text:
\section{Weighted Information Measures}
\mbox{\quad}
\def\mathfrak B}\def\fM{\mathfrak M}\def\fX{\mathfrak X{\mathfrak B}\def\fM{\mathfrak M}\def\fX{\mathfrak X}
 \def\mathcal B}\def\cM{\mathcal M}\def\cX{\mathcal X{\mathcal B}\def\cM{\mathcal M}\def\cX{\mathcal X}
\def\mathbf u}\def\bv{\mathbf v}\def\bx{\mathbf x}\def\by{\mathbf y{\mathbf u}\def\bv{\mathbf v}\def\bx{\mathbf x}\def\by{\mathbf y}
\def\omega} \def\Om{\Omega{\omega} \def\Om{\Omega}
\def\mathbb P} \def\hw{h^{\rm w}} \def\hwphi{{h^{\rm w}_\phi}} \def\bbR{\mathbb R{\mathbb P} \def\hw{h^{\rm w}} \def\hwphi{{h^{\rm w}_\phi}} \def\bbR{\mathbb R}
\def\begin{eqnarray}} \def\eeq{\end{eqnarray}{\begin{eqnarray}} \def\eeq{\end{eqnarray}}
\def\begin{eqnarray*}} \def\eeqq{\end{eqnarray*}{\begin{eqnarray*}} \def\eeqq{\end{eqnarray*}}
\def{\rm d}} \def\Dwphi{{D^{\rm w}_\phi}{{\rm d}} \def\Dwphi{{D^{\rm w}_\phi}}
\def\mathbf{X}{\mathbf{X}}
\def{D^{\rm w}_\phi}{{D^{\rm w}_\phi}}
\def{D^{\rm w}_\Phi}} \def\iw{i^{\rm w}_{\phi}{{D^{\rm w}_\Phi}} \def\iw{i^{\rm w}_{\phi}}
\def\mathbb{E}{\mathbb{E}}
\def{\mathbf 1}} \def\fB{{\mathfrak B}}  \def\fM{{\mathfrak M}{{\mathbf 1}} \def\mathfrak B}\def\fM{\mathfrak M}\def\fX{\mathfrak X{{\mathfrak B}}  \def\fM{{\mathfrak M}}
\def\displaystyle} \def\bbE{{\mathbb E}} \def\bbP{{\mathbb P}{\displaystyle} \def\bbE{{\mathbb E}} \def\mathbb P} \def\hw{h^{\rm w}} \def\hwphi{{h^{\rm w}_\phi}} \def\bbR{\mathbb R{{\mathbb P}}
\def\bar{F}{\bar{F}}
\def{\mathcal{E}^{\rm w}_{\phi}}{{\mathcal{E}^{\rm w}_{\phi}}}
\def{\overline{\mathcal{E}}^{\rm w}_{\phi}}{{\overline{\mathcal{E}}^{\rm w}_{\phi}}}
\def\lambda{\lambda}

{\bf Abstract.} We generalize the weighted cumulative entropies (WCRE and WCE), introduced in \cite{M}, for a system or component lifetime. Representing properties of cumulative entropies, several bounds and inequalities for the WCRE is proposed.
\vskip .5 truecm
\textbf{2000 MSC.} 62N05, 62B10\\

\textbf{Keywords:} weighted cumulative residual entropy, weighted cumulative entropy, Shannon entropy, statistical estimation, Weibull distribution, weight function.

\section{Introduction. The weighted cumulative entropies}

 An important measure of the uncertainty is entropy, commonly termed the Shannon information measure, \cite{Sh}. The current paper deals with {\it weighted} entropy; for the definition and initial results on weighted entropy the reader is referred to \cite{BG,G}.

Furthermore, the entropy of the residual lifetime $X_t=[X-t|X>t]$ as a dynamic measure of uncertainty was considered in \cite{E}. Recently, further progress as introducing cumulative entropy, cumulative residual entropy and  weighted cumulative entropy was made in \cite{RCVW,CL,M}.

The purpose of this work is to obtain a number of results for weighted cumulative entropies in the case where the weight is a general non-negative function.

Let $X$ be a non-negative absolutely continuous random variable describing a component failure time, with the probability density function (PDF), $f(x)$, the cumulative distribution function (CDF), $F(x)=P(X\leq x)$, and the survival function (SF), $\bar{F}(x)=P(X>x)$.
\begin{defn}
 Given a function $x\in \bbR\mapsto\phi (x )\geq 0$, and an RV
$X:\;\Om\to\bbR$, with a PDF $f$,
the {\bf weighted cumulative residual entropy} (WCRE) of $X$ (or $F$) with  weight function (WF) $\phi$ is defined by
\begin{eqnarray}} \def\eeq{\end{eqnarray}\label{eq:WCRE}
{\mathcal{E}^{\rm w}_{\phi}} (X)={\mathcal{E}^{\rm w}_{\phi}} (F) =-\int_{\bbR_+}\phi (x )\mathbb P} \def\hw{h^{\rm w}} \def\hwphi{{h^{\rm w}_\phi}} \def\bbR{\mathbb R(|X|>x)\log\,\mathbb P} \def\hw{h^{\rm w}} \def\hwphi{{h^{\rm w}_\phi}} \def\bbR{\mathbb R(|X|>x){\rm d}} \def\Dwphi{{D^{\rm w}_\phi} x. \eeq
Note that a standard agreement $0=0\cdot\log\,0=0\cdot\log\,\infty$ is adopted throughout the paper.\\
Given the CDF, $\bx\in\bbR_+ \mapsto F(\bx )\in [0,1]$, with WF $\phi$, the {\bf weighted cumulative entropy} (WCE) of non-negative random lifetime $X$ is defined by
\begin{eqnarray}} \def\eeq{\end{eqnarray}\label{eq:WCE}
{\overline{\mathcal{E}}^{\rm w}_{\phi}} (X)={\overline{\mathcal{E}}^{\rm w}_{\phi}} (F) =-\int_{\bbR_+}\phi (x )\mathbb P} \def\hw{h^{\rm w}} \def\hwphi{{h^{\rm w}_\phi}} \def\bbR{\mathbb R(|X|\leq x)\log\,\mathbb P} \def\hw{h^{\rm w}} \def\hwphi{{h^{\rm w}_\phi}} \def\bbR{\mathbb R(|X|\leq x){\rm d}} \def\Dwphi{{D^{\rm w}_\phi} x. \eeq
Particularly when $\phi(x)=x$ the WCRE and WCE in (\ref{eq:WCRE}) and (\ref{eq:WCE}) can be turned out as (8) and (9) in \cite{M}. In what follows, we intend to use the same abbreviation as in \cite{M} for the weighted cumulative residual and weighted cumulative entropies.
\end{defn}
\vskip .5 truecm
\begin{exam}
{\rm{(WCRE of the uniform distribution)}} Consider an RV $X$ with uniform distribution in the interval $a<b$, $f(x)=\displaystyle} \def\bbE{{\mathbb E}} \def\bbP{{\mathbb P} \frac{1}{(b-a)}$. Then the WCRE is the following:
\begin{eqnarray*}} \def\eeqq{\end{eqnarray*} \begin{array}{ccl}{\mathcal{E}^{\rm w}_{\phi}}(F)&=&-\displaystyle} \def\bbE{{\mathbb E}} \def\bbP{{\mathbb P}\int_a^b \phi(x) (1-\frac{x}{b-a})\log (1-\frac{x}{b-a}) {\rm d}} \def\Dwphi{{D^{\rm w}_\phi} x\\
&=& \displaystyle} \def\bbE{{\mathbb E}} \def\bbP{{\mathbb P} (b-a)\int_{\frac{b-2a}{b-a}}^{\frac{-a}{b-a}} \phi((b-a)(1-y))y \log y {\rm d}} \def\Dwphi{{D^{\rm w}_\phi} y.
\end{array}\eeqq
In particular, with $\phi(x)=x$, one obtains:
\begin{eqnarray*}} \def\eeqq{\end{eqnarray*} \begin{array}{l}{\mathcal{E}^{\rm w}_{\phi}}(F)=\displaystyle} \def\bbE{{\mathbb E}} \def\bbP{{\mathbb P}\frac{1}{36(a-b)}\bigg[24 a^3\log\left(\frac{2a-b}{a-b}\right)-15a^2b-a^3-5b^3+6\log\left(\frac{2a-b}{a-b}\right)b^3\\
\qquad\displaystyle} \def\bbE{{\mathbb E}} \def\bbP{{\mathbb P} -18 ab^2\log\left(\frac{2a-b}{a-b}\right)+21ab^2+6a^3\log\left(\frac{a}{a-b}\right)-18a^2b\log\left(\frac{a}{a-b}\right)\bigg].\end{array}\eeqq
\end{exam}
\vskip .5 truecm
\begin{exam}
{\rm{(WCRE of the Gaussian distribution)}} Let $g(x)$ be the Gaussian PDF with mean $\mu$ and variance $\sigma^2$. Therefore, the SF is obtained as $\bar{G}(x)=erfc(\frac{x-\mu}{\sigma})$. Here $erfc(x)$ is the complementary error function:
\begin{eqnarray*}} \def\eeqq{\end{eqnarray*} erfc(x)=\frac{1}{\sqrt{2\pi}}\int_x^\infty \exp(-\frac{t^2}{2}){\rm d}} \def\Dwphi{{D^{\rm w}_\phi} t.\eeqq
In accordance with (\ref{eq:WCRE}) we obtain:
\begin{eqnarray*}} \def\eeqq{\end{eqnarray*} {\mathcal{E}^{\rm w}_{\phi}}(G)=-\int_0^\infty\phi(x)erfc(\frac{x-\mu}{\sigma})\log erfc(\frac{x-\mu}{\sigma}) {\rm d}} \def\Dwphi{{D^{\rm w}_\phi} x.\eeqq
\end{exam}
\vskip .5 truecm

Given an RV $X$ with CDF $F(x)$ and SF $\bar{F}(x)$, set,
\begin{eqnarray*}} \def\eeqq{\end{eqnarray*} m^{\rm w}_F(t)=\frac{1}{\bar{F}(t)}\int_t^\infty \phi(x)\bar{F}(x){\rm d}} \def\Dwphi{{D^{\rm w}_\phi} x\quad \textrm{and} \quad \overline{m}^{\rm w}_F(t)=\frac{1}{F(t)}\int_0^t \phi(x) F(x) {\rm d}} \def\Dwphi{{D^{\rm w}_\phi} x.\eeqq
Pictorially $m^{\rm w}_F(t)$ represents the weighted mean inactivity time (WMIT) and then $\overline{m}^{\rm w}_F(t)$ the weighted mean residual time (WMRT).
\begin{lem}
{\rm{(Cf. Proposition 2.1 from \cite{M}.)}} Let $X$ be an absolutely continuous RV. Then
\begin{eqnarray}} \def\eeq{\end{eqnarray} {\mathcal{E}^{\rm w}_{\phi}}(F)=\bbE\big(m^{\rm w}_F(X)\big),\eeq
and
\begin{eqnarray}} \def\eeq{\end{eqnarray} {\overline{\mathcal{E}}^{\rm w}_{\phi}}(F)=\bbE\big(\overline{m}^{\rm w}_F(X)\big).\eeq
\end{lem}

{\bf Proof.}\;The proof follows directly with the same methodology in \cite{M} but replacing $\phi(x)$ in $x$.\\
\def\bar{G}{\bar{G}}
\begin{defn}
Given two functions $x\in \bbR^+\mapsto \bar{F}(x )\in[0,1]$ and $x\in \bbR^+\mapsto \bar{G}(x )\in[0,1]$, the {\bf relative} WCRE of $\bar{G}$ relative to $\bar{F}$ for given WF $\phi$ is defined by
\begin{eqnarray}} \def\eeq{\end{eqnarray} \label{eq:4.01} D^{\rm w}_{\phi}(\bar{F}\|\bar{G})=\displaystyle} \def\bbE{{\mathbb E}} \def\bbP{{\mathbb P}\int\limits_{\bbR_+} \phi(x) \bar{F}(x)\log\;\frac{\bar{F}(x)}{\bar{G}(x)} {\rm d}} \def\Dwphi{{D^{\rm w}_\phi} x. \eeq
Alternatively, $D^{\rm w}_{\phi}(\bar{F}\|\bar{G})$ can be termed as weighted Kullback Leibler divergence between SFs $\bar{F}$, $\bar{G}$ with WF $\phi$.
\end{defn}
Paying homage to the Theorem 1.1 in \cite{SY}, with similar methodology the following assertion holds true. We omit the proof.
\begin{thm}
Given SFs $\bar{F}$ and $\bar{G}$ in $[0,1]$, assume that a WF $x\in \bbR_+\mapsto \phi(x )\geq 0$ obeys
\begin{eqnarray*}} \def\eeqq{\end{eqnarray*} \int\limits_{\bbR_+} \phi(x)\big[\bar{F}(x)-\bar{G}(x)\big]{\rm d}} \def\Dwphi{{D^{\rm w}_\phi} x \geq 0.\eeqq
Then
\begin{eqnarray}} \def\eeq{\end{eqnarray} D^{\rm w}_{\phi}(\bar{F}\|\bar{G}) \geq 0.\eeq
The equally occurs iff  $\displaystyle} \def\bbE{{\mathbb E}} \def\bbP{{\mathbb P}\big[\frac{\bar{G}}{\bar{F}}-1\big]\phi=0$ for $\bar{F}$-almost all $x\in \bbR_+$.
\end{thm}
\vskip .5 truecm

