number $1/\sqrt{2}$ is sharp.
\end{cor}

\begin{cor}\label{KP3-cor4}
Suppose that $f=h+\overline{g}$ is  a harmonic mapping in $\ID$ such that $|f(z)| \leq 1$ in $\ID$, where
$h(z)=\sum_{k=0}^{\infty}a_{k}z^{k}$ and $g(z)=\sum_{k=1}^{\infty}b_{k}z^{k}$. If $p \in [1,2]$ and $|a_0|<1$, then
\be\label{KP3-eq7}
|a_0|+\sum_{k=1}^\infty \left (|a_k|^p +|b_k|^p\right )^{1/p}r^{k} \leq 1 \mbox{ for } r \leq r_p(|a_0|)
=\sqrt{\frac{1-|a_0|}{2^{(2/p)-1}+1+(2^{(2/p)-1}-1)|a_0|}}.
\ee
In the case $p \ge 2$ the following inequality holds:
\begin{equation} \label{Sharp}
|a_0|+\sum_{k=1}^\infty \left (|a_k|^p +|b_k|^p\right )^{1/p}r^{k} \leq 1 \mbox{ for } r \leq \sqrt{\frac{1-|a_0|}{2}}.
\end{equation}
\end{cor}
\bpf
Clearly, for $p \in [1,2]$ and $|a_0|<1$,  Theorem \ref{KP3-th3} gives that
$$|a_0|+\sum_{k=1}^\infty \left (|a_k|^p +|b_k|^p\right )^{1/p}r^{k} \leq |a_0|+
2^{(1/p)-1/2}\,\sqrt{1-|a_0|^2}\frac{r}{\sqrt{1-r^2}}
$$
which is less than or equal to $1$ if  $ r \leq r_p(|a_0|)$. This gives the condition \eqref{KP3-eq7}. The case  $p \ge 2$
follows similarly.
\epf

For the harmonic functions, the case $p=1$ of \eqref{KP3-eq7} gives
$$|a_0|+\sum_{k=1}^\infty (|a_k| +|b_k|)r^{k} \leq 1 \mbox{ for } r \leq r_1(|a_0|)
=\sqrt{\frac{1-|a_0|}{3 +|a_0|}}.
$$
Also, we remark that in the case $|a_0|=1$ in Corollary \ref{KP3-cor4} we have $a_k=b_k=0$ for all $k \ge 1$.

\br
{\rm
The example
$$f(z)=\frac{z+a_0}{1+\overline{a_0} z}=\sum_{k=0}^\infty a_k z^k
$$
shows that in the case $p \ge 2$ and $|a_0|=1/2$ the inequality (\ref{Sharp}) is sharp for the analytic case. This fact follows from the identity
$$\sum_{k=0}^\infty |a_k| r^k=|a_0|+\frac{r(1-|a_0|^2)}{1-r|a_0|}
$$
which implies that the Bohr radius for this function is $1/(1+2|a_0|).$
}
\er

We know that in the classical case the Bohr radius is $1/3$. In harmonic case, the $p$-Bohr radius can be also $1/3$
under an additional assumption on the constant coefficient $a_0$.

\begin{cor}\label{KP3-cor5}
Let $f=h+\overline{g}$ be a harmonic mapping in $\ID$ such that $|f(z)| \leq 1$ in $\ID$, where
$h(z)=\sum_{k=0}^{\infty}a_{k}z^{k}$ and $g(z)=\sum_{k=1}^{\infty}b_{k}z^{k}$. If $p \in [1, 2]$ and
\be\label{KP2-eq5}
|a_0| \leq A(p)=\frac{8-2^{(2/p)-1}}{8+2^{(2/p)-1}},
\ee
then
$$|a_0|+\sum_{k=1}^\infty (|a_k|^p +|b_k|^p)^{1/p}r^{k} \leq 1 \mbox{ for } r \leq 1/3.
$$
\end{cor}
\bpf Suffices to observe that $r_p(|a_0|)$ given in Corollary \ref{KP3-cor4} shows that
$r_p(|a_0|)\geq 1/3$ whenever $|a_0|$ satisfies the inequality  \eqref{KP2-eq5}. The desired conclusion follows.
\epf

\br
{\rm
Since $|a_0|\leq 1$, the question is whether the range for $a_0$ in Corollary \ref{KP3-cor5} could be enlarged.
Note also that $A(p)$ is an increasing function of $p$ and thus, $A(p)\in [3/5, 7/9]$ for $p \in [1, 2]$. We now show that
in the case of $p=1$, the value $A(1)$ cannot be greater than $0.67404$ so that the number $3/5=0.6$ is quite close to be sharp.
To see this, we will use an example from the paper \cite{Abu}:
$$ f(z)= \frac{2}{\pi}\mbox{\rm Im}\left(\log\frac{1+ z}{1-z}\right)\sin \mu + i\cos\mu
$$
which may be rewritten as
$$ f(z)= a_0+\sum_{k=1}^{\infty}a_{k}z^{k}  +\sum_{k=1}^{\infty}b_{k}\overline{z^{k}}
$$
where
$$a_0= i\cos\mu ~\mbox{ and }~ a_k=\frac{2\sin \mu}{\pi (2k-1)}=-b_k \mbox{ for $k\geq 1$}.
$$
Consequently,
$$|a_0|+\sum_{k=1}^\infty (|a_k|^p +|b_k|^p)^{1/p}r^{k}= |\cos\mu|+ \frac{2^{1/p}}{\pi}\log\left(\frac{1+ r}{1-r}\right)|\sin \mu| .
 $$
Thus, for the case $r=1/3$, the last expression is greater than or equal to $1$ whenever
$$ |a_0|+ \frac{2^{1/p}\,\log 2}{\pi} \sqrt{1-|a_0|^2}   \geq 1
$$
which gives the condition
$$|a_0|=|\cos\mu| \ge \frac{\pi^2-2^{2/p} \log^2 2}{\pi^2+2^{2/p} \log^2 2}.
$$
This observation means that
$$A(p) \leq \frac{\pi^2-2^{2/p} \log^2 2}{\pi^2+2^{2/p} \log^2 2}.
$$
For instance, $0.6 \leq A(1) \leq 0.67404$ and $\frac{7}{9} \leq A(2) \leq 0.82256$.
}
\er