\begin{thm}
{\rm{(Estimating the WCRE via a uniform distribution, cf. Theorem 1.2 in \cite{SY}.)}}\;Assume that RV $X$ takes at most $m$ values and set $p_i=\mathbb P} \def\hw{h^{\rm w}} \def\hwphi{{h^{\rm w}_\phi}} \def\bbR{\mathbb R (X=i)$ and $\overline{p}_i=\sum\limits_{j=1}^i p_j$. Suppose that for given $0<\beta\leq 1$
\begin{eqnarray*}} \def\eeqq{\end{eqnarray*} \sum_{i=1}^m \phi(i)\big[\overline{p}_i-\beta i\big]\geq 0,\eeqq
Then
\begin{eqnarray*}} \def\eeqq{\end{eqnarray*} -\sum_{i=1}^m \phi(i) \overline{p}_i \log\;\overline{p}_i\leq -\log\;\beta \sum_{i=1}^m \phi(i) \overline{p}_i -\sum_{i=1}^m \phi(i) \overline{p}_i \log\;i,\eeqq
with equality iff for all $i=1\dots m$, $\phi(i)\big[\overline{p}_i-\beta i\big]=0$.\\
Furthermore, assume for given $\alpha,\beta\in \bbR$:
\begin{eqnarray*}} \def\eeqq{\end{eqnarray*} \int_{\bbR_+} \phi(x) \big[\bar{F}(x)-(\alpha-\beta x)\big]{\rm d}} \def\Dwphi{{D^{\rm w}_\phi} x \geq 0,\eeqq
The following assertion for non-negative RV $X$ holds true:
\begin{eqnarray*}} \def\eeqq{\end{eqnarray*} {\mathcal{E}^{\rm w}_{\phi}}(X)\leq -\int_{\bbR_+}\phi(x)\bar{F}(x) \log (\alpha-\beta x){\rm d}} \def\Dwphi{{D^{\rm w}_\phi} x.\eeqq
Here the equality holds iff  $\phi(x) \big[\bar{F}(x)-(\alpha-\beta x)\big]=0$.
\end{thm}
\vskip .5 truecm
\begin{defn}
Consider a random vector $\mathbf{X}=(X_1,\dots,X_n): \Om\to\bbR^n$ with join survival function $\bar{F}(\bx)=\mathbb P} \def\hw{h^{\rm w}} \def\hwphi{{h^{\rm w}_\phi}} \def\bbR{\mathbb R[X_1>x_1,\dots,X_n>x_n]$. The WCRE and WCE for given WF $\phi$, are defined by
\begin{eqnarray}} \def\eeq{\end{eqnarray}\label{Multi.WCRE.WCE} \begin{array}{c}
{\mathcal{E}^{\rm w}_{\phi}}(\mathbf{X})=-\displaystyle} \def\bbE{{\mathbb E}} \def\bbP{{\mathbb P}\int\limits_{\bbR_+^n}\phi(\bx)P[|\mathbf{X}|>\bx]\log P[|\mathbf{X}|>\bx]{\rm d}} \def\Dwphi{{D^{\rm w}_\phi} \bx,\\
{\overline{\mathcal{E}}^{\rm w}_{\phi}}(\mathbf{X})=-\displaystyle} \def\bbE{{\mathbb E}} \def\bbP{{\mathbb P}\int \limits_{\bbR_+^n}\phi(\bx)P[|\mathbf{X}|\leq\bx]\log P[|\mathbf{X}|\leq\bx]{\rm d}} \def\Dwphi{{D^{\rm w}_\phi} \bx.\end{array}\eeq
Here $\bx=(x_1,\dots,x_n)$ and $\bbR_+^n=\big(x_i\in\bbR^n, x_i\geq 0\big)$.\\
Let $(X_1,X_2)\in \bbR^2\mapsto\phi (x_1,x_2)$ be a given bivariate WF. The {\bf conditional} WCRE of $X_1$ given $X_2$ is defined by
\begin{eqnarray}} \def\eeq{\end{eqnarray}\label{conWCRE}\begin{array}{l}{\mathcal{E}^{\rm w}_{\phi}}(X_1|X_2)\\
\quad=-\displaystyle} \def\bbE{{\mathbb E}} \def\bbP{{\mathbb P}\int_{\bbR_+^2} \phi(x_1,x_2)\mathbb P} \def\hw{h^{\rm w}} \def\hwphi{{h^{\rm w}_\phi}} \def\bbR{\mathbb R(|X_1|> x_1,|X_2|> x_2)\log \frac{\mathbb P} \def\hw{h^{\rm w}} \def\hwphi{{h^{\rm w}_\phi}} \def\bbR{\mathbb R(|X_1|> x_1,|X_2|> x_2)}{\mathbb P} \def\hw{h^{\rm w}} \def\hwphi{{h^{\rm w}_\phi}} \def\bbR{\mathbb R(|X_2|>x_2)}\;{\rm d}} \def\Dwphi{{D^{\rm w}_\phi} x_1\; {\rm d}} \def\Dwphi{{D^{\rm w}_\phi} x_2.\end{array}\eeq
and the {\bf mutual} WCRE between non-negative random vectors $\mathbf{X}$ with joint SF $\bar{F}$ and marginal $\bar{F}_i$, $i=1,\dots,n$ by
\begin{eqnarray}} \def\eeq{\end{eqnarray} \begin{array}{l}\tau^{\rm w}_{\phi}(\mathbf{X}):=\displaystyle} \def\bbE{{\mathbb E}} \def\bbP{{\mathbb P} D^{\rm w}_{\phi}(\bar{F}\|\bar{F}_1\otimes\dots\otimes\bar{F}_n)\\
\quad=\displaystyle} \def\bbE{{\mathbb E}} \def\bbP{{\mathbb P}\int\limits_{\bbR_+^n}\phi(\bx)\bar{F}(\bx) \log \frac{\bar{F}(\bx)}{\bar{F}_1(x_1)\dots \bar{F}_n(x_n)}\; {\rm d}} \def\Dwphi{{D^{\rm w}_\phi} \bx. \end{array}\eeq
\end{defn}
\vskip .5 truecm
\begin{lem}
{\rm{(Bounding on conditional WCRE, Cf. Lemma 1.1 in \cite{SY}.)}}\;Let $\mathbf{X}_1^2=(X_1,X_2)$ be a pair of RVs with the joint SF $\bar{F}(x_1,x_2)$ and marginal SFs $\bar{F}_1(x_1)$, $\bar{F}_2(x_2)$. Suppose that the WF $\phi$ obeys
\begin{eqnarray}} \def\eeq{\end{eqnarray} \int\limits_{\bbR_+^2}\phi(\bx_1^2)\bar{F}(\bx_1^2)\big[\bar{F}(x_1|x_2)-1\big]{\rm d}} \def\Dwphi{{D^{\rm w}_\phi} \bx_1^2 \leq 0\eeq
Then
\begin{eqnarray}} \def\eeq{\end{eqnarray} {\mathcal{E}^{\rm w}_{\phi}}(\mathbf{X}_1^2)\leq \mathcal{E}^{\rm w}_{\psi_2}(X_2) \;\hbox{ or, equivalently, }\; {\mathcal{E}^{\rm w}_{\phi}}(X_1|X_2)\geq 0.\eeq
Here $\psi_2=\displaystyle} \def\bbE{{\mathbb E}} \def\bbP{{\mathbb P}\int\limits_{\bbR_+}\phi(\bx_1^2)\bar{F}(x_1|x_2){\rm d}} \def\Dwphi{{D^{\rm w}_\phi} x_1$ and the equality holds true iff $\phi(\bx_1^2)\big[\bar{F}(x_1|x_2)-1\big]=0$ for all $\bx\in \bbR_+^2$.\\
Furthermore, consider triple RVs $\mathbf{X}_1^3=(X_1,X_2,X_3)$ and assume that
\begin{eqnarray}} \def\eeq{\end{eqnarray} \int\limits_{\bbR_+^3}\phi(\bx_1^3)\bar{F}(\bx_1^3)\big[\bar{F}(x_1|\bx_2^3)-1\big]{\rm d}} \def\Dwphi{{D^{\rm w}_\phi} \bx_1^3.\eeq
Then
\begin{eqnarray}} \def\eeq{\end{eqnarray}\label{Buond.1} {\mathcal{E}^{\rm w}_{\phi}}(\mathbf{X}_1^3)\leq \mathcal{E}^{\rm w}_{\psi_{23}}(\mathbf{X}_2^3)\;\hbox{ or, equivalently, }\; {\mathcal{E}^{\rm w}_{\phi}}(X_1|\mathbf{X}_2^3)\geq 0.\eeq
Here $\psi_{23}=\displaystyle} \def\bbE{{\mathbb E}} \def\bbP{{\mathbb P}\int\limits_{\bbR_+}\phi(\bx_1^3)\bar{F}(x_1|\bx_2^3){\rm d}} \def\Dwphi{{D^{\rm w}_\phi} x_1$.
In (\ref{Buond.1}) the equality holds true iff $\phi(\bx_1^3)\big[\bar{F}(x_1|\bx_2^3)-1\big]=0$ for all $\bx_1^3\in \bbR_+^3$.
\end{lem}
\vskip .5 truecm
\def\mathcal{E}^{\rm w}{\mathcal{E}^{\rm w}}
By subscribing $\bar{F}$ in $f$ in Theorem 1.3 in \cite{SY} with the same arguments, the following assertion, omitting the proof, is achieved.
\begin{thm}
{\rm{(Sub-additivity of the WCRE, Cf. Theorem 1.3 in \cite{SY}.)}}\;Let $\mathbf{X}_1^2=(X_1,X_2)$ be a pair of RVs with join SF $\bar{F}(\bx_1^2)$ and marginal survival function $\bar{F}_1(x_1)$, $\bar{F}_2(x_2)$. Moreover suppose that the WF $(x_1,x_2)\in \bbR_+^2 \mapsto \phi(x_1,x_2)\geq 0$ obeys
\begin{eqnarray}} \def\eeq{\end{eqnarray} \label{eqthm3.1}\int\limits_{\bbR_+^2}\phi(\bx_1^2)\big[\bar{F}(\bx_1^2)-\bar{F}_1(x_1)\bar{F}_2(x_2)\big]{\rm d}} \def\Dwphi{{D^{\rm w}_\phi} \bx_1^2\geq 0.\eeq
Then
\begin{eqnarray}} \def\eeq{\end{eqnarray} \begin{array}{c} {\mathcal{E}^{\rm w}_{\phi}}(\mathbf{X}_1^2)\leq
\mathcal{E}^{\rm w}_{\psi _1}(X_1)+\mathcal{E}^{\rm w}_{\psi _2}(X_2),\;\hbox{ or, equivalently, }\;{\mathcal{E}^{\rm w}_{\phi}} (X_1|X_2)\leq
\mathcal{E}^{\rm w}_{\psi_1^{\,}}(X_1),\\
\;\hbox{ or, equivalently, }\;\tau^{\rm w}_\phi (X_1:X_2)\geq 0.\end{array}\eeq
The equality occurs iff $X_1$, $X_2$ are independent modulo $\phi$ i.e. $\phi(\bx_1^2)\bigg[1-\displaystyle} \def\bbE{{\mathbb E}} \def\bbP{{\mathbb P}\frac{\bar{F}_1(x_1)\bar{F}_2(x_2)}{\bar{F}(\bx_1^2)}\bigg]=0$ for all $\bx_1^2\in \bbR_+^2$. Here $\psi_1$ and $\psi_2$ are emerging from conditional survival functions:
\begin{eqnarray}} \def\eeq{\end{eqnarray}\psi_i=\displaystyle} \def\bbE{{\mathbb E}} \def\bbP{{\mathbb P} \int\limits_{\bbR_+} \phi(\bx_1^2)\bar{F}(x_j|x_i){\rm d}} \def\Dwphi{{D^{\rm w}_\phi} x_j,\;\;\; i,j=1,2,\;\; i\neq j.\eeq
\end{thm}
For given WF $\mathbf{X}_1^3\in\bbR_+^3\mapsto \phi(\bx_1^2)\geq 0$, define
\begin{eqnarray}} \def\eeq{\end{eqnarray}\begin{array}{cl} \psi_{12}(\bx_1^2)=\displaystyle} \def\bbE{{\mathbb E}} \def\bbP{{\mathbb P} \int\limits_{\bbR_+}\phi(\bx_1^3)\bar{F}(x_3|\bx_1^2){\rm d}} \def\Dwphi{{D^{\rm w}_\phi} x_3,\;\;\; \bx_1^2\in \bbR_+^2.\\
\psi_1^{23}(x_1)=\displaystyle} \def\bbE{{\mathbb E}} \def\bbP{{\mathbb P} \int\limits_{\bbR_+^2} \phi(\bx_1^3)\bar{F}(\bx_2^3|x_1){\rm d}} \def\Dwphi{{D^{\rm w}_\phi} \bx_2^3,\;\;\; x_1\in \bbR_+.\end{array}\eeq
and similarly define $\psi_k^{ij}$ and $\psi_{ij}$ for distinct labels $1\leq i,j,k\leq 3$. Then as in \cite{SY}, use $\psi_{12}$ in (\ref{eqthm3.1}) if the assumption
\begin{eqnarray}} \def\eeq{\end{eqnarray} \int\limits_{\bbR_+^3}\phi(\bx_1^3)\big[\bar{F}(\bx_1^2)-\bar{F}_1(x_1)\bar{F}_2(x_2)\big]\bar{F}(x_3|\bx_1^2) {\rm d}} \def\Dwphi{{D^{\rm w}_\phi} \bx_1^2\geq 0.\eeq
holds true, Then
\begin{eqnarray}} \def\eeq{\end{eqnarray}  \displaystyle} \def\bbE{{\mathbb E}} \def\bbP{{\mathbb P} \mathcal{E}^{\rm w}_{\psi_{12}}(X_1|X_2)\leq \mathcal{E}^{\rm w}_{\psi_1^{23}}(X_1).\eeq

Following the given assertions in \cite{SY}, the analogue inequalities each requiring its own assumption are represented. Note that in the list of assumptions (1.15),(1.17),(1,22),(1.27) in \cite{SY} swap $\bar{F}$ in $f$:
\begin{eqnarray}} \def\eeq{\end{eqnarray}\label{eq:1.27}\begin{array}{l}\hbox{by Lemma 1.1, \cite{SY}:}\;\;0\leq\;{\mathcal{E}^{\rm w}_{\phi}} (X_1|\mathbf{X}_2^3),
\quad\hbox{assuming 1.17 (a modified form of 1.15),}\\
\;\;\displaystyle} \def\bbE{{\mathbb E}} \def\bbP{{\mathbb P}\begin{array}{ll}
\hbox{by Lemma 1.3, \cite{SY}:}&{\mathcal{E}^{\rm w}_{\phi}} (X_1|\mathbf{X}_2^3)\leq \mathcal{E}^{\rm w}_{\psi_{12}}(X_1|X_2), \;
\hbox{assuming (1.27),}\\
\hbox{by Theorem 1.3, \cite{SY}:}&\mathcal{E}^{\rm w}_{\psi_{12}}(X_1|X_2)\leq \mathcal{E}^{\rm w}_{\psi_1^{23}}(X_1),
 \;\hbox{assuming (1.22),}\\
 \hbox{by Lemma 1.2, \cite{SY}:}&\mathcal{E}^{\rm w}_{\psi_{12}^{\,}} (X_1|X_2)\leq \mathcal{E}^{\rm w}_\phi(\mathbf{X}_{1,3}|X_2), \;
 \hbox{assuming (1.26),}\\
\hbox{by Theorem 1.4, \cite{SY}:} &\mathcal{E}^{\rm w}_\phi(\mathbf{X}_{13}|X_2)\leq \mathcal{E}^{\rm w}_{\psi_{12}^{\,}}(X_1|X_2)
+\mathcal{E}^{\rm w}_{\psi_{23}^{\,}}(X_3|X_2),\;\hbox{assuming (1.27),}\\
\hbox{by Theorem 1.5, \cite{SY}:} &{\mathcal{E}^{\rm w}_{\phi}}(\mathbf{X}_1^3)+\mathcal{E}^{\rm w}_{\psi_2^{13}}(X_2)\leq \mathcal{E}^{\rm w}_{\psi_{12}^{\,}}(\mathbf{X}_1^2)+
\mathcal{E}^{\rm w}_{\psi_{23}^{\,}}(\mathbf{X}_2^3), \; \hbox{assuming (1.27).}\end{array}\end{array}\eeq
\vskip 0.5 truecm
Next we represent a number of results which are analogue assertions in \cite{SY}, hence proofs omitted.
\begin{thm}
{\rm (Strong sub-additivity of the WCRE).} Given a triple of RVs $\mathbf{X}_1^3=(X_1,X_2,X_3)$, assume that
\begin{eqnarray}} \def\eeq{\end{eqnarray}\label{assum.St.sub}\displaystyle} \def\bbE{{\mathbb E}} \def\bbP{{\mathbb P}\int\limits_{\bbR_+^3}\phi(\bx_1^3)\Big[\bar{F}(\bx_1^3)-\bar{F}(x_2)\prod\limits_{i=1,3} \bar{F}(x_i|x_2)\Big]{\rm d}} \def\Dwphi{{D^{\rm w}_\phi} \bx_1^3 \geq 0.\eeq
is fulfilled. Then
\begin{eqnarray}} \def\eeq{\end{eqnarray} {\mathcal{E}^{\rm w}_{\phi}}(\mathbf{X}_1^3)-\mathcal{E}^{\rm w}_{\psi_{2}^{13}}(X_2)\leq \mathcal{E}^{\rm w}_{\psi_{12}}(\mathbf{X}_1^2)+\mathcal{E}^{\rm w}_{\psi_{23}}(\mathbf{X}_2^3).\eeq
The equality holds iff RVs $X_1$ and $X_3$ are conditionally independent $X_2$.
\end{thm}
\begin{thm}
{\rm (a) (Convexity of relative WCRE).} Given a WF $x\in\bbR_+\mapsto\phi (x)$ and $\lambda_1\lambda_2 \in(0,1)$ with $\lambda_1+\lambda_2=1$, then
\begin{eqnarray}} \def\eeq{\end{eqnarray} \lambda_1 \Dwphi(\bar{F}_1\|\bar{G}_1)+\lambda_2\Dwphi(\bar{F}_2\|\bar{G}_2)\geq \Dwphi(\lambda_1\bar{F}_1+
\lambda_2 \bar{F}_2\|\lambda_1\bar{G}_1+\lambda_2\bar{G}_2),\eeq
with equality iff $\lambda_1\lambda_2 =0$ or $\bar{F}_1=\bar{F}_2$ and $\bar{G}_1=\bar{G}_2$ modulo $\phi$.

{\rm{(b)}} {\rm{(Data-processing inequality for relative WCRE).}} Let $(\bar{F},\bar{G})$ be the pair of SFs and $\phi$ a WF in $\bbR_+$. For given stochastic kernel ${\mbox{\boldmath${\Pi}$}}=(\Pi (x,y), x,y\in\bbR_+)$, set $\Psi(u)=\displaystyle} \def\bbE{{\mathbb E}} \def\bbP{{\mathbb P}\int_{\bbR_+} \phi(x)\Pi (u,x){\rm d}} \def\Dwphi{{D^{\rm w}_\phi} x$.
Then
\begin{eqnarray}} \def\eeq{\end{eqnarray}\label{eq:2.3}D^{\rm{w}}_{\Psi} (\bar{F}||\bar{G})\geq \Dwphi(\bar{F}{\mbox{\boldmath${\Pi}$}}\,\|\,\bar{G}{\mbox{\boldmath${\Pi}$}})\eeq
where $\big(\bar{F}{\mbox{\boldmath${\Pi}$}}\big)(x)=\displaystyle} \def\bbE{{\mathbb E}} \def\bbP{{\mathbb P}\int_{\bbR_+}\bar{F}(u)\Pi (u,x){\rm d}} \def\Dwphi{{D^{\rm w}_\phi} u$ and
$\big(\bar{G}{\mbox{\boldmath${\Pi}$}}\big)(x)=\displaystyle} \def\bbE{{\mathbb E}} \def\bbP{{\mathbb P}\int_{\bbR_+}\bar{G}(u)\Pi (u,x){\rm d}} \def\Dwphi{{D^{\rm w}_\phi} u$. The equality occurs iff
$\bar{F}{\mbox{\boldmath${\Pi}$}}=\bar{F}$ and $\bar{G}{\mbox{\boldmath${\Pi}$}}=\bar{G}$.
\end{thm}
\begin{thm}
Let $\mathbf{X}_1^3$ be a triple of  RVs with joint SF $\bar{F}(\bx_1^3)$. Let $\bx_1^3=(x_1,x_2,x_3)\in\bbR_+^3\mapsto \phi(\bx_1^3)$ be a WF such that $X_1$ and $X_3$ are conditionally independent given $X_2$ modulo $\phi$.\\

{\rm{(a)}} {\rm (Data-processing inequality for conditional WCRE).} Assume inequality \eqref{assum.St.sub} by swapping $X_2$ with $X_1$. Then
the following assertion for conditional WCREs holds true:
\begin{eqnarray}} \def\eeq{\end{eqnarray} \label{eq:2.5}\mathcal{E}^{\rm w}_{\psi_{23}}(X_3|X_2) \leq \mathcal{E}^{\rm w}_{\psi_{13}}(X_3|X_1),\eeq
with equality iff $X_2$ and $X_3$ are independent modulo $\phi$. In addition assume that given WF $\phi$ and triple of RVs $\mathbf{X}_1^3$ obey
\begin{eqnarray}} \def\eeq{\end{eqnarray} \int_{\bbR_+^3}\phi(\bx_1^3)\bar{F}(\bx_1^3)\Big[\bar{F}_{2|13}(x_2|\bx_{13}-1\Big]{\rm d}} \def\Dwphi{{D^{\rm w}_\phi} \bx_1^3\leq 0\eeq
Then
\begin{eqnarray}} \def\eeq{\end{eqnarray} \mathcal{E}^{\rm w}_{\psi_{13}}(X_3|X_1)\leq 2 \mathcal{E}^{\rm w}_{\psi_{23}}(X_3|X_2);\eeq
{\rm{(b)}} {\rm (Data-processing inequality for mutual WCRE).}
Assume inequality \eqref{eqWCR:2.8}:
\begin{eqnarray}} \def\eeq{\end{eqnarray}\label{eqWCR:2.8}
\displaystyle} \def\bbE{{\mathbb E}} \def\bbP{{\mathbb P}\int_{\bbR_+^3}\phi (\bx_1^3)\Big[\bar{F}(\bx_1^3)-\bar{F}_3(x_3)\prod\limits_{i=1,2}\bar{F}_{i|3}(x_i|x_3)
\Big]{\rm d}} \def\Dwphi{{D^{\rm w}_\phi}\bx_1^3\geq 0\eeq
Then
\begin{eqnarray}} \def\eeq{\end{eqnarray}\label{eqWCR:2.9}\tau^{\rm w}_{\psi_{13}^{\,}}(X_1:X_3) \leq \tau^{\rm w}_{\psi_{12}^{\,}}(X_1:X_2).\eeq
Here, equality in \eqref{eqWCR:2.9} holds iff, modulo $\phi$, RVs $X_1$ and $X_2$ are conditionally
independent given $X_3$.
\end{thm}

\begin{thm}\label{thm:concavity.WCRE}
{\rm (Concavity of the WCRE).} Given WF $\phi$, set $\phi'(x)=\displaystyle} \def\bbE{{\mathbb E}} \def\bbP{{\mathbb P}\frac{{\rm d}} \def\Dwphi{{D^{\rm w}_\phi}}{{\rm d}} \def\Dwphi{{D^{\rm w}_\phi} x}\phi(x)$ and $\phi''(x)=\displaystyle} \def\bbE{{\mathbb E}} \def\bbP{{\mathbb P} \frac{{\rm d}} \def\Dwphi{{D^{\rm w}_\phi}^2}{{\rm d}} \def\Dwphi{{D^{\rm w}_\phi} x^2}\phi(x)$. The functional $\bar{F}\mapsto {\mathcal{E}^{\rm w}_{\phi}}(1-\bar{F})$ is concave function in $\bar{F}$ under following suppositions:\\
\\
{\rm (i)} The WF $\phi$ is non-increasing (non-decreasing) for $x\in[e^{-1},1]\;(x\in [0,e^{-1}])$.\\
{\rm (ii)} For $x\in[0,1]$
\begin{eqnarray}} \def\eeq{\end{eqnarray} \displaystyle} \def\bbE{{\mathbb E}} \def\bbP{{\mathbb P} \phi''(\bar{F}^{-1}(x))-\displaystyle} \def\bbE{{\mathbb E}} \def\bbP{{\mathbb P}\frac{f'(\bar{F}^{-1}(x))}{f(\bar{F}^{-1}(x))} \phi'(\bar{F}^{-1}(x)) \leq 0,\eeq
here $f'$ denotes the derivative of $f$ w.r.t. $x$.
\end{thm}

{\bf Proof.}\; Set $g(x)=x\log x,\; x\in[0,1]$. To implement the concavity property for WCRE, it is sufficient to prove the function $\phi(\bar{F}^{-1}(x)).g(x)$, $x\in[0,1]$ is convex. Therefore we compute
\begin{eqnarray}} \def\eeq{\end{eqnarray} \displaystyle} \def\bbE{{\mathbb E}} \def\bbP{{\mathbb P}\frac{{\rm d}} \def\Dwphi{{D^{\rm w}_\phi}}{{\rm d}} \def\Dwphi{{D^{\rm w}_\phi} x} \phi(\bar{F}^{-1}(x)).g(x)=\displaystyle} \def\bbE{{\mathbb E}} \def\bbP{{\mathbb P} \big(\frac{{\rm d}} \def\Dwphi{{D^{\rm w}_\phi}}{{\rm d}} \def\Dwphi{{D^{\rm w}_\phi} x} \phi(\bar{F}^{-1}(x)). g(x)+\phi(\bar{F}^{-1}(x)).\frac{{\rm d}} \def\Dwphi{{D^{\rm w}_\phi}}{{\rm d}} \def\Dwphi{{D^{\rm w}_\phi} x} g(x).\eeq
and
\begin{eqnarray}} \def\eeq{\end{eqnarray} \label{eq.second.deriv}\begin{array}{l} \displaystyle} \def\bbE{{\mathbb E}} \def\bbP{{\mathbb P}\frac{{\rm d}} \def\Dwphi{{D^{\rm w}_\phi}^2}{{\rm d}} \def\Dwphi{{D^{\rm w}_\phi} x^2} \phi(\bar{F}^{-1}(x)).g(x)\\
\quad=\displaystyle} \def\bbE{{\mathbb E}} \def\bbP{{\mathbb P} \big(\frac{{\rm d}} \def\Dwphi{{D^{\rm w}_\phi}^2}{{\rm d}} \def\Dwphi{{D^{\rm w}_\phi} x^2}\phi(\bar{F}^{-1}(x))\big).g(x)+\big(\frac{{\rm d}} \def\Dwphi{{D^{\rm w}_\phi}^2}{{\rm d}} \def\Dwphi{{D^{\rm w}_\phi} x^2} g(x)\big). \phi(\bar{F}^{-1}(x))+2\frac{{\rm d}} \def\Dwphi{{D^{\rm w}_\phi}}{{\rm d}} \def\Dwphi{{D^{\rm w}_\phi} x}g(x). \frac{{\rm d}} \def\Dwphi{{D^{\rm w}_\phi}}{{\rm d}} \def\Dwphi{{D^{\rm w}_\phi} x}\phi(\bar{F}^{-1}(x)).\end{array}\eeq
Evidently the middle expression in RHS of above inequality is non-negative. Furthermore, note that
\begin{eqnarray}} \def\eeq{\end{eqnarray}\label{eq.1}\begin{array}{cl} \displaystyle} \def\bbE{{\mathbb E}} \def\bbP{{\mathbb P}\frac{{\rm d}} \def\Dwphi{{D^{\rm w}_\phi}}{{\rm d}} \def\Dwphi{{D^{\rm w}_\phi} x}\phi(\bar{F}^{-1}(x))=-\displaystyle} \def\bbE{{\mathbb E}} \def\bbP{{\mathbb P}\frac{\phi'(\bar{F}^{-1}(x))}{f(\bar{F}^{-1}(x))},\\
\displaystyle} \def\bbE{{\mathbb E}} \def\bbP{{\mathbb P} \frac{{\rm d}} \def\Dwphi{{D^{\rm w}_\phi}^2}{{\rm d}} \def\Dwphi{{D^{\rm w}_\phi} x^2}\phi(\bar{F}^{-1}(x))=\frac{1}{f^2(\bar{F}^{-1}(x))}\Big[\phi''(\bar{F}^{-1}(x))-\displaystyle} \def\bbE{{\mathbb E}} \def\bbP{{\mathbb P}\frac{f'(\bar{F}^{-1}(x))}{f(\bar{F}^{-1}(x))} \phi'(\bar{F}^{-1}(x))\Big].\end{array}\eeq
Combining (\ref{eq.second.deriv}) and (\ref{eq.1}), under assumptions (i) and (ii) we conclude the result. $\quad$ $\Box$

\section{Additional results}

Following steps in the proof of Theorem 1 from \cite{RCVW}, we propose the following theorem.
\begin{thm}\label{Finity.WCRE}
Assume for given $0<a<\infty$ the following integrals are finite:
\begin{eqnarray}} \def\eeq{\end{eqnarray}\label{cond:1}\displaystyle} \def\bbE{{\mathbb E}} \def\bbP{{\mathbb P} \int\limits_{(0,a)^n}\phi(\bx){\rm d}} \def\Dwphi{{D^{\rm w}_\phi} \bx <\infty\;\; \textrm{and}\;\; \displaystyle} \def\bbE{{\mathbb E}} \def\bbP{{\mathbb P}\int\limits_{\bbR_+^n/(0,a)^n} \phi(\bx) \prod\limits_{i=1}^n x_i^{\frac{-p\alpha}{n}}{\rm d}} \def\Dwphi{{D^{\rm w}_\phi} \bx<\infty\eeq
Then ${\mathcal{E}^{\rm w}_{\phi}}(\mathbf{X})<\infty$ if for all $i$, $p$ and some $0\leq \alpha \leq 1$, $\bbE[X_i^p]<\;\infty$.

Furthermore, set $\psi(x)=\displaystyle} \def\bbE{{\mathbb E}} \def\bbP{{\mathbb P}\int_0^x\phi(t){\rm d}} \def\Dwphi{{D^{\rm w}_\phi} t$, in particular, $\phi(\bx)=\prod\limits_{i=1}^n \phi(x_i)$. Then for all $a>0$ the assumptions (\ref{cond:1}) take the form:
\begin{eqnarray*}} \def\eeqq{\end{eqnarray*} \displaystyle} \def\bbE{{\mathbb E}} \def\bbP{{\mathbb P} \int_a^\infty \phi(x_i)\; x_i^{\frac{-p\alpha}{n}}{\rm d}} \def\Dwphi{{D^{\rm w}_\phi} x_i <\; \infty,\;\;\; \psi(a)-\psi(0)<\infty.\eeqq
\end{thm}

{\bf Proof.}\;Following arguments given in \cite{RCVW}, we using H\"{o}lder's inequality. Recall Step 2 in the Theorem 1 of \cite{RCVW}. For $0\leq \alpha \leq 1$ we have
\begin{eqnarray*}} \def\eeqq{\end{eqnarray*} P[X_i>x_i,1\leq i \leq n]|\log P[X_i>x_i,1\leq i \leq n]| \leq \frac{e^{-1}}{1-\alpha}\prod\limits_{i=1}^n P[X_i>x_i]^{\frac{\alpha}{n}}.\eeqq
By multiplying both sides of above inequality in $\phi(\bx)$ and then integrating on $\bbR_+^n$, we obtain
\begin{eqnarray*}} \def\eeqq{\end{eqnarray*} {\mathcal{E}^{\rm w}_{\phi}}(\mathbf{X})\leq \frac{e^{-1}}{1-\alpha} \int\limits_{\bbR_+^n}\phi(\bx) \prod\limits_{i=1}^n P[X_i>x_i]^{\frac{\alpha}{n}} {\rm d}} \def\Dwphi{{D^{\rm w}_\phi} \bx.\eeqq
Furthermore,
\begin{eqnarray}} \def\eeq{\end{eqnarray} {\mathcal{E}^{\rm w}_{\phi}}(\mathbf{X})\label{eqq:1} \leq \frac{e^{-1}}{1-\alpha} \Big[\int\limits_{(0,a)^n} \phi(\bx){\rm d}} \def\Dwphi{{D^{\rm w}_\phi} \bx +\int\limits_{\bbR_+^n/(0,a)^n} \phi(\bx) \prod\limits_{i=1}^n P[X_i>x_i]^{\frac{\alpha}{n}} {\rm d}} \def\Dwphi{{D^{\rm w}_\phi} \bx\Big].\eeq
Owing to Markov inequality for $p\geq 0$, the last term of RHS in (\ref{eqq:1}) is less and equal than:
\begin{eqnarray*}} \def\eeqq{\end{eqnarray*} \frac{e^{-1}}{1-\alpha}\bigg[ \prod\limits_{i=1}^n \bbE[X_i^p]^{\frac{\alpha}{n}}\bigg]\int_{\bbR_+^n/(0,a)^n}\phi(\bx)\prod\limits_{i=1}^n x_i^{-\frac{p\alpha}{n}}{\rm d}} \def\Dwphi{{D^{\rm w}_\phi} \bx.\eeqq
By virtue of (\ref{cond:1}) this leads directly to the result. $\quad$ $\Box$\\
\vskip .5 truecm
{\bf Remark.}\;Note that in case $\phi(\bx)=\prod\limits_{i=1}^n \phi(x_i)$, the (\ref{eqq:1}) reads
\begin{eqnarray*}} \def\eeqq{\end{eqnarray*} \begin{array}{l}
 {\mathcal{E}^{\rm w}_{\phi}}(X)\leq \displaystyle} \def\bbE{{\mathbb E}} \def\bbP{{\mathbb P}\frac{e^{-1}}{1-\alpha}\prod\limits_{i=1}^n\int_0^\infty \phi(x_i) P[X_i>x_i]^{\frac{\alpha}{n}}{\rm d}} \def\Dwphi{{D^{\rm w}_\phi} x_i\\
 \quad\leq \displaystyle} \def\bbE{{\mathbb E}} \def\bbP{{\mathbb P}\frac{e^{-1}}{1-\alpha}\prod\limits_{i=1}^n\bigg\{ (\psi(a)-\psi(0))+\bbE[X_i^p]^{\frac{\alpha}{n}}\int_a^\infty \phi(x_i) x_i^{\frac{-p\alpha}{n}}{\rm d}} \def\Dwphi{{D^{\rm w}_\phi} x_i\bigg\}.
 \end{array}\eeqq
\vskip .5 truecm
Using the method in Theorem 5, \cite{R} and arguments in Theorem \ref{Finity.WCRE}, the following result is given. The proof of Theorem \ref{thm:convergence} is similar to Theorem 5 in \cite{R} and omitted.
\begin{thm}\label{thm:convergence}
{\rm{(Cf. Theorem 5 from \cite{R}.)}}\; Let the random vector $\mathbf{X}_k$ converges in distribution to the random vector $\mathbf{X}$. Also suppose that $\phi$ is a WF whereas (\ref{cond:1}) holds true. If all $\mathbf{X}_k$ are bounded in $L^p$ then
\begin{eqnarray}} \def\eeq{\end{eqnarray} \lim\limits_{k} {\mathcal{E}^{\rm w}_{\phi}}(\mathbf{X}_k)={\mathcal{E}^{\rm w}_{\phi}}(\mathbf{X}).\eeq
\end{thm}
\vskip .5 truecm
Now we focus on the sum of independent RVs: The standard Shannon and cumulative entropies of a sum of independent RVs is larger than and equal of each . We show as analogues as Theorem 2 in \cite{RCVW}, the same result is fulfilled for WCRE either.
\begin{thm}
Consider two non-negative and independent RVs $X$ and $Y$ with PDFs $f_X$ and $f_Y$, respectively. Then
\begin{eqnarray*}} \def\eeqq{\end{eqnarray*} \max\bigg\{\mathcal{E}^{\rm w}_{\psi_Y}(X),\mathcal{E}^{\rm w}_{\psi_X}(Y)\bigg\} \leq {\mathcal{E}^{\rm w}_{\phi}} (X+Y).\eeqq
Here $\psi_Y(x)=\displaystyle} \def\bbE{{\mathbb E}} \def\bbP{{\mathbb P}\int f_Y(y) \phi(x+y){\rm d}} \def\Dwphi{{D^{\rm w}_\phi} y$ and swap $X$ with $Y$ in $\psi_X$.
\end{thm}