Finally, as a harmonic analog of Theorem \Ref{KP2-odd1}, we have the following result in a general form and the proof of
it follows from the lines of proof of Theorem \ref{KP3-th3}. So we omit its detail.

\begin{thm}\label{KP3-th4}
Let $f(z)=\sum_{k=0}^{\infty}a_{2k+1}z^{2k+1} + \sum_{k=0}^{\infty}\overline{b_{2k+1}z^{2k+1}}$ be
an odd harmonic function in $\ID$ such that $|f(z)| <1$ in $\ID$.
Then for $p \in [1,2]$ we have the following estimate:
$$\sum_{k=0}^\infty (|a_{2k+1}|^p +|b_{2k+1}|^p)^{1/p}r^{2k+1}\leq  2^{(1/p)-1/2}\frac{r}{\sqrt{1-r^4}} \leq 1
~\mbox{ for }~ r \leq \rho_p
$$
where $\rho_p =\sqrt{\sqrt{4^{(2/p)-2}+1}-2^{(2/p)-2}}. $ In the case $p \ge 2$ the following inequality holds:
$$\sum_{k=0}^\infty (|a_{2k+1}|^p +|b_{2k+1}|^p)^{1/p}r^{2k+1}\leq  \frac{r}{\sqrt{1-r^4}} \leq 1 ~\mbox{ for }~
r \leq \rho_2=\sqrt{\frac{\sqrt{5}-1}{2}}.
$$
\end{thm}

Let us remark that the number $\rho_1=\sqrt{\sqrt{2}-1}=0.6435 \ldots $ cannot be replaced by a number greater than
$$r_0=\frac{e^{\pi/2}-1}{e^{\pi/2}+1}=\tanh \frac{\pi}{4}=0.65579\ldots
$$
Thus, it is natural to conjecture that the Bohr radius in the case of harmonic odd function is
$r_0\approx 0.65579\ldots .$

Also, let us remark that $\rho_2=\sqrt{(\sqrt{5}-1)/2}=0.7861\ldots $. This number cannot be more than  $0.789991$.

\subsection*{Acknowledgements}
The research of the first author was supported by Russian foundation for basic research, Proj. 17-01-00282, and the
research of the second author was supported by the project RUS/RFBR/P-163 under Department of Science \& Technology (India).
The second author is currently on leave from  IIT Madras.