{\bf Proof.}\; We again follow the argument from \cite{RCVW}. By using Jensen's inequality, write:
\begin{eqnarray}} \def\eeq{\end{eqnarray}\label{eq:09} \begin{array}{l} P[X+Y>w]\log P[X+Y>w]\\
\qquad\leq\displaystyle} \def\bbE{{\mathbb E}} \def\bbP{{\mathbb P} \int f_Y(y)P[X>w-y]\log P[X>w-y]{\rm d}} \def\Dwphi{{D^{\rm w}_\phi} y.\end{array}\eeq
Multiply both sides by $\phi(w)$ and then integrate with respect to $w$ from $0$ to $\infty$:
\begin{eqnarray*}} \def\eeqq{\end{eqnarray*}\begin{array}{l}
-{\mathcal{E}^{\rm w}_{\phi}}(X+Y) \leq \displaystyle} \def\bbE{{\mathbb E}} \def\bbP{{\mathbb P}\int f_Y(y) \int_0^\infty \phi(w)P[X>w-y]\log P[X>w-y]{\rm d}} \def\Dwphi{{D^{\rm w}_\phi} w {\rm d}} \def\Dwphi{{D^{\rm w}_\phi} y\\
=\displaystyle} \def\bbE{{\mathbb E}} \def\bbP{{\mathbb P}  \int f_Y(y) \int_y^\infty \phi(w)P[X>w-y]\log P[X>w-y]{\rm d}} \def\Dwphi{{D^{\rm w}_\phi} w {\rm d}} \def\Dwphi{{D^{\rm w}_\phi} y\\
=\displaystyle} \def\bbE{{\mathbb E}} \def\bbP{{\mathbb P}  \int f_Y(y) \int_0^\infty \phi(w+y)P[X>w]\log P[X>w]{\rm d}} \def\Dwphi{{D^{\rm w}_\phi} w {\rm d}} \def\Dwphi{{D^{\rm w}_\phi} y\\
=\displaystyle} \def\bbE{{\mathbb E}} \def\bbP{{\mathbb P} \int_0^\infty \bigg[\int \phi(w+y)f_Y(y){\rm d}} \def\Dwphi{{D^{\rm w}_\phi} y\bigg] P[X>w] \log P[X>w]{\rm d}} \def\Dwphi{{D^{\rm w}_\phi} w=-\mathcal{E}^{\rm w}_{\psi_Y}(X).\end{array}\eeqq
The first equality here is obtained because $X$ is a non-negative RV. Consequently, for $w<y$, $P[X>w-y]=1$. $\quad$ $\Box$
\vskip .5 truecm
\vskip .5 truecm
In addition, the following extended assertion of Theorem 4 from \cite{RCVW} holds true (and is straightforward).
\begin{thm}
For given independent RV $X_i$, we have
\begin{eqnarray}} \def\eeq{\end{eqnarray} {\mathcal{E}^{\rm w}_{\phi}}(\mathbf{X})=\displaystyle} \def\bbE{{\mathbb E}} \def\bbP{{\mathbb P}\sum\limits_{i} \mathcal{E}^{\rm w}_{\phi^*_i}(X_i),\eeq
where $\phi^*_i=\displaystyle} \def\bbE{{\mathbb E}} \def\bbP{{\mathbb P}\int \phi(\bx)\prod\limits_{j\ne i}\bar{F}(x_j){\rm d}} \def\Dwphi{{D^{\rm w}_\phi} \bx_1^{i-1}{\rm d}} \def\Dwphi{{D^{\rm w}_\phi} \bx_{i+1}^n$. In a particular case $\phi(\bx)=\prod\limits_{j=1}^n \phi_j(x_j)$, set $\psi_j(x_j)=\displaystyle} \def\bbE{{\mathbb E}} \def\bbP{{\mathbb P}\int_0^{x_j} \phi_j(t){\rm d}} \def\Dwphi{{D^{\rm w}_\phi} t$. Then ${\mathcal{E}^{\rm w}_{\phi}}(\mathbf{X})$ in (\ref{Multi.WCRE.WCE}) becomes:
\begin{eqnarray*}} \def\eeqq{\end{eqnarray*} {\mathcal{E}^{\rm w}_{\phi}}(\mathbf{X})= \sum\limits_{i=1}^n \Bigg(\prod\limits_{j=1,j\neq i}^n \bigg[\bbE_{X_j}(\psi_j(X_j))-\psi_j(0)\bigg]\Bigg) \mathcal{E}^{\rm w}_{\phi_i}(X_i). \eeqq
\end{thm}

\section{Bounds for the WCRE}
In this section, our goal is to establish additional bounds for the WCRE. First, let us show how the WCRE can be dominated by the standard entropy, as well as by the CRE; cf. \cite{RCVW}.
\begin{thm}\label{thm:Bound1}
Let $h(X)$ be the Shannon entropy of a non-negative RV $X$ having PDF $f$ and SF $\bar{F}$. Then
\begin{eqnarray}} \def\eeq{\end{eqnarray}\label{eq:09} {\mathcal{E}^{\rm w}_{\phi}}(X) \geq \alpha_\phi\; \exp\{h(X)\},\eeq
where
\begin{eqnarray}} \def\eeq{\end{eqnarray}\label{eq:2.03}\alpha_\phi=\displaystyle} \def\bbE{{\mathbb E}} \def\bbP{{\mathbb P} \exp\bigg\{\bbE[\log \phi(X)]+\int_0^1\log \bigg(x|\log x|\bigg){\rm d}} \def\Dwphi{{D^{\rm w}_\phi} x\bigg\}.\eeq
\end{thm}
{\bf Proof.}\; The proof follows directly from log-sum inequality:
\begin{eqnarray*}} \def\eeqq{\end{eqnarray*}\begin{array}{l}
\displaystyle} \def\bbE{{\mathbb E}} \def\bbP{{\mathbb P}\int f(x)\log \frac{f(x)}{\phi(x) \bar{F}(x)|\log \bar{F}(x)|}{\rm d}} \def\Dwphi{{D^{\rm w}_\phi} x\\
\quad \geq \displaystyle} \def\bbE{{\mathbb E}} \def\bbP{{\mathbb P} \Big(\int f(x) {\rm d}} \def\Dwphi{{D^{\rm w}_\phi} x\Big)\log\Bigg(\frac{\int f(x) {\rm d}} \def\Dwphi{{D^{\rm w}_\phi} x}{\int \phi(x)\bar{F}(x)|\log \bar{F}(x)|{\rm d}} \def\Dwphi{{D^{\rm w}_\phi} x}\Bigg)=\log \frac{1}{{\mathcal{E}^{\rm w}_{\phi}}(X)}.
\end{array}\eeqq
Note that if $\displaystyle} \def\bbE{{\mathbb E}} \def\bbP{{\mathbb P} \int \phi(x) \bar{F}(x) \log \bar{F}(x) {\rm d}} \def\Dwphi{{D^{\rm w}_\phi} x=\infty$, the proof is trivial. The LHS leads:
$$-h(X)-\bbE[\log \phi(X)]-\displaystyle} \def\bbE{{\mathbb E}} \def\bbP{{\mathbb P} \int_0^1 \log x|\log x|{\rm d}} \def\Dwphi{{D^{\rm w}_\phi} x.$$
Consequently,
$$ \log {\mathcal{E}^{\rm w}_{\phi}}(X)\geq h(X)+\bbE[\log \phi(X)]+\displaystyle} \def\bbE{{\mathbb E}} \def\bbP{{\mathbb P} \int_0^1 \log x|\log x|{\rm d}} \def\Dwphi{{D^{\rm w}_\phi} x.$$
This completes the proof. $\quad$ $\Box$\\

Let $(X,Y)$ be a pair of RVs with a conditional SF $\bar{F}(x|y)$. Moreover assume an additional WF $(x,y)\in \bbR_+^2\mapsto\varphi(x,y)\geq 0$. The WCRE of RV $X$ given $Y=y$ with WF $\varphi(x,y)$ is defined by
\begin{eqnarray}} \def\eeq{\end{eqnarray}\label{CWCRE} \mathcal{E}^{\rm w}_{\varphi}(X|Y=y)=-\int \varphi(x,y) \bar{F}(x|y)\log \bar{F}(x|y)\;{\rm d}} \def\Dwphi{{D^{\rm w}_\phi} x. \eeq
Later, owing to (\ref{CWCRE}), the generalized statement of Theorem \ref{thm:Bound1} with similar proof is driven, therefore we omit the proof.

\begin{lem}
For a non-negative RV $X$, let $f(x|y)$ be the conditional PDF $X$ given $Y=y$. Set
\begin{eqnarray*}} \def\eeqq{\end{eqnarray*} \alpha_\varphi(y)=\displaystyle} \def\bbE{{\mathbb E}} \def\bbP{{\mathbb P} \exp\bigg\{\bbE_{X|Y=y}[\log \varphi(X,y)]+\int_0^1\log \bigg(x|\log x|\bigg){\rm d}} \def\Dwphi{{D^{\rm w}_\phi} x\bigg\}\eeqq
Then
\begin{eqnarray}} \def\eeq{\end{eqnarray}\label{eq:2.02} \mathcal{E}^{\rm w}_\varphi(X|Y=y) \geq \alpha_\varphi(y)\; \exp\{h(X|Y=y)\}.\eeq
where $h(X|Y=y)$ is Shannon entropy of $X$ given $Y=y$.
\end{lem}
\vskip .5 truecm
\begin{defn}
Given WF $(x,y)\in \bbR_+^2\mapsto\varphi(x,y)\geq 0$, set $\overline{\phi}(x)=\displaystyle} \def\bbE{{\mathbb E}} \def\bbP{{\mathbb P} \int_0^\infty \varphi(x,y) f(y|x){\rm d}} \def\Dwphi{{D^{\rm w}_\phi} y$. The {\bf Cross} WCRE is introduced by
\begin{eqnarray}} \def\eeq{\end{eqnarray} \mathcal{E}^{\rm w}_{\varphi}(X,Y)=\mathcal{E}^{\rm w}_{\overline{\phi}}(X)+\bbE_X\big[\mathcal{E}^{\rm w}_{\varphi}(Y|X=x)\big].\eeq
\end{defn}
\vskip .5 truecm
\begin{lem}
Assume WF $(x,y)\in \bbR_+^2\mapsto\varphi(x,y)\geq 1$ and set
\begin{eqnarray}} \def\eeq{\end{eqnarray}\label{eq:21.03}\alpha^*_\varphi=\displaystyle} \def\bbE{{\mathbb E}} \def\bbP{{\mathbb P} \exp\bigg\{\bbE_{X,Y}[\log \phi(X,Y)]+\int_0^1\log \bigg(x|\log x|\bigg){\rm d}} \def\Dwphi{{D^{\rm w}_\phi} x\bigg\}.\eeq
Then
\begin{eqnarray*}} \def\eeqq{\end{eqnarray*} {\mathcal{E}^{\rm w}_{\phi}}(X,Y)\geq 2\alpha^*_\varphi \exp\{\frac{h(X,Y)}{2}\}.\eeqq
\end{lem}

{\bf Proof.}\;Following the assumption $\varphi\geq 1$, we observe
\begin{eqnarray}} \def\eeq{\end{eqnarray}\label{eq:22:02} \bbE_X\big[\log \overline{\phi}(X)\big]\geq \bbE_{X,Y}\big[\log \varphi(X,Y)\big].\eeq
Owing to the convexity of $e^x$ and Jensen inequality, we have
\begin{eqnarray}} \def\eeq{\end{eqnarray}\label{eq:2.04} \bbE\big[\mathcal{E}^{\rm w}_{\varphi}(Y|X=x)\big]\geq \alpha^*_\varphi \exp\{h(Y|X)\}.\eeq
where $h(Y|X)$ is denoted for conditional standard entropy.\\
\begin{eqnarray}} \def\eeq{\end{eqnarray}\label{eq22:01} \begin{array}{ccl}
\mathcal{E}^{\rm w}_{\varphi}(X,Y)&=&\mathcal{E}^{\rm w}_{\overline{\phi}}(X)+\bbE_X\big[\mathcal{E}^{\rm w}_{\varphi}(Y|X=x)\big]\\
&\geq& \alpha_{\overline{\phi}}\exp\{h(X)\}+\alpha^*_{\varphi}\exp\{h(Y|X)\}\\
&\geq& 2\alpha^*_{\varphi}\exp\{\frac{h(X,Y)}{2}\}.\end{array}\eeq
Here $\alpha_{\overline{\phi}}$ is defined as (\ref{eq:2.03}) by replacing $\overline{\phi}$ in $\phi$. The first inequality in (\ref{eq22:01}) drives from (\ref{eq:09}) and (\ref{eq:2.04}). The second inequality holds by using (\ref{eq:22:02}) and $2\exp(\displaystyle} \def\bbE{{\mathbb E}} \def\bbP{{\mathbb P}\frac{t+s}{2})\leq \exp(t)+\exp(s)$.  $\quad$ $\Box$
\vskip .5 truecm
\begin{lem}
{\rm{(Cf. Proposition 4 from \cite{RCVW}.)}} Let $X$ be a non-negative continuous RV. Given WF $\phi$, suppose that $\psi(x)=\displaystyle} \def\bbE{{\mathbb E}} \def\bbP{{\mathbb P}\int_0^x\phi(t){\rm d}} \def\Dwphi{{D^{\rm w}_\phi} t$ is bounded. There exist a function $Y=g(X)$ such that:
\begin{itemize}
\item[] (i) The WCRE and the weighted entropy (WE) are related by
\begin{eqnarray*}} \def\eeqq{\end{eqnarray*} h^{\rm w}_{\phi}(Y)=\frac{{\mathcal{E}^{\rm w}_{\phi}}(X)}{\bbE(X)}+\frac{\bbE(\psi(X))-\psi(0)}{\bbE(X)}\log\;\bbE(X),\eeqq
\item[] (ii) Assume $\psi(0)=0$, then the WCRE and the Shannon entropy (SE) are related by
\begin{eqnarray*}} \def\eeqq{\end{eqnarray*} h(Y)=\frac{{\mathcal{E}^{\rm w}_{\phi}}(X)}{\bbE(\psi(X))}-\frac{\Theta}{\bbE(\psi(X))}+\log\; \bbE(\psi(X)),\eeqq
here $\Theta=\displaystyle} \def\bbE{{\mathbb E}} \def\bbP{{\mathbb P}\int_0^\infty \phi(x) \log \phi(x)\; \bar{F}(x){\rm d}} \def\Dwphi{{D^{\rm w}_\phi} x$.
\end{itemize}
\end{lem}