redpajama_set_name: RedPajamaArXiv


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text:
Sculpted by Evolution
by David Schmitt, Ph.D., Peter Jaret
It's time for a grown-up conversation about sex. No, not that one. The other one, about the minds of men and women. Converging lines of empirical evidence—from developmental neuroscience, medical genetics, evolutionary biology, cross-cultural psychology, and new studies of transsexuality—along with our evolutionary heritage, all point to the same conclusion: There are psychological differences between men and women. And they affect matters as trivial as sensitivity to smelly socks and as significant as susceptibility to disorders such as depression and autism. In addition, women and men appear to experience pain differently. The male hormone testosterone may protect against pain, some research suggests. The female hormone estrogen, on the other hand, may intensify pain. And when they're in pain, men and women seem to react differently, too. Studies suggest that women have greater sensitivity to pain than men and are more likely to dwell on it. Women are more likely to be diagnosed with painful conditions such as rheumatoid arthritis, multiple sclerosis, irritable bowel syndrome, cystitis, migraines, and fibromyalgia. A large study that looked at the medical records of more than 11,000 patients found that women reported more pain than men across a wide range of medical conditions.
Then again, men may simply be less willing to admit that they're in pain—toughing it out because they think that's the manly thing to do. Yet when men do complain about being in pain, they may be taken more seriously than women. In an older (2001) review article in the Journal of Law, Medicine & Ethics, for example, researchers found that women were less likely to receive aggressive treatment when diagnosed with a painful condition. They were also more likely than men to have their pain dismissed as "emotional" or "all in their heads." And a 2008 study in Academic Emergency Medicine found that women who visited emergency rooms with abdominal pain were less likely than men to be given high-strength pain medications for abdominal pain.
The dramatic physical and behavioral differences between men and women, including strength and size, pubertal timing, consistent patterns around the world of hunting versus gathering and childrearing, as well as pervasive differences in risk-taking, mortality, and reproductive requirements, attest to the likelihood that evolution sculpted adaptations into men and women that make us somewhat different creatures. Psychologically, this sculpting by evolution has left men and women with particular approaches to life and love built upon a common core of human nature.
Ironically, just as the evidence is mounting that psychological sex differences are real, denial of differences has become rampant. Attempts at respectful and productive conversations about biological sex differences often end with name-calling (genetic determinist!) or outright cancellation of events—not to mention the very public firing of a Google software engineer for writing a memo on the topic.
One reason that conversations about sex differences run aground is the widespread lack of foundational knowledge about sex and gender. Then there's the huge array of influences, from both inside and out, that shape all our emotions, thoughts, and behaviors. Still, denying men's and women's different psychologies is not merely a denial of reality; it has serious health consequences for significant segments of the population.
Sex vs. Gender Differences
It's most logical to term the differences between men and women sex differences, not gender differences. After all, our species has biological sexes—typically defined by gamete size, genital morphology, the presence or absence of a Y chromosome, and normative sex hormone levels. For sure, there are atypical (and uncommon) variations in sex chromosomes and in pivotal hormonal experiences during sexual development that can make defining one's sex unclear. The International Olympic Committee has struggled for decades to define biological sex—and it's still struggling.
Whether you identify as a man or a woman is your sexual identity. When they study differences in the way self-identified men and women think (such as how they read a map), feel (the degree to which they experience empathy), and behave (say, their likelihood of committing homicide), psychologists are said to be investigating psychological sex differences.
Gender, or gender psychology, according to the American Psychological Association, reflects the attitudes, feelings, and behaviors that a given culture associates with biological sex. The term "gender identity" is often conflated with sexual identity, but "gender" refers to whether a person is typically masculine and/or feminine as defined by their local culture. (Sometimes this is called gender-role or sex-role orientation or gender expression. I know, it's confusing, which is why so many find it difficult to be clear when discussing sex differences.)
Some psychologists contend that we should call most differences between men and women gender differences, not sex differences—because they feel that such differences are culturally constructed and the term "sex" should be reserved for differences that are primarily biological in origin. But this is a dangerous game to play, as social psychologist Alice Eagly has pointed out: It presupposes the ultimate source of observable differences between men and women.
There are other variations in sex-related identities that complicate discussions. For instance, people's sexual orientation can come in many forms across varieties of sex and gender, including androphilia (finding male bodies erotic), gynephilia (finding female bodies erotic), bisexuality, asexuality, and more. Sex and gender and orientations come in many varieties. But to compare the psychologies of self-identified men and women is to discuss psychological sex differences—whatever their origins.
It is a common, but erroneous, notion that men and women differ because of the simple presence (or absence) of a Y chromosome and its accompanying effects on testosterone levels. The fact that there are many degrees of difference between men and women contradicts that belief. But what makes any sex difference worth talking about in the first place is the size of the difference.
When analyzing data from large numbers of previous studies—a procedure called meta-analysis—psychologists have found several sex differences in the realm of emotions. They have shown that women are somewhat more empathic than men, whereas men tend to more powerfully experience the emotion of sexual jealousy. In the cognitive realm, men tend to be better able to rotate a dimensional object in their mind and to recognize, say, an upside-down character, whereas women excel at locating an object in a visual field and remembering exactly where Big Ben is on a map of London. Behaviorally, men are more physically aggressive and homicidal with same-sex others than women are, and women tend to choose older and wealthier partners for marriage compared with men. What kind of difference, how big a difference, and how pervasive a difference must there be to support a claim that men are psychologically different from women?
Take one of the most obvious and noncontroversial differences between men and women. The typical man is much taller than the typical woman. But the sex difference in height (or any other sex difference) is not a statement about individual men and women; it pertains only to group averages.
Therein resides a key problem in discussing sex differences: Research almost always approaches sex differences as group averages, but most people think about individual men and women. When researchers talk about the average man being taller than the average woman, they are not implying that every member of the group "men" is taller than all members of the group "women." They are talking about height differences as distributed across many individuals from both groups.
If you were to line up 100 women and 100 men and peruse the heights of the two groups, you would easily notice differences in their group distributions of heights. In the United States, the average adult woman is only about as tall as the average 14-year-old boy. Too often, when scientists claim that there exists a sex difference in height, it gets translated by others as meaning that the feature of height is biologically essential to being a man (taller) or a woman (shorter). The tendency to misconstrue as a necessary difference what shows up only as a group difference is a huge barrier to respectful and productive conversations about sex differences.
One way to clarify discussions about differences in group averages is to put a specific number to them. Psychologists often use a precise number to express the size of sex differences, referred to as an "effect size," with the most common usage being the dstatistic. A positive d value typically indicates that men are higher on a particular attribute; a negative value indicates that women are higher. The size of the d value establishes exactly how big the average sex difference is.
A d value near zero means that the sex difference is trivial. Once a d value reaches +/- 0.20, psychologists take notice. A d value of -0.20, for instance, indicates that 58 percent of women are higher than the average man on a psychological trait. These are considered "small" effect sizes. Sex differences in interpersonal trust, conformity, and general verbal ability reside in this range.
A d value of +0.50 is considered "moderate" and indicates that 69 percent of men are higher than the average woman on a particular attribute. Sex differences in spatial rotation skills, certain mathematics abilities (3-dimensional geometry and calculus), and task-oriented leadership (focusing on accomplishing a group goal rather than maintaining harmony within the group) reside within this size range.
A d value of -0.80 is considered "large" and indicates that 79 percent of women are higher than the average man. Sex differences in tender-mindedness, being interested more in people than in things, and lack of interest in casual sex reside in this size range.
Larger d values are less common in psychology, but a value of +1.00 indicates that 84 percent of men are higher than the average woman. Sex differences of this magnitude include differences in height, in expressing interest in engineering as an occupation, and in absence of sexual disgust (such as not feeling grossed out when hearing the neighbors having sex).
A d value of +2.00 indicates that 98 percent of men are higher than the average woman in a trait, about as close as researchers can get to finding a truly dimorphic difference. Sex differences in throwing ability, grip strength, and voice pitch are in this range.
No matter how big or small a sex difference, there is almost always significant overlap across distributions of men and women. Some women are able to throw farther than some men. Psychological sex differences are about group distributions, not dichotomous binaries of all men versus all women.
Biology Matters
Neurologically, all humans start as female. During prenatal development, the Y chromosome of human males initiates a series of masculinizing events of both body (during the first two months) and brain (after the first trimester). One of the most critical periods in the second trimester of gestation occurs when male brains, but typically not female brains, are permanently altered by exposure to androgens.
According to the organizational hypothesis of sexual differentiation, the prenatal exposure to androgens masculinizes the brain in ways that influence psychological sex differences. There is substantial evidence that these effects are real. For instance, prenatal androgen exposure within normal levels predicts sex differences in postnatal play preferences (rough-and-tumble play), personality traits (thrill-seeking, aggression), and cognitive abilities (mental rotation ability).
Evidence supporting the organizational effects of androgen also arises from studies of male and female children who, for various reasons, have atypical hormonal profiles. Studies of girls prenatally exposed to male-typical levels of androgens during the second trimester of gestation display more male-typical psychology (in play preferences, personality traits, and cognitive abilities) compared with their unaffected sisters. Studies of infants as young as a few months consistently find that some psychological sex differences emerge before extensive gender socialization has taken place. Exhibit A is spatial rotation ability, as seen in preverbal, five-month-old infants rotating and recognizing mirror-image objects.
Studies of adults experiencing hormone treatments, gender dysphoria, and transsexualism similarly indicate that to some degree, biology contributes to psychological sex differences between men and women. For instance, several studies have found that male-to-female transsexuals show signs of feminine psychological and physical traits before they transition. After undergoing hormone treatments to reduce their testosterone, more pronounced feminine thoughts, feelings, and behaviors are evident.
Development Matters
Biological predisposition to develop a masculine or feminine psychology in no way implies that men's and women's psychologies form a simple binary; the overlap in distribution of men's and women's traits refutes such dichotomous thinking. Nor does the existence of biological predispositions mean that sex differences are fixed and unchangeable after birth—a kind of genetically deterministic thinking. Nearly all biological mechanisms that have arisen over the course of history have been designed to be responsive to key elements in the environment, developmentally sensitive to features of family, social structures, and local ecologies. And the input from the environment almost always affects the degree of sexual differentiation.
Many psychological sex differences emerge long after prenatal experiences. Sex differences appear during puberty or other critical periods when genes become sensitive to activation by major maturational events, such as sexual debut, parenting, and menopause. Sex differences in the personality trait of neuroticism, including sensitivity to negative emotions and vulnerability to stress, do not reach their full adult form until around age 14, for example, suggesting that pubertal factors influence their development. In other words, some psychological sex differences are specially designed by evolution to arise developmentally and only after particular milestones.
Besides the early organizational effects and the later activational ones, some psychological sex differences result from direct effects of genes—apart from the sex chromosomes and their related hormones—functioning differently in men's and women's brains. For example, psychologist Janet Hyde and her colleagues have found that a serotonin-transporter gene, 5-HTTLPR, which exists in short and long versions, the short version being associated with higher negative emotionality, is more closely linked to the emergence of neuroticism-related traits in women than in men. The researchers identify gene variants affecting other neurotransmitters that are similarly disproportionately expressed in women. And they cite several mechanisms—such as X-linkage, or the involvement of genes found on the X chromosome, and estrogen-induced gene expression—that may play key causal roles in the emergence of one of the most notable psychological sex differences: the higher prevalence of depression among women than men.