{\bf Proof.}\; The proof is straightforward by considering the CDF, $F$, as an RV having PDF $\displaystyle} \def\bbE{{\mathbb E}} \def\bbP{{\mathbb P}\frac{P(X>x)}{\bbE(X)}$ and $\displaystyle} \def\bbE{{\mathbb E}} \def\bbP{{\mathbb P}\frac{\phi(x)P(X>x)}{\bbE(\psi(X))}$, respectively. Next use the definitions of SE and WE.\\
Note that simply by choosing $g(x)=F^{-1}(F(x))$ we can find a $g$. $\quad$ $\Box$
\vskip .5 truecm
Next we present a Lower bound for WCR, the origin of this Lemma goes back to Proposition 1 from \cite{R}.
\begin{lem}\label{lemma2.4}
Let $X$ and $Y$ be two iid RVs. Also For given WF $\phi$ set $\psi(x)=\displaystyle} \def\bbE{{\mathbb E}} \def\bbP{{\mathbb P} \int_0^x \phi(t){\rm d}} \def\Dwphi{{D^{\rm w}_\phi} t$. We obtain
\begin{eqnarray}} \def\eeq{\end{eqnarray}\label{eq:3.02} 2 {\mathcal{E}^{\rm w}_{\phi}}(X) \geq \bbE\big[|\psi(X)-\psi(Y)|\big].\eeq
In particular, suppose that $X$ is a non-negative RV, then
\begin{eqnarray}} \def\eeq{\end{eqnarray}\label{eq:3.03} 2 {\mathcal{E}^{\rm w}_{\phi}}(X) \geq \bbE\big[|\psi(X)-\bbE[\psi(X)]|\big].\eeq
\end{lem}

{\bf Proof.}\;According to Proposition 1, \cite{R}, similarly we derive:
\begin{eqnarray}} \def\eeq{\end{eqnarray}\label{eq:3.01} 2\bar{F}(x)-2\bar{F}^2(x)= \mathbb P} \def\hw{h^{\rm w}} \def\hwphi{{h^{\rm w}_\phi}} \def\bbR{\mathbb R\big[\max\{X,Y\}>x\big]-\mathbb P} \def\hw{h^{\rm w}} \def\hwphi{{h^{\rm w}_\phi}} \def\bbR{\mathbb R\big[\min\{X,Y\}>x\big],\eeq
multiplying both sides of (\ref{eq:3.01}) in $\phi(x)$ and then integrating from zero to infinity:
\begin{eqnarray*}} \def\eeqq{\end{eqnarray*} \begin{array}{l} 2 \displaystyle} \def\bbE{{\mathbb E}} \def\bbP{{\mathbb P}\int_0^\infty \phi(x) \bar{F}(x)(1-\bar{F}(x)){\rm d}} \def\Dwphi{{D^{\rm w}_\phi} x\\
\quad=\displaystyle} \def\bbE{{\mathbb E}} \def\bbP{{\mathbb P}\int_0^\infty \phi(x) \mathbb P} \def\hw{h^{\rm w}} \def\hwphi{{h^{\rm w}_\phi}} \def\bbR{\mathbb R\big[\max\{X,Y\}>x\big]- \displaystyle} \def\bbE{{\mathbb E}} \def\bbP{{\mathbb P}\int_0^\infty \phi(x) \mathbb P} \def\hw{h^{\rm w}} \def\hwphi{{h^{\rm w}_\phi}} \def\bbR{\mathbb R\big[\min\{X,Y\}>x\big].\end{array}\eeqq
Next using integrate by part in RHS , we can write
\begin{eqnarray}} \def\eeq{\end{eqnarray}\label{eq:Lemma 2.4}2\int_0^\infty \bar{F}(x)|\log \bar{F}(x)|{\rm d}} \def\Dwphi{{D^{\rm w}_\phi} x \geq \bbE\big[|\psi(X)-\psi(Y)|\big].\eeq
The LHS can be modified becaue of $x(1-x)\leq x|\log x|$.
The inequality (\ref{eq:Lemma 2.4}) proves (\ref{eq:3.02}). Moreover the assertion (\ref{eq:3.03}) follows directly from:
\begin{eqnarray*}} \def\eeqq{\end{eqnarray*} \bbE\big[|\psi(X)-\psi(Y)|\big]\geq \bbE\big[|\psi(X)-\bbE[\psi(X)]|\big].\qquad \Box \eeqq
\vskip .5 truecm
\begin{lem}
{\rm{(Cf. Proposition 2 from \cite{R}.)}} Let $X$ be a non-negative RV. Then for function $\psi$ defined as in Lemma \ref{lemma2.4}:
\begin{eqnarray}} \def\eeq{\end{eqnarray} {\mathcal{E}^{\rm w}_{\phi}}(X)=\bbE\bigg[(\psi(0)-\psi(X))(1+\log \bar{F}(X))\bigg].\eeq
\end{lem}

{\bf Proof.}\;The proof is straightforward and based on the equality:
\begin{eqnarray*}} \def\eeqq{\end{eqnarray*} \bar{F}(x)\log \bar{F}(x)=-\int_x^\infty (1+\log \bar{F}(t)){\rm d}} \def\Dwphi{{D^{\rm w}_\phi} \bar{F}(t).\qquad \Box\eeqq
\vskip .5 truecm
{\bf Remark:}\; More application of conjugate or the Fenchel Transform of the convex function $x\log x$ is $\exp(y-1)$, that is
\begin{eqnarray*}} \def\eeqq{\end{eqnarray*} \exp(y-1)=\sup\left[xy-x\log x:0<x<\infty\right].\eeqq
Consequently, for non-negative RVs $X$ and $Y$:
\begin{eqnarray*}} \def\eeqq{\end{eqnarray*} xy\leq x\log x +\exp(y-1).\eeqq
If we use this inequality, emerging the definition WCRE, an upper bound for WCRE in terms of $|\psi(X)-\bbE[\psi(X)]|$ is given:
\begin{eqnarray*}} \def\eeqq{\end{eqnarray*} {\mathcal{E}^{\rm w}_{\phi}}(X) \leq 2\bbE\bigg[|\psi(X)-\bbE[\psi(X)]|\log |\psi(X)-\bbE[\psi(X)]|\bigg]+\frac{4}{e}.\eeqq
Here $\psi$ is defined as before.
\vskip .5 truecm
\begin{thm}
{\rm{(Cf. Theorem 1 from \cite{R}.)}}\;Suppose that $X$ is a non-negative RV. Set $\psi(x)=\displaystyle} \def\bbE{{\mathbb E}} \def\bbP{{\mathbb P}\int_0^x \phi(t){\rm d}} \def\Dwphi{{D^{\rm w}_\phi} t$ and $\psi^{-1}$ the inverse function of $\psi$. Then
\begin{eqnarray}} \def\eeq{\end{eqnarray} \label{maineq:2.1} \begin{array}{l}\bbE\big[\psi(X)\log^+\psi(x)\big]\\
\qquad \displaystyle} \def\bbE{{\mathbb E}} \def\bbP{{\mathbb P}\leq {\mathcal{E}^{\rm w}_{\phi}}(X)+\bbE\big[\psi(X){\mathbf 1}(X>\psi^{-1}(1))\big]\log \bigg(e\;\bbE\big[\psi(X){\mathbf 1}(X>\psi^{-1}(1))\big]\bigg).\end{array}\eeq
This implies: $\bbE\big[\psi(X)\log^+\psi(x)\big]<\infty$ if WCRE is finite.
\end{thm}

{\bf Proof.}\;Following standard calculations, (see \cite{R}), we can write
\begin{eqnarray}} \def\eeq{\end{eqnarray} \label{eq:2.05} \begin{array}{l}\bbE\big[\psi(X)\log^+\psi(x)\big]\\
\qquad=\displaystyle} \def\bbE{{\mathbb E}} \def\bbP{{\mathbb P}\bbE\big[\psi(X){\mathbf 1}(X>\psi^{-1}(1))\big]-\bar{F}(\psi^{-1}(1))+\displaystyle} \def\bbE{{\mathbb E}} \def\bbP{{\mathbb P}\int_{\psi^{-1}(1)}^\infty \phi(x)\bar{F}(x) \log \psi(x){\rm d}} \def\Dwphi{{D^{\rm w}_\phi} x. \end{array}\eeq
Moreover, for $t>\psi^{-1}(1)$ one yields:
\begin{eqnarray*}} \def\eeqq{\end{eqnarray*}\psi(t)\mathbb P} \def\hw{h^{\rm w}} \def\hwphi{{h^{\rm w}_\phi}} \def\bbR{\mathbb R(X>t)\leq \bbE\big[\psi(X){\mathbf 1}(X>t)\big]\leq \bbE\big[\psi(X){\mathbf 1}(X>\psi^{-1}(1))\big].\eeqq
Therefore, we obtain
\begin{eqnarray*}} \def\eeqq{\end{eqnarray*} \begin{array}{l} \displaystyle} \def\bbE{{\mathbb E}} \def\bbP{{\mathbb P} \int_{\psi^{-1}(1)}^\infty \phi(x)\bar{F}(x) \log \psi(x){\rm d}} \def\Dwphi{{D^{\rm w}_\phi} x\\
\quad \leq \displaystyle} \def\bbE{{\mathbb E}} \def\bbP{{\mathbb P} {\mathcal{E}^{\rm w}_{\phi}}(X)+\log \bbE\big[\psi(X){\mathbf 1}(X>\psi^{-1}(1))\big]\displaystyle} \def\bbE{{\mathbb E}} \def\bbP{{\mathbb P} \int_{\psi^{-1}(1)}^\infty \phi(x)\bar{F}(x){\rm d}} \def\Dwphi{{D^{\rm w}_\phi} x.\end{array}\eeqq
Finally according to (\ref{eq:2.05}), we get
\begin{eqnarray*}} \def\eeqq{\end{eqnarray*} \bbE\big[\psi(X)\log^+\psi(x)\big] \leq {\mathcal{E}^{\rm w}_{\phi}}(X)+\varsigma\Bigg(1+ \log \bbE\big[\psi(X){\mathbf 1}(X>\psi^{-1}(1))\big]\Bigg).\eeqq
Here $\varsigma=\displaystyle} \def\bbE{{\mathbb E}} \def\bbP{{\mathbb P}\bbE\big[\psi(X){\mathbf 1}(X>\psi^{-1}(1))\big]-\bar{F}(\psi^{-1}(1))$. The inequality (\ref{maineq:2.1}) holds true then. $\quad$ $\Box$
\vskip .5 truecm

\section{Maximum WCRE properties}

\begin{thm}\label{thm3:1}
Suppose $x\in \bbR_+\mapsto \phi(x)\geq 0$ is given WF. Then $\bar{F}^{\rm m}$ maximizes the WCRE ${\mathcal{E}^{\rm w}_{\phi}}(\bar{F})$, modulo $\phi$, uniquely when the following constrains are fulfilled:
\begin{eqnarray}} \def\eeq{\end{eqnarray} \int\limits_{\bbR_+}\phi(x)\big[\bar{F}(x)-\bar{F}^{\rm m}(x)\big]{\rm d}} \def\Dwphi{{D^{\rm w}_\phi} x\geq 0 \quad \textrm{and} \quad \int\limits_{\bbR_+}\phi(x)\big[\bar{F}(x)-\bar{F}^{\rm m}(x)\big]\log \bar{F}^{\rm m}(x) {\rm d}} \def\Dwphi{{D^{\rm w}_\phi} x\geq 0.\eeq
\end{thm}
\vskip .5 truecm
\def\mathbf{t}} \def\bmu{\mathbf{\mu}} \def\BC{\mathbf{C}{\mathbf{t}} \def\bmu{\mathbf{\mu}} \def\BC{\mathbf{C}}
\begin{exam}\label{exam:max.G}
Consider a random vector $\mathbf{X}_1^n=(X_1,\ldots ,X_n):\Om\to\bbR^n$ with PDF $f$, the PDF $F$ and the SF $\bar{F}$, mean vector $\bmu =(\mu_1,\ldots ,\mu_n)$ with
 $\bbE X_i=\mu_i$ and covariance matrix $\BC=(C_{ij})$ with $C_{ij}=\bbE\Big[(X_i-\mu_i)(X_j-\mu_j)\Big]$,
$1\leq i,j\leq n$.  Let $f^{\rm{No}}$ be the normal PDF with the same  ${\mbox{\boldmath${\mu}$}}$ and $\BC$ and $\bar{F}^{\rm{No}}$ be the normal SF. Introduce
\begin{eqnarray}} \def\eeq{\end{eqnarray} \label{defn.alpha*}\alpha^*(\bx)=\displaystyle} \def\bbE{{\mathbb E}} \def\bbP{{\mathbb P}\int\limits_{\bx}^{\infty} \exp\Big\{-\frac{1}{2}(\mathbf{t}} \def\bmu{\mathbf{\mu}} \def\BC{\mathbf{C}-\bmu)^T \BC^{-1}(\mathbf{t}} \def\bmu{\mathbf{\mu}} \def\BC{\mathbf{C}-\bmu)\Big\} {\rm d}} \def\Dwphi{{D^{\rm w}_\phi}\mathbf{t}} \def\bmu{\mathbf{\mu}} \def\BC{\mathbf{C}.\eeq
Then
\begin{eqnarray}} \def\eeq{\end{eqnarray} \label{CDF.MNormal} \displaystyle} \def\bbE{{\mathbb E}} \def\bbP{{\mathbb P}\rho(\BC):=\bar{F}^{\rm{No}}(\bx)=\displaystyle} \def\bbE{{\mathbb E}} \def\bbP{{\mathbb P}(2\pi)^{-n/2}({\rm det}\; \BC)^{-1/2} \alpha^*(\bx). \eeq
Given a WF $\bx_1^n=(x_1,\ldots ,x_n)\in\bbR^n\mapsto\phi (\bx_1^n)\geq 0$, suppose that
\begin{eqnarray}} \def\eeq{\end{eqnarray} \begin{array}{l}
\displaystyle} \def\bbE{{\mathbb E}} \def\bbP{{\mathbb P}\int\limits_{\bbR_+^n}\phi(\bx)\Big[\bar{F}(\bx)-\bar{F}^{\rm{No}}(\bx)\Big] {\rm d}} \def\Dwphi{{D^{\rm w}_\phi} \bx \geq 0\;\; \hbox{and}\\
\displaystyle} \def\bbE{{\mathbb E}} \def\bbP{{\mathbb P} \log\big[(2\pi)^{n/2} ({\rm det}\;\BC)^{1/2}\big]\int\limits_{\bbR_+^n} \phi(\bx)\big[\bar{F}(\bx)-\bar{F}^{\rm No}(\bx)\big]{\rm d}} \def\Dwphi{{D^{\rm w}_\phi} \bx -
\int\limits_{\bbR_+^n} \phi(\bx)\big[\bar{F}(\bx)-\bar{F}^{\rm No}(\bx)\big] \log \alpha^*(\bx)  {\rm d}} \def\Dwphi{{D^{\rm w}_\phi} \bx \leq 0.\end{array}\eeq
Then
\begin{eqnarray}} \def\eeq{\end{eqnarray} \begin{array}{l}{\mathcal{E}^{\rm w}_{\phi}}(F)\leq {\mathcal{E}^{\rm w}_{\phi}}(F^{\rm No})\\
\quad=\displaystyle} \def\bbE{{\mathbb E}} \def\bbP{{\mathbb P} \frac{1}{2} \log \big[(2\pi)^n ({\rm det}\;\BC)\big]\int\limits_{\bbR_+^n}\phi(\bx)\bar{F}^{\rm No}(\bx){\rm d}} \def\Dwphi{{D^{\rm w}_\phi} \bx
-\int\limits_{\bbR_+^n}\phi(\bx)\bar{F}^{\rm No}(\bx) \log \alpha^*(\bx) {\rm d}} \def\Dwphi{{D^{\rm w}_\phi}\bx.\end{array}\eeq
with equality iff $\bar{F}=\bar{F}^{\rm{No}}$ modulo $\phi$.
\end{exam}
\vskip .5 truecm
\vskip .5 truecm
\begin{exam}
Let $F^{Exp}$ and $\bar{F}^{Exp}$ be respectively CDF and SF on $\bbR_{+}$ with mean $\frac{1}{\lambda}$. Suppose the following constrains are fulfilled:
\begin{eqnarray}} \def\eeq{\end{eqnarray}\displaystyle} \def\bbE{{\mathbb E}} \def\bbP{{\mathbb P}\int\limits_{\bbR_+}\phi(x)\bigg[\bar{F}(x)-\bar{F}^{Exp}(x)\bigg]{\rm d}} \def\Dwphi{{D^{\rm w}_\phi} x\geq 0 \;\; \textrm{and}\;\; \displaystyle} \def\bbE{{\mathbb E}} \def\bbP{{\mathbb P}\int\limits_{\bbR_+}x\;\phi(x)\bigg[\bar{F}(x)-\bar{F}^{Exp}(x)\bigg]{\rm d}} \def\Dwphi{{D^{\rm w}_\phi} x\leq 0,\eeq
where $x\in \bbR_+\mapsto\phi(x)\geq0$ is a given WF. Then
\begin{eqnarray}} \def\eeq{\end{eqnarray} {\mathcal{E}^{\rm w}_{\phi}}(F)\leq {\mathcal{E}^{\rm w}_{\phi}}(F^{Exp})=\lambda \displaystyle} \def\bbE{{\mathbb E}} \def\bbP{{\mathbb P}\int\limits_{\bbR_+} x\;\phi(x) e^{-\lambda x} {\rm d}} \def\Dwphi{{D^{\rm w}_\phi} x.\eeq
and $\bar{F}^{Exp}$ is a unique maximizer modulo $\phi$.
\end{exam}