Even if they originate from direct genetic effects, psychological sex differences do not take precisely the same form or manifest to the same degree across all cultures. Human psychology is highly sensitive to developmental and socioecological contexts. Such environmental factors as the threat of malaria and other infections, the number of men relative to the number of women in a local population, the degree to which men and women must compete in order to find a mate and reproduce—all magnify or minimize sex differences.
For instance, across all cultures men more than women tend to favor physical attractiveness in potential marriage partners, especially such cues to youth and fertility as a small waist and curvy hips. However, when pathogens are prevalent and diseases such as malaria are a constant danger (say, in India and Brazil as opposed to New Zealand and Germany), both men and women tend to emphasize physical attractiveness in a potential mate—possibly because attractiveness is generally a reliable indicator of good health. In these environments, the amplification of desire for physical attractiveness is more pronounced among women, leading to smaller sex differences in mate preference for attractiveness in societies with higher pathogen burdens.
Culture Matters (But Not How You Think)
Fact: As a percentage of enrollment, there are more female science majors in Burma, Oman, and Morocco than in the countries of Scandinavia.
Fact: American women are 15 percent less likely to reach a managerial position in the workplace than are men—but in Sweden women are 48 percent less likely, in Norway 52 percent, in Finland 56 percent, and in Denmark 63 percent.
Whatever the differences in men's and women's psyches—empathy, jealousy, cognitive abilities, mate preferences—many theories in psychology assume that they result primarily from direct gender socialization by parents, media, and societal institutions. As a result, it is often expected that sex differences will be smaller in cultures with higher levels of gender-related egalitarianism, as in Scandinavia, where socialization and roles are more balanced between men and women and sociopolitical gender equity prevails.
Surprisingly, several large cross-cultural studies have found this is not at all the case. Whether scientists measure Big Five personality traits, such as neuroticism; Dark Triad traits, such as psychopathy; or self-esteem, subjective well-being, or depression, empirical evidence shows that most sex differences are conspicuously larger in cultures with more egalitarian gender roles—as in Scandinavia.
The same holds true for cognitive attributes, including mental rotation and location ability, objectively measured on tests, as well as for physical traits such as height and blood pressure (both greater in men). And among such differences as preferring physically attractive mates, some of the largest psychological variances of all occur among the most progressive people: Scandinavians. The phenomenon is called the gender equality paradox.
Culture matters in explaining psychological sex differences, but not in the way most people think. It's not harsher gender socialization by parents and media, stringent societal gender roles, or institutional sociopolitical forces that widen the differences between men and women in the most progressive nations in the world. When you treat everyone the same, as in the Nordic countries, it's only genetic predispositions that produce the most observable individual differences. Extremes of sexual freedom beget larger psychological sex differences. Or as explained by Israeli psychologists Shalom Schwartz and Tammy Rubel-Lifshitz, it may be that having fewer gendered restrictions in a culture allows "both sexes to pursue more freely the values they inherently care about more."
Dials, Not Switches
If your sexual identity is "I am a man," it is likely that you also have a deeper voice and a stronger sex drive than most women. But not all women. Evolution in sexually reproducing species allows for a lot of variation along sex and gender dimensions. Think about sex differences for myriad traits as dozens of interconnected sex/gender dials with endpoints ranging from extreme male-typicality/masculine to extreme female-typicality/feminine. Our psychological sex difference dials do not all need to be turned up to 11 for men and women to be significantly different from one another and for evolution to have played a role in producing human sexual diversity.
Viewing sex differences as dimensional sex/gender dials makes it clear that there is not one simple sex adaptation that gives rise to male and female psychologies. Rather, there are likely dozens, if not hundreds, of evolved functional mechanisms generating physical sex differences in height, strength, voice, and hirsuteness and psychological differences in personality, play preferences, mate selection, erotic desires, personal values, and cognitive abilities. Each adaptation turns the respective sex/gender dials of men and women in oblique, context-sensitive ways; each contributes a small part in generating the panoply of sex differences seen in our species all around the world.
It is ironic that just when science is rapidly improving its fundamental understanding of sex differences and documenting the sometimes subtle ways that biology and culture interact, the progress has come under assault. The uproar over the Google memo was just one example. Another manifestation is the recent decision by England's Royal Society to name as science book of the year Testosterone Rex: Unmaking the Myths of Our Gendered Minds by Cordelia Fine. Healthy backlash points out the dangers of wholly biologizing our sexual selves, but Fine repudiates psychological sex differences altogether. Perhaps significantly, the judges were primarily media members, not scientists.
Despite Fine's increasingly vocal perspective, substantial evidence attests to the existence of many psychological sex differences. But even a difference clearly stemming from prenatal hormone exposure—say, preference for rough-and-tumble play—does not imply genetic determinism; the feature is still modifiable by future developmental experiences.
And sometimes we will want to do all that we can to modify sex differences. It is critical to acknowledge that biological sex differences are not necessarily morally good or justifiable. There are sex differences whose development society needs to actively redress, such as the greater risk of severe autism in males and depression in females. There is only one way to develop the tools necessary for subduing such undesirable developments—understanding their biological provenance. And that starts with recognizing their existence.
Divergent Thinking
Some traits are more prevalent in women, some in men; listed by magnitude of sex difference.
General verbal ability
Indirect aggression (gossip)
Interpersonal trust
Sensitivity to negative emotions
Spatial location ability
Tendency to smile
MEDIUM/LARGE:
Preference for status in mate
Tender-mindedness
Cooking, among foragers
Early onset of puberty
Interest in people over things
Preference for female-typical toys
Preference for taller mate
Primary caretaker of children, among foragers
Sexual disgust
Vulnerability to depression
SMALL/MEDIUM:
Sexual jealousy
3-D geometry ability
Task-oriented leadership
Mental rotation ability
Physical aggression
Preference for physically attractive mate
Deep voice pitch
Early mortality
Likelihood of homicide
Preference for rough-and-tumble play
Throwing ability
Upper-body strength
Vulnerability to psychopathy
10 Ways Depression is Different in Men and Women, by Jody Smith
The Crucial Difference Between Men and Women in Relationships, by Dr. Seth Meyers
Men and Women's Differences Not Distinct, by Emma Gray
← Fetishization of Young Girls
Philosophy of Daddy Doms →

redpajama_set_name: RedPajamaCommonCrawl


text:
Tag: Weddings
c. Aaron Almendral
Plan Your Prospect Park Wedding—Virtually!
The pandemic has thrown a wrench into the wedding season, with nuptials postponed while we wait for the all-good to gather in person. While the parties are pending, many couples are moving ahead in planning their special day. In Prospect Park, we are helping couples plan their weddings—virtually!
Contact us to set up your virtual Picnic House
Prospect Park holds one of Brooklyn's favorite wedding venues, the Prospect Park Picnic House. We spoke to Cat Feliciano, Prospect Park Alliance's Rentals & Event Planning Coordinator for the Prospect Park Picnic House, about the current state of weddings in the park and her virtual tours.
Hi Cat! What is the new process for setting up a virtual tour of the Picnic House?
Couples can get in touch with us through our website contact form, and then we will contact you to set up a virtual tour of the Prospect Park Picnic House, usually on Thursdays between 11 am – 2 pm.
Great, once you set up a tour, what happens next?
We will send you a link to a video meeting, and we do a walk through of the space using the 360 degree tour we have of the facility. I start at out front of the building, just like I would if we were in person, and walk you through the whole space, pointing out amenities and the outdoor areas that are part of the rental. During the tour, I take questions about everything you could want to know about having a wedding at the Prospect Park Picnic House. After the tour, I follow up with photos of previous weddings and celebrations, and give you lots of information about our caterers and preferred vendors.
What questions always come up on these tours?
Everyone is asking, "when are you going to be open?" and of course we don't know, but we're optimistic that it will be soon since we're already in Phase 2 of the reopening. More intimate gatherings may come back first before we are able to be at full capacity. People like to know about the amenities at the Picnic House, about the House Manager that comes with the space, and about our vendors and caterers.
When are people booking their weddings for?
Right now we're seeing a lot of people booking dates in 2021 or 2022. We know that is far off, and if you want to hold a date, we give you a courtesy hold of five days so that you can reach out to vendors, speak with your family, and figure out if it works for you.
Do you anticipate holding in-person showings of the Prospect Park Picnic House soon?
Yes, the in-person showings will soon be starting back up. We're following all precautions to keep our staff and our patrons safe and we'll keep holding the virtual tours so that people don't have to worry about putting themselves at risk.
Learn more about having your wedding or special event at the Prospect Park Picnic House.
c. Brooke Fitts
More Reasons to Celebrate at the Picnic House
Prospect Park Alliance has expanded its catering options at the Picnic House, its wedding and special event venue situated on the Long Meadow in Prospect Park.
This bucolic destination provides a flexible and scenic setting for weddings and special events of all styles and budgets. Proceeds support Prospect Park Alliance, the non-profit that sustains the park, in the care of Brooklyn's Backyard, keeping the park green and vibrant for today and decades to come.
The following is the roster of Picnic House caterers. Learn more about having your wedding or special event at the Prospect Park Picnic House.
Bon Soir Caterers
Founded in 1977, Bon Soir offers our clients a wide range of custom menus and services. Reflecting the global and contemporary Brooklyn community which we serve, our cuisine is simple yet tasteful, affordable yet elegant, and of course locally sourced and sustainable.
Emma's Torch
Emma's Torch is a non-profit social enterprise that empowers refugees through culinary education. We collaborate with our students on techniques and ingredients to craft flavor-forward American cuisine with an international twist.
Fig & Pig
With over 15 years of catering and restaurant experience, Culinary Institute of America graduate chef Holly Sheppard provides creative, quality-driven catering devoted to showcasing seasonal ingredients and producing exceptional, food-focused events.
Brought to you by the team behind Rucola, Metta and June, our food is ingredient-driven with a slow food ethos and a vegetable-focused menu. As New York City's only trash-free and carbon-neutral caterer, our primary mission is to provide food that is delicious, sustainable and beautifully executed.
The Raging Skillet
For over 30 years, The Raging Skillet has catered weddings and special occasions with fusion cuisine reflecting Asian, Latin American, Middle Eastern, Southern, Southwestern and Cajun traditions, as well as Chef Rossi's mother's Hungarian Yiddish home cooking. Vegan, vegetarian, gluten free menus available.
Saffron by Miriam and 1803NYC
Saffron Catering is a full-service catering company that specializes in Middle Eastern and Israeli, and New Orleans influenced Cajun cuisine. We offer a range of options from hors d'oeuvres, small plates, full menu, and buffet style catering, for any palate, including dietary restrictions, allergies and kid's menus.
Steven Brown Caterers
Serving New York since 1991, we provide unmatched creative cuisine, artistic presentation, exceptional value, and professional service from first contact through execution of your event. Our menus range from traditional fare to the latest culinary trends, reflecting the diverse cultures of the city we live in.
Kosher Catering Options
Simply Divine
Simply Divine creates delicious, extraordinary memories for discerning clients who require a glatt kosher event along with a sophisticated secular experience. Known for more than 25 years for our outstanding tasting menus, seasoned style, exciting décor, and superb staff, we're still getting raves year after year.
Above & Beyond Catering
Above & Beyond Catering is an OU-certified, full-service, off-premise catering and event planning. With over 15 years of experience, we know how much planning and precision is needed to successfully host a seamless event. Whatever the celebration, Above & Beyond will exceed all expectations.
Bension Kohen Caterers
Since 1979, the family-owned, Glatt Kosher Bension Kohen Caterers has served the Brooklyn community with dedication and warmth. We strive to ensure that your event is beautiful, efficient and elegant.
About the Picnic House in Prospect Park
The Picnic House in Prospect Park is a charming and historic building in the heart of Brooklyn's premier park. With scenic views of the 90-acre Long Meadow, outdoor spaces for ceremonies and cocktails, floor-to-ceiling windows, hardwood floors and a working wood fireplace, it provides a bucolic setting for your perfect day. The Picnic House can be tailored for every taste and budget, and is a blank canvas to create the ideal wedding no matter what the season. The Picnic House comfortably seats 175 guests for dinner and dancing or 240 for a cocktail-style event. Proceeds benefit Prospect Park Alliance, the non-profit organization that sustains Prospect Park. For more information, visit prospectpark.org/weddings.
Prospect Park's Scandalous First Wedding
It probably wouldn't surprise you to know that Prospect Park is a popular destination for weddings. Couples who love the park and want a slice of nature in their Brooklyn nuptials choose the Prospect Park Picnic House and Boathouse as the place to tie the knot every year. But this was not always the case. The first Prospect Park wedding, which took place in the park's former Rose Garden in 1923, caused quite a stir.
Let's go back to the 1870s, the early days of Prospect Park when famed designers Olmsted and Vaux were first plotting out the features of the park. The area, known now as the Rose Garden, was first conceived as a children's playground, with a carousel powered by a real horse, seesaws and swings. Despite the attractions, the playground was not a popular destination. The area's geographic features made it too hot and exposed to be a playground. It was, however, an excellent climate for growing roses. And so the park's Rose Garden came to be.
In the 1890s, the landscape was transformed into a botanical destination. Three water basins—reconstructions of which are still visible at the site—filled with aquatic plants and fish were installed by the design firm of McKim, Mead and White. The landscape was planted with an assortment of exotic flowers and roses, and in 1901, the Brooklyn Daily Eagle exclaimed, "some of the roses are larger than teacups," adding, "there can be no finer sight in the domain of floriculture in the United States."
In 1923, local residents Elizabeth Hoyt Senarens and Owen Morton Gunderson applied for a permit to be married in Prospect Park. The Park Commissioner issued the first permit of its kind for a wedding to be held at 7:45 am, "so that there might be no interruption from a crowd of romping children or unsympathetic grownups." The wedding was considered a novelty and a scandal, and was widely covered in the press by publications such as Brooklyn Life, Brooklyn Standard Union, the Brooklyn Daily Eagle, and even The New York Times.
The bride, Elizabeth Senarens, explained to the Brooklyn Daily Eagle that, "she had always yearned for roses and even as a little girl had thought the Prospect Park rose gardens the most romantic place for a wedding in the world." But the media attention surrounding the nuptials took its toll when the couple's pastor decided at the last minute that he would not officiate the wedding. In an interview with the Times, the reverend was quoted saying, "I never did consent to perform the ceremony in the park," and that when he learned the details he, "refused at once."
The Rose Garden in a vintage postcard, c. Prospect Park Archives/Bob Levine Collection
The couple had difficulty finding another pastor to perform their ceremony, until finally Rev. Ernest J. Marvin of the Fenimore Street M.E. Church agreed on the condition of anonymity. The press did not honor this request, publishing his name as the minister who presided over this historic first wedding in Prospect Park.
The morning of the wedding came, according to the Standard Union, "blowing a chilly breeze… not at all conducive to romance." Police were on hand to contain the throngs of curious onlookers, and the wedding proceeded, attended by the family and friends of the wedding party, newspaper reporters and photographers, and a "few stragglers on their way to work," as per a report from the Times.
Today, the former Rose Garden is entering a new phase, as Prospect Park Alliance begins plans to reimagine this northeastern area of the park that has seen little use in past decades. In recent years, the Alliance has invited the community to help shape the future of this area through community visioning and feedback surveys. Thanks to the input of the local community, Alliance architects are now in the process of designing a space that will serve the entire community, and encourage many more memorable occasions for years to come.
This story comes to us from Turnstile Tours: learn about this and other amazing tales of park history on a tour of the Prospect Park presented by the Alliance in partnership with Turnstile Tours. Tours for families and park lovers of all ages explore the parks' little known treasures, architectural and cultural history.
Learn more and take a park tour this spring!