The next Theorem is a direct result of Theorem \ref{thm:concavity.WCRE} and Example \ref{exam:max.G}.
\begin{thm}
{\rm (The weighted Ky Fan inequality using the WCRE; cf. \cite{SY}, Theorem 3.2).}\;Assume for given $\lambda_1,\lambda_2\in[0,1]$ with $\lambda_1+\lambda_2=1$ and positive definite matrices $\BC_1$, $\BC_2$ and $\BC=\BC_1+\BC_2$ the assumption in Theorem \ref{thm:concavity.WCRE}, (i) and (ii) hold true. Furthermore
\begin{eqnarray}} \def\eeq{\end{eqnarray}\begin{array}{l}
\displaystyle} \def\bbE{{\mathbb E}} \def\bbP{{\mathbb P}\int\limits_{\bbR_+^n}\phi(\bx)\big[\lambda_1 \bar{F}_{\BC_1}^0(\bx)+\lambda_2\bar{F}_{\BC_2}^0(\bx)-\bar{F}_{\BC}^0(\bx)\big]{\rm d}} \def\Dwphi{{D^{\rm w}_\phi} \bx\geq 0\;\;\; \hbox{and}\\
\displaystyle} \def\bbE{{\mathbb E}} \def\bbP{{\mathbb P} \log\big[(2\pi)^{n/2} ({\rm det}\;\BC)^{1/2}\big]\int\limits_{\bbR_+^n} \phi(\bx)\big[\lambda_1 \bar{F}_{\BC_1}^0(\bx)+\lambda_2\bar{F}_{\BC_2}^0(\bx)-\bar{F}_{\BC}^0(\bx)\big]{\rm d}} \def\Dwphi{{D^{\rm w}_\phi} \bx \\
\displaystyle} \def\bbE{{\mathbb E}} \def\bbP{{\mathbb P}-\int\limits_{\bbR_+^n} \phi(\bx)\big[\lambda_1 \bar{F}_{\BC_1}^0(\bx)+\lambda_2\bar{F}_{\BC_2}^0(\bx)-\bar{F}_{\BC}^0(\bx)\big] \log \alpha_{\BC}^*(\bx)  {\rm d}} \def\Dwphi{{D^{\rm w}_\phi} \bx \leq 0.\end{array}\eeq
are fulfilled. Then
\begin{eqnarray}} \def\eeq{\end{eqnarray} \rho(\lambda_1\BC_1+\lambda_2\BC_2)-\lambda_1\rho(\BC_1)-\lambda_2\rho(\BC_2)\geq 0.\eeq
with equality iff $\lambda_1\lambda_2=0$ or $\BC_1=\BC_2$.
\end{thm}

\vskip .5 truecm

\begin{lem}\label{lem4.1}
Let $\mathbf{X}_1^n=(X_1,\ldots ,X_n)$ be a random vector, with components $X_i:\Om\to\cX_i$, $1\leq i\leq n$,
and the joint SF $\bar{F}$. Introduce the random vector $\overline{\bx}_i=(\bx_1^{i-1},\bx_{i+1}^n)$, $\bar{F}_i(x_i)$ the marginal SF for RV $X_i$:
\begin{eqnarray*}} \def\eeqq{\end{eqnarray*} \bar{F}_i(x_i)=\lim _{\overline{\bx}_i\rightarrow \infty} \bar{F}(\bx)\;\;\; \hbox{and}\;\;\; \bar{F}_{|i}(\bx_1^n|x_i)=\displaystyle} \def\bbE{{\mathbb E}} \def\bbP{{\mathbb P}\frac{\bar{F}(\bx_1^n)}{\bar{F}_i(x_i)}.\eeqq
For given a WF $\phi$, suppose that
\begin{eqnarray}} \def\eeq{\end{eqnarray} \displaystyle} \def\bbE{{\mathbb E}} \def\bbP{{\mathbb P} \int\limits_{\bbR_+^n} \phi(\bx)\Big[\bar{F}(\bx)-\prod\limits_{i=1}^n \bar{F}_i(x_i)\Big]{\rm d}} \def\Dwphi{{D^{\rm w}_\phi}\bx \geq 0.\eeq
Then
\begin{eqnarray}} \def\eeq{\end{eqnarray}\label{Ineq.M} {\mathcal{E}^{\rm w}_{\phi}}(\mathbf{X})\leq \sum\limits_{i=1}^n \mathcal{E}^{\rm w}_{\psi_i} (X_i).\eeq
where
\begin{eqnarray}} \def\eeq{\end{eqnarray} \psi_i(x_i)=\int \limits_{\bbR_+^{n-1}}\phi(\bx_1^n)\bar{F}_{|i}(\bx_1^n|x_i){\rm d}} \def\Dwphi{{D^{\rm w}_\phi} \bx_1^{i-1}{\rm d}} \def\Dwphi{{D^{\rm w}_\phi} \bx _{i+1}^n.\eeq
The equality in (\ref{Ineq.M}) holds true holds iff, modulo $\phi$, components $X_1, 1dots, X_n$ are independent.
\end{lem}
In the following theorem a straightforward application of Lemma \ref{lem4.1} is given.
\begin{thm}
{\rm (The weighted Hadamard inequality using the WCRE; cf. \cite{SY}, Theorem 3.3).}\;Let  $\BC=(C_{ij})$ be a positive definite $n\times n$ matrix
and $\bar{F}^{\rm{No}}_{\BC}$ the normal SF with the zero mean vector and the covariance matrix $\BC$. For given WF $\bx_1^n=(x_1,\ldots ,x_n)\in\bbR^n\mapsto\phi (\bx_1^n)$, introduce $\alpha^*(\bx)$ by (\ref{defn.alpha*}) and
\begin{eqnarray}} \def\eeq{\end{eqnarray} \alpha^*_i(x)=\displaystyle} \def\bbE{{\mathbb E}} \def\bbP{{\mathbb P}\int\limits_{x}^\infty e^{-t^2\big/2C_{ii}}{\rm d}} \def\Dwphi{{D^{\rm w}_\phi} t\;\;\;\; \hbox{and}\;\;\;\; \alpha=\displaystyle} \def\bbE{{\mathbb E}} \def\bbP{{\mathbb P} \int\limits_{\bbR_+^n} \phi(\bx) \bar{F}^{\rm{No}}(\bx){\rm d}} \def\Dwphi{{D^{\rm w}_\phi} \bx.\eeq
Suppose that
\begin{eqnarray}} \def\eeq{\end{eqnarray} \displaystyle} \def\bbE{{\mathbb E}} \def\bbP{{\mathbb P} \int\limits_{\bbR_+^n} \phi(\bx)\Big[\bar{F}^{\rm {No}}(\bx)-\prod\limits_{i=1}^n \bar{F}^{\rm {No}}_i(x_i)\Big]{\rm d}} \def\Dwphi{{D^{\rm w}_\phi}\bx \geq 0.\eeq
Then
\begin{eqnarray}} \def\eeq{\end{eqnarray} \displaystyle} \def\bbE{{\mathbb E}} \def\bbP{{\mathbb P} \frac{\alpha}{2}\log \bigg[\prod\limits_i C_{ii}\Big/ {\rm det}\;\BC\bigg]+\int\limits_{\bbR_+^n} \phi(\bx) \bar{F}^{\rm{No}}(\bx) \log \bigg[\alpha^*(\bx)\Big/\prod\limits_i \alpha^*_i(x_i)\bigg] {\rm d}} \def\Dwphi{{D^{\rm w}_\phi} \bx \geq 0,\eeq
with equality iff $\BC$ is diagonal.
\end{thm}

\vskip .5 truecm
Next, we provide a characterization of the Weibull distribution using the maximum WCRE.
\begin{thm}
{\rm{(Cf. Theorem 2 from \cite{R}.)}}\;Suppose $\psi^*_p(x)=\displaystyle} \def\bbE{{\mathbb E}} \def\bbP{{\mathbb P}\int_0^x t^p\phi(t){\rm d}} \def\Dwphi{{D^{\rm w}_\phi} t$ is a non-negative WF such that $\psi(x)=\displaystyle} \def\bbE{{\mathbb E}} \def\bbP{{\mathbb P}\int_0^x \phi(t){\rm d}} \def\Dwphi{{D^{\rm w}_\phi} t$ and $x\in \bbR^+\mapsto \phi(x)\in[0,1]$. Among all non-negative RVs with given $\bbE[\psi(X)]$ and $\bbE[\psi^*_p(X)]$ the Weibull distribution $W$ with SF $\bar{F}_{\rm Wib}(t)=\exp(-\lambda^q t^q)$, has the maximal WCRE.\\
Here the parameters $q=p$ and
\begin{eqnarray}} \def\eeq{\end{eqnarray}\label{lambda.Weibull} \lambda^q=\bigg(\frac{c_p}{\bbE[\psi(X)]-\psi(0)}\bigg)^p.\bigg(\frac{\bbE[\psi^*_p(X)]-\psi^*_p(0)}{\bbE_{\rm Wib}[\psi^*_q(X)]-\psi^*_q(0)}\bigg);\eeq
where $c_p=\Gamma(1+\frac{1}{p})$.
\end{thm}

{\bf Proof.}\; The subsequence argument works by using Log-sum inequality once more. According to (21) in \cite{R} but replacing $\bar{G}(x)=\bar{F}_{\rm Wib}(x)=\exp(-\mu^p x^p)$ in (\ref{eq:4.01}), we get
\begin{eqnarray*}} \def\eeqq{\end{eqnarray*} {\mathcal{E}^{\rm w}_{\phi}}(X) \leq \bigg[\bbE[\psi(X)]-\psi(0)\bigg] \log \frac{\bbE[\psi(X)]-\psi(0)}{\mu^{-1}c_p} +\mu^p \int_0^\infty \phi(t) t^p \bar{F}(t) {\rm d}} \def\Dwphi{{D^{\rm w}_\phi} t.\eeqq
Now choose $\mu^{-1}c_p=\bbE[\psi(X)]-\psi(0)$:
\begin{eqnarray*}} \def\eeqq{\end{eqnarray*} {\mathcal{E}^{\rm w}_{\phi}}(X)\leq \frac{c_p^p \big(\bbE[\psi^*_p(X)]-\psi^*_p(0)\big)}{\bbE[\psi(X)]-\psi(0)}.\eeqq
Finally let $q=p$ and $\lambda$ as in (\ref{lambda.Weibull}), therefore we have
\begin{eqnarray*}} \def\eeqq{\end{eqnarray*} {\mathcal{E}^{\rm w}_{\phi}}(X) \leq \lambda^q \bigg(\bbE_{\rm Wib}[\psi^*_q(X)]-\psi^*_q(0)\bigg)={\mathcal{E}^{\rm w}_{\phi}}(Wib).\eeqq
This completes the proof. $\quad$ $\Box$

\vskip .5 truecm
\begin{thm}
Suppose that functions $\psi$ and $\phi$ are as in Theorem 3.2: $\psi (x)=\displaystyle} \def\bbE{{\mathbb E}} \def\bbP{{\mathbb P}\int_0^x \phi(t){\rm d}} \def\Dwphi{{D^{\rm w}_\phi} t)$, and $0\leq\phi (x)\leq 1$.
Let $X$ be a given non-negative RV. In addition assume $Z:=X(\lambda)$ is an exponentially distributed RV with mean
$\lambda^{-1}=\bbE[\psi(X)]-\psi(0)$. If the constraints
\begin{eqnarray}} \def\eeq{\end{eqnarray}\label{assum3.1} \displaystyle} \def\bbE{{\mathbb E}} \def\bbP{{\mathbb P}\int\limits_{\bbR_+}x\; \phi(x) \big[\bar{F}(x)-\bar{F}^{\rm Exp}(x)\big]{\rm d}} \def\Dwphi{{D^{\rm w}_\phi} x \geq 0.\eeq
holds true, then
\begin{eqnarray}} \def\eeq{\end{eqnarray}\label{maxi.Exp} {\mathcal{E}^{\rm w}_{\phi}}(X)\leq {\mathcal{E}^{\rm w}_{\phi}}(X(\lambda))= \lambda\displaystyle} \def\bbE{{\mathbb E}} \def\bbP{{\mathbb P}\int\limits_{\bbR_+} x\;\phi(x)\bar{F}^{\rm Exp}(x) {\rm d}} \def\Dwphi{{D^{\rm w}_\phi} x.\eeq
\end{thm}

{\bf Proof.}\; Using log-sum inequality we obtain
\begin{eqnarray}} \def\eeq{\end{eqnarray}\label{eq3:1} \begin{array}{l}
\displaystyle} \def\bbE{{\mathbb E}} \def\bbP{{\mathbb P}\int\limits_{\bbR_+} \phi(x) \bar{F}(x) \log \big(\phi(x)\bar{F}(x) e^{\lambda\;x}\big){\rm d}} \def\Dwphi{{D^{\rm w}_\phi} x\\
\qquad \geq \displaystyle} \def\bbE{{\mathbb E}} \def\bbP{{\mathbb P}\bigg(\int\limits_{\bbR_+}\phi(x) \bar{F}(x) {\rm d}} \def\Dwphi{{D^{\rm w}_\phi} x\bigg)\log \bigg(\lambda \int\limits_{\bbR_+} \phi(x) \bar{F}(x) {\rm d}} \def\Dwphi{{D^{\rm w}_\phi} x\bigg).\end{array}\eeq
We also can write
$$\bbE[\psi(X)]-\psi(0)=\displaystyle} \def\bbE{{\mathbb E}} \def\bbP{{\mathbb P}\int\limits_{\bbR_+}\phi(x)\bar{F}(x){\rm d}} \def\Dwphi{{D^{\rm w}_\phi} x, $$
therefore the expression (\ref{eq3:1}) becomes
\begin{eqnarray*}} \def\eeqq{\end{eqnarray*} \begin{array}{l}
-{\mathcal{E}^{\rm w}_{\phi}}(F)+\displaystyle} \def\bbE{{\mathbb E}} \def\bbP{{\mathbb P}\int\limits_{\bbR_+} \bar{F}(x) \phi(x) \log \phi(x){\rm d}} \def\Dwphi{{D^{\rm w}_\phi} x+\lambda \int\limits_{\bbR_+}x\;\phi(x) \bar{F}(x){\rm d}} \def\Dwphi{{D^{\rm w}_\phi} x\\
\qquad \geq \displaystyle} \def\bbE{{\mathbb E}} \def\bbP{{\mathbb P}\big(\bbE[\psi(X)]-\psi(0)\big)\bigg\{\log \lambda+\log \big(\bbE[\psi(X)]-\psi(0)\big)\bigg\}.\end{array}\eeqq
Equivalently
\begin{eqnarray}} \def\eeq{\end{eqnarray} \label{eq3.2}\begin{array}{ccl}
-{\mathcal{E}^{\rm w}_{\phi}}(F)\geq \displaystyle} \def\bbE{{\mathbb E}} \def\bbP{{\mathbb P} &-&\int\limits_{\bbR_+} \bar{F}(x) \phi(x) \log \phi(x){\rm d}} \def\Dwphi{{D^{\rm w}_\phi} x-\lambda \int\limits_{\bbR_+}x\;\phi(x) \bar{F}(x){\rm d}} \def\Dwphi{{D^{\rm w}_\phi} x\\
&+& \displaystyle} \def\bbE{{\mathbb E}} \def\bbP{{\mathbb P} \big(\bbE[\psi(X)]-\psi(0)\big)\bigg\{\log \lambda+\log \big(\bbE[\psi(X)]-\psi(0)\big)\bigg\}.\end{array}\eeq
Now, set
$$\varpi=\bigg(\bbE[\psi(X)]-\psi(0)\bigg)^2 \Big(\displaystyle} \def\bbE{{\mathbb E}} \def\bbP{{\mathbb P}\int\limits_{\bbR_+} x\;\phi(x)\bar{F}(x) {\rm d}} \def\Dwphi{{D^{\rm w}_\phi} x\Big)^{-1},\;\; \lambda^*=\frac{\varpi}{\bbE[\psi(X)]-\psi(0)}$$
It is admissible (\ref{eq3.2}) is fulfilled for all positive $\lambda$, so is also valid for maximum value of $\lambda=\lambda^*$. This represents the formula:
\begin{eqnarray*}} \def\eeqq{\end{eqnarray*}\begin{array}{ccl}
-{\mathcal{E}^{\rm w}_{\phi}}(F)\\
\quad&\geq& -\displaystyle} \def\bbE{{\mathbb E}} \def\bbP{{\mathbb P}\int\limits_{\bbR_+} \bar{F}(x) \phi(x) \log \phi(x){\rm d}} \def\Dwphi{{D^{\rm w}_\phi} x-\big[\bbE[\psi(X)]-\psi(0)\big]+\displaystyle} \def\bbE{{\mathbb E}} \def\bbP{{\mathbb P}\big[\bbE[\psi(X)]-\psi(0)\big]\log \varpi\\
&\geq& -\displaystyle} \def\bbE{{\mathbb E}} \def\bbP{{\mathbb P}\big[\bbE[\psi(X)]-\psi(0)\big]+\big[\bbE[\psi(X)]-\psi(0)\big]\log\varpi\\
&\geq& -\displaystyle} \def\bbE{{\mathbb E}} \def\bbP{{\mathbb P}\big[\bbE[\psi(X)]-\psi(0)\big]+\big[\bbE[\psi(X)]-\psi(0)\big]\displaystyle} \def\bbE{{\mathbb E}} \def\bbP{{\mathbb P}\bigg\{1-\frac{1}{\varpi}\bigg\}\\
&=& -\displaystyle} \def\bbE{{\mathbb E}} \def\bbP{{\mathbb P} \big[\bbE[\psi(X)]-\psi(0)\big] \varpi^{-1}.
\end{array}\eeqq
The second inequality holds true owing to $\phi\in[0,1]$ and the last inequality is satisfied by using $\log x\geq 1-\frac{1}{x}$, $x\in \bbR_+$.\\
Recalling assumption (\ref{assum3.1}), leads to (\ref{maxi.Exp}). $\quad$ $\Box$
\vskip .5 truecm

{\emph{Acknowledgements --}}
YS thanks the Math Department, Penn State University,  for the financial support and hospitality
during the academic year 2014-5. SYS thanks the CAPES PNPD-UFSCAR Foundation
for the financial support in the year 2014-5. SYS thanks
the Federal University of Sao Carlos, Department of Statistics, for hospitality during the year 2014-5.

\vskip .5 truecm

redpajama_set_name: RedPajamaArXiv


text:
13th High School in Wrocław
Monday, 16 January 2023 / Published in Facilities
This modern sports complex on Haukego-Bosaka Street in Wrocław belongs to the Aleksander Fredro High School No. XIII.
More than 60 thousand pieces of white Venus bricks from the LHL Klinkier range were used for its façade.
The 810m2 building includes a multi-purpose sports hall, gym, fitness and dance rooms, as well as a fencing room.
The gymnasiums are stacked on top of each other like blocks to make use of the small plot space. And the sports block is tied together by a link at first-floor level with the high school buildings.
Importantly, the halls can be used not only by students, but also – after school hours and at weekends – by local residents.
The project is designed by the ARCH_IT studio from Wrocław and has already received the Brick Award 2017.
The construction of the complex is also one of the stages of the revitalization of this part of the Oławskie Przedmieście.
Facade: Wenus clinker brick from the LHL Klinkier range
Location: Wrocław, Lower Silesia
Design: Arch_it / http://archit.pl/
Total usable area of the building: 810m2
Number of bricks: 60 thousand pieces
FILM SCHOOL IN KATOWICE
DETACHED HOUSE IN WROCŁAW
THE BRIDGE HOTEL WROCŁAW

redpajama_set_name: RedPajamaCommonCrawl


text:
<?php

namespace DCarbone\PHPFHIRGenerated\DSTU2\PHPFHIRTests\FHIRElement;

use PHPUnit\Framework\TestCase;
use DCarbone\PHPFHIRGenerated\DSTU2\FHIRElement\FHIRDeviceMetricColor;

/**
 * Class FHIRDeviceMetricColorTest
 * @package \DCarbone\PHPFHIRGenerated\DSTU2\PHPFHIRTests\FHIRElement
 */
class FHIRDeviceMetricColorTest extends TestCase
{
    public function testCanConstructTypeNoArgs()
    {
        $type = new FHIRDeviceMetricColor();
        $this->assertInstanceOf('\DCarbone\PHPFHIRGenerated\DSTU2\FHIRElement\FHIRDeviceMetricColor', $type);
    }
}

redpajama_set_name: RedPajamaGithub


EXCLUSIVE: Feature director James Mangold has come on board to direct the CBS/CBS Studios drama pilot Rookies, an ensemble cop show from The Color of Money writer Richard Price, which is executive produced by Robert De Niro and Jane Rosenthal through their Tribeca production banner. Price also is exec producing the pilot, about six NYPD rookies who balance their personal lives with learning the beat on the streets of Manhattan. De Niro and Mangold worked together on the 1997 movie Cop Land, also about a group of NYPD policemen. Mangold directed the movie, which co-starred De Niro. Because of the duo's great experience on Cop Land, De Niro and Rosenthal approached Mangold to direct Rookies. This is the second TV directing gig for CAA-repped Mangold, who also directed the pilot for Men in Trees and served as an executive producer on the ABC series.
Meanwhile, Mad Men helmer Lesli Linka Glatter has signed on to direct the ABC drama pilot Grace, from Grey's Anatomy executive producer Krista Vernoff, a dysfunctional family drama set in the world of professional dance. WME-repped Glatter was nominated for an Emmy and won a DGA Award for the Guy Walks Into an Advertising Agency episode of Mad Men. She recently directed the pilot for ABC Family's Pretty Little Liars.
And Adam Bernstein, who directed the pilots for NBC's 30 Rock and Scrubs, has been tapped to direct the Fox single-camera comedy pilot Little in Common, from Veronica Mars creator Rob Thomas. The project, produced by Warner Bros. TV, is about 3 families united through their kids' little league sports.


text:
The leafy canopy amidst sharply angled branches of the towering oak cast a cool shadow over the center of the town square. The tall and gaunt priest once again returns to give sermon.
"Come Family, gather and listen to a warning," the Priest calls out to the crowd in the town square.
"I have seen with my own eyes something troublesome," he points at his eyes to draw attention to his gaze and continues. "An army of lizardmen encroaching less than a day's journey through the woods to the south."
"It was there where my companion Duffy MacNeil, a fine and noble warrior, and I saw these lizardmen."
— Cut to a room in the Cask & Wick Waywards Inn. The room is blanketed in shadow. A candle silhouettes the kneeling priest. —
The priest mutters a prayer, "My queen, I have found another killer that is eager to soil his axe and hammer in blood. You will not believe his first words to me were that he wished to swing his axe and hammer into something. He meant something of flesh and blood."
He grins, waving a hand over the shadowy flame, "Ah, he is ripe to send you more souls, my queen. With your bidding, I will give this man aid to split and crush countless foes."
— Cut back to the Town Square. —
The priest continues, "If you see this brave man, give him thanks for his duty to Nyrheim in keeping our Family safe."
— Cut to the forest in a ring of burnt out trees. —
A large centipede leaps from a tree, mandibles clicking audibly as it lunges towards Duffy. The agile fighter lifts his axe and bisects the centipede right between its fangs. The centipede's legs flail as it falls to the ground. The thing curls upon itself twitching lifelessly.
"Yes, give this man thanks for ridding the forest of blood-thirsty creatures that stalk in the night," the priest gives emphasis to the word blood-thirsty. "He fought bravely against centipedes the size of men and later against a far more powerful foe."
"We came upon this more powerful foe as we happened upon a small pond in the forest. There we discovered what we first thought were harmless beasts of burden, like the herds of rothe that aid the farmers in tilling the soil so they may grow their crops and in turn give our Family sustenance. But these were not these, but we did not know it yet."
The priest holds out a welcoming hand, "We didn't see the beasts' handlers but figured they were nomads just as we were all once nomads in search of a brighter future and new home."
"Just as we were nearly extending an open hand of welcome, a lizardman erupted out of the pond with spear in hand and began hissing and grunting in our direction. A second lizardman (what must have been a barbarous shaman) did not erupt from the water, but slinked out, licking his teeth in hunger. It wore a suit of bones upon its chest. I dare not imagine who its victims were, but I hope they were granted a quick death and now bask in the warm light of the Eternal Sun."
— Cut to the forest. —
A path is swathed out, leading two directions. Telgaunt and Duffy stealthily follow the two lizardmen who drive their beasts through the path. The rear lizardman realizes it is being followed, turns to face the two, and and waves a bony shield in the air.
Telgaunt charges forward with his fingers gathering passing shadow. He places his hand upon the chest of the lizard man shouting, "Shadow envelop and steal you."
Black wisps of shadow pour from Telgaunt's hand onto the chest of the lizardman and spread throughout its body. His eyes fill with black and give it a vision of the Shadowrealm where it will live out an eternity in the service of the Dark Queen. A moment later, its chest bursts open to no blood pouring out, but instead tendrils of shadow cascading on the forest floor around its lifeless corpse.
The second lizardman wearing bone armor rushes forward, grasps its claws at Telgaunt's neck but a shroud of shadow protects him. It then swings a powerful club at Telgaunt's head but this time the shadow reaches out to shield the incoming blow. A tail from one of the beasts slashes through the path that would have easily killed a man, but passes right through the shadowy priest.
— Cut to the Town Square. —
The priest looks to the sky in reverence to those who might have been the lizardman's victims and now make up its bone armor, "I am grateful that the gods have delivered to us a man so brave. I am grateful to have been accompanied by the brave Duffy. For if it weren't for him, we would not have defeated the vile lizardmen and, what we thought were harmless beasts of burden, were actually vicious beasts themselves that when commanded charged, bit, and swiped their massive whiplike tails at us."
"I will spare you the details of the gruesome sight that it was. I'm sure you can well imagine the lizardmen were ought to wreck the stuff of nightmares upon us."
— Cut to the same forest path. —
Duffy sidesteps the shaman lizardman's claw and parries it's club effortlessly before bringing his axe and hammer upon its head crushing it instantly.
The priests holds out a single finger to give warning, "Through courage and duty to Family, we defeated these vile lizardmen and their beasts. In our victory we delayed their plans to do the people of Nyrheim harm. We found that in their packs they had all manner of construction supplies. Supplies to build a tower less than a day's journey from our home! I cannot fathom to think of what they might do if they were to gain a foothold so near to us."
Townsolk mutter to each other, some calling out in fear or disbelief.
"But what can we do, you ask?"
"Let us continue to pray to the gods that they deliver us safely. And, if you are able, give whatever aid you can to the Nyrgärde so they may continue to protect our Family from harm that brews outside of these walls."
The priest digs out of his pack an alms box and places it at his feet before lowering his head and whispering an inaudible prayer to his goddess.
This entry was posted in Characters, D&D, Telgaunt, West Marches. Bookmark the permalink.
← A New Recruit
The Making of a Priest →


text:
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It is no secret we love to make sensory doughs and playdough. In fact, it is something we have been doing for the last two years every month and sharing. This month we are super excited to share an edible playdough recipe that not only taste absolutely delicious, it is super fun too. This edible jelly bean play dough will have your kids begging to make it every single day.
I have wanted to make edible jelly bean playdough for two years and never seem to get around to it. This year, I was determined to get ahead of the game and make something we could all play with.
Before you can even get started, you have to sort and prep your jelly beans. Take it from us when we tell you we tried everything before we decided on this method. We tried putting the jelly beans in the microwave with nothing else… don't do that!
Then, we tried adding 1/4 cup water to our cup with 10-20 jelly beans. We microwaved for 20 seconds and got a really nice colored water.
We should have stopped there, but we tried to use coconut oil instead of the water to nix the water and lower the amount of liquid we used. Yep… that didn't work either. The oil and sugar never mix.
So, the end story is… add 1/4 cup water to a glass or bowl with about 20 jelly beans and microwave for 30 seconds. Strain the jelly beans out of the liquid and set aside.
Add 2 cups of powdered sugar to a bowl and pour your jelly bean liquid and 1 tbs of coconut oil on top. Stirring as you go. Once it is well mixed, add more cups of sugar 1/4 cup at a time until it is the perfect consistency.
If you find the dough is too wet, simply add more powdered sugar.
Once you have the perfect consistency, you can roll the dough and knead it until it is smooth. Now it is perfect for play time and loads of sensory input!
I am not exaggerating when I say the kids LOVED this playdough. Between wanting to eat the whole bowl and loving the texture, my kids were sold!
You can use the dough for molding, creating, or even just free play.
Without any promting, Legoman decided to work on fine motor skills by hiding and searching for hidden treasures in the dough.
We definitely have to give this one a go! It looks so beautiful!