text:
Sydney Theatre Awards, Marika Aubrey, Matilda, Glenn HAzeldine
NEWCASTLE-raised actress Marika Aubrey has been nominated in the 2015 Sydney Theatre Awards for Best Actress in a Supporting Role in a Musical for her role in Matilda as Mrs Wormwood, the mother of the title character. She's up against three actresses, including another Matilda cast member, Elise McCann, who plays Miss Honey, a sympathetic teacher of the troubled Matilda. The Sydney Theatre Awards, with the nominees and winners chosen by 10 theatre critics, will be announced on January 15. Another former Newcastle actor, Glenn Hazeldine, was a member of the casts of Sydney Theatre Company's After Dinner and Love and Information, both of which are nominated for the Best Ensemble Award. Hazeldine appeared in a 2000 Newcastle production of After Dinner, winning a CONDA for his performance as Stephen, a man waiting with a mate for a band to start in a pub. In the STC production he played the mate, Gordon, desperate to talk to someone about his relationship breakdown. Matilda, which plays at Sydney's Lyric Theatre until February 28, already has a quartet of winners, the girls who alternate in the role of the title character. Molly Barwick, Sasha Rose, Georgia Taplin and Bella Thomas will receive a Special Achievement Award. The show's other nominees, both in the Actor in a Supporting Role in a Musical category, are Daniel Frederiksen as Mr Wormwood, and James Millar as headmistress Miss Trunchbull.
https://nnimgt-a.akamaihd.net/transform/v1/crop/frm/eAHvuQGs6RUXufjt9tidNR/d5201c15-f2e9-4604-bca1-eda7638e9f86.jpg/r0_1617_3456_3570_w1200_h678_fmax.jpg
December 23 2015 - 4:00PM
Newcastle thespians up for awards
KEN LONGWORTH
NEWCASTLE-raised actress Marika Aubrey has been nominated in the 2015 Sydney Theatre Awards for Best Actress in a Supporting Role in a Musical for her role in Matilda as Mrs Wormwood, the mother of the title character.
She's up against three actresses, including another Matilda cast member, Elise McCann, who plays Miss Honey, a sympathetic teacher of the troubled Matilda.
The Sydney Theatre Awards, with the nominees and winners chosen by 10 theatre critics, will be announced on January 15.
Another former Newcastle actor, Glenn Hazeldine, was a member of the casts of Sydney Theatre Company's After Dinner and Love and Information, both of which are nominated for the Best Ensemble Award.
Hazeldine appeared in a 2000 Newcastle production of After Dinner, winning a CONDA for his performance as Stephen, a man waiting with a mate for a band to start in a pub. In the STC production he played the mate, Gordon, desperate to talk to someone about his relationship breakdown.
Matilda, which plays at Sydney's Lyric Theatre until February 28, already has a quartet of winners, the girls who alternate in the role of the title character.
Molly Barwick, Sasha Rose, Georgia Taplin and Bella Thomas will receive a Special Achievement Award.
The show's other nominees, both in the Actor in a Supporting Role in a Musical category, are Daniel Frederiksen as Mr Wormwood, and James Millar as headmistress Miss Trunchbull.


text:
parts[at]trunkmonkeyparts.com
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Camaro Packages
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Home Corvette SUV in the Works?
Corvette SUV in the Works?
Does the world need a Corvette SUV? The article seems to state that there is some traction amongst people who make the Corvette or, at least, former executives think that it would be a good money maker. The case for a Corvette SUV is made in the article below. It also details some of the obstacles to making the Corvette SUV.
Photo from: http://gmauthority.com/blog/wp-content/uploads/2014/01/2014-Chevrolet-Corvette-C7-Stingray-logo.jpg
1. Former executives were brought together to talk about the possibity of the Corvette SUV.
2. They agreed that it would be a profitable venture for the car company if pursued.
3. A new architecture would have to be developed to make the idea viable.
Quote: "The Detroit News spoke with analysts and former General Motors executives to get an idea surrounding the potential success a Corvette SUV could hold. Most agreed it would be a smash hit." (Szymkowski, 2017)
Link to Article: http://gmauthority.com/blog/2017/03/the-case-for-a-corvette-suv-resurfaces/#ixzz4cdakcHiR
Suburban Chevrolet
Sandy, OR 97055


text:
Q: Remove single word attribute in JQuery I have a Bootstrap 4 button on my html webpage. w3schools (https://www.w3schools.com/bootstrap4/bootstrap_buttons.asp) says that including a button the following way will have it appear disabled, i.e. unclickable.
<button type="button" class="btn btn-primary" disabled>Disabled Primary</button>

I'd like to set up the button's form so that when it's clicked, it becomes disabled. Here is the relevant portion of what I have so far:
 $(document).on('submit', '.Validate', function(e){
        e.preventDefault();
        var form = $( this );
        form.attr('disabled')
        $.ajax({
            type:'POST',
            url:'/workflowautomator/setready/',
            data:{
                name: (form.find('input[class="name"]')).val()
            },
            success:function(){
                form.removeAttr('disabled')
            }
        });
 });

However, the usage I see for .attr is more like
$( "#greatphoto" ).attr( "title", "Photo by Kelly Clark" );

where there is a name for the attribute and a value, instead of up above, where the word "disabled" is alone in the button start tag. Is the "disabled" word not actually a proper attribute? w3schools describes it as such. How might I make this code work?