text:
Did a Boggart or Headless Lady cause traffic accident?
Barry McCann
BARRY McCANN describes how a mysterious spirit may behind an Lancashire apprentice falling off his bike in 1947
We have all heard tales of motorists picking up ghostly hitch hikers, but knocking down a phantom road victim? Among the stories researched by Paranormal Investigations in Lancashire is a strange incident of the cyclist who collided with a pedestrian… who wasn't there.
The Lancastrian town of Leyland was a place better known for its car manufacturing industry. It was during a rainy night in 1947 that Ken Swift hurriedly cycled to his night school class at Wellfield Senior School, as part of his apprenticeship. The journey was always a mad dash as he would not set off until the latest episode of Dick Barton – Special Agent finished on the radio, and then just make it for 7pm registration.
Despite battling the driving rain, Ken was in the town centre within minutes. He had passed the old Regent cinema and was approaching the final turn onto School Lane when, suddenly, a figure stepped out into the road and into his path. Braking at the last moment, he was certain of hitting the unfortunate pedestrian before tumbling over the handle bars and onto the pavement.
A relatively unscathed Ken got himself up, concerned the person he was certain of just colliding with. To his amazement, he was alone with nobody else there. Looking up and down the road, the only people visible was a small number gathered outside the cinema back down the road, but no one else within the vicinity. Thought the accident had happened within a split second, Ken was sure the figure was dark and hooded, which fostered his speculation as to whether it could have been a ghost.
Next day at work, he related his experience to a fellow apprentice called Jack. Instead of being met with scepticism, Jack informed him that particular section of Leyland was reputed to be haunted by the ghost of a lady. It was a tale known to older residents of the town, many of whom deliberately walked the opposite side of School Lane to avoid the area and its spectre.
Ken later related the incident to Paranormal Investigations, who undertook further research into the history of the area. They discovered an old narrow lane used to stand just north of where the incident took place. It was called Boggart Lane, so called as local legend spoke of it being haunted by the ghost of a headless woman.
What are Boggarts?
Boggarts are mischievous spirits that haunt old country lanes and sometimes attack travellers, their name being derivative of the old English "bar-gheist" meaning "gate ghost." And one of the popular Lancastrian folk tales concerning them is the boggart of Longridge who takes the form of a hooded woman, and then reveals to victims that the hood is disguising she has no head.
From Ken's testimony, he clearly had no prior knowledge of Boggart Lane or its ghost which lends his story more credence.
The fact he thought the mysterious figure to be hooded is also intriguing given the area's background, and the fact Leyland is not that far from Longridge. Was it simply a ghost? Or was the young apprentice the victim of a boggart? He can only be thankful at completing his night school course without being troubled by the figure a second time.
References: Strange Goings on in Lancashire by Stuart Leadbetter. Edited by Carol Anne Strange.
BARRY McCANN is a writer, speaker, performer and broadcaster. He regularly writes short stories for the Lancashire Evening Post and was recently included in the anthology His Red Eyes Again, published by The Dracula Society (and available on Amazon.) He regularly appears on BBC Radio Lancashire and is the Folklore Correspondent on BBC Radio Cumbria. He has also hosted two Ingrid Pitt - Queen of Horror Festivals in Hastings, having previously worked for the legendary actress as her researcher.
9 Weird Welsh Mythical Creatures that will blow your mind!
These weird Welsh Mythical Creatures will blow your mind, reports NIA JONES!
North Dublin's 7 spookiest places to visit
North Dublin native and guest writer DOMINIC McELROY tells us of his 7 favourite spooky locations on the dark side of the River Liffey!
Cryptid Worms: Legend and Lore
CLAIRE BARRAND explores the disturbing stories of Cryptid Mutant Worms from folklore
Durham: Top 5 Haunted Places to Visit
LEE D. MUNRO explores five haunted places to visit in Durham in North-East England
Rosemary Timperley, The Greatest Horror Writer You've Never Heard Of
Rosemary Timperley is a forgotten English horror/ghost writer who deserves more attention that she gets, says MICHAEL S COLLINS


text:
Susan Youens - Retracing a Winter Journey
"I like these songs better than all the rest, and someday you will too", Franz Schubert told the friends who were the first to hear his song cycle, Winterreise. These lieder have always found admiring audiences, but the poetry he chose to set them to has been widely regarded as weak and trivial. In Retracing a Winter's Journey, Susan Youens looks not only at Schubert's music but at the poetry, drawn from the works of Wilhelm Müller, who once wrote in his diary, "perhaps there is a kindred spirit somewhere who will hear the tunes behind the words and give them back to me!"
Youens maintains that Müller, in depicting the wanderings of the alienated lover, produced poetry that was simple but not simple-minded, poetry that embraced simplicity as part of its meaning. In her view, Müller used the ruder folk forms to give his verse greater immediacy, to convey more powerfully the wanderer's complex inner state. Youens addresses many different aspects of Winterreise: the cultural milieu to which it belonged, the genesis of both the poetry and the music, Schubert's transformation of poetic cycle into music, the philosophical dimension of the work, and its musical structure.
© Cornell University Press
Youens – Schubert's Late Lieder
Wersin – Schubert hören


I have a very early version of the TT program. (v3.0) I tried to load it onto my new computer with Windows XP, without success. I was wondering if there was an update that would make it run on my new computer.
What happens when you try to install it on your computer?
The only solution with this older software that will work is to use a computer (virtual, dual-boot, or true second machine) with an older version of Windows installed. If you have Win7 Pro or Ultimate (or are willing to upgrade to one of these versions), you can create a virtual XP computer for free. Download and install Virtual PC and XP mode from Microsoft. Install them in order. Then install the XP software in the virtual computer and it will work. (Other virtual computer programs are Parallels and VMWare.) All of the other options will require a licensed version of the older Windows OS. Please note that the virtual computer can't share files directly with the main computer. You'll need to save files to external or removable media (optical media or flash drive, for example) to share them. You can't have a printer installed on both the main computer and the virtual one at the same time. You will be asked to uninstall the printer if you try.
Your other option is to upgrade to PrintMaster Platinum 18.1, see http://www.broderbund.com/ for details, which is compatible with Windows 7. This will import your Printmaster 10 projects.
I have windows xp professional logged off and cannot remember user name and password. Is it possible to bypass? Can I still get in my own laptop without reinstalling disk?
Will not function with Windows 7, what is wrong?
Not all programs that worked in Windows XP will run in Windows Vista or Windows 7.
I suggest you try compatibility mode to see if it helps.
If the program is not compatible, then you may try to install and run the program in compatibility mode.
4) Select Run this program in compatibility mode and select Windows Vista or whatever operating system the program was running successfully.
1. Open the Program Compatibility troubleshooter by clicking theStart button.
Then click Control Panel. In the search box, type troubleshooter, and then clickTroubleshooting. Under Programs, clickRun programs made for previous versions of Windows.
2. Follow the instructions in the troubleshooter.
If you cannot install a program, insert the installation disc for the program and, using the troubleshooter, browse to the program's setup file, usually called Setup.exe, Install.exe, or something similar. The troubleshooter is not designed to work on programs that have an .msi file name extension.
If you run Windows 7 Professional or Ultimate version, install Windows Virtual PC and Windows XP mode (http://www.microsoft.com/windows/virtual-pc/default.aspx) and execute your software in Windows XP.
If you run Windows 7 Home Premium, try VirtualBox (www.virtualbox.org) or VMware Player (www.vmware.com) and install a separate copy of Windows XP (either Home Edition or Professional) for that purpose.You can find virtualization software free of charge that will run on Windows 7 Home Premium.
1. File is corrupt, bad, or missing.
2. File is not designed for your version of Windows.
3. File is a virus, worm, or other malware file.
Step 1: Verify that the program installation CD is clean. If the CD is dirty or bad, you could get this error during the install and/or get this error after the program has been installed because it was not copied properly onto the computer.
Step 2: Make sure the program or file you're attempting to run does not contain the same name as the long file name directory it is contained in. For example, executing a file named "program" in the "program files" directory can cause errors.
Step 3: If you're trying to run a program that is not designed for your version of Windows you may get this error. Although many older programs designed to work in older versions of Windows will work with new versions of Windows, unfortunately, not all programs will work.
· If the program is an older MS-DOS program or early Windows program you may get this error.
· If the program is designed for a 64-bit version of Windows and you're running it in a 32-bit version of Windows, it will not work and generate this error.
You cannot upgrade (as you say load ) from Windows XP to Windows XP. You simply cannot do this and there is no point in doing it as I explain. You can upgrade Windows XP to Windows Vista easily. In the case that you want to keep Windows XP and want to enjoy better security and new features than what you do is you go to "My computer" icon and right click it and then take properties of it. The new page that will open you carefully check for these SP1(Service Pack 1), SP2 (Service Pack 2) or likewise SP3.
If you do not find any (SP)Service Packs than according to your requirement you can download and install SP1 or SP2.
Caution: The third level service Pack for Windows XP is SP3. Microsoft has declared that it is not for everyone. I strongly discourage you to install SP3 unless you have a strong requirement for it. After that you can put your XP on normal regular updates from Microsoft.
Trying to download my pictures from Canon Powershot A430 onto my new laptop Windows 7. Downloaded the disk and now trying without success to download my pictures onto the computer. Help??
Windows XP SP 1 and SP 2 Canon PowerShot A430 has no update about the new driver software for windows 7 Try doing this direct transfer in your computer 1. Connect the camera to your computer using the supplied USB cable of your camera. 2. Set the mode dial to (playback) and turn on the power.
The camera and computer will be able to communicate. 3. If there is no pop up window that has Canon CameraWindow, Click the [Start] menu and select [all Programs] or [Programs]. followed by [Canon Utilities], [Camera Window], [PowerShot-IXY-IXUS-DV 6] and [CameraWindow] 4. Download the images.
* Download the images using the camera or the computer.
* By default, the downloaded images are saved into the [My Pictures] folder. ** If the above process does not work you can just use the memory card of your camera and connect that on your laptop. (some laptop has built in card reader) The memory card works like a flash drive and you can just transfer files like transferring from one folder to another. ** Issue on the operating system compatibility.
windows ten is free and can be loaded onto a flash drive and you can use this to install. Your programs might not be met be your system specifications. Try install or upgrade computer.
MGI Photosuite on Vista, THE FINAL SOLUTION!!! First you need to get hold of an MGI disc or a download. I have been trying to get MGI Photosuite III SE to install on Vista for months without success and then when I thought I'd finally sorted it I could not launch the program due to a script error. You do not need Virtual PC, Virtual XP or any add-ons. Simply find and right click the the MGI setup file, check the run in compatibility mode as Windows XP SP2 and install. After install, send a program launch shortcut to your desktop, right click, go to properties and check the Windows XP compatibility mode once again. Finally, while you still have the compatibility screen open add a tick check to Run As Administrator. Hey presto my beloved MGI Photosuite is back! MGI is fantastic for being user friendly and simple, and although I know other users insist there are better programs, I've tried loads and I still think it's a winner.
3. Go into 'search' and type in 'hdaud' and delete those files that come up.
4. Reboot and double check 2 and 3 above.
5. Reboot and go into CMOS and enable your onboard audio.
6. Load your audio drivers from your setup disc-DO NOT LET THE COMPUTER LOOK FOR A DRIVER THRU THE 'FOUND NEW HARDWARE' PROMPT-PRESS CANCEL.
7. Reboot and hopefully, enjoy the sound.
Can i install Windows Media Player 11 without checking my Windows Geniune verification?
Your friend will have to settle for an older version of Media Player, or download a copy using an internet connection and a his valid copy of Windows XP. The software you downloaded is single-user type, which Microsoft authorized for the computer you downloaded it with; and won't install onto another machine.
If Teacher's toolbox 4.0 does not install on windows 7 64-bit system, system can I get to replace it? If Teacher's toolbox 4.0 does not install on windows 7 64-bit system, system can I get to replace it?
we are trying to download teachers toolbox 4.0 to windows 7 64 bit with no success...is there a download somewhere?
When I go to print my gradebook, the Class, Subject, and Semester Tabs all list every single task rather that just the final average. This makes printing the report a nightmare... these categories end up being 20+ pages long.
Have a manual for AbleSoft Teacher's Toolbox 4.0 (700193-015-01) for PC?


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Enrica Garzilli, Ahmed Rashid: "New initiatives and collaboration between the USA and Pakistan on Afghanistan" — An Interview
Alessandro Cisilin, Deconstructing Social Media in India
Sheetal Sharma, European Union and Empowerment of Women in Afghanistan: Issues and Challenges
Daniela Bandelli, Indian Women in Community Radio: The Case Studies of Radio Namaskar in Orissa and Radio Dhadkan in Madhya Pradesh
Shaireen Rasheed, The Politics of Neo-liberalism, Sexuality and Islam
Enrica Garzilli, Afghanistan, Issues at stake and Viable Solutions: An Interview with H.R.H. Princess India of Afghanistan
Vol. 2 No. 3 (December 1, 1996)
From 'Baylan' to 'Bruha': Hispanic Impact on the Animist Priestess in the Philippines
The Hispanization of the Word.
The Search.
From baylan to bruha.
Women, Information, and the Future: Collecting and Sharing Resources Worldwide
The views expressed in articles and reviews are the authors' own and should not be attributed to the Asiatica Association.
by Carolyn Brewer
In the Philippines, over the last two decades, with the worldwide spread of scholarship that focuses on previously marginalized groups, those interested in sixteenth century Philippine history have begun re-reading manuscripts written at the beginning of the Spanish period. These scholars have been somewhat surprised to learn that, before the coming of the Spanish to the archipelago, women, and a few men dressed as women (known as asog), were the original leaders within the traditional animist religion. The knowledge that women once had direct access to, and exercised control within the spiritual realms has excited some. Consequently, there has been a move towards liberation hovering on the discursive edges of mainstream historical scholarship which is determined to acknowledge what has previously been repressed and to uncover the acts and criteria of the exclusions by which these women vanished into a historical abyss.2
In the colonization process, some colonisers, intent on imposing their own world view, banished indigenous languages to the margins, and throughout the world many aboriginal languages were lost completely. In the Philippines, at the time of the Spanish conquest, rather than impose the Spanish language, the priests were instructed by the first bishop, Salazar, to learn and preach in the languages of the inhabitants. However, certain clusters of words, especially those involving animist priestesses (as the Spanish called them) were altered, negated and then marginalized almost to extinction. This movement parallelled the demonization and eventual disappearance of the priestess from historical texts.
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Q: How to fix this file related problem i m reading from file line by line but when i read some garbage character like  space /r is being added i m nt getting why it is being added although there is  no such character in file from where i m reading ..i have used fread and fgets both from both i m getting the same problem please reply if u have solution for this problem

A: The file was probably edited/created on Windows. Windows uses \r\n as a line delimiter. When you read the file, you must strip the \r manually. Since most editors treat \r\n as a single character (line end), you can't "see" it but it's still in the file. Use a hex editor if you want to see it or a tool like od.

A: Open the file in text mode.
/* ... */
fopen(filename, "r"); /* notice no 'b' in mode */
/* ... */

Supposing you're on Windows ... on reading operations, the library is responsible for translating the literal "\r\n" present on disk to "\n"; and on writing operation, the library translates "\n" to "\r\n".

redpajama_set_name: RedPajamaStackExchange


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National Geographic Explorer Follows "Diving With a Purpose" in New Podcast
January 19, 2022 January 19, 2022 by Hillary Bonner
Good Morning All! If you recall, I posted an article regarding a shipwreck in Coral Bay a few months back with mention of the dive group, Diving With a Purpose (DWP). Well, that spring issue of National Geographic feature (also mentioned in the article) will be available online on February 7, but a six episode podcast featuring the work and "purpose" of this incredible dive group will be live on January 27. And I am so incredibly excited to "dive" into these stories.
National Geographic Explorer, Tara Roberts, has spent the last few years working alongside DWP, a group of divers who visited St. John in July to explore the mysterious Coral Bay shipwreck. This incredible group is comprised of black scuba divers, scientists, historians and archeologists who aim to explore the 1000 missing ships around the globe that carried enslaved Africans. In the podcast, Roberts explores the past of the Henrietta Marie (a British slave ship that sank near the coast of Florida during the 18th century), a 16th Century Tudor shipwreck and the Clodita which was discovered off the coast of Alabama in 2019. This audio journey follows Roberts and DWP as they explore the lost shipwrecks of the transatlantic slave trade.
The March Issue of National Geographic will be available online on February 7. And, I have it on good authority that Coral Bay will have a mention 😉
She takes a scientific approach to these wrecks in the Episode Two interview with DWP founder, Ken Stewart "and with him delves into the history of the Spanish pirate ship, the Guerrero, as well as the technical approach to investigating underwater archeological sites."
Her personal connection to these stories takes her to a village in Mozambique to recount the impact of the slave trade on the villages there. And, to her home in North Carolina and then to Mobile, Alabama where she connects with direct descendants of Africans who were taken to the United States on the slave ship Clodita. (See Full Episode Guide)
Now, I reference the Coral Bay shipwreck in this article, not because it is featured in this series (according to the episode guide), but because this series focuses on the importance of uncovering these truths from the depths and on the dive group who explored our local wreck in July. A wreck that may help translate the yet to be told stories of the past for native St. Johnians. The series stresses the necessity of discovering and documenting these stories for present day descendants of enslaved Africans. This podcast, and the magazine to follow, may help to paint a picture of the struggles of descendants of slavery to connect the dots of their generational past.
The long forgotten anchor of Coral Bay 1 – The shipwreck that DWP explored in Coral Bay this summer
The trailer for the podcast is below, but you can hear a snippet on the National Geographic – Into the Depths website. Episode one airs on January 27 and will be available anywhere you get your podcasts.
Coming Soon to a Liquor Store Near You – Pusser's Painkillers in a Can!
Beautiful Boats and Community Members Gather at Lovango to Honor Captain Phil Chalker


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