A: This is not the right one you need to use:
form.attr('disabled')

Replace it with:
form.find(".btn-primary").addClass("btn-disabled").prop("disabled", true);

You have to use .prop() to add and remove the property attributes. Check it out in the below snippet:

$(function () {
  $("button").click(function () {
    $(this).addClass("btn-disabled").prop("disabled", true);
  });
});
body {padding: 50px;}
<script src="https://ajax.googleapis.com/ajax/libs/jquery/2.1.1/jquery.min.js"></script>
<link href="https://stackpath.bootstrapcdn.com/bootstrap/4.1.3/css/bootstrap.min.css" rel="stylesheet"/>
<button class="btn btn-primary">Disable Me</button>

redpajama_set_name: RedPajamaStackExchange


Complete reception of all the files of your project through the export utility that includes the Xailer IDE.
Adaptation by our technical department of all your sources to ensure proper compilation and linking with Xailer for Harbour.
Return the modified project for further testing by the customer.
Assistance or resolution for any errors that may have been found in the previous phase.
Xailer guarantees the absolute confidentiality and proper disposition of all the sources received. As a result is not acceptable projects whose content could be competitive with Xailer or are intended to be used especially for xBase developers.
The minimum charge for this service is € 300 per project up to 1 Mega sources size. For each additional mega € 150 will be charged in half mega fractions.
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The customer agrees to perform all Phase IV controls within 30 days after delivery of the project by Xailer team.


Microsoft plans to improve Outlook Web App e-mail experiences by adding two new features to its commercial Office 365 service offerings.
The features, expected to arrive "later this year," include one that will help reduce e-mail clutter plus a collaboration enhancement for Outlook Web App, which is Microsoft's browser-enabled e-mail client. In addition, Microsoft indicated that it will release an Outlook Web App for Android-based smartphones, possibly this year. Last year, Microsoft released Outlook Web Apps for Apple's iPhone and iPad.
Microsoft also plans to integrate its new Groups capability, introduced as a SharePoint feature earlier this month, more broadly across its Office 365 services. Groups show public and private groups for employees to join, as well as individual conversations (called "cards") in a Yammer-like social networking fashion. The idea is to support a more collaborative business environment.
"Conversations that happen in public groups can lead to better information sharing, and free exchange of ideas connects people and makes the org more productive," wrote Steve Chew, senior product marketing manager in the Exchange group, in Microsoft's announcement.
It's an idea that may, or may not, be welcome in today's work environments, but it reflects Microsoft's embrace of the Yammer enterprise social networking solution that it purchased more than a year ago. Microsoft had indicated back then that it planned a broader integration of Yammer across Office 365, and it is now forecasting that the integration will start to take place sometime this year.
A forthcoming "Clutter" Outlook Web App feature will aim to unburden users from their less relevant e-mail. It uses Microsoft's new Office Graph technology to learn what users value or don't value in their in-boxes. The valued mail gets displayed in the Outlook Web App, while the rest gets archived at the bottom of the screen in a link called "Clutter," along with the number of archived e-mails. Users can use a right-mouse-button click to change a message's status if it was incorrectly marked as clutter. The Clutter name is currently considered to be a code name, so Microsoft may call it something else when it gets released.
When you send an attachment from your computer or device you can now automatically upload the file to your OneDrive cloud drive and send it as a link.
You can also easily attach a file directly from your OneDrive cloud drive when sending an email in Outlook Web App.
Permissions can be set on the files, allowing document editing or be viewing only. The document editing or viewing experience can be accessed side by side directly from within the Outlook Web App pane. Microsoft describes that capability as the integration of Office Online in Outlook Web App. Recipients will get an indication if they can edit the file.
These changes will apply to Microsoft's commercial Office 365 services. It's not clear if they will apply also to Office Web Apps Server or to the consumer Office Online experience.


This entry was posted in Institute of Special Education Advocacy ISEA and tagged ISEA, PELE Clinic, PELE Special Education Advocacy Clinic, W&M Institute of Special Education Advocacy by Wrightslaw. Bookmark the permalink.
This may sound like a chicken or the egg type of question.
I have seen data where there is a high number of individuals serving time who have a below average IQ. I have often wondered how our public school's pipeline is impacting this figure.
With "The Mathew Effect", the enviornment and resources can significantly effect an IQ.
Are the masses in prison due to a lower IQ or is their lower IQ attributed to the educational path their school officials elected to put them on, eliminating other options for their lives?
Parents need to be aware that the School to Prison Pipline is not just a notion that a paranoid parent dreamed up. It exists and it is utilized every day.
I am really concerned about the school to prison pipeline Mr. Tulman speaks about. He is so right about the districts not meeting academic needs, not complying with IDEA and an absolute disregard for transition plans. Our school district is so openly hostile and they seem to want to provoke the children, like they want them to become disciplinary problems. I wish I was at the seminar. It sounds very informative! Have a great day!
We are teaching advocates, attorneys, and law students how to understand tests and measurements, not to administer and interpret them. There is a big difference.
It is necessary for advocates, attorneys, and law students to understand basic psychometrics and what tests are designed to measure in order to be effective advocates for children with special needs.
I hope it is a wonderful fruitful week there, and I wish I could be with you!
I am not understanding how during a training session potential advocates are expected to learn how to understand evaluation "tests". I am speaking primarily of intellectual functioning instruments. Psychologists and Diagnosticians take several classes (and a masters degree is required) and many hours of continuing education to accomplish what you say you can teach in a training seminar? I am having difficulty determining the reliability of this